Finite Group Databasebeta

← #79200

Subgroups — #79200

Order 384.

484 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#909C211yesmaximal, characteristic
2#5522C211yesmaximal, characteristic
3C2×#887C211yesmaximal, characteristic, complemented
4C3×#428C211yesmaximal, characteristic, complemented
5Dic3×M16C211yesmaximal, characteristic, complemented
6#5004C211yesmaximal, characteristic, complemented
7#4966C211yesmaximal, characteristic
8#4889C212yesmaximal, complemented
9#4873C212yesmaximal, complemented
10#4853C212yesmaximal, complemented
11#4852C212yesmaximal, complemented
12C4×#137—31nomaximal, Sylow
13C2×#235C2211yescharacteristic
14C22×Dic6C2211yescharacteristic
15C2×C3×M16C2211yescharacteristic
16C3×#117C2211yescharacteristic
17C2×C3×C4⋊C4C2211yescharacteristic
18C2×Dic3⋊C4—21no—
19C2×C4×Dic3—21no—
20C2×C4×Dic3C2211yescharacteristic
21C2×#208C2211yescharacteristic
22#909C418yescomplemented
23C2×Dic12C2212yes—
24C2×#204C2212yes—
25#893C2211yescharacteristic
26#893C2211yescharacteristic
27#887—21no—
28#887—21no—
29#887C2211yescharacteristic
30#887C2212yes—
31#887C2212yes—
32#887C2211yescharacteristic
33#67⋊C4—21no—
34#67⋊C4—21no—
35#872—21no—
36C42×D3—21no—
37C3×#107C2212yes—
38C3×#106C2212yes—
39#815C2212yes—
40C8×Dic3C2212yes—
41#810C2212yes—
42#810C2212yes—
43#810C2212yes—
44#809C2212yes—
45C2×#137—31no—
46C4×#41—31no—
47C2×C4×Q8—31no—
48#441—32no—
49#440—32no—
50C4×Q16—32no—
51C4×SD16—32no—
52#428D311yescharacteristic, complemented
53#419—31no—
54#417—31no—
55C4×M16—31no—
56C22×C3×C4—21no—
57C22×C3×C4C2311yescharacteristic
58C2×#67—21no—
59C22×Dic3—21no—
60C22×Dic3D411yescharacteristic
61C22×Dic3D411yescharacteristic
62#235—22no—
63#235C2×C414yes—
64C2×D12C2311yescharacteristic
65C2×C4×D3—21no—
66C2×C4×D3—21no—
67C2×C4×D3—21no—
68C2×Dic6—21no—
69C2×Dic6—22no—
70C2×Dic6C2×C414yescomplemented
71C2×Dic6C2312yes—
72C2×Dic6C2311yescharacteristic
73C3×M16C2×C414yescomplemented
74C2×C3×C8C2312yes—
75C3×C4⋊C4—21no—
76C3×C4⋊C4C2312yes—
77C3×C4⋊C4C2311yescharacteristic
78C3×C4⋊C4C2311yescharacteristic
79C3×C22⋊C4—21no—
80C3×C42—21no—
81(C2×D3)⋊C4—21no—
82(C2×D3)⋊C4—21no—
83Dic3⋊C4—21no—
84Dic3⋊C4—21no—
85Dic3⋊C4—22no—
86Dic3⋊C4—22no—
87C4×Dic3—21no—
88C4×Dic3—21no—
89C4×Dic3—22no—
90C4×Dic3—22no—
91C4×Dic3D411yescharacteristic
92C4×Dic3D411yescharacteristic
93C4×Dic3C2311yescharacteristic
94C4×Dic3C2311yescharacteristic
95#208—22no—
96C2×C3⋊C8C2312yes—
97Dic12C2×C418yescomplemented
98#204C2×C418yescomplemented
99C2×#41—31no—
100C22×Q8—31no—
101#137—38no—
102C2×Q16—32no—
103C2×SD16—32no—
104C2×M16—31no—
105C2×M16C2×D311yescharacteristic
106C4×Q8—32no—
107C4×Q8—32no—
108C4×Q8—31no—
109C4×Q8—31no—
110C4×Q8—61no—
111C4×Q8—61no—
112C4×D4—31no—
113C4×D4—31no—
114C4×D4—61no—
115C4×D4—61no—
116#117—61no—
117#117C2×D311yescharacteristic
118C2×C4⋊C4—61no—
119C2×C4⋊C4C2×D311yescharacteristic
120C2×C42—31no—
121C2×C42—61no—
122C2×C42—61no—
123#107C2×D312yes—
124#106C2×D312yes—
125#103—32no—
126#103—32no—
127#103—32no—
128#102—32no—
129#97—32no—
130C4×C8—32no—
131C23×C3C2×D411yescharacteristic
132C22×D3—21no—
133C2×C3×C4—22no—
134C2×C3×C4—21no—
135C2×C3×C4—21no—
136C2×C3×C4—42no—
137C2×C3×C4C22×C412yes—
138C2×C3×C4C2×D411yescharacteristic
139C2×C3×C4#4112yes—
140C2×C3×C4C2411yescharacteristic
141#67—24no—
142C2×Dic3—24no—
143C2×Dic3—22no—
144C2×Dic3—21no—
145C2×Dic3—22no—
146C2×Dic3—41no—
147C2×Dic3C2×D411yescharacteristic
148C2×Dic3C2×D411yescharacteristic
149C2×Dic3C2×D411yescharacteristic
150C2×Dic3C2×D411yescharacteristic
151D12—21no—
152D12C22×C412yes—
153C4×D3—24no—
154C4×D3—42no—
155C4×D3—42no—
156Dic6—24no—
157Dic6—22no—
158Dic6—21no—
159Dic6C22×C414yes—
160Dic6C22×C412yes—
161C3×C8C22×C414yes—
162C3⋊C8—22no—
163#41—34no—
164#41—62no—
165C2×Q8—34no—
166C2×Q8—32no—
167C2×Q8—31no—
168C2×Q8—61no—
169C2×Q8—62no—
170C2×D4—31no—
171C2×D4—61no—
172C22×C4—21no—
173C22×C4—31no—
174C22×C4—31no—
175C22×C4—61no—
176C22×C4—61no—
177C22×C4—61no—
178C22×C4—61no—
179C22×C4C22×D311yescharacteristic
180Q16—38no—
181SD16—38no—
182M16—62no—
183M16C4×D314yescomplemented
184C2×C8—32no—
185C2×C8C22×D312yes—
186C4⋊C4—21no—
187C4⋊C4—62no—
188C4⋊C4—62no—
189C4⋊C4—61no—
190C4⋊C4—61no—
191C4⋊C4C22×D312yes—
192C4⋊C4C22×D311yescharacteristic
193C4⋊C4C22×D311yescharacteristic
194C22⋊C4—21no—
195C22⋊C4—61no—
196C22⋊C4—61no—
197C42—21no—
198C42—31no—
199C42—31no—
200C42—31no—
201C42—31no—
202C42—61no—
203C42—61no—
204C42—62no—
205C42—62no—
206C22×C3—21no—
207C22×C3—21no—
208C22×C3C4×D412yes—
209C22×C3C22×D411yescharacteristic
210C2×D3—22no—
211C2×D3—41no—
212C2×D3—41no—
213C3×C4—41no—
214C3×C4—41no—
215C3×C4—42no—
216C3×C4C4×D412yes—
217C3×C4C23×C411yescharacteristic
218C3×C4C2×#4111yescharacteristic
219Dic3—24no—
220Dic3—22no—
221Dic3—21no—
222Dic3—21no—
223Dic3—21no—
224Dic3#13712yescomplemented
225C23—61no—
226C23D3×D411yescharacteristic
227Q8—34no—
228Q8—32no—
229Q8—64no—
230Q8—62no—
231Q8—61no—
232D4—32no—
233D4—64no—
234D4—61no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—31no—
239C2×C4—31no—
240C2×C4—31no—
241C2×C4—31no—
242C2×C4—42no—
243C2×C4—62no—
244C2×C4—64no—
245C2×C4—61no—
246C2×C4—62no—
247C2×C4—64no—
248C2×C4—122no—
249C2×C4—122no—
250C2×C4—121no—
251C2×C4C2×C4×D312yes—
252C2×C4D3×D411yescharacteristic
253C2×C4#23912yes—
254C2×C4C23×D311yescharacteristic
255C8—62no—
256C8C2×C4×D314yes—
257C2×C3—22no—
258C2×C3C2×C4×D411yescharacteristic
259C2×C3C2×#13711yescharacteristic
260C2×C3#57511yescharacteristic
261D3—42no—
262C22—21no—
263C22—21no—
264C22—62no—
265C22—121no—
266C22—121no—
267C22#89312yes—
268C22C2×D3×D411yescharacteristic
269C4—32no—
270C4—42no—
271C4—41no—
272C4—41no—
273C4—62no—
274C4—64no—
275C4—61no—
276C4—61no—
277C4—61no—
278C4#89312yes—
279C4C22×C4×D311yescharacteristic
280C4C2×#23911yescharacteristic
281C3C4×#13711yescharacteristic, complemented, Sylow
282C2—22no—
283C2—122no—
284C2C2×#89311yescharacteristic
285C2#577311yescharacteristic
286C2D3×#13711yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutatorC3×C4C23×C4
FrattiniC2×C4C23×D3
FittingC3×#428C2
socleC22×C3C22×D4
prime radicalC2×#235C22
cosocleC2×C3×C4C24
hypercenter#428D3
hypocenterC3C4×#137
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
3. C2×#887C2semidirect
4. C3×#428C2semidirect
5. Dic3×M16C2semidirect
6. #5004C2semidirect
8. #4889C2semidirect
9. #4873C2semidirect
10. #4853C2semidirect
11. #4852C2semidirect
22. #909C4semidirect
52. #428D3semidirect
70. C2×Dic6C2×C4semidirect
73. C3×M16C2×C4semidirect
97. Dic12C2×C4semidirect
98. #204C2×C4semidirect
183. M16C4×D3semidirect
224. Dic3#137semidirect
281. C3C4×#137semidirect

Maximal subgroups

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

subgroupindex
C2×#9092
#55222
C2×#8872
C3×#4282
Dic3×M162
#50042
#49662
#48892
#48732
#48532
#48522
C4×#1373