Finite Group Databasebeta

← #79098

Subgroups — #79098

Order 384.

346 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 117 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#5534C212yesmaximal, complemented
2C3×#424C211yesmaximal, characteristic, complemented
3#5137C211yesmaximal, characteristic, complemented
4#5135C211yesmaximal, characteristic
5#5114C212yesmaximal, complemented
6#4851C212yesmaximal, complemented
7#4845C212yesmaximal, complemented
8#4844C212yesmaximal, complemented
9#4842C212yesmaximal, complemented
10#3263—31nomaximal, Sylow
11C2×C3×M16C2211yescharacteristic
12C22×C3×C8C2211yescharacteristic
13C3×#117C2211yescharacteristic
14#930C2212yes—
15#924C2212yes—
16C2×Dic12C2211yescharacteristic
17C2×D24C2211yescharacteristic
18C2×#204C2212yes—
19#898—22no—
20#896—22no—
21#885—22no—
22#883—22no—
23#875C2212yes—
24C4×D12C2212yes—
25#869C2212yes—
26C4×Dic6C2212yes—
27C3×C4⋊C8C2212yes—
28C3×C4⋊C8C2212yes—
29#821C2212yes—
30#821C2212yes—
31#817C2211yescharacteristic
32#817C2211yescharacteristic
33#817C2212yes—
34#816C2212yes—
35#816C2212yes—
36#542—32no—
37#476—32no—
38#475—32no—
39#470—31no—
40#469—31no—
41#465—32no—
42#464—32no—
43#463—32no—
44#424D311yescharacteristic, complemented
45C22×C3×C4C2311yescharacteristic
46C2×#67—22no—
47C2×D12C2312yes—
48C2×C4×D3—22no—
49C2×Dic6—22no—
50C2×Dic6C2312yes—
51C3×M16—41no—
52C2×C3×C8—41no—
53C2×C3×C8C2311yescharacteristic
54C2×C3×C8C2312yes—
55C2×C3×C8C2311yescharacteristic
56C3×C4⋊C4D412yes—
57C3×C22⋊C4D411yescharacteristic
58C3×C22⋊C4D411yescharacteristic
59C3×C42C2312yes—
60#217—22no—
61(C2×D3)⋊C4—22no—
62(C2×D3)⋊C4—22no—
63(C2×D3)⋊C4—22no—
64(C2×D3)⋊C4—22no—
65C4⋊Dic3—22no—
66C4⋊Dic3C2312yes—
67C4⋊Dic3C2312yes—
68Dic3⋊C4—22no—
69Dic3⋊C4—22no—
70C4×Dic3—22no—
71Dic12—41no—
72D24—41no—
73#204—42no—
74C2×Q16—31no—
75C2×SD16—32no—
76C2×D8—31no—
77C2×M16C2×D311yescharacteristic
78C22×C8C2×D311yescharacteristic
79C4⋊Q8—32no—
80#126—62no—
81#124—32no—
82#124—62no—
83#123—62no—
84C22⋊Q8—32no—
85C22⋊Q8—62no—
86#121—32no—
87C4×Q8—32no—
88C4×D4—32no—
89#117C2×D311yescharacteristic
90#106—32no—
91#106—31no—
92#106—31no—
93C4⋊C8C2×D312yes—
94C4⋊C8C2×D312yes—
95#103—32no—
96#103—32no—
97#102—32no—
98#102—32no—
99C23×C3C2×D411yescharacteristic
100C22×D3—22no—
101C2×C3×C4—21no—
102C2×C3×C4—21no—
103C2×C3×C4C2×D412yes—
104C2×C3×C4C2×D411yescharacteristic
105C2×C3×C4C2×D412yes—
106C2×C3×C4C2411yescharacteristic
107#67—82no—
108C2×Dic3—22no—
109C2×Dic3—22no—
110C2×Dic3—22no—
111D12—22no—
112D12—22no—
113C4×D3—82no—
114Dic6—22no—
115Dic6—22no—
116Dic6—82no—
117C3×C8—22no—
118C3×C8—21no—
119C3×C8—21no—
120C2×Q8—32no—
121C2×Q8—62no—
122C2×D4—32no—
123C2×D4—62no—
124C22×C4—62no—
125C22×C4C22×D311yescharacteristic
126Q16—121no—
127SD16—122no—
128D8—121no—
129M16—41no—
130C2×C8—41no—
131C2×C8C22×D311yescharacteristic
132C2×C8C22×D312yes—
133C2×C8C22×D311yescharacteristic
134C4⋊C4—32no—
135C4⋊C4—32no—
136C4⋊C4—62no—
137C4⋊C4—62no—
138C4⋊C4—62no—
139C4⋊C4D1212yes—
140C22⋊C4—62no—
141C22⋊C4—62no—
142C22⋊C4—62no—
143C22⋊C4—62no—
144C22⋊C4—62no—
145C22⋊C4D1211yescharacteristic
146C22⋊C4D1211yescharacteristic
147C42—62no—
148C42C22×D312yes—
149C22×C3—21no—
150C22×C3—22no—
151C22×C3C22×D411yescharacteristic
152C2×D3—42no—
153C2×D3—42no—
154C2×D3—42no—
155C3×C4—21no—
156C3×C4—22no—
157C3×C4—22no—
158C3×C4H(5,𝔽2)11yescharacteristic
159C3×C42−1+411yescharacteristic
160Dic3—42no—
161Dic3—42no—
162Dic3—42no—
163C23—62no—
164C23C2×D1211yescharacteristic
165Q8—62no—
166Q8—62no—
167Q8—242no—
168D4—62no—
169D4—62no—
170D4—242no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—62no—
174C2×C4—62no—
175C2×C4—62no—
176C2×C4—242no—
177C2×C4C2×D1212yes—
178C2×C4C2×D1212yes—
179C2×C4C2×D1211yescharacteristic
180C2×C4C23×D311yescharacteristic
181C8—22no—
182C8—21no—
183C8—21no—
184C2×C3—41no—
185C2×C3#53711yescharacteristic
186C2×C3#57712yes—
187D3—82no—
188C22—22no—
189C22—21no—
190C22—122no—
191C22—122no—
192C22—122no—
193C22C22×D1211yescharacteristic
194C4—22no—
195C4—22no—
196C4—21no—
197C4—122no—
198C4—122no—
199C4—122no—
200C4#100411yescharacteristic
201C4#100711yescharacteristic
202C3#326311yescharacteristic, complemented, Sylow
203C2—41no—
204C2—242no—
205C2#551011yescharacteristic
206C2#575112yes—
subgroupG/H
centerC22C22×D12
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#424C2
socleC22×C3C22×D4
prime radical#5135C2
hypercenter#424D3
hypocenterC3#3263
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ #117 ◁ #424

Complements

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

NH ≅ G/Ntype
1. #5534C2semidirect
2. C3×#424C2semidirect
3. #5137C2semidirect
5. #5114C2semidirect
6. #4851C2semidirect
7. #4845C2semidirect
8. #4844C2semidirect
9. #4842C2semidirect
44. #424D3semidirect
202. C3#3263semidirect

Maximal subgroups

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

subgroupindex
#55342
C3×#4242
#51372
#51352
#51142
#48512
#48452
#48442
#48422
#32633