Finite Group Databasebeta

← #16854966

Subgroups — #16854966

Order 3968.

754 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 754 conjugacy classes of subgroups — too many to draw. Only its 141 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×#92497C211yesmaximal, characteristic
2C2×#11758492C212yesmaximal, complemented
3C22⋊Q8×D31C211yesmaximal, characteristic, complemented
4#14521212C212yesmaximal, complemented
5#14521208C212yesmaximal, complemented
6#14521207C211yesmaximal, characteristic, complemented
7#14521204C211yesmaximal, characteristic, complemented
8#14521200C211yesmaximal, characteristic, complemented
9#14521047C212yesmaximal, complemented
10#14521011C211yesmaximal, characteristic, complemented
11C31×#450C211yesmaximal, characteristic, complemented
12C22×D4×C31C2211yescharacteristic
13C22×C22⋊D31—21no—
14C23×Dic31—21no—
15C2×Q8×D31C2211yescharacteristic, complemented
16C2×#92497—28no—
17C2×#92497—21no—
18C2×#92497—21no—
19C2×#92497C2212yes—
20C2×#92497C2212yes—
21C22×C4×D31C2211yescharacteristic
22C22×Dic62C2211yescharacteristic
23C2×SD16×C31C2212yescomplemented
24C31×C22⋊Q8C2211yescharacteristic
25#11758523C2211yescharacteristic, complemented
26C2×#92482C2212yescomplemented
27C2×#92480C2212yescomplemented
28#11758501C2211yescharacteristic
29#11758492—28no—
30C2×Dic124C2212yescomplemented
31C2×#92469C2212yescomplemented
32#11758474C2211yescharacteristic, complemented
33#11758473—21no—
34C4⋊C4×D31—21no—
35C4⋊C4×D31C2211yescharacteristic, complemented
36C22⋊C4×D31—21no—
37#11758455—21no—
38C31×#102C2212yescomplemented
39C31×C22⋊C8C2211yescharacteristic
40#11758420C2212yescomplemented
41C22⋊(C31⋊C8)C2211yescharacteristic
42(C2×D31)⋊C8C2212yescomplemented
43#11758404C2212yescomplemented
44#11758398C2212yescomplemented
45C24×C31—21no—
46C23×D31D411yescharacteristic
47C2×Q8×C31C2311yescharacteristic
48C2×D4×C31—24no—
49C2×D4×C31—21no—
50C2×D4×C31C2312yes—
51C22×C4×C31C2311yescharacteristic
52C2×C22⋊D31—28no—
53C2×C22⋊D31—22no—
54C2×C22⋊D31—22no—
55C22×Dic31—28no—
56C22×Dic31—21no—
57C22×Dic31—22no—
58C22×Dic31—22no—
59C22×Dic31D411yescharacteristic
60Q8×D31—24no—
61#92497—28no—
62#92497—28no—
63#92497—24no—
64#92497—24no—
65#92497—24no—
66#92497D418yes—
67C2×C4×D31—24no—
68C2×C4×D31—21no—
69C2×C4×D31—21no—
70C2×C4×D31D411yescharacteristic
71C2×C4×D31D411yescharacteristic
72C2×C4×D31C2311yescharacteristic
73C2×C4×D31C2311yescharacteristic
74C2×Dic62—21no—
75C2×Dic62—24no—
76C2×Dic62—21no—
77C2×Dic62C2311yescharacteristic
78C2×Dic62C2312yes—
79SD16×C31—24no—
80C2×C8×C31C2312yes—
81C4⋊C4×C31—21no—
82C4⋊C4×C31C2311yescharacteristic
83C22⋊C4×C31—21no—
84C22⋊Dic31—21no—
85#92482—24no—
86#92480—24no—
87(C2×D31)⋊C4—21no—
88(C2×D31)⋊C4—21no—
89C4⋊Dic31—21no—
90C4⋊Dic31C2311yescharacteristic
91Dic31⋊C4—21no—
92Dic31⋊C4—21no—
93Dic31⋊C4—21no—
94C2×C31⋊C8C2312yes—
95Dic124—24no—
96#92469—24no—
97C23×C31—24no—
98C23×C31—22no—
99C23×C31—42no—
100C23×C31—42no—
101C23×C31C2×D411yescharacteristic
102C22×D31—22no—
103C22×D31—22no—
104C22×D31—41no—
105C22×D31C2×D411yescharacteristic
106C22×D31C2×D411yescharacteristic
107Q8×C31—22no—
108D4×C31—24no—
109D4×C31—22no—
110D4×C31C2×D414yes—
111C2×C4×C31—22no—
112C2×C4×C31—21no—
113C2×C4×C31—21no—
114C2×C4×C31C2×D411yescharacteristic
115C2×C4×C31C2411yescharacteristic
116C22⋊D31—28no—
117C22⋊D31—28no—
118C2×Dic31—24no—
119C2×Dic31—28no—
120C2×Dic31—22no—
121C2×Dic31—22no—
122C2×Dic31—28no—
123C2×Dic31—21no—
124C2×Dic31—21no—
125C2×Dic31C2×D411yescharacteristic
126C2×Dic31C2×D411yescharacteristic
127C4×D31—22no—
128C4×D31—22no—
129C4×D31—22no—
130C4×D31—42no—
131C4×D31—42no—
132C4×D31C2×D414yes—
133Dic62—24no—
134Dic62—22no—
135Dic62—22no—
136Dic62—42no—
137Dic62C2×D414yes—
138C8×C31—22no—
139C31⋊C8—22no—
140#3007—311nomaximal, Sylow
141C22×C31—21no—
142C22×C31—21no—
143C22×C31—24no—
144C22×C31—48no—
145C22×C31—42no—
146C22×C31—42no—
147C22×C31#13712yescomplemented
148C22×C31C22×D411yescharacteristic
149C2×D31—21no—
150C2×D31—21no—
151C2×D31—21no—
152C2×D31—21no—
153C2×D31—21no—
154C2×D31—44no—
155C2×D31#12012yes—
156C4×C31—21no—
157C4×C31—41no—
158C4×C31—41no—
159C4×C31C22×D412yes—
160Dic31—24no—
161Dic31—21no—
162Dic31—41no—
163Dic31—41no—
164Dic31#12012yes—
165C22×#41—311no—
166C2×#137—312no—
167C2×C22⋊Q8—311no—
168#452—312no—
169#452—312no—
170#451—311no—
171#451—311no—
172#450—311no—
173#450D3111yescharacteristic, complemented
174#417—312no—
175#407—311no—
176C2×C31—22no—
177C2×C31—44no—
178C2×C31C2×#12011yescharacteristic
179C2×C31C2×#13711yescharacteristic
180C2×C31#57511yescharacteristic
181D31—22no—
182D31—41no—
183C2×#41—312no—
184C2×#41—312no—
185C2×#41—621no—
186C2×#41—621no—
187C2×#41—628no—
188C22×Q8—311no—
189C22×Q8—311no—
190C22×D4—621no—
191C22×D4C2×D3111yescharacteristic
192C23×C4—311no—
193C23×C4—621no—
194#137—628no—
195C2×Q16—312no—
196C2×Q16—312no—
197C2×SD16—312no—
198C2×SD16C2×D3112yescomplemented
199C2×M16—312no—
200C22⋊Q8—311no—
201C22⋊Q8—311no—
202C22⋊Q8—311no—
203C22⋊Q8—621no—
204C22⋊Q8—621no—
205C22⋊Q8C2×D3111yescharacteristic
206C2×C4⋊C4—311no—
207C2×C4⋊C4—621no—
208C2×C22⋊C4—621no—
209#103—312no—
210#103—312no—
211#102—312no—
212#102C2×D3112yescomplemented
213C22⋊C8—311no—
214C22⋊C8—312no—
215C22⋊C8C2×D3111yescharacteristic
216C31#300711yescharacteristic, complemented, Sylow
217C24—21no—
218C24—311no—
219#41—318no—
220#41—624no—
221#41—624no—
222#41—628no—
223#41—628no—
224#41—624no—
225C2×Q8—312no—
226C2×Q8—311no—
227C2×Q8—621no—
228C2×Q8—624no—
229C2×Q8—624no—
230C2×Q8—621no—
231C2×Q8C22×D3111yescharacteristic
232C2×D4—24no—
233C2×D4—21no—
234C2×D4—622no—
235C2×D4—622no—
236C2×D4—628no—
237C2×D4C22×D3112yes—
238C22×C4—311no—
239C22×C4—311no—
240C22×C4—311no—
241C22×C4—311no—
242C22×C4—311no—
243C22×C4—624no—
244C22×C4—621no—
245C22×C4—621no—
246C22×C4—622no—
247C22×C4—622no—
248C22×C4—628no—
249C22×C4—621no—
250C22×C4C22×D3111yescharacteristic
251Q16—624no—
252Q16—624no—
253SD16—24no—
254SD16—624no—
255M16—624no—
256C2×C8—312no—
257C2×C8C22×D3112yes—
258C4⋊C4—21no—
259C4⋊C4—311no—
260C4⋊C4—621no—
261C4⋊C4—621no—
262C4⋊C4—621no—
263C4⋊C4—621no—
264C4⋊C4C22×D3111yescharacteristic
265C22⋊C4—21no—
266C22⋊C4—621no—
267C22⋊C4—621no—
268C22⋊C4—621no—
269C23—24no—
270C23—22no—
271C23—42no—
272C23—42no—
273C23—311no—
274C23—311no—
275C23—622no—
276C23—622no—
277C23—1241no—
278C23D4×D3111yescharacteristic
279Q8—22no—
280Q8—314no—
281Q8—624no—
282Q8—622no—
283Q8—622no—
284Q8—1242no—
285D4—24no—
286D4—22no—
287D4—628no—
288D4—628no—
289D4D4×D3114yes—
290C2×C4—22no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—311no—
294C2×C4—314no—
295C2×C4—311no—
296C2×C4—622no—
297C2×C4—622no—
298C2×C4—622no—
299C2×C4—628no—
300C2×C4—624no—
301C2×C4—628no—
302C2×C4—622no—
303C2×C4—622no—
304C2×C4—621no—
305C2×C4—621no—
306C2×C4—1242no—
307C2×C4—1242no—
308C2×C4D4×D3111yescharacteristic
309C2×C4C23×D3111yescharacteristic
310C8—22no—
311C8—622no—
312C22—21no—
313C22—21no—
314C22—24no—
315C22—48no—
316C22—42no—
317C22—42no—
318C22—312no—
319C22—621no—
320C22—621no—
321C22—621no—
322C22—621no—
323C22—621no—
324C22—1244no—
325C22#1175849212yescomplemented
326C22C2×D4×D3111yescharacteristic
327C4—21no—
328C4—41no—
329C4—41no—
330C4—312no—
331C4—621no—
332C4—624no—
333C4—1241no—
334C4—1241no—
335C4C2×D4×D3112yes—
336C2—22no—
337C2—44no—
338C2—622no—
339C2—1241no—
340C2#120×D3111yescharacteristic
341C2C2×#1175849211yescharacteristic
342C2#1452184511yescharacteristic
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#450C2
socleC22×C31C22×D4
prime radicalC22×#92497C2
hypercenter#450D31
hypocenterC31#3007
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
2. C2×#11758492C2semidirect
3. C22⋊Q8×D31C2semidirect
4. #14521212C2semidirect
5. #14521208C2semidirect
6. #14521207C2semidirect
7. #14521204C2semidirect
8. #14521200C2semidirect
9. #14521047C2semidirect
10. #14521011C2semidirect
11. C31×#450C2semidirect
15. C2×Q8×D31C22semidirect
23. C2×SD16×C31C22semidirect
25. #11758523C22semidirect
26. C2×#92482C22semidirect
27. C2×#92480C22semidirect
30. C2×Dic124C22semidirect
31. C2×#92469C22semidirect
32. #11758474C22semidirect
35. C4⋊C4×D31C22semidirect
38. C31×#102C22semidirect
40. #11758420C22semidirect
42. (C2×D31)⋊C8C22semidirect
43. #11758404C22semidirect
44. #11758398C22semidirect
147. C22×C31#137semidirect
173. #450D31semidirect
198. C2×SD16C2×D31semidirect
212. #102C2×D31semidirect
216. C31#3007semidirect
325. C22#11758492semidirect

Maximal subgroups

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

subgroupindex
C22×#924972
C2×#117584922
C22⋊Q8×D312
#145212122
#145212082
#145212072
#145212042
#145212002
#145210472
#145210112
C31×#4502
#300731