Finite Group Databasebeta

← #42119805

Subgroups — #42119805

Order 27584.

220 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 91 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#28352179C211yesmaximal, characteristic, complemented
2#28352178C211yesmaximal, characteristic, complemented
3#28352175C211yesmaximal, characteristic, complemented
4#28352174C211yesmaximal, characteristic, complemented
5#28352174C211yesmaximal, characteristic, complemented
6#28352172C211yesmaximal, characteristic, complemented
7#28352171C211yesmaximal, characteristic, complemented
8#28352169C211yesmaximal, characteristic, complemented
9#28352167C211yesmaximal, characteristic, complemented
10#28352165C211yesmaximal, characteristic, complemented
11#28352164C211yesmaximal, characteristic, complemented
12#28352163C211yesmaximal, characteristic, complemented
13#28352162C211yesmaximal, characteristic, complemented
14#28352162C211yesmaximal, characteristic, complemented
15C22⋊Q8×C431C211yesmaximal, characteristic, complemented
16C2×Dic862C2211yescharacteristic
17C2×D1724C2211yescharacteristic
18C2×D1724C2211yescharacteristic
19C2×D1724C2211yescharacteristic
20C2×C22⋊D431C2211yescharacteristic
21C2×C22⋊D431C2211yescharacteristic
22C2×C22⋊D431C2211yescharacteristic
23C2×C4×D431C2211yescharacteristic
24C2×C4×D431C2211yescharacteristic
25C2×C4×D431C2211yescharacteristic
26C4⋊Dic431C2211yescharacteristic
27C4⋊Dic431C2211yescharacteristic
28Dic431⋊C4C2211yescharacteristic
29Dic431⋊C4C2211yescharacteristic
30Dic431⋊C4C2211yescharacteristic
31Dic431⋊C4C2211yescharacteristic
32Dic431⋊C4C2211yescharacteristic
33C22⋊Dic431C2211yescharacteristic
34(C2×D431)⋊C4C2211yescharacteristic
35(C2×D431)⋊C4C2211yescharacteristic
36(C2×D431)⋊C4C2211yescharacteristic
37(C2×D431)⋊C4C2211yescharacteristic
38(C2×D431)⋊C4C2211yescharacteristic
39(C2×D431)⋊C4C2211yescharacteristic
40(C2×D431)⋊C4C2211yescharacteristic
41(C2×D431)⋊C4C2211yescharacteristic
42(C2×D431)⋊C4C2211yescharacteristic
43C4×Dic431C2211yescharacteristic
44C2×Q8×C431C2211yescharacteristic
45C22×C4×C431C2211yescharacteristic
46C4⋊C4×C431C2211yescharacteristic
47C4⋊C4×C431C2211yescharacteristic
48C4⋊C4×C431C2211yescharacteristic
49C22⋊C4×C431C2211yescharacteristic
50C22⋊C4×C431C2211yescharacteristic
51C22×D431C2311yescharacteristic
52C22×D431C2311yescharacteristic
53C22×D431C2311yescharacteristic
54Dic862—41no—
55D1724—41no—
56D1724—41no—
57D1724—41no—
58C22⋊D431—41no—
59C22⋊D431—41no—
60C22⋊D431—41no—
61C2×Dic431C2311yescharacteristic
62C2×Dic431C2311yescharacteristic
63C2×Dic431C2311yescharacteristic
64C2×Dic431C2311yescharacteristic
65C2×Dic431C2311yescharacteristic
66C4×D431—41no—
67C4×D431—41no—
68C4×D431—41no—
69C23×C431C2311yescharacteristic
70Q8×C431—41no—
71C2×C4×C431—41no—
72C2×C4×C431C2311yescharacteristic
73C2×C4×C431C2311yescharacteristic
74C2×C4×C431C2311yescharacteristic
75C2×C4×C431C2311yescharacteristic
76C2×C4×C431C2311yescharacteristic
77C2×C4×C431C2311yescharacteristic
78C22×C431—21no—
79C22×C431—21no—
80C22×C431—21no—
81C22×C431C2411yescharacteristic
82C2×D431—21no—
83C2×D431—21no—
84C2×D431—21no—
85C2×D431—21no—
86C2×D431—21no—
87C2×D431—21no—
88C2×D431—21no—
89C2×D431—21no—
90C2×D431—21no—
91C4×C431—21no—
92C4×C431—21no—
93C4×C431—21no—
94C4×C431—21no—
95C4×C431—21no—
96C4×C431—21no—
97Dic431—21no—
98Dic431—21no—
99Dic431—21no—
100Dic431—21no—
101Dic431—21no—
102C2×C431—41no—
103C2×C431H(5,𝔽2)11yescharacteristic
104C2×C431H(5,𝔽2)11yescharacteristic
105C2×C4312−1+411yescharacteristic
106D431—41no—
107D431—41no—
108D431—41no—
109C431#56211yescharacteristic, complemented, Sylow
110#562—4311nomaximal, Sylow
111#125—4311no—
112#124—4311no—
113#124—4311no—
114#123—4311no—
115#123—4311no—
116#123—4311no—
117#123—4311no—
118C22⋊Q8—4311no—
119C22⋊Q8—4311no—
120C22⋊Q8—4311no—
121C22⋊Q8D43111yescharacteristic, complemented
122#121—4311no—
123#121—4311no—
124#121—4311no—
125#121—4311no—
126C2×Q8—4311no—
127C2×Q8C2×D43111yescharacteristic
128C2×D4—4311no—
129C2×D4—4311no—
130C2×D4—4311no—
131C2×D4—4311no—
132C2×D4—4311no—
133C2×D4—4311no—
134C22×C4—4311no—
135C22×C4—4311no—
136C22×C4—4311no—
137C22×C4C2×D43111yescharacteristic
138C4⋊C4—4311no—
139C4⋊C4—4311no—
140C4⋊C4—4311no—
141C4⋊C4—4311no—
142C4⋊C4—4311no—
143C4⋊C4—4311no—
144C4⋊C4—4311no—
145C4⋊C4C2×D43111yescharacteristic
146C4⋊C4C2×D43111yescharacteristic
147C4⋊C4C2×D43111yescharacteristic
148C22⋊C4—4311no—
149C22⋊C4—4311no—
150C22⋊C4—4311no—
151C22⋊C4—4311no—
152C22⋊C4—4311no—
153C22⋊C4—4311no—
154C22⋊C4—4311no—
155C22⋊C4—4311no—
156C22⋊C4—4311no—
157C22⋊C4—4311no—
158C22⋊C4C2×D43111yescharacteristic
159C22⋊C4C2×D43111yescharacteristic
160C42—4311no—
161C23—4311no—
162C23—4311no—
163C23—4311no—
164C23C22×D43111yescharacteristic
165Q8—41no—
166Q8—17241no—
167D4—17241no—
168D4—17241no—
169D4—17241no—
170D4—17241no—
171D4—17241no—
172D4—17241no—
173C2×C4—41no—
174C2×C4—4311no—
175C2×C4—4311no—
176C2×C4—4311no—
177C2×C4—4311no—
178C2×C4—4311no—
179C2×C4—17241no—
180C2×C4—17241no—
181C2×C4—17241no—
182C2×C4C22×D43111yescharacteristic
183C2×C4C22×D43111yescharacteristic
184C2×C4C22×D43111yescharacteristic
185C2×C4C22×D43111yescharacteristic
186C2×C4C22×D43111yescharacteristic
187C2×C4C22×D43111yescharacteristic
188C22—21no—
189C22—21no—
190C22—21no—
191C22—8621no—
192C22—8621no—
193C22—8621no—
194C22—8621no—
195C22—8621no—
196C22—8621no—
197C22—8621no—
198C22—8621no—
199C22—8621no—
200C22C23×D43111yescharacteristic
201C4—21no—
202C4—21no—
203C4—21no—
204C4—21no—
205C4—21no—
206C4—21no—
207C4—8621no—
208C4—8621no—
209C4—8621no—
210C4—8621no—
211C4—8621no—
212C2—41no—
213C2—17241no—
214C2—17241no—
215C2—17241no—
216C2#2835223411yescharacteristic
217C2#2835223511yescharacteristic
218C2#2835223711yescharacteristic
subgroupG/H
center, FrattiniC22C23×D431
commutator, socle, cosocleC22×C431C24
FittingC22⋊Q8×C431C2
hypercenterC22⋊Q8D431
hypocenterC431#562
LCSG ▷ C22×C431 ▷ C431
UCSC1 ◁ C22 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. #28352179C2semidirect
2. #28352178C2semidirect
3. #28352175C2semidirect
4. #28352174C2semidirect
5. #28352174C2semidirect
6. #28352172C2semidirect
7. #28352171C2semidirect
8. #28352169C2semidirect
9. #28352167C2semidirect
10. #28352165C2semidirect
11. #28352164C2semidirect
12. #28352163C2semidirect
13. #28352162C2semidirect
14. #28352162C2semidirect
15. C22⋊Q8×C431C2semidirect
109. C431#562semidirect
121. C22⋊Q8D431semidirect

Maximal subgroups

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

subgroupindex
#283521792
#283521782
#283521752
#283521742
#283521722
#283521712
#283521692
#283521672
#283521652
#283521642
#283521632
#283521622
C22⋊Q8×C4312
#562431