Finite Group Databasebeta

← #16854970

Subgroups — #16854970

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×#92499C211yesmaximal, characteristic
2C2×#11758495C212yesmaximal, complemented
3C22⋊Q8×D31C211yesmaximal, characteristic, complemented
4#14521214C212yesmaximal, complemented
5#14521210C212yesmaximal, complemented
6#14521205C211yesmaximal, characteristic, complemented
7#14521202C211yesmaximal, characteristic, complemented
8#14521199C211yesmaximal, characteristic, complemented
9#14521049C212yesmaximal, complemented
10#14521011C211yesmaximal, characteristic, complemented
11C31×#451C211yesmaximal, characteristic, complemented
12C22×Q8×C31C2211yescharacteristic
13C2×#92499—28no—
14C2×#92499—21no—
15C2×#92499—21no—
16C2×#92499C2212yes—
17C2×#92499C2212yes—
18C2×Q8×D31C2211yescharacteristic, complemented
19C22×D124—21no—
20C22×D124C2211yescharacteristic
21C22×C4×D31—21no—
22C22×C4×D31C2211yescharacteristic
23C2×Q16×C31C2212yescomplemented
24C31×C22⋊Q8C2211yescharacteristic
25#11758523C2211yescharacteristic, complemented
26C2×#92482C2212yescomplemented
27C2×#92481C2212yescomplemented
28#11758501C2211yescharacteristic
29#11758495—28no—
30C2×#92470C2212yescomplemented
31C2×#92469C2212yescomplemented
32#11758474C2211yescharacteristic, complemented
33#11758473—21no—
34C4⋊C4×D31—21no—
35C4⋊C4×D31C2211yescharacteristic, complemented
36C22⋊C4×D31—21no—
37#11758455—21no—
38C31×#103C2212yescomplemented
39C31×C22⋊C8C2211yescharacteristic
40#11758423C2212yescomplemented
41C22⋊(C31⋊C8)C2211yescharacteristic
42#11758409C2212yescomplemented
43(C2×D31)⋊C8C2212yescomplemented
44#11758397C2212yescomplemented
45C23×D31—21no—
46C23×D31D411yescharacteristic
47C2×Q8×C31—24no—
48C2×Q8×C31—21no—
49C2×Q8×C31C2311yescharacteristic
50C2×Q8×C31C2312yes—
51C22×C4×C31—21no—
52C22×C4×C31C2311yescharacteristic
53C22×Dic31D411yescharacteristic
54#92499—28no—
55#92499—28no—
56#92499—24no—
57#92499—24no—
58#92499—24no—
59#92499D418yes—
60Q8×D31—24no—
61C2×D124—28no—
62C2×D124—22no—
63C2×D124—22no—
64C2×D124—21no—
65C2×D124—24no—
66C2×D124C2312yes—
67C2×C4×D31—28no—
68C2×C4×D31—22no—
69C2×C4×D31—22no—
70C2×C4×D31—24no—
71C2×C4×D31—21no—
72C2×C4×D31—21no—
73C2×C4×D31D411yescharacteristic
74C2×C4×D31D411yescharacteristic
75C2×C4×D31C2311yescharacteristic
76C2×C4×D31C2311yescharacteristic
77C2×Dic62—21no—
78C2×Dic62C2311yescharacteristic
79Q16×C31—24no—
80C2×C8×C31C2312yes—
81C4⋊C4×C31—21no—
82C4⋊C4×C31C2311yescharacteristic
83C22⋊C4×C31—21no—
84C22⋊Dic31—21no—
85#92482—24no—
86#92481—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—
95#92470—24no—
96#92469—24no—
97C23×C31C2×D411yescharacteristic
98C22×D31—24no—
99C22×D31—22no—
100C22×D31—22no—
101C22×D31—22no—
102C22×D31—41no—
103C22×D31—42no—
104C22×D31—42no—
105C22×D31C2×D411yescharacteristic
106C22×D31C2×D411yescharacteristic
107Q8×C31—24no—
108Q8×C31—22no—
109Q8×C31—22no—
110Q8×C31C2×D414yes—
111C2×C4×C31—24no—
112C2×C4×C31—22no—
113C2×C4×C31—22no—
114C2×C4×C31—21no—
115C2×C4×C31—21no—
116C2×C4×C31C2×D411yescharacteristic
117C2×C4×C31C2411yescharacteristic
118C2×Dic31—22no—
119C2×Dic31—21no—
120C2×Dic31—21no—
121C2×Dic31C2×D411yescharacteristic
122C2×Dic31C2×D411yescharacteristic
123D124—28no—
124D124—24no—
125D124—28no—
126D124—22no—
127D124C2×D414yes—
128C4×D31—28no—
129C4×D31—22no—
130C4×D31—22no—
131C4×D31—28no—
132C4×D31—22no—
133C4×D31—42no—
134C4×D31—42no—
135C4×D31C2×D414yes—
136Dic62—22no—
137Dic62—42no—
138C8×C31—22no—
139C31⋊C8—22no—
140#3007—311nomaximal, Sylow
141C22×C31—21no—
142C22×C31—21no—
143C22×C31#13712yescomplemented
144C22×C31C22×D411yescharacteristic
145C2×D31—21no—
146C2×D31—21no—
147C2×D31—21no—
148C2×D31—24no—
149C2×D31—21no—
150C2×D31—21no—
151C2×D31—44no—
152C2×D31—48no—
153C2×D31—42no—
154C2×D31—42no—
155C2×D31#12012yes—
156C4×C31—24no—
157C4×C31—21no—
158C4×C31—41no—
159C4×C31—41no—
160C4×C31C22×D412yes—
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#451D3111yescharacteristic, complemented
172#450—311no—
173#450—311no—
174#417—312no—
175#407—311no—
176C2×C31—22no—
177C2×C31C2×#12011yescharacteristic
178C2×C31C2×#13711yescharacteristic
179C2×C31#57511yescharacteristic
180D31—22no—
181D31—41no—
182D31—44no—
183C2×#41—312no—
184C2×#41—312no—
185C2×#41—621no—
186C2×#41—621no—
187C2×#41—628no—
188C22×Q8—311no—
189C22×Q8C2×D3111yescharacteristic
190C22×D4—311no—
191C22×D4—621no—
192C23×C4—311no—
193C23×C4—621no—
194#137—628no—
195C2×Q16—312no—
196C2×Q16C2×D3112yescomplemented
197C2×SD16—312no—
198C2×SD16—312no—
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#103C2×D3112yescomplemented
211#102—312no—
212#102—312no—
213C22⋊C8—311no—
214C22⋊C8—312no—
215C22⋊C8C2×D3111yescharacteristic
216C31#300711yescharacteristic, complemented, Sylow
217C24—311no—
218C24—621no—
219#41—318no—
220#41—624no—
221#41—624no—
222#41—628no—
223#41—628no—
224#41—624no—
225C2×Q8—24no—
226C2×Q8—21no—
227C2×Q8—311no—
228C2×Q8—621no—
229C2×Q8—624no—
230C2×Q8C22×D3111yescharacteristic
231C2×Q8C22×D3112yes—
232C2×D4—312no—
233C2×D4—624no—
234C2×D4—622no—
235C2×D4—622no—
236C2×D4—628no—
237C2×D4—621no—
238C22×C4—21no—
239C22×C4—311no—
240C22×C4—311no—
241C22×C4—311no—
242C22×C4—311no—
243C22×C4—311no—
244C22×C4—624no—
245C22×C4—622no—
246C22×C4—622no—
247C22×C4—628no—
248C22×C4—621no—
249C22×C4—621no—
250C22×C4C22×D3111yescharacteristic
251Q16—24no—
252Q16—624no—
253SD16—624no—
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—311no—
270C23—311no—
271C23—622no—
272C23—624no—
273C23—622no—
274C23—622no—
275C23—1241no—
276C23—1242no—
277C23—1242no—
278C23D4×D3111yescharacteristic
279Q8—24no—
280Q8—22no—
281Q8—22no—
282Q8—622no—
283Q8—1242no—
284Q8D4×D3114yes—
285D4—314no—
286D4—628no—
287D4—628no—
288D4—624no—
289D4—622no—
290C2×C4—24no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—311no—
296C2×C4—314no—
297C2×C4—311no—
298C2×C4—622no—
299C2×C4—628no—
300C2×C4—622no—
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—312no—
315C22—621no—
316C22—621no—
317C22—624no—
318C22—621no—
319C22—621no—
320C22—621no—
321C22—1244no—
322C22—1248no—
323C22—1242no—
324C22—1242no—
325C22#1175849512yescomplemented
326C22C2×D4×D3111yescharacteristic
327C4—24no—
328C4—21no—
329C4—41no—
330C4—41no—
331C4—312no—
332C4—621no—
333C4—1241no—
334C4—1241no—
335C4C2×D4×D3112yes—
336C2—22no—
337C2—622no—
338C2—1241no—
339C2—1244no—
340C2#120×D3111yescharacteristic
341C2C2×#1175849511yescharacteristic
342C2#1452184811yescharacteristic
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#451C2
socleC22×C31C22×D4
prime radicalC22×#92499C2
hypercenter#451D31
hypocenterC31#3007
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
2. C2×#11758495C2semidirect
3. C22⋊Q8×D31C2semidirect
4. #14521214C2semidirect
5. #14521210C2semidirect
6. #14521205C2semidirect
7. #14521202C2semidirect
8. #14521199C2semidirect
9. #14521049C2semidirect
10. #14521011C2semidirect
11. C31×#451C2semidirect
18. C2×Q8×D31C22semidirect
23. C2×Q16×C31C22semidirect
25. #11758523C22semidirect
26. C2×#92482C22semidirect
27. C2×#92481C22semidirect
30. C2×#92470C22semidirect
31. C2×#92469C22semidirect
32. #11758474C22semidirect
35. C4⋊C4×D31C22semidirect
38. C31×#103C22semidirect
40. #11758423C22semidirect
42. #11758409C22semidirect
43. (C2×D31)⋊C8C22semidirect
44. #11758397C22semidirect
143. C22×C31#137semidirect
171. #451D31semidirect
196. C2×Q16C2×D31semidirect
210. #103C2×D31semidirect
216. C31#3007semidirect
325. C22#11758495semidirect

Maximal subgroups

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

subgroupindex
C22×#924992
C2×#117584952
C22⋊Q8×D312
#145212142
#145212102
#145212052
#145212022
#145211992
#145210492
#145210112
C31×#4512
#300731