Finite Group Databasebeta

← #83390

Subgroups — #83390

Order 384.

732 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 732 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C2×#1010C211yesmaximal, characteristic, complemented
2C3×#535C211yesmaximal, characteristic, complemented
3C22×#214C211yesmaximal, characteristic, complemented
4C2×#932C211yesmaximal, characteristic, complemented
5#5234C211yesmaximal, characteristic, complemented
6#5040C214yesmaximal, complemented
7#5037C214yesmaximal, complemented
8#5004C211yesmaximal, characteristic, complemented
9#5002C211yesmaximal, characteristic, complemented
10#3010—31nomaximal, Sylow
11C2×C3×#41C2211yescharacteristic
12C22×C3×D4C2211yescharacteristic
13#1010—24no—
14#1010—24no—
15C2×D3×Q8—21no—
16C2×#237—21no—
17C2×#237—22no—
18C2×#235C2211yescharacteristic
19C2×#235C2212yescomplemented
20C22×Dic6C2211yescharacteristic
21C22×Dic6C2212yescomplemented
22C3×#127—21no—
23C3×#124—21no—
24C3×#121C2214yes—
25C3×#120—22no—
26C3×#117C2211yescharacteristic
27C2×#214—24no—
28C2×#214C2212yes—
29C2×#214C2214yes—
30#932—24no—
31C2×#213C2212yes—
32C2×#208C2211yescharacteristic
33C22×C3⋊C8C2211yescharacteristic
34#830C2214yescomplemented
35#810C2212yescomplemented
36#810C2214yescomplemented
37#809C2212yescomplemented
38C2×2−1+4—31no—
39C2×AGL(1,ℤ/8)—31no—
40C22×SD16—31no—
41#535D311yescharacteristic, complemented
42#449—34no—
43#448—34no—
44#419—31no—
45#416—31no—
46#408—31no—
47C24×C3—21no—
48C3×#41—22no—
49C3×#41—22no—
50C2×C3×Q8D411yescharacteristic
51C2×C3×D4—24no—
52C2×C3×D4—22no—
53C2×C3×D4—22no—
54C2×C3×D4—21no—
55C2×C3×D4D412yes—
56C2×C3×D4D411yescharacteristic
57C2×C3×D4C2312yes—
58C22×C3×C4C2312yes—
59C22×C3×C4C2311yescharacteristic
60C2×#67—21no—
61C2×#67—22no—
62C22×Dic3—21no—
63C22×Dic3—22no—
64D3×Q8—28no—
65#237—216no—
66#237—28no—
67#235—22no—
68#235—24no—
69#235—24no—
70#235D414yescomplemented
71C2×D12C2311yescharacteristic
72C2×C4×D3—21no—
73C2×C4×D3—22no—
74C2×Dic6—21no—
75C2×Dic6—22no—
76C2×Dic6—22no—
77C2×Dic6—24no—
78C2×Dic6—24no—
79C2×Dic6D414yescomplemented
80C2×Dic6C2312yes—
81C2×Dic6C2311yescharacteristic
82C3×C4⋊C4C2312yes—
83C3×C22⋊C4—21no—
84C3×C22⋊C4—24no—
85C3×C42—21no—
86#214—24no—
87#214—28no—
88#213—24no—
89#208—22no—
90C2×C3⋊C8—22no—
91C2×C3⋊C8C2312yes—
92C2×C3⋊C8C2312yes—
932−1+4—64no—
942−1+4—64no—
95C2×#41—31no—
96C2×#41—32no—
97C2×#41—62no—
98C2×#41—61no—
99C2×#41C2×D311yescharacteristic
100C22×Q8—31no—
101C22×Q8—32no—
102C22×Q8—61no—
103C22×D4C2×D311yescharacteristic
104AGL(1,ℤ/8)—64no—
105C2×SD16—34no—
106C2×SD16—32no—
107C2×SD16—64no—
108C2×D8—32no—
109C2×M16—31no—
110C22×C8—31no—
111#127—21no—
112#124—21no—
113#121C2×D314yes—
114#120—22no—
115#117C2×D311yescharacteristic
116#103—32no—
117#103—34no—
118#102—32no—
119C22⋊C8—34no—
120C23×C3—22no—
121C23×C3—42no—
122C23×C3—82no—
123C23×C3C2×D412yes—
124C23×C3C2×D411yescharacteristic
125C22×D3—21no—
126C3×Q8—22no—
127C3×D4—24no—
128C3×D4—22no—
129C3×D4—22no—
130C3×D4—22no—
131C3×D4—42no—
132C3×D4—84no—
133C2×C3×C4—22no—
134C2×C3×C4—22no—
135C2×C3×C4—22no—
136C2×C3×C4C2×D412yes—
137C2×C3×C4C2×D412yes—
138C2×C3×C4C2×D412yes—
139C2×C3×C4C2×D411yescharacteristic
140C2×C3×C4C2411yescharacteristic
141#67—28no—
142#67—24no—
143C2×Dic3—28no—
144C2×Dic3—24no—
145C2×Dic3—22no—
146C2×Dic3—21no—
147D12—21no—
148D12C2×D412yes—
149C4×D3—28no—
150C4×D3—24no—
151Dic6—28no—
152Dic6—24no—
153Dic6—22no—
154Dic6—21no—
155Dic6C2×D412yes—
156Dic6C2×D414yes—
157C3⋊C8—22no—
158C3⋊C8—22no—
159C24—21no—
160#41—22no—
161#41—22no—
162#41—34no—
163#41—64no—
164#41—64no—
165#41—616no—
166#41—68no—
167#41—62no—
168C2×Q8—34no—
169C2×Q8—32no—
170C2×Q8—31no—
171C2×Q8—64no—
172C2×Q8—64no—
173C2×Q8—68no—
174C2×Q8—62no—
175C2×Q8—62no—
176C2×Q8—61no—
177C2×Q8#6711yescharacteristic
178C2×D4—24no—
179C2×D4—22no—
180C2×D4—22no—
181C2×D4—21no—
182C2×D4—31no—
183C2×D4—62no—
184C2×D4—61no—
185C2×D4#6712yes—
186C2×D4#6711yescharacteristic
187C2×D4C22×D312yes—
188C22×C4—62no—
189C22×C4—61no—
190C22×C4—62no—
191C22×C4—61no—
192C22×C4C22×D312yes—
193C22×C4C22×D311yescharacteristic
194SD16—68no—
195SD16—64no—
196D8—64no—
197M16—62no—
198C2×C8—32no—
199C2×C8—32no—
200C2×C8—62no—
201C4⋊C4C22×D312yes—
202C22⋊C4—21no—
203C22⋊C4—24no—
204C42—21no—
205C22×C3—22no—
206C22×C3—22no—
207C22×C3—24no—
208C22×C3—42no—
209C22×C3—44no—
210C22×C3—84no—
211C22×C3#12012yes—
212C22×C3C22×D411yescharacteristic
213C2×D3—22no—
214C2×D3—42no—
215C3×C4—22no—
216C3×C4—42no—
217C3×C4#12012yes—
218C3×C4C22×D411yescharacteristic
219C3×C4C22×D411yescharacteristic
220Dic3—22no—
221Dic3—24no—
222C23—22no—
223C23—42no—
224C23—61no—
225C23—82no—
226C23C2×#6712yes—
227C23C2×#6711yescharacteristic
228Q8—22no—
229Q8—34no—
230Q8—32no—
231Q8—68no—
232Q8—64no—
233Q8—62no—
234Q8—61no—
235D4—24no—
236D4—22no—
237D4—22no—
238D4—22no—
239D4—32no—
240D4—42no—
241D4—68no—
242D4—64no—
243D4—61no—
244D4—84no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—68no—
249C2×C4—68no—
250C2×C4—64no—
251C2×C4—62no—
252C2×C4—61no—
253C2×C4—64no—
254C2×C4D3×D412yes—
255C2×C4D3×D412yes—
256C2×C4C2×#6712yes—
257C2×C4C2×#6711yescharacteristic
258C2×C4C23×D311yescharacteristic
259C8—62no—
260C8—62no—
261C2×C3—22no—
262C2×C3—42no—
263C2×C3—82no—
264C2×C3C2×#12011yescharacteristic
265C2×C3#57712yes—
266D3—42no—
267C22—22no—
268C22—22no—
269C22—24no—
270C22—42no—
271C22—44no—
272C22—62no—
273C22—84no—
274C22—122no—
275C22#93712yes—
276C22C22×#6711yescharacteristic
277C4—22no—
278C4—42no—
279C4—64no—
280C4—62no—
281C4#93712yes—
282C4C2×D3×D411yescharacteristic
283C4C2×D3×D411yescharacteristic
284C3#301011yescharacteristic, complemented, Sylow
285C2—22no—
286C2—42no—
287C2—82no—
288C2—122no—
289C2C2×#93711yescharacteristic
290C2#583612yes—
subgroupG/H
centerC22C22×#67
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#535C2
socleC22×C3C22×D4
hypercenter#535D3
hypocenterC3#3010
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×#41 ◁ #535

Complements

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

NH ≅ G/Ntype
1. C2×#1010C2semidirect
2. C3×#535C2semidirect
3. C22×#214C2semidirect
4. C2×#932C2semidirect
5. #5234C2semidirect
6. #5040C2semidirect
7. #5037C2semidirect
8. #5004C2semidirect
9. #5002C2semidirect
19. C2×#235C22semidirect
21. C22×Dic6C22semidirect
34. #830C22semidirect
35. #810C22semidirect
36. #810C22semidirect
37. #809C22semidirect
41. #535D3semidirect
70. #235D4semidirect
79. C2×Dic6D4semidirect
284. C3#3010semidirect

Maximal subgroups

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

subgroupindex
C2×#10102
C3×#5352
C22×#2142
C2×#9322
#52342
#50402
#50372
#50042
#50022
#30103