Finite Group Databasebeta

← #77822

Subgroups — #77822

Order 384.

512 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 129 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
1C2×#893C211yesmaximal, characteristic
2C2×#889C211yesmaximal, characteristic, complemented
3C3×#398C211yesmaximal, characteristic, complemented
4#4993C212yesmaximal, complemented
5#4991C211yesmaximal, characteristic
6#4989C212yesmaximal, complemented
7#4983C211yesmaximal, characteristic, complemented
8#4675C212yesmaximal, complemented
9#4672C211yesmaximal, characteristic
10#4668C212yesmaximal, complemented
11#4651C211yesmaximal, characteristic, complemented
12#2633—31nomaximal, Sylow
13C22×D12C2211yescharacteristic
14C22×C4×D3C2212yes—
15C2×C3×C4⋊C4C2211yescharacteristic
16C2×C3×C4⋊C4C2211yescharacteristic
17C2×C3×C4⋊C4C2212yes—
18C2×(C2×D3)⋊C4C2212yes—
19C2×(C2×D3)⋊C4C2212yes—
20C2×(C2×D3)⋊C4C2212yes—
21C2×(C2×D3)⋊C4C2212yes—
22C2×(C2×D3)⋊C4C2212yes—
23C2×C4⋊Dic3C2211yescharacteristic
24C2×Dic3⋊C4C2211yescharacteristic
25C2×Dic3⋊C4C2211yescharacteristic
26C2×Dic3⋊C4C2211yescharacteristic
27C2×Dic3⋊C4C2211yescharacteristic
28C2×Dic3⋊C4C2212yes—
29C2×C4×Dic3C2211yescharacteristic
30#893—24no—
31#889—24no—
32C3×#95C2211yescharacteristic
33C3×#95C2212yes—
34C3⋊#95C2212yes—
35C3⋊#95C2212yes—
36C3⋊#95C2211yescharacteristic
37C3⋊#95C2212yes—
38C2×#125—31no—
39C2×C4×D4—31no—
40#399—32no—
41#398—31no—
42#398D311yescharacteristic, complemented
43#397—32no—
44#394—31no—
45#394—31no—
46#385—32no—
47#385—32no—
48#383—31no—
49C23×D3C2312yes—
50C22×C3×C4C2312yes—
51C22×C3×C4C2312yes—
52C22×C3×C4C2311yescharacteristic
53C22×C3×C4C2311yescharacteristic
54C22×C3×C4C2311yescharacteristic
55C22×Dic3C2312yes—
56C22×Dic3C2311yescharacteristic
57C22×Dic3C2311yescharacteristic
58C22×Dic3C2311yescharacteristic
59C22×Dic3C2311yescharacteristic
60C2×D12—24no—
61C2×D12—22no—
62C2×C4×D3—24no—
63C2×C4×D3—42no—
64C2×C4×D3—42no—
65C3×C4⋊C4—22no—
66C3×C4⋊C4—22no—
67C3×C4⋊C4—42no—
68(C2×D3)⋊C4—24no—
69(C2×D3)⋊C4—24no—
70(C2×D3)⋊C4—24no—
71(C2×D3)⋊C4—42no—
72(C2×D3)⋊C4—42no—
73C4⋊Dic3—22no—
74Dic3⋊C4—22no—
75Dic3⋊C4—22no—
76Dic3⋊C4—22no—
77Dic3⋊C4—22no—
78Dic3⋊C4—42no—
79C4×Dic3D414yes—
80C22×D4—31no—
81C23×C4—32no—
82#125—64no—
83C4×D4—64no—
84C2×C4⋊C4—31no—
85C2×C4⋊C4—31no—
86C2×C4⋊C4—31no—
87C2×C4⋊C4—31no—
88C2×C4⋊C4—31no—
89C2×C4⋊C4—32no—
90C2×C4⋊C4C2×D311yescharacteristic
91C2×C4⋊C4C2×D311yescharacteristic
92C2×C4⋊C4C2×D312yes—
93C2×C22⋊C4—32no—
94C2×C22⋊C4—32no—
95C2×C22⋊C4—32no—
96C2×C22⋊C4—32no—
97C2×C22⋊C4—32no—
98C2×C42—31no—
99#95—32no—
100#95—31no—
101#95—32no—
102#95—32no—
103#95C2×D311yescharacteristic
104#95C2×D312yes—
105C23×C3C2411yescharacteristic
106C22×D3—22no—
107C22×D3—22no—
108C22×D3—22no—
109C22×D3—22no—
110C22×D3—22no—
111C22×D3—22no—
112C22×D3#4114yes—
113C2×C3×C4—22no—
114C2×C3×C4—21no—
115C2×C3×C4—21no—
116C2×C3×C4—22no—
117C2×C3×C4—22no—
118C2×C3×C4—22no—
119C2×C3×C4—22no—
120C2×C3×C4—21no—
121C2×C3×C4—22no—
122C2×C3×C4—21no—
123C2×C3×C4—22no—
124C2×C3×C4—22no—
125C2×C3×C4C2×D412yes—
126C2×Dic3—21no—
127C2×Dic3—21no—
128C2×Dic3—22no—
129C2×Dic3—21no—
130C2×Dic3—21no—
131C2×Dic3—22no—
132C2×Dic3—22no—
133C2×Dic3—21no—
134C2×Dic3—22no—
135C2×Dic3—21no—
136C2×Dic3—22no—
137C2×Dic3C2×D412yes—
138C2×Dic3C2×D412yes—
139D12—28no—
140C4×D3—48no—
141C24—32no—
142C2×D4—64no—
143C2×D4—62no—
144C22×C4—31no—
145C22×C4—31no—
146C22×C4—32no—
147C22×C4—31no—
148C22×C4—31no—
149C22×C4—64no—
150C22×C4—122no—
151C22×C4—122no—
152C22×C4C22×D312yes—
153C22×C4C22×D312yes—
154C22×C4C22×D311yescharacteristic
155C22×C4C22×D311yescharacteristic
156C22×C4C22×D311yescharacteristic
157C4⋊C4—22no—
158C4⋊C4—22no—
159C4⋊C4—42no—
160C4⋊C4—62no—
161C4⋊C4—62no—
162C4⋊C4—62no—
163C4⋊C4—62no—
164C4⋊C4—62no—
165C4⋊C4—122no—
166C22⋊C4—64no—
167C22⋊C4—64no—
168C22⋊C4—64no—
169C22⋊C4—122no—
170C22⋊C4—122no—
171C42—34no—
172C22×C3C22×D411yescharacteristic
173C22×C3C2×#4112yes—
174C22×C3H(5,𝔽2)11yescharacteristic
175C22×C3H(5,𝔽2)11yescharacteristic
176C22×C32−1+412yes—
177C2×D3—24no—
178C2×D3—24no—
179C2×D3—24no—
180C2×D3—42no—
181C2×D3—42no—
182C2×D3—42no—
183C2×D3—42no—
184C3×C4—22no—
185C3×C4—41no—
186C3×C4—42no—
187C3×C4—41no—
188C3×C4—42no—
189Dic3—22no—
190Dic3—22no—
191Dic3—41no—
192Dic3—41no—
193Dic3—42no—
194C23—34no—
195C23—62no—
196C23—62no—
197C23—62no—
198C23—62no—
199C23—62no—
200C23—62no—
201C23C23×D311yescharacteristic
202D4—68no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—22no—
208C2×C4—21no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—21no—
214C2×C4—22no—
215C2×C4—32no—
216C2×C4—32no—
217C2×C4—61no—
218C2×C4—62no—
219C2×C4—61no—
220C2×C4—61no—
221C2×C4—61no—
222C2×C4—61no—
223C2×C4—62no—
224C2×C4—62no—
225C2×C4—62no—
226C2×C4—62no—
227C2×C4—61no—
228C2×C4—128no—
229C2×C4D3×D412yes—
230C2×C3#52511yescharacteristic
231C2×C3#53511yescharacteristic
232C2×C3#53611yescharacteristic
233C2×C3#53912yes—
234C2×C3#53912yes—
235D3—44no—
236C22—64no—
237C22—64no—
238C22—64no—
239C22—122no—
240C22—122no—
241C22—122no—
242C22—122no—
243C22C2×D3×D411yescharacteristic
244C22#100411yescharacteristic
245C22#100712yes—
246C22D3×#4112yes—
247C22#100911yescharacteristic
248C4—22no—
249C4—42no—
250C4—41no—
251C4—42no—
252C4—41no—
253C4—62no—
254C4—62no—
255C4—121no—
256C4—122no—
257C4—121no—
258C3#263311yescharacteristic, complemented, Sylow
259C2—124no—
260C2#560711yescharacteristic
261C2#563812yes—
262C2#565611yescharacteristic
263C2#569212yes—
264C2#572411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#398C2
prime radicalC22×D12C22
hypercenter#398D3
hypocenterC3#2633
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
2. C2×#889C2semidirect
3. C3×#398C2semidirect
4. #4993C2semidirect
6. #4989C2semidirect
7. #4983C2semidirect
8. #4675C2semidirect
10. #4668C2semidirect
11. #4651C2semidirect
42. #398D3semidirect
258. C3#2633semidirect

Maximal subgroups

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

subgroupindex
C2×#8932
C2×#8892
C3×#3982
#49932
#49912
#49892
#49832
#46752
#46722
#46682
#46512
#26333