Finite Group Databasebeta

← #79725

Subgroups — #79725

Order 384.

368 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 129 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#5578C212yesmaximal
2C3×C4×Q16C211yesmaximal, characteristic, complemented
3#5188C211yesmaximal, characteristic
4#5183C211yesmaximal, characteristic, complemented
5C4×#216C212yesmaximal, complemented
6#4887C211yesmaximal, characteristic
7#4874C211yesmaximal, characteristic, complemented
8#4813C212yesmaximal
9#4792C212yesmaximal, complemented
10C4×Dic12C211yesmaximal, characteristic, complemented
11C4×D3×C8C211yesmaximal, characteristic
12#3170—31nomaximal, Sylow
13C2×C3×Q16C2211yescharacteristic
14C3×C4×Q8C2212yes—
15#946C2212yes—
16#944C2212yes—
17C2×#216C2212yes—
18C2×Dic12C2211yescharacteristic
19C2×D3×C8C2211yescharacteristic
20C2×D3×C8C2211yescharacteristic
21#897C2212yes—
22#896—22no—
23#889—22no—
24#888C2212yes—
25#872—22no—
26C42×D3C2211yescharacteristic
27C4×Dic6—22no—
28C4×Dic6C2212yes—
29C3×#107C2211yescharacteristic
30C3×#103C2212yes—
31C3×C4×C8C2211yescharacteristic
32#835C2212yes—
33#818C2211yescharacteristic
34#816C2212yes—
35C8×Dic3C2211yescharacteristic
36C8×Dic3C2211yescharacteristic
37#810C2212yes—
38#807C2212yes—
39C4×C3⋊C8C2211yescharacteristic
40#533—32no—
41#500—31no—
42#494—31no—
43#487—32no—
44#467—31no—
45#467—31no—
46#467—32no—
47C4×Q16—32no—
48C4×Q16—31no—
49C4×Q16D311yescharacteristic, complemented
50C2×C4×C8—31no—
51C2×C3×Q8C2312yes—
52C2×C4×D3D411yescharacteristic
53C2×C4×D3D411yescharacteristic
54C2×C4×D3C2311yescharacteristic
55C2×Dic6—22no—
56C2×Dic6C2312yes—
57C3×Q16—22no—
58C2×C3×C8C2311yescharacteristic
59C2×C3×C8C2311yescharacteristic
60C3×C4⋊C4—22no—
61C3×C4⋊C4C2312yes—
62C3×C42—22no—
63C3×C42C2311yescharacteristic
64#216—24no—
65(C2×D3)⋊C4—22no—
66(C2×D3)⋊C4—22no—
67C4⋊Dic3—22no—
68C4⋊Dic3C2312yes—
69Dic3⋊C4—22no—
70Dic3⋊C4—22no—
71Dic3⋊C4—22no—
72Dic3⋊C4—22no—
73Dic3⋊C4—22no—
74C4×Dic3—22no—
75C4×Dic3D411yescharacteristic
76C4×Dic3D411yescharacteristic
77C4×Dic3C2311yescharacteristic
78C2×C3⋊C8C2311yescharacteristic
79C2×C3⋊C8C2311yescharacteristic
80Dic12—22no—
81D3×C8—22no—
82D3×C8—22no—
83C2×Q16—32no—
84C2×Q16—31no—
85C2×Q16C2×D311yescharacteristic
86C22×C8—31no—
87C22×C8—31no—
88C4⋊Q8—32no—
89C4⋊Q8—32no—
90#125—62no—
91C22⋊Q8—32no—
92C22⋊Q8—32no—
93C22⋊Q8—62no—
94C4×Q8—32no—
95C4×Q8—62no—
96C4×Q8C2×D312yes—
97#117—62no—
98C2×C42—31no—
99#107—31no—
100#107—32no—
101#107C2×D311yescharacteristic
102#103—32no—
103#103—32no—
104#103—32no—
105#103C2×D312yes—
106C4×C8—31no—
107C4×C8—31no—
108C4×C8—31no—
109C4×C8C2×D311yescharacteristic
110C22×D3C2×D411yescharacteristic
111C3×Q8—22no—
112C3×Q8—22no—
113C2×C3×C4—22no—
114C2×C3×C4—22no—
115C2×C3×C4C2×D411yescharacteristic
116C2×C3×C4C2×D411yescharacteristic
117C2×C3×C4C2411yescharacteristic
118C2×Dic3—22no—
119C2×Dic3—22no—
120C2×Dic3C2×D411yescharacteristic
121C2×Dic3C2×D411yescharacteristic
122C2×Dic3C2×D411yescharacteristic
123C4×D3—21no—
124C4×D3—21no—
125C4×D3—22no—
126C4×D3Q1614yescomplemented
127Dic6—22no—
128Dic6—22no—
129Dic6—44no—
130C3×C8—21no—
131C3×C8#4112yes—
132C3⋊C8—21no—
133C3⋊C8#4112yes—
134C2×Q8—32no—
135C2×Q8—62no—
136C2×Q8C22×D312yes—
137C22×C4—31no—
138C22×C4—31no—
139C22×C4—31no—
140Q16—22no—
141Q16—64no—
142Q16—62no—
143C2×C8—31no—
144C2×C8—31no—
145C2×C8—62no—
146C2×C8—62no—
147C2×C8C22×D311yescharacteristic
148C2×C8C22×D311yescharacteristic
149C4⋊C4—22no—
150C4⋊C4—32no—
151C4⋊C4—62no—
152C4⋊C4—62no—
153C4⋊C4—62no—
154C4⋊C4—62no—
155C4⋊C4—62no—
156C4⋊C4—62no—
157C4⋊C4C22×D312yes—
158C22⋊C4—62no—
159C22⋊C4—62no—
160C42—22no—
161C42—31no—
162C42—31no—
163C42—31no—
164C42—62no—
165C42C22×D311yescharacteristic
166C22×C3C22×D411yescharacteristic
167C2×D3—21no—
168C2×D3—21no—
169C2×D3C2×Q1612yes—
170C3×C4—21no—
171C3×C4—42no—
172C3×C4—42no—
173C3×C4C2×Q1612yes—
174C3×C4C2×#4111yescharacteristic
175C3×C4C2×#4111yescharacteristic
176Dic3—21no—
177Dic3—42no—
178Dic3—42no—
179Dic3C2×Q1612yes—
180Dic3#13512yes—
181C23—31no—
182Q8—22no—
183Q8—22no—
184Q8—62no—
185Q8—62no—
186Q8—124no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—31no—
190C2×C4—31no—
191C2×C4—34no—
192C2×C4—31no—
193C2×C4—62no—
194C2×C4—61no—
195C2×C4—61no—
196C2×C4—62no—
197C2×C4—62no—
198C2×C4D3×D411yescharacteristic
199C2×C4D3×D411yescharacteristic
200C2×C4C23×D311yescharacteristic
201C8—21no—
202C8—32no—
203C8—61no—
204C8#23512yes—
205C2×C3#53211yescharacteristic
206C2×C3C22×Q1611yescharacteristic
207C2×C3C2×#13511yescharacteristic
208D3—22no—
209C22—32no—
210C22—61no—
211C22—61no—
212C22C2×D3×D411yescharacteristic
213C4—21no—
214C4—32no—
215C4—32no—
216C4—42no—
217C4—42no—
218C4—61no—
219C4—122no—
220C4—122no—
221C4D3×Q1612yes—
222C4C2×#23511yescharacteristic
223C4D3×#4111yescharacteristic
224C3#317011yescharacteristic, complemented, Sylow
225C2—62no—
226C2#554811yescharacteristic
227C2C2×D3×Q1611yescharacteristic
228C2D3×#13511yescharacteristic
subgroupG/H
centerC2×C4D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×C4×Q16C2
socleC22×C3C22×D4
prime radicalC22×D3C2×D4
hypercenterC4×Q16D3
hypocenterC3#3170
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C42 ◁ C4×Q16

Complements

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

NH ≅ G/Ntype
2. C3×C4×Q16C2semidirect
4. #5183C2semidirect
5. C4×#216C2semidirect
7. #4874C2semidirect
9. #4792C2semidirect
10. C4×Dic12C2semidirect
49. C4×Q16D3semidirect
126. C4×D3Q16semidirect
224. C3#3170semidirect

Maximal subgroups

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

subgroupindex
#55782
C3×C4×Q162
#51882
#51832
C4×#2162
#48872
#48742
#48132
#47922
C4×Dic122
C4×D3×C82
#31703