Finite Group Databasebeta

← #78840

Subgroups — #78840

Order 384.

504 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 133 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
1#5833C211yesmaximal, characteristic, complemented
2C2×#949C211yesmaximal, characteristic, complemented
3C2×#901C211yesmaximal, characteristic, complemented
4#5525C211yesmaximal, characteristic, complemented
5C3×#416C211yesmaximal, characteristic, complemented
6C2×#821C211yesmaximal, characteristic, complemented
7#4995C211yesmaximal, characteristic, complemented
8#4812C212yesmaximal, complemented
9#4807C212yesmaximal, complemented
10#4782C212yesmaximal, complemented
11#4775C212yesmaximal, complemented
12#3062—31nomaximal, Sylow
13C2×C3×#41C2211yescharacteristic
14C2×#235C2211yescharacteristic
15C22×D12C2211yescharacteristic
16C22×C3×C8C2211yescharacteristic
17C3×#117C2211yescharacteristic
18#949—24no—
19#946C2212yescomplemented
20C2×#215C2212yes—
21#939—22no—
22#938C2212yescomplemented
23C2×#213C2212yes—
24#930—21no—
25C2×C4⋊Dic3C2211yescharacteristic
26#924—21no—
27C2×#208C2211yescharacteristic
28#901—24no—
29C2×ΓL(1,𝔽25)C2212yescomplemented
30C2×D3×C8C2212yescomplemented
31#895C2212yes—
32#895C2212yes—
33#882—22no—
34#875—21no—
35#874—21no—
36C3×#103C2211yescharacteristic, complemented
37C3×#103C2211yescharacteristic, complemented
38C3×#102C2211yescharacteristic, complemented
39C3×#102C2211yescharacteristic, complemented
40#821C2212yes—
41#821C2212yes—
42#808C2212yescomplemented
43#807C2212yescomplemented
44C2×AGL(1,ℤ/8)—31no—
45C2×#131—31no—
46#537—31no—
47#535—31no—
48#469—32no—
49#468—32no—
50#466—32no—
51#465—32no—
52#428—31no—
53#416D311yescharacteristic, complemented
54C2×#102—31no—
55C23×D3—21no—
56C3×#41—22no—
57C3×#41—22no—
58C2×C3×Q8C2311yescharacteristic
59C2×C3×D4—21no—
60C2×C3×D4C2311yescharacteristic
61C22×C3×C4—21no—
62C22×C3×C4C2311yescharacteristic
63C2×#67—22no—
64C2×#67D412yes—
65C22×Dic3—21no—
66#235—22no—
67#235—22no—
68C2×D12—22no—
69C2×D12—22no—
70C2×D12—21no—
71C2×D12—22no—
72C2×D12—22no—
73C2×D12D411yescharacteristic
74C2×D12C2312yes—
75C2×C4×D3C2312yes—
76C2×Dic6D411yescharacteristic
77C2×C3×C8—22no—
78C2×C3×C8C2311yescharacteristic
79C2×C3×C8C2311yescharacteristic
80C3×C4⋊C4C2311yescharacteristic
81C3×C4⋊C4C2311yescharacteristic
82C3×C22⋊C4—21no—
83C3×C42—21no—
84#217—22no—
85#215—24no—
86#213—24no—
87(C2×D3)⋊C4—22no—
88(C2×D3)⋊C4—22no—
89(C2×D3)⋊C4—22no—
90C4⋊Dic3—21no—
91C4⋊Dic3C2312yes—
92Dic3⋊C4—22no—
93#208D414yescomplemented
94C2×C3⋊C8C2312yes—
95ΓL(1,𝔽25)—24no—
96D3×C8—24no—
97C2×#41—31no—
98C2×#41C2×D311yescharacteristic
99C22×D4—31no—
100AGL(1,ℤ/8)—64no—
101C2×SD16—32no—
102C2×D8—32no—
103#131—64no—
104C2×M16—31no—
105C2×M16—32no—
106C22×C8—32no—
107C22×C8C2×D311yescharacteristic
108#127—61no—
109#124—61no—
110C22⋊Q8—32no—
111C22⋊Q8—61no—
112#121—32no—
113#121—32no—
114#121—32no—
115#121—61no—
116#121—62no—
117#120—62no—
118#117C2×D311yescharacteristic
119C2×C4⋊C4—31no—
120#107—32no—
121#106—32no—
122#103C2×D311yescharacteristic, complemented
123#103C2×D311yescharacteristic, complemented
124#102—32no—
125#102—32no—
126#102C2×D311yescharacteristic, complemented
127#102C2×D311yescharacteristic, complemented
128C23×C3—21no—
129C23×C3C2×D411yescharacteristic
130C22×D3—22no—
131C22×D3—42no—
132C22×D3—81no—
133C22×D3—81no—
134C22×D3C2×D412yes—
135C3×Q8—21no—
136C3×Q8—21no—
137C3×D4—21no—
138C3×D4—21no—
139C3×D4—42no—
140C2×C3×C4—21no—
141C2×C3×C4—21no—
142C2×C3×C4—21no—
143C2×C3×C4—42no—
144C2×C3×C4C2×D411yescharacteristic
145C2×C3×C4C2×D412yes—
146C2×C3×C4#4112yes—
147C2×C3×C4C2411yescharacteristic
148#67—24no—
149#67—82no—
150C2×Dic3—22no—
151C2×Dic3—81no—
152C2×Dic3C2×D412yes—
153D12—22no—
154D12—22no—
155D12—22no—
156D12—42no—
157D12—82no—
158D12—82no—
159C4×D3—22no—
160C4×D3—22no—
161Dic6—22no—
162C3×C8—21no—
163C3×C8—21no—
164C3⋊C8C2×D414yes—
165C24—61no—
166#41—22no—
167#41—22no—
168#41—62no—
169#41—62no—
170C2×Q8—31no—
171C2×Q8C22×D311yescharacteristic
172C2×D4—21no—
173C2×D4—32no—
174C2×D4—31no—
175C2×D4—32no—
176C2×D4—62no—
177C2×D4—62no—
178C2×D4—62no—
179C2×D4—62no—
180C2×D4—62no—
181C2×D4—61no—
182C2×D4C22×D311yescharacteristic
183C22×C4—21no—
184C22×C4—32no—
185C22×C4—61no—
186C22×C4C22×D311yescharacteristic
187SD16—64no—
188D8—64no—
189M16—34no—
190M16—64no—
191C2×C8—22no—
192C2×C8—32no—
193C2×C8—64no—
194C2×C8C22×D311yescharacteristic
195C2×C8C22×D311yescharacteristic
196C4⋊C4—32no—
197C4⋊C4—62no—
198C4⋊C4—61no—
199C4⋊C4C22×D311yescharacteristic
200C4⋊C4C22×D311yescharacteristic
201C22⋊C4—21no—
202C22⋊C4—62no—
203C22⋊C4—62no—
204C22⋊C4—62no—
205C22⋊C4—62no—
206C42—21no—
207C22×C3—21no—
208C22×C3—21no—
209C22×C3—41no—
210C22×C3—41no—
211C22×C3—41no—
212C22×C3#12112yes—
213C22×C3C22×D411yescharacteristic
214C2×D3—22no—
215C2×D3—22no—
216C2×D3—22no—
217C2×D3—42no—
218C2×D3—42no—
219C2×D3—42no—
220C2×D3—84no—
221C3×C4—41no—
222C3×C4—41no—
223C3×C4—41no—
224C3×C4#12112yes—
225C3×C4C22×D411yescharacteristic
226C3×C4C2×#4111yescharacteristic
227Dic3—22no—
228Dic3—42no—
229C23—21no—
230C23—32no—
231C23—62no—
232C23—122no—
233C23—241no—
234C23—241no—
235C23D3×D411yescharacteristic
236Q8—21no—
237Q8—21no—
238Q8—62no—
239D4—21no—
240D4—21no—
241D4—42no—
242D4—64no—
243D4—62no—
244D4—62no—
245D4—62no—
246D4—122no—
247D4—242no—
248D4—242no—
249D4—242no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—32no—
254C2×C4—42no—
255C2×C4—62no—
256C2×C4—62no—
257C2×C4—62no—
258C2×C4—241no—
259C2×C4#23512yes—
260C2×C4D3×D411yescharacteristic
261C2×C4D3×D412yes—
262C2×C4C23×D311yescharacteristic
263C8—21no—
264C8—21no—
265C8—34no—
266C2×C3—22no—
267C2×C3—81no—
268C2×C3C2×#12111yescharacteristic
269C2×C3#57611yescharacteristic
270C2×C3#57711yescharacteristic
271D3—42no—
272D3—82no—
273C22—21no—
274C22—21no—
275C22—41no—
276C22—41no—
277C22—41no—
278C22—62no—
279C22—62no—
280C22—62no—
281C22—122no—
282C22—122no—
283C22—122no—
284C22—244no—
285C22#88412yes—
286C22C2×D3×D411yescharacteristic
287C4—41no—
288C4—41no—
289C4—41no—
290C4—62no—
291C4—122no—
292C4#88412yes—
293C4C2×#23511yescharacteristic
294C4C2×D3×D411yescharacteristic
295C3#306211yescharacteristic, complemented, Sylow
296C2—22no—
297C2—81no—
298C2—122no—
299C2—242no—
300C2C2×#88411yescharacteristic
301C2#576911yescharacteristic
302C2#577011yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#416C2
socleC22×C3C22×D4
hypercenter#416D3
hypocenterC3#3062
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
1. #5833C2semidirect
2. C2×#949C2semidirect
3. C2×#901C2semidirect
4. #5525C2semidirect
5. C3×#416C2semidirect
6. C2×#821C2semidirect
7. #4995C2semidirect
8. #4812C2semidirect
9. #4807C2semidirect
10. #4782C2semidirect
11. #4775C2semidirect
19. #946C22semidirect
22. #938C22semidirect
29. C2×ΓL(1,𝔽25)C22semidirect
30. C2×D3×C8C22semidirect
36. C3×#103C22semidirect
37. C3×#103C22semidirect
38. C3×#102C22semidirect
39. C3×#102C22semidirect
42. #808C22semidirect
43. #807C22semidirect
53. #416D3semidirect
93. #208D4semidirect
122. #103C2×D3semidirect
123. #103C2×D3semidirect
126. #102C2×D3semidirect
127. #102C2×D3semidirect
295. C3#3062semidirect

Maximal subgroups

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

subgroupindex
#58332
C2×#9492
C2×#9012
#55252
C3×#4162
C2×#8212
#49952
#48122
#48072
#47822
#47752
#30623