Finite Group Databasebeta

← #10901412

Subgroups — #10901412

Order 768.

1056 conjugacy classes of subgroups.

The subgroup lattice has 714 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5624C211yesmaximal, characteristic
2C2×#4891C211yesmaximal, characteristic, complemented
3C2×#4796C211yesmaximal, characteristic, complemented
4#73998C211yesmaximal, characteristic, complemented
5C3×#2036C211yesmaximal, characteristic, complemented
6#71694C211yesmaximal, characteristic, complemented
7#70618C211yesmaximal, characteristic, complemented
8#70279C211yesmaximal, characteristic, complemented
9#70245C211yesmaximal, characteristic, complemented
10#70222C211yesmaximal, characteristic, complemented
11#69934C211yesmaximal, characteristic
12#69908C211yesmaximal, characteristic, complemented
13#69524C211yesmaximal, characteristic
14#68893C211yesmaximal, characteristic, complemented
15#68829C211yesmaximal, characteristic, complemented
16#29693—31nomaximal, Sylow
17C2×C3×C22⋊Q8C2211yescharacteristic
18C2×#947C2211yescharacteristic
19C2×Q8×Dic3C2211yescharacteristic
20C2×#930C2211yescharacteristic
21C22×C4×Dic3C2211yescharacteristic
22#5624—24no—
23#5624—24no—
24C2×#898—21no—
25C2×#893C2211yescharacteristic
26C2×#892—21no—
27C2×#890C2211yescharacteristic
28C2×#890C2211yescharacteristic
29C2×#886—21no—
30C2×#67⋊C4—21no—
31C2×C3×#107C2211yescharacteristic
32C2×C3×#103C2211yescharacteristic
33C2×C3×#103C2211yescharacteristic
34C2×C3×C22⋊C8C2211yescharacteristic
35C3×#380C2211yescharacteristic
36C2×#835C2211yescharacteristic
37C2×#835C2211yescharacteristic
38#5212C2211yescharacteristic
39C2×#830C2211yescharacteristic
40C2×#821C2211yescharacteristic
41C2×#821C2211yescharacteristic
42C2×#818C2211yescharacteristic
43C2×Dic3⋊C8C2211yescharacteristic
44C2×Dic3⋊C8C2211yescharacteristic
45#4984C2211yescharacteristic
46#4977C2211yescharacteristic
47#4972C2211yescharacteristic
48C2×#809C2211yescharacteristic
49C2×#809C2211yescharacteristic
50C2×#808C2211yescharacteristic
51C2×#808C2211yescharacteristic
52#4950—21no—
53#4946—21no—
54#4891—24no—
55#4796—24no—
56#4655—21no—
57#4654—21no—
58C3×#340C2211yescharacteristic
59#4551C2211yescharacteristic
60#4550C2211yescharacteristic
61#4550C2211yescharacteristic
62#4529C2211yescharacteristic
63#4529C2211yescharacteristic
64C2×#529—31no—
65C2×#479—31no—
66C2×#478—31no—
67#2851—31no—
68#2059—31no—
69#2059—31no—
70#2042—31no—
71#2042—31no—
72#2036D311yescharacteristic, complemented
73#2035—31no—
74#2034—31no—
75#2033—31no—
76#1805—31no—
77#1804—31no—
78#1797—31no—
79C22×C3×Q8C2311yescharacteristic
80C23×C3×C4C2311yescharacteristic
81C22×#67—21no—
82C23×Dic3D411yescharacteristic
83C23×Dic3D411yescharacteristic
84C22×D12C2311yescharacteristic
85C22×C4×D3—21no—
86C22×C3×C8C2311yescharacteristic
87C22×C3×C8C2311yescharacteristic
88C3×C22⋊Q8—22no—
89C3×C22⋊Q8—22no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C22⋊C4—21no—
95C2×C3×C22⋊C4—21no—
96C2×#217—21no—
97#947—22no—
98#947—22no—
99Q8×Dic3—22no—
100Q8×Dic3D414yes—
101#930—22no—
102#930—22no—
103C2×(C2×D3)⋊C4—21no—
104C2×(C2×D3)⋊C4—21no—
105C2×(C2×D3)⋊C4—21no—
106C2×(C2×D3)⋊C4—21no—
107C2×C4⋊Dic3—21no—
108C2×C4⋊Dic3—21no—
109C2×C4⋊Dic3—21no—
110C2×C4⋊Dic3—21no—
111C2×C4⋊Dic3C2311yescharacteristic
112C2×C4⋊Dic3C2311yescharacteristic
113C2×C4⋊Dic3C2311yescharacteristic
114C2×Dic3⋊C4—21no—
115C2×Dic3⋊C4—21no—
116C2×Dic3⋊C4—21no—
117C2×C4×Dic3—21no—
118C2×C4×Dic3—22no—
119C2×C4×Dic3—22no—
120C2×C4×Dic3D411yescharacteristic
121C2×C4×Dic3D411yescharacteristic
122C2×C4×Dic3C2311yescharacteristic
123C2×C4×Dic3C2311yescharacteristic
124C22×C3⋊C8C2311yescharacteristic
125C22×C3⋊C8C2311yescharacteristic
126#898—28no—
127#893—22no—
128#893D414yes—
129#892—28no—
130#890—22no—
131#890—22no—
132#890—22no—
133#890—22no—
134#886—28no—
135#67⋊C4—28no—
136C3×#107—22no—
137C3×#103—22no—
138C3×#103—41no—
139C3×C22⋊C8—41no—
140C3×#95—21no—
141#835—22no—
142#835—41no—
143C3⋊#95—21no—
144C3⋊#95—21no—
145C3⋊#95—21no—
146C3⋊#95—21no—
147C3⋊#95—21no—
148#830—41no—
149#821—22no—
150#821—41no—
151#818—22no—
152Dic3⋊C8—22no—
153Dic3⋊C8—22no—
154#809—22no—
155#809—41no—
156#808—22no—
157#808—22no—
158#529—64no—
159#529—64no—
160C2×#126—61no—
161C2×#125—31no—
162C2×#125—31no—
163C2×#124—31no—
164C2×#123—61no—
165C2×C22⋊Q8C2×D311yescharacteristic
166C2×#121—31no—
167C2×C4×Q8—31no—
168C2×C4×D4—31no—
169C2×C4×D4—61no—
170C2×#117—61no—
171C22×C42—31no—
172#479—64no—
173#478—64no—
174C2×#107—31no—
175C2×#107C2×D311yescharacteristic
176C2×#106—31no—
177C2×#106—31no—
178C2×C4⋊C8—31no—
179C2×C4⋊C8—31no—
180C2×#103—31no—
181C2×#103—31no—
182C2×#103C2×D311yescharacteristic
183C2×#103C2×D311yescharacteristic
184C2×#102—31no—
185C2×#102—31no—
186C2×#102—31no—
187C2×#102—31no—
188C2×C22⋊C8—31no—
189C2×C22⋊C8C2×D311yescharacteristic
190#389—31no—
191#389—31no—
192#387—61no—
193#387—61no—
194#385—61no—
195#385—61no—
196#382—31no—
197#380—31no—
198#380C2×D311yescharacteristic
199#340—31no—
200#340—31no—
201#340—31no—
202#340C2×D311yescharacteristic
203#336—31no—
204#336—31no—
205C24×C3C2×D411yescharacteristic
206C23×D3—21no—
207C2×C3×Q8—21no—
208C2×C3×Q8—21no—
209C2×C3×Q8—21no—
210C2×C3×Q8—21no—
211C2×C3×Q8—21no—
212C2×C3×Q8C2×D412yes—
213C22×C3×C4—21no—
214C22×C3×C4—21no—
215C22×C3×C4—21no—
216C22×C3×C4—21no—
217C22×C3×C4—21no—
218C22×C3×C4—21no—
219C22×C3×C4—21no—
220C22×C3×C4—21no—
221C22×C3×C4—21no—
222C22×C3×C4—21no—
223C22×C3×C4C2×D411yescharacteristic
224C22×C3×C4C2411yescharacteristic
225C2×#67—24no—
226C2×#67—42no—
227C2×#67—42no—
228C22×Dic3—22no—
229C22×Dic3—21no—
230C22×Dic3—22no—
231C22×Dic3—22no—
232C22×Dic3—21no—
233C22×Dic3—21no—
234C22×Dic3—41no—
235C22×Dic3—41no—
236C22×Dic3—41no—
237C22×Dic3C2×D411yescharacteristic
238C22×Dic3C2×D411yescharacteristic
239C22×Dic3C2×D411yescharacteristic
240C22×Dic3C2×D411yescharacteristic
241C2×D12—21no—
242C2×D12—21no—
243C2×D12—21no—
244C2×D12—21no—
245C2×D12—21no—
246C2×D12C2×D412yes—
247C2×C4×D3—24no—
248C2×C4×D3—42no—
249C2×C4×D3—42no—
250C2×C3×C8—21no—
251C2×C3×C8—21no—
252C2×C3×C8—21no—
253C2×C3×C8—21no—
254C2×C3×C8—21no—
255C2×C3×C8—21no—
256C3×C4⋊C4—24no—
257C3×C4⋊C4—21no—
258C3×C4⋊C4—21no—
259C3×C4⋊C4—21no—
260C3×C4⋊C4—21no—
261C3×C4⋊C4—21no—
262C3×C4⋊C4C2×D412yes—
263C3×C22⋊C4—24no—
264C3×C22⋊C4—42no—
265#217—42no—
266(C2×D3)⋊C4—24no—
267(C2×D3)⋊C4—24no—
268(C2×D3)⋊C4—24no—
269(C2×D3)⋊C4—24no—
270C4⋊Dic3—24no—
271C4⋊Dic3—21no—
272C4⋊Dic3—24no—
273C4⋊Dic3—21no—
274C4⋊Dic3—21no—
275C4⋊Dic3—21no—
276C4⋊Dic3—21no—
277C4⋊Dic3—42no—
278C4⋊Dic3—42no—
279C4⋊Dic3C2×D412yes—
280Dic3⋊C4—24no—
281Dic3⋊C4—42no—
282Dic3⋊C4—42no—
283C4×Dic3—21no—
284C4×Dic3—21no—
285C4×Dic3—24no—
286C4×Dic3—22no—
287C4×Dic3—22no—
288C4×Dic3C2×D412yes—
289C4×Dic3C2×D412yes—
290C2×C3⋊C8—21no—
291C2×C3⋊C8—21no—
292C2×C3⋊C8—21no—
293C2×C3⋊C8—21no—
294C2×C3⋊C8—21no—
295C2×C3⋊C8—21no—
296C22×Q8C22×D311yescharacteristic
297C22×D4—31no—
298C22×D4—61no—
299C23×C4—31no—
300C23×C4—31no—
301C23×C4—61no—
302C23×C4C22×D311yescharacteristic
303C22×C8—31no—
304C22×C8—31no—
305C22×C8C22×D311yescharacteristic
306C22×C8C22×D311yescharacteristic
307#126—68no—
308#125—62no—
309#125—62no—
310#125—62no—
311#125—62no—
312#124—62no—
313#124—62no—
314#123—68no—
315C22⋊Q8—22no—
316C22⋊Q8—22no—
317#121—62no—
318#121—62no—
319C4×Q8—34no—
320C4×Q8—62no—
321C4×D4—34no—
322C4×D4—62no—
323C4×D4—68no—
324#117—68no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—31no—
327C2×C4⋊C4—31no—
328C2×C4⋊C4—31no—
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⋊C4C22×D311yescharacteristic
337C2×C4⋊C4C22×D311yescharacteristic
338C2×C4⋊C4C22×D311yescharacteristic
339C2×C22⋊C4—21no—
340C2×C22⋊C4—21no—
341C2×C22⋊C4—61no—
342C2×C22⋊C4—61no—
343C2×C22⋊C4—61no—
344C2×C22⋊C4—61no—
345C2×C22⋊C4—61no—
346C2×C42—31no—
347C2×C42—31no—
348C2×C42—31no—
349C2×C42—31no—
350C2×C42—62no—
351C2×C42—62no—
352C2×C42—61no—
353#107—22no—
354#107—62no—
355#106—62no—
356#106—62no—
357C4⋊C8—62no—
358C4⋊C8—62no—
359#103—22no—
360#103—41no—
361#103—62no—
362#103—121no—
363#102—62no—
364#102—62no—
365#102—121no—
366#102—121no—
367C22⋊C8—41no—
368C22⋊C8—121no—
369#95—21no—
370#95—61no—
371#95—61no—
372#95—61no—
373#95—61no—
374#95—61no—
375C23×C3—21no—
376C23×C3—21no—
377C23×C3—21no—
378C23×C3—21no—
379C23×C3—21no—
380C23×C3—21no—
381C23×C3#13512yes—
382C23×C3C22×D411yescharacteristic
383C22×D3—22no—
384C22×D3—41no—
385C22×D3—41no—
386C22×D3—41no—
387C22×D3—41no—
388C22×D3—41no—
389C22×D3—41no—
390C3×Q8—22no—
391C3×Q8—22no—
392C3×Q8—41no—
393C3×Q8—41no—
394C2×C3×C4—21no—
395C2×C3×C4—22no—
396C2×C3×C4—22no—
397C2×C3×C4—21no—
398C2×C3×C4—21no—
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×C4—41no—
406C2×C3×C4—41no—
407C2×C3×C4—41no—
408C2×C3×C4—41no—
409C2×C3×C4—41no—
410C2×C3×C4—41no—
411C2×C3×C4—41no—
412C2×C3×C4—41no—
413C2×C3×C4C22×D411yescharacteristic
414C2×C3×C4C22×D411yescharacteristic
415C2×C3×C4H(5,𝔽2)11yescharacteristic
416C2×C3×C42−1+411yescharacteristic
417C2×C3×C42−1+411yescharacteristic
418C2×C3×C42−1+411yescharacteristic
419#67—48no—
420C2×Dic3—21no—
421C2×Dic3—21no—
422C2×Dic3—21no—
423C2×Dic3—21no—
424C2×Dic3—21no—
425C2×Dic3—21no—
426C2×Dic3—22no—
427C2×Dic3—21no—
428C2×Dic3—21no—
429C2×Dic3—21no—
430C2×Dic3—21no—
431C2×Dic3—44no—
432C2×Dic3—44no—
433C2×Dic3—41no—
434C2×Dic3—41no—
435C2×Dic3—41no—
436C2×Dic3—41no—
437C2×Dic3—41no—
438C2×Dic3—41no—
439C2×Dic3—41no—
440C2×Dic3—41no—
441C2×Dic3#12012yes—
442C2×Dic3#12012yes—
443D12—22no—
444D12—22no—
445D12—41no—
446D12—41no—
447C4×D3—48no—
448C3×C8—41no—
449C3×C8—41no—
450C3⋊C8—41no—
451C3⋊C8—41no—
452C24—61no—
453C24D3×D411yescharacteristic
454C2×Q8—21no—
455C2×Q8—21no—
456C2×Q8—21no—
457C2×Q8—21no—
458C2×Q8—21no—
459C2×Q8D3×D412yes—
460C2×D4—32no—
461C2×D4—64no—
462C2×D4—61no—
463C2×D4—61no—
464C2×D4—61no—
465C2×D4—61no—
466C2×D4—61no—
467C2×D4—122no—
468C2×D4—122no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—31no—
480C22×C4—31no—
481C22×C4—31no—
482C22×C4—31no—
483C22×C4—64no—
484C22×C4—62no—
485C22×C4—62no—
486C22×C4—62no—
487C22×C4—61no—
488C22×C4—61no—
489C22×C4—61no—
490C22×C4—122no—
491C22×C4—122no—
492C22×C4—121no—
493C22×C4—121no—
494C22×C4—121no—
495C22×C4D3×D411yescharacteristic
496C22×C4C23×D311yescharacteristic
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—61no—
504C2×C8—61no—
505C2×C8—61no—
506C2×C8—61no—
507C2×C8—61no—
508C2×C8—61no—
509C4⋊C4—24no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—32no—
516C4⋊C4—64no—
517C4⋊C4—64no—
518C4⋊C4—64no—
519C4⋊C4—61no—
520C4⋊C4—61no—
521C4⋊C4—61no—
522C4⋊C4—61no—
523C4⋊C4—61no—
524C4⋊C4—122no—
525C4⋊C4—122no—
526C4⋊C4—122no—
527C4⋊C4—122no—
528C4⋊C4D3×D412yes—
529C22⋊C4—24no—
530C22⋊C4—42no—
531C22⋊C4—64no—
532C22⋊C4—64no—
533C22⋊C4—64no—
534C22⋊C4—64no—
535C22⋊C4—122no—
536C42—32no—
537C42—32no—
538C42—62no—
539C42—62no—
540C42—64no—
541C42—61no—
542C42—61no—
543C22×C3—22no—
544C22×C3—22no—
545C22×C3—22no—
546C22×C3—41no—
547C22×C3—41no—
548C22×C3—41no—
549C22×C3—41no—
550C22×C3C2×#12011yescharacteristic
551C22×C3#53611yescharacteristic
552C22×C3#53711yescharacteristic
553C22×C3C2×#13511yescharacteristic
554C22×C3#57511yescharacteristic
555C22×C3#57511yescharacteristic
556C22×C3#57511yescharacteristic
557C2×D3—42no—
558C2×D3—42no—
559C2×D3—42no—
560C2×D3—81no—
561C2×D3—81no—
562C2×D3—81no—
563C2×D3—81no—
564C3×C4—41no—
565C3×C4—42no—
566C3×C4—42no—
567C3×C4—81no—
568C3×C4—81no—
569C3×C4#53611yescharacteristic
570C3×C4#53611yescharacteristic
571C3×C4#53711yescharacteristic
572C3×C4#53711yescharacteristic
573Dic3—22no—
574Dic3—22no—
575Dic3—41no—
576Dic3—42no—
577Dic3—41no—
578Dic3—81no—
579Dic3—81no—
580C23—21no—
581C23—21no—
582C23—21no—
583C23—21no—
584C23—21no—
585C23—21no—
586C23—62no—
587C23—121no—
588C23—121no—
589C23—121no—
590C23—121no—
591C23—121no—
592C23—121no—
593C23#91912yes—
594C23C2×D3×D411yescharacteristic
595Q8—22no—
596Q8—22no—
597Q8—41no—
598Q8—41no—
599D4—62no—
600D4—62no—
601D4—128no—
602D4—121no—
603D4—121no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—21no—
609C2×C4—32no—
610C2×C4—32no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—61no—
626C2×C4—61no—
627C2×C4—61no—
628C2×C4—61no—
629C2×C4—61no—
630C2×C4—61no—
631C2×C4—61no—
632C2×C4—61no—
633C2×C4—61no—
634C2×C4—61no—
635C2×C4—62no—
636C2×C4—128no—
637C2×C4—124no—
638C2×C4—124no—
639C2×C4—121no—
640C2×C4—121no—
641C2×C4—121no—
642C2×C4—121no—
643C2×C4—121no—
644C2×C4—121no—
645C2×C4—121no—
646C2×C4—121no—
647C2×C4C2×D3×D411yescharacteristic
648C2×C4C2×D3×D411yescharacteristic
649C2×C4#100411yescharacteristic
650C2×C4#101011yescharacteristic
651C2×C4#101011yescharacteristic
652C2×C4#101011yescharacteristic
653C8—41no—
654C8—41no—
655C8—121no—
656C8—121no—
657C2×C3—42no—
658C2×C3#261611yescharacteristic
659C2×C3#300211yescharacteristic
660C2×C3#300511yescharacteristic
661C2×C3#323211yescharacteristic
662C2×C3#324011yescharacteristic
663C2×C3#325011yescharacteristic
664C2×C3#325611yescharacteristic
665D3—82no—
666C22—22no—
667C22—22no—
668C22—22no—
669C22—41no—
670C22—41no—
671C22—41no—
672C22—41no—
673C22—122no—
674C22—122no—
675C22—122no—
676C22—241no—
677C22—241no—
678C22—241no—
679C22—241no—
680C22D3×#12011yescharacteristic
681C22#559711yescharacteristic
682C22#564811yescharacteristic
683C22C2×#91911yescharacteristic
684C22#576811yescharacteristic
685C22#577311yescharacteristic
686C22#577711yescharacteristic
687C4—41no—
688C4—42no—
689C4—42no—
690C4—62no—
691C4—62no—
692C4—81no—
693C4—81no—
694C4—121no—
695C4—122no—
696C4—121no—
697C4—241no—
698C4—241no—
699C4#559711yescharacteristic
700C4#560811yescharacteristic
701C4#564811yescharacteristic
702C4#567511yescharacteristic
703C3#2969311yescharacteristic, complemented, Sylow
704C2—42no—
705C2—242no—
706C2#7697011yescharacteristic
707C2#8008311yescharacteristic
708C2#8018211yescharacteristic
709C2#8074811yescharacteristic
710C2#8079511yescharacteristic
711C2#8146211yescharacteristic
712C2#8150311yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2036C2
socleC23×C3C22×D4
prime radicalC2×#930C22
hypercenter#2036D3
hypocenterC3#29693
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2036

Complements

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

NH ≅ G/Ntype
2. C2×#4891C2semidirect
3. C2×#4796C2semidirect
4. #73998C2semidirect
5. C3×#2036C2semidirect
6. #71694C2semidirect
7. #70618C2semidirect
8. #70279C2semidirect
9. #70245C2semidirect
10. #70222C2semidirect
12. #69908C2semidirect
14. #68893C2semidirect
15. #68829C2semidirect
72. #2036D3semidirect
703. C3#29693semidirect

Maximal subgroups

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

subgroupindex
C2×#56242
C2×#48912
C2×#47962
#739982
C3×#20362
#716942
#706182
#702792
#702452
#702222
#699342
#699082
#695242
#688932
#688292
#296933