Finite Group Databasebeta

← #77784

Subgroups — #77784

Order 384.

604 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 604 conjugacy classes of subgroups — too many to draw. Only its 147 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×#896C212yesmaximal, complemented
2C2×#893C211yesmaximal, characteristic
3C2×#887C211yesmaximal, characteristic, complemented
4C3×#398C211yesmaximal, characteristic, complemented
5#4991C211yesmaximal, characteristic
6#4990C212yesmaximal, complemented
7#4983C211yesmaximal, characteristic, complemented
8#4672C211yesmaximal, characteristic
9#4669C212yesmaximal, complemented
10#4667C212yesmaximal, complemented
11#4650C211yesmaximal, characteristic, complemented
12#2536—31nomaximal, Sylow
13C22×D12C2211yescharacteristic
14C22×C4×D3C2212yes—
15C22×Dic6C2211yescharacteristic
16C2×C3×C4⋊C4C2211yescharacteristic
17C2×C3×C4⋊C4C2211yescharacteristic
18C2×C3×C4⋊C4C2212yes—
19C2×(C2×D3)⋊C4C2212yes—
20C2×(C2×D3)⋊C4C2212yes—
21C2×(C2×D3)⋊C4C2212yes—
22C2×(C2×D3)⋊C4C2212yes—
23C2×(C2×D3)⋊C4C2212yes—
24C2×Dic3⋊C4C2212yes—
25C2×Dic3⋊C4C2211yescharacteristic
26C2×Dic3⋊C4C2212yes—
27C2×Dic3⋊C4C2211yescharacteristic
28C2×C4×Dic3C2211yescharacteristic
29C2×C4×Dic3C2211yescharacteristic
30C2×C4×Dic3C2211yescharacteristic
31#896—28no—
32#893—24no—
33#887—24no—
34C3×#95C2211yescharacteristic
35C3×#95C2212yes—
36C3⋊#95C2212yes—
37C3⋊#95C2211yescharacteristic
38C3⋊#95C2212yes—
39C2×C22⋊Q8—32no—
40C2×C4×Q8—31no—
41C2×C4×D4—31no—
42#398—31no—
43#398D311yescharacteristic, complemented
44#394—31no—
45#394—31no—
46#388—32no—
47#388—32no—
48#382—31no—
49#380—32no—
50C23×D3C2312yes—
51C22×C3×C4C2312yes—
52C22×C3×C4C2312yes—
53C22×C3×C4C2311yescharacteristic
54C22×C3×C4C2311yescharacteristic
55C22×C3×C4C2311yescharacteristic
56C22×Dic3C2312yes—
57C22×Dic3C2311yescharacteristic
58C22×Dic3C2311yescharacteristic
59C22×Dic3C2311yescharacteristic
60C22×Dic3C2311yescharacteristic
61C2×D12—24no—
62C2×D12—22no—
63C2×C4×D3—24no—
64C2×C4×D3—24no—
65C2×C4×D3—24no—
66C2×Dic6—22no—
67C2×Dic6—22no—
68C2×Dic6D414yescomplemented
69C3×C4⋊C4—22no—
70C3×C4⋊C4—22no—
71C3×C4⋊C4—24no—
72(C2×D3)⋊C4—24no—
73(C2×D3)⋊C4—24no—
74(C2×D3)⋊C4—24no—
75(C2×D3)⋊C4—24no—
76(C2×D3)⋊C4—24no—
77Dic3⋊C4—22no—
78Dic3⋊C4—22no—
79Dic3⋊C4—24no—
80Dic3⋊C4—24no—
81C4×Dic3—22no—
82C4×Dic3—22no—
83C4×Dic3D414yes—
84C22×Q8—31no—
85C22×D4—31no—
86C23×C4—32no—
87C22⋊Q8—68no—
88C4×Q8—64no—
89C4×D4—64no—
90C2×C4⋊C4—31no—
91C2×C4⋊C4—31no—
92C2×C4⋊C4—32no—
93C2×C4⋊C4—32no—
94C2×C4⋊C4C2×D311yescharacteristic
95C2×C4⋊C4C2×D311yescharacteristic
96C2×C4⋊C4C2×D312yes—
97C2×C22⋊C4—32no—
98C2×C22⋊C4—32no—
99C2×C22⋊C4—32no—
100C2×C22⋊C4—32no—
101C2×C22⋊C4—32no—
102C2×C42—31no—
103C2×C42—31no—
104C2×C42—31no—
105#95—32no—
106#95—32no—
107#95—31no—
108#95C2×D312yes—
109#95C2×D311yescharacteristic
110C23×C3C2411yescharacteristic
111C22×D3—22no—
112C22×D3—22no—
113C22×D3—22no—
114C22×D3—22no—
115C22×D3—22no—
116C22×D3—22no—
117C22×D3#4114yes—
118C2×C3×C4—21no—
119C2×C3×C4—21no—
120C2×C3×C4—22no—
121C2×C3×C4—21no—
122C2×C3×C4—22no—
123C2×C3×C4—22no—
124C2×C3×C4—22no—
125C2×C3×C4—22no—
126C2×C3×C4—22no—
127C2×C3×C4—22no—
128C2×C3×C4—22no—
129C2×C3×C4C2×D412yes—
130C2×C3×C4C2×D412yes—
131C2×Dic3—21no—
132C2×Dic3—21no—
133C2×Dic3—22no—
134C2×Dic3—21no—
135C2×Dic3—22no—
136C2×Dic3—22no—
137C2×Dic3—22no—
138C2×Dic3—22no—
139C2×Dic3C2×D412yes—
140C2×Dic3C2×D412yes—
141C2×Dic3C2×D412yes—
142C2×Dic3C2×D412yes—
143C2×Dic3#4112yes—
144D12—28no—
145C4×D3—48no—
146Dic6—28no—
147C24—32no—
148C2×Q8—34no—
149C2×Q8—62no—
150C2×Q8—62no—
151C2×D4—64no—
152C2×D4—62no—
153C22×C4—31no—
154C22×C4—31no—
155C22×C4—31no—
156C22×C4—32no—
157C22×C4—31no—
158C22×C4—64no—
159C22×C4—64no—
160C22×C4—64no—
161C22×C4C22×D312yes—
162C22×C4C22×D312yes—
163C22×C4C22×D311yescharacteristic
164C22×C4C22×D311yescharacteristic
165C22×C4C22×D311yescharacteristic
166C4⋊C4—22no—
167C4⋊C4—22no—
168C4⋊C4—24no—
169C4⋊C4—62no—
170C4⋊C4—64no—
171C4⋊C4—64no—
172C4⋊C4—62no—
173C22⋊C4—64no—
174C22⋊C4—64no—
175C22⋊C4—64no—
176C22⋊C4—64no—
177C22⋊C4—64no—
178C42—34no—
179C42—62no—
180C42—62no—
181C22×C3C22×D411yescharacteristic
182C22×C3C22×D411yescharacteristic
183C22×C3C2×#4111yescharacteristic
184C22×C3C2×#4112yes—
185C22×C32−1+412yes—
186C2×D3—24no—
187C2×D3—24no—
188C2×D3—24no—
189C2×D3—42no—
190C2×D3—42no—
191C2×D3—42no—
192C2×D3—42no—
193C3×C4—22no—
194C3×C4—22no—
195C3×C4—42no—
196C3×C4—41no—
197C3×C4—42no—
198Dic3—22no—
199Dic3—22no—
200Dic3—41no—
201Dic3—42no—
202Dic3#12114yes—
203C23—34no—
204C23—62no—
205C23—62no—
206C23—62no—
207C23—62no—
208C23—62no—
209C23—62no—
210C23C23×D311yescharacteristic
211Q8—68no—
212D4—68no—
213C2×C4—21no—
214C2×C4—22no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—32no—
225C2×C4—32no—
226C2×C4—32no—
227C2×C4—32no—
228C2×C4—32no—
229C2×C4—61no—
230C2×C4—62no—
231C2×C4—61no—
232C2×C4—61no—
233C2×C4—62no—
234C2×C4—62no—
235C2×C4—62no—
236C2×C4—62no—
237C2×C4—128no—
238C2×C4D3×D412yes—
239C2×C4D3×D412yes—
240C2×C3C2×#12111yescharacteristic
241C2×C3#52511yescharacteristic
242C2×C3#53611yescharacteristic
243C2×C3#54812yes—
244C2×C3#55212yes—
245D3—44no—
246C22—64no—
247C22—64no—
248C22—64no—
249C22—122no—
250C22—122no—
251C22—122no—
252C22—122no—
253C22C2×D3×D411yescharacteristic
254C22C2×D3×D411yescharacteristic
255C22#100712yes—
256C22D3×#4112yes—
257C22D3×#4111yescharacteristic
258C4—22no—
259C4—22no—
260C4—34no—
261C4—42no—
262C4—41no—
263C4—42no—
264C4—62no—
265C4—62no—
266C4—122no—
267C4—121no—
268C3#253611yescharacteristic, complemented, Sylow
269C2—124no—
270C2D3×#12111yescharacteristic
271C2#563312yes—
272C2#565611yescharacteristic
273C2#569312yes—
274C2#572411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#398C2
prime radicalC22×D12C22
hypercenter#398D3
hypocenterC3#2536
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
1. C2×#896C2semidirect
3. C2×#887C2semidirect
4. C3×#398C2semidirect
6. #4990C2semidirect
7. #4983C2semidirect
9. #4669C2semidirect
10. #4667C2semidirect
11. #4650C2semidirect
43. #398D3semidirect
68. C2×Dic6D4semidirect
268. C3#2536semidirect

Maximal subgroups

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

subgroupindex
C2×#8962
C2×#8932
C2×#8872
C3×#3982
#49912
#49902
#49832
#46722
#46692
#46672
#46502
#25363