Finite Group Databasebeta

← #10876086

Subgroups — #10876086

Order 768.

738 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#82334C211yesmaximal, characteristic
2#74173C211yesmaximal, characteristic, complemented
3C3×#1804C211yesmaximal, characteristic
4#71612C211yesmaximal, characteristic
5#70488C211yesmaximal, characteristic, complemented
6#70448C211yesmaximal, characteristic, complemented
7#70171C211yesmaximal, characteristic
8#70171C211yesmaximal, characteristic
9#70170C211yesmaximal, characteristic
10#70152C211yesmaximal, characteristic
11#69542C211yesmaximal, characteristic, complemented
12#68915C211yesmaximal, characteristic, complemented
13#68854C211yesmaximal, characteristic, complemented
14#68845C211yesmaximal, characteristic, complemented
15#68816C211yesmaximal, characteristic, complemented
16#27579—31nomaximal, Sylow
17C2×C3×C4×D4C2211yescharacteristic
18C2×D4×Dic3C2211yescharacteristic
19C2×D4×Dic3C2211yescharacteristic
20C22×#214C2211yescharacteristic
21C2×C3×C4⋊C8C2211yescharacteristic
22C2×C3×#102C2211yescharacteristic
23C2×C3×#102C2211yescharacteristic
24C3×#382C2211yescharacteristic
25C2×#832C2211yescharacteristic
26#5210—21no—
27C2×C3⋊#95—21no—
28C2×#817C2211yescharacteristic
29C2×#816C2211yescharacteristic
30C2×#816C2211yescharacteristic
31C2×#815C2211yescharacteristic
32#4984C2211yescharacteristic
33#4977C2211yescharacteristic
34Dic3×C4⋊C4C2211yescharacteristic
35#4973C2211yescharacteristic
36#4973C2211yescharacteristic
37#4972C2211yescharacteristic
38C2×#810C2211yescharacteristic
39#4946—21no—
40#4942—21no—
41C4×Dic3⋊C4C2211yescharacteristic
42#4929C2211yescharacteristic
43#4928C2211yescharacteristic
44C3⋊#401—21no—
45#4660—21no—
46#4654—21no—
47#4651—21no—
48C3×#340C2211yescharacteristic
49C3×#340C2211yescharacteristic
50#4551C2211yescharacteristic
51#4550C2211yescharacteristic
52#4550C2211yescharacteristic
53#4550C2211yescharacteristic
54#4529C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4524C2211yescharacteristic
60#2514—31no—
61#2349—31no—
62#2101—31no—
63#2101—31no—
64#2074—31no—
65#2066—31no—
66#1935—31no—
67#1919—31no—
68#1917—31no—
69#1877—31no—
70#1877—31no—
71#1877—31no—
72#1864—31no—
73#1804D311yescharacteristic
74#1762—31no—
75C22×C3×D4C2311yescharacteristic
76C23×C3×C4—21no—
77C23×Dic3—21no—
78C23×Dic3—21no—
79C22×Dic6C2311yescharacteristic
80C22×C3×C8C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C3×C4×D4—22no—
83C3×C4×D4—41no—
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C22⋊C4—21no—
88C2×C3×C42C2311yescharacteristic
89C2×#217—21no—
90C2×#217—21no—
91D4×Dic3—22no—
92D4×Dic3—22no—
93D4×Dic3—22no—
94D4×Dic3—41no—
95C2×#214—22no—
96C2×#214—22no—
97C2×#214—22no—
98C2×C4⋊Dic3—21no—
99C2×C4⋊Dic3C2311yescharacteristic
100C2×C4⋊Dic3C2311yescharacteristic
101C2×C4⋊Dic3C2311yescharacteristic
102C2×Dic3⋊C4—21no—
103C2×Dic3⋊C4—21no—
104C2×Dic3⋊C4D411yescharacteristic
105C2×Dic3⋊C4D411yescharacteristic
106C2×C4×Dic3—21no—
107C2×C4×Dic3C2311yescharacteristic
108C2×C4×Dic3C2311yescharacteristic
109C22×C3⋊C8C2311yescharacteristic
110C22×C3⋊C8C2311yescharacteristic
111C3×C4⋊C8—41no—
112C3×#102—22no—
113C3×#102—22no—
114C3×#95—21no—
115C3×#95—21no—
116#832—22no—
117C3⋊#95—21no—
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⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95D411yescharacteristic
130C3⋊#95D411yescharacteristic
131#817—22no—
132#816—22no—
133#816—41no—
134#815—41no—
135#810—41no—
136C22×SD16—31no—
137C2×C4×D4—31no—
138C2×C4×D4—31no—
139C2×C4×D4C2×D311yescharacteristic
140C2×#106—31no—
141C2×C4⋊C8C2×D311yescharacteristic
142C2×#103—31no—
143C2×#103—31no—
144C2×#103—31no—
145C2×#102—31no—
146C2×#102C2×D311yescharacteristic
147C2×#102C2×D311yescharacteristic
148C2×#97—31no—
149#401—61no—
150#400—61no—
151#391—31no—
152#391—31no—
153#391—31no—
154#389—31no—
155#389—31no—
156#389—31no—
157#387—61no—
158#385—61no—
159#385—61no—
160#383—61no—
161#382—31no—
162#382C2×D311yescharacteristic
163#380—61no—
164C4×C4⋊C4—31no—
165C4×C4⋊C4—31no—
166C2×#95—61no—
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#336—31no—
176#336—31no—
177#336—31no—
178#336—31no—
179C24×C3—21no—
180C2×C3×D4—21no—
181C2×C3×D4—21no—
182C2×C3×D4—21no—
183C2×C3×D4#4112yes—
184C2×C3×D4#4112yes—
185C2×C3×D4#4112yes—
186C22×C3×C4—21no—
187C22×C3×C4—21no—
188C22×C3×C4—21no—
189C22×C3×C4—42no—
190C22×C3×C4—42no—
191C22×C3×C4—81no—
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4C2×D411yescharacteristic
194C22×C3×C4C2411yescharacteristic
195C22×Dic3—21no—
196C22×Dic3—24no—
197C22×Dic3—21no—
198C22×Dic3—21no—
199C22×Dic3—21no—
200C22×Dic3—42no—
201C22×Dic3—42no—
202C22×Dic3—81no—
203C22×Dic3—81no—
204C22×Dic3—81no—
205C22×Dic3C2×D411yescharacteristic
206C22×Dic3C2×D411yescharacteristic
207C22×Dic3C2×D411yescharacteristic
208C22×Dic3C2×D411yescharacteristic
209C2×Dic6—21no—
210C2×Dic6—21no—
211C2×Dic6—21no—
212C2×Dic6—21no—
213C2×Dic6—21no—
214C2×Dic6—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—
221C3×C4⋊C4—21no—
222C3×C4⋊C4—21no—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—21no—
225C3×C4⋊C4—21no—
226C3×C4⋊C4—21no—
227C3×C22⋊C4—42no—
228C3×C42—21no—
229C3×C42—21no—
230#217—24no—
231#217—42no—
232#214—44no—
233C4⋊Dic3—21no—
234C4⋊Dic3—21no—
235C4⋊Dic3—21no—
236C4⋊Dic3—21no—
237C4⋊Dic3—21no—
238C4⋊Dic3—42no—
239C4⋊Dic3#4112yes—
240Dic3⋊C4—41no—
241Dic3⋊C4—41no—
242Dic3⋊C4—81no—
243Dic3⋊C4—81no—
244C4×Dic3—21no—
245C4×Dic3—21no—
246C4×Dic3—21no—
247C4×Dic3—21no—
248C4×Dic3—42no—
249C2×C3⋊C8—21no—
250C2×C3⋊C8—21no—
251C2×C3⋊C8—21no—
252C2×C3⋊C8—21no—
253C2×C3⋊C8—21no—
254C2×C3⋊C8—21no—
255C22×Q8—31no—
256C22×D4C22×D311yescharacteristic
257C23×C4—21no—
258C23×C4—61no—
259C23×C4—61no—
260C2×SD16—62no—
261C2×SD16—62no—
262C2×SD16—62no—
263C22×C8—31no—
264C22×C8—31no—
265C22×C8C22×D311yescharacteristic
266C22×C8C22×D311yescharacteristic
267C4×D4—22no—
268C4×D4—41no—
269C4×D4—62no—
270C4×D4—62no—
271C4×D4—62no—
272C4×D4—121no—
273C2×C4⋊C4—31no—
274C2×C4⋊C4—31no—
275C2×C4⋊C4—31no—
276C2×C4⋊C4—31no—
277C2×C4⋊C4—31no—
278C2×C4⋊C4—61no—
279C2×C4⋊C4—61no—
280C2×C4⋊C4—61no—
281C2×C4⋊C4C22×D311yescharacteristic
282C2×C4⋊C4C22×D311yescharacteristic
283C2×C4⋊C4C22×D311yescharacteristic
284C2×C22⋊C4—21no—
285C2×C22⋊C4—61no—
286C2×C22⋊C4—61no—
287C2×C42—31no—
288C2×C42—31no—
289C2×C42—61no—
290C2×C42C22×D311yescharacteristic
291#106—62no—
292C4⋊C8—41no—
293#103—62no—
294#103—121no—
295#103—121no—
296#102—22no—
297#102—22no—
298#102—62no—
299#97—121no—
300#95—21no—
301#95—21no—
302#95—31no—
303#95—31no—
304#95—61no—
305#95—61no—
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—
315#95—61no—
316C23×C3—22no—
317C23×C3—22no—
318C23×C3—22no—
319C23×C3—41no—
320C23×C3—41no—
321C23×C3—41no—
322C23×C3—41no—
323C23×C3C22×D411yescharacteristic
324C3×D4—22no—
325C3×D4—22no—
326C3×D4—22no—
327C3×D4#12614yes—
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—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—41no—
342C2×C3×C4—41no—
343C2×C3×C4—84no—
344C2×C3×C4C2×#4111yescharacteristic
345C2×C3×C4C2×#4111yescharacteristic
346C2×C3×C4C2×#4111yescharacteristic
347C2×C3×C4C2×#4111yescharacteristic
348C2×C3×C4H(5,𝔽2)11yescharacteristic
349C2×C3×C42−1+411yescharacteristic
350C2×Dic3—21no—
351C2×Dic3—21no—
352C2×Dic3—22no—
353C2×Dic3—21no—
354C2×Dic3—21no—
355C2×Dic3—21no—
356C2×Dic3—21no—
357C2×Dic3—21no—
358C2×Dic3—21no—
359C2×Dic3—21no—
360C2×Dic3—21no—
361C2×Dic3—21no—
362C2×Dic3—21no—
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—84no—
375C2×Dic3—84no—
376Dic6—41no—
377Dic6—41no—
378Dic6—41no—
379Dic6—41no—
380C3×C8—41no—
381C3×C8—41no—
382C3⋊C8—41no—
383C3⋊C8—41no—
384C24—21no—
385C2×Q8—61no—
386C2×Q8—61no—
387C2×Q8—61no—
388C2×Q8—61no—
389C2×Q8—61no—
390C2×Q8—61no—
391C2×D4—21no—
392C2×D4—21no—
393C2×D4—21no—
394C2×D4#23512yes—
395C2×D4#23712yes—
396C2×D4#23912yes—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—31no—
401C22×C4—31no—
402C22×C4—31no—
403C22×C4—31no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—61no—
407C22×C4—64no—
408C22×C4—61no—
409C22×C4—61no—
410C22×C4—61no—
411C22×C4—81no—
412C22×C4—122no—
413C22×C4—122no—
414C22×C4—241no—
415C22×C4—241no—
416C22×C4—241no—
417C22×C4D3×D411yescharacteristic
418C22×C4D3×D411yescharacteristic
419C22×C4C23×D311yescharacteristic
420SD16—124no—
421C2×C8—21no—
422C2×C8—21no—
423C2×C8—21no—
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—21no—
427C2×C8—61no—
428C2×C8—61no—
429C2×C8—61no—
430C2×C8—61no—
431C2×C8—61no—
432C2×C8—61no—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—32no—
440C4⋊C4—61no—
441C4⋊C4—61no—
442C4⋊C4—61no—
443C4⋊C4—61no—
444C4⋊C4—61no—
445C4⋊C4—121no—
446C4⋊C4—121no—
447C4⋊C4—122no—
448C4⋊C4—241no—
449C4⋊C4—241no—
450C22⋊C4—42no—
451C22⋊C4—64no—
452C22⋊C4—122no—
453C42—21no—
454C42—21no—
455C42—61no—
456C42—61no—
457C42—61no—
458C42—61no—
459C42—122no—
460C22×C3—24no—
461C22×C3—42no—
462C22×C3—42no—
463C22×C3—42no—
464C22×C3—42no—
465C22×C3—42no—
466C22×C3—42no—
467C22×C3#52511yescharacteristic
468C22×C3#54611yescharacteristic
469C22×C3#54811yescharacteristic
470C22×C3#57511yescharacteristic
471C22×C3#57511yescharacteristic
472C22×C3#57711yescharacteristic
473C22×C3#57711yescharacteristic
474C3×C4—41no—
475C3×C4—41no—
476C3×C4—81no—
477C3×C4—81no—
478C3×C4—81no—
479C3×C4C2×#12611yescharacteristic
480C3×C4#53911yescharacteristic
481C3×C4#55211yescharacteristic
482C3×C4#55311yescharacteristic
483Dic3—41no—
484Dic3—41no—
485Dic3—42no—
486Dic3—41no—
487Dic3—41no—
488Dic3—81no—
489Dic3—81no—
490Dic3—81no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23C2×D3×D411yescharacteristic
499Q8—121no—
500Q8—121no—
501Q8—121no—
502Q8—121no—
503D4—22no—
504D4—22no—
505D4—22no—
506D4#89814yes—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—61no—
523C2×C4—61no—
524C2×C4—61no—
525C2×C4—61no—
526C2×C4—61no—
527C2×C4—61no—
528C2×C4—61no—
529C2×C4—61no—
530C2×C4—61no—
531C2×C4—61no—
532C2×C4—61no—
533C2×C4—61no—
534C2×C4—62no—
535C2×C4—84no—
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—121no—
545C2×C4—121no—
546C2×C4—121no—
547C2×C4—244no—
548C2×C4—244no—
549C2×C4C2×#23511yescharacteristic
550C2×C4C2×#23711yescharacteristic
551C2×C4#100411yescharacteristic
552C2×C4C2×#23911yescharacteristic
553C2×C4D3×#4111yescharacteristic
554C2×C4#101011yescharacteristic
555C8—41no—
556C8—41no—
557C8—121no—
558C8—121no—
559C2×C3—44no—
560C2×C3#248111yescharacteristic
561C2×C3#311111yescharacteristic
562C2×C3#312311yescharacteristic
563C2×C3#331111yescharacteristic
564C2×C3#332311yescharacteristic
565C2×C3#337611yescharacteristic
566C2×C3#337611yescharacteristic
567C22—24no—
568C22—42no—
569C22—42no—
570C22—42no—
571C22—42no—
572C22—42no—
573C22—42no—
574C22#549011yescharacteristic
575C22#559811yescharacteristic
576C22#562711yescharacteristic
577C22#576811yescharacteristic
578C22#577011yescharacteristic
579C22#577311yescharacteristic
580C22#577511yescharacteristic
581C4—41no—
582C4—41no—
583C4—81no—
584C4—81no—
585C4—81no—
586C4—121no—
587C4—121no—
588C4—121no—
589C4—121no—
590C4—122no—
591C4—241no—
592C4—241no—
593C4—241no—
594C4C2×#89811yescharacteristic
595C4#553111yescharacteristic
596C4#560111yescharacteristic
597C4#564511yescharacteristic
598C3#2757911yescharacteristic, complemented, Sylow
599C2—44no—
600C2#7391011yescharacteristic
601C2#7885511yescharacteristic
602C2#7893911yescharacteristic
603C2#8052211yescharacteristic
604C2#8052311yescharacteristic
605C2#8114211yescharacteristic
606C2#8114711yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1804C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
hypercenter#1804D3
hypocenterC3#27579
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1804

Complements

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

NH ≅ G/Ntype
2. #74173C2semidirect
5. #70488C2semidirect
6. #70448C2semidirect
11. #69542C2semidirect
12. #68915C2semidirect
13. #68854C2semidirect
14. #68845C2semidirect
15. #68816C2semidirect
598. C3#27579semidirect

Maximal subgroups

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

subgroupindex
#823342
#741732
C3×#18042
#716122
#704882
#704482
#701712
#701702
#701522
#695422
#689152
#688542
#688452
#688162
#275793