Finite Group Databasebeta

← #83402

Subgroups — #83402

Order 384.

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

normal subgroups only

This group has 564 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
1C3×#536C211yesmaximal, characteristic, complemented
2C2×#950C211yesmaximal, characteristic, complemented
3C2×#949C211yesmaximal, characteristic, complemented
4C2×#942C211yesmaximal, characteristic, complemented
5C22×#215C211yesmaximal, characteristic, complemented
6#5525C211yesmaximal, characteristic, complemented
7#5086C212yesmaximal, complemented
8#5085C212yesmaximal, complemented
9#5059C212yesmaximal, complemented
10#5058C212yesmaximal, complemented
11#4997C211yesmaximal, characteristic, complemented
12#3154—31nomaximal, Sylow
13C2×C3×#41C2211yescharacteristic
14C22×C3×Q8C2211yescharacteristic
15C2×#235C2211yescharacteristic
16C22×D12C2211yescharacteristic
17C3×C4⋊Q8C2212yescomplemented
18C3×#124C2212yescomplemented
19C3×#123—22no—
20C3×C22⋊Q8—22no—
21C3×#117C2211yescharacteristic
22#950—24no—
23#949—24no—
24C2×#216C2212yescomplemented
25C2×#216C2211yescharacteristic, complemented
26#942—24no—
27C2×#215—24no—
28C2×#215C2212yes—
29C2×#215C2212yes—
30C2×#215C2212yes—
31C2×#215C2212yescomplemented
32C2×#215C2211yescharacteristic, complemented
33C2×#214C2211yescharacteristic, complemented
34C2×#213C2212yes—
35C2×#213C2211yescharacteristic, complemented
36C2×#208C2211yescharacteristic
37C22×C3⋊C8C2211yescharacteristic
38#895—22no—
39#882—22no—
40#875C2212yes—
41#874C2212yes—
42#803C2212yescomplemented
43C4×C3⋊C8C2212yescomplemented
44C2×#137—31no—
45C2×AGL(1,ℤ/8)—31no—
46C2×#135—31no—
47C22×SD16—31no—
48#536D311yescharacteristic, complemented
49#535—31no—
50#497—32no—
51#496—32no—
52#495—32no—
53#492—32no—
54#405—31no—
55C23×D3—21no—
56C3×#41—22no—
57C3×#41—22no—
58C2×C3×Q8—21no—
59C2×C3×Q8—22no—
60C2×C3×Q8—22no—
61C2×C3×Q8C2312yes—
62C2×C3×Q8C2311yescharacteristic
63C2×C3×D4—21no—
64C2×C3×D4C2311yescharacteristic
65C22×C3×C4—21no—
66C22×C3×C4—21no—
67C22×C3×C4C2311yescharacteristic
68C2×#67—21no—
69#235—22no—
70#235—22no—
71C2×D12—22no—
72C2×D12—22no—
73C2×D12—21no—
74C2×D12—24no—
75C2×D12C2311yescharacteristic
76C2×D12C2312yes—
77C2×C4×D3—21no—
78C2×Dic6C2311yescharacteristic
79C3×C4⋊C4—24no—
80C3×C4⋊C4D412yes—
81C3×C22⋊C4—24no—
82C3×C22⋊C4D412yes—
83C3×C42C2312yes—
84#216—24no—
85#216—22no—
86#215—24no—
87#215—24no—
88#215—24no—
89#215—24no—
90#215—22no—
91#214—22no—
92#213—24no—
93#213—22no—
94(C2×D3)⋊C4—24no—
95#208D414yescomplemented
96C2×C3⋊C8D414yescomplemented
97C2×C3⋊C8C2311yescharacteristic
98C2×C3⋊C8C2312yes—
99C2×C3⋊C8C2311yescharacteristic
100C2×#41—31no—
101C2×#41C2×D311yescharacteristic
102C22×Q8C2×D311yescharacteristic
103C22×D4—31no—
104#137—64no—
105AGL(1,ℤ/8)—64no—
106#135—64no—
107C2×Q16—31no—
108C2×Q16—32no—
109C2×SD16—32no—
110C2×SD16—32no—
111C2×SD16—31no—
112C2×SD16—32no—
113C2×SD16—32no—
114C2×SD16—31no—
115C2×SD16—64no—
116C2×D8—31no—
117C2×D8—32no—
118C2×M16—31no—
119C22×C8—31no—
120C4⋊Q8C2×D312yescomplemented
121#127—32no—
122#124—32no—
123#124C2×D312yescomplemented
124#123—22no—
125C22⋊Q8—22no—
126#121—62no—
127#120—62no—
128#117C2×D311yescharacteristic
129#97—32no—
130C4×C8—32no—
131C23×C3—21no—
132C23×C3C2×D411yescharacteristic
133C22×D3—22no—
134C22×D3—21no—
135C22×D3—42no—
136C22×D3—82no—
137C3×Q8—22no—
138C3×Q8—21no—
139C3×Q8—22no—
140C3×Q8—21no—
141C3×Q8—42no—
142C3×D4—21no—
143C3×D4—21no—
144C3×D4—42no—
145C2×C3×C4—22no—
146C2×C3×C4—21no—
147C2×C3×C4—42no—
148C2×C3×C4—42no—
149C2×C3×C4C2×D412yes—
150C2×C3×C4C2×D412yes—
151C2×C3×C4C2×D414yes—
152C2×C3×C4C2×D411yescharacteristic
153C2×C3×C4C2411yescharacteristic
154#67—42no—
155C2×Dic3—21no—
156D12—22no—
157D12—21no—
158D12—22no—
159D12—21no—
160D12—42no—
161D12—84no—
162C4×D3—42no—
163Dic6—21no—
164Dic6—21no—
165C3⋊C8C2×D412yes—
166C3⋊C8C2×D412yes—
167C3⋊C8C2×D414yes—
168C24—61no—
169#41—22no—
170#41—22no—
171#41—62no—
172#41—62no—
173C2×Q8—21no—
174C2×Q8—22no—
175C2×Q8—22no—
176C2×Q8—31no—
177C2×Q8C22×D312yes—
178C2×Q8C22×D311yescharacteristic
179C2×D4—21no—
180C2×D4—32no—
181C2×D4—31no—
182C2×D4—61no—
183C2×D4—62no—
184C2×D4—62no—
185C2×D4—61no—
186C2×D4—64no—
187C2×D4C22×D311yescharacteristic
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—61no—
191C22×C4C22×D311yescharacteristic
192Q16—62no—
193Q16—64no—
194SD16—62no—
195SD16—64no—
196SD16—64no—
197SD16—62no—
198SD16—64no—
199SD16—64no—
200D8—62no—
201D8—64no—
202M16—34no—
203C2×C8—34no—
204C2×C8—31no—
205C2×C8—32no—
206C2×C8—31no—
207C4⋊C4—24no—
208C4⋊C4#6712yes—
209C22⋊C4—24no—
210C22⋊C4—64no—
211C22⋊C4#6712yes—
212C42C22×D312yes—
213C22×C3—22no—
214C22×C3—41no—
215C22×C3—42no—
216C22×C3#12712yes—
217C22×C3C22×D411yescharacteristic
218C2×D3—41no—
219C2×D3—42no—
220C2×D3—42no—
221C2×D3—44no—
222C2×D3—84no—
223C3×C4—24no—
224C3×C4—41no—
225C3×C4—42no—
226C3×C4#12712yes—
227C3×C4C22×D411yescharacteristic
228C3×C4C22×D411yescharacteristic
229Dic3—41no—
230C23—21no—
231C23—62no—
232C23—61no—
233C23—122no—
234C23—242no—
235C23C2×#6711yescharacteristic
236Q8—22no—
237Q8—21no—
238Q8—22no—
239Q8—21no—
240Q8—42no—
241Q8—61no—
242Q8—61no—
243D4—21no—
244D4—21no—
245D4—42no—
246D4—62no—
247D4—61no—
248D4—62no—
249D4—61no—
250D4—122no—
251D4—122no—
252D4—244no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—61no—
258C2×C4—122no—
259C2×C4D3×D412yes—
260C2×C4D3×D412yes—
261C2×C4C2×#6714yes—
262C2×C4C2×#6711yescharacteristic
263C2×C4C23×D311yescharacteristic
264C8—32no—
265C8—32no—
266C8—34no—
267C2×C3—22no—
268C2×C3—81no—
269C2×C3C2×#12711yescharacteristic
270C2×C3#57712yes—
271D3—81no—
272D3—82no—
273C22—22no—
274C22—41no—
275C22—42no—
276C22—121no—
277C22—122no—
278C22—122no—
279C22—124no—
280C22—244no—
281C22#94012yes—
282C22C22×#6711yescharacteristic
283C4—24no—
284C4—41no—
285C4—42no—
286C4—121no—
287C4#94012yes—
288C4C2×D3×D411yescharacteristic
289C4C2×D3×D411yescharacteristic
290C3#315411yescharacteristic, complemented, Sylow
291C2—22no—
292C2—81no—
293C2—242no—
294C2—241no—
295C2C2×#94011yescharacteristic
296C2#583712yes—
subgroupG/H
centerC22C22×#67
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#536C2
socleC22×C3C22×D4
hypercenter#536D3
hypocenterC3#3154
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ #117 ◁ #536

Complements

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

NH ≅ G/Ntype
1. C3×#536C2semidirect
2. C2×#950C2semidirect
3. C2×#949C2semidirect
4. C2×#942C2semidirect
5. C22×#215C2semidirect
6. #5525C2semidirect
7. #5086C2semidirect
8. #5085C2semidirect
9. #5059C2semidirect
10. #5058C2semidirect
11. #4997C2semidirect
17. C3×C4⋊Q8C22semidirect
18. C3×#124C22semidirect
24. C2×#216C22semidirect
25. C2×#216C22semidirect
31. C2×#215C22semidirect
32. C2×#215C22semidirect
33. C2×#214C22semidirect
35. C2×#213C22semidirect
42. #803C22semidirect
43. C4×C3⋊C8C22semidirect
48. #536D3semidirect
95. #208D4semidirect
96. C2×C3⋊C8D4semidirect
120. C4⋊Q8C2×D3semidirect
123. #124C2×D3semidirect
290. C3#3154semidirect

Maximal subgroups

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

subgroupindex
C3×#5362
C2×#9502
C2×#9492
C2×#9422
C22×#2152
#55252
#50862
#50852
#50592
#50582
#49972
#31543