Finite Group Databasebeta

← #30620336

Subgroups — #30620336

Order 15616.

1068 conjugacy classes of subgroups.

The subgroup lattice has 704 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D4×#11758217C212yesmaximal
2#21462013C212yesmaximal, complemented
3#21461846C212yesmaximal, complemented
4#21461846C212yesmaximal, complemented
5#21461721C211yesmaximal, characteristic, complemented
6#21461718C211yesmaximal, characteristic, complemented
7#21461686C211yesmaximal, characteristic, complemented
8#21461572C211yesmaximal, characteristic, complemented
9#21461525C211yesmaximal, characteristic, complemented
10C2×#16840642C211yesmaximal, characteristic, complemented
11C2×#16840449C211yesmaximal, characteristic
12C22×#11758217C2212yes—
13C22×#11758217C2212yes—
14#16840797—22no—
15#16840793—22no—
16#16840793—22no—
17D4×#92420—22no—
18#16840765C2212yes—
19C2×#14519838C2212yes—
20C2×#14519838C2212yes—
21#16840732C2212yes—
22#16840732C2212yes—
23C4×#11758217C2212yes—
24#16840642C2211yescharacteristic
25#16840642C2212yes—
26#16840642C2211yescharacteristic
27#16840642C2212yes—
28#16840642C2211yescharacteristic
29#16840642C2211yescharacteristic
30#16840642C2212yes—
31#16840642C2212yes—
32#16840449—22no—
33#16840449—22no—
34#16840449—22no—
35#16840449—22no—
36C2×#14519770C411yescharacteristic
37C2×#14519770C411yescharacteristic
38C2×#14519768C411yescharacteristic
39C2×#14519767C411yescharacteristic
40C2×#14519767C2211yescharacteristic
41C2×#14519758C411yescharacteristic
42C2×#14519754C2211yescharacteristic
43C2×#14519754C2211yescharacteristic
44C2×#14519742C2211yescharacteristic
45C2×D4×Dic61C2211yescharacteristic
46C2×#14519736C411yescharacteristic
47C2×#14519733C411yescharacteristic
48C2×#14519733C2211yescharacteristic
49C22×#11758196C2211yescharacteristic
50C2×#123×C61C411yescharacteristic
51C2×#11758217—22no—
52C2×#11758217—22no—
53C2×#11758217—24no—
54C2×#11758217—24no—
55C2×#11758217—42no—
56C2×#11758217—42no—
57C2×#11758217C2312yes—
58C2×#11758217C2312yes—
59C2×#11758217C2312yes—
60C2×#11758217C2312yes—
61C22×#92420—22no—
62C22×#92420—22no—
63#14519844—22no—
64C4⋊#92420—22no—
65#14519838—24no—
66#14519838—42no—
67C22⋊#92420—22no—
68C22⋊#92420—22no—
69C22⋊#92420—22no—
70C22⋊#92420—22no—
71#14519832—22no—
72C4×#92420—22no—
73C22×Dic122C2311yescharacteristic
74C23×Dic61C2×C411yescharacteristic
75C23×Dic61C2311yescharacteristic
76C23×Dic61C2311yescharacteristic
77#14519770—22no—
78#14519770—22no—
79#14519770—22no—
80#14519770—22no—
81#14519768—22no—
82#14519768—22no—
83#14519767—22no—
84#14519767—22no—
85#14519767—22no—
86#14519767—22no—
87#14519758—22no—
88#14519758—22no—
89#14519754—22no—
90#14519754—22no—
91#14519754—22no—
92#14519754—22no—
93#14519742—22no—
94#14519742—22no—
95D4×Dic61D412yes—
96D4×Dic61D412yes—
97D4×Dic61D412yes—
98D4×Dic61D412yes—
99#14519736—21no—
100#14519736—21no—
101#14519736—21no—
102#14519736—21no—
103#14519733—22no—
104#14519733—22no—
105#14519733—21no—
106#14519733—21no—
107#14519733—21no—
108#14519733—21no—
109C2×C4⋊Dic61C2×C411yescharacteristic
110C2×C4⋊Dic61C2×C411yescharacteristic
111C2×C4⋊Dic61C2×C411yescharacteristic
112C2×C4⋊Dic61C2×C411yescharacteristic
113C2×#11758196—22no—
114C2×#11758196—22no—
115C2×#11758196C2×C411yescharacteristic
116C2×#11758196C2×C411yescharacteristic
117C2×#11758196C2×C411yescharacteristic
118C2×#11758196C2×C411yescharacteristic
119C2×#11758196C2×C411yescharacteristic
120C2×#11758196C2×C411yescharacteristic
121C2×#11758196C2×C411yescharacteristic
122C2×#11758196C2311yescharacteristic
123C2×#11758196C2311yescharacteristic
124C2×#11758196C2311yescharacteristic
125C2×C22⋊Dic61C2×C411yescharacteristic
126C2×C22⋊Dic61C2×C411yescharacteristic
127C2×C22⋊Dic61C2×C411yescharacteristic
128C2×C22⋊Dic61C2×C411yescharacteristic
129C2×C22⋊Dic61C2×C411yescharacteristic
130C2×C4×Dic61C2×C411yescharacteristic
131C2×C4×Dic61C2×C411yescharacteristic
132C2×C4×Dic61C2×C411yescharacteristic
133C2×C4×Dic61C2×C411yescharacteristic
134C2×C4×Dic61C2311yescharacteristic
135C22×D4×C61C2×C411yescharacteristic
136C23×C4×C61C2×C411yescharacteristic
137#123×C61—22no—
138#123×C61—22no—
139C2×C4⋊C4×C61C2×C411yescharacteristic
140C2×C4⋊C4×C61C2×C411yescharacteristic
141C2×C22⋊C4×C61C2×C411yescharacteristic
142C2×C22⋊C4×C61C2×C411yescharacteristic
143C2×C22⋊C4×C61C2×C411yescharacteristic
144C24×C61C22×C411yescharacteristic
145C24×C61C22×C411yescharacteristic
146C2×D4×C61—42no—
147C2×D4×C61C22⋊C412yes—
148C2×D4×C61C22⋊C412yes—
149C22×C4×C61—21no—
150C22×C4×C61—21no—
151C22×C4×C61—42no—
152C22×C4×C61C22×C411yescharacteristic
153C22×C4×C61C22×C411yescharacteristic
154C22×C4×C61C22×C411yescharacteristic
155C22×C4×C61C22×C411yescharacteristic
156C22×C4×C61C22×C411yescharacteristic
157C22×Dic61—22no—
158C22×Dic61—22no—
159C22×Dic61—21no—
160C22×Dic61—21no—
161C22×Dic61—21no—
162C22×Dic61—21no—
163C22×Dic61—21no—
164C22×Dic61—21no—
165C22×Dic61—42no—
166C22×Dic61C22×C411yescharacteristic
167C22×Dic61C22×C411yescharacteristic
168C22×Dic61C22×C411yescharacteristic
169C22×Dic61C22×C411yescharacteristic
170C22×Dic61C22×C411yescharacteristic
171C22×Dic61C22×C411yescharacteristic
172C22×Dic61C22×C411yescharacteristic
173C22×Dic61C2×D412yes—
174C22×Dic61C2×D412yes—
175C22×Dic61C2×D412yes—
176C22×Dic61C2×D412yes—
177C22×Dic61C2411yescharacteristic
178C2×Dic122—22no—
179C2×Dic122—22no—
180C2×Dic122—21no—
181C2×Dic122—21no—
182#11758217—22no—
183#11758217—22no—
184#11758217—22no—
185#11758217—22no—
186#11758217—22no—
187#11758217—22no—
188#11758217—22no—
189#11758217—44no—
190#11758217—44no—
191#11758217#4114yes—
192C2×#92420—22no—
193C2×#92420—22no—
194C2×#92420—22no—
195C2×#92420—22no—
196C2×#92420—44no—
197C2×#92420—44no—
198C4⋊C4×C61—22no—
199C4⋊C4×C61—22no—
200C22⋊C4×C61—21no—
201C22⋊C4×C61—21no—
202C22⋊C4×C61—22no—
203C22⋊C4×C61—22no—
204C22⋊Dic61—22no—
205C22⋊Dic61—21no—
206C22⋊Dic61—21no—
207C22⋊Dic61—22no—
208C22⋊Dic61C22⋊C411yescharacteristic
209C22⋊Dic61C22⋊C411yescharacteristic
210C22⋊Dic61C22⋊C411yescharacteristic
211C22⋊Dic61C22⋊C411yescharacteristic
212C22⋊Dic61C22⋊C412yes—
213C22⋊Dic61C22⋊C412yes—
214C4⋊Dic61—21no—
215C4⋊Dic61—21no—
216C4⋊Dic61—21no—
217C4⋊Dic61—21no—
218C4⋊Dic61—22no—
219C4⋊Dic61C22⋊C412yes—
220C4⋊Dic61C22⋊C412yes—
221#11758196—22no—
222#11758196—22no—
223#11758196—22no—
224#11758196—22no—
225#11758196—21no—
226#11758196—21no—
227#11758196—21no—
228#11758196—21no—
229#11758196—22no—
230#11758196—22no—
231#11758196—22no—
232#11758196—22no—
233C4×Dic61—21no—
234C4×Dic61—21no—
235C4×Dic61—21no—
236C4×Dic61—21no—
237C4×Dic61—21no—
238C4×Dic61—21no—
239C4×Dic61—21no—
240C4×Dic61—21no—
241C4×Dic61C2×D412yes—
242C4×Dic61C2×D412yes—
243C23×C61—21no—
244C23×C61—21no—
245C23×C61—22no—
246C23×C61—21no—
247C23×C61—21no—
248C23×C61—41no—
249C23×C61—41no—
250C23×C61—41no—
251C23×C61C2×C22⋊C412yes—
252C23×C61C2×C22⋊C411yescharacteristic
253C23×C61C2×C22⋊C411yescharacteristic
254C23×C61C23×C411yescharacteristic
255D4×C61—24no—
256D4×C61—42no—
257C2×C4×C61—21no—
258C2×C4×C61—21no—
259C2×C4×C61—21no—
260C2×C4×C61—21no—
261C2×C4×C61—21no—
262C2×C4×C61—21no—
263C2×C4×C61—41no—
264C2×C4×C61—42no—
265C2×C4×C61—42no—
266C2×C4×C61—41no—
267C2×C4×C61—41no—
268C2×C4×C61—41no—
269C2×C4×C61C2×C22⋊C412yes—
270C2×Dic61—21no—
271C2×Dic61—21no—
272C2×Dic61—21no—
273C2×Dic61—21no—
274C2×Dic61—22no—
275C2×Dic61—22no—
276C2×Dic61—22no—
277C2×Dic61—22no—
278C2×Dic61—22no—
279C2×Dic61—22no—
280C2×Dic61—22no—
281C2×Dic61—22no—
282C2×Dic61—21no—
283C2×Dic61—21no—
284C2×Dic61—21no—
285C2×Dic61—21no—
286C2×Dic61—21no—
287C2×Dic61—21no—
288C2×Dic61—41no—
289C2×Dic61—42no—
290C2×Dic61—42no—
291C2×Dic61—41no—
292C2×Dic61—41no—
293C2×Dic61—41no—
294C2×Dic61C2×C22⋊C411yescharacteristic
295C2×Dic61C2×C22⋊C411yescharacteristic
296C2×Dic61C2×C22⋊C411yescharacteristic
297C2×Dic61C2×C22⋊C411yescharacteristic
298C2×Dic61C2×C22⋊C412yes—
299C2×Dic61C22×D412yes—
300C2×Dic61C2×#4112yes—
301C2×Dic61H(5,𝔽2)11yescharacteristic
302C2×Dic612−1+411yescharacteristic
303Dic122—42no—
304Dic122—42no—
305#92420—24no—
306#92420—42no—
307#92420—42no—
308#92420—42no—
309#22480—611nomaximal, Sylow
310C22×C61—21no—
311C22×C61—21no—
312C22×C61—21no—
313C22×C61—21no—
314C22×C61—21no—
315C22×C61—22no—
316C22×C61—21no—
317C22×C61—22no—
318C22×C61—21no—
319C22×C61—22no—
320C22×C61—42no—
321C22×C61—42no—
322C22×C61—81no—
323C22×C61—81no—
324C22×C61#41312yes—
325C22×C61C22×C22⋊C411yescharacteristic
326C22×C61C4×#4111yescharacteristic
327C22×C61#52011yescharacteristic
328C4×C61—22no—
329C4×C61—41no—
330C4×C61—41no—
331C4×C61—41no—
332C4×C61—41no—
333Dic61—21no—
334Dic61—22no—
335Dic61—21no—
336Dic61—21no—
337Dic61—21no—
338Dic61—41no—
339Dic61—41no—
340Dic61—41no—
341Dic61—41no—
342Dic61#54612yes—
343Dic61#54712yes—
344C2×#552—611no—
345D4×M16—612no—
346#1998—612no—
347#1901—612no—
348#1901—612no—
349#1822—611no—
350#1821—611no—
351#1801—611no—
352C4×#101—611no—
353#1742—611no—
354C2×#343—611no—
355C2×C61—21no—
356C2×C61—21no—
357C2×C61—22no—
358C2×C61—41no—
359C2×C61—42no—
360C2×C61#233911yescharacteristic
361C2×C61C2×#41311yescharacteristic
362C2×C61#288911yescharacteristic
363C22×M16—612no—
364C22×M16—612no—
365#552—1222no—
366#552—1222no—
367#552—1222no—
368#552—1222no—
369C2×#126—611no—
370C2×#126—611no—
371C2×#125—611no—
372C2×#125—611no—
373C2×#125—611no—
374C2×#123—611no—
375C2×#123#692311yescharacteristic
376C2×C22⋊Q8—611no—
377C2×C22⋊Q8—611no—
378C2×C4×Q8—611no—
379C2×C4×D4—611no—
380C2×#117—611no—
381C2×#117—611no—
382C2×#117—611no—
383C22×C4⋊C4—611no—
384#436—1222no—
385#435—1222no—
386#435—1222no—
387C8×D4—1222no—
388#423—612no—
389C2×#101—612no—
390C2×#101—612no—
391#407—612no—
392#407—612no—
393C4×M16—612no—
394#343—611no—
395#343—611no—
396#343—611no—
397#343—612no—
398#343—612no—
399#343—612no—
400#343—612no—
401#343—611no—
402C61#2248011yescharacteristic, complemented, Sylow
403C22×Q8—611no—
404C22×D4C2×#692311yescharacteristic
405C23×C4—611no—
406C23×C4—611no—
407C23×C4—611no—
408C23×C4C2×#692311yescharacteristic
409C2×M16—612no—
410C2×M16—612no—
411C2×M16—612no—
412C2×M16—612no—
413C2×M16—1222no—
414C2×M16—1222no—
415C2×M16—1224no—
416C2×M16—1224no—
417C2×M16—2442no—
418C2×M16—2442no—
419C22×C8—1222no—
420C22×C8—1222no—
421#126—1222no—
422#126—1222no—
423#126—1222no—
424#126—1222no—
425#125—1222no—
426#125—1222no—
427#125—1222no—
428#125—1222no—
429#125—1222no—
430#125—1222no—
431#123—22no—
432#123—22no—
433#123—1222no—
434#123—1222no—
435C22⋊Q8—1222no—
436C22⋊Q8—1222no—
437C22⋊Q8—1222no—
438C22⋊Q8—1222no—
439C4×Q8—1222no—
440C4×Q8—1222no—
441C4×D4—612no—
442C4×D4—612no—
443C4×D4—612no—
444C4×D4—612no—
445#117—1221no—
446#117—1221no—
447#117—1221no—
448#117—1221no—
449#117—1222no—
450#117—1222no—
451#117—1221no—
452#117—1221no—
453#117—1221no—
454#117—1221no—
455C2×C4⋊C4—611no—
456C2×C4⋊C4—611no—
457C2×C4⋊C4—611no—
458C2×C4⋊C4—611no—
459C2×C4⋊C4—611no—
460C2×C4⋊C4—611no—
461C2×C4⋊C4—611no—
462C2×C4⋊C4—611no—
463C2×C4⋊C4—611no—
464C2×C4⋊C4—611no—
465C2×C4⋊C4—611no—
466C2×C4⋊C4—611no—
467C2×C4⋊C4—611no—
468C2×C4⋊C4—611no—
469C2×C4⋊C4—1222no—
470C2×C4⋊C4—1222no—
471C2×C4⋊C4C2×#692311yescharacteristic
472C2×C4⋊C4C2×#692311yescharacteristic
473C2×C22⋊C4—611no—
474C2×C22⋊C4—611no—
475C2×C22⋊C4—611no—
476C2×C22⋊C4—611no—
477C2×C22⋊C4—611no—
478C2×C22⋊C4C2×#692311yescharacteristic
479C2×C22⋊C4C2×#692311yescharacteristic
480C2×C22⋊C4C2×#692311yescharacteristic
481C2×C42—611no—
482C2×C42—611no—
483C2×C42—611no—
484C2×C42—611no—
485C2×C42—611no—
486C4⋊C8—1222no—
487C4⋊C8—1222no—
488#101—1224no—
489#101—2442no—
490C22⋊C8—1222no—
491C22⋊C8—1222no—
492C22⋊C8—1222no—
493C22⋊C8—1222no—
494#97—1222no—
495C4×C8—1222no—
496C24C22×#692311yescharacteristic
497C24C22×#692311yescharacteristic
498C2×Q8—1222no—
499C2×Q8—1221no—
500C2×Q8—1222no—
501C2×Q8—1221no—
502C2×D4—42no—
503C2×D4C22⋊#692312yes—
504C2×D4C22⋊#692312yes—
505C22×C4—21no—
506C22×C4—21no—
507C22×C4—42no—
508C22×C4—611no—
509C22×C4—612no—
510C22×C4—612no—
511C22×C4—612no—
512C22×C4—612no—
513C22×C4—611no—
514C22×C4—611no—
515C22×C4—611no—
516C22×C4—611no—
517C22×C4—611no—
518C22×C4—611no—
519C22×C4—611no—
520C22×C4—1222no—
521C22×C4—1222no—
522C22×C4—1221no—
523C22×C4—1221no—
524C22×C4—1221no—
525C22×C4—1221no—
526C22×C4—1221no—
527C22×C4—1221no—
528C22×C4—2442no—
529C22×C4C22×#692311yescharacteristic
530C22×C4C22×#692311yescharacteristic
531C22×C4C22×#692311yescharacteristic
532C22×C4C22×#692311yescharacteristic
533C22×C4C22×#692311yescharacteristic
534M16—614no—
535M16—1222no—
536M16—1222no—
537M16—1222no—
538M16—1222no—
539M16—1222no—
540M16—1222no—
541M16—1222no—
542M16—2444no—
543M16—2444no—
544C2×C8—1222no—
545C2×C8—1222no—
546C2×C8—1222no—
547C2×C8—1222no—
548C2×C8—2444no—
549C2×C8—2444no—
550C4⋊C4—22no—
551C4⋊C4—22no—
552C4⋊C4—612no—
553C4⋊C4—612no—
554C4⋊C4—1222no—
555C4⋊C4—1222no—
556C4⋊C4—1222no—
557C4⋊C4—1222no—
558C4⋊C4—1222no—
559C4⋊C4—1221no—
560C4⋊C4—1221no—
561C4⋊C4—1221no—
562C4⋊C4—1221no—
563C4⋊C4—1222no—
564C4⋊C4—1221no—
565C4⋊C4—1221no—
566C4⋊C4—1221no—
567C4⋊C4—1221no—
568C4⋊C4—1222no—
569C4⋊C4—1222no—
570C4⋊C4—1222no—
571C22⋊C4—21no—
572C22⋊C4—21no—
573C22⋊C4—22no—
574C22⋊C4—22no—
575C22⋊C4—611no—
576C22⋊C4—611no—
577C22⋊C4—611no—
578C22⋊C4—611no—
579C22⋊C4—612no—
580C22⋊C4—612no—
581C22⋊C4—1221no—
582C22⋊C4—1221no—
583C22⋊C4—1222no—
584C22⋊C4—1222no—
585C42—612no—
586C42—612no—
587C42—1221no—
588C42—1221no—
589C42—1221no—
590C42—1221no—
591C42—1221no—
592C42—1221no—
593C42—1221no—
594C42—1221no—
595C23—22no—
596C23—21no—
597C23—21no—
598C23—21no—
599C23—21no—
600C23—41no—
601C23—41no—
602C23—41no—
603C23C2×C22⋊#692312yes—
604C23C2×C22⋊#692311yescharacteristic
605C23C2×C22⋊#692311yescharacteristic
606C23C23×#692311yescharacteristic
607Q8—2442no—
608Q8—2442no—
609D4—24no—
610D4—42no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—41no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—612no—
624C2×C4—612no—
625C2×C4—611no—
626C2×C4—611no—
627C2×C4—611no—
628C2×C4—611no—
629C2×C4—611no—
630C2×C4—611no—
631C2×C4—612no—
632C2×C4—1222no—
633C2×C4—1222no—
634C2×C4—1222no—
635C2×C4—1222no—
636C2×C4—1222no—
637C2×C4—1222no—
638C2×C4—1222no—
639C2×C4—1222no—
640C2×C4—1221no—
641C2×C4—1221no—
642C2×C4—1221no—
643C2×C4—1221no—
644C2×C4—1221no—
645C2×C4—1221no—
646C2×C4—1221no—
647C2×C4—1221no—
648C2×C4—1221no—
649C2×C4—1221no—
650C2×C4—2441no—
651C2×C4—2442no—
652C2×C4—2442no—
653C2×C4—2441no—
654C2×C4—2441no—
655C2×C4—2441no—
656C2×C4C2×C22⋊#692312yes—
657C8—1224no—
658C8—2442no—
659C8—2442no—
660C8—2442no—
661C22—21no—
662C22—21no—
663C22—21no—
664C22—21no—
665C22—21no—
666C22—22no—
667C22—21no—
668C22—22no—
669C22—21no—
670C22—22no—
671C22—42no—
672C22—42no—
673C22—81no—
674C22—81no—
675C22#1684074912yes—
676C22C22×C22⋊#692311yescharacteristic
677C22#41×#692311yescharacteristic
678C22#1684083111yescharacteristic
679C4—22no—
680C4—41no—
681C4—41no—
682C4—41no—
683C4—41no—
684C4—612no—
685C4—612no—
686C4—1221no—
687C4—1221no—
688C4—1222no—
689C4—1221no—
690C4—1221no—
691C4—2441no—
692C4—2441no—
693C4—2441no—
694C4—2441no—
695C2—21no—
696C2—21no—
697C2—22no—
698C2—41no—
699C2—42no—
700C2#2146232211yescharacteristic
701C2C2×#1684074911yescharacteristic
702C2#2146243111yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC23×C61C23×C4
FrattiniC23C23×#6923
FittingC2×#123×C61C4
socleC22×C61C22×C22⋊C4
prime radicalC22×D4×C61C2×C4
cosocleC22×Dic61C24
hypercenterC2×#123#6923
hypocenterC61#22480
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×D4 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
2. #21462013C2semidirect
3. #21461846C2semidirect
4. #21461846C2semidirect
5. #21461721C2semidirect
6. #21461718C2semidirect
7. #21461686C2semidirect
8. #21461572C2semidirect
9. #21461525C2semidirect
10. C2×#16840642C2semidirect
402. C61#22480semidirect

Maximal subgroups

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

subgroupindex
D4×#117582172
#214620132
#214618462
#214617212
#214617182
#214616862
#214615722
#214615252
C2×#168406422
C2×#168404492
#2248061