Finite Group Databasebeta

← #41003422

Subgroups — #41003422

Order 26944.

250 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 93 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#27212361C211yesmaximal, characteristic, complemented
2#27212359C211yesmaximal, characteristic, complemented
3#27212353C211yesmaximal, characteristic, complemented
4#27212352C211yesmaximal, characteristic, complemented
5#27212349C211yesmaximal, characteristic, complemented
6#27212347C211yesmaximal, characteristic, complemented
7#27212344C211yesmaximal, characteristic, complemented
8#27212338C211yesmaximal, characteristic, complemented
9#27212337C211yesmaximal, characteristic, complemented
10#27212334C211yesmaximal, characteristic, complemented
11#27212334C211yesmaximal, characteristic, complemented
12C4×C22⋊D421C211yesmaximal, characteristic, complemented
13#27212326C211yesmaximal, characteristic, complemented
14C22⋊C4×D421C211yesmaximal, characteristic, complemented
15#121×C421C211yesmaximal, characteristic, complemented
16C23×D421C2211yescharacteristic
17C2×Dic842C2211yescharacteristic, complemented
18C2×C22⋊D421C2211yescharacteristic
19C2×C22⋊D421C2211yescharacteristic
20C2×C22⋊D421C2211yescharacteristic
21C2×C22⋊D421C2211yescharacteristic
22C22×Dic421C2211yescharacteristic, complemented
23C2×C4×D421C2211yescharacteristic
24C2×C4×D421C2211yescharacteristic
25C4⋊Dic421C2211yescharacteristic, complemented
26#20269750C2211yescharacteristic, complemented
27#20269750C2211yescharacteristic, complemented
28#20269750C2211yescharacteristic, complemented
29#20269750C2211yescharacteristic, complemented
30C22⋊Dic421C2211yescharacteristic
31C22⋊Dic421C2211yescharacteristic
32C22⋊Dic421C2211yescharacteristic
33C22⋊Dic421C2211yescharacteristic, complemented
34C22⋊Dic421C2211yescharacteristic, complemented
35C22⋊Dic421C2211yescharacteristic, complemented
36#20269748C2211yescharacteristic
37#20269748C2211yescharacteristic
38#20269748C2211yescharacteristic
39#20269748C2211yescharacteristic
40#20269748C2211yescharacteristic
41#20269748C2211yescharacteristic
42C4×Dic421C2211yescharacteristic, 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—22no—
52C22×D421—41no—
53C22×D421—41no—
54C22×D421C2311yescharacteristic
55C22×D421C2311yescharacteristic
56Dic842—41no—
57C22⋊D421—22no—
58C22⋊D421—22no—
59C22⋊D421—41no—
60C22⋊D421—41no—
61C2×Dic421—41no—
62C2×Dic421C2311yescharacteristic
63C2×Dic421C2311yescharacteristic
64C2×Dic421C2311yescharacteristic
65C2×Dic421C2311yescharacteristic
66C2×Dic421C2311yescharacteristic
67C2×Dic421C2311yescharacteristic
68C4×D421—41no—
69C4×D421—41no—
70C23×C421C2311yescharacteristic
71C23×C421C2311yescharacteristic
72C23×C421C2311yescharacteristic
73D4×C421—41no—
74D4×C421—41no—
75D4×C421—41no—
76C2×C4×C421—41no—
77C2×C4×C421C2311yescharacteristic
78C2×C4×C421C2311yescharacteristic
79C2×C4×C421C2311yescharacteristic
80C2×C4×C421C2311yescharacteristic
81C22×C421—21no—
82C22×C421—21no—
83C22×C421—21no—
84C22×C421—21no—
85C22×C421—21no—
86C22×C421—21no—
87C22×C421—21no—
88C22×C421—21no—
89C22×C421—21no—
90C22×C421C2411yescharacteristic
91C2×D421—21no—
92C2×D421—21no—
93C2×D421—21no—
94C2×D421—21no—
95C2×D421—21no—
96C2×D421—44no—
97C2×D421#4112yes—
98C4×C421—21no—
99C4×C421—21no—
100C4×C421—21no—
101C4×C421—21no—
102Dic421—21no—
103Dic421—21no—
104Dic421—21no—
105Dic421—21no—
106Dic421—21no—
107Dic421—21no—
108C2×C421—41no—
109C2×C421—41no—
110C2×C421—41no—
111C2×C421C2×#4111yescharacteristic
112C2×C421H(5,𝔽2)11yescharacteristic
113C2×C421H(5,𝔽2)11yescharacteristic
114D421—22no—
115D421—41no—
116C421#53811yescharacteristic, complemented, Sylow
117#538—4211nomaximal, Sylow
118#126—4211no—
119#126—4211no—
120#124—4211no—
121#124—4211no—
122#123—4211no—
123#123—4211no—
124C22⋊Q8—4211no—
125#121—4211no—
126#121—4211no—
127#121D42111yescharacteristic, complemented
128#120—4211no—
129#120—4211no—
130C4×D4—4211no—
131C4×D4—4211no—
132C2×C22⋊C4—4211no—
133C24—4211no—
134C2×Q8—4211no—
135C2×D4—4211no—
136C2×D4—4211no—
137C2×D4—4211no—
138C2×D4—4211no—
139C2×D4C2×D42111yescharacteristic
140C2×D4C2×D42111yescharacteristic
141C2×D4C2×D42111yescharacteristic, complemented
142C22×C4—4211no—
143C22×C4—4211no—
144C22×C4—4211no—
145C22×C4C2×D42111yescharacteristic, complemented
146C4⋊C4—4211no—
147C4⋊C4—4211no—
148C4⋊C4—4211no—
149C4⋊C4—4211no—
150C4⋊C4—4211no—
151C4⋊C4C2×D42111yescharacteristic, complemented
152C22⋊C4—4211no—
153C22⋊C4—4211no—
154C22⋊C4—4211no—
155C22⋊C4—4211no—
156C22⋊C4—4211no—
157C22⋊C4—4211no—
158C22⋊C4—4211no—
159C22⋊C4—4211no—
160C22⋊C4—4211no—
161C22⋊C4—4211no—
162C22⋊C4—4211no—
163C22⋊C4—4211no—
164C22⋊C4C2×D42111yescharacteristic
165C22⋊C4C2×D42111yescharacteristic, complemented
166C42—4211no—
167C42—4211no—
168C23—8422no—
169C23—16841no—
170C23—16841no—
171C23—4211no—
172C23—4211no—
173C23C22×D42111yescharacteristic
174C23C22×D42111yescharacteristic
175C23C22×D42111yescharacteristic
176Q8—16841no—
177D4—41no—
178D4—41no—
179D4—41no—
180D4—8422no—
181D4—8422no—
182D4—16841no—
183D4—16841no—
184C2×C4—41no—
185C2×C4—16841no—
186C2×C4—16841no—
187C2×C4—16841no—
188C2×C4—4211no—
189C2×C4—4211no—
190C2×C4—4211no—
191C2×C4—4211no—
192C2×C4—4211no—
193C2×C4—4211no—
194C2×C4C22×D42111yescharacteristic
195C2×C4C22×D42111yescharacteristic
196C2×C4C22×D42111yescharacteristic
197C2×C4C22×D42111yescharacteristic
198C22—21no—
199C22—21no—
200C22—21no—
201C22—21no—
202C22—21no—
203C22—21no—
204C22—21no—
205C22—21no—
206C22—21no—
207C22—8421no—
208C22—8421no—
209C22—8421no—
210C22—8421no—
211C22—8421no—
212C22—16844no—
213C22—4212no—
214C22C23×D42111yescharacteristic
215C4—21no—
216C4—21no—
217C4—21no—
218C4—21no—
219C4—8421no—
220C4—8421no—
221C4—8421no—
222C4—8421no—
223C4—8421no—
224C4—8421no—
225C2—41no—
226C2—41no—
227C2—41no—
228C2—8422no—
229C2—16841no—
230C2#41×D42111yescharacteristic
231C2#2721241111yescharacteristic
232C2#2721241111yescharacteristic
subgroupG/H
center, FrattiniC22C23×D421
commutator, socle, cosocleC22×C421C24
Fitting#121×C421C2
hypercenter#121D421
hypocenterC421#538
LCSG ▷ C22×C421 ▷ C421
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. #27212361C2semidirect
2. #27212359C2semidirect
3. #27212353C2semidirect
4. #27212352C2semidirect
5. #27212349C2semidirect
6. #27212347C2semidirect
7. #27212344C2semidirect
8. #27212338C2semidirect
9. #27212337C2semidirect
10. #27212334C2semidirect
11. #27212334C2semidirect
12. C4×C22⋊D421C2semidirect
13. #27212326C2semidirect
14. C22⋊C4×D421C2semidirect
15. #121×C421C2semidirect
17. C2×Dic842C22semidirect
22. C22×Dic421C22semidirect
25. C4⋊Dic421C22semidirect
26. #20269750C22semidirect
27. #20269750C22semidirect
28. #20269750C22semidirect
29. #20269750C22semidirect
33. C22⋊Dic421C22semidirect
34. C22⋊Dic421C22semidirect
35. C22⋊Dic421C22semidirect
42. C4×Dic421C22semidirect
43. C4×Dic421C22semidirect
46. C2×D4×C421C22semidirect
47. C22×C4×C421C22semidirect
48. C4⋊C4×C421C22semidirect
50. C22⋊C4×C421C22semidirect
116. C421#538semidirect
127. #121D421semidirect
141. C2×D4C2×D421semidirect
145. C22×C4C2×D421semidirect
151. C4⋊C4C2×D421semidirect
165. C22⋊C4C2×D421semidirect

Maximal subgroups

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

subgroupindex
#272123612
#272123592
#272123532
#272123522
#272123492
#272123472
#272123442
#272123382
#272123372
#272123342
C4×C22⋊D4212
#272123262
C22⋊C4×D4212
#121×C4212
#538421