Finite Group Databasebeta

← #10877468

Subgroups — #10877468

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#82256C211yesmaximal, characteristic
2#82249C211yesmaximal, characteristic
3C3×#1735C211yesmaximal, characteristic
4#71972C211yesmaximal, characteristic
5C8×#217C211yesmaximal, characteristic
6#71585C211yesmaximal, characteristic
7#71575C211yesmaximal, characteristic
8#68914C211yesmaximal, characteristic, complemented
9#68911C211yesmaximal, characteristic, complemented
10#68851C211yesmaximal, characteristic, complemented
11#68850C211yesmaximal, characteristic, complemented
12#68847C211yesmaximal, characteristic, complemented
13#68844C211yesmaximal, characteristic, complemented
14#68842C211yesmaximal, characteristic, complemented
15#68804C211yesmaximal, characteristic, complemented
16#27700—31nomaximal, Sylow
17C23×C3×C8C2211yescharacteristic
18C2×C3×#117C2211yescharacteristic
19C2×C3×#117C2211yescharacteristic
20C2×#924C2211yescharacteristic
21C2×#890C2211yescharacteristic
22#5212C2211yescharacteristic
23C4×#217C2211yescharacteristic
24C2×#830C2211yescharacteristic
25C2×#816C2211yescharacteristic
26C2×#816C2211yescharacteristic
27C2×C8×Dic3C2211yescharacteristic
28C2×C8×Dic3C2211yescharacteristic
29#4984C2211yescharacteristic
30#4977C2211yescharacteristic
31#4975C2211yescharacteristic
32#4975C2211yescharacteristic
33#4973C2211yescharacteristic
34#4973C2211yescharacteristic
35#4972C2211yescharacteristic
36C2×#810C2211yescharacteristic
37C2×#810C2211yescharacteristic
38#4952—21no—
39#4952—21no—
40#4949—21no—
41#4948—21no—
42#4933—21no—
43#4933—21no—
44#4933—21no—
45#4928—21no—
46C3×#340C2211yescharacteristic
47C3×#340C2211yescharacteristic
48C3×#340C2211yescharacteristic
49C3×#340C2211yescharacteristic
50#4551C2211yescharacteristic
51#4551C2211yescharacteristic
52#4550C2211yescharacteristic
53#4550C2211yescharacteristic
54#4529C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#2499—31no—
61#2441—31no—
62#2101—31no—
63#2099—31no—
64#2072—31no—
65#2026—31no—
66#1991—31no—
67#1933—31no—
68#1933—31no—
69#1892—31no—
70#1873—31no—
71#1834—31no—
72#1825—31no—
73C8×C22⋊C4—31no—
74#1735D311yescharacteristic
75C23×C3×C4C2311yescharacteristic
76C22×Dic6C2311yescharacteristic
77C22×C3×C8—22no—
78C22×C3×C8—22no—
79C22×C3×C8—41no—
80C22×C3×C8C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C3×#117—22no—
83C3×#117—22no—
84C3×#117—22no—
85C3×#117—41no—
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C22⋊C4—21no—
91C2×C3×C22⋊C4—21no—
92C2×C3×C42—21no—
93C2×C3×C42—21no—
94C2×#217—21no—
95C2×#217—21no—
96C2×#217D411yescharacteristic
97C2×#217D411yescharacteristic
98C2×C4⋊Dic3—21no—
99C2×C4⋊Dic3—21no—
100C2×C4⋊Dic3C2311yescharacteristic
101C2×C4⋊Dic3C2311yescharacteristic
102C2×C4⋊Dic3C2311yescharacteristic
103#924—22no—
104#924—22no—
105C2×Dic3⋊C4—21no—
106C2×Dic3⋊C4—21no—
107C2×Dic3⋊C4—21no—
108C2×Dic3⋊C4—21no—
109C2×Dic3⋊C4—21no—
110C2×Dic3⋊C4—21no—
111C2×Dic3⋊C4—21no—
112C2×C4×Dic3C2311yescharacteristic
113C2×C4×Dic3C2311yescharacteristic
114C22×C3⋊C8C2311yescharacteristic
115C22×C3⋊C8C2311yescharacteristic
116#890—22no—
117#890—41no—
118C3⋊#95—21no—
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⋊#95D411yescharacteristic
128C3⋊#95D411yescharacteristic
129#830—22no—
130#816—22no—
131#816—41no—
132C8×Dic3—22no—
133C8×Dic3—41no—
134#810—22no—
135#810—41no—
136C23×C8C2×D311yescharacteristic
137C2×#125—31no—
138C2×C22⋊Q8—31no—
139C2×#117C2×D311yescharacteristic
140C2×#117C2×D311yescharacteristic
141C2×#103—31no—
142C2×#103—31no—
143C2×#103—31no—
144C2×#103—31no—
145C2×C22⋊C8—31no—
146C2×C4×C8—31no—
147C2×C4×C8—31no—
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#382—31no—
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—
179C24×C3C2×D411yescharacteristic
180C22×C3×C4—21no—
181C22×C3×C4—21no—
182C22×C3×C4—21no—
183C22×C3×C4—21no—
184C22×C3×C4—21no—
185C22×C3×C4—21no—
186C22×C3×C4—21no—
187C22×C3×C4—21no—
188C22×C3×C4C2×D411yescharacteristic
189C22×C3×C4#4112yes—
190C22×C3×C4#4112yes—
191C22×C3×C4C2411yescharacteristic
192C22×Dic3—21no—
193C22×Dic3—21no—
194C22×Dic3—21no—
195C22×Dic3—21no—
196C22×Dic3C2×D411yescharacteristic
197C22×Dic3C2×D411yescharacteristic
198C22×Dic3C2×D411yescharacteristic
199C22×Dic3C2×D411yescharacteristic
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—44no—
212C2×C3×C8#4112yes—
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—21no—
220C3×C4⋊C4#4112yes—
221C3×C22⋊C4—42no—
222C3×C22⋊C4—42no—
223C3×C42—42no—
224C3×C42—42no—
225#217—22no—
226#217—22no—
227#217—42no—
228#217—81no—
229C4⋊Dic3—21no—
230C4⋊Dic3—21no—
231C4⋊Dic3—21no—
232C4⋊Dic3—21no—
233C4⋊Dic3—21no—
234C4⋊Dic3—21no—
235C4⋊Dic3—42no—
236C4⋊Dic3—42no—
237Dic3⋊C4—42no—
238Dic3⋊C4—42no—
239Dic3⋊C4—42no—
240Dic3⋊C4—42no—
241Dic3⋊C4—42no—
242Dic3⋊C4—81no—
243Dic3⋊C4—81no—
244C4×Dic3—21no—
245C4×Dic3—21no—
246C4×Dic3—21no—
247C4×Dic3—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—
254C22×Q8—31no—
255C23×C4C22×D311yescharacteristic
256C22×C8—22no—
257C22×C8—22no—
258C22×C8—31no—
259C22×C8—31no—
260C22×C8—41no—
261C22×C8C22×D311yescharacteristic
262C22×C8C22×D311yescharacteristic
263#125—62no—
264#125—121no—
265C22⋊Q8—62no—
266C22⋊Q8—62no—
267#117—22no—
268#117—22no—
269#117—22no—
270#117—41no—
271C2×C4⋊C4—31no—
272C2×C4⋊C4—31no—
273C2×C4⋊C4—31no—
274C2×C4⋊C4—61no—
275C2×C4⋊C4—61no—
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—21no—
288C2×C22⋊C4—21no—
289C2×C22⋊C4—31no—
290C2×C22⋊C4—31no—
291C2×C22⋊C4—61no—
292C2×C22⋊C4—61no—
293C2×C42—21no—
294C2×C42—21no—
295C2×C42—31no—
296C2×C42—31no—
297#103—62no—
298#103—62no—
299#103—121no—
300#103—121no—
301C22⋊C8—62no—
302C4×C8—62no—
303C4×C8—121no—
304#95—31no—
305#95—31no—
306#95—61no—
307#95—61no—
308#95—61no—
309#95—61no—
310#95—61no—
311#95—61no—
312#95—61no—
313#95—61no—
314#95—61no—
315C23×C3—21no—
316C23×C3—21no—
317C23×C3—21no—
318C23×C3—21no—
319C23×C3—21no—
320C23×C3—21no—
321C23×C3#12412yes—
322C23×C3C22×D411yescharacteristic
323C2×C3×C4—21no—
324C2×C3×C4—21no—
325C2×C3×C4—22no—
326C2×C3×C4—22no—
327C2×C3×C4—22no—
328C2×C3×C4—22no—
329C2×C3×C4—22no—
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×C4#12412yes—
342C2×C3×C4C2×#4111yescharacteristic
343C2×C3×C4C2×#4111yescharacteristic
344C2×C3×C4C2×#4111yescharacteristic
345C2×C3×C4C2×#4111yescharacteristic
346C2×C3×C4H(5,𝔽2)11yescharacteristic
347C2×C3×C42−1+411yescharacteristic
348C2×Dic3—21no—
349C2×Dic3—21no—
350C2×Dic3—21no—
351C2×Dic3—21no—
352C2×Dic3—21no—
353C2×Dic3—21no—
354C2×Dic3—21no—
355C2×Dic3—21no—
356C2×Dic3—21no—
357C2×Dic3—21no—
358C2×Dic3—41no—
359C2×Dic3—41no—
360C2×Dic3—41no—
361C2×Dic3—41no—
362C2×Dic3—41no—
363C2×Dic3—41no—
364C2×Dic3—41no—
365C2×Dic3—41no—
366C2×Dic3—41no—
367C2×Dic3—41no—
368C2×Dic3—41no—
369C2×Dic3—41no—
370C2×Dic3#13512yes—
371C2×Dic3#13512yes—
372Dic6—41no—
373Dic6—41no—
374Dic6—41no—
375Dic6—41no—
376C3×C8—22no—
377C3×C8—41no—
378C3⋊C8—41no—
379C3⋊C8—41no—
380C24D3×D411yescharacteristic
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—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—31no—
396C22×C4—31no—
397C22×C4—31no—
398C22×C4—31no—
399C22×C4—61no—
400C22×C4—61no—
401C22×C4—61no—
402C22×C4—61no—
403C22×C4#23512yes—
404C22×C4D3×D411yescharacteristic
405C22×C4#23712yes—
406C22×C4C23×D311yescharacteristic
407C2×C8—21no—
408C2×C8—21no—
409C2×C8—21no—
410C2×C8—21no—
411C2×C8—21no—
412C2×C8—44no—
413C2×C8—61no—
414C2×C8—61no—
415C2×C8—61no—
416C2×C8—61no—
417C2×C8—61no—
418C2×C8—61no—
419C2×C8#23512yes—
420C4⋊C4—21no—
421C4⋊C4—21no—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—61no—
428C4⋊C4—61no—
429C4⋊C4—61no—
430C4⋊C4—61no—
431C4⋊C4—61no—
432C4⋊C4—61no—
433C4⋊C4—122no—
434C4⋊C4—122no—
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—42no—
444C22⋊C4—42no—
445C22⋊C4—62no—
446C22⋊C4—62no—
447C22⋊C4—122no—
448C22⋊C4—241no—
449C42—42no—
450C42—42no—
451C42—61no—
452C42—61no—
453C42—61no—
454C42—61no—
455C22×C3—22no—
456C22×C3—22no—
457C22×C3—22no—
458C22×C3—41no—
459C22×C3—41no—
460C22×C3—41no—
461C22×C3—41no—
462C22×C3C2×#12411yescharacteristic
463C22×C3#54611yescharacteristic
464C22×C3#54711yescharacteristic
465C22×C3C2×#13511yescharacteristic
466C22×C3C2×#13511yescharacteristic
467C22×C3#57711yescharacteristic
468C22×C3#57811yescharacteristic
469C3×C4—22no—
470C3×C4—42no—
471C3×C4—81no—
472C3×C4—81no—
473C3×C4—81no—
474C3×C4#52911yescharacteristic
475C3×C4#53911yescharacteristic
476C3×C4#55111yescharacteristic
477C3×C4#55211yescharacteristic
478Dic3—22no—
479Dic3—22no—
480Dic3—41no—
481Dic3—41no—
482Dic3—81no—
483Dic3—81no—
484Dic3—81no—
485Dic3—81no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23#88512yes—
493C23C2×D3×D411yescharacteristic
494Q8—121no—
495Q8—121no—
496Q8—121no—
497Q8—121no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—32no—
506C2×C4—32no—
507C2×C4—41no—
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—61no—
519C2×C4—61no—
520C2×C4—61no—
521C2×C4—61no—
522C2×C4—61no—
523C2×C4—61no—
524C2×C4—61no—
525C2×C4—61no—
526C2×C4—61no—
527C2×C4—61no—
528C2×C4—121no—
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#88512yes—
541C2×C4C2×#23511yescharacteristic
542C2×C4C2×#23511yescharacteristic
543C2×C4C2×#23511yescharacteristic
544C2×C4C2×#23711yescharacteristic
545C2×C4#100411yescharacteristic
546C2×C4#100711yescharacteristic
547C8—22no—
548C8—41no—
549C8—121no—
550C8—121no—
551C2×C3—42no—
552C2×C3#248811yescharacteristic
553C2×C3#313111yescharacteristic
554C2×C3#313711yescharacteristic
555C2×C3#330211yescharacteristic
556C2×C3#330411yescharacteristic
557C2×C3#333611yescharacteristic
558C2×C3#333711yescharacteristic
559C22—22no—
560C22—22no—
561C22—22no—
562C22—41no—
563C22—41no—
564C22—41no—
565C22—41no—
566C22#549011yescharacteristic
567C22C2×#88511yescharacteristic
568C22#550711yescharacteristic
569C22D3×#13511yescharacteristic
570C22D3×#13511yescharacteristic
571C22#577011yescharacteristic
572C22#577111yescharacteristic
573C4—22no—
574C4—42no—
575C4—62no—
576C4—62no—
577C4—81no—
578C4—81no—
579C4—81no—
580C4—121no—
581C4—121no—
582C4—241no—
583C4—241no—
584C4—241no—
585C4—241no—
586C4#547911yescharacteristic
587C4#549511yescharacteristic
588C4#551311yescharacteristic
589C4#551511yescharacteristic
590C3#2770011yescharacteristic, complemented, Sylow
591C2—42no—
592C2#7335211yescharacteristic
593C2#7882511yescharacteristic
594C2#7882811yescharacteristic
595C2#7884911yescharacteristic
596C2#7885111yescharacteristic
597C2#7918611yescharacteristic
598C2#7918711yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1735C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenter#1735D3
hypocenterC3#27700
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1735

Complements

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

NH ≅ G/Ntype
8. #68914C2semidirect
9. #68911C2semidirect
10. #68851C2semidirect
11. #68850C2semidirect
12. #68847C2semidirect
13. #68844C2semidirect
14. #68842C2semidirect
15. #68804C2semidirect
590. C3#27700semidirect

Maximal subgroups

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

subgroupindex
#822562
#822492
C3×#17352
#719722
C8×#2172
#715852
#715752
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