Finite Group Databasebeta

← #81033

Subgroups — #81033

Order 384.

342 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 115 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#5732C211yesmaximal, characteristic
2#5579C211yesmaximal, characteristic
3C3×#475C211yesmaximal, characteristic, complemented
4#5080C211yesmaximal, characteristic, complemented
5C4×#216C211yesmaximal, characteristic, complemented
6#4862C212yesmaximal, complemented
7#4861C212yesmaximal, complemented
8#4850C211yesmaximal, characteristic, complemented
9#4838C211yesmaximal, characteristic
10#4816C212yesmaximal, complemented
11#4814C212yesmaximal, complemented
12#3356—31nomaximal, Sylow
13C3×C4⋊Q8C2211yescharacteristic
14C3×C4×Q8C2211yescharacteristic
15#947—21no—
16#946C2212yes—
17Q8×Dic3—21no—
18C2×#216C2211yescharacteristic
19C2×ΓL(1,𝔽25)C2212yes—
20#898—22no—
21#897C2212yes—
22#896—22no—
23#892C2212yes—
24D3×C4⋊C4C2212yes—
25C4×D12—21no—
26C4×D12C2211yescharacteristic
27#870—21no—
28#869C2211yescharacteristic
29C4×Dic6C2211yescharacteristic
30C3×#106C2212yes—
31C3×C4⋊C8C2211yescharacteristic
32C3×#103C2212yes—
33#835C2211yescharacteristic
34(C2×D3)⋊C8C2212yes—
35#818C2212yes—
36#816C2212yes—
37#810C2211yescharacteristic
38#808C2212yes—
39#808C2212yes—
40#807C2211yescharacteristic
41C4×C3⋊C8C2211yescharacteristic
42#556—31no—
43#554—31no—
44#498—31no—
45#485—32no—
46#483—32no—
47#477—31no—
48#475D311yescharacteristic, complemented
49#470—32no—
50C4×Q16—31no—
51#436—31no—
52#428—32no—
53C2×C3×Q8—21no—
54C2×C3×Q8C2311yescharacteristic
55C2×D12D411yescharacteristic
56C2×C4×D3—22no—
57C2×C4×D3C2312yes—
58C2×Dic6—21no—
59C2×Dic6C2311yescharacteristic
60C2×C3×C8C2312yes—
61C3×C4⋊C4—21no—
62C3×C4⋊C4—21no—
63C3×C4⋊C4C2311yescharacteristic
64C3×C4⋊C4C2312yes—
65C3×C42—21no—
66C3×C42C2311yescharacteristic
67#216—22no—
68(C2×D3)⋊C4—22no—
69(C2×D3)⋊C4—22no—
70(C2×D3)⋊C4—22no—
71(C2×D3)⋊C4D412yes—
72C4⋊Dic3—21no—
73C4⋊Dic3—22no—
74C4⋊Dic3D411yescharacteristic
75C4⋊Dic3C2311yescharacteristic
76C4⋊Dic3C2312yes—
77Dic3⋊C4—21no—
78Dic3⋊C4—22no—
79Dic3⋊C4—22no—
80C4×Dic3—21no—
81C4×Dic3—22no—
82C2×C3⋊C8C2311yescharacteristic
83C2×C3⋊C8C2311yescharacteristic
84ΓL(1,𝔽25)—24no—
85C2×Q16—31no—
86C2×M16—32no—
87C4⋊Q8—31no—
88C4⋊Q8C2×D311yescharacteristic
89#126—62no—
90#125—61no—
91#124—61no—
92C22⋊Q8—32no—
93C22⋊Q8—32no—
94C22⋊Q8—62no—
95C4×Q8—31no—
96C4×Q8—61no—
97C4×Q8C2×D311yescharacteristic
98C4×D4—31no—
99C4×D4—61no—
100#117—32no—
101C2×C4⋊C4—32no—
102#107—32no—
103#107—31no—
104#106—32no—
105#106—32no—
106#106C2×D312yes—
107C4⋊C8C2×D311yescharacteristic
108#103—31no—
109#103—32no—
110#103—31no—
111#103C2×D312yes—
112C22⋊C8—32no—
113C4×C8—31no—
114C22×D3C2×D412yes—
115C3×Q8—21no—
116C3×Q8—21no—
117C3×Q8—42no—
118C2×C3×C4—22no—
119C2×C3×C4—21no—
120C2×C3×C4—21no—
121C2×C3×C4C2×D411yescharacteristic
122C2×C3×C4C2×D411yescharacteristic
123C2×C3×C4C2411yescharacteristic
124C2×Dic3—21no—
125C2×Dic3—21no—
126C2×Dic3—22no—
127C2×Dic3C2×D412yes—
128D12—22no—
129C4×D3—22no—
130C4×D3—22no—
131C4×D3—82no—
132Dic6—21no—
133Dic6—21no—
134Dic6—81no—
135C3×C8—22no—
136C3⋊C8—21no—
137C3⋊C8#4112yes—
138C2×Q8—21no—
139C2×Q8—31no—
140C2×Q8—61no—
141C2×Q8C22×D311yescharacteristic
142C2×D4—31no—
143C22×C4—32no—
144C22×C4—62no—
145Q16—62no—
146M16—64no—
147C2×C8—31no—
148C2×C8—31no—
149C2×C8C22×D312yes—
150C4⋊C4—21no—
151C4⋊C4—21no—
152C4⋊C4—31no—
153C4⋊C4—32no—
154C4⋊C4—31no—
155C4⋊C4—62no—
156C4⋊C4—62no—
157C4⋊C4—62no—
158C4⋊C4—61no—
159C4⋊C4—61no—
160C4⋊C4C22×D311yescharacteristic
161C4⋊C4C22×D312yes—
162C22⋊C4—32no—
163C22⋊C4—62no—
164C22⋊C4—62no—
165C22⋊C4—62no—
166C42—21no—
167C42—62no—
168C42—61no—
169C42C22×D311yescharacteristic
170C22×C3C22×D411yescharacteristic
171C2×D3—22no—
172C2×D3—22no—
173C2×D3—22no—
174C3×C4—21no—
175C3×C4—41no—
176C3×C4—42no—
177C3×C4—41no—
178C3×C4#13712yes—
179C3×C4C2×#4111yescharacteristic
180C3×C42−1+411yescharacteristic
181Dic3—22no—
182Dic3—41no—
183Dic3—42no—
184Dic3—41no—
185C23—32no—
186Q8—21no—
187Q8—21no—
188Q8—42no—
189Q8—61no—
190Q8—61no—
191Q8—241no—
192D4—62no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—22no—
196C2×C4—32no—
197C2×C4—62no—
198C2×C4—62no—
199C2×C4—61no—
200C2×C4—61no—
201C2×C4—62no—
202C2×C4—242no—
203C2×C4D3×D411yescharacteristic
204C2×C4D3×D411yescharacteristic
205C2×C4C23×D311yescharacteristic
206C8—22no—
207C8—32no—
208C8—61no—
209C2×C3#54711yescharacteristic
210C2×C3C2×#13711yescharacteristic
211C2×C3#57711yescharacteristic
212D3—42no—
213C22—62no—
214C22—62no—
215C22—62no—
216C22C2×D3×D411yescharacteristic
217C4—21no—
218C4—42no—
219C4—41no—
220C4—41no—
221C4—62no—
222C4—122no—
223C4—121no—
224C4—121no—
225C4#91512yes—
226C4#100711yescharacteristic
227C4D3×#4111yescharacteristic
228C3#335611yescharacteristic, complemented, Sylow
229C2—122no—
230C2#563111yescharacteristic
231C2C2×#91511yescharacteristic
232C2#577811yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#475C2
socleC22×C3C22×D4
prime radicalC2×D12D4
hypercenter#475D3
hypocenterC3#3356
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
3. C3×#475C2semidirect
4. #5080C2semidirect
5. C4×#216C2semidirect
6. #4862C2semidirect
7. #4861C2semidirect
8. #4850C2semidirect
10. #4816C2semidirect
11. #4814C2semidirect
48. #475D3semidirect
228. C3#3356semidirect

Maximal subgroups

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

subgroupindex
#57322
#55792
C3×#4752
#50802
C4×#2162
#48622
#48612
#48502
#48382
#48162
#48142
#33563