Finite Group Databasebeta

← #79270

Subgroups — #79270

Order 384.

370 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 125 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×#431C211yesmaximal, characteristic, complemented
4#4847C214yesmaximal, complemented
5#4737C214yesmaximal, complemented
6#4694C211yesmaximal, characteristic, complemented
7#4694C211yesmaximal, characteristic, complemented
8#4682C211yesmaximal, characteristic, complemented
9#4680C211yesmaximal, characteristic, complemented
10#3228—31nomaximal, Sylow
11C2×#235—21no—
12C2×C3×C42C2211yescharacteristic
13#930—21no—
14#930C2212yes—
15#926C2212yes—
16#924—21no—
17#924—21no—
18#875—21no—
19#874C2211yescharacteristic
20C4×D12—21no—
21C4×D12C2212yes—
22#870C2212yes—
23#869C2211yescharacteristic
24#869C2211yescharacteristic
25#869C2211yescharacteristic
26C4×Dic6—21no—
27C4×Dic6—21no—
28C3×C4⋊C8C2212yes—
29C3×C22⋊C8C2212yes—
30C3×C4×C8C2211yescharacteristic
31C3×C4×C8C2211yescharacteristic
32#821C2214yes—
33#821C2214yes—
34#818C2214yes—
35#817C2214yes—
36#533—31no—
37#532—31no—
38#500—31no—
39#498—31no—
40#486—31no—
41#486—31no—
42#482—34no—
43#478—34no—
44#431D311yescharacteristic, complemented
45C22×C3×C4D411yescharacteristic
46C22×C3×C4D411yescharacteristic
47C22×C3×C4C2311yescharacteristic
48C2×#67—22no—
49#235—44no—
50C2×D12—21no—
51C2×D12—21no—
52C2×D12C2312yes—
53C2×C4×D3—22no—
54C2×Dic6—21no—
55C2×Dic6—21no—
56C2×Dic6—21no—
57C2×C3×C8C2312yes—
58C2×C3×C8C2312yes—
59C3×C42D411yescharacteristic
60C3×C42D411yescharacteristic
61C3×C42C2311yescharacteristic
62C3×C42C2311yescharacteristic
63#217—22no—
64(C2×D3)⋊C4—22no—
65(C2×D3)⋊C4—22no—
66C4⋊Dic3—21no—
67C4⋊Dic3—21no—
68C4⋊Dic3—21no—
69C4⋊Dic3C2312yes—
70C4⋊Dic3C2312yes—
71C4⋊Dic3C2312yes—
72Dic3⋊C4—22no—
73Dic3⋊C4—22no—
74C4×Dic3—22no—
75C2×#41—61no—
76C4⋊Q8—31no—
77C4⋊Q8—31no—
78C4⋊Q8—31no—
79#127—31no—
80#125—32no—
81#124—61no—
82C22⋊Q8—61no—
83C22⋊Q8—61no—
84#121—32no—
85#121—61no—
86C4×Q8—61no—
87C4×Q8—61no—
88C4×D4—32no—
89C4×D4—61no—
90#117—32no—
91C2×C42C2×D311yescharacteristic
92#107—34no—
93#106—34no—
94C4⋊C8C2×D312yes—
95#102—34no—
96#102—34no—
97C22⋊C8C2×D312yes—
98C4×C8C2×D311yescharacteristic
99C4×C8C2×D311yescharacteristic
100C23×C3C2×D411yescharacteristic
101C22×D3—22no—
102C2×C3×C4—22no—
103C2×C3×C4—22no—
104C2×C3×C4—22no—
105C2×C3×C4C2×D411yescharacteristic
106C2×C3×C4C2×D411yescharacteristic
107C2×C3×C4C2×D411yescharacteristic
108C2×C3×C4C2×D411yescharacteristic
109C2×C3×C4C2×D411yescharacteristic
110C2×C3×C4C2411yescharacteristic
111#67—44no—
112C2×Dic3—22no—
113C2×Dic3—22no—
114C2×Dic3—22no—
115D12—24no—
116D12—42no—
117D12—42no—
118C4×D3—44no—
119Dic6—42no—
120Dic6—42no—
121Dic6—42no—
122C3×C8—22no—
123C3×C8—22no—
124#41—124no—
125C2×Q8—61no—
126C2×Q8—61no—
127C2×Q8—61no—
128C2×D4—32no—
129C2×D4—62no—
130C2×D4—61no—
131C2×D4—61no—
132C22×C4—62no—
133C22×C4D1211yescharacteristic
134C22×C4D1211yescharacteristic
135C22×C4C22×D311yescharacteristic
136C2×C8C22×D312yes—
137C2×C8C22×D312yes—
138C4⋊C4—32no—
139C4⋊C4—32no—
140C4⋊C4—32no—
141C4⋊C4—61no—
142C4⋊C4—62no—
143C4⋊C4—62no—
144C4⋊C4—61no—
145C4⋊C4—61no—
146C22⋊C4—62no—
147C22⋊C4—62no—
148C22⋊C4—62no—
149C42—62no—
150C42D1211yescharacteristic
151C42D1211yescharacteristic
152C42C22×D311yescharacteristic
153C42C22×D311yescharacteristic
154C22×C3—21no—
155C22×C3—21no—
156C22×C3—21no—
157C22×C3C22×D411yescharacteristic
158C2×D3—42no—
159C2×D3—42no—
160C2×D3—42no—
161C3×C4—21no—
162C3×C4—21no—
163C3×C4—21no—
164C3×C4#13512yes—
165C3×C4AGL(1,ℤ/8)12yes—
166C3×C42−1+412yes—
167Dic3—42no—
168Dic3—42no—
169Dic3—42no—
170C23—62no—
171C23C2×D1211yescharacteristic
172Q8—122no—
173Q8—122no—
174Q8—122no—
175D4—64no—
176D4—124no—
177D4—122no—
178D4—122no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—62no—
183C2×C4—62no—
184C2×C4—62no—
185C2×C4—124no—
186C2×C4C2×D1211yescharacteristic
187C2×C4C2×D1211yescharacteristic
188C2×C4C2×D1211yescharacteristic
189C2×C4C2×D1211yescharacteristic
190C2×C4C2×D1211yescharacteristic
191C2×C4C23×D311yescharacteristic
192C8—22no—
193C8—22no—
194C2×C3—41no—
195C2×C3#53611yescharacteristic
196C2×C3C2×#13511yescharacteristic
197C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
198D3—82no—
199C22—21no—
200C22—21no—
201C22—21no—
202C22—122no—
203C22—122no—
204C22—122no—
205C22C22×D1211yescharacteristic
206C4—21no—
207C4—21no—
208C4—21no—
209C4—122no—
210C4—122no—
211C4—122no—
212C4#90412yes—
213C4#90812yes—
214C4#101012yes—
215C3#322811yescharacteristic, complemented, Sylow
216C2—41no—
217C2—242no—
218C2#552611yescharacteristic
219C2C2×#90411yescharacteristic
220C2C2×#90811yescharacteristic
subgroupG/H
centerC22C22×D12
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#431C2
socleC22×C3C22×D4
prime radical#5477C2
hypercenter#431D3
hypocenterC3#3228
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #431

Complements

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

NH ≅ G/Ntype
2. #5470C2semidirect
3. C3×#431C2semidirect
4. #4847C2semidirect
5. #4737C2semidirect
6. #4694C2semidirect
7. #4694C2semidirect
8. #4682C2semidirect
9. #4680C2semidirect
44. #431D3semidirect
215. C3#3228semidirect

Maximal subgroups

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

subgroupindex
#54772
#54702
C3×#4312
#48472
#47372
#46942
#46822
#46802
#32283