Finite Group Databasebeta

← #10849300

Subgroups — #10849300

Order 768.

980 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5721C211yesmaximal, characteristic
2C2×#4797C211yesmaximal, characteristic
3C2×#4796C211yesmaximal, characteristic
4#76078C211yesmaximal, characteristic
5C3×#2043C211yesmaximal, characteristic
6#71781C211yesmaximal, characteristic
7#70284C211yesmaximal, characteristic
8#70246C211yesmaximal, characteristic
9#70237C211yesmaximal, characteristic
10#70236C211yesmaximal, characteristic
11#70229C211yesmaximal, characteristic
12#70220C211yesmaximal, characteristic
13#69632C211yesmaximal, characteristic
14#68683C211yesmaximal, characteristic
15#68620C211yesmaximal, characteristic
16#25657—31nomaximal, Sylow
17C2×C3×C4⋊Q8C2211yescharacteristic
18C2×Q8×Dic3C2211yescharacteristic
19C2×Q8×Dic3C2211yescharacteristic
20C2×#944C2211yescharacteristic
21C2×#944C2211yescharacteristic
22#5721—24no—
23#5721—24no—
24C2×#890—21no—
25C2×#890C2211yescharacteristic
26C2×#889—21no—
27C2×#889—21no—
28C2×#887—21no—
29C2×C4×Dic6C2211yescharacteristic
30C2×C4×Dic6C2211yescharacteristic
31C2×C3×#103C2211yescharacteristic
32C2×C3×#103C2211yescharacteristic
33C2×C3×#103C2211yescharacteristic
34C2×C3×#103C2211yescharacteristic
35C3×#389C2211yescharacteristic
36C2×#835C2211yescharacteristic
37C2×#835C2211yescharacteristic
38C2×#835C2211yescharacteristic
39C2×#835C2211yescharacteristic
40C2×#818C2211yescharacteristic
41C2×#816C2211yescharacteristic
42C2×#816C2211yescharacteristic
43C2×#815C2211yescharacteristic
44C2×Dic3⋊C8C2211yescharacteristic
45#4984—21no—
46#4984C2211yescharacteristic
47#4979—21no—
48Dic3×C4⋊C4C2211yescharacteristic
49#4972C2211yescharacteristic
50C2×#810C2211yescharacteristic
51C2×#810C2211yescharacteristic
52C2×#808C2211yescharacteristic
53#4928C2211yescharacteristic
54#4797—24no—
55#4796—24no—
56#4657—21no—
57#4647—21no—
58C3×#336C2211yescharacteristic
59#4551C2211yescharacteristic
60#4550C2211yescharacteristic
61#4550C2211yescharacteristic
62#4529C2211yescharacteristic
63#4524C2211yescharacteristic
64C2×#557—31no—
65C2×#491—31no—
66C2×#479—31no—
67#2584—31no—
68#2043—31no—
69#2043—31no—
70#2043—31no—
71#2043D311yescharacteristic
72#1985—31no—
73#1981—31no—
74#1865—31no—
75#1865—31no—
76#1863—31no—
77#1863—31no—
78#1763—31no—
79C22×C3×Q8C2311yescharacteristic
80C22×C3×Q8C2311yescharacteristic
81C22×Dic6D411yescharacteristic
82C22×Dic6C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C3×C4⋊Q8—22no—
86C3×C4⋊Q8—22no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C42C2311yescharacteristic
93Q8×Dic3—22no—
94Q8×Dic3—22no—
95Q8×Dic3—22no—
96Q8×Dic3D414yes—
97#944—22no—
98#944—22no—
99#944—22no—
100#944D414yes—
101C2×C4⋊Dic3—21no—
102C2×C4⋊Dic3—21no—
103C2×C4⋊Dic3—21no—
104C2×C4⋊Dic3—21no—
105C2×C4⋊Dic3D411yescharacteristic
106C2×C4⋊Dic3C2311yescharacteristic
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⋊C4—21no—
116C2×Dic3⋊C4—21no—
117C2×Dic3⋊C4D411yescharacteristic
118C2×Dic3⋊C4D411yescharacteristic
119C2×C4×Dic3—21no—
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#890—28no—
128#890—22no—
129#890—22no—
130#889—28no—
131#889—28no—
132#887—28no—
133C4×Dic6—22no—
134C4×Dic6—22no—
135C4×Dic6—22no—
136C4×Dic6—22no—
137C3×#103—22no—
138C3×#103—22no—
139C3×#103—22no—
140C3×#103—22no—
141C3×#95—21no—
142#835—22no—
143#835—22no—
144#835—22no—
145#835—22no—
146C3⋊#95—21no—
147C3⋊#95—21no—
148C3⋊#95—21no—
149C3⋊#95—21no—
150#818—22no—
151#816—22no—
152#816—22no—
153#815—22no—
154Dic3⋊C8—22no—
155#810—22no—
156#810—22no—
157#808—22no—
158#557—64no—
159#557—64no—
160C2×C4⋊Q8—31no—
161C2×C4⋊Q8—31no—
162C2×C4⋊Q8C2×D311yescharacteristic
163C2×#125—31no—
164C2×#125—61no—
165C2×#125—61no—
166C2×#125—61no—
167C2×C4×Q8—31no—
168C2×C4×Q8—31no—
169C2×C4×Q8—31no—
170C2×C4×Q8—31no—
171C2×C4×Q8—61no—
172#491—64no—
173#479—64no—
174C2×#107—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×#103—31no—
184C2×#103—31no—
185C2×#103C2×D311yescharacteristic
186C2×#103C2×D311yescharacteristic
187C2×#103C2×D311yescharacteristic
188C2×#103C2×D311yescharacteristic
189C2×#97—31no—
190#398—61no—
191#395—61no—
192#389—31no—
193#389—31no—
194#389—31no—
195#389—61no—
196#389—61no—
197#389C2×D311yescharacteristic
198C4×C4⋊C4—31no—
199#340—31no—
200#340—31no—
201#336—31no—
202#336—31no—
203#336—31no—
204#336C2×D311yescharacteristic
205C2×C3×Q8—21no—
206C2×C3×Q8—21no—
207C2×C3×Q8—21no—
208C2×C3×Q8—21no—
209C2×C3×Q8—21no—
210C2×C3×Q8—21no—
211C2×C3×Q8—21no—
212C2×C3×Q8—21no—
213C2×C3×Q8—21no—
214C2×C3×Q8—21no—
215C2×C3×Q8C2×D412yes—
216C2×C3×Q8C2×D412yes—
217C22×C3×C4—21no—
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—21no—
225C22×Dic3—21no—
226C22×Dic3—21no—
227C22×Dic3—21no—
228C22×Dic3C2×D411yescharacteristic
229C22×Dic3C2×D411yescharacteristic
230C22×Dic3C2×D411yescharacteristic
231C22×Dic3C2×D411yescharacteristic
232C2×Dic6—22no—
233C2×Dic6—22no—
234C2×Dic6—22no—
235C2×Dic6—21no—
236C2×Dic6—21no—
237C2×Dic6—21no—
238C2×Dic6—21no—
239C2×Dic6—21no—
240C2×Dic6C2×D412yes—
241C2×C3×C8—21no—
242C2×C3×C8—21no—
243C2×C3×C8—21no—
244C2×C3×C8—21no—
245C2×C3×C8—21no—
246C2×C3×C8—21no—
247C3×C4⋊C4—24no—
248C3×C4⋊C4—24no—
249C3×C4⋊C4—21no—
250C3×C4⋊C4—21no—
251C3×C4⋊C4—21no—
252C3×C4⋊C4—42no—
253C3×C4⋊C4#4112yes—
254C3×C42—21no—
255C3×C42—21no—
256C4⋊Dic3—24no—
257C4⋊Dic3—21no—
258C4⋊Dic3—21no—
259C4⋊Dic3—22no—
260C4⋊Dic3—24no—
261C4⋊Dic3—21no—
262C4⋊Dic3—21no—
263C4⋊Dic3—42no—
264C4⋊Dic3—81no—
265C4⋊Dic3C2×D412yes—
266C4⋊Dic3#4112yes—
267Dic3⋊C4—24no—
268Dic3⋊C4—24no—
269Dic3⋊C4—24no—
270Dic3⋊C4—22no—
271Dic3⋊C4—22no—
272Dic3⋊C4—24no—
273Dic3⋊C4—24no—
274Dic3⋊C4—42no—
275Dic3⋊C4—42no—
276Dic3⋊C4—81no—
277C4×Dic3—24no—
278C4×Dic3—24no—
279C4×Dic3—21no—
280C4×Dic3—21no—
281C4×Dic3—24no—
282C4×Dic3—81no—
283C4×Dic3C2×D412yes—
284C4×Dic3C2×D412yes—
285C2×C3⋊C8—21no—
286C2×C3⋊C8—21no—
287C2×C3⋊C8—21no—
288C2×C3⋊C8—21no—
289C2×C3⋊C8—21no—
290C2×C3⋊C8—21no—
291C22×Q8—31no—
292C22×Q8—31no—
293C22×Q8C22×D311yescharacteristic
294C22×Q8C22×D311yescharacteristic
295C22×C8—31no—
296C22×C8—31no—
297C22×C8C22×D311yescharacteristic
298C22×C8C22×D311yescharacteristic
299C4⋊Q8—22no—
300C4⋊Q8—22no—
301C4⋊Q8—34no—
302C4⋊Q8—62no—
303C4⋊Q8—62no—
304C4⋊Q8—62no—
305#125—62no—
306#125—62no—
307#125—68no—
308#125—68no—
309#125—68no—
310C4×Q8—34no—
311C4×Q8—62no—
312C4×Q8—62no—
313C4×Q8—62no—
314C4×Q8—62no—
315C4×Q8—62no—
316C4×Q8—68no—
317C4×Q8—62no—
318C4×Q8—62no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—31no—
323C2×C4⋊C4—31no—
324C2×C4⋊C4—31no—
325C2×C4⋊C4—31no—
326C2×C4⋊C4—31no—
327C2×C4⋊C4—31no—
328C2×C4⋊C4—61no—
329C2×C4⋊C4—61no—
330C2×C4⋊C4—61no—
331C2×C4⋊C4—61no—
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×C42—31no—
343C2×C42—31no—
344C2×C42—61no—
345C2×C42—61no—
346C2×C42—61no—
347C2×C42—61no—
348C2×C42C22×D311yescharacteristic
349#107—62no—
350#106—62no—
351C4⋊C8—62no—
352#103—22no—
353#103—22no—
354#103—22no—
355#103—22no—
356#103—62no—
357#103—62no—
358#103—62no—
359#103—62no—
360#103—62no—
361#103—62no—
362#103—62no—
363#103—62no—
364#97—62no—
365#95—21no—
366#95—61no—
367#95—61no—
368#95—61no—
369#95—61no—
370C23×C3C22×D411yescharacteristic
371C3×Q8—22no—
372C3×Q8—22no—
373C3×Q8—22no—
374C3×Q8—22no—
375C3×Q8—41no—
376C3×Q8—41no—
377C3×Q8—41no—
378C3×Q8—41no—
379C2×C3×C4—21no—
380C2×C3×C4—21no—
381C2×C3×C4—22no—
382C2×C3×C4—22no—
383C2×C3×C4—21no—
384C2×C3×C4—22no—
385C2×C3×C4—21no—
386C2×C3×C4—21no—
387C2×C3×C4—41no—
388C2×C3×C4—41no—
389C2×C3×C4—41no—
390C2×C3×C4—41no—
391C2×C3×C4—41no—
392C2×C3×C4—41no—
393C2×C3×C4—41no—
394C2×C3×C4—41no—
395C2×C3×C4—41no—
396C2×C3×C4#13712yes—
397C2×C3×C4C22×D411yescharacteristic
398C2×C3×C4C22×D411yescharacteristic
399C2×C3×C4C2×#4111yescharacteristic
400C2×C3×C4C2×#4111yescharacteristic
401C2×C3×C42−1+411yescharacteristic
402C2×C3×C42−1+411yescharacteristic
403C2×Dic3—21no—
404C2×Dic3—21no—
405C2×Dic3—21no—
406C2×Dic3—21no—
407C2×Dic3—21no—
408C2×Dic3—21no—
409C2×Dic3—22no—
410C2×Dic3—22no—
411C2×Dic3—21no—
412C2×Dic3—22no—
413C2×Dic3—21no—
414C2×Dic3—21no—
415C2×Dic3—41no—
416C2×Dic3—41no—
417C2×Dic3—41no—
418C2×Dic3—41no—
419C2×Dic3—41no—
420C2×Dic3—41no—
421C2×Dic3—41no—
422C2×Dic3—41no—
423C2×Dic3—41no—
424C2×Dic3#12012yes—
425C2×Dic3#12012yes—
426C2×Dic3#13712yes—
427Dic6—28no—
428Dic6—22no—
429Dic6—22no—
430Dic6—41no—
431Dic6—41no—
432C3×C8—41no—
433C3×C8—41no—
434C3⋊C8—41no—
435C3⋊C8—41no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8—21no—
444C2×Q8—21no—
445C2×Q8—21no—
446C2×Q8—32no—
447C2×Q8—62no—
448C2×Q8—62no—
449C2×Q8—62no—
450C2×Q8—61no—
451C2×Q8—61no—
452C2×Q8—61no—
453C2×Q8—61no—
454C2×Q8—61no—
455C2×Q8D3×D412yes—
456C2×Q8D3×D412yes—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—31no—
462C22×C4—31no—
463C22×C4—31no—
464C22×C4—31no—
465C22×C4—61no—
466C22×C4—61no—
467C22×C4—61no—
468C22×C4—61no—
469C22×C4D3×D411yescharacteristic
470C22×C4D3×D411yescharacteristic
471C22×C4C23×D311yescharacteristic
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—61no—
479C2×C8—61no—
480C2×C8—61no—
481C2×C8—61no—
482C2×C8—61no—
483C2×C8—61no—
484C4⋊C4—24no—
485C4⋊C4—24no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—32no—
490C4⋊C4—32no—
491C4⋊C4—42no—
492C4⋊C4—62no—
493C4⋊C4—62no—
494C4⋊C4—62no—
495C4⋊C4—64no—
496C4⋊C4—64no—
497C4⋊C4—64no—
498C4⋊C4—64no—
499C4⋊C4—64no—
500C4⋊C4—64no—
501C4⋊C4—64no—
502C4⋊C4—61no—
503C4⋊C4—61no—
504C4⋊C4—61no—
505C4⋊C4—61no—
506C4⋊C4—122no—
507C4⋊C4—122no—
508C4⋊C4—122no—
509C4⋊C4—241no—
510C4⋊C4—241no—
511C4⋊C4#23712yes—
512C42—21no—
513C42—21no—
514C42—32no—
515C42—32no—
516C42—64no—
517C42—64no—
518C42—64no—
519C42—61no—
520C42—61no—
521C42—241no—
522C22×C3C2×#12011yescharacteristic
523C22×C3#54811yescharacteristic
524C22×C3#54811yescharacteristic
525C22×C3C2×#13711yescharacteristic
526C22×C3C2×#13711yescharacteristic
527C22×C3#57711yescharacteristic
528C22×C3#57811yescharacteristic
529C3×C4—22no—
530C3×C4—42no—
531C3×C4—42no—
532C3×C4—42no—
533C3×C4—41no—
534C3×C4—81no—
535C3×C4#53211yescharacteristic
536C3×C4#53611yescharacteristic
537C3×C4#54711yescharacteristic
538C3×C4#54711yescharacteristic
539Dic3—22no—
540Dic3—22no—
541Dic3—22no—
542Dic3—42no—
543Dic3—42no—
544Dic3—41no—
545Dic3—42no—
546Dic3—81no—
547C23C2×D3×D411yescharacteristic
548Q8—22no—
549Q8—22no—
550Q8—22no—
551Q8—22no—
552Q8—41no—
553Q8—41no—
554Q8—41no—
555Q8—41no—
556Q8—68no—
557Q8—62no—
558Q8—62no—
559Q8—121no—
560Q8—121no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—32no—
570C2×C4—32no—
571C2×C4—32no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—61no—
582C2×C4—61no—
583C2×C4—61no—
584C2×C4—61no—
585C2×C4—61no—
586C2×C4—61no—
587C2×C4—62no—
588C2×C4—61no—
589C2×C4—61no—
590C2×C4—62no—
591C2×C4—61no—
592C2×C4—62no—
593C2×C4—121no—
594C2×C4—121no—
595C2×C4—121no—
596C2×C4—121no—
597C2×C4—121no—
598C2×C4—121no—
599C2×C4—121no—
600C2×C4—121no—
601C2×C4—121no—
602C2×C4#91512yes—
603C2×C4C2×D3×D411yescharacteristic
604C2×C4C2×D3×D411yescharacteristic
605C2×C4C2×#23711yescharacteristic
606C2×C4D3×#4111yescharacteristic
607C2×C4#101011yescharacteristic
608C2×C4#101011yescharacteristic
609C8—41no—
610C8—41no—
611C8—121no—
612C8—121no—
613C2×C3#243211yescharacteristic
614C2×C3#300411yescharacteristic
615C2×C3#301311yescharacteristic
616C2×C3#337211yescharacteristic
617C2×C3#337211yescharacteristic
618C2×C3#337411yescharacteristic
619C2×C3#337411yescharacteristic
620C22D3×#12011yescharacteristic
621C22#559811yescharacteristic
622C22#567711yescharacteristic
623C22C2×#91511yescharacteristic
624C22#576611yescharacteristic
625C22D3×#13711yescharacteristic
626C22#577811yescharacteristic
627C4—22no—
628C4—42no—
629C4—41no—
630C4—42no—
631C4—42no—
632C4—62no—
633C4—62no—
634C4—62no—
635C4—81no—
636C4—121no—
637C4—122no—
638C4—122no—
639C4—122no—
640C4—241no—
641C4#559611yescharacteristic
642C4#561111yescharacteristic
643C4#564811yescharacteristic
644C4#565511yescharacteristic
645C3#2565711yescharacteristic, complemented, Sylow
646C2#7690411yescharacteristic
647C2#8003911yescharacteristic
648C2#8018611yescharacteristic
649C2#8041711yescharacteristic
650C2#8046611yescharacteristic
651C2#8162711yescharacteristic
652C2#8173211yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2043C2
socle, prime radicalC23×C3C22×D4
hypercenter#2043D3
hypocenterC3#25657
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2043

Complements

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

NH ≅ G/Ntype
645. C3#25657semidirect

Maximal subgroups

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

subgroupindex
C2×#57212
C2×#47972
C2×#47962
#760782
C3×#20432
#717812
#702842
#702462
#702372
#702362
#702292
#702202
#696322
#686832
#686202
#256573