Finite Group Databasebeta

← #16855107

Subgroups — #16855107

Order 3968.

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

normal subgroups only

This group has 800 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#14521608C212yesmaximal, complemented
3#14521238C214yesmaximal, complemented
4#14521235C214yesmaximal, complemented
5C2×#11758422C211yesmaximal, characteristic, complemented
6C2×#11758394C212yesmaximal, complemented
7C31×#534C211yesmaximal, characteristic, complemented
8C22×D4×C31C2211yescharacteristic
9C22×D4×C31C2211yescharacteristic, complemented
10C22×C22⋊D31—21no—
11C22×C22⋊D31C2212yes—
12#11758561—22no—
13#11758561—24no—
14#11758561C2214yes—
15C2×#92497—22no—
16C2×D4×D31—22no—
17C2×#92495—21no—
18C31×#123—21no—
19C31×#123C2212yescomplemented
20C31×#121—22no—
21C31×#120—21no—
22C31×#120C2212yescomplemented
23C2×C22⋊C4×C31C2211yescharacteristic, complemented
24#11758530C2214yescomplemented
25C2×C22⋊Dic31C2212yescomplemented
26#11758516—24no—
27#11758514—22no—
28#11758512—22no—
29#11758506C2214yescomplemented
30#11758422C2214yescomplemented
31#11758394C2218yescomplemented
32C24×C31D411yescharacteristic, complemented
33C24×C31C2311yescharacteristic
34C24×C31C2311yescharacteristic, complemented
35C23×D31—21no—
36C23×D31D412yescomplemented
37C2×D4×C31—22no—
38C2×D4×C31—22no—
39C2×D4×C31—22no—
40C2×D4×C31—22no—
41C2×D4×C31—21no—
42C2×D4×C31—21no—
43C2×D4×C31—22no—
44C2×D4×C31D412yescomplemented
45C2×D4×C31C2312yes—
46C22×C4×C31—21no—
47C22×C4×C31D411yescharacteristic, complemented
48C22×C4×C31C2311yescharacteristic, complemented
49C2×C22⋊D31—22no—
50C2×C22⋊D31—24no—
51C2×C22⋊D31—22no—
52C2×C22⋊D31—22no—
53C2×C22⋊D31—22no—
54C2×C22⋊D31—22no—
55C2×C22⋊D31—24no—
56C2×C22⋊D31—24no—
57C2×C22⋊D31D414yescomplemented
58C2×C22⋊D31C2314yes—
59C22×Dic31—21no—
60C22×Dic31—22no—
61C22×Dic31D412yescomplemented
62#92497—28no—
63#92497—28no—
64D4×D31—28no—
65D4×D31—28no—
66#92495—24no—
67#92495—24no—
68C2×D124—21no—
69C2×C4×D31—22no—
70C2×Dic62—21no—
71C4⋊C4×C31—22no—
72C22⋊C4×C31—21no—
73C22⋊C4×C31—22no—
74C22⋊C4×C31—22no—
75C22⋊C4×C31C2312yescomplemented
76C22⋊Dic31—22no—
77C22⋊Dic31—24no—
78C22⋊Dic31C2314yescomplemented
79(C2×D31)⋊C4—24no—
80Dic31⋊C4—24no—
81C23×C31—21no—
82C23×C31—21no—
83C23×C31—22no—
84C23×C31—21no—
85C23×C31—81no—
86C23×C31—81no—
87C23×C31—81no—
88C23×C31C2×D412yes—
89C23×C31C2×D412yes—
90C23×C31C2×D412yes—
91C23×C31C2×D411yescharacteristic
92C23×C31C2×D411yescharacteristic
93C23×C31C2411yescharacteristic
94C22×D31—22no—
95C22×D31—24no—
96C22×D31—22no—
97C22×D31—82no—
98C22×D31—82no—
99C22×D31C2×D414yes—
100D4×C31—24no—
101D4×C31—24no—
102D4×C31—41no—
103D4×C31—42no—
104D4×C31—41no—
105D4×C31—82no—
106C2×C4×C31—21no—
107C2×C4×C31—21no—
108C2×C4×C31—22no—
109C2×C4×C31—22no—
110C2×C4×C31—21no—
111C2×C4×C31—81no—
112C2×C4×C31C2×D412yes—
113C22⋊D31—28no—
114C22⋊D31—24no—
115C22⋊D31—24no—
116C22⋊D31—24no—
117C22⋊D31—24no—
118C22⋊D31—84no—
119C2×Dic31—24no—
120C2×Dic31—22no—
121C2×Dic31—22no—
122C2×Dic31—24no—
123C2×Dic31—82no—
124C2×Dic31C2×D414yes—
125D124—24no—
126C4×D31—28no—
127Dic62—24no—
128#3026—311nomaximal, Sylow
129C22×C31—22no—
130C22×C31—42no—
131C22×C31—41no—
132C22×C31—42no—
133C22×C31—41no—
134C22×C31—41no—
135C22×C31—42no—
136C22×C31—42no—
137C22×C31—44no—
138C22×C31—42no—
139C22×C31—41no—
140C22×C31—81no—
141C22×C31#12011yescharacteristic
142C22×C31#12012yes—
143C22×C31#12011yescharacteristic
144C22×C31C22×D412yes—
145C22×C31C22×D411yescharacteristic
146C2×D31—24no—
147C2×D31—44no—
148C2×D31—44no—
149C2×D31—82no—
150C2×D31—84no—
151C4×C31—22no—
152C4×C31—42no—
153C4×C31—41no—
154Dic31—24no—
155Dic31—44no—
156C2×H(5,𝔽2)—311no—
157#534—312no—
158#534D3111yescharacteristic, complemented
159#458—314no—
160B(4,𝔽2)—314no—
161C2×#99—311no—
162C2×#99—312no—
163C2×C31—22no—
164C2×C31—21no—
165C2×C31—21no—
166C2×C31—22no—
167C2×C31—21no—
168C2×C31—42no—
169C2×C31—81no—
170C2×C31C2×#12011yescharacteristic
171D31—44no—
172H(5,𝔽2)—314no—
173H(5,𝔽2)—624no—
174H(5,𝔽2)—622no—
175C2×#41—622no—
176C2×#41—621no—
177C22×D4—312no—
178C22×D4—622no—
179C22×D4—621no—
180C22×D4C2×D3111yescharacteristic
181C22×D4C2×D3111yescharacteristic, complemented
182#123—21no—
183#123—314no—
184#123—622no—
185#123C2×D3112yescomplemented
186#121—22no—
187#121—624no—
188#120—21no—
189#120—314no—
190#120—622no—
191#120C2×D3112yescomplemented
192C2×C22⋊C4—312no—
193C2×C22⋊C4C2×D3111yescharacteristic, complemented
194#99—314no—
195#99—318no—
196C31#302611yescharacteristic, complemented, Sylow
197C24—312no—
198C24—621no—
199C24C22⋊D3111yescharacteristic, complemented
200C24C22×D3111yescharacteristic
201C24C22×D3111yescharacteristic, complemented
202#41—624no—
203#41—628no—
204#41—628no—
205#41—624no—
206C2×Q8—621no—
207C2×D4—22no—
208C2×D4—22no—
209C2×D4—22no—
210C2×D4—22no—
211C2×D4—21no—
212C2×D4—21no—
213C2×D4—22no—
214C2×D4—314no—
215C2×D4—314no—
216C2×D4—621no—
217C2×D4—628no—
218C2×D4—628no—
219C2×D4—622no—
220C2×D4—624no—
221C2×D4—622no—
222C2×D4—622no—
223C2×D4—622no—
224C2×D4—622no—
225C2×D4—624no—
226C2×D4—624no—
227C2×D4C22⋊D3112yescomplemented
228C2×D4C22×D3112yes—
229C22×C4—21no—
230C22×C4—312no—
231C22×C4—622no—
232C22×C4—621no—
233C22×C4—622no—
234C22×C4C22⋊D3111yescharacteristic, complemented
235C22×C4C22×D3111yescharacteristic, complemented
236C4⋊C4—22no—
237C4⋊C4—624no—
238C22⋊C4—21no—
239C22⋊C4—22no—
240C22⋊C4—22no—
241C22⋊C4—314no—
242C22⋊C4—624no—
243C22⋊C4—624no—
244C22⋊C4—622no—
245C22⋊C4C22×D3112yescomplemented
246C23—21no—
247C23—21no—
248C23—22no—
249C23—21no—
250C23—81no—
251C23—81no—
252C23—81no—
253C23—314no—
254C23—624no—
255C23—622no—
256C23—622no—
257C23—2482no—
258C23—2482no—
259C23D4×D3112yes—
260C23D4×D3112yes—
261C23C2×C22⋊D3112yes—
262C23C2×C22⋊D3111yescharacteristic
263C23C2×C22⋊D3111yescharacteristic
264C23C23×D3111yescharacteristic
265Q8—624no—
266D4—24no—
267D4—24no—
268D4—41no—
269D4—41no—
270D4—42no—
271D4—82no—
272D4—624no—
273D4—628no—
274D4—624no—
275D4—624no—
276D4—624no—
277D4—624no—
278D4—2484no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—21no—
284C2×C4—81no—
285C2×C4—314no—
286C2×C4—628no—
287C2×C4—624no—
288C2×C4—622no—
289C2×C4—622no—
290C2×C4—624no—
291C2×C4—2482no—
292C2×C4C2×C22⋊D3112yes—
293C22—22no—
294C22—41no—
295C22—42no—
296C22—41no—
297C22—41no—
298C22—42no—
299C22—42no—
300C22—42no—
301C22—42no—
302C22—44no—
303C22—41no—
304C22—81no—
305C22—624no—
306C22—1244no—
307C22—1244no—
308C22—2482no—
309C22—2484no—
310C22#1175851411yescharacteristic
311C22#1175851411yescharacteristic
312C22#1175851412yes—
313C22C2×D4×D3112yes—
314C22C22×C22⋊D3111yescharacteristic
315C4—22no—
316C4—42no—
317C4—41no—
318C4—624no—
319C4—1244no—
320C2—21no—
321C2—22no—
322C2—21no—
323C2—21no—
324C2—22no—
325C2—42no—
326C2—81no—
327C2—1244no—
328C2C2×#1175851411yescharacteristic
subgroupG/H
centerC2C2×#11758514
commutator, cosocleC23×C31C24
FrattiniC23C23×D31
FittingC31×#534C2
socleC2×C31C2×#120
hypercenter#534D31
hypocenterC31#3026
LCSG ▷ C23×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C2 ◁ C24 ◁ #534

Complements

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

NH ≅ G/Ntype
1. C2×#11758561C2semidirect
2. #14521608C2semidirect
3. #14521238C2semidirect
4. #14521235C2semidirect
5. C2×#11758422C2semidirect
6. C2×#11758394C2semidirect
7. C31×#534C2semidirect
9. C22×D4×C31C22semidirect
19. C31×#123C22semidirect
22. C31×#120C22semidirect
23. C2×C22⋊C4×C31C22semidirect
24. #11758530C22semidirect
25. C2×C22⋊Dic31C22semidirect
29. #11758506C22semidirect
30. #11758422C22semidirect
31. #11758394C22semidirect
32. C24×C31D4semidirect
34. C24×C31C23semidirect
36. C23×D31D4semidirect
44. C2×D4×C31D4semidirect
47. C22×C4×C31D4semidirect
48. C22×C4×C31C23semidirect
57. C2×C22⋊D31D4semidirect
61. C22×Dic31D4semidirect
75. C22⋊C4×C31C23semidirect
78. C22⋊Dic31C23semidirect
158. #534D31semidirect
181. C22×D4C2×D31semidirect
185. #123C2×D31semidirect
191. #120C2×D31semidirect
193. C2×C22⋊C4C2×D31semidirect
196. C31#3026semidirect
199. C24C22⋊D31semidirect
201. C24C22×D31semidirect
227. C2×D4C22⋊D31semidirect
234. C22×C4C22⋊D31semidirect
235. C22×C4C22×D31semidirect
245. C22⋊C4C22×D31semidirect

Maximal subgroups

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

subgroupindex
C2×#117585612
#145216082
#145212382
#145212352
C2×#117584222
C2×#117583942
C31×#5342
#302631