Finite Group Databasebeta

← #80896

Subgroups — #80896

Order 384.

354 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 109 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#5624C212yesmaximal, complemented
2C3×#470C211yesmaximal, characteristic, complemented
3#5185C212yesmaximal, complemented
4#5135C211yesmaximal, characteristic
5#5118C211yesmaximal, characteristic, complemented
6#5049C212yesmaximal, complemented
7#4871C212yesmaximal, complemented
8#4819C212yesmaximal, complemented
9#4799C212yesmaximal, complemented
10#3256—31nomaximal, Sylow
11C2×C3×Q16C2211yescharacteristic
12C2×C3×M16C2211yescharacteristic
13C3×C22⋊Q8C2212yes—
14#947C2212yes—
15Q8×Dic3C2212yes—
16C2×#215C2212yes—
17#930C2212yes—
18C2×C4×Dic3C2211yescharacteristic
19C2×#208C2211yescharacteristic
20C2×D24C2211yescharacteristic
21#898—22no—
22#893C2212yes—
23#892—22no—
24#890C2212yes—
25#886—22no—
26#67⋊C4—22no—
27C3×#106C2211yescharacteristic
28C3×#103C2212yes—
29#835C2212yes—
30#821C2212yes—
31#817C2211yescharacteristic
32Dic3⋊C8C2212yes—
33Dic3⋊C8C2212yes—
34#809C2212yes—
35#807C2212yes—
36#529—32no—
37#479—32no—
38#478—32no—
39#470D311yescharacteristic, complemented
40#469—31no—
41#468—32no—
42#464—32no—
43#463—32no—
44#423—31no—
45C2×C3×Q8C2312yes—
46C22×C3×C4C2311yescharacteristic
47C2×#67—22no—
48C22×Dic3D412yes—
49C2×D12C2312yes—
50C2×C4×D3—22no—
51C3×Q16—41no—
52C3×M16—41no—
53C2×C3×C8C2311yescharacteristic
54C2×C3×C8C2311yescharacteristic
55C3×C4⋊C4—22no—
56C3×C4⋊C4C2312yes—
57C3×C22⋊C4—22no—
58#215—42no—
59(C2×D3)⋊C4—22no—
60(C2×D3)⋊C4—22no—
61(C2×D3)⋊C4—22no—
62(C2×D3)⋊C4—22no—
63C4⋊Dic3—22no—
64C4⋊Dic3—22no—
65C4⋊Dic3C2312yes—
66Dic3⋊C4—22no—
67C4×Dic3—22no—
68C4×Dic3D411yescharacteristic
69C4×Dic3D411yescharacteristic
70C4×Dic3C2312yes—
71#208—41no—
72C2×C3⋊C8C2312yes—
73D24—41no—
74C2×Q16C2×D311yescharacteristic
75C2×SD16—32no—
76C2×D8—31no—
77C2×M16—31no—
78C2×M16C2×D311yescharacteristic
79#126—62no—
80#125—32no—
81#124—32no—
82#123—62no—
83C22⋊Q8C2×D312yes—
84#121—32no—
85C4×Q8—32no—
86C4×D4—32no—
87C4×D4—62no—
88#117—62no—
89C2×C42—31no—
90#107—32no—
91#106—31no—
92#106C2×D311yescharacteristic
93C4⋊C8—32no—
94C4⋊C8—32no—
95#103—32no—
96#103C2×D312yes—
97#102—32no—
98#102—32no—
99C23×C3C2×D411yescharacteristic
100C22×D3—22no—
101C3×Q8—22no—
102C3×Q8—22no—
103C2×C3×C4—21no—
104C2×C3×C4—21no—
105C2×C3×C4—22no—
106C2×C3×C4—22no—
107C2×C3×C4C2×D411yescharacteristic
108C2×C3×C4C2411yescharacteristic
109#67—44no—
110C2×Dic3—22no—
111C2×Dic3—42no—
112C2×Dic3C2×D412yes—
113C2×Dic3C2×D412yes—
114D12—22no—
115D12—22no—
116C4×D3—44no—
117C3×C8—21no—
118C3×C8—21no—
119C3⋊C8—22no—
120C2×Q8C22×D312yes—
121C2×D4—32no—
122C2×D4—62no—
123C22×C4—32no—
124C22×C4—62no—
125C22×C4C22×D311yescharacteristic
126Q16—41no—
127SD16—122no—
128D8—121no—
129M16—41no—
130M16—121no—
131C2×C8—32no—
132C2×C8C22×D311yescharacteristic
133C2×C8C22×D311yescharacteristic
134C4⋊C4—22no—
135C4⋊C4—32no—
136C4⋊C4—62no—
137C4⋊C4—62no—
138C4⋊C4—62no—
139C4⋊C4C22×D312yes—
140C22⋊C4—22no—
141C22⋊C4—62no—
142C22⋊C4—62no—
143C22⋊C4—62no—
144C22⋊C4—62no—
145C42—31no—
146C42—31no—
147C42—32no—
148C42—62no—
149C22×C3—21no—
150C22×C3—22no—
151C22×C3C22×D411yescharacteristic
152C2×D3—42no—
153C2×D3—42no—
154C2×D3—42no—
155C3×C4—21no—
156C3×C4—42no—
157C3×C4—42no—
158C3×C4H(5,𝔽2)11yescharacteristic
159C3×C42−1+411yescharacteristic
160Dic3—22no—
161Dic3—22no—
162Dic3—42no—
163C23—62no—
164C23D3×D411yescharacteristic
165Q8—22no—
166Q8—22no—
167D4—62no—
168D4—62no—
169D4—124no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—32no—
175C2×C4—32no—
176C2×C4—62no—
177C2×C4—124no—
178C2×C4—122no—
179C2×C4D3×D411yescharacteristic
180C2×C4C23×D311yescharacteristic
181C8—21no—
182C8—21no—
183C8—62no—
184C2×C3—41no—
185C2×C3#53711yescharacteristic
186C2×C3#57512yes—
187D3—82no—
188C22—22no—
189C22—21no—
190C22—122no—
191C22—122no—
192C22—122no—
193C22C2×D3×D411yescharacteristic
194C4—21no—
195C4—42no—
196C4—42no—
197C4—62no—
198C4—62no—
199C4—122no—
200C4#100411yescharacteristic
201C4#101011yescharacteristic
202C3#325611yescharacteristic, complemented, Sylow
203C2—41no—
204C2—242no—
205C2#559711yescharacteristic
206C2#577712yes—
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#470C2
socleC22×C3C22×D4
prime radical#5135C2
hypercenter#470D3
hypocenterC3#3256
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
1. #5624C2semidirect
2. C3×#470C2semidirect
3. #5185C2semidirect
5. #5118C2semidirect
6. #5049C2semidirect
7. #4871C2semidirect
8. #4819C2semidirect
9. #4799C2semidirect
39. #470D3semidirect
202. C3#3256semidirect

Maximal subgroups

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

subgroupindex
#56242
C3×#4702
#51852
#51352
#51182
#50492
#48712
#48192
#47992
#32563