Finite Group Databasebeta

← #82507

Subgroups — #82507

Order 384.

388 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C4×#41C211yesmaximal, characteristic
2#5470C211yesmaximal, characteristic, complemented
3#5233C212yesmaximal
4#5022C212yesmaximal, complemented
5#5021C211yesmaximal, characteristic, complemented
6#5020C211yesmaximal, characteristic, complemented
7#5009C212yesmaximal, complemented
8#5008C211yesmaximal, characteristic, complemented
9#5007C211yesmaximal, characteristic, complemented
10#4998C212yesmaximal, complemented
11C2×C4⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
12#3076—31nomaximal, Sylow
13C2×C3×#41C2211yescharacteristic
14C3×C4×Q8C2211yescharacteristic
15C3×C4×Q8C2211yescharacteristic
16C3×C4×D4—21no—
17C3×C4×D4—21no—
18C3×C4×D4C2211yescharacteristic
19C3×C4×D4C2211yescharacteristic
20C3×#117—21no—
21C3×#117C2211yescharacteristic
22C2×C3×C42—21no—
23C2×C3×C42C2211yescharacteristic
24#926C2212yes—
25#924—21no—
26#924—21no—
27C22×C3⋊C8C2212yes—
28#870C2212yes—
29#869C2211yescharacteristic
30#869C2211yescharacteristic
31C4×Dic6—21no—
32C4×Dic6—21no—
33#835C2212yes—
34#835C2212yes—
35#832C2212yes—
36#832C2212yes—
37#808C2212yes—
38#808C2212yes—
39#807C2212yes—
40#807C2212yes—
41C4⋊(C3⋊C8)C2211yescharacteristic
42C4⋊(C3⋊C8)C2211yescharacteristic
43C4⋊(C3⋊C8)C2212yes—
44#533—31no—
45C4×#41D311yescharacteristic
46#479—32no—
47#478—32no—
48#477—31no—
49#476—31no—
50#475—31no—
51#474—31no—
52#427—32no—
53C2×C4⋊C8—31no—
54#416—32no—
55C3×#41—22no—
56C3×#41Q814yescomplemented
57C2×C3×Q8C2311yescharacteristic
58C2×C3×D4—21no—
59C2×C3×D4C2311yescharacteristic
60C22×C3×C4—21no—
61C22×C3×C4—21no—
62C22×C3×C4—21no—
63C22×C3×C4D411yescharacteristic
64C22×C3×C4D411yescharacteristic
65C22×C3×C4C2311yescharacteristic
66C2×Dic6—21no—
67C2×Dic6—21no—
68C3×C4⋊C4—21no—
69C3×C4⋊C4—21no—
70C3×C4⋊C4C2311yescharacteristic
71C3×C4⋊C4C2311yescharacteristic
72C3×C22⋊C4—21no—
73C3×C22⋊C4—21no—
74C3×C22⋊C4—21no—
75C3×C42—21no—
76C3×C42—21no—
77C3×C42—21no—
78C3×C42D411yescharacteristic
79C3×C42D411yescharacteristic
80C3×C42C2311yescharacteristic
81C3×C42C2311yescharacteristic
82#217—22no—
83C4⋊Dic3—21no—
84C4⋊Dic3—21no—
85C4⋊Dic3C2312yes—
86C4⋊Dic3C2312yes—
87Dic3⋊C4—22no—
88Dic3⋊C4—22no—
89C4×Dic3—22no—
90C2×C3⋊C8—24no—
91C2×C3⋊C8C2312yes—
92C2×C3⋊C8C2312yes—
93C2×#41C2×D311yescharacteristic
94C22×C8—32no—
95C4⋊Q8—31no—
96C4⋊Q8—31no—
97#125—32no—
98C22⋊Q8—61no—
99C22⋊Q8—61no—
100C4×Q8—61no—
101C4×Q8—61no—
102C4×Q8C2×D311yescharacteristic
103C4×Q8C2×D311yescharacteristic
104C4×D4—21no—
105C4×D4—21no—
106C4×D4C2×D311yescharacteristic
107C4×D4C2×D311yescharacteristic
108#117—21no—
109#117—32no—
110#117C2×D311yescharacteristic
111C2×C42—21no—
112C2×C42C2×D311yescharacteristic
113#107—32no—
114#107—32no—
115#106—32no—
116#106—32no—
117C4⋊C8—31no—
118C4⋊C8—31no—
119C4⋊C8—32no—
120#103—32no—
121#103—32no—
122#102—32no—
123#102—32no—
124C23×C3—21no—
125C23×C3C2×D411yescharacteristic
126C3×Q8—21no—
127C3×Q8C2×Q812yes—
128C3×D4—24no—
129C3×D4—21no—
130C3×D4C2×Q812yes—
131C2×C3×C4—21no—
132C2×C3×C4—24no—
133C2×C3×C4—22no—
134C2×C3×C4—21no—
135C2×C3×C4—21no—
136C2×C3×C4—41no—
137C2×C3×C4—42no—
138C2×C3×C4—42no—
139C2×C3×C4C2×D411yescharacteristic
140C2×C3×C4C2×D411yescharacteristic
141C2×C3×C4C2×D411yescharacteristic
142C2×C3×C4C2×D411yescharacteristic
143C2×C3×C4C2×D411yescharacteristic
144C2×C3×C4C2×Q812yes—
145C2×C3×C4#4112yes—
146C2×C3×C4C2411yescharacteristic
147C2×Dic3—22no—
148C2×Dic3—22no—
149Dic6—42no—
150Dic6—42no—
151C3⋊C8—22no—
152C3⋊C8—22no—
153#41—22no—
154#41Dic614yescomplemented
155C2×Q8—61no—
156C2×Q8—61no—
157C2×Q8C22×D311yescharacteristic
158C2×D4—21no—
159C2×D4C22×D311yescharacteristic
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4#6711yescharacteristic
164C22×C4#6711yescharacteristic
165C22×C4C22×D311yescharacteristic
166C2×C8—32no—
167C2×C8—32no—
168C2×C8—64no—
169C4⋊C4—21no—
170C4⋊C4—21no—
171C4⋊C4—32no—
172C4⋊C4—32no—
173C4⋊C4—62no—
174C4⋊C4—62no—
175C4⋊C4—61no—
176C4⋊C4—61no—
177C4⋊C4C22×D311yescharacteristic
178C4⋊C4C22×D311yescharacteristic
179C22⋊C4—21no—
180C22⋊C4—21no—
181C22⋊C4—21no—
182C22⋊C4—62no—
183C42—21no—
184C42—21no—
185C42—21no—
186C42—62no—
187C42#6711yescharacteristic
188C42#6711yescharacteristic
189C42C22×D311yescharacteristic
190C42C22×D311yescharacteristic
191C22×C3—21no—
192C22×C3—21no—
193C22×C3—22no—
194C22×C3—41no—
195C22×C3—41no—
196C22×C3C22⋊Q812yes—
197C22×C3C22×D411yescharacteristic
198C3×C4—22no—
199C3×C4—21no—
200C3×C4—21no—
201C3×C4—21no—
202C3×C4—41no—
203C3×C4—41no—
204C3×C4C22⋊Q812yes—
205C3×C4#13512yes—
206C3×C4C22×Q811yescharacteristic
207C3×C4C2×#4111yescharacteristic
208Dic3—42no—
209Dic3—42no—
210C23—21no—
211C23C2×#6711yescharacteristic
212Q8—21no—
213Q8—122no—
214Q8—122no—
215Q8C2×Dic612yes—
216D4—24no—
217D4—21no—
218D4C2×Dic612yes—
219C2×C4—21no—
220C2×C4—24no—
221C2×C4—22no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—41no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—62no—
228C2×C4—62no—
229C2×C4C2×Dic612yes—
230C2×C4#23512yes—
231C2×C4C2×#6711yescharacteristic
232C2×C4C2×#6711yescharacteristic
233C2×C4C2×#6711yescharacteristic
234C2×C4C2×#6711yescharacteristic
235C2×C4C2×#6711yescharacteristic
236C2×C4C23×D311yescharacteristic
237C8—62no—
238C8—62no—
239C2×C3—22no—
240C2×C3—42no—
241C2×C3C2×C22⋊Q811yescharacteristic
242C2×C3C2×#13511yescharacteristic
243C2×C3#57511yescharacteristic
244C22—21no—
245C22—21no—
246C22—22no—
247C22—41no—
248C22—41no—
249C22#92412yes—
250C22C22×#6711yescharacteristic
251C4—22no—
252C4—21no—
253C4—21no—
254C4—21no—
255C4—41no—
256C4—41no—
257C4—122no—
258C4—122no—
259C4#92412yes—
260C4#95012yes—
261C4C22×Dic611yescharacteristic
262C4C2×#23511yescharacteristic
263C3#307611yescharacteristic, complemented, Sylow
264C2—22no—
265C2—42no—
266C2C2×#92411yescharacteristic
267C2C2×#95011yescharacteristic
268C2#582211yescharacteristic
subgroupG/H
centerC2×C4C2×#67
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×C4×#41C2
socleC22×C3C22×D4
prime radicalC2×C3×#41C22
hypercenterC4×#41D3
hypocenterC3#3076
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C4×#41

Complements

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

NH ≅ G/Ntype
2. #5470C2semidirect
4. #5022C2semidirect
5. #5021C2semidirect
6. #5020C2semidirect
7. #5009C2semidirect
8. #5008C2semidirect
9. #5007C2semidirect
10. #4998C2semidirect
11. C2×C4⋊(C3⋊C8)C2semidirect
56. C3×#41Q8semidirect
154. #41Dic6semidirect
263. C3#3076semidirect

Maximal subgroups

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

subgroupindex
C3×C4×#412
#54702
#52332
#50222
#50212
#50202
#50092
#50082
#50072
#49982
C2×C4⋊(C3⋊C8)2
#30763