Finite Group Databasebeta

← #28411990

Subgroups — #28411990

Order 14656.

330 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 109 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×#16814666C211yesmaximal, characteristic, complemented
2C22×C4×D229C211yesmaximal, characteristic, complemented
3#20300973C211yesmaximal, characteristic, complemented
4#20300970C211yesmaximal, characteristic, complemented
5#20300966C211yesmaximal, characteristic, complemented
6#20300965C211yesmaximal, characteristic, complemented
7#20300961C211yesmaximal, characteristic, complemented
8#20300960C211yesmaximal, characteristic, complemented
9#20300958C211yesmaximal, characteristic, complemented
10#20300956C211yesmaximal, characteristic, complemented
11C4×D916C211yesmaximal, characteristic, complemented
12C4×C22⋊D229C211yesmaximal, characteristic, complemented
13C4×C22⋊D229C211yesmaximal, characteristic, complemented
14#20300945C211yesmaximal, characteristic, complemented
15C4×D4×C229C211yesmaximal, characteristic, complemented
16C23×D229C2211yescharacteristic
17#16814666—24no—
18C2×Dic458C2211yescharacteristic, complemented
19C2×D916C2211yescharacteristic
20C2×D916C2211yescharacteristic
21C2×C22⋊D229C2211yescharacteristic
22C2×C22⋊D229C2211yescharacteristic
23C2×C22⋊D229C2211yescharacteristic
24C2×C22⋊D229C2211yescharacteristic, complemented
25C22×Dic229C2211yescharacteristic
26C2×C4×D229—24no—
27C2×C4×D229C2211yescharacteristic
28C2×C4×D229C2211yescharacteristic
29C2×C4×D229C2211yescharacteristic
30C2×C4×D229C2211yescharacteristic
31C2×C4×D229C2211yescharacteristic, complemented
32C4⋊Dic229C2211yescharacteristic, complemented
33#16814645C2211yescharacteristic, complemented
34#16814645C2211yescharacteristic, complemented
35#16814645C2211yescharacteristic, complemented
36#16814645C2211yescharacteristic, complemented
37C22⋊Dic229C2211yescharacteristic
38C22⋊Dic229C2211yescharacteristic, complemented
39#16814643C2211yescharacteristic
40#16814643C2211yescharacteristic
41#16814643C2211yescharacteristic
42#16814643C2211yescharacteristic
43#16814643C2211yescharacteristic, complemented
44#16814643C2211yescharacteristic, complemented
45C4×Dic229C2211yescharacteristic, complemented
46C2×D4×C229C2211yescharacteristic
47C22×C4×C229C2211yescharacteristic
48C22×C4×C229C2211yescharacteristic, complemented
49C4⋊C4×C229C2211yescharacteristic, complemented
50C22⋊C4×C229C2211yescharacteristic
51C22⋊C4×C229C2211yescharacteristic, complemented
52C42×C229C2211yescharacteristic, complemented
53C22×D229—22no—
54C22×D229—22no—
55C22×D229—22no—
56C22×D229C2311yescharacteristic
57C22×D229C2311yescharacteristic
58C22×D229C2311yescharacteristic
59Dic458—22no—
60D916—22no—
61D916—22no—
62C22⋊D229—22no—
63C22⋊D229—22no—
64C22⋊D229—22no—
65C22⋊D229—22no—
66C2×Dic229—22no—
67C2×Dic229C2311yescharacteristic
68C2×Dic229C2311yescharacteristic
69C2×Dic229C2311yescharacteristic
70C2×Dic229C2311yescharacteristic
71C2×Dic229C2311yescharacteristic
72C4×D229—22no—
73C4×D229—22no—
74C4×D229—22no—
75C4×D229—22no—
76C4×D229D414yescomplemented
77C23×C229C2311yescharacteristic
78C23×C229C2311yescharacteristic
79D4×C229—22no—
80C2×C4×C229—22no—
81C2×C4×C229—22no—
82C2×C4×C229C2311yescharacteristic
83C2×C4×C229C2311yescharacteristic
84C2×C4×C229C2311yescharacteristic
85C2×C4×C229C2311yescharacteristic
86C2×C4×C229C2311yescharacteristic
87C22×C229—21no—
88C22×C229—21no—
89C22×C229—21no—
90C22×C229—21no—
91C22×C229—21no—
92C22×C229#4112yescomplemented
93C22×C229C2411yescharacteristic
94C2×D229—28no—
95C2×D229—21no—
96C2×D229—21no—
97C2×D229—21no—
98C2×D229—21no—
99C2×D229—21no—
100C2×D229—21no—
101C2×D229—21no—
102C2×D229C2×D412yes—
103C2×D229#4112yescomplemented
104C4×C229—21no—
105C4×C229—21no—
106C4×C229—21no—
107C4×C229—21no—
108C4×C229C2×D412yes—
109Dic229—21no—
110Dic229—21no—
111Dic229—21no—
112Dic229—21no—
113Dic229C2×D412yes—
114C2×C229—22no—
115C2×C229—41no—
116C2×C229C22×D411yescharacteristic
117C2×C229C2×#4111yescharacteristic
118C2×C229C2×#4111yescharacteristic
119D229—22no—
120D229—22no—
121D229—41no—
122C229#52511yescharacteristic, complemented, Sylow
123#525—2291nomaximal, Sylow
124C2×#41—2291no—
125C23×C4—2291no—
126#123—2291no—
127#123—2291no—
128C22⋊Q8—2291no—
129C22⋊Q8—2291no—
130#121—2291no—
131#121—2291no—
132#120—2291no—
133#120—2291no—
134C4×D4—2291no—
135C4×D4—2291no—
136C4×D4—2291no—
137C4×D4D22911yescharacteristic, complemented
138#117—2291no—
139C24—2291no—
140#41—4584no—
141C2×Q8—2291no—
142C2×D4—2291no—
143C2×D4—2291no—
144C2×D4—2291no—
145C2×D4—2291no—
146C2×D4—2291no—
147C2×D4—2291no—
148C2×D4C2×D22911yescharacteristic
149C22×C4—4584no—
150C22×C4—2291no—
151C22×C4—2291no—
152C22×C4—2291no—
153C22×C4—2291no—
154C22×C4—2291no—
155C22×C4—2291no—
156C22×C4C2×D22911yescharacteristic
157C22×C4C2×D22911yescharacteristic, complemented
158C4⋊C4—2291no—
159C4⋊C4—2291no—
160C4⋊C4—2291no—
161C4⋊C4—2291no—
162C4⋊C4—2291no—
163C4⋊C4C2×D22911yescharacteristic, complemented
164C22⋊C4—2291no—
165C22⋊C4—2291no—
166C22⋊C4—2291no—
167C22⋊C4—2291no—
168C22⋊C4—2291no—
169C22⋊C4—2291no—
170C22⋊C4—2291no—
171C22⋊C4—2291no—
172C22⋊C4C2×D22911yescharacteristic
173C22⋊C4C2×D22911yescharacteristic, complemented
174C42—2291no—
175C42C2×D22911yescharacteristic, complemented
176C23—4582no—
177C23—4582no—
178C23—4582no—
179C23—2291no—
180C23—2291no—
181C23—2291no—
182C23C22×D22911yescharacteristic
183C23C22×D22911yescharacteristic
184Q8—4582no—
185D4—22no—
186D4—4582no—
187D4—4582no—
188D4—4582no—
189D4—4582no—
190D4—4582no—
191D4—4582no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—4582no—
195C2×C4—4582no—
196C2×C4—4582no—
197C2×C4—4582no—
198C2×C4—4582no—
199C2×C4—2291no—
200C2×C4—2291no—
201C2×C4—2291no—
202C2×C4—2291no—
203C2×C4—2291no—
204C2×C4—2294no—
205C2×C4C22×D22911yescharacteristic
206C2×C4C22×D22911yescharacteristic
207C2×C4C22×D22911yescharacteristic
208C2×C4C22×D22911yescharacteristic
209C2×C4C22×D22911yescharacteristic
210C22—21no—
211C22—21no—
212C22—21no—
213C22—21no—
214C22—21no—
215C22—4588no—
216C22—4581no—
217C22—4581no—
218C22—4581no—
219C22—4581no—
220C22—4581no—
221C22—4581no—
222C22—4581no—
223C22—2292no—
224C22—2292no—
225C22#1681466612yescomplemented
226C22C23×D22911yescharacteristic
227C4—21no—
228C4—21no—
229C4—21no—
230C4—21no—
231C4—4581no—
232C4—4581no—
233C4—4581no—
234C4—4581no—
235C4—2292no—
236C4D4×D22912yes—
237C2—22no—
238C2—41no—
239C2—9161no—
240C2—4582no—
241C2—4582no—
242C2C2×D4×D22911yescharacteristic
243C2#41×D22911yescharacteristic
244C2C2×#1681466611yescharacteristic
subgroupG/H
centerC2×C4C22×D229
commutator, socle, cosocleC22×C229C24
FrattiniC22C23×D229
FittingC4×D4×C229C2
hypercenterC4×D4D229
hypocenterC229#525
LCSG ▷ C22×C229 ▷ C229
UCSC1 ◁ C2×C4 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. C2×#16814666C2semidirect
2. C22×C4×D229C2semidirect
3. #20300973C2semidirect
4. #20300970C2semidirect
5. #20300966C2semidirect
6. #20300965C2semidirect
7. #20300961C2semidirect
8. #20300960C2semidirect
9. #20300958C2semidirect
10. #20300956C2semidirect
11. C4×D916C2semidirect
12. C4×C22⋊D229C2semidirect
13. C4×C22⋊D229C2semidirect
14. #20300945C2semidirect
15. C4×D4×C229C2semidirect
18. C2×Dic458C22semidirect
24. C2×C22⋊D229C22semidirect
31. C2×C4×D229C22semidirect
32. C4⋊Dic229C22semidirect
33. #16814645C22semidirect
34. #16814645C22semidirect
35. #16814645C22semidirect
36. #16814645C22semidirect
38. C22⋊Dic229C22semidirect
43. #16814643C22semidirect
44. #16814643C22semidirect
45. C4×Dic229C22semidirect
48. C22×C4×C229C22semidirect
49. C4⋊C4×C229C22semidirect
51. C22⋊C4×C229C22semidirect
52. C42×C229C22semidirect
76. C4×D229D4semidirect
92. C22×C229#41semidirect
103. C2×D229#41semidirect
122. C229#525semidirect
137. C4×D4D229semidirect
157. C22×C4C2×D229semidirect
163. C4⋊C4C2×D229semidirect
173. C22⋊C4C2×D229semidirect
175. C42C2×D229semidirect
225. C22#16814666semidirect

Maximal subgroups

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

subgroupindex
C2×#168146662
C22×C4×D2292
#203009732
#203009702
#203009662
#203009652
#203009612
#203009602
#203009582
#203009562
C4×D9162
C4×C22⋊D2292
#203009452
C4×D4×C2292
#525229