Finite Group Databasebeta

← #76978

Subgroups — #76978

Order 384.

512 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 133 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×D4×Dic3C211yesmaximal, characteristic
2C2×#889C211yesmaximal, characteristic, complemented
3C3×#394C211yesmaximal, characteristic
4#5222C211yesmaximal, characteristic
5#4981C211yesmaximal, characteristic, complemented
6#4952C212yesmaximal, complemented
7#4950C212yesmaximal, complemented
8#4949C212yesmaximal, complemented
9#4946C212yesmaximal, complemented
10#4661C211yesmaximal, characteristic, complemented
11#4651C211yesmaximal, characteristic, complemented
12#2633—31nomaximal, Sylow
13C22×C3×D4C2211yescharacteristic
14C23×Dic3C2212yes—
15C2×C3×C4⋊C4C2211yescharacteristic
16C2×C3×C22⋊C4C2212yes—
17C2×C3×C22⋊C4C2212yes—
18C2×#217C2212yes—
19C2×#217C2212yes—
20C2×#217C2212yes—
21D4×Dic3—24no—
22C2×C4⋊Dic3C2211yescharacteristic
23C2×C4⋊Dic3C2211yescharacteristic
24C2×C4⋊Dic3C2212yes—
25C2×Dic3⋊C4C2211yescharacteristic
26C2×Dic3⋊C4C2211yescharacteristic
27C2×Dic3⋊C4C2211yescharacteristic
28C2×Dic3⋊C4C2211yescharacteristic
29C2×Dic3⋊C4C2212yes—
30C2×C4×Dic3C2211yescharacteristic
31#889—24no—
32C3×#95C2211yescharacteristic
33C3⋊#95C2212yes—
34C3⋊#95C2212yes—
35C3⋊#95C2211yescharacteristic
36C3⋊#95C2212yes—
37C3⋊#95C2212yes—
38C2×#125—31no—
39C2×C4×D4—31no—
40#399—32no—
41#398—31no—
42#398—31no—
43#397—32no—
44#394—31no—
45#394D311yescharacteristic
46#385—32no—
47#385—32no—
48#383—31no—
49C24×C3C2312yes—
50C2×C3×D4—24no—
51C2×C3×D4—22no—
52C22×C3×C4C2312yes—
53C22×C3×C4C2311yescharacteristic
54C22×C3×C4C2311yescharacteristic
55C22×C3×C4C2311yescharacteristic
56C22×Dic3—24no—
57C22×Dic3—42no—
58C22×Dic3—42no—
59C22×Dic3C2312yes—
60C22×Dic3C2311yescharacteristic
61C22×Dic3C2312yes—
62C22×Dic3C2311yescharacteristic
63C22×Dic3C2311yescharacteristic
64C22×Dic3C2311yescharacteristic
65C3×C4⋊C4—22no—
66C3×C22⋊C4—24no—
67C3×C22⋊C4—42no—
68#217—24no—
69#217—24no—
70#217—42no—
71C4⋊Dic3—22no—
72C4⋊Dic3—22no—
73C4⋊Dic3—42no—
74Dic3⋊C4—22no—
75Dic3⋊C4—22no—
76Dic3⋊C4—22no—
77Dic3⋊C4—22no—
78Dic3⋊C4—42no—
79C4×Dic3D414yes—
80C22×D4C2×D311yescharacteristic
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—31no—
90C2×C4⋊C4—32no—
91C2×C4⋊C4—32no—
92C2×C4⋊C4C2×D311yescharacteristic
93C2×C22⋊C4—32no—
94C2×C22⋊C4—32no—
95C2×C22⋊C4—32no—
96C2×C22⋊C4C2×D312yes—
97C2×C22⋊C4C2×D312yes—
98C2×C42—31no—
99#95—32no—
100#95—31no—
101#95—32no—
102#95—32no—
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×C3#4114yes—
112C23×C3C2411yescharacteristic
113C3×D4—28no—
114C2×C3×C4—22no—
115C2×C3×C4—21no—
116C2×C3×C4—21no—
117C2×C3×C4—22no—
118C2×C3×C4—22no—
119C2×C3×C4—21no—
120C2×C3×C4—22no—
121C2×C3×C4—21no—
122C2×C3×C4—22no—
123C2×C3×C4C2×D412yes—
124C2×Dic3—21no—
125C2×Dic3—21no—
126C2×Dic3—22no—
127C2×Dic3—21no—
128C2×Dic3—21no—
129C2×Dic3—22no—
130C2×Dic3—22no—
131C2×Dic3—22no—
132C2×Dic3—22no—
133C2×Dic3—22no—
134C2×Dic3—21no—
135C2×Dic3—22no—
136C2×Dic3—21no—
137C2×Dic3—22no—
138C2×Dic3—48no—
139C2×Dic3C2×D412yes—
140C2×Dic3C2×D412yes—
141C24C22×D312yes—
142C2×D4—24no—
143C2×D4—22no—
144C22×C4—31no—
145C22×C4—31no—
146C22×C4—32no—
147C22×C4—31no—
148C22×C4—32no—
149C22×C4—31no—
150C22×C4—64no—
151C22×C4—122no—
152C22×C4—122no—
153C22×C4C22×D312yes—
154C22×C4C22×D311yescharacteristic
155C22×C4C22×D311yescharacteristic
156C22×C4C22×D311yescharacteristic
157C4⋊C4—22no—
158C4⋊C4—62no—
159C4⋊C4—62no—
160C4⋊C4—62no—
161C4⋊C4—62no—
162C4⋊C4—62no—
163C4⋊C4—62no—
164C4⋊C4—122no—
165C4⋊C4—122no—
166C22⋊C4—24no—
167C22⋊C4—42no—
168C22⋊C4—64no—
169C22⋊C4—64no—
170C22⋊C4—122no—
171C42—34no—
172C22×C3—24no—
173C22×C3—24no—
174C22×C3—24no—
175C22×C3—42no—
176C22×C3—42no—
177C22×C3—42no—
178C22×C3—42no—
179C22×C3C22×D411yescharacteristic
180C22×C3C2×#4112yes—
181C22×C3H(5,𝔽2)11yescharacteristic
182C22×C3H(5,𝔽2)11yescharacteristic
183C22×C32−1+412yes—
184C3×C4—22no—
185C3×C4—41no—
186C3×C4—41no—
187C3×C4—42no—
188Dic3—22no—
189Dic3—22no—
190Dic3—41no—
191Dic3—41no—
192Dic3—42no—
193Dic3—42no—
194C23—22no—
195C23—22no—
196C23—22no—
197C23—22no—
198C23—22no—
199C23—22no—
200C23#23714yes—
201C23C23×D311yescharacteristic
202D4—28no—
203C2×C4—22no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—21no—
209C2×C4—22no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—32no—
213C2×C4—32no—
214C2×C4—61no—
215C2×C4—61no—
216C2×C4—62no—
217C2×C4—62no—
218C2×C4—61no—
219C2×C4—61no—
220C2×C4—62no—
221C2×C4—62no—
222C2×C4—61no—
223C2×C4—62no—
224C2×C4—61no—
225C2×C4—62no—
226C2×C4—62no—
227C2×C4—62no—
228C2×C4—128no—
229C2×C4D3×D412yes—
230C2×C3—44no—
231C2×C3#52511yescharacteristic
232C2×C3#53511yescharacteristic
233C2×C3#53611yescharacteristic
234C2×C3#53912yes—
235C2×C3#53912yes—
236C22—24no—
237C22—24no—
238C22—24no—
239C22—42no—
240C22—42no—
241C22—42no—
242C22—42no—
243C22C2×D3×D411yescharacteristic
244C22C2×#23712yes—
245C22#100411yescharacteristic
246C22#100911yescharacteristic
247C22#101012yes—
248C4—22no—
249C4—41no—
250C4—42no—
251C4—41no—
252C4—62no—
253C4—62no—
254C4—122no—
255C4—121no—
256C4—122no—
257C4—121no—
258C3#263311yescharacteristic, complemented, Sylow
259C2—44no—
260C2#558611yescharacteristic
261C2#560212yes—
262C2#560711yescharacteristic
263C2#566512yes—
264C2#567511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#394C2
prime radicalC22×C3×D4C22
hypercenter#394D3
hypocenterC3#2633
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
2. C2×#889C2semidirect
5. #4981C2semidirect
6. #4952C2semidirect
7. #4950C2semidirect
8. #4949C2semidirect
9. #4946C2semidirect
10. #4661C2semidirect
11. #4651C2semidirect
258. C3#2633semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Dic32
C2×#8892
C3×#3942
#52222
#49812
#49522
#49502
#49492
#49462
#46612
#46512
#26333