Finite Group Databasebeta

← #10692252

Subgroups — #10692252

Order 768.

704 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3006C211yesmaximal, characteristic
2#84075C211yesmaximal, characteristic, complemented
3#72135C211yesmaximal, characteristic, complemented
4#71676C211yesmaximal, characteristic, complemented
5#71675C211yesmaximal, characteristic, complemented
6#67688C211yesmaximal, characteristic, complemented
7#67684C211yesmaximal, characteristic, complemented
8#9347—31nomaximal, Sylow
9C22×C3×#41C2211yescharacteristic
10C2×C3×AGL(1,ℤ/8)C411yescharacteristic, complemented
11C2×C3×AGL(1,ℤ/8)C411yescharacteristic, complemented
12C2×C3×#121C2211yescharacteristic
13C2×#948—21no—
14C3×#449—21no—
15C3×#449—21no—
16C3×#448—21no—
17C3×#447—21no—
18C3×#417C411yescharacteristic, complemented
19C3×#417C411yescharacteristic, complemented
20C3×#407C2211yescharacteristic
21#5234—21no—
22#5234—22no—
23C2×#835—21no—
24C2×#832—21no—
25#5212C2211yescharacteristic
26#5207—21no—
27#5207C2211yescharacteristic
28C2×#830—21no—
29C2×#830C2211yescharacteristic
30C2×#830C2211yescharacteristic
31#4537—21no—
32#4537—21no—
33#4525—22no—
34#3006D311yescharacteristic
35#2880—31no—
36#1788—31no—
37#1519—31no—
38#1471—31no—
39#1305—31no—
40#1298—31no—
41C2×C3×#41—21no—
42C2×C3×#41—21no—
43C2×C3×#41—42no—
44C2×C3×#41—42no—
45C2×C3×#41C2×C411yescharacteristic
46C2×C3×#41C2×C411yescharacteristic
47C2×C3×#41D411yescharacteristic
48C2×C3×#41D411yescharacteristic
49C22×C3×Q8D411yescharacteristic
50C22×C3×D4—21no—
51C22×C3×D4—21no—
52C22×C3×D4C2×C411yescharacteristic
53C22×C3×D4D411yescharacteristic
54C23×C3×C4—21no—
55C23×C3×C4C2311yescharacteristic
56C3×AGL(1,ℤ/8)—42no—
57C3×AGL(1,ℤ/8)—42no—
58C2×C3×SD16—21no—
59C2×C3×SD16—21no—
60C2×C3×D8—21no—
61C2×C3×D8—21no—
62C2×C3×M16C2×C411yescharacteristic
63C2×C3×M16C2×C411yescharacteristic
64C3×#121—21no—
65C3×#121—21no—
66C3×#121—21no—
67C3×#121—21no—
68C2×C3×C4⋊C4C2×C411yescharacteristic
69C2×C3×C22⋊C4—21no—
70C2×#217—21no—
71#948—42no—
72#948—42no—
73C2×C4⋊Dic3—21no—
74C2×#208—22no—
75C2×#208—21no—
76C22×C3⋊C8—22no—
77C22×C3⋊C8—21no—
78C22×C3⋊C8—21no—
79C3×#103—21no—
80C3×#103—21no—
81C3×#102—21no—
82C3×#102—21no—
83C3×C22⋊C8—21no—
84C3×C22⋊C8—21no—
85#835—42no—
86#832—42no—
87C3⋊#95—21no—
88#830—22no—
89#830—22no—
90#830—21no—
91#830—21no—
92#830—21no—
93#830—21no—
94#830D411yescharacteristic
95#830D411yescharacteristic
96#830D412yes—
97C22×#41C2×D311yescharacteristic
98C2×AGL(1,ℤ/8)Dic311yescharacteristic, complemented
99C2×AGL(1,ℤ/8)Dic311yescharacteristic, complemented
100C2×#131—61no—
101C2×#121C2×D311yescharacteristic
102#449—21no—
103#449—21no—
104#448—21no—
105#447—21no—
106#417Dic311yescharacteristic, complemented
107#417Dic311yescharacteristic, complemented
108C2×#103—61no—
109C2×#102—61no—
110#408—62no—
111#408—61no—
112#407—31no—
113#407—61no—
114#407C2×D311yescharacteristic
115C2×C22⋊C8—31no—
116C2×C22⋊C8—31no—
117C2×C22⋊C8—61no—
118#380—31no—
119#327—61no—
120#327—61no—
121#323—62no—
122C24×C3—21no—
123C24×C3—21no—
124C24×C3#4111yescharacteristic
125C3×#41—22no—
126C3×#41—22no—
127C3×#41—22no—
128C3×#41—22no—
129C3×#41—41no—
130C3×#41—41no—
131C3×#41—41no—
132C3×#41—41no—
133C3×#41—44no—
134C3×#41—44no—
135C2×C3×Q8—21no—
136C2×C3×Q8—22no—
137C2×C3×Q8—41no—
138C2×C3×Q8C22⋊C411yescharacteristic
139C2×C3×Q8C22⋊C411yescharacteristic
140C2×C3×D4—21no—
141C2×C3×D4—22no—
142C2×C3×D4—21no—
143C2×C3×D4—21no—
144C2×C3×D4—21no—
145C2×C3×D4—21no—
146C2×C3×D4—21no—
147C2×C3×D4—21no—
148C2×C3×D4—41no—
149C2×C3×D4—41no—
150C2×C3×D4—41no—
151C2×C3×D4—44no—
152C2×C3×D4—41no—
153C2×C3×D4—41no—
154C2×C3×D4—81no—
155C2×C3×D4—81no—
156C2×C3×D4C22⋊C411yescharacteristic
157C2×C3×D4C22⋊C411yescharacteristic
158C22×C3×C4—21no—
159C22×C3×C4—21no—
160C22×C3×C4—21no—
161C22×C3×C4—21no—
162C22×C3×C4—21no—
163C22×C3×C4—21no—
164C22×C3×C4—41no—
165C22×C3×C4—41no—
166C22×C3×C4—44no—
167C22×C3×C4C22×C411yescharacteristic
168C22×C3×C4C2×D411yescharacteristic
169C22×C3×C4C2×D411yescharacteristic
170C22×C3×C4C2×D411yescharacteristic
171C22×C3×C4C2×D411yescharacteristic
172C22×C3×C4#4111yescharacteristic
173C22×Dic3—41no—
174C3×SD16—42no—
175C3×SD16—42no—
176C3×D8—42no—
177C3×D8—42no—
178C3×M16—41no—
179C3×M16—41no—
180C2×C3×C8—21no—
181C2×C3×C8—21no—
182C3×C4⋊C4—21no—
183C3×C4⋊C4—21no—
184C3×C22⋊C4—41no—
185C3×C22⋊C4—41no—
186#217—81no—
187C4⋊Dic3—41no—
188C4⋊Dic3—41no—
189#208—44no—
190#208—41no—
191#208—41no—
192C2×C3⋊C8—21no—
193C2×C3⋊C8—22no—
194C2×C3⋊C8—21no—
195C2×C3⋊C8—44no—
196C2×C3⋊C8—41no—
197C2×C3⋊C8—41no—
198C2×C3⋊C8—41no—
199C2×C3⋊C8—41no—
200C2×C3⋊C8—41no—
201C2×#41—21no—
202C2×#41—21no—
203C2×#41—42no—
204C2×#41—42no—
205C2×#41C2×Dic311yescharacteristic
206C2×#41C2×Dic311yescharacteristic
207C2×#41#6711yescharacteristic
208C2×#41#6711yescharacteristic
209C22×Q8#6711yescharacteristic
210C22×D4—21no—
211C22×D4—21no—
212C22×D4C2×Dic311yescharacteristic
213C22×D4#6711yescharacteristic
214C23×C4—21no—
215C23×C4C22×D311yescharacteristic
216AGL(1,ℤ/8)—42no—
217AGL(1,ℤ/8)—42no—
218C2×SD16—21no—
219C2×SD16—21no—
220C2×D8—21no—
221C2×D8—21no—
222#131—122no—
223#131—122no—
224C2×M16—62no—
225C2×M16—61no—
226C2×M16C2×Dic311yescharacteristic
227C2×M16C2×Dic311yescharacteristic
228C22×C8—62no—
229C22×C8—61no—
230C22×C8—61no—
231#121—21no—
232#121—21no—
233#121—21no—
234#121—21no—
235C2×C4⋊C4—61no—
236C2×C4⋊C4C2×Dic311yescharacteristic
237C2×C22⋊C4—21no—
238C2×C22⋊C4—61no—
239#103—21no—
240#103—21no—
241#103—122no—
242#102—21no—
243#102—21no—
244#102—122no—
245C22⋊C8—21no—
246C22⋊C8—21no—
247C22⋊C8—32no—
248C22⋊C8—31no—
249C22⋊C8—31no—
250C22⋊C8—62no—
251C22⋊C8—62no—
252C22⋊C8—61no—
253C22⋊C8—61no—
254C22⋊C8—61no—
255C22⋊C8—61no—
256#95—61no—
257C23×C3—21no—
258C23×C3—21no—
259C23×C3—21no—
260C23×C3—21no—
261C23×C3—41no—
262C23×C3—41no—
263C23×C3—42no—
264C23×C3—42no—
265C23×C3—41no—
266C23×C3—41no—
267C23×C3—41no—
268C23×C3—81no—
269C23×C3—81no—
270C23×C3—81no—
271C23×C3C4×D411yescharacteristic
272C23×C3#12111yescharacteristic
273C23×C3#12311yescharacteristic
274C3×Q8—21no—
275C3×Q8—21no—
276C3×Q8—21no—
277C3×Q8—42no—
278C3×Q8#9912yes—
279C3×D4—24no—
280C3×D4—21no—
281C3×D4—21no—
282C3×D4—21no—
283C3×D4—42no—
284C3×D4—42no—
285C3×D4—42no—
286C3×D4—42no—
287C3×D4—41no—
288C3×D4—41no—
289C3×D4—81no—
290C3×D4—82no—
291C3×D4—81no—
292C3×D4—81no—
293C3×D4#9912yes—
294C2×C3×C4—24no—
295C2×C3×C4—21no—
296C2×C3×C4—21no—
297C2×C3×C4—21no—
298C2×C3×C4—21no—
299C2×C3×C4—21no—
300C2×C3×C4—21no—
301C2×C3×C4—21no—
302C2×C3×C4—21no—
303C2×C3×C4—41no—
304C2×C3×C4—41no—
305C2×C3×C4—41no—
306C2×C3×C4—42no—
307C2×C3×C4—42no—
308C2×C3×C4—42no—
309C2×C3×C4—42no—
310C2×C3×C4—41no—
311C2×C3×C4—81no—
312C2×C3×C4C2×C22⋊C411yescharacteristic
313C2×C3×C4C4×D411yescharacteristic
314C2×C3×C4#12011yescharacteristic
315C2×C3×C4#12111yescharacteristic
316C2×Dic3—81no—
317C2×Dic3—81no—
318C2×Dic3—81no—
319C3×C8—41no—
320C3×C8—41no—
321C3⋊C8—41no—
322C3⋊C8—42no—
323C3⋊C8—41no—
324C3⋊C8—81no—
325C24—21no—
326C24—21no—
327C24#23711yescharacteristic
328#41—22no—
329#41—22no—
330#41—22no—
331#41—22no—
332#41—41no—
333#41—41no—
334#41—41no—
335#41—41no—
336#41—44no—
337#41—44no—
338C2×Q8—21no—
339C2×Q8—22no—
340C2×Q8—41no—
341C2×Q8#21711yescharacteristic
342C2×Q8#21711yescharacteristic
343C2×D4—21no—
344C2×D4—22no—
345C2×D4—21no—
346C2×D4—21no—
347C2×D4—21no—
348C2×D4—21no—
349C2×D4—21no—
350C2×D4—21no—
351C2×D4—41no—
352C2×D4—41no—
353C2×D4—41no—
354C2×D4—44no—
355C2×D4—41no—
356C2×D4—41no—
357C2×D4—81no—
358C2×D4—81no—
359C2×D4#21711yescharacteristic
360C2×D4#21711yescharacteristic
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—44no—
370C22×C4—121no—
371C22×C4D3×D411yescharacteristic
372C22×C4D3×D411yescharacteristic
373C22×C4#23711yescharacteristic
374C22×C4C22×Dic311yescharacteristic
375C22×C4C2×#6711yescharacteristic
376C22×C4C2×#6711yescharacteristic
377SD16—42no—
378SD16—42no—
379D8—42no—
380D8—42no—
381M16—41no—
382M16—41no—
383M16—124no—
384M16—121no—
385M16—121no—
386C2×C8—21no—
387C2×C8—21no—
388C2×C8—61no—
389C2×C8—62no—
390C2×C8—61no—
391C2×C8—124no—
392C2×C8—121no—
393C2×C8—121no—
394C2×C8—121no—
395C2×C8—121no—
396C2×C8—121no—
397C4⋊C4—21no—
398C4⋊C4—21no—
399C4⋊C4—121no—
400C4⋊C4—121no—
401C22⋊C4—41no—
402C22⋊C4—41no—
403C22⋊C4—241no—
404C22×C3—21no—
405C22×C3—21no—
406C22×C3—21no—
407C22×C3—21no—
408C22×C3—21no—
409C22×C3—22no—
410C22×C3—21no—
411C22×C3—21no—
412C22×C3—21no—
413C22×C3—41no—
414C22×C3—41no—
415C22×C3—41no—
416C22×C3—41no—
417C22×C3—41no—
418C22×C3—81no—
419C22×C3—81no—
420C22×C3—82no—
421C22×C3—82no—
422C22×C3—81no—
423C22×C3—81no—
424C22×C3—81no—
425C22×C3—162no—
426C22×C3#38611yescharacteristic
427C22×C3#47111yescharacteristic
428C22×C3#47211yescharacteristic
429C3×C4—21no—
430C3×C4—21no—
431C3×C4—22no—
432C3×C4—21no—
433C3×C4—41no—
434C3×C4—81no—
435C3×C4C2×#9911yescharacteristic
436C3×C4C2×#9911yescharacteristic
437Dic3—161no—
438C23—21no—
439C23—21no—
440C23—21no—
441C23—21no—
442C23—41no—
443C23—41no—
444C23—42no—
445C23—42no—
446C23—41no—
447C23—41no—
448C23—41no—
449C23—81no—
450C23—81no—
451C23—81no—
452C23D4×Dic311yescharacteristic
453C23#93511yescharacteristic
454C23#93911yescharacteristic
455Q8—21no—
456Q8—21no—
457Q8—21no—
458Q8—42no—
459Q8#83412yes—
460D4—24no—
461D4—21no—
462D4—21no—
463D4—21no—
464D4—42no—
465D4—42no—
466D4—42no—
467D4—42no—
468D4—41no—
469D4—41no—
470D4—81no—
471D4—82no—
472D4—81no—
473D4—81no—
474D4#83412yes—
475C2×C4—24no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—41no—
492C2×C4—81no—
493C2×C4—241no—
494C2×C4—241no—
495C2×C4—241no—
496C2×C4D4×Dic311yescharacteristic
497C2×C4#93711yescharacteristic
498C2×C4#93911yescharacteristic
499C2×C4C2×#21711yescharacteristic
500C8—41no—
501C8—41no—
502C8—121no—
503C8—122no—
504C8—121no—
505C8—241no—
506C2×C3—21no—
507C2×C3—21no—
508C2×C3—41no—
509C2×C3—41no—
510C2×C3—42no—
511C2×C3—81no—
512C2×C3—161no—
513C2×C3#188111yescharacteristic
514C2×C3#189811yescharacteristic
515C2×C3#190711yescharacteristic
516C22—21no—
517C22—21no—
518C22—21no—
519C22—21no—
520C22—21no—
521C22—21no—
522C22—21no—
523C22—22no—
524C22—21no—
525C22—41no—
526C22—41no—
527C22—41no—
528C22—41no—
529C22—41no—
530C22—81no—
531C22—81no—
532C22—82no—
533C22—82no—
534C22—81no—
535C22—81no—
536C22—81no—
537C22—162no—
538C22#519911yescharacteristic
539C22#520011yescharacteristic
540C22#522011yescharacteristic
541C4—21no—
542C4—21no—
543C4—22no—
544C4—21no—
545C4—41no—
546C4—81no—
547C4—481no—
548C4C2×#83411yescharacteristic
549C4C2×#83411yescharacteristic
550C3#934711yescharacteristic, complemented, Sylow
551C2—21no—
552C2—21no—
553C2—41no—
554C2—41no—
555C2—42no—
556C2—81no—
557C2—161no—
558C2#7167011yescharacteristic
559C2#7209111yescharacteristic
560C2#7211611yescharacteristic
subgroupG/H
centerC22#5220
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
Fitting, prime radicalC3×#3006C2
socleC22×C3#386
cosocleC23×C3×C4C23
hypercenter#3006D3
hypocenterC3#9347
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3006

Complements

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

NH ≅ G/Ntype
2. #84075C2semidirect
3. #72135C2semidirect
4. #71676C2semidirect
5. #71675C2semidirect
6. #67688C2semidirect
7. #67684C2semidirect
10. C2×C3×AGL(1,ℤ/8)C4semidirect
11. C2×C3×AGL(1,ℤ/8)C4semidirect
18. C3×#417C4semidirect
19. C3×#417C4semidirect
98. C2×AGL(1,ℤ/8)Dic3semidirect
99. C2×AGL(1,ℤ/8)Dic3semidirect
106. #417Dic3semidirect
107. #417Dic3semidirect
550. C3#9347semidirect

Maximal subgroups

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

subgroupindex
C3×#30062
#840752
#721352
#716762
#716752
#676882
#676842
#93473