Finite Group Databasebeta

← #79582

Subgroups — #79582

Order 384.

380 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 113 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#5559C212yesmaximal, complemented
2C3×C4×D8C211yesmaximal, characteristic, complemented
3#5154C212yesmaximal, complemented
4#5016C212yesmaximal, complemented
5#4889C212yesmaximal, complemented
6#4786C212yesmaximal, complemented
7#4767C212yesmaximal, complemented
8#4696C211yesmaximal, characteristic, complemented
9#4679C211yesmaximal, characteristic, complemented
10#3403—31nomaximal, Sylow
11C2×C3×D8C2211yescharacteristic
12C3×C4×D4C2212yes—
13#940C2212yes—
14#936C2212yes—
15C2×#214C2212yes—
16C2×#213C2212yes—
17#929—22no—
18C4×#67—22no—
19C2×Dic12C2211yescharacteristic
20C2×D24C2211yescharacteristic
21C2×#204C2212yes—
22#893C2212yes—
23#887C2212yes—
24#885—22no—
25#883—22no—
26#875C2212yes—
27C4×Dic6C2211yescharacteristic
28C3×#107C2211yescharacteristic
29C3×#102C2212yes—
30C3×C4×C8C2211yescharacteristic
31#815C2212yes—
32Dic3⋊C8C2212yes—
33#810C2212yes—
34#809C2212yes—
35C4⋊(C3⋊C8)C2211yescharacteristic
36#559—32no—
37#496—32no—
38#495—31no—
39#464—32no—
40#463—32no—
41#459—32no—
42#446—31no—
43#440—32no—
44C4×D8D311yescharacteristic, complemented
45C2×C3×D4C2312yes—
46C22×C3×C4—22no—
47C2×#67—22no—
48C2×#67—22no—
49C2×D12C2312yes—
50C2×C4×D3—22no—
51C2×Dic6D411yescharacteristic
52C2×Dic6C2312yes—
53C3×D8—22no—
54C2×C3×C8C2311yescharacteristic
55C2×C3×C8C2311yescharacteristic
56C3×C4⋊C4C2312yes—
57C3×C22⋊C4—22no—
58C3×C42C2311yescharacteristic
59#217—22no—
60#217—22no—
61#214—42no—
62#213—42no—
63(C2×D3)⋊C4—22no—
64(C2×D3)⋊C4—22no—
65(C2×D3)⋊C4—22no—
66C4⋊Dic3D411yescharacteristic
67Dic3⋊C4—22no—
68Dic3⋊C4D412yes—
69C4×Dic3—22no—
70C4×Dic3C2312yes—
71C2×C3⋊C8C2312yes—
72Dic12—41no—
73D24—41no—
74#204—24no—
75C2×Q16—31no—
76C2×SD16—32no—
77C2×SD16—32no—
78C2×D8—32no—
79C2×D8—31no—
80C2×D8C2×D311yescharacteristic
81#127—32no—
82#124—32no—
83#124—32no—
84#124—62no—
85#123—62no—
86#123—62no—
87C4×Q8—31no—
88C4×Q8—32no—
89C4×D4—32no—
90C4×D4—62no—
91C4×D4C2×D312yes—
92#107C2×D311yescharacteristic
93C4⋊C8—32no—
94C4⋊C8—31no—
95#103—32no—
96#102—32no—
97#102C2×D312yes—
98#97—32no—
99C4×C8C2×D311yescharacteristic
100C23×C3—22no—
101C22×D3—22no—
102C3×D4—22no—
103C3×D4—22no—
104C2×C3×C4—22no—
105C2×C3×C4—82no—
106C2×C3×C4C2×D411yescharacteristic
107C2×C3×C4C2×D411yescharacteristic
108C2×C3×C4C2411yescharacteristic
109#67—44no—
110#67—82no—
111C2×Dic3—22no—
112C2×Dic3C2×D412yes—
113C2×Dic3C2×D412yes—
114D12—22no—
115D12—22no—
116C4×D3—82no—
117Dic6—22no—
118Dic6—22no—
119Dic6—41no—
120C3×C8—21no—
121C3×C8#4112yes—
122C3⋊C8—22no—
123C2×Q8—31no—
124C2×Q8—32no—
125C2×D4—32no—
126C2×D4—62no—
127C2×D4—62no—
128C2×D4C22×D312yes—
129C22×C4—22no—
130C22×C4—62no—
131Q16—121no—
132SD16—64no—
133SD16—122no—
134D8—22no—
135D8—121no—
136D8—122no—
137C2×C8—32no—
138C2×C8C22×D311yescharacteristic
139C2×C8C22×D311yescharacteristic
140C4⋊C4—32no—
141C4⋊C4—31no—
142C4⋊C4—62no—
143C4⋊C4C22×D312yes—
144C22⋊C4—22no—
145C22⋊C4—62no—
146C22⋊C4—62no—
147C22⋊C4—62no—
148C22⋊C4—62no—
149C22⋊C4—62no—
150C42—32no—
151C42—62no—
152C42C22×D311yescharacteristic
153C22×C3—42no—
154C22×C3—42no—
155C22×C3—42no—
156C22×C3C22×D411yescharacteristic
157C2×D3—42no—
158C2×D3—42no—
159C2×D3—42no—
160C3×C4—21no—
161C3×C4—21no—
162C3×C4—42no—
163C3×C4C2×#4111yescharacteristic
164C3×C4H(5,𝔽2)11yescharacteristic
165Dic3—22no—
166Dic3—22no—
167Dic3—42no—
168C23—22no—
169C23—62no—
170Q8—62no—
171Q8—62no—
172Q8—121no—
173D4—22no—
174D4—22no—
175D4—62no—
176D4—62no—
177D4—124no—
178D4—242no—
179C2×C4—22no—
180C2×C4—32no—
181C2×C4—32no—
182C2×C4—62no—
183C2×C4—82no—
184C2×C4—242no—
185C2×C4D3×D411yescharacteristic
186C2×C4D3×D411yescharacteristic
187C2×C4C23×D311yescharacteristic
188C8—21no—
189C8—62no—
190C8#23512yes—
191C2×C3—82no—
192C2×C3#55011yescharacteristic
193C2×C3#57511yescharacteristic
194C2×C3#57611yescharacteristic
195D3—82no—
196C22—42no—
197C22—42no—
198C22—42no—
199C22—122no—
200C22—122no—
201C22—122no—
202C22C2×D3×D411yescharacteristic
203C4—21no—
204C4—21no—
205C4—42no—
206C4—62no—
207C4—62no—
208C4—122no—
209C4C2×#23511yescharacteristic
210C4#100911yescharacteristic
211C3#340311yescharacteristic, complemented, Sylow
212C2—82no—
213C2—242no—
214C2#555311yescharacteristic
215C2#575711yescharacteristic
216C2#576811yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×C4×D8C2
socleC22×C3C22×D4
hypercenterC4×D8D3
hypocenterC3#3403
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ C4×D8

Complements

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

NH ≅ G/Ntype
1. #5559C2semidirect
2. C3×C4×D8C2semidirect
3. #5154C2semidirect
4. #5016C2semidirect
5. #4889C2semidirect
6. #4786C2semidirect
7. #4767C2semidirect
8. #4696C2semidirect
9. #4679C2semidirect
44. C4×D8D3semidirect
211. C3#3403semidirect

Maximal subgroups

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

subgroupindex
#55592
C3×C4×D82
#51542
#50162
#48892
#47862
#47672
#46962
#46792
#34033