Finite Group Databasebeta

← #81006

Subgroups — #81006

Order 384.

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

normal subgroups only

This group has 324 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#5721C211yesmaximal, characteristic
2#5570C211yesmaximal, characteristic
3C3×#475C211yesmaximal, characteristic
4#5078C211yesmaximal, characteristic
5C4×#216C211yesmaximal, characteristic
6#4858C212yesmaximal
7#4857C212yesmaximal
8#4851C211yesmaximal, characteristic
9#4830C211yesmaximal, characteristic
10#4799C212yesmaximal
11#4790C212yesmaximal
12#3383—31nomaximal, Sylow
13C3×C4⋊Q8C2211yescharacteristic
14C3×C4×Q8C2211yescharacteristic
15Q8×Dic3—21no—
16Q8×Dic3C2212yes—
17#944—21no—
18C2×#216C2211yescharacteristic
19#890—22no—
20#890C2212yes—
21#890C2212yes—
22#889—22no—
23#887C2212yes—
24#869—21no—
25#869C2211yescharacteristic
26C4×Dic6—21no—
27C4×Dic6C2211yescharacteristic
28C4×Dic6C2211yescharacteristic
29C3×#106C2212yes—
30C3×C4⋊C8C2211yescharacteristic
31C3×#103C2212yes—
32#835C2211yescharacteristic
33#817C2212yes—
34#816C2212yes—
35Dic3⋊C8C2212yes—
36C8×Dic3C2212yes—
37#810C2211yescharacteristic
38#807C2211yescharacteristic
39#807C2212yes—
40#807C2212yes—
41C4×C3⋊C8C2211yescharacteristic
42#557—31no—
43#557—31no—
44#500—31no—
45#499—32no—
46#479—32no—
47#479—32no—
48#475—31no—
49#475D311yescharacteristic
50C8×Q8—31no—
51C4×Q16—31no—
52C4×Q16—32no—
53C2×C3×Q8—21no—
54C2×C3×Q8C2311yescharacteristic
55C2×Dic6—21no—
56C2×Dic6D411yescharacteristic
57C2×Dic6C2311yescharacteristic
58C2×C3×C8C2312yes—
59C3×C4⋊C4—21no—
60C3×C4⋊C4—21no—
61C3×C4⋊C4C2312yes—
62C3×C4⋊C4C2311yescharacteristic
63C3×C42—21no—
64C3×C42C2311yescharacteristic
65#216Q814yescomplemented
66C4⋊Dic3—21no—
67C4⋊Dic3—22no—
68C4⋊Dic3—22no—
69C4⋊Dic3—22no—
70C4⋊Dic3D411yescharacteristic
71C4⋊Dic3C2312yes—
72C4⋊Dic3C2311yescharacteristic
73Dic3⋊C4—21no—
74Dic3⋊C4—22no—
75Dic3⋊C4—22no—
76Dic3⋊C4—22no—
77Dic3⋊C4D412yes—
78C4×Dic3—21no—
79C4×Dic3—22no—
80C4×Dic3—22no—
81C4×Dic3C2312yes—
82C2×C3⋊C8C2311yescharacteristic
83C2×C3⋊C8C2311yescharacteristic
84C2×Q16—31no—
85C4⋊Q8—31no—
86C4⋊Q8—61no—
87C4⋊Q8—61no—
88C4⋊Q8C2×D311yescharacteristic
89#125—32no—
90#125—32no—
91#125—62no—
92#125—62no—
93C4×Q8—31no—
94C4×Q8—32no—
95C4×Q8—32no—
96C4×Q8—31no—
97C4×Q8—61no—
98C4×Q8—61no—
99C4×Q8C2×D311yescharacteristic
100#107—31no—
101#107—32no—
102#107—32no—
103#106—32no—
104#106C2×D312yes—
105C4⋊C8—32no—
106C4⋊C8C2×D311yescharacteristic
107#103—31no—
108#103—32no—
109#103—31no—
110#103C2×D312yes—
111C4×C8—32no—
112C4×C8—31no—
113C3×Q8—21no—
114C3×Q8—42no—
115C3×Q8C2×Q812yes—
116C2×C3×C4—21no—
117C2×C3×C4—22no—
118C2×C3×C4—21no—
119C2×C3×C4C2×D411yescharacteristic
120C2×C3×C4C2×D411yescharacteristic
121C2×C3×C4C2411yescharacteristic
122C2×Dic3—21no—
123C2×Dic3—21no—
124C2×Dic3—22no—
125C2×Dic3C2×D412yes—
126C2×Dic3C2×D412yes—
127Dic6—22no—
128Dic6—21no—
129Dic6—42no—
130Dic6C2×Q812yes—
131C3×C8—22no—
132C3⋊C8—21no—
133C3⋊C8C2×Q812yes—
134C2×Q8—21no—
135C2×Q8—31no—
136C2×Q8—31no—
137C2×Q8—61no—
138C2×Q8C22×D311yescharacteristic
139Q16—34no—
140C2×C8—31no—
141C2×C8—31no—
142C2×C8C22×D312yes—
143C4⋊C4—21no—
144C4⋊C4—21no—
145C4⋊C4—31no—
146C4⋊C4—32no—
147C4⋊C4—31no—
148C4⋊C4—32no—
149C4⋊C4—62no—
150C4⋊C4—62no—
151C4⋊C4—62no—
152C4⋊C4—62no—
153C4⋊C4—62no—
154C4⋊C4—62no—
155C4⋊C4—61no—
156C4⋊C4—61no—
157C4⋊C4C22×D311yescharacteristic
158C4⋊C4C22×D312yes—
159C42—21no—
160C42—32no—
161C42—62no—
162C42—62no—
163C42—61no—
164C42C22×D311yescharacteristic
165C22×C3C22×D411yescharacteristic
166C3×C4—22no—
167C3×C4—21no—
168C3×C4—42no—
169C3×C4—41no—
170C3×C4#13512yes—
171C3×C4C22×Q811yescharacteristic
172C3×C42−1+411yescharacteristic
173Dic3—22no—
174Dic3—22no—
175Dic3—22no—
176Dic3—42no—
177Dic3—41no—
178Q8—21no—
179Q8—32no—
180Q8—42no—
181Q8—62no—
182Q8—61no—
183Q8—122no—
184Q8D3×Q812yes—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—22no—
188C2×C4—32no—
189C2×C4—32no—
190C2×C4—61no—
191C2×C4—61no—
192C2×C4—62no—
193C2×C4D3×D411yescharacteristic
194C2×C4D3×D411yescharacteristic
195C2×C4C23×D311yescharacteristic
196C8—22no—
197C8—32no—
198C8—61no—
199C2×C3D4×Q811yescharacteristic
200C2×C3C2×#13511yescharacteristic
201C2×C3#57811yescharacteristic
202C22C2×D3×D411yescharacteristic
203C4—22no—
204C4—21no—
205C4—42no—
206C4—41no—
207C4—62no—
208C4—62no—
209C4—62no—
210C4—122no—
211C4—121no—
212C4#91612yes—
213C4C2×D3×Q811yescharacteristic
214C4#101011yescharacteristic
215C3#338311yescharacteristic, complemented, Sylow
216C2#563211yescharacteristic
217C2C2×#91611yescharacteristic
218C2#577911yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#475C2
socle, prime radicalC22×C3C22×D4
hypercenter#475D3
hypocenterC3#3383
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #475

Complements

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

NH ≅ G/Ntype
65. #216Q8semidirect
215. C3#3383semidirect

Maximal subgroups

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

subgroupindex
#57212
#55702
C3×#4752
#50782
C4×#2162
#48582
#48572
#48512
#48302
#47992
#47902
#33833