Finite Group Databasebeta

← #83979

Subgroups — #83979

Order 384.

738 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 738 conjugacy classes of subgroups — too many to draw. Only its 153 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
1C22×D3×Q8C211yesmaximal, characteristic, complemented
2C2×C3×#137C211yesmaximal, characteristic, complemented
3#5828C211yesmaximal, characteristic
4C2×#951C211yesmaximal, characteristic, complemented
5#5225C211yesmaximal, characteristic, complemented
6#5186C214yesmaximal, complemented
7#5170C214yesmaximal, complemented
8#5126C211yesmaximal, characteristic, complemented
9#5120C211yesmaximal, characteristic, complemented
10#3004—31nomaximal, Sylow
11C2×C3×#41C2211yescharacteristic
12C22×C3×Q8C2211yescharacteristic
13C2×D3×Q8—24no—
14C2×D3×Q8—22no—
15C2×D3×Q8—24no—
16C2×D3×Q8C2214yes—
17C22×C4×D3—21no—
18C22×C4×D3C2211yescharacteristic
19C22×Dic6—21no—
20C22×Dic6C2211yescharacteristic
21C3×#137—24no—
22C2×C3×Q16C2212yescomplemented
23C2×C3×SD16C2212yescomplemented
24C2×C3×M16C2211yescharacteristic
25#951—24no—
26#946C2212yes—
27C2×#216C2212yescomplemented
28#938C2212yes—
29#937—21no—
30#935—21no—
31C2×#214C2212yescomplemented
32C4×#67—22no—
33#926C2211yescharacteristic
34C2×#208C2211yescharacteristic
35#835C2214yescomplemented
36(C2×D3)⋊C8C2214yes—
37#816C2214yescomplemented
38C23×Q8—31no—
39C2×#137—31no—
40C2×#137D311yescharacteristic, complemented
41#525—31no—
42#451—34no—
43#448—34no—
44#419—31no—
45#419—31no—
46#407—31no—
47C23×D3D411yescharacteristic
48C3×#41—22no—
49C3×#41—22no—
50C2×C3×Q8—22no—
51C2×C3×Q8—21no—
52C2×C3×Q8D414yescomplemented
53C2×C3×Q8C2311yescharacteristic
54C2×C3×Q8C2312yes—
55C2×C3×D4—21no—
56C2×C3×D4C2311yescharacteristic
57C22×C3×C4—21no—
58C22×C3×C4—21no—
59C22×C3×C4C2311yescharacteristic
60C2×#67—22no—
61C22×Dic3—21no—
62C22×Dic3D411yescharacteristic
63D3×Q8—216no—
64D3×Q8—28no—
65D3×Q8—28no—
66C2×C4×D3—28no—
67C2×C4×D3—24no—
68C2×C4×D3—22no—
69C2×C4×D3—22no—
70C2×C4×D3D412yes—
71C2×C4×D3C2312yes—
72C2×Dic6—28no—
73C2×Dic6—21no—
74C2×Dic6—22no—
75C2×Dic6—24no—
76C2×Dic6D414yescomplemented
77C2×Dic6C2312yes—
78C3×Q16—24no—
79C3×SD16—24no—
80C3×M16—22no—
81C2×C3×C8C2312yes—
82#217—21no—
83#217—22no—
84#216—24no—
85#214—24no—
86(C2×D3)⋊C4—22no—
87C4⋊Dic3C2312yes—
88Dic3⋊C4—22no—
89C4×Dic3—21no—
90#208—22no—
91C2×C3⋊C8C2312yes—
92C2×#41C2×D311yescharacteristic
93C22×Q8—34no—
94C22×Q8—31no—
95C22×Q8—61no—
96C22×Q8—62no—
97C22×Q8—64no—
98C22×Q8—64no—
99C22×Q8C2×D311yescharacteristic
100C23×C4—31no—
101C23×C4—61no—
102#137—24no—
103#137—64no—
104C2×Q16—32no—
105C2×Q16C2×D312yescomplemented
106C2×SD16—32no—
107C2×SD16C2×D312yescomplemented
108C2×M16—31no—
109C2×M16C2×D311yescharacteristic
110#123—61no—
111C22⋊Q8—32no—
112#121—32no—
113#120—61no—
114C4×D4—62no—
115#117—31no—
116#103—34no—
117#103—34no—
118C22⋊C8—34no—
119C23×C3—21no—
120C23×C3C2×D411yescharacteristic
121C22×D3—22no—
122C22×D3—24no—
123C22×D3C2×D412yes—
124C3×Q8—24no—
125C3×Q8—22no—
126C3×Q8—21no—
127C3×Q8—21no—
128C3×Q8C2×D414yes—
129C3×D4—21no—
130C3×D4—21no—
131C3×D4—42no—
132C2×C3×C4—24no—
133C2×C3×C4—22no—
134C2×C3×C4—21no—
135C2×C3×C4—42no—
136C2×C3×C4C2×D411yescharacteristic
137C2×C3×C4C2×D412yes—
138C2×C3×C4C2×D412yes—
139C2×C3×C4C2411yescharacteristic
140#67—44no—
141C2×Dic3—22no—
142C2×Dic3—24no—
143C2×Dic3—22no—
144C2×Dic3—22no—
145C2×Dic3C2×D412yes—
146C4×D3—216no—
147C4×D3—24no—
148C4×D3—22no—
149C4×D3—22no—
150Dic6—216no—
151Dic6—24no—
152Dic6—22no—
153Dic6C2×D414yes—
154C3×C8—22no—
155C3⋊C8—22no—
156C24—31no—
157#41—22no—
158#41—22no—
159C2×Q8—22no—
160C2×Q8—21no—
161C2×Q8—34no—
162C2×Q8—32no—
163C2×Q8—64no—
164C2×Q8—68no—
165C2×Q8—68no—
166C2×Q8—616no—
167C2×Q8—68no—
168C2×Q8—61no—
169C2×Q8—62no—
170C2×Q8#6714yescomplemented
171C2×Q8C22×D311yescharacteristic
172C2×Q8C22×D312yes—
173C2×D4—21no—
174C2×D4—62no—
175C2×D4C22×D311yescharacteristic
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—31no—
179C22×C4—32no—
180C22×C4—32no—
181C22×C4—64no—
182C22×C4—68no—
183C22×C4—61no—
184C22×C4—62no—
185C22×C4—62no—
186C22×C4C22×D311yescharacteristic
187Q16—24no—
188Q16—64no—
189SD16—24no—
190SD16—64no—
191M16—22no—
192M16—62no—
193C2×C8—32no—
194C2×C8C22×D312yes—
195C4⋊C4—32no—
196C4⋊C4—62no—
197C22⋊C4—62no—
198C22⋊C4—62no—
199C22⋊C4—61no—
200C42—61no—
201C22×C3—22no—
202C22×C3—41no—
203C22×C3—42no—
204C22×C3#12012yes—
205C22×C3C22×D411yescharacteristic
206C2×D3—28no—
207C2×D3—22no—
208C2×D3—22no—
209C2×D3#13714yescomplemented
210C3×C4—24no—
211C3×C4—41no—
212C3×C4#12012yes—
213C3×C4C22×D411yescharacteristic
214C3×C4C22×D411yescharacteristic
215Dic3—22no—
216Dic3—24no—
217Dic3—42no—
218C23—21no—
219C23—32no—
220C23—64no—
221C23—62no—
222C23D3×D411yescharacteristic
223Q8—24no—
224Q8—22no—
225Q8—21no—
226Q8—21no—
227Q8—34no—
228Q8—616no—
229Q8—64no—
230Q8—62no—
231Q8C2×#6714yes—
232D4—21no—
233D4—21no—
234D4—42no—
235D4—124no—
236C2×C4—24no—
237C2×C4—22no—
238C2×C4—21no—
239C2×C4—32no—
240C2×C4—42no—
241C2×C4—616no—
242C2×C4—62no—
243C2×C4—64no—
244C2×C4—62no—
245C2×C4—64no—
246C2×C4—62no—
247C2×C4—62no—
248C2×C4—62no—
249C2×C4D3×D411yescharacteristic
250C2×C4D3×D412yes—
251C2×C4C2×#6712yes—
252C2×C4C23×D311yescharacteristic
253C8—22no—
254C8—62no—
255C2×C3—22no—
256C2×C3—81no—
257C2×C3C2×#12011yescharacteristic
258C2×C3C2×#13712yes—
259D3—24no—
260C22—22no—
261C22—34no—
262C22—42no—
263C22—41no—
264C22—68no—
265C22—62no—
266C22—62no—
267C22#93712yes—
268C22C2×D3×D411yescharacteristic
269C4—24no—
270C4—41no—
271C4—62no—
272C4—64no—
273C4—122no—
274C4#93712yes—
275C4C2×D3×D411yescharacteristic
276C4C22×#6711yescharacteristic
277C3#300411yescharacteristic, complemented, Sylow
278C2—22no—
279C2—64no—
280C2—81no—
281C2D3×#13712yes—
282C2C2×#93711yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC2×C3×#137C2
socleC22×C3C22×D4
prime radical#5828C2
hypercenterC2×#137D3
hypocenterC3#3004
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
1. C22×D3×Q8C2semidirect
2. C2×C3×#137C2semidirect
4. C2×#951C2semidirect
5. #5225C2semidirect
6. #5186C2semidirect
7. #5170C2semidirect
8. #5126C2semidirect
9. #5120C2semidirect
22. C2×C3×Q16C22semidirect
23. C2×C3×SD16C22semidirect
27. C2×#216C22semidirect
31. C2×#214C22semidirect
35. #835C22semidirect
37. #816C22semidirect
40. C2×#137D3semidirect
52. C2×C3×Q8D4semidirect
76. C2×Dic6D4semidirect
105. C2×Q16C2×D3semidirect
107. C2×SD16C2×D3semidirect
170. C2×Q8#67semidirect
209. C2×D3#137semidirect
277. C3#3004semidirect

Maximal subgroups

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

subgroupindex
C22×D3×Q82
C2×C3×#1372
#58282
C2×#9512
#52252
#51862
#51702
#51262
#51202
#30043