Finite Group Databasebeta

← #10858792

Subgroups — #10858792

Order 768.

818 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2360C211yesmaximal, characteristic
2#73493C211yesmaximal, characteristic
3#71696C211yesmaximal, characteristic
4#71692C211yesmaximal, characteristic
5#71629C211yesmaximal, characteristic, complemented
6#71626C211yesmaximal, characteristic, complemented
7C2×#4529C211yesmaximal, characteristic
8#71556C211yesmaximal, characteristic
9#69682C211yesmaximal, characteristic, complemented
10#69668C211yesmaximal, characteristic, complemented
11#69611C211yesmaximal, characteristic, complemented
12#69610C211yesmaximal, characteristic, complemented
13#69490C211yesmaximal, characteristic, complemented
14#69459C211yesmaximal, characteristic, complemented
15#69449C211yesmaximal, characteristic
16#26412—31nomaximal, Sylow
17C2×C3×C22⋊Q8C2211yescharacteristic
18C2×C3×C4×Q8C2211yescharacteristic
19C22×C3×C4⋊C4C2211yescharacteristic
20C22×C4⋊Dic3C2211yescharacteristic
21C23×C3⋊C8C2211yescharacteristic
22C2×#878—21no—
23C2×#869C2211yescharacteristic
24C3×#391C2211yescharacteristic
25C3×#389—21no—
26C3×#389C2211yescharacteristic
27C3×#387—21no—
28C3×#385—21no—
29C3×C4×C4⋊C4C2211yescharacteristic
30C3×C4×C22⋊C4—21no—
31C3×C4×C22⋊C4C2211yescharacteristic
32C2×#835C2211yescharacteristic
33C2×#835C2211yescharacteristic
34C2×#835C2211yescharacteristic
35C2×#835C2211yescharacteristic
36#5212C2211yescharacteristic
37C2×#830C2211yescharacteristic
38C2×#808C2211yescharacteristic
39C2×#808C2211yescharacteristic
40#4950—21no—
41#4935C2211yescharacteristic
42#4928C2211yescharacteristic
43#4928C2211yescharacteristic
44C2×C4⋊(C3⋊C8)C2211yescharacteristic
45C2×C4⋊(C3⋊C8)C2211yescharacteristic
46#4661—21no—
47#4656—21no—
48#4529C2211yescharacteristic
49#4529C2211yescharacteristic
50#4529C2211yescharacteristic
51#4529C2211yescharacteristic
52#4529C2211yescharacteristic
53#4529C2211yescharacteristic
54#4529C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4524C2211yescharacteristic
59#4524C2211yescharacteristic
60#2440—31no—
61#2360D311yescharacteristic
62#2099—31no—
63#2088—31no—
64#2066—31no—
65#2024—31no—
66#1988—31no—
67#1919—31no—
68#1892—31no—
69#1872—31no—
70#1865—31no—
71#1853—31no—
72#1850—31no—
73#1803—31no—
74C2×#340—31no—
75C22×C3×Q8C2311yescharacteristic
76C23×C3×C4—21no—
77C23×C3×C4C2311yescharacteristic
78C23×Dic3—21no—
79C22×Dic6—21no—
80C3×C22⋊Q8—22no—
81C3×C22⋊Q8Q814yescomplemented
82C3×C4×Q8—22no—
83C3×C4×Q8—41no—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4—22no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4—41no—
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C22⋊C4—21no—
94C2×C3×C22⋊C4D411yescharacteristic
95C2×C3×C22⋊C4D411yescharacteristic
96C2×C3×C42—21no—
97C2×C3×C42—21no—
98C2×C3×C42C2311yescharacteristic
99C2×C3×C42C2311yescharacteristic
100C2×#217—21no—
101C2×C4⋊Dic3—22no—
102C2×C4⋊Dic3—21no—
103C2×C4⋊Dic3—21no—
104C2×C4⋊Dic3—21no—
105C2×C4⋊Dic3—21no—
106C2×C4⋊Dic3—41no—
107C2×C4⋊Dic3C2311yescharacteristic
108C2×C4⋊Dic3C2311yescharacteristic
109C2×C4⋊Dic3C2311yescharacteristic
110C2×C4⋊Dic3C2311yescharacteristic
111C2×Dic3⋊C4—21no—
112C2×Dic3⋊C4—21no—
113C22×C3⋊C8—22no—
114C22×C3⋊C8—41no—
115C22×C3⋊C8—41no—
116C22×C3⋊C8C2311yescharacteristic
117C22×C3⋊C8C2311yescharacteristic
118C22×C3⋊C8C2311yescharacteristic
119C22×C3⋊C8C2311yescharacteristic
120#878—44no—
121#869—22no—
122#869—41no—
123C3×#95—21no—
124C3×#95—21no—
125C3×#95—21no—
126C3×#95—21no—
127C3×#95D411yescharacteristic
128C3×#95D411yescharacteristic
129#835—22no—
130#835—41no—
131#835—41no—
132#835—41no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136#830—22no—
137#808—22no—
138#808—41no—
139C4⋊(C3⋊C8)—41no—
140C4⋊(C3⋊C8)—41no—
141C23×C8—31no—
142C2×C4⋊Q8—31no—
143C2×C22⋊Q8—61no—
144C2×C22⋊Q8C2×D311yescharacteristic
145C2×C4×Q8C2×D311yescharacteristic
146C22×C4⋊C4—31no—
147C22×C4⋊C4C2×D311yescharacteristic
148C2×#106—31no—
149C2×#106—31no—
150C2×C4⋊C8—31no—
151C2×C4⋊C8—31no—
152C2×#103—31no—
153C2×#103—31no—
154C2×#103—31no—
155C2×#103—31no—
156C2×C22⋊C8—31no—
157#398—61no—
158#395—61no—
159#391C2×D311yescharacteristic
160#389—21no—
161#389—31no—
162#389—31no—
163#389C2×D311yescharacteristic
164#387—21no—
165#385—21no—
166#385—61no—
167#384—31no—
168#380—31no—
169C4×C4⋊C4C2×D311yescharacteristic
170C4×C22⋊C4—21no—
171C4×C22⋊C4C2×D311yescharacteristic
172#340—31no—
173#340—31no—
174#340—31no—
175#340—31no—
176#340—31no—
177#340—31no—
178#340—31no—
179#340—31no—
180#340—31no—
181#340—31no—
182#336—31no—
183#336—31no—
184C24×C3C2×D411yescharacteristic
185C2×C3×Q8—21no—
186C2×C3×Q8—21no—
187C2×C3×Q8—21no—
188C2×C3×Q8—21no—
189C2×C3×Q8—21no—
190C2×C3×Q8C2×Q812yes—
191C22×C3×C4—21no—
192C22×C3×C4—21no—
193C22×C3×C4—21no—
194C22×C3×C4—21no—
195C22×C3×C4—21no—
196C22×C3×C4—21no—
197C22×C3×C4—21no—
198C22×C3×C4—21no—
199C22×C3×C4—42no—
200C22×C3×C4—42no—
201C22×C3×C4—81no—
202C22×C3×C4C2×D411yescharacteristic
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2×Q812yes—
208C22×C3×C4#4112yes—
209C22×C3×C4C2411yescharacteristic
210C22×Dic3—21no—
211C22×Dic3—21no—
212C22×Dic3—21no—
213C22×Dic3—21no—
214C22×Dic3—42no—
215C22×Dic3—42no—
216C22×Dic3—81no—
217C2×Dic6—42no—
218C2×Dic6—42no—
219C2×Dic6—81no—
220C3×C4⋊C4—24no—
221C3×C4⋊C4—21no—
222C3×C4⋊C4—21no—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—21no—
225C3×C4⋊C4—42no—
226C3×C4⋊C4—42no—
227C3×C4⋊C4—81no—
228C3×C4⋊C4C2×Q812yes—
229C3×C4⋊C4#4112yes—
230C3×C22⋊C4—24no—
231C3×C22⋊C4—22no—
232C3×C22⋊C4SD1614yes—
233C3×C42—21no—
234C3×C42—21no—
235C3×C42—21no—
236C3×C42—21no—
237C3×C42—42no—
238C3×C42—81no—
239#217—42no—
240C4⋊Dic3—21no—
241C4⋊Dic3—21no—
242C4⋊Dic3—21no—
243C4⋊Dic3—21no—
244C4⋊Dic3—21no—
245C4⋊Dic3—21no—
246C4⋊Dic3—21no—
247C4⋊Dic3—42no—
248C4⋊Dic3—42no—
249C4⋊Dic3—42no—
250C4⋊Dic3—81no—
251C4⋊Dic3#4112yes—
252Dic3⋊C4—42no—
253Dic3⋊C4—42no—
254C2×C3⋊C8—21no—
255C2×C3⋊C8—21no—
256C2×C3⋊C8—21no—
257C2×C3⋊C8—21no—
258C2×C3⋊C8—21no—
259C2×C3⋊C8—21no—
260C2×C3⋊C8—21no—
261C2×C3⋊C8—21no—
262C2×C3⋊C8—21no—
263C2×C3⋊C8—21no—
264C2×C3⋊C8—21no—
265C2×C3⋊C8—21no—
266C2×C3⋊C8—44no—
267C22×Q8—61no—
268C22×Q8C22×D311yescharacteristic
269C23×C4—21no—
270C23×C4—61no—
271C23×C4C22×D311yescharacteristic
272C22×C8—31no—
273C22×C8—31no—
274C22×C8—31no—
275C22×C8—31no—
276C22×C8—62no—
277C22×C8—121no—
278C22×C8—121no—
279C4⋊Q8—62no—
280C4⋊Q8—121no—
281C22⋊Q8—22no—
282C22⋊Q8—124no—
283C22⋊Q8Dic614yescomplemented
284C4×Q8—22no—
285C4×Q8—41no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—31no—
292C2×C4⋊C4—31no—
293C2×C4⋊C4—31no—
294C2×C4⋊C4—31no—
295C2×C4⋊C4—41no—
296C2×C4⋊C4—62no—
297C2×C4⋊C4—61no—
298C2×C4⋊C4—61no—
299C2×C4⋊C4—61no—
300C2×C4⋊C4—61no—
301C2×C4⋊C4—61no—
302C2×C4⋊C4—61no—
303C2×C4⋊C4—121no—
304C2×C4⋊C4C22×D311yescharacteristic
305C2×C4⋊C4C22×D311yescharacteristic
306C2×C4⋊C4C22×D311yescharacteristic
307C2×C22⋊C4—21no—
308C2×C22⋊C4—61no—
309C2×C22⋊C4#6711yescharacteristic
310C2×C22⋊C4#6711yescharacteristic
311C2×C42—21no—
312C2×C42—21no—
313C2×C42C22×D311yescharacteristic
314C2×C42C22×D311yescharacteristic
315#106—62no—
316#106—121no—
317C4⋊C8—121no—
318C4⋊C8—121no—
319#103—62no—
320#103—121no—
321#103—121no—
322#103—121no—
323C22⋊C8—62no—
324#95—21no—
325#95—21no—
326#95—21no—
327#95—21no—
328#95—61no—
329#95—61no—
330#95—61no—
331#95#6711yescharacteristic
332#95#6711yescharacteristic
333C23×C3—21no—
334C23×C3—21no—
335C23×C3—21no—
336C23×C3—21no—
337C23×C3C22⋊Q812yes—
338C23×C3C2×SD1612yes—
339C23×C3#13712yes—
340C23×C3C22×D411yescharacteristic
341C3×Q8—22no—
342C3×Q8—22no—
343C3×Q8—41no—
344C3×Q8—41no—
345C2×C3×C4—21no—
346C2×C3×C4—21no—
347C2×C3×C4—21no—
348C2×C3×C4—21no—
349C2×C3×C4—21no—
350C2×C3×C4—21no—
351C2×C3×C4—22no—
352C2×C3×C4—22no—
353C2×C3×C4—22no—
354C2×C3×C4—22no—
355C2×C3×C4—22no—
356C2×C3×C4—22no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—22no—
360C2×C3×C4—21no—
361C2×C3×C4—21no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
364C2×C3×C4—41no—
365C2×C3×C4—41no—
366C2×C3×C4—41no—
367C2×C3×C4—41no—
368C2×C3×C4—41no—
369C2×C3×C4—41no—
370C2×C3×C4—41no—
371C2×C3×C4—41no—
372C2×C3×C4—41no—
373C2×C3×C4—84no—
374C2×C3×C4C22⋊Q812yes—
375C2×C3×C4C2×SD1612yes—
376C2×C3×C4C2×SD1612yes—
377C2×C3×C4C22×Q811yescharacteristic
378C2×C3×C4C2×#4111yescharacteristic
379C2×C3×C4C2×#4111yescharacteristic
380C2×C3×C4C2×#4111yescharacteristic
381C2×C3×C4H(5,𝔽2)11yescharacteristic
382C2×C3×C4H(5,𝔽2)11yescharacteristic
383C2×Dic3—22no—
384C2×Dic3—41no—
385C2×Dic3—41no—
386C2×Dic3—41no—
387C2×Dic3—41no—
388C2×Dic3—41no—
389C2×Dic3—41no—
390C2×Dic3—41no—
391C2×Dic3—41no—
392C2×Dic3—41no—
393C2×Dic3—41no—
394C2×Dic3—41no—
395C2×Dic3—84no—
396Dic6—84no—
397C3⋊C8—41no—
398C3⋊C8—41no—
399C3⋊C8—41no—
400C3⋊C8—41no—
401C24C2×#6711yescharacteristic
402C2×Q8—21no—
403C2×Q8—21no—
404C2×Q8—21no—
405C2×Q8—21no—
406C2×Q8—21no—
407C2×Q8—122no—
408C2×Q8—122no—
409C2×Q8—241no—
410C2×Q8C2×Dic612yes—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—61no—
422C22×C4—61no—
423C22×C4—61no—
424C22×C4—61no—
425C22×C4—81no—
426C22×C4—122no—
427C22×C4—122no—
428C22×C4—241no—
429C22×C4C2×Dic612yes—
430C22×C4#23512yes—
431C22×C4C2×#6711yescharacteristic
432C22×C4C2×#6711yescharacteristic
433C22×C4C2×#6711yescharacteristic
434C22×C4C2×#6711yescharacteristic
435C22×C4C2×#6711yescharacteristic
436C22×C4C23×D311yescharacteristic
437C2×C8—61no—
438C2×C8—61no—
439C2×C8—61no—
440C2×C8—61no—
441C2×C8—61no—
442C2×C8—61no—
443C2×C8—61no—
444C2×C8—61no—
445C2×C8—61no—
446C2×C8—61no—
447C2×C8—61no—
448C2×C8—61no—
449C2×C8—124no—
450C4⋊C4—24no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—32no—
456C4⋊C4—42no—
457C4⋊C4—42no—
458C4⋊C4—61no—
459C4⋊C4—61no—
460C4⋊C4—61no—
461C4⋊C4—61no—
462C4⋊C4—61no—
463C4⋊C4—61no—
464C4⋊C4—61no—
465C4⋊C4—81no—
466C4⋊C4—122no—
467C4⋊C4—122no—
468C4⋊C4—122no—
469C4⋊C4—122no—
470C4⋊C4—122no—
471C4⋊C4—241no—
472C4⋊C4C2×Dic612yes—
473C4⋊C4#23712yes—
474C22⋊C4—24no—
475C22⋊C4—22no—
476C22⋊C4—122no—
477C22⋊C4#21514yes—
478C42—21no—
479C42—21no—
480C42—21no—
481C42—21no—
482C42—42no—
483C42—81no—
484C22×C3—22no—
485C22×C3—22no—
486C22×C3—22no—
487C22×C3—22no—
488C22×C3—22no—
489C22×C3—22no—
490C22×C3#47514yes—
491C22×C3C2×C22⋊Q811yescharacteristic
492C22×C3#54611yescharacteristic
493C22×C3#54611yescharacteristic
494C22×C3C22×SD1611yescharacteristic
495C22×C3C2×#13711yescharacteristic
496C22×C3#57611yescharacteristic
497C22×C3#57811yescharacteristic
498C3×C4—22no—
499C3×C4—22no—
500C3×C4—22no—
501C3×C4—42no—
502C3×C4—42no—
503C3×C4—41no—
504C3×C4—41no—
505C3×C4—42no—
506C3×C4—81no—
507C3×C4#52911yescharacteristic
508C3×C4#53811yescharacteristic
509C3×C4#55411yescharacteristic
510C3×C4#55411yescharacteristic
511Dic3—42no—
512Dic3—81no—
513Dic3—81no—
514Dic3—81no—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—21no—
519C23#92412yes—
520C23C2×#21512yes—
521C23#95112yes—
522C23C22×#6711yescharacteristic
523Q8—22no—
524Q8—22no—
525Q8—41no—
526Q8—41no—
527Q8—244no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—21no—
541C2×C4—22no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—62no—
557C2×C4—84no—
558C2×C4—121no—
559C2×C4—121no—
560C2×C4—121no—
561C2×C4—121no—
562C2×C4—121no—
563C2×C4—121no—
564C2×C4—121no—
565C2×C4—121no—
566C2×C4—121no—
567C2×C4—121no—
568C2×C4—121no—
569C2×C4—244no—
570C2×C4#92412yes—
571C2×C4C2×#21512yes—
572C2×C4C2×#21512yes—
573C2×C4C22×Dic611yescharacteristic
574C2×C4C2×#23511yescharacteristic
575C2×C4C2×#23711yescharacteristic
576C2×C4#100411yescharacteristic
577C2×C4D3×#4111yescharacteristic
578C2×C4#100911yescharacteristic
579C8—121no—
580C8—121no—
581C8—121no—
582C8—121no—
583C2×C3—24no—
584C2×C3#245011yescharacteristic
585C2×C3C2×#47511yescharacteristic
586C2×C3#308311yescharacteristic
587C2×C3#329511yescharacteristic
588C2×C3#329811yescharacteristic
589C2×C3#331211yescharacteristic
590C2×C3#331411yescharacteristic
591C22—22no—
592C22—22no—
593C22—22no—
594C22—22no—
595C22—22no—
596C22—22no—
597C22#502014yes—
598C22C2×#92411yescharacteristic
599C22#580611yescharacteristic
600C22C22×#21511yescharacteristic
601C22C2×#95111yescharacteristic
602C22#582911yescharacteristic
603C22#583511yescharacteristic
604C22#583811yescharacteristic
605C4—22no—
606C4—22no—
607C4—22no—
608C4—41no—
609C4—42no—
610C4—42no—
611C4—41no—
612C4—42no—
613C4—81no—
614C4—122no—
615C4—241no—
616C4—241no—
617C4—241no—
618C4#553811yescharacteristic
619C4#553911yescharacteristic
620C4#554311yescharacteristic
621C4#556011yescharacteristic
622C3#2641211yescharacteristic, complemented, Sylow
623C2—24no—
624C2#8236111yescharacteristic
625C2C2×#502011yescharacteristic
626C2#8255911yescharacteristic
627C2#8349811yescharacteristic
628C2#8355011yescharacteristic
629C2#8360311yescharacteristic
630C2#8363411yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2360C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenter#2360D3
hypocenterC3#26412
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2360

Complements

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

NH ≅ G/Ntype
5. #71629C2semidirect
6. #71626C2semidirect
9. #69682C2semidirect
10. #69668C2semidirect
11. #69611C2semidirect
12. #69610C2semidirect
13. #69490C2semidirect
14. #69459C2semidirect
81. C3×C22⋊Q8Q8semidirect
283. C22⋊Q8Dic6semidirect
622. C3#26412semidirect

Maximal subgroups

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

subgroupindex
C3×#23602
#734932
#716962
#716922
#716292
#716262
C2×#45292
#715562
#696822
#696682
#696112
#696102
#694902
#694592
#694492
#264123