Finite Group Databasebeta

← #10849291

Subgroups — #10849291

Order 768.

1012 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×#5671C211yesmaximal, characteristic
2C2×#4797C211yesmaximal, characteristic, complemented
3C2×#4763C211yesmaximal, characteristic
4#76078C211yesmaximal, characteristic, complemented
5C3×#2042C211yesmaximal, characteristic
6#71722C211yesmaximal, characteristic
7#70238C211yesmaximal, characteristic, complemented
8#70235C211yesmaximal, characteristic, complemented
9#70229C211yesmaximal, characteristic, complemented
10#70216C211yesmaximal, characteristic
11#70178C211yesmaximal, characteristic
12#70152C211yesmaximal, characteristic
13#69631C211yesmaximal, characteristic, complemented
14#68684C211yesmaximal, characteristic, complemented
15#68620C211yesmaximal, characteristic, complemented
16#25656—31nomaximal, Sylow
17C2×C3×#124C2211yescharacteristic
18C2×Q8×Dic3C2211yescharacteristic
19C2×#944C2211yescharacteristic
20C2×#936C2211yescharacteristic
21C2×D4×Dic3C2211yescharacteristic
22#5671—24no—
23#5671—24no—
24C2×#890C2211yescharacteristic
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#4984—21no—
46#4984C2211yescharacteristic
47#4979—21no—
48Dic3×C4⋊C4C2211yescharacteristic
49#4972C2211yescharacteristic
50C2×#810C2211yescharacteristic
51C2×#810C2211yescharacteristic
52C2×#807C2211yescharacteristic
53#4928C2211yescharacteristic
54#4797—24no—
55#4763—24no—
56#4657—21no—
57#4647—21no—
58C3×#336C2211yescharacteristic
59#4551C2211yescharacteristic
60#4550C2211yescharacteristic
61#4550C2211yescharacteristic
62#4529C2211yescharacteristic
63#4524C2211yescharacteristic
64C2×#556—31no—
65C2×#491—31no—
66C2×#478—31no—
67#2584—31no—
68#2042—31no—
69#2042—31no—
70#2042—31no—
71#2042D311yescharacteristic
72#1985—31no—
73#1981—31no—
74#1865—31no—
75#1865—31no—
76#1863—31no—
77#1863—31no—
78#1762—31no—
79C22×C3×Q8C2311yescharacteristic
80C22×C3×D4C2311yescharacteristic
81C23×Dic3—21no—
82C22×Dic6D411yescharacteristic
83C22×Dic6C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C3×#124—22no—
87C3×#124—22no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C22⋊C4—21no—
92C2×C3×C22⋊C4—21no—
93C2×C3×C42C2311yescharacteristic
94C2×#217—21no—
95C2×#217—21no—
96C2×#217—21no—
97Q8×Dic3—22no—
98Q8×Dic3—22no—
99#944—22no—
100#944D414yes—
101#936—22no—
102#936—22no—
103D4×Dic3—22no—
104D4×Dic3D414yes—
105C2×C4⋊Dic3—21no—
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#890—22no—
128#890—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#807—22no—
158#556—64no—
159#556—64no—
160C2×C4⋊Q8—31no—
161C2×#126—61no—
162C2×#126—61no—
163C2×#125—31no—
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#491—64no—
173#478—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×#103C2×D311yescharacteristic
184C2×#103C2×D311yescharacteristic
185C2×#102—31no—
186C2×#102—31no—
187C2×#102C2×D311yescharacteristic
188C2×#102C2×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
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—21no—
225C22×Dic3—21no—
226C22×Dic3—21no—
227C22×Dic3—24no—
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—41no—
242C2×Dic6—41no—
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—
250C3×C4⋊C4—21no—
251C3×C4⋊C4—21no—
252C3×C4⋊C4—21no—
253C3×C4⋊C4—42no—
254C3×C4⋊C4#4112yes—
255C3×C22⋊C4—24no—
256C3×C22⋊C4—24no—
257C3×C42—21no—
258C3×C42—21no—
259#217—24no—
260#217—24no—
261#217—24no—
262C4⋊Dic3—24no—
263C4⋊Dic3—21no—
264C4⋊Dic3—21no—
265C4⋊Dic3—22no—
266C4⋊Dic3—21no—
267C4⋊Dic3—21no—
268C4⋊Dic3—42no—
269C4⋊Dic3—81no—
270C4⋊Dic3C2×D412yes—
271C4⋊Dic3#4112yes—
272Dic3⋊C4—24no—
273Dic3⋊C4—22no—
274Dic3⋊C4—22no—
275Dic3⋊C4—24no—
276Dic3⋊C4—24no—
277Dic3⋊C4—42no—
278Dic3⋊C4—42no—
279Dic3⋊C4—81no—
280C4×Dic3—24no—
281C4×Dic3—24no—
282C4×Dic3—21no—
283C4×Dic3—21no—
284C4×Dic3—81no—
285C4×Dic3C2×D412yes—
286C4×Dic3C2×D412yes—
287C2×C3⋊C8—21no—
288C2×C3⋊C8—21no—
289C2×C3⋊C8—21no—
290C2×C3⋊C8—21no—
291C2×C3⋊C8—21no—
292C2×C3⋊C8—21no—
293C22×Q8—31no—
294C22×Q8—31no—
295C22×Q8C22×D311yescharacteristic
296C22×D4C22×D311yescharacteristic
297C23×C4—61no—
298C22×C8—31no—
299C22×C8—31no—
300C22×C8C22×D311yescharacteristic
301C22×C8C22×D311yescharacteristic
302C4⋊Q8—34no—
303C4⋊Q8—62no—
304#126—68no—
305#126—68no—
306#125—62no—
307#125—62no—
308#124—22no—
309#124—22no—
310#124—62no—
311#124—62no—
312C22⋊Q8—68no—
313C4×Q8—62no—
314C4×Q8—62no—
315C4×Q8—62no—
316C4×Q8—62no—
317C4×Q8—62no—
318C4×Q8—62no—
319C4×D4—34no—
320C4×D4—62no—
321#117—68no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—31no—
324C2×C4⋊C4—31no—
325C2×C4⋊C4—31no—
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⋊C4—61no—
337C2×C4⋊C4—61no—
338C2×C4⋊C4C22×D311yescharacteristic
339C2×C4⋊C4C22×D311yescharacteristic
340C2×C22⋊C4—21no—
341C2×C22⋊C4—21no—
342C2×C22⋊C4—61no—
343C2×C22⋊C4—61no—
344C2×C22⋊C4—61no—
345C2×C42—31no—
346C2×C42—31no—
347C2×C42—61no—
348C2×C42—61no—
349C2×C42—61no—
350C2×C42C22×D311yescharacteristic
351#107—62no—
352#106—62no—
353C4⋊C8—62no—
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#102—22no—
363#102—22no—
364#102—62no—
365#102—62no—
366#97—62no—
367#95—21no—
368#95—61no—
369#95—61no—
370#95—61no—
371#95—61no—
372C23×C3—22no—
373C23×C3—41no—
374C23×C3—41no—
375C23×C3—41no—
376C23×C3—41no—
377C23×C3—41no—
378C23×C3—41no—
379C23×C3C22×D411yescharacteristic
380C3×Q8—22no—
381C3×Q8—22no—
382C3×Q8—41no—
383C3×Q8—41no—
384C3×D4—22no—
385C3×D4—22no—
386C3×D4—41no—
387C3×D4—41no—
388C2×C3×C4—21no—
389C2×C3×C4—22no—
390C2×C3×C4—21no—
391C2×C3×C4—22no—
392C2×C3×C4—21no—
393C2×C3×C4—21no—
394C2×C3×C4—21no—
395C2×C3×C4—21no—
396C2×C3×C4—41no—
397C2×C3×C4—41no—
398C2×C3×C4—41no—
399C2×C3×C4—41no—
400C2×C3×C4—41no—
401C2×C3×C4—41no—
402C2×C3×C4—41no—
403C2×C3×C4C22×D411yescharacteristic
404C2×C3×C4C22×D411yescharacteristic
405C2×C3×C4C2×#4111yescharacteristic
406C2×C3×C4C2×#4111yescharacteristic
407C2×C3×C42−1+411yescharacteristic
408C2×C3×C42−1+411yescharacteristic
409C2×Dic3—21no—
410C2×Dic3—21no—
411C2×Dic3—21no—
412C2×Dic3—21no—
413C2×Dic3—21no—
414C2×Dic3—22no—
415C2×Dic3—22no—
416C2×Dic3—21no—
417C2×Dic3—22no—
418C2×Dic3—21no—
419C2×Dic3—21no—
420C2×Dic3—21no—
421C2×Dic3—21no—
422C2×Dic3—48no—
423C2×Dic3—41no—
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#12012yes—
433C2×Dic3#12012yes—
434Dic6—22no—
435Dic6—22no—
436Dic6—44no—
437Dic6—41no—
438Dic6—41no—
439C3×C8—41no—
440C3×C8—41no—
441C3⋊C8—41no—
442C3⋊C8—41no—
443C24—21no—
444C2×Q8—21no—
445C2×Q8—21no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—32no—
450C2×Q8—62no—
451C2×Q8—61no—
452C2×Q8—61no—
453C2×Q8—61no—
454C2×Q8—61no—
455C2×Q8—61no—
456C2×Q8—121no—
457C2×Q8—121no—
458C2×Q8D3×D412yes—
459C2×D4—21no—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4—21no—
463C2×D4—21no—
464C2×D4D3×D412yes—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—31no—
469C22×C4—31no—
470C22×C4—31no—
471C22×C4—31no—
472C22×C4—64no—
473C22×C4—61no—
474C22×C4—61no—
475C22×C4—61no—
476C22×C4—61no—
477C22×C4—122no—
478C22×C4—122no—
479C22×C4D3×D411yescharacteristic
480C22×C4D3×D411yescharacteristic
481C22×C4C23×D311yescharacteristic
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—61no—
489C2×C8—61no—
490C2×C8—61no—
491C2×C8—61no—
492C2×C8—61no—
493C2×C8—61no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—32no—
498C4⋊C4—32no—
499C4⋊C4—42no—
500C4⋊C4—62no—
501C4⋊C4—62no—
502C4⋊C4—62no—
503C4⋊C4—64no—
504C4⋊C4—64no—
505C4⋊C4—64no—
506C4⋊C4—64no—
507C4⋊C4—61no—
508C4⋊C4—61no—
509C4⋊C4—61no—
510C4⋊C4—61no—
511C4⋊C4—122no—
512C4⋊C4—122no—
513C4⋊C4—122no—
514C4⋊C4—241no—
515C4⋊C4—241no—
516C4⋊C4#23712yes—
517C22⋊C4—24no—
518C22⋊C4—24no—
519C22⋊C4—64no—
520C22⋊C4—64no—
521C22⋊C4—64no—
522C42—21no—
523C42—21no—
524C42—32no—
525C42—32no—
526C42—64no—
527C42—64no—
528C42—61no—
529C42—61no—
530C42—241no—
531C22×C3—42no—
532C22×C3—42no—
533C22×C3—42no—
534C22×C3—81no—
535C22×C3—81no—
536C22×C3—81no—
537C22×C3—81no—
538C22×C3C2×#12011yescharacteristic
539C22×C3#54811yescharacteristic
540C22×C3#54811yescharacteristic
541C22×C3#57511yescharacteristic
542C22×C3#57511yescharacteristic
543C22×C3#57711yescharacteristic
544C22×C3#57811yescharacteristic
545C3×C4—42no—
546C3×C4—42no—
547C3×C4—41no—
548C3×C4—41no—
549C3×C4—81no—
550C3×C4#53211yescharacteristic
551C3×C4#53611yescharacteristic
552C3×C4#54711yescharacteristic
553C3×C4#54711yescharacteristic
554Dic3—22no—
555Dic3—22no—
556Dic3—42no—
557Dic3—42no—
558Dic3—41no—
559Dic3—42no—
560Dic3—41no—
561Dic3—81no—
562C23—22no—
563C23—41no—
564C23—41no—
565C23—41no—
566C23—41no—
567C23—41no—
568C23—41no—
569C23C2×D3×D411yescharacteristic
570Q8—22no—
571Q8—22no—
572Q8—41no—
573Q8—41no—
574Q8—62no—
575Q8—62no—
576Q8—124no—
577Q8—121no—
578Q8—121no—
579D4—22no—
580D4—22no—
581D4—41no—
582D4—41no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—32no—
592C2×C4—32no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—61no—
601C2×C4—61no—
602C2×C4—61no—
603C2×C4—61no—
604C2×C4—62no—
605C2×C4—61no—
606C2×C4—61no—
607C2×C4—61no—
608C2×C4—61no—
609C2×C4—61no—
610C2×C4—62no—
611C2×C4—61no—
612C2×C4—62no—
613C2×C4—128no—
614C2×C4—121no—
615C2×C4—121no—
616C2×C4—121no—
617C2×C4—121no—
618C2×C4—121no—
619C2×C4—121no—
620C2×C4—121no—
621C2×C4—121no—
622C2×C4—121no—
623C2×C4C2×D3×D411yescharacteristic
624C2×C4C2×D3×D411yescharacteristic
625C2×C4C2×#23711yescharacteristic
626C2×C4D3×#4111yescharacteristic
627C2×C4#101011yescharacteristic
628C2×C4#101011yescharacteristic
629C8—41no—
630C8—41no—
631C8—121no—
632C8—121no—
633C2×C3—82no—
634C2×C3#243211yescharacteristic
635C2×C3#300511yescharacteristic
636C2×C3#301311yescharacteristic
637C2×C3#337611yescharacteristic
638C2×C3#337611yescharacteristic
639C2×C3#337711yescharacteristic
640C2×C3#337711yescharacteristic
641C22—42no—
642C22—42no—
643C22—42no—
644C22—81no—
645C22—81no—
646C22—81no—
647C22—81no—
648C22D3×#12011yescharacteristic
649C22#559811yescharacteristic
650C22#567711yescharacteristic
651C22#576811yescharacteristic
652C22#577011yescharacteristic
653C22#577311yescharacteristic
654C22#577911yescharacteristic
655C4—42no—
656C4—41no—
657C4—42no—
658C4—41no—
659C4—62no—
660C4—62no—
661C4—81no—
662C4—122no—
663C4—121no—
664C4—121no—
665C4—122no—
666C4—122no—
667C4—241no—
668C4#559611yescharacteristic
669C4#561111yescharacteristic
670C4#564811yescharacteristic
671C4#565511yescharacteristic
672C3#2565611yescharacteristic, complemented, Sylow
673C2—82no—
674C2#7690411yescharacteristic
675C2#8008311yescharacteristic
676C2#8023211yescharacteristic
677C2#8052311yescharacteristic
678C2#8058611yescharacteristic
679C2#8166111yescharacteristic
680C2#8167711yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2042C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
hypercenter#2042D3
hypocenterC3#25656
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×#4797C2semidirect
4. #76078C2semidirect
7. #70238C2semidirect
8. #70235C2semidirect
9. #70229C2semidirect
13. #69631C2semidirect
14. #68684C2semidirect
15. #68620C2semidirect
672. C3#25656semidirect

Maximal subgroups

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

subgroupindex
C2×#56712
C2×#47972
C2×#47632
#760782
C3×#20422
#717222
#702382
#702352
#702292
#702162
#701782
#701522
#696312
#686842
#686202
#256563