Finite Group Databasebeta

← #74172

Subgroups — #74172

Order 384.

440 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 137 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×#382C211yesmaximal, characteristic, complemented
2#4988C211yesmaximal, characteristic
3#4988C211yesmaximal, characteristic
4#4984C211yesmaximal, characteristic, complemented
5Dic3×C4⋊C4C211yesmaximal, characteristic, complemented
6#4939C211yesmaximal, characteristic
7C4×Dic3⋊C4C211yesmaximal, characteristic, complemented
8#4675C212yesmaximal, complemented
9#4674C212yesmaximal, complemented
10#4670C212yesmaximal, complemented
11#4669C212yesmaximal, complemented
12#2513—31nomaximal, Sylow
13C22×D12C2211yescharacteristic
14C2×C3×C4⋊C4C2211yescharacteristic
15C2×C3×C4⋊C4C2211yescharacteristic
16C2×C3×C42C2211yescharacteristic
17C2×(C2×D3)⋊C4C2212yes—
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⋊Dic3C2212yes—
25C2×Dic3⋊C4C2211yescharacteristic
26C2×Dic3⋊C4C2211yescharacteristic
27C2×C4×Dic3C2211yescharacteristic
28C2×C4×Dic3C2211yescharacteristic
29C2×C4×Dic3C2212yes—
30C3×#95C2212yes—
31C3×#95C2212yes—
32C3⋊#95C2212yes—
33C3⋊#95C2211yescharacteristic
34C3⋊#95C2211yescharacteristic
35C3⋊#95C2212yes—
36#397—32no—
37#397—32no—
38#390—31no—
39#390—31no—
40#390—31no—
41#389—31no—
42#388—32no—
43#388—32no—
44#382D311yescharacteristic, complemented
45C4×C4⋊C4—31no—
46C4×C4⋊C4—31no—
47C23×D3C2312yes—
48C22×C3×C4C2312yes—
49C22×C3×C4C2311yescharacteristic
50C22×C3×C4C2311yescharacteristic
51C22×C3×C4C2312yes—
52C22×C3×C4C2311yescharacteristic
53C22×Dic3C2311yescharacteristic
54C22×Dic3C2311yescharacteristic
55C22×Dic3C2312yes—
56C22×Dic3C2311yescharacteristic
57C22×Dic3C2311yescharacteristic
58C2×D12—22no—
59C2×D12—22no—
60C2×D12—22no—
61C3×C4⋊C4—22no—
62C3×C4⋊C4—22no—
63C3×C42—22no—
64(C2×D3)⋊C4—24no—
65(C2×D3)⋊C4—24no—
66(C2×D3)⋊C4—42no—
67(C2×D3)⋊C4—42no—
68(C2×D3)⋊C4—42no—
69(C2×D3)⋊C4—42no—
70C4⋊Dic3—22no—
71C4⋊Dic3—42no—
72Dic3⋊C4—22no—
73Dic3⋊C4—22no—
74C4×Dic3—22no—
75C4×Dic3—22no—
76C4×Dic3—42no—
77C22×D4—31no—
78C2×C4⋊C4—31no—
79C2×C4⋊C4—31no—
80C2×C4⋊C4—31no—
81C2×C4⋊C4—32no—
82C2×C4⋊C4C2×D311yescharacteristic
83C2×C4⋊C4C2×D311yescharacteristic
84C2×C22⋊C4—32no—
85C2×C22⋊C4—32no—
86C2×C22⋊C4—32no—
87C2×C22⋊C4—32no—
88C2×C22⋊C4—32no—
89C2×C22⋊C4—32no—
90C2×C42—31no—
91C2×C42—31no—
92C2×C42—32no—
93C2×C42C2×D311yescharacteristic
94#95—31no—
95#95—32no—
96#95—31no—
97#95—32no—
98#95C2×D312yes—
99#95C2×D312yes—
100C23×C3C2411yescharacteristic
101C22×D3—22no—
102C22×D3—22no—
103C22×D3—22no—
104C22×D3—22no—
105C22×D3—22no—
106C22×D3—22no—
107C22×D3—22no—
108C2×C3×C4—21no—
109C2×C3×C4—21no—
110C2×C3×C4—22no—
111C2×C3×C4—22no—
112C2×C3×C4—21no—
113C2×C3×C4—21no—
114C2×C3×C4—22no—
115C2×C3×C4—22no—
116C2×C3×C4—22no—
117C2×C3×C4—22no—
118C2×C3×C4—21no—
119C2×C3×C4—21no—
120C2×C3×C4#4112yes—
121C2×C3×C4#4112yes—
122C2×C3×C4#4112yes—
123C2×Dic3—21no—
124C2×Dic3—21no—
125C2×Dic3—21no—
126C2×Dic3—21no—
127C2×Dic3—21no—
128C2×Dic3—21no—
129C2×Dic3—21no—
130C2×Dic3—22no—
131C2×Dic3—22no—
132C2×Dic3—21no—
133C2×Dic3—22no—
134C2×Dic3—21no—
135C2×Dic3—21no—
136C2×Dic3#4112yes—
137C2×Dic3#4112yes—
138D12—44no—
139C24—32no—
140C2×D4—62no—
141C2×D4—62no—
142C2×D4—62no—
143C22×C4—31no—
144C22×C4—31no—
145C22×C4—31no—
146C22×C4—31no—
147C22×C4—32no—
148C22×C4C22×D312yes—
149C22×C4C22×D311yescharacteristic
150C22×C4C22×D311yescharacteristic
151C22×C4C22×D312yes—
152C22×C4C22×D311yescharacteristic
153C4⋊C4—22no—
154C4⋊C4—22no—
155C4⋊C4—62no—
156C4⋊C4—62no—
157C4⋊C4—62no—
158C4⋊C4—122no—
159C22⋊C4—64no—
160C22⋊C4—64no—
161C22⋊C4—122no—
162C22⋊C4—122no—
163C22⋊C4—122no—
164C22⋊C4—122no—
165C42—22no—
166C42—62no—
167C42—62no—
168C42—122no—
169C22×C3C2×#4111yescharacteristic
170C22×C3C2×#4111yescharacteristic
171C22×C3C2×#4111yescharacteristic
172C22×C3C2×#4111yescharacteristic
173C22×C3C2×#4111yescharacteristic
174C22×C3H(5,𝔽2)11yescharacteristic
175C22×C3H(5,𝔽2)11yescharacteristic
176C2×D3—42no—
177C2×D3—42no—
178C2×D3—42no—
179C2×D3—42no—
180C2×D3—42no—
181C2×D3—42no—
182C2×D3—42no—
183C3×C4—41no—
184C3×C4—41no—
185C3×C4—42no—
186C3×C4—42no—
187C3×C4#12614yes—
188Dic3—22no—
189Dic3—22no—
190Dic3—41no—
191Dic3—42no—
192Dic3—41no—
193C23—62no—
194C23—62no—
195C23—62no—
196C23—62no—
197C23—62no—
198C23—62no—
199C23—62no—
200C23C23×D311yescharacteristic
201D4—124no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—22no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—22no—
214C2×C4—32no—
215C2×C4—32no—
216C2×C4—61no—
217C2×C4—61no—
218C2×C4—61no—
219C2×C4—61no—
220C2×C4—61no—
221C2×C4—61no—
222C2×C4—62no—
223C2×C4—61no—
224C2×C4—62no—
225C2×C4—61no—
226C2×C4—61no—
227C2×C4—62no—
228C2×C4—61no—
229C2×C4#23512yes—
230C2×C4#23712yes—
231C2×C4#23912yes—
232C2×C3C2×#12611yescharacteristic
233C2×C3#52911yescharacteristic
234C2×C3#54011yescharacteristic
235C2×C3#55311yescharacteristic
236C2×C3#55511yescharacteristic
237C2×C3#55911yescharacteristic
238C2×C3#55911yescharacteristic
239D3—82no—
240C22—122no—
241C22—122no—
242C22—122no—
243C22—122no—
244C22—122no—
245C22—122no—
246C22—122no—
247C22C2×#23511yescharacteristic
248C22C2×#23711yescharacteristic
249C22C2×#23911yescharacteristic
250C22D3×#4111yescharacteristic
251C22D3×#4111yescharacteristic
252C22#100911yescharacteristic
253C22#100911yescharacteristic
254C4—42no—
255C4—42no—
256C4—41no—
257C4—41no—
258C4—62no—
259C4—62no—
260C4—121no—
261C4—122no—
262C4—121no—
263C4#89814yes—
264C3#251311yescharacteristic, complemented, Sylow
265C2—242no—
266C2C2×#89811yescharacteristic
267C2#553011yescharacteristic
268C2#553211yescharacteristic
269C2#568111yescharacteristic
270C2#568111yescharacteristic
271C2#569111yescharacteristic
272C2#569711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#382C2
prime radicalC22×D12C22
hypercenter#382D3
hypocenterC3#2513
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #382

Complements

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

NH ≅ G/Ntype
1. C3×#382C2semidirect
4. #4984C2semidirect
5. Dic3×C4⋊C4C2semidirect
7. C4×Dic3⋊C4C2semidirect
8. #4675C2semidirect
9. #4674C2semidirect
10. #4670C2semidirect
11. #4669C2semidirect
44. #382D3semidirect
264. C3#2513semidirect

Maximal subgroups

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

subgroupindex
C3×#3822
#49882
#49842
Dic3×C4⋊C42
#49392
C4×Dic3⋊C42
#46752
#46742
#46702
#46692
#25133