Finite Group Databasebeta

← #41003482

Subgroups — #41003482

Order 26944.

334 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 105 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×#20269771C211yesmaximal, characteristic, complemented
2C2×D4×D421C211yesmaximal, characteristic, complemented
3#27212352C211yesmaximal, characteristic, complemented
4#27212351C211yesmaximal, characteristic, complemented
5#27212344C211yesmaximal, characteristic, complemented
6#27212341C211yesmaximal, characteristic, complemented
7#27212339C211yesmaximal, characteristic, complemented
8#27212338C211yesmaximal, characteristic, complemented
9#27212336C211yesmaximal, characteristic, complemented
10#27212336C211yesmaximal, characteristic, complemented
11#27212328C211yesmaximal, characteristic, complemented
12#27212326C211yesmaximal, characteristic, complemented
13C2×#20269748C211yesmaximal, characteristic, complemented
14C22⋊C4×D421C211yesmaximal, characteristic, complemented
15#123×C421C211yesmaximal, characteristic, complemented
16C23×D421C2211yescharacteristic
17C23×D421C2211yescharacteristic
18#20269771—24no—
19C2×Dic842C2211yescharacteristic, complemented
20C2×D1684C2211yescharacteristic
21C2×D1684C2211yescharacteristic
22C2×D1684C2211yescharacteristic, complemented
23C2×D1684C2211yescharacteristic, complemented
24C2×C22⋊D421C2211yescharacteristic
25C2×C22⋊D421C2211yescharacteristic
26C2×C22⋊D421C2211yescharacteristic, complemented
27C2×C22⋊D421C2211yescharacteristic, complemented
28D4×D421—24no—
29C22×Dic421C2211yescharacteristic, complemented
30C2×C4×D421C2211yescharacteristic
31C2×C4×D421C2211yescharacteristic, complemented
32C2×C4×D421C2211yescharacteristic, complemented
33C2×C4×D421C2211yescharacteristic, complemented
34#20269750C2211yescharacteristic, complemented
35#20269750C2211yescharacteristic, complemented
36C22⋊Dic421C2211yescharacteristic, complemented
37#20269748C2211yescharacteristic
38#20269748C2211yescharacteristic
39#20269748C2211yescharacteristic
40#20269748C2211yescharacteristic, complemented
41#20269748C2211yescharacteristic, complemented
42#20269748C2211yescharacteristic, complemented
43#20269748C2211yescharacteristic, complemented
44#20269748C2211yescharacteristic, complemented
45C4×Dic421C2211yescharacteristic, complemented
46C2×D4×C421C2211yescharacteristic
47C22×C4×C421C2211yescharacteristic, complemented
48C4⋊C4×C421C2211yescharacteristic, complemented
49C4⋊C4×C421C2211yescharacteristic, complemented
50C22⋊C4×C421C2211yescharacteristic, complemented
51C22⋊C4×C421C2211yescharacteristic, complemented
52C22⋊C4×C421C2211yescharacteristic, complemented
53C22×D421—22no—
54C22×D421—22no—
55C22×D421—22no—
56C22×D421—22no—
57C22×D421—41no—
58C22×D421—41no—
59C22×D421C2311yescharacteristic
60C22×D421C2311yescharacteristic
61C22×D421C2311yescharacteristic
62C22×D421C2311yescharacteristic
63Dic842—22no—
64D1684—22no—
65D1684—22no—
66D1684—22no—
67D1684—41no—
68C22⋊D421—22no—
69C22⋊D421—22no—
70C22⋊D421—41no—
71C22⋊D421D414yescomplemented
72C2×Dic421—41no—
73C2×Dic421C2311yescharacteristic
74C2×Dic421C2311yescharacteristic
75C2×Dic421C2311yescharacteristic
76C2×Dic421C2311yescharacteristic
77C4×D421—22no—
78C4×D421—22no—
79C4×D421—22no—
80C4×D421—41no—
81C23×C421C2311yescharacteristic
82C23×C421C2311yescharacteristic
83D4×C421—22no—
84C2×C4×C421—22no—
85C2×C4×C421C2311yescharacteristic
86C2×C4×C421C2311yescharacteristic
87C2×C4×C421C2311yescharacteristic
88C2×C4×C421C2311yescharacteristic
89C2×C4×C421C2311yescharacteristic
90C22×C421—21no—
91C22×C421—21no—
92C22×C421—21no—
93C22×C421—21no—
94C22×C421—21no—
95C22×C421C2×D412yes—
96C22×C421C2411yescharacteristic
97C2×D421—21no—
98C2×D421—21no—
99C2×D421—21no—
100C2×D421—21no—
101C2×D421—21no—
102C2×D421—21no—
103C2×D421—21no—
104C2×D421—21no—
105C2×D421—21no—
106C2×D421—21no—
107C2×D421—44no—
108C2×D421—44no—
109C2×D421C2×D412yes—
110C2×D421#4112yescomplemented
111C4×C421—21no—
112C4×C421—21no—
113C4×C421—21no—
114C4×C421—21no—
115C4×C421—21no—
116Dic421—21no—
117Dic421—21no—
118Dic421—21no—
119Dic421C2×D412yes—
120C2×C421—22no—
121C2×C421—41no—
122C2×C421C22×D411yescharacteristic
123C2×C421C2×#4111yescharacteristic
124C2×C421H(5,𝔽2)11yescharacteristic
125D421—22no—
126D421—22no—
127D421—41no—
128D421—41no—
129C421#54611yescharacteristic, complemented, Sylow
130#546—4211nomaximal, Sylow
131C2×#41—4211no—
132C22×D4—4211no—
133#124—4211no—
134#123—4211no—
135#123D42111yescharacteristic, complemented
136C22⋊Q8—4211no—
137#121—4211no—
138#121—4211no—
139#121—4211no—
140#120—4211no—
141#120—4211no—
142C4×D4—4211no—
143C4×D4—4211no—
144C2×C22⋊C4—4211no—
145C2×C22⋊C4—4211no—
146C24—4211no—
147C24—4211no—
148#41—8424no—
149C2×Q8—4211no—
150C2×D4—8424no—
151C2×D4—4211no—
152C2×D4—4211no—
153C2×D4—4211no—
154C2×D4—4211no—
155C2×D4—4211no—
156C2×D4—4211no—
157C2×D4—4211no—
158C2×D4—4211no—
159C2×D4C2×D42111yescharacteristic
160C22×C4—4211no—
161C22×C4—4211no—
162C22×C4—4211no—
163C22×C4—4211no—
164C22×C4—4211no—
165C22×C4C2×D42111yescharacteristic, complemented
166C4⋊C4—4211no—
167C4⋊C4—4211no—
168C4⋊C4C2×D42111yescharacteristic, complemented
169C4⋊C4C2×D42111yescharacteristic, complemented
170C22⋊C4—4211no—
171C22⋊C4—4211no—
172C22⋊C4—4211no—
173C22⋊C4—4211no—
174C22⋊C4—4211no—
175C22⋊C4—4211no—
176C22⋊C4—4211no—
177C22⋊C4—4211no—
178C22⋊C4—4211no—
179C22⋊C4C2×D42111yescharacteristic, complemented
180C22⋊C4C2×D42111yescharacteristic, complemented
181C22⋊C4C2×D42111yescharacteristic, complemented
182C42—4211no—
183C23—8422no—
184C23—8422no—
185C23—8422no—
186C23—8422no—
187C23—16841no—
188C23—16841no—
189C23—4211no—
190C23—4211no—
191C23—4211no—
192C23—4211no—
193C23C22×D42111yescharacteristic
194C23C22×D42111yescharacteristic
195Q8—8422no—
196D4—22no—
197D4—8422no—
198D4—8422no—
199D4—8422no—
200D4—8422no—
201D4—8422no—
202D4—16841no—
203D4—16841no—
204D4—4214no—
205C2×C4—22no—
206C2×C4—8422no—
207C2×C4—8422no—
208C2×C4—8422no—
209C2×C4—16841no—
210C2×C4—16841no—
211C2×C4—4211no—
212C2×C4—4211no—
213C2×C4—4211no—
214C2×C4—4211no—
215C2×C4C22×D42111yescharacteristic
216C2×C4C22×D42111yescharacteristic
217C2×C4C22×D42111yescharacteristic
218C2×C4C22×D42111yescharacteristic
219C2×C4C22×D42111yescharacteristic
220C22—21no—
221C22—21no—
222C22—21no—
223C22—21no—
224C22—21no—
225C22—8421no—
226C22—8421no—
227C22—8421no—
228C22—8421no—
229C22—8421no—
230C22—8421no—
231C22—8421no—
232C22—8421no—
233C22—8421no—
234C22—8421no—
235C22—16844no—
236C22—16844no—
237C22—4212no—
238C22—4212no—
239C22D4×D42112yes—
240C22C23×D42111yescharacteristic
241C4—21no—
242C4—21no—
243C4—21no—
244C4—21no—
245C4—21no—
246C4—8421no—
247C4—8421no—
248C4—8421no—
249C4—4212no—
250C2—22no—
251C2—41no—
252C2—8422no—
253C2—8422no—
254C2—16841no—
255C2—16841no—
256C2C2×D4×D42111yescharacteristic
257C2#41×D42111yescharacteristic
258C2#2721241211yescharacteristic
subgroupG/H
center, FrattiniC22C23×D421
commutator, socle, cosocleC22×C421C24
Fitting#123×C421C2
hypercenter#123D421
hypocenterC421#546
LCSG ▷ C22×C421 ▷ C421
UCSC1 ◁ C22 ◁ #123

Complements

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

NH ≅ G/Ntype
1. C2×#20269771C2semidirect
2. C2×D4×D421C2semidirect
3. #27212352C2semidirect
4. #27212351C2semidirect
5. #27212344C2semidirect
6. #27212341C2semidirect
7. #27212339C2semidirect
8. #27212338C2semidirect
9. #27212336C2semidirect
10. #27212336C2semidirect
11. #27212328C2semidirect
12. #27212326C2semidirect
13. C2×#20269748C2semidirect
14. C22⋊C4×D421C2semidirect
15. #123×C421C2semidirect
19. C2×Dic842C22semidirect
22. C2×D1684C22semidirect
23. C2×D1684C22semidirect
26. C2×C22⋊D421C22semidirect
27. C2×C22⋊D421C22semidirect
29. C22×Dic421C22semidirect
31. C2×C4×D421C22semidirect
32. C2×C4×D421C22semidirect
33. C2×C4×D421C22semidirect
34. #20269750C22semidirect
35. #20269750C22semidirect
36. C22⋊Dic421C22semidirect
40. #20269748C22semidirect
41. #20269748C22semidirect
42. #20269748C22semidirect
43. #20269748C22semidirect
44. #20269748C22semidirect
45. C4×Dic421C22semidirect
47. C22×C4×C421C22semidirect
48. C4⋊C4×C421C22semidirect
49. C4⋊C4×C421C22semidirect
50. C22⋊C4×C421C22semidirect
51. C22⋊C4×C421C22semidirect
52. C22⋊C4×C421C22semidirect
71. C22⋊D421D4semidirect
110. C2×D421#41semidirect
129. C421#546semidirect
135. #123D421semidirect
165. C22×C4C2×D421semidirect
168. C4⋊C4C2×D421semidirect
169. C4⋊C4C2×D421semidirect
179. C22⋊C4C2×D421semidirect
180. C22⋊C4C2×D421semidirect
181. C22⋊C4C2×D421semidirect

Maximal subgroups

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

subgroupindex
C2×#202697712
C2×D4×D4212
#272123522
#272123512
#272123442
#272123412
#272123392
#272123382
#272123362
#272123282
#272123262
C2×#202697482
C22⋊C4×D4212
#123×C4212
#546421