Finite Group Databasebeta

← #10849267

Subgroups — #10849267

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×#5542C211yesmaximal, characteristic
2C2×#4877C211yesmaximal, characteristic, complemented
3C2×#4768C211yesmaximal, characteristic
4#76078C211yesmaximal, characteristic, complemented
5C3×#1862C211yesmaximal, characteristic
6#71607C211yesmaximal, characteristic
7#70642C211yesmaximal, characteristic, complemented
8#70216C211yesmaximal, characteristic
9#70178C211yesmaximal, characteristic
10#70162C211yesmaximal, characteristic
11#69633C211yesmaximal, characteristic, complemented
12#68907C211yesmaximal, characteristic, complemented
13#68781C211yesmaximal, characteristic, complemented
14#68631C211yesmaximal, characteristic, complemented
15#68619C211yesmaximal, characteristic, complemented
16#25654—31nomaximal, Sylow
17C2×C3×C4×D4C2211yescharacteristic
18C2×#936C2211yescharacteristic
19C2×#936C2211yescharacteristic
20C2×#926—21no—
21C2×#924—21no—
22#5542—24no—
23#5542—24no—
24C2×#890C2211yescharacteristic
25C2×#887C2211yescharacteristic
26C2×#887C2211yescharacteristic
27C2×#879—21no—
28C2×#879—21no—
29C2×#869C2211yescharacteristic
30C2×C4×Dic6C2211yescharacteristic
31C2×C3×#107C2211yescharacteristic
32C2×C3×#102C2211yescharacteristic
33C2×C3×#102C2211yescharacteristic
34C3×#389C2211yescharacteristic
35C2×#832C2211yescharacteristic
36C2×#832C2211yescharacteristic
37C2×#816C2211yescharacteristic
38C2×#816C2211yescharacteristic
39C2×#816C2211yescharacteristic
40C2×#816C2211yescharacteristic
41C2×Dic3⋊C8C2211yescharacteristic
42C2×C8×Dic3C2211yescharacteristic
43#4984—21no—
44#4984C2211yescharacteristic
45#4979—21no—
46Dic3×C4⋊C4C2211yescharacteristic
47#4972C2211yescharacteristic
48C2×#810C2211yescharacteristic
49C2×#810C2211yescharacteristic
50C2×#810C2211yescharacteristic
51C2×#810C2211yescharacteristic
52C2×#808C2211yescharacteristic
53#4928C2211yescharacteristic
54#4877—24no—
55#4768—24no—
56#4657—21no—
57#4647—21no—
58C3×#340C2211yescharacteristic
59C3×#336C2211yescharacteristic
60#4550C2211yescharacteristic
61#4550C2211yescharacteristic
62#4529C2211yescharacteristic
63#4524C2211yescharacteristic
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#1984—31no—
73#1983—31no—
74#1865—31no—
75#1864—31no—
76#1863—31no—
77#1862D311yescharacteristic
78C4×#103—31no—
79C22×C3×D4C2311yescharacteristic
80C23×C3×C4—21no—
81C22×Dic6D411yescharacteristic
82C22×Dic6C2311yescharacteristic
83C22×Dic6C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C3×C4×D4—22no—
87C3×C4×D4—22no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C22⋊C4—21no—
93C2×C3×C42C2311yescharacteristic
94C2×#217—21no—
95C2×#217—21no—
96C2×#217—21no—
97C2×#217—21no—
98#936—22no—
99#936—22no—
100#936—22no—
101#936D414yes—
102#926—28no—
103C2×C4⋊Dic3—21no—
104C2×C4⋊Dic3—21no—
105C2×C4⋊Dic3—21no—
106C2×C4⋊Dic3—21no—
107C2×C4⋊Dic3D411yescharacteristic
108C2×C4⋊Dic3C2311yescharacteristic
109C2×C4⋊Dic3C2311yescharacteristic
110#924—28no—
111C2×Dic3⋊C4—21no—
112C2×Dic3⋊C4—21no—
113C2×Dic3⋊C4—21no—
114C2×Dic3⋊C4—21no—
115C2×Dic3⋊C4—21no—
116C2×Dic3⋊C4D411yescharacteristic
117C2×Dic3⋊C4D411yescharacteristic
118C2×C4×Dic3—21no—
119C2×C4×Dic3—21no—
120C2×C4×Dic3—21no—
121C2×C4×Dic3C2311yescharacteristic
122C2×C4×Dic3C2311yescharacteristic
123C22×C3⋊C8C2311yescharacteristic
124C22×C3⋊C8C2311yescharacteristic
125#890—22no—
126#890—22no—
127#887—22no—
128#887—22no—
129#887—22no—
130#887D414yes—
131#879—28no—
132#879—28no—
133#869—22no—
134#869—22no—
135C4×Dic6—22no—
136C4×Dic6—22no—
137C3×#107—22no—
138C3×#102—22no—
139C3×#102—22no—
140C3×#95—21no—
141#832—22no—
142#832—22no—
143C3⋊#95—21no—
144C3⋊#95—21no—
145C3⋊#95—21no—
146C3⋊#95—21no—
147#816—22no—
148#816—22no—
149#816—22no—
150#816—22no—
151Dic3⋊C8—22no—
152C8×Dic3—22no—
153#810—22no—
154#810—22no—
155#810—22no—
156#810—22no—
157#808—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×Q8—31no—
170C2×C4×D4C2×D311yescharacteristic
171C2×#117—61no—
172#488—64no—
173#479—64no—
174C2×#107C2×D311yescharacteristic
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×#102—31no—
186C2×#102—31no—
187C2×#102C2×D311yescharacteristic
188C2×#102C2×D311yescharacteristic
189C2×C4×C8—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#340C2×D311yescharacteristic
201#336—31no—
202#336—31no—
203#336—31no—
204#336C2×D311yescharacteristic
205C24×C3—21no—
206C2×C3×D4—21no—
207C2×C3×D4—21no—
208C2×C3×D4—21no—
209C2×C3×D4—21no—
210C2×C3×D4—21no—
211C2×C3×D4C2×D412yes—
212C22×C3×C4—24no—
213C22×C3×C4—21no—
214C22×C3×C4—21no—
215C22×C3×C4—21no—
216C22×C3×C4—42no—
217C22×C3×C4—42no—
218C22×C3×C4C2×D411yescharacteristic
219C22×C3×C4C2×D411yescharacteristic
220C22×C3×C4C2411yescharacteristic
221C22×Dic3—21no—
222C22×Dic3—21no—
223C22×Dic3—21no—
224C22×Dic3—21no—
225C22×Dic3C2×D411yescharacteristic
226C22×Dic3C2×D411yescharacteristic
227C22×Dic3C2×D411yescharacteristic
228C22×Dic3C2×D411yescharacteristic
229C2×Dic6—21no—
230C2×Dic6—21no—
231C2×Dic6—21no—
232C2×Dic6—21no—
233C2×Dic6—21no—
234C2×Dic6—21no—
235C2×Dic6—21no—
236C2×Dic6—21no—
237C2×Dic6—22no—
238C2×Dic6—21no—
239C2×Dic6—21no—
240C2×Dic6—22no—
241C2×Dic6—41no—
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—
250C3×C4⋊C4—21no—
251C3×C4⋊C4—21no—
252C3×C4⋊C4—21no—
253C3×C4⋊C4—21no—
254C3×C4⋊C4—42no—
255C3×C4⋊C4C2×D412yes—
256C3×C4⋊C4#4112yes—
257C3×C22⋊C4—24no—
258C3×C42—21no—
259C3×C42—21no—
260#217—24no—
261#217—24no—
262#217—24no—
263#217—24no—
264C4⋊Dic3—24no—
265C4⋊Dic3—24no—
266C4⋊Dic3—21no—
267C4⋊Dic3—22no—
268C4⋊Dic3—21no—
269C4⋊Dic3—21no—
270C4⋊Dic3—42no—
271C4⋊Dic3—81no—
272C4⋊Dic3#4112yes—
273Dic3⋊C4—24no—
274Dic3⋊C4—22no—
275Dic3⋊C4—22no—
276Dic3⋊C4—24no—
277Dic3⋊C4—42no—
278Dic3⋊C4—42no—
279Dic3⋊C4—81no—
280C4×Dic3—24no—
281C4×Dic3—21no—
282C4×Dic3—21no—
283C4×Dic3—24no—
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×Q8—31no—
296C22×D4C22×D311yescharacteristic
297C23×C4—21no—
298C22×C8—31no—
299C22×C8—31no—
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—34no—
314C4×Q8—62no—
315C4×Q8—62no—
316C4×Q8—62no—
317C4×Q8—62no—
318C4×Q8—62no—
319C4×D4—22no—
320C4×D4—22no—
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—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⋊C4C22×D311yescharacteristic
338C2×C4⋊C4C22×D311yescharacteristic
339C2×C4⋊C4C22×D311yescharacteristic
340C2×C22⋊C4—21no—
341C2×C22⋊C4—61no—
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—22no—
352#106—62no—
353C4⋊C8—62no—
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—22no—
363#102—22no—
364#102—62no—
365#102—62no—
366C4×C8—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×D4—22no—
381C3×D4—22no—
382C3×D4—41no—
383C3×D4—41no—
384C2×C3×C4—21no—
385C2×C3×C4—21no—
386C2×C3×C4—22no—
387C2×C3×C4—21no—
388C2×C3×C4—22no—
389C2×C3×C4—21no—
390C2×C3×C4—21no—
391C2×C3×C4—48no—
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×C4—41no—
399C2×C3×C4#13512yes—
400C2×C3×C4C22×D411yescharacteristic
401C2×C3×C4C22×D411yescharacteristic
402C2×C3×C4C2×#4111yescharacteristic
403C2×C3×C4C2×#4111yescharacteristic
404C2×C3×C42−1+411yescharacteristic
405C2×C3×C42−1+411yescharacteristic
406C2×Dic3—21no—
407C2×Dic3—21no—
408C2×Dic3—21no—
409C2×Dic3—21no—
410C2×Dic3—21no—
411C2×Dic3—21no—
412C2×Dic3—22no—
413C2×Dic3—22no—
414C2×Dic3—22no—
415C2×Dic3—21no—
416C2×Dic3—21no—
417C2×Dic3—21no—
418C2×Dic3—21no—
419C2×Dic3—41no—
420C2×Dic3—41no—
421C2×Dic3—41no—
422C2×Dic3—41no—
423C2×Dic3—41no—
424C2×Dic3—41no—
425C2×Dic3—41no—
426C2×Dic3—41no—
427C2×Dic3—41no—
428C2×Dic3#12012yes—
429C2×Dic3#12012yes—
430Dic6—22no—
431Dic6—22no—
432Dic6—22no—
433Dic6—22no—
434Dic6—44no—
435Dic6—41no—
436Dic6—41no—
437Dic6—41no—
438Dic6—41no—
439C3×C8—41no—
440C3×C8—41no—
441C3⋊C8—41no—
442C3⋊C8—41no—
443C24—21no—
444C2×Q8—32no—
445C2×Q8—32no—
446C2×Q8—62no—
447C2×Q8—62no—
448C2×Q8—61no—
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—121no—
459C2×D4—21no—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4—21no—
463C2×D4—21no—
464C2×D4D3×D412yes—
465C22×C4—24no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—31no—
470C22×C4—31no—
471C22×C4—31no—
472C22×C4—31no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—61no—
476C22×C4—61no—
477C22×C4—61no—
478C22×C4—61no—
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—21no—
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—122no—
511C4⋊C4—122no—
512C4⋊C4—122no—
513C4⋊C4—241no—
514C4⋊C4—241no—
515C4⋊C4#23512yes—
516C4⋊C4D3×D412yes—
517C22⋊C4—24no—
518C22⋊C4—64no—
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×C3C2×#13511yescharacteristic
542C22×C3#57511yescharacteristic
543C22×C3#57711yescharacteristic
544C22×C3#57811yescharacteristic
545C3×C4—22no—
546C3×C4—42no—
547C3×C4—42no—
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—42no—
559Dic3—41no—
560Dic3—41no—
561Dic3—81no—
562C23—22no—
563C23—41no—
564C23—41no—
565C23—41no—
566C23—41no—
567C23—41no—
568C23—41no—
569C23C2×D3×D411yescharacteristic
570Q8—62no—
571Q8—62no—
572Q8—62no—
573Q8—62no—
574Q8—124no—
575Q8—121no—
576Q8—121no—
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—22no—
588C2×C4—21no—
589C2×C4—22no—
590C2×C4—32no—
591C2×C4—32no—
592C2×C4—48no—
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—61no—
611C2×C4—62no—
612C2×C4—62no—
613C2×C4—121no—
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#91212yes—
623C2×C4C2×#23511yescharacteristic
624C2×C4C2×D3×D411yescharacteristic
625C2×C4C2×D3×D411yescharacteristic
626C2×C4#100711yescharacteristic
627C2×C4D3×#4111yescharacteristic
628C2×C4#101011yescharacteristic
629C8—41no—
630C8—41no—
631C8—121no—
632C8—121no—
633C2×C3—82no—
634C2×C3#243211yescharacteristic
635C2×C3#300211yescharacteristic
636C2×C3#301311yescharacteristic
637C2×C3#336011yescharacteristic
638C2×C3#336411yescharacteristic
639C2×C3#337611yescharacteristic
640C2×C3#337711yescharacteristic
641C22—42no—
642C22—42no—
643C22—42no—
644C22—81no—
645C22—81no—
646C22—81no—
647C22—81no—
648C22#555211yescharacteristic
649C22D3×#12011yescharacteristic
650C22#563311yescharacteristic
651C22C2×#91211yescharacteristic
652C22#577011yescharacteristic
653C22#577311yescharacteristic
654C22#577911yescharacteristic
655C4—22no—
656C4—41no—
657C4—42no—
658C4—42no—
659C4—62no—
660C4—62no—
661C4—81no—
662C4—122no—
663C4—121no—
664C4—121no—
665C4—122no—
666C4—122no—
667C4—241no—
668C4#550711yescharacteristic
669C4#554811yescharacteristic
670C4#562811yescharacteristic
671C4#565511yescharacteristic
672C3#2565411yescharacteristic, complemented, Sylow
673C2—82no—
674C2#7441111yescharacteristic
675C2#7956811yescharacteristic
676C2#7975811yescharacteristic
677C2#8000111yescharacteristic
678C2#8023211yescharacteristic
679C2#8099411yescharacteristic
680C2#8115411yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1862C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
hypercenter#1862D3
hypocenterC3#25654
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1862

Complements

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

NH ≅ G/Ntype
2. C2×#4877C2semidirect
4. #76078C2semidirect
7. #70642C2semidirect
11. #69633C2semidirect
12. #68907C2semidirect
13. #68781C2semidirect
14. #68631C2semidirect
15. #68619C2semidirect
672. C3#25654semidirect

Maximal subgroups

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

subgroupindex
C2×#55422
C2×#48772
C2×#47682
#760782
C3×#18622
#716072
#706422
#702162
#701782
#701622
#696332
#689072
#687812
#686312
#686192
#256543