Finite Group Databasebeta

← #10831754

Subgroups — #10831754

Order 768.

1064 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5690C211yesmaximal, characteristic
2C2×#4791C211yesmaximal, characteristic, complemented
3C2×#4817C211yesmaximal, characteristic
4#76547C211yesmaximal, characteristic, complemented
5C3×#2026C211yesmaximal, characteristic, complemented
6#71739C211yesmaximal, characteristic, complemented
7#70276C211yesmaximal, characteristic
8#70232C211yesmaximal, characteristic, complemented
9#69687C211yesmaximal, characteristic
10#68895C211yesmaximal, characteristic
11#68893C211yesmaximal, characteristic
12#68828C211yesmaximal, characteristic, complemented
13#68806C211yesmaximal, characteristic, complemented
14#68671C211yesmaximal, characteristic
15#68620C211yesmaximal, characteristic, complemented
16#24343—31nomaximal, Sylow
17C2×C3×#125C2211yescharacteristic
18C2×Q8×Dic3C2211yescharacteristic
19C22×#215C2211yescharacteristic
20#5690—24no—
21#5690—24no—
22C2×#897—21no—
23C2×#896—21no—
24C2×#896—21no—
25C2×#893C2211yescharacteristic
26C2×#893C2211yescharacteristic
27C2×D3×C4⋊C4—21no—
28C2×#890C2211yescharacteristic
29C2×#888C2211yescharacteristic
30C2×C4×D12C2211yescharacteristic
31C2×C4×Dic6C2211yescharacteristic
32C2×C3×#103C2211yescharacteristic
33C2×C3×#103C2211yescharacteristic
34C3×#391C2211yescharacteristic
35#5229C2211yescharacteristic
36C2×#835C2211yescharacteristic
37C2×#821C2211yescharacteristic
38C2×#821C2211yescharacteristic
39C2×#817C2211yescharacteristic
40C2×#815C2211yescharacteristic
41C2×Dic3⋊C8C2211yescharacteristic
42#4984C2211yescharacteristic
43#4977C2211yescharacteristic
44C2×#809C2211yescharacteristic
45C2×#808C2211yescharacteristic
46C2×#807C2211yescharacteristic
47#4928C2211yescharacteristic
48#4817—24no—
49#4791—24no—
50#4660—21no—
51#4656—21no—
52#4655—21no—
53#4654—21no—
54C3×#340C2211yescharacteristic
55C3×#340C2211yescharacteristic
56C3×#336C2211yescharacteristic
57#4551C2211yescharacteristic
58#4550C2211yescharacteristic
59#4550C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62#4529C2211yescharacteristic
63#4524C2211yescharacteristic
64C2×#554—31no—
65C2×#475—31no—
66C2×#440—31no—
67#2581—31no—
68#2059—31no—
69#2059—31no—
70#2026—31no—
71#2026D311yescharacteristic, complemented
72#1993—31no—
73#1985—31no—
74#1877—31no—
75#1877—31no—
76#1864—31no—
77#1864—31no—
78#1804—31no—
79C22×C3×Q8C2311yescharacteristic
80C22×D12C2311yescharacteristic
81C22×C4×D3—21no—
82C22×C4×D3—21no—
83C22×C4×D3—21no—
84C22×Dic6—21no—
85C22×Dic6D411yescharacteristic
86C22×C3×C8C2311yescharacteristic
87C22×C3×C8C2311yescharacteristic
88C3×#125—22no—
89C3×#125—22no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C42C2311yescharacteristic
96Q8×Dic3—22no—
97Q8×Dic3—22no—
98C2×#215—22no—
99C2×#215—22no—
100C2×#215Q814yescomplemented
101C2×(C2×D3)⋊C4—21no—
102C2×(C2×D3)⋊C4—21no—
103C2×(C2×D3)⋊C4—21no—
104C2×C4⋊Dic3—21no—
105C2×C4⋊Dic3—21no—
106C2×C4⋊Dic3—21no—
107C2×C4⋊Dic3D411yescharacteristic
108C2×C4⋊Dic3C2311yescharacteristic
109C2×C4⋊Dic3C2311yescharacteristic
110C2×C4⋊Dic3C2311yescharacteristic
111C2×Dic3⋊C4—21no—
112C2×Dic3⋊C4—21no—
113C2×Dic3⋊C4—21no—
114C2×Dic3⋊C4D411yescharacteristic
115C2×Dic3⋊C4D411yescharacteristic
116C2×C4×Dic3—21no—
117C2×C4×Dic3C2311yescharacteristic
118C2×C4×Dic3C2311yescharacteristic
119C22×C3⋊C8C2311yescharacteristic
120C22×C3⋊C8C2311yescharacteristic
121#897—28no—
122#896—28no—
123#896—28no—
124#893—22no—
125#893—22no—
126#893—22no—
127#893D414yes—
128D3×C4⋊C4—28no—
129#890—22no—
130#890—22no—
131#888—22no—
132#888D414yes—
133C4×D12—22no—
134C4×D12—22no—
135C4×Dic6—22no—
136C4×Dic6—22no—
137C3×#103—22no—
138C3×#103—22no—
139C3×#95—21no—
140C3×#95—21no—
141#835—22no—
142C3⋊#95—21no—
143C3⋊#95—21no—
144C3⋊#95—21no—
145C3⋊#95—21no—
146C3⋊#95—21no—
147C3⋊#95—21no—
148#821—22no—
149#821—22no—
150#817—22no—
151#815—22no—
152Dic3⋊C8—22no—
153#809—22no—
154#808—22no—
155#807—22no—
156C22×SD16—31no—
157#554—64no—
158#554—64no—
159C2×C4⋊Q8—31no—
160C2×#125—31no—
161C2×#125C2×D311yescharacteristic
162C2×C22⋊Q8—61no—
163C2×C22⋊Q8—61no—
164C2×C22⋊Q8—61no—
165C2×C4×Q8—31no—
166C2×C4×Q8—31no—
167C2×C4×D4—31no—
168C2×C4×D4—31no—
169C2×C4×D4—31no—
170C22×C4⋊C4—61no—
171#475—64no—
172#440—64no—
173C2×#107—31no—
174C2×#106—31no—
175C2×#106—31no—
176C2×C4⋊C8—31no—
177C2×#103—31no—
178C2×#103C2×D311yescharacteristic
179C2×#103C2×D311yescharacteristic
180C2×#102—31no—
181C2×#102—31no—
182C2×#102—31no—
183C2×#97—31no—
184#400—61no—
185#395—61no—
186#391—31no—
187#391C2×D311yescharacteristic
188#389—31no—
189#389—31no—
190#387—61no—
191#387—61no—
192#382—31no—
193#340—31no—
194#340—31no—
195#340—31no—
196#340—31no—
197#340C2×D311yescharacteristic
198#340C2×D311yescharacteristic
199#336—31no—
200#336—31no—
201#336—31no—
202#336C2×D311yescharacteristic
203C23×D3—21no—
204C2×C3×Q8—21no—
205C2×C3×Q8—21no—
206C2×C3×Q8—21no—
207C2×C3×Q8—21no—
208C2×C3×Q8—21no—
209C2×C3×Q8C2×Q812yes—
210C22×C3×C4—21no—
211C22×C3×C4—21no—
212C22×C3×C4—21no—
213C22×C3×C4—21no—
214C22×C3×C4C2×D411yescharacteristic
215C22×C3×C4C2×D411yescharacteristic
216C22×C3×C4C2411yescharacteristic
217C22×Dic3—21no—
218C22×Dic3—21no—
219C22×Dic3—21no—
220C22×Dic3C2×D411yescharacteristic
221C22×Dic3C2×D411yescharacteristic
222C22×Dic3C2×D411yescharacteristic
223C22×Dic3C2×D411yescharacteristic
224C2×D12—21no—
225C2×D12—21no—
226C2×D12—21no—
227C2×D12C2×D412yes—
228C2×D12C2×Q812yes—
229C2×D12#4112yes—
230C2×C4×D3—24no—
231C2×C4×D3—24no—
232C2×C4×D3—24no—
233C2×C4×D3—42no—
234C2×C4×D3—42no—
235C2×C4×D3—81no—
236C2×C4×D3—81no—
237C2×C4×D3—81no—
238C2×C4×D3—81no—
239C2×Dic6—22no—
240C2×Dic6—24no—
241C2×Dic6—41no—
242C2×Dic6—41no—
243C2×Dic6—81no—
244C2×Dic6—81no—
245C2×C3×C8—21no—
246C2×C3×C8—21no—
247C2×C3×C8—21no—
248C2×C3×C8—21no—
249C2×C3×C8—21no—
250C2×C3×C8—21no—
251C3×C4⋊C4—24no—
252C3×C4⋊C4—24no—
253C3×C4⋊C4—21no—
254C3×C4⋊C4—21no—
255C3×C4⋊C4—21no—
256C3×C4⋊C4—21no—
257C3×C4⋊C4C2×D412yes—
258C3×C4⋊C4C2×D412yes—
259C3×C42—21no—
260C3×C42—21no—
261#215—28no—
262(C2×D3)⋊C4—24no—
263(C2×D3)⋊C4—24no—
264(C2×D3)⋊C4—24no—
265C4⋊Dic3—24no—
266C4⋊Dic3—21no—
267C4⋊Dic3—22no—
268C4⋊Dic3—21no—
269C4⋊Dic3—21no—
270C4⋊Dic3—21no—
271C4⋊Dic3—21no—
272C4⋊Dic3—42no—
273C4⋊Dic3—42no—
274C4⋊Dic3C2×D412yes—
275Dic3⋊C4—24no—
276Dic3⋊C4—22no—
277Dic3⋊C4—22no—
278Dic3⋊C4—24no—
279Dic3⋊C4—42no—
280C4×Dic3—21no—
281C4×Dic3—21no—
282C4×Dic3—42no—
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⋊C8C2×Q812yes—
291C22×Q8—31no—
292C22×Q8—61no—
293C22×Q8C22×D311yescharacteristic
294C22×D4—31no—
295C23×C4—61no—
296C23×C4—61no—
297C23×C4—61no—
298C2×SD16—34no—
299C2×SD16—62no—
300C2×SD16—62no—
301C22×C8—31no—
302C22×C8—31no—
303C22×C8C22×D311yescharacteristic
304C22×C8C22×D311yescharacteristic
305C4⋊Q8—34no—
306C4⋊Q8—62no—
307#125—22no—
308#125—22no—
309#125—62no—
310#125—62no—
311C22⋊Q8—68no—
312C22⋊Q8—68no—
313C22⋊Q8—68no—
314C4×Q8—62no—
315C4×Q8—62no—
316C4×Q8—62no—
317C4×Q8—62no—
318C4×D4—34no—
319C4×D4—62no—
320C4×D4—62no—
321C4×D4—62no—
322C4×D4—62no—
323C4×D4—62no—
324C2×C4⋊C4—21no—
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—68no—
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×C42—31no—
346C2×C42—31no—
347C2×C42—61no—
348C2×C42C22×D311yescharacteristic
349#107—62no—
350#106—62no—
351#106—62no—
352C4⋊C8—62no—
353#103—22no—
354#103—22no—
355#103—62no—
356#102—62no—
357#102—62no—
358#102—62no—
359#97—62no—
360#95—21no—
361#95—21no—
362#95—61no—
363#95—61no—
364#95—61no—
365#95—61no—
366#95—61no—
367#95—61no—
368C23×C3C22×D411yescharacteristic
369C22×D3—22no—
370C22×D3—22no—
371C22×D3—22no—
372C22×D3—41no—
373C22×D3—41no—
374C22×D3—41no—
375C22×D3—41no—
376C3×Q8—22no—
377C3×Q8—22no—
378C3×Q8—41no—
379C3×Q8—41no—
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—21no—
388C2×C3×C4—21no—
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—41no—
397C2×C3×C4—41no—
398C2×C3×C4C22×D411yescharacteristic
399C2×C3×C4C22×D411yescharacteristic
400C2×C3×C4C22×Q811yescharacteristic
401C2×C3×C4C2×#4111yescharacteristic
402C2×C3×C42−1+411yescharacteristic
403C2×C3×C42−1+411yescharacteristic
404C2×Dic3—21no—
405C2×Dic3—21no—
406C2×Dic3—21no—
407C2×Dic3—21no—
408C2×Dic3—21no—
409C2×Dic3—21no—
410C2×Dic3—22no—
411C2×Dic3—21no—
412C2×Dic3—21no—
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#12012yes—
424C2×Dic3#12012yes—
425D12—22no—
426D12—22no—
427D12—22no—
428D12C22⋊Q814yes—
429C4×D3—48no—
430C4×D3—84no—
431C4×D3—84no—
432Dic6—44no—
433Dic6—84no—
434C3×C8—41no—
435C3×C8—41no—
436C3⋊C8—22no—
437C3⋊C8—41no—
438C24—61no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8—21no—
444C2×Q8—62no—
445C2×Q8—64no—
446C2×Q8—121no—
447C2×Q8—121no—
448C2×Q8—241no—
449C2×Q8—241no—
450C2×Q8D3×Q812yes—
451C2×D4—32no—
452C2×D4—32no—
453C2×D4—32no—
454C2×D4—61no—
455C2×D4—61no—
456C2×D4—61no—
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—64no—
466C22×C4—64no—
467C22×C4—64no—
468C22×C4—61no—
469C22×C4—61no—
470C22×C4—61no—
471C22×C4—122no—
472C22×C4—122no—
473C22×C4—241no—
474C22×C4—241no—
475C22×C4—241no—
476C22×C4—241no—
477C22×C4D3×D411yescharacteristic
478C22×C4D3×D411yescharacteristic
479C22×C4C23×D311yescharacteristic
480SD16—68no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—32no—
488C2×C8—61no—
489C2×C8—61no—
490C2×C8—61no—
491C2×C8—61no—
492C2×C8—61no—
493C4⋊C4—24no—
494C4⋊C4—24no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—32no—
500C4⋊C4—62no—
501C4⋊C4—62no—
502C4⋊C4—62no—
503C4⋊C4—64no—
504C4⋊C4—64no—
505C4⋊C4—64no—
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⋊C4D3×D412yes—
515C4⋊C4D3×D412yes—
516C22⋊C4—64no—
517C22⋊C4—64no—
518C22⋊C4—64no—
519C42—21no—
520C42—21no—
521C42—32no—
522C42—32no—
523C42—61no—
524C42—61no—
525C42—122no—
526C22×C3C2×#12011yescharacteristic
527C22×C3#54811yescharacteristic
528C22×C3D4×Q811yescharacteristic
529C22×C3#57511yescharacteristic
530C22×C3#57511yescharacteristic
531C22×C3#57711yescharacteristic
532C22×C3#57711yescharacteristic
533C2×D3—24no—
534C2×D3—42no—
535C2×D3—42no—
536C2×D3—42no—
537C2×D3—42no—
538C2×D3—42no—
539C2×D3—42no—
540C3×C4—42no—
541C3×C4—41no—
542C3×C4—42no—
543C3×C4—42no—
544C3×C4—41no—
545C3×C4—81no—
546C3×C4C2×C22⋊Q811yescharacteristic
547C3×C4#53611yescharacteristic
548C3×C4#54811yescharacteristic
549C3×C4D4×Q811yescharacteristic
550Dic3—22no—
551Dic3—22no—
552Dic3—41no—
553Dic3—42no—
554Dic3—41no—
555Dic3—81no—
556Dic3—81no—
557C23—62no—
558C23—62no—
559C23—62no—
560C23—121no—
561C23—121no—
562C23—121no—
563C23—121no—
564C23C2×D3×D411yescharacteristic
565Q8—22no—
566Q8—22no—
567Q8—41no—
568Q8—41no—
569Q8—124no—
570Q8—244no—
571D4—34no—
572D4—62no—
573D4—62no—
574D4—62no—
575C2×C4—22no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—32no—
585C2×C4—32no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—61no—
596C2×C4—61no—
597C2×C4—61no—
598C2×C4—61no—
599C2×C4—61no—
600C2×C4—61no—
601C2×C4—61no—
602C2×C4—61no—
603C2×C4—61no—
604C2×C4—61no—
605C2×C4—62no—
606C2×C4—128no—
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—244no—
616C2×C4—244no—
617C2×C4C2×D3×D411yescharacteristic
618C2×C4C2×D3×D411yescharacteristic
619C2×C4C2×D3×Q811yescharacteristic
620C2×C4D3×#4111yescharacteristic
621C2×C4#101011yescharacteristic
622C2×C4#101011yescharacteristic
623C8—41no—
624C8—41no—
625C8—62no—
626C8—121no—
627C2×C3#240911yescharacteristic
628C2×C3#300511yescharacteristic
629C2×C3#300911yescharacteristic
630C2×C3#337611yescharacteristic
631C2×C3#337611yescharacteristic
632C2×C3#338811yescharacteristic
633C2×C3#338811yescharacteristic
634D3—44no—
635C22—64no—
636C22—122no—
637C22—122no—
638C22—122no—
639C22—122no—
640C22—122no—
641C22—122no—
642C22D3×#12011yescharacteristic
643C22#563211yescharacteristic
644C22#567711yescharacteristic
645C22#576811yescharacteristic
646C22#577011yescharacteristic
647C22#577711yescharacteristic
648C22#577811yescharacteristic
649C4—41no—
650C4—42no—
651C4—41no—
652C4—42no—
653C4—42no—
654C4—62no—
655C4—62no—
656C4—81no—
657C4—121no—
658C4—121no—
659C4—122no—
660C4—241no—
661C4—241no—
662C4D3×C22⋊Q811yescharacteristic
663C4#563211yescharacteristic
664C4#567511yescharacteristic
665C4#567711yescharacteristic
666C3#2434311yescharacteristic, complemented, Sylow
667C2—124no—
668C2#7678011yescharacteristic
669C2#8022711yescharacteristic
670C2#8022911yescharacteristic
671C2#8122511yescharacteristic
672C2#8126611yescharacteristic
673C2#8166111yescharacteristic
674C2#8173411yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2026C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#2026D3
hypocenterC3#24343
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2026

Complements

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

NH ≅ G/Ntype
2. C2×#4791C2semidirect
4. #76547C2semidirect
5. C3×#2026C2semidirect
6. #71739C2semidirect
8. #70232C2semidirect
12. #68828C2semidirect
13. #68806C2semidirect
15. #68620C2semidirect
71. #2026D3semidirect
100. C2×#215Q8semidirect
666. C3#24343semidirect

Maximal subgroups

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

subgroupindex
C2×#56902
C2×#47912
C2×#48172
#765472
C3×#20262
#717392
#702762
#702322
#696872
#688952
#688932
#688282
#688062
#686712
#686202
#243433