Finite Group Databasebeta

← #41001038

Subgroups — #41001038

Order 26816.

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#27211713C211yesmaximal, characteristic, complemented
2#27211713C211yesmaximal, characteristic, complemented
3#27211707C211yesmaximal, characteristic, complemented
4#27211703C211yesmaximal, characteristic, complemented
5#27211703C211yesmaximal, characteristic, complemented
6#27211702C211yesmaximal, characteristic, complemented
7#27211694C211yesmaximal, characteristic, complemented
8#27211688C211yesmaximal, characteristic
9#27211680C211yesmaximal, characteristic, complemented
10C4×C22⋊D419C211yesmaximal, characteristic, complemented
11#27211676C211yesmaximal, characteristic, complemented
12#27211674C211yesmaximal, characteristic, complemented
13C2×(C2×D419)⋊C4C211yesmaximal, characteristic, complemented
14C22⋊C4×D419C211yesmaximal, characteristic, complemented
15C22⋊Q8×C419C211yesmaximal, characteristic, complemented
16C23×D419C2211yescharacteristic
17C2×D1676C2211yescharacteristic
18C2×D1676C2211yescharacteristic
19C2×C22⋊D419C2211yescharacteristic
20C22×Dic419C2211yescharacteristic
21C2×C4×D419C2211yescharacteristic
22C2×C4×D419C2211yescharacteristic
23C2×C4×D419C2211yescharacteristic, complemented
24C4⋊Dic419C2211yescharacteristic, complemented
25C4⋊Dic419C2211yescharacteristic, complemented
26Dic419⋊C4C2211yescharacteristic, complemented
27Dic419⋊C4C2211yescharacteristic, complemented
28Dic419⋊C4C2211yescharacteristic, complemented
29C22⋊Dic419C2211yescharacteristic
30(C2×D419)⋊C4C2211yescharacteristic
31(C2×D419)⋊C4C2211yescharacteristic
32(C2×D419)⋊C4C2211yescharacteristic
33(C2×D419)⋊C4C2211yescharacteristic
34(C2×D419)⋊C4C2211yescharacteristic
35(C2×D419)⋊C4C2211yescharacteristic
36(C2×D419)⋊C4C2211yescharacteristic
37(C2×D419)⋊C4C2211yescharacteristic
38(C2×D419)⋊C4C2211yescharacteristic, complemented
39(C2×D419)⋊C4C2211yescharacteristic, complemented
40(C2×D419)⋊C4C2211yescharacteristic, complemented
41C4×Dic419C2211yescharacteristic, complemented
42C4×Dic419C2211yescharacteristic, complemented
43C4×Dic419C2211yescharacteristic, complemented
44C2×Q8×C419C2211yescharacteristic, complemented
45C22×C4×C419C2211yescharacteristic
46C4⋊C4×C419C2211yescharacteristic, complemented
47C4⋊C4×C419C2211yescharacteristic, complemented
48C4⋊C4×C419C2211yescharacteristic, complemented
49C22⋊C4×C419C2211yescharacteristic
50C22⋊C4×C419C2211yescharacteristic
51C22×D419—22no—
52C22×D419—22no—
53C22×D419—41no—
54C22×D419C2311yescharacteristic
55C22×D419C2311yescharacteristic
56C22×D419C2311yescharacteristic
57D1676—22no—
58D1676—41no—
59C22⋊D419—22no—
60C2×Dic419—22no—
61C2×Dic419C2311yescharacteristic
62C2×Dic419C2311yescharacteristic
63C2×Dic419C2311yescharacteristic
64C2×Dic419C2311yescharacteristic
65C2×Dic419C2311yescharacteristic
66C4×D419—22no—
67C4×D419—41no—
68C4×D419—41no—
69C23×C419C2311yescharacteristic
70Q8×C419—41no—
71C2×C4×C419—41no—
72C2×C4×C419C2311yescharacteristic
73C2×C4×C419C2311yescharacteristic
74C2×C4×C419C2311yescharacteristic
75C2×C4×C419C2311yescharacteristic
76C2×C4×C419C2311yescharacteristic
77C2×C4×C419C2311yescharacteristic
78C22×C419—21no—
79C22×C419—21no—
80C22×C419#4112yes—
81C22×C419C2411yescharacteristic
82C2×D419—21no—
83C2×D419—21no—
84C2×D419—21no—
85C2×D419—21no—
86C2×D419—21no—
87C2×D419—21no—
88C2×D419—21no—
89C2×D419—21no—
90C2×D419—44no—
91C2×D419#4112yes—
92C4×C419—21no—
93C4×C419—21no—
94C4×C419—21no—
95C4×C419—21no—
96C4×C419—21no—
97C4×C419—21no—
98Dic419—21no—
99Dic419—21no—
100Dic419—21no—
101Dic419—21no—
102Dic419—21no—
103C2×C419—22no—
104C2×C419C2×#4111yescharacteristic
105C2×C419C2×#4111yescharacteristic
106C2×C419H(5,𝔽2)11yescharacteristic
107D419—22no—
108D419—41no—
109D419—41no—
110C419#55111yescharacteristic, complemented, Sylow
111#551—4191nomaximal, Sylow
112#126—4191no—
113#126—4191no—
114#124—4191no—
115#123—4191no—
116#123—4191no—
117#123—4191no—
118C22⋊Q8—4191no—
119C22⋊Q8D41911yescharacteristic, complemented
120#120—4191no—
121C4×D4—4191no—
122C4×D4—4191no—
123#117—4191no—
124#117—4191no—
125C2×C22⋊C4—4191no—
126C2×C22⋊C4—4191no—
127C24—4191no—
128C2×Q8C2×D41911yescharacteristic, complemented
129C2×D4—4191no—
130C2×D4—4191no—
131C2×D4—4191no—
132C22×C4—4191no—
133C22×C4—4191no—
134C22×C4—4191no—
135C22×C4—4191no—
136C22×C4C2×D41911yescharacteristic
137C4⋊C4—4191no—
138C4⋊C4—4191no—
139C4⋊C4—4191no—
140C4⋊C4—4191no—
141C4⋊C4—4191no—
142C4⋊C4C2×D41911yescharacteristic, complemented
143C4⋊C4C2×D41911yescharacteristic, complemented
144C4⋊C4C2×D41911yescharacteristic, complemented
145C22⋊C4—4191no—
146C22⋊C4—4191no—
147C22⋊C4—4191no—
148C22⋊C4—4191no—
149C22⋊C4—4191no—
150C22⋊C4—4191no—
151C22⋊C4—4191no—
152C22⋊C4—4191no—
153C22⋊C4—4191no—
154C22⋊C4—4191no—
155C22⋊C4—4191no—
156C22⋊C4—4191no—
157C22⋊C4C2×D41911yescharacteristic
158C22⋊C4C2×D41911yescharacteristic
159C42—4191no—
160C42—4191no—
161C42—4191no—
162C23—8382no—
163C23—8382no—
164C23—16761no—
165C23—4191no—
166C23—4191no—
167C23—4191no—
168C23C22×D41911yescharacteristic
169Q8—41no—
170D4—8382no—
171D4—8382no—
172D4—16761no—
173C2×C4—41no—
174C2×C4—8382no—
175C2×C4—8382no—
176C2×C4—16761no—
177C2×C4—16761no—
178C2×C4—4191no—
179C2×C4—4191no—
180C2×C4—4191no—
181C2×C4—4191no—
182C2×C4—4191no—
183C2×C4C22×D41911yescharacteristic
184C2×C4C22×D41911yescharacteristic
185C2×C4C22×D41911yescharacteristic
186C2×C4C22×D41911yescharacteristic
187C2×C4C22×D41911yescharacteristic
188C2×C4C22×D41911yescharacteristic
189C22—21no—
190C22—21no—
191C22—8381no—
192C22—8381no—
193C22—8381no—
194C22—8381no—
195C22—8381no—
196C22—8381no—
197C22—8381no—
198C22—8381no—
199C22—16764no—
200C22—4192no—
201C22#2026941212yes—
202C22C23×D41911yescharacteristic
203C4—21no—
204C4—21no—
205C4—21no—
206C4—21no—
207C4—21no—
208C4—21no—
209C4—8381no—
210C4—8381no—
211C4—8381no—
212C4—8381no—
213C4—8381no—
214C2—22no—
215C2—8382no—
216C2—16761no—
217C2—16761no—
218C2#41×D41911yescharacteristic
219C2C2×#2026941211yescharacteristic
220C2#2721176311yescharacteristic
subgroupG/H
center, FrattiniC22C23×D419
commutator, socle, cosocleC22×C419C24
FittingC22⋊Q8×C419C2
prime radical#27211688C2
hypercenterC22⋊Q8D419
hypocenterC419#551
LCSG ▷ C22×C419 ▷ C419
UCSC1 ◁ C22 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. #27211713C2semidirect
2. #27211713C2semidirect
3. #27211707C2semidirect
4. #27211703C2semidirect
5. #27211703C2semidirect
6. #27211702C2semidirect
7. #27211694C2semidirect
9. #27211680C2semidirect
10. C4×C22⋊D419C2semidirect
11. #27211676C2semidirect
12. #27211674C2semidirect
13. C2×(C2×D419)⋊C4C2semidirect
14. C22⋊C4×D419C2semidirect
15. C22⋊Q8×C419C2semidirect
23. C2×C4×D419C22semidirect
24. C4⋊Dic419C22semidirect
25. C4⋊Dic419C22semidirect
26. Dic419⋊C4C22semidirect
27. Dic419⋊C4C22semidirect
28. Dic419⋊C4C22semidirect
38. (C2×D419)⋊C4C22semidirect
39. (C2×D419)⋊C4C22semidirect
40. (C2×D419)⋊C4C22semidirect
41. C4×Dic419C22semidirect
42. C4×Dic419C22semidirect
43. C4×Dic419C22semidirect
44. C2×Q8×C419C22semidirect
46. C4⋊C4×C419C22semidirect
47. C4⋊C4×C419C22semidirect
48. C4⋊C4×C419C22semidirect
110. C419#551semidirect
119. C22⋊Q8D419semidirect
128. C2×Q8C2×D419semidirect
142. C4⋊C4C2×D419semidirect
143. C4⋊C4C2×D419semidirect
144. C4⋊C4C2×D419semidirect

Maximal subgroups

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

subgroupindex
#272117132
#272117072
#272117032
#272117022
#272116942
#272116882
#272116802
C4×C22⋊D4192
#272116762
#272116742
C2×(C2×D419)⋊C42
C22⋊C4×D4192
C22⋊Q8×C4192
#551419