Finite Group Databasebeta

← #10868296

Subgroups — #10868296

Order 768.

680 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82464C211yesmaximal, characteristic
2#76239C211yesmaximal, characteristic
3C3×#1865C211yesmaximal, characteristic, complemented
4#71631C211yesmaximal, characteristic, complemented
5#70580C211yesmaximal, characteristic
6#70558C211yesmaximal, characteristic, complemented
7#70274C211yesmaximal, characteristic
8#70262C211yesmaximal, characteristic, complemented
9#70243C211yesmaximal, characteristic
10#70233C211yesmaximal, characteristic, complemented
11#69621C211yesmaximal, characteristic, complemented
12#68813C211yesmaximal, characteristic
13#68796C211yesmaximal, characteristic, complemented
14#68658C211yesmaximal, characteristic
15#68623C211yesmaximal, characteristic, complemented
16#27060—31nomaximal, Sylow
17C2×C3×C4×Q8C2211yescharacteristic
18C2×#946C2211yescharacteristic
19C2×#892C2211yescharacteristic
20C2×C3×#106C2211yescharacteristic
21C2×C3×#103C2211yescharacteristic
22C2×C3×#103C2211yescharacteristic
23C3×#389C2211yescharacteristic
24#5229C2211yescharacteristic
25C2×#835C2211yescharacteristic
26C2×#835C2211yescharacteristic
27C2×(C2×D3)⋊C8C2211yescharacteristic
28C2×(C2×D3)⋊C8C2211yescharacteristic
29C2×#817C2211yescharacteristic
30C2×Dic3⋊C8C2211yescharacteristic
31#4994—21no—
32#4990—21no—
33#4987C2211yescharacteristic
34#4987C2211yescharacteristic
35#4985—21no—
36#4984C2211yescharacteristic
37#4978—21no—
38#4975C2211yescharacteristic
39Dic3×C4⋊C4C2211yescharacteristic
40C2×#808C2211yescharacteristic
41C2×#807C2211yescharacteristic
42#4940—21no—
43C4×(C2×D3)⋊C4C2211yescharacteristic
44C4×C4⋊Dic3C2211yescharacteristic
45C4×Dic3⋊C4—21no—
46#4928C2211yescharacteristic
47C2×#803C2211yescharacteristic
48#4669—21no—
49#4654—21no—
50C3×#340C2211yescharacteristic
51C3×#336C2211yescharacteristic
52#4551C2211yescharacteristic
53#4551C2211yescharacteristic
54#4551C2211yescharacteristic
55#4550C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#2497—31no—
61#2362—31no—
62#2072—31no—
63#2068—31no—
64#2035—31no—
65#2035—31no—
66#1948—31no—
67#1944—31no—
68#1924—31no—
69#1920—31no—
70#1920—31no—
71#1865D311yescharacteristic, complemented
72#1854—31no—
73#1825—31no—
74#1763—31no—
75C22×C3×Q8C2311yescharacteristic
76C22×C4×D3C2311yescharacteristic
77C22×C3×C8C2311yescharacteristic
78C22×C3×C8C2311yescharacteristic
79C3×C4×Q8—22no—
80C3×C4×Q8—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×C42—21no—
87C2×C3×C42C2311yescharacteristic
88#946—22no—
89#946—22no—
90C2×(C2×D3)⋊C4—21no—
91C2×(C2×D3)⋊C4—21no—
92C2×(C2×D3)⋊C4—21no—
93C2×(C2×D3)⋊C4—21no—
94C2×(C2×D3)⋊C4D411yescharacteristic
95C2×(C2×D3)⋊C4D411yescharacteristic
96C2×C4⋊Dic3—21no—
97C2×C4⋊Dic3C2311yescharacteristic
98C2×C4⋊Dic3C2311yescharacteristic
99C2×C4⋊Dic3C2311yescharacteristic
100C2×C4⋊Dic3C2311yescharacteristic
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×Dic3C2311yescharacteristic
109C22×C3⋊C8C2311yescharacteristic
110C22×C3⋊C8C2311yescharacteristic
111#892—22no—
112#892—41no—
113C3×#106—22no—
114C3×#103—22no—
115C3×#103—41no—
116C3×#95—21no—
117#835—22no—
118#835—22no—
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(C2×D3)⋊C8—22no—
130(C2×D3)⋊C8—41no—
131#817—22no—
132Dic3⋊C8—22no—
133#808—22no—
134#807—41no—
135#803—41no—
136C2×C22⋊Q8—31no—
137C2×C4×Q8C2×D311yescharacteristic
138C2×#117—31no—
139C2×#107—31no—
140C2×#106—31no—
141C2×#106—31no—
142C2×#106C2×D311yescharacteristic
143C2×C4⋊C8—31no—
144C2×#103—31no—
145C2×#103—31no—
146C2×#103C2×D311yescharacteristic
147C2×#103C2×D311yescharacteristic
148C2×C22⋊C8—31no—
149C2×C22⋊C8—31no—
150C2×#97—31no—
151#400—61no—
152#396—61no—
153#391—31no—
154#389—31no—
155#389—31no—
156#389C2×D311yescharacteristic
157#388—61no—
158#388—61no—
159#388—61no—
160#387—61no—
161#387—61no—
162#384—31no—
163#380—31no—
164#380—31no—
165C4×C4⋊C4—31no—
166C4×C4⋊C4—31no—
167C4×C4⋊C4—61no—
168C4×C22⋊C4—31no—
169#340—31no—
170#340—31no—
171#340—31no—
172#340—31no—
173#340—31no—
174#340—31no—
175#340—31no—
176#340C2×D311yescharacteristic
177#336—31no—
178#336C2×D311yescharacteristic
179C23×D3C2×D411yescharacteristic
180C2×C3×Q8—21no—
181C2×C3×Q8—21no—
182C2×C3×Q8—21no—
183C2×C3×Q8—21no—
184C2×C3×Q8—21no—
185C2×C3×Q8#4112yes—
186C22×C3×C4—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
200C2×C4×D3—21no—
201C2×C4×D3—21no—
202C2×C4×D3—21no—
203C2×C4×D3—21no—
204C2×C4×D3—21no—
205C2×C4×D3—21no—
206C2×C3×C8—21no—
207C2×C3×C8—21no—
208C2×C3×C8—21no—
209C2×C3×C8—21no—
210C2×C3×C8—21no—
211C2×C3×C8—21no—
212C3×C4⋊C4—21no—
213C3×C4⋊C4—21no—
214C3×C4⋊C4—21no—
215C3×C4⋊C4—21no—
216C3×C4⋊C4—21no—
217C3×C4⋊C4—42no—
218C3×C4⋊C4—81no—
219C3×C4⋊C4#4112yes—
220C3×C42—21no—
221C3×C42—21no—
222C3×C42—42no—
223(C2×D3)⋊C4—24no—
224(C2×D3)⋊C4—22no—
225(C2×D3)⋊C4—41no—
226(C2×D3)⋊C4—42no—
227(C2×D3)⋊C4—81no—
228(C2×D3)⋊C4—81no—
229C4⋊Dic3—21no—
230C4⋊Dic3—21no—
231C4⋊Dic3—21no—
232C4⋊Dic3—21no—
233C4⋊Dic3—21no—
234C4⋊Dic3—21no—
235C4⋊Dic3—21no—
236C4⋊Dic3—42no—
237C4⋊Dic3#4112yes—
238Dic3⋊C4—24no—
239Dic3⋊C4—42no—
240Dic3⋊C4—81no—
241Dic3⋊C4—81no—
242C4×Dic3—21no—
243C4×Dic3—21no—
244C4×Dic3—42no—
245C4×Dic3—42no—
246C4×Dic3—81no—
247C2×C3⋊C8—21no—
248C2×C3⋊C8—21no—
249C2×C3⋊C8—21no—
250C2×C3⋊C8—21no—
251C2×C3⋊C8—21no—
252C2×C3⋊C8#4112yes—
253C22×Q8C22×D311yescharacteristic
254C23×C4—31no—
255C22×C8—31no—
256C22×C8—31no—
257C22×C8C22×D311yescharacteristic
258C22×C8C22×D311yescharacteristic
259C22⋊Q8—62no—
260C22⋊Q8—62no—
261C4×Q8—22no—
262C4×Q8—41no—
263#117—62no—
264#117—121no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—31no—
268C2×C4⋊C4—31no—
269C2×C4⋊C4—31no—
270C2×C4⋊C4—31no—
271C2×C4⋊C4—61no—
272C2×C4⋊C4—61no—
273C2×C4⋊C4—61no—
274C2×C4⋊C4—61no—
275C2×C4⋊C4—61no—
276C2×C4⋊C4C22×D311yescharacteristic
277C2×C4⋊C4C22×D311yescharacteristic
278C2×C4⋊C4C22×D311yescharacteristic
279C2×C22⋊C4—31no—
280C2×C22⋊C4—31no—
281C2×C22⋊C4—61no—
282C2×C22⋊C4—61no—
283C2×C22⋊C4—61no—
284C2×C22⋊C4—61no—
285C2×C42—21no—
286C2×C42—31no—
287C2×C42—61no—
288C2×C42—61no—
289C2×C42—61no—
290C2×C42C22×D311yescharacteristic
291#107—121no—
292#106—22no—
293#106—62no—
294#106—62no—
295C4⋊C8—62no—
296#103—22no—
297#103—41no—
298#103—62no—
299#103—62no—
300C22⋊C8—62no—
301C22⋊C8—121no—
302#97—121no—
303#95—21no—
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—
314C23×C3C22×D411yescharacteristic
315C22×D3—21no—
316C22×D3—21no—
317C22×D3—21no—
318C22×D3—21no—
319C22×D3—21no—
320C22×D3—21no—
321C22×D3—21no—
322C3×Q8—22no—
323C3×Q8—22no—
324C3×Q8—41no—
325C3×Q8—41no—
326C2×C3×C4—22no—
327C2×C3×C4—22no—
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—41no—
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—21no—
353C2×Dic3—22no—
354C2×Dic3—21no—
355C2×Dic3—21no—
356C2×Dic3—21no—
357C2×Dic3—21no—
358C2×Dic3—21no—
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#13712yes—
371C4×D3—41no—
372C4×D3—41no—
373C4×D3—41no—
374C4×D3—41no—
375C3×C8—41no—
376C3×C8—41no—
377C3⋊C8—22no—
378C3⋊C8—41no—
379C24—31no—
380C2×Q8—21no—
381C2×Q8—21no—
382C2×Q8—21no—
383C2×Q8—21no—
384C2×Q8—21no—
385C2×Q8#23512yes—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—31no—
391C22×C4—31no—
392C22×C4—31no—
393C22×C4—61no—
394C22×C4—61no—
395C22×C4—61no—
396C22×C4—61no—
397C22×C4—61no—
398C22×C4—61no—
399C22×C4—61no—
400C22×C4—61no—
401C22×C4—61no—
402C22×C4—61no—
403C22×C4D3×D411yescharacteristic
404C22×C4D3×D411yescharacteristic
405C22×C4C23×D311yescharacteristic
406C2×C8—21no—
407C2×C8—21no—
408C2×C8—21no—
409C2×C8—21no—
410C2×C8—21no—
411C2×C8—21no—
412C2×C8—32no—
413C2×C8—61no—
414C2×C8—61no—
415C2×C8—61no—
416C2×C8—61no—
417C2×C8—61no—
418C4⋊C4—21no—
419C4⋊C4—21no—
420C4⋊C4—21no—
421C4⋊C4—21no—
422C4⋊C4—21no—
423C4⋊C4—32no—
424C4⋊C4—42no—
425C4⋊C4—64no—
426C4⋊C4—61no—
427C4⋊C4—61no—
428C4⋊C4—61no—
429C4⋊C4—61no—
430C4⋊C4—61no—
431C4⋊C4—61no—
432C4⋊C4—61no—
433C4⋊C4—81no—
434C4⋊C4—122no—
435C4⋊C4—122no—
436C4⋊C4—241no—
437C4⋊C4—241no—
438C4⋊C4#23712yes—
439C22⋊C4—62no—
440C22⋊C4—64no—
441C22⋊C4—121no—
442C22⋊C4—122no—
443C22⋊C4—241no—
444C22⋊C4—241no—
445C42—21no—
446C42—21no—
447C42—42no—
448C42—61no—
449C42—61no—
450C42—122no—
451C42—122no—
452C42—241no—
453C22×C3#52511yescharacteristic
454C22×C3#54611yescharacteristic
455C22×C3#54711yescharacteristic
456C22×C3C2×#13711yescharacteristic
457C22×C3#57511yescharacteristic
458C22×C3#57711yescharacteristic
459C22×C3#57711yescharacteristic
460C2×D3—41no—
461C2×D3—41no—
462C2×D3—41no—
463C2×D3—41no—
464C2×D3—41no—
465C2×D3—41no—
466C2×D3—41no—
467C3×C4—41no—
468C3×C4—42no—
469C3×C4—42no—
470C3×C4—41no—
471C3×C4—81no—
472C3×C4—81no—
473C3×C4#52911yescharacteristic
474C3×C4#53911yescharacteristic
475C3×C4#55311yescharacteristic
476C3×C4#55611yescharacteristic
477Dic3—22no—
478Dic3—41no—
479Dic3—42no—
480Dic3—41no—
481Dic3—81no—
482Dic3—81no—
483Dic3—81no—
484C23—61no—
485C23—61no—
486C23—61no—
487C23—61no—
488C23—61no—
489C23—61no—
490C23—61no—
491C23C2×D3×D411yescharacteristic
492Q8—22no—
493Q8—22no—
494Q8—41no—
495Q8—41no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—22no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—22no—
504C2×C4—32no—
505C2×C4—41no—
506C2×C4—41no—
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—61no—
516C2×C4—61no—
517C2×C4—61no—
518C2×C4—61no—
519C2×C4—62no—
520C2×C4—61no—
521C2×C4—61no—
522C2×C4—61no—
523C2×C4—61no—
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×C4C2×#23511yescharacteristic
540C2×C4C2×#23711yescharacteristic
541C2×C4#100411yescharacteristic
542C2×C4#100711yescharacteristic
543C2×C4D3×#4111yescharacteristic
544C2×C4D3×#4111yescharacteristic
545C8—41no—
546C8—41no—
547C8—62no—
548C8—121no—
549C2×C3#246611yescharacteristic
550C2×C3#311611yescharacteristic
551C2×C3#312611yescharacteristic
552C2×C3#332111yescharacteristic
553C2×C3#333111yescharacteristic
554C2×C3#334411yescharacteristic
555C2×C3#335011yescharacteristic
556D3—81no—
557C22—121no—
558C22—121no—
559C22—121no—
560C22—121no—
561C22—121no—
562C22—121no—
563C22—121no—
564C22#554711yescharacteristic
565C22#559111yescharacteristic
566C22#563111yescharacteristic
567C22#576211yescharacteristic
568C22#577011yescharacteristic
569C22D3×#13711yescharacteristic
570C22#577711yescharacteristic
571C4—41no—
572C4—42no—
573C4—41no—
574C4—42no—
575C4—62no—
576C4—81no—
577C4—81no—
578C4—121no—
579C4—121no—
580C4—122no—
581C4—241no—
582C4—241no—
583C4—241no—
584C4#551511yescharacteristic
585C4#553811yescharacteristic
586C4#564111yescharacteristic
587C4#566111yescharacteristic
588C3#2706011yescharacteristic, complemented, Sylow
589C2—241no—
590C2#7443411yescharacteristic
591C2#7966111yescharacteristic
592C2#7984811yescharacteristic
593C2#8005411yescharacteristic
594C2#8026011yescharacteristic
595C2#8103011yescharacteristic
596C2#8125811yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1865C2
socleC23×C3C22×D4
prime radicalC23×D3C2×D4
hypercenter#1865D3
hypocenterC3#27060
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1865

Complements

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

NH ≅ G/Ntype
3. C3×#1865C2semidirect
4. #71631C2semidirect
6. #70558C2semidirect
8. #70262C2semidirect
10. #70233C2semidirect
11. #69621C2semidirect
13. #68796C2semidirect
15. #68623C2semidirect
71. #1865D3semidirect
588. C3#27060semidirect

Maximal subgroups

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

subgroupindex
#824642
#762392
C3×#18652
#716312
#705802
#705582
#702742
#702622
#702432
#702332
#696212
#688132
#687962
#686582
#686232
#270603