Finite Group Databasebeta

← #16828717

Subgroups — #16828717

Order 3712.

994 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 994 conjugacy classes of subgroups — too many to draw. Only its 153 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×D4×D29C211yesmaximal, characteristic, complemented
2C2×#11745311C211yesmaximal, characteristic, complemented
3#14271670C211yesmaximal, characteristic, complemented
4#14271312C214yesmaximal, complemented
5#14271295C214yesmaximal, complemented
6#14271169C211yesmaximal, characteristic, complemented
7#14271168C211yesmaximal, characteristic, complemented
8#14271126C211yesmaximal, characteristic, complemented
9C2×C29×AGL(1,ℤ/8)C211yesmaximal, characteristic, complemented
10C24×D29—21no—
11C2×#41×C29C2211yescharacteristic, complemented
12C22×D4×C29C2211yescharacteristic
13C22×C22⋊D29—21no—
14C2×D4×D29—24no—
15C2×D4×D29—24no—
16C2×D4×D29—22no—
17C2×D4×D29C2214yes—
18C22×D116C2211yescharacteristic
19C22×C4×D29C2211yescharacteristic
20C29×AGL(1,ℤ/8)—24no—
21C2×SD16×C29C2212yescomplemented
22C2×D8×C29C2212yescomplemented
23C2×M16×C29C2211yescharacteristic, complemented
24#11745311—24no—
25#11745308C2212yescomplemented
26C2×#90784C2212yescomplemented
27#11745300C2212yescomplemented
28#11745299—21no—
29#11745297—21no—
30C2×#90782C2212yescomplemented
31C4×C22⋊D29—22no—
32#11745288C2211yescharacteristic, complemented
33C2×#90777C2211yescharacteristic, complemented
34#11745180C2214yescomplemented
35#11745169C2214yescomplemented
36#11745168C2214yescomplemented
37C24×C29—21no—
38C23×D29—28no—
39C23×D29—21no—
40C23×D29—24no—
41C23×D29—48no—
42C23×D29D411yescharacteristic
43#41×C29—22no—
44#41×C29—22no—
45C2×Q8×C29C2311yescharacteristic, complemented
46C2×D4×C29—21no—
47C2×D4×C29—21no—
48C2×D4×C29—22no—
49C2×D4×C29D414yescomplemented
50C2×D4×C29C2312yes—
51C2×D4×C29C2311yescharacteristic, complemented
52C22×C4×C29—21no—
53C22×C4×C29C2311yescharacteristic
54C2×C22⋊D29—28no—
55C2×C22⋊D29—22no—
56C2×C22⋊D29—24no—
57C22×Dic29D411yescharacteristic
58D4×D29—216no—
59D4×D29—28no—
60D4×D29—28no—
61C2×D116—21no—
62C2×D116—22no—
63C2×D116D414yescomplemented
64C2×D116C2312yes—
65C2×C4×D29—22no—
66C2×C4×D29—22no—
67C2×C4×D29D412yes—
68C2×C4×D29C2312yes—
69SD16×C29—24no—
70D8×C29—24no—
71M16×C29—22no—
72C2×C8×C29C2312yescomplemented
73C22⋊Dic29—21no—
74C22⋊Dic29—22no—
75#90784—24no—
76#90782—24no—
77#90781—22no—
78C4⋊Dic29C2312yescomplemented
79#90779—22no—
80C4×Dic29—21no—
81#90777—22no—
82C2×#6597C2312yescomplemented
83C23×C29—24no—
84C23×C29—22no—
85C23×C29—21no—
86C23×C29—44no—
87C23×C29C2×D411yescharacteristic
88C22×D29—22no—
89C22×D29—24no—
90C22×D29—216no—
91C22×D29—24no—
92C22×D29—22no—
93C22×D29—432no—
94C22×D29—48no—
95C22×D29—44no—
96C22×D29—48no—
97C22×D29C2×D412yes—
98Q8×C29—21no—
99Q8×C29—21no—
100D4×C29—24no—
101D4×C29—22no—
102D4×C29—21no—
103D4×C29—21no—
104D4×C29—42no—
105D4×C29C2×D414yes—
106C2×C4×C29—21no—
107C2×C4×C29—42no—
108C2×C4×C29C2×D411yescharacteristic
109C2×C4×C29C2×D412yes—
110C2×C4×C29C2×D412yes—
111C2×C4×C29C2411yescharacteristic
112C22⋊D29—216no—
113C22⋊D29—44no—
114C2×Dic29—22no—
115C2×Dic29—22no—
116C2×Dic29C2×D412yes—
117D116—24no—
118D116—22no—
119D116C2×D414yes—
120C4×D29—24no—
121C4×D29—22no—
122C4×D29—22no—
123C8×C29—22no—
124#6597—22no—
125#3003—291nomaximal, Sylow
126C22×C29—22no—
127C22×C29—24no—
128C22×C29—48no—
129C22×C29—44no—
130C22×C29—41no—
131C22×C29—42no—
132C22×C29#12012yes—
133C22×C29C22×D411yescharacteristic
134C2×D29—28no—
135C2×D29—22no—
136C2×D29—22no—
137C2×D29—24no—
138C2×D29—432no—
139C2×D29—48no—
140C2×D29—44no—
141C2×D29AGL(1,ℤ/8)14yescomplemented
142C4×C29—41no—
143C4×C29#12012yes—
144C4×C29C22×D411yescharacteristic
145C4×C29C22×D411yescharacteristic
146Dic29—22no—
147Dic29—42no—
148C23×D4—291no—
149C2×AGL(1,ℤ/8)—291no—
150C2×AGL(1,ℤ/8)D2911yescharacteristic, complemented
151#525—291no—
152#450—294no—
153#447—294no—
154#418—291no—
155#418—291no—
156#407—291no—
157C2×C29—22no—
158C2×C29—44no—
159C2×C29—81no—
160C2×C29C2×#12011yescharacteristic
161C2×C29C2×AGL(1,ℤ/8)12yes—
162D29—24no—
163D29—44no—
164C25—581no—
165C2×#41C2×D2911yescharacteristic, complemented
166C22×D4—294no—
167C22×D4—291no—
168C22×D4—582no—
169C22×D4—581no—
170C22×D4—584no—
171C22×D4—584no—
172C22×D4C2×D2911yescharacteristic
173C23×C4—291no—
174AGL(1,ℤ/8)—24no—
175AGL(1,ℤ/8)—584no—
176C2×SD16—292no—
177C2×SD16C2×D2912yescomplemented
178C2×D8—292no—
179C2×D8C2×D2912yescomplemented
180C2×M16—291no—
181C2×M16C2×D2911yescharacteristic, complemented
182#123—581no—
183C22⋊Q8—292no—
184#121—292no—
185#120—581no—
186C4×D4—582no—
187#117—291no—
188#102—294no—
189#102—294no—
190C22⋊C8—294no—
191C29#300311yescharacteristic, complemented, Sylow
192C24—21no—
193C24—291no—
194C24—584no—
195C24—588no—
196C24—581no—
197C24—1168no—
198#41—22no—
199#41—22no—
200C2×Q8C22×D2911yescharacteristic, complemented
201C2×D4—21no—
202C2×D4—21no—
203C2×D4—22no—
204C2×D4—294no—
205C2×D4—292no—
206C2×D4—584no—
207C2×D4—588no—
208C2×D4—5816no—
209C2×D4—588no—
210C2×D4—588no—
211C2×D4—582no—
212C2×D4—581no—
213C2×D4—582no—
214C2×D4C22⋊D2914yescomplemented
215C2×D4C22×D2912yes—
216C2×D4C22×D2911yescharacteristic, complemented
217C22×C4—21no—
218C22×C4—291no—
219C22×C4—292no—
220C22×C4—292no—
221C22×C4—582no—
222C22×C4—582no—
223C22×C4C22×D2911yescharacteristic
224SD16—24no—
225SD16—584no—
226D8—24no—
227D8—584no—
228M16—22no—
229M16—582no—
230C2×C8—292no—
231C2×C8C22×D2912yescomplemented
232C4⋊C4—292no—
233C4⋊C4—582no—
234C22⋊C4—582no—
235C22⋊C4—582no—
236C22⋊C4—581no—
237C42—581no—
238C23—24no—
239C23—22no—
240C23—21no—
241C23—44no—
242C23—292no—
243C23—584no—
244C23—5816no—
245C23—582no—
246C23—584no—
247C23—582no—
248C23—1164no—
249C23—1168no—
250C23—11632no—
251C23—1168no—
252C23D4×D2911yescharacteristic
253Q8—21no—
254Q8—21no—
255D4—24no—
256D4—22no—
257D4—21no—
258D4—21no—
259D4—42no—
260D4—294no—
261D4—5816no—
262D4—584no—
263D4—582no—
264D4—1164no—
265D4C2×C22⋊D2914yes—
266C2×C4—21no—
267C2×C4—42no—
268C2×C4—292no—
269C2×C4—582no—
270C2×C4—584no—
271C2×C4—582no—
272C2×C4—582no—
273C2×C4—582no—
274C2×C4D4×D2911yescharacteristic
275C2×C4D4×D2912yes—
276C2×C4C2×C22⋊D2912yes—
277C2×C4C23×D2911yescharacteristic
278C8—22no—
279C8—582no—
280C22—22no—
281C22—24no—
282C22—48no—
283C22—44no—
284C22—41no—
285C22—42no—
286C22—294no—
287C22—588no—
288C22—582no—
289C22—582no—
290C22—584no—
291C22—1168no—
292C22—11632no—
293C22—1164no—
294C22#1174529912yes—
295C22C2×D4×D2911yescharacteristic
296C4—41no—
297C4—582no—
298C4—1162no—
299C4#1174529912yes—
300C4C2×D4×D2911yescharacteristic
301C4C22×C22⋊D2911yescharacteristic
302C2—22no—
303C2—44no—
304C2—81no—
305C2—584no—
306C2—1164no—
307C2C2×#1174529911yescharacteristic
308C2AGL(1,ℤ/8)×D2912yes—
subgroupG/H
centerC22C2×D4×D29
commutator, cosocleC2×C4×C29C24
FrattiniC2×C4C23×D29
FittingC2×C29×AGL(1,ℤ/8)C2
socleC22×C29C22×D4
hypercenterC2×AGL(1,ℤ/8)D29
hypocenterC29#3003
LCSG ▷ C2×C4×C29 ▷ C2×C29 ▷ C29
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C22×D4×D29C2semidirect
2. C2×#11745311C2semidirect
3. #14271670C2semidirect
4. #14271312C2semidirect
5. #14271295C2semidirect
6. #14271169C2semidirect
7. #14271168C2semidirect
8. #14271126C2semidirect
9. C2×C29×AGL(1,ℤ/8)C2semidirect
11. C2×#41×C29C22semidirect
21. C2×SD16×C29C22semidirect
22. C2×D8×C29C22semidirect
23. C2×M16×C29C22semidirect
25. #11745308C22semidirect
26. C2×#90784C22semidirect
27. #11745300C22semidirect
30. C2×#90782C22semidirect
32. #11745288C22semidirect
33. C2×#90777C22semidirect
34. #11745180C22semidirect
35. #11745169C22semidirect
36. #11745168C22semidirect
45. C2×Q8×C29C23semidirect
49. C2×D4×C29D4semidirect
51. C2×D4×C29C23semidirect
63. C2×D116D4semidirect
72. C2×C8×C29C23semidirect
78. C4⋊Dic29C23semidirect
82. C2×#6597C23semidirect
141. C2×D29AGL(1,ℤ/8)semidirect
150. C2×AGL(1,ℤ/8)D29semidirect
165. C2×#41C2×D29semidirect
177. C2×SD16C2×D29semidirect
179. C2×D8C2×D29semidirect
181. C2×M16C2×D29semidirect
191. C29#3003semidirect
200. C2×Q8C22×D29semidirect
214. C2×D4C22⋊D29semidirect
216. C2×D4C22×D29semidirect
231. C2×C8C22×D29semidirect

Maximal subgroups

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

subgroupindex
C22×D4×D292
C2×#117453112
#142716702
#142713122
#142712952
#142711692
#142711682
#142711262
C2×C29×AGL(1,ℤ/8)2
#300329