Finite Group Databasebeta

← #79319

Subgroups — #79319

Order 384.

366 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 121 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
1#5477C211yesmaximal, characteristic
2#5470C211yesmaximal, characteristic, complemented
3C3×#432C211yesmaximal, characteristic, complemented
4#4846C212yesmaximal, complemented
5#4842C212yesmaximal, complemented
6#4737C214yesmaximal, complemented
7#4713C212yesmaximal, complemented
8#4702C212yesmaximal, complemented
9#3235—31nomaximal, Sylow
10C2×#235—21no—
11C2×C3×C42C2211yescharacteristic
12#930—21no—
13#930C2212yes—
14#926C2212yes—
15#924—21no—
16#924—21no—
17#875—21no—
18#874C2211yescharacteristic
19C4×D12—21no—
20C4×D12C2212yes—
21#870C2212yes—
22#869C2211yescharacteristic
23#869C2211yescharacteristic
24#869C2211yescharacteristic
25C4×Dic6—21no—
26C4×Dic6—21no—
27C3×C4⋊C8C2212yes—
28C3×C22⋊C8C2212yes—
29C3×#97C2212yes—
30#821C2214yes—
31#821C2214yes—
32#818C2214yes—
33#817C2214yes—
34#533—31no—
35#532—31no—
36#501—32no—
37#490—32no—
38#482—34no—
39#476—32no—
40#474—32no—
41#432D311yescharacteristic, complemented
42C22×C3×C4D411yescharacteristic
43C22×C3×C4D411yescharacteristic
44C22×C3×C4C2311yescharacteristic
45C2×#67—22no—
46#235—44no—
47C2×D12—21no—
48C2×D12—21no—
49C2×D12C2312yes—
50C2×C4×D3—22no—
51C2×Dic6—21no—
52C2×Dic6—21no—
53C2×Dic6—21no—
54C2×C3×C8C2314yes—
55C3×C42D411yescharacteristic
56C3×C42D411yescharacteristic
57C3×C42C2311yescharacteristic
58C3×C42C2311yescharacteristic
59#217—22no—
60(C2×D3)⋊C4—22no—
61(C2×D3)⋊C4—22no—
62C4⋊Dic3—21no—
63C4⋊Dic3—21no—
64C4⋊Dic3—21no—
65C4⋊Dic3C2312yes—
66C4⋊Dic3C2312yes—
67C4⋊Dic3C2312yes—
68Dic3⋊C4—22no—
69Dic3⋊C4—22no—
70C4×Dic3—22no—
71C2×#41—61no—
72C4⋊Q8—31no—
73C4⋊Q8—31no—
74C4⋊Q8—31no—
75#127—31no—
76#125—32no—
77#124—61no—
78C22⋊Q8—61no—
79C22⋊Q8—61no—
80#121—32no—
81#121—61no—
82C4×Q8—61no—
83C4×Q8—61no—
84C4×D4—32no—
85C4×D4—61no—
86#117—32no—
87C2×C42C2×D311yescharacteristic
88#107—34no—
89#106—34no—
90C4⋊C8C2×D312yes—
91#102—34no—
92#102—34no—
93C22⋊C8C2×D312yes—
94#97C2×D312yes—
95C23×C3C2×D411yescharacteristic
96C22×D3—22no—
97C2×C3×C4—22no—
98C2×C3×C4—22no—
99C2×C3×C4—41no—
100C2×C3×C4C2×D411yescharacteristic
101C2×C3×C4C2×D411yescharacteristic
102C2×C3×C4C2×D411yescharacteristic
103C2×C3×C4C2×D411yescharacteristic
104C2×C3×C4C2×D411yescharacteristic
105C2×C3×C4C2411yescharacteristic
106#67—44no—
107C2×Dic3—22no—
108C2×Dic3—22no—
109C2×Dic3—22no—
110D12—24no—
111D12—42no—
112D12—42no—
113C4×D3—44no—
114Dic6—42no—
115Dic6—42no—
116Dic6—42no—
117C3×C8—24no—
118#41—124no—
119C2×Q8—61no—
120C2×Q8—61no—
121C2×Q8—61no—
122C2×D4—32no—
123C2×D4—62no—
124C2×D4—61no—
125C2×D4—61no—
126C22×C4—62no—
127C22×C4D1211yescharacteristic
128C22×C4D1211yescharacteristic
129C22×C4C22×D311yescharacteristic
130C2×C8C22×D314yes—
131C4⋊C4—32no—
132C4⋊C4—32no—
133C4⋊C4—32no—
134C4⋊C4—61no—
135C4⋊C4—62no—
136C4⋊C4—62no—
137C4⋊C4—61no—
138C4⋊C4—61no—
139C22⋊C4—62no—
140C22⋊C4—62no—
141C22⋊C4—62no—
142C42—62no—
143C42D1211yescharacteristic
144C42D1211yescharacteristic
145C42C22×D311yescharacteristic
146C42C22×D311yescharacteristic
147C22×C3—21no—
148C22×C3—21no—
149C22×C3—21no—
150C22×C3C22×D411yescharacteristic
151C2×D3—42no—
152C2×D3—42no—
153C2×D3—42no—
154C3×C4—21no—
155C3×C4—21no—
156C3×C4—21no—
157C3×C4—21no—
158C3×C4AGL(1,ℤ/8)12yes—
159C3×C42−1+412yes—
160Dic3—42no—
161Dic3—42no—
162Dic3—42no—
163C23—62no—
164C23C2×D1211yescharacteristic
165Q8—122no—
166Q8—122no—
167Q8—122no—
168D4—64no—
169D4—122no—
170D4—122no—
171D4—124no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—41no—
175C2×C4—62no—
176C2×C4—62no—
177C2×C4—62no—
178C2×C4—124no—
179C2×C4C2×D1211yescharacteristic
180C2×C4C2×D1211yescharacteristic
181C2×C4C2×D1211yescharacteristic
182C2×C4C2×D1211yescharacteristic
183C2×C4C2×D1211yescharacteristic
184C2×C4C23×D311yescharacteristic
185C8—24no—
186C2×C3—41no—
187C2×C3#53611yescharacteristic
188C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
189C2×C3#57511yescharacteristic
190D3—82no—
191C22—21no—
192C22—21no—
193C22—21no—
194C22—122no—
195C22—122no—
196C22—122no—
197C22C22×D1211yescharacteristic
198C4—21no—
199C4—21no—
200C4—21no—
201C4—21no—
202C4—122no—
203C4—122no—
204C4—122no—
205C4#90812yes—
206C4#101012yes—
207C3#323511yescharacteristic, complemented, Sylow
208C2—41no—
209C2—242no—
210C2#552611yescharacteristic
211C2C2×#90811yescharacteristic
212C2#574811yescharacteristic
subgroupG/H
centerC22C22×D12
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#432C2
socleC22×C3C22×D4
prime radical#5477C2
hypercenter#432D3
hypocenterC3#3235
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #432

Complements

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

NH ≅ G/Ntype
2. #5470C2semidirect
3. C3×#432C2semidirect
4. #4846C2semidirect
5. #4842C2semidirect
6. #4737C2semidirect
7. #4713C2semidirect
8. #4702C2semidirect
41. #432D3semidirect
207. C3#3235semidirect

Maximal subgroups

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

subgroupindex
#54772
#54702
C3×#4322
#48462
#48422
#47372
#47132
#47022
#32353