Finite Group Databasebeta

← #82383

Subgroups — #82383

Order 384.

532 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 532 conjugacy classes of subgroups — too many to draw. Only its 157 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×#870C211yesmaximal, characteristic, complemented
3#5222C212yesmaximal
4#5212C212yesmaximal, complemented
5#5212C212yesmaximal, complemented
6#4985C212yesmaximal, complemented
7#4945C214yesmaximal, complemented
8C4×C4⋊Dic3C211yesmaximal, characteristic, complemented
9#2562—31nomaximal, Sylow
10C22×C3×D4C2211yescharacteristic
11C23×C3×C4C2212yes—
12C3×C4×D4—24no—
13C2×C3×C4⋊C4C2211yescharacteristic
14C2×C3×C22⋊C4C2212yes—
15C2×C3×C42C2211yescharacteristic
16C2×#217C2214yes—
17C2×#217C2214yes—
18C2×C4⋊Dic3C2212yes—
19C2×C4⋊Dic3C2211yescharacteristic
20C2×C4⋊Dic3C2211yescharacteristic
21C2×Dic3⋊C4C2212yes—
22C2×Dic3⋊C4C2212yes—
23C2×C4×Dic3C2212yes—
24#870—24no—
25C3⋊#95C2212yes—
26C3⋊#95C2214yes—
27C3⋊#95C2214yes—
28C2×#125—31no—
29C2×C4×D4D311yescharacteristic
30#400—32no—
31#394—32no—
32#388—34no—
33#380—32no—
34#380—32no—
35C4×C4⋊C4—31no—
36C24×C3C2312yes—
37C2×C3×D4—24no—
38C2×C3×D4—22no—
39C22×C3×C4—24no—
40C22×C3×C4—24no—
41C22×C3×C4—24no—
42C22×C3×C4C2312yes—
43C22×C3×C4C2311yescharacteristic
44C22×C3×C4C2311yescharacteristic
45C22×C3×C4C2311yescharacteristic
46C22×Dic3C2314yes—
47C22×Dic3C2312yes—
48C22×Dic3C2312yes—
49C3×C4⋊C4—22no—
50C3×C22⋊C4—24no—
51C3×C42D414yes—
52#217—28no—
53#217—44no—
54C4⋊Dic3—22no—
55C4⋊Dic3—22no—
56C4⋊Dic3—24no—
57Dic3⋊C4—24no—
58Dic3⋊C4—24no—
59C4×Dic3—42no—
60C22×D4C2×D311yescharacteristic
61C23×C4C2×D312yes—
62#125—64no—
63C4×D4—24no—
64C2×C4⋊C4—31no—
65C2×C4⋊C4—31no—
66C2×C4⋊C4—32no—
67C2×C4⋊C4—32no—
68C2×C4⋊C4—32no—
69C2×C4⋊C4C2×D311yescharacteristic
70C2×C22⋊C4—34no—
71C2×C22⋊C4—34no—
72C2×C22⋊C4C2×D312yes—
73C2×C42—32no—
74C2×C42C2×D311yescharacteristic
75#95—32no—
76#95—34no—
77#95—34no—
78C23×C3—22no—
79C23×C3—22no—
80C23×C3—22no—
81C23×C3—22no—
82C23×C3—22no—
83C23×C3—22no—
84C23×C3#4114yes—
85C23×C3C2411yescharacteristic
86C3×D4—28no—
87C2×C3×C4—22no—
88C2×C3×C4—21no—
89C2×C3×C4—21no—
90C2×C3×C4—22no—
91C2×C3×C4—22no—
92C2×C3×C4—22no—
93C2×C3×C4—48no—
94C2×C3×C4C2×D412yes—
95C2×C3×C4C2×D412yes—
96C2×C3×C4C2×D412yes—
97C2×C3×C4#4114yes—
98C2×Dic3—22no—
99C2×Dic3—22no—
100C2×Dic3—24no—
101C2×Dic3—24no—
102C2×Dic3—24no—
103C2×Dic3—24no—
104C2×Dic3—22no—
105C2×Dic3—22no—
106C24C22×D312yes—
107C2×D4—24no—
108C2×D4—22no—
109C22×C4—24no—
110C22×C4—24no—
111C22×C4—24no—
112C22×C4—34no—
113C22×C4—32no—
114C22×C4—32no—
115C22×C4C22×D312yes—
116C22×C4C22×D311yescharacteristic
117C22×C4C22×D311yescharacteristic
118C22×C4C22×D311yescharacteristic
119C4⋊C4—22no—
120C4⋊C4—64no—
121C4⋊C4—64no—
122C4⋊C4—64no—
123C4⋊C4—62no—
124C4⋊C4—62no—
125C22⋊C4—24no—
126C22⋊C4—68no—
127C22⋊C4—124no—
128C42—122no—
129C42#6714yes—
130C22×C3—24no—
131C22×C3—24no—
132C22×C3—24no—
133C22×C3—42no—
134C22×C3—42no—
135C22×C3—42no—
136C22×C3—42no—
137C22×C3C22×D411yescharacteristic
138C22×C3C2×#4112yes—
139C22×C3C2×#4112yes—
140C22×C3H(5,𝔽2)11yescharacteristic
141C22×C32−1+411yescharacteristic
142C3×C4—22no—
143C3×C4—22no—
144C3×C4—42no—
145C3×C4#12414yes—
146Dic3—42no—
147Dic3—42no—
148Dic3—44no—
149C23—22no—
150C23—22no—
151C23—22no—
152C23—22no—
153C23—22no—
154C23—22no—
155C23#23514yes—
156C23C23×D311yescharacteristic
157D4—28no—
158C2×C4—22no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—48no—
165C2×C4—64no—
166C2×C4—64no—
167C2×C4—64no—
168C2×C4—64no—
169C2×C4—62no—
170C2×C4—62no—
171C2×C4—62no—
172C2×C4—62no—
173C2×C4#23714yes—
174C2×C4C2×#6712yes—
175C2×C4C2×#6712yes—
176C2×C4C2×#6712yes—
177C2×C3—44no—
178C2×C3#52511yescharacteristic
179C2×C3C2×#12411yescharacteristic
180C2×C3#53711yescharacteristic
181C2×C3#55212yes—
182C2×C3#55312yes—
183C22—24no—
184C22—24no—
185C22—24no—
186C22—42no—
187C22—42no—
188C22—42no—
189C22—42no—
190C22C2×#23512yes—
191C22C2×#23712yes—
192C22#100911yescharacteristic
193C22#101011yescharacteristic
194C22C22×#6711yescharacteristic
195C4—22no—
196C4—22no—
197C4—42no—
198C4—122no—
199C4—124no—
200C4—122no—
201C4#93614yes—
202C3#256211yescharacteristic, complemented, Sylow
203C2—44no—
204C2#554112yes—
205C2#556512yes—
206C2#579111yescharacteristic
207C2C2×#93611yescharacteristic
208C2#583311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC2×C3×C4×D4C2
prime radicalC22×C3×D4C22
hypercenterC2×C4×D4D3
hypocenterC3#2562
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×#870C2semidirect
4. #5212C2semidirect
5. #5212C2semidirect
6. #4985C2semidirect
7. #4945C2semidirect
8. C4×C4⋊Dic3C2semidirect
202. C3#2562semidirect

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×#8702
#52222
#52122
#49852
#49452
C4×C4⋊Dic32
#25623