Finite Group Databasebeta

← #14521726

Subgroups — #14521726

Order 1984.

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
1C31×C22⋊Q8C211yesmaximal, characteristic, complemented
2#11758524C211yesmaximal, characteristic, complemented
3#11758523C211yesmaximal, characteristic, complemented
4C4×C22⋊D31C211yesmaximal, characteristic, complemented
5C2×(C2×D31)⋊C4C211yesmaximal, characteristic, complemented
6#11758475C211yesmaximal, characteristic, complemented
7#11758475C211yesmaximal, characteristic, complemented
8#11758471C211yesmaximal, characteristic, complemented
9#11758471C211yesmaximal, characteristic, complemented
10#11758470C211yesmaximal, characteristic, complemented
11#11758469C211yesmaximal, characteristic, complemented
12#11758463C211yesmaximal, characteristic, complemented
13#11758459C211yesmaximal, characteristic
14C22⋊C4×D31C211yesmaximal, characteristic, complemented
15#11758454C211yesmaximal, characteristic, complemented
16C23×D31C2211yescharacteristic
17C2×Q8×C31C2211yescharacteristic, complemented
18C22×C4×C31C2211yescharacteristic
19C2×C22⋊D31C2211yescharacteristic
20C22×Dic31C2211yescharacteristic
21C2×D124C2211yescharacteristic
22C2×D124C2211yescharacteristic
23C2×C4×D31C2211yescharacteristic
24C2×C4×D31C2211yescharacteristic
25C2×C4×D31C2211yescharacteristic, complemented
26C4⋊C4×C31C2211yescharacteristic, complemented
27C4⋊C4×C31C2211yescharacteristic, complemented
28C4⋊C4×C31C2211yescharacteristic, complemented
29C22⋊C4×C31C2211yescharacteristic
30C22⋊C4×C31C2211yescharacteristic
31C22⋊Dic31C2211yescharacteristic
32(C2×D31)⋊C4C2211yescharacteristic
33(C2×D31)⋊C4C2211yescharacteristic
34(C2×D31)⋊C4C2211yescharacteristic
35(C2×D31)⋊C4C2211yescharacteristic
36(C2×D31)⋊C4C2211yescharacteristic
37(C2×D31)⋊C4C2211yescharacteristic
38(C2×D31)⋊C4C2211yescharacteristic
39(C2×D31)⋊C4C2211yescharacteristic
40(C2×D31)⋊C4C2211yescharacteristic, complemented
41(C2×D31)⋊C4C2211yescharacteristic, complemented
42(C2×D31)⋊C4C2211yescharacteristic, complemented
43C4⋊Dic31C2211yescharacteristic, complemented
44C4⋊Dic31C2211yescharacteristic, complemented
45Dic31⋊C4C2211yescharacteristic, complemented
46Dic31⋊C4C2211yescharacteristic, complemented
47Dic31⋊C4C2211yescharacteristic, complemented
48C4×Dic31C2211yescharacteristic, complemented
49C4×Dic31C2211yescharacteristic, complemented
50C4×Dic31C2211yescharacteristic, complemented
51C23×C31C2311yescharacteristic
52C22×D31—22no—
53C22×D31—22no—
54C22×D31—41no—
55C22×D31C2311yescharacteristic
56C22×D31C2311yescharacteristic
57C22×D31C2311yescharacteristic
58Q8×C31—41no—
59C2×C4×C31—41no—
60C2×C4×C31C2311yescharacteristic
61C2×C4×C31C2311yescharacteristic
62C2×C4×C31C2311yescharacteristic
63C2×C4×C31C2311yescharacteristic
64C2×C4×C31C2311yescharacteristic
65C2×C4×C31C2311yescharacteristic
66C22⋊D31—22no—
67C2×Dic31—22no—
68C2×Dic31C2311yescharacteristic
69C2×Dic31C2311yescharacteristic
70C2×Dic31C2311yescharacteristic
71C2×Dic31C2311yescharacteristic
72C2×Dic31C2311yescharacteristic
73D124—22no—
74D124—41no—
75C4×D31—22no—
76C4×D31—41no—
77C4×D31—41no—
78C22×C31—21no—
79C22×C31—21no—
80C22×C31#4112yes—
81C22×C31C2411yescharacteristic
82C2×D31—21no—
83C2×D31—21no—
84C2×D31—21no—
85C2×D31—21no—
86C2×D31—21no—
87C2×D31—21no—
88C2×D31—21no—
89C2×D31—21no—
90C2×D31—44no—
91C2×D31#4112yes—
92C4×C31—21no—
93C4×C31—21no—
94C4×C31—21no—
95C4×C31—21no—
96C4×C31—21no—
97C4×C31—21no—
98Dic31—21no—
99Dic31—21no—
100Dic31—21no—
101Dic31—21no—
102Dic31—21no—
103#551—311nomaximal, Sylow
104C2×C31—22no—
105C2×C31C2×#4111yescharacteristic
106C2×C31C2×#4111yescharacteristic
107C2×C31H(5,𝔽2)11yescharacteristic
108D31—22no—
109D31—41no—
110D31—41no—
111#126—311no—
112#126—311no—
113#124—311no—
114#123—311no—
115#123—311no—
116#123—311no—
117C22⋊Q8—311no—
118C22⋊Q8D3111yescharacteristic, complemented
119#120—311no—
120C4×D4—311no—
121C4×D4—311no—
122#117—311no—
123#117—311no—
124C2×C22⋊C4—311no—
125C2×C22⋊C4—311no—
126C31#55111yescharacteristic, complemented, Sylow
127C24—311no—
128C2×Q8C2×D3111yescharacteristic, complemented
129C2×D4—311no—
130C2×D4—311no—
131C2×D4—311no—
132C22×C4—311no—
133C22×C4—311no—
134C22×C4—311no—
135C22×C4—311no—
136C22×C4C2×D3111yescharacteristic
137C4⋊C4—311no—
138C4⋊C4—311no—
139C4⋊C4—311no—
140C4⋊C4—311no—
141C4⋊C4—311no—
142C4⋊C4C2×D3111yescharacteristic, complemented
143C4⋊C4C2×D3111yescharacteristic, complemented
144C4⋊C4C2×D3111yescharacteristic, complemented
145C22⋊C4—311no—
146C22⋊C4—311no—
147C22⋊C4—311no—
148C22⋊C4—311no—
149C22⋊C4—311no—
150C22⋊C4—311no—
151C22⋊C4—311no—
152C22⋊C4—311no—
153C22⋊C4—311no—
154C22⋊C4—311no—
155C22⋊C4—311no—
156C22⋊C4—311no—
157C22⋊C4C2×D3111yescharacteristic
158C22⋊C4C2×D3111yescharacteristic
159C42—311no—
160C42—311no—
161C42—311no—
162C23—311no—
163C23—311no—
164C23—311no—
165C23—622no—
166C23—622no—
167C23—1241no—
168C23C22×D3111yescharacteristic
169Q8—41no—
170D4—622no—
171D4—622no—
172D4—1241no—
173C2×C4—41no—
174C2×C4—311no—
175C2×C4—311no—
176C2×C4—311no—
177C2×C4—311no—
178C2×C4—311no—
179C2×C4—622no—
180C2×C4—622no—
181C2×C4—1241no—
182C2×C4—1241no—
183C2×C4C22×D3111yescharacteristic
184C2×C4C22×D3111yescharacteristic
185C2×C4C22×D3111yescharacteristic
186C2×C4C22×D3111yescharacteristic
187C2×C4C22×D3111yescharacteristic
188C2×C4C22×D3111yescharacteristic
189C22—21no—
190C22—21no—
191C22—312no—
192C22—621no—
193C22—621no—
194C22—621no—
195C22—621no—
196C22—621no—
197C22—621no—
198C22—621no—
199C22—621no—
200C22—1244no—
201C22#9249912yes—
202C22C23×D3111yescharacteristic
203C4—21no—
204C4—21no—
205C4—21no—
206C4—21no—
207C4—21no—
208C4—21no—
209C4—621no—
210C4—621no—
211C4—621no—
212C4—621no—
213C4—621no—
214C2—22no—
215C2—622no—
216C2—1241no—
217C2—1241no—
218C2#1175856111yescharacteristic
219C2C2×#9249911yescharacteristic
220C2#41×D3111yescharacteristic
subgroupG/H
center, FrattiniC22C23×D31
commutator, socle, cosocleC22×C31C24
FittingC31×C22⋊Q8C2
prime radical#11758459C2
hypercenterC22⋊Q8D31
hypocenterC31#551
LCSG ▷ C22×C31 ▷ C31
UCSC1 ◁ C22 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C31×C22⋊Q8C2semidirect
2. #11758524C2semidirect
3. #11758523C2semidirect
4. C4×C22⋊D31C2semidirect
5. C2×(C2×D31)⋊C4C2semidirect
6. #11758475C2semidirect
7. #11758475C2semidirect
8. #11758471C2semidirect
9. #11758471C2semidirect
10. #11758470C2semidirect
11. #11758469C2semidirect
12. #11758463C2semidirect
14. C22⋊C4×D31C2semidirect
15. #11758454C2semidirect
17. C2×Q8×C31C22semidirect
25. C2×C4×D31C22semidirect
26. C4⋊C4×C31C22semidirect
27. C4⋊C4×C31C22semidirect
28. C4⋊C4×C31C22semidirect
40. (C2×D31)⋊C4C22semidirect
41. (C2×D31)⋊C4C22semidirect
42. (C2×D31)⋊C4C22semidirect
43. C4⋊Dic31C22semidirect
44. C4⋊Dic31C22semidirect
45. Dic31⋊C4C22semidirect
46. Dic31⋊C4C22semidirect
47. Dic31⋊C4C22semidirect
48. C4×Dic31C22semidirect
49. C4×Dic31C22semidirect
50. C4×Dic31C22semidirect
118. C22⋊Q8D31semidirect
126. C31#551semidirect
128. C2×Q8C2×D31semidirect
142. C4⋊C4C2×D31semidirect
143. C4⋊C4C2×D31semidirect
144. C4⋊C4C2×D31semidirect

Maximal subgroups

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

subgroupindex
C31×C22⋊Q82
#117585242
#117585232
C4×C22⋊D312
C2×(C2×D31)⋊C42
#117584752
#117584712
#117584702
#117584692
#117584632
#117584592
C22⋊C4×D312
#117584542
#55131