Finite Group Databasebeta

← #11675018

Subgroups — #11675018

Order 768.

2064 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3480C211yesmaximal, characteristic, complemented
2C2×Q8×D12C211yesmaximal, characteristic, complemented
3#83715C216yesmaximal, complemented
4#83688C212yesmaximal, complemented
5#83668C216yesmaximal, complemented
6#83580C212yesmaximal, complemented
7#83297C213yesmaximal, complemented
8#83283C213yesmaximal, complemented
9C4×#949C213yesmaximal, complemented
10#82508C213yesmaximal, complemented
11Q8×#208C211yesmaximal, characteristic, complemented
12#61317—31nomaximal, Sylow
13C22×D3×Q8—21no—
14C3×#559C2213yescomplemented
15C3×#557C2211yescharacteristic, complemented
16C3×#556C2213yescomplemented
17C3×#553—23no—
18C3×#550C2211yescharacteristic, complemented
19C3×#548—21no—
20C3×#532C2213yes—
21C3×#529—23no—
22C3×C4×#41—21no—
23C3×C4×#41C2213yes—
24C2×C3×C4×Q8C2211yescharacteristic
25C2×#949C2213yes—
26Q8×D12—21no—
27Q8×D12—23no—
28Q8×D12C2216yes—
29Q8×D12C2212yes—
30C2×#897—23no—
31C2×C4×D12C2213yes—
32C2×#869C2213yes—
33#5232C2216yes—
34#5232C2213yes—
35#5083C2216yes—
36#5079C2216yescomplemented
37#5072C2216yes—
38#5070C2216yescomplemented
39#5057C22112yes—
40#5055C22112yescomplemented
41#5026C2216yes—
42C4×#215C2216yes—
43#5024C2216yescomplemented
44Q8×C3⋊C8C2212yescomplemented
45#5021C2216yescomplemented
46#5020C2216yescomplemented
47#5014C2216yes—
48C4×#213C2216yes—
49#5008C2216yescomplemented
50#5007C2216yescomplemented
51#5001C2211yescharacteristic
52#4995C2213yes—
53#4924C2213yes—
54C4×#208C2213yes—
55#3480D311yescharacteristic, complemented
56C2×D4×Q8—31no—
57#3373—36no—
58#3367—36no—
59#3359—32no—
60#3358—32no—
61#3141—33no—
62#3079—33no—
63#3036—33no—
64Q8×M16—31no—
65C4×AGL(1,ℤ/8)—33no—
66C2×C3×#41—21no—
67C2×C3×#41C2313yes—
68C22×C3×Q8D411yescharacteristic
69C2×D3×Q8—26no—
70C2×D3×Q8—22no—
71C2×D3×Q8—48no—
72C22×D12C2311yescharacteristic
73C22×C4×D3—23no—
74C22×Dic6—23no—
75C3×C4⋊Q8C2313yes—
76C3×#127C2313yes—
77C3×#126—26no—
78C3×#125—23no—
79C3×#124—23no—
80C3×#124C2316yes—
81C3×#123—26no—
82C3×C22⋊Q8—23no—
83C3×#121—23no—
84C3×#121—23no—
85C3×#121—23no—
86C3×C4×Q8—21no—
87C3×C4×Q8D411yescharacteristic
88C3×C4×Q8D413yes—
89C3×C4×Q8C2313yes—
90C3×C4×Q8C2313yes—
91C3×C4×Q8C2313yes—
92C3×C4×Q8C2311yescharacteristic
93C3×C4×D4—23no—
94C3×C4×D4—23no—
95C3×C4×D4—23no—
96C3×C4×D4—23no—
97C3×C4×D4—23no—
98C3×C4×D4C2313yes—
99C3×C4×D4C2313yes—
100C3×#117—23no—
101C3×#117—23no—
102C3×#117C2311yescharacteristic
103C2×C3×C4⋊C4D413yes—
104C2×C3×C42—23no—
105C2×C3×C42C2313yes—
106#949—212no—
107C2×#215C2316yes—
108C2×#213C2316yes—
109C2×(C2×D3)⋊C4—23no—
110C2×C4⋊Dic3—23no—
111C2×C4⋊Dic3C2313yes—
112C2×#208C2311yescharacteristic
113C2×#208C2313yes—
114#897—26no—
115#897—26no—
116#897—212no—
117C4×D12—23no—
118C4×D12—23no—
119C4×D12C2316yes—
120C4×D12C2316yes—
121#869—23no—
122#869—23no—
123#869C2316yes—
124#869C2316yes—
125#835C23112yes—
126#835C2316yes—
127#832C23112yes—
128#832C2316yes—
129#809C2312yes—
130#809C2312yes—
131#808C2316yes—
132#807C2316yes—
133C4⋊(C3⋊C8)C2316yes—
134C4⋊(C3⋊C8)C2316yes—
135#803C2316yes—
136C4×C3⋊C8C2316yes—
137C23×Q8—61no—
138C2×AGL(1,ℤ/8)—33no—
139#559C2×D313yescomplemented
140#557C2×D311yescharacteristic, complemented
141#556C2×D313yescomplemented
142#553—23no—
143#550C2×D311yescharacteristic, complemented
144D4×Q8—32no—
145D4×Q8—36no—
146D4×Q8—61no—
147D4×Q8—63no—
148#548—21no—
149#532C2×D313yes—
150C2×C4⋊Q8—33no—
151#529—23no—
152C2×C22⋊Q8—63no—
153C4×#41—21no—
154C4×#41C2×D313yes—
155C2×C4×Q8C2×D311yescharacteristic
156C2×C4×D4—33no—
157#489—312no—
158#487—36no—
159#486—36no—
160#477—36no—
161#476—36no—
162#475—36no—
163#474—36no—
164#463—312no—
165#461—36no—
166#459—36no—
167#446—36no—
168C8×Q8—32no—
169#442—36no—
170#440—36no—
171C4×SD16—36no—
172C4×D8—36no—
173#428—33no—
174#423—33no—
175#418—31no—
176#417—36no—
177#417—33no—
178C4×M16—33no—
179C23×D3—21no—
180C3×#41—26no—
181C3×#41—26no—
182C3×#41—44no—
183C2×C3×Q8—24no—
184C2×C3×Q8C2×D413yes—
185C2×C3×Q8C2×D411yescharacteristic
186C2×C3×Q8C2413yes—
187C2×C3×D4—23no—
188C2×C3×D4—23no—
189C2×C3×D4—23no—
190C2×C3×D4C2413yes—
191C22×C3×C4—23no—
192C22×C3×C4—23no—
193C22×C3×C4—23no—
194C22×C3×C4C2×D413yes—
195C22×C3×C4C2×D413yes—
196C22×C3×C4C2411yescharacteristic
197C22×Dic3—23no—
198D3×Q8—424no—
199D3×Q8—48no—
200C2×D12—21no—
201C2×D12—22no—
202C2×D12#4114yescomplemented
203C2×D12C2412yes—
204C2×C4×D3—26no—
205C2×C4×D3—26no—
206C2×C4×D3—412no—
207C2×Dic6—26no—
208C2×Dic6—26no—
209C2×Dic6—412no—
210C3×C4⋊C4—23no—
211C3×C4⋊C4—23no—
212C3×C4⋊C4—23no—
213C3×C4⋊C4—23no—
214C3×C4⋊C4C2×D413yes—
215C3×C4⋊C4C2×D413yes—
216C3×C4⋊C4C2×D416yes—
217C3×C4⋊C4C2411yescharacteristic
218C3×C4⋊C4C2411yescharacteristic
219C3×C22⋊C4—21no—
220C3×C22⋊C4—26no—
221C3×C22⋊C4—23no—
222C3×C22⋊C4—23no—
223C3×C22⋊C4—26no—
224C3×C42—21no—
225C3×C42—23no—
226C3×C42—23no—
227C3×C42C2×D413yes—
228C3×C42C2×D413yes—
229C3×C42C2413yes—
230C3×C42C2413yes—
231#215—212no—
232#213—212no—
233(C2×D3)⋊C4—26no—
234(C2×D3)⋊C4—26no—
235C4⋊Dic3—23no—
236C4⋊Dic3—26no—
237C4⋊Dic3—26no—
238C4⋊Dic3C2416yes—
239#208—26no—
240#208—22no—
241C2×C3⋊C8C2412yes—
242C2×C3⋊C8C2416yes—
243C2×#41—21no—
244C2×#41C22×D313yes—
245C22×Q8—66no—
246C22×Q8—62no—
247C22×Q8—63no—
248C22×Q8—128no—
249C22×Q8#6711yescharacteristic
250C22×D4—31no—
251C23×C4—63no—
252AGL(1,ℤ/8)—612no—
253C2×SD16—36no—
254C2×D8—36no—
255C2×M16—33no—
256C2×M16—31no—
257C4⋊Q8—36no—
258C4⋊Q8—36no—
259C4⋊Q8—63no—
260C4⋊Q8—63no—
261C4⋊Q8C22×D313yes—
262#127C22×D313yes—
263#126—26no—
264#125—23no—
265#124—23no—
266#124C22×D316yes—
267#123—26no—
268C22⋊Q8—23no—
269C22⋊Q8—66no—
270C22⋊Q8—66no—
271C22⋊Q8—612no—
272#121—23no—
273#121—23no—
274#121—23no—
275C4×Q8—21no—
276C4×Q8#6711yescharacteristic
277C4×Q8#6713yes—
278C4×Q8C22×D313yes—
279C4×Q8C22×D313yes—
280C4×Q8C22×D311yescharacteristic
281C4×Q8C22×D313yes—
282C4×D4—23no—
283C4×D4—23no—
284C4×D4—23no—
285C4×D4—23no—
286C4×D4—23no—
287C4×D4—36no—
288C4×D4—36no—
289C4×D4—63no—
290C4×D4—63no—
291C4×D4C22×D313yes—
292C4×D4C22×D313yes—
293#117—23no—
294#117—23no—
295#117C22×D311yescharacteristic
296C2×C4⋊C4—33no—
297C2×C4⋊C4—63no—
298C2×C4⋊C4#6713yes—
299C2×C22⋊C4—63no—
300C2×C42—23no—
301C2×C42C22×D313yes—
302#107—36no—
303#106—36no—
304C4⋊C8—36no—
305C4⋊C8—36no—
306#103—36no—
307#103—312no—
308#102—36no—
309#102—312no—
310#102—32no—
311#102—32no—
312#97—36no—
313C4×C8—36no—
314C23×C3—23no—
315C23×C3C22×D411yescharacteristic
316C22×D3—24no—
317C22×D3—22no—
318C22×D3—42no—
319C22×D3—42no—
320C3×Q8—26no—
321C3×Q8—23no—
322C3×Q8—23no—
323C3×Q8AGL(1,ℤ/8)14yescomplemented
324C3×D4—23no—
325C3×D4—23no—
326C3×D4—46no—
327C3×D4—46no—
328C3×D4—83no—
329C2×C3×C4—26no—
330C2×C3×C4—21no—
331C2×C3×C4—21no—
332C2×C3×C4—23no—
333C2×C3×C4—43no—
334C2×C3×C4—46no—
335C2×C3×C4—46no—
336C2×C3×C4—83no—
337C2×C3×C4C22×D413yes—
338C2×C3×C4C22×D413yes—
339C2×C3×C4C22×D413yes—
340C2×C3×C4C22×D411yescharacteristic
341C2×C3×C4C22×D413yes—
342C2×C3×C4C2×#4112yes—
343C2×C3×C42−1+412yes—
344C2×C3×C4C2511yescharacteristic
345C2×Dic3—26no—
346C2×Dic3—46no—
347D12—24no—
348D12—22no—
349D12C2×#4114yes—
350C4×D3—412no—
351C4×D3—412no—
352Dic6—412no—
353Dic6—412no—
354C3⋊C8—26no—
355C3⋊C8—22no—
356C24—61no—
357#41—26no—
358#41—26no—
359#41—44no—
360C2×Q8—24no—
361C2×Q8—66no—
362C2×Q8—66no—
363C2×Q8—1212no—
364C2×Q8—1224no—
365C2×Q8—128no—
366C2×Q8C2×#6713yes—
367C2×Q8C2×#6711yescharacteristic
368C2×Q8C23×D313yes—
369C2×D4—23no—
370C2×D4—23no—
371C2×D4—23no—
372C2×D4—34no—
373C2×D4—32no—
374C2×D4—62no—
375C2×D4—61no—
376C2×D4C23×D313yes—
377C22×C4—23no—
378C22×C4—23no—
379C22×C4—23no—
380C22×C4—63no—
381C22×C4—66no—
382C22×C4—66no—
383C22×C4—1212no—
384C22×C4C2×#6713yes—
385C22×C4C2×#6713yes—
386C22×C4C23×D311yescharacteristic
387SD16—612no—
388D8—612no—
389M16—66no—
390M16—62no—
391C2×C8—32no—
392C2×C8—36no—
393C4⋊C4—23no—
394C4⋊C4—23no—
395C4⋊C4—23no—
396C4⋊C4—23no—
397C4⋊C4—36no—
398C4⋊C4—63no—
399C4⋊C4—66no—
400C4⋊C4—66no—
401C4⋊C4C2×#6713yes—
402C4⋊C4C2×#6713yes—
403C4⋊C4C2×#6716yes—
404C4⋊C4C23×D311yescharacteristic
405C4⋊C4C23×D311yescharacteristic
406C22⋊C4—21no—
407C22⋊C4—26no—
408C22⋊C4—23no—
409C22⋊C4—23no—
410C22⋊C4—26no—
411C22⋊C4—66no—
412C22⋊C4—66no—
413C42—21no—
414C42—23no—
415C42—23no—
416C42C2×#6713yes—
417C42C2×#6713yes—
418C42C23×D313yes—
419C42C23×D313yes—
420C22×C3—21no—
421C22×C3—21no—
422C22×C3—43no—
423C22×C3—43no—
424C22×C3—43no—
425C22×C3#54812yes—
426C22×C3C23×D411yescharacteristic
427C2×D3—24no—
428C2×D3—48no—
429C2×D3—42no—
430C2×D3—42no—
431C3×C4—23no—
432C3×C4—23no—
433C3×C4—23no—
434C3×C4—41no—
435C3×C4—43no—
436C3×C4—41no—
437C3×C4#54812yes—
438C3×C4C2×AGL(1,ℤ/8)16yes—
439C3×C4C22×#4111yescharacteristic
440C3×C4C2×2−1+411yescharacteristic
441Dic3—46no—
442C23—23no—
443C23—62no—
444C23—64no—
445C23—122no—
446C23—122no—
447C23C22×#6711yescharacteristic
448Q8—26no—
449Q8—23no—
450Q8—23no—
451Q8—1212no—
452Q8—1212no—
453Q8#94914yescomplemented
454D4—23no—
455D4—23no—
456D4—34no—
457D4—46no—
458D4—46no—
459D4—62no—
460D4—64no—
461D4—83no—
462C2×C4—26no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—23no—
466C2×C4—43no—
467C2×C4—46no—
468C2×C4—46no—
469C2×C4—66no—
470C2×C4—83no—
471C2×C4—1212no—
472C2×C4—126no—
473C2×C4—1212no—
474C2×C4D3×#4112yes—
475C2×C4#101012yes—
476C2×C4C22×#6713yes—
477C2×C4C22×#6713yes—
478C2×C4C22×#6713yes—
479C2×C4C22×#6713yes—
480C2×C4C22×#6711yescharacteristic
481C2×C4C24×D311yescharacteristic
482C8—66no—
483C8—62no—
484C2×C3—22no—
485C2×C3—83no—
486C2×C3C2×#54811yescharacteristic
487C2×C3C22×AGL(1,ℤ/8)11yescharacteristic
488C2×C3#358611yescharacteristic
489D3—44no—
490C22—21no—
491C22—21no—
492C22—43no—
493C22—43no—
494C22—43no—
495C22—64no—
496C22—128no—
497C22—122no—
498C22—122no—
499C22#583112yes—
500C22C23×#6711yescharacteristic
501C4—23no—
502C4—23no—
503C4—23no—
504C4—41no—
505C4—41no—
506C4—43no—
507C4—126no—
508C4C2×#94916yes—
509C4#583112yes—
510C4C2×D3×#4111yescharacteristic
511C4C2×#101011yescharacteristic
512C3#6131711yescharacteristic, complemented, Sylow
513C2—22no—
514C2—83no—
515C2—124no—
516C2C22×#94911yescharacteristic
517C2C2×#583111yescharacteristic
518C2#8729911yescharacteristic
subgroupG/H
centerC22C23×#67
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC3×#3480C2
socleC22×C3C23×D4
hypercenter#3480D3
hypocenterC3#61317
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C4×Q8 ◁ #3480

Complements

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

NH ≅ G/Ntype
1. C3×#3480C2semidirect
2. C2×Q8×D12C2semidirect
3. #83715C2semidirect
4. #83688C2semidirect
5. #83668C2semidirect
6. #83580C2semidirect
7. #83297C2semidirect
8. #83283C2semidirect
9. C4×#949C2semidirect
10. #82508C2semidirect
11. Q8×#208C2semidirect
14. C3×#559C22semidirect
15. C3×#557C22semidirect
16. C3×#556C22semidirect
18. C3×#550C22semidirect
36. #5079C22semidirect
38. #5070C22semidirect
40. #5055C22semidirect
43. #5024C22semidirect
44. Q8×C3⋊C8C22semidirect
45. #5021C22semidirect
46. #5020C22semidirect
49. #5008C22semidirect
50. #5007C22semidirect
55. #3480D3semidirect
139. #559C2×D3semidirect
140. #557C2×D3semidirect
141. #556C2×D3semidirect
143. #550C2×D3semidirect
202. C2×D12#41semidirect
323. C3×Q8AGL(1,ℤ/8)semidirect
453. Q8#949semidirect
512. C3#61317semidirect

Maximal subgroups

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

subgroupindex
C3×#34802
C2×Q8×D122
#837152
#836882
#836682
#835802
#832972
#832832
C4×#9492
#825082
Q8×#2082
#613173