Finite Group Databasebeta

← #36374653

Subgroups — #36374653

Order 22336.

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×#19150233C211yesmaximal, characteristic, complemented
2C22×C4×D349C211yesmaximal, characteristic, complemented
3#26013624C211yesmaximal, characteristic, complemented
4#26013621C211yesmaximal, characteristic, complemented
5#26013617C211yesmaximal, characteristic, complemented
6#26013616C211yesmaximal, characteristic, complemented
7#26013612C211yesmaximal, characteristic, complemented
8#26013611C211yesmaximal, characteristic, complemented
9#26013609C211yesmaximal, characteristic, complemented
10#26013607C211yesmaximal, characteristic, complemented
11C4×D1396C211yesmaximal, characteristic, complemented
12C4×C22⋊D349C211yesmaximal, characteristic, complemented
13C4×C22⋊D349C211yesmaximal, characteristic, complemented
14#26013596C211yesmaximal, characteristic, complemented
15C4×D4×C349C211yesmaximal, characteristic, complemented
16C23×D349C2211yescharacteristic
17#19150233—24no—
18C2×Dic698C2211yescharacteristic, complemented
19C2×D1396C2211yescharacteristic
20C2×D1396C2211yescharacteristic
21C2×C22⋊D349C2211yescharacteristic
22C2×C22⋊D349C2211yescharacteristic
23C2×C22⋊D349C2211yescharacteristic
24C2×C22⋊D349C2211yescharacteristic, complemented
25C22×Dic349C2211yescharacteristic
26C2×C4×D349—24no—
27C2×C4×D349C2211yescharacteristic
28C2×C4×D349C2211yescharacteristic
29C2×C4×D349C2211yescharacteristic
30C2×C4×D349C2211yescharacteristic
31C2×C4×D349C2211yescharacteristic, complemented
32C4⋊Dic349C2211yescharacteristic, complemented
33#19150212C2211yescharacteristic, complemented
34#19150212C2211yescharacteristic, complemented
35#19150212C2211yescharacteristic, complemented
36#19150212C2211yescharacteristic, complemented
37C22⋊Dic349C2211yescharacteristic
38C22⋊Dic349C2211yescharacteristic, complemented
39#19150210C2211yescharacteristic
40#19150210C2211yescharacteristic
41#19150210C2211yescharacteristic
42#19150210C2211yescharacteristic
43#19150210C2211yescharacteristic, complemented
44#19150210C2211yescharacteristic, complemented
45C4×Dic349C2211yescharacteristic, complemented
46C2×D4×C349C2211yescharacteristic
47C22×C4×C349C2211yescharacteristic
48C22×C4×C349C2211yescharacteristic, complemented
49C4⋊C4×C349C2211yescharacteristic, complemented
50C22⋊C4×C349C2211yescharacteristic
51C22⋊C4×C349C2211yescharacteristic, complemented
52C42×C349C2211yescharacteristic, complemented
53C22×D349—22no—
54C22×D349—22no—
55C22×D349—22no—
56C22×D349C2311yescharacteristic
57C22×D349C2311yescharacteristic
58C22×D349C2311yescharacteristic
59Dic698—22no—
60D1396—22no—
61D1396—22no—
62C22⋊D349—22no—
63C22⋊D349—22no—
64C22⋊D349—22no—
65C22⋊D349—22no—
66C2×Dic349—22no—
67C2×Dic349C2311yescharacteristic
68C2×Dic349C2311yescharacteristic
69C2×Dic349C2311yescharacteristic
70C2×Dic349C2311yescharacteristic
71C2×Dic349C2311yescharacteristic
72C4×D349—22no—
73C4×D349—22no—
74C4×D349—22no—
75C4×D349—22no—
76C4×D349D414yescomplemented
77C23×C349C2311yescharacteristic
78C23×C349C2311yescharacteristic
79D4×C349—22no—
80C2×C4×C349—22no—
81C2×C4×C349—22no—
82C2×C4×C349C2311yescharacteristic
83C2×C4×C349C2311yescharacteristic
84C2×C4×C349C2311yescharacteristic
85C2×C4×C349C2311yescharacteristic
86C2×C4×C349C2311yescharacteristic
87C22×C349—21no—
88C22×C349—21no—
89C22×C349—21no—
90C22×C349—21no—
91C22×C349—21no—
92C22×C349#4112yescomplemented
93C22×C349C2411yescharacteristic
94C2×D349—28no—
95C2×D349—21no—
96C2×D349—21no—
97C2×D349—21no—
98C2×D349—21no—
99C2×D349—21no—
100C2×D349—21no—
101C2×D349—21no—
102C2×D349C2×D412yes—
103C2×D349#4112yescomplemented
104C4×C349—21no—
105C4×C349—21no—
106C4×C349—21no—
107C4×C349—21no—
108C4×C349C2×D412yes—
109Dic349—21no—
110Dic349—21no—
111Dic349—21no—
112Dic349—21no—
113Dic349C2×D412yes—
114C2×C349—22no—
115C2×C349—41no—
116C2×C349C22×D411yescharacteristic
117C2×C349C2×#4111yescharacteristic
118C2×C349C2×#4111yescharacteristic
119D349—22no—
120D349—22no—
121D349—41no—
122C349#52511yescharacteristic, complemented, Sylow
123#525—3491nomaximal, Sylow
124C2×#41—3491no—
125C23×C4—3491no—
126#123—3491no—
127#123—3491no—
128C22⋊Q8—3491no—
129C22⋊Q8—3491no—
130#121—3491no—
131#121—3491no—
132#120—3491no—
133#120—3491no—
134C4×D4—3491no—
135C4×D4—3491no—
136C4×D4—3491no—
137C4×D4D34911yescharacteristic, complemented
138#117—3491no—
139C24—3491no—
140#41—6984no—
141C2×Q8—3491no—
142C2×D4—3491no—
143C2×D4—3491no—
144C2×D4—3491no—
145C2×D4—3491no—
146C2×D4—3491no—
147C2×D4—3491no—
148C2×D4C2×D34911yescharacteristic
149C22×C4—3491no—
150C22×C4—3491no—
151C22×C4—3491no—
152C22×C4—3491no—
153C22×C4—3491no—
154C22×C4—3491no—
155C22×C4—6984no—
156C22×C4C2×D34911yescharacteristic
157C22×C4C2×D34911yescharacteristic, complemented
158C4⋊C4—3491no—
159C4⋊C4—3491no—
160C4⋊C4—3491no—
161C4⋊C4—3491no—
162C4⋊C4—3491no—
163C4⋊C4C2×D34911yescharacteristic, complemented
164C22⋊C4—3491no—
165C22⋊C4—3491no—
166C22⋊C4—3491no—
167C22⋊C4—3491no—
168C22⋊C4—3491no—
169C22⋊C4—3491no—
170C22⋊C4—3491no—
171C22⋊C4—3491no—
172C22⋊C4C2×D34911yescharacteristic
173C22⋊C4C2×D34911yescharacteristic, complemented
174C42—3491no—
175C42C2×D34911yescharacteristic, complemented
176C23—3491no—
177C23—3491no—
178C23—3491no—
179C23—6982no—
180C23—6982no—
181C23—6982no—
182C23C22×D34911yescharacteristic
183C23C22×D34911yescharacteristic
184Q8—6982no—
185D4—22no—
186D4—6982no—
187D4—6982no—
188D4—6982no—
189D4—6982no—
190D4—6982no—
191D4—6982no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—3491no—
195C2×C4—3494no—
196C2×C4—3491no—
197C2×C4—3491no—
198C2×C4—3491no—
199C2×C4—3491no—
200C2×C4—6982no—
201C2×C4—6982no—
202C2×C4—6982no—
203C2×C4—6982no—
204C2×C4—6982no—
205C2×C4C22×D34911yescharacteristic
206C2×C4C22×D34911yescharacteristic
207C2×C4C22×D34911yescharacteristic
208C2×C4C22×D34911yescharacteristic
209C2×C4C22×D34911yescharacteristic
210C22—21no—
211C22—21no—
212C22—21no—
213C22—21no—
214C22—21no—
215C22—3492no—
216C22—3492no—
217C22—6988no—
218C22—6981no—
219C22—6981no—
220C22—6981no—
221C22—6981no—
222C22—6981no—
223C22—6981no—
224C22—6981no—
225C22#1915023312yescomplemented
226C22C23×D34911yescharacteristic
227C4—21no—
228C4—21no—
229C4—21no—
230C4—21no—
231C4—3492no—
232C4—6981no—
233C4—6981no—
234C4—6981no—
235C4—6981no—
236C4D4×D34912yes—
237C2—22no—
238C2—41no—
239C2—13961no—
240C2—6982no—
241C2—6982no—
242C2C2×D4×D34911yescharacteristic
243C2#41×D34911yescharacteristic
244C2C2×#1915023311yescharacteristic
subgroupG/H
centerC2×C4C22×D349
commutator, socle, cosocleC22×C349C24
FrattiniC22C23×D349
FittingC4×D4×C349C2
hypercenterC4×D4D349
hypocenterC349#525
LCSG ▷ C22×C349 ▷ C349
UCSC1 ◁ C2×C4 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. C2×#19150233C2semidirect
2. C22×C4×D349C2semidirect
3. #26013624C2semidirect
4. #26013621C2semidirect
5. #26013617C2semidirect
6. #26013616C2semidirect
7. #26013612C2semidirect
8. #26013611C2semidirect
9. #26013609C2semidirect
10. #26013607C2semidirect
11. C4×D1396C2semidirect
12. C4×C22⋊D349C2semidirect
13. C4×C22⋊D349C2semidirect
14. #26013596C2semidirect
15. C4×D4×C349C2semidirect
18. C2×Dic698C22semidirect
24. C2×C22⋊D349C22semidirect
31. C2×C4×D349C22semidirect
32. C4⋊Dic349C22semidirect
33. #19150212C22semidirect
34. #19150212C22semidirect
35. #19150212C22semidirect
36. #19150212C22semidirect
38. C22⋊Dic349C22semidirect
43. #19150210C22semidirect
44. #19150210C22semidirect
45. C4×Dic349C22semidirect
48. C22×C4×C349C22semidirect
49. C4⋊C4×C349C22semidirect
51. C22⋊C4×C349C22semidirect
52. C42×C349C22semidirect
76. C4×D349D4semidirect
92. C22×C349#41semidirect
103. C2×D349#41semidirect
122. C349#525semidirect
137. C4×D4D349semidirect
157. C22×C4C2×D349semidirect
163. C4⋊C4C2×D349semidirect
173. C22⋊C4C2×D349semidirect
175. C42C2×D349semidirect
225. C22#19150233semidirect

Maximal subgroups

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

subgroupindex
C2×#191502332
C22×C4×D3492
#260136242
#260136212
#260136172
#260136162
#260136122
#260136112
#260136092
#260136072
C4×D13962
C4×C22⋊D3492
#260135962
C4×D4×C3492
#525349