Finite Group Databasebeta

← #76944

Subgroups — #76944

Order 384.

788 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 788 conjugacy classes of subgroups — too many to draw. Only its 161 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×#939C212yesmaximal, complemented
2C2×#884C212yesmaximal, complemented
3C2×#67⋊C4C212yesmaximal, complemented
4C3×#394C211yesmaximal, characteristic, complemented
5#4991C211yesmaximal, characteristic, complemented
6#4975C211yesmaximal, characteristic, complemented
7#4961C212yesmaximal, complemented
8#4942C212yesmaximal, complemented
9#4667C212yesmaximal, complemented
10#2537—31nomaximal, Sylow
11C22×C3×D4C2211yescharacteristic
12C22×#67C2212yes—
13C22×#67C2212yes—
14C23×Dic3C2212yes—
15C22×D12C2211yescharacteristic
16C22×C4×D3C2212yes—
17C2×C3×C4⋊C4C2211yescharacteristic
18C2×C3×C22⋊C4C2212yes—
19C2×C3×C22⋊C4C2212yes—
20C2×#217C2212yes—
21#939—28no—
22C2×(C2×D3)⋊C4C2212yes—
23C2×(C2×D3)⋊C4C2212yes—
24C2×(C2×D3)⋊C4C2212yes—
25C2×C4⋊Dic3C2211yescharacteristic
26C2×Dic3⋊C4C2212yes—
27C2×Dic3⋊C4C2212yes—
28C2×Dic3⋊C4C2212yes—
29C2×C4×Dic3C2211yescharacteristic
30#884—28no—
31#67⋊C4—28no—
32C3×#95C2211yescharacteristic
33C3⋊#95C2212yes—
34C3⋊#95C2211yescharacteristic
35C2×#121—32no—
36C2×#121—32no—
37C2×C4×D4—32no—
38#394—31no—
39#394—32no—
40#394D311yescharacteristic, complemented
41#384—31no—
42#380—32no—
43#380—32no—
44C24×C3C2312yes—
45C23×D3C2312yes—
46C2×C3×D4—24no—
47C2×C3×D4—41no—
48C22×C3×C4C2312yes—
49C22×C3×C4C2311yescharacteristic
50C22×C3×C4C2311yescharacteristic
51C22×C3×C4C2311yescharacteristic
52C2×#67—24no—
53C2×#67—24no—
54C2×#67—24no—
55C2×#67—24no—
56C2×#67—42no—
57C2×#67D418yescomplemented
58C22×Dic3—24no—
59C22×Dic3—24no—
60C22×Dic3—24no—
61C22×Dic3C2312yes—
62C22×Dic3C2312yes—
63C22×Dic3C2312yes—
64C2×D12—24no—
65C2×D12—41no—
66C2×C4×D3—24no—
67C2×C4×D3—24no—
68C2×C4×D3—24no—
69C3×C4⋊C4—41no—
70C3×C22⋊C4—24no—
71C3×C22⋊C4—24no—
72#217—24no—
73(C2×D3)⋊C4—24no—
74(C2×D3)⋊C4—24no—
75(C2×D3)⋊C4—24no—
76C4⋊Dic3—41no—
77Dic3⋊C4—24no—
78Dic3⋊C4—24no—
79Dic3⋊C4—24no—
80C4×Dic3D414yescomplemented
81C22×D4—32no—
82C22×D4—32no—
83C22×D4—31no—
84C22×D4C2×D311yescharacteristic
85C23×C4—32no—
86C23×C4—32no—
87#121—68no—
88#121—68no—
89C4×D4—68no—
90C2×C4⋊C4—32no—
91C2×C4⋊C4—32no—
92C2×C4⋊C4—32no—
93C2×C4⋊C4—31no—
94C2×C4⋊C4C2×D311yescharacteristic
95C2×C22⋊C4—32no—
96C2×C22⋊C4—32no—
97C2×C22⋊C4—32no—
98C2×C22⋊C4—32no—
99C2×C22⋊C4C2×D312yes—
100C2×C22⋊C4C2×D312yes—
101C2×C42—31no—
102#95—31no—
103#95—32no—
104#95C2×D311yescharacteristic
105C23×C3—22no—
106C23×C3—22no—
107C23×C3—22no—
108C23×C3—22no—
109C23×C3—22no—
110C23×C3—22no—
111C23×C3C2×D414yes—
112C23×C3C2411yescharacteristic
113C22×D3—22no—
114C22×D3—22no—
115C22×D3—22no—
116C22×D3—22no—
117C22×D3—22no—
118C22×D3—22no—
119C22×D3C2×D414yes—
120C3×D4—44no—
121C2×C3×C4—21no—
122C2×C3×C4—21no—
123C2×C3×C4—22no—
124C2×C3×C4—22no—
125C2×C3×C4—22no—
126C2×C3×C4—22no—
127C2×C3×C4—22no—
128C2×C3×C4—21no—
129C2×C3×C4—21no—
130C2×C3×C4C2×D412yes—
131#67—216no—
132#67—48no—
133C2×Dic3—22no—
134C2×Dic3—22no—
135C2×Dic3—22no—
136C2×Dic3—22no—
137C2×Dic3—22no—
138C2×Dic3—22no—
139C2×Dic3—48no—
140C2×Dic3C2×D414yes—
141C2×Dic3C2×D414yes—
142C2×Dic3#4114yes—
143D12—44no—
144C4×D3—48no—
145C24—32no—
146C24C22×D312yes—
147C2×D4—24no—
148C2×D4—38no—
149C2×D4—41no—
150C2×D4—64no—
151C2×D4—64no—
152C2×D4—64no—
153C2×D4—64no—
154C2×D4—64no—
155C2×D4—121no—
156C2×D4—122no—
157C22×C4—32no—
158C22×C4—32no—
159C22×C4—32no—
160C22×C4—64no—
161C22×C4—64no—
162C22×C4—64no—
163C22×C4—64no—
164C22×C4—64no—
165C22×C4—64no—
166C22×C4C22×D312yes—
167C22×C4C22×D311yescharacteristic
168C22×C4C22×D311yescharacteristic
169C22×C4C22×D311yescharacteristic
170C4⋊C4—41no—
171C4⋊C4—64no—
172C4⋊C4—64no—
173C4⋊C4—64no—
174C4⋊C4—121no—
175C22⋊C4—24no—
176C22⋊C4—24no—
177C22⋊C4—64no—
178C22⋊C4—64no—
179C22⋊C4—64no—
180C22⋊C4—64no—
181C42—34no—
182C22×C3—24no—
183C22×C3—24no—
184C22×C3—24no—
185C22×C3—42no—
186C22×C3—42no—
187C22×C3—42no—
188C22×C3—42no—
189C22×C3C22×D411yescharacteristic
190C22×C3C22×D412yes—
191C22×C3C2×#4112yes—
192C22×C3H(5,𝔽2)11yescharacteristic
193C22×C32−1+411yescharacteristic
194C2×D3—24no—
195C2×D3—24no—
196C2×D3—24no—
197C2×D3—42no—
198C2×D3—42no—
199C2×D3—42no—
200C2×D3—42no—
201C3×C4—22no—
202C3×C4—42no—
203C3×C4—41no—
204C3×C4—41no—
205Dic3—42no—
206Dic3—42no—
207Dic3#12118yescomplemented
208C23—22no—
209C23—22no—
210C23—22no—
211C23—22no—
212C23—22no—
213C23—22no—
214C23—34no—
215C23—62no—
216C23—62no—
217C23—62no—
218C23—62no—
219C23—62no—
220C23—62no—
221C23D3×D414yes—
222C23C23×D311yescharacteristic
223D4—44no—
224D4—616no—
225D4—124no—
226D4—128no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—22no—
236C2×C4—34no—
237C2×C4—34no—
238C2×C4—34no—
239C2×C4—62no—
240C2×C4—62no—
241C2×C4—62no—
242C2×C4—62no—
243C2×C4—62no—
244C2×C4—62no—
245C2×C4—128no—
246C2×C4—128no—
247C2×C4D3×D412yes—
248C2×C3—44no—
249C2×C3C2×#12112yes—
250C2×C3#53711yescharacteristic
251C2×C3D4211yescharacteristic
252C2×C3#54712yes—
253C2×C3#55511yescharacteristic
254D3—44no—
255C22—24no—
256C22—24no—
257C22—24no—
258C22—42no—
259C22—42no—
260C22—42no—
261C22—42no—
262C22—64no—
263C22—64no—
264C22—64no—
265C22—122no—
266C22—122no—
267C22—122no—
268C22—122no—
269C22C2×D3×D412yes—
270C22C2×D3×D411yescharacteristic
271C22#100411yescharacteristic
272C22D3×#4112yes—
273C22#101011yescharacteristic
274C4—22no—
275C4—38no—
276C4—41no—
277C4—41no—
278C4—42no—
279C4—122no—
280C4—122no—
281C3#253711yescharacteristic, complemented, Sylow
282C2—44no—
283C2—124no—
284C2#559011yescharacteristic
285C2#559711yescharacteristic
286C2D3×#12112yes—
287C2#565512yes—
288C2#567011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#394C2
hypercenter#394D3
hypocenterC3#2537
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C2×#939C2semidirect
2. C2×#884C2semidirect
3. C2×#67⋊C4C2semidirect
4. C3×#394C2semidirect
5. #4991C2semidirect
6. #4975C2semidirect
7. #4961C2semidirect
8. #4942C2semidirect
9. #4667C2semidirect
40. #394D3semidirect
57. C2×#67D4semidirect
80. C4×Dic3D4semidirect
207. Dic3#121semidirect
281. C3#2537semidirect

Maximal subgroups

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

subgroupindex
C2×#9392
C2×#8842
C2×#67⋊C42
C3×#3942
#49912
#49752
#49612
#49422
#46672
#25373