Finite Group Databasebeta

← #76599

Subgroups — #76599

Order 384.

392 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 149 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×#391C211yesmaximal, characteristic
2#5229C212yesmaximal
3#4982C212yesmaximal
4#4929C211yesmaximal, characteristic
5#4928C211yesmaximal, characteristic
6#4661C212yesmaximal
7#4656C212yesmaximal
8#4654C212yesmaximal
9#4647C212yesmaximal
10#2806—31nomaximal, Sylow
11C22×C3×Q8C2211yescharacteristic
12C22×Dic6C2211yescharacteristic
13C2×C3×C4⋊C4C2211yescharacteristic
14C2×C3×C42C2211yescharacteristic
15C2×C4⋊Dic3C2211yescharacteristic
16C2×C4⋊Dic3C2212yes—
17C2×C4⋊Dic3C2212yes—
18C2×C4⋊Dic3C2211yescharacteristic
19C2×C4⋊Dic3C2211yescharacteristic
20C2×Dic3⋊C4C2212yes—
21C2×Dic3⋊C4C2212yes—
22C2×Dic3⋊C4C2212yes—
23C2×C4×Dic3C2212yes—
24C3×#95C2212yes—
25C3×#95C2212yes—
26C3⋊#95C2212yes—
27C3⋊#95C2212yes—
28C3⋊#95C2212yes—
29C3⋊#95C2212yes—
30C3⋊#95C2212yes—
31C3⋊#95C2212yes—
32#398—32no—
33#398—32no—
34#395—32no—
35#391—32no—
36#391—31no—
37#391D311yescharacteristic
38#389—31no—
39#389—32no—
40#387—32no—
41C2×C3×Q8—41no—
42C2×C3×Q8—41no—
43C2×C3×Q8Q814yescomplemented
44C22×C3×C4C2312yes—
45C22×C3×C4C2311yescharacteristic
46C22×C3×C4C2312yes—
47C22×C3×C4C2311yescharacteristic
48C22×C3×C4C2311yescharacteristic
49C22×Dic3C2312yes—
50C22×Dic3C2312yes—
51C22×Dic3C2312yes—
52C22×Dic3C2312yes—
53C2×Dic6—22no—
54C2×Dic6—41no—
55C2×Dic6—41no—
56C3×C4⋊C4—41no—
57C3×C42—41no—
58C4⋊Dic3—41no—
59C4⋊Dic3—42no—
60C4⋊Dic3—41no—
61C4⋊Dic3—42no—
62C4⋊Dic3Q814yescomplemented
63Dic3⋊C4—24no—
64Dic3⋊C4—42no—
65Dic3⋊C4—42no—
66C4×Dic3—42no—
67C22×Q8—31no—
68C22×Q8C2×D311yescharacteristic
69C2×C4⋊C4—31no—
70C2×C4⋊C4—32no—
71C2×C4⋊C4—32no—
72C2×C4⋊C4—31no—
73C2×C4⋊C4—32no—
74C2×C4⋊C4—31no—
75C2×C4⋊C4—32no—
76C2×C4⋊C4—32no—
77C2×C4⋊C4C2×D311yescharacteristic
78C2×C42—32no—
79C2×C42C2×D311yescharacteristic
80#95—32no—
81#95—32no—
82#95—32no—
83#95—32no—
84#95—32no—
85#95—32no—
86#95C2×D312yes—
87#95C2×D312yes—
88C23×C3C2411yescharacteristic
89C3×Q8—44no—
90C2×C3×C4—22no—
91C2×C3×C4—21no—
92C2×C3×C4—21no—
93C2×C3×C4—22no—
94C2×C3×C4—21no—
95C2×C3×C4—21no—
96C2×C3×C4—21no—
97C2×C3×C4—21no—
98C2×C3×C4—22no—
99C2×C3×C4—22no—
100C2×C3×C4—22no—
101C2×C3×C4C2×Q812yes—
102C2×C3×C4C2×Q814yes—
103C2×C3×C4C2×Q812yes—
104C2×C3×C4#4112yes—
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—22no—
114C2×Dic3—22no—
115C2×Dic3—22no—
116C2×Dic3C2×Q814yes—
117Dic6—44no—
118C2×Q8—41no—
119C2×Q8—41no—
120C2×Q8—62no—
121C2×Q8—121no—
122C2×Q8—121no—
123C2×Q8Dic614yescomplemented
124C22×C4—32no—
125C22×C4—32no—
126C22×C4—32no—
127C22×C4—32no—
128C22×C4C22×D312yes—
129C22×C4C22×D311yescharacteristic
130C22×C4C22×D312yes—
131C22×C4C22×D311yescharacteristic
132C22×C4C22×D311yescharacteristic
133C4⋊C4—34no—
134C4⋊C4—41no—
135C4⋊C4—64no—
136C4⋊C4—122no—
137C4⋊C4—122no—
138C4⋊C4—121no—
139C4⋊C4—122no—
140C4⋊C4—121no—
141C4⋊C4—122no—
142C42—41no—
143C42—122no—
144C22×C3C22×Q811yescharacteristic
145C22×C3C22×Q811yescharacteristic
146C22×C3C2×#4111yescharacteristic
147C22×C3H(5,𝔽2)11yescharacteristic
148C22×C3H(5,𝔽2)11yescharacteristic
149C22×C3H(5,𝔽2)11yescharacteristic
150C22×C32−1+411yescharacteristic
151C3×C4—22no—
152C3×C4—24no—
153C3×C4—22no—
154C3×C4—22no—
155C3×C4—42no—
156Dic3—24no—
157Dic3—42no—
158Dic3—42no—
159Dic3—42no—
160C23C23×D311yescharacteristic
161Q8—44no—
162Q8—124no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—22no—
167C2×C4—21no—
168C2×C4—22no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—34no—
175C2×C4—62no—
176C2×C4—62no—
177C2×C4—62no—
178C2×C4—62no—
179C2×C4—62no—
180C2×C4—62no—
181C2×C4—62no—
182C2×C4—62no—
183C2×C4—62no—
184C2×C4—62no—
185C2×C4—62no—
186C2×C4C2×Dic612yes—
187C2×C4C2×Dic614yes—
188C2×C4D3×Q812yes—
189C2×C4#23912yes—
190C2×C3#53811yescharacteristic
191C2×C3#54311yescharacteristic
192C2×C3#54411yescharacteristic
193C2×C3#55411yescharacteristic
194C2×C3#55711yescharacteristic
195C2×C3Q8211yescharacteristic
196C2×C3#56211yescharacteristic
197C22C22×Dic611yescharacteristic
198C22#100411yescharacteristic
199C22#100411yescharacteristic
200C22C2×D3×Q811yescharacteristic
201C22C2×#23911yescharacteristic
202C22#100911yescharacteristic
203C22#101011yescharacteristic
204C4—22no—
205C4—22no—
206C4—24no—
207C4—22no—
208C4—42no—
209C4—64no—
210C4—122no—
211C4—122no—
212C4—122no—
213C3#280611yescharacteristic, complemented, Sylow
214C2#548311yescharacteristic
215C2Q8×Dic611yescharacteristic
216C2#557011yescharacteristic
217C2#562311yescharacteristic
218C2#564411yescharacteristic
219C2#568411yescharacteristic
220C2#573011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, prime radical, cosocleC23×C3C24
FittingC3×#391C2
hypercenter#391D3
hypocenterC3#2806
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
43. C2×C3×Q8Q8semidirect
62. C4⋊Dic3Q8semidirect
123. C2×Q8Dic6semidirect
213. C3#2806semidirect

Maximal subgroups

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

subgroupindex
C3×#3912
#52292
#49822
#49292
#49282
#46612
#46562
#46542
#46472
#28063