Finite Group Databasebeta

← #76814

Subgroups — #76814

Order 384.

520 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 131 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×#946C211yesmaximal, characteristic, complemented
2C2×#67⋊C4C211yesmaximal, characteristic
3C3×#393C211yesmaximal, characteristic, complemented
4#5230C211yesmaximal, characteristic, complemented
5#4961C212yesmaximal
6#4953C211yesmaximal, characteristic, complemented
7#4946C212yesmaximal, complemented
8#4675C212yesmaximal, complemented
9#4664C211yesmaximal, characteristic, complemented
10#4661C211yesmaximal, characteristic, complemented
11#4654C212yesmaximal, complemented
12#2638—31nomaximal, Sylow
13C22×C3×Q8C2211yescharacteristic
14C22×#67C2211yescharacteristic
15C23×Dic3C2211yescharacteristic
16C22×C4×D3C2211yescharacteristic
17C2×C3×C22⋊C4C2211yescharacteristic
18C2×C3×C22⋊C4C2212yes—
19C2×#217C2212yes—
20#946—24no—
21C2×(C2×D3)⋊C4C2211yescharacteristic
22C2×(C2×D3)⋊C4C2212yes—
23C2×(C2×D3)⋊C4C2212yes—
24C2×C4⋊Dic3C2212yes—
25C2×C4⋊Dic3C2211yescharacteristic
26C2×Dic3⋊C4C2212yes—
27C2×Dic3⋊C4C2211yescharacteristic
28C2×Dic3⋊C4C2212yes—
29C2×C4×Dic3C2211yescharacteristic
30#67⋊C4—24no—
31C3×#95C2212yes—
32C3×#95C2211yescharacteristic
33C3⋊#95C2212yes—
34C3⋊#95C2212yes—
35C3⋊#95C2212yes—
36C3⋊#95C2212yes—
37C3⋊#95C2211yescharacteristic
38C2×C22⋊Q8—31no—
39C2×C4×D4—31no—
40#398—31no—
41#397—31no—
42#397—32no—
43#395—31no—
44#394—32no—
45#393D311yescharacteristic, complemented
46#387—32no—
47#385—32no—
48#381—31no—
49C24×C3C2311yescharacteristic
50C23×D3C2311yescharacteristic
51C2×C3×Q8—22no—
52C2×C3×Q8—42no—
53C22×C3×C4C2311yescharacteristic
54C22×C3×C4C2312yes—
55C22×C3×C4C2312yes—
56C22×C3×C4C2311yescharacteristic
57C2×#67—22no—
58C2×#67—22no—
59C2×#67—22no—
60C22×Dic3—22no—
61C22×Dic3—42no—
62C22×Dic3C2312yes—
63C22×Dic3C2311yescharacteristic
64C22×Dic3C2312yes—
65C22×Dic3C2311yescharacteristic
66C22×Dic3C2311yescharacteristic
67C2×C4×D3—42no—
68C2×C4×D3D414yes—
69C3×C22⋊C4—22no—
70C3×C22⋊C4—42no—
71#217—24no—
72(C2×D3)⋊C4—22no—
73(C2×D3)⋊C4—24no—
74(C2×D3)⋊C4—42no—
75C4⋊Dic3—22no—
76C4⋊Dic3—24no—
77Dic3⋊C4—22no—
78Dic3⋊C4—24no—
79Dic3⋊C4—42no—
80C4×Dic3—22no—
81C22×Q8C2×D311yescharacteristic
82C22×D4—31no—
83C23×C4—31no—
84C23×C4—31no—
85C22⋊Q8—64no—
86C4×D4—64no—
87C2×C4⋊C4—31no—
88C2×C4⋊C4—32no—
89C2×C4⋊C4—31no—
90C2×C4⋊C4—32no—
91C2×C4⋊C4—32no—
92C2×C22⋊C4—32no—
93C2×C22⋊C4—31no—
94C2×C22⋊C4—32no—
95C2×C22⋊C4—32no—
96C2×C22⋊C4C2×D311yescharacteristic
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
105#95C2×D312yes—
106C23×C3—21no—
107C23×C3—21no—
108C23×C3—22no—
109C23×C3—22no—
110C23×C3#4112yes—
111C23×C3C2411yescharacteristic
112C22×D3—21no—
113C22×D3—21no—
114C22×D3—22no—
115C22×D3—22no—
116C22×D3C2×D412yes—
117C3×Q8—44no—
118C2×C3×C4—22no—
119C2×C3×C4—21no—
120C2×C3×C4—22no—
121C2×C3×C4—22no—
122C2×C3×C4—22no—
123C2×C3×C4—22no—
124C2×C3×C4—22no—
125C2×C3×C4—22no—
126C2×C3×C4—22no—
127C2×C3×C4C2×D412yes—
128#67—28no—
129C2×Dic3—21no—
130C2×Dic3—21no—
131C2×Dic3—22no—
132C2×Dic3—22no—
133C2×Dic3—21no—
134C2×Dic3—22no—
135C2×Dic3—22no—
136C2×Dic3—22no—
137C2×Dic3—22no—
138C2×Dic3—22no—
139C2×Dic3—22no—
140C2×Dic3—44no—
141C2×Dic3C2×D412yes—
142C2×Dic3#4112yes—
143C4×D3—44no—
144C24—31no—
145C24C22×D311yescharacteristic
146C2×Q8—22no—
147C2×Q8—42no—
148C2×D4—62no—
149C2×D4—62no—
150C2×D4—62no—
151C22×C4—31no—
152C22×C4—31no—
153C22×C4—34no—
154C22×C4—32no—
155C22×C4—32no—
156C22×C4—31no—
157C22×C4—62no—
158C22×C4—122no—
159C22×C4—122no—
160C22×C4C22×D311yescharacteristic
161C22×C4C22×D312yes—
162C22×C4C22×D312yes—
163C22×C4C22×D311yescharacteristic
164C4⋊C4—64no—
165C4⋊C4—64no—
166C4⋊C4—62no—
167C4⋊C4—62no—
168C4⋊C4—122no—
169C22⋊C4—22no—
170C22⋊C4—42no—
171C22⋊C4—62no—
172C22⋊C4—64no—
173C22⋊C4—64no—
174C22⋊C4—122no—
175C42—62no—
176C22×C3—22no—
177C22×C3—24no—
178C22×C3—41no—
179C22×C3—41no—
180C22×C3—42no—
181C22×C3C22×D411yescharacteristic
182C22×C3C2×#4111yescharacteristic
183C22×C3C2×#4111yescharacteristic
184C22×C3H(5,𝔽2)12yes—
185C22×C32−1+412yes—
186C2×D3—22no—
187C2×D3—22no—
188C2×D3—22no—
189C2×D3—42no—
190C2×D3—42no—
191C3×C4—22no—
192C3×C4—42no—
193C3×C4—41no—
194C3×C4—42no—
195Dic3—22no—
196Dic3—22no—
197Dic3—42no—
198Dic3—41no—
199Dic3—42no—
200C23—21no—
201C23—22no—
202C23—21no—
203C23—22no—
204C23—32no—
205C23—62no—
206C23—61no—
207C23—61no—
208C23—62no—
209C23#23912yes—
210C23C23×D311yescharacteristic
211Q8—44no—
212D4—68no—
213C2×C4—22no—
214C2×C4—21no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—32no—
223C2×C4—32no—
224C2×C4—61no—
225C2×C4—62no—
226C2×C4—61no—
227C2×C4—62no—
228C2×C4—61no—
229C2×C4—62no—
230C2×C4—62no—
231C2×C4—62no—
232C2×C4—62no—
233C2×C4—62no—
234C2×C4—62no—
235C2×C4—124no—
236C2×C4—124no—
237C2×C4D3×D412yes—
238C2×C3—42no—
239C2×C3#52511yescharacteristic
240C2×C3#53411yescharacteristic
241C2×C3#53611yescharacteristic
242C2×C3#53912yes—
243C2×C3#54212yes—
244D3—42no—
245C22—24no—
246C22—22no—
247C22—42no—
248C22—41no—
249C22—41no—
250C22—62no—
251C22—62no—
252C22—62no—
253C22—122no—
254C22—122no—
255C22C2×D3×D411yescharacteristic
256C22#100412yes—
257C22C2×#23911yescharacteristic
258C22D3×#4111yescharacteristic
259C22#101012yes—
260C4—22no—
261C4—42no—
262C4—41no—
263C4—42no—
264C4—62no—
265C4—62no—
266C4—122no—
267C4—121no—
268C4—122no—
269C3#263811yescharacteristic, complemented, Sylow
270C2—42no—
271C2—122no—
272C2#558911yescharacteristic
273C2#562711yescharacteristic
274C2#564512yes—
275C2#566912yes—
276C2#567511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#393C2
prime radicalC22×#67C22
hypercenter#393D3
hypocenterC3#2638
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. C2×#946C2semidirect
3. C3×#393C2semidirect
4. #5230C2semidirect
6. #4953C2semidirect
7. #4946C2semidirect
8. #4675C2semidirect
9. #4664C2semidirect
10. #4661C2semidirect
11. #4654C2semidirect
45. #393D3semidirect
269. C3#2638semidirect

Maximal subgroups

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

subgroupindex
C2×#9462
C2×#67⋊C42
C3×#3932
#52302
#49612
#49532
#49462
#46752
#46642
#46612
#46542
#26383