Finite Group Databasebeta

← #16828756

Subgroups — #16828756

Order 3712.

784 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 784 conjugacy classes of subgroups — too many to draw. Only its 149 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, complemented
2C2×#11745294C211yesmaximal, characteristic, complemented
3#14271730C211yesmaximal, characteristic, complemented
4#14271305C214yesmaximal, complemented
5#14271295C214yesmaximal, complemented
6#14271168C211yesmaximal, characteristic, complemented
7#14271160C211yesmaximal, characteristic, complemented
8#14271129C211yesmaximal, characteristic, complemented
9C22×D8×C29C211yesmaximal, characteristic, complemented
10C22×D4×C29C2211yescharacteristic
11C22×D4×C29C2211yescharacteristic, complemented
12C22×C22⋊D29—22no—
13#11745359—24no—
14#11745359—24no—
15C2×#90807—21no—
16C2×#90807C2212yescomplemented
17C2×D4×D29—21no—
18C2×D4×D29C2212yescomplemented
19C2×#90805C2211yescharacteristic
20C2×#90805C2211yescharacteristic
21C2×D8×C29—24no—
22C2×D8×C29C2214yescomplemented
23C22×C8×C29C2211yescharacteristic, complemented
24#11745315—22no—
25#11745302—21no—
26#11745300C2214yescomplemented
27#11745298—21no—
28C2×#90783C2212yescomplemented
29#11745294—24no—
30C2×#90782C2212yescomplemented
31#11745288C2211yescharacteristic, complemented
32C2×#90777C2211yescharacteristic, complemented
33#11745180C2214yescomplemented
34#11745169C2212yescomplemented
35#11745168C2214yescomplemented
36#11745164C2212yescomplemented
37C24×C29—21no—
38C24×C29—21no—
39C23×D29—22no—
40C2×D4×C29—21no—
41C2×D4×C29—22no—
42C2×D4×C29—22no—
43C2×D4×C29—22no—
44C2×D4×C29—21no—
45C2×D4×C29D414yescomplemented
46C2×D4×C29C2312yes—
47C2×D4×C29C2312yes—
48C22×C4×C29C2311yescharacteristic
49C2×C22⋊D29—216no—
50C2×C22⋊D29—21no—
51C2×C22⋊D29—22no—
52C2×C22⋊D29—24no—
53C2×C22⋊D29—22no—
54C2×C22⋊D29D412yes—
55C22×Dic29—22no—
56#90807—28no—
57#90807—24no—
58#90807—24no—
59D4×D29—28no—
60D4×D29—24no—
61D4×D29—24no—
62#90805—22no—
63#90805—22no—
64#90805—22no—
65#90805D414yescomplemented
66C2×D116D411yescharacteristic
67C2×D116C2311yescharacteristic
68C2×C4×D29—21no—
69C2×C4×D29C2312yes—
70C2×Dic58D411yescharacteristic
71C2×Dic58C2311yescharacteristic
72D8×C29—28no—
73C2×C8×C29—22no—
74C2×C8×C29C2312yes—
75C22⋊Dic29—21no—
76C22⋊Dic29—24no—
77#90783—24no—
78#90782—24no—
79C4⋊Dic29C2312yes—
80C4×Dic29—21no—
81#90777—22no—
82C2×#6597C2312yes—
83C23×C29—22no—
84C23×C29—24no—
85C23×C29—22no—
86C23×C29—42no—
87C23×C29—82no—
88C23×C29—82no—
89C23×C29C2×D411yescharacteristic
90C22×D29—21no—
91C22×D29—28no—
92C22×D29—84no—
93C22×D29C2×D412yes—
94D4×C29—24no—
95D4×C29—22no—
96D4×C29—22no—
97D4×C29—22no—
98D4×C29—42no—
99D4×C29C2×D414yes—
100C2×C4×C29C2×D411yescharacteristic
101C2×C4×C29C2×D412yes—
102C2×C4×C29C2×D412yes—
103C2×C4×C29C2411yescharacteristic
104C22⋊D29—28no—
105C22⋊D29—28no—
106C22⋊D29—24no—
107C22⋊D29—24no—
108C22⋊D29—84no—
109C2×Dic29—28no—
110C2×Dic29—21no—
111C2×Dic29—22no—
112C2×Dic29C2×D412yes—
113D116—22no—
114D116—21no—
115D116C2×D412yes—
116C4×D29—24no—
117C4×D29—22no—
118C4×D29—22no—
119Dic58—22no—
120Dic58—21no—
121Dic58C2×D412yes—
122C8×C29—22no—
123#6597—22no—
124#3008—291nomaximal, Sylow
125C22×C29—22no—
126C22×C29—24no—
127C22×C29—44no—
128C22×C29—42no—
129C22×C29—44no—
130C22×C29—84no—
131C22×C29—84no—
132C22×C29#12012yes—
133C22×C29C22×D411yescharacteristic
134C2×D29—22no—
135C2×D29—22no—
136C2×D29—24no—
137C2×D29—42no—
138C2×D29—88no—
139C4×C29#12012yes—
140C4×C29C22×D411yescharacteristic
141C4×C29C22×D411yescharacteristic
142Dic29—22no—
143Dic29—22no—
144Dic29—42no—
145C2×H(5,𝔽2)—291no—
146C2×AGL(1,ℤ/8)—291no—
147C22×D8D2911yescharacteristic, complemented
148#535—291no—
149#449—294no—
150#447—294no—
151#418—291no—
152#416—291no—
153#408—291no—
154C2×C29—22no—
155C2×C29—44no—
156C2×C29—82no—
157C2×C29C2×#12011yescharacteristic
158C2×C29#57612yes—
159D29—42no—
160D29—42no—
161H(5,𝔽2)—584no—
162H(5,𝔽2)—584no—
163C2×#41—291no—
164C2×#41—292no—
165C2×#41—291no—
166C2×#41—581no—
167C22×D4—292no—
168C22×D4—581no—
169C22×D4—582no—
170C22×D4C2×D2911yescharacteristic
171C22×D4C2×D2911yescharacteristic, complemented
172AGL(1,ℤ/8)—584no—
173C2×SD16—292no—
174C2×D8—24no—
175C2×D8—292no—
176C2×D8C2×D2914yescomplemented
177C2×M16—291no—
178C22×C8C2×D2911yescharacteristic, complemented
179#127—581no—
180#124—581no—
181#121—294no—
182#120—582no—
183#117—291no—
184#103—292no—
185#102—294no—
186#102—292no—
187C22⋊C8—294no—
188C29#300811yescharacteristic, complemented, Sylow
189C24—21no—
190C24—21no—
191C24—582no—
192#41—294no—
193#41—584no—
194#41—588no—
195#41—584no—
196#41—582no—
197#41—582no—
198#41—582no—
199C2×Q8—291no—
200C2×Q8—291no—
201C2×D4—21no—
202C2×D4—22no—
203C2×D4—22no—
204C2×D4—21no—
205C2×D4—22no—
206C2×D4—292no—
207C2×D4—291no—
208C2×D4—291no—
209C2×D4—582no—
210C2×D4—584no—
211C2×D4—588no—
212C2×D4—582no—
213C2×D4—5816no—
214C2×D4—581no—
215C2×D4—584no—
216C2×D4—584no—
217C2×D4C22⋊D2914yescomplemented
218C2×D4C22×D2912yes—
219C2×D4C22×D2912yes—
220C22×C4—292no—
221C22×C4—582no—
222C22×C4—581no—
223C22×C4C22×D2911yescharacteristic
224SD16—584no—
225D8—28no—
226D8—584no—
227M16—582no—
228C2×C8—22no—
229C2×C8—292no—
230C2×C8C22×D2912yes—
231C4⋊C4—292no—
232C22⋊C4—584no—
233C22⋊C4—581no—
234C42—581no—
235C23—22no—
236C23—24no—
237C23—22no—
238C23—42no—
239C23—82no—
240C23—82no—
241C23—292no—
242C23—588no—
243C23—581no—
244C23—2324no—
245C23D4×D2911yescharacteristic
246Q8—292no—
247Q8—582no—
248Q8—581no—
249D4—24no—
250D4—22no—
251D4—22no—
252D4—22no—
253D4—42no—
254D4—292no—
255D4—588no—
256D4—588no—
257D4—584no—
258D4—584no—
259D4—582no—
260D4—581no—
261D4—2324no—
262D4C2×C22⋊D2914yes—
263C2×C4—292no—
264C2×C4—588no—
265C2×C4—581no—
266C2×C4—584no—
267C2×C4—582no—
268C2×C4—582no—
269C2×C4—582no—
270C2×C4D4×D2911yescharacteristic
271C2×C4D4×D2912yes—
272C2×C4C2×C22⋊D2912yes—
273C2×C4C23×D2911yescharacteristic
274C8—22no—
275C8—582no—
276C22—22no—
277C22—24no—
278C22—44no—
279C22—42no—
280C22—44no—
281C22—84no—
282C22—84no—
283C22—582no—
284C22—582no—
285C22—584no—
286C22—1162no—
287C22—2328no—
288C22#1174529912yes—
289C22C2×D4×D2911yescharacteristic
290C4—582no—
291C4—582no—
292C4—1162no—
293C4#1174529912yes—
294C4C2×D4×D2911yescharacteristic
295C4C22×C22⋊D2911yescharacteristic
296C2—22no—
297C2—44no—
298C2—82no—
299C2—1162no—
300C2—1162no—
301C2C2×#1174529911yescharacteristic
302C2#1427196412yes—
subgroupG/H
centerC22C2×D4×D29
commutator, cosocleC2×C4×C29C24
FrattiniC2×C4C23×D29
FittingC22×D8×C29C2
socleC22×C29C22×D4
hypercenterC22×D8D29
hypocenterC29#3008
LCSG ▷ C2×C4×C29 ▷ C2×C29 ▷ C29
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×D8

Complements

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

NH ≅ G/Ntype
1. C2×#11745359C2semidirect
2. C2×#11745294C2semidirect
3. #14271730C2semidirect
4. #14271305C2semidirect
5. #14271295C2semidirect
6. #14271168C2semidirect
7. #14271160C2semidirect
8. #14271129C2semidirect
9. C22×D8×C29C2semidirect
11. C22×D4×C29C22semidirect
16. C2×#90807C22semidirect
18. C2×D4×D29C22semidirect
22. C2×D8×C29C22semidirect
23. C22×C8×C29C22semidirect
26. #11745300C22semidirect
28. C2×#90783C22semidirect
30. C2×#90782C22semidirect
31. #11745288C22semidirect
32. C2×#90777C22semidirect
33. #11745180C22semidirect
34. #11745169C22semidirect
35. #11745168C22semidirect
36. #11745164C22semidirect
45. C2×D4×C29D4semidirect
65. #90805D4semidirect
147. C22×D8D29semidirect
171. C22×D4C2×D29semidirect
176. C2×D8C2×D29semidirect
178. C22×C8C2×D29semidirect
188. C29#3008semidirect
217. C2×D4C22⋊D29semidirect

Maximal subgroups

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

subgroupindex
C2×#117453592
C2×#117452942
#142717302
#142713052
#142712952
#142711682
#142711602
#142711292
C22×D8×C292
#300829