Finite Group Databasebeta

← #83382

Subgroups — #83382

Order 384.

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

normal subgroups only

This group has 596 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×#535C211yesmaximal, characteristic, complemented
2C2×#950C211yesmaximal, characteristic, complemented
3C2×#949C211yesmaximal, characteristic, complemented
4C2×#932C211yesmaximal, characteristic, complemented
5C22×#213C211yesmaximal, characteristic, complemented
6#5525C211yesmaximal, characteristic, complemented
7#5074C212yesmaximal, complemented
8#5073C212yesmaximal, complemented
9#5059C212yesmaximal, complemented
10#5058C212yesmaximal, complemented
11#4997C211yesmaximal, characteristic, complemented
12#3155—31nomaximal, Sylow
13C2×C3×#41C2211yescharacteristic
14C22×C3×D4C2211yescharacteristic
15C2×#235C2211yescharacteristic
16C22×D12C2211yescharacteristic
17C3×#127C2212yescomplemented
18C3×#124C2212yescomplemented
19C3×#121—22no—
20C3×#120—22no—
21C3×#117C2211yescharacteristic
22#950—24no—
23#949—24no—
24C2×#216C2211yescharacteristic, complemented
25C2×#215C2212yescomplemented
26C2×#215C2211yescharacteristic, complemented
27C2×#214C2212yescomplemented
28C2×#214C2211yescharacteristic, complemented
29#932—24no—
30C2×#213—24no—
31C2×#213C2212yes—
32C2×#213C2212yes—
33C2×#213C2212yescomplemented
34C2×#213C2212yescomplemented
35C2×#213C2211yescharacteristic, complemented
36C2×#208C2211yescharacteristic
37C22×C3⋊C8C2211yescharacteristic
38#895—22no—
39#882—22no—
40#875C2212yescomplemented
41#874C2212yescomplemented
42#803C2212yescomplemented
43C4×C3⋊C8C2212yescomplemented
44C2×AGL(1,ℤ/8)—31no—
45C2×AGL(1,ℤ/8)—31no—
46C2×#135—31no—
47C22×D8—31no—
48#535—31no—
49#535D311yescharacteristic, complemented
50#496—32no—
51#496—32no—
52#495—32no—
53#493—32no—
54#405—31no—
55C24×C3—21no—
56C23×D3—21no—
57C3×#41—22no—
58C3×#41—22no—
59C2×C3×Q8C2311yescharacteristic
60C2×C3×D4—21no—
61C2×C3×D4—22no—
62C2×C3×D4—22no—
63C2×C3×D4—21no—
64C2×C3×D4—24no—
65C2×C3×D4C2312yes—
66C2×C3×D4C2311yescharacteristic
67C22×C3×C4—21no—
68C22×C3×C4C2311yescharacteristic
69C2×#67—21no—
70#235—22no—
71#235—22no—
72C2×D12—22no—
73C2×D12—22no—
74C2×D12—21no—
75C2×D12—24no—
76C2×D12C2311yescharacteristic
77C2×D12C2312yes—
78C2×C4×D3—21no—
79C2×Dic6C2311yescharacteristic
80C3×C4⋊C4D412yes—
81C3×C22⋊C4—24no—
82C3×C22⋊C4D412yes—
83C3×C42C2312yes—
84#216—22no—
85#215—24no—
86#215—22no—
87#214—24no—
88#214—22no—
89#213—24no—
90#213—24no—
91#213—24no—
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×D4—31no—
103C22×D4C2×D311yescharacteristic
104AGL(1,ℤ/8)—64no—
105AGL(1,ℤ/8)—64no—
106#135—64no—
107C2×Q16—31no—
108C2×SD16—31no—
109C2×SD16—32no—
110C2×SD16—31no—
111C2×SD16—32no—
112C2×D8—32no—
113C2×D8—32no—
114C2×D8—31no—
115C2×D8—32no—
116C2×D8—32no—
117C2×D8—64no—
118C2×M16—31no—
119C22×C8—31no—
120#127—32no—
121#127C2×D312yescomplemented
122#124—32no—
123#124C2×D312yescomplemented
124#121—22no—
125#121—62no—
126#120—22no—
127#120—62no—
128#117C2×D311yescharacteristic
129#97—32no—
130C4×C8—32no—
131C23×C3—22no—
132C23×C3—21no—
133C23×C3—42no—
134C23×C3—82no—
135C23×C3C2×D411yescharacteristic
136C22×D3—22no—
137C22×D3—21no—
138C22×D3—42no—
139C22×D3—82no—
140C3×Q8—21no—
141C3×Q8—21no—
142C3×D4—22no—
143C3×D4—21no—
144C3×D4—22no—
145C3×D4—21no—
146C3×D4—42no—
147C3×D4—42no—
148C3×D4—84no—
149C2×C3×C4—21no—
150C2×C3×C4—42no—
151C2×C3×C4C2×D412yes—
152C2×C3×C4C2×D412yes—
153C2×C3×C4C2×D414yes—
154C2×C3×C4C2×D411yescharacteristic
155C2×C3×C4C2411yescharacteristic
156#67—42no—
157C2×Dic3—21no—
158D12—22no—
159D12—21no—
160D12—22no—
161D12—21no—
162D12—42no—
163D12—84no—
164C4×D3—42no—
165Dic6—21no—
166Dic6—21no—
167C3⋊C8C2×D412yes—
168C3⋊C8C2×D412yes—
169C3⋊C8C2×D414yes—
170C24—21no—
171C24—61no—
172#41—22no—
173#41—22no—
174#41—62no—
175#41—62no—
176C2×Q8—31no—
177C2×Q8C22×D311yescharacteristic
178C2×D4—21no—
179C2×D4—22no—
180C2×D4—22no—
181C2×D4—21no—
182C2×D4—24no—
183C2×D4—32no—
184C2×D4—31no—
185C2×D4—61no—
186C2×D4—62no—
187C2×D4—62no—
188C2×D4—61no—
189C2×D4—64no—
190C2×D4C22×D312yes—
191C2×D4C22×D311yescharacteristic
192C22×C4—21no—
193C22×C4—61no—
194C22×C4C22×D311yescharacteristic
195Q16—62no—
196SD16—62no—
197SD16—64no—
198SD16—62no—
199SD16—64no—
200D8—64no—
201D8—64no—
202D8—62no—
203D8—64no—
204D8—64no—
205M16—34no—
206C2×C8—34no—
207C2×C8—31no—
208C2×C8—32no—
209C2×C8—31no—
210C4⋊C4#6712yes—
211C22⋊C4—24no—
212C22⋊C4—64no—
213C22⋊C4#6712yes—
214C42C22×D312yes—
215C22×C3—22no—
216C22×C3—41no—
217C22×C3—42no—
218C22×C3—42no—
219C22×C3—44no—
220C22×C3—84no—
221C22×C3#12712yes—
222C22×C3C22×D411yescharacteristic
223C2×D3—41no—
224C2×D3—42no—
225C2×D3—42no—
226C2×D3—44no—
227C2×D3—84no—
228C3×C4—24no—
229C3×C4—41no—
230C3×C4#12712yes—
231C3×C4C22×D411yescharacteristic
232C3×C4C22×D411yescharacteristic
233Dic3—41no—
234C23—22no—
235C23—21no—
236C23—42no—
237C23—62no—
238C23—61no—
239C23—82no—
240C23—122no—
241C23—242no—
242C23C2×#6711yescharacteristic
243Q8—21no—
244Q8—21no—
245Q8—61no—
246Q8—61no—
247D4—22no—
248D4—21no—
249D4—22no—
250D4—21no—
251D4—42no—
252D4—42no—
253D4—62no—
254D4—61no—
255D4—62no—
256D4—61no—
257D4—84no—
258D4—122no—
259D4—122no—
260D4—244no—
261C2×C4—21no—
262C2×C4—42no—
263C2×C4—61no—
264C2×C4—122no—
265C2×C4D3×D412yes—
266C2×C4D3×D412yes—
267C2×C4C2×#6714yes—
268C2×C4C2×#6711yescharacteristic
269C2×C4C23×D311yescharacteristic
270C8—32no—
271C8—32no—
272C8—34no—
273C2×C3—22no—
274C2×C3—81no—
275C2×C3—82no—
276C2×C3C2×#12711yescharacteristic
277C2×C3#57612yes—
278D3—81no—
279D3—82no—
280C22—22no—
281C22—41no—
282C22—42no—
283C22—42no—
284C22—44no—
285C22—84no—
286C22—121no—
287C22—122no—
288C22—122no—
289C22—124no—
290C22—244no—
291C22#94012yes—
292C22C22×#6711yescharacteristic
293C4—24no—
294C4—41no—
295C4—121no—
296C4#94012yes—
297C4C2×D3×D411yescharacteristic
298C4C2×D3×D411yescharacteristic
299C3#315511yescharacteristic, complemented, Sylow
300C2—22no—
301C2—82no—
302C2—81no—
303C2—242no—
304C2—241no—
305C2C2×#94011yescharacteristic
306C2#583512yes—
subgroupG/H
centerC22C22×#67
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#535C2
socleC22×C3C22×D4
hypercenter#535D3
hypocenterC3#3155
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ #117 ◁ #535

Complements

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

NH ≅ G/Ntype
1. C3×#535C2semidirect
2. C2×#950C2semidirect
3. C2×#949C2semidirect
4. C2×#932C2semidirect
5. C22×#213C2semidirect
6. #5525C2semidirect
7. #5074C2semidirect
8. #5073C2semidirect
9. #5059C2semidirect
10. #5058C2semidirect
11. #4997C2semidirect
17. C3×#127C22semidirect
18. C3×#124C22semidirect
24. C2×#216C22semidirect
25. C2×#215C22semidirect
26. C2×#215C22semidirect
27. C2×#214C22semidirect
28. C2×#214C22semidirect
33. C2×#213C22semidirect
34. C2×#213C22semidirect
35. C2×#213C22semidirect
40. #875C22semidirect
41. #874C22semidirect
42. #803C22semidirect
43. C4×C3⋊C8C22semidirect
49. #535D3semidirect
95. #208D4semidirect
96. C2×C3⋊C8D4semidirect
121. #127C2×D3semidirect
123. #124C2×D3semidirect
299. C3#3155semidirect

Maximal subgroups

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

subgroupindex
C3×#5352
C2×#9502
C2×#9492
C2×#9322
C22×#2132
#55252
#50742
#50732
#50592
#50582
#49972
#31553