Finite Group Databasebeta

← #41003514

Subgroups — #41003514

Order 26944.

248 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 97 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#27212360C211yesmaximal, characteristic, complemented
3#27212354C211yesmaximal, characteristic, complemented
4#27212351C211yesmaximal, characteristic, complemented
5#27212351C211yesmaximal, characteristic, complemented
6#27212349C211yesmaximal, characteristic, complemented
7#27212344C211yesmaximal, characteristic, complemented
8#27212343C211yesmaximal, characteristic, complemented
9#27212336C211yesmaximal, characteristic
10C4×D1684C211yesmaximal, characteristic, complemented
11#27212326C211yesmaximal, characteristic, complemented
12#27212323C211yesmaximal, characteristic, complemented
13C2×#20269748C211yesmaximal, characteristic, complemented
14C22⋊C4×D421C211yesmaximal, characteristic, complemented
15#117×C421C211yesmaximal, characteristic, complemented
16C23×D421C2211yescharacteristic
17C2×Dic842C2211yescharacteristic, complemented
18C2×D1684C2211yescharacteristic
19C2×D1684C2211yescharacteristic
20C2×C22⋊D421C2211yescharacteristic
21C22×Dic421C2211yescharacteristic
22C2×C4×D421C2211yescharacteristic
23C2×C4×D421C2211yescharacteristic
24C2×C4×D421C2211yescharacteristic, complemented
25C4⋊Dic421C2211yescharacteristic, complemented
26#20269750C2211yescharacteristic, complemented
27#20269750C2211yescharacteristic, complemented
28#20269750C2211yescharacteristic, complemented
29#20269750C2211yescharacteristic, complemented
30#20269750C2211yescharacteristic, complemented
31C22⋊Dic421C2211yescharacteristic
32#20269748C2211yescharacteristic
33#20269748C2211yescharacteristic
34#20269748C2211yescharacteristic
35#20269748C2211yescharacteristic
36#20269748C2211yescharacteristic
37#20269748C2211yescharacteristic
38#20269748C2211yescharacteristic
39#20269748C2211yescharacteristic
40#20269748C2211yescharacteristic, complemented
41#20269748C2211yescharacteristic, complemented
42#20269748C2211yescharacteristic, complemented
43C4×Dic421C2211yescharacteristic, complemented
44C22×C4×C421C2211yescharacteristic
45C4⋊C4×C421C2211yescharacteristic, complemented
46C4⋊C4×C421C2211yescharacteristic, complemented
47C22⋊C4×C421C2211yescharacteristic
48C22⋊C4×C421C2211yescharacteristic
49C42×C421C2211yescharacteristic, complemented
50C42×C421C2211yescharacteristic, complemented
51C22×D421—22no—
52C22×D421—22no—
53C22×D421—41no—
54C22×D421C2311yescharacteristic
55C22×D421C2311yescharacteristic
56C22×D421C2311yescharacteristic
57Dic842—41no—
58D1684—22no—
59D1684—41no—
60C22⋊D421—22no—
61C2×Dic421—41no—
62C2×Dic421C2311yescharacteristic
63C2×Dic421C2311yescharacteristic
64C2×Dic421C2311yescharacteristic
65C2×Dic421C2311yescharacteristic
66C2×Dic421C2311yescharacteristic
67C4×D421—22no—
68C4×D421—41no—
69C4×D421—41no—
70C23×C421C2311yescharacteristic
71C2×C4×C421—22no—
72C2×C4×C421C2311yescharacteristic
73C2×C4×C421C2311yescharacteristic
74C2×C4×C421C2311yescharacteristic
75C2×C4×C421C2311yescharacteristic
76C2×C4×C421C2311yescharacteristic
77C2×C4×C421C2311yescharacteristic
78C22×C421—21no—
79C22×C421—21no—
80C22×C421#4112yes—
81C22×C421C2411yescharacteristic
82C2×D421—21no—
83C2×D421—21no—
84C2×D421—21no—
85C2×D421—21no—
86C2×D421—21no—
87C2×D421—21no—
88C2×D421—21no—
89C2×D421—21no—
90C2×D421—44no—
91C2×D421#4112yes—
92C4×C421—21no—
93C4×C421—21no—
94C4×C421—21no—
95C4×C421—21no—
96C4×C421—21no—
97C4×C421—21no—
98Dic421—21no—
99Dic421—21no—
100Dic421—21no—
101Dic421—21no—
102Dic421—21no—
103C2×C421—22no—
104C2×C421C2×#4111yescharacteristic
105C2×C421C2×#4111yescharacteristic
106C2×C421H(5,𝔽2)11yescharacteristic
107D421—22no—
108D421—41no—
109D421—41no—
110C421#55111yescharacteristic, complemented, Sylow
111#551—4211nomaximal, Sylow
112#126—4211no—
113#126—4211no—
114#124—4211no—
115#123—4211no—
116#123—4211no—
117#123—4211no—
118C22⋊Q8—4211no—
119C22⋊Q8—4211no—
120#120—4211no—
121C4×D4—4211no—
122C4×D4—4211no—
123#117—4211no—
124#117D42111yescharacteristic, complemented
125C2×C22⋊C4—4211no—
126C2×C22⋊C4—4211no—
127C24—4211no—
128C2×Q8—4211no—
129C2×D4—4211no—
130C2×D4—4211no—
131C2×D4—4211no—
132C22×C4—4211no—
133C22×C4—4211no—
134C22×C4—4211no—
135C22×C4—4211no—
136C22×C4C2×D42111yescharacteristic
137C4⋊C4—4211no—
138C4⋊C4—4211no—
139C4⋊C4—4211no—
140C4⋊C4—4211no—
141C4⋊C4—4211no—
142C4⋊C4—4211no—
143C4⋊C4C2×D42111yescharacteristic, complemented
144C4⋊C4C2×D42111yescharacteristic, complemented
145C22⋊C4—4211no—
146C22⋊C4—4211no—
147C22⋊C4—4211no—
148C22⋊C4—4211no—
149C22⋊C4—4211no—
150C22⋊C4—4211no—
151C22⋊C4—4211no—
152C22⋊C4—4211no—
153C22⋊C4—4211no—
154C22⋊C4—4211no—
155C22⋊C4—4211no—
156C22⋊C4—4211no—
157C22⋊C4C2×D42111yescharacteristic
158C22⋊C4C2×D42111yescharacteristic
159C42—4211no—
160C42C2×D42111yescharacteristic, complemented
161C42C2×D42111yescharacteristic, complemented
162C23—8422no—
163C23—8422no—
164C23—16841no—
165C23—4211no—
166C23—4211no—
167C23—4211no—
168C23C22×D42111yescharacteristic
169Q8—16841no—
170D4—8422no—
171D4—8422no—
172D4—16841no—
173C2×C4—22no—
174C2×C4—8422no—
175C2×C4—16841no—
176C2×C4—16841no—
177C2×C4—16841no—
178C2×C4—4211no—
179C2×C4—4211no—
180C2×C4—4211no—
181C2×C4—4211no—
182C2×C4—4211no—
183C2×C4C22×D42111yescharacteristic
184C2×C4C22×D42111yescharacteristic
185C2×C4C22×D42111yescharacteristic
186C2×C4C22×D42111yescharacteristic
187C2×C4C22×D42111yescharacteristic
188C2×C4C22×D42111yescharacteristic
189C22—21no—
190C22—21no—
191C22—8421no—
192C22—8421no—
193C22—8421no—
194C22—8421no—
195C22—8421no—
196C22—8421no—
197C22—8421no—
198C22—8421no—
199C22—16844no—
200C22—4212no—
201C22#2026977112yes—
202C22C23×D42111yescharacteristic
203C4—21no—
204C4—21no—
205C4—21no—
206C4—21no—
207C4—21no—
208C4—21no—
209C4—8421no—
210C4—8421no—
211C4—8421no—
212C4—8421no—
213C4—8421no—
214C2—22no—
215C2—8422no—
216C2—16841no—
217C2—16841no—
218C2#41×D42111yescharacteristic
219C2C2×#2026977111yescharacteristic
220C2#2721241211yescharacteristic
subgroupG/H
center, FrattiniC22C23×D421
commutator, socle, cosocleC22×C421C24
Fitting#117×C421C2
prime radical#27212336C2
hypercenter#117D421
hypocenterC421#551
LCSG ▷ C22×C421 ▷ C421
UCSC1 ◁ C22 ◁ #117

Complements

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

NH ≅ G/Ntype
1. #27212361C2semidirect
2. #27212360C2semidirect
3. #27212354C2semidirect
4. #27212351C2semidirect
5. #27212351C2semidirect
6. #27212349C2semidirect
7. #27212344C2semidirect
8. #27212343C2semidirect
10. C4×D1684C2semidirect
11. #27212326C2semidirect
12. #27212323C2semidirect
13. C2×#20269748C2semidirect
14. C22⋊C4×D421C2semidirect
15. #117×C421C2semidirect
17. C2×Dic842C22semidirect
24. C2×C4×D421C22semidirect
25. C4⋊Dic421C22semidirect
26. #20269750C22semidirect
27. #20269750C22semidirect
28. #20269750C22semidirect
29. #20269750C22semidirect
30. #20269750C22semidirect
40. #20269748C22semidirect
41. #20269748C22semidirect
42. #20269748C22semidirect
43. C4×Dic421C22semidirect
45. C4⋊C4×C421C22semidirect
46. C4⋊C4×C421C22semidirect
49. C42×C421C22semidirect
50. C42×C421C22semidirect
110. C421#551semidirect
124. #117D421semidirect
143. C4⋊C4C2×D421semidirect
144. C4⋊C4C2×D421semidirect
160. C42C2×D421semidirect
161. C42C2×D421semidirect

Maximal subgroups

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

subgroupindex
#272123612
#272123602
#272123542
#272123512
#272123492
#272123442
#272123432
#272123362
C4×D16842
#272123262
#272123232
C2×#202697482
C22⋊C4×D4212
#117×C4212
#551421