Finite Group Databasebeta

← #82375

Subgroups — #82375

Order 384.

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

normal subgroups only

This group has 604 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×C3×C4×D4C211yesmaximal, characteristic
2C2×#924C212yesmaximal, complemented
3C2×C4×Dic6C211yesmaximal, characteristic, complemented
4#5222C212yesmaximal
5#5212C212yesmaximal, complemented
6#4983C212yesmaximal, complemented
7#4944C214yesmaximal, complemented
8#4936C211yesmaximal, characteristic, complemented
9#2536—31nomaximal, Sylow
10C22×C3×D4C2211yescharacteristic
11C23×C3×C4C2212yes—
12C22×Dic6C2211yescharacteristic
13C3×C4×D4—24no—
14C2×C3×C4⋊C4C2211yescharacteristic
15C2×C3×C22⋊C4C2212yes—
16C2×C3×C42C2211yescharacteristic
17C2×#217C2214yes—
18C2×#217C2214yes—
19C2×C4⋊Dic3C2212yes—
20C2×C4⋊Dic3C2211yescharacteristic
21#924—28no—
22C2×Dic3⋊C4C2214yes—
23C2×Dic3⋊C4C2212yes—
24C2×C4×Dic3C2212yes—
25C4×Dic6—24no—
26C3⋊#95C2212yes—
27C3⋊#95C2212yes—
28C3⋊#95C2214yes—
29C2×C22⋊Q8—32no—
30C2×C4×Q8—31no—
31C2×C4×D4D311yescharacteristic
32#398—32no—
33#394—32no—
34#388—34no—
35#382—31no—
36#380—32no—
37C24×C3C2312yes—
38C2×C3×D4—24no—
39C2×C3×D4—22no—
40C22×C3×C4—24no—
41C22×C3×C4—24no—
42C22×C3×C4—24no—
43C22×C3×C4C2312yes—
44C22×C3×C4C2311yescharacteristic
45C22×C3×C4C2311yescharacteristic
46C22×C3×C4C2311yescharacteristic
47C22×Dic3C2314yes—
48C22×Dic3C2312yes—
49C22×Dic3C2312yes—
50C2×Dic6—22no—
51C2×Dic6—22no—
52C2×Dic6D414yescomplemented
53C3×C4⋊C4—22no—
54C3×C22⋊C4—24no—
55C3×C42D414yes—
56#217—28no—
57#217—28no—
58C4⋊Dic3—22no—
59C4⋊Dic3—24no—
60Dic3⋊C4—24no—
61Dic3⋊C4—28no—
62C4×Dic3—24no—
63C22×Q8—31no—
64C22×D4C2×D311yescharacteristic
65C23×C4C2×D312yes—
66C22⋊Q8—68no—
67C4×Q8—64no—
68C4×D4—24no—
69C2×C4⋊C4—31no—
70C2×C4⋊C4—32no—
71C2×C4⋊C4—32no—
72C2×C4⋊C4—34no—
73C2×C4⋊C4C2×D311yescharacteristic
74C2×C22⋊C4—34no—
75C2×C22⋊C4—34no—
76C2×C22⋊C4C2×D312yes—
77C2×C42—32no—
78C2×C42C2×D311yescharacteristic
79#95—34no—
80#95—32no—
81#95—32no—
82C23×C3—22no—
83C23×C3—22no—
84C23×C3—22no—
85C23×C3—22no—
86C23×C3—22no—
87C23×C3—22no—
88C23×C3#4114yes—
89C23×C3C2411yescharacteristic
90C3×D4—28no—
91C2×C3×C4—22no—
92C2×C3×C4—22no—
93C2×C3×C4—22no—
94C2×C3×C4—22no—
95C2×C3×C4—22no—
96C2×C3×C4—48no—
97C2×C3×C4C2×D412yes—
98C2×C3×C4C2×D412yes—
99C2×C3×C4C2×D412yes—
100C2×C3×C4C2×D412yes—
101C2×C3×C4#4112yes—
102C2×Dic3—22no—
103C2×Dic3—22no—
104C2×Dic3—22no—
105C2×Dic3—24no—
106C2×Dic3—24no—
107C2×Dic3—24no—
108C2×Dic3—24no—
109C2×Dic3C2×D414yes—
110Dic6—28no—
111C24C22×D312yes—
112C2×Q8—34no—
113C2×Q8—62no—
114C2×Q8—62no—
115C2×D4—24no—
116C2×D4—22no—
117C22×C4—24no—
118C22×C4—24no—
119C22×C4—24no—
120C22×C4—32no—
121C22×C4—34no—
122C22×C4—32no—
123C22×C4C22×D312yes—
124C22×C4C22×D311yescharacteristic
125C22×C4C22×D311yescharacteristic
126C22×C4C22×D311yescharacteristic
127C4⋊C4—22no—
128C4⋊C4—68no—
129C4⋊C4—64no—
130C4⋊C4—64no—
131C4⋊C4—62no—
132C22⋊C4—24no—
133C22⋊C4—68no—
134C22⋊C4—68no—
135C42—64no—
136C42#6714yes—
137C22×C3—24no—
138C22×C3—24no—
139C22×C3—24no—
140C22×C3—42no—
141C22×C3—42no—
142C22×C3—42no—
143C22×C3—42no—
144C22×C3C22×D411yescharacteristic
145C22×C3C22×D411yescharacteristic
146C22×C3C2×#4112yes—
147C22×C3C2×#4111yescharacteristic
148C22×C32−1+412yes—
149C3×C4—22no—
150C3×C4—22no—
151C3×C4—42no—
152C3×C4#12114yes—
153Dic3—24no—
154Dic3—42no—
155Dic3—44no—
156C23—22no—
157C23—22no—
158C23—22no—
159C23—22no—
160C23—22no—
161C23—22no—
162C23#23514yes—
163C23C23×D311yescharacteristic
164Q8—68no—
165D4—28no—
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—34no—
172C2×C4—48no—
173C2×C4—62no—
174C2×C4—62no—
175C2×C4—64no—
176C2×C4—64no—
177C2×C4—64no—
178C2×C4—64no—
179C2×C4—62no—
180C2×C4D3×D412yes—
181C2×C4#23712yes—
182C2×C4C2×#6712yes—
183C2×C4C2×#6712yes—
184C2×C4C2×#6712yes—
185C2×C3—44no—
186C2×C3C2×#12111yescharacteristic
187C2×C3#52511yescharacteristic
188C2×C3#53611yescharacteristic
189C2×C3#54812yes—
190C2×C3#55212yes—
191C22—24no—
192C22—24no—
193C22—24no—
194C22—42no—
195C22—42no—
196C22—42no—
197C22—42no—
198C22C2×#23512yes—
199C22C2×D3×D411yescharacteristic
200C22C2×#23711yescharacteristic
201C22#101012yes—
202C22C22×#6711yescharacteristic
203C4—22no—
204C4—22no—
205C4—42no—
206C4—64no—
207C4—122no—
208C4—124no—
209C4#93814yes—
210C3#253611yescharacteristic, complemented, Sylow
211C2—44no—
212C2#554112yes—
213C2#555212yes—
214C2#579111yescharacteristic
215C2C2×#93811yescharacteristic
216C2#582511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC2×C3×C4×D4C2
prime radicalC22×C3×D4C22
hypercenterC2×C4×D4D3
hypocenterC3#2536
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
2. C2×#924C2semidirect
3. C2×C4×Dic6C2semidirect
5. #5212C2semidirect
6. #4983C2semidirect
7. #4944C2semidirect
8. #4936C2semidirect
52. C2×Dic6D4semidirect
210. C3#2536semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×D42
C2×#9242
C2×C4×Dic62
#52222
#52122
#49832
#49442
#49362
#25363