Finite Group Databasebeta

← #86759

Subgroups — #86759

Order 384.

1492 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1007C211yesmaximal, characteristic, complemented
2C22×D3×Q8C211yesmaximal, characteristic, complemented
3C3×D4×Q8C211yesmaximal, characteristic, complemented
4#5817C211yesmaximal, characteristic, complemented
5Q8×#67C211yesmaximal, characteristic, complemented
6#5727C213yesmaximal, complemented
7#5724C213yesmaximal, complemented
8#5633C213yesmaximal, complemented
9#5632C213yesmaximal, complemented
10#5628C213yesmaximal, complemented
11#5627C213yesmaximal, complemented
12#5576C211yesmaximal, characteristic, complemented
13#5572C211yesmaximal, characteristic, complemented
14#5552C213yesmaximal, complemented
15#5547C213yesmaximal, complemented
16#3486—31nomaximal, Sylow
17C22×C3×Q8C2211yescharacteristic
18C22×C3×Q8C2211yescharacteristic, complemented
19#1007—28no—
20C2×#239C2213yes—
21C2×#239C2211yescharacteristic, complemented
22C2×D3×Q8—28no—
23C2×D3×Q8C2213yes—
24C2×D3×Q8C2213yes—
25C2×D3×Q8C2211yescharacteristic
26C2×D3×Q8C2211yescharacteristic
27C2×D3×Q8C2213yescomplemented
28C2×#235C2213yes—
29C2×#235C2213yescomplemented
30C22×C4×D3C2213yes—
31C22×Dic6C2213yes—
32C3×C4⋊Q8C2213yescomplemented
33C3×C22⋊Q8C2213yes—
34C3×C22⋊Q8C2213yescomplemented
35C3×C4×Q8C2211yescharacteristic, complemented
36C3×C4×D4C2213yes—
37#947C2213yescomplemented
38#946C2213yes—
39#946C2213yescomplemented
40Q8×Dic3C2211yescharacteristic, complemented
41#944C2213yescomplemented
42#939C2213yes—
43#937C2211yescharacteristic
44#930C2213yes—
45C4×#67C2213yes—
46C4×#67C2213yes—
47#924C2213yescomplemented
48#897C2213yescomplemented
49#896C2213yes—
50#896C2216yescomplemented
51#896C2216yescomplemented
52#895C2213yes—
53#894C2213yes—
54#894C2216yes—
55#893C2216yes—
56#892C2213yes—
57#888C2216yescomplemented
58#887C2213yescomplemented
59#885C2216yescomplemented
60#884C2213yes—
61#883C2213yes—
62#882C2213yes—
63#67⋊C4C2216yes—
64#878C2213yes—
65#875C2213yescomplemented
66C4×D12C2213yes—
67#872C2213yes—
68C4×Dic6C2213yescomplemented
69C2×2−1+4—31no—
70C23×Q8—31no—
71D4×Q8—31no—
72D4×Q8—33no—
73D4×Q8—33no—
74D4×Q8D311yescharacteristic, complemented
75#548—31no—
76#548—33no—
77#548—31no—
78#548—33no—
79#536—33no—
80#536—33no—
81#525—33no—
82#525—33no—
83#519—31no—
84C23×D3C2311yescharacteristic
85C2×C3×Q8—24no—
86C2×C3×Q8—24no—
87C2×C3×Q8C2313yes—
88C2×C3×Q8C2313yes—
89C2×C3×Q8C2311yescharacteristic
90C2×C3×D4C2311yescharacteristic
91C22×C3×C4C2313yes—
92C22×C3×C4C2313yes—
93C2×#67C2313yes—
94C2×#67C2311yescharacteristic
95C2×#67C2311yescharacteristic
96C2×#67C2313yes—
97C22×Dic3C2313yes—
98#239—24no—
99#239—212no—
100D3×Q8—24no—
101D3×Q8—212no—
102D3×Q8—212no—
103D3×Q8—212no—
104D3×Q8D418yescomplemented
105#235—212no—
106#235—212no—
107C2×D12C2313yes—
108C2×D12C2313yes—
109C2×D12C2313yes—
110C2×C4×D3—212no—
111C2×C4×D3C2313yes—
112C2×C4×D3C2313yes—
113C2×C4×D3C2316yes—
114C2×C4×D3C2313yes—
115C2×C4×D3C2313yes—
116C2×C4×D3C2313yes—
117C2×Dic6—212no—
118C2×Dic6C2313yes—
119C2×Dic6C2313yes—
120C2×Dic6C2316yes—
121C2×Dic6C2313yes—
122C3×C4⋊C4C2313yes—
123C3×C4⋊C4C2316yes—
124C3×C4⋊C4C2313yes—
125C3×C22⋊C4C2313yes—
126C3×C22⋊C4C2313yes—
127C3×C42C2313yes—
128#217C2311yescharacteristic
129#217C2313yes—
130(C2×D3)⋊C4C2311yescharacteristic
131(C2×D3)⋊C4C2313yes—
132(C2×D3)⋊C4C2311yescharacteristic
133(C2×D3)⋊C4C2313yes—
134(C2×D3)⋊C4C2316yes—
135(C2×D3)⋊C4C2313yes—
136(C2×D3)⋊C4C2313yes—
137(C2×D3)⋊C4C2316yes—
138C4⋊Dic3C2313yes—
139C4⋊Dic3C2313yes—
140Dic3⋊C4C2313yes—
141Dic3⋊C4C2313yes—
142Dic3⋊C4C2313yes—
143Dic3⋊C4C2316yes—
144Dic3⋊C4C2313yes—
145C4×Dic3C2316yes—
146C4×Dic3C2313yes—
1472−1+4—68no—
148C2×#41—33no—
149C2×#41—31no—
150C2×#41—33no—
151C2×#41—33no—
152C22×Q8—31no—
153C22×Q8—33no—
154C22×Q8—33no—
155C22×Q8—31no—
156C22×Q8—33no—
157C22×Q8—33no—
158C22×Q8—68no—
159C22×Q8C2×D311yescharacteristic
160C22×Q8C2×D311yescharacteristic, complemented
161C23×C4—33no—
162C4⋊Q8—33no—
163C4⋊Q8—36no—
164C4⋊Q8C2×D313yescomplemented
165#124—33no—
166#124—36no—
167#124—33no—
168#123—36no—
169#123—33no—
170#123—33no—
171C22⋊Q8—33no—
172C22⋊Q8—33no—
173C22⋊Q8—33no—
174C22⋊Q8—33no—
175C22⋊Q8—36no—
176C22⋊Q8—36no—
177C22⋊Q8—33no—
178C22⋊Q8—33no—
179C22⋊Q8C2×D313yes—
180C22⋊Q8C2×D313yescomplemented
181#121—33no—
182#121—33no—
183#121—33no—
184#121—33no—
185#120—31no—
186#120—33no—
187C4×Q8—31no—
188C4×Q8—33no—
189C4×Q8—33no—
190C4×Q8C2×D311yescharacteristic, complemented
191C4×D4—33no—
192C4×D4—33no—
193C4×D4—36no—
194C4×D4—36no—
195C4×D4—33no—
196C4×D4C2×D313yes—
197#117—33no—
198#117—33no—
199C23×C3C2411yescharacteristic
200C23×C3C2411yescharacteristic
201C22×D3—22no—
202C22×D3—22no—
203C22×D3—22no—
204C22×D3C2413yes—
205C22×D3C2411yescharacteristic
206C22×D3C2411yescharacteristic
207C3×Q8—26no—
208C3×Q8—26no—
209C3×Q8C2×D414yes—
210C3×D4—22no—
211C2×C3×C4—26no—
212C2×C3×C4—26no—
213C2×C3×C4C2413yes—
214C2×C3×C4C2411yescharacteristic
215C2×C3×C4C2413yes—
216C2×C3×C4C2413yes—
217C2×C3×C4C2413yes—
218#67—26no—
219#67—22no—
220#67—22no—
221#67—26no—
222C2×Dic3—26no—
223C2×Dic3C2411yescharacteristic
224C2×Dic3C2413yes—
225C2×Dic3C2411yescharacteristic
226C2×Dic3C2413yes—
227C2×Dic3C2413yes—
228D12—26no—
229D12—26no—
230D12—26no—
231C4×D3—26no—
232C4×D3—26no—
233C4×D3—26no—
234C4×D3—26no—
235C4×D3—212no—
236C4×D3C2×D4112yes—
237Dic6—26no—
238Dic6—212no—
239Dic6—26no—
240Dic6C2×D4112yes—
241C24—31no—
242#41—612no—
243#41—612no—
244#41—612no—
245#41—64no—
246C2×Q8—24no—
247C2×Q8—24no—
248C2×Q8—38no—
249C2×Q8—33no—
250C2×Q8—36no—
251C2×Q8—33no—
252C2×Q8—33no—
253C2×Q8—612no—
254C2×Q8—612no—
255C2×Q8—612no—
256C2×Q8—612no—
257C2×Q8—64no—
258C2×Q8C22×D313yes—
259C2×Q8C22×D313yes—
260C2×Q8C22×D311yescharacteristic
261C2×D4—31no—
262C2×D4—33no—
263C2×D4—31no—
264C2×D4—33no—
265C2×D4—33no—
266C2×D4—33no—
267C2×D4—33no—
268C2×D4C22×D311yescharacteristic
269C22×C4—33no—
270C22×C4—33no—
271C22×C4—33no—
272C22×C4—33no—
273C22×C4—36no—
274C22×C4—33no—
275C22×C4—33no—
276C22×C4—612no—
277C22×C4C22×D313yes—
278C22×C4C22×D313yes—
279C4⋊C4—33no—
280C4⋊C4—33no—
281C4⋊C4—33no—
282C4⋊C4—33no—
283C4⋊C4—36no—
284C4⋊C4—33no—
285C4⋊C4—33no—
286C4⋊C4C22×D313yes—
287C4⋊C4C22×D316yes—
288C4⋊C4C22×D313yes—
289C22⋊C4—31no—
290C22⋊C4—33no—
291C22⋊C4—33no—
292C22⋊C4—31no—
293C22⋊C4—33no—
294C22⋊C4—31no—
295C22⋊C4—33no—
296C22⋊C4—36no—
297C22⋊C4—36no—
298C22⋊C4—33no—
299C22⋊C4C22×D313yes—
300C22⋊C4C22×D313yes—
301C42—36no—
302C42—33no—
303C42C22×D313yes—
304C22×C3—21no—
305C22×C3—21no—
306C22×C3—21no—
307C22×C3—21no—
308C22×C3—21no—
309C22×C32−1+412yescomplemented
310C22×C3C2511yescharacteristic
311C2×D3—28no—
312C2×D3—21no—
313C2×D3—21no—
314C2×D3—21no—
315C2×D3—21no—
316C2×D3—23no—
317C2×D3—23no—
318C2×D3—23no—
319C2×D3C22×D412yes—
320C2×D32−1+412yescomplemented
321C3×C4—23no—
322C3×C4—23no—
323C3×C4—23no—
324C3×C4—21no—
325C3×C4C22×D416yes—
326Dic3—23no—
327Dic3—21no—
328Dic3—23no—
329Dic3—21no—
330Dic3C22×D416yes—
331C23—31no—
332C23—31no—
333C23—33no—
334C23—62no—
335C23—62no—
336C23—62no—
337C23C23×D311yescharacteristic
338C23C23×D311yescharacteristic
339Q8—26no—
340Q8—26no—
341Q8—312no—
342Q8—612no—
343Q8—66no—
344Q8—66no—
345Q8D3×D414yes—
346D4—22no—
347D4—66no—
348D4—66no—
349D4—66no—
350D4—62no—
351D4—62no—
352D4—66no—
353D4—66no—
354C2×C4—26no—
355C2×C4—26no—
356C2×C4—33no—
357C2×C4—312no—
358C2×C4—33no—
359C2×C4—31no—
360C2×C4—33no—
361C2×C4—31no—
362C2×C4—66no—
363C2×C4—612no—
364C2×C4—66no—
365C2×C4—66no—
366C2×C4—66no—
367C2×C4—66no—
368C2×C4C23×D313yes—
369C2×C4C23×D311yescharacteristic
370C2×C4C23×D313yes—
371C2×C4C23×D313yes—
372C2×C4C23×D313yes—
373C2×C3—22no—
374C2×C3—41no—
375C2×C3C23×D411yescharacteristic
376C2×C3C2×2−1+411yescharacteristic
377C2×C3C2×2−1+411yescharacteristic
378D3—22no—
379D3—22no—
380D3—43no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—32no—
387C22—32no—
388C22—68no—
389C22—61no—
390C22—61no—
391C22—61no—
392C22—61no—
393C22—63no—
394C22—63no—
395C22—63no—
396C22#100712yescomplemented
397C22C24×D311yescharacteristic
398C4—23no—
399C4—23no—
400C4—21no—
401C4—23no—
402C4—36no—
403C4—61no—
404C4—63no—
405C4—63no—
406C4—61no—
407C4C2×D3×D416yes—
408C3#348611yescharacteristic, complemented, Sylow
409C2—22no—
410C2—41no—
411C2—62no—
412C2—62no—
413C2—123no—
414C2C22×D3×D411yescharacteristic
415C2C2×#100711yescharacteristic
416C2D3×2−1+411yescharacteristic
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×D4×Q8C2
hypercenterD4×Q8D3
hypocenterC3#3486
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ D4×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#1007C2semidirect
2. C22×D3×Q8C2semidirect
3. C3×D4×Q8C2semidirect
4. #5817C2semidirect
5. Q8×#67C2semidirect
6. #5727C2semidirect
7. #5724C2semidirect
8. #5633C2semidirect
9. #5632C2semidirect
10. #5628C2semidirect
11. #5627C2semidirect
12. #5576C2semidirect
13. #5572C2semidirect
14. #5552C2semidirect
15. #5547C2semidirect
18. C22×C3×Q8C22semidirect
21. C2×#239C22semidirect
27. C2×D3×Q8C22semidirect
29. C2×#235C22semidirect
32. C3×C4⋊Q8C22semidirect
34. C3×C22⋊Q8C22semidirect
35. C3×C4×Q8C22semidirect
37. #947C22semidirect
39. #946C22semidirect
40. Q8×Dic3C22semidirect
41. #944C22semidirect
47. #924C22semidirect
48. #897C22semidirect
50. #896C22semidirect
51. #896C22semidirect
57. #888C22semidirect
58. #887C22semidirect
59. #885C22semidirect
65. #875C22semidirect
68. C4×Dic6C22semidirect
74. D4×Q8D3semidirect
104. D3×Q8D4semidirect
160. C22×Q8C2×D3semidirect
164. C4⋊Q8C2×D3semidirect
180. C22⋊Q8C2×D3semidirect
190. C4×Q8C2×D3semidirect
309. C22×C32−1+4semidirect
320. C2×D32−1+4semidirect
396. C22#1007semidirect
408. C3#3486semidirect

Maximal subgroups

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

subgroupindex
C2×#10072
C22×D3×Q82
C3×D4×Q82
#58172
Q8×#672
#57272
#57242
#56332
#56322
#56282
#56272
#55762
#55722
#55522
#55472
#34863