Finite Group Databasebeta

← #30615994

Subgroups — #30615994

Order 15616.

948 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840787C211yesmaximal, characteristic, complemented
2C22×C4×#92420C211yesmaximal, characteristic, complemented
3#21461891C211yesmaximal, characteristic, complemented
4#21461890C211yesmaximal, characteristic, complemented
5#21461875C211yesmaximal, characteristic, complemented
6#21461872C211yesmaximal, characteristic, complemented
7#21461794C211yesmaximal, characteristic
8#21461794C211yesmaximal, characteristic
9#21461763C211yesmaximal, characteristic, complemented
10#21461762C211yesmaximal, characteristic, complemented
11#21461607C211yesmaximal, characteristic, complemented
12#21461548C211yesmaximal, characteristic, complemented
13#21461545C211yesmaximal, characteristic, complemented
14#21461538C211yesmaximal, characteristic, complemented
15#21448255C211yesmaximal, characteristic
16C23×#92420C2211yescharacteristic
17C23×#92420C2211yescharacteristic
18#16840787—24no—
19C2×#14519844C2211yescharacteristic
20C2×#14519844C2211yescharacteristic
21C2×C4⋊#92420C2211yescharacteristic
22C2×C4⋊#92420C2211yescharacteristic
23C2×#14519834C2211yescharacteristic
24C2×#14519834C2211yescharacteristic
25C2×#14519834C2211yescharacteristic
26C2×#14519834C2211yescharacteristic
27C2×C22⋊#92420C2211yescharacteristic
28C2×C22⋊#92420C2211yescharacteristic
29C2×#14519832C2211yescharacteristic
30C2×#14519832C2211yescharacteristic
31C2×C4×#92420—24no—
32C2×C4×#92420C2211yescharacteristic
33C2×C4×#92420C2211yescharacteristic
34C2×C4×#92420C2211yescharacteristic
35C2×C4×#92420C2211yescharacteristic
36#16840628C2211yescharacteristic
37#16840628C2211yescharacteristic
38#16840628C2211yescharacteristic
39#16840628C2211yescharacteristic
40#16840628C2211yescharacteristic
41#16840628C2211yescharacteristic
42#16840628C2211yescharacteristic
43#16840628C2211yescharacteristic
44#16840628C2211yescharacteristic
45#16840628C2211yescharacteristic
46C2×#14519737C2211yescharacteristic
47C2×#14519735C2211yescharacteristic
48C22×C4×Dic61C2211yescharacteristic
49#16839654C411yescharacteristic
50#16839652C411yescharacteristic
51#16839649C411yescharacteristic
52#16839643C411yescharacteristic
53#16839641C411yescharacteristic
54#16839639C411yescharacteristic
55#16839637C411yescharacteristic
56C4⋊C4×Dic61C2211yescharacteristic
57C22⋊C4×Dic61C2211yescharacteristic
58#16839609C2211yescharacteristic
59#16839607C2211yescharacteristic
60C61×#385C411yescharacteristic
61C22×#92420—22no—
62C22×#92420—22no—
63C22×#92420—22no—
64C22×#92420—22no—
65C22×#92420—22no—
66C22×#92420—22no—
67C22×#92420C2311yescharacteristic
68C22×#92420C2311yescharacteristic
69C22×#92420C2311yescharacteristic
70C22×#92420C2311yescharacteristic
71C22×#92420C2311yescharacteristic
72C22×#92420C2311yescharacteristic
73C22×#92420C2311yescharacteristic
74C22×#92420C2311yescharacteristic
75#14519844—22no—
76#14519844—22no—
77C4⋊#92420—22no—
78C4⋊#92420—22no—
79#14519834—22no—
80#14519834—22no—
81#14519834—22no—
82#14519834—22no—
83C22⋊#92420—22no—
84C22⋊#92420—22no—
85#14519832—22no—
86#14519832—22no—
87C4×#92420—22no—
88C4×#92420—22no—
89C4×#92420—22no—
90C4×#92420D414yescomplemented
91C22×C22⋊D61C2×C411yescharacteristic
92C23×Dic61C2×C411yescharacteristic
93C23×Dic61C2311yescharacteristic
94C22×C4×D61C2311yescharacteristic
95#14519737—22no—
96#14519737—22no—
97#14519735—22no—
98#14519735—22no—
99C2×C4⋊Dic61C2×C411yescharacteristic
100C2×C4⋊Dic61C2×C411yescharacteristic
101C2×#11758196C2×C411yescharacteristic
102C2×C22⋊Dic61C2×C411yescharacteristic
103C2×#11758198C2×C411yescharacteristic
104C2×#11758198C2×C411yescharacteristic
105C2×#11758198C2×C411yescharacteristic
106C2×#11758198C2×C411yescharacteristic
107C2×#11758198C2×C411yescharacteristic
108C2×C4×Dic61—22no—
109C2×C4×Dic61—22no—
110C2×C4×Dic61C2×C411yescharacteristic
111C2×C4×Dic61C2×C411yescharacteristic
112C2×C4×Dic61C2311yescharacteristic
113C2×C4×Dic61C2311yescharacteristic
114C2×C4×Dic61C2311yescharacteristic
115C2×C4×Dic61C2311yescharacteristic
116C2×C4×Dic61C2311yescharacteristic
117#14519684C2×C411yescharacteristic
118#14519684C2×C411yescharacteristic
119#14519684C2×C411yescharacteristic
120#14519684C2×C411yescharacteristic
121#14519684C2×C411yescharacteristic
122#14519684C2×C411yescharacteristic
123#14519684C2×C411yescharacteristic
124#14519684C2×C411yescharacteristic
125C23×C4×C61C2×C411yescharacteristic
126C2×C4⋊C4×C61C2×C411yescharacteristic
127C2×C4⋊C4×C61C2×C411yescharacteristic
128C2×C22⋊C4×C61C2×C411yescharacteristic
129C2×C22⋊C4×C61C2×C411yescharacteristic
130#95×C61C2×C411yescharacteristic
131#95×C61C2×C411yescharacteristic
132C24×C61C22×C411yescharacteristic
133C23×D61C22×C411yescharacteristic
134C22×C4×C61—22no—
135C22×C4×C61—22no—
136C22×C4×C61—22no—
137C22×C4×C61C22×C411yescharacteristic
138C22×C4×C61C22×C411yescharacteristic
139C22×C4×C61C22×C411yescharacteristic
140C22×C4×C61C22×C411yescharacteristic
141C22×C4×C61C22×C411yescharacteristic
142C22×C4×C61C22×C411yescharacteristic
143C2×C22⋊D61—22no—
144C2×C22⋊D61—22no—
145C2×C22⋊D61—22no—
146C22×Dic61—22no—
147C22×Dic61—22no—
148C22×Dic61—22no—
149C22×Dic61—21no—
150C22×Dic61—21no—
151C22×Dic61—21no—
152C22×Dic61—21no—
153C22×Dic61C22×C411yescharacteristic
154C22×Dic61C22×C411yescharacteristic
155C22×Dic61C22×C411yescharacteristic
156C22×Dic61C22×C411yescharacteristic
157C22×Dic61C22×C411yescharacteristic
158C22×Dic61C22×C411yescharacteristic
159C22×Dic61#4112yes—
160C22×Dic61#4112yes—
161C22×Dic61C2411yescharacteristic
162C2×C4×D61—21no—
163C2×C4×D61—21no—
164C2×C4×D61—21no—
165C2×C4×D61—21no—
166C2×C4×D61—21no—
167C2×C4×D61—21no—
168C2×#92420—28no—
169C2×#92420—28no—
170C2×#92420—21no—
171C2×#92420—21no—
172C2×#92420—21no—
173C2×#92420—21no—
174C2×#92420—21no—
175C2×#92420—21no—
176C2×#92420—21no—
177C2×#92420—21no—
178C2×#92420—21no—
179C2×#92420—21no—
180C2×#92420—21no—
181C2×#92420—21no—
182C2×#92420—21no—
183C2×#92420—21no—
184C2×#92420—21no—
185C2×#92420—21no—
186C2×#92420—21no—
187C2×#92420—21no—
188C2×#92420—21no—
189C2×#92420—21no—
190C2×#92420C2×D412yes—
191C2×#92420C2×D412yes—
192C2×#92420#4112yes—
193C2×#92420#4112yes—
194C4⋊C4×C61—22no—
195C4⋊C4×C61—22no—
196C22⋊C4×C61—22no—
197C22⋊C4×C61—22no—
198C22⋊Dic61—22no—
199#11758198—22no—
200#11758198—22no—
201#11758198—22no—
202#11758198—22no—
203#11758198—22no—
204C4⋊Dic61—22no—
205C4⋊Dic61—22no—
206#11758196—22no—
207C4×Dic61—22no—
208C4×Dic61—22no—
209C4×Dic61—21no—
210C4×Dic61—21no—
211C4×Dic61—21no—
212C4×Dic61—21no—
213C4×Dic61—21no—
214C4×Dic61—21no—
215C4×Dic61—21no—
216C4×Dic61—21no—
217C4×Dic61C2×D412yes—
218C4×Dic61#4112yes—
219C23×C61—21no—
220C23×C61—21no—
221C23×C61—21no—
222C23×C61—21no—
223C23×C61#11712yes—
224C23×C61#13112yes—
225C23×C61#13112yes—
226C23×C61C23×C411yescharacteristic
227C22×D61—21no—
228C22×D61—21no—
229C22×D61—21no—
230C22×D61—21no—
231C22×D61—21no—
232C22×D61—21no—
233C22×D61—21no—
234C2×C4×C61—28no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61—21no—
239C2×C4×C61—21no—
240C2×C4×C61—21no—
241C2×C4×C61—21no—
242C2×C4×C61—21no—
243C2×C4×C61—21no—
244C2×C4×C61—21no—
245C2×C4×C61—21no—
246C2×C4×C61—21no—
247C2×C4×C61—21no—
248C2×C4×C61—21no—
249C2×C4×C61—21no—
250C2×C4×C61—21no—
251C2×C4×C61C4×D412yes—
252C2×C4×C61#13112yes—
253C22⋊D61—44no—
254C2×Dic61—28no—
255C2×Dic61—21no—
256C2×Dic61—21no—
257C2×Dic61—21no—
258C2×Dic61—21no—
259C2×Dic61—21no—
260C2×Dic61—21no—
261C2×Dic61—22no—
262C2×Dic61—22no—
263C2×Dic61—22no—
264C2×Dic61—22no—
265C2×Dic61—21no—
266C2×Dic61—21no—
267C2×Dic61—21no—
268C2×Dic61—21no—
269C2×Dic61—21no—
270C2×Dic61—21no—
271C2×Dic61—21no—
272C2×Dic61—21no—
273C2×Dic61—21no—
274C2×Dic61#11712yes—
275C2×Dic61C4×D412yes—
276C2×Dic61#13112yes—
277C2×Dic61C22×D411yescharacteristic
278C2×Dic61C2×#4111yescharacteristic
279C2×Dic61C2×#4111yescharacteristic
280C2×Dic61C2×#4111yescharacteristic
281C2×Dic61C2×#4111yescharacteristic
282C2×Dic61C2×#4111yescharacteristic
283C4×D61—41no—
284C4×D61—41no—
285C4×D61—41no—
286C4×D61—41no—
287#92420—22no—
288#92420—22no—
289#92420—22no—
290#92420—22no—
291#92420—41no—
292#92420—41no—
293#92420—41no—
294#92420—41no—
295#21103—611nomaximal, Sylow
296C22×C61—22no—
297C22×C61—22no—
298C22×C61—22no—
299C22×C61—22no—
300C22×C61—22no—
301C22×C61—22no—
302C22×C61#43314yescomplemented
303C22×C61C2×#11711yescharacteristic
304C22×C61C2×C4×D411yescharacteristic
305C22×C61C4×#4111yescharacteristic
306C22×C61C2×#13111yescharacteristic
307C22×C61C2×#13111yescharacteristic
308C22×C61C2×#13111yescharacteristic
309C22×C61C2×#13111yescharacteristic
310C2×D61—41no—
311C2×D61—41no—
312C2×D61—41no—
313C2×D61—41no—
314C2×D61—41no—
315C2×D61—41no—
316C2×D61—41no—
317C4×C61—22no—
318C4×C61—22no—
319C4×C61—41no—
320C4×C61—41no—
321C4×C61—41no—
322C4×C61—41no—
323Dic61—22no—
324Dic61—22no—
325Dic61—22no—
326Dic61—41no—
327Dic61—41no—
328Dic61—41no—
329Dic61#52511yescharacteristic
330Dic61#52511yescharacteristic
331Dic61#52911yescharacteristic
332Dic61#52911yescharacteristic
333C2×#433—611no—
334C22×C4×C8—611no—
335#2290—611no—
336#1923—611no—
337#1923—611no—
338#1916—611no—
339#1916—611no—
340#1869—611no—
341#1869—611no—
342#1850—611no—
343#1850—611no—
344#1768—611no—
345#1750—611no—
346#1749—611no—
347#1746—611no—
348C2×C61—24no—
349C2×C61#229011yescharacteristic
350C2×C61C2×#43311yescharacteristic
351C2×C61#292211yescharacteristic
352C2×C61#292911yescharacteristic
353C2×C61#292911yescharacteristic
354C2×C61#296611yescharacteristic
355C2×C61#296611yescharacteristic
356D61—81no—
357C23×C8—611no—
358C23×C8—611no—
359C2×C4×D4—611no—
360C2×#117—611no—
361C22×C42—611no—
362#433—1224no—
363C2×C4⋊C8—611no—
364C2×C4⋊C8—611no—
365C2×C4⋊C8—611no—
366C2×C4⋊C8—611no—
367C2×C22⋊C8—611no—
368C2×C22⋊C8—611no—
369C2×C22⋊C8—611no—
370C2×C22⋊C8—611no—
371C2×C22⋊C8—611no—
372C2×C22⋊C8—611no—
373C2×#97—611no—
374C2×#97—611no—
375C2×C4×C8—611no—
376C2×C4×C8—611no—
377C2×C4×C8—611no—
378C2×C4×C8—611no—
379C2×C4×C8—1224no—
380#388—611no—
381#388—611no—
382#387—611no—
383#387—611no—
384#386—611no—
385#386—611no—
386#385—611no—
387#385#692311yescharacteristic
388C4×C4⋊C4—611no—
389C4×C22⋊C4—611no—
390C4×C22⋊C4—611no—
391#376—611no—
392#336—611no—
393#336—611no—
394#336—611no—
395#336—611no—
396#336—611no—
397#336—611no—
398#336—611no—
399#336—611no—
400#336—611no—
401#336—611no—
402C61#2110311yescharacteristic, complemented, Sylow
403C22×D4—611no—
404C23×C4—611no—
405C23×C4—611no—
406C23×C4—611no—
407C23×C4C2×#692311yescharacteristic
408C22×C8—611no—
409C22×C8—611no—
410C22×C8—611no—
411C22×C8—611no—
412C22×C8—611no—
413C22×C8—611no—
414C22×C8—611no—
415C22×C8—611no—
416C22×C8—1222no—
417C22×C8—1222no—
418C22×C8—1222no—
419C22×C8—1222no—
420C22×C8—1222no—
421C22×C8—1222no—
422C4×D4—1222no—
423C4×D4—1222no—
424#117—1222no—
425#117—1222no—
426C2×C4⋊C4—611no—
427C2×C4⋊C4—611no—
428C2×C4⋊C4—611no—
429C2×C4⋊C4C2×#692311yescharacteristic
430C2×C4⋊C4C2×#692311yescharacteristic
431C2×C22⋊C4—611no—
432C2×C22⋊C4—611no—
433C2×C22⋊C4—611no—
434C2×C22⋊C4—611no—
435C2×C22⋊C4—611no—
436C2×C22⋊C4—611no—
437C2×C22⋊C4C2×#692311yescharacteristic
438C2×C22⋊C4C2×#692311yescharacteristic
439C2×C42—611no—
440C2×C42—611no—
441C2×C42—611no—
442C2×C42—611no—
443C2×C42—611no—
444C2×C42—611no—
445C2×C42—611no—
446C2×C42—1222no—
447C2×C42—1222no—
448C4⋊C8—1222no—
449C4⋊C8—1222no—
450C4⋊C8—1222no—
451C4⋊C8—1222no—
452C22⋊C8—1222no—
453C22⋊C8—1222no—
454C22⋊C8—1222no—
455C22⋊C8—1222no—
456C22⋊C8—1222no—
457C22⋊C8—1222no—
458#97—1222no—
459#97—1222no—
460C4×C8—614no—
461C4×C8—1222no—
462C4×C8—1222no—
463C4×C8—1222no—
464#95—611no—
465#95—611no—
466#95—611no—
467#95—611no—
468#95—611no—
469#95—611no—
470#95—611no—
471#95—611no—
472#95C2×#692311yescharacteristic
473#95C2×#692311yescharacteristic
474C24—611no—
475C24C22×#692311yescharacteristic
476C2×D4—1222no—
477C2×D4—1222no—
478C2×D4—1222no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—611no—
483C22×C4—612no—
484C22×C4—612no—
485C22×C4—611no—
486C22×C4—611no—
487C22×C4—611no—
488C22×C4—611no—
489C22×C4—611no—
490C22×C4—611no—
491C22×C4—1221no—
492C22×C4—1221no—
493C22×C4—1221no—
494C22×C4—1221no—
495C22×C4—1221no—
496C22×C4—1221no—
497C22×C4—1221no—
498C22×C4—1221no—
499C22×C4—1222no—
500C22×C4—1221no—
501C22×C4—1221no—
502C22×C4—1222no—
503C22×C4—1222no—
504C22×C4C22×#692311yescharacteristic
505C22×C4C22×#692311yescharacteristic
506C22×C4C22×#692311yescharacteristic
507C22×C4C22×#692311yescharacteristic
508C22×C4C22×#692311yescharacteristic
509C22×C4C22×#692311yescharacteristic
510C2×C8—612no—
511C2×C8—612no—
512C2×C8—612no—
513C2×C8—612no—
514C2×C8—1221no—
515C2×C8—1221no—
516C2×C8—1221no—
517C2×C8—1221no—
518C2×C8—1221no—
519C2×C8—1221no—
520C2×C8—1221no—
521C2×C8—1221no—
522C2×C8—1228no—
523C2×C8—1228no—
524C2×C8—1221no—
525C2×C8—1221no—
526C2×C8—1221no—
527C2×C8—1221no—
528C2×C8—1221no—
529C2×C8—1221no—
530C2×C8—1221no—
531C2×C8—1221no—
532C2×C8—1221no—
533C2×C8—1221no—
534C2×C8—1221no—
535C2×C8—1221no—
536C4⋊C4—22no—
537C4⋊C4—22no—
538C4⋊C4—1222no—
539C4⋊C4—1222no—
540C4⋊C4—1222no—
541C22⋊C4—22no—
542C22⋊C4—22no—
543C22⋊C4—1222no—
544C22⋊C4—1222no—
545C22⋊C4—1222no—
546C22⋊C4—1222no—
547C22⋊C4—1222no—
548C22⋊C4—1222no—
549C42—612no—
550C42—612no—
551C42—1221no—
552C42—1221no—
553C42—1221no—
554C42—1221no—
555C42—1221no—
556C42—1222no—
557C42—1221no—
558C42—1221no—
559C42—1221no—
560C42—1222no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—1221no—
566C23—1221no—
567C23—1221no—
568C23—1221no—
569C23—1221no—
570C23—1221no—
571C23—1221no—
572C23#1451985612yes—
573C23#1451986412yes—
574C23#1451986512yes—
575C23C23×#692311yescharacteristic
576D4—2444no—
577C2×C4—28no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—611no—
595C2×C4—611no—
596C2×C4—611no—
597C2×C4—611no—
598C2×C4—611no—
599C2×C4—611no—
600C2×C4—612no—
601C2×C4—612no—
602C2×C4—612no—
603C2×C4—1222no—
604C2×C4—1222no—
605C2×C4—1222no—
606C2×C4—1222no—
607C2×C4—1228no—
608C2×C4—1221no—
609C2×C4—1221no—
610C2×C4—1221no—
611C2×C4—1221no—
612C2×C4—1221no—
613C2×C4—1221no—
614C2×C4—1221no—
615C2×C4—1221no—
616C2×C4—1221no—
617C2×C4—1221no—
618C2×C4—1221no—
619C2×C4—1221no—
620C2×C4—1221no—
621C2×C4—1221no—
622C2×C4—1221no—
623C2×C4—2441no—
624C2×C4—2441no—
625C2×C4—2441no—
626C2×C4—2441no—
627C2×C4D4×#692312yes—
628C2×C4#1451986412yes—
629C8—1222no—
630C8—1222no—
631C8—1222no—
632C8—1222no—
633C8—2441no—
634C8—2441no—
635C8—2441no—
636C8—2441no—
637C22—22no—
638C22—22no—
639C22—22no—
640C22—22no—
641C22—22no—
642C22—22no—
643C22—2441no—
644C22—2441no—
645C22—2441no—
646C22—2441no—
647C22—2441no—
648C22—2441no—
649C22—2441no—
650C22#1684078714yescomplemented
651C22C2×#1451985611yescharacteristic
652C22C2×D4×#692311yescharacteristic
653C22#41×#692311yescharacteristic
654C22C2×#1451986411yescharacteristic
655C22C2×#1451986411yescharacteristic
656C22D61⋊#13111yescharacteristic
657C22C2×#1451986511yescharacteristic
658C4—22no—
659C4—22no—
660C4—41no—
661C4—41no—
662C4—41no—
663C4—41no—
664C4—611no—
665C4—611no—
666C4—611no—
667C4—611no—
668C4—1222no—
669C4—1222no—
670C4—1222no—
671C4—2441no—
672C4—2441no—
673C4—2441no—
674C2—24no—
675C2—4881no—
676C2#2146227311yescharacteristic
677C2C2×#1684078711yescharacteristic
678C2#2146250211yescharacteristic
679C2#2146253411yescharacteristic
680C2#2146253511yescharacteristic
681C2#2146263411yescharacteristic
682C2#2146263511yescharacteristic
subgroupG/H
center, FrattiniC23C23×#6923
commutator, socleC23×C61C23×C4
FittingC61×#385C4
prime radicalC22×C22⋊D61C2×C4
cosocleC22×Dic61C24
hypercenter#385#6923
hypocenterC61#21103
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C23 ◁ #385

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#16840787C2semidirect
2. C22×C4×#92420C2semidirect
3. #21461891C2semidirect
4. #21461890C2semidirect
5. #21461875C2semidirect
6. #21461872C2semidirect
9. #21461763C2semidirect
10. #21461762C2semidirect
11. #21461607C2semidirect
12. #21461548C2semidirect
13. #21461545C2semidirect
14. #21461538C2semidirect
90. C4×#92420D4semidirect
302. C22×C61#433semidirect
402. C61#21103semidirect
650. C22#16840787semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#168407872
C22×C4×#924202
#214618912
#214618902
#214618752
#214618722
#214617942
#214617632
#214617622
#214616072
#214615482
#214615452
#214615382
#214482552
#2110361