Finite Group Databasebeta

← #77797

Subgroups — #77797

Order 384.

468 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 157 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×#889C211yesmaximal, characteristic
2C2×#887C211yesmaximal, characteristic
3C3×#398C211yesmaximal, characteristic
4#4983C211yesmaximal, characteristic
5#4979C211yesmaximal, characteristic
6#4976C212yesmaximal
7#4972C212yesmaximal
8#4656C211yesmaximal, characteristic
9#4649C211yesmaximal, characteristic
10#4647C212yesmaximal
11#4647C212yesmaximal
12#2584—31nomaximal, Sylow
13C22×Dic6C2211yescharacteristic
14C2×C3×C4⋊C4C2211yescharacteristic
15C2×C3×C4⋊C4C2211yescharacteristic
16C2×C3×C4⋊C4C2212yes—
17C2×C4⋊Dic3C2211yescharacteristic
18C2×C4⋊Dic3C2212yes—
19C2×Dic3⋊C4C2212yes—
20C2×Dic3⋊C4C2211yescharacteristic
21C2×Dic3⋊C4C2212yes—
22C2×Dic3⋊C4C2211yescharacteristic
23C2×Dic3⋊C4C2211yescharacteristic
24C2×Dic3⋊C4C2211yescharacteristic
25C2×Dic3⋊C4C2212yes—
26C2×Dic3⋊C4C2212yes—
27C2×C4×Dic3C2212yes—
28C2×C4×Dic3C2212yes—
29C2×C4×Dic3C2211yescharacteristic
30#889—24no—
31#887—24no—
32C3×#95C2212yes—
33C3×#95C2211yescharacteristic
34C3⋊#95C2212yes—
35C3⋊#95C2212yes—
36C3⋊#95C2211yescharacteristic
37C3⋊#95C2212yes—
38C2×#125—31no—
39C2×C4×Q8—31no—
40#398—31no—
41#398D311yescharacteristic
42#395—31no—
43#395—31no—
44#389—32no—
45#389—32no—
46#389—32no—
47#389—32no—
48C4×C4⋊C4—31no—
49C22×C3×C4C2311yescharacteristic
50C22×C3×C4C2312yes—
51C22×C3×C4C2311yescharacteristic
52C22×C3×C4C2311yescharacteristic
53C22×C3×C4C2312yes—
54C22×Dic3C2311yescharacteristic
55C22×Dic3C2311yescharacteristic
56C22×Dic3C2311yescharacteristic
57C22×Dic3C2312yes—
58C22×Dic3C2312yes—
59C22×Dic3C2311yescharacteristic
60C2×Dic6—22no—
61C2×Dic6—24no—
62C3×C4⋊C4—22no—
63C3×C4⋊C4—22no—
64C3×C4⋊C4—42no—
65C4⋊Dic3—22no—
66C4⋊Dic3—24no—
67Dic3⋊C4—22no—
68Dic3⋊C4—24no—
69Dic3⋊C4—22no—
70Dic3⋊C4—22no—
71Dic3⋊C4—22no—
72Dic3⋊C4—24no—
73Dic3⋊C4—42no—
74Dic3⋊C4Q818yescomplemented
75C4×Dic3—24no—
76C4×Dic3—42no—
77C4×Dic3D414yes—
78C22×Q8—31no—
79#125—64no—
80C4×Q8—64no—
81C2×C4⋊C4—32no—
82C2×C4⋊C4—32no—
83C2×C4⋊C4—31no—
84C2×C4⋊C4—31no—
85C2×C4⋊C4—31no—
86C2×C4⋊C4—31no—
87C2×C4⋊C4—31no—
88C2×C4⋊C4—32no—
89C2×C4⋊C4—32no—
90C2×C4⋊C4—32no—
91C2×C4⋊C4C2×D311yescharacteristic
92C2×C4⋊C4C2×D311yescharacteristic
93C2×C4⋊C4C2×D312yes—
94C2×C42—31no—
95C2×C42—32no—
96C2×C42—32no—
97#95—31no—
98#95—32no—
99#95—32no—
100#95—32no—
101#95C2×D311yescharacteristic
102#95C2×D312yes—
103C23×C3C2411yescharacteristic
104C2×C3×C4—22no—
105C2×C3×C4—22no—
106C2×C3×C4—22no—
107C2×C3×C4—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×C4C2×D412yes—
116C2×C3×C4C2×Q814yes—
117C2×Dic3—22no—
118C2×Dic3—22no—
119C2×Dic3—22no—
120C2×Dic3—21no—
121C2×Dic3—22no—
122C2×Dic3—21no—
123C2×Dic3—21no—
124C2×Dic3—21no—
125C2×Dic3—22no—
126C2×Dic3—21no—
127C2×Dic3—21no—
128C2×Dic3C2×D412yes—
129C2×Dic3C2×D412yes—
130C2×Dic3C2×Q814yes—
131C2×Dic3C2×Q814yes—
132C2×Dic3#4114yes—
133Dic6—28no—
134C2×Q8—64no—
135C2×Q8—62no—
136C22×C4—31no—
137C22×C4—32no—
138C22×C4—32no—
139C22×C4—31no—
140C22×C4—31no—
141C22×C4—31no—
142C22×C4C22×D311yescharacteristic
143C22×C4C22×D312yes—
144C22×C4C22×D311yescharacteristic
145C22×C4C22×D311yescharacteristic
146C22×C4C22×D312yes—
147C4⋊C4—22no—
148C4⋊C4—22no—
149C4⋊C4—38no—
150C4⋊C4—42no—
151C4⋊C4—64no—
152C4⋊C4—62no—
153C4⋊C4—62no—
154C4⋊C4—62no—
155C4⋊C4—62no—
156C4⋊C4—64no—
157C4⋊C4—64no—
158C4⋊C4—62no—
159C4⋊C4—122no—
160C42—34no—
161C42—64no—
162C42—122no—
163C22×C3C22×D411yescharacteristic
164C22×C3C22×Q812yes—
165C22×C3C2×#4112yes—
166C22×C3H(5,𝔽2)11yescharacteristic
167C22×C32−1+411yescharacteristic
168C3×C4—22no—
169C3×C4—24no—
170C3×C4—41no—
171C3×C4—42no—
172C3×C4—41no—
173Dic3—24no—
174Dic3—22no—
175Dic3—24no—
176Dic3—41no—
177Dic3—41no—
178Dic3C4⋊Q814yes—
179C23C23×D311yescharacteristic
180Q8—68no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—21no—
191C2×C4—22no—
192C2×C4—32no—
193C2×C4—34no—
194C2×C4—32no—
195C2×C4—34no—
196C2×C4—34no—
197C2×C4—62no—
198C2×C4—61no—
199C2×C4—62no—
200C2×C4—62no—
201C2×C4—62no—
202C2×C4—61no—
203C2×C4—61no—
204C2×C4—61no—
205C2×C4—61no—
206C2×C4—62no—
207C2×C4—61no—
208C2×C4D3×D412yes—
209C2×C4D3×Q814yes—
210C2×C3#52511yescharacteristic
211C2×C3C2×C4⋊Q811yescharacteristic
212C2×C3#53711yescharacteristic
213C2×C3#55412yes—
214C2×C3#55712yes—
215C22C2×D3×D411yescharacteristic
216C22#100411yescharacteristic
217C22C2×D3×Q812yes—
218C22D3×#4112yes—
219C22#101011yescharacteristic
220C4—24no—
221C4—22no—
222C4—34no—
223C4—41no—
224C4—42no—
225C4—41no—
226C4—64no—
227C4—64no—
228C4—62no—
229C4—121no—
230C4—121no—
231C3#258411yescharacteristic, complemented, Sylow
232C2#560811yescharacteristic
233C2#563612yes—
234C2#565611yescharacteristic
235C2#568512yes—
236C2D3×C4⋊Q811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, prime radical, cosocleC23×C3C24
FittingC3×#398C2
hypercenter#398D3
hypocenterC3#2584
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
74. Dic3⋊C4Q8semidirect
231. C3#2584semidirect

Maximal subgroups

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

subgroupindex
C2×#8892
C2×#8872
C3×#3982
#49832
#49792
#49762
#49722
#46562
#46492
#46472
#25843