Finite Group Databasebeta

← #16857521

Subgroups — #16857521

Order 3968.

494 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 125 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#14521702C211yesmaximal, characteristic, complemented
2D4×D124C211yesmaximal, characteristic, complemented
3#14521397C211yesmaximal, characteristic, complemented
4#14521364C212yesmaximal, complemented
5#14521286C212yesmaximal, complemented
6#14521245C211yesmaximal, characteristic, complemented
7#14521184C212yesmaximal, complemented
8C4×#92479C211yesmaximal, characteristic, complemented
9#14521114C211yesmaximal, characteristic, complemented
10#102×D31C212yesmaximal, complemented
11C31×#474C211yesmaximal, characteristic, complemented
12C2×#92499—21no—
13C2×D4×D31C2212yes—
14C22×D124—21no—
15C31×C4⋊Q8C2211yescharacteristic, complemented
16C4×D4×C31C2211yescharacteristic
17#11758523—21no—
18C2×#92479C2211yescharacteristic
19#11758507—21no—
20C2×D248C2212yescomplemented
21C2×C8×D31C2212yescomplemented
22#11758472C2212yes—
23#11758472C2212yescomplemented
24#11758471—22no—
25C4⋊C4×D31C2212yescomplemented
26#11758459—22no—
27#11758451C2211yescharacteristic
28C4×D124C2211yescharacteristic
29C4×D124C2211yescharacteristic, complemented
30C31×#107C2212yescomplemented
31C31×C4⋊C8C2211yescharacteristic, complemented
32C31×#102C2212yes—
33#11758420C2211yescharacteristic
34#11758409C2212yescomplemented
35(C2×D31)⋊C8C2212yescomplemented
36#11758397C2212yescomplemented
37#11758397C2211yescharacteristic, complemented
38#11758397C2212yescomplemented
39#11758395C2211yescharacteristic, complemented
40C4×C31⋊C8C2211yescharacteristic, complemented
41C23×D31—22no—
42C2×Q8×C31—21no—
43C2×D4×C31C2311yescharacteristic
44C22×C4×C31—21no—
45C2×C22⋊D31—22no—
46#92499—44no—
47D4×D31—24no—
48D4×D31—42no—
49C2×D124—22no—
50C2×D124—22no—
51C2×D124—21no—
52C2×D124—21no—
53C2×D124—44no—
54C2×D124D411yescharacteristic
55C2×D124C2312yes—
56C2×D124C2311yescharacteristic
57C2×C4×D31—22no—
58C2×C4×D31—21no—
59C2×C4×D31C2312yes—
60C2×C8×C31C2312yes—
61C4⋊C4×C31—21no—
62C4⋊C4×C31C2311yescharacteristic
63C4⋊C4×C31C2312yes—
64C22⋊C4×C31—21no—
65C42×C31C2311yescharacteristic
66#92479—22no—
67(C2×D31)⋊C4—21no—
68(C2×D31)⋊C4—22no—
69(C2×D31)⋊C4—22no—
70(C2×D31)⋊C4D412yes—
71C4⋊Dic31D411yescharacteristic
72C4⋊Dic31C2311yescharacteristic
73Dic31⋊C4—22no—
74C2×C31⋊C8C2311yescharacteristic
75C2×C31⋊C8C2311yescharacteristic
76D248—42no—
77C8×D31—42no—
78C23×C31—21no—
79C22×D31—21no—
80C22×D31—22no—
81C22×D31—44no—
82C22×D31—44no—
83C22×D31—82no—
84C22×D31C2×D412yes—
85Q8×C31—42no—
86D4×C31—21no—
87D4×C31#4112yescomplemented
88C2×C4×C31—22no—
89C2×C4×C31—21no—
90C2×C4×C31—42no—
91C2×C4×C31C2×D411yescharacteristic
92C2×C4×C31C2×D411yescharacteristic
93C2×C4×C31C2411yescharacteristic
94C22⋊D31—44no—
95C2×Dic31—21no—
96C2×Dic31C2×D412yes—
97D124—21no—
98D124—22no—
99D124—21no—
100D124—22no—
101D124—44no—
102D124—44no—
103D124—42no—
104D124—81no—
105D124D814yescomplemented
106C4×D31—22no—
107C4×D31—22no—
108C4×D31—44no—
109C4×D31—42no—
110C8×C31—22no—
111C31⋊C8—21no—
112C31⋊C8—21no—
113#3294—311nomaximal, Sylow
114C22×C31—22no—
115C22×C31—41no—
116C22×C31—41no—
117C22×C31C22×D411yescharacteristic
118C2×D31—22no—
119C2×D31—22no—
120C2×D31—42no—
121C2×D31—41no—
122C2×D31—41no—
123C2×D31—42no—
124C2×D31—42no—
125C2×D31—41no—
126C2×D31—88no—
127C2×D31C2×D814yes—
128C4×C31—21no—
129C4×C31—42no—
130C4×C31—41no—
131C4×C31C2×D812yes—
132C4×C31C2×#4111yescharacteristic
133C4×C31H(5,𝔽2)11yescharacteristic
134Dic31—22no—
135Dic31—41no—
136#547—311no—
137D42—311no—
138#486—311no—
139#480—312no—
140#474D3111yescharacteristic, complemented
141#466—312no—
142#459—311no—
143#447—312no—
144C4×D8—311no—
145C8×D4—311no—
146C2×#102—312no—
147C2×C31—42no—
148C2×C31#54611yescharacteristic
149C2×C31C22×D811yescharacteristic
150C2×C31#57711yescharacteristic
151D31—24no—
152D31—81no—
153D31—82no—
154C2×#41—621no—
155C22×D4—312no—
156C22×D4—621no—
157C2×D8—311no—
158C2×D8—312no—
159C22×C8—312no—
160C4⋊Q8C2×D3111yescharacteristic, complemented
161#127—311no—
162#123—622no—
163C22⋊Q8—621no—
164#121—312no—
165#121—312no—
166#121—621no—
167#120—622no—
168C4×D4—311no—
169C4×D4—311no—
170C4×D4C2×D3111yescharacteristic
171C2×C4⋊C4—312no—
172#107—311no—
173#107C2×D3112yescomplemented
174C4⋊C8C2×D3111yescharacteristic, complemented
175#102—311no—
176#102—312no—
177#102—312no—
178#102—311no—
179#102—312no—
180#102C2×D3112yes—
181C22⋊C8—312no—
182C4×C8—311no—
183C31#329411yescharacteristic, complemented, Sylow
184C24—622no—
185#41—1244no—
186C2×Q8—21no—
187C2×D4—311no—
188C2×D4—312no—
189C2×D4—311no—
190C2×D4—622no—
191C2×D4—622no—
192C2×D4—622no—
193C2×D4—624no—
194C2×D4—621no—
195C2×D4—621no—
196C2×D4—1244no—
197C2×D4—1242no—
198C2×D4C22×D3111yescharacteristic
199C22×C4—21no—
200C22×C4—312no—
201C22×C4—622no—
202C22×C4—621no—
203D8—622no—
204D8—1242no—
205C2×C8—311no—
206C2×C8—311no—
207C2×C8—1242no—
208C2×C8C22×D3112yes—
209C4⋊C4—21no—
210C4⋊C4—311no—
211C4⋊C4—311no—
212C4⋊C4—622no—
213C4⋊C4C22×D3112yes—
214C4⋊C4C22×D3111yescharacteristic
215C22⋊C4—21no—
216C22⋊C4—312no—
217C22⋊C4—622no—
218C22⋊C4—622no—
219C22⋊C4—621no—
220C42C22×D3111yescharacteristic
221C23—21no—
222C23—312no—
223C23—621no—
224C23—622no—
225C23—1244no—
226C23—1244no—
227C23—2482no—
228Q8—42no—
229D4—21no—
230D4—314no—
231D4—621no—
232D4—622no—
233D4—621no—
234D4—622no—
235D4—1244no—
236D4—1244no—
237D4—1244no—
238D4—1242no—
239D4—2481no—
240D4#9249912yescomplemented
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—42no—
244C2×C4—312no—
245C2×C4—622no—
246C2×C4—622no—
247C2×C4—621no—
248C2×C4—1244no—
249C2×C4—1242no—
250C2×C4D4×D3111yescharacteristic
251C2×C4D4×D3111yescharacteristic
252C2×C4C23×D3111yescharacteristic
253C8—22no—
254C8—621no—
255C8—621no—
256C22—22no—
257C22—41no—
258C22—41no—
259C22—314no—
260C22—622no—
261C22—622no—
262C22—1242no—
263C22—1241no—
264C22—1242no—
265C22—1241no—
266C22—1241no—
267C22—1242no—
268C22—2488no—
269C22C2×D4×D3111yescharacteristic
270C4—21no—
271C4—41no—
272C4—42no—
273C4—622no—
274C4—1241no—
275C4D8×D3112yes—
276C4C2×#9249911yescharacteristic
277C4#1175856611yescharacteristic
278C2—42no—
279C2—624no—
280C2—2482no—
281C2—2481no—
282C2#1452169011yescharacteristic
283C2C2×D8×D3111yescharacteristic
284C2#1452185811yescharacteristic
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#474C2
socleC22×C31C22×D4
hypercenter#474D31
hypocenterC31#3294
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

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

NH ≅ G/Ntype
1. #14521702C2semidirect
2. D4×D124C2semidirect
3. #14521397C2semidirect
4. #14521364C2semidirect
5. #14521286C2semidirect
6. #14521245C2semidirect
7. #14521184C2semidirect
8. C4×#92479C2semidirect
9. #14521114C2semidirect
10. #102×D31C2semidirect
11. C31×#474C2semidirect
15. C31×C4⋊Q8C22semidirect
20. C2×D248C22semidirect
21. C2×C8×D31C22semidirect
23. #11758472C22semidirect
25. C4⋊C4×D31C22semidirect
29. C4×D124C22semidirect
30. C31×#107C22semidirect
31. C31×C4⋊C8C22semidirect
34. #11758409C22semidirect
35. (C2×D31)⋊C8C22semidirect
36. #11758397C22semidirect
37. #11758397C22semidirect
38. #11758397C22semidirect
39. #11758395C22semidirect
40. C4×C31⋊C8C22semidirect
87. D4×C31#41semidirect
105. D124D8semidirect
140. #474D31semidirect
160. C4⋊Q8C2×D31semidirect
173. #107C2×D31semidirect
174. C4⋊C8C2×D31semidirect
183. C31#3294semidirect
240. D4#92499semidirect

Maximal subgroups

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

subgroupindex
#145217022
D4×D1242
#145213972
#145213642
#145212862
#145212452
#145211842
C4×#924792
#145211142
#102×D312
C31×#4742
#329431