Finite Group Databasebeta

← #79096

Subgroups — #79096

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#5535C211yesmaximal, characteristic
2#5518C211yesmaximal, characteristic, complemented
3C3×#424C211yesmaximal, characteristic, complemented
4#5134C212yesmaximal, complemented
5#5114C212yesmaximal, complemented
6#4848C212yesmaximal, complemented
7#4847C212yesmaximal, complemented
8#4841C212yesmaximal, complemented
9#4829C212yesmaximal, complemented
10#3260—31nomaximal, Sylow
11C2×C3×M16C2211yescharacteristic
12C22×C3×C8C2211yescharacteristic
13C3×#117C2211yescharacteristic
14#930C2212yes—
15#924C2212yes—
16C2×#204C2212yes—
17C2×#204C2212yes—
18#894—22no—
19#889—22no—
20#884—22no—
21#879—22no—
22#874C2211yescharacteristic
23C4×D12C2212yes—
24#870C2211yescharacteristic
25#870C2211yescharacteristic
26#869C2211yescharacteristic
27C4×Dic6C2212yes—
28C3×C4⋊C8C2212yes—
29C3×C4⋊C8C2212yes—
30#821C2212yes—
31#821C2212yes—
32#818C2212yes—
33#817C2212yes—
34#816C2212yes—
35#816C2212yes—
36#541—31no—
37#540—31no—
38#479—32no—
39#478—32no—
40#468—32no—
41#465—32no—
42#461—32no—
43#460—32no—
44#424D311yescharacteristic, complemented
45C22×C3×C4C2311yescharacteristic
46C2×#67—22no—
47C2×D12—22no—
48C2×D12C2312yes—
49C2×C4×D3—22no—
50C2×Dic6C2312yes—
51C3×M16—41no—
52C2×C3×C8—41no—
53C2×C3×C8C2312yes—
54C2×C3×C8C2312yes—
55C3×C4⋊C4D412yes—
56C3×C22⋊C4D412yes—
57C3×C42C2311yescharacteristic
58C3×C42C2311yescharacteristic
59#217—22no—
60(C2×D3)⋊C4—22no—
61(C2×D3)⋊C4—22no—
62C4⋊Dic3—22no—
63C4⋊Dic3C2312yes—
64C4⋊Dic3C2312yes—
65Dic3⋊C4—22no—
66Dic3⋊C4—22no—
67Dic3⋊C4—22no—
68Dic3⋊C4—22no—
69C4×Dic3—22no—
70#204—42no—
71#204—42no—
72C2×SD16—32no—
73C2×SD16—32no—
74C2×M16C2×D311yescharacteristic
75C22×C8C2×D311yescharacteristic
76C4⋊Q8—31no—
77#127—31no—
78#126—62no—
79#125—31no—
80#125—31no—
81#125—62no—
82#123—62no—
83C22⋊Q8—32no—
84#121—32no—
85#121—62no—
86C4×Q8—32no—
87C4×D4—32no—
88#117C2×D311yescharacteristic
89#107—32no—
90#106—32no—
91C4⋊C8C2×D312yes—
92C4⋊C8C2×D312yes—
93#103—32no—
94#103—32no—
95#102—32no—
96#102—32no—
97C23×C3C2×D411yescharacteristic
98C22×D3—22no—
99C2×C3×C4—21no—
100C2×C3×C4—21no—
101C2×C3×C4C2×D412yes—
102C2×C3×C4C2×D411yescharacteristic
103C2×C3×C4C2×D412yes—
104C2×C3×C4C2411yescharacteristic
105#67—82no—
106C2×Dic3—22no—
107C2×Dic3—22no—
108C2×Dic3—22no—
109D12—22no—
110D12—22no—
111D12—82no—
112C4×D3—82no—
113Dic6—22no—
114Dic6—22no—
115C3×C8—22no—
116C3×C8—22no—
117C2×Q8—32no—
118C2×D4—32no—
119C2×D4—62no—
120C2×D4—62no—
121C22×C4—62no—
122C22×C4C22×D311yescharacteristic
123SD16—122no—
124SD16—122no—
125M16—41no—
126C2×C8—41no—
127C2×C8C22×D312yes—
128C2×C8C22×D312yes—
129C4⋊C4—32no—
130C4⋊C4—32no—
131C4⋊C4—62no—
132C4⋊C4—62no—
133C4⋊C4—62no—
134C4⋊C4—62no—
135C4⋊C4—62no—
136C4⋊C4D1212yes—
137C22⋊C4—62no—
138C22⋊C4—62no—
139C22⋊C4—62no—
140C22⋊C4D1212yes—
141C42—62no—
142C42C22×D311yescharacteristic
143C42C22×D311yescharacteristic
144C22×C3—21no—
145C22×C3—22no—
146C22×C3C22×D411yescharacteristic
147C2×D3—42no—
148C2×D3—42no—
149C2×D3—42no—
150C3×C4—21no—
151C3×C4—22no—
152C3×C4—22no—
153C3×C4H(5,𝔽2)11yescharacteristic
154C3×C42−1+411yescharacteristic
155Dic3—42no—
156Dic3—42no—
157Dic3—42no—
158C23—62no—
159C23C2×D1211yescharacteristic
160Q8—62no—
161Q8—62no—
162D4—62no—
163D4—62no—
164D4—242no—
165D4—242no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—62no—
169C2×C4—62no—
170C2×C4—62no—
171C2×C4—242no—
172C2×C4C2×D1212yes—
173C2×C4C2×D1212yes—
174C2×C4C2×D1211yescharacteristic
175C2×C4C23×D311yescharacteristic
176C8—22no—
177C8—22no—
178C2×C3—41no—
179C2×C3#53711yescharacteristic
180C2×C3#57712yes—
181D3—82no—
182C22—22no—
183C22—21no—
184C22—122no—
185C22—122no—
186C22—122no—
187C22C22×D1211yescharacteristic
188C4—22no—
189C4—22no—
190C4—21no—
191C4—122no—
192C4—122no—
193C4—122no—
194C4#100411yescharacteristic
195C4#100711yescharacteristic
196C3#326011yescharacteristic, complemented, Sylow
197C2—41no—
198C2—242no—
199C2#551011yescharacteristic
200C2#575112yes—
subgroupG/H
centerC22C22×D12
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#424C2
socleC22×C3C22×D4
prime radical#5535C2
hypercenter#424D3
hypocenterC3#3260
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
2. #5518C2semidirect
3. C3×#424C2semidirect
4. #5134C2semidirect
5. #5114C2semidirect
6. #4848C2semidirect
7. #4847C2semidirect
8. #4841C2semidirect
9. #4829C2semidirect
44. #424D3semidirect
196. C3#3260semidirect

Maximal subgroups

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

subgroupindex
#55352
#55182
C3×#4242
#51342
#51142
#48482
#48472
#48412
#48292
#32603