Finite Group Databasebeta

← #76134

Subgroups — #76134

Order 384.

528 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 141 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
2C3×#389C211yesmaximal, characteristic, complemented
3#4994C212yesmaximal, complemented
4#4993C212yesmaximal, complemented
5#4989C212yesmaximal, complemented
6#4939C211yesmaximal, characteristic
7#4929C211yesmaximal, characteristic, complemented
8#4665C212yesmaximal
9#4657C212yesmaximal, complemented
10#2732—31nomaximal, Sylow
11C22×D12C2211yescharacteristic
12C22×C4×D3C2212yes—
13C22×Dic6C2211yescharacteristic
14C2×C3×C4⋊C4C2211yescharacteristic
15C2×C3×C4⋊C4C2212yes—
16C2×C3×C4⋊C4C2211yescharacteristic
17C2×C3×C42C2211yescharacteristic
18C2×(C2×D3)⋊C4C2212yes—
19C2×(C2×D3)⋊C4C2212yes—
20C2×(C2×D3)⋊C4C2212yes—
21C2×(C2×D3)⋊C4C2212yes—
22C2×(C2×D3)⋊C4C2212yes—
23C2×C4⋊Dic3C2211yescharacteristic
24C2×C4⋊Dic3C2211yescharacteristic
25C2×Dic3⋊C4C2212yes—
26C2×Dic3⋊C4C2212yes—
27C2×Dic3⋊C4C2212yes—
28#896—28no—
29C3×#95C2212yes—
30C3⋊#95C2212yes—
31C3⋊#95C2212yes—
32C3⋊#95C2212yes—
33C2×C22⋊Q8—32no—
34#399—32no—
35#398—32no—
36#396—32no—
37#391—31no—
38#390—31no—
39#389D311yescharacteristic, complemented
40#386—32no—
41#385—32no—
42C23×D3C2312yes—
43C22×C3×C4C2312yes—
44C22×C3×C4C2312yes—
45C22×C3×C4C2311yescharacteristic
46C22×C3×C4C2311yescharacteristic
47C22×C3×C4C2311yescharacteristic
48C22×Dic3C2312yes—
49C22×Dic3C2312yes—
50C22×Dic3C2312yes—
51C2×D12—41no—
52C2×D12—41no—
53C2×D12Q814yescomplemented
54C2×C4×D3—24no—
55C2×C4×D3—42no—
56C2×C4×D3—42no—
57C2×Dic6—41no—
58C2×Dic6—41no—
59C2×Dic6D414yescomplemented
60C3×C4⋊C4—24no—
61C3×C4⋊C4—22no—
62C3×C4⋊C4—41no—
63C3×C42—41no—
64(C2×D3)⋊C4—24no—
65(C2×D3)⋊C4—24no—
66(C2×D3)⋊C4—24no—
67(C2×D3)⋊C4—42no—
68(C2×D3)⋊C4—42no—
69C4⋊Dic3—41no—
70C4⋊Dic3—41no—
71Dic3⋊C4—24no—
72Dic3⋊C4—24no—
73Dic3⋊C4—42no—
74C22×Q8—31no—
75C22×D4—31no—
76C23×C4—32no—
77C22⋊Q8—68no—
78C2×C4⋊C4—32no—
79C2×C4⋊C4—32no—
80C2×C4⋊C4—31no—
81C2×C4⋊C4—31no—
82C2×C4⋊C4—32no—
83C2×C4⋊C4C2×D312yes—
84C2×C4⋊C4C2×D311yescharacteristic
85C2×C4⋊C4C2×D311yescharacteristic
86C2×C22⋊C4—32no—
87C2×C22⋊C4—32no—
88C2×C22⋊C4—32no—
89C2×C22⋊C4—32no—
90C2×C22⋊C4—32no—
91C2×C42C2×D311yescharacteristic
92#95—32no—
93#95—32no—
94#95—32no—
95#95C2×D312yes—
96C23×C3C2411yescharacteristic
97C22×D3—22no—
98C22×D3—22no—
99C22×D3—22no—
100C22×D3—22no—
101C22×D3—22no—
102C22×D3—22no—
103C22×D3C2×Q814yes—
104C2×C3×C4—21no—
105C2×C3×C4—21no—
106C2×C3×C4—21no—
107C2×C3×C4—21no—
108C2×C3×C4—21no—
109C2×C3×C4—21no—
110C2×C3×C4—22no—
111C2×C3×C4—22no—
112C2×C3×C4—22no—
113C2×C3×C4—22no—
114C2×C3×C4—22no—
115C2×C3×C4—22no—
116C2×C3×C4C2×D412yes—
117C2×C3×C4C2×Q812yes—
118C2×C3×C4#4112yes—
119C2×Dic3—22no—
120C2×Dic3—22no—
121C2×Dic3—22no—
122C2×Dic3—22no—
123C2×Dic3—22no—
124C2×Dic3—22no—
125C2×Dic3—22no—
126C2×Dic3—22no—
127C2×Dic3C2×D414yes—
128D12—44no—
129C4×D3—48no—
130Dic6—44no—
131C24—32no—
132C2×Q8—34no—
133C2×Q8—121no—
134C2×Q8—121no—
135C2×D4—34no—
136C2×D4—121no—
137C2×D4—121no—
138C22×C4—32no—
139C22×C4—32no—
140C22×C4—32no—
141C22×C4—64no—
142C22×C4—122no—
143C22×C4—122no—
144C22×C4C22×D312yes—
145C22×C4C22×D312yes—
146C22×C4C22×D311yescharacteristic
147C22×C4C22×D311yescharacteristic
148C22×C4C22×D311yescharacteristic
149C4⋊C4—22no—
150C4⋊C4—24no—
151C4⋊C4—41no—
152C4⋊C4—64no—
153C4⋊C4—64no—
154C4⋊C4—122no—
155C4⋊C4—121no—
156C4⋊C4—121no—
157C22⋊C4—64no—
158C22⋊C4—64no—
159C22⋊C4—64no—
160C22⋊C4—122no—
161C22⋊C4—122no—
162C42—41no—
163C22×C3C22×D411yescharacteristic
164C22×C3C22×Q811yescharacteristic
165C22×C3C2×#4111yescharacteristic
166C22×C3H(5,𝔽2)11yescharacteristic
167C22×C3H(5,𝔽2)11yescharacteristic
168C22×C3H(5,𝔽2)11yescharacteristic
169C22×C32−1+411yescharacteristic
170C2×D3—24no—
171C2×D3—24no—
172C2×D3—24no—
173C2×D3—42no—
174C2×D3—42no—
175C2×D3—42no—
176C2×D3—42no—
177C3×C4—22no—
178C3×C4—22no—
179C3×C4—22no—
180C3×C4—42no—
181C3×C4—42no—
182Dic3—24no—
183Dic3—42no—
184Dic3—42no—
185C23—34no—
186C23—62no—
187C23—62no—
188C23—62no—
189C23—62no—
190C23—62no—
191C23—62no—
192C23C23×D311yescharacteristic
193Q8—124no—
194D4—124no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—34no—
208C2×C4—62no—
209C2×C4—62no—
210C2×C4—62no—
211C2×C4—62no—
212C2×C4—62no—
213C2×C4—62no—
214C2×C4—62no—
215C2×C4—62no—
216C2×C4—128no—
217C2×C4#23512yes—
218C2×C4D3×D412yes—
219C2×C4D3×Q812yes—
220C2×C3#53411yescharacteristic
221C2×C3#53911yescharacteristic
222C2×C3#54311yescharacteristic
223C2×C3D4×Q811yescharacteristic
224C2×C3#55011yescharacteristic
225C2×C3#55411yescharacteristic
226C2×C3#56211yescharacteristic
227D3—44no—
228C22—64no—
229C22—64no—
230C22—64no—
231C22—122no—
232C22—122no—
233C22—122no—
234C22—122no—
235C22C2×#23511yescharacteristic
236C22C2×D3×D411yescharacteristic
237C22#100411yescharacteristic
238C22C2×D3×Q811yescharacteristic
239C22#100711yescharacteristic
240C22#100911yescharacteristic
241C22#100911yescharacteristic
242C4—22no—
243C4—22no—
244C4—22no—
245C4—42no—
246C4—42no—
247C4—64no—
248C4—122no—
249C4—122no—
250C3#273211yescharacteristic, complemented, Sylow
251C2—124no—
252C2#551511yescharacteristic
253C2#555311yescharacteristic
254C2#557911yescharacteristic
255C2#560511yescharacteristic
256C2#563411yescharacteristic
257C2#570011yescharacteristic
258C2#572711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#389C2
prime radicalC22×D12C22
hypercenter#389D3
hypocenterC3#2732
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #389

Complements

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

NH ≅ G/Ntype
1. C2×#896C2semidirect
2. C3×#389C2semidirect
3. #4994C2semidirect
4. #4993C2semidirect
5. #4989C2semidirect
7. #4929C2semidirect
9. #4657C2semidirect
39. #389D3semidirect
53. C2×D12Q8semidirect
59. C2×Dic6D4semidirect
250. C3#2732semidirect

Maximal subgroups

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

subgroupindex
C2×#8962
C3×#3892
#49942
#49932
#49892
#49392
#49292
#46652
#46572
#27323