Finite Group Databasebeta

← #41003578

Subgroups — #41003578

Order 26944.

252 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 91 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
1#27212362C211yesmaximal, characteristic, complemented
2#27212361C211yesmaximal, characteristic, complemented
3#27212359C211yesmaximal, characteristic, complemented
4#27212351C211yesmaximal, characteristic, complemented
5#27212350C211yesmaximal, characteristic, complemented
6#27212349C211yesmaximal, characteristic, complemented
7#27212340C211yesmaximal, characteristic, complemented
8#27212339C211yesmaximal, characteristic, complemented
9#27212338C211yesmaximal, characteristic, complemented
10#27212338C211yesmaximal, characteristic, complemented
11#27212337C211yesmaximal, characteristic, complemented
12#27212336C211yesmaximal, characteristic, complemented
13#27212334C211yesmaximal, characteristic, complemented
14#27212334C211yesmaximal, characteristic, complemented
15#121×C421C211yesmaximal, characteristic, complemented
16C23×D421C2211yescharacteristic
17C2×D1684C2211yescharacteristic
18C2×D1684C2211yescharacteristic
19C2×C22⋊D421C2211yescharacteristic
20C2×C22⋊D421C2211yescharacteristic
21C2×C22⋊D421C2211yescharacteristic
22C2×C22⋊D421C2211yescharacteristic
23C2×C22⋊D421C2211yescharacteristic
24C2×C22⋊D421C2211yescharacteristic, complemented
25C2×C22⋊D421C2211yescharacteristic, complemented
26C22×Dic421C2211yescharacteristic, complemented
27C2×C4×D421C2211yescharacteristic, complemented
28C4⋊Dic421C2211yescharacteristic, complemented
29C4⋊Dic421C2211yescharacteristic, complemented
30#20269750C2211yescharacteristic, complemented
31#20269750C2211yescharacteristic, complemented
32#20269750C2211yescharacteristic, complemented
33C22⋊Dic421C2211yescharacteristic
34C22⋊Dic421C2211yescharacteristic
35C22⋊Dic421C2211yescharacteristic, complemented
36#20269748C2211yescharacteristic
37#20269748C2211yescharacteristic
38#20269748C2211yescharacteristic
39#20269748C2211yescharacteristic
40#20269748C2211yescharacteristic
41#20269748C2211yescharacteristic
42#20269748C2211yescharacteristic, complemented
43C4×Dic421C2211yescharacteristic, complemented
44C2×D4×C421C2211yescharacteristic
45C2×D4×C421C2211yescharacteristic
46C2×D4×C421C2211yescharacteristic, complemented
47C22×C4×C421C2211yescharacteristic, complemented
48C4⋊C4×C421C2211yescharacteristic, complemented
49C22⋊C4×C421C2211yescharacteristic
50C22⋊C4×C421C2211yescharacteristic, complemented
51C22×D421—41no—
52C22×D421—41no—
53C22×D421—41no—
54C22×D421C2311yescharacteristic
55C22×D421C2311yescharacteristic
56C22×D421C2311yescharacteristic
57D1684—41no—
58D1684—41no—
59C22⋊D421—41no—
60C22⋊D421—41no—
61C22⋊D421—41no—
62C22⋊D421—41no—
63C22⋊D421—41no—
64C22⋊D421—41no—
65C22⋊D421—41no—
66C2×Dic421—41no—
67C2×Dic421C2311yescharacteristic
68C2×Dic421C2311yescharacteristic
69C2×Dic421C2311yescharacteristic
70C2×Dic421C2311yescharacteristic
71C2×Dic421C2311yescharacteristic
72C4×D421—41no—
73C23×C421C2311yescharacteristic
74C23×C421C2311yescharacteristic
75C23×C421C2311yescharacteristic
76D4×C421—41no—
77D4×C421—41no—
78D4×C421—41no—
79C2×C4×C421—41no—
80C2×C4×C421C2311yescharacteristic
81C2×C4×C421C2311yescharacteristic
82C2×C4×C421C2311yescharacteristic
83C2×C4×C421C2311yescharacteristic
84C22×C421—21no—
85C22×C421—21no—
86C22×C421—21no—
87C22×C421—21no—
88C22×C421—21no—
89C22×C421—21no—
90C22×C421—21no—
91C22×C421—21no—
92C22×C421—21no—
93C22×C421C2411yescharacteristic
94C2×D421—21no—
95C2×D421—21no—
96C2×D421—21no—
97C2×D421—21no—
98C2×D421—21no—
99C2×D421—21no—
100C2×D421—21no—
101C2×D421—21no—
102C2×D421—21no—
103C2×D421—44no—
104C4×C421—21no—
105C4×C421—21no—
106C4×C421—21no—
107C4×C421—21no—
108Dic421—21no—
109Dic421—21no—
110Dic421—21no—
111Dic421—21no—
112Dic421—21no—
113C2×C421—41no—
114C2×C421—41no—
115C2×C421—41no—
116C2×C421H(5,𝔽2)11yescharacteristic
117C2×C421H(5,𝔽2)11yescharacteristic
118C2×C421H(5,𝔽2)11yescharacteristic
119D421—41no—
120D421—41no—
121D421—41no—
122C421#56011yescharacteristic, complemented, Sylow
123#560—4211nomaximal, Sylow
124#127—4211no—
125#126—4211no—
126#126—4211no—
127#123—4211no—
128#123—4211no—
129#123—4211no—
130#121—4211no—
131#121—4211no—
132#121—4211no—
133#121—4211no—
134#121—4211no—
135#121D42111yescharacteristic, complemented
136#120—4211no—
137#120—4211no—
138#120—4211no—
139C24—4211no—
140C2×D4—4211no—
141C2×D4—4211no—
142C2×D4—4211no—
143C2×D4—4211no—
144C2×D4—4211no—
145C2×D4—4211no—
146C2×D4—4211no—
147C2×D4—4211no—
148C2×D4—4211no—
149C2×D4C2×D42111yescharacteristic
150C2×D4C2×D42111yescharacteristic
151C2×D4C2×D42111yescharacteristic, complemented
152C22×C4—4211no—
153C22×C4—4211no—
154C22×C4C2×D42111yescharacteristic, complemented
155C4⋊C4—4211no—
156C4⋊C4—4211no—
157C4⋊C4—4211no—
158C4⋊C4—4211no—
159C4⋊C4—4211no—
160C4⋊C4C2×D42111yescharacteristic, complemented
161C22⋊C4—4211no—
162C22⋊C4—4211no—
163C22⋊C4—4211no—
164C22⋊C4—4211no—
165C22⋊C4—4211no—
166C22⋊C4—4211no—
167C22⋊C4—4211no—
168C22⋊C4—4211no—
169C22⋊C4—4211no—
170C22⋊C4—4211no—
171C22⋊C4C2×D42111yescharacteristic
172C22⋊C4C2×D42111yescharacteristic, complemented
173C42—4211no—
174C23—16841no—
175C23—16841no—
176C23—16841no—
177C23—4211no—
178C23—4211no—
179C23—4211no—
180C23C22×D42111yescharacteristic
181C23C22×D42111yescharacteristic
182C23C22×D42111yescharacteristic
183D4—41no—
184D4—41no—
185D4—41no—
186D4—16841no—
187D4—16841no—
188D4—16841no—
189D4—16841no—
190D4—16841no—
191D4—16841no—
192D4—16841no—
193D4—16841no—
194D4—16841no—
195C2×C4—41no—
196C2×C4—16841no—
197C2×C4—16841no—
198C2×C4—4211no—
199C2×C4—4211no—
200C2×C4—4211no—
201C2×C4—4211no—
202C2×C4—4211no—
203C2×C4C22×D42111yescharacteristic
204C2×C4C22×D42111yescharacteristic
205C2×C4C22×D42111yescharacteristic
206C2×C4C22×D42111yescharacteristic
207C22—21no—
208C22—21no—
209C22—21no—
210C22—21no—
211C22—21no—
212C22—21no—
213C22—21no—
214C22—21no—
215C22—21no—
216C22—8421no—
217C22—8421no—
218C22—8421no—
219C22—8421no—
220C22—8421no—
221C22—8421no—
222C22—8421no—
223C22—8421no—
224C22—8421no—
225C22—16844no—
226C22C23×D42111yescharacteristic
227C4—21no—
228C4—21no—
229C4—21no—
230C4—21no—
231C4—8421no—
232C4—8421no—
233C4—8421no—
234C4—8421no—
235C4—8421no—
236C2—41no—
237C2—41no—
238C2—41no—
239C2—16841no—
240C2—16841no—
241C2—16841no—
242C2#2721241111yescharacteristic
243C2#2721241111yescharacteristic
244C2#2721241211yescharacteristic
subgroupG/H
center, FrattiniC22C23×D421
commutator, socle, cosocleC22×C421C24
Fitting#121×C421C2
hypercenter#121D421
hypocenterC421#560
LCSG ▷ C22×C421 ▷ C421
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. #27212362C2semidirect
2. #27212361C2semidirect
3. #27212359C2semidirect
4. #27212351C2semidirect
5. #27212350C2semidirect
6. #27212349C2semidirect
7. #27212340C2semidirect
8. #27212339C2semidirect
9. #27212338C2semidirect
10. #27212338C2semidirect
11. #27212337C2semidirect
12. #27212336C2semidirect
13. #27212334C2semidirect
14. #27212334C2semidirect
15. #121×C421C2semidirect
24. C2×C22⋊D421C22semidirect
25. C2×C22⋊D421C22semidirect
26. C22×Dic421C22semidirect
27. C2×C4×D421C22semidirect
28. C4⋊Dic421C22semidirect
29. C4⋊Dic421C22semidirect
30. #20269750C22semidirect
31. #20269750C22semidirect
32. #20269750C22semidirect
35. C22⋊Dic421C22semidirect
42. #20269748C22semidirect
43. C4×Dic421C22semidirect
46. C2×D4×C421C22semidirect
47. C22×C4×C421C22semidirect
48. C4⋊C4×C421C22semidirect
50. C22⋊C4×C421C22semidirect
122. C421#560semidirect
135. #121D421semidirect
151. C2×D4C2×D421semidirect
154. C22×C4C2×D421semidirect
160. C4⋊C4C2×D421semidirect
172. C22⋊C4C2×D421semidirect

Maximal subgroups

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

subgroupindex
#272123622
#272123612
#272123592
#272123512
#272123502
#272123492
#272123402
#272123392
#272123382
#272123372
#272123362
#272123342
#121×C4212
#560421