Finite Group Databasebeta

← #30615162

Subgroups — #30615162

Order 15616.

1342 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×#16840787C212yesmaximal, complemented
2#21461711C211yesmaximal, characteristic, complemented
3#21461711C211yesmaximal, characteristic, complemented
4#21461678C212yesmaximal, complemented
5#21461676C211yesmaximal, characteristic, complemented
6#21461674C211yesmaximal, characteristic, complemented
7#21461644C212yesmaximal
8#21461548C212yesmaximal, complemented
9#21461546C211yesmaximal, characteristic, complemented
10C4×#14519834C211yesmaximal, characteristic, complemented
11C2×C4×#11758226C211yesmaximal, characteristic
12#16840787—28no—
13C2×#14519844C2212yes—
14C2×C4⋊#92420C2212yes—
15C2×#14519834C2212yes—
16C2×#14519834C2212yes—
17C2×#14519834C2212yes—
18C2×#14519834C2212yes—
19C2×#14519834C2212yes—
20C2×#14519834C2212yes—
21C2×#14519832C2212yes—
22C2×C4×#92420C2212yes—
23#16840628C2211yescharacteristic
24#16840628C2211yescharacteristic
25#16840628C2211yescharacteristic
26#16840628C2211yescharacteristic
27#16840628C2212yes—
28#16840628C2212yes—
29C22×#11758226C2211yescharacteristic
30C4×#11758226—28no—
31C2×Q8×Dic61C411yescharacteristic
32C2×C4×D244C411yescharacteristic
33C2×C4×D244C411yescharacteristic
34C2×C4×D244C411yescharacteristic
35C2×#14519739C411yescharacteristic
36C2×#14519739C411yescharacteristic
37C2×#14519739C411yescharacteristic
38C2×#14519735C2211yescharacteristic
39C2×#14519735C2211yescharacteristic
40C2×#14519735C2211yescharacteristic
41C2×C42×D61C2211yescharacteristic
42C2×C42×D61C2211yescharacteristic
43C2×C42×D61C2211yescharacteristic
44C2×C4×Q8×C61C411yescharacteristic
45C22×#92420C2312yes—
46C22×#92420C2312yes—
47C22×#92420C2312yes—
48C22×#92420C2312yes—
49#14519844—24no—
50C4⋊#92420—24no—
51#14519834—24no—
52#14519834—24no—
53#14519834—24no—
54#14519834—24no—
55#14519834—24no—
56#14519834—24no—
57#14519832—24no—
58C4×#92420—24no—
59C2×#11758226—24no—
60C2×#11758226—24no—
61C2×#11758226—22no—
62C2×#11758226—22no—
63C22×D244C2×C411yescharacteristic
64C22×D244C2×C411yescharacteristic
65C22×D244C2×C411yescharacteristic
66C22×C4×D61C2×C411yescharacteristic
67C22×C4×D61C2×C411yescharacteristic
68C22×C4×D61C2×C411yescharacteristic
69C22×C4×D61C2×C411yescharacteristic
70C22×C4×D61C2×C411yescharacteristic
71C22×C4×D61C2×C411yescharacteristic
72C22×C4×D61C2311yescharacteristic
73C22×C4×D61C2311yescharacteristic
74C22×C4×D61C2311yescharacteristic
75Q8×Dic61—24no—
76C4×D244—24no—
77C4×D244—24no—
78C4×D244—24no—
79#14519739—24no—
80#14519739—24no—
81#14519739—24no—
82#14519735—24no—
83#14519735—24no—
84#14519735D418yes—
85C2×C4⋊Dic61C2×C411yescharacteristic
86C2×C4⋊Dic61C2×C411yescharacteristic
87C2×C4⋊Dic61C2×C411yescharacteristic
88C2×#11758198C2×C411yescharacteristic
89C2×#11758198C2×C411yescharacteristic
90C2×#11758198C2×C411yescharacteristic
91C2×#11758198C2×C411yescharacteristic
92C2×#11758198C2×C411yescharacteristic
93C2×#11758198C2×C411yescharacteristic
94C2×C4×Dic61C2×C411yescharacteristic
95C2×C4×Dic61C2×C411yescharacteristic
96C2×C4×Dic61C2×C411yescharacteristic
97C2×C4×Dic61C2311yescharacteristic
98C2×C4×Dic61C2311yescharacteristic
99C2×C4×Dic61C2311yescharacteristic
100C2×C4×Dic61C2311yescharacteristic
101C42×D61—24no—
102C42×D61—24no—
103C42×D61—24no—
104C22×Q8×C61C2×C411yescharacteristic
105C4×Q8×C61—24no—
106C2×C4⋊C4×C61C2×C411yescharacteristic
107C2×C4⋊C4×C61C2×C411yescharacteristic
108C2×C4⋊C4×C61C2×C411yescharacteristic
109C2×C42×C61C2×C411yescharacteristic
110C2×C42×C61C2×C411yescharacteristic
111C2×C42×C61C2×C411yescharacteristic
112C23×D61C22×C411yescharacteristic
113C23×D61C22×C411yescharacteristic
114C23×D61C22×C411yescharacteristic
115C2×Q8×C61—22no—
116C2×Q8×C61—22no—
117C2×Q8×C61—22no—
118C22×C4×C61C22×C411yescharacteristic
119C22×C4×C61C22×C411yescharacteristic
120C22×C4×C61C22×C411yescharacteristic
121C22×C4×C61C22×C411yescharacteristic
122C22×C4×C61C22×C411yescharacteristic
123C22×C4×C61C22×C411yescharacteristic
124C22×C4×C61C22×C411yescharacteristic
125C22×Dic61C22×C411yescharacteristic
126C22×Dic61C22×C411yescharacteristic
127C22×Dic61C22×C411yescharacteristic
128C22×Dic61C22×C411yescharacteristic
129C22×Dic61C2411yescharacteristic
130#11758226—216no—
131#11758226—216no—
132C2×D244—22no—
133C2×D244—22no—
134C2×D244—22no—
135C2×D244—22no—
136C2×D244—22no—
137C2×D244—22no—
138C2×D244—22no—
139C2×D244—22no—
140C2×D244—22no—
141C2×C4×D61—22no—
142C2×C4×D61—22no—
143C2×C4×D61—22no—
144C2×C4×D61—21no—
145C2×C4×D61—21no—
146C2×C4×D61—22no—
147C2×C4×D61—21no—
148C2×C4×D61—21no—
149C2×C4×D61—22no—
150C2×C4×D61—22no—
151C2×C4×D61—22no—
152C2×C4×D61—22no—
153C2×C4×D61—22no—
154C2×C4×D61—22no—
155C2×C4×D61—22no—
156C2×C4×D61—22no—
157C2×C4×D61—22no—
158C2×C4×D61—22no—
159C2×C4×D61—22no—
160C2×C4×D61—22no—
161C2×C4×D61—41no—
162C2×C4×D61—41no—
163C2×C4×D61—41no—
164C2×C4×D61—41no—
165C2×C4×D61—41no—
166C2×C4×D61—41no—
167C2×C4×D61C2×D414yes—
168C2×C4×D61#4114yes—
169C2×C4×D61#4112yes—
170C2×C4×D61#4112yes—
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—22no—
183C4⋊C4×C61—22no—
184C4⋊C4×C61—22no—
185C4⋊C4×C61C22⋊C414yes—
186C42×C61—22no—
187C42×C61—22no—
188C42×C61—22no—
189#11758198—22no—
190#11758198—22no—
191#11758198—22no—
192#11758198—22no—
193#11758198C22⋊C414yes—
194#11758198C22⋊C414yes—
195C4⋊Dic61—22no—
196C4⋊Dic61—22no—
197C4⋊Dic61C22⋊C414yes—
198C4×Dic61—22no—
199C4×Dic61—22no—
200C4×Dic61—22no—
201C4×Dic61—22no—
202C4×Dic61—22no—
203C4×Dic61C2×D414yes—
204C4×Dic61C2×D414yes—
205C23×C61C23×C411yescharacteristic
206C22×D61—21no—
207C22×D61—21no—
208C22×D61—21no—
209C22×D61—21no—
210C22×D61—21no—
211C22×D61—21no—
212C22×D61—21no—
213C22×D61—21no—
214C22×D61—21no—
215C22×D61—21no—
216C22×D61—21no—
217C22×D61—21no—
218C22×D61—21no—
219C22×D61—21no—
220C22×D61—21no—
221C22×D61—21no—
222C22×D61—21no—
223C22×D61—21no—
224C22×D61C2×C22⋊C412yes—
225C22×D61#11712yes—
226C22×D61#11712yes—
227Q8×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—21no—
240C2×C4×C61—21no—
241C2×C4×C61—21no—
242C2×C4×C61—21no—
243C2×C4×C61C2×C22⋊C412yes—
244C2×C4×C61C2×C22⋊C412yes—
245C2×C4×C61C2×C22⋊C412yes—
246C2×C4×C61#11712yes—
247C2×C4×C61#11712yes—
248C2×C4×C61#13112yes—
249C2×Dic61—21no—
250C2×Dic61—21no—
251C2×Dic61—21no—
252C2×Dic61—21no—
253C2×Dic61—21no—
254C2×Dic61—21no—
255C2×Dic61—21no—
256C2×Dic61—21no—
257C2×Dic61—21no—
258C2×Dic61C2×C22⋊C412yes—
259C2×Dic61C2×C22⋊C412yes—
260C2×Dic61#13112yes—
261C2×Dic61C22×D412yes—
262C2×Dic61C2×#4112yes—
263C2×Dic61C2×#4111yescharacteristic
264C2×Dic61C2×#4111yescharacteristic
265D244—28no—
266D244—28no—
267D244—28no—
268C4×D61—24no—
269C4×D61—24no—
270C4×D61—24no—
271C4×D61—24no—
272C4×D61—24no—
273C4×D61—24no—
274C4×D61—44no—
275C4×D61—44no—
276C4×D61—44no—
277C4×D61—44no—
278C4×D61—44no—
279C4×D61—44no—
280#92420—42no—
281#92420—42no—
282#92420—42no—
283#92420—42no—
284#20720—611nomaximal, Sylow
285C22×C61C22×C22⋊C411yescharacteristic
286C22×C61C2×#11711yescharacteristic
287C22×C61C2×#11711yescharacteristic
288C22×C61C2×#13111yescharacteristic
289C22×C61C2×#13111yescharacteristic
290C22×C61#56811yescharacteristic
291C22×C61#56811yescharacteristic
292C2×D61—22no—
293C2×D61—22no—
294C2×D61—22no—
295C2×D61—22no—
296C2×D61—22no—
297C2×D61—22no—
298C2×D61—22no—
299C2×D61—22no—
300C2×D61—22no—
301C2×D61—41no—
302C2×D61—41no—
303C2×D61—41no—
304C2×D61—41no—
305C2×D61—41no—
306C2×D61—41no—
307C2×D61—41no—
308C2×D61—41no—
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—22no—
318C4×C61—22no—
319C4×C61—41no—
320Dic61—22no—
321Dic61—22no—
322Dic61—22no—
323Dic61—41no—
324Dic61#52512yes—
325Dic61#52512yes—
326C2×C4×#41—611no—
327C2×#433—612no—
328#1817—611no—
329#1817—611no—
330#1798—612no—
331#1797—611no—
332#1797—611no—
333#1780—612no—
334#1750—612no—
335#1749—611no—
336C4×C22⋊C8—611no—
337C2×C61#228011yescharacteristic
338C2×C61#288011yescharacteristic
339C2×C61#288011yescharacteristic
340C2×C61#292211yescharacteristic
341C2×C61#292311yescharacteristic
342C2×C61#292511yescharacteristic
343C2×C61#292511yescharacteristic
344D61—42no—
345D61—42no—
346D61—42no—
347C22×#41—611no—
348C4×#41—1228no—
349C2×C4×Q8—611no—
350C2×C4×Q8#692311yescharacteristic
351C2×C4×D4—611no—
352C2×C4×D4—611no—
353C2×C4×D4—611no—
354C2×C4×D4—611no—
355C2×C4×D4—611no—
356C2×C4×D4—611no—
357C2×#117—611no—
358C2×#117—611no—
359C2×#117—611no—
360C22×C42—611no—
361C22×C42—611no—
362C22×C42—611no—
363#433—1228no—
364C2×C4⋊C8—612no—
365C2×C4⋊C8—612no—
366C2×C22⋊C8—612no—
367C2×C22⋊C8—612no—
368C2×C22⋊C8—612no—
369C2×C22⋊C8—612no—
370C2×C22⋊C8—612no—
371C2×C22⋊C8—612no—
372C2×#97—612no—
373C2×C4×C8—612no—
374#336—611no—
375#336—611no—
376#336—611no—
377#336—611no—
378#336—612no—
379#336—612no—
380C61#2072011yescharacteristic, complemented, Sylow
381C2×#41—1224no—
382C2×#41—1224no—
383C2×#41—1222no—
384C2×#41—1222no—
385C22×Q8C2×#692311yescharacteristic
386C22×D4—611no—
387C22×D4—611no—
388C22×D4—611no—
389C23×C4—611no—
390C23×C4—611no—
391C23×C4—611no—
392C23×C4—611no—
393C23×C4—611no—
394C23×C4—611no—
395C23×C4—611no—
396C23×C4—611no—
397C23×C4—611no—
398C22×C8—612no—
399C22×C8—612no—
400C22×C8—612no—
401C22×C8—612no—
402C4×Q8—24no—
403C4×Q8—1224no—
404C4×D4—1224no—
405C4×D4—1224no—
406C4×D4—1224no—
407C4×D4—1224no—
408C4×D4—1224no—
409C4×D4—1224no—
410#117—618no—
411#117—1224no—
412#117—1224no—
413C2×C4⋊C4—611no—
414C2×C4⋊C4—611no—
415C2×C4⋊C4—611no—
416C2×C4⋊C4C2×#692311yescharacteristic
417C2×C4⋊C4C2×#692311yescharacteristic
418C2×C4⋊C4C2×#692311yescharacteristic
419C2×C22⋊C4—611no—
420C2×C22⋊C4—611no—
421C2×C22⋊C4—611no—
422C2×C22⋊C4—611no—
423C2×C22⋊C4—611no—
424C2×C22⋊C4—611no—
425C2×C42—611no—
426C2×C42—611no—
427C2×C42—611no—
428C2×C42—611no—
429C2×C42—611no—
430C2×C42—611no—
431C2×C42—611no—
432C2×C42—1224no—
433C2×C42—1224no—
434C2×C42—1224no—
435C2×C42C2×#692311yescharacteristic
436C2×C42C2×#692311yescharacteristic
437C2×C42C2×#692311yescharacteristic
438C4⋊C8—1224no—
439C4⋊C8—1224no—
440C22⋊C8—1224no—
441C22⋊C8—1224no—
442C22⋊C8—1224no—
443C22⋊C8—1224no—
444C22⋊C8—1224no—
445C22⋊C8—1224no—
446#97—1224no—
447C4×C8—1224no—
448C24—611no—
449C24—611no—
450C24—611no—
451#41—12216no—
452#41—12216no—
453C2×Q8—22no—
454C2×Q8—22no—
455C2×Q8—22no—
456C2×D4—1222no—
457C2×D4—1222no—
458C2×D4—1222no—
459C2×D4—1222no—
460C2×D4—1222no—
461C2×D4—1222no—
462C2×D4—1222no—
463C2×D4—1222no—
464C2×D4—1222no—
465C22×C4—611no—
466C22×C4—611no—
467C22×C4—611no—
468C22×C4—611no—
469C22×C4—614no—
470C22×C4—614no—
471C22×C4—612no—
472C22×C4—612no—
473C22×C4—611no—
474C22×C4—1222no—
475C22×C4—1222no—
476C22×C4—1222no—
477C22×C4—1221no—
478C22×C4—1221no—
479C22×C4—1222no—
480C22×C4—1221no—
481C22×C4—1221no—
482C22×C4—1222no—
483C22×C4—1222no—
484C22×C4—1222no—
485C22×C4—1222no—
486C22×C4—1222no—
487C22×C4—1222no—
488C22×C4—1222no—
489C22×C4—1222no—
490C22×C4—1222no—
491C22×C4—1222no—
492C22×C4—1222no—
493C22×C4—1222no—
494C22×C4—2441no—
495C22×C4—2441no—
496C22×C4—2441no—
497C22×C4—2441no—
498C22×C4—2441no—
499C22×C4—2441no—
500C22×C4C22×#692311yescharacteristic
501C22×C4C22×#692311yescharacteristic
502C22×C4C22×#692311yescharacteristic
503C22×C4C22×#692311yescharacteristic
504C22×C4C22×#692311yescharacteristic
505C22×C4C22×#692311yescharacteristic
506C22×C4C22×#692311yescharacteristic
507C2×C8—1222no—
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—22no—
520C4⋊C4—22no—
521C4⋊C4—614no—
522C4⋊C4—1222no—
523C4⋊C4—1222no—
524C4⋊C4C22⋊#692314yes—
525C22⋊C4—614no—
526C22⋊C4—614no—
527C22⋊C4—1222no—
528C22⋊C4—1222no—
529C22⋊C4—1222no—
530C22⋊C4—1222no—
531C42—22no—
532C42—22no—
533C42—22no—
534C42—614no—
535C42—614no—
536C42—1222no—
537C42—1222no—
538C42—1222no—
539C42—1222no—
540C42—1222no—
541C23—612no—
542C23—612no—
543C23—612no—
544C23—1221no—
545C23—1221no—
546C23—1221no—
547C23—1221no—
548C23—1221no—
549C23—1221no—
550C23—1221no—
551C23—1221no—
552C23—1221no—
553C23—1221no—
554C23—1221no—
555C23—1221no—
556C23—1221no—
557C23—1221no—
558C23—1221no—
559C23—1221no—
560C23—1221no—
561C23—1221no—
562C23C23×#692311yescharacteristic
563Q8—28no—
564D4—1228no—
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—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—612no—
583C2×C4—612no—
584C2×C4—611no—
585C2×C4—611no—
586C2×C4—612no—
587C2×C4—612no—
588C2×C4—612no—
589C2×C4—1224no—
590C2×C4—1224no—
591C2×C4—1224no—
592C2×C4—1224no—
593C2×C4—1224no—
594C2×C4—1224no—
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—2444no—
605C2×C4—2444no—
606C2×C4—2444no—
607C2×C4—2444no—
608C2×C4—2444no—
609C2×C4—2444no—
610C2×C4C2×C22⋊#692312yes—
611C2×C4C2×C22⋊#692312yes—
612C2×C4C2×C22⋊#692312yes—
613C2×C4#1451985612yes—
614C2×C4#1451985612yes—
615C2×C4#1451986512yes—
616C8—2442no—
617C8—2442no—
618C8—2442no—
619C8—2442no—
620C22—1222no—
621C22—1222no—
622C22—1222no—
623C22—1222no—
624C22—1222no—
625C22—1222no—
626C22—1222no—
627C22—1222no—
628C22—1222no—
629C22—2441no—
630C22—2441no—
631C22—2441no—
632C22—2441no—
633C22—2441no—
634C22—2441no—
635C22—2441no—
636C22—2441no—
637C22—2441no—
638C22—2441no—
639C22—2441no—
640C22—2441no—
641C22C22×C22⋊#692311yescharacteristic
642C22C2×#1451985611yescharacteristic
643C22C2×#1451985611yescharacteristic
644C22D61⋊#13111yescharacteristic
645C22C2×#1451986511yescharacteristic
646C22#1684084111yescharacteristic
647C22#1684084211yescharacteristic
648C4—22no—
649C4—22no—
650C4—22no—
651C4—22no—
652C4—22no—
653C4—22no—
654C4—41no—
655C4—612no—
656C4—612no—
657C4—1222no—
658C4—1222no—
659C4—1222no—
660C4—2441no—
661C2—2442no—
662C2—2442no—
663C2—2442no—
664C2#2146226311yescharacteristic
665C2#2146240811yescharacteristic
666C2#2146240911yescharacteristic
667C2#2146250411yescharacteristic
668C2#2146250711yescharacteristic
669C2#2146251811yescharacteristic
670C2#2146251911yescharacteristic
subgroupG/H
center, FrattiniC23C23×#6923
commutator, socleC23×C61C23×C4
FittingC2×C4×Q8×C61C4
prime radicalC22×#11758226C22
cosocleC22×Dic61C24
hypercenterC2×C4×Q8#6923
hypocenterC61#20720
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#16840787C2semidirect
2. #21461711C2semidirect
3. #21461711C2semidirect
4. #21461678C2semidirect
5. #21461676C2semidirect
6. #21461674C2semidirect
8. #21461548C2semidirect
9. #21461546C2semidirect
10. C4×#14519834C2semidirect
380. C61#20720semidirect

Maximal subgroups

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

subgroupindex
C2×#168407872
#214617112
#214616782
#214616762
#214616742
#214616442
#214615482
#214615462
C4×#145198342
C2×C4×#117582262
#2072061