Finite Group Databasebeta

← #82913

Subgroups — #82913

Order 384.

420 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
1C3×#525C211yesmaximal, characteristic
2C2×#926C211yesmaximal, characteristic, complemented
3#5233C212yesmaximal, complemented
4C2×#830C211yesmaximal, characteristic, complemented
5#5045C212yesmaximal, complemented
6#5044C211yesmaximal, characteristic, complemented
7#5043C211yesmaximal, characteristic, complemented
8#5035C212yesmaximal, complemented
9#5034C211yesmaximal, characteristic, complemented
10#5033C211yesmaximal, characteristic, complemented
11#4998C212yesmaximal, complemented
12#3091—31nomaximal, Sylow
13C2×C3×#41C2211yescharacteristic
14C23×C3×C4C2211yescharacteristic
15C3×#123—21no—
16C3×C22⋊Q8C2211yescharacteristic
17C3×C22⋊Q8C2211yescharacteristic
18C3×#121C2211yescharacteristic
19C3×#121C2211yescharacteristic
20C3×#120—21no—
21C3×C4×D4—21no—
22C3×C4×D4—21no—
23C3×#117C2211yescharacteristic
24C2×#217—21no—
25#926—21no—
26#926—21no—
27#926C2212yes—
28#926C2212yes—
29C2×C4⋊Dic3C2211yescharacteristic
30C2×C4⋊Dic3C2211yescharacteristic
31C2×C4×Dic3—21no—
32C22×C3⋊C8C2212yes—
33#835C2212yescomplemented
34#835C2212yescomplemented
35#832C2212yescomplemented
36#832C2212yescomplemented
37#830C2211yescharacteristic
38#830C2211yescharacteristic
39#830C2212yes—
40#808C2212yescomplemented
41#808C2212yescomplemented
42#807C2212yescomplemented
43#807C2212yescomplemented
44#525D311yescharacteristic
45C2×#117—31no—
46#485—32no—
47#484—31no—
48#483—31no—
49#482—32no—
50#481—31no—
51#480—31no—
52#427—32no—
53#416—32no—
54C2×C22⋊C8—31no—
55C24×C3D411yescharacteristic
56C3×#41—24no—
57C2×C3×Q8C2311yescharacteristic
58C2×C3×D4—21no—
59C2×C3×D4—21no—
60C2×C3×D4—21no—
61C2×C3×D4C2311yescharacteristic
62C22×C3×C4—24no—
63C22×C3×C4—21no—
64C22×C3×C4D411yescharacteristic
65C22×C3×C4D411yescharacteristic
66C22×C3×C4D411yescharacteristic
67C22×C3×C4C2311yescharacteristic
68C22×C3×C4C2311yescharacteristic
69C22×C3×C4C2311yescharacteristic
70C22×Dic3—21no—
71C22×Dic3—21no—
72C3×C4⋊C4—21no—
73C3×C4⋊C4—21no—
74C3×C4⋊C4C2311yescharacteristic
75C3×C4⋊C4C2311yescharacteristic
76C3×C22⋊C4—21no—
77C3×C22⋊C4—21no—
78C3×C22⋊C4—21no—
79C3×C22⋊C4—21no—
80C3×C22⋊C4—21no—
81C3×C42—21no—
82#217—22no—
83#217—22no—
84C4⋊Dic3—21no—
85C4⋊Dic3—21no—
86C4⋊Dic3C2312yes—
87C4⋊Dic3C2312yes—
88C4×Dic3—22no—
89C4×Dic3—22no—
90C2×C3⋊C8—24no—
91C2×C3⋊C8C2312yes—
92C2×C3⋊C8C2312yes—
93C2×#41C2×D311yescharacteristic
94C23×C4C2×D311yescharacteristic
95C22×C8—32no—
96#123—21no—
97C22⋊Q8C2×D311yescharacteristic
98C22⋊Q8C2×D311yescharacteristic
99#121C2×D311yescharacteristic
100#121C2×D311yescharacteristic
101#120—21no—
102C4×D4—21no—
103C4×D4—21no—
104#117—32no—
105#117—32no—
106#117—61no—
107#117—61no—
108#117C2×D311yescharacteristic
109C2×C4⋊C4—31no—
110C2×C4⋊C4—31no—
111C2×C22⋊C4—61no—
112C2×C42—61no—
113#107—32no—
114#107—32no—
115#106—32no—
116#106—32no—
117#103—32no—
118#103—32no—
119#102—32no—
120#102—32no—
121C22⋊C8—31no—
122C22⋊C8—31no—
123C22⋊C8—32no—
124C23×C3—22no—
125C23×C3—22no—
126C23×C3—21no—
127C23×C3—41no—
128C23×C3C2×D411yescharacteristic
129C23×C3C2×D411yescharacteristic
130C23×C3C2×D411yescharacteristic
131C3×Q8—22no—
132C3×D4—22no—
133C3×D4—42no—
134C3×D4—42no—
135C3×D4—42no—
136C2×C3×C4—22no—
137C2×C3×C4—22no—
138C2×C3×C4—22no—
139C2×C3×C4—22no—
140C2×C3×C4—22no—
141C2×C3×C4—21no—
142C2×C3×C4—21no—
143C2×C3×C4—21no—
144C2×C3×C4—42no—
145C2×C3×C4C2×D411yescharacteristic
146C2×C3×C4C2×D411yescharacteristic
147C2×C3×C4C2×D411yescharacteristic
148C2×C3×C4#4114yes—
149C2×C3×C4C2411yescharacteristic
150C2×Dic3—22no—
151C2×Dic3—22no—
152C2×Dic3—42no—
153C2×Dic3—42no—
154C3⋊C8—22no—
155C3⋊C8—22no—
156C24#6711yescharacteristic
157#41—24no—
158C2×Q8C22×D311yescharacteristic
159C2×D4—21no—
160C2×D4—21no—
161C2×D4—21no—
162C2×D4C22×D311yescharacteristic
163C22×C4—24no—
164C22×C4—21no—
165C22×C4—61no—
166C22×C4—61no—
167C22×C4#6711yescharacteristic
168C22×C4#6711yescharacteristic
169C22×C4#6711yescharacteristic
170C22×C4C22×D311yescharacteristic
171C22×C4C22×D311yescharacteristic
172C22×C4C22×D311yescharacteristic
173C2×C8—32no—
174C2×C8—32no—
175C2×C8—64no—
176C4⋊C4—21no—
177C4⋊C4—21no—
178C4⋊C4—32no—
179C4⋊C4—32no—
180C4⋊C4—61no—
181C4⋊C4—61no—
182C4⋊C4C22×D311yescharacteristic
183C4⋊C4C22×D311yescharacteristic
184C22⋊C4—21no—
185C22⋊C4—21no—
186C22⋊C4—21no—
187C22⋊C4—21no—
188C22⋊C4—21no—
189C22⋊C4—62no—
190C22⋊C4—62no—
191C42—21no—
192C42—62no—
193C42—62no—
194C22×C3—21no—
195C22×C3—21no—
196C22×C3—21no—
197C22×C3—21no—
198C22×C3—21no—
199C22×C3—21no—
200C22×C3—21no—
201C22×C3—44no—
202C22×C3—41no—
203C22×C3—41no—
204C22×C3—41no—
205C22×C3#12312yes—
206C22×C3#13512yes—
207C22×C3C22×D411yescharacteristic
208C3×C4—21no—
209C3×C4—21no—
210C3×C4—41no—
211C3×C4—41no—
212C3×C4—41no—
213C3×C4#12312yes—
214C3×C4C2×#4112yes—
215Dic3—42no—
216Dic3—42no—
217C23—22no—
218C23—22no—
219C23—21no—
220C23—41no—
221C23C2×#6711yescharacteristic
222C23C2×#6711yescharacteristic
223C23C2×#6711yescharacteristic
224Q8—22no—
225D4—22no—
226D4—42no—
227D4—42no—
228D4—42no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—42no—
238C2×C4—62no—
239C2×C4—62no—
240C2×C4—122no—
241C2×C4—122no—
242C2×C4#23714yes—
243C2×C4C2×#6711yescharacteristic
244C2×C4C2×#6711yescharacteristic
245C2×C4C2×#6711yescharacteristic
246C2×C4C23×D311yescharacteristic
247C8—62no—
248C8—62no—
249C2×C3—22no—
250C2×C3—22no—
251C2×C3—41no—
252C2×C3—81no—
253C2×C3C2×#12311yescharacteristic
254C2×C3C2×#13511yescharacteristic
255C2×C3#57511yescharacteristic
256C22—21no—
257C22—21no—
258C22—21no—
259C22—21no—
260C22—21no—
261C22—21no—
262C22—21no—
263C22—44no—
264C22—41no—
265C22—41no—
266C22—41no—
267C22#93512yes—
268C22#95012yes—
269C22C22×#6711yescharacteristic
270C4—21no—
271C4—21no—
272C4—41no—
273C4—41no—
274C4—41no—
275C4—122no—
276C4—122no—
277C4#93512yes—
278C4C2×#23712yes—
279C3#309111yescharacteristic, complemented, Sylow
280C2—22no—
281C2—22no—
282C2—41no—
283C2—81no—
284C2C2×#93511yescharacteristic
285C2C2×#95011yescharacteristic
286C2#582211yescharacteristic
subgroupG/H
centerC2×C4C2×#67
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
Fitting, prime radicalC3×#525C2
socleC22×C3C22×D4
hypercenter#525D3
hypocenterC3#3091
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
2. C2×#926C2semidirect
3. #5233C2semidirect
4. C2×#830C2semidirect
5. #5045C2semidirect
6. #5044C2semidirect
7. #5043C2semidirect
8. #5035C2semidirect
9. #5034C2semidirect
10. #5033C2semidirect
11. #4998C2semidirect
33. #835C22semidirect
34. #835C22semidirect
35. #832C22semidirect
36. #832C22semidirect
40. #808C22semidirect
41. #808C22semidirect
42. #807C22semidirect
43. #807C22semidirect
279. C3#3091semidirect

Maximal subgroups

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

subgroupindex
C3×#5252
C2×#9262
#52332
C2×#8302
#50452
#50442
#50432
#50352
#50342
#50332
#49982
#30913