Finite Group Databasebeta

← #16855067

Subgroups — #16855067

Order 3968.

762 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 762 conjugacy classes of subgroups — too many to draw. Only its 145 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
1C2×#11758561C211yesmaximal, characteristic, complemented
2#14521847C212yesmaximal, complemented
3#14521226C214yesmaximal, complemented
4#14521216C214yesmaximal, complemented
5#14521067C212yesmaximal, complemented
6C2×#11758421C211yesmaximal, characteristic, complemented
7C31×#535C211yesmaximal, characteristic, complemented
8C2×#41×C31C2211yescharacteristic, complemented
9C22×D4×C31C2211yescharacteristic
10C22×C22⋊D31—22no—
11#11758561—22no—
12#11758561—24no—
13#11758561C2214yes—
14C2×#92497—22no—
15C2×D4×D31—22no—
16C2×#92495C2212yes—
17C31×#127C2212yescomplemented
18C31×#124C2212yescomplemented
19C31×#121—22no—
20C31×#120—22no—
21C31×#117C2211yescharacteristic, complemented
22#11758528—22no—
23#11758527—24no—
24#11758526—22no—
25#11758519C2214yescomplemented
26#11758509C2214yescomplemented
27C2×#92474C2212yescomplemented
28#11758421C2214yescomplemented
29#11758393C2218yescomplemented
30C24×C31D412yes—
31C23×D31—22no—
32#41×C31—22no—
33#41×C31—22no—
34C2×Q8×C31C2311yescharacteristic, complemented
35C2×D4×C31—21no—
36C2×D4×C31—24no—
37C2×D4×C31—22no—
38C2×D4×C31—22no—
39C2×D4×C31D412yes—
40C2×D4×C31C2312yes—
41C2×D4×C31C2311yescharacteristic, complemented
42C22×C4×C31—21no—
43C22×C4×C31C2311yescharacteristic
44C2×C22⋊D31—24no—
45C2×C22⋊D31—28no—
46C2×C22⋊D31—22no—
47C2×C22⋊D31—28no—
48C22×Dic31—22no—
49#92497—28no—
50#92497—28no—
51D4×D31—28no—
52D4×D31—28no—
53#92495—22no—
54#92495—22no—
55#92495D414yescomplemented
56#92495C2314yes—
57C2×D124D412yescomplemented
58C2×C4×D31—22no—
59C2×Dic62D412yescomplemented
60C4⋊C4×C31—21no—
61C22⋊C4×C31—21no—
62C22⋊C4×C31—24no—
63C42×C31C2312yescomplemented
64#92482—24no—
65#92481—24no—
66#92480—24no—
67#92479—24no—
68#92474—22no—
69#92474C2314yescomplemented
70C2×C31⋊C8—22no—
71C23×C31—22no—
72C23×C31—22no—
73C23×C31—21no—
74C23×C31—82no—
75C23×C31C2×D414yes—
76C23×C31C2×D411yescharacteristic
77C22×D31—22no—
78C22×D31—28no—
79C22×D31—84no—
80Q8×C31—22no—
81D4×C31—24no—
82D4×C31—24no—
83D4×C31—22no—
84D4×C31—41no—
85D4×C31—41no—
86D4×C31—84no—
87C2×C4×C31—22no—
88C2×C4×C31—21no—
89C2×C4×C31—42no—
90C2×C4×C31C2×D412yes—
91C2×C4×C31C2×D412yes—
92C2×C4×C31C2×D411yescharacteristic
93C2×C4×C31C2411yescharacteristic
94C22⋊D31—216no—
95C22⋊D31—24no—
96C22⋊D31—24no—
97C2×Dic31—28no—
98C2×Dic31—22no—
99D124—22no—
100D124C2×D414yes—
101C4×D31—24no—
102C4×D31—24no—
103Dic62—22no—
104Dic62C2×D414yes—
105C31⋊C8—24no—
106#3019—311nomaximal, Sylow
107C22×C31—24no—
108C22×C31—41no—
109C22×C31—44no—
110C22×C31—44no—
111C22×C31—42no—
112C22×C31—41no—
113C22×C31—82no—
114C22×C31#12011yescharacteristic
115C22×C31#12011yescharacteristic
116C22×C31C22×D411yescharacteristic
117C2×D31—24no—
118C2×D31—44no—
119C2×D31—88no—
120C4×C31—42no—
121C4×C31—41no—
122C4×C31#12012yes—
123C4×C31C22×D412yes—
124Dic31—24no—
125C2×H(5,𝔽2)—311no—
126#575—312no—
127#535D3111yescharacteristic, complemented
128#455—314no—
129#453—314no—
130#421—312no—
131C2×#100—311no—
132C2×C31—21no—
133C2×C31—21no—
134C2×C31—21no—
135C2×C31—44no—
136C2×C31—81no—
137C2×C31C2×#12011yescharacteristic
138D31—44no—
139H(5,𝔽2)—314no—
140H(5,𝔽2)—624no—
141H(5,𝔽2)—622no—
142C2×#41—312no—
143C2×#41—622no—
144C2×#41C2×D3111yescharacteristic, complemented
145C22×D4—622no—
146C22×D4—622no—
147C22×D4C2×D3111yescharacteristic
148#137—314no—
149#137—622no—
150AGL(1,ℤ/8)—314no—
151AGL(1,ℤ/8)—622no—
152#135—624no—
153C2×M16—312no—
154#127C2×D3112yescomplemented
155#124C2×D3112yescomplemented
156#121—22no—
157#120—22no—
158#117C2×D3111yescharacteristic, complemented
159#104—318no—
160#100—314no—
161C31#301911yescharacteristic, complemented, Sylow
162C24—622no—
163C24C22⋊D3112yes—
164#41—22no—
165#41—22no—
166#41—314no—
167#41—314no—
168#41—628no—
169#41—628no—
170#41—622no—
171#41—622no—
172C2×Q8—312no—
173C2×Q8C22×D3111yescharacteristic, complemented
174C2×D4—21no—
175C2×D4—24no—
176C2×D4—22no—
177C2×D4—22no—
178C2×D4—312no—
179C2×D4—628no—
180C2×D4—628no—
181C2×D4—624no—
182C2×D4—628no—
183C2×D4—622no—
184C2×D4—628no—
185C2×D4C22⋊D3112yes—
186C2×D4C22×D3112yes—
187C2×D4C22×D3111yescharacteristic, complemented
188C22×C4—21no—
189C22×C4—622no—
190C22×C4—622no—
191C22×C4C22×D3111yescharacteristic
192Q16—624no—
193SD16—624no—
194SD16—624no—
195D8—624no—
196M16—314no—
197M16—622no—
198C2×C8—622no—
199C4⋊C4—21no—
200C22⋊C4—21no—
201C22⋊C4—24no—
202C42C22×D3112yescomplemented
203C23—22no—
204C23—22no—
205C23—21no—
206C23—82no—
207C23—628no—
208C23—622no—
209C23—2484no—
210C23C2×C22⋊D3114yes—
211C23C2×C22⋊D3111yescharacteristic
212Q8—22no—
213Q8—314no—
214Q8—622no—
215D4—24no—
216D4—24no—
217D4—22no—
218D4—41no—
219D4—41no—
220D4—84no—
221D4—314no—
222D4—6216no—
223D4—624no—
224D4—624no—
225D4—622no—
226C2×C4—22no—
227C2×C4—21no—
228C2×C4—42no—
229C2×C4—628no—
230C2×C4—622no—
231C2×C4—624no—
232C2×C4—624no—
233C2×C4D4×D3112yes—
234C2×C4D4×D3112yes—
235C2×C4C2×C22⋊D3111yescharacteristic
236C2×C4C23×D3111yescharacteristic
237C8—624no—
238C22—24no—
239C22—41no—
240C22—44no—
241C22—44no—
242C22—42no—
243C22—41no—
244C22—82no—
245C22—624no—
246C22—1244no—
247C22—2488no—
248C22#1175851411yescharacteristic
249C22#1175851411yescharacteristic
250C22C22×C22⋊D3111yescharacteristic
251C4—42no—
252C4—41no—
253C4—624no—
254C4#1175851412yes—
255C4C2×D4×D3112yes—
256C2—21no—
257C2—21no—
258C2—21no—
259C2—44no—
260C2—81no—
261C2—1244no—
262C2C2×#1175851411yescharacteristic
subgroupG/H
centerC2C2×#11758514
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#535C2
socleC2×C31C2×#120
hypercenter#535D31
hypocenterC31#3019
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C2 ◁ C22×C4 ◁ #535

Complements

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

NH ≅ G/Ntype
1. C2×#11758561C2semidirect
2. #14521847C2semidirect
3. #14521226C2semidirect
4. #14521216C2semidirect
5. #14521067C2semidirect
6. C2×#11758421C2semidirect
7. C31×#535C2semidirect
8. C2×#41×C31C22semidirect
17. C31×#127C22semidirect
18. C31×#124C22semidirect
21. C31×#117C22semidirect
25. #11758519C22semidirect
26. #11758509C22semidirect
27. C2×#92474C22semidirect
28. #11758421C22semidirect
29. #11758393C22semidirect
34. C2×Q8×C31C23semidirect
41. C2×D4×C31C23semidirect
55. #92495D4semidirect
57. C2×D124D4semidirect
59. C2×Dic62D4semidirect
63. C42×C31C23semidirect
69. #92474C23semidirect
127. #535D31semidirect
144. C2×#41C2×D31semidirect
154. #127C2×D31semidirect
155. #124C2×D31semidirect
158. #117C2×D31semidirect
161. C31#3019semidirect
173. C2×Q8C22×D31semidirect
187. C2×D4C22×D31semidirect
202. C42C22×D31semidirect

Maximal subgroups

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

subgroupindex
C2×#117585612
#145218472
#145212262
#145212162
#145210672
C2×#117584212
C31×#5352
#301931