Finite Group Databasebeta

← #81939

Subgroups — #81939

Order 384.

340 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 137 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#5729C212yesmaximal
2C3×#500C211yesmaximal, characteristic, complemented
3#5081C212yesmaximal, complemented
4#4881C212yesmaximal
5#4880C211yesmaximal, characteristic
6D3×#107C211yesmaximal, characteristic
7#4878C211yesmaximal, characteristic, complemented
8#4876C212yesmaximal, complemented
9#4874C211yesmaximal, characteristic, complemented
10#4689C211yesmaximal, characteristic
11#4683C211yesmaximal, characteristic, complemented
12#3164—31nomaximal, Sylow
13C3×C4⋊Q8C2212yes—
14#946—22no—
15#944—22no—
16C2×ΓL(1,𝔽25)C2211yescharacteristic
17C2×D3×C8C2211yescharacteristic
18#896—22no—
19#892C2212yes—
20D3×C4⋊C4C2212yes—
21#890C2212yes—
22#889—22no—
23#888C2212yes—
24#872C2211yescharacteristic
25#870C2212yes—
26C3×#107C2211yescharacteristic
27C3×#107C2212yes—
28C3×#107C2211yescharacteristic
29C3×C4×C8C2211yescharacteristic
30#818C2211yescharacteristic
31#818C2211yescharacteristic
32#817C2212yes—
33#815C2211yescharacteristic
34C8×Dic3C2211yescharacteristic
35#808C2212yes—
36#808C2212yes—
37#807C2212yes—
38#807C2212yes—
39#803C2211yescharacteristic
40#544—32no—
41#501—32no—
42#501—32no—
43#500—31no—
44#500D311yescharacteristic, complemented
45#499—31no—
46#499—31no—
47#428—32no—
48#427—31no—
49C2×#107—31no—
50#405—31no—
51C2×C3×Q8—22no—
52C2×C4×D3—22no—
53C2×C4×D3C2311yescharacteristic
54C2×Dic6—22no—
55C2×C3×C8C2311yescharacteristic
56C2×C3×C8C2311yescharacteristic
57C3×C4⋊C4—22no—
58C3×C4⋊C4C2312yes—
59C3×C4⋊C4C2312yes—
60C3×C42C2311yescharacteristic
61(C2×D3)⋊C4—22no—
62(C2×D3)⋊C4D411yescharacteristic
63(C2×D3)⋊C4D411yescharacteristic
64C4⋊Dic3—22no—
65C4⋊Dic3C2312yes—
66C4⋊Dic3C2312yes—
67Dic3⋊C4—22no—
68Dic3⋊C4—22no—
69Dic3⋊C4—22no—
70Dic3⋊C4—22no—
71Dic3⋊C4—22no—
72Dic3⋊C4D411yescharacteristic
73Dic3⋊C4D411yescharacteristic
74C4×Dic3—22no—
75C4×Dic3C2311yescharacteristic
76C2×C3⋊C8C2311yescharacteristic
77C2×C3⋊C8C2311yescharacteristic
78ΓL(1,𝔽25)Q814yescomplemented
79D3×C8Q814yescomplemented
80C2×M16—31no—
81C22×C8—31no—
82C4⋊Q8—32no—
83C4⋊Q8—62no—
84C4⋊Q8C2×D312yes—
85#125—32no—
86#125—32no—
87#125—62no—
88C22⋊Q8—62no—
89C22⋊Q8—62no—
90#117—31no—
91#117—32no—
92C2×C4⋊C4—32no—
93#107—31no—
94#107—31no—
95#107—32no—
96#107—32no—
97#107C2×D311yescharacteristic
98#107C2×D311yescharacteristic
99#107C2×D312yes—
100#106—32no—
101#106—32no—
102#106—32no—
103#97—31no—
104#97—31no—
105C4×C8—31no—
106C4×C8C2×D311yescharacteristic
107C22×D3C2×D411yescharacteristic
108C3×Q8—82no—
109C2×C3×C4—22no—
110C2×C3×C4—22no—
111C2×C3×C4C2×D411yescharacteristic
112C2×C3×C4C2×D411yescharacteristic
113C2×C3×C4C2411yescharacteristic
114C2×Dic3—22no—
115C2×Dic3—22no—
116C2×Dic3C2×D411yescharacteristic
117C2×Dic3C2×D411yescharacteristic
118C2×Dic3C2×D411yescharacteristic
119C4×D3—82no—
120C4×D3C2×Q812yes—
121C4×D3C2×Q812yes—
122Dic6—82no—
123C3×C8C2×Q812yes—
124C3×C8C2×Q812yes—
125C3⋊C8C2×Q812yes—
126C3⋊C8C2×Q812yes—
127C2×Q8—22no—
128C2×Q8—62no—
129C22×C4—31no—
130C22×C4—62no—
131M16—34no—
132C2×C8—31no—
133C2×C8—31no—
134C2×C8—34no—
135C2×C8C22×D311yescharacteristic
136C2×C8C22×D311yescharacteristic
137C4⋊C4—22no—
138C4⋊C4—31no—
139C4⋊C4—31no—
140C4⋊C4—32no—
141C4⋊C4—32no—
142C4⋊C4—62no—
143C4⋊C4—62no—
144C4⋊C4—62no—
145C4⋊C4—62no—
146C4⋊C4—62no—
147C4⋊C4—62no—
148C4⋊C4C22×D312yes—
149C4⋊C4C22×D312yes—
150C22⋊C4—31no—
151C22⋊C4—31no—
152C22⋊C4—62no—
153C42—31no—
154C42—62no—
155C42C22×D311yescharacteristic
156C22×C3C22×D411yescharacteristic
157C2×D3—21no—
158C2×D3—21no—
159C2×D3C4⋊Q812yes—
160C3×C4—21no—
161C3×C4—21no—
162C3×C4—42no—
163C3×C4—42no—
164C3×C4C22×Q811yescharacteristic
165C3×C4C22×Q811yescharacteristic
166Dic3—21no—
167Dic3—21no—
168Dic3—42no—
169Dic3—42no—
170Dic3C4⋊Q812yes—
171C23—31no—
172Q8—82no—
173Q8—242no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—31no—
177C2×C4—32no—
178C2×C4—32no—
179C2×C4—31no—
180C2×C4—31no—
181C2×C4—62no—
182C2×C4—62no—
183C2×C4—242no—
184C2×C4D3×D411yescharacteristic
185C2×C4D3×D411yescharacteristic
186C2×C4C23×D311yescharacteristic
187C8—32no—
188C8—32no—
189C8D3×Q812yes—
190C8D3×Q812yes—
191C2×C3C2×C4⋊Q811yescharacteristic
192C2×C3#57611yescharacteristic
193C2×C3#57811yescharacteristic
194D3—22no—
195C22—32no—
196C22—61no—
197C22—61no—
198C22C2×D3×D411yescharacteristic
199C4—21no—
200C4—21no—
201C4—32no—
202C4—42no—
203C4—42no—
204C4—61no—
205C4—61no—
206C4—122no—
207C4—122no—
208C4C2×D3×Q811yescharacteristic
209C4C2×D3×Q811yescharacteristic
210C3#316411yescharacteristic, complemented, Sylow
211C2—62no—
212C2D3×C4⋊Q811yescharacteristic
213C2#575711yescharacteristic
214C2#576611yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#500C2
socleC22×C3C22×D4
prime radicalC22×D3C2×D4
hypercenter#500D3
hypocenterC3#3164
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #500

Complements

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

NH ≅ G/Ntype
2. C3×#500C2semidirect
3. #5081C2semidirect
7. #4878C2semidirect
8. #4876C2semidirect
9. #4874C2semidirect
11. #4683C2semidirect
44. #500D3semidirect
78. ΓL(1,𝔽25)Q8semidirect
79. D3×C8Q8semidirect
210. C3#3164semidirect

Maximal subgroups

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

subgroupindex
#57292
C3×#5002
#50812
#48812
#48802
D3×#1072
#48782
#48762
#48742
#46892
#46832
#31643