Finite Group Databasebeta

← #82923

Subgroups — #82923

Order 384.

500 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 141 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×#525C211yesmaximal, characteristic, complemented
2C2×#950C211yesmaximal, characteristic, complemented
3#5791C211yesmaximal, characteristic, complemented
4C22×#208C211yesmaximal, characteristic, complemented
5#5233C211yesmaximal, characteristic, complemented
6#5052C212yesmaximal, complemented
7#5049C212yesmaximal, complemented
8#5042C212yesmaximal, complemented
9#5039C212yesmaximal, complemented
10#5002C211yesmaximal, characteristic, complemented
11#4998C211yesmaximal, characteristic, complemented
12#3054—31nomaximal, Sylow
13C2×C3×#41C2211yescharacteristic
14C23×C3×C4C2211yescharacteristic
15C2×#235C2211yescharacteristic
16C3×#123—21no—
17C3×C22⋊Q8C2212yes—
18C3×#121C2212yes—
19C3×#120—21no—
20C3×C4×D4—22no—
21C3×#117C2211yescharacteristic
22#953—21no—
23#950—24no—
24C2×#216C2211yescharacteristic, complemented
25C2×#215C2211yescharacteristic, complemented
26C2×#214C2211yescharacteristic, complemented
27C2×#213C2211yescharacteristic, complemented
28#930C2212yes—
29#929—21no—
30C4×#67—22no—
31#926C2211yescharacteristic
32#924C2212yes—
33C2×#208—24no—
34C2×#208C2212yes—
35C2×#208C2212yes—
36C22×C3⋊C8C2211yescharacteristic
37C22×C3⋊C8C2211yescharacteristic
38#835C2212yescomplemented
39#832C2212yescomplemented
40#810C2212yescomplemented
41#809C2212yescomplemented
42#808C2212yescomplemented
43#807C2212yescomplemented
44C2×#135—31no—
45C22×M16—31no—
46#525—31no—
47#525D311yescharacteristic, complemented
48#470—32no—
49#469—32no—
50#468—32no—
51#468—32no—
52#427—31no—
53#416—31no—
54#416—31no—
55C24×C3D411yescharacteristic
56C3×#41—22no—
57C3×#41—22no—
58C2×C3×Q8C2311yescharacteristic
59C2×C3×D4—22no—
60C2×C3×D4—21no—
61C2×C3×D4C2311yescharacteristic
62C22×C3×C4—22no—
63C22×C3×C4—22no—
64C22×C3×C4—21no—
65C22×C3×C4D412yes—
66C22×C3×C4D411yescharacteristic
67C22×C3×C4C2312yes—
68C22×C3×C4C2311yescharacteristic
69C2×#67—22no—
70C2×#67—21no—
71#235—22no—
72#235—22no—
73C2×D12C2311yescharacteristic
74C2×C4×D3—21no—
75C2×Dic6C2311yescharacteristic
76C3×C4⋊C4—22no—
77C3×C4⋊C4C2312yes—
78C3×C22⋊C4—21no—
79C3×C22⋊C4—22no—
80C3×C22⋊C4—22no—
81C3×C42—21no—
82#217—22no—
83#217—21no—
84#216—22no—
85#215—22no—
86#214—22no—
87#213—22no—
88(C2×D3)⋊C4—22no—
89C4⋊Dic3C2312yes—
90Dic3⋊C4—22no—
91C4×Dic3—21no—
92#208—24no—
93#208—24no—
94C2×C3⋊C8—22no—
95C2×C3⋊C8D414yescomplemented
96C2×C3⋊C8C2312yes—
97C2×C3⋊C8C2311yescharacteristic
98C2×C3⋊C8C2311yescharacteristic
99C2×#41—31no—
100C2×#41C2×D311yescharacteristic
101C23×C4C2×D311yescharacteristic
102#135—64no—
103C2×Q16—31no—
104C2×SD16—31no—
105C2×SD16—31no—
106C2×D8—31no—
107C2×M16—32no—
108C2×M16—32no—
109C2×M16—64no—
110C22×C8—31no—
111C22×C8—31no—
112#123—21no—
113#123—61no—
114C22⋊Q8—32no—
115C22⋊Q8C2×D312yes—
116#121—32no—
117#121C2×D312yes—
118#120—21no—
119#120—61no—
120C4×D4—22no—
121C4×D4—62no—
122#117—31no—
123#117C2×D311yescharacteristic
124#107—32no—
125#106—32no—
126#103—32no—
127#103—32no—
128#102—32no—
129#102—32no—
130C23×C3—22no—
131C23×C3—21no—
132C23×C3—42no—
133C23×C3C2×D412yes—
134C23×C3C2×D411yescharacteristic
135C22×D3—21no—
136C3×Q8—21no—
137C3×Q8—21no—
138C3×D4—21no—
139C3×D4—21no—
140C3×D4—44no—
141C3×D4—42no—
142C2×C3×C4—22no—
143C2×C3×C4—24no—
144C2×C3×C4—22no—
145C2×C3×C4—22no—
146C2×C3×C4—22no—
147C2×C3×C4—21no—
148C2×C3×C4—42no—
149C2×C3×C4C2×D412yes—
150C2×C3×C4C2×D412yes—
151C2×C3×C4C2×D411yescharacteristic
152C2×C3×C4#4112yes—
153C2×C3×C4C2411yescharacteristic
154#67—44no—
155#67—42no—
156C2×Dic3—22no—
157C2×Dic3—21no—
158D12—21no—
159D12—21no—
160C4×D3—42no—
161Dic6—21no—
162Dic6—21no—
163C3⋊C8—22no—
164C3⋊C8C2×D412yes—
165C3⋊C8C2×D412yes—
166C24#6711yescharacteristic
167#41—22no—
168#41—22no—
169#41—62no—
170#41—62no—
171C2×Q8—31no—
172C2×Q8C22×D311yescharacteristic
173C2×D4—22no—
174C2×D4—21no—
175C2×D4—31no—
176C2×D4—62no—
177C2×D4—61no—
178C2×D4C22×D311yescharacteristic
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—21no—
182C22×C4—61no—
183C22×C4#6711yescharacteristic
184C22×C4#6712yes—
185C22×C4C22×D312yes—
186C22×C4C22×D311yescharacteristic
187Q16—62no—
188SD16—62no—
189SD16—62no—
190D8—62no—
191M16—64no—
192M16—64no—
193C2×C8—34no—
194C2×C8—31no—
195C2×C8—31no—
196C2×C8—32no—
197C2×C8—62no—
198C4⋊C4—22no—
199C4⋊C4—32no—
200C4⋊C4—62no—
201C4⋊C4C22×D312yes—
202C22⋊C4—21no—
203C22⋊C4—22no—
204C22⋊C4—22no—
205C22⋊C4—61no—
206C22⋊C4—62no—
207C22⋊C4—62no—
208C42—21no—
209C42—61no—
210C22×C3—22no—
211C22×C3—22no—
212C22×C3—22no—
213C22×C3—22no—
214C22×C3—44no—
215C22×C3—41no—
216C22×C3—42no—
217C22×C3#12112yes—
218C22×C3C22×D411yescharacteristic
219C2×D3—41no—
220C2×D3—42no—
221C3×C4—22no—
222C3×C4—41no—
223C3×C4—42no—
224C3×C4#12112yes—
225C3×C4C22×D411yescharacteristic
226C3×C4C2×#4111yescharacteristic
227Dic3—41no—
228Dic3—42no—
229C23—22no—
230C23—21no—
231C23—42no—
232C23—61no—
233C23C2×#6712yes—
234C23C2×#6711yescharacteristic
235Q8—21no—
236Q8—21no—
237Q8—61no—
238Q8—61no—
239D4—21no—
240D4—21no—
241D4—44no—
242D4—42no—
243D4—61no—
244D4—61no—
245D4—124no—
246D4—122no—
247C2×C4—22no—
248C2×C4—24no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—21no—
253C2×C4—42no—
254C2×C4—62no—
255C2×C4—61no—
256C2×C4—122no—
257C2×C4D3×D412yes—
258C2×C4#23712yes—
259C2×C4C2×#6712yes—
260C2×C4C2×#6711yescharacteristic
261C2×C4C23×D311yescharacteristic
262C8—32no—
263C8—32no—
264C8—62no—
265C2×C3—22no—
266C2×C3—42no—
267C2×C3—81no—
268C2×C3C2×#12111yescharacteristic
269C2×C3#57512yes—
270D3—81no—
271C22—22no—
272C22—22no—
273C22—22no—
274C22—22no—
275C22—44no—
276C22—41no—
277C22—42no—
278C22—121no—
279C22—122no—
280C22#93912yes—
281C22C22×#6711yescharacteristic
282C4—22no—
283C4—41no—
284C4—42no—
285C4—121no—
286C4—122no—
287C4#93912yes—
288C4C2×D3×D411yescharacteristic
289C4C2×#23711yescharacteristic
290C3#305411yescharacteristic, complemented, Sylow
291C2—22no—
292C2—42no—
293C2—81no—
294C2—241no—
295C2C2×#93911yescharacteristic
296C2#582212yes—
subgroupG/H
centerC2×C4C2×#67
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#525C2
socleC22×C3C22×D4
hypercenter#525D3
hypocenterC3#3054
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. C3×#525C2semidirect
2. C2×#950C2semidirect
3. #5791C2semidirect
4. C22×#208C2semidirect
5. #5233C2semidirect
6. #5052C2semidirect
7. #5049C2semidirect
8. #5042C2semidirect
9. #5039C2semidirect
10. #5002C2semidirect
11. #4998C2semidirect
24. C2×#216C22semidirect
25. C2×#215C22semidirect
26. C2×#214C22semidirect
27. C2×#213C22semidirect
38. #835C22semidirect
39. #832C22semidirect
40. #810C22semidirect
41. #809C22semidirect
42. #808C22semidirect
43. #807C22semidirect
47. #525D3semidirect
95. C2×C3⋊C8D4semidirect
290. C3#3054semidirect

Maximal subgroups

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

subgroupindex
C3×#5252
C2×#9502
#57912
C22×#2082
#52332
#50522
#50492
#50422
#50392
#50022
#49982
#30543