Finite Group Databasebeta

← #10859047

Subgroups — #10859047

Order 768.

784 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2362C211yesmaximal, characteristic
2#73493C211yesmaximal, characteristic
3#71697C211yesmaximal, characteristic
4#71692C211yesmaximal, characteristic
5#71628C211yesmaximal, characteristic, complemented
6#71626C211yesmaximal, characteristic, complemented
7#71581C211yesmaximal, characteristic
8#71576C211yesmaximal, characteristic
9#69682C211yesmaximal, characteristic, complemented
10#69668C211yesmaximal, characteristic, complemented
11#69624C211yesmaximal, characteristic, complemented
12#69612C211yesmaximal, characteristic, complemented
13#69490C211yesmaximal, characteristic, complemented
14#69458C211yesmaximal, characteristic, complemented
15#69450C211yesmaximal, characteristic
16#26429—31nomaximal, Sylow
17C2×C3×C22⋊Q8C2211yescharacteristic
18C2×C3×C4×Q8C2211yescharacteristic
19C2×C3×#117C2211yescharacteristic
20C22×C4⋊Dic3C2211yescharacteristic
21C22×#208C2211yescharacteristic
22C2×#878—21no—
23C2×#869C2211yescharacteristic
24C3×#391C2211yescharacteristic
25C3×#389C2211yescharacteristic
26C3×#388—21no—
27C3×#388—21no—
28C3×#387—21no—
29C3×C4×C4⋊C4—21no—
30C3×C4×C4⋊C4C2211yescharacteristic
31C3×C4×C22⋊C4C2211yescharacteristic
32C2×#835C2211yescharacteristic
33C2×#835C2211yescharacteristic
34C2×#835C2211yescharacteristic
35C2×#835C2211yescharacteristic
36#5212C2211yescharacteristic
37C2×#830C2211yescharacteristic
38C2×#808C2211yescharacteristic
39C2×#807C2211yescharacteristic
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#2362D311yescharacteristic
62#2100—31no—
63#2088—31no—
64#2066—31no—
65#2024—31no—
66#1988—31no—
67#1918—31no—
68#1894—31no—
69#1872—31no—
70#1863—31no—
71#1854—31no—
72#1851—31no—
73#1803—31no—
74#1736—31no—
75C22×C3×Q8C2311yescharacteristic
76C23×C3×C4C2311yescharacteristic
77C23×Dic3—21no—
78C22×Dic6—21no—
79C3×C22⋊Q8—22no—
80C3×C22⋊Q8Q814yescomplemented
81C3×C4×Q8—22no—
82C3×C4×Q8—41no—
83C3×#117—22no—
84C3×#117—41no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C22⋊C4—21no—
92C2×C3×C22⋊C4—21no—
93C2×C3×C22⋊C4D411yescharacteristic
94C2×C3×C22⋊C4D411yescharacteristic
95C2×C3×C42—21no—
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—
113C2×#208—22no—
114C2×#208—41no—
115C2×#208—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—41no—
138#807—22no—
139C4⋊(C3⋊C8)—41no—
140C4⋊(C3⋊C8)—41no—
141C22×M16—31no—
142C2×C4⋊Q8—31no—
143C2×C22⋊Q8—61no—
144C2×C22⋊Q8C2×D311yescharacteristic
145C2×C4×Q8C2×D311yescharacteristic
146C2×#117C2×D311yescharacteristic
147C22×C4⋊C4—31no—
148C2×#107—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—31no—
161#389—31no—
162#389C2×D311yescharacteristic
163#388—21no—
164#388—21no—
165#387—21no—
166#385—61no—
167#384—31no—
168#380—31no—
169C4×C4⋊C4—21no—
170C4×C4⋊C4C2×D311yescharacteristic
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×C4C2×D411yescharacteristic
200C22×C3×C4C2×D411yescharacteristic
201C22×C3×C4C2×D411yescharacteristic
202C22×C3×C4C2×D411yescharacteristic
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4C2×Q812yes—
205C22×C3×C4#4112yes—
206C22×C3×C4C2411yescharacteristic
207C22×Dic3—21no—
208C22×Dic3—21no—
209C22×Dic3—21no—
210C22×Dic3—21no—
211C22×Dic3—42no—
212C22×Dic3—42no—
213C22×Dic3—81no—
214C2×Dic6—42no—
215C2×Dic6—42no—
216C2×Dic6—81no—
217C3×C4⋊C4—24no—
218C3×C4⋊C4—21no—
219C3×C4⋊C4—21no—
220C3×C4⋊C4—21no—
221C3×C4⋊C4—21no—
222C3×C4⋊C4—42no—
223C3×C4⋊C4—42no—
224C3×C4⋊C4C2×Q812yes—
225C3×C4⋊C4#4112yes—
226C3×C22⋊C4—24no—
227C3×C22⋊C4—22no—
228C3×C22⋊C4—22no—
229C3×C22⋊C4—42no—
230C3×C42—21no—
231C3×C42—21no—
232C3×C42—21no—
233C3×C42—21no—
234C3×C42—42no—
235C3×C42—42no—
236C3×C42—42no—
237#217—42no—
238C4⋊Dic3—21no—
239C4⋊Dic3—21no—
240C4⋊Dic3—21no—
241C4⋊Dic3—21no—
242C4⋊Dic3—21no—
243C4⋊Dic3—21no—
244C4⋊Dic3—21no—
245C4⋊Dic3—42no—
246C4⋊Dic3—42no—
247C4⋊Dic3—42no—
248C4⋊Dic3—81no—
249C4⋊Dic3#4112yes—
250Dic3⋊C4—42no—
251Dic3⋊C4—42no—
252#208—44no—
253C2×C3⋊C8—21no—
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—
265C22×Q8—61no—
266C22×Q8C22×D311yescharacteristic
267C23×C4—61no—
268C23×C4C22×D311yescharacteristic
269C2×M16—62no—
270C2×M16—121no—
271C2×M16—121no—
272C22×C8—31no—
273C22×C8—31no—
274C22×C8—31no—
275C22×C8—31no—
276C4⋊Q8—62no—
277C4⋊Q8—121no—
278C22⋊Q8—22no—
279C22⋊Q8—124no—
280C22⋊Q8Dic614yescomplemented
281C4×Q8—22no—
282C4×Q8—41no—
283#117—22no—
284#117—41no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—31no—
289C2×C4⋊C4—31no—
290C2×C4⋊C4—31no—
291C2×C4⋊C4—31no—
292C2×C4⋊C4—62no—
293C2×C4⋊C4—61no—
294C2×C4⋊C4—61no—
295C2×C4⋊C4—61no—
296C2×C4⋊C4—61no—
297C2×C4⋊C4—61no—
298C2×C4⋊C4—61no—
299C2×C4⋊C4—121no—
300C2×C4⋊C4C22×D311yescharacteristic
301C2×C4⋊C4C22×D311yescharacteristic
302C2×C4⋊C4C22×D311yescharacteristic
303C2×C22⋊C4—21no—
304C2×C22⋊C4—21no—
305C2×C22⋊C4—61no—
306C2×C22⋊C4#6711yescharacteristic
307C2×C22⋊C4#6711yescharacteristic
308C2×C42—21no—
309C2×C42—21no—
310C2×C42—21no—
311C2×C42C22×D311yescharacteristic
312C2×C42C22×D311yescharacteristic
313#107—62no—
314#106—121no—
315C4⋊C8—121no—
316C4⋊C8—121no—
317#103—62no—
318#103—121no—
319#103—121no—
320#103—121no—
321C22⋊C8—62no—
322#95—21no—
323#95—21no—
324#95—21no—
325#95—21no—
326#95—61no—
327#95—61no—
328#95—61no—
329#95#6711yescharacteristic
330#95#6711yescharacteristic
331C23×C3—21no—
332C23×C3—21no—
333C23×C3—21no—
334C23×C3—21no—
335C23×C3—21no—
336C23×C3—21no—
337C23×C3C22⋊Q812yes—
338C23×C3C22×D411yescharacteristic
339C3×Q8—22no—
340C3×Q8—22no—
341C3×Q8—41no—
342C3×Q8—41no—
343C2×C3×C4—21no—
344C2×C3×C4—21no—
345C2×C3×C4—21no—
346C2×C3×C4—21no—
347C2×C3×C4—21no—
348C2×C3×C4—21no—
349C2×C3×C4—22no—
350C2×C3×C4—22no—
351C2×C3×C4—22no—
352C2×C3×C4—22no—
353C2×C3×C4—22no—
354C2×C3×C4—22no—
355C2×C3×C4—21no—
356C2×C3×C4—21no—
357C2×C3×C4—22no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—21no—
361C2×C3×C4—21no—
362C2×C3×C4—41no—
363C2×C3×C4—41no—
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×C4C22⋊Q812yes—
372C2×C3×C4AGL(1,ℤ/8)12yes—
373C2×C3×C4#13712yes—
374C2×C3×C4C22×Q811yescharacteristic
375C2×C3×C4C2×#4111yescharacteristic
376C2×C3×C4C2×#4111yescharacteristic
377C2×C3×C4C2×#4111yescharacteristic
378C2×C3×C4H(5,𝔽2)11yescharacteristic
379C2×C3×C4H(5,𝔽2)11yescharacteristic
380C2×Dic3—22no—
381C2×Dic3—41no—
382C2×Dic3—41no—
383C2×Dic3—41no—
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—84no—
393Dic6—84no—
394C3⋊C8—41no—
395C3⋊C8—41no—
396C3⋊C8—41no—
397C3⋊C8—41no—
398C24C2×#6711yescharacteristic
399C2×Q8—21no—
400C2×Q8—21no—
401C2×Q8—21no—
402C2×Q8—21no—
403C2×Q8—21no—
404C2×Q8—122no—
405C2×Q8—122no—
406C2×Q8—241no—
407C2×Q8C2×Dic612yes—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—61no—
417C22×C4—61no—
418C22×C4—61no—
419C22×C4—61no—
420C22×C4—122no—
421C22×C4—122no—
422C22×C4—241no—
423C22×C4C2×Dic612yes—
424C22×C4#23512yes—
425C22×C4C2×#6711yescharacteristic
426C22×C4C2×#6711yescharacteristic
427C22×C4C2×#6711yescharacteristic
428C22×C4C2×#6711yescharacteristic
429C22×C4C2×#6711yescharacteristic
430C22×C4C23×D311yescharacteristic
431M16—124no—
432C2×C8—61no—
433C2×C8—61no—
434C2×C8—61no—
435C2×C8—61no—
436C2×C8—61no—
437C2×C8—61no—
438C2×C8—61no—
439C2×C8—61no—
440C2×C8—61no—
441C2×C8—61no—
442C2×C8—61no—
443C2×C8—61no—
444C4⋊C4—24no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—32no—
450C4⋊C4—42no—
451C4⋊C4—42no—
452C4⋊C4—61no—
453C4⋊C4—61no—
454C4⋊C4—61no—
455C4⋊C4—61no—
456C4⋊C4—61no—
457C4⋊C4—61no—
458C4⋊C4—61no—
459C4⋊C4—122no—
460C4⋊C4—122no—
461C4⋊C4—122no—
462C4⋊C4—122no—
463C4⋊C4—122no—
464C4⋊C4—241no—
465C4⋊C4C2×Dic612yes—
466C4⋊C4#23712yes—
467C22⋊C4—24no—
468C22⋊C4—22no—
469C22⋊C4—22no—
470C22⋊C4—42no—
471C22⋊C4—122no—
472C42—21no—
473C42—21no—
474C42—21no—
475C42—21no—
476C42—42no—
477C42—42no—
478C42—42no—
479C22×C3—22no—
480C22×C3—22no—
481C22×C3—22no—
482C22×C3—41no—
483C22×C3—41no—
484C22×C3—41no—
485C22×C3—41no—
486C22×C3C2×C22⋊Q811yescharacteristic
487C22×C3#54611yescharacteristic
488C22×C3#54611yescharacteristic
489C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
490C22×C3C2×#13711yescharacteristic
491C22×C3#57711yescharacteristic
492C22×C3#57811yescharacteristic
493C3×C4—22no—
494C3×C4—22no—
495C3×C4—22no—
496C3×C4—41no—
497C3×C4—41no—
498C3×C4—42no—
499C3×C4—42no—
500C3×C4—42no—
501C3×C4—81no—
502C3×C4#52911yescharacteristic
503C3×C4#53811yescharacteristic
504C3×C4#55411yescharacteristic
505C3×C4#55411yescharacteristic
506Dic3—42no—
507Dic3—81no—
508Dic3—81no—
509Dic3—81no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23#92412yes—
517C23C22×#6711yescharacteristic
518Q8—22no—
519Q8—22no—
520Q8—41no—
521Q8—41no—
522Q8—244no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—62no—
552C2×C4—121no—
553C2×C4—121no—
554C2×C4—121no—
555C2×C4—121no—
556C2×C4—121no—
557C2×C4—121no—
558C2×C4—121no—
559C2×C4—121no—
560C2×C4—121no—
561C2×C4—121no—
562C2×C4—121no—
563C2×C4—244no—
564C2×C4#92412yes—
565C2×C4#94912yes—
566C2×C4#95112yes—
567C2×C4C22×Dic611yescharacteristic
568C2×C4C2×#23511yescharacteristic
569C2×C4C2×#23711yescharacteristic
570C2×C4#100411yescharacteristic
571C2×C4D3×#4111yescharacteristic
572C2×C4#100911yescharacteristic
573C8—121no—
574C8—121no—
575C8—121no—
576C8—121no—
577C2×C3—42no—
578C2×C3#245011yescharacteristic
579C2×C3#307911yescharacteristic
580C2×C3#308211yescharacteristic
581C2×C3#331011yescharacteristic
582C2×C3#331611yescharacteristic
583C2×C3#332811yescharacteristic
584C2×C3#333011yescharacteristic
585C22—22no—
586C22—22no—
587C22—22no—
588C22—41no—
589C22—41no—
590C22—41no—
591C22—41no—
592C22C2×#92411yescharacteristic
593C22#580611yescharacteristic
594C22C2×#94911yescharacteristic
595C22C2×#95111yescharacteristic
596C22#582911yescharacteristic
597C22#583711yescharacteristic
598C22#583811yescharacteristic
599C4—22no—
600C4—22no—
601C4—22no—
602C4—41no—
603C4—42no—
604C4—42no—
605C4—42no—
606C4—41no—
607C4—81no—
608C4—122no—
609C4—241no—
610C4—241no—
611C4—241no—
612C4#553811yescharacteristic
613C4#553911yescharacteristic
614C4#554311yescharacteristic
615C4#556011yescharacteristic
616C3#2642911yescharacteristic, complemented, Sylow
617C2—42no—
618C2#8236111yescharacteristic
619C2#8250811yescharacteristic
620C2#8255011yescharacteristic
621C2#8351411yescharacteristic
622C2#8353811yescharacteristic
623C2#8359911yescharacteristic
624C2#8367511yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2362C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenter#2362D3
hypocenterC3#26429
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2362

Complements

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

NH ≅ G/Ntype
5. #71628C2semidirect
6. #71626C2semidirect
9. #69682C2semidirect
10. #69668C2semidirect
11. #69624C2semidirect
12. #69612C2semidirect
13. #69490C2semidirect
14. #69458C2semidirect
80. C3×C22⋊Q8Q8semidirect
280. C22⋊Q8Dic6semidirect
616. C3#26429semidirect

Maximal subgroups

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

subgroupindex
C3×#23622
#734932
#716972
#716922
#716282
#716262
#715812
#715762
#696822
#696682
#696242
#696122
#694902
#694582
#694502
#264293