Finite Group Databasebeta

← #77172

Subgroups — #77172

Order 384.

840 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 840 conjugacy classes of subgroups — too many to draw. Only its 171 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D3×Q8C211yesmaximal, characteristic, complemented
2C2×#896C212yesmaximal, complemented
3C2×#893C211yesmaximal, characteristic
4C3×#395C211yesmaximal, characteristic, complemented
5#5231C211yesmaximal, characteristic, complemented
6#4989C212yesmaximal, complemented
7#4979C211yesmaximal, characteristic, complemented
8#4673C211yesmaximal, characteristic, complemented
9#4667C211yesmaximal, characteristic, complemented
10#4665C212yesmaximal
11#4646C212yesmaximal, complemented
12#2409—31nomaximal, Sylow
13C22×C3×Q8C2211yescharacteristic
14C2×D3×Q8—24no—
15C2×D3×Q8—28no—
16C22×D12C2211yescharacteristic
17C22×C4×D3C2212yes—
18C22×C4×D3C2211yescharacteristic
19C22×C4×D3C2211yescharacteristic
20C22×Dic6C2211yescharacteristic
21C22×Dic6C2212yes—
22C2×C3×C4⋊C4C2211yescharacteristic
23C2×C3×C4⋊C4C2212yes—
24C2×(C2×D3)⋊C4C2212yes—
25C2×(C2×D3)⋊C4C2212yes—
26C2×(C2×D3)⋊C4C2212yes—
27C2×(C2×D3)⋊C4C2211yescharacteristic
28C2×(C2×D3)⋊C4C2211yescharacteristic
29C2×Dic3⋊C4C2212yes—
30C2×Dic3⋊C4C2212yes—
31C2×C4×Dic3C2211yescharacteristic
32#896—28no—
33#893—24no—
34C3×#95C2212yes—
35C3×#95C2211yescharacteristic
36C3⋊#95C2212yes—
37C3⋊#95C2212yes—
38C3⋊#95C2212yes—
39C3⋊#95C2211yescharacteristic
40C23×Q8—31no—
41C2×C22⋊Q8—32no—
42C2×C4×D4—31no—
43#395—31no—
44#395D311yescharacteristic, complemented
45#393—31no—
46#393—31no—
47#391—32no—
48#386—32no—
49#385—32no—
50#380—31no—
51C23×D3C2311yescharacteristic
52C23×D3C2311yescharacteristic
53C2×C3×Q8—22no—
54C2×C3×Q8—24no—
55C22×C3×C4C2312yes—
56C22×C3×C4C2311yescharacteristic
57C22×C3×C4C2311yescharacteristic
58C22×C3×C4C2312yes—
59C22×C3×C4C2311yescharacteristic
60C22×Dic3C2311yescharacteristic
61C22×Dic3C2312yes—
62C22×Dic3C2312yes—
63C22×Dic3C2311yescharacteristic
64D3×Q8—232no—
65C2×D12—22no—
66C2×D12—22no—
67C2×D12Q814yescomplemented
68C2×C4×D3—24no—
69C2×C4×D3—24no—
70C2×C4×D3—24no—
71C2×C4×D3—24no—
72C2×C4×D3—22no—
73C2×C4×D3—24no—
74C2×C4×D3D414yes—
75C2×Dic6—22no—
76C2×Dic6—24no—
77C2×Dic6—24no—
78C2×Dic6—24no—
79C2×Dic6D418yescomplemented
80C3×C4⋊C4—22no—
81C3×C4⋊C4—24no—
82(C2×D3)⋊C4—22no—
83(C2×D3)⋊C4—24no—
84(C2×D3)⋊C4—22no—
85(C2×D3)⋊C4—24no—
86(C2×D3)⋊C4—24no—
87Dic3⋊C4—24no—
88Dic3⋊C4—24no—
89C4×Dic3—22no—
90C22×Q8—32no—
91C22×Q8—31no—
92C22×Q8—64no—
93C22×Q8—68no—
94C22×Q8C2×D311yescharacteristic
95C22×D4—31no—
96C23×C4—31no—
97C23×C4—31no—
98C23×C4—32no—
99C22⋊Q8—68no—
100C4×D4—64no—
101C2×C4⋊C4—32no—
102C2×C4⋊C4—32no—
103C2×C4⋊C4C2×D311yescharacteristic
104C2×C4⋊C4C2×D312yes—
105C2×C22⋊C4—31no—
106C2×C22⋊C4—31no—
107C2×C22⋊C4—32no—
108C2×C22⋊C4—32no—
109C2×C22⋊C4—32no—
110C2×C42—31no—
111#95—31no—
112#95—32no—
113#95—32no—
114#95—32no—
115#95C2×D311yescharacteristic
116#95C2×D312yes—
117C23×C3C2411yescharacteristic
118C22×D3—22no—
119C22×D3—22no—
120C22×D3—22no—
121C22×D3—21no—
122C22×D3—21no—
123C22×D3—22no—
124C22×D3C2×D412yes—
125C22×D3C2×Q812yes—
126C22×D3C2×Q812yescomplemented
127C22×D3#4112yes—
128C3×Q8—28no—
129C2×C3×C4—21no—
130C2×C3×C4—22no—
131C2×C3×C4—22no—
132C2×C3×C4—22no—
133C2×C3×C4—21no—
134C2×C3×C4—22no—
135C2×C3×C4—22no—
136C2×C3×C4—21no—
137C2×C3×C4—22no—
138C2×C3×C4—22no—
139C2×C3×C4C2×D412yes—
140C2×C3×C4C2×D414yes—
141C2×C3×C4C2×Q812yescomplemented
142C2×Dic3—22no—
143C2×Dic3—22no—
144C2×Dic3—21no—
145C2×Dic3—22no—
146C2×Dic3—22no—
147C2×Dic3—22no—
148C2×Dic3—22no—
149C2×Dic3C2×D414yes—
150C2×Dic3C2×D412yes—
151C2×Dic3C2×D414yes—
152D12—28no—
153C4×D3—216no—
154C4×D3—28no—
155C4×D3—44no—
156Dic6—28no—
157Dic6—216no—
158C24—31no—
159C24—31no—
160C2×Q8—22no—
161C2×Q8—24no—
162C2×Q8—38no—
163C2×Q8—64no—
164C2×Q8—64no—
165C2×Q8—64no—
166C2×Q8—632no—
167C2×Q8—62no—
168C2×D4—34no—
169C2×D4—62no—
170C2×D4—62no—
171C22×C4—31no—
172C22×C4—32no—
173C22×C4—34no—
174C22×C4—31no—
175C22×C4—32no—
176C22×C4—64no—
177C22×C4—64no—
178C22×C4—62no—
179C22×C4—64no—
180C22×C4—64no—
181C22×C4—64no—
182C22×C4C22×D312yes—
183C22×C4C22×D311yescharacteristic
184C22×C4C22×D311yescharacteristic
185C22×C4C22×D312yes—
186C22×C4C22×D311yescharacteristic
187C4⋊C4—22no—
188C4⋊C4—24no—
189C4⋊C4—64no—
190C4⋊C4—64no—
191C22⋊C4—64no—
192C22⋊C4—62no—
193C22⋊C4—64no—
194C22⋊C4—62no—
195C22⋊C4—64no—
196C42—62no—
197C22×C3C22×D411yescharacteristic
198C22×C3C22×D412yes—
199C22×C3C22×Q811yescharacteristic
200C22×C3C2×#4111yescharacteristic
201C22×C32−1+412yes—
202C2×D3—22no—
203C2×D3—22no—
204C2×D3—24no—
205C2×D3—24no—
206C2×D3—22no—
207C2×D3—24no—
208C2×D3—41no—
209C2×D3—41no—
210C2×D3—42no—
211C2×D3C22⋊Q814yescomplemented
212C3×C4—22no—
213C3×C4—24no—
214C3×C4—22no—
215C3×C4—41no—
216C3×C4—42no—
217Dic3—24no—
218Dic3—41no—
219Dic3—42no—
220Dic3#12014yes—
221C23—32no—
222C23—32no—
223C23—32no—
224C23—32no—
225C23—62no—
226C23—62no—
227C23—62no—
228C23—61no—
229C23—62no—
230C23—61no—
231C23C23×D311yescharacteristic
232Q8—28no—
233Q8—616no—
234Q8—68no—
235D4—68no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—32no—
247C2×C4—34no—
248C2×C4—34no—
249C2×C4—68no—
250C2×C4—616no—
251C2×C4—62no—
252C2×C4—62no—
253C2×C4—61no—
254C2×C4—62no—
255C2×C4—62no—
256C2×C4—62no—
257C2×C4—62no—
258C2×C4—124no—
259C2×C4D3×D412yes—
260C2×C4D3×D414yes—
261C2×C4D3×Q812yescomplemented
262C2×C3C2×#12011yescharacteristic
263C2×C3C2×C22⋊Q811yescharacteristic
264C2×C3#53611yescharacteristic
265C2×C3#54812yes—
266C2×C3D4×Q812yes—
267D3—24no—
268D3—42no—
269C22—34no—
270C22—64no—
271C22—62no—
272C22—64no—
273C22—62no—
274C22—64no—
275C22—62no—
276C22—122no—
277C22—121no—
278C22—121no—
279C22C2×D3×D412yes—
280C22C2×D3×D411yescharacteristic
281C22C2×D3×Q811yescharacteristic
282C22#100712yes—
283C22D3×#4111yescharacteristic
284C4—22no—
285C4—24no—
286C4—22no—
287C4—34no—
288C4—42no—
289C4—41no—
290C4—64no—
291C4—122no—
292C4—121no—
293C3#240911yescharacteristic, complemented, Sylow
294C2—64no—
295C2—122no—
296C2D3×#12011yescharacteristic
297C2D3×C22⋊Q811yescharacteristic
298C2#563312yes—
299C2#572411yescharacteristic
300C2#572712yes—
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#395C2
prime radicalC22×D12C22
hypercenter#395D3
hypocenterC3#2409
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
1. C22×D3×Q8C2semidirect
2. C2×#896C2semidirect
4. C3×#395C2semidirect
5. #5231C2semidirect
6. #4989C2semidirect
7. #4979C2semidirect
8. #4673C2semidirect
9. #4667C2semidirect
11. #4646C2semidirect
44. #395D3semidirect
67. C2×D12Q8semidirect
79. C2×Dic6D4semidirect
126. C22×D3C2×Q8semidirect
141. C2×C3×C4C2×Q8semidirect
211. C2×D3C22⋊Q8semidirect
261. C2×C4D3×Q8semidirect
293. C3#2409semidirect

Maximal subgroups

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

subgroupindex
C22×D3×Q82
C2×#8962
C2×#8932
C3×#3952
#52312
#49892
#49792
#46732
#46672
#46652
#46462
#24093