Finite Group Databasebeta

← #10849358

Subgroups — #10849358

Order 768.

1028 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×#5568C211yesmaximal, characteristic
2C2×#4854C211yesmaximal, characteristic, complemented
3C2×#4794C211yesmaximal, characteristic
4#76071C211yesmaximal, characteristic, complemented
5C3×#1865C211yesmaximal, characteristic, complemented
6#71629C211yesmaximal, characteristic, complemented
7#70584C211yesmaximal, characteristic
8#70284C211yesmaximal, characteristic, complemented
9#70246C211yesmaximal, characteristic, complemented
10#70230C211yesmaximal, characteristic
11#69628C211yesmaximal, characteristic
12#68895C211yesmaximal, characteristic
13#68783C211yesmaximal, characteristic, complemented
14#68630C211yesmaximal, characteristic
15#68620C211yesmaximal, characteristic, complemented
16#25662—31nomaximal, Sylow
17C2×C3×C4×Q8C2211yescharacteristic
18C2×#947C2211yescharacteristic
19C2×#944C2211yescharacteristic
20#5568—24no—
21#5568—24no—
22C2×#898—21no—
23C2×#896—21no—
24C2×#893C2211yescharacteristic
25C2×#888C2211yescharacteristic
26C2×#887C2211yescharacteristic
27C2×#876—21no—
28C2×#875C2211yescharacteristic
29C2×#872—21no—
30C2×C4×Dic6C2211yescharacteristic
31C2×C3×#106C2211yescharacteristic
32C2×C3×#103C2211yescharacteristic
33C2×C3×#103C2211yescharacteristic
34C3×#389C2211yescharacteristic
35C2×#835C2211yescharacteristic
36C2×#835C2211yescharacteristic
37C2×#821C2211yescharacteristic
38C2×#821C2211yescharacteristic
39C2×#816C2211yescharacteristic
40C2×#816C2211yescharacteristic
41C2×#815C2211yescharacteristic
42C2×Dic3⋊C8C2211yescharacteristic
43#4984C2211yescharacteristic
44#4979—21no—
45#4975—21no—
46Dic3×C4⋊C4C2211yescharacteristic
47#4972C2211yescharacteristic
48C2×#810C2211yescharacteristic
49C2×#810C2211yescharacteristic
50C2×#809C2211yescharacteristic
51C2×#809C2211yescharacteristic
52C2×#808C2211yescharacteristic
53#4928C2211yescharacteristic
54#4854—24no—
55#4794—24no—
56#4656—21no—
57#4646—21no—
58C3×#340C2211yescharacteristic
59C3×#336C2211yescharacteristic
60#4550C2211yescharacteristic
61#4550C2211yescharacteristic
62#4529C2211yescharacteristic
63#4524C2211yescharacteristic
64C2×#556—31no—
65C2×#489—31no—
66C2×#475—31no—
67#2567—31no—
68#2043—31no—
69#2043—31no—
70#2042—31no—
71#2042—31no—
72#1985—31no—
73#1983—31no—
74#1865—31no—
75#1865D311yescharacteristic, complemented
76#1864—31no—
77#1864—31no—
78#1763—31no—
79C22×C3×Q8C2311yescharacteristic
80C22×D12C2311yescharacteristic
81C22×C4×D3—21no—
82C22×Dic6—21no—
83C22×Dic6D411yescharacteristic
84C22×Dic6C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C22×C3×C8C2311yescharacteristic
87C3×C4×Q8—22no—
88C3×C4×Q8—22no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C42—21no—
95C2×C3×C42C2311yescharacteristic
96#947—22no—
97#947D414yes—
98#944—22no—
99#944—22no—
100C2×(C2×D3)⋊C4—21no—
101C2×(C2×D3)⋊C4—21no—
102C2×(C2×D3)⋊C4—21no—
103C2×(C2×D3)⋊C4—21no—
104C2×(C2×D3)⋊C4—21no—
105C2×C4⋊Dic3—21no—
106C2×C4⋊Dic3D411yescharacteristic
107C2×C4⋊Dic3C2311yescharacteristic
108C2×C4⋊Dic3C2311yescharacteristic
109C2×Dic3⋊C4—21no—
110C2×Dic3⋊C4—21no—
111C2×Dic3⋊C4—21no—
112C2×Dic3⋊C4—21no—
113C2×Dic3⋊C4—21no—
114C2×Dic3⋊C4—21no—
115C2×Dic3⋊C4D411yescharacteristic
116C2×Dic3⋊C4D411yescharacteristic
117C2×C4×Dic3—21no—
118C2×C4×Dic3—21no—
119C2×C4×Dic3C2311yescharacteristic
120C2×C4×Dic3C2311yescharacteristic
121C22×C3⋊C8C2311yescharacteristic
122C22×C3⋊C8C2311yescharacteristic
123#898—28no—
124#896—28no—
125#893—22no—
126#893—22no—
127#888—22no—
128#888—22no—
129#887—22no—
130#887D414yes—
131#876—28no—
132#875—22no—
133#875—22no—
134#872—28no—
135C4×Dic6—22no—
136C4×Dic6—22no—
137C3×#106—22no—
138C3×#103—22no—
139C3×#103—22no—
140C3×#95—21no—
141#835—22no—
142#835—22no—
143C3⋊#95—21no—
144C3⋊#95—21no—
145C3⋊#95—21no—
146C3⋊#95—21no—
147#821—22no—
148#821—22no—
149#816—22no—
150#816—22no—
151#815—22no—
152Dic3⋊C8—22no—
153#810—22no—
154#810—22no—
155#809—22no—
156#809—22no—
157#808—22no—
158#556—64no—
159#556—64no—
160C2×C4⋊Q8—31no—
161C2×C4⋊Q8—31no—
162C2×#126—61no—
163C2×#126—61no—
164C2×#124—31no—
165C2×#124—31no—
166C2×C22⋊Q8—61no—
167C2×C4×Q8—31no—
168C2×C4×Q8—31no—
169C2×C4×Q8C2×D311yescharacteristic
170C2×C4×D4—31no—
171C2×#117—61no—
172#489—64no—
173#475—64no—
174C2×#106—31no—
175C2×#106C2×D311yescharacteristic
176C2×C4⋊C8—31no—
177C2×#103—31no—
178C2×#103—31no—
179C2×#103—31no—
180C2×#103—31no—
181C2×#103—31no—
182C2×#103—31no—
183C2×#103C2×D311yescharacteristic
184C2×#103C2×D311yescharacteristic
185C2×#102—31no—
186C2×#102—31no—
187C2×#102—31no—
188C2×#102—31no—
189C2×#97—31no—
190#395—61no—
191#395—61no—
192#391—61no—
193#389—31no—
194#389—31no—
195#389—31no—
196#389C2×D311yescharacteristic
197#384—61no—
198C4×C4⋊C4—31no—
199#340—31no—
200#340C2×D311yescharacteristic
201#336—31no—
202#336—31no—
203#336—31no—
204#336C2×D311yescharacteristic
205C23×D3—21no—
206C2×C3×Q8—21no—
207C2×C3×Q8—21no—
208C2×C3×Q8—21no—
209C2×C3×Q8—21no—
210C2×C3×Q8—21no—
211C2×C3×Q8C2×D412yes—
212C22×C3×C4—21no—
213C22×C3×C4—21no—
214C22×C3×C4—21no—
215C22×C3×C4—21no—
216C22×C3×C4C2×D411yescharacteristic
217C22×C3×C4C2×D411yescharacteristic
218C22×C3×C4C2411yescharacteristic
219C22×Dic3—21no—
220C22×Dic3—21no—
221C22×Dic3—21no—
222C22×Dic3C2×D411yescharacteristic
223C22×Dic3C2×D411yescharacteristic
224C22×Dic3C2×D411yescharacteristic
225C22×Dic3C2×D411yescharacteristic
226C2×D12—21no—
227C2×D12—21no—
228C2×D12—21no—
229C2×D12—21no—
230C2×D12—21no—
231C2×D12C2×D412yes—
232C2×C4×D3—24no—
233C2×C4×D3—42no—
234C2×C4×D3—42no—
235C2×Dic6—21no—
236C2×Dic6—21no—
237C2×Dic6—21no—
238C2×Dic6—21no—
239C2×Dic6—21no—
240C2×Dic6—22no—
241C2×Dic6—22no—
242C2×Dic6—41no—
243C2×Dic6—42no—
244C2×Dic6—81no—
245C2×Dic6—81no—
246C2×Dic6C2×D412yes—
247C2×C3×C8—21no—
248C2×C3×C8—21no—
249C2×C3×C8—21no—
250C2×C3×C8—21no—
251C2×C3×C8—21no—
252C2×C3×C8—21no—
253C3×C4⋊C4—24no—
254C3×C4⋊C4—21no—
255C3×C4⋊C4—21no—
256C3×C4⋊C4—21no—
257C3×C4⋊C4—21no—
258C3×C4⋊C4—81no—
259C3×C4⋊C4C2×D412yes—
260C3×C4⋊C4#4112yes—
261C3×C42—24no—
262C3×C42—21no—
263C3×C42—21no—
264(C2×D3)⋊C4—24no—
265(C2×D3)⋊C4—24no—
266(C2×D3)⋊C4—24no—
267(C2×D3)⋊C4—24no—
268(C2×D3)⋊C4—24no—
269C4⋊Dic3—21no—
270C4⋊Dic3—22no—
271C4⋊Dic3—21no—
272C4⋊Dic3—21no—
273C4⋊Dic3—81no—
274C4⋊Dic3#4112yes—
275Dic3⋊C4—24no—
276Dic3⋊C4—24no—
277Dic3⋊C4—22no—
278Dic3⋊C4—22no—
279Dic3⋊C4—24no—
280Dic3⋊C4—42no—
281Dic3⋊C4—81no—
282Dic3⋊C4—81no—
283C4×Dic3—21no—
284C4×Dic3—21no—
285C4×Dic3—24no—
286C4×Dic3—42no—
287C4×Dic3C2×D412yes—
288C4×Dic3C2×D412yes—
289C2×C3⋊C8—21no—
290C2×C3⋊C8—21no—
291C2×C3⋊C8—21no—
292C2×C3⋊C8—21no—
293C2×C3⋊C8—21no—
294C2×C3⋊C8—21no—
295C22×Q8—31no—
296C22×Q8—31no—
297C22×Q8—61no—
298C22×Q8C22×D311yescharacteristic
299C22×D4—31no—
300C23×C4—61no—
301C22×C8—31no—
302C22×C8—31no—
303C22×C8C22×D311yescharacteristic
304C22×C8C22×D311yescharacteristic
305C4⋊Q8—62no—
306C4⋊Q8—62no—
307C4⋊Q8—62no—
308C4⋊Q8—62no—
309#126—68no—
310#126—68no—
311#124—34no—
312#124—62no—
313#124—62no—
314#124—62no—
315C22⋊Q8—68no—
316C4×Q8—22no—
317C4×Q8—22no—
318C4×Q8—34no—
319C4×Q8—62no—
320C4×Q8—62no—
321C4×Q8—62no—
322C4×D4—62no—
323C4×D4—62no—
324#117—68no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—31no—
328C2×C4⋊C4—31no—
329C2×C4⋊C4—31no—
330C2×C4⋊C4—31no—
331C2×C4⋊C4—31no—
332C2×C4⋊C4—61no—
333C2×C4⋊C4—61no—
334C2×C4⋊C4—61no—
335C2×C4⋊C4—61no—
336C2×C4⋊C4—61no—
337C2×C4⋊C4—61no—
338C2×C4⋊C4—61no—
339C2×C4⋊C4C22×D311yescharacteristic
340C2×C4⋊C4C22×D311yescharacteristic
341C2×C4⋊C4C22×D311yescharacteristic
342C2×C22⋊C4—61no—
343C2×C22⋊C4—61no—
344C2×C22⋊C4—61no—
345C2×C22⋊C4—61no—
346C2×C22⋊C4—61no—
347C2×C42—21no—
348C2×C42—31no—
349C2×C42—31no—
350C2×C42—61no—
351C2×C42—61no—
352C2×C42C22×D311yescharacteristic
353#106—22no—
354#106—62no—
355C4⋊C8—62no—
356#103—22no—
357#103—22no—
358#103—62no—
359#103—62no—
360#103—62no—
361#103—62no—
362#103—62no—
363#103—62no—
364#102—62no—
365#102—62no—
366#102—62no—
367#102—62no—
368#97—62no—
369#95—21no—
370#95—61no—
371#95—61no—
372#95—61no—
373#95—61no—
374C23×C3C22×D411yescharacteristic
375C22×D3—22no—
376C22×D3—41no—
377C22×D3—41no—
378C22×D3—41no—
379C22×D3—41no—
380C22×D3—41no—
381C22×D3—41no—
382C3×Q8—22no—
383C3×Q8—22no—
384C3×Q8—41no—
385C3×Q8—41no—
386C2×C3×C4—22no—
387C2×C3×C4—21no—
388C2×C3×C4—22no—
389C2×C3×C4—21no—
390C2×C3×C4—22no—
391C2×C3×C4—21no—
392C2×C3×C4—21no—
393C2×C3×C4—21no—
394C2×C3×C4—21no—
395C2×C3×C4—41no—
396C2×C3×C4—41no—
397C2×C3×C4—41no—
398C2×C3×C4—41no—
399C2×C3×C4—41no—
400C2×C3×C4—41no—
401C2×C3×C4—41no—
402C2×C3×C4—41no—
403C2×C3×C4—41no—
404C2×C3×C4C22×D411yescharacteristic
405C2×C3×C4C22×D411yescharacteristic
406C2×C3×C4C2×#4111yescharacteristic
407C2×C3×C4C2×#4111yescharacteristic
408C2×C3×C42−1+411yescharacteristic
409C2×C3×C42−1+411yescharacteristic
410C2×Dic3—21no—
411C2×Dic3—21no—
412C2×Dic3—21no—
413C2×Dic3—21no—
414C2×Dic3—21no—
415C2×Dic3—21no—
416C2×Dic3—21no—
417C2×Dic3—22no—
418C2×Dic3—22no—
419C2×Dic3—21no—
420C2×Dic3—21no—
421C2×Dic3—41no—
422C2×Dic3—41no—
423C2×Dic3—41no—
424C2×Dic3—41no—
425C2×Dic3—41no—
426C2×Dic3—41no—
427C2×Dic3—41no—
428C2×Dic3#12012yes—
429C2×Dic3#12012yes—
430C2×Dic3#13712yes—
431D12—22no—
432D12—22no—
433D12—41no—
434D12—41no—
435C4×D3—48no—
436Dic6—22no—
437Dic6—22no—
438Dic6—44no—
439Dic6—41no—
440Dic6—41no—
441Dic6—84no—
442C3×C8—41no—
443C3×C8—41no—
444C3⋊C8—41no—
445C3⋊C8—41no—
446C24—61no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8—21no—
452C2×Q8—32no—
453C2×Q8—62no—
454C2×Q8—62no—
455C2×Q8—61no—
456C2×Q8—61no—
457C2×Q8—61no—
458C2×Q8—61no—
459C2×Q8—61no—
460C2×Q8—121no—
461C2×Q8—122no—
462C2×Q8—241no—
463C2×Q8—241no—
464C2×Q8D3×D412yes—
465C2×D4—32no—
466C2×D4—61no—
467C2×D4—61no—
468C2×D4—61no—
469C2×D4—61no—
470C2×D4—61no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—31no—
476C22×C4—31no—
477C22×C4—31no—
478C22×C4—31no—
479C22×C4—64no—
480C22×C4—61no—
481C22×C4—61no—
482C22×C4—61no—
483C22×C4—122no—
484C22×C4—122no—
485C22×C4D3×D411yescharacteristic
486C22×C4D3×D411yescharacteristic
487C22×C4C23×D311yescharacteristic
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—61no—
495C2×C8—61no—
496C2×C8—61no—
497C2×C8—61no—
498C2×C8—61no—
499C2×C8—61no—
500C4⋊C4—24no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—32no—
506C4⋊C4—62no—
507C4⋊C4—62no—
508C4⋊C4—62no—
509C4⋊C4—64no—
510C4⋊C4—64no—
511C4⋊C4—64no—
512C4⋊C4—61no—
513C4⋊C4—61no—
514C4⋊C4—61no—
515C4⋊C4—81no—
516C4⋊C4—122no—
517C4⋊C4—241no—
518C4⋊C4—241no—
519C4⋊C4—241no—
520C4⋊C4#23512yes—
521C4⋊C4D3×D412yes—
522C22⋊C4—64no—
523C22⋊C4—64no—
524C22⋊C4—64no—
525C22⋊C4—64no—
526C22⋊C4—64no—
527C42—24no—
528C42—21no—
529C42—21no—
530C42—32no—
531C42—32no—
532C42—64no—
533C42—61no—
534C42—61no—
535C42—122no—
536C22×C3C2×#12011yescharacteristic
537C22×C3#54811yescharacteristic
538C22×C3#54811yescharacteristic
539C22×C3C2×#13711yescharacteristic
540C22×C3#57511yescharacteristic
541C22×C3#57711yescharacteristic
542C22×C3#57711yescharacteristic
543C2×D3—42no—
544C2×D3—42no—
545C2×D3—42no—
546C2×D3—81no—
547C2×D3—81no—
548C2×D3—81no—
549C2×D3—81no—
550C3×C4—42no—
551C3×C4—42no—
552C3×C4—41no—
553C3×C4—42no—
554C3×C4—41no—
555C3×C4—81no—
556C3×C4#53211yescharacteristic
557C3×C4#53611yescharacteristic
558C3×C4#54711yescharacteristic
559C3×C4#54711yescharacteristic
560Dic3—22no—
561Dic3—22no—
562Dic3—22no—
563Dic3—42no—
564Dic3—42no—
565Dic3—41no—
566Dic3—81no—
567C23—62no—
568C23—121no—
569C23—121no—
570C23—121no—
571C23—121no—
572C23—121no—
573C23—121no—
574C23C2×D3×D411yescharacteristic
575Q8—22no—
576Q8—22no—
577Q8—41no—
578Q8—41no—
579Q8—62no—
580Q8—62no—
581Q8—124no—
582Q8—121no—
583Q8—121no—
584Q8—244no—
585D4—62no—
586D4—62no—
587D4—121no—
588D4—121no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—22no—
594C2×C4—21no—
595C2×C4—22no—
596C2×C4—21no—
597C2×C4—22no—
598C2×C4—32no—
599C2×C4—32no—
600C2×C4—32no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—61no—
611C2×C4—61no—
612C2×C4—61no—
613C2×C4—61no—
614C2×C4—61no—
615C2×C4—61no—
616C2×C4—62no—
617C2×C4—61no—
618C2×C4—61no—
619C2×C4—61no—
620C2×C4—62no—
621C2×C4—128no—
622C2×C4—121no—
623C2×C4—121no—
624C2×C4—121no—
625C2×C4—121no—
626C2×C4—121no—
627C2×C4—121no—
628C2×C4—121no—
629C2×C4C2×#23511yescharacteristic
630C2×C4C2×D3×D411yescharacteristic
631C2×C4C2×D3×D411yescharacteristic
632C2×C4#100711yescharacteristic
633C2×C4D3×#4111yescharacteristic
634C2×C4#101011yescharacteristic
635C8—41no—
636C8—41no—
637C8—121no—
638C8—121no—
639C2×C3#243211yescharacteristic
640C2×C3#300711yescharacteristic
641C2×C3#301011yescharacteristic
642C2×C3#336911yescharacteristic
643C2×C3#336911yescharacteristic
644C2×C3#337611yescharacteristic
645C2×C3#337611yescharacteristic
646D3—82no—
647C22—122no—
648C22—122no—
649C22—122no—
650C22—241no—
651C22—241no—
652C22—241no—
653C22—241no—
654C22#555211yescharacteristic
655C22D3×#12011yescharacteristic
656C22#563311yescharacteristic
657C22#576211yescharacteristic
658C22#576811yescharacteristic
659C22D3×#13711yescharacteristic
660C22#577811yescharacteristic
661C4—42no—
662C4—41no—
663C4—42no—
664C4—41no—
665C4—42no—
666C4—62no—
667C4—62no—
668C4—62no—
669C4—81no—
670C4—122no—
671C4—121no—
672C4—122no—
673C4—241no—
674C4#550711yescharacteristic
675C4#554811yescharacteristic
676C4#562811yescharacteristic
677C4#565511yescharacteristic
678C3#2566211yescharacteristic, complemented, Sylow
679C2—242no—
680C2#7441111yescharacteristic
681C2#7963011yescharacteristic
682C2#7981511yescharacteristic
683C2#8004111yescharacteristic
684C2#8022811yescharacteristic
685C2#8100411yescharacteristic
686C2#8126511yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1865C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#1865D3
hypocenterC3#25662
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1865

Complements

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

NH ≅ G/Ntype
2. C2×#4854C2semidirect
4. #76071C2semidirect
5. C3×#1865C2semidirect
6. #71629C2semidirect
8. #70284C2semidirect
9. #70246C2semidirect
13. #68783C2semidirect
15. #68620C2semidirect
75. #1865D3semidirect
678. C3#25662semidirect

Maximal subgroups

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

subgroupindex
C2×#55682
C2×#48542
C2×#47942
#760712
C3×#18652
#716292
#705842
#702842
#702462
#702302
#696282
#688952
#687832
#686302
#686202
#256623