Finite Group Databasebeta

← #78307

Subgroups — #78307

Order 384.

376 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 129 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×#400C211yesmaximal, characteristic
2#4979C211yesmaximal, characteristic
3#4978C212yesmaximal
4#4973C211yesmaximal, characteristic
5#4659C212yesmaximal
6#4657C212yesmaximal
7#4655C212yesmaximal
8#4646C212yesmaximal
9#4646C212yesmaximal
10#2799—31nomaximal, Sylow
11C22×Dic6C2211yescharacteristic
12C22×Dic6C2211yescharacteristic
13C2×C3×C4⋊C4C2211yescharacteristic
14C2×C3×C4⋊C4C2211yescharacteristic
15C2×C4⋊Dic3C2212yes—
16C2×Dic3⋊C4C2212yes—
17C2×Dic3⋊C4C2212yes—
18C2×Dic3⋊C4C2212yes—
19C2×C4×Dic3C2212yes—
20C2×C4×Dic3C2211yescharacteristic
21C3×#95C2212yes—
22C3×#95C2212yes—
23C3×#95C2211yescharacteristic
24C3⋊#95C2212yes—
25C3⋊#95C2212yes—
26C3⋊#95C2212yes—
27C3⋊#95C2212yes—
28C3⋊#95C2212yes—
29C3⋊#95C2212yes—
30C3⋊#95C2212yes—
31C3⋊#95C2211yescharacteristic
32#400—32no—
33#400D311yescharacteristic
34#398—32no—
35#395—31no—
36#391—31no—
37#391—32no—
38#391—32no—
39#387—32no—
40#387—32no—
41C22×C3×C4C2312yes—
42C22×C3×C4C2311yescharacteristic
43C22×C3×C4C2311yescharacteristic
44C22×C3×C4C2312yes—
45C22×C3×C4C2311yescharacteristic
46C22×Dic3C2312yes—
47C22×Dic3C2312yes—
48C22×Dic3C2312yes—
49C22×Dic3C2312yes—
50C2×Dic6—24no—
51C2×Dic6—41no—
52C2×Dic6—41no—
53C2×Dic6—41no—
54C2×Dic6Q814yescomplemented
55C3×C4⋊C4—41no—
56C3×C4⋊C4—41no—
57C4⋊Dic3—42no—
58Dic3⋊C4—24no—
59Dic3⋊C4—24no—
60Dic3⋊C4—42no—
61C4×Dic3—42no—
62C4×Dic3—41no—
63C22×Q8—31no—
64C22×Q8—31no—
65C2×C4⋊C4—32no—
66C2×C4⋊C4—32no—
67C2×C4⋊C4—32no—
68C2×C4⋊C4—32no—
69C2×C4⋊C4C2×D311yescharacteristic
70C2×C4⋊C4C2×D311yescharacteristic
71C2×C42—31no—
72C2×C42—32no—
73#95—32no—
74#95—32no—
75#95—32no—
76#95—32no—
77#95—32no—
78#95—32no—
79#95—32no—
80#95—31no—
81#95C2×D312yes—
82#95C2×D312yes—
83#95C2×D311yescharacteristic
84C23×C3C2411yescharacteristic
85C2×C3×C4—21no—
86C2×C3×C4—21no—
87C2×C3×C4—22no—
88C2×C3×C4—22no—
89C2×C3×C4—22no—
90C2×C3×C4—22no—
91C2×C3×C4—22no—
92C2×C3×C4—21no—
93C2×C3×C4—21no—
94C2×C3×C4—22no—
95C2×C3×C4—24no—
96C2×C3×C4C2×Q812yes—
97C2×Dic3—22no—
98C2×Dic3—22no—
99C2×Dic3—22no—
100C2×Dic3—22no—
101C2×Dic3—22no—
102C2×Dic3—22no—
103C2×Dic3—22no—
104C2×Dic3—22no—
105C2×Dic3—22no—
106C2×Dic3—22no—
107C2×Dic3C2×Q814yes—
108C2×Dic3#4114yes—
109Dic6—44no—
110Dic6—44no—
111C2×Q8—34no—
112C2×Q8—64no—
113C2×Q8—121no—
114C2×Q8—121no—
115C2×Q8—121no—
116C22×C4—32no—
117C22×C4—32no—
118C22×C4—32no—
119C22×C4—32no—
120C22×C4C22×D312yes—
121C22×C4C22×D311yescharacteristic
122C22×C4C22×D311yescharacteristic
123C22×C4C22×D312yes—
124C22×C4C22×D311yescharacteristic
125C4⋊C4—41no—
126C4⋊C4—41no—
127C4⋊C4—64no—
128C4⋊C4—64no—
129C4⋊C4—122no—
130C4⋊C4—122no—
131C42—121no—
132C42—122no—
133C22×C3C22×Q811yescharacteristic
134C22×C3C2×#4112yes—
135C22×C3H(5,𝔽2)11yescharacteristic
136C22×C3H(5,𝔽2)11yescharacteristic
137C22×C32−1+412yes—
138C3×C4—22no—
139C3×C4—42no—
140C3×C4—41no—
141C3×C4—41no—
142C3×C4—42no—
143Dic3—24no—
144Dic3—24no—
145Dic3—42no—
146Dic3—42no—
147C23C23×D311yescharacteristic
148Q8—124no—
149Q8—124no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—22no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—22no—
156C2×C4—22no—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—24no—
160C2×C4—22no—
161C2×C4—34no—
162C2×C4—34no—
163C2×C4—62no—
164C2×C4—62no—
165C2×C4—62no—
166C2×C4—62no—
167C2×C4—62no—
168C2×C4—62no—
169C2×C4—62no—
170C2×C4—62no—
171C2×C4—62no—
172C2×C4—62no—
173C2×C4D3×Q812yes—
174C2×C3#54212yes—
175C2×C3#54311yescharacteristic
176C2×C3#55111yescharacteristic
177C2×C3#55712yes—
178C2×C3#56311yescharacteristic
179C22#100411yescharacteristic
180C22C2×D3×Q811yescharacteristic
181C22D3×#4112yes—
182C22#100911yescharacteristic
183C22#101012yes—
184C4—22no—
185C4—41no—
186C4—41no—
187C4—42no—
188C4—42no—
189C4—64no—
190C4—64no—
191C4—122no—
192C4—122no—
193C3#279911yescharacteristic, complemented, Sylow
194C2#563411yescharacteristic
195C2#565911yescharacteristic
196C2#567211yescharacteristic
197C2#568512yes—
198C2#571012yes—
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, prime radical, cosocleC23×C3C24
FittingC3×#400C2
hypercenter#400D3
hypocenterC3#2799
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
54. C2×Dic6Q8semidirect
193. C3#2799semidirect

Maximal subgroups

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

subgroupindex
C3×#4002
#49792
#49782
#49732
#46592
#46572
#46552
#46462
#27993