Finite Group Databasebeta

← #16857518

Subgroups — #16857518

Order 3968.

494 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 139 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
1D4×D124C211yesmaximal, characteristic, complemented
2#14521397C211yesmaximal, characteristic, complemented
3#14521362C212yesmaximal, complemented
4#14521326C211yesmaximal, characteristic, complemented
5#14521283C212yesmaximal, complemented
6#14521244C211yesmaximal, characteristic, complemented
7#14521182C212yesmaximal, complemented
8C4×#92479C211yesmaximal, characteristic, complemented
9D4×C31⋊C8C211yesmaximal, characteristic, complemented
10C2×#11758397C212yesmaximal, complemented
11C31×#547C211yesmaximal, characteristic, complemented
12C2×#41×C31—21no—
13C2×D4×D31—21no—
14C22×D124C2212yes—
15C31×C4⋊Q8C2211yescharacteristic, complemented
16C31×#123—22no—
17C31×C22⋊Q8—21no—
18C31×#121C2212yescomplemented
19C4×D4×C31C2211yescharacteristic
20C4×D4×C31C2211yescharacteristic, complemented
21C2×C4⋊C4×C31C2212yescomplemented
22C2×#92479C2211yescharacteristic
23C2×#92479C2212yescomplemented
24#11758507C2212yes—
25C22×C31⋊C8C2212yescomplemented
26#11758472—21no—
27#11758459—22no—
28#11758451C2211yescharacteristic
29C4×D124C2211yescharacteristic
30#11758420C2211yescharacteristic, complemented
31C22⋊(C31⋊C8)C2212yescomplemented
32#11758397C2211yescharacteristic
33#11758397C2212yes—
34#11758397C2212yescomplemented
35#11758397C2212yescomplemented
36#11758397C2212yescomplemented
37#11758395C2211yescharacteristic, complemented
38#11758395C2212yescomplemented
39C4⋊(C31⋊C8)C2211yescharacteristic, complemented
40C4×C31⋊C8C2211yescharacteristic, complemented
41C23×D31—22no—
42#41×C31—44no—
43C2×Q8×C31—21no—
44C2×D4×C31—22no—
45C2×D4×C31D411yescharacteristic
46C2×D4×C31C2311yescharacteristic
47C22×C4×C31—22no—
48C22×C4×C31—21no—
49C22×C4×C31C2312yes—
50C2×C22⋊D31—22no—
51D4×D31—44no—
52C2×D124—24no—
53C2×D124—21no—
54C2×D124—22no—
55C2×D124—21no—
56C2×D124—42no—
57C2×D124C2312yes—
58C2×D124C2311yescharacteristic
59C2×C4×D31—21no—
60C4⋊C4×C31—22no—
61C4⋊C4×C31—21no—
62C4⋊C4×C31D411yescharacteristic
63C4⋊C4×C31C2311yescharacteristic
64C4⋊C4×C31C2312yes—
65C22⋊C4×C31—21no—
66C22⋊C4×C31—22no—
67C22⋊C4×C31D412yes—
68C42×C31C2311yescharacteristic
69#92479—22no—
70#92479—42no—
71(C2×D31)⋊C4—21no—
72(C2×D31)⋊C4—22no—
73C4⋊Dic31C2311yescharacteristic
74C2×C31⋊C8—42no—
75C2×C31⋊C8C2311yescharacteristic
76C2×C31⋊C8C2311yescharacteristic
77C2×C31⋊C8C2312yes—
78C23×C31—21no—
79C23×C31C2×D412yes—
80C22×D31—21no—
81C22×D31—22no—
82C22×D31—44no—
83C22×D31—44no—
84C22×D31—82no—
85Q8×C31—42no—
86D4×C31—21no—
87D4×C31—21no—
88D4×C31—44no—
89D4×C31D814yescomplemented
90C2×C4×C31—22no—
91C2×C4×C31—22no—
92C2×C4×C31—21no—
93C2×C4×C31—22no—
94C2×C4×C31—44no—
95C2×C4×C31—42no—
96C2×C4×C31C2×D411yescharacteristic
97C2×C4×C31C2×D412yes—
98C2×C4×C31C2×D411yescharacteristic
99C2×C4×C31C2411yescharacteristic
100C22⋊D31—44no—
101C2×Dic31—21no—
102D124—22no—
103D124—21no—
104D124—22no—
105D124—44no—
106D124—42no—
107D124—81no—
108D124#4112yescomplemented
109C4×D31—42no—
110C31⋊C8—21no—
111C31⋊C8—21no—
112C31⋊C8—22no—
113#3294—311nomaximal, Sylow
114C22×C31—22no—
115C22×C31—22no—
116C22×C31—41no—
117C22×C31—41no—
118C22×C31—41no—
119C22×C31C2×D814yes—
120C22×C31C22×D411yescharacteristic
121C2×D31—22no—
122C2×D31—41no—
123C2×D31—41no—
124C2×D31—42no—
125C2×D31—42no—
126C2×D31—42no—
127C2×D31—88no—
128C4×C31—22no—
129C4×C31—21no—
130C4×C31—41no—
131C4×C31—42no—
132C4×C31C2×D812yes—
133C4×C31C2×#4111yescharacteristic
134C4×C31H(5,𝔽2)11yescharacteristic
135Dic31—41no—
136#547D3111yescharacteristic, complemented
137D42—311no—
138#486—311no—
139#480—312no—
140#474—311no—
141#466—312no—
142#459—311no—
143#447—312no—
144C4×D8—311no—
145C8×D4—311no—
146C2×#102—312no—
147C2×C31—24no—
148C2×C31—81no—
149C2×C31#54611yescharacteristic
150C2×C31C22×D811yescharacteristic
151C2×C31#57711yescharacteristic
152D31—42no—
153D31—82no—
154C2×#41—21no—
155C22×D4—312no—
156C22×D4—621no—
157C2×D8—311no—
158C2×D8—312no—
159C22×C8—312no—
160C4⋊Q8C2×D3111yescharacteristic, complemented
161#127—311no—
162#123—22no—
163C22⋊Q8—21no—
164#121—312no—
165#121—621no—
166#121C2×D3112yescomplemented
167#120—622no—
168C4×D4—311no—
169C4×D4C2×D3111yescharacteristic
170C4×D4C2×D3111yescharacteristic, complemented
171C2×C4⋊C4C2×D3112yescomplemented
172#107—311no—
173#107—312no—
174C4⋊C8—311no—
175#102—311no—
176#102—311no—
177#102—312no—
178#102—312no—
179#102—312no—
180#102—312no—
181C22⋊C8—312no—
182C4×C8—311no—
183C31#329411yescharacteristic, complemented, Sylow
184C24—622no—
185#41—44no—
186C2×Q8—21no—
187C2×D4—22no—
188C2×D4—311no—
189C2×D4—312no—
190C2×D4—624no—
191C2×D4—622no—
192C2×D4—622no—
193C2×D4—621no—
194C2×D4—621no—
195C2×D4—1242no—
196C2×D4—1244no—
197C2×D4C22⋊D3111yescharacteristic
198C2×D4C22×D3111yescharacteristic
199C22×C4—22no—
200C22×C4—21no—
201C22×C4—621no—
202C22×C4C22×D3112yes—
203D8—622no—
204D8—1242no—
205C2×C8—311no—
206C2×C8—311no—
207C2×C8—312no—
208C2×C8—1242no—
209C4⋊C4—22no—
210C4⋊C4—21no—
211C4⋊C4—311no—
212C4⋊C4C22⋊D3111yescharacteristic
213C4⋊C4C22×D3111yescharacteristic
214C4⋊C4C22×D3112yes—
215C22⋊C4—21no—
216C22⋊C4—22no—
217C22⋊C4—621no—
218C22⋊C4—622no—
219C22⋊C4C22⋊D3112yes—
220C42C22×D3111yescharacteristic
221C23—21no—
222C23—621no—
223C23—622no—
224C23—1244no—
225C23—1244no—
226C23—2482no—
227C23C2×C22⋊D3112yes—
228Q8—42no—
229D4—21no—
230D4—21no—
231D4—44no—
232D4—312no—
233D4—622no—
234D4—621no—
235D4—622no—
236D4—1244no—
237D4—1244no—
238D4—1242no—
239D4—2481no—
240D4#9247914yescomplemented
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—44no—
246C2×C4—42no—
247C2×C4—621no—
248C2×C4—1242no—
249C2×C4C2×C22⋊D3111yescharacteristic
250C2×C4C2×C22⋊D3112yes—
251C2×C4C2×C22⋊D3111yescharacteristic
252C2×C4C23×D3111yescharacteristic
253C8—621no—
254C8—621no—
255C8—622no—
256C22—22no—
257C22—22no—
258C22—41no—
259C22—41no—
260C22—41no—
261C22—622no—
262C22—1241no—
263C22—1241no—
264C22—1242no—
265C22—1242no—
266C22—1242no—
267C22—2488no—
268C22C2×#9247914yes—
269C22C22×C22⋊D3111yescharacteristic
270C4—21no—
271C4—22no—
272C4—41no—
273C4—42no—
274C4—1241no—
275C4C2×#9247912yes—
276C4#41×D3111yescharacteristic
277C4#1175856611yescharacteristic
278C2—24no—
279C2—81no—
280C2—1242no—
281C2—2482no—
282C2#1452168711yescharacteristic
283C2C22×#9247911yescharacteristic
284C2#1452185711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#547C2
socleC22×C31C22×D4
hypercenter#547D31
hypocenterC31#3294
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C4×D4 ◁ #547

Complements

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

NH ≅ G/Ntype
1. D4×D124C2semidirect
2. #14521397C2semidirect
3. #14521362C2semidirect
4. #14521326C2semidirect
5. #14521283C2semidirect
6. #14521244C2semidirect
7. #14521182C2semidirect
8. C4×#92479C2semidirect
9. D4×C31⋊C8C2semidirect
10. C2×#11758397C2semidirect
11. C31×#547C2semidirect
15. C31×C4⋊Q8C22semidirect
18. C31×#121C22semidirect
20. C4×D4×C31C22semidirect
21. C2×C4⋊C4×C31C22semidirect
23. C2×#92479C22semidirect
25. C22×C31⋊C8C22semidirect
30. #11758420C22semidirect
31. C22⋊(C31⋊C8)C22semidirect
34. #11758397C22semidirect
35. #11758397C22semidirect
36. #11758397C22semidirect
37. #11758395C22semidirect
38. #11758395C22semidirect
39. C4⋊(C31⋊C8)C22semidirect
40. C4×C31⋊C8C22semidirect
89. D4×C31D8semidirect
108. D124#41semidirect
136. #547D31semidirect
160. C4⋊Q8C2×D31semidirect
166. #121C2×D31semidirect
170. C4×D4C2×D31semidirect
171. C2×C4⋊C4C2×D31semidirect
183. C31#3294semidirect
240. D4#92479semidirect

Maximal subgroups

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

subgroupindex
D4×D1242
#145213972
#145213622
#145213262
#145212832
#145212442
#145211822
C4×#924792
D4×C31⋊C82
C2×#117583972
C31×#5472
#329431