Finite Group Databasebeta

← #76361

Subgroups — #76361

Order 384.

564 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 165 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×D4×Dic3C211yesmaximal, characteristic
2C2×#888C211yesmaximal, characteristic, complemented
3C3×#390C211yesmaximal, characteristic
4#5221C211yesmaximal, characteristic
5#4977C211yesmaximal, characteristic, complemented
6#4947C214yesmaximal, complemented
7#4942C214yesmaximal, complemented
8#4936C211yesmaximal, characteristic, complemented
9C4×Dic3⋊C4C211yesmaximal, characteristic, complemented
10#2563—31nomaximal, Sylow
11C22×C3×D4C2211yescharacteristic
12C23×Dic3C2212yes—
13C22×Dic6C2212yes—
14C2×C3×C4⋊C4C2211yescharacteristic
15C2×C3×C22⋊C4C2214yes—
16C2×C3×C42C2211yescharacteristic
17C2×#217C2212yes—
18C2×#217C2214yes—
19D4×Dic3—24no—
20C2×C4⋊Dic3C2211yescharacteristic
21C2×C4⋊Dic3C2211yescharacteristic
22C2×Dic3⋊C4C2212yes—
23C2×Dic3⋊C4C2212yes—
24C2×C4×Dic3C2211yescharacteristic
25C2×C4×Dic3C2211yescharacteristic
26#888—24no—
27C3⋊#95C2212yes—
28C3⋊#95C2214yes—
29C3⋊#95C2214yes—
30C2×C4⋊Q8—31no—
31C2×C4×D4—31no—
32#393—34no—
33#390—31no—
34#390D311yescharacteristic
35#382—31no—
36#382—31no—
37#380—34no—
38C4×C4⋊C4—31no—
39C24×C3C2312yes—
40C2×C3×D4—22no—
41C2×C3×D4—22no—
42C2×C3×D4Q814yescomplemented
43C22×C3×C4C2312yes—
44C22×C3×C4C2312yes—
45C22×C3×C4C2311yescharacteristic
46C22×Dic3—24no—
47C22×Dic3—28no—
48C22×Dic3C2312yes—
49C22×Dic3C2312yes—
50C22×Dic3C2312yes—
51C22×Dic3C2311yescharacteristic
52C22×Dic3C2311yescharacteristic
53C2×Dic6—24no—
54C2×Dic6—44no—
55C3×C4⋊C4—22no—
56C3×C22⋊C4—44no—
57C3×C42—22no—
58#217—24no—
59#217—44no—
60C4⋊Dic3—22no—
61C4⋊Dic3—22no—
62Dic3⋊C4—24no—
63Dic3⋊C4—24no—
64C4×Dic3—22no—
65C4×Dic3D414yes—
66C22×Q8—32no—
67C22×D4C2×D311yescharacteristic
68C23×C4—32no—
69C4⋊Q8—64no—
70C4×D4—64no—
71C2×C4⋊C4—31no—
72C2×C4⋊C4—32no—
73C2×C4⋊C4—31no—
74C2×C4⋊C4—32no—
75C2×C4⋊C4C2×D311yescharacteristic
76C2×C22⋊C4—32no—
77C2×C22⋊C4—34no—
78C2×C22⋊C4C2×D314yes—
79C2×C42—31no—
80C2×C42—31no—
81C2×C42C2×D311yescharacteristic
82#95—32no—
83#95—34no—
84#95—34no—
85C23×C3—22no—
86C23×C3—24no—
87C23×C3—22no—
88C23×C3—24no—
89C23×C3C2×Q814yes—
90C23×C3C2411yescharacteristic
91C3×D4—28no—
92C2×C3×C4—22no—
93C2×C3×C4—22no—
94C2×C3×C4—22no—
95C2×C3×C4—22no—
96C2×C3×C4—24no—
97C2×C3×C4C2×D412yes—
98C2×C3×C4C2×Q812yes—
99C2×C3×C4#4112yes—
100C2×Dic3—22no—
101C2×Dic3—22no—
102C2×Dic3—22no—
103C2×Dic3—22no—
104C2×Dic3—22no—
105C2×Dic3—24no—
106C2×Dic3—22no—
107C2×Dic3—24no—
108C2×Dic3—48no—
109C2×Dic3C2×D412yes—
110C2×Dic3C2×D412yes—
111C2×Dic3#4114yes—
112Dic6—48no—
113C24C22×D312yes—
114C2×Q8—64no—
115C2×Q8—124no—
116C2×D4—22no—
117C2×D4—22no—
118C2×D4Dic614yescomplemented
119C22×C4—31no—
120C22×C4—31no—
121C22×C4—32no—
122C22×C4—32no—
123C22×C4—32no—
124C22×C4—64no—
125C22×C4—68no—
126C22×C4C22×D312yes—
127C22×C4C22×D312yes—
128C22×C4C22×D311yescharacteristic
129C4⋊C4—22no—
130C4⋊C4—64no—
131C4⋊C4—64no—
132C4⋊C4—62no—
133C4⋊C4—62no—
134C22⋊C4—44no—
135C22⋊C4—64no—
136C22⋊C4—124no—
137C42—22no—
138C42—34no—
139C42—62no—
140C22×C3—24no—
141C22×C3—28no—
142C22×C3—42no—
143C22×C3—42no—
144C22×C3—44no—
145C22×C3C22×D411yescharacteristic
146C22×C3C22×Q811yescharacteristic
147C22×C3C2×#4111yescharacteristic
148C22×C3C2×#4112yes—
149C22×C3H(5,𝔽2)12yes—
150C3×C4—42no—
151C3×C4—42no—
152C3×C4C22⋊Q814yes—
153Dic3—22no—
154Dic3—42no—
155Dic3—42no—
156Dic3—42no—
157Dic3#12414yes—
158C23—22no—
159C23—22no—
160C23—24no—
161C23—24no—
162C23C2×Dic614yes—
163C23C23×D311yescharacteristic
164Q8—128no—
165D4—28no—
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—24no—
170C2×C4—22no—
171C2×C4—32no—
172C2×C4—34no—
173C2×C4—32no—
174C2×C4—62no—
175C2×C4—62no—
176C2×C4—62no—
177C2×C4—62no—
178C2×C4—62no—
179C2×C4—64no—
180C2×C4—62no—
181C2×C4—64no—
182C2×C4—128no—
183C2×C4C2×Dic612yes—
184C2×C4D3×D412yes—
185C2×C4#23712yes—
186C2×C3—44no—
187C2×C3C2×C22⋊Q811yescharacteristic
188C2×C3C2×#12411yescharacteristic
189C2×C3#53511yescharacteristic
190C2×C3#55412yes—
191C2×C3#55512yes—
192C22—24no—
193C22—28no—
194C22—42no—
195C22—42no—
196C22—44no—
197C22C22×Dic611yescharacteristic
198C22C2×D3×D411yescharacteristic
199C22C2×#23711yescharacteristic
200C22#100412yes—
201C22D3×#4112yes—
202C4—34no—
203C4—42no—
204C4—42no—
205C4—62no—
206C4—122no—
207C4—122no—
208C4—122no—
209C4#87814yes—
210C3#256311yescharacteristic, complemented, Sylow
211C2—44no—
212C2C2×#87811yescharacteristic
213C2#553912yes—
214C2#561612yes—
215C2D3×#12411yescharacteristic
216C2#571511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#390C2
prime radicalC22×C3×D4C22
hypercenter#390D3
hypocenterC3#2563
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
2. C2×#888C2semidirect
5. #4977C2semidirect
6. #4947C2semidirect
7. #4942C2semidirect
8. #4936C2semidirect
9. C4×Dic3⋊C4C2semidirect
42. C2×C3×D4Q8semidirect
118. C2×D4Dic6semidirect
210. C3#2563semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Dic32
C2×#8882
C3×#3902
#52212
#49772
#49472
#49422
#49362
C4×Dic3⋊C42
#25633