Finite Group Databasebeta

← #81088

Subgroups — #81088

Order 384.

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

normal subgroups only

This group has 396 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#5730C211yesmaximal, characteristic, complemented
2#5553C211yesmaximal, characteristic, complemented
3C3×#476C211yesmaximal, characteristic, complemented
4#5088C211yesmaximal, characteristic, complemented
5C4×#213C211yesmaximal, characteristic, complemented
6#4871C212yesmaximal, complemented
7#4869C212yesmaximal, complemented
8#4843C211yesmaximal, characteristic, complemented
9#4831C211yesmaximal, characteristic, complemented
10#4786C212yesmaximal, complemented
11#4764C212yesmaximal, complemented
12#3366—31nomaximal, Sylow
13C2×#235—21no—
14C3×C4⋊Q8C2211yescharacteristic
15C3×C4×D4C2211yescharacteristic
16#946—22no—
17#940C2212yes—
18C2×#213C2211yescharacteristic
19#930—21no—
20C2×D24C2212yes—
21#897—21no—
22#893C2212yes—
23#893C2212yes—
24D3×C4⋊C4—21no—
25#890C2212yes—
26#884—22no—
27#874C2211yescharacteristic
28C4×D12C2211yescharacteristic
29C4×Dic6C2211yescharacteristic
30C3×#106C2212yes—
31C3×C4⋊C8C2211yescharacteristic
32C3×#102C2212yes—
33#832C2211yescharacteristic
34#821C2212yes—
35#815C2212yes—
36Dic3⋊C8C2212yes—
37#809C2211yescharacteristic
38#809C2212yes—
39#809C2212yes—
40#807C2211yescharacteristic
41C4×C3⋊C8C2211yescharacteristic
42#554—31no—
43#550—31no—
44#490—32no—
45#486—31no—
46#478—32no—
47#476D311yescharacteristic, complemented
48#459—31no—
49#459—32no—
50#446—31no—
51#442—32no—
52C4×D8—31no—
53C2×C3×Q8—21no—
54C2×C3×D4C2311yescharacteristic
55C22×C3×C4—21no—
56C2×#67—22no—
57C2×#67—22no—
58#235—44no—
59C2×D12—21no—
60C2×D12—21no—
61C2×D12C2312yes—
62C2×D12C2311yescharacteristic
63C2×C4×D3—21no—
64C2×C4×D3—22no—
65C2×C4×D3—22no—
66C2×Dic6D411yescharacteristic
67C2×C3×C8C2312yes—
68C3×C4⋊C4—21no—
69C3×C4⋊C4C2311yescharacteristic
70C3×C4⋊C4C2312yes—
71C3×C22⋊C4—21no—
72C3×C42C2311yescharacteristic
73#213—22no—
74(C2×D3)⋊C4—21no—
75(C2×D3)⋊C4—22no—
76(C2×D3)⋊C4—22no—
77C4⋊Dic3—22no—
78C4⋊Dic3D411yescharacteristic
79C4⋊Dic3C2311yescharacteristic
80Dic3⋊C4—22no—
81Dic3⋊C4D412yes—
82C4×Dic3C2312yes—
83C2×C3⋊C8C2311yescharacteristic
84C2×C3⋊C8C2311yescharacteristic
85D24—24no—
86C2×#41—61no—
87C2×D8—31no—
88C2×D8—32no—
89C4⋊Q8C2×D311yescharacteristic
90#127—32no—
91#127—31no—
92#125—32no—
93C22⋊Q8—61no—
94C22⋊Q8—62no—
95#121—62no—
96#121—61no—
97C4×Q8—31no—
98C4×D4—32no—
99C4×D4—32no—
100C4×D4—31no—
101C4×D4C2×D311yescharacteristic
102C2×C4⋊C4—61no—
103#107—31no—
104#106C2×D312yes—
105C4⋊C8—32no—
106C4⋊C8C2×D311yescharacteristic
107#102—31no—
108#102—32no—
109#102—31no—
110#102—32no—
111#102—32no—
112#102C2×D312yes—
113#97—32no—
114C4×C8—31no—
115C23×C3—21no—
116C22×D3—21no—
117C22×D3—22no—
118C3×Q8—81no—
119C3×D4—21no—
120C3×D4—21no—
121C2×C3×C4—22no—
122C2×C3×C4—21no—
123C2×C3×C4—42no—
124C2×C3×C4C2×D411yescharacteristic
125C2×C3×C4C2×D411yescharacteristic
126C2×C3×C4C2411yescharacteristic
127#67—44no—
128#67—82no—
129C2×Dic3—21no—
130C2×Dic3C2×D412yes—
131C2×Dic3C2×D412yes—
132D12—22no—
133D12—21no—
134D12—22no—
135D12—42no—
136D12—81no—
137D12#4112yescomplemented
138C4×D3—44no—
139C4×D3—42no—
140C4×D3—82no—
141Dic6—22no—
142C3×C8—22no—
143C3⋊C8—21no—
144C3⋊C8—21no—
145#41—124no—
146C2×Q8—21no—
147C2×Q8—31no—
148C2×D4—31no—
149C2×D4—32no—
150C2×D4—62no—
151C2×D4—62no—
152C2×D4—61no—
153C2×D4—61no—
154C2×D4C22×D311yescharacteristic
155C22×C4—21no—
156C22×C4—62no—
157C22×C4—62no—
158C22×C4—61no—
159D8—64no—
160D8—62no—
161C2×C8—31no—
162C2×C8—31no—
163C2×C8C22×D312yes—
164C4⋊C4—21no—
165C4⋊C4—31no—
166C4⋊C4—32no—
167C4⋊C4—31no—
168C4⋊C4—62no—
169C4⋊C4—62no—
170C4⋊C4C22×D312yes—
171C4⋊C4C22×D311yescharacteristic
172C22⋊C4—21no—
173C22⋊C4—62no—
174C22⋊C4—62no—
175C22⋊C4—61no—
176C42—32no—
177C42C22×D311yescharacteristic
178C22×C3—41no—
179C22×C3—41no—
180C22×C3—41no—
181C22×C3C22×D411yescharacteristic
182C2×D3—22no—
183C2×D3—41no—
184C2×D3—41no—
185C2×D3—42no—
186C2×D3—42no—
187C2×D3—42no—
188C3×C4—21no—
189C3×C4—41no—
190C3×C4—42no—
191C3×C4AGL(1,ℤ/8)12yes—
192C3×C4C2×#4111yescharacteristic
193C3×C42−1+411yescharacteristic
194Dic3—22no—
195Dic3—22no—
196Dic3—41no—
197C23—21no—
198C23—61no—
199C23—62no—
200Q8—62no—
201Q8—81no—
202D4—21no—
203D4—21no—
204D4—32no—
205D4—61no—
206D4—62no—
207D4—62no—
208D4—124no—
209D4—122no—
210D4—241no—
211D4—242no—
212C2×C4—21no—
213C2×C4—22no—
214C2×C4—32no—
215C2×C4—32no—
216C2×C4—42no—
217C2×C4—61no—
218C2×C4—124no—
219C2×C4—122no—
220C2×C4—242no—
221C2×C4D3×D411yescharacteristic
222C2×C4D3×D411yescharacteristic
223C2×C4C23×D311yescharacteristic
224C8—22no—
225C8—61no—
226C8—61no—
227C2×C3—81no—
228C2×C3#54811yescharacteristic
229C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
230C2×C3#57611yescharacteristic
231D3—42no—
232D3—82no—
233C22—41no—
234C22—41no—
235C22—41no—
236C22—62no—
237C22—121no—
238C22—121no—
239C22—122no—
240C22—122no—
241C22—122no—
242C22C2×D3×D411yescharacteristic
243C4—21no—
244C4—41no—
245C4—42no—
246C4—62no—
247C4—62no—
248C4—121no—
249C4#91412yes—
250C4#100711yescharacteristic
251C4D3×#4111yescharacteristic
252C3#336611yescharacteristic, complemented, Sylow
253C2—81no—
254C2—122no—
255C2—242no—
256C2#563311yescharacteristic
257C2C2×#91411yescharacteristic
258C2#577411yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#476C2
socleC22×C3C22×D4
hypercenter#476D3
hypocenterC3#3366
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #476

Complements

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

NH ≅ G/Ntype
1. #5730C2semidirect
2. #5553C2semidirect
3. C3×#476C2semidirect
4. #5088C2semidirect
5. C4×#213C2semidirect
6. #4871C2semidirect
7. #4869C2semidirect
8. #4843C2semidirect
9. #4831C2semidirect
10. #4786C2semidirect
11. #4764C2semidirect
47. #476D3semidirect
137. D12#41semidirect
252. C3#3366semidirect

Maximal subgroups

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

subgroupindex
#57302
#55532
C3×#4762
#50882
C4×#2132
#48712
#48692
#48432
#48312
#47862
#47642
#33663