Finite Group Databasebeta

← #30620468

Subgroups — #30620468

Order 15616.

1364 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840814C212yesmaximal, complemented
2#21461944C212yesmaximal, complemented
3#21461943C212yesmaximal, complemented
4#21461942C212yesmaximal, complemented
5#21461676C211yesmaximal, characteristic, complemented
6#21461675C211yesmaximal, characteristic, complemented
7#21461673C211yesmaximal, characteristic, complemented
8#21461546C211yesmaximal, characteristic, complemented
9#21461545C211yesmaximal, characteristic, complemented
10C4×C22⋊#92420C211yesmaximal, characteristic, complemented
11C2×C4×#11758224C211yesmaximal, characteristic
12#16840814—28no—
13C2×#14519844C2212yes—
14C2×#14519844C2212yes—
15C2×C4⋊#92420C2212yes—
16C2×#14519834C2212yes—
17C2×#14519834C2212yes—
18C2×C22⋊#92420C2212yes—
19C2×C22⋊#92420C2212yes—
20C2×C22⋊#92420C2212yes—
21C2×C22⋊#92420C2212yes—
22C2×#14519832C2212yes—
23C2×#14519832C2212yes—
24C2×C4×#92420C2212yes—
25#16840628C2211yescharacteristic
26#16840628C2211yescharacteristic
27#16840628C2211yescharacteristic
28#16840628C2211yescharacteristic
29C22×#11758224C411yescharacteristic
30C4×#11758224—28no—
31C2×C4×Dic122C411yescharacteristic
32C2×#14519742C2211yescharacteristic
33C2×D4×Dic61C2211yescharacteristic
34C2×C4×C22⋊D61C411yescharacteristic
35C2×C4×C22⋊D61C411yescharacteristic
36C2×#14519737C2211yescharacteristic
37C2×#14519737C2211yescharacteristic
38C2×#14519735C411yescharacteristic
39C2×#14519733C411yescharacteristic
40C2×#14519733C411yescharacteristic
41C22×C4×Dic61C2211yescharacteristic
42C22×C4×Dic61C2211yescharacteristic
43C2×C42×D61C2211yescharacteristic
44C2×C4×D4×C61C411yescharacteristic
45C22×#92420C2312yes—
46C22×#92420C2312yes—
47C22×#92420C2312yes—
48C22×#92420C2312yes—
49#14519844—24no—
50#14519844—24no—
51C4⋊#92420—24no—
52#14519834—24no—
53#14519834—42no—
54C22⋊#92420—24no—
55C22⋊#92420—24no—
56C22⋊#92420—42no—
57C22⋊#92420—42no—
58#14519832—24no—
59#14519832—24no—
60C4×#92420—24no—
61C2×#11758224—24no—
62C2×#11758224—24no—
63C2×#11758224—24no—
64C22×Dic122C2×C411yescharacteristic
65C22×C22⋊D61C2×C411yescharacteristic
66C22×C22⋊D61C2×C411yescharacteristic
67C23×Dic61C2×C411yescharacteristic
68C23×Dic61C2×C411yescharacteristic
69C23×Dic61C2311yescharacteristic
70C23×Dic61C2311yescharacteristic
71C22×C4×D61C2×C411yescharacteristic
72C22×C4×D61C2×C411yescharacteristic
73C22×C4×D61C2311yescharacteristic
74C4×Dic122—24no—
75#14519742D418yes—
76D4×Dic61—24no—
77C4×C22⋊D61—24no—
78C4×C22⋊D61—24no—
79#14519737—24no—
80#14519737—24no—
81#14519735—24no—
82#14519733—24no—
83#14519733—24no—
84C2×C4⋊Dic61C2×C411yescharacteristic
85C2×#11758196C2×C411yescharacteristic
86C2×#11758196C2×C411yescharacteristic
87C2×#11758196C2×C411yescharacteristic
88C2×#11758196C2×C411yescharacteristic
89C2×C22⋊Dic61C2×C411yescharacteristic
90C2×C22⋊Dic61C2×C411yescharacteristic
91C2×#11758198C2×C411yescharacteristic
92C2×#11758198C2×C411yescharacteristic
93C2×C4×Dic61—24no—
94C2×C4×Dic61—24no—
95C2×C4×Dic61C2×C411yescharacteristic
96C2×C4×Dic61C2×C411yescharacteristic
97C2×C4×Dic61C2×C411yescharacteristic
98C2×C4×Dic61C2×C411yescharacteristic
99C2×C4×Dic61C2×C411yescharacteristic
100C2×C4×Dic61C2311yescharacteristic
101C2×C4×Dic61C2311yescharacteristic
102C2×C4×Dic61C2311yescharacteristic
103C2×C4×Dic61C2311yescharacteristic
104C42×D61—24no—
105C22×D4×C61C2×C411yescharacteristic
106C23×C4×C61C2×C411yescharacteristic
107C23×C4×C61C2×C411yescharacteristic
108C4×D4×C61—24no—
109C2×C4⋊C4×C61C2×C411yescharacteristic
110C2×C22⋊C4×C61C2×C411yescharacteristic
111C2×C22⋊C4×C61C2×C411yescharacteristic
112C2×C42×C61C2×C411yescharacteristic
113C24×C61C22×C411yescharacteristic
114C24×C61C22×C411yescharacteristic
115C23×D61C22×C411yescharacteristic
116C2×D4×C61—22no—
117C2×D4×C61—22no—
118C2×D4×C61—22no—
119C22×C4×C61—22no—
120C22×C4×C61—22no—
121C22×C4×C61—22no—
122C22×C4×C61—22no—
123C22×C4×C61—41no—
124C22×C4×C61—41no—
125C22×C4×C61C22×C411yescharacteristic
126C22×C4×C61C22×C411yescharacteristic
127C22×C4×C61C22×C411yescharacteristic
128C22×C4×C61C22×C411yescharacteristic
129C22×C4×C61C22×C411yescharacteristic
130C2×C22⋊D61—22no—
131C2×C22⋊D61—22no—
132C2×C22⋊D61—22no—
133C2×C22⋊D61—22no—
134C2×C22⋊D61—22no—
135C2×C22⋊D61M1614yes—
136C22×Dic61—22no—
137C22×Dic61—22no—
138C22×Dic61—22no—
139C22×Dic61—22no—
140C22×Dic61—22no—
141C22×Dic61—22no—
142C22×Dic61—22no—
143C22×Dic61—22no—
144C22×Dic61—22no—
145C22×Dic61—21no—
146C22×Dic61—21no—
147C22×Dic61—21no—
148C22×Dic61—21no—
149C22×Dic61M1614yes—
150C22×Dic61C22×C411yescharacteristic
151C22×Dic61C22×C411yescharacteristic
152C22×Dic61C22×C411yescharacteristic
153C22×Dic61C22×C411yescharacteristic
154C22×Dic61C22×C411yescharacteristic
155C22×Dic61C22×C411yescharacteristic
156C22×Dic61C2411yescharacteristic
157#11758224—232no—
158C2×C4×D61—22no—
159C2×C4×D61—22no—
160C2×C4×D61—22no—
161C2×C4×D61—22no—
162C2×C4×D61—22no—
163C2×C4×D61—22no—
164C2×C4×D61—22no—
165C2×C4×D61—21no—
166C2×C4×D61—21no—
167C2×C4×D61—41no—
168C2×Dic122—22no—
169C2×Dic122—22no—
170C2×Dic122C22⋊C414yes—
171C2×#92420—22no—
172C2×#92420—22no—
173C2×#92420—22no—
174C2×#92420—22no—
175C2×#92420—22no—
176C2×#92420—22no—
177C2×#92420—22no—
178C2×#92420—22no—
179C2×#92420—22no—
180C2×#92420—22no—
181C2×#92420—22no—
182C2×#92420#4114yescomplemented
183C4⋊C4×C61C22⋊C414yes—
184C22⋊C4×C61—22no—
185C22⋊C4×C61—22no—
186C42×C61—22no—
187C22⋊Dic61—22no—
188C22⋊Dic61—22no—
189#11758198—22no—
190#11758198—22no—
191C4⋊Dic61—22no—
192#11758196—22no—
193#11758196—22no—
194#11758196C22⋊C414yes—
195#11758196C22⋊C414yes—
196C4×Dic61—22no—
197C4×Dic61—22no—
198C4×Dic61—22no—
199C4×Dic61—22no—
200C4×Dic61—22no—
201C4×Dic61—22no—
202C4×Dic61C2×D414yes—
203C4×Dic61C2×D414yes—
204C4×Dic61C2×D414yes—
205C23×C61—21no—
206C23×C61—21no—
207C23×C61—21no—
208C23×C61—21no—
209C23×C61—21no—
210C23×C61—21no—
211C23×C61—21no—
212C23×C61—21no—
213C23×C61—21no—
214C23×C61—21no—
215C23×C61—21no—
216C23×C61—21no—
217C23×C61C2×M1612yes—
218C23×C61C2×M1612yes—
219C23×C61C23×C411yescharacteristic
220C22×D61—21no—
221C22×D61—21no—
222C22×D61—21no—
223C22×D61—21no—
224C22×D61—21no—
225C22×D61—21no—
226C22×D61C2×M1612yes—
227D4×C61—28no—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61—21no—
239C2×C4×C61—44no—
240C2×C4×C61—44no—
241C2×C4×C61C2×C22⋊C412yes—
242C2×C4×C61C2×C22⋊C412yes—
243C2×C4×C61C2×C22⋊C412yes—
244C2×C4×C61#13112yes—
245C22⋊D61—28no—
246C22⋊D61—28no—
247C2×Dic61—28no—
248C2×Dic61—21no—
249C2×Dic61—21no—
250C2×Dic61—21no—
251C2×Dic61—21no—
252C2×Dic61—21no—
253C2×Dic61—21no—
254C2×Dic61—28no—
255C2×Dic61—21no—
256C2×Dic61—21no—
257C2×Dic61—21no—
258C2×Dic61—21no—
259C2×Dic61—21no—
260C2×Dic61—21no—
261C2×Dic61—42no—
262C2×Dic61—42no—
263C2×Dic61—42no—
264C2×Dic61—42no—
265C2×Dic61C2×C22⋊C412yes—
266C2×Dic61C2×C22⋊C412yes—
267C2×Dic61C2×C22⋊C412yes—
268C2×Dic61C2×M1612yes—
269C2×Dic61C2×M1612yes—
270C2×Dic61C2×M1612yes—
271C2×Dic61C22×D412yes—
272C2×Dic61C2×#4112yes—
273C2×Dic61H(5,𝔽2)11yescharacteristic
274C2×Dic612−1+411yescharacteristic
275C4×D61—28no—
276C4×D61—44no—
277C4×D61—42no—
278C4×D61—42no—
279Dic122—28no—
280#92420—24no—
281#92420—42no—
282#92420—42no—
283#92420—42no—
284#22532—611nomaximal, Sylow
285C22×C61—22no—
286C22×C61—22no—
287C22×C61—22no—
288C22×C61—22no—
289C22×C61—22no—
290C22×C61—22no—
291C22×C61—41no—
292C22×C61—41no—
293C22×C61—41no—
294C22×C61—41no—
295C22×C61—41no—
296C22×C61—41no—
297C22×C61—41no—
298C22×C61—41no—
299C22×C61C22×C22⋊C411yescharacteristic
300C22×C61C4×#4111yescharacteristic
301C22×C61#52011yescharacteristic
302C22×C61C22×M1611yescharacteristic
303C22×C61C22×M1611yescharacteristic
304C22×C61C2×#13111yescharacteristic
305C22×C61#56811yescharacteristic
306C2×D61—22no—
307C2×D61—22no—
308C2×D61—22no—
309C2×D61—41no—
310C2×D61—41no—
311C2×D61—41no—
312C2×D61—41no—
313C4×C61—22no—
314C4×C61—22no—
315C4×C61—22no—
316C4×C61—22no—
317C4×C61—41no—
318Dic61—22no—
319Dic61—22no—
320Dic61—22no—
321Dic61—41no—
322Dic61—41no—
323Dic61#40714yes—
324Dic61#54812yes—
325Dic61#55012yes—
326C2×C4×#41—611no—
327C2×#446—612no—
328#1955—612no—
329#1955—612no—
330#1954—612no—
331#1797—611no—
332#1797—611no—
333#1797—611no—
334#1749—611no—
335#1749—611no—
336C4×C22⋊C8—611no—
337C2×C61—42no—
338C2×C61—42no—
339C2×C61#234111yescharacteristic
340C2×C61C2×#40711yescharacteristic
341C2×C61#288011yescharacteristic
342C2×C61#296511yescharacteristic
343C2×C61#297011yescharacteristic
344C2×C61#298911yescharacteristic
345C2×C61#299011yescharacteristic
346D61—42no—
347C22×#41—611no—
348C4×#41—1228no—
349C2×C4×Q8—611no—
350C2×C4×Q8—611no—
351C2×C4×D4—611no—
352C2×C4×D4—611no—
353C2×C4×D4—611no—
354C2×C4×D4—611no—
355C2×C4×D4—611no—
356C2×C4×D4#692311yescharacteristic
357C2×#117—611no—
358C2×#117—611no—
359C2×#117—611no—
360C22×C42—611no—
361C22×C42—611no—
362C22×C42—611no—
363#446—1228no—
364C2×C4⋊C8—612no—
365C2×C4⋊C8—612no—
366C2×C4⋊C8—612no—
367C2×C22⋊C8—612no—
368C2×C22⋊C8—612no—
369C2×C22⋊C8—612no—
370C2×C22⋊C8—612no—
371C2×C22⋊C8—612no—
372C2×C22⋊C8—612no—
373C2×#97—612no—
374C2×#97—612no—
375C2×C4×C8—612no—
376#336—611no—
377#336—611no—
378#336—611no—
379#336—611no—
380C61#2253211yescharacteristic, complemented, Sylow
381C2×#41—1224no—
382C2×#41—1224no—
383C2×#41—1224no—
384C22×Q8—611no—
385C22×D4—611no—
386C22×D4—611no—
387C22×D4C2×#692311yescharacteristic
388C23×C4—611no—
389C23×C4—611no—
390C23×C4—611no—
391C23×C4—611no—
392C23×C4—611no—
393C23×C4—611no—
394C23×C4—611no—
395C23×C4C2×#692311yescharacteristic
396C23×C4C2×#692311yescharacteristic
397C22×C8—612no—
398C22×C8—612no—
399C22×C8—612no—
400C22×C8—612no—
401C4×Q8—618no—
402C4×Q8—1224no—
403C4×D4—24no—
404C4×D4—1224no—
405C4×D4—1224no—
406C4×D4—1224no—
407C4×D4—1224no—
408C4×D4—1224no—
409#117—1224no—
410#117—1224no—
411#117—1224no—
412C2×C4⋊C4—611no—
413C2×C4⋊C4—611no—
414C2×C4⋊C4—611no—
415C2×C4⋊C4—611no—
416C2×C4⋊C4—611no—
417C2×C4⋊C4C2×#692311yescharacteristic
418C2×C22⋊C4—611no—
419C2×C22⋊C4—611no—
420C2×C22⋊C4—611no—
421C2×C22⋊C4—611no—
422C2×C22⋊C4C2×#692311yescharacteristic
423C2×C22⋊C4C2×#692311yescharacteristic
424C2×C42—611no—
425C2×C42—611no—
426C2×C42—611no—
427C2×C42—611no—
428C2×C42—611no—
429C2×C42—611no—
430C2×C42—611no—
431C2×C42—611no—
432C2×C42—611no—
433C2×C42—1224no—
434C2×C42—1224no—
435C2×C42—1224no—
436C2×C42C2×#692311yescharacteristic
437C4⋊C8—1224no—
438C4⋊C8—1224no—
439C4⋊C8—1224no—
440C22⋊C8—1224no—
441C22⋊C8—1224no—
442C22⋊C8—1224no—
443C22⋊C8—2442no—
444C22⋊C8—2442no—
445C22⋊C8—2442no—
446#97—1224no—
447#97—1224no—
448C4×C8—1224no—
449C24—611no—
450C24C22×#692311yescharacteristic
451C24C22×#692311yescharacteristic
452#41—12232no—
453C2×Q8—614no—
454C2×Q8—1222no—
455C2×Q8—1222no—
456C2×D4—22no—
457C2×D4—22no—
458C2×D4—22no—
459C2×D4—614no—
460C2×D4—1222no—
461C2×D4—1222no—
462C2×D4—1222no—
463C2×D4—1222no—
464C2×D4—1222no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—41no—
470C22×C4—41no—
471C22×C4—611no—
472C22×C4—611no—
473C22×C4—614no—
474C22×C4—611no—
475C22×C4—611no—
476C22×C4—611no—
477C22×C4—611no—
478C22×C4—611no—
479C22×C4—1222no—
480C22×C4—1222no—
481C22×C4—1221no—
482C22×C4—1222no—
483C22×C4—1222no—
484C22×C4—1221no—
485C22×C4—1222no—
486C22×C4—1222no—
487C22×C4—1222no—
488C22×C4—1222no—
489C22×C4—1222no—
490C22×C4—1222no—
491C22×C4—1222no—
492C22×C4—1221no—
493C22×C4—1221no—
494C22×C4—1222no—
495C22×C4—1222no—
496C22×C4—1221no—
497C22×C4—1221no—
498C22×C4—1222no—
499C22×C4—1222no—
500C22×C4—1222no—
501C22×C4—2441no—
502C22×C4C22×#692311yescharacteristic
503C22×C4C22×#692311yescharacteristic
504C22×C4C22×#692311yescharacteristic
505C22×C4C22×#692311yescharacteristic
506C22×C4C22×#692311yescharacteristic
507C2×C8—614no—
508C2×C8—1222no—
509C2×C8—1222no—
510C2×C8—1222no—
511C2×C8—1222no—
512C2×C8—1222no—
513C2×C8—1222no—
514C2×C8—1222no—
515C2×C8—1222no—
516C2×C8—1222no—
517C2×C8—1222no—
518C2×C8—1222no—
519C4⋊C4—614no—
520C4⋊C4—614no—
521C4⋊C4—1222no—
522C4⋊C4—1222no—
523C4⋊C4—1222no—
524C4⋊C4C22⋊#692314yes—
525C22⋊C4—22no—
526C22⋊C4—22no—
527C22⋊C4—1222no—
528C22⋊C4—1222no—
529C22⋊C4—1222no—
530C22⋊C4—1222no—
531C42—22no—
532C42—614no—
533C42—614no—
534C42—614no—
535C42—1222no—
536C42—1222no—
537C42—1222no—
538C42—1222no—
539C42—1222no—
540C42—1222no—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—612no—
554C23—1221no—
555C23—1221no—
556C23—1221no—
557C23—1221no—
558C23—1221no—
559C23—1221no—
560C23D61⋊M1612yes—
561C23D61⋊M1612yes—
562C23C23×#692311yescharacteristic
563Q8—1228no—
564D4—28no—
565D4—1228no—
566D4—1228no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—612no—
581C2×C4—612no—
582C2×C4—612no—
583C2×C4—611no—
584C2×C4—611no—
585C2×C4—612no—
586C2×C4—612no—
587C2×C4—612no—
588C2×C4—612no—
589C2×C4—612no—
590C2×C4—1228no—
591C2×C4—1228no—
592C2×C4—1228no—
593C2×C4—1221no—
594C2×C4—1221no—
595C2×C4—1221no—
596C2×C4—1221no—
597C2×C4—1221no—
598C2×C4—1221no—
599C2×C4—1221no—
600C2×C4—1221no—
601C2×C4—1221no—
602C2×C4—1221no—
603C2×C4—1221no—
604C2×C4—1221no—
605C2×C4—2444no—
606C2×C4—2442no—
607C2×C4—2442no—
608C2×C4—2442no—
609C2×C4—2442no—
610C2×C4—2442no—
611C2×C4—2442no—
612C2×C4C2×C22⋊#692312yes—
613C2×C4C2×C22⋊#692312yes—
614C2×C4C2×C22⋊#692312yes—
615C2×C4#1451986412yes—
616C8—1224no—
617C8—2442no—
618C8—2442no—
619C8—2442no—
620C22—22no—
621C22—22no—
622C22—22no—
623C22—22no—
624C22—22no—
625C22—22no—
626C22—41no—
627C22—41no—
628C22—41no—
629C22—41no—
630C22—41no—
631C22—41no—
632C22—41no—
633C22—41no—
634C22—1222no—
635C22—1222no—
636C22—1222no—
637C22—2441no—
638C22—2441no—
639C22—2441no—
640C22—2441no—
641C22C22×C22⋊#692311yescharacteristic
642C22#41×#692311yescharacteristic
643C22#1684083111yescharacteristic
644C22C2×D61⋊M1611yescharacteristic
645C22C2×D61⋊M1611yescharacteristic
646C22C2×#1451986411yescharacteristic
647C22#1684084011yescharacteristic
648C4—22no—
649C4—22no—
650C4—22no—
651C4—22no—
652C4—41no—
653C4—612no—
654C4—612no—
655C4—614no—
656C4—1222no—
657C4—1222no—
658C4—1222no—
659C4—2441no—
660C4—2441no—
661C2—42no—
662C2—42no—
663C2—2442no—
664C2#2146232411yescharacteristic
665C2D61⋊#40711yescharacteristic
666C2#2146240711yescharacteristic
667C2#2146262911yescharacteristic
668C2#2146265911yescharacteristic
669C2#2146273911yescharacteristic
670C2#2146274411yescharacteristic
subgroupG/H
center, FrattiniC23C23×#6923
commutator, socleC23×C61C23×C4
FittingC2×C4×D4×C61C4
prime radicalC22×#11758224C4
cosocleC22×Dic61C24
hypercenterC2×C4×D4#6923
hypocenterC61#22532
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. C2×#16840814C2semidirect
2. #21461944C2semidirect
3. #21461943C2semidirect
4. #21461942C2semidirect
5. #21461676C2semidirect
6. #21461675C2semidirect
7. #21461673C2semidirect
8. #21461546C2semidirect
9. #21461545C2semidirect
10. C4×C22⋊#92420C2semidirect
182. C2×#92420#41semidirect
380. C61#22532semidirect

Maximal subgroups

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

subgroupindex
C2×#168408142
#214619442
#214619432
#214619422
#214616762
#214616752
#214616732
#214615462
#214615452
C4×C22⋊#924202
C2×C4×#117582242
#2253261