Finite Group Databasebeta

← #10849268

Subgroups — #10849268

Order 768.

1018 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5584C211yesmaximal, characteristic
2#82483C211yesmaximal, characteristic, complemented
3C2×#4829C211yesmaximal, characteristic, complemented
4C2×#4697C211yesmaximal, characteristic
5C3×#1984C211yesmaximal, characteristic, complemented
6#70458C211yesmaximal, characteristic
7C4×#816C211yesmaximal, characteristic, complemented
8#68684C211yesmaximal, characteristic, complemented
9#68683C211yesmaximal, characteristic, complemented
10#68671C211yesmaximal, characteristic
11#68670C211yesmaximal, characteristic
12#68631C211yesmaximal, characteristic, complemented
13#68631C211yesmaximal, characteristic, complemented
14#68630C211yesmaximal, characteristic
15#68630C211yesmaximal, characteristic
16#25654—31nomaximal, Sylow
17C2×C3×C4×Q8C2211yescharacteristic
18C2×#946—21no—
19#5584—24no—
20#5584—24no—
21C2×#898—21no—
22C2×#898—21no—
23C2×#892—21no—
24C2×#875C2211yescharacteristic
25C2×#875C2211yescharacteristic
26C2×C4×D12C2211yescharacteristic
27C2×#870C2211yescharacteristic
28C2×#869C2211yescharacteristic
29C2×C4×Dic6C2211yescharacteristic
30C2×C4×Dic6C2211yescharacteristic
31C2×C3×C4⋊C8C2211yescharacteristic
32C2×C3×C4×C8C2211yescharacteristic
33#5230—21no—
34C2×#821C2211yescharacteristic
35C2×#821C2211yescharacteristic
36C2×#821C2211yescharacteristic
37C2×#821C2211yescharacteristic
38C2×#818C2211yescharacteristic
39C2×#817C2211yescharacteristic
40C2×#816C2211yescharacteristic
41C2×#816C2211yescharacteristic
42C2×#816C2211yescharacteristic
43C2×#816C2211yescharacteristic
44C2×#816C2211yescharacteristic
45C2×#816C2211yescharacteristic
46C2×#816C2211yescharacteristic
47C2×#816C2211yescharacteristic
48#4983—21no—
49#4976—21no—
50#4976—21no—
51C4×C4⋊Dic3C2211yescharacteristic
52#4928C2211yescharacteristic
53#4928C2211yescharacteristic
54#4928C2211yescharacteristic
55#4928C2211yescharacteristic
56#4829—24no—
57#4697—24no—
58C3×#336C2211yescharacteristic
59C3×#336C2211yescharacteristic
60C3×#336C2211yescharacteristic
61C3×#336C2211yescharacteristic
62#4551C2211yescharacteristic
63#4551C2211yescharacteristic
64C2×#556—31no—
65C2×#488—31no—
66C2×#479—31no—
67#2584—31no—
68#2043—31no—
69#2043—31no—
70#2042—31no—
71#2042—31no—
72#1984D311yescharacteristic, complemented
73#1983—31no—
74#1865—31no—
75#1864—31no—
76#1863—31no—
77#1862—31no—
78C4×#103—31no—
79C22×C3×Q8D411yescharacteristic
80C22×D12C2311yescharacteristic
81C22×C4×D3—21no—
82C22×Dic6C2311yescharacteristic
83C22×Dic6C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C22×C3×C8C2311yescharacteristic
87C22×C3×C8C2311yescharacteristic
88C3×C4×Q8—22no—
89C3×C4×Q8—22no—
90C2×C3×C4⋊C4D411yescharacteristic
91C2×C3×C4⋊C4D411yescharacteristic
92C2×C3×C4⋊C4D411yescharacteristic
93C2×C3×C42C2311yescharacteristic
94C2×C3×C42C2311yescharacteristic
95C2×C3×C42C2311yescharacteristic
96#946—28no—
97C2×(C2×D3)⋊C4—21no—
98C2×(C2×D3)⋊C4—21no—
99C2×(C2×D3)⋊C4—21no—
100C2×(C2×D3)⋊C4—21no—
101C2×(C2×D3)⋊C4—21no—
102C2×C4⋊Dic3—21no—
103C2×C4⋊Dic3—21no—
104C2×C4⋊Dic3C2311yescharacteristic
105C2×C4⋊Dic3C2311yescharacteristic
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×C4×Dic3—21no—
118C2×C4×Dic3—21no—
119C2×C4×Dic3—21no—
120#898—28no—
121#898—28no—
122#892—28no—
123#875—22no—
124#875—22no—
125#875—22no—
126#875D414yes—
127C4×D12—22no—
128C4×D12—22no—
129#870—22no—
130#870—22no—
131#869—22no—
132#869—22no—
133C4×Dic6—22no—
134C4×Dic6—22no—
135C4×Dic6—22no—
136C4×Dic6D414yes—
137C3×C4⋊C8—22no—
138C3×C4×C8—22no—
139C3⋊#95—21no—
140C3⋊#95—21no—
141C3⋊#95—21no—
142C3⋊#95—21no—
143C3⋊#95—21no—
144#821—22no—
145#821—22no—
146#821—22no—
147#821—22no—
148#818—22no—
149#817—22no—
150#816—22no—
151#816—22no—
152#816—22no—
153#816—22no—
154#816—22no—
155#816—22no—
156#816—22no—
157#816—22no—
158#556—64no—
159#556—64no—
160C2×C4⋊Q8—31no—
161C2×#126—61no—
162C2×#126—61no—
163C2×#125—31no—
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#488—64no—
173#479—64no—
174C2×#107—31no—
175C2×#106—31no—
176C2×C4⋊C8C2×D311yescharacteristic
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×#102—31no—
186C2×#102—31no—
187C2×#102—31no—
188C2×#102—31no—
189C2×C4×C8C2×D311yescharacteristic
190#398—61no—
191#395—61no—
192#389—31no—
193#389—31no—
194#389—31no—
195#389—31no—
196#389—61no—
197#389—61no—
198C4×C4⋊C4—31no—
199#340—31no—
200#340—31no—
201#336C2×D311yescharacteristic
202#336C2×D311yescharacteristic
203#336C2×D311yescharacteristic
204#336C2×D311yescharacteristic
205C23×D3—21no—
206C2×C3×Q8—22no—
207C2×C3×Q8—22no—
208C2×C3×Q8—41no—
209C22×C3×C4C2×D411yescharacteristic
210C22×C3×C4C2×D411yescharacteristic
211C22×C3×C4C2×D411yescharacteristic
212C22×C3×C4C2×D411yescharacteristic
213C22×C3×C4C2×D411yescharacteristic
214C22×C3×C4C2×D411yescharacteristic
215C22×C3×C4C2411yescharacteristic
216C22×Dic3—21no—
217C22×Dic3—21no—
218C22×Dic3—21no—
219C22×Dic3—21no—
220C22×Dic3—21no—
221C22×Dic3—21no—
222C22×Dic3—21no—
223C2×D12—21no—
224C2×D12—21no—
225C2×D12—21no—
226C2×D12—21no—
227C2×D12—21no—
228C2×D12C2×D412yes—
229C2×C4×D3—24no—
230C2×C4×D3—42no—
231C2×C4×D3—42no—
232C2×Dic6—21no—
233C2×Dic6—21no—
234C2×Dic6—21no—
235C2×Dic6—21no—
236C2×Dic6—21no—
237C2×Dic6—21no—
238C2×Dic6—21no—
239C2×Dic6—21no—
240C2×Dic6—21no—
241C2×Dic6—21no—
242C2×Dic6C2×D412yes—
243C2×Dic6C2×D412yes—
244C2×C3×C8—21no—
245C2×C3×C8—21no—
246C2×C3×C8—21no—
247C2×C3×C8—21no—
248C2×C3×C8—21no—
249C2×C3×C8—21no—
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—22no—
257C3×C4⋊C4—22no—
258C3×C4⋊C4—22no—
259C3×C42—21no—
260C3×C42—21no—
261C3×C42—21no—
262C3×C42—21no—
263C3×C42C2×D412yes—
264C3×C42C2×D412yes—
265(C2×D3)⋊C4—24no—
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—24no—
272C4⋊Dic3—21no—
273C4⋊Dic3—21no—
274C4⋊Dic3—21no—
275C4⋊Dic3—21no—
276C4⋊Dic3—21no—
277C4⋊Dic3—21no—
278C4⋊Dic3—21no—
279C4⋊Dic3C2×D412yes—
280C4⋊Dic3#4112yes—
281C4⋊Dic3#4112yes—
282Dic3⋊C4—24no—
283Dic3⋊C4—24no—
284Dic3⋊C4—42no—
285Dic3⋊C4—42no—
286Dic3⋊C4—42no—
287Dic3⋊C4—42no—
288Dic3⋊C4—81no—
289Dic3⋊C4—81no—
290C4×Dic3—24no—
291C4×Dic3—24no—
292C4×Dic3—81no—
293C22×Q8—31no—
294C22×Q8—31no—
295C22×Q8D1211yescharacteristic
296C22×D4—31no—
297C23×C4—61no—
298C22×C8C22×D311yescharacteristic
299C22×C8C22×D311yescharacteristic
300C22×C8C22×D311yescharacteristic
301C22×C8C22×D311yescharacteristic
302C4⋊Q8—62no—
303C4⋊Q8—62no—
304#126—68no—
305#126—68no—
306#125—62no—
307#125—62no—
308#124—34no—
309#124—62no—
310#124—62no—
311#124—62no—
312C22⋊Q8—68no—
313C4×Q8—22no—
314C4×Q8—22no—
315C4×Q8—34no—
316C4×Q8—62no—
317C4×Q8—62no—
318C4×Q8—62no—
319C4×D4—62no—
320C4×D4—62no—
321#117—68no—
322C2×C4⋊C4—31no—
323C2×C4⋊C4—31no—
324C2×C4⋊C4—31no—
325C2×C4⋊C4—31no—
326C2×C4⋊C4—31no—
327C2×C4⋊C4—61no—
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⋊C4D1211yescharacteristic
338C2×C4⋊C4D1211yescharacteristic
339C2×C4⋊C4D1211yescharacteristic
340C2×C22⋊C4—61no—
341C2×C22⋊C4—61no—
342C2×C22⋊C4—61no—
343C2×C22⋊C4—61no—
344C2×C22⋊C4—61no—
345C2×C42—61no—
346C2×C42—61no—
347C2×C42—61no—
348C2×C42C22×D311yescharacteristic
349C2×C42C22×D311yescharacteristic
350C2×C42C22×D311yescharacteristic
351#107—62no—
352#106—62no—
353C4⋊C8—22no—
354#103—62no—
355#103—62no—
356#103—62no—
357#103—62no—
358#103—62no—
359#103—62no—
360#103—62no—
361#103—62no—
362#102—62no—
363#102—62no—
364#102—62no—
365#102—62no—
366C4×C8—22no—
367#95—61no—
368#95—61no—
369#95—61no—
370#95—61no—
371#95—61no—
372C23×C3C22×D411yescharacteristic
373C22×D3—22no—
374C22×D3—41no—
375C22×D3—41no—
376C22×D3—41no—
377C22×D3—41no—
378C22×D3—41no—
379C22×D3—41no—
380C3×Q8—44no—
381C2×C3×C4—21no—
382C2×C3×C4—21no—
383C2×C3×C4—21no—
384C2×C3×C4—21no—
385C2×C3×C4—21no—
386C2×C3×C4—21no—
387C2×C3×C4—21no—
388C2×C3×C4—21no—
389C2×C3×C4—21no—
390C2×C3×C4—21no—
391C2×C3×C4—21no—
392C2×C3×C4—21no—
393C2×C3×C4—21no—
394C2×C3×C4—21no—
395C2×C3×C4—21no—
396C2×C3×C4#12012yes—
397C2×C3×C4#12012yes—
398C2×C3×C4#13512yes—
399C2×C3×C4C22×D411yescharacteristic
400C2×C3×C4C22×D411yescharacteristic
401C2×C3×C4C2×#4111yescharacteristic
402C2×C3×C4C2×#4111yescharacteristic
403C2×C3×C42−1+411yescharacteristic
404C2×C3×C42−1+411yescharacteristic
405C2×Dic3—22no—
406C2×Dic3—22no—
407C2×Dic3—22no—
408C2×Dic3—22no—
409C2×Dic3—22no—
410C2×Dic3—41no—
411C2×Dic3—41no—
412C2×Dic3—41no—
413C2×Dic3—41no—
414C2×Dic3—41no—
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—41no—
425C2×Dic3—41no—
426D12—22no—
427D12—22no—
428D12—41no—
429D12—41no—
430C4×D3—48no—
431Dic6—22no—
432Dic6—22no—
433Dic6—22no—
434Dic6—22no—
435Dic6—41no—
436Dic6—41no—
437Dic6—41no—
438Dic6—41no—
439C3×C8—41no—
440C3×C8—41no—
441C3×C8—41no—
442C3×C8—41no—
443C24—61no—
444C2×Q8—22no—
445C2×Q8—22no—
446C2×Q8—32no—
447C2×Q8—32no—
448C2×Q8—41no—
449C2×Q8—61no—
450C2×Q8—61no—
451C2×Q8—61no—
452C2×Q8—61no—
453C2×Q8—61no—
454C2×Q8—61no—
455C2×Q8—61no—
456C2×Q8—61no—
457C2×Q8—61no—
458C2×Q8—61no—
459C2×D4—32no—
460C2×D4—61no—
461C2×D4—61no—
462C2×D4—61no—
463C2×D4—61no—
464C2×D4—61no—
465C22×C4—64no—
466C22×C4—61no—
467C22×C4—61no—
468C22×C4—61no—
469C22×C4—61no—
470C22×C4—61no—
471C22×C4—61no—
472C22×C4—61no—
473C22×C4—122no—
474C22×C4—122no—
475C22×C4C2×D1211yescharacteristic
476C22×C4C2×D1211yescharacteristic
477C22×C4C2×D1211yescharacteristic
478C22×C4C2×D1211yescharacteristic
479C22×C4C2×D1211yescharacteristic
480C22×C4C2×D1211yescharacteristic
481C22×C4C23×D311yescharacteristic
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—21no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—32no—
498C4⋊C4—32no—
499C4⋊C4—32no—
500C4⋊C4—64no—
501C4⋊C4—64no—
502C4⋊C4—64no—
503C4⋊C4—64no—
504C4⋊C4—61no—
505C4⋊C4—61no—
506C4⋊C4—61no—
507C4⋊C4—61no—
508C4⋊C4—61no—
509C4⋊C4—61no—
510C4⋊C4—61no—
511C4⋊C4—122no—
512C4⋊C4—122no—
513C4⋊C4—122no—
514C4⋊C4—122no—
515C4⋊C4—241no—
516C4⋊C4—241no—
517C22⋊C4—64no—
518C22⋊C4—64no—
519C22⋊C4—64no—
520C22⋊C4—64no—
521C22⋊C4—64no—
522C42—21no—
523C42—21no—
524C42—21no—
525C42—21no—
526C42—64no—
527C42—64no—
528C42—241no—
529C42D3×D412yes—
530C42D3×D412yes—
531C22×C3C2×#12011yescharacteristic
532C22×C3#54811yescharacteristic
533C22×C3#54811yescharacteristic
534C22×C3C2×#13511yescharacteristic
535C22×C3#57511yescharacteristic
536C22×C3#57711yescharacteristic
537C22×C3#57811yescharacteristic
538C2×D3—42no—
539C2×D3—42no—
540C2×D3—42no—
541C2×D3—81no—
542C2×D3—81no—
543C2×D3—81no—
544C2×D3—81no—
545C3×C4—22no—
546C3×C4—22no—
547C3×C4—22no—
548C3×C4—41no—
549C3×C4—41no—
550C3×C4—41no—
551C3×C4#53211yescharacteristic
552C3×C4#53611yescharacteristic
553C3×C4#54711yescharacteristic
554C3×C4#54711yescharacteristic
555Dic3—42no—
556Dic3—42no—
557Dic3—42no—
558Dic3—42no—
559Dic3—42no—
560Dic3—81no—
561Dic3—81no—
562C23—62no—
563C23—121no—
564C23—121no—
565C23—121no—
566C23—121no—
567C23—121no—
568C23—121no—
569C23C22×D1211yescharacteristic
570Q8—44no—
571Q8—62no—
572Q8—62no—
573Q8—62no—
574Q8—62no—
575Q8—121no—
576Q8—121no—
577Q8—121no—
578Q8—121no—
579D4—62no—
580D4—62no—
581D4—121no—
582D4—121no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—62no—
599C2×C4—62no—
600C2×C4—62no—
601C2×C4—62no—
602C2×C4—62no—
603C2×C4—128no—
604C2×C4—121no—
605C2×C4—121no—
606C2×C4—121no—
607C2×C4—121no—
608C2×C4—121no—
609C2×C4—121no—
610C2×C4—121no—
611C2×C4—121no—
612C2×C4—121no—
613C2×C4—121no—
614C2×C4—121no—
615C2×C4—121no—
616C2×C4—121no—
617C2×C4—121no—
618C2×C4—121no—
619C2×C4—121no—
620C2×C4#88212yes—
621C2×C4#88212yes—
622C2×C4#90412yes—
623C2×C4C2×D3×D411yescharacteristic
624C2×C4C2×D3×D411yescharacteristic
625C2×C4#100711yescharacteristic
626C2×C4#100711yescharacteristic
627C2×C4D3×#4111yescharacteristic
628C2×C4D3×#4111yescharacteristic
629C8—41no—
630C8—41no—
631C8—41no—
632C8—41no—
633C2×C3#243211yescharacteristic
634C2×C3#300211yescharacteristic
635C2×C3#301311yescharacteristic
636C2×C3#336011yescharacteristic
637C2×C3#336411yescharacteristic
638C2×C3#337611yescharacteristic
639C2×C3#337711yescharacteristic
640D3—82no—
641C22—122no—
642C22—122no—
643C22—122no—
644C22—241no—
645C22—241no—
646C22—241no—
647C22—241no—
648C22C2×#88211yescharacteristic
649C22#557611yescharacteristic
650C22#557611yescharacteristic
651C22C2×#90411yescharacteristic
652C22#574811yescharacteristic
653C22#575111yescharacteristic
654C22#575311yescharacteristic
655C4—22no—
656C4—22no—
657C4—22no—
658C4—41no—
659C4—41no—
660C4—41no—
661C4—122no—
662C4—122no—
663C4—122no—
664C4—122no—
665C4—122no—
666C4—241no—
667C4—241no—
668C4#563111yescharacteristic
669C4#563111yescharacteristic
670C4#566811yescharacteristic
671C4#572411yescharacteristic
672C3#2565411yescharacteristic, complemented, Sylow
673C2—242no—
674C2#7649811yescharacteristic
675C2#7861011yescharacteristic
676C2#7868011yescharacteristic
677C2#7992611yescharacteristic
678C2#7993411yescharacteristic
679C2#7997411yescharacteristic
680C2#7998511yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1984C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#1984D3
hypocenterC3#25654
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #1984

Complements

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

NH ≅ G/Ntype
2. #82483C2semidirect
3. C2×#4829C2semidirect
5. C3×#1984C2semidirect
7. C4×#816C2semidirect
8. #68684C2semidirect
9. #68683C2semidirect
12. #68631C2semidirect
13. #68631C2semidirect
72. #1984D3semidirect
672. C3#25654semidirect

Maximal subgroups

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

subgroupindex
C2×#55842
#824832
C2×#48292
C2×#46972
C3×#19842
#704582
C4×#8162
#686842
#686832
#686712
#686702
#686312
#686302
#256543