Finite Group Databasebeta

← #74093

Subgroups — #74093

Order 384.

408 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 137 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
1C3×#381C211yesmaximal, characteristic
2#5212C211yesmaximal, characteristic
3C4×#217C211yesmaximal, characteristic
4#4949C212yesmaximal
5#4948C212yesmaximal
6#4661C212yesmaximal, complemented
7#4657C212yesmaximal, complemented
8#4655C212yesmaximal, complemented
9#4647C212yesmaximal, complemented
10#2667—31nomaximal, Sylow
11C23×C3×C4C2211yescharacteristic
12C2×C3×C22⋊C4C2212yes—
13C2×#217C2212yes—
14C2×#217C2211yescharacteristic
15C2×#217C2211yescharacteristic
16C2×C4⋊Dic3C2212yes—
17C2×C4⋊Dic3C2212yes—
18C2×C4⋊Dic3C2212yes—
19C2×Dic3⋊C4C2212yes—
20C2×Dic3⋊C4C2212yes—
21C2×Dic3⋊C4C2212yes—
22C2×Dic3⋊C4C2212yes—
23C2×C4×Dic3C2212yes—
24C3×#95C2211yescharacteristic
25C3×#95C2211yescharacteristic
26C3×#95C2212yes—
27C3⋊#95C2212yes—
28C3⋊#95C2212yes—
29C3⋊#95C2212yes—
30C3⋊#95C2211yescharacteristic
31C3⋊#95C2211yescharacteristic
32#399—32no—
33#398—32no—
34#398—32no—
35#396—32no—
36#389—32no—
37#387—32no—
38#381D311yescharacteristic
39#380—31no—
40C4×C22⋊C4—31no—
41C24×C3C2311yescharacteristic
42C22×C3×C4—42no—
43C22×C3×C4Q814yes—
44C22×C3×C4C2312yes—
45C22×C3×C4C2312yes—
46C22×C3×C4C2311yescharacteristic
47C22×C3×C4C2311yescharacteristic
48C22×Dic3C2312yes—
49C22×Dic3C2312yes—
50C22×Dic3C2312yes—
51C22×Dic3C2312yes—
52C3×C22⋊C4—42no—
53#217—24no—
54#217—41no—
55#217—41no—
56C4⋊Dic3—42no—
57C4⋊Dic3—42no—
58C4⋊Dic3—42no—
59Dic3⋊C4—24no—
60Dic3⋊C4—24no—
61Dic3⋊C4—42no—
62Dic3⋊C4—42no—
63C4×Dic3—42no—
64C23×C4C2×D311yescharacteristic
65C2×C4⋊C4—32no—
66C2×C4⋊C4—32no—
67C2×C4⋊C4—32no—
68C2×C4⋊C4—32no—
69C2×C4⋊C4—32no—
70C2×C4⋊C4—32no—
71C2×C4⋊C4—32no—
72C2×C22⋊C4—32no—
73C2×C22⋊C4—31no—
74C2×C22⋊C4—31no—
75C2×C22⋊C4C2×D312yes—
76C2×C42—32no—
77#95—31no—
78#95—31no—
79#95—32no—
80#95—32no—
81#95—32no—
82#95C2×D311yescharacteristic
83#95C2×D311yescharacteristic
84#95C2×D312yes—
85C23×C3—22no—
86C23×C3—21no—
87C23×C3—22no—
88C23×C3—21no—
89C23×C3C2×Q812yes—
90C23×C3C2411yescharacteristic
91C2×C3×C4—22no—
92C2×C3×C4—22no—
93C2×C3×C4—22no—
94C2×C3×C4—22no—
95C2×C3×C4—22no—
96C2×C3×C4—22no—
97C2×C3×C4—22no—
98C2×C3×C4—22no—
99C2×C3×C4—44no—
100C2×C3×C4C2×Q812yes—
101C2×C3×C4C2×Q812yes—
102C2×Dic3—22no—
103C2×Dic3—22no—
104C2×Dic3—22no—
105C2×Dic3—22no—
106C2×Dic3—22no—
107C2×Dic3—22no—
108C2×Dic3—22no—
109C2×Dic3—22no—
110C2×Dic3—22no—
111C2×Dic3—22no—
112C2×Dic3—22no—
113C2×Dic3#4114yes—
114C24C22×D311yescharacteristic
115C22×C4—32no—
116C22×C4—32no—
117C22×C4—32no—
118C22×C4—32no—
119C22×C4—42no—
120C22×C4Dic614yes—
121C22×C4C22×D312yes—
122C22×C4C22×D312yes—
123C22×C4C22×D311yescharacteristic
124C22×C4C22×D311yescharacteristic
125C4⋊C4—64no—
126C4⋊C4—64no—
127C4⋊C4—122no—
128C4⋊C4—122no—
129C4⋊C4—122no—
130C4⋊C4—122no—
131C4⋊C4—122no—
132C22⋊C4—42no—
133C22⋊C4—64no—
134C22⋊C4—121no—
135C22⋊C4—121no—
136C42—122no—
137C22×C3—22no—
138C22×C3—22no—
139C22×C3—22no—
140C22×C3—42no—
141C22×C3—42no—
142C22×C3C22×Q811yescharacteristic
143C22×C3C2×#4112yes—
144C22×C3H(5,𝔽2)12yes—
145C22×C3H(5,𝔽2)11yescharacteristic
146C22×C32−1+411yescharacteristic
147C3×C4—22no—
148C3×C4—22no—
149C3×C4—42no—
150C3×C4—42no—
151Dic3—24no—
152Dic3—42no—
153Dic3—42no—
154Dic3—42no—
155C23—22no—
156C23—22no—
157C23—21no—
158C23—21no—
159C23C2×Dic612yes—
160C23C23×D311yescharacteristic
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—34no—
170C2×C4—44no—
171C2×C4—62no—
172C2×C4—62no—
173C2×C4—62no—
174C2×C4—62no—
175C2×C4—62no—
176C2×C4—62no—
177C2×C4—62no—
178C2×C4—62no—
179C2×C4—62no—
180C2×C4—62no—
181C2×C4—62no—
182C2×C4C2×Dic612yes—
183C2×C4C2×Dic612yes—
184C2×C3—42no—
185C2×C3#53311yescharacteristic
186C2×C3#53912yes—
187C2×C3#54012yes—
188C2×C3#54311yescharacteristic
189C2×C3#54411yescharacteristic
190C22—22no—
191C22—22no—
192C22—22no—
193C22—42no—
194C22—42no—
195C22C22×Dic611yescharacteristic
196C22#100412yes—
197C22D3×#4112yes—
198C22#100911yescharacteristic
199C22#101011yescharacteristic
200C4—22no—
201C4—22no—
202C4—42no—
203C4—42no—
204C4—64no—
205C4—122no—
206C4—122no—
207C4—122no—
208C3#266711yescharacteristic, complemented, Sylow
209C2—42no—
210C2#548311yescharacteristic
211C2#551711yescharacteristic
212C2#551911yescharacteristic
213C2#566012yes—
214C2#566312yes—
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#381C2
prime radicalC24×C3C23
hypercenter#381D3
hypocenterC3#2667
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #381

Complements

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

NH ≅ G/Ntype
6. #4661C2semidirect
7. #4657C2semidirect
8. #4655C2semidirect
9. #4647C2semidirect
208. C3#2667semidirect

Maximal subgroups

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

subgroupindex
C3×#3812
#52122
C4×#2172
#49492
#49482
#46612
#46572
#46552
#46472
#26673