Finite Group Databasebeta

← #16855059

Subgroups — #16855059

Order 3968.

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

normal subgroups only

This group has 724 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
1#14521885C211yesmaximal, characteristic, complemented
2#14521847C212yesmaximal, complemented
3#14521231C212yesmaximal, complemented
4#14521226C212yesmaximal, complemented
5#14521221C212yesmaximal, complemented
6#14521216C212yesmaximal, complemented
7C2×#11758393C212yesmaximal, complemented
8#14521033C211yesmaximal, characteristic, complemented
9C31×#532C211yesmaximal, characteristic, complemented
10C2×#41×C31C2211yescharacteristic
11C2×#41×C31C2211yescharacteristic, complemented
12#11758567—22no—
13#11758566—22no—
14#41×D31—24no—
15#11758564C2212yes—
16#11758561—22no—
17#11758561C2212yes—
18C2×#92495—22no—
19C2×#92495C2212yes—
20C31×C4⋊Q8C2211yescharacteristic, complemented
21C31×#127C2211yescharacteristic, complemented
22C31×#124C2211yescharacteristic, complemented
23C31×#124C2211yescharacteristic, complemented
24C31×#121—22no—
25C4×D4×C31—22no—
26C2×C42×C31C2211yescharacteristic, complemented
27#11758528—22no—
28#11758527—24no—
29#11758526—22no—
30#11758519C2214yescomplemented
31#11758509C2214yescomplemented
32C2×#92474C2212yescomplemented
33#11758424C2212yescomplemented
34#11758421C2212yescomplemented
35#11758393C2214yescomplemented
36#11758393C2214yescomplemented
37#41×C31—22no—
38#41×C31—22no—
39#41×C31—22no—
40#41×C31—22no—
41C2×Q8×C31C2311yescharacteristic
42C2×Q8×C31C2311yescharacteristic
43C2×D4×C31—22no—
44C2×D4×C31—21no—
45C2×D4×C31D412yes—
46C2×D4×C31C2311yescharacteristic
47C2×D4×C31C2311yescharacteristic
48C22×C4×C31—21no—
49C22×C4×C31—21no—
50C22×C4×C31D412yes—
51C22×C4×C31C2311yescharacteristic
52C2×C22⋊D31—24no—
53C2×C22⋊D31—22no—
54#92499—24no—
55Q8×D31—24no—
56#92497—28no—
57#92497—24no—
58D4×D31—28no—
59D4×D31—24no—
60#92495—24no—
61#92495—24no—
62#92495—22no—
63#92495—24no—
64#92495—22no—
65#92495D414yescomplemented
66#92495C2314yes—
67C2×D124—22no—
68C2×D124D412yescomplemented
69C2×C4×D31—24no—
70C2×C4×D31—22no—
71C2×Dic62—22no—
72C2×Dic62D412yescomplemented
73C4⋊C4×C31—22no—
74C22⋊C4×C31—22no—
75C22⋊C4×C31—22no—
76C42×C31—21no—
77C42×C31C2311yescharacteristic
78C42×C31C2311yescharacteristic
79#92482—24no—
80#92481—24no—
81#92480—24no—
82#92479—24no—
83#92474—22no—
84#92474C2314yes—
85C2×C31⋊C8—22no—
86C23×C31—21no—
87C23×C31C2×D412yes—
88C23×C31C2×D411yescharacteristic
89C22×D31—24no—
90C22×D31—22no—
91Q8×C31—22no—
92Q8×C31—22no—
93D4×C31—24no—
94D4×C31—22no—
95D4×C31—22no—
96D4×C31—44no—
97D4×C31—42no—
98C2×C4×C31—21no—
99C2×C4×C31—22no—
100C2×C4×C31—22no—
101C2×C4×C31—21no—
102C2×C4×C31—21no—
103C2×C4×C31—42no—
104C2×C4×C31—41no—
105C2×C4×C31—41no—
106C2×C4×C31C2×D412yes—
107C2×C4×C31C2×D412yes—
108C2×C4×C31C2×D412yes—
109C2×C4×C31C2×D411yescharacteristic
110C2×C4×C31C2411yescharacteristic
111C22⋊D31—28no—
112C22⋊D31—24no—
113C22⋊D31—24no—
114C2×Dic31—24no—
115C2×Dic31—22no—
116D124—24no—
117D124—22no—
118D124C2×D414yes—
119C4×D31—28no—
120C4×D31—24no—
121C4×D31—24no—
122Dic62—24no—
123Dic62—22no—
124Dic62C2×D414yes—
125C31⋊C8—24no—
126#3018—311nomaximal, Sylow
127C22×C31—22no—
128C22×C31—41no—
129C22×C31—42no—
130C22×C31—42no—
131C22×C31—41no—
132C22×C31#12012yes—
133C22×C31C22×D411yescharacteristic
134C2×D31—24no—
135C2×D31—82no—
136C2×D31—84no—
137C4×C31—22no—
138C4×C31—22no—
139C4×C31—41no—
140C4×C31—41no—
141C4×C31#12011yescharacteristic
142C4×C31#12011yescharacteristic
143C4×C31C22×D412yes—
144Dic31—24no—
145#585—311no—
146#575—312no—
147#532D3111yescharacteristic, complemented
148#456—312no—
149#455—312no—
150#454—312no—
151#453—312no—
152C2×#104—312no—
153#413—311no—
154C2×C31—21no—
155C2×C31—22no—
156C2×C31—42no—
157C2×C31—81no—
158C2×C31C2×#12011yescharacteristic
159D31—44no—
1602−1+4—312no—
1612−1+4—622no—
162H(5,𝔽2)—312no—
163H(5,𝔽2)—622no—
164H(5,𝔽2)—622no—
165C2×#41—312no—
166C2×#41—624no—
167C2×#41—622no—
168C2×#41C2×D3111yescharacteristic
169C2×#41C2×D3111yescharacteristic, complemented
170#137—314no—
171#137—622no—
172AGL(1,ℤ/8)—314no—
173AGL(1,ℤ/8)—622no—
174#135—624no—
175C2×M16—312no—
176C4⋊Q8C2×D3111yescharacteristic, complemented
177#127C2×D3111yescharacteristic, complemented
178#124C2×D3111yescharacteristic, complemented
179#124C2×D3111yescharacteristic, complemented
180#121—22no—
181C4×D4—22no—
182C2×C42C2×D3111yescharacteristic, complemented
183#104—314no—
184#104—314no—
185#101—312no—
186#100—312no—
187C31#301811yescharacteristic, complemented, Sylow
188#41—22no—
189#41—22no—
190#41—22no—
191#41—22no—
192#41—314no—
193#41—314no—
194#41—624no—
195#41—624no—
196#41—624no—
197#41—628no—
198#41—624no—
199#41—622no—
200#41—622no—
201#41—624no—
202C2×Q8—312no—
203C2×Q8—622no—
204C2×Q8—624no—
205C2×Q8C22×D3111yescharacteristic
206C2×Q8C22×D3111yescharacteristic
207C2×D4—22no—
208C2×D4—21no—
209C2×D4—312no—
210C2×D4—622no—
211C2×D4—624no—
212C2×D4—628no—
213C2×D4—622no—
214C2×D4—624no—
215C2×D4C22⋊D3112yes—
216C2×D4C22×D3111yescharacteristic
217C2×D4C22×D3111yescharacteristic
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—624no—
221C22×C4—622no—
222C22×C4C22⋊D3112yes—
223C22×C4C22×D3111yescharacteristic
224Q16—624no—
225SD16—624no—
226SD16—624no—
227D8—624no—
228M16—314no—
229M16—622no—
230C2×C8—622no—
231C4⋊C4—22no—
232C22⋊C4—22no—
233C22⋊C4—22no—
234C42—21no—
235C42C22×D3111yescharacteristic
236C42C22×D3111yescharacteristic
237C23—21no—
238C23—624no—
239C23—622no—
240C23C2×C22⋊D3112yes—
241C23C2×C22⋊D3111yescharacteristic
242Q8—22no—
243Q8—22no—
244Q8—314no—
245Q8—624no—
246Q8—622no—
247D4—24no—
248D4—22no—
249D4—22no—
250D4—44no—
251D4—42no—
252D4—314no—
253D4—624no—
254D4—628no—
255D4—624no—
256D4—624no—
257D4—622no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—42no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—628no—
267C2×C4—624no—
268C2×C4—622no—
269C2×C4—624no—
270C2×C4—624no—
271C2×C4D4×D3112yes—
272C2×C4D4×D3112yes—
273C2×C4C2×C22⋊D3112yes—
274C2×C4C2×C22⋊D3111yescharacteristic
275C2×C4C23×D3111yescharacteristic
276C8—624no—
277C22—22no—
278C22—41no—
279C22—42no—
280C22—42no—
281C22—41no—
282C22—624no—
283C22—2482no—
284C22—2484no—
285C22#1175851412yes—
286C22C22×C22⋊D3111yescharacteristic
287C4—22no—
288C4—22no—
289C4—41no—
290C4—41no—
291C4—624no—
292C4#1175851411yescharacteristic
293C4#1175851411yescharacteristic
294C4C2×D4×D3112yes—
295C2—21no—
296C2—22no—
297C2—42no—
298C2—81no—
299C2—1244no—
300C2C2×#1175851411yescharacteristic
subgroupG/H
centerC4#11758514
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#532C2
socleC2×C31C2×#120
hypercenter#532D31
hypocenterC31#3018
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C4 ◁ C22×C4 ◁ #532

Complements

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

NH ≅ G/Ntype
1. #14521885C2semidirect
2. #14521847C2semidirect
3. #14521231C2semidirect
4. #14521226C2semidirect
5. #14521221C2semidirect
6. #14521216C2semidirect
7. C2×#11758393C2semidirect
8. #14521033C2semidirect
9. C31×#532C2semidirect
11. C2×#41×C31C22semidirect
20. C31×C4⋊Q8C22semidirect
21. C31×#127C22semidirect
22. C31×#124C22semidirect
23. C31×#124C22semidirect
26. C2×C42×C31C22semidirect
30. #11758519C22semidirect
31. #11758509C22semidirect
32. C2×#92474C22semidirect
33. #11758424C22semidirect
34. #11758421C22semidirect
35. #11758393C22semidirect
36. #11758393C22semidirect
65. #92495D4semidirect
68. C2×D124D4semidirect
72. C2×Dic62D4semidirect
147. #532D31semidirect
169. C2×#41C2×D31semidirect
176. C4⋊Q8C2×D31semidirect
177. #127C2×D31semidirect
178. #124C2×D31semidirect
179. #124C2×D31semidirect
182. C2×C42C2×D31semidirect
187. C31#3018semidirect

Maximal subgroups

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

subgroupindex
#145218852
#145218472
#145212312
#145212262
#145212212
#145212162
C2×#117583932
#145210332
C31×#5322
#301831