Finite Group Databasebeta

← #86714

Subgroups — #86714

Order 384.

1522 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D3×#41C211yesmaximal, characteristic, complemented
2C2×#1007C211yesmaximal, characteristic, complemented
3C3×#548C211yesmaximal, characteristic, complemented
4#5831C211yesmaximal, characteristic, complemented
5Q8×#67C211yesmaximal, characteristic, complemented
6#5676C213yesmaximal, complemented
7#5675C213yesmaximal, complemented
8#5631C213yesmaximal, complemented
9D3×C22⋊Q8C213yesmaximal, complemented
10#5612C213yesmaximal, complemented
11#5608C213yesmaximal, complemented
12Q8×D12C211yesmaximal, characteristic, complemented
13#5574C211yesmaximal, characteristic, complemented
14#5551C213yesmaximal, complemented
15#5547C213yesmaximal, complemented
16#3489—31nomaximal, Sylow
17C2×C3×#41C2211yescharacteristic
18C22×C3×Q8C2211yescharacteristic, complemented
19D3×#41—28no—
20#1007—28no—
21C2×#239C2211yescharacteristic
22C2×#239C2213yescomplemented
23C2×D3×Q8C2211yescharacteristic
24C2×D3×Q8C2213yescomplemented
25C2×D3×Q8C2211yescharacteristic, complemented
26C2×#237C2213yes—
27C2×D3×D4C2213yes—
28C2×#235C2213yes—
29C2×#235C2213yescomplemented
30C2×#235C2213yescomplemented
31C22×C4×D3C2213yes—
32C3×#124C2213yescomplemented
33C3×C22⋊Q8C2213yescomplemented
34C3×#121C2213yescomplemented
35C3×C4×Q8C2211yescharacteristic, complemented
36C3×C4×D4C2213yescomplemented
37#946C2213yes—
38Q8×Dic3C2211yescharacteristic, complemented
39#944C2213yescomplemented
40#939C2213yescomplemented
41#938C2213yescomplemented
42#938C2216yescomplemented
43#937C2213yes—
44#936C2213yescomplemented
45C4×#67C2213yes—
46C4×#67C2213yescomplemented
47#924C2213yescomplemented
48#924C2213yescomplemented
49#897C2213yes—
50#897C2213yescomplemented
51#896C2213yes—
52#894C2213yescomplemented
53#893C2213yescomplemented
54D3×C4⋊C4C2216yescomplemented
55D3×C4⋊C4C2213yescomplemented
56#884C2211yescharacteristic
57#884C2216yescomplemented
58#883C2216yescomplemented
59#883C2216yescomplemented
60#883C2213yescomplemented
61#882C2213yes—
62D3×C22⋊C4C2216yes—
63#878C2213yescomplemented
64#878C2216yescomplemented
65C4×D12C2213yes—
66C4×D12C2213yescomplemented
67#872C2213yescomplemented
68#869C2213yescomplemented
69C2×2−1+4—31no—
70C22×#41—31no—
71D4×Q8—31no—
72D4×Q8—31no—
73#548—31no—
74#548D311yescharacteristic, complemented
75#547—33no—
76#547—33no—
77#546—33no—
78#546—33no—
79#537—33no—
80#536—33no—
81#525—33no—
82C2×C22⋊Q8—33no—
83#520—31no—
84C23×D3C2313yes—
85C3×#41—24no—
86C2×C3×Q8—24no—
87C2×C3×Q8C2313yes—
88C2×C3×Q8C2311yescharacteristic
89C2×C3×D4C2313yes—
90C2×C3×D4C2313yes—
91C22×C3×C4C2313yes—
92C22×C3×C4C2313yes—
93C2×#67C2316yes—
94C2×#67C2313yes—
95C2×#67C2311yescharacteristic
96C2×#67C2311yescharacteristic
97C2×#67C2313yes—
98C22×Dic3C2313yes—
99#239—212no—
100#239D418yescomplemented
101D3×Q8—24no—
102D3×Q8—24no—
103D3×Q8—212no—
104#237—212no—
105D3×D4—212no—
106#235—212no—
107#235—212no—
108#235—212no—
109C2×D12C2311yescharacteristic
110C2×D12C2313yes—
111C2×D12C2313yes—
112C2×C4×D3—212no—
113C2×C4×D3C2313yes—
114C2×C4×D3C2313yes—
115C2×C4×D3C2316yes—
116C2×C4×D3C2313yes—
117C2×C4×D3C2311yescharacteristic
118C2×C4×D3C2316yes—
119C2×C4×D3C2313yes—
120C2×Dic6C2313yes—
121C2×Dic6C2313yes—
122C2×Dic6C2313yes—
123C3×C4⋊C4C2313yes—
124C3×C4⋊C4C2313yes—
125C3×C22⋊C4C2316yes—
126C3×C22⋊C4C2313yes—
127C3×C22⋊C4C2311yescharacteristic
128C3×C42C2313yes—
129#217C2316yes—
130#217C2313yes—
131#217C2313yes—
132(C2×D3)⋊C4C2313yes—
133(C2×D3)⋊C4C2313yes—
134(C2×D3)⋊C4C2313yes—
135(C2×D3)⋊C4C2313yes—
136(C2×D3)⋊C4C2313yes—
137(C2×D3)⋊C4C2311yescharacteristic
138C4⋊Dic3C2313yes—
139C4⋊Dic3C2316yes—
140C4⋊Dic3C2313yes—
141Dic3⋊C4C2311yescharacteristic
142Dic3⋊C4C2313yes—
143Dic3⋊C4C2313yes—
144Dic3⋊C4C2316yes—
145Dic3⋊C4C2313yes—
146C4×Dic3C2313yes—
1472−1+4—68no—
148C2×#41—31no—
149C2×#41—33no—
150C2×#41—33no—
151C2×#41—33no—
152C2×#41—33no—
153C2×#41—33no—
154C2×#41—68no—
155C2×#41C2×D311yescharacteristic
156C22×Q8—33no—
157C22×Q8—31no—
158C22×Q8—31no—
159C22×Q8C2×D311yescharacteristic, complemented
160C22×D4—33no—
161C23×C4—33no—
162C4⋊Q8—33no—
163C4⋊Q8—33no—
164#124—33no—
165#124C2×D313yescomplemented
166#123—33no—
167#123—33no—
168#123—36no—
169#123—36no—
170C22⋊Q8—33no—
171C22⋊Q8—33no—
172C22⋊Q8—33no—
173C22⋊Q8—33no—
174C22⋊Q8—33no—
175C22⋊Q8—33no—
176C22⋊Q8—36no—
177C22⋊Q8—33no—
178C22⋊Q8C2×D313yescomplemented
179#121—33no—
180#121—36no—
181#121—33no—
182#121—31no—
183#121—36no—
184#121C2×D313yescomplemented
185#120—33no—
186#120—33no—
187C4×Q8—31no—
188C4×Q8C2×D311yescharacteristic, complemented
189C4×D4—33no—
190C4×D4—33no—
191C4×D4—33no—
192C4×D4—33no—
193C4×D4—33no—
194C4×D4C2×D313yescomplemented
195#117—33no—
196C2×C4⋊C4—33no—
197C2×C4⋊C4—36no—
198C2×C22⋊C4—36no—
199C23×C3C2413yes—
200C23×C3C2411yescharacteristic
201C22×D3—26no—
202C22×D3—26no—
203C22×D3—43no—
204C22×D3C2411yescharacteristic
205C22×D3C2413yes—
206C22×D3C2411yescharacteristic
207C3×Q8—26no—
208C3×Q8C2×D414yes—
209C3×D4—26no—
210C3×D4—43no—
211C2×C3×C4—26no—
212C2×C3×C4—26no—
213C2×C3×C4C2413yes—
214C2×C3×C4C2411yescharacteristic
215C2×C3×C4C2411yescharacteristic
216C2×C3×C4C2413yes—
217C2×C3×C4C2413yes—
218#67—26no—
219#67—212no—
220#67—22no—
221#67—22no—
222#67—43no—
223C2×Dic3—26no—
224C2×Dic3C2411yescharacteristic
225C2×Dic3C2413yes—
226C2×Dic3C2413yes—
227C2×Dic3C2413yes—
228C2×Dic3C2411yescharacteristic
229D12—26no—
230D12—22no—
231D12C2×D4112yes—
232C4×D3—26no—
233C4×D3—212no—
234C4×D3—26no—
235C4×D3—26no—
236C4×D3—22no—
237C4×D3—46no—
238C4×D3C2×D4112yes—
239Dic6—26no—
240Dic6—26no—
241Dic6—26no—
242C24—33no—
243#41—24no—
244#41—38no—
245#41—612no—
246#41—612no—
247#41—612no—
248#41—612no—
249#41—612no—
250C2×Q8—24no—
251C2×Q8—33no—
252C2×Q8—33no—
253C2×Q8—33no—
254C2×Q8—612no—
255C2×Q8—64no—
256C2×Q8—64no—
257C2×Q8C22×D313yes—
258C2×Q8C22×D311yescharacteristic
259C2×D4—31no—
260C2×D4—33no—
261C2×D4—31no—
262C2×D4—36no—
263C2×D4—33no—
264C2×D4—33no—
265C2×D4—31no—
266C2×D4—33no—
267C2×D4—612no—
268C2×D4C22×D313yes—
269C2×D4C22×D313yes—
270C22×C4—33no—
271C22×C4—33no—
272C22×C4—36no—
273C22×C4—33no—
274C22×C4—33no—
275C22×C4—33no—
276C22×C4—31no—
277C22×C4—36no—
278C22×C4—612no—
279C22×C4C22×D313yes—
280C22×C4C22×D313yes—
281C4⋊C4—33no—
282C4⋊C4—33no—
283C4⋊C4—33no—
284C4⋊C4—36no—
285C4⋊C4—36no—
286C4⋊C4—31no—
287C4⋊C4—33no—
288C4⋊C4—33no—
289C4⋊C4C22×D313yes—
290C4⋊C4C22×D313yes—
291C22⋊C4—36no—
292C22⋊C4—33no—
293C22⋊C4—33no—
294C22⋊C4—33no—
295C22⋊C4—33no—
296C22⋊C4—33no—
297C22⋊C4—33no—
298C22⋊C4—31no—
299C22⋊C4—33no—
300C22⋊C4C22×D316yes—
301C22⋊C4C22×D311yescharacteristic
302C22⋊C4C22×D313yes—
303C42—33no—
304C42C22×D313yes—
305C22×C3—23no—
306C22×C3—21no—
307C22×C3—21no—
308C22×C3—23no—
309C22×C3—23no—
310C22×C3—21no—
311C22×C3C2511yescharacteristic
312C2×D3—21no—
313C2×D3—23no—
314C2×D3—23no—
315C2×D3—21no—
316C2×D3—21no—
317C2×D3—21no—
318C2×D3—21no—
319C2×D3—412no—
320C2×D3C22×D416yes—
321C2×D32−1+412yescomplemented
322C3×C4—23no—
323C3×C4—21no—
324C3×C4—21no—
325C3×C4—23no—
326C3×C4C22×D416yes—
327Dic3—23no—
328Dic3—23no—
329Dic3—21no—
330Dic3—23no—
331Dic3C22×D412yes—
332C23—31no—
333C23—33no—
334C23—31no—
335C23—66no—
336C23—66no—
337C23—123no—
338C23C23×D313yes—
339C23C23×D311yescharacteristic
340Q8—26no—
341Q8—66no—
342Q8—66no—
343Q8—66no—
344Q8D3×D414yes—
345D4—26no—
346D4—312no—
347D4—43no—
348D4—66no—
349D4—62no—
350D4—62no—
351D4—612no—
352D4—66no—
353D4—62no—
354D4—123no—
355C2×C4—26no—
356C2×C4—26no—
357C2×C4—31no—
358C2×C4—312no—
359C2×C4—31no—
360C2×C4—33no—
361C2×C4—33no—
362C2×C4—33no—
363C2×C4—62no—
364C2×C4—66no—
365C2×C4—612no—
366C2×C4—66no—
367C2×C4—66no—
368C2×C4—66no—
369C2×C4—126no—
370C2×C4C23×D311yescharacteristic
371C2×C4C23×D313yes—
372C2×C4C23×D313yes—
373C2×C4C23×D311yescharacteristic
374C2×C4C23×D313yes—
375C2×C3—43no—
376C2×C3—41no—
377C2×C3C23×D411yescharacteristic
378C2×C3C2×2−1+411yescharacteristic
379C2×C3#58511yescharacteristic
380D3—22no—
381D3—26no—
382D3—41no—
383C22—21no—
384C22—23no—
385C22—21no—
386C22—23no—
387C22—21no—
388C22—23no—
389C22—36no—
390C22—32no—
391C22—61no—
392C22—63no—
393C22—61no—
394C22—63no—
395C22—61no—
396C22—61no—
397C22—61no—
398C22—1212no—
399C22C24×D311yescharacteristic
400C4—21no—
401C4—23no—
402C4—23no—
403C4—21no—
404C4—32no—
405C4—63no—
406C4—63no—
407C4—63no—
408C4—61no—
409C4C2×D3×D416yes—
410C3#348911yescharacteristic, complemented, Sylow
411C2—43no—
412C2—41no—
413C2—62no—
414C2—66no—
415C2—121no—
416C2C22×D3×D411yescharacteristic
417C2#596411yescharacteristic
418C2D3×2−1+411yescharacteristic
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×#548C2
hypercenter#548D3
hypocenterC3#3489
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ #548

Complements

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

NH ≅ G/Ntype
1. C2×D3×#41C2semidirect
2. C2×#1007C2semidirect
3. C3×#548C2semidirect
4. #5831C2semidirect
5. Q8×#67C2semidirect
6. #5676C2semidirect
7. #5675C2semidirect
8. #5631C2semidirect
9. D3×C22⋊Q8C2semidirect
10. #5612C2semidirect
11. #5608C2semidirect
12. Q8×D12C2semidirect
13. #5574C2semidirect
14. #5551C2semidirect
15. #5547C2semidirect
18. C22×C3×Q8C22semidirect
22. C2×#239C22semidirect
24. C2×D3×Q8C22semidirect
25. C2×D3×Q8C22semidirect
29. C2×#235C22semidirect
30. C2×#235C22semidirect
32. C3×#124C22semidirect
33. C3×C22⋊Q8C22semidirect
34. C3×#121C22semidirect
35. C3×C4×Q8C22semidirect
36. C3×C4×D4C22semidirect
38. Q8×Dic3C22semidirect
39. #944C22semidirect
40. #939C22semidirect
41. #938C22semidirect
42. #938C22semidirect
44. #936C22semidirect
46. C4×#67C22semidirect
47. #924C22semidirect
48. #924C22semidirect
50. #897C22semidirect
52. #894C22semidirect
53. #893C22semidirect
54. D3×C4⋊C4C22semidirect
55. D3×C4⋊C4C22semidirect
57. #884C22semidirect
58. #883C22semidirect
59. #883C22semidirect
60. #883C22semidirect
63. #878C22semidirect
64. #878C22semidirect
66. C4×D12C22semidirect
67. #872C22semidirect
68. #869C22semidirect
74. #548D3semidirect
100. #239D4semidirect
159. C22×Q8C2×D3semidirect
165. #124C2×D3semidirect
178. C22⋊Q8C2×D3semidirect
184. #121C2×D3semidirect
188. C4×Q8C2×D3semidirect
194. C4×D4C2×D3semidirect
321. C2×D32−1+4semidirect
410. C3#3489semidirect

Maximal subgroups

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

subgroupindex
C2×D3×#412
C2×#10072
C3×#5482
#58312
Q8×#672
#56762
#56752
#56312
D3×C22⋊Q82
#56122
#56082
Q8×D122
#55742
#55512
#55472
#34893