Finite Group Databasebeta

← #69537

Subgroups — #69537

Order 384.

612 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 101 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×#235C211yesmaximal, characteristic
2C3×#381C211yesmaximal, characteristic, complemented
3#5211C211yesmaximal, characteristic, complemented
4C2×#806C214yesmaximal, complemented
5#1782—31nomaximal, Sylow
6C23×C3×C4C2211yescharacteristic
7C22×#67C2212yes—
8C2×#235—22no—
9C2×#235—22no—
10C2×#235—28no—
11C22×D12C411yescharacteristic, complemented
12C22×C4×D3C412yescomplemented
13C22×Dic6C411yescharacteristic, complemented
14C2×C3×C22⋊C4C2212yes—
15C2×#217C2212yes—
16C3×#95—22no—
17C3⋊#95—22no—
18#806—28no—
19C22×#41—31no—
20C2×#99—34no—
21#381—31no—
22#381D311yescharacteristic, complemented
23C24×C3C2311yescharacteristic
24C23×D3C2×C412yes—
25C22×C3×C4—24no—
26C22×C3×C4—22no—
27C22×C3×C4C2×C411yescharacteristic
28C22×C3×C4C2×C411yescharacteristic
29C22×C3×C4D412yes—
30C22×C3×C4D412yes—
31C2×#67—22no—
32C2×#67—24no—
33C2×#67—22no—
34C2×#67D418yes—
35C22×Dic3—22no—
36C22×Dic3C2×C412yes—
37#235—216no—
38#235—216no—
39C2×D12—24no—
40C2×D12—22no—
41C2×C4×D3—24no—
42C2×C4×D3—24no—
43C2×C4×D3—24no—
44C2×Dic6—22no—
45C2×Dic6—24no—
46C3×C22⋊C4—24no—
47#217—24no—
48C2×#41—68no—
49C2×#41—62no—
50C2×#41—62no—
51C22×Q8—31no—
52C22×D4—32no—
53C22×D4—31no—
54C23×C4—32no—
55C23×C4C2×D311yescharacteristic
56C2×C22⋊C4—32no—
57C2×C22⋊C4C2×D312yes—
58#99—68no—
59#95—22no—
60#95—62no—
61C23×C3—42no—
62C23×C3C22×C411yescharacteristic
63C23×C3C2×D411yescharacteristic
64C23×C3C2×D414yes—
65C23×C3C2×D411yescharacteristic
66C22×D3—22no—
67C22×D3—22no—
68C22×D3—44no—
69C22×D3C22⋊C414yes—
70C2×C3×C4—22no—
71C2×C3×C4—22no—
72C2×C3×C4—24no—
73C2×C3×C4—24no—
74C2×C3×C4—44no—
75C2×C3×C4—42no—
76C2×C3×C4C22⋊C412yes—
77C2×C3×C4C22⋊C412yes—
78#67—28no—
79#67—28no—
80C2×Dic3—22no—
81C2×Dic3—22no—
82C2×Dic3—44no—
83C2×Dic3—42no—
84C2×Dic3C22⋊C414yes—
85D12—28no—
86C4×D3—216no—
87Dic6—28no—
88C24—32no—
89C24C22×D311yescharacteristic
90#41—616no—
91#41—616no—
92C2×Q8—64no—
93C2×Q8—62no—
94C2×D4—38no—
95C2×D4—64no—
96C2×D4—62no—
97C2×D4—64no—
98C2×D4—62no—
99C2×D4—62no—
100C22×C4—24no—
101C22×C4—22no—
102C22×C4—32no—
103C22×C4—64no—
104C22×C4—64no—
105C22×C4—64no—
106C22×C4—62no—
107C22×C4C4×D311yescharacteristic
108C22×C4C4×D311yescharacteristic
109C22×C4D1212yes—
110C22×C4#6712yes—
111C22⋊C4—24no—
112C22⋊C4—64no—
113C22×C3—22no—
114C22×C3—22no—
115C22×C3—22no—
116C22×C3—44no—
117C22×C3C2×C22⋊C412yes—
118C22×C3C2×C22⋊C411yescharacteristic
119C22×C3#12012yes—
120C22×C3#12012yes—
121C2×D3—24no—
122C2×D3—44no—
123C2×D3—44no—
124C2×D3—44no—
125C3×C4—22no—
126C3×C4—22no—
127C3×C4—82no—
128Dic3—24no—
129Dic3—82no—
130C23—34no—
131C23—42no—
132C23—62no—
133C23—62no—
134C23—124no—
135C23C2×C4×D311yescharacteristic
136C23C2×D1211yescharacteristic
137C23D3×D414yes—
138C23C2×#6711yescharacteristic
139Q8—68no—
140D4—68no—
141D4—68no—
142D4—68no—
143C2×C4—22no—
144C2×C4—22no—
145C2×C4—24no—
146C2×C4—24no—
147C2×C4—34no—
148C2×C4—44no—
149C2×C4—42no—
150C2×C4—616no—
151C2×C4—62no—
152C2×C4—62no—
153C2×C4—124no—
154C2×C4—122no—
155C2×C4(C2×D3)⋊C412yes—
156C2×C4(C2×D3)⋊C412yes—
157C2×C3—22no—
158C2×C3—22no—
159C2×C3—22no—
160C2×C3#37911yescharacteristic
161C2×C3#41012yes—
162D3—44no—
163C22—22no—
164C22—22no—
165C22—22no—
166C22—44no—
167C22—64no—
168C22—124no—
169C22—124no—
170C22—124no—
171C22D3×C22⋊C412yes—
172C22#88212yes—
173C22C2×(C2×D3)⋊C411yescharacteristic
174C22#93712yes—
175C4—22no—
176C4—22no—
177C4—64no—
178C4—82no—
179C4—242no—
180C3#178211yescharacteristic, complemented, Sylow
181C2—22no—
182C2—22no—
183C2—22no—
184C2—124no—
185C2#495511yescharacteristic
186C2#500612yes—
subgroupG/H
centerC22C2×(C2×D3)⋊C4
commutatorC23×C3C22×C4
FrattiniC24C22×D3
FittingC3×#381C2
socleC22×C3C2×C22⋊C4
prime radicalC22×#235C2
cosocleC24×C3C23
hypercenter#381D3
hypocenterC3#1782
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #381

Complements

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

NH ≅ G/Ntype
2. C3×#381C2semidirect
3. #5211C2semidirect
4. C2×#806C2semidirect
11. C22×D12C4semidirect
12. C22×C4×D3C4semidirect
13. C22×Dic6C4semidirect
22. #381D3semidirect
180. C3#1782semidirect

Maximal subgroups

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

subgroupindex
C22×#2352
C3×#3812
#52112
C2×#8062
#17823