Finite Group Databasebeta

← #16855110

Subgroups — #16855110

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#14521611C212yesmaximal, complemented
3#14521608C211yesmaximal, characteristic, complemented
4#14521240C214yesmaximal, complemented
5#14521237C214yesmaximal, complemented
6C2×#11758394C212yesmaximal, complemented
7C2×C31×#99C211yesmaximal, characteristic, complemented
8C22×D4×C31C2211yescharacteristic
9C22×C22⋊D31—21no—
10C22×C22⋊D31C2212yes—
11C22×C22⋊D31C2211yescharacteristic, complemented
12#11758561—22no—
13#11758561—24no—
14#11758561C2214yes—
15C2×#92497—22no—
16C2×D4×D31—22no—
17C2×#92495—21no—
18C2×C22⋊C4×C31C2212yescomplemented
19#11758530—21no—
20C2×C22⋊Dic31C2211yescharacteristic, complemented
21#11758516—22no—
22#11758514C2212yescomplemented
23#11758512C2212yescomplemented
24#11758507—24no—
25#11758506—21no—
26#11758463—22no—
27#11758463C2214yescomplemented
28#11758459—22no—
29#11758459C2214yescomplemented
30C31×#99C2214yescomplemented
31#11758394C2218yescomplemented
32C24×C31D411yescharacteristic, complemented
33C24×C31C2311yescharacteristic
34C23×D31—21no—
35C23×D31D412yescomplemented
36C23×D31C2311yescharacteristic, complemented
37C2×D4×C31—22no—
38C2×D4×C31—22no—
39C2×D4×C31D412yescomplemented
40C2×D4×C31C2312yes—
41C22×C4×C31—22no—
42C22×C4×C31D411yescharacteristic, complemented
43C2×C22⋊D31—22no—
44C2×C22⋊D31—24no—
45C2×C22⋊D31—22no—
46C2×C22⋊D31—22no—
47C2×C22⋊D31—22no—
48C2×C22⋊D31—22no—
49C2×C22⋊D31—24no—
50C2×C22⋊D31—22no—
51C2×C22⋊D31—21no—
52C2×C22⋊D31—22no—
53C2×C22⋊D31—21no—
54C2×C22⋊D31—22no—
55C2×C22⋊D31D414yescomplemented
56C2×C22⋊D31C2314yes—
57C22×Dic31—21no—
58C22×Dic31—21no—
59C22×Dic31D412yescomplemented
60C22×Dic31C2311yescharacteristic, complemented
61#92497—28no—
62#92497—28no—
63D4×D31—28no—
64D4×D31—28no—
65#92495—24no—
66#92495—24no—
67C2×D124—21no—
68C2×D124—24no—
69C2×C4×D31—22no—
70C2×Dic62—21no—
71C22⋊C4×C31—22no—
72C22⋊C4×C31C2314yescomplemented
73C22⋊Dic31—21no—
74C22⋊Dic31—22no—
75C22⋊Dic31C2312yescomplemented
76(C2×D31)⋊C4—24no—
77(C2×D31)⋊C4—22no—
78(C2×D31)⋊C4—24no—
79C4⋊Dic31—24no—
80Dic31⋊C4—22no—
81C23×C31—21no—
82C23×C31—21no—
83C23×C31—81no—
84C23×C31—81no—
85C23×C31C2×D412yes—
86C23×C31C2×D411yescharacteristic
87C23×C31C2×D411yescharacteristic
88C23×C31C2×D412yes—
89C23×C31C2×D412yes—
90C23×C31C2411yescharacteristic
91C22×D31—22no—
92C22×D31—24no—
93C22×D31—22no—
94C22×D31—22no—
95C22×D31—21no—
96C22×D31—81no—
97C22×D31—82no—
98C22×D31—82no—
99C22×D31C2×D414yes—
100D4×C31—24no—
101D4×C31—24no—
102C2×C4×C31—21no—
103C2×C4×C31—21no—
104C2×C4×C31—24no—
105C2×C4×C31—82no—
106C2×C4×C31C2×D412yes—
107C22⋊D31—28no—
108C22⋊D31—24no—
109C22⋊D31—24no—
110C22⋊D31—24no—
111C22⋊D31—24no—
112C22⋊D31—41no—
113C22⋊D31—42no—
114C22⋊D31—41no—
115C22⋊D31—82no—
116C2×Dic31—24no—
117C2×Dic31—22no—
118C2×Dic31—22no—
119C2×Dic31—21no—
120C2×Dic31—22no—
121C2×Dic31—22no—
122C2×Dic31—81no—
123C2×Dic31C2×D414yes—
124D124—24no—
125D124—84no—
126C4×D31—28no—
127Dic62—24no—
128#3026—311nomaximal, Sylow
129C22×C31—22no—
130C22×C31—41no—
131C22×C31—42no—
132C22×C31—42no—
133C22×C31—41no—
134C22×C31—41no—
135C22×C31—42no—
136C22×C31—42no—
137C22×C31—81no—
138C22×C31#12011yescharacteristic
139C22×C31#12012yes—
140C22×C31#12011yescharacteristic
141C22×C31C22×D411yescharacteristic
142C22×C31C22×D412yes—
143C2×D31—24no—
144C2×D31—44no—
145C2×D31—44no—
146C2×D31—44no—
147C2×D31—42no—
148C2×D31—41no—
149C2×D31—82no—
150C2×D31—84no—
151C4×C31—22no—
152C4×C31—44no—
153Dic31—24no—
154Dic31—42no—
155Dic31—41no—
156C2×H(5,𝔽2)—311no—
157#534—311no—
158#534—312no—
159#458—314no—
160B(4,𝔽2)—314no—
161C2×#99—312no—
162C2×#99D3111yescharacteristic, complemented
163C2×C31—21no—
164C2×C31—22no—
165C2×C31—21no—
166C2×C31—22no—
167C2×C31—21no—
168C2×C31—42no—
169C2×C31C2×#12011yescharacteristic
170D31—44no—
171D31—81no—
172H(5,𝔽2)—314no—
173H(5,𝔽2)—624no—
174H(5,𝔽2)—622no—
175C2×#41—622no—
176C2×#41—621no—
177C22×D4—312no—
178C22×D4—311no—
179C22×D4—622no—
180C22×D4—621no—
181C22×D4C2×D3111yescharacteristic
182#123—312no—
183#123—314no—
184#123—622no—
185#123—621no—
186#121—624no—
187#121—622no—
188#120—312no—
189#120—314no—
190#120—622no—
191#120—621no—
192C2×C22⋊C4—311no—
193C2×C22⋊C4C2×D3112yescomplemented
194#99—318no—
195#99C2×D3114yescomplemented
196C31#302611yescharacteristic, complemented, Sylow
197C24—312no—
198C24—311no—
199C24—621no—
200C24D12411yescharacteristic, complemented
201C24C22×D3111yescharacteristic
202#41—624no—
203#41—628no—
204#41—628no—
205#41—624no—
206C2×Q8—621no—
207C2×D4—22no—
208C2×D4—22no—
209C2×D4—314no—
210C2×D4—314no—
211C2×D4—621no—
212C2×D4—624no—
213C2×D4—628no—
214C2×D4—628no—
215C2×D4—622no—
216C2×D4—624no—
217C2×D4—622no—
218C2×D4—622no—
219C2×D4—624no—
220C2×D4—622no—
221C2×D4—622no—
222C2×D4—621no—
223C2×D4—621no—
224C2×D4—622no—
225C2×D4—622no—
226C2×D4—622no—
227C2×D4D12412yescomplemented
228C2×D4C22×D3112yes—
229C22×C4—22no—
230C22×C4—312no—
231C22×C4—311no—
232C22×C4—622no—
233C22×C4—621no—
234C22×C4—621no—
235C22×C4D12411yescharacteristic, complemented
236C4⋊C4—624no—
237C4⋊C4—622no—
238C22⋊C4—22no—
239C22⋊C4—312no—
240C22⋊C4—624no—
241C22⋊C4—624no—
242C22⋊C4—621no—
243C22⋊C4—622no—
244C22⋊C4—622no—
245C22⋊C4C22×D3114yescomplemented
246C23—21no—
247C23—21no—
248C23—81no—
249C23—81no—
250C23—314no—
251C23—624no—
252C23—622no—
253C23—622no—
254C23—621no—
255C23—622no—
256C23—2481no—
257C23—2482no—
258C23—2482no—
259C23C2×D12412yes—
260C23C2×D12411yescharacteristic
261C23C2×D12411yescharacteristic
262C23D4×D3112yes—
263C23D4×D3112yes—
264C23C23×D3111yescharacteristic
265Q8—624no—
266D4—24no—
267D4—24no—
268D4—624no—
269D4—628no—
270D4—624no—
271D4—624no—
272D4—624no—
273D4—624no—
274D4—1241no—
275D4—1241no—
276D4—1242no—
277D4—2484no—
278D4—2482no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—24no—
282C2×C4—82no—
283C2×C4—314no—
284C2×C4—628no—
285C2×C4—624no—
286C2×C4—622no—
287C2×C4—622no—
288C2×C4—622no—
289C2×C4—621no—
290C2×C4—622no—
291C2×C4—2481no—
292C2×C4C2×D12412yes—
293C22—22no—
294C22—42no—
295C22—41no—
296C22—41no—
297C22—42no—
298C22—41no—
299C22—42no—
300C22—42no—
301C22—81no—
302C22—624no—
303C22—1244no—
304C22—1244no—
305C22—1244no—
306C22—1242no—
307C22—1241no—
308C22—2482no—
309C22—2484no—
310C22#1175845911yescharacteristic
311C22#1175845911yescharacteristic
312C22#1175845912yes—
313C22C22×D12411yescharacteristic
314C22C2×D4×D3112yes—
315C4—22no—
316C4—44no—
317C4—624no—
318C4—1242no—
319C4—1241no—
320C2—22no—
321C2—21no—
322C2—21no—
323C2—21no—
324C2—22no—
325C2—42no—
326C2—1244no—
327C2—2481no—
328C2C2×#1175845911yescharacteristic
subgroupG/H
centerC2C2×#11758459
commutator, cosocleC23×C31C24
FrattiniC23C23×D31
FittingC2×C31×#99C2
socleC2×C31C2×#120
hypercenterC2×#99D31
hypocenterC31#3026
LCSG ▷ C23×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C2 ◁ C24 ◁ C2×#99

Complements

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

NH ≅ G/Ntype
1. C2×#11758561C2semidirect
2. #14521611C2semidirect
3. #14521608C2semidirect
4. #14521240C2semidirect
5. #14521237C2semidirect
6. C2×#11758394C2semidirect
7. C2×C31×#99C2semidirect
11. C22×C22⋊D31C22semidirect
18. C2×C22⋊C4×C31C22semidirect
20. C2×C22⋊Dic31C22semidirect
22. #11758514C22semidirect
23. #11758512C22semidirect
27. #11758463C22semidirect
29. #11758459C22semidirect
30. C31×#99C22semidirect
31. #11758394C22semidirect
32. C24×C31D4semidirect
35. C23×D31D4semidirect
36. C23×D31C23semidirect
39. C2×D4×C31D4semidirect
42. C22×C4×C31D4semidirect
55. C2×C22⋊D31D4semidirect
59. C22×Dic31D4semidirect
60. C22×Dic31C23semidirect
72. C22⋊C4×C31C23semidirect
75. C22⋊Dic31C23semidirect
162. C2×#99D31semidirect
193. C2×C22⋊C4C2×D31semidirect
195. #99C2×D31semidirect
196. C31#3026semidirect
200. C24D124semidirect
227. C2×D4D124semidirect
235. C22×C4D124semidirect
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
#145216112
#145216082
#145212402
#145212372
C2×#117583942
C2×C31×#992
#302631