Finite Group Databasebeta

← #16854946

Subgroups — #16854946

Order 3968.

738 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 738 conjugacy classes of subgroups — too many to draw. Only its 143 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
1C22×Q8×D31C211yesmaximal, characteristic, complemented
2C2×#11758495C212yesmaximal, complemented
3#14521562C211yesmaximal, characteristic
4#14521204C211yesmaximal, characteristic, complemented
5#14521203C212yesmaximal, complemented
6#14521190C211yesmaximal, characteristic, complemented
7#14521189C212yesmaximal, complemented
8#14521188C211yesmaximal, characteristic, complemented
9#14521056C212yesmaximal, complemented
10#14521011C211yesmaximal, characteristic, complemented
11C31×#451C211yesmaximal, characteristic, complemented
12C22×Q8×C31C2211yescharacteristic
13C2×#92499C2211yescharacteristic
14C2×Q8×D31—28no—
15C2×Q8×D31—21no—
16C2×Q8×D31—21no—
17C2×Q8×D31C2212yes—
18C2×Q8×D31C2212yes—
19C22×C4×D31—21no—
20C22×C4×D31C2211yescharacteristic
21C22×Dic62—21no—
22C22×Dic62C2211yescharacteristic
23C2×Q16×C31C2212yescomplemented
24C31×C22⋊Q8C2211yescharacteristic
25#11758523C2211yescharacteristic
26C2×#92482C2212yescomplemented
27C2×#92481C2212yescomplemented
28#11758507C2211yescharacteristic
29#11758495—28no—
30C2×#92470C2212yescomplemented
31C2×#92469C2212yes—
32#11758472C2211yescharacteristic
33#11758471—21no—
34#11758470—21no—
35#11758469C2211yescharacteristic
36#11758459—21no—
37(C22⋊D31)⋊C4—21no—
38C31×#103C2212yescomplemented
39C31×C22⋊C8C2211yescharacteristic
40#11758423C2212yescomplemented
41C22⋊(C31⋊C8)C2211yescharacteristic
42(C2×D31)⋊C8C2212yes—
43#11758404C2212yescomplemented
44#11758398C2212yescomplemented
45C23×D31D411yescharacteristic
46C2×Q8×C31—24no—
47C2×Q8×C31—21no—
48C2×Q8×C31C2311yescharacteristic
49C2×Q8×C31C2312yes—
50C22×C4×C31—21no—
51C22×C4×C31C2311yescharacteristic
52C2×C22⋊D31—21no—
53C22×Dic31—21no—
54C22×Dic31D411yescharacteristic
55#92499—24no—
56Q8×D31—28no—
57Q8×D31—28no—
58Q8×D31—24no—
59Q8×D31—24no—
60Q8×D31—24no—
61Q8×D31D418yescomplemented
62C2×D124—21no—
63C2×D124—21no—
64C2×D124C2311yescharacteristic
65C2×C4×D31—28no—
66C2×C4×D31—22no—
67C2×C4×D31—22no—
68C2×C4×D31—24no—
69C2×C4×D31—21no—
70C2×C4×D31D411yescharacteristic
71C2×C4×D31D411yescharacteristic
72C2×C4×D31C2311yescharacteristic
73C2×C4×D31C2311yescharacteristic
74C2×Dic62—28no—
75C2×Dic62—21no—
76C2×Dic62—22no—
77C2×Dic62—22no—
78C2×Dic62—24no—
79C2×Dic62C2312yes—
80Q16×C31—24no—
81C2×C8×C31C2312yes—
82C4⋊C4×C31—21no—
83C4⋊C4×C31C2311yescharacteristic
84C22⋊C4×C31—21no—
85#92482—24no—
86#92481—24no—
87(C2×D31)⋊C4—21no—
88(C2×D31)⋊C4—21no—
89(C2×D31)⋊C4—21no—
90(C2×D31)⋊C4—21no—
91C4⋊Dic31C2311yescharacteristic
92Dic31⋊C4—21no—
93C4×Dic31—21no—
94C2×C31⋊C8C2312yes—
95#92470—24no—
96#92469—24no—
97C23×C31C2×D411yescharacteristic
98C22×D31—22no—
99C22×D31—22no—
100C22×D31—22no—
101C22×D31—21no—
102C22×D31C2×D411yescharacteristic
103C22×D31C2×D411yescharacteristic
104Q8×C31—24no—
105Q8×C31—22no—
106Q8×C31—22no—
107Q8×C31C2×D414yes—
108C2×C4×C31—24no—
109C2×C4×C31—22no—
110C2×C4×C31—22no—
111C2×C4×C31—21no—
112C2×C4×C31—21no—
113C2×C4×C31C2×D411yescharacteristic
114C2×C4×C31C2411yescharacteristic
115C22⋊D31—42no—
116C2×Dic31—24no—
117C2×Dic31—22no—
118C2×Dic31—22no—
119C2×Dic31—21no—
120C2×Dic31C2×D411yescharacteristic
121C2×Dic31C2×D411yescharacteristic
122D124—22no—
123D124—42no—
124D124—42no—
125C4×D31—28no—
126C4×D31—28no—
127C4×D31—22no—
128C4×D31—22no—
129C4×D31—22no—
130C4×D31—42no—
131C4×D31C2×D414yes—
132Dic62—28no—
133Dic62—24no—
134Dic62—22no—
135Dic62—28no—
136Dic62C2×D414yes—
137C8×C31—22no—
138C31⋊C8—22no—
139#3004—311nomaximal, Sylow
140C22×C31—21no—
141C22×C31—21no—
142C22×C31#13712yescomplemented
143C22×C31C22×D411yescharacteristic
144C2×D31—28no—
145C2×D31—21no—
146C2×D31—21no—
147C2×D31—21no—
148C2×D31—21no—
149C2×D31—41no—
150C2×D31—41no—
151C2×D31—41no—
152C2×D31#12012yes—
153C2×D31#13712yescomplemented
154C4×C31—24no—
155C4×C31—21no—
156C4×C31—41no—
157C4×C31—41no—
158C4×C31C22×D412yes—
159Dic31—21no—
160Dic31—24no—
161Dic31—41no—
162Dic31#12012yes—
163C23×Q8—311no—
164C2×#137—312no—
165#525—311no—
166#451—312no—
167#451—311no—
168#451D3111yescharacteristic, complemented
169#448—311no—
170#448—312no—
171#448—311no—
172#419—312no—
173#407—311no—
174C2×C31—22no—
175C2×C31C2×#12011yescharacteristic
176C2×C31C2×#13711yescharacteristic
177C2×C31C2×#13711yescharacteristic
178D31—22no—
179D31—22no—
180D31—81no—
181C2×#41—311no—
182C22×Q8—312no—
183C22×Q8—312no—
184C22×Q8—311no—
185C22×Q8—621no—
186C22×Q8—621no—
187C22×Q8—621no—
188C22×Q8—628no—
189C22×Q8C2×D3111yescharacteristic
190C23×C4—311no—
191C23×C4—621no—
192#137—628no—
193C2×Q16—312no—
194C2×Q16C2×D3112yescomplemented
195C2×SD16—312no—
196C2×SD16—312no—
197C2×M16—312no—
198#123—621no—
199C22⋊Q8—311no—
200C22⋊Q8C2×D3111yescharacteristic
201#121—311no—
202#121—311no—
203#120—621no—
204C4×D4—621no—
205C4×D4—621no—
206#117—311no—
207#103—312no—
208#103—312no—
209#103—312no—
210#103C2×D3112yescomplemented
211C22⋊C8—311no—
212C22⋊C8—312no—
213C22⋊C8C2×D3111yescharacteristic
214C31#300411yescharacteristic, complemented, Sylow
215C24—311no—
216#41—624no—
217C2×Q8—24no—
218C2×Q8—21no—
219C2×Q8—318no—
220C2×Q8—312no—
221C2×Q8—622no—
222C2×Q8—622no—
223C2×Q8—624no—
224C2×Q8—628no—
225C2×Q8—624no—
226C2×Q8—624no—
227C2×Q8—628no—
228C2×Q8—628no—
229C2×Q8—624no—
230C2×Q8—621no—
231C2×Q8C22×D3111yescharacteristic
232C2×Q8C22×D3112yes—
233C2×D4—311no—
234C2×D4—621no—
235C2×D4—621no—
236C2×D4—621no—
237C22×C4—21no—
238C22×C4—311no—
239C22×C4—311no—
240C22×C4—311no—
241C22×C4—311no—
242C22×C4—311no—
243C22×C4—624no—
244C22×C4—621no—
245C22×C4—622no—
246C22×C4—622no—
247C22×C4—628no—
248C22×C4—621no—
249C22×C4C22×D3111yescharacteristic
250Q16—24no—
251Q16—624no—
252SD16—624no—
253SD16—624no—
254M16—624no—
255C2×C8—312no—
256C2×C8C22×D3112yes—
257C4⋊C4—21no—
258C4⋊C4—311no—
259C4⋊C4—621no—
260C4⋊C4C22×D3111yescharacteristic
261C22⋊C4—21no—
262C22⋊C4—621no—
263C22⋊C4—621no—
264C22⋊C4—621no—
265C22⋊C4—621no—
266C42—621no—
267C23—311no—
268C23—311no—
269C23—622no—
270C23—622no—
271C23—622no—
272C23—621no—
273C23D4×D3111yescharacteristic
274Q8—24no—
275Q8—22no—
276Q8—22no—
277Q8—314no—
278Q8—628no—
279Q8—628no—
280Q8—624no—
281Q8—622no—
282Q8D4×D3114yes—
283D4—622no—
284D4—1242no—
285D4—1242no—
286D4—1242no—
287C2×C4—24no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—311no—
293C2×C4—314no—
294C2×C4—311no—
295C2×C4—622no—
296C2×C4—622no—
297C2×C4—628no—
298C2×C4—628no—
299C2×C4—622no—
300C2×C4—624no—
301C2×C4—622no—
302C2×C4—622no—
303C2×C4—621no—
304C2×C4—1242no—
305C2×C4D4×D3111yescharacteristic
306C2×C4C23×D3111yescharacteristic
307C8—22no—
308C8—622no—
309C22—21no—
310C22—21no—
311C22—312no—
312C22—312no—
313C22—628no—
314C22—621no—
315C22—621no—
316C22—621no—
317C22—621no—
318C22—1241no—
319C22—1241no—
320C22—1241no—
321C22#1175849512yescomplemented
322C22C2×D4×D3111yescharacteristic
323C4—24no—
324C4—21no—
325C4—41no—
326C4—41no—
327C4—312no—
328C4—621no—
329C4—624no—
330C4—1241no—
331C4C2×D4×D3112yes—
332C2—22no—
333C2—622no—
334C2—622no—
335C2—2481no—
336C2#120×D3111yescharacteristic
337C2#137×D3111yescharacteristic
338C2C2×#1175849511yescharacteristic
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#451C2
socleC22×C31C22×D4
prime radical#14521562C2
hypercenter#451D31
hypocenterC31#3004
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
1. C22×Q8×D31C2semidirect
2. C2×#11758495C2semidirect
4. #14521204C2semidirect
5. #14521203C2semidirect
6. #14521190C2semidirect
7. #14521189C2semidirect
8. #14521188C2semidirect
9. #14521056C2semidirect
10. #14521011C2semidirect
11. C31×#451C2semidirect
23. C2×Q16×C31C22semidirect
26. C2×#92482C22semidirect
27. C2×#92481C22semidirect
30. C2×#92470C22semidirect
38. C31×#103C22semidirect
40. #11758423C22semidirect
43. #11758404C22semidirect
44. #11758398C22semidirect
61. Q8×D31D4semidirect
142. C22×C31#137semidirect
153. C2×D31#137semidirect
168. #451D31semidirect
194. C2×Q16C2×D31semidirect
210. #103C2×D31semidirect
214. C31#3004semidirect
321. C22#11758495semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×D312
C2×#117584952
#145215622
#145212042
#145212032
#145211902
#145211892
#145211882
#145210562
#145210112
C31×#4512
#300431