Finite Group Databasebeta

← #10877475

Subgroups — #10877475

Order 768.

702 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83058C211yesmaximal, characteristic
2#74165C211yesmaximal, characteristic
3C3×#1991C211yesmaximal, characteristic, complemented
4#71744C211yesmaximal, characteristic, complemented
5C8×(C2×D3)⋊C4C211yesmaximal, characteristic
6#69767C211yesmaximal, characteristic, complemented
7#69542C211yesmaximal, characteristic, complemented
8#68914C211yesmaximal, characteristic, complemented
9#68882C211yesmaximal, characteristic
10#68879C211yesmaximal, characteristic
11#68866C211yesmaximal, characteristic
12#68851C211yesmaximal, characteristic, complemented
13#68847C211yesmaximal, characteristic, complemented
14#68842C211yesmaximal, characteristic, complemented
15#68813C211yesmaximal, characteristic
16#27700—31nomaximal, Sylow
17C2×C3×#125C2211yescharacteristic
18C22×D3×C8C2211yescharacteristic
19C2×#897C2211yescharacteristic
20C2×#892C2211yescharacteristic
21C2×#892C2211yescharacteristic
22C2×C3×C4×C8C2211yescharacteristic
23C3×#382C2211yescharacteristic
24C2×(C2×D3)⋊C8C2211yescharacteristic
25C2×#816C2211yescharacteristic
26C2×#816C2211yescharacteristic
27C2×C8×Dic3C2211yescharacteristic
28#4994—21no—
29#4993—21no—
30#4987C2211yescharacteristic
31#4984C2211yescharacteristic
32#4978—21no—
33#4975C2211yescharacteristic
34#4975C2211yescharacteristic
35#4973C2211yescharacteristic
36#4972—21no—
37C2×#810C2211yescharacteristic
38C2×#810C2211yescharacteristic
39C4×(C2×D3)⋊C4C2211yescharacteristic
40#4929C2211yescharacteristic
41#4928C2211yescharacteristic
42#4675—21no—
43#4675—21no—
44#4655—21no—
45#4655—21no—
46C3×#340C2211yescharacteristic
47C3×#340C2211yescharacteristic
48C3×#340C2211yescharacteristic
49C3×#340C2211yescharacteristic
50#4551C2211yescharacteristic
51#4551C2211yescharacteristic
52#4550C2211yescharacteristic
53#4529C2211yescharacteristic
54#4529C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4524C2211yescharacteristic
60#2499—31no—
61#2441—31no—
62#2101—31no—
63#2099—31no—
64#2072—31no—
65#2026—31no—
66#1991D311yescharacteristic, complemented
67#1933—31no—
68#1933—31no—
69#1892—31no—
70#1873—31no—
71#1834—31no—
72#1825—31no—
73C8×C22⋊C4—31no—
74#1735—31no—
75C22×C4×D3C2311yescharacteristic
76C22×Dic6C2311yescharacteristic
77C22×C3×C8C2311yescharacteristic
78C22×C3×C8C2311yescharacteristic
79C3×#125—22no—
80C3×#125—41no—
81C2×C3×C4⋊C4—21no—
82C2×C3×C4⋊C4—21no—
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C42C2311yescharacteristic
88C2×(C2×D3)⋊C4—21no—
89C2×(C2×D3)⋊C4—21no—
90C2×(C2×D3)⋊C4—21no—
91C2×(C2×D3)⋊C4—21no—
92C2×(C2×D3)⋊C4D411yescharacteristic
93C2×(C2×D3)⋊C4D411yescharacteristic
94C2×C4⋊Dic3—21no—
95C2×C4⋊Dic3—21no—
96C2×C4⋊Dic3C2311yescharacteristic
97C2×C4⋊Dic3C2311yescharacteristic
98C2×C4⋊Dic3C2311yescharacteristic
99C2×Dic3⋊C4—21no—
100C2×Dic3⋊C4—21no—
101C2×Dic3⋊C4—21no—
102C2×Dic3⋊C4—21no—
103C2×Dic3⋊C4—21no—
104C2×C4×Dic3—21no—
105C2×C4×Dic3—21no—
106C2×C4×Dic3C2311yescharacteristic
107C22×C3⋊C8C2311yescharacteristic
108C22×C3⋊C8C2311yescharacteristic
109C2×D3×C8—22no—
110C2×D3×C8—22no—
111C2×D3×C8—41no—
112#897—22no—
113#897—22no—
114#892—22no—
115#892—22no—
116#892—22no—
117#892—41no—
118C3×C4×C8—41no—
119C3×#95—21no—
120C3×#95—21no—
121C3⋊#95—21no—
122C3⋊#95—21no—
123C3⋊#95—21no—
124C3⋊#95—21no—
125C3⋊#95—21no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128C3⋊#95D411yescharacteristic
129C3⋊#95D411yescharacteristic
130(C2×D3)⋊C8—22no—
131#816—22no—
132#816—41no—
133C8×Dic3—22no—
134#810—22no—
135#810—41no—
136C23×C8—31no—
137C2×#125C2×D311yescharacteristic
138C2×C22⋊Q8—31no—
139C2×#117—31no—
140C2×#117—31no—
141C2×#103—31no—
142C2×#103—31no—
143C2×#103—31no—
144C2×#103—31no—
145C2×C22⋊C8—31no—
146C2×C4×C8—31no—
147C2×C4×C8C2×D311yescharacteristic
148#399—61no—
149#397—61no—
150#397—61no—
151#396—61no—
152#391—31no—
153#391—31no—
154#389—31no—
155#389—31no—
156#389—61no—
157#387—61no—
158#387—61no—
159#387—61no—
160#384—31no—
161#384—31no—
162#382C2×D311yescharacteristic
163#380—31no—
164C4×C22⋊C4—31no—
165#340—31no—
166#340—31no—
167#340—31no—
168#340—31no—
169#340—31no—
170#340—31no—
171#340—31no—
172#340—31no—
173#340C2×D311yescharacteristic
174#340C2×D311yescharacteristic
175#340C2×D311yescharacteristic
176#340C2×D311yescharacteristic
177#336—31no—
178#336—31no—
179C23×D3C2×D411yescharacteristic
180C22×C3×C4—21no—
181C22×C3×C4—21no—
182C22×C3×C4—21no—
183C22×C3×C4—21no—
184C22×C3×C4C2×D411yescharacteristic
185C22×C3×C4C2×D411yescharacteristic
186C22×C3×C4C2411yescharacteristic
187C22×Dic3—21no—
188C22×Dic3—21no—
189C22×Dic3—21no—
190C22×Dic3—21no—
191C22×Dic3C2×D411yescharacteristic
192C22×Dic3C2×D411yescharacteristic
193C22×Dic3C2×D411yescharacteristic
194C2×C4×D3—21no—
195C2×C4×D3—21no—
196C2×C4×D3—21no—
197C2×C4×D3—21no—
198C2×C4×D3#4112yes—
199C2×C4×D3#4112yes—
200C2×Dic6—21no—
201C2×Dic6—21no—
202C2×Dic6—21no—
203C2×Dic6—21no—
204C2×Dic6—21no—
205C2×Dic6—21no—
206C2×C3×C8—21no—
207C2×C3×C8—21no—
208C2×C3×C8—21no—
209C2×C3×C8—21no—
210C2×C3×C8—21no—
211C2×C3×C8#4112yes—
212C3×C4⋊C4—21no—
213C3×C4⋊C4—21no—
214C3×C4⋊C4—21no—
215C3×C4⋊C4—21no—
216C3×C4⋊C4—21no—
217C3×C4⋊C4—21no—
218C3×C4⋊C4—21no—
219C3×C4⋊C4—42no—
220C3×C4⋊C4—42no—
221C3×C4⋊C4#4112yes—
222C3×C42—21no—
223C3×C42—21no—
224(C2×D3)⋊C4—22no—
225(C2×D3)⋊C4—22no—
226(C2×D3)⋊C4—42no—
227(C2×D3)⋊C4—42no—
228(C2×D3)⋊C4—42no—
229(C2×D3)⋊C4—81no—
230C4⋊Dic3—21no—
231C4⋊Dic3—21no—
232C4⋊Dic3—21no—
233C4⋊Dic3—21no—
234C4⋊Dic3—21no—
235C4⋊Dic3—21no—
236C4⋊Dic3—42no—
237C4⋊Dic3—42no—
238Dic3⋊C4—42no—
239Dic3⋊C4—42no—
240Dic3⋊C4—42no—
241Dic3⋊C4—81no—
242Dic3⋊C4—81no—
243C4×Dic3—21no—
244C4×Dic3—21no—
245C4×Dic3—42no—
246C4×Dic3—42no—
247C2×C3⋊C8—21no—
248C2×C3⋊C8—21no—
249C2×C3⋊C8—21no—
250C2×C3⋊C8—21no—
251C2×C3⋊C8—21no—
252C2×C3⋊C8—21no—
253D3×C8—44no—
254C22×Q8—31no—
255C23×C4—31no—
256C22×C8—31no—
257C22×C8—31no—
258C22×C8—62no—
259C22×C8—62no—
260C22×C8—121no—
261C22×C8C22×D311yescharacteristic
262C22×C8C22×D311yescharacteristic
263#125—22no—
264#125—41no—
265C22⋊Q8—62no—
266C22⋊Q8—62no—
267#117—62no—
268#117—62no—
269#117—62no—
270#117—121no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—31no—
274C2×C4⋊C4—31no—
275C2×C4⋊C4—31no—
276C2×C4⋊C4—61no—
277C2×C4⋊C4—61no—
278C2×C4⋊C4—61no—
279C2×C4⋊C4—61no—
280C2×C4⋊C4—61no—
281C2×C4⋊C4—61no—
282C2×C4⋊C4—61no—
283C2×C4⋊C4C22×D311yescharacteristic
284C2×C4⋊C4C22×D311yescharacteristic
285C2×C4⋊C4C22×D311yescharacteristic
286C2×C4⋊C4C22×D311yescharacteristic
287C2×C22⋊C4—31no—
288C2×C22⋊C4—31no—
289C2×C22⋊C4—61no—
290C2×C22⋊C4—61no—
291C2×C22⋊C4—61no—
292C2×C22⋊C4—61no—
293C2×C42—31no—
294C2×C42—61no—
295C2×C42—61no—
296C2×C42C22×D311yescharacteristic
297#103—62no—
298#103—62no—
299#103—121no—
300#103—121no—
301C22⋊C8—62no—
302C4×C8—41no—
303C4×C8—62no—
304#95—21no—
305#95—21no—
306#95—31no—
307#95—31no—
308#95—61no—
309#95—61no—
310#95—61no—
311#95—61no—
312#95—61no—
313#95—61no—
314#95—61no—
315C23×C3C22×D411yescharacteristic
316C22×D3—21no—
317C22×D3—21no—
318C22×D3—21no—
319C22×D3—21no—
320C22×D3—21no—
321C22×D3—21no—
322C22×D3#12412yes—
323C2×C3×C4—22no—
324C2×C3×C4—21no—
325C2×C3×C4—21no—
326C2×C3×C4—21no—
327C2×C3×C4—21no—
328C2×C3×C4—21no—
329C2×C3×C4—21no—
330C2×C3×C4—41no—
331C2×C3×C4—41no—
332C2×C3×C4—41no—
333C2×C3×C4—41no—
334C2×C3×C4—41no—
335C2×C3×C4—41no—
336C2×C3×C4—41no—
337C2×C3×C4—41no—
338C2×C3×C4—41no—
339C2×C3×C4—41no—
340C2×C3×C4—41no—
341C2×C3×C4C2×#4111yescharacteristic
342C2×C3×C4C2×#4111yescharacteristic
343C2×C3×C4C2×#4111yescharacteristic
344C2×C3×C4C2×#4111yescharacteristic
345C2×C3×C4H(5,𝔽2)11yescharacteristic
346C2×C3×C42−1+411yescharacteristic
347C2×Dic3—21no—
348C2×Dic3—21no—
349C2×Dic3—21no—
350C2×Dic3—21no—
351C2×Dic3—21no—
352C2×Dic3—21no—
353C2×Dic3—41no—
354C2×Dic3—41no—
355C2×Dic3—41no—
356C2×Dic3—41no—
357C2×Dic3—41no—
358C2×Dic3—41no—
359C2×Dic3—41no—
360C2×Dic3—41no—
361C2×Dic3—41no—
362C2×Dic3—41no—
363C2×Dic3—41no—
364C2×Dic3—41no—
365C2×Dic3#12412yes—
366C2×Dic3#13512yes—
367C2×Dic3#13512yes—
368C4×D3—22no—
369C4×D3—22no—
370C4×D3—22no—
371C4×D3—22no—
372Dic6—41no—
373Dic6—41no—
374Dic6—41no—
375Dic6—41no—
376C3×C8—22no—
377C3×C8—41no—
378C3⋊C8—41no—
379C3⋊C8—41no—
380C24—31no—
381C2×Q8—61no—
382C2×Q8—61no—
383C2×Q8—61no—
384C2×Q8—61no—
385C2×Q8—61no—
386C2×Q8—61no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—31no—
392C22×C4—31no—
393C22×C4—31no—
394C22×C4—32no—
395C22×C4—32no—
396C22×C4—61no—
397C22×C4—61no—
398C22×C4—61no—
399C22×C4—61no—
400C22×C4—61no—
401C22×C4—61no—
402C22×C4—61no—
403C22×C4—61no—
404C22×C4D3×D411yescharacteristic
405C22×C4D3×D411yescharacteristic
406C22×C4C23×D311yescharacteristic
407C2×C8—21no—
408C2×C8—21no—
409C2×C8—21no—
410C2×C8—21no—
411C2×C8—21no—
412C2×C8—61no—
413C2×C8—61no—
414C2×C8—61no—
415C2×C8—61no—
416C2×C8—61no—
417C2×C8—61no—
418C2×C8—124no—
419C2×C8#23912yes—
420C4⋊C4—21no—
421C4⋊C4—21no—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C4⋊C4—61no—
430C4⋊C4—61no—
431C4⋊C4—61no—
432C4⋊C4—61no—
433C4⋊C4—61no—
434C4⋊C4—61no—
435C4⋊C4—122no—
436C4⋊C4—122no—
437C4⋊C4—122no—
438C4⋊C4—122no—
439C4⋊C4—122no—
440C4⋊C4—241no—
441C4⋊C4—241no—
442C4⋊C4#23512yes—
443C22⋊C4—62no—
444C22⋊C4—62no—
445C22⋊C4—122no—
446C22⋊C4—122no—
447C22⋊C4—122no—
448C22⋊C4—241no—
449C42—21no—
450C42—21no—
451C42—61no—
452C42—61no—
453C42—122no—
454C42—122no—
455C22×C3C2×#12411yescharacteristic
456C22×C3#54611yescharacteristic
457C22×C3#54711yescharacteristic
458C22×C3C2×#13511yescharacteristic
459C22×C3C2×#13511yescharacteristic
460C22×C3#57711yescharacteristic
461C22×C3#57811yescharacteristic
462C2×D3—22no—
463C2×D3—22no—
464C2×D3—22no—
465C2×D3—41no—
466C2×D3—41no—
467C2×D3—41no—
468C2×D3—41no—
469C3×C4—41no—
470C3×C4—42no—
471C3×C4—41no—
472C3×C4—81no—
473C3×C4—81no—
474C3×C4—81no—
475C3×C4#52911yescharacteristic
476C3×C4#53911yescharacteristic
477C3×C4#55111yescharacteristic
478C3×C4#55211yescharacteristic
479Dic3—22no—
480Dic3—22no—
481Dic3—22no—
482Dic3—81no—
483Dic3—81no—
484Dic3—81no—
485Dic3—81no—
486C23—32no—
487C23—61no—
488C23—61no—
489C23—61no—
490C23—61no—
491C23—61no—
492C23—61no—
493C23C2×D3×D411yescharacteristic
494Q8—121no—
495Q8—121no—
496Q8—121no—
497Q8—121no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—22no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—32no—
506C2×C4—32no—
507C2×C4—32no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—61no—
520C2×C4—61no—
521C2×C4—61no—
522C2×C4—61no—
523C2×C4—61no—
524C2×C4—61no—
525C2×C4—62no—
526C2×C4—62no—
527C2×C4—62no—
528C2×C4—62no—
529C2×C4—121no—
530C2×C4—121no—
531C2×C4—121no—
532C2×C4—121no—
533C2×C4—121no—
534C2×C4—121no—
535C2×C4—121no—
536C2×C4—121no—
537C2×C4—121no—
538C2×C4—121no—
539C2×C4—121no—
540C2×C4—121no—
541C2×C4C2×#23511yescharacteristic
542C2×C4#100411yescharacteristic
543C2×C4C2×#23911yescharacteristic
544C2×C4#100711yescharacteristic
545C2×C4D3×#4111yescharacteristic
546C2×C4D3×#4111yescharacteristic
547C8—22no—
548C8—41no—
549C8—121no—
550C8—121no—
551C2×C3#248811yescharacteristic
552C2×C3#313111yescharacteristic
553C2×C3#313711yescharacteristic
554C2×C3#330211yescharacteristic
555C2×C3#330411yescharacteristic
556C2×C3#333611yescharacteristic
557C2×C3#333711yescharacteristic
558D3—42no—
559C22—62no—
560C22—62no—
561C22—62no—
562C22—121no—
563C22—121no—
564C22—121no—
565C22—121no—
566C22#549011yescharacteristic
567C22D3×#12411yescharacteristic
568C22#572611yescharacteristic
569C22D3×#13511yescharacteristic
570C22D3×#13511yescharacteristic
571C22#577011yescharacteristic
572C22#577111yescharacteristic
573C4—41no—
574C4—42no—
575C4—41no—
576C4—62no—
577C4—62no—
578C4—62no—
579C4—81no—
580C4—81no—
581C4—81no—
582C4—241no—
583C4—241no—
584C4—241no—
585C4—241no—
586C4#558011yescharacteristic
587C4#558111yescharacteristic
588C4#563711yescharacteristic
589C4#563811yescharacteristic
590C3#2770011yescharacteristic, complemented, Sylow
591C2—122no—
592C2#7662911yescharacteristic
593C2#7884911yescharacteristic
594C2#7885111yescharacteristic
595C2#8163711yescharacteristic
596C2#8163911yescharacteristic
597C2#8192411yescharacteristic
598C2#8192511yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1991C2
socleC23×C3C22×D4
prime radicalC23×D3C2×D4
hypercenter#1991D3
hypocenterC3#27700
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1991

Complements

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

NH ≅ G/Ntype
3. C3×#1991C2semidirect
4. #71744C2semidirect
6. #69767C2semidirect
7. #69542C2semidirect
8. #68914C2semidirect
12. #68851C2semidirect
13. #68847C2semidirect
14. #68842C2semidirect
66. #1991D3semidirect
590. C3#27700semidirect

Maximal subgroups

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

subgroupindex
#830582
#741652
C3×#19912
#717442
C8×(C2×D3)⋊C42
#697672
#695422
#689142
#688822
#688792
#688662
#688512
#688472
#688422
#688132
#277003