Finite Group Databasebeta

← #78318

Subgroups — #78318

Order 384.

380 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 125 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×#890C211yesmaximal, characteristic
2C2×#889C211yesmaximal, characteristic
3C3×#400C211yesmaximal, characteristic
4#4985C211yesmaximal, characteristic
5#4981C211yesmaximal, characteristic
6#4661C211yesmaximal, characteristic
7#4661C212yesmaximal
8#4658C212yesmaximal
9#4657C212yesmaximal
10#4657C212yesmaximal
11#4650C211yesmaximal, characteristic
12#2822—31nomaximal, Sylow
13C2×C3×C4⋊C4C2211yescharacteristic
14C2×C3×C4⋊C4C2211yescharacteristic
15C2×C4⋊Dic3C2212yes—
16C2×C4⋊Dic3C2211yescharacteristic
17C2×C4⋊Dic3C2211yescharacteristic
18C2×C4⋊Dic3C2212yes—
19C2×Dic3⋊C4C2211yescharacteristic
20C2×Dic3⋊C4C2211yescharacteristic
21C2×Dic3⋊C4C2212yes—
22C2×Dic3⋊C4C2211yescharacteristic
23C2×Dic3⋊C4C2211yescharacteristic
24C2×Dic3⋊C4C2212yes—
25C2×Dic3⋊C4C2212yes—
26C2×C4×Dic3C2211yescharacteristic
27#890—24no—
28#889—24no—
29C3×#95C2211yescharacteristic
30C3×#95C2212yes—
31C3×#95C2212yes—
32C3⋊#95C2212yes—
33C3⋊#95C2211yescharacteristic
34C3⋊#95C2212yes—
35C3⋊#95C2212yes—
36C3⋊#95C2212yes—
37C3⋊#95C2212yes—
38C2×#125—31no—
39C2×#125—31no—
40#400—31no—
41#400—32no—
42#400D311yescharacteristic
43#398—31no—
44#398—32no—
45#398—32no—
46#398—31no—
47#398—32no—
48#382—31no—
49C22×C3×C4C2312yes—
50C22×C3×C4C2312yes—
51C22×C3×C4C2311yescharacteristic
52C22×C3×C4C2311yescharacteristic
53C22×C3×C4C2311yescharacteristic
54C22×Dic3C2312yes—
55C22×Dic3C2311yescharacteristic
56C22×Dic3C2311yescharacteristic
57C22×Dic3C2312yes—
58C22×Dic3C2311yescharacteristic
59C22×Dic3C2311yescharacteristic
60C3×C4⋊C4—22no—
61C3×C4⋊C4—22no—
62C4⋊Dic3—22no—
63C4⋊Dic3—22no—
64C4⋊Dic3—24no—
65C4⋊Dic3—42no—
66Dic3⋊C4—22no—
67Dic3⋊C4—22no—
68Dic3⋊C4—22no—
69Dic3⋊C4—22no—
70Dic3⋊C4—24no—
71Dic3⋊C4—42no—
72Dic3⋊C4—42no—
73C4×Dic3D414yes—
74#125—64no—
75#125—64no—
76C2×C4⋊C4—31no—
77C2×C4⋊C4—31no—
78C2×C4⋊C4—31no—
79C2×C4⋊C4—32no—
80C2×C4⋊C4—31no—
81C2×C4⋊C4—32no—
82C2×C4⋊C4—31no—
83C2×C4⋊C4—31no—
84C2×C4⋊C4—32no—
85C2×C4⋊C4—32no—
86C2×C4⋊C4—32no—
87C2×C4⋊C4C2×D311yescharacteristic
88C2×C4⋊C4C2×D311yescharacteristic
89C2×C42—31no—
90#95—32no—
91#95—31no—
92#95—32no—
93#95—32no—
94#95—32no—
95#95—32no—
96#95C2×D311yescharacteristic
97#95C2×D312yes—
98#95C2×D312yes—
99C23×C3C2411yescharacteristic
100C2×C3×C4—22no—
101C2×C3×C4—21no—
102C2×C3×C4—21no—
103C2×C3×C4—22no—
104C2×C3×C4—21no—
105C2×C3×C4—22no—
106C2×C3×C4—21no—
107C2×C3×C4—22no—
108C2×C3×C4—24no—
109C2×C3×C4—22no—
110C2×C3×C4—22no—
111C2×C3×C4C2×D412yes—
112C2×Dic3—21no—
113C2×Dic3—21no—
114C2×Dic3—22no—
115C2×Dic3—21no—
116C2×Dic3—21no—
117C2×Dic3—22no—
118C2×Dic3—22no—
119C2×Dic3—22no—
120C2×Dic3—21no—
121C2×Dic3—21no—
122C2×Dic3—22no—
123C2×Dic3—22no—
124C2×Dic3—22no—
125C2×Dic3—22no—
126C2×Dic3C2×D412yes—
127C2×Dic3C2×D412yes—
128C22×C4—31no—
129C22×C4—31no—
130C22×C4—32no—
131C22×C4—31no—
132C22×C4—31no—
133C22×C4—32no—
134C22×C4C22×D312yes—
135C22×C4C22×D311yescharacteristic
136C22×C4C22×D312yes—
137C22×C4C22×D311yescharacteristic
138C22×C4C22×D311yescharacteristic
139C4⋊C4—22no—
140C4⋊C4—22no—
141C4⋊C4—62no—
142C4⋊C4—62no—
143C4⋊C4—64no—
144C4⋊C4—62no—
145C4⋊C4—62no—
146C4⋊C4—64no—
147C4⋊C4—62no—
148C4⋊C4—62no—
149C4⋊C4—122no—
150C4⋊C4—122no—
151C4⋊C4—122no—
152C42—34no—
153C22×C3C22×D411yescharacteristic
154C22×C3H(5,𝔽2)11yescharacteristic
155C22×C32−1+411yescharacteristic
156C22×C32−1+412yes—
157C22×C32−1+412yes—
158C3×C4—22no—
159C3×C4—41no—
160C3×C4—42no—
161C3×C4—41no—
162C3×C4—42no—
163Dic3—22no—
164Dic3—22no—
165Dic3—41no—
166Dic3—41no—
167Dic3—42no—
168Dic3—42no—
169C23C23×D311yescharacteristic
170C2×C4—22no—
171C2×C4—21no—
172C2×C4—22no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—22no—
176C2×C4—21no—
177C2×C4—22no—
178C2×C4—24no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—32no—
182C2×C4—32no—
183C2×C4—61no—
184C2×C4—61no—
185C2×C4—62no—
186C2×C4—61no—
187C2×C4—61no—
188C2×C4—62no—
189C2×C4—61no—
190C2×C4—62no—
191C2×C4—62no—
192C2×C4—62no—
193C2×C4—61no—
194C2×C4—62no—
195C2×C4—62no—
196C2×C4—62no—
197C2×C4D3×D412yes—
198C2×C3#53611yescharacteristic
199C2×C3#53611yescharacteristic
200C2×C3#53711yescharacteristic
201C2×C3#56312yes—
202C2×C3#56412yes—
203C22C2×D3×D411yescharacteristic
204C22#100711yescharacteristic
205C22#100911yescharacteristic
206C22#101012yes—
207C22#101012yes—
208C4—22no—
209C4—41no—
210C4—42no—
211C4—42no—
212C4—41no—
213C4—62no—
214C4—62no—
215C4—121no—
216C4—122no—
217C4—121no—
218C4—122no—
219C3#282211yescharacteristic, complemented, Sylow
220C2#562911yescharacteristic
221C2#564811yescharacteristic
222C2#567511yescharacteristic
223C2C3⋊#56412yes—
224C2#571212yes—
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, prime radical, cosocleC23×C3C24
FittingC3×#400C2
hypercenter#400D3
hypocenterC3#2822
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
219. C3#2822semidirect

Maximal subgroups

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

subgroupindex
C2×#8902
C2×#8892
C3×#4002
#49852
#49812
#46612
#46582
#46572
#46502
#28223