Finite Group Databasebeta

← #77171

Subgroups — #77171

Order 384.

504 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 167 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
2C2×#887C211yesmaximal, characteristic
3C3×#395C211yesmaximal, characteristic
4#5230C211yesmaximal, characteristic
5#4982C212yesmaximal
6#4981C211yesmaximal, characteristic
7Dic3×C4⋊C4C212yesmaximal
8#4661C211yesmaximal, characteristic
9#4656C212yesmaximal
10#4649C212yesmaximal
11#4648C211yesmaximal, characteristic
12#2399—31nomaximal, Sylow
13C22×C3×Q8C2211yescharacteristic
14C22×Dic6C2211yescharacteristic
15C2×C3×C4⋊C4C2212yes—
16C2×C3×C4⋊C4C2211yescharacteristic
17Q8×Dic3—24no—
18C2×C4⋊Dic3C2212yes—
19C2×C4⋊Dic3C2212yes—
20C2×C4⋊Dic3C2211yescharacteristic
21C2×Dic3⋊C4C2212yes—
22C2×Dic3⋊C4C2212yes—
23C2×Dic3⋊C4C2211yescharacteristic
24C2×Dic3⋊C4C2211yescharacteristic
25C2×C4×Dic3C2211yescharacteristic
26C2×C4×Dic3C2212yes—
27C2×C4×Dic3C2211yescharacteristic
28C2×C4×Dic3C2211yescharacteristic
29C2×C4×Dic3C2212yes—
30#887—24no—
31C3×#95C2211yescharacteristic
32C3×#95C2212yes—
33C3⋊#95C2212yes—
34C3⋊#95C2211yescharacteristic
35C3⋊#95C2212yes—
36C3⋊#95C2212yes—
37C3⋊#95C2212yes—
38C2×C4×Q8—31no—
39C2×C4×Q8—31no—
40#398—31no—
41#398—32no—
42#398—31no—
43#395—31no—
44#395—32no—
45#395D311yescharacteristic
46C4×C4⋊C4—32no—
47C4×C4⋊C4—32no—
48#376—31no—
49C2×C3×Q8—22no—
50C2×C3×Q8—24no—
51C22×C3×C4C2311yescharacteristic
52C22×C3×C4C2311yescharacteristic
53C22×C3×C4C2312yes—
54C22×C3×C4C2312yes—
55C22×C3×C4C2311yescharacteristic
56C22×Dic3C2311yescharacteristic
57C22×Dic3C2312yes—
58C22×Dic3C2311yescharacteristic
59C22×Dic3C2312yes—
60C22×Dic3C2311yescharacteristic
61C22×Dic3C2311yescharacteristic
62C2×Dic6—22no—
63C2×Dic6—22no—
64C2×Dic6—22no—
65C3×C4⋊C4—22no—
66C3×C4⋊C4—24no—
67C4⋊Dic3—22no—
68C4⋊Dic3—24no—
69C4⋊Dic3—24no—
70Dic3⋊C4—22no—
71Dic3⋊C4—24no—
72Dic3⋊C4—24no—
73Dic3⋊C4—22no—
74C4×Dic3—22no—
75C4×Dic3—22no—
76C4×Dic3—24no—
77C4×Dic3D414yes—
78C4×Dic3Q818yescomplemented
79C22×Q8—31no—
80C22×Q8C2×D311yescharacteristic
81C4×Q8—64no—
82C4×Q8—64no—
83C2×C4⋊C4—31no—
84C2×C4⋊C4—32no—
85C2×C4⋊C4—31no—
86C2×C4⋊C4—31no—
87C2×C4⋊C4—32no—
88C2×C4⋊C4—32no—
89C2×C4⋊C4—32no—
90C2×C4⋊C4C2×D311yescharacteristic
91C2×C4⋊C4C2×D312yes—
92C2×C42—31no—
93C2×C42—32no—
94C2×C42—31no—
95C2×C42—32no—
96C2×C42—31no—
97#95—31no—
98#95—32no—
99#95—32no—
100#95—32no—
101#95—32no—
102#95C2×D311yescharacteristic
103#95C2×D312yes—
104C23×C3C2411yescharacteristic
105C3×Q8—28no—
106C2×C3×C4—22no—
107C2×C3×C4—21no—
108C2×C3×C4—22no—
109C2×C3×C4—22no—
110C2×C3×C4—21no—
111C2×C3×C4—22no—
112C2×C3×C4—22no—
113C2×C3×C4—21no—
114C2×C3×C4—22no—
115C2×C3×C4C2×D412yes—
116C2×C3×C4C2×Q814yes—
117C2×C3×C4#4114yes—
118C2×C3×C4#4112yes—
119C2×Dic3—21no—
120C2×Dic3—22no—
121C2×Dic3—22no—
122C2×Dic3—21no—
123C2×Dic3—21no—
124C2×Dic3—21no—
125C2×Dic3—21no—
126C2×Dic3—22no—
127C2×Dic3—22no—
128C2×Dic3—22no—
129C2×Dic3—22no—
130C2×Dic3C2×D412yes—
131C2×Dic3C2×D412yes—
132C2×Dic3C2×Q814yes—
133C2×Dic3C2×Q814yes—
134C2×Dic3#4112yes—
135Dic6—28no—
136C2×Q8—22no—
137C2×Q8—24no—
138C2×Q8—62no—
139C2×Q8—62no—
140C2×Q8—62no—
141C22×C4—31no—
142C22×C4—32no—
143C22×C4—31no—
144C22×C4—31no—
145C22×C4—31no—
146C22×C4—32no—
147C22×C4C22×D311yescharacteristic
148C22×C4C22×D311yescharacteristic
149C22×C4C22×D312yes—
150C22×C4C22×D312yes—
151C22×C4C22×D311yescharacteristic
152C4⋊C4—22no—
153C4⋊C4—24no—
154C4⋊C4—62no—
155C4⋊C4—64no—
156C4⋊C4—64no—
157C4⋊C4—62no—
158C4⋊C4—62no—
159C4⋊C4—64no—
160C4⋊C4—64no—
161C42—34no—
162C42—38no—
163C42—62no—
164C42—64no—
165C42—62no—
166C22×C3C22×D411yescharacteristic
167C22×C3C22×Q812yes—
168C22×C3C2×#4112yes—
169C22×C3C2×#4111yescharacteristic
170C22×C3C2×#4111yescharacteristic
171C3×C4—22no—
172C3×C4—22no—
173C3×C4—24no—
174C3×C4—24no—
175C3×C4—41no—
176Dic3—22no—
177Dic3—22no—
178Dic3—24no—
179Dic3—41no—
180Dic3—42no—
181Dic3C4⋊Q814yes—
182C23C23×D311yescharacteristic
183Q8—28no—
184Q8—68no—
185C2×C4—21no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—22no—
194C2×C4—32no—
195C2×C4—34no—
196C2×C4—32no—
197C2×C4—34no—
198C2×C4—32no—
199C2×C4—61no—
200C2×C4—62no—
201C2×C4—62no—
202C2×C4—61no—
203C2×C4—61no—
204C2×C4—61no—
205C2×C4—61no—
206C2×C4—62no—
207C2×C4—62no—
208C2×C4—62no—
209C2×C4—62no—
210C2×C4D3×D412yes—
211C2×C4#23714yes—
212C2×C4D3×Q814yes—
213C2×C4#23912yes—
214C2×C3#52511yescharacteristic
215C2×C3#52511yescharacteristic
216C2×C3C2×C4⋊Q811yescharacteristic
217C2×C3#53312yes—
218C2×C3#53312yes—
219C22C2×D3×D411yescharacteristic
220C22C2×#23712yes—
221C22C2×D3×Q812yes—
222C22C2×#23911yescharacteristic
223C22D3×#4111yescharacteristic
224C4—22no—
225C4—24no—
226C4—24no—
227C4—22no—
228C4—34no—
229C4—41no—
230C4—64no—
231C4—62no—
232C4—62no—
233C4—121no—
234C4—122no—
235C3#239911yescharacteristic, complemented, Sylow
236C2#558611yescharacteristic
237C2#562212yes—
238C2#562711yescharacteristic
239C2D3×C4⋊Q811yescharacteristic
240C2#572812yes—
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, prime radical, cosocleC23×C3C24
FittingC3×#395C2
hypercenter#395D3
hypocenterC3#2399
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
78. C4×Dic3Q8semidirect
235. C3#2399semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×Dic32
C2×#8872
C3×#3952
#52302
#49822
#49812
Dic3×C4⋊C42
#46612
#46562
#46492
#46482
#23993