Finite Group Databasebeta

← #30564775

Subgroups — #30564775

Order 15616.

2194 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840433C211yesmaximal, characteristic, complemented
2#21457763C214yesmaximal, complemented
3#21457752C214yesmaximal, complemented
4#21457575C214yesmaximal, complemented
5#21457545C214yesmaximal, complemented
6#21456757C211yesmaximal, characteristic, complemented
7#21456734C211yesmaximal, characteristic, complemented
8#21456580C214yesmaximal, complemented
9#21455854C211yesmaximal, characteristic, complemented
10#21455846C211yesmaximal, characteristic, complemented
11#21455035C214yesmaximal, complemented
12#21454843C211yesmaximal, characteristic, complemented
13C61×#3506C211yesmaximal, characteristic, complemented
14C22×#11758222C2212yes—
15C2×#14519797C2218yescomplemented
16#16840433—22no—
17#16840433—22no—
18#16840433C2214yes—
19#16840433C2214yes—
20C2×#14519774C2211yescharacteristic
21C2×#14519773—21no—
22C2×#14519751C2212yes—
23C2×#14519750—22no—
24C2×C4×Dic122C2211yescharacteristic
25C2×#14519739—21no—
26#16840138C2212yescomplemented
27#16840136C2212yescomplemented
28#16840110C2212yescomplemented
29#16840110C2212yescomplemented
30#16840095C2212yes—
31#16840086C2218yescomplemented
32#16840078C2212yes—
33#16840054C2212yescomplemented
34#16840044C2212yescomplemented
35#16839996C2218yescomplemented
36#16839993C2218yescomplemented
37#16839969C2212yescomplemented
38#16839962C2214yescomplemented
39#16839959C2212yescomplemented
40#16839957C2214yescomplemented
41#16839922C2218yescomplemented
42#16839917C2212yes—
43#16839912C2214yes—
44#16839907C2218yescomplemented
45#16839905C2218yescomplemented
46#16839900C2212yes—
47#16839895C2214yes—
48C4×#11758202C2214yescomplemented
49C4×#11758200C2214yescomplemented
50D4×#92418C2214yescomplemented
51D4×#92418C2214yescomplemented
52#16839815C2211yescharacteristic
53#16839815C2211yescharacteristic
54#16839756C2218yescomplemented
55C2×C22⋊#92418C2214yes—
56C61×#559—22no—
57C61×#556C2212yescomplemented
58C61×#555C2212yescomplemented
59C61×#553—22no—
60C61×#550C2212yescomplemented
61C61×#547C2212yescomplemented
62C61×#538—22no—
63C61×#535—21no—
64C61×#533C2211yescharacteristic
65C61×#532C2211yescharacteristic
66C61×#525C2214yes—
67C61×#520—21no—
68C2×C4×D4×C61C2211yescharacteristic
69C2×#11758222—24no—
70C2×#11758222—48no—
71C2×#11758222C2318yes—
72C22×Dic122C2311yescharacteristic
73C22×D244—21no—
74C22×D244—21no—
75C22×D244C2312yes—
76C22×C22⋊D61—22no—
77C22×C22⋊D61—22no—
78C22×C4×D61—22no—
79#14519797—416no—
80C2×#11758202C2314yes—
81C2×#11758200C2314yes—
82C2×#11758201C2318yes—
83C2×#11758199C2318yes—
84C22×#92418C2318yes—
85#14519774—21no—
86#14519774—21no—
87#14519774C2312yes—
88#14519774C2312yes—
89#14519773—24no—
90#14519773—24no—
91#14519751—24no—
92#14519751C2318yes—
93#14519750—28no—
94#14519750—28no—
95C4×Dic122—21no—
96C4×Dic122—21no—
97C4×Dic122C2312yes—
98C4×Dic122C2312yes—
99#14519739—24no—
100#14519739—24no—
101C2×C4⋊Dic61C2311yescharacteristic
102C2×#11758196—21no—
103C2×#11758198—22no—
104C2×C4×Dic61—21no—
105#14519710C2314yes—
106#14519708C2314yes—
107C4⋊#92418C2312yes—
108C4⋊#92418C2312yes—
109#14519701C2314yes—
110#14519700C2314yes—
111#14519700C2314yes—
112#14519698C2314yes—
113#14519697C2318yes—
114#14519697C2318yes—
115C22⋊#92418C2312yes—
116C22⋊#92418C2312yes—
117C22⋊#92418C2318yes—
118C4×#92418C2312yes—
119C4×#92418C2312yes—
120C2×#41×C61—21no—
121C2×#41×C61C2314yes—
122C22×D4×C61D411yescharacteristic
123C23×C4×C61C2312yes—
124C4⋊Q8×C61C2311yescharacteristic
125C4⋊Q8×C61C2311yescharacteristic
126C4⋊Q8×C61C2311yescharacteristic
127#127×C61—22no—
128#127×C61C2311yescharacteristic
129#126×C61—24no—
130#125×C61—21no—
131#124×C61—22no—
132#124×C61—24no—
133#124×C61—21no—
134#123×C61—24no—
135#123×C61—24no—
136C22⋊Q8×C61—21no—
137C22⋊Q8×C61C2312yes—
138C22⋊Q8×C61C2314yes—
139#121×C61—22no—
140#121×C61—22no—
141#121×C61—21no—
142#121×C61—24no—
143#121×C61C2312yes—
144#121×C61C2314yes—
145#120×C61—24no—
146C4×Q8×C61—22no—
147C4×Q8×C61—21no—
148C4×Q8×C61C2312yes—
149C4×D4×C61—24no—
150C4×D4×C61—24no—
151C4×D4×C61—22no—
152C4×D4×C61—21no—
153C4×D4×C61D412yes—
154C4×D4×C61D412yes—
155C4×D4×C61C2312yes—
156C4×D4×C61C2312yes—
157C4×D4×C61C2312yes—
158#117×C61—21no—
159#117×C61—22no—
160#117×C61C2314yes—
161C2×C4⋊C4×C61—22no—
162C2×C4⋊C4×C61D411yescharacteristic
163C2×C22⋊C4×C61D412yes—
164C2×C42×C61C2311yescharacteristic
165C24×C61C2×D412yes—
166C23×D61—22no—
167#41×C61—48no—
168#41×C61—44no—
169C2×Q8×C61—21no—
170C2×Q8×C61—21no—
171C2×Q8×C61C2412yes—
172C2×D4×C61—22no—
173C2×D4×C61—24no—
174C2×D4×C61—24no—
175C2×D4×C61—24no—
176C2×D4×C61—21no—
177C2×D4×C61—21no—
178C2×D4×C61C2×D412yes—
179C2×D4×C61C2×D412yes—
180C2×D4×C61C2412yes—
181C22×C4×C61—22no—
182C22×C4×C61—24no—
183C22×C4×C61—44no—
184C22×C4×C61C2×D412yes—
185C22×C4×C61C2×D411yescharacteristic
186C22×C4×C61C2×D414yes—
187C22×C4×C61C2×D411yescharacteristic
188C22×C4×C61C2411yescharacteristic
189C22×C4×C61C2414yes—
190C2×C22⋊D61—28no—
191C2×C22⋊D61—28no—
192C2×C22⋊D61—48no—
193C2×C22⋊D61—84no—
194C22×Dic61—21no—
195C22×Dic61—21no—
196#11758222—216no—
197#11758222—48no—
198#11758222—416no—
199C2×D244—22no—
200C2×D244—22no—
201C2×D244—22no—
202C2×D244—22no—
203C2×D244—22no—
204C2×D244—44no—
205C2×D244—44no—
206C2×D244—82no—
207C2×D244C2414yes—
208C2×C4×D61—28no—
209C2×C4×D61—48no—
210C2×Dic122—21no—
211C2×Dic122—24no—
212C2×Dic122C2412yes—
213C4⋊C4×C61—21no—
214C4⋊C4×C61—24no—
215C4⋊C4×C61—21no—
216C4⋊C4×C61—22no—
217C4⋊C4×C61—21no—
218C4⋊C4×C61—21no—
219C4⋊C4×C61—22no—
220C4⋊C4×C61C2×D412yes—
221C4⋊C4×C61C2×D412yes—
222C4⋊C4×C61C2412yes—
223C4⋊C4×C61C2412yes—
224C22⋊C4×C61—22no—
225C22⋊C4×C61—24no—
226C22⋊C4×C61—24no—
227C22⋊C4×C61—24no—
228C22⋊C4×C61—22no—
229C22⋊C4×C61—22no—
230C22⋊C4×C61C2×D414yes—
231C22⋊C4×C61C2×D414yes—
232C42×C61—24no—
233C42×C61—22no—
234C42×C61C2×D411yescharacteristic
235C42×C61C2×D411yescharacteristic
236C42×C61C2411yescharacteristic
237C42×C61C2411yescharacteristic
238#11758202—28no—
239#11758201—48no—
240#11758200—28no—
241#11758199—48no—
242#11758198—28no—
243C4⋊Dic61—21no—
244C4⋊Dic61C2412yes—
245#11758196—22no—
246#11758196—22no—
247C4×Dic61—22no—
248C4×Dic61—22no—
249C2×#92418—48no—
250C2×#92418C2414yes—
251C2×#92418C2414yes—
252C23×C61—24no—
253C23×C61—22no—
254C23×C61—42no—
255C23×C61—42no—
256C23×C61C22×D414yes—
257C23×C61C22×D411yescharacteristic
258C22×D61—24no—
259C22×D61—44no—
260C22×D61—44no—
261C22×D61—44no—
262Q8×C61—24no—
263Q8×C61—42no—
264Q8×C61—81no—
265D4×C61—24no—
266D4×C61—24no—
267D4×C61—24no—
268D4×C61—44no—
269D4×C61—42no—
270D4×C61—81no—
271D4×C61—84no—
272D4×C61—84no—
273C2×C4×C61—28no—
274C2×C4×C61—22no—
275C2×C4×C61—22no—
276C2×C4×C61—22no—
277C2×C4×C61—22no—
278C2×C4×C61—42no—
279C2×C4×C61—41no—
280C2×C4×C61—44no—
281C2×C4×C61—44no—
282C2×C4×C61—84no—
283C2×C4×C61#13514yescomplemented
284C2×C4×C61C22×D411yescharacteristic
285C2×C4×C61C22×D411yescharacteristic
286C2×C4×C61C22×D411yescharacteristic
287C2×C4×C61C22×D414yes—
288C2×C4×C61C22×D411yescharacteristic
289C2×C4×C61C22×D411yescharacteristic
290C2×C4×C61C2511yescharacteristic
291C22⋊D61—416no—
292C22⋊D61—88no—
293C2×Dic61—24no—
294C2×Dic61—22no—
295C2×Dic61—22no—
296C2×Dic61—42no—
297D244—28no—
298D244—44no—
299D244—44no—
300D244—44no—
301D244—82no—
302D244—82no—
303C4×D61—416no—
304Dic122—24no—
305Dic122—22no—
306Dic122H(5,𝔽2)14yes—
307#92418—24no—
308#92418—24no—
309#61566—611nomaximal, Sylow
310C22×C61—21no—
311C22×C61—21no—
312C22×C61—24no—
313C22×C61—24no—
314C22×C61—24no—
315C22×C61—42no—
316C22×C61—42no—
317C22×C61—42no—
318C22×C61—84no—
319C22×C61C2×#13512yes—
320C22×C61C23×D411yescharacteristic
321C2×D61—44no—
322C2×D61—44no—
323C2×D61—44no—
324C2×D61—88no—
325C4×C61—21no—
326C4×C61—21no—
327C4×C61—21no—
328C4×C61—24no—
329C4×C61—42no—
330C4×C61—42no—
331C4×C61—42no—
332C4×C61C2×#13512yes—
333C4×C61C2×#13512yes—
334C4×C61C2×H(5,𝔽2)12yes—
335Dic61—24no—
336Dic61—42no—
337#3506D6111yescharacteristic, complemented
338C2×#550—611no—
339#3305—614no—
340#3304—614no—
341#3287—614no—
342#3284—614no—
343#3208—611no—
344#3205—611no—
345#3188—614no—
346#3104—611no—
347#3103—611no—
348#3002—614no—
349#2972—611no—
350C2×C61—22no—
351C2×C61—44no—
352C2×C61—82no—
353C2×C61#348411yescharacteristic
354C2×C61C22×#13511yescharacteristic
355C2×C61#358511yescharacteristic
356D61—84no—
357C22×#41—612no—
358C2×#135—618no—
359#559—22no—
360#556C2×D6112yescomplemented
361#555C2×D6112yescomplemented
362#553—22no—
363#550—614no—
364#550—614no—
365#550—1222no—
366#550—1222no—
367#550C2×D6112yescomplemented
368#547C2×D6112yescomplemented
369#538—22no—
370#535—21no—
371#533C2×D6111yescharacteristic
372#532C2×D6111yescharacteristic
373C2×#127—611no—
374C2×#127—1221no—
375#525C2×D6114yes—
376C2×#121—612no—
377C2×#121—1222no—
378#520—21no—
379C2×C4×Q8—611no—
380C2×C4×D4—1221no—
381C2×C4×D4C2×D6111yescharacteristic
382#493—612no—
383#492—612no—
384#486—612no—
385#486—612no—
386#484—612no—
387#482—618no—
388#481—612no—
389#477—612no—
390#475—612no—
391#466—618no—
392#465—618no—
393#462—612no—
394#460—612no—
395#460—614no—
396#459—614no—
397#452—618no—
398#451—612no—
399#450—614no—
400#449—618no—
401#449—618no—
402#448—612no—
403#447—614no—
404C4×Q16—614no—
405C4×SD16—614no—
406C8×D4—614no—
407C8×D4—614no—
408#431—611no—
409#431—611no—
410#416—618no—
411C2×C22⋊C8—614no—
412C61#6156611yescharacteristic, complemented, Sylow
413C2×#41—21no—
414C2×#41—618no—
415C2×#41—1224no—
416C2×#41—2448no—
417C2×#41C22×D6114yes—
418C22×Q8—611no—
419C22×D4—612no—
420C22×D4—1222no—
421C22×D4—1222no—
422C22×D4—1221no—
423C22×D4—1221no—
424C22×D4C22⋊D6111yescharacteristic
425C23×C4—1222no—
426C23×C4C22×D6112yes—
427#135—24416no—
428C2×Q16—614no—
429C2×SD16—614no—
430C2×SD16—618no—
431C2×D8—618no—
432C22×C8—618no—
433C4⋊Q8C22×D6111yescharacteristic
434C4⋊Q8C22×D6111yescharacteristic
435C4⋊Q8C22×D6111yescharacteristic
436#127—22no—
437#127—612no—
438#127—612no—
439#127—1224no—
440#127—1224no—
441#127—1221no—
442#127—1221no—
443#127C22×D6111yescharacteristic
444#126—24no—
445#125—21no—
446#124—22no—
447#124—24no—
448#124—21no—
449#123—24no—
450#123—24no—
451C22⋊Q8—21no—
452C22⋊Q8C22×D6112yes—
453C22⋊Q8C22×D6114yes—
454#121—22no—
455#121—22no—
456#121—21no—
457#121—24no—
458#121—618no—
459#121—1224no—
460#121—1228no—
461#121—1228no—
462#121C22×D6112yes—
463#121C22×D6114yes—
464#120—24no—
465C4×Q8—22no—
466C4×Q8—21no—
467C4×Q8—612no—
468C4×Q8—612no—
469C4×Q8—1221no—
470C4×Q8—1221no—
471C4×Q8C22×D6112yes—
472C4×D4—24no—
473C4×D4—24no—
474C4×D4—22no—
475C4×D4—21no—
476C4×D4—1224no—
477C4×D4—1224no—
478C4×D4C22⋊D6112yes—
479C4×D4C22⋊D6112yes—
480C4×D4C22×D6112yes—
481C4×D4C22×D6112yes—
482C4×D4C22×D6112yes—
483#117—21no—
484#117—22no—
485#117C22×D6114yes—
486C2×C4⋊C4—22no—
487C2×C4⋊C4—611no—
488C2×C4⋊C4—1221no—
489C2×C4⋊C4C22⋊D6111yescharacteristic
490C2×C22⋊C4—1222no—
491C2×C22⋊C4C22⋊D6112yes—
492C2×C42—1221no—
493C2×C42C22×D6111yescharacteristic
494#107—614no—
495#106—614no—
496C4⋊C8—612no—
497C4⋊C8—612no—
498#103—614no—
499#103—614no—
500#103—614no—
501#102—614no—
502#102—618no—
503#102—618no—
504C22⋊C8—612no—
505C22⋊C8—612no—
506C22⋊C8—618no—
507C4×C8—612no—
508C4×C8—612no—
509C24—1222no—
510C24C2×C22⋊D6112yes—
511#41—48no—
512#41—44no—
513#41—12216no—
514#41—2448no—
515#41—24416no—
516C2×Q8—21no—
517C2×Q8—21no—
518C2×Q8—612no—
519C2×Q8—1224no—
520C2×Q8—1221no—
521C2×Q8C23×D6112yes—
522C2×D4—22no—
523C2×D4—24no—
524C2×D4—24no—
525C2×D4—21no—
526C2×D4—21no—
527C2×D4—24no—
528C2×D4—614no—
529C2×D4—1228no—
530C2×D4—1228no—
531C2×D4—1222no—
532C2×D4—1222no—
533C2×D4—1222no—
534C2×D4—1222no—
535C2×D4—1222no—
536C2×D4—4882no—
537C2×D4—4884no—
538C2×D4—2444no—
539C2×D4—2448no—
540C2×D4—2444no—
541C2×D4C2×C22⋊D6112yes—
542C2×D4C2×C22⋊D6112yes—
543C2×D4C23×D6112yes—
544C22×C4—22no—
545C22×C4—24no—
546C22×C4—44no—
547C22×C4—1221no—
548C22×C4—1221no—
549C22×C4—1228no—
550C22×C4—2448no—
551C22×C4C2×C22⋊D6111yescharacteristic
552C22×C4C2×C22⋊D6112yes—
553C22×C4C2×C22⋊D6114yes—
554C22×C4C2×C22⋊D6111yescharacteristic
555C22×C4C23×D6111yescharacteristic
556C22×C4C23×D6114yes—
557Q16—1228no—
558SD16—1228no—
559SD16—2448no—
560D8—2448no—
561C2×C8—614no—
562C2×C8—614no—
563C2×C8—2448no—
564C4⋊C4—21no—
565C4⋊C4—24no—
566C4⋊C4—21no—
567C4⋊C4—22no—
568C4⋊C4—21no—
569C4⋊C4—21no—
570C4⋊C4—22no—
571C4⋊C4—612no—
572C4⋊C4—1221no—
573C4⋊C4—1222no—
574C4⋊C4—1222no—
575C4⋊C4C2×C22⋊D6112yes—
576C4⋊C4C2×C22⋊D6112yes—
577C4⋊C4C23×D6112yes—
578C4⋊C4C23×D6112yes—
579C22⋊C4—22no—
580C22⋊C4—24no—
581C22⋊C4—24no—
582C22⋊C4—24no—
583C22⋊C4—22no—
584C22⋊C4—22no—
585C22⋊C4—1228no—
586C22⋊C4C2×C22⋊D6114yes—
587C22⋊C4C2×C22⋊D6114yes—
588C42—24no—
589C42—22no—
590C42—1222no—
591C42—1222no—
592C42C2×C22⋊D6111yescharacteristic
593C42C2×C22⋊D6111yescharacteristic
594C42C23×D6111yescharacteristic
595C42C23×D6111yescharacteristic
596C23—24no—
597C23—22no—
598C23—42no—
599C23—42no—
600C23—1224no—
601C23—2444no—
602C23—2444no—
603C23—2444no—
604C23C22×C22⋊D6114yes—
605C23C22×C22⋊D6111yescharacteristic
606Q8—24no—
607Q8—42no—
608Q8—81no—
609Q8—614no—
610Q8—1224no—
611Q8—1222no—
612D4—24no—
613D4—24no—
614D4—24no—
615D4—44no—
616D4—42no—
617D4—81no—
618D4—84no—
619D4—84no—
620D4—1228no—
621D4—4882no—
622D4—4882no—
623D4—4888no—
624D4—2444no—
625D4—24416no—
626D4—2444no—
627D4—2444no—
628C2×C4—28no—
629C2×C4—22no—
630C2×C4—22no—
631C2×C4—22no—
632C2×C4—22no—
633C2×C4—42no—
634C2×C4—44no—
635C2×C4—41no—
636C2×C4—44no—
637C2×C4—84no—
638C2×C4—1224no—
639C2×C4—1222no—
640C2×C4—1222no—
641C2×C4—24416no—
642C2×C4—2442no—
643C2×C4#1451979714yescomplemented
644C2×C4C22×C22⋊D6114yes—
645C2×C4C22×C22⋊D6111yescharacteristic
646C2×C4C22×C22⋊D6111yescharacteristic
647C2×C4C22×C22⋊D6111yescharacteristic
648C2×C4C22×C22⋊D6111yescharacteristic
649C2×C4C22×C22⋊D6111yescharacteristic
650C2×C4C24×D6111yescharacteristic
651C8—1224no—
652C8—1224no—
653C22—21no—
654C22—21no—
655C22—24no—
656C22—24no—
657C22—24no—
658C22—42no—
659C22—42no—
660C22—42no—
661C22—84no—
662C22—4888no—
663C22—2444no—
664C22—2444no—
665C22—2444no—
666C22C2×#1451979712yes—
667C22C23×C22⋊D6111yescharacteristic
668C4—21no—
669C4—21no—
670C4—21no—
671C4—24no—
672C4—42no—
673C4—42no—
674C4—42no—
675C4—1224no—
676C4—2442no—
677C4C2×#1451979712yes—
678C4C2×#1451979712yes—
679C4H(5,𝔽2)×D6112yes—
680C2—22no—
681C2—44no—
682C2—82no—
683C2—4884no—
684C2#2145945011yescharacteristic
685C2C22×#1451979711yescharacteristic
686C2#2146059811yescharacteristic
subgroupG/H
centerC22C23×C22⋊D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC61×#3506C2
socleC22×C61C23×D4
hypercenter#3506D61
hypocenterC61#61566
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ #3506

Complements

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

NH ≅ G/Ntype
1. C2×#16840433C2semidirect
2. #21457763C2semidirect
3. #21457752C2semidirect
4. #21457575C2semidirect
5. #21457545C2semidirect
6. #21456757C2semidirect
7. #21456734C2semidirect
8. #21456580C2semidirect
9. #21455854C2semidirect
10. #21455846C2semidirect
11. #21455035C2semidirect
12. #21454843C2semidirect
13. C61×#3506C2semidirect
15. C2×#14519797C22semidirect
26. #16840138C22semidirect
27. #16840136C22semidirect
28. #16840110C22semidirect
29. #16840110C22semidirect
31. #16840086C22semidirect
33. #16840054C22semidirect
34. #16840044C22semidirect
35. #16839996C22semidirect
36. #16839993C22semidirect
37. #16839969C22semidirect
38. #16839962C22semidirect
39. #16839959C22semidirect
40. #16839957C22semidirect
41. #16839922C22semidirect
44. #16839907C22semidirect
45. #16839905C22semidirect
48. C4×#11758202C22semidirect
49. C4×#11758200C22semidirect
50. D4×#92418C22semidirect
51. D4×#92418C22semidirect
54. #16839756C22semidirect
57. C61×#556C22semidirect
58. C61×#555C22semidirect
60. C61×#550C22semidirect
61. C61×#547C22semidirect
283. C2×C4×C61#135semidirect
337. #3506D61semidirect
360. #556C2×D61semidirect
361. #555C2×D61semidirect
367. #550C2×D61semidirect
368. #547C2×D61semidirect
412. C61#61566semidirect
643. C2×C4#14519797semidirect

Maximal subgroups

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

subgroupindex
C2×#168404332
#214577632
#214577522
#214575752
#214575452
#214567572
#214567342
#214565802
#214558542
#214558462
#214550352
#214548432
C61×#35062
#6156661