Finite Group Databasebeta

← #76778

Subgroups — #76778

Order 384.

528 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 155 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×Q8×Dic3C211yesmaximal, characteristic, complemented
2C2×#877C211yesmaximal, characteristic
3C3×#393C211yesmaximal, characteristic
4#5230C211yesmaximal, characteristic, complemented
5#4949C211yesmaximal, characteristic
6#4948C212yesmaximal
7Dic3×C22⋊C4C212yesmaximal
8#4659C211yesmaximal, characteristic, complemented
9#4657C212yesmaximal, complemented
10#4649C212yesmaximal, complemented
11#4649C211yesmaximal, characteristic, complemented
12#2396—31nomaximal, Sylow
13C22×C3×Q8C2211yescharacteristic
14C23×Dic3C2211yescharacteristic
15C2×C3×C22⋊C4C2212yes—
16C2×C3×C22⋊C4C2211yescharacteristic
17C2×#217C2212yes—
18C2×#217C2211yescharacteristic
19Q8×Dic3—24no—
20C2×C4⋊Dic3C2212yes—
21C2×C4⋊Dic3C2211yescharacteristic
22C2×Dic3⋊C4C2212yes—
23C2×Dic3⋊C4C2212yes—
24C2×Dic3⋊C4C2212yes—
25C2×Dic3⋊C4C2211yescharacteristic
26C2×Dic3⋊C4C2211yescharacteristic
27C2×C4×Dic3C2212yes—
28C2×C4×Dic3C2211yescharacteristic
29C2×C4×Dic3C2212yes—
30C2×C4×Dic3C2211yescharacteristic
31#877—24no—
32C3×#95C2212yes—
33C3×#95C2211yescharacteristic
34C3⋊#95C2212yes—
35C3⋊#95C2211yescharacteristic
36C3⋊#95C2212yes—
37C3⋊#95C2212yes—
38C2×C4×Q8—31no—
39C2×#117—31no—
40#400—31no—
41#399—31no—
42#398—32no—
43#396—32no—
44#395—31no—
45#393D311yescharacteristic
46C4×C4⋊C4—31no—
47C4×C4⋊C4—32no—
48C4×C22⋊C4—32no—
49C24×C3C2311yescharacteristic
50C2×C3×Q8—22no—
51C2×C3×Q8—24no—
52C22×C3×C4C2312yes—
53C22×C3×C4C2311yescharacteristic
54C22×C3×C4C2311yescharacteristic
55C22×C3×C4C2312yes—
56C22×Dic3—22no—
57C22×Dic3—22no—
58C22×Dic3Q814yescomplemented
59C22×Dic3C2311yescharacteristic
60C22×Dic3C2312yes—
61C22×Dic3C2311yescharacteristic
62C22×Dic3C2311yescharacteristic
63C22×Dic3C2312yes—
64C22×Dic3C2311yescharacteristic
65C3×C22⋊C4—22no—
66C3×C22⋊C4—24no—
67#217—22no—
68#217—24no—
69C4⋊Dic3—22no—
70C4⋊Dic3—24no—
71Dic3⋊C4—22no—
72Dic3⋊C4—24no—
73Dic3⋊C4—22no—
74Dic3⋊C4—24no—
75Dic3⋊C4—24no—
76C4×Dic3—22no—
77C4×Dic3—24no—
78C4×Dic3—24no—
79C4×Dic3D414yes—
80C22×Q8C2×D311yescharacteristic
81C23×C4—31no—
82C4×Q8—64no—
83#117—64no—
84C2×C4⋊C4—31no—
85C2×C4⋊C4—31no—
86C2×C4⋊C4—32no—
87C2×C4⋊C4—31no—
88C2×C4⋊C4—32no—
89C2×C4⋊C4—32no—
90C2×C4⋊C4—32no—
91C2×C22⋊C4—31no—
92C2×C22⋊C4—32no—
93C2×C22⋊C4C2×D311yescharacteristic
94C2×C22⋊C4C2×D312yes—
95C2×C42—31no—
96C2×C42—31no—
97C2×C42—32no—
98C2×C42—32no—
99#95—32no—
100#95—32no—
101#95—31no—
102#95—32no—
103#95C2×D312yes—
104#95C2×D311yescharacteristic
105C23×C3—21no—
106C23×C3—22no—
107C23×C3—22no—
108C23×C3—21no—
109C23×C3C2×Q812yes—
110C23×C3C2411yescharacteristic
111C3×Q8—28no—
112C2×C3×C4—22no—
113C2×C3×C4—22no—
114C2×C3×C4—22no—
115C2×C3×C4—21no—
116C2×C3×C4—22no—
117C2×C3×C4—22no—
118C2×C3×C4—22no—
119C2×C3×C4—22no—
120C2×C3×C4C2×D412yes—
121C2×C3×C4#4114yes—
122C2×Dic3—21no—
123C2×Dic3—22no—
124C2×Dic3—22no—
125C2×Dic3—22no—
126C2×Dic3—21no—
127C2×Dic3—28no—
128C2×Dic3—21no—
129C2×Dic3—22no—
130C2×Dic3—22no—
131C2×Dic3—22no—
132C2×Dic3—22no—
133C2×Dic3C2×D412yes—
134C2×Dic3C2×D412yes—
135C2×Dic3C2×Q812yes—
136C2×Dic3C2×Q812yes—
137C2×Dic3#4114yes—
138C2×Dic3#4112yes—
139C24C22×D311yescharacteristic
140C2×Q8—22no—
141C2×Q8—24no—
142C22×C4—31no—
143C22×C4—34no—
144C22×C4—31no—
145C22×C4—32no—
146C22×C4—31no—
147C22×C4—31no—
148C22×C4—32no—
149C22×C4—62no—
150C22×C4—62no—
151C22×C4C22×D312yes—
152C22×C4C22×D311yescharacteristic
153C22×C4C22×D311yescharacteristic
154C22×C4C22×D312yes—
155C4⋊C4—62no—
156C4⋊C4—64no—
157C4⋊C4—64no—
158C4⋊C4—64no—
159C4⋊C4—62no—
160C4⋊C4—62no—
161C4⋊C4—64no—
162C22⋊C4—22no—
163C22⋊C4—24no—
164C22⋊C4—62no—
165C22⋊C4—64no—
166C42—34no—
167C42—62no—
168C42—64no—
169C42—64no—
170C22×C3—22no—
171C22×C3—24no—
172C22×C3—41no—
173C22×C3—42no—
174C22×C3—41no—
175C22×C3C22×D411yescharacteristic
176C22×C3C22×Q811yescharacteristic
177C22×C3C2×#4112yes—
178C22×C3C2×#4111yescharacteristic
179C22×C3C2×#4112yes—
180C3×C4—22no—
181C3×C4—24no—
182C3×C4—42no—
183C3×C4—41no—
184Dic3—22no—
185Dic3—24no—
186Dic3—22no—
187Dic3—41no—
188Dic3—42no—
189Dic3C22⋊Q814yes—
190C23—21no—
191C23—22no—
192C23—22no—
193C23—21no—
194C23D3×Q812yes—
195C23C23×D311yescharacteristic
196Q8—28no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—21no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—32no—
206C2×C4—32no—
207C2×C4—32no—
208C2×C4—32no—
209C2×C4—32no—
210C2×C4—34no—
211C2×C4—68no—
212C2×C4—61no—
213C2×C4—62no—
214C2×C4—62no—
215C2×C4—62no—
216C2×C4—61no—
217C2×C4—61no—
218C2×C4—62no—
219C2×C4—62no—
220C2×C4—62no—
221C2×C4—62no—
222C2×C4D3×D412yes—
223C2×C4#23714yes—
224C2×C3—42no—
225C2×C3C2×C22⋊Q811yescharacteristic
226C2×C3#52511yescharacteristic
227C2×C3#52912yes—
228C2×C3#53211yescharacteristic
229C2×C3#53312yes—
230C22—24no—
231C22—22no—
232C22—41no—
233C22—42no—
234C22—41no—
235C22C2×D3×D411yescharacteristic
236C22C2×#23712yes—
237C22C2×D3×Q811yescharacteristic
238C22D3×#4112yes—
239C22D3×#4111yescharacteristic
240C4—22no—
241C4—24no—
242C4—34no—
243C4—41no—
244C4—42no—
245C4—62no—
246C4—64no—
247C4—62no—
248C4—122no—
249C4—121no—
250C3#239611yescharacteristic, complemented, Sylow
251C2—42no—
252C2#558611yescharacteristic
253C2#562212yes—
254C2D3×C22⋊Q811yescharacteristic
255C2#566811yescharacteristic
256C2#568012yes—
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#393C2
prime radicalC24×C3C23
hypercenter#393D3
hypocenterC3#2396
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. C2×Q8×Dic3C2semidirect
4. #5230C2semidirect
8. #4659C2semidirect
9. #4657C2semidirect
10. #4649C2semidirect
11. #4649C2semidirect
58. C22×Dic3Q8semidirect
250. C3#2396semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×Dic32
C2×#8772
C3×#3932
#52302
#49492
#49482
Dic3×C22⋊C42
#46592
#46572
#46492
#23963