Finite Group Databasebeta

← #77258

Subgroups — #77258

Order 384.

524 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 153 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
2C2×#896C211yesmaximal, characteristic
3C3×#395C211yesmaximal, characteristic, complemented
4#5230C211yesmaximal, characteristic, complemented
5#4987C212yesmaximal
6#4981C211yesmaximal, characteristic, complemented
7#4979C212yesmaximal, complemented
8#4667C212yesmaximal
9D3×#95C211yesmaximal, characteristic
10#4661C211yesmaximal, characteristic, complemented
11#4657C212yesmaximal, complemented
12#2659—31nomaximal, Sylow
13C22×C3×Q8C2211yescharacteristic
14C22×C4×D3C2211yescharacteristic
15C22×C4×D3C2212yes—
16C22×Dic6C2211yescharacteristic
17C2×C3×C4⋊C4C2211yescharacteristic
18C2×C3×C4⋊C4C2212yes—
19#946—24no—
20C2×(C2×D3)⋊C4C2211yescharacteristic
21C2×(C2×D3)⋊C4C2212yes—
22C2×(C2×D3)⋊C4C2211yescharacteristic
23C2×C4⋊Dic3C2211yescharacteristic
24C2×C4⋊Dic3C2212yes—
25C2×Dic3⋊C4C2211yescharacteristic
26C2×Dic3⋊C4C2212yes—
27C2×Dic3⋊C4C2211yescharacteristic
28C2×Dic3⋊C4C2212yes—
29C2×Dic3⋊C4C2212yes—
30#896—24no—
31C3×#95C2212yes—
32C3×#95C2211yescharacteristic
33C3⋊#95C2212yes—
34C3⋊#95C2212yes—
35C3⋊#95C2212yes—
36C3⋊#95C2211yescharacteristic
37C3⋊#95C2212yes—
38C2×C22⋊Q8—31no—
39C2×C22⋊Q8—31no—
40#398—31no—
41#398—32no—
42#398—31no—
43#395—31no—
44#395—32no—
45#395D311yescharacteristic, complemented
46#380—32no—
47#380—32no—
48C2×#95—31no—
49C23×D3C2311yescharacteristic
50C2×C3×Q8—22no—
51C2×C3×Q8—42no—
52C22×C3×C4C2311yescharacteristic
53C22×C3×C4C2312yes—
54C22×C3×C4C2311yescharacteristic
55C22×C3×C4C2311yescharacteristic
56C22×C3×C4C2312yes—
57C22×Dic3C2311yescharacteristic
58C22×Dic3C2311yescharacteristic
59C22×Dic3C2312yes—
60C22×Dic3C2311yescharacteristic
61C22×Dic3C2312yes—
62C2×C4×D3—42no—
63C2×C4×D3—42no—
64C2×C4×D3—42no—
65C2×C4×D3D414yes—
66C2×C4×D3Q818yescomplemented
67C2×Dic6—22no—
68C2×Dic6—42no—
69C3×C4⋊C4—22no—
70C3×C4⋊C4—42no—
71(C2×D3)⋊C4—22no—
72(C2×D3)⋊C4—22no—
73(C2×D3)⋊C4—24no—
74C4⋊Dic3—22no—
75C4⋊Dic3—42no—
76Dic3⋊C4—22no—
77Dic3⋊C4—22no—
78Dic3⋊C4—24no—
79Dic3⋊C4—42no—
80Dic3⋊C4—42no—
81C22×Q8—31no—
82C22×Q8C2×D311yescharacteristic
83C23×C4—31no—
84C23×C4—32no—
85C22⋊Q8—64no—
86C22⋊Q8—64no—
87C2×C4⋊C4—32no—
88C2×C4⋊C4—31no—
89C2×C4⋊C4—31no—
90C2×C4⋊C4—31no—
91C2×C4⋊C4—32no—
92C2×C4⋊C4—32no—
93C2×C4⋊C4—32no—
94C2×C4⋊C4C2×D311yescharacteristic
95C2×C4⋊C4C2×D312yes—
96C2×C22⋊C4—31no—
97C2×C22⋊C4—32no—
98C2×C22⋊C4—31no—
99#95—31no—
100#95—32no—
101#95—32no—
102#95—32no—
103#95—32no—
104#95C2×D311yescharacteristic
105#95C2×D312yes—
106C23×C3C2411yescharacteristic
107C22×D3—22no—
108C22×D3—21no—
109C22×D3—21no—
110C22×D3C2×D412yes—
111C22×D3C2×Q814yes—
112C3×Q8—44no—
113C2×C3×C4—22no—
114C2×C3×C4—22no—
115C2×C3×C4—22no—
116C2×C3×C4—21no—
117C2×C3×C4—22no—
118C2×C3×C4—21no—
119C2×C3×C4—21no—
120C2×C3×C4—22no—
121C2×C3×C4—22no—
122C2×C3×C4—22no—
123C2×C3×C4—21no—
124C2×C3×C4C2×D412yes—
125C2×C3×C4C2×Q814yes—
126C2×Dic3—22no—
127C2×Dic3—22no—
128C2×Dic3—22no—
129C2×Dic3—21no—
130C2×Dic3—22no—
131C2×Dic3—21no—
132C2×Dic3—22no—
133C2×Dic3—22no—
134C2×Dic3—21no—
135C2×Dic3—21no—
136C2×Dic3—22no—
137C2×Dic3C2×D412yes—
138C2×Dic3C2×Q814yes—
139C4×D3—44no—
140C4×D3—48no—
141Dic6—44no—
142C24—31no—
143C2×Q8—22no—
144C2×Q8—42no—
145C2×Q8—62no—
146C2×Q8—122no—
147C22×C4—32no—
148C22×C4—31no—
149C22×C4—34no—
150C22×C4—38no—
151C22×C4—31no—
152C22×C4—32no—
153C22×C4—31no—
154C22×C4—122no—
155C22×C4—122no—
156C22×C4—122no—
157C22×C4C22×D311yescharacteristic
158C22×C4C22×D312yes—
159C22×C4C22×D311yescharacteristic
160C22×C4C22×D311yescharacteristic
161C22×C4C22×D312yes—
162C4⋊C4—22no—
163C4⋊C4—42no—
164C4⋊C4—62no—
165C4⋊C4—64no—
166C4⋊C4—62no—
167C4⋊C4—62no—
168C4⋊C4—122no—
169C4⋊C4—122no—
170C4⋊C4—122no—
171C22⋊C4—62no—
172C22⋊C4—62no—
173C22⋊C4—64no—
174C22×C3C22×D411yescharacteristic
175C22×C3C22×Q812yes—
176C22×C3H(5,𝔽2)12yes—
177C22×C3H(5,𝔽2)11yescharacteristic
178C22×C32−1+411yescharacteristic
179C2×D3—22no—
180C2×D3—22no—
181C2×D3—24no—
182C2×D3—24no—
183C2×D3C4⋊Q814yes—
184C3×C4—22no—
185C3×C4—24no—
186C3×C4—41no—
187C3×C4—42no—
188C3×C4—41no—
189Dic3—22no—
190Dic3—24no—
191Dic3—41no—
192Dic3—41no—
193Dic3—42no—
194C23—32no—
195C23—34no—
196C23—62no—
197C23—61no—
198C23—61no—
199C23C23×D311yescharacteristic
200Q8—44no—
201Q8—124no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—32no—
214C2×C4—34no—
215C2×C4—62no—
216C2×C4—62no—
217C2×C4—62no—
218C2×C4—61no—
219C2×C4—62no—
220C2×C4—61no—
221C2×C4—62no—
222C2×C4—61no—
223C2×C4—62no—
224C2×C4—62no—
225C2×C4—61no—
226C2×C4—128no—
227C2×C4—124no—
228C2×C4D3×D412yes—
229C2×C4D3×Q814yes—
230C2×C3C2×C4⋊Q811yescharacteristic
231C2×C3#53411yescharacteristic
232C2×C3#53711yescharacteristic
233C2×C3#54312yes—
234C2×C3#54412yes—
235D3—24no—
236C22—34no—
237C22—64no—
238C22—64no—
239C22—62no—
240C22—62no—
241C22C2×D3×D411yescharacteristic
242C22#100412yes—
243C22C2×D3×Q812yes—
244C22#100711yescharacteristic
245C22#100911yescharacteristic
246C4—24no—
247C4—22no—
248C4—42no—
249C4—41no—
250C4—41no—
251C4—64no—
252C4—62no—
253C4—121no—
254C4—121no—
255C4—122no—
256C3#265911yescharacteristic, complemented, Sylow
257C2—64no—
258C2#558911yescharacteristic
259C2#562911yescharacteristic
260C2#563412yes—
261C2D3×C4⋊Q811yescharacteristic
262C2#572912yes—
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#395C2
prime radicalC23×D3C23
hypercenter#395D3
hypocenterC3#2659
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
3. C3×#395C2semidirect
4. #5230C2semidirect
6. #4981C2semidirect
7. #4979C2semidirect
10. #4661C2semidirect
11. #4657C2semidirect
45. #395D3semidirect
66. C2×C4×D3Q8semidirect
256. C3#2659semidirect

Maximal subgroups

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

subgroupindex
C2×#9462
C2×#8962
C3×#3952
#52302
#49872
#49812
#49792
#46672
D3×#952
#46612
#46572
#26593