Finite Group Databasebeta

← #16855060

Subgroups — #16855060

Order 3968.

724 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 724 conjugacy classes of subgroups — too many to draw. Only its 161 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#14521847C212yesmaximal, complemented
2#14521595C211yesmaximal, characteristic, complemented
3#14521232C212yesmaximal, complemented
4#14521227C212yesmaximal, complemented
5#14521222C212yesmaximal, complemented
6#14521217C212yesmaximal, complemented
7C2×#11758425C212yesmaximal, complemented
8#14521033C211yesmaximal, characteristic, complemented
9C31×#585C211yesmaximal, characteristic, complemented
10C31×2−1+4—22no—
11C31×2−1+4C2212yes—
12C31×H(5,𝔽2)—22no—
13C31×H(5,𝔽2)—22no—
14C31×H(5,𝔽2)C2212yes—
15C2×#41×C31—24no—
16C2×#41×C31—22no—
17C2×#41×C31C2212yes—
18C2×#41×C31C2211yescharacteristic
19C2×#92495C2211yescharacteristic, complemented
20#11758528C2214yescomplemented
21#11758527—24no—
22#11758526C2214yescomplemented
23#11758524C2211yescharacteristic, complemented
24#11758521C2211yescharacteristic, complemented
25#11758519—22no—
26#11758517C2211yescharacteristic, complemented
27#11758516—22no—
28#11758513C2211yescharacteristic, complemented
29#11758509—22no—
30C4×C22⋊D31—22no—
31C2×C4×Dic31C2211yescharacteristic, complemented
32C2×#92474C2212yescomplemented
33#11758425C2214yescomplemented
34#11758425C2214yescomplemented
35#11758424C2212yescomplemented
36#11758421C2212yescomplemented
37#41×C31—24no—
38#41×C31—28no—
39#41×C31—24no—
40#41×C31—24no—
41#41×C31—24no—
42#41×C31—22no—
43#41×C31—22no—
44#41×C31—24no—
45#41×C31—22no—
46#41×C31—22no—
47#41×C31D414yescomplemented
48#41×C31C2314yes—
49C2×Q8×C31—22no—
50C2×Q8×C31—24no—
51C2×Q8×C31D412yescomplemented
52C2×Q8×C31C2311yescharacteristic
53C2×D4×C31—28no—
54C2×D4×C31—22no—
55C2×D4×C31—24no—
56C2×D4×C31—22no—
57C2×D4×C31—24no—
58C2×D4×C31D412yes—
59C2×D4×C31D412yescomplemented
60C2×D4×C31C2311yescharacteristic
61C22×C4×C31—24no—
62C22×C4×C31—22no—
63C22×C4×C31D412yes—
64C22×C4×C31C2311yescharacteristic
65C2×C22⋊D31—22no—
66C2×C22⋊D31—21no—
67C22×Dic31—21no—
68#92495—22no—
69#92495—22no—
70C2×D124C2311yescharacteristic
71C2×C4×D31—21no—
72C2×Dic62C2311yescharacteristic
73C22⋊Dic31—22no—
74#92482—24no—
75#92481—24no—
76#92480—24no—
77#92479—24no—
78(C2×D31)⋊C4—22no—
79Dic31⋊C4—22no—
80C4×Dic31—21no—
81C4×Dic31C2311yescharacteristic
82C4×Dic31C2311yescharacteristic
83#92474—22no—
84#92474C2314yes—
85C2×C31⋊C8—22no—
86C23×C31—24no—
87C23×C31—22no—
88C23×C31C2×D412yes—
89C23×C31C2×D411yescharacteristic
90C22×D31—21no—
91Q8×C31—24no—
92Q8×C31—22no—
93Q8×C31—22no—
94Q8×C31C2×D414yes—
95D4×C31—28no—
96D4×C31—24no—
97D4×C31—24no—
98D4×C31—24no—
99D4×C31—24no—
100D4×C31—22no—
101D4×C31—22no—
102D4×C31C2×D414yes—
103C2×C4×C31—24no—
104C2×C4×C31—28no—
105C2×C4×C31—22no—
106C2×C4×C31—24no—
107C2×C4×C31—22no—
108C2×C4×C31—22no—
109C2×C4×C31—24no—
110C2×C4×C31C2×D412yes—
111C2×C4×C31C2×D411yescharacteristic
112C2×C4×C31C2×D412yes—
113C2×C4×C31C2×D412yes—
114C2×C4×C31C2411yescharacteristic
115C22⋊D31—44no—
116C22⋊D31—42no—
117C2×Dic31—21no—
118C2×Dic31—21no—
119C2×Dic31—21no—
120C2×Dic31—42no—
121D124—22no—
122C4×D31—41no—
123C4×D31—41no—
124Dic62—22no—
125C31⋊C8—24no—
126#3018—311nomaximal, Sylow
127C22×C31—22no—
128C22×C31—24no—
129C22×C31—41no—
130C22×C31—42no—
131C22×C31—82no—
132C22×C31—84no—
133C22×C31#12012yes—
134C22×C31C22×D411yescharacteristic
135C2×D31—42no—
136C2×D31—41no—
137C4×C31—24no—
138C4×C31—22no—
139C4×C31#12011yescharacteristic
140C4×C31#12011yescharacteristic
141C4×C31C22×D412yes—
142Dic31—22no—
143Dic31—41no—
144Dic31—41no—
145#585D3111yescharacteristic, complemented
146#575—312no—
147#532—311no—
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—44no—
158C2×C31C2×#12011yescharacteristic
159D31—81no—
1602−1+4—22no—
1612−1+4C2×D3112yes—
162H(5,𝔽2)—22no—
163H(5,𝔽2)—22no—
164H(5,𝔽2)C2×D3112yes—
165C2×#41—24no—
166C2×#41—22no—
167C2×#41—311no—
168C2×#41C2×D3112yes—
169C2×#41C2×D3111yescharacteristic
170#137—314no—
171#137—622no—
172AGL(1,ℤ/8)—314no—
173AGL(1,ℤ/8)—622no—
174#135—624no—
175C2×M16—312no—
176C4⋊Q8—311no—
177#127—311no—
178#124—311no—
179#124—311no—
180#121—622no—
181C4×D4—622no—
182C2×C42—311no—
183#104—314no—
184#104—314no—
185#101—312no—
186#100—312no—
187C31#301811yescharacteristic, complemented, Sylow
188#41—24no—
189#41—28no—
190#41—24no—
191#41—24no—
192#41—24no—
193#41—22no—
194#41—22no—
195#41—24no—
196#41—22no—
197#41—22no—
198#41—622no—
199#41—622no—
200#41C22⋊D3114yescomplemented
201#41C22×D3114yes—
202C2×Q8—22no—
203C2×Q8—24no—
204C2×Q8—311no—
205C2×Q8C22⋊D3112yescomplemented
206C2×Q8C22×D3111yescharacteristic
207C2×D4—28no—
208C2×D4—22no—
209C2×D4—24no—
210C2×D4—22no—
211C2×D4—24no—
212C2×D4—311no—
213C2×D4—622no—
214C2×D4—621no—
215C2×D4C22⋊D3112yes—
216C2×D4C22⋊D3112yescomplemented
217C2×D4C22×D3111yescharacteristic
218C22×C4—24no—
219C22×C4—22no—
220C22×C4—621no—
221C22×C4—621no—
222C22×C4C22⋊D3112yes—
223C22×C4C22×D3111yescharacteristic
224Q16—624no—
225SD16—624no—
226SD16—624no—
227D8—624no—
228M16—314no—
229M16—622no—
230C2×C8—622no—
231C4⋊C4—622no—
232C22⋊C4—622no—
233C22⋊C4—622no—
234C42—311no—
235C42—311no—
236C42—621no—
237C23—24no—
238C23—22no—
239C23—621no—
240C23C2×C22⋊D3112yes—
241C23C2×C22⋊D3111yescharacteristic
242Q8—24no—
243Q8—22no—
244Q8—22no—
245Q8—622no—
246Q8C2×C22⋊D3114yes—
247D4—28no—
248D4—24no—
249D4—24no—
250D4—24no—
251D4—24no—
252D4—22no—
253D4—22no—
254D4—622no—
255D4—1244no—
256D4—1242no—
257D4C2×C22⋊D3114yes—
258C2×C4—24no—
259C2×C4—28no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—24no—
263C2×C4—22no—
264C2×C4—24no—
265C2×C4—621no—
266C2×C4—621no—
267C2×C4—621no—
268C2×C4—1242no—
269C2×C4—1241no—
270C2×C4—1241no—
271C2×C4C2×C22⋊D3112yes—
272C2×C4C2×C22⋊D3111yescharacteristic
273C2×C4C2×C22⋊D3112yes—
274C2×C4C2×C22⋊D3112yes—
275C2×C4C23×D3111yescharacteristic
276C8—624no—
277C22—22no—
278C22—24no—
279C22—41no—
280C22—42no—
281C22—82no—
282C22—84no—
283C22—1242no—
284C22—1241no—
285C22#1175853012yes—
286C22C22×C22⋊D3111yescharacteristic
287C4—24no—
288C4—22no—
289C4—622no—
290C4—1241no—
291C4—1241no—
292C4#1175853011yescharacteristic
293C4#1175853011yescharacteristic
294C4C22×C22⋊D3112yes—
295C2—21no—
296C2—22no—
297C2—42no—
298C2—44no—
299C2—2481no—
300C2C2×#1175853011yescharacteristic
subgroupG/H
centerC4#11758530
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#585C2
socleC2×C31C2×#120
hypercenter#585D31
hypocenterC31#3018
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C4 ◁ C22×C4 ◁ #585

Complements

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

NH ≅ G/Ntype
1. #14521847C2semidirect
2. #14521595C2semidirect
3. #14521232C2semidirect
4. #14521227C2semidirect
5. #14521222C2semidirect
6. #14521217C2semidirect
7. C2×#11758425C2semidirect
8. #14521033C2semidirect
9. C31×#585C2semidirect
19. C2×#92495C22semidirect
20. #11758528C22semidirect
22. #11758526C22semidirect
23. #11758524C22semidirect
24. #11758521C22semidirect
26. #11758517C22semidirect
28. #11758513C22semidirect
31. C2×C4×Dic31C22semidirect
32. C2×#92474C22semidirect
33. #11758425C22semidirect
34. #11758425C22semidirect
35. #11758424C22semidirect
36. #11758421C22semidirect
47. #41×C31D4semidirect
51. C2×Q8×C31D4semidirect
59. C2×D4×C31D4semidirect
145. #585D31semidirect
187. C31#3018semidirect
200. #41C22⋊D31semidirect
205. C2×Q8C22⋊D31semidirect
216. C2×D4C22⋊D31semidirect

Maximal subgroups

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

subgroupindex
#145218472
#145215952
#145212322
#145212272
#145212222
#145212172
C2×#117584252
#145210332
C31×#5852
#301831