Finite Group Databasebeta

← #86219

Subgroups — #86219

Order 384.

1212 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#532C211yesmaximal, characteristic, complemented
2#5828C212yesmaximal, complemented
3Dic3×#41C212yesmaximal, complemented
4#5728C212yesmaximal, complemented
5#5716C212yesmaximal, complemented
6#5680C214yesmaximal, complemented
7#5606C214yesmaximal, complemented
8#5600C214yesmaximal, complemented
9#5557C214yesmaximal, complemented
10C4×#237C214yesmaximal, complemented
11C2×C42×D3C211yesmaximal, characteristic, complemented
12#5470C211yesmaximal, characteristic, complemented
13#3444—31nomaximal, Sylow
14C2×C3×#41C2212yes—
15C2×#237C2214yes—
16C22×C4×D3C2211yescharacteristic
17C22×C4×D3C2212yes—
18C3×C4⋊Q8C2211yescharacteristic, complemented
19C3×#127C2211yescharacteristic, complemented
20C3×#124C2212yescomplemented
21C3×#121C2214yes—
22C3×C4×D4C2214yescomplemented
23C2×C3×C42C2211yescharacteristic
24#946C2214yes—
25Q8×Dic3C2214yescomplemented
26#938C2214yes—
27#938C2214yes—
28#937C2214yes—
29#936C2214yescomplemented
30#935C2214yes—
31D4×Dic3C2218yes—
32D4×Dic3C2214yescomplemented
33D4×Dic3C2214yescomplemented
34C4×#67C2218yes—
35#926C2212yes—
36#926C2214yescomplemented
37#924C2214yes—
38C2×C4×Dic3C2211yescharacteristic
39C2×C4×Dic3C2212yes—
40C2×C4×Dic3C2214yescomplemented
41#897C2214yes—
42#892C2214yes—
43#890C2214yescomplemented
44#887C2214yescomplemented
45#883C2218yes—
46#67⋊C4C2218yes—
47#879C2218yescomplemented
48#877C2218yescomplemented
49C42×D3C2212yes—
50C42×D3C2212yes—
51C42×D3C2212yes—
52C42×D3C2212yes—
53#870C2212yescomplemented
54#869C2212yescomplemented
55C4×Dic6C2214yescomplemented
56#533—32no—
57#533—31no—
58#532—32no—
59#532D311yescharacteristic, complemented
60#529—34no—
61#529—34no—
62#529—34no—
63#525—32no—
64#525—34no—
65C4×#41—32no—
66C4×#41—34no—
67C22×C42—31no—
68C23×D3C2311yescharacteristic
69C3×#41—28no—
70C2×C3×Q8C2312yes—
71C2×C3×D4C2314yes—
72C2×C3×D4C2312yes—
73C2×C3×D4C2314yes—
74C22×C3×C4C2312yes—
75C22×C3×C4C2314yes—
76C22×C3×C4C2311yescharacteristic
77C2×#67C2318yes—
78C22×Dic3C2318yes—
79C22×Dic3C2312yes—
80C22×Dic3C2311yescharacteristic
81#237—216no—
82C2×C4×D3—24no—
83C2×C4×D3—28no—
84C2×C4×D3C2312yes—
85C2×C4×D3C2312yes—
86C2×C4×D3C2312yes—
87C2×C4×D3C2312yes—
88C2×C4×D3C2312yes—
89C2×C4×D3C2312yes—
90C2×Dic6C2314yes—
91C3×C4⋊C4C2314yes—
92C3×C22⋊C4C2318yes—
93C3×C42C2312yes—
94C3×C42C2311yescharacteristic
95C3×C42C2311yescharacteristic
96#217C2318yes—
97#217C2318yes—
98#217C2314yes—
99(C2×D3)⋊C4C2318yes—
100C4⋊Dic3C2314yes—
101C4⋊Dic3C2314yes—
102C4⋊Dic3C2318yes—
103Dic3⋊C4C2318yes—
104Dic3⋊C4C2318yes—
105C4×Dic3C2314yes—
106C4×Dic3C2312yes—
107C4×Dic3C2312yes—
108C4×Dic3C2312yes—
109C4×Dic3C2312yes—
110C4×Dic3C2311yescharacteristic
111C4×Dic3C2311yescharacteristic
112C4×Dic3C2312yes—
113C4×Dic3C2318yes—
114C2×#41—34no—
115C2×#41C2×D312yes—
116C23×C4—32no—
117C23×C4—31no—
118C4⋊Q8—32no—
119C4⋊Q8C2×D311yescharacteristic, complemented
120#127C2×D311yescharacteristic, complemented
121#126—38no—
122#125—34no—
123#125—32no—
124#124—34no—
125#124C2×D312yescomplemented
126#123—34no—
127#123—38no—
128C22⋊Q8—34no—
129C22⋊Q8—34no—
130C22⋊Q8—34no—
131#121—34no—
132#121—34no—
133#121C2×D314yes—
134#120—34no—
135C4×Q8—34no—
136C4×Q8—34no—
137C4×Q8—34no—
138C4×D4—38no—
139C4×D4—34no—
140C4×D4—34no—
141C4×D4—38no—
142C4×D4—38no—
143C4×D4C2×D314yescomplemented
144#117—32no—
145#117—34no—
146#117—34no—
147#117—38no—
148C2×C42—34no—
149C2×C42—31no—
150C2×C42—32no—
151C2×C42—32no—
152C2×C42—32no—
153C2×C42—32no—
154C2×C42—32no—
155C2×C42C2×D311yescharacteristic
156C23×C3C2414yes—
157C23×C3C2411yescharacteristic
158C22×D3—22no—
159C22×D3—24no—
160C22×D3C2412yes—
161C3×Q8—24no—
162C3×D4—28no—
163C3×D4—28no—
164C3×D4—24no—
165C2×C3×C4—24no—
166C2×C3×C4—28no—
167C2×C3×C4#4114yescomplemented
168C2×C3×C4C2412yes—
169C2×C3×C4C2414yes—
170C2×C3×C4C2411yescharacteristic
171C2×C3×C4C2412yes—
172C2×C3×C4C2411yescharacteristic
173#67—216no—
174C2×Dic3—22no—
175C2×Dic3—24no—
176C2×Dic3—216no—
177C2×Dic3C2412yes—
178C2×Dic3C2412yes—
179C2×Dic3C2418yes—
180C2×Dic3C2412yes—
181C4×D3—24no—
182C4×D3—24no—
183C4×D3—24no—
184C4×D3—24no—
185C4×D3—24no—
186C4×D3#4118yescomplemented
187Dic6—28no—
188C24—31no—
189#41—28no—
190#41—616no—
191C2×Q8—34no—
192C2×Q8C22×D312yes—
193C2×D4—38no—
194C2×D4C22×D314yes—
195C2×D4C22×D312yes—
196C2×D4C22×D314yes—
197C22×C4—32no—
198C22×C4—32no—
199C22×C4—31no—
200C22×C4—38no—
201C22×C4—32no—
202C22×C4—32no—
203C22×C4—32no—
204C22×C4—32no—
205C22×C4—32no—
206C22×C4—68no—
207C22×C4—64no—
208C22×C4C22×D312yes—
209C22×C4C22×D314yes—
210C22×C4C22×D311yescharacteristic
211C4⋊C4—34no—
212C4⋊C4—34no—
213C4⋊C4—38no—
214C4⋊C4—38no—
215C4⋊C4—38no—
216C4⋊C4C22×D314yes—
217C22⋊C4—34no—
218C22⋊C4—38no—
219C22⋊C4—38no—
220C22⋊C4—38no—
221C22⋊C4C22×D318yes—
222C42—32no—
223C42—32no—
224C42—32no—
225C42—32no—
226C42—32no—
227C42—38no—
228C42—31no—
229C42—31no—
230C42—34no—
231C42C22×D311yescharacteristic
232C42C22×D311yescharacteristic
233C42C22×D312yes—
234C22×C3—22no—
235C22×C3—28no—
236C22×C3—24no—
237C22×C3C2×#4112yes—
238C22×C3C2511yescharacteristic
239C2×D3—28no—
240C2×D3—22no—
241C2×D3—22no—
242C2×D3C2×#4114yes—
243C3×C4—22no—
244C3×C4—24no—
245C3×C4C2×#4112yes—
246C3×C4C2×#4112yes—
247C3×C4C2×#4114yes—
248Dic3—28no—
249Dic3—22no—
250Dic3—22no—
251Dic3C2×#4114yes—
252C23—32no—
253C23—64no—
254C23—62no—
255C23C23×D314yes—
256C23C23×D311yescharacteristic
257Q8—24no—
258Q8—68no—
259D4—28no—
260D4—24no—
261D4—28no—
262D4—616no—
263C2×C4—28no—
264C2×C4—24no—
265C2×C4—32no—
266C2×C4—38no—
267C2×C4—32no—
268C2×C4—32no—
269C2×C4—38no—
270C2×C4—64no—
271C2×C4—64no—
272C2×C4—64no—
273C2×C4—616no—
274C2×C4—62no—
275C2×C4—64no—
276C2×C4—64no—
277C2×C4—64no—
278C2×C4#23714yescomplemented
279C2×C4C23×D312yes—
280C2×C4C23×D314yes—
281C2×C4C23×D311yescharacteristic
282C2×C4C23×D312yes—
283C2×C4C23×D311yescharacteristic
284C2×C3—22no—
285C2×C3—44no—
286C2×C3C22×#4111yescharacteristic
287C2×C3C22×#4112yes—
288D3—24no—
289C22—22no—
290C22—24no—
291C22—28no—
292C22—34no—
293C22—68no—
294C22—62no—
295C22—62no—
296C22C2×#23712yes—
297C22C24×D311yescharacteristic
298C4—24no—
299C4—22no—
300C4—34no—
301C4—62no—
302C4—62no—
303C4—68no—
304C4C2×#23712yes—
305C4C2×#23712yes—
306C4D3×#4114yes—
307C3#344411yescharacteristic, complemented, Sylow
308C2—22no—
309C2—44no—
310C2—64no—
311C2C22×#23711yescharacteristic
312C2C2×D3×#4112yes—
subgroupG/H
centerC2×C4C23×D3
commutator, socle, cosocleC22×C3C25
FrattiniC22C24×D3
FittingC3×#532C2
hypercenter#532D3
hypocenterC3#3444
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ #532

Complements

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

NH ≅ G/Ntype
1. C3×#532C2semidirect
2. #5828C2semidirect
3. Dic3×#41C2semidirect
4. #5728C2semidirect
5. #5716C2semidirect
6. #5680C2semidirect
7. #5606C2semidirect
8. #5600C2semidirect
9. #5557C2semidirect
10. C4×#237C2semidirect
11. C2×C42×D3C2semidirect
12. #5470C2semidirect
18. C3×C4⋊Q8C22semidirect
19. C3×#127C22semidirect
20. C3×#124C22semidirect
22. C3×C4×D4C22semidirect
25. Q8×Dic3C22semidirect
29. #936C22semidirect
32. D4×Dic3C22semidirect
33. D4×Dic3C22semidirect
36. #926C22semidirect
40. C2×C4×Dic3C22semidirect
43. #890C22semidirect
44. #887C22semidirect
47. #879C22semidirect
48. #877C22semidirect
53. #870C22semidirect
54. #869C22semidirect
55. C4×Dic6C22semidirect
59. #532D3semidirect
119. C4⋊Q8C2×D3semidirect
120. #127C2×D3semidirect
125. #124C2×D3semidirect
143. C4×D4C2×D3semidirect
167. C2×C3×C4#41semidirect
186. C4×D3#41semidirect
278. C2×C4#237semidirect
307. C3#3444semidirect

Maximal subgroups

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

subgroupindex
C3×#5322
#58282
Dic3×#412
#57282
#57162
#56802
#56062
#56002
#55572
C4×#2372
C2×C42×D32
#54702
#34443