Finite Group Databasebeta

← #76252

Subgroups — #76252

Order 384.

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

normal subgroups only

This group has 420 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
1C3×#389C211yesmaximal, characteristic, complemented
2#4994C212yesmaximal, complemented
3#4992C212yesmaximal, complemented
4#4988C211yesmaximal, characteristic
5#4988C211yesmaximal, characteristic
6#4984C211yesmaximal, characteristic, complemented
7#4972C211yesmaximal, characteristic, complemented
8#4939C211yesmaximal, characteristic
9#4932C211yesmaximal, characteristic, complemented
10#4675C212yesmaximal, complemented
11#4674C212yesmaximal, complemented
12#2644—31nomaximal, Sylow
13C22×D12C2211yescharacteristic
14C2×C3×C4⋊C4C2211yescharacteristic
15C2×C3×C4⋊C4C2211yescharacteristic
16C2×C3×C4⋊C4C2212yes—
17C2×C3×C42C2211yescharacteristic
18C2×(C2×D3)⋊C4C2212yes—
19C2×(C2×D3)⋊C4C2212yes—
20C2×(C2×D3)⋊C4C2212yes—
21C2×(C2×D3)⋊C4C2212yes—
22C2×(C2×D3)⋊C4C2212yes—
23C2×(C2×D3)⋊C4C2212yes—
24C2×C4⋊Dic3C2211yescharacteristic
25C2×C4⋊Dic3C2212yes—
26C2×Dic3⋊C4C2212yes—
27C2×C4×Dic3C2211yescharacteristic
28C2×C4×Dic3C2211yescharacteristic
29C3×#95C2212yes—
30C3⋊#95C2211yescharacteristic
31C3⋊#95C2212yes—
32C3⋊#95C2211yescharacteristic
33C3⋊#95C2212yes—
34C3⋊#95C2211yescharacteristic
35C3⋊#95C2211yescharacteristic
36#397—32no—
37#397—32no—
38#397—32no—
39#396—32no—
40#390—31no—
41#390—31no—
42#390—31no—
43#389—31no—
44#389—31no—
45#389D311yescharacteristic, complemented
46#376—31no—
47C23×D3C2312yes—
48C22×C3×C4C2311yescharacteristic
49C22×C3×C4C2311yescharacteristic
50C22×C3×C4C2312yes—
51C22×C3×C4C2312yes—
52C22×C3×C4C2311yescharacteristic
53C22×Dic3C2311yescharacteristic
54C22×Dic3C2311yescharacteristic
55C22×Dic3C2311yescharacteristic
56C22×Dic3C2311yescharacteristic
57C22×Dic3C2312yes—
58C2×D12—22no—
59C2×D12—22no—
60C2×D12—22no—
61C3×C4⋊C4—22no—
62C3×C4⋊C4—22no—
63C3×C4⋊C4—42no—
64C3×C42—22no—
65(C2×D3)⋊C4—42no—
66(C2×D3)⋊C4—42no—
67(C2×D3)⋊C4—42no—
68(C2×D3)⋊C4—42no—
69(C2×D3)⋊C4—42no—
70(C2×D3)⋊C4—42no—
71C4⋊Dic3—22no—
72C4⋊Dic3—42no—
73Dic3⋊C4—42no—
74C4×Dic3—22no—
75C4×Dic3—22no—
76C22×D4—31no—
77C2×C4⋊C4—31no—
78C2×C4⋊C4—32no—
79C2×C4⋊C4—32no—
80C2×C4⋊C4C2×D311yescharacteristic
81C2×C4⋊C4C2×D311yescharacteristic
82C2×C4⋊C4C2×D312yes—
83C2×C22⋊C4—32no—
84C2×C22⋊C4—32no—
85C2×C22⋊C4—32no—
86C2×C22⋊C4—32no—
87C2×C22⋊C4—32no—
88C2×C22⋊C4—32no—
89C2×C42—31no—
90C2×C42—31no—
91C2×C42C2×D311yescharacteristic
92#95—31no—
93#95—31no—
94#95—31no—
95#95—31no—
96#95—32no—
97#95—32no—
98#95C2×D312yes—
99C23×C3C2411yescharacteristic
100C22×D3—22no—
101C22×D3—22no—
102C22×D3—22no—
103C22×D3—22no—
104C22×D3—22no—
105C22×D3—22no—
106C22×D3—22no—
107C2×C3×C4—21no—
108C2×C3×C4—21no—
109C2×C3×C4—22no—
110C2×C3×C4—22no—
111C2×C3×C4—22no—
112C2×C3×C4—22no—
113C2×C3×C4—21no—
114C2×C3×C4—21no—
115C2×C3×C4—22no—
116C2×C3×C4—22no—
117C2×C3×C4—21no—
118C2×C3×C4—21no—
119C2×C3×C4#4112yes—
120C2×C3×C4#4112yes—
121C2×C3×C4#4112yes—
122C2×Dic3—21no—
123C2×Dic3—21no—
124C2×Dic3—21no—
125C2×Dic3—21no—
126C2×Dic3—22no—
127C2×Dic3—21no—
128C2×Dic3—22no—
129C2×Dic3—21no—
130C2×Dic3—22no—
131C2×Dic3—21no—
132C2×Dic3—21no—
133C2×Dic3—21no—
134C2×Dic3—21no—
135C2×Dic3—21no—
136C2×Dic3—21no—
137D12—44no—
138C24—32no—
139C2×D4—62no—
140C2×D4—62no—
141C2×D4—62no—
142C22×C4—31no—
143C22×C4—31no—
144C22×C4—31no—
145C22×C4—31no—
146C22×C4—32no—
147C22×C4C22×D311yescharacteristic
148C22×C4C22×D311yescharacteristic
149C22×C4C22×D312yes—
150C22×C4C22×D312yes—
151C22×C4C22×D311yescharacteristic
152C4⋊C4—22no—
153C4⋊C4—22no—
154C4⋊C4—42no—
155C4⋊C4—62no—
156C4⋊C4—122no—
157C4⋊C4—122no—
158C22⋊C4—122no—
159C22⋊C4—122no—
160C22⋊C4—122no—
161C22⋊C4—122no—
162C22⋊C4—122no—
163C22⋊C4—122no—
164C42—22no—
165C42—62no—
166C42—62no—
167C22×C3C2×#4111yescharacteristic
168C22×C3C2×#4111yescharacteristic
169C22×C3C2×#4111yescharacteristic
170C22×C3H(5,𝔽2)11yescharacteristic
171C22×C3H(5,𝔽2)11yescharacteristic
172C22×C3H(5,𝔽2)11yescharacteristic
173C22×C32−1+411yescharacteristic
174C2×D3—42no—
175C2×D3—42no—
176C2×D3—42no—
177C2×D3—42no—
178C2×D3—42no—
179C2×D3—42no—
180C2×D3—42no—
181C3×C4—42no—
182C3×C4—41no—
183C3×C4—42no—
184C3×C4—41no—
185C3×C4#12614yes—
186Dic3—41no—
187Dic3—42no—
188Dic3—41no—
189Dic3—41no—
190Dic3—41no—
191C23—62no—
192C23—62no—
193C23—62no—
194C23—62no—
195C23—62no—
196C23—62no—
197C23—62no—
198C23C23×D311yescharacteristic
199D4—124no—
200C2×C4—22no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—61no—
213C2×C4—61no—
214C2×C4—61no—
215C2×C4—61no—
216C2×C4—61no—
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—61no—
226C2×C4—61no—
227C2×C4#23512yes—
228C2×C4#23712yes—
229C2×C4#23912yes—
230C2×C3C2×#12611yescharacteristic
231C2×C3#54011yescharacteristic
232C2×C3#54011yescharacteristic
233C2×C3#54011yescharacteristic
234C2×C3#54211yescharacteristic
235C2×C3#54211yescharacteristic
236C2×C3#54211yescharacteristic
237D3—82no—
238C22—122no—
239C22—122no—
240C22—122no—
241C22—122no—
242C22—122no—
243C22—122no—
244C22—122no—
245C22C2×#23511yescharacteristic
246C22C2×#23711yescharacteristic
247C22#100411yescharacteristic
248C22C2×#23911yescharacteristic
249C22#100711yescharacteristic
250C22#100911yescharacteristic
251C22#100911yescharacteristic
252C4—41no—
253C4—42no—
254C4—41no—
255C4—42no—
256C4—121no—
257C4—122no—
258C4—121no—
259C4—121no—
260C4—121no—
261C4#89814yes—
262C3#264411yescharacteristic, complemented, Sylow
263C2—242no—
264C2C2×#89811yescharacteristic
265C2#556211yescharacteristic
266C2#558211yescharacteristic
267C2#561911yescharacteristic
268C2#564311yescharacteristic
269C2#569711yescharacteristic
270C2#573311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#389C2
prime radicalC22×D12C22
hypercenter#389D3
hypocenterC3#2644
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #389

Complements

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

NH ≅ G/Ntype
1. C3×#389C2semidirect
2. #4994C2semidirect
3. #4992C2semidirect
6. #4984C2semidirect
7. #4972C2semidirect
9. #4932C2semidirect
10. #4675C2semidirect
11. #4674C2semidirect
45. #389D3semidirect
262. C3#2644semidirect

Maximal subgroups

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

subgroupindex
C3×#3892
#49942
#49922
#49882
#49842
#49722
#49392
#49322
#46752
#46742
#26443