Finite Group Databasebeta

← #16827950

Subgroups — #16827950

Order 3712.

800 conjugacy classes of subgroups; only the 209 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 209 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×#11745359C211yesmaximal, characteristic
2#14271636C212yesmaximal, complemented
3#14271135C214yesmaximal, complemented
4C2×#11745182C211yesmaximal, characteristic, complemented
5C2×#11745154C212yesmaximal, complemented
6#99×D29C214yesmaximal, complemented
7C2×C29×#99C211yesmaximal, characteristic, complemented
8C22×D4×C29C2211yescharacteristic
9C22×C22⋊D29C412yescomplemented
10C22×C22⋊D29C2211yescharacteristic
11C22×C22⋊D29C2211yescharacteristic
12#11745359—24no—
13#11745359—24no—
14C2×#90807C412yescomplemented
15C2×#90807C2212yes—
16C2×D4×D29C412yescomplemented
17C2×D4×D29C2212yes—
18C2×#90805C412yescomplemented
19C2×C22⋊C4×C29C2212yescomplemented
20C2×C22⋊Dic29C2212yescomplemented
21C4×C22⋊D29—24no—
22#11745243—24no—
23C22⋊C4×D29C2214yescomplemented
24#11745239C2214yescomplemented
25C29×#99C2214yescomplemented
26#11745182C2214yescomplemented
27#11745154C2218yescomplemented
28C24×C29C2×C411yescharacteristic, complemented
29C24×C29C2311yescharacteristic
30C23×D29C2×C411yescharacteristic, complemented
31C23×D29C2×C412yescomplemented
32C23×D29C2311yescharacteristic
33C2×D4×C29—22no—
34C2×D4×C29—22no—
35C2×D4×C29C2×C412yescomplemented
36C2×D4×C29C2312yes—
37C22×C4×C29—22no—
38C22×C4×C29C2×C411yescharacteristic, complemented
39C2×C22⋊D29—22no—
40C2×C22⋊D29—22no—
41C2×C22⋊D29—24no—
42C2×C22⋊D29—24no—
43C2×C22⋊D29C2×C412yes—
44C2×C22⋊D29C2×C412yes—
45C2×C22⋊D29C2×C412yescomplemented
46C2×C22⋊D29C2×C412yescomplemented
47C2×C22⋊D29C2×C412yescomplemented
48C2×C22⋊D29D414yes—
49C2×C22⋊D29D414yes—
50C2×C22⋊D29C2312yes—
51C22×Dic29—22no—
52C22×Dic29C2×C411yescharacteristic, complemented
53C22×Dic29C2×C412yescomplemented
54C22×Dic29C2311yescharacteristic
55#90807—28no—
56#90807—28no—
57D4×D29—28no—
58D4×D29—28no—
59#90805—24no—
60#90805—24no—
61C2×D116C2×C412yescomplemented
62C2×C4×D29—24no—
63C2×C4×D29C2×C412yescomplemented
64C2×C4×D29C2×C412yescomplemented
65C2×Dic58C2×C412yescomplemented
66C22⋊C4×C29—22no—
67C22⋊C4×C29C2314yes—
68C22⋊Dic29—22no—
69C22⋊Dic29C2314yes—
70#90781—24no—
71#90779—24no—
72C4×Dic29—24no—
73C23×C29—21no—
74C23×C29—21no—
75C23×C29—81no—
76C23×C29—81no—
77C23×C29C22×C412yes—
78C23×C29C22×C411yescharacteristic
79C23×C29C22×C411yescharacteristic
80C23×C29C2×D412yes—
81C23×C29C2×D412yes—
82C23×C29C2411yescharacteristic
83C22×D29—22no—
84C22×D29—24no—
85C22×D29—81no—
86C22×D29—82no—
87C22×D29—81no—
88C22×D29C22×C412yes—
89C22×D29C22×C412yes—
90C22×D29C2×D414yes—
91D4×C29—24no—
92D4×C29—24no—
93C2×C4×C29—21no—
94C2×C4×C29—21no—
95C2×C4×C29—24no—
96C2×C4×C29—82no—
97C2×C4×C29C22×C412yes—
98C22⋊D29—22no—
99C22⋊D29—22no—
100C22⋊D29—24no—
101C22⋊D29—24no—
102C22⋊D29—24no—
103C22⋊D29C22⋊C414yes—
104C22⋊D29C22⋊C414yes—
105C22⋊D29C22⋊C418yes—
106C2×Dic29—22no—
107C2×Dic29—24no—
108C2×Dic29—24no—
109C2×Dic29—82no—
110C2×Dic29C22×C412yes—
111C2×Dic29C22×C412yes—
112C2×Dic29C2×D414yes—
113D116—24no—
114C4×D29—24no—
115C4×D29—24no—
116C4×D29—84no—
117Dic58—24no—
118#2883—291nomaximal, Sylow
119C22×C29—22no—
120C22×C29—42no—
121C22×C29—42no—
122C22×C29—41no—
123C22×C29—41no—
124C22×C29—41no—
125C22×C29—42no—
126C22×C29—81no—
127C22×C29—81no—
128C22×C29C2×C22⋊C412yes—
129C22×C29C2×C22⋊C411yescharacteristic
130C22×C29C2×C22⋊C411yescharacteristic
131C22×C29C23×C411yescharacteristic
132C22×C29C22×D412yes—
133C2×D29—22no—
134C2×D29—44no—
135C2×D29—42no—
136C2×D29—42no—
137C2×D29—84no—
138C2×D29—82no—
139C2×D29C2×C22⋊C414yes—
140C4×C29—22no—
141C4×C29—44no—
142Dic29—22no—
143Dic29—44no—
144Dic29C2×C22⋊C414yes—
145C2×H(5,𝔽2)—291no—
146#518—292no—
147#410—294no—
148C2×#99—291no—
149C2×#99—292no—
150C2×#99—294no—
151C2×#99D2911yescharacteristic, complemented
152C2×C29—22no—
153C2×C29—21no—
154C2×C29—22no—
155C2×C29—21no—
156C2×C29—21no—
157C2×C29—42no—
158C2×C29C22×C22⋊C411yescharacteristic
159D29—24no—
160D29—42no—
161H(5,𝔽2)—584no—
162H(5,𝔽2)—584no—
163C2×#41—292no—
164C2×#41—292no—
165C2×#41—292no—
166C22×D4—291no—
167C22×D4—292no—
168C22×D4—291no—
169C22×D4—292no—
170C22×D4—292no—
171C22×D4C2×D2911yescharacteristic
172C4×D4—584no—
173C4×D4—584no—
174#117—294no—
175C2×C22⋊C4—292no—
176C2×C22⋊C4—294no—
177C2×C22⋊C4C2×D2912yescomplemented
178#99—294no—
179#99—298no—
180#99C2×D2914yescomplemented
181C29#288311yescharacteristic, complemented, Sylow
182C24—291no—
183C24—292no—
184C24—291no—
185C24C4×D2911yescharacteristic, complemented
186C24C22×D2911yescharacteristic
187#41—584no—
188#41—584no—
189#41—588no—
190#41—588no—
191C2×Q8—292no—
192C2×D4—22no—
193C2×D4—22no—
194C2×D4—292no—
195C2×D4—292no—
196C2×D4—294no—
197C2×D4—294no—
198C2×D4—292no—
199C2×D4—292no—
200C2×D4—292no—
201C2×D4—292no—
202C2×D4—292no—
203C2×D4—588no—
204C2×D4—588no—
205C2×D4—584no—
206C2×D4—584no—
207C2×D4—582no—
208C2×D4—582no—
209C2×D4C4×D2912yescomplemented
210C2×D4C22×D2912yes—
211C22×C4—22no—
212C22×C4—291no—
213C22×C4—292no—
214C22×C4—291no—
215C22×C4—292no—
216C22×C4—292no—
217C22×C4—584no—
218C22×C4—582no—
219C22×C4C4×D2911yescharacteristic, complemented
220C4⋊C4—584no—
221C22⋊C4—22no—
222C22⋊C4—294no—
223C22⋊C4—584no—
224C22⋊C4—582no—
225C22⋊C4C22×D2914yes—
226C42—584no—
227C23—21no—
228C23—21no—
229C23—81no—
230C23—81no—
231C23—292no—
232C23—294no—
233C23—292no—
234C23—584no—
235C23—582no—
236C23—2321no—
237C23—2321no—
238C23—2322no—
239C23C2×C4×D2912yes—
240C23C2×C4×D2911yescharacteristic
241C23C2×C4×D2911yescharacteristic
242C23D4×D2912yes—
243C23D4×D2912yes—
244C23C23×D2911yescharacteristic
245Q8—584no—
246D4—24no—
247D4—24no—
248D4—294no—
249D4—294no—
250D4—298no—
251D4—584no—
252D4—584no—
253D4—584no—
254D4—584no—
255D4—582no—
256D4—582no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—24no—
260C2×C4—82no—
261C2×C4—292no—
262C2×C4—294no—
263C2×C4—292no—
264C2×C4—584no—
265C2×C4—584no—
266C2×C4—584no—
267C2×C4—582no—
268C2×C4—584no—
269C2×C4—2324no—
270C2×C4—2322no—
271C2×C4C2×C4×D2912yes—
272C22—22no—
273C22—42no—
274C22—42no—
275C22—41no—
276C22—41no—
277C22—41no—
278C22—42no—
279C22—81no—
280C22—81no—
281C22—294no—
282C22—582no—
283C22—1164no—
284C22—1162no—
285C22—1162no—
286C22—2322no—
287C22—2324no—
288C22C22⋊C4×D2911yescharacteristic
289C22C22⋊C4×D2911yescharacteristic
290C22C22⋊C4×D2912yes—
291C22C22×C4×D2911yescharacteristic
292C22C2×D4×D2912yes—
293C4—22no—
294C4—44no—
295C4—294no—
296C4—582no—
297C4—1164no—
298C2—22no—
299C2—21no—
300C2—21no—
301C2—21no—
302C2—22no—
303C2—42no—
304C2—584no—
305C2—1162no—
306C2C2×C22⋊C4×D2911yescharacteristic
subgroupG/H
centerC2C2×C22⋊C4×D29
commutatorC22×C29C23×C4
FrattiniC23C23×D29
FittingC2×C29×#99C2
socleC2×C29C22×C22⋊C4
prime radicalC2×#11745359C2
cosocleC23×C29C24
hypercenterC2×#99D29
hypocenterC29#2883
LCSG ▷ C22×C29 ▷ C2×C29 ▷ C29
UCSC1 ◁ C2 ◁ C24 ◁ C2×#99

Complements

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

NH ≅ G/Ntype
2. #14271636C2semidirect
3. #14271135C2semidirect
4. C2×#11745182C2semidirect
5. C2×#11745154C2semidirect
6. #99×D29C2semidirect
7. C2×C29×#99C2semidirect
9. C22×C22⋊D29C4semidirect
14. C2×#90807C4semidirect
16. C2×D4×D29C4semidirect
18. C2×#90805C4semidirect
19. C2×C22⋊C4×C29C22semidirect
20. C2×C22⋊Dic29C22semidirect
23. C22⋊C4×D29C22semidirect
24. #11745239C22semidirect
25. C29×#99C22semidirect
26. #11745182C22semidirect
27. #11745154C22semidirect
28. C24×C29C2×C4semidirect
30. C23×D29C2×C4semidirect
31. C23×D29C2×C4semidirect
35. C2×D4×C29C2×C4semidirect
38. C22×C4×C29C2×C4semidirect
45. C2×C22⋊D29C2×C4semidirect
46. C2×C22⋊D29C2×C4semidirect
47. C2×C22⋊D29C2×C4semidirect
52. C22×Dic29C2×C4semidirect
53. C22×Dic29C2×C4semidirect
61. C2×D116C2×C4semidirect
63. C2×C4×D29C2×C4semidirect
64. C2×C4×D29C2×C4semidirect
65. C2×Dic58C2×C4semidirect
151. C2×#99D29semidirect
177. C2×C22⋊C4C2×D29semidirect
180. #99C2×D29semidirect
181. C29#2883semidirect
185. C24C4×D29semidirect
209. C2×D4C4×D29semidirect
219. C22×C4C4×D29semidirect

Maximal subgroups

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

subgroupindex
C2×#117453592
#142716362
#142711352
C2×#117451822
C2×#117451542
#99×D292
C2×C29×#992
#288329