Finite Group Databasebeta

← #10867902

Subgroups — #10867902

Order 768.

728 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#73512C211yesmaximal, characteristic
2#73465C211yesmaximal, characteristic
3C3×#1923C211yesmaximal, characteristic
4#70488C211yesmaximal, characteristic, complemented
5#70454C211yesmaximal, characteristic, complemented
6#69936C211yesmaximal, characteristic
7#69936C211yesmaximal, characteristic
8#69928C211yesmaximal, characteristic
9#69925C211yesmaximal, characteristic
10#69769C211yesmaximal, characteristic, complemented
11C4×#816C211yesmaximal, characteristic, complemented
12#68684C211yesmaximal, characteristic, complemented
13#68636C211yesmaximal, characteristic, complemented
14#68623C211yesmaximal, characteristic, complemented
15#68622C211yesmaximal, characteristic, complemented
16#27034—31nomaximal, Sylow
17C22×C4⋊Dic3C2211yescharacteristic
18C2×#924C2211yescharacteristic
19C2×#879—21no—
20C2×C4×Dic6C2211yescharacteristic
21C2×C3×C4⋊C8C2211yescharacteristic
22C2×C3×C22⋊C8C2211yescharacteristic
23C2×C3×C22⋊C8C2211yescharacteristic
24C2×C3×C4×C8C2211yescharacteristic
25C3×C4×C22⋊C4C2211yescharacteristic
26#5212C2211yescharacteristic
27#5212C2211yescharacteristic
28C2×#817C2211yescharacteristic
29C2×#817C2211yescharacteristic
30C2×#817C2211yescharacteristic
31C2×#817C2211yescharacteristic
32C2×#816C2211yescharacteristic
33C2×#816C2211yescharacteristic
34C2×#816C2211yescharacteristic
35C2×#816C2211yescharacteristic
36#4953—21no—
37#4946—21no—
38Dic3×C22⋊C4—21no—
39#4935C2211yescharacteristic
40C4×C4⋊Dic3C2211yescharacteristic
41C4×C4⋊Dic3C2211yescharacteristic
42#4929C2211yescharacteristic
43#4928C2211yescharacteristic
44#4928C2211yescharacteristic
45#4928C2211yescharacteristic
46#4659—21no—
47#4655—21no—
48#4653—21no—
49#4647—21no—
50C3×#336C2211yescharacteristic
51C3×#336C2211yescharacteristic
52#4551C2211yescharacteristic
53#4551C2211yescharacteristic
54#4551C2211yescharacteristic
55#4551C2211yescharacteristic
56#4551C2211yescharacteristic
57#4551C2211yescharacteristic
58#4551C2211yescharacteristic
59#4551C2211yescharacteristic
60#2496—31no—
61#2360—31no—
62#2072—31no—
63#2066—31no—
64#2035—31no—
65#2035—31no—
66#1948—31no—
67#1941—31no—
68#1923D311yescharacteristic
69#1920—31no—
70#1919—31no—
71#1865—31no—
72#1853—31no—
73#1824—31no—
74C4×#103—31no—
75C23×C3×C4C2311yescharacteristic
76C23×Dic3—21no—
77C22×Dic6C2311yescharacteristic
78C22×C3×C8C2311yescharacteristic
79C22×C3×C8C2311yescharacteristic
80C22×C3×C8C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C2×C3×C22⋊C4D411yescharacteristic
83C2×C3×C22⋊C4D411yescharacteristic
84C2×C3×C42C2311yescharacteristic
85C2×C3×C42C2311yescharacteristic
86C2×#217—21no—
87C2×#217—21no—
88C2×#217—21no—
89C2×C4⋊Dic3—22no—
90C2×C4⋊Dic3—21no—
91C2×C4⋊Dic3—21no—
92C2×C4⋊Dic3—21no—
93C2×C4⋊Dic3—41no—
94C2×C4⋊Dic3C2311yescharacteristic
95C2×C4⋊Dic3C2311yescharacteristic
96C2×C4⋊Dic3C2311yescharacteristic
97C2×C4⋊Dic3C2311yescharacteristic
98C2×C4⋊Dic3C2311yescharacteristic
99C2×C4⋊Dic3C2311yescharacteristic
100C2×C4⋊Dic3C2311yescharacteristic
101#924—22no—
102#924—22no—
103C2×Dic3⋊C4—21no—
104C2×Dic3⋊C4—21no—
105C2×Dic3⋊C4—21no—
106C2×Dic3⋊C4—21no—
107C2×Dic3⋊C4—21no—
108C2×C4×Dic3—21no—
109C2×C4×Dic3—21no—
110C2×C4×Dic3—21no—
111#879—44no—
112C4×Dic6—22no—
113C4×Dic6—41no—
114C3×C4⋊C8—22no—
115C3×C22⋊C8—22no—
116C3×C22⋊C8—41no—
117C3×C4×C8—41no—
118C3×#95D411yescharacteristic
119C3×#95D411yescharacteristic
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⋊#95—21no—
129#817—22no—
130#817—22no—
131#817—22no—
132#817—41no—
133#816—22no—
134#816—22no—
135#816—22no—
136#816—41no—
137C2×#126—61no—
138C2×C22⋊Q8—31no—
139C2×C4×Q8—31no—
140C22×C4⋊C4—31no—
141C2×#106—31no—
142C2×#106—31no—
143C2×#106—31no—
144C2×#106—31no—
145C2×C4⋊C8C2×D311yescharacteristic
146C2×#103—31no—
147C2×#103—31no—
148C2×#103—31no—
149C2×#103—31no—
150C2×C22⋊C8C2×D311yescharacteristic
151C2×C22⋊C8C2×D311yescharacteristic
152C2×C4×C8C2×D311yescharacteristic
153#400—61no—
154#397—61no—
155#391—31no—
156#389—31no—
157#389—31no—
158#389—31no—
159#389—61no—
160#387—61no—
161#387—61no—
162#385—61no—
163#384—31no—
164#380—31no—
165#380—31no—
166C4×C4⋊C4—31no—
167C4×C4⋊C4—31no—
168C4×C22⋊C4—61no—
169C4×C22⋊C4C2×D311yescharacteristic
170#340—31no—
171#340—31no—
172#340—31no—
173#340—31no—
174#340—31no—
175#340—31no—
176#340—31no—
177#340—31no—
178#336C2×D311yescharacteristic
179#336C2×D311yescharacteristic
180C24×C3C2×D411yescharacteristic
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×C4C2×D411yescharacteristic
188C22×C3×C4C2×D411yescharacteristic
189C22×C3×C4C2×D411yescharacteristic
190C22×C3×C4C2×D411yescharacteristic
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2411yescharacteristic
193C22×Dic3—21no—
194C22×Dic3—21no—
195C22×Dic3—21no—
196C22×Dic3—21no—
197C22×Dic3—21no—
198C22×Dic3—21no—
199C22×Dic3—21no—
200C22×Dic3—21no—
201C22×Dic3—42no—
202C22×Dic3—42no—
203C22×Dic3—81no—
204C2×Dic6—21no—
205C2×Dic6—21no—
206C2×Dic6—21no—
207C2×Dic6—21no—
208C2×Dic6—21no—
209C2×Dic6#4112yes—
210C2×C3×C8—21no—
211C2×C3×C8—21no—
212C2×C3×C8—21no—
213C2×C3×C8—21no—
214C2×C3×C8—21no—
215C2×C3×C8—21no—
216C2×C3×C8—21no—
217C2×C3×C8—21no—
218C2×C3×C8—21no—
219C2×C3×C8—21no—
220C2×C3×C8—21no—
221C2×C3×C8#4112yes—
222C3×C22⋊C4—41no—
223C3×C22⋊C4SD1614yes—
224C3×C42—21no—
225C3×C42—21no—
226C3×C42—21no—
227C3×C42—21no—
228#217—24no—
229#217—42no—
230#217—42no—
231C4⋊Dic3—21no—
232C4⋊Dic3—21no—
233C4⋊Dic3—21no—
234C4⋊Dic3—21no—
235C4⋊Dic3—21no—
236C4⋊Dic3—21no—
237C4⋊Dic3—21no—
238C4⋊Dic3—21no—
239C4⋊Dic3—21no—
240C4⋊Dic3—21no—
241C4⋊Dic3—21no—
242C4⋊Dic3—21no—
243C4⋊Dic3—42no—
244C4⋊Dic3—42no—
245C4⋊Dic3—42no—
246C4⋊Dic3#4112yes—
247C4⋊Dic3#4112yes—
248Dic3⋊C4—24no—
249Dic3⋊C4—42no—
250Dic3⋊C4—42no—
251Dic3⋊C4—42no—
252Dic3⋊C4—81no—
253C4×Dic3—42no—
254C4×Dic3—42no—
255C4×Dic3—81no—
256C22×Q8—31no—
257C23×C4—61no—
258C23×C4C22×D311yescharacteristic
259C22×C8C22×D311yescharacteristic
260C22×C8C22×D311yescharacteristic
261C22×C8C22×D311yescharacteristic
262C22×C8C22×D311yescharacteristic
263#126—124no—
264C22⋊Q8—62no—
265C22⋊Q8—62no—
266C4×Q8—62no—
267C4×Q8—121no—
268C2×C4⋊C4—31no—
269C2×C4⋊C4—31no—
270C2×C4⋊C4—31no—
271C2×C4⋊C4—31no—
272C2×C4⋊C4—31no—
273C2×C4⋊C4—31no—
274C2×C4⋊C4—31no—
275C2×C4⋊C4—62no—
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⋊C4—61no—
284C2×C4⋊C4—121no—
285C2×C22⋊C4—61no—
286C2×C22⋊C4—61no—
287C2×C22⋊C4—61no—
288C2×C22⋊C4D1211yescharacteristic
289C2×C22⋊C4D1211yescharacteristic
290C2×C42—61no—
291C2×C42—61no—
292C2×C42—61no—
293C2×C42C22×D311yescharacteristic
294C2×C42C22×D311yescharacteristic
295#106—62no—
296#106—62no—
297#106—62no—
298#106—121no—
299C4⋊C8—22no—
300#103—62no—
301#103—62no—
302#103—62no—
303#103—121no—
304C22⋊C8—22no—
305C22⋊C8—41no—
306C4×C8—41no—
307#95—61no—
308#95—61no—
309#95—61no—
310#95—61no—
311#95—61no—
312#95—61no—
313#95—61no—
314#95—61no—
315#95—61no—
316#95D1211yescharacteristic
317#95D1211yescharacteristic
318C23×C3—21no—
319C23×C3—21no—
320C23×C3—21no—
321C23×C3—21no—
322C23×C3—21no—
323C23×C3—21no—
324C23×C3C2×SD1612yes—
325C23×C3C22×D411yescharacteristic
326C2×C3×C4—21no—
327C2×C3×C4—21no—
328C2×C3×C4—21no—
329C2×C3×C4—21no—
330C2×C3×C4—21no—
331C2×C3×C4—21no—
332C2×C3×C4—21no—
333C2×C3×C4—21no—
334C2×C3×C4—21no—
335C2×C3×C4—21no—
336C2×C3×C4—21no—
337C2×C3×C4—21no—
338C2×C3×C4—21no—
339C2×C3×C4—41no—
340C2×C3×C4—41no—
341C2×C3×C4—41no—
342C2×C3×C4—41no—
343C2×C3×C4C2×SD1612yes—
344C2×C3×C4C2×SD1612yes—
345C2×C3×C4C2×#4111yescharacteristic
346C2×C3×C4C2×#4111yescharacteristic
347C2×C3×C4C2×#4111yescharacteristic
348C2×C3×C4C2×#4111yescharacteristic
349C2×C3×C4H(5,𝔽2)11yescharacteristic
350C2×C3×C42−1+411yescharacteristic
351C2×Dic3—22no—
352C2×Dic3—22no—
353C2×Dic3—22no—
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—41no—
366C2×Dic3—41no—
367C2×Dic3—41no—
368C2×Dic3—41no—
369C2×Dic3—41no—
370C2×Dic3—41no—
371C2×Dic3—41no—
372C2×Dic3—41no—
373C2×Dic3—41no—
374C2×Dic3—41no—
375C2×Dic3—84no—
376Dic6—22no—
377Dic6—22no—
378Dic6—41no—
379Dic6—41no—
380C3×C8—22no—
381C3×C8—41no—
382C3×C8—41no—
383C3×C8—41no—
384C24C2×D1211yescharacteristic
385C2×Q8—32no—
386C2×Q8—61no—
387C2×Q8—61no—
388C2×Q8—61no—
389C2×Q8—61no—
390C2×Q8—61no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—61no—
398C22×C4—61no—
399C22×C4—61no—
400C22×C4—61no—
401C22×C4—61no—
402C22×C4—61no—
403C22×C4—61no—
404C22×C4—61no—
405C22×C4—122no—
406C22×C4—122no—
407C22×C4—241no—
408C22×C4C2×D1211yescharacteristic
409C22×C4C2×D1211yescharacteristic
410C22×C4C2×D1211yescharacteristic
411C22×C4C2×D1211yescharacteristic
412C22×C4C2×D1211yescharacteristic
413C22×C4C23×D311yescharacteristic
414C2×C8—21no—
415C2×C8—21no—
416C2×C8—21no—
417C2×C8—21no—
418C2×C8—21no—
419C2×C8—21no—
420C2×C8—21no—
421C2×C8—21no—
422C2×C8—21no—
423C2×C8—21no—
424C2×C8—21no—
425C2×C8#23712yes—
426C4⋊C4—32no—
427C4⋊C4—32no—
428C4⋊C4—64no—
429C4⋊C4—61no—
430C4⋊C4—61no—
431C4⋊C4—61no—
432C4⋊C4—61no—
433C4⋊C4—61no—
434C4⋊C4—61no—
435C4⋊C4—61no—
436C4⋊C4—61no—
437C4⋊C4—61no—
438C4⋊C4—61no—
439C4⋊C4—61no—
440C4⋊C4—61no—
441C4⋊C4—122no—
442C4⋊C4—122no—
443C4⋊C4—122no—
444C4⋊C4—122no—
445C4⋊C4—122no—
446C4⋊C4—122no—
447C4⋊C4—241no—
448C22⋊C4—41no—
449C22⋊C4—64no—
450C22⋊C4—122no—
451C22⋊C4—122no—
452C22⋊C4#20414yes—
453C42—21no—
454C42—21no—
455C42—21no—
456C42—21no—
457C42—122no—
458C42—122no—
459C42—241no—
460C22×C3—22no—
461C22×C3—22no—
462C22×C3—22no—
463C22×C3—41no—
464C22×C3—41no—
465C22×C3—41no—
466C22×C3—41no—
467C22×C3#52511yescharacteristic
468C22×C3#54611yescharacteristic
469C22×C3#54711yescharacteristic
470C22×C3C22×SD1611yescharacteristic
471C22×C3#57511yescharacteristic
472C22×C3#57711yescharacteristic
473C22×C3#57811yescharacteristic
474C3×C4—22no—
475C3×C4—22no—
476C3×C4—41no—
477C3×C4—41no—
478C3×C4—41no—
479C3×C4#52911yescharacteristic
480C3×C4#53911yescharacteristic
481C3×C4#55311yescharacteristic
482C3×C4#55611yescharacteristic
483Dic3—42no—
484Dic3—42no—
485Dic3—42no—
486Dic3—81no—
487Dic3—81no—
488Dic3—81no—
489Dic3—81no—
490Dic3—81no—
491C23—21no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23C2×#20412yes—
498C23C22×D1211yescharacteristic
499Q8—62no—
500Q8—62no—
501Q8—121no—
502Q8—121no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—62no—
521C2×C4—62no—
522C2×C4—62no—
523C2×C4—121no—
524C2×C4—121no—
525C2×C4—121no—
526C2×C4—121no—
527C2×C4—121no—
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—121no—
541C2×C4—121no—
542C2×C4—121no—
543C2×C4—121no—
544C2×C4—244no—
545C2×C4C2×#20412yes—
546C2×C4C2×#20412yes—
547C2×C4C2×#23711yescharacteristic
548C2×C4#100411yescharacteristic
549C2×C4D3×#4111yescharacteristic
550C2×C4D3×#4111yescharacteristic
551C2×C4D3×#4111yescharacteristic
552C2×C4#101011yescharacteristic
553C8—22no—
554C8—41no—
555C8—41no—
556C8—41no—
557C2×C3—42no—
558C2×C3#246611yescharacteristic
559C2×C3#310311yescharacteristic
560C2×C3#312011yescharacteristic
561C2×C3#329811yescharacteristic
562C2×C3#331111yescharacteristic
563C2×C3#333511yescharacteristic
564C2×C3#335211yescharacteristic
565C22—22no—
566C22—22no—
567C22—22no—
568C22—41no—
569C22—41no—
570C22—41no—
571C22—41no—
572C22#552411yescharacteristic
573C22#555411yescharacteristic
574C22#555511yescharacteristic
575C22C22×#20411yescharacteristic
576C22#574811yescharacteristic
577C22#575111yescharacteristic
578C22#575311yescharacteristic
579C4—22no—
580C4—22no—
581C4—41no—
582C4—41no—
583C4—41no—
584C4—122no—
585C4—122no—
586C4—122no—
587C4—241no—
588C4—241no—
589C4—241no—
590C4—241no—
591C4—241no—
592C4#560111yescharacteristic
593C4#566011yescharacteristic
594C4#568011yescharacteristic
595C4#570211yescharacteristic
596C3#2703411yescharacteristic, complemented, Sylow
597C2—42no—
598C2#7551611yescharacteristic
599C2#7930911yescharacteristic
600C2#7938511yescharacteristic
601C2#7941211yescharacteristic
602C2#7944711yescharacteristic
603C2#7950211yescharacteristic
604C2#7953011yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1923C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenter#1923D3
hypocenterC3#27034
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1923

Complements

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

NH ≅ G/Ntype
4. #70488C2semidirect
5. #70454C2semidirect
10. #69769C2semidirect
11. C4×#816C2semidirect
12. #68684C2semidirect
13. #68636C2semidirect
14. #68623C2semidirect
15. #68622C2semidirect
596. C3#27034semidirect

Maximal subgroups

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

subgroupindex
#735122
#734652
C3×#19232
#704882
#704542
#699362
#699282
#699252
#697692
C4×#8162
#686842
#686362
#686232
#686222
#270343