Finite Group Databasebeta

← #81600

Subgroups — #81600

Order 384.

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

normal subgroups only

This group has 380 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#5728C211yesmaximal, characteristic
2D3×C4⋊Q8C211yesmaximal, characteristic
3C3×#487C211yesmaximal, characteristic, complemented
4#5093C211yesmaximal, characteristic, complemented
5#5079C211yesmaximal, characteristic, complemented
6#4802C214yesmaximal
7#4797C214yesmaximal, complemented
8C4×ΓL(1,𝔽25)C211yesmaximal, characteristic
9#4681C211yesmaximal, characteristic, complemented
10#3139—31nomaximal, Sylow
11C2×D3×Q8C2212yes—
12C3×C4⋊Q8C2211yescharacteristic
13C3×C4⋊Q8C2211yescharacteristic
14#946—21no—
15Q8×Dic3—21no—
16#944C2212yes—
17C2×ΓL(1,𝔽25)C2212yes—
18#897—21no—
19#892C2212yes—
20D3×C4⋊C4—21no—
21D3×C4⋊C4—21no—
22#890C2212yes—
23#888—21no—
24#888—21no—
25#887—21no—
26C42×D3C2211yescharacteristic
27#869C2211yescharacteristic
28#869C2211yescharacteristic
29C3×#103C2214yes—
30C3×C4×C8C2211yescharacteristic
31#835C2214yes—
32#816C2214yes—
33#815C2212yes—
34#810C2214yes—
35C4×C3⋊C8C2211yescharacteristic
36#533—31no—
37C2×C4⋊Q8—31no—
38#491—34no—
39#487—31no—
40#487—31no—
41#487—31no—
42#487D311yescharacteristic, complemented
43#419—34no—
44C4×M16—31no—
45C2×C3×Q8—21no—
46C2×C3×Q8C2312yes—
47D3×Q8—28no—
48C2×C4×D3—22no—
49C2×C4×D3D411yescharacteristic
50C2×C4×D3D411yescharacteristic
51C2×C4×D3C2311yescharacteristic
52C2×Dic6—21no—
53C2×Dic6—22no—
54C2×Dic6C2312yes—
55C2×C3×C8C2312yes—
56C3×C4⋊C4—21no—
57C3×C4⋊C4—21no—
58C3×C4⋊C4—21no—
59C3×C4⋊C4C2312yes—
60C3×C42C2311yescharacteristic
61(C2×D3)⋊C4—22no—
62C4⋊Dic3—21no—
63C4⋊Dic3—21no—
64C4⋊Dic3—21no—
65C4⋊Dic3—22no—
66C4⋊Dic3C2312yes—
67Dic3⋊C4—22no—
68Dic3⋊C4—22no—
69Dic3⋊C4—22no—
70C4×Dic3—22no—
71C4×Dic3D411yescharacteristic
72C4×Dic3D411yescharacteristic
73C4×Dic3C2311yescharacteristic
74C2×C3⋊C8C2312yes—
75ΓL(1,𝔽25)—24no—
76C22×Q8—32no—
77C2×M16—32no—
78C4⋊Q8—32no—
79C4⋊Q8—31no—
80C4⋊Q8—31no—
81C4⋊Q8—61no—
82C4⋊Q8—61no—
83C4⋊Q8C2×D311yescharacteristic
84C4⋊Q8C2×D311yescharacteristic
85#125—32no—
86C22⋊Q8—61no—
87C22⋊Q8—61no—
88C4×Q8—61no—
89C4×Q8—61no—
90#117—32no—
91C2×C4⋊C4—61no—
92C2×C4⋊C4—61no—
93C2×C42—31no—
94#103—34no—
95#103—34no—
96#103—34no—
97#103C2×D314yes—
98#97—32no—
99C4×C8—31no—
100C4×C8C2×D311yescharacteristic
101C22×D3C2×D411yescharacteristic
102C3×Q8—24no—
103C3×Q8—42no—
104C2×C3×C4—22no—
105C2×C3×C4—22no—
106C2×C3×C4C2×D411yescharacteristic
107C2×C3×C4C2×D411yescharacteristic
108C2×C3×C4C2411yescharacteristic
109C2×Dic3—22no—
110C2×Dic3—22no—
111C2×Dic3C2×D411yescharacteristic
112C2×Dic3C2×D411yescharacteristic
113C2×Dic3C2×D411yescharacteristic
114C4×D3—22no—
115C4×D3—22no—
116C4×D3—44no—
117C4×D3#4114yes—
118Dic6—24no—
119Dic6—42no—
120Dic6—44no—
121C3×C8—22no—
122C3⋊C8—22no—
123C2×Q8—21no—
124C2×Q8—32no—
125C2×Q8—62no—
126C2×Q8—68no—
127C2×Q8—61no—
128C2×Q8C22×D312yes—
129C22×C4—31no—
130C22×C4—31no—
131C22×C4—31no—
132C22×C4—62no—
133M16—64no—
134C2×C8—32no—
135C2×C8C22×D312yes—
136C4⋊C4—21no—
137C4⋊C4—21no—
138C4⋊C4—21no—
139C4⋊C4—32no—
140C4⋊C4—62no—
141C4⋊C4—62no—
142C4⋊C4—62no—
143C4⋊C4—62no—
144C4⋊C4—61no—
145C4⋊C4—61no—
146C4⋊C4—61no—
147C4⋊C4C22×D312yes—
148C22⋊C4—62no—
149C42—31no—
150C42—31no—
151C42—31no—
152C42—62no—
153C42C22×D311yescharacteristic
154C22×C3C22×D411yescharacteristic
155C2×D3—21no—
156C2×D3—21no—
157C2×D3#12412yes—
158C3×C4—42no—
159C3×C4—42no—
160C3×C4#13712yes—
161C3×C4#13712yes—
162C3×C4C2×#4112yes—
163Dic3—21no—
164Dic3—21no—
165Dic3—42no—
166Dic3—42no—
167Dic3#12412yes—
168C23—31no—
169Q8—24no—
170Q8—42no—
171Q8—64no—
172Q8—124no—
173Q8—122no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—31no—
177C2×C4—34no—
178C2×C4—31no—
179C2×C4—31no—
180C2×C4—62no—
181C2×C4—62no—
182C2×C4—62no—
183C2×C4—62no—
184C2×C4—124no—
185C2×C4D3×D411yescharacteristic
186C2×C4D3×D411yescharacteristic
187C2×C4C23×D311yescharacteristic
188C8—22no—
189C8—62no—
190C2×C3C2×#12411yescharacteristic
191C2×C3C2×#13711yescharacteristic
192C2×C3C2×#13711yescharacteristic
193D3—22no—
194C22—32no—
195C22—61no—
196C22—61no—
197C22C2×D3×D411yescharacteristic
198C4—32no—
199C4—42no—
200C4—42no—
201C4—61no—
202C4—61no—
203C4—122no—
204C4—122no—
205C4#91512yes—
206C4#91812yes—
207C4D3×#4112yes—
208C3#313911yescharacteristic, complemented, Sylow
209C2—62no—
210C2D3×#12411yescharacteristic
211C2C2×#91511yescharacteristic
212C2C2×#91811yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#487C2
socleC22×C3C22×D4
prime radicalC22×D3C2×D4
hypercenter#487D3
hypocenterC3#3139
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
3. C3×#487C2semidirect
4. #5093C2semidirect
5. #5079C2semidirect
7. #4797C2semidirect
9. #4681C2semidirect
42. #487D3semidirect
208. C3#3139semidirect

Maximal subgroups

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

subgroupindex
#57282
D3×C4⋊Q82
C3×#4872
#50932
#50792
#48022
#47972
C4×ΓL(1,𝔽25)2
#46812
#31393