Finite Group Databasebeta

← #77208

Subgroups — #77208

Order 384.

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

normal subgroups only

This group has 448 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×#944C211yesmaximal, characteristic
2C2×#889C211yesmaximal, characteristic
3C3×#395C211yesmaximal, characteristic
4#5230C211yesmaximal, characteristic
5#4982C212yesmaximal
6#4979C211yesmaximal, characteristic
7#4975C212yesmaximal
8#4657C212yesmaximal
9#4656C211yesmaximal, characteristic
10#4650C212yesmaximal
11#4648C211yesmaximal, characteristic
12#2660—31nomaximal, Sylow
13C22×C3×Q8C2211yescharacteristic
14C22×Dic6C2211yescharacteristic
15C2×C3×C4⋊C4C2211yescharacteristic
16C2×C3×C4⋊C4C2212yes—
17#944—24no—
18C2×C4⋊Dic3C2211yescharacteristic
19C2×C4⋊Dic3C2212yes—
20C2×Dic3⋊C4C2211yescharacteristic
21C2×Dic3⋊C4C2212yes—
22C2×Dic3⋊C4C2211yescharacteristic
23C2×Dic3⋊C4C2211yescharacteristic
24C2×Dic3⋊C4C2212yes—
25C2×Dic3⋊C4C2211yescharacteristic
26C2×Dic3⋊C4C2212yes—
27C2×Dic3⋊C4C2212yes—
28C2×C4×Dic3C2211yescharacteristic
29C2×C4×Dic3C2212yes—
30#889—24no—
31C3×#95C2211yescharacteristic
32C3×#95C2212yes—
33C3⋊#95C2212yes—
34C3⋊#95C2211yescharacteristic
35C3⋊#95C2212yes—
36C3⋊#95C2212yes—
37C3⋊#95C2212yes—
38C2×C4⋊Q8—31no—
39C2×#125—31no—
40#398—32no—
41#398—32no—
42#395—31no—
43#395—31no—
44#395—31no—
45#395D311yescharacteristic
46#384—32no—
47#382—32no—
48#376—31no—
49C2×C3×Q8—22no—
50C2×C3×Q8—42no—
51C22×C3×C4C2311yescharacteristic
52C22×C3×C4C2312yes—
53C22×C3×C4C2311yescharacteristic
54C22×C3×C4C2311yescharacteristic
55C22×C3×C4C2312yes—
56C22×Dic3C2311yescharacteristic
57C22×Dic3C2312yes—
58C22×Dic3C2311yescharacteristic
59C22×Dic3C2311yescharacteristic
60C22×Dic3C2312yes—
61C22×Dic3C2311yescharacteristic
62C2×Dic6—22no—
63C2×Dic6—42no—
64C3×C4⋊C4—22no—
65C3×C4⋊C4—42no—
66C4⋊Dic3—22no—
67C4⋊Dic3—42no—
68Dic3⋊C4—22no—
69Dic3⋊C4—22no—
70Dic3⋊C4—22no—
71Dic3⋊C4—22no—
72Dic3⋊C4—24no—
73Dic3⋊C4—24no—
74Dic3⋊C4—42no—
75Dic3⋊C4—42no—
76C4×Dic3D414yes—
77C4×Dic3Q818yescomplemented
78C22×Q8—31no—
79C22×Q8C2×D311yescharacteristic
80C4⋊Q8—64no—
81#125—64no—
82C2×C4⋊C4—31no—
83C2×C4⋊C4—31no—
84C2×C4⋊C4—31no—
85C2×C4⋊C4—31no—
86C2×C4⋊C4—31no—
87C2×C4⋊C4—32no—
88C2×C4⋊C4—32no—
89C2×C4⋊C4—32no—
90C2×C4⋊C4—32no—
91C2×C4⋊C4—32no—
92C2×C4⋊C4C2×D311yescharacteristic
93C2×C4⋊C4C2×D312yes—
94C2×C42—31no—
95C2×C42—32no—
96#95—31no—
97#95—32no—
98#95—32no—
99#95—32no—
100#95—32no—
101#95C2×D311yescharacteristic
102#95C2×D312yes—
103C23×C3C2411yescharacteristic
104C3×Q8—44no—
105C2×C3×C4—22no—
106C2×C3×C4—22no—
107C2×C3×C4—22no—
108C2×C3×C4—21no—
109C2×C3×C4—22no—
110C2×C3×C4—21no—
111C2×C3×C4—22no—
112C2×C3×C4—22no—
113C2×C3×C4—22no—
114C2×C3×C4—21no—
115C2×C3×C4—21no—
116C2×C3×C4C2×D412yes—
117C2×C3×C4C2×Q814yes—
118C2×Dic3—22no—
119C2×Dic3—21no—
120C2×Dic3—22no—
121C2×Dic3—21no—
122C2×Dic3—21no—
123C2×Dic3—21no—
124C2×Dic3—22no—
125C2×Dic3—22no—
126C2×Dic3—22no—
127C2×Dic3—22no—
128C2×Dic3—21no—
129C2×Dic3—21no—
130C2×Dic3C2×D412yes—
131C2×Dic3C2×D412yes—
132C2×Dic3C2×Q814yes—
133C2×Dic3C2×Q814yes—
134Dic6—44no—
135C2×Q8—22no—
136C2×Q8—42no—
137C2×Q8—62no—
138C2×Q8—122no—
139C22×C4—31no—
140C22×C4—32no—
141C22×C4—31no—
142C22×C4—31no—
143C22×C4—32no—
144C22×C4—31no—
145C22×C4C22×D311yescharacteristic
146C22×C4C22×D312yes—
147C22×C4C22×D311yescharacteristic
148C22×C4C22×D311yescharacteristic
149C22×C4C22×D312yes—
150C4⋊C4—22no—
151C4⋊C4—42no—
152C4⋊C4—62no—
153C4⋊C4—62no—
154C4⋊C4—62no—
155C4⋊C4—62no—
156C4⋊C4—64no—
157C4⋊C4—64no—
158C4⋊C4—62no—
159C4⋊C4—122no—
160C4⋊C4—122no—
161C4⋊C4—122no—
162C42—34no—
163C42—38no—
164C22×C3C22×D411yescharacteristic
165C22×C3C22×Q812yes—
166C22×C3H(5,𝔽2)12yes—
167C22×C32−1+411yescharacteristic
168C22×C32−1+411yescharacteristic
169C3×C4—22no—
170C3×C4—24no—
171C3×C4—41no—
172C3×C4—42no—
173C3×C4—41no—
174Dic3—22no—
175Dic3—24no—
176Dic3—42no—
177Dic3—41no—
178Dic3—41no—
179Dic3C4⋊Q814yes—
180C23C23×D311yescharacteristic
181Q8—44no—
182Q8—124no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—32no—
195C2×C4—34no—
196C2×C4—32no—
197C2×C4—34no—
198C2×C4—62no—
199C2×C4—62no—
200C2×C4—61no—
201C2×C4—61no—
202C2×C4—61no—
203C2×C4—61no—
204C2×C4—62no—
205C2×C4—62no—
206C2×C4—62no—
207C2×C4—62no—
208C2×C4—61no—
209C2×C4—61no—
210C2×C4D3×D412yes—
211C2×C4D3×Q814yes—
212C2×C3C2×C4⋊Q811yescharacteristic
213C2×C3#53511yescharacteristic
214C2×C3#53611yescharacteristic
215C2×C3#54412yes—
216C2×C3#54412yes—
217C22C2×D3×D411yescharacteristic
218C22#100412yes—
219C22C2×D3×Q812yes—
220C22#100711yescharacteristic
221C22#101011yescharacteristic
222C4—22no—
223C4—24no—
224C4—34no—
225C4—42no—
226C4—41no—
227C4—41no—
228C4—64no—
229C4—62no—
230C4—122no—
231C4—121no—
232C4—121no—
233C3#266011yescharacteristic, complemented, Sylow
234C2#559211yescharacteristic
235C2#562312yes—
236C2#562811yescharacteristic
237C2D3×C4⋊Q811yescharacteristic
238C2#572912yes—
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, prime radical, cosocleC23×C3C24
FittingC3×#395C2
hypercenter#395D3
hypocenterC3#2660
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
77. C4×Dic3Q8semidirect
233. C3#2660semidirect

Maximal subgroups

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

subgroupindex
C2×#9442
C2×#8892
C3×#3952
#52302
#49822
#49792
#49752
#46572
#46562
#46502
#46482
#26603