Finite Group Databasebeta

← #82075

Subgroups — #82075

Order 384.

564 conjugacy classes of subgroups; only the 189 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 189 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
1C22×C3×C42C211yesmaximal, characteristic
2C2×#924C211yesmaximal, characteristic
3#5212C211yesmaximal, characteristic
4#5211C212yesmaximal
5#5210C212yesmaximal
6#4933C214yesmaximal, complemented
7#4929C212yesmaximal, complemented
8#4928C212yesmaximal, complemented
9#2855—31nomaximal, Sylow
10C23×C3×C4C2211yescharacteristic
11C23×C3×C4C2211yescharacteristic
12C23×C3×C4C2211yescharacteristic
13C22×Dic6C2211yescharacteristic
14C2×C3×C42—24no—
15C2×C3×C42C2212yes—
16C2×C3×C42C2212yes—
17C2×#217C2212yes—
18C2×#217C2212yes—
19C2×C4⋊Dic3C2211yescharacteristic
20C2×C4⋊Dic3C2212yes—
21#924—24no—
22C2×Dic3⋊C4C2214yes—
23C2×Dic3⋊C4C2212yes—
24C3⋊#95C2214yes—
25C3⋊#95C2214yes—
26C3⋊#95C2214yes—
27C3⋊#95C2212yes—
28C2×C22⋊Q8—31no—
29C22×C42D311yescharacteristic
30#391—32no—
31#389—32no—
32#387—34no—
33#385—32no—
34#381—32no—
35#380—31no—
36C24×C3C2311yescharacteristic
37C22×C3×C4—22no—
38C22×C3×C4—22no—
39C22×C3×C4—22no—
40C22×C3×C4—24no—
41C22×C3×C4—24no—
42C22×C3×C4D414yes—
43C22×C3×C4Q814yescomplemented
44C22×C3×C4C2311yescharacteristic
45C22×C3×C4C2312yes—
46C22×C3×C4C2312yes—
47C22×C3×C4C2311yescharacteristic
48C22×Dic3C2314yes—
49C22×Dic3C2312yes—
50C22×Dic3C2312yes—
51C2×Dic6—22no—
52C2×Dic6—24no—
53C3×C42—24no—
54C3×C42—24no—
55#217—24no—
56#217—24no—
57C4⋊Dic3—22no—
58C4⋊Dic3—24no—
59Dic3⋊C4—24no—
60Dic3⋊C4—28no—
61C22×Q8—31no—
62C23×C4C2×D311yescharacteristic
63C23×C4C2×D311yescharacteristic
64C23×C4C2×D311yescharacteristic
65C22⋊Q8—64no—
66C2×C4⋊C4—32no—
67C2×C4⋊C4—31no—
68C2×C4⋊C4—32no—
69C2×C4⋊C4—34no—
70C2×C22⋊C4—32no—
71C2×C22⋊C4—32no—
72C2×C42—24no—
73C2×C42C2×D312yes—
74C2×C42C2×D312yes—
75#95—32no—
76#95—34no—
77#95—34no—
78#95—34no—
79C23×C3—22no—
80C23×C3—22no—
81C23×C3C2×D412yes—
82C23×C3C2×Q812yes—
83C23×C3#4112yes—
84C23×C3C2411yescharacteristic
85C2×C3×C4—28no—
86C2×C3×C4—28no—
87C2×C3×C4—22no—
88C2×C3×C4—22no—
89C2×C3×C4—22no—
90C2×C3×C4—22no—
91C2×C3×C4—22no—
92C2×C3×C4—28no—
93C2×C3×C4C2×D412yes—
94C2×C3×C4C2×D412yes—
95C2×C3×C4C2×Q814yes—
96C2×C3×C4#4114yes—
97C2×C3×C4#4114yes—
98C2×Dic3—22no—
99C2×Dic3—22no—
100C2×Dic3—24no—
101C2×Dic3—24no—
102C2×Dic3—24no—
103C2×Dic3—24no—
104C2×Dic3—24no—
105Dic6—44no—
106C24C22×D311yescharacteristic
107C2×Q8—64no—
108C2×Q8—62no—
109C22×C4—22no—
110C22×C4—22no—
111C22×C4—22no—
112C22×C4—24no—
113C22×C4—24no—
114C22×C4—34no—
115C22×C4—32no—
116C22×C4—32no—
117C22×C4Dic614yescomplemented
118C22×C4#6714yes—
119C22×C4C22×D312yes—
120C22×C4C22×D312yes—
121C22×C4C22×D311yescharacteristic
122C22×C4C22×D311yescharacteristic
123C4⋊C4—68no—
124C4⋊C4—64no—
125C4⋊C4—64no—
126C4⋊C4—62no—
127C22⋊C4—64no—
128C22⋊C4—64no—
129C42—24no—
130C42—24no—
131C22×C3—22no—
132C22×C3—22no—
133C22×C3—24no—
134C22×C3—24no—
135C22×C3C22⋊Q814yes—
136C22×C3C22×D411yescharacteristic
137C22×C3C22×Q811yescharacteristic
138C22×C3C2×#4111yescharacteristic
139C22×C3C2×#4112yes—
140C22×C3C2×#4112yes—
141C3×C4—24no—
142C3×C4—22no—
143C3×C4—24no—
144C3×C4#12314yes—
145Dic3—42no—
146Dic3—42no—
147Dic3—44no—
148C23—22no—
149C23—22no—
150C23C2×Dic612yes—
151C23#23512yes—
152C23C2×#6712yes—
153C23C23×D311yescharacteristic
154Q8—124no—
155C2×C4—28no—
156C2×C4—28no—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—28no—
163C2×C4—64no—
164C2×C4—64no—
165C2×C4—64no—
166C2×C4—64no—
167C2×C4—64no—
168C2×C4—62no—
169C2×C4—62no—
170C2×C4C2×Dic614yes—
171C2×C4#23514yes—
172C2×C4#23514yes—
173C2×C4C2×#6712yes—
174C2×C4C2×#6712yes—
175C2×C3—24no—
176C2×C3C2×C22⋊Q811yescharacteristic
177C2×C3C2×#12311yescharacteristic
178C2×C3#52511yescharacteristic
179C2×C3#52912yes—
180C2×C3#53312yes—
181C22—24no—
182C22—24no—
183C22—22no—
184C22—22no—
185C22#92414yes—
186C22C22×Dic611yescharacteristic
187C22C2×#23512yes—
188C22C2×#23512yes—
189C22C2×#23511yescharacteristic
190C22C22×#6711yescharacteristic
191C4—24no—
192C4—22no—
193C4—24no—
194C4—124no—
195C4—122no—
196C4—122no—
197C4#92914yes—
198C3#285511yescharacteristic, complemented, Sylow
199C2—24no—
200C2#547012yes—
201C2#547912yes—
202C2C2×#92411yescharacteristic
203C2C2×#92911yescharacteristic
204C2#579111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC22×C3×C42C2
prime radicalC24×C3C23
hypercenterC22×C42D3
hypocenterC3#2855
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42

Complements

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

NH ≅ G/Ntype
6. #4933C2semidirect
7. #4929C2semidirect
8. #4928C2semidirect
43. C22×C3×C4Q8semidirect
117. C22×C4Dic6semidirect
198. C3#2855semidirect

Maximal subgroups

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

subgroupindex
C22×C3×C422
C2×#9242
#52122
#52112
#52102
#49332
#49292
#49282
#28553