Finite Group Databasebeta

← #10849355

Subgroups — #10849355

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×#5671C211yesmaximal, characteristic
2C2×#4791C211yesmaximal, characteristic, complemented
3C2×#4765C211yesmaximal, characteristic
4#76071C211yesmaximal, characteristic, complemented
5C3×#2042C211yesmaximal, characteristic
6#71722C211yesmaximal, characteristic
7#70284C211yesmaximal, characteristic, complemented
8#70246C211yesmaximal, characteristic, complemented
9#70220C211yesmaximal, characteristic, complemented
10#70170C211yesmaximal, characteristic
11#70169C211yesmaximal, characteristic
12#70162C211yesmaximal, characteristic
13#69632C211yesmaximal, characteristic, complemented
14#68684C211yesmaximal, characteristic, complemented
15#68620C211yesmaximal, characteristic, complemented
16#25662—31nomaximal, Sylow
17C2×C3×#124C2211yescharacteristic
18C2×Q8×Dic3C2211yescharacteristic
19C2×#944C2211yescharacteristic
20C2×#936C2211yescharacteristic
21C2×D4×Dic3C2211yescharacteristic
22#5671—24no—
23#5671—24no—
24C2×#888C2211yescharacteristic
25C2×#879—21no—
26C2×#879—21no—
27C2×#878—21no—
28C2×#877—21no—
29C2×C4×Dic6C2211yescharacteristic
30C2×C4×Dic6C2211yescharacteristic
31C2×C3×#103C2211yescharacteristic
32C2×C3×#103C2211yescharacteristic
33C2×C3×#102C2211yescharacteristic
34C2×C3×#102C2211yescharacteristic
35C3×#389C2211yescharacteristic
36C2×#835C2211yescharacteristic
37C2×#835C2211yescharacteristic
38C2×#832C2211yescharacteristic
39C2×#832C2211yescharacteristic
40C2×#817C2211yescharacteristic
41C2×#816C2211yescharacteristic
42C2×#816C2211yescharacteristic
43C2×#815C2211yescharacteristic
44C2×Dic3⋊C8C2211yescharacteristic
45#4984C2211yescharacteristic
46#4979—21no—
47#4975—21no—
48Dic3×C4⋊C4C2211yescharacteristic
49#4972C2211yescharacteristic
50C2×#810C2211yescharacteristic
51C2×#810C2211yescharacteristic
52C2×#808C2211yescharacteristic
53#4928C2211yescharacteristic
54#4791—24no—
55#4765—24no—
56#4656—21no—
57#4646—21no—
58C3×#336C2211yescharacteristic
59#4551C2211yescharacteristic
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#2042D311yescharacteristic
72#1985—31no—
73#1983—31no—
74#1865—31no—
75#1865—31no—
76#1864—31no—
77#1864—31no—
78#1763—31no—
79C22×C3×Q8C2311yescharacteristic
80C22×C3×D4C2311yescharacteristic
81C23×Dic3—21no—
82C22×Dic6—21no—
83C22×Dic6D411yescharacteristic
84C22×Dic6C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C22×C3×C8C2311yescharacteristic
87C3×#124—22no—
88C3×#124—22no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C22⋊C4—21no—
93C2×C3×C22⋊C4—21no—
94C2×C3×C42C2311yescharacteristic
95C2×#217—21no—
96C2×#217—21no—
97C2×#217—21no—
98Q8×Dic3—22no—
99Q8×Dic3D414yes—
100#944—22no—
101#944—22no—
102#936—22no—
103#936D414yes—
104D4×Dic3—22no—
105D4×Dic3—22no—
106C2×C4⋊Dic3—21no—
107C2×C4⋊Dic3—21no—
108C2×C4⋊Dic3D411yescharacteristic
109C2×C4⋊Dic3C2311yescharacteristic
110C2×C4⋊Dic3C2311yescharacteristic
111C2×C4⋊Dic3C2311yescharacteristic
112C2×Dic3⋊C4—21no—
113C2×Dic3⋊C4—21no—
114C2×Dic3⋊C4—21no—
115C2×Dic3⋊C4—21no—
116C2×Dic3⋊C4—21no—
117C2×Dic3⋊C4—21no—
118C2×Dic3⋊C4D411yescharacteristic
119C2×Dic3⋊C4D411yescharacteristic
120C2×C4×Dic3—21no—
121C2×C4×Dic3—21no—
122C2×C4×Dic3—21no—
123C2×C4×Dic3C2311yescharacteristic
124C2×C4×Dic3C2311yescharacteristic
125C22×C3⋊C8C2311yescharacteristic
126C22×C3⋊C8C2311yescharacteristic
127#888—22no—
128#888—22no—
129#879—28no—
130#879—28no—
131#878—28no—
132#877—28no—
133C4×Dic6—22no—
134C4×Dic6—22no—
135C4×Dic6—22no—
136C4×Dic6—22no—
137C3×#103—22no—
138C3×#103—22no—
139C3×#102—22no—
140C3×#102—22no—
141C3×#95—21no—
142#835—22no—
143#835—22no—
144#832—22no—
145#832—22no—
146C3⋊#95—21no—
147C3⋊#95—21no—
148C3⋊#95—21no—
149C3⋊#95—21no—
150#817—22no—
151#816—22no—
152#816—22no—
153#815—22no—
154Dic3⋊C8—22no—
155#810—22no—
156#810—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×#124C2×D311yescharacteristic
166C2×C22⋊Q8—61no—
167C2×C4×Q8—31no—
168C2×C4×Q8—31no—
169C2×C4×Q8—31no—
170C2×C4×D4—31no—
171C2×#117—61no—
172#489—64no—
173#475—64no—
174C2×#106—31no—
175C2×#106—31no—
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×#102C2×D311yescharacteristic
188C2×#102C2×D311yescharacteristic
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#340—31no—
201#336—31no—
202#336—31no—
203#336—31no—
204#336C2×D311yescharacteristic
205C24×C3—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—
212C2×C3×D4—21no—
213C2×C3×D4—21no—
214C2×C3×D4—21no—
215C2×C3×D4—21no—
216C2×C3×D4—21no—
217C2×C3×D4C2×D412yes—
218C22×C3×C4—21no—
219C22×C3×C4—21no—
220C22×C3×C4—21no—
221C22×C3×C4C2×D411yescharacteristic
222C22×C3×C4C2×D411yescharacteristic
223C22×C3×C4C2411yescharacteristic
224C22×Dic3—24no—
225C22×Dic3—21no—
226C22×Dic3—21no—
227C22×Dic3—21no—
228C22×Dic3—21no—
229C22×Dic3—42no—
230C22×Dic3—42no—
231C22×Dic3C2×D411yescharacteristic
232C22×Dic3C2×D411yescharacteristic
233C22×Dic3C2×D411yescharacteristic
234C22×Dic3C2×D411yescharacteristic
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—21no—
254C3×C4⋊C4—21no—
255C3×C4⋊C4—21no—
256C3×C4⋊C4—81no—
257C3×C4⋊C4#4112yes—
258C3×C22⋊C4—24no—
259C3×C22⋊C4—24no—
260C3×C42—21no—
261C3×C42—21no—
262#217—24no—
263#217—24no—
264#217—24no—
265C4⋊Dic3—21no—
266C4⋊Dic3—21no—
267C4⋊Dic3—22no—
268C4⋊Dic3—24no—
269C4⋊Dic3—21no—
270C4⋊Dic3—21no—
271C4⋊Dic3—81no—
272C4⋊Dic3C2×D412yes—
273C4⋊Dic3#4112yes—
274Dic3⋊C4—24no—
275Dic3⋊C4—24no—
276Dic3⋊C4—24no—
277Dic3⋊C4—22no—
278Dic3⋊C4—22no—
279Dic3⋊C4—42no—
280Dic3⋊C4—81no—
281Dic3⋊C4—81no—
282C4×Dic3—24no—
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×D4C22×D311yescharacteristic
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—22no—
312#124—22no—
313#124—34no—
314#124—62no—
315C22⋊Q8—68no—
316C4×Q8—34no—
317C4×Q8—62no—
318C4×Q8—62no—
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—31no—
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⋊C4—61no—
340C2×C4⋊C4C22×D311yescharacteristic
341C2×C4⋊C4C22×D311yescharacteristic
342C2×C22⋊C4—21no—
343C2×C22⋊C4—21no—
344C2×C22⋊C4—61no—
345C2×C22⋊C4—61no—
346C2×C22⋊C4—61no—
347C2×C42—31no—
348C2×C42—31no—
349C2×C42—61no—
350C2×C42—61no—
351C2×C42—61no—
352C2×C42C22×D311yescharacteristic
353#106—62no—
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—22no—
365#102—22no—
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×C3—22no—
375C23×C3—41no—
376C23×C3—41no—
377C23×C3—41no—
378C23×C3—41no—
379C23×C3—41no—
380C23×C3—41no—
381C23×C3C22×D411yescharacteristic
382C3×Q8—22no—
383C3×Q8—22no—
384C3×Q8—41no—
385C3×Q8—41no—
386C3×D4—22no—
387C3×D4—22no—
388C3×D4—41no—
389C3×D4—41no—
390C2×C3×C4—21no—
391C2×C3×C4—22no—
392C2×C3×C4—21no—
393C2×C3×C4—22no—
394C2×C3×C4—21no—
395C2×C3×C4—21no—
396C2×C3×C4—21no—
397C2×C3×C4—21no—
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×C4—41no—
405C2×C3×C4C22×D411yescharacteristic
406C2×C3×C4C22×D411yescharacteristic
407C2×C3×C4C2×#4111yescharacteristic
408C2×C3×C4C2×#4111yescharacteristic
409C2×C3×C42−1+411yescharacteristic
410C2×C3×C42−1+411yescharacteristic
411C2×Dic3—21no—
412C2×Dic3—21no—
413C2×Dic3—21no—
414C2×Dic3—21no—
415C2×Dic3—21no—
416C2×Dic3—21no—
417C2×Dic3—22no—
418C2×Dic3—21no—
419C2×Dic3—22no—
420C2×Dic3—22no—
421C2×Dic3—21no—
422C2×Dic3—21no—
423C2×Dic3—48no—
424C2×Dic3—41no—
425C2×Dic3—41no—
426C2×Dic3—41no—
427C2×Dic3—41no—
428C2×Dic3—41no—
429C2×Dic3—41no—
430C2×Dic3—41no—
431C2×Dic3—41no—
432C2×Dic3—41no—
433C2×Dic3#12012yes—
434C2×Dic3#12012yes—
435C2×Dic3#13712yes—
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—21no—
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—21no—
466C2×D4—21no—
467C2×D4—21no—
468C2×D4—21no—
469C2×D4—21no—
470C2×D4D3×D412yes—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—31no—
475C22×C4—31no—
476C22×C4—31no—
477C22×C4—31no—
478C22×C4—64no—
479C22×C4—61no—
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—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—32no—
504C4⋊C4—32no—
505C4⋊C4—62no—
506C4⋊C4—62no—
507C4⋊C4—62no—
508C4⋊C4—64no—
509C4⋊C4—64no—
510C4⋊C4—64no—
511C4⋊C4—64no—
512C4⋊C4—61no—
513C4⋊C4—61no—
514C4⋊C4—61no—
515C4⋊C4—61no—
516C4⋊C4—81no—
517C4⋊C4—122no—
518C4⋊C4—241no—
519C4⋊C4—241no—
520C4⋊C4—241no—
521C4⋊C4#23712yes—
522C22⋊C4—24no—
523C22⋊C4—24no—
524C22⋊C4—64no—
525C22⋊C4—64no—
526C22⋊C4—64no—
527C42—21no—
528C42—21no—
529C42—32no—
530C42—32no—
531C42—64no—
532C42—64no—
533C42—61no—
534C42—61no—
535C42—122no—
536C22×C3—42no—
537C22×C3—42no—
538C22×C3—42no—
539C22×C3—81no—
540C22×C3—81no—
541C22×C3—81no—
542C22×C3—81no—
543C22×C3C2×#12011yescharacteristic
544C22×C3#54811yescharacteristic
545C22×C3#54811yescharacteristic
546C22×C3C2×#13711yescharacteristic
547C22×C3#57511yescharacteristic
548C22×C3#57711yescharacteristic
549C22×C3#57711yescharacteristic
550C3×C4—42no—
551C3×C4—42no—
552C3×C4—41no—
553C3×C4—41no—
554C3×C4—81no—
555C3×C4#53211yescharacteristic
556C3×C4#53611yescharacteristic
557C3×C4#54711yescharacteristic
558C3×C4#54711yescharacteristic
559Dic3—22no—
560Dic3—22no—
561Dic3—22no—
562Dic3—42no—
563Dic3—42no—
564Dic3—42no—
565Dic3—41no—
566Dic3—81no—
567C23—22no—
568C23—41no—
569C23—41no—
570C23—41no—
571C23—41no—
572C23—41no—
573C23—41no—
574C23C2×D3×D411yescharacteristic
575Q8—22no—
576Q8—22no—
577Q8—41no—
578Q8—41no—
579Q8—62no—
580Q8—62no—
581Q8—124no—
582Q8—121no—
583Q8—121no—
584Q8—244no—
585D4—22no—
586D4—22no—
587D4—41no—
588D4—41no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—32no—
598C2×C4—32no—
599C2×C4—32no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—61no—
608C2×C4—61no—
609C2×C4—61no—
610C2×C4—61no—
611C2×C4—61no—
612C2×C4—61no—
613C2×C4—62no—
614C2×C4—61no—
615C2×C4—61no—
616C2×C4—62no—
617C2×C4—61no—
618C2×C4—62no—
619C2×C4—128no—
620C2×C4—121no—
621C2×C4—121no—
622C2×C4—121no—
623C2×C4—121no—
624C2×C4—121no—
625C2×C4—121no—
626C2×C4—121no—
627C2×C4—121no—
628C2×C4—121no—
629C2×C4C2×D3×D411yescharacteristic
630C2×C4C2×D3×D411yescharacteristic
631C2×C4C2×#23711yescharacteristic
632C2×C4D3×#4111yescharacteristic
633C2×C4#101011yescharacteristic
634C2×C4#101011yescharacteristic
635C8—41no—
636C8—41no—
637C8—121no—
638C8—121no—
639C2×C3—82no—
640C2×C3#243211yescharacteristic
641C2×C3#300711yescharacteristic
642C2×C3#301011yescharacteristic
643C2×C3#336911yescharacteristic
644C2×C3#336911yescharacteristic
645C2×C3#337611yescharacteristic
646C2×C3#337611yescharacteristic
647C22—42no—
648C22—42no—
649C22—42no—
650C22—81no—
651C22—81no—
652C22—81no—
653C22—81no—
654C22D3×#12011yescharacteristic
655C22#559811yescharacteristic
656C22#567711yescharacteristic
657C22#576811yescharacteristic
658C22#577011yescharacteristic
659C22#577511yescharacteristic
660C22D3×#13711yescharacteristic
661C4—41no—
662C4—42no—
663C4—42no—
664C4—41no—
665C4—62no—
666C4—62no—
667C4—62no—
668C4—81no—
669C4—122no—
670C4—122no—
671C4—121no—
672C4—122no—
673C4—241no—
674C4#559611yescharacteristic
675C4#561111yescharacteristic
676C4#564811yescharacteristic
677C4#565511yescharacteristic
678C3#2566211yescharacteristic, complemented, Sylow
679C2—82no—
680C2#7690411yescharacteristic
681C2#8008611yescharacteristic
682C2#8022811yescharacteristic
683C2#8052211yescharacteristic
684C2#8058211yescharacteristic
685C2#8166111yescharacteristic
686C2#8167111yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2042C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
hypercenter#2042D3
hypocenterC3#25662
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2042

Complements

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

NH ≅ G/Ntype
2. C2×#4791C2semidirect
4. #76071C2semidirect
7. #70284C2semidirect
8. #70246C2semidirect
9. #70220C2semidirect
13. #69632C2semidirect
14. #68684C2semidirect
15. #68620C2semidirect
678. C3#25662semidirect

Maximal subgroups

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

subgroupindex
C2×#56712
C2×#47912
C2×#47652
#760712
C3×#20422
#717222
#702842
#702462
#702202
#701702
#701692
#701622
#696322
#686842
#686202
#256623