Finite Group Databasebeta

← #10876059

Subgroups — #10876059

Order 768.

724 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#76339C211yesmaximal, characteristic
2#74157C211yesmaximal, characteristic, complemented
3C3×#2073C211yesmaximal, characteristic
4#70213C211yesmaximal, characteristic
5#70213C211yesmaximal, characteristic
6#70188C211yesmaximal, characteristic
7#70180C211yesmaximal, characteristic
8#69660C211yesmaximal, characteristic
9#69542C211yesmaximal, characteristic, complemented
10#68904C211yesmaximal, characteristic, complemented
11#68820C211yesmaximal, characteristic, complemented
12#68810C211yesmaximal, characteristic, complemented
13#68783C211yesmaximal, characteristic, complemented
14#68683C211yesmaximal, characteristic, complemented
15#68613C211yesmaximal, characteristic, complemented
16#27578—31nomaximal, Sylow
17C22×#214C2211yescharacteristic
18C2×#887C2211yescharacteristic
19C2×#887C2211yescharacteristic
20C2×#879—21no—
21C2×C4×Dic6C2211yescharacteristic
22C2×C3×#102C2211yescharacteristic
23C2×C3×#102C2211yescharacteristic
24C3×#390C2211yescharacteristic
25C3×#382C2211yescharacteristic
26#5221C2211yescharacteristic
27#5221C2211yescharacteristic
28C2×#832C2211yescharacteristic
29C2×#818C2211yescharacteristic
30C2×#816C2211yescharacteristic
31C2×#816C2211yescharacteristic
32C2×#815C2211yescharacteristic
33C2×Dic3⋊C8C2211yescharacteristic
34#4975C2211yescharacteristic
35Dic3×C4⋊C4C2211yescharacteristic
36#4972C2211yescharacteristic
37#4972C2211yescharacteristic
38C2×#810C2211yescharacteristic
39#4952—21no—
40#4948—21no—
41#4945—21no—
42C4×C4⋊Dic3C2211yescharacteristic
43#4928C2211yescharacteristic
44#4650—21no—
45#4649—21no—
46#4647—21no—
47#4647—21no—
48C3×#340C2211yescharacteristic
49C3×#340C2211yescharacteristic
50C3×#336C2211yescharacteristic
51#4551C2211yescharacteristic
52#4550C2211yescharacteristic
53#4550C2211yescharacteristic
54#4529C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4524C2211yescharacteristic
60#2512—31no—
61#2351—31no—
62#2101—31no—
63#2099—31no—
64#2073D311yescharacteristic
65#2067—31no—
66#1936—31no—
67#1921—31no—
68#1917—31no—
69#1880—31no—
70#1880—31no—
71#1880—31no—
72#1863—31no—
73#1803—31no—
74#1763—31no—
75C22×C3×D4C2311yescharacteristic
76C22×Dic6C2311yescharacteristic
77C22×C3×C8C2311yescharacteristic
78C22×C3×C8C2311yescharacteristic
79C2×C3×C4⋊C4C2311yescharacteristic
80C2×C3×C4⋊C4C2311yescharacteristic
81C2×C3×C4⋊C4C2311yescharacteristic
82C2×C3×C22⋊C4—21no—
83C2×C3×C22⋊C4—21no—
84C2×C3×C42C2311yescharacteristic
85C2×#217—21no—
86C2×#217—21no—
87C2×#217—21no—
88C2×#217—21no—
89C2×#214—22no—
90C2×#214—22no—
91C2×#214—22no—
92C2×C4⋊Dic3D411yescharacteristic
93C2×C4⋊Dic3D411yescharacteristic
94C2×C4⋊Dic3C2311yescharacteristic
95C2×C4⋊Dic3C2311yescharacteristic
96C2×C4⋊Dic3C2311yescharacteristic
97C2×Dic3⋊C4—21no—
98C2×Dic3⋊C4—21no—
99C2×Dic3⋊C4—21no—
100C2×Dic3⋊C4—21no—
101C2×Dic3⋊C4—21no—
102C2×Dic3⋊C4—21no—
103C2×Dic3⋊C4—21no—
104C2×Dic3⋊C4—21no—
105C2×C4×Dic3—21no—
106C2×C4×Dic3—21no—
107C2×C4×Dic3—21no—
108C2×C4×Dic3—21no—
109C2×C4×Dic3C2311yescharacteristic
110C2×C4×Dic3C2311yescharacteristic
111C22×C3⋊C8C2311yescharacteristic
112C22×C3⋊C8C2311yescharacteristic
113#887—22no—
114#887—22no—
115#887—41no—
116#887—41no—
117#879—44no—
118C4×Dic6—22no—
119C4×Dic6—22no—
120C3×#102—41no—
121C3×#102—41no—
122C3×#95—21no—
123C3×#95—21no—
124#832—22no—
125C3⋊#95—21no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95D411yescharacteristic
130C3⋊#95D411yescharacteristic
131#818—22no—
132#816—22no—
133#816—22no—
134#815—41no—
135Dic3⋊C8—41no—
136#810—22no—
137C22×SD16—31no—
138C2×#126—61no—
139C2×C4×Q8—31no—
140C2×C4×Q8—31no—
141C2×C4×Q8—31no—
142C2×#107—31no—
143C2×C4⋊C8—31no—
144C2×#103—31no—
145C2×#103—31no—
146C2×#103—31no—
147C2×#102—31no—
148C2×#102C2×D311yescharacteristic
149C2×#102C2×D311yescharacteristic
150C2×#97—31no—
151#397—61no—
152#396—61no—
153#390—31no—
154#390—31no—
155#390C2×D311yescharacteristic
156#389—31no—
157#389—31no—
158#389—31no—
159#389—61no—
160#389—61no—
161#388—61no—
162#384—31no—
163#382—61no—
164#382C2×D311yescharacteristic
165C4×C4⋊C4—31no—
166C4×C4⋊C4—31no—
167C4×C4⋊C4—61no—
168#340—31no—
169#340—31no—
170#340—31no—
171#340—31no—
172#340—31no—
173#340—31no—
174#340C2×D311yescharacteristic
175#340C2×D311yescharacteristic
176#336—31no—
177#336—31no—
178#336—31no—
179#336C2×D311yescharacteristic
180C24×C3—21no—
181C2×C3×D4—21no—
182C2×C3×D4—21no—
183C2×C3×D4—21no—
184C2×C3×D4—21no—
185C2×C3×D4—21no—
186C2×C3×D4—21no—
187C22×C3×C4—21no—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4C2×D411yescharacteristic
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2411yescharacteristic
193C22×Dic3—21no—
194C22×Dic3—21no—
195C22×Dic3—21no—
196C22×Dic3—21no—
197C22×Dic3C2×D411yescharacteristic
198C22×Dic3C2×D411yescharacteristic
199C22×Dic3C2×D411yescharacteristic
200C22×Dic3C2×D411yescharacteristic
201C2×Dic6—21no—
202C2×Dic6—21no—
203C2×Dic6—21no—
204C2×Dic6#4112yes—
205C2×Dic6#4112yes—
206C2×Dic6#4112yes—
207C2×C3×C8—21no—
208C2×C3×C8—21no—
209C2×C3×C8—21no—
210C2×C3×C8—21no—
211C2×C3×C8—21no—
212C2×C3×C8—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×C22⋊C4—42no—
220C3×C22⋊C4—81no—
221C3×C42—21no—
222C3×C42—21no—
223#217—42no—
224#217—42no—
225#217—42no—
226#217—81no—
227#214—44no—
228C4⋊Dic3—21no—
229C4⋊Dic3—22no—
230C4⋊Dic3—21no—
231C4⋊Dic3—21no—
232C4⋊Dic3—21no—
233C4⋊Dic3—21no—
234C4⋊Dic3—41no—
235C4⋊Dic3#4112yes—
236Dic3⋊C4—24no—
237Dic3⋊C4—42no—
238Dic3⋊C4—42no—
239Dic3⋊C4—42no—
240Dic3⋊C4—42no—
241Dic3⋊C4—42no—
242Dic3⋊C4—42no—
243Dic3⋊C4—81no—
244C4×Dic3—24no—
245C4×Dic3—21no—
246C4×Dic3—21no—
247C4×Dic3—21no—
248C4×Dic3—21no—
249C4×Dic3—42no—
250C4×Dic3—42no—
251C4×Dic3—42no—
252C2×C3⋊C8—21no—
253C2×C3⋊C8—21no—
254C2×C3⋊C8—21no—
255C2×C3⋊C8—21no—
256C2×C3⋊C8—21no—
257C2×C3⋊C8—21no—
258C22×Q8—31no—
259C22×D4C22×D311yescharacteristic
260C2×SD16—62no—
261C2×SD16—62no—
262C2×SD16—62no—
263C22×C8—31no—
264C22×C8—31no—
265C22×C8C22×D311yescharacteristic
266C22×C8C22×D311yescharacteristic
267#126—124no—
268C4×Q8—62no—
269C4×Q8—62no—
270C4×Q8—62no—
271C4×Q8—62no—
272C4×Q8—121no—
273C4×Q8—121no—
274C2×C4⋊C4—31no—
275C2×C4⋊C4—31no—
276C2×C4⋊C4—31no—
277C2×C4⋊C4—31no—
278C2×C4⋊C4—31no—
279C2×C4⋊C4—61no—
280C2×C4⋊C4—61no—
281C2×C4⋊C4—61no—
282C2×C4⋊C4—61no—
283C2×C4⋊C4—61no—
284C2×C4⋊C4—61no—
285C2×C4⋊C4—61no—
286C2×C4⋊C4—61no—
287C2×C4⋊C4C22×D311yescharacteristic
288C2×C4⋊C4C22×D311yescharacteristic
289C2×C4⋊C4C22×D311yescharacteristic
290C2×C22⋊C4—21no—
291C2×C22⋊C4—21no—
292C2×C22⋊C4—61no—
293C2×C22⋊C4—61no—
294C2×C22⋊C4—61no—
295C2×C22⋊C4—61no—
296C2×C42—31no—
297C2×C42—31no—
298C2×C42—61no—
299C2×C42—61no—
300C2×C42—61no—
301C2×C42—61no—
302C2×C42C22×D311yescharacteristic
303#107—62no—
304C4⋊C8—121no—
305#103—62no—
306#103—62no—
307#103—62no—
308#102—41no—
309#102—41no—
310#102—62no—
311#97—121no—
312#95—21no—
313#95—21no—
314#95—31no—
315#95—31no—
316#95—61no—
317#95—61no—
318#95—61no—
319#95—61no—
320C23×C3—41no—
321C23×C3—41no—
322C23×C3—41no—
323C23×C3—41no—
324C23×C3—41no—
325C23×C3—41no—
326C23×C3—41no—
327C23×C3C22×D411yescharacteristic
328C3×D4—41no—
329C3×D4—41no—
330C3×D4—41no—
331C3×D4—41no—
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—41no—
339C2×C3×C4—41no—
340C2×C3×C4—41no—
341C2×C3×C4—41no—
342C2×C3×C4—41no—
343C2×C3×C4—41no—
344C2×C3×C4—41no—
345C2×C3×C4—41no—
346C2×C3×C4—41no—
347C2×C3×C4C2×#4111yescharacteristic
348C2×C3×C4C2×#4111yescharacteristic
349C2×C3×C4C2×#4111yescharacteristic
350C2×C3×C4C2×#4111yescharacteristic
351C2×C3×C4H(5,𝔽2)11yescharacteristic
352C2×C3×C42−1+411yescharacteristic
353C2×Dic3—22no—
354C2×Dic3—21no—
355C2×Dic3—22no—
356C2×Dic3—22no—
357C2×Dic3—21no—
358C2×Dic3—22no—
359C2×Dic3—21no—
360C2×Dic3—21no—
361C2×Dic3—21no—
362C2×Dic3—21no—
363C2×Dic3—21no—
364C2×Dic3—21no—
365C2×Dic3—21no—
366C2×Dic3—21no—
367C2×Dic3—21no—
368C2×Dic3—41no—
369C2×Dic3—41no—
370C2×Dic3—41no—
371C2×Dic3—41no—
372C2×Dic3—41no—
373C2×Dic3—41no—
374C2×Dic3—41no—
375C2×Dic3—41no—
376C2×Dic3AGL(1,ℤ/8)12yes—
377Dic6—22no—
378Dic6—22no—
379Dic6—22no—
380Dic6#12614yescomplemented
381C3×C8—41no—
382C3×C8—41no—
383C3⋊C8—41no—
384C3⋊C8—41no—
385C24—21no—
386C2×Q8—32no—
387C2×Q8—32no—
388C2×Q8—32no—
389C2×Q8—61no—
390C2×Q8—61no—
391C2×Q8—61no—
392C2×D4—21no—
393C2×D4—21no—
394C2×D4—21no—
395C2×D4—21no—
396C2×D4—21no—
397C2×D4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—31no—
402C22×C4—31no—
403C22×C4—31no—
404C22×C4—31no—
405C22×C4—61no—
406C22×C4—61no—
407C22×C4—61no—
408C22×C4—61no—
409C22×C4D3×D411yescharacteristic
410C22×C4D3×D411yescharacteristic
411C22×C4C23×D311yescharacteristic
412SD16—124no—
413C2×C8—21no—
414C2×C8—21no—
415C2×C8—21no—
416C2×C8—21no—
417C2×C8—21no—
418C2×C8—21no—
419C2×C8—61no—
420C2×C8—61no—
421C2×C8—61no—
422C2×C8—61no—
423C2×C8—61no—
424C2×C8—61no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—32no—
432C4⋊C4—62no—
433C4⋊C4—64no—
434C4⋊C4—61no—
435C4⋊C4—61no—
436C4⋊C4—61no—
437C4⋊C4—61no—
438C4⋊C4—61no—
439C4⋊C4—121no—
440C4⋊C4—122no—
441C4⋊C4—122no—
442C4⋊C4—122no—
443C4⋊C4—122no—
444C4⋊C4—122no—
445C4⋊C4—122no—
446C4⋊C4—241no—
447C22⋊C4—42no—
448C22⋊C4—81no—
449C22⋊C4—122no—
450C22⋊C4—122no—
451C22⋊C4—122no—
452C22⋊C4—241no—
453C42—21no—
454C42—21no—
455C42—64no—
456C42—61no—
457C42—61no—
458C42—61no—
459C42—61no—
460C42—122no—
461C42—122no—
462C42—122no—
463C22×C3—81no—
464C22×C3—81no—
465C22×C3—81no—
466C22×C3—81no—
467C22×C3—81no—
468C22×C3—81no—
469C22×C3—81no—
470C22×C3#52511yescharacteristic
471C22×C3#54611yescharacteristic
472C22×C3#54811yescharacteristic
473C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
474C22×C3#57511yescharacteristic
475C22×C3#57711yescharacteristic
476C22×C3#57811yescharacteristic
477C3×C4—41no—
478C3×C4—41no—
479C3×C4—81no—
480C3×C4—81no—
481C3×C4—81no—
482C3×C4C2×#12611yescharacteristic
483C3×C4#53911yescharacteristic
484C3×C4#55211yescharacteristic
485C3×C4#55311yescharacteristic
486Dic3—24no—
487Dic3—22no—
488Dic3—42no—
489Dic3—41no—
490Dic3—41no—
491Dic3—41no—
492Dic3—81no—
493Dic3—81no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23C2×D3×D411yescharacteristic
502Q8—34no—
503Q8—62no—
504Q8—62no—
505Q8—62no—
506D4—41no—
507D4—41no—
508D4—41no—
509D4—41no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—32no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—62no—
527C2×C4—62no—
528C2×C4—61no—
529C2×C4—61no—
530C2×C4—62no—
531C2×C4—62no—
532C2×C4—61no—
533C2×C4—61no—
534C2×C4—61no—
535C2×C4—61no—
536C2×C4—61no—
537C2×C4—61no—
538C2×C4—61no—
539C2×C4—61no—
540C2×C4—61no—
541C2×C4—121no—
542C2×C4—121no—
543C2×C4—121no—
544C2×C4—121no—
545C2×C4—121no—
546C2×C4—121no—
547C2×C4—121no—
548C2×C4—121no—
549C2×C4#100411yescharacteristic
550C2×C4D3×#4111yescharacteristic
551C2×C4D3×#4111yescharacteristic
552C2×C4D3×#4111yescharacteristic
553C2×C4D3×#4111yescharacteristic
554C2×C4#101011yescharacteristic
555C8—41no—
556C8—41no—
557C8—121no—
558C8—121no—
559C2×C3—161no—
560C2×C3#248111yescharacteristic
561C2×C3#311511yescharacteristic
562C2×C3#312111yescharacteristic
563C2×C3#330811yescharacteristic
564C2×C3#332611yescharacteristic
565C2×C3#336811yescharacteristic
566C2×C3#337711yescharacteristic
567C22—81no—
568C22—81no—
569C22—81no—
570C22—81no—
571C22—81no—
572C22—81no—
573C22—81no—
574C22#561211yescharacteristic
575C22#565611yescharacteristic
576C22#567711yescharacteristic
577C22#576811yescharacteristic
578C22#577111yescharacteristic
579C22D3×AGL(1,ℤ/8)11yescharacteristic
580C22#577511yescharacteristic
581C4—41no—
582C4—41no—
583C4—64no—
584C4—62no—
585C4—81no—
586C4—81no—
587C4—81no—
588C4—121no—
589C4—121no—
590C4—121no—
591C4—122no—
592C4—241no—
593C4—241no—
594C4#566011yescharacteristic
595C4#566111yescharacteristic
596C4D3×#12611yescharacteristic
597C4#570811yescharacteristic
598C3#2757811yescharacteristic, complemented, Sylow
599C2—161no—
600C2#7748311yescharacteristic
601C2#8056211yescharacteristic
602C2#8056611yescharacteristic
603C2#8135111yescharacteristic
604C2#8135311yescharacteristic
605C2#8166311yescharacteristic
606C2#8170911yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2073C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
hypercenter#2073D3
hypocenterC3#27578
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2073

Complements

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

NH ≅ G/Ntype
2. #74157C2semidirect
9. #69542C2semidirect
10. #68904C2semidirect
11. #68820C2semidirect
12. #68810C2semidirect
13. #68783C2semidirect
14. #68683C2semidirect
15. #68613C2semidirect
380. Dic6#126semidirect
598. C3#27578semidirect

Maximal subgroups

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

subgroupindex
#763392
#741572
C3×#20732
#702132
#701882
#701802
#696602
#695422
#689042
#688202
#688102
#687832
#686832
#686132
#275783