Finite Group Databasebeta

← #83982

Subgroups — #83982

Order 384.

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

normal subgroups only

This group has 752 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×#1009C211yesmaximal, characteristic
2C2×C3×#137C211yesmaximal, characteristic, complemented
3#5816C211yesmaximal, characteristic, complemented
4C22×#215C211yesmaximal, characteristic, complemented
5#5233C211yesmaximal, characteristic, complemented
6#5187C214yesmaximal, complemented
7#5169C214yesmaximal, complemented
8#5131C211yesmaximal, characteristic, complemented
9#5129C211yesmaximal, characteristic, complemented
10#3009—31nomaximal, Sylow
11C2×C3×#41C2211yescharacteristic
12C22×C3×Q8C2211yescharacteristic, complemented
13#1009—24no—
14#1009—24no—
15C2×#239C2212yes—
16C2×D3×D4—22no—
17C2×D3×D4C2212yes—
18C2×#235—21no—
19C2×#235C2211yescharacteristic
20C22×D12—21no—
21C22×D12C2211yescharacteristic
22C3×#137—24no—
23C2×C3×Q16C2212yescomplemented
24C2×C3×SD16C2212yescomplemented
25C2×C3×M16C2211yescharacteristic, complemented
26#947—21no—
27#946C2214yescomplemented
28#944—21no—
29C2×#215—24no—
30C2×#215C2214yescomplemented
31#929—22no—
32#926C2211yescharacteristic, complemented
33C22×C3⋊C8C2211yescharacteristic, complemented
34#835C2212yescomplemented
35#832C2212yescomplemented
36#821C2214yescomplemented
37(C2×D3)⋊C8C2214yescomplemented
38C2×H(5,𝔽2)—31no—
39C2×#137D311yescharacteristic, complemented
40C22×SD16—31no—
41#536—31no—
42#452—34no—
43#450—34no—
44#418—31no—
45#416—31no—
46#408—31no—
47C23×D3—21no—
48C23×D3—22no—
49C3×#41—22no—
50C3×#41D414yes—
51C2×C3×Q8—21no—
52C2×C3×Q8—22no—
53C2×C3×Q8—22no—
54C2×C3×Q8C2312yes—
55C2×C3×Q8C2311yescharacteristic
56C2×C3×D4—21no—
57C2×C3×D4C2311yescharacteristic
58C22×C3×C4—21no—
59C22×C3×C4—21no—
60C22×C3×C4C2311yescharacteristic
61C2×#67—22no—
62C2×#67D412yes—
63#239—24no—
64#239—24no—
65D3×D4—216no—
66D3×D4—24no—
67D3×D4—24no—
68#235—28no—
69#235—22no—
70#235—22no—
71C2×D12—28no—
72C2×D12—22no—
73C2×D12—21no—
74C2×D12—22no—
75C2×D12D411yescharacteristic
76C2×D12D414yes—
77C2×D12C2312yes—
78C2×C4×D3—22no—
79C2×C4×D3C2312yes—
80C2×Dic6D411yescharacteristic
81C3×Q16—24no—
82C3×SD16—24no—
83C3×M16—22no—
84C2×C3×C8C2312yes—
85#217—21no—
86#215—28no—
87(C2×D3)⋊C4—24no—
88C4⋊Dic3C2312yes—
89Dic3⋊C4—24no—
90C4×Dic3—21no—
91C2×C3⋊C8—22no—
92C2×C3⋊C8C2312yes—
93H(5,𝔽2)—64no—
94H(5,𝔽2)—64no—
95C2×#41—31no—
96C2×#41—32no—
97C2×#41—61no—
98C2×#41C2×D311yescharacteristic
99C22×Q8C2×D311yescharacteristic, complemented
100C22×D4—32no—
101C22×D4—31no—
102C22×D4—61no—
103C22×D4—62no—
104#137—24no—
105C2×Q16C2×D312yescomplemented
106C2×SD16—34no—
107C2×SD16—64no—
108C2×SD16C2×D312yescomplemented
109C2×M16C2×D311yescharacteristic, complemented
110C22×C8—31no—
111C4⋊Q8—61no—
112#124—61no—
113#123—62no—
114C22⋊Q8—34no—
115#117—31no—
116#103—32no—
117#102—32no—
118#102—34no—
119C22⋊C8—34no—
120C23×C3—21no—
121C23×C3C2×D411yescharacteristic
122C22×D3—22no—
123C22×D3—28no—
124C22×D3—24no—
125C22×D3—84no—
126C22×D3—82no—
127C22×D3C2×D412yes—
128C3×Q8—21no—
129C3×Q8—22no—
130C3×Q8—22no—
131C3×Q8—42no—
132C3×Q8C2×D412yes—
133C3×D4—24no—
134C3×D4—21no—
135C3×D4C2×D412yes—
136C2×C3×C4—21no—
137C2×C3×C4—24no—
138C2×C3×C4—22no—
139C2×C3×C4—42no—
140C2×C3×C4C2×D411yescharacteristic
141C2×C3×C4C2×D412yes—
142C2×C3×C4C2×D412yes—
143C2×C3×C4C2411yescharacteristic
144#67—28no—
145#67—24no—
146C2×Dic3—22no—
147C2×Dic3C2×D412yes—
148D12—28no—
149D12—22no—
150D12—24no—
151D12—22no—
152D12C2×D414yes—
153C4×D3—28no—
154C4×D3—22no—
155C4×D3—22no—
156Dic6—22no—
157C3×C8—22no—
158C3⋊C8—22no—
159C24—62no—
160C24—61no—
161#41—22no—
162#41—68no—
163#41—64no—
164#41—64no—
165#41—62no—
166#41—62no—
167#41#6714yes—
168C2×Q8—21no—
169C2×Q8—22no—
170C2×Q8—22no—
171C2×Q8—31no—
172C2×Q8C22×D312yes—
173C2×Q8C22×D311yescharacteristic
174C2×D4—21no—
175C2×D4—32no—
176C2×D4—34no—
177C2×D4—31no—
178C2×D4—32no—
179C2×D4—62no—
180C2×D4—68no—
181C2×D4—62no—
182C2×D4—64no—
183C2×D4—616no—
184C2×D4—64no—
185C2×D4—61no—
186C2×D4—62no—
187C2×D4C22×D311yescharacteristic
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—32no—
191C22×C4—62no—
192C22×C4C22×D311yescharacteristic
193Q16—24no—
194SD16—24no—
195SD16—68no—
196M16—22no—
197C2×C8—32no—
198C2×C8—62no—
199C2×C8C22×D312yes—
200C4⋊C4—32no—
201C4⋊C4—64no—
202C22⋊C4—64no—
203C22⋊C4—61no—
204C42—61no—
205C22×C3—22no—
206C22×C3—22no—
207C22×C3—42no—
208C22×C3#12012yes—
209C22×C3C22×D411yescharacteristic
210C2×D3—22no—
211C2×D3—24no—
212C2×D3—24no—
213C2×D3—44no—
214C2×D3—84no—
215C2×D3—88no—
216C3×C4—22no—
217C3×C4—42no—
218C3×C4#12012yes—
219C3×C4C22×D411yescharacteristic
220C3×C4C22×D411yescharacteristic
221Dic3—22no—
222Dic3—42no—
223C23—21no—
224C23—32no—
225C23—68no—
226C23—62no—
227C23—64no—
228C23—242no—
229C23—244no—
230C23D3×D411yescharacteristic
231Q8—21no—
232Q8—22no—
233Q8—22no—
234Q8—42no—
235Q8—62no—
236Q8C2×#6712yes—
237D4—24no—
238D4—21no—
239D4—34no—
240D4—68no—
241D4—68no—
242D4—64no—
243D4—64no—
244D4—62no—
245D4—62no—
246D4C2×#6712yes—
247C2×C4—21no—
248C2×C4—24no—
249C2×C4—22no—
250C2×C4—32no—
251C2×C4—42no—
252C2×C4—68no—
253C2×C4—62no—
254C2×C4—62no—
255C2×C4—62no—
256C2×C4D3×D411yescharacteristic
257C2×C4D3×D412yes—
258C2×C4C2×#6712yes—
259C2×C4C23×D311yescharacteristic
260C8—22no—
261C8—62no—
262C2×C3—22no—
263C2×C3—42no—
264C2×C3C2×#12011yescharacteristic
265C2×C3#57712yes—
266D3—42no—
267D3—44no—
268C22—22no—
269C22—22no—
270C22—42no—
271C22—64no—
272C22—64no—
273C22—62no—
274C22—124no—
275C22—244no—
276C22—248no—
277C22#93712yes—
278C22C2×D3×D411yescharacteristic
279C4—22no—
280C4—42no—
281C4—62no—
282C4—122no—
283C4#93712yes—
284C4C2×D3×D411yescharacteristic
285C4C22×#6711yescharacteristic
286C3#300911yescharacteristic, complemented, Sylow
287C2—22no—
288C2—42no—
289C2—122no—
290C2—124no—
291C2#577812yes—
292C2C2×#93711yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC2×C3×#137C2
socleC22×C3C22×D4
prime radicalC2×#1009C2
hypercenterC2×#137D3
hypocenterC3#3009
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
2. C2×C3×#137C2semidirect
3. #5816C2semidirect
4. C22×#215C2semidirect
5. #5233C2semidirect
6. #5187C2semidirect
7. #5169C2semidirect
8. #5131C2semidirect
9. #5129C2semidirect
12. C22×C3×Q8C22semidirect
23. C2×C3×Q16C22semidirect
24. C2×C3×SD16C22semidirect
25. C2×C3×M16C22semidirect
27. #946C22semidirect
30. C2×#215C22semidirect
32. #926C22semidirect
33. C22×C3⋊C8C22semidirect
34. #835C22semidirect
35. #832C22semidirect
36. #821C22semidirect
37. (C2×D3)⋊C8C22semidirect
39. C2×#137D3semidirect
99. C22×Q8C2×D3semidirect
105. C2×Q16C2×D3semidirect
108. C2×SD16C2×D3semidirect
109. C2×M16C2×D3semidirect
286. C3#3009semidirect

Maximal subgroups

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

subgroupindex
C2×#10092
C2×C3×#1372
#58162
C22×#2152
#52332
#51872
#51692
#51312
#51292
#30093