Finite Group Databasebeta

← #37527189

Subgroups — #37527189

Order 22976.

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#26040005C211yesmaximal, characteristic, complemented
2#26040004C211yesmaximal, characteristic, complemented
3#26040001C211yesmaximal, characteristic, complemented
4#26040000C211yesmaximal, characteristic, complemented
5#26040000C211yesmaximal, characteristic, complemented
6#26039998C211yesmaximal, characteristic, complemented
7#26039997C211yesmaximal, characteristic, complemented
8#26039995C211yesmaximal, characteristic, complemented
9#26039993C211yesmaximal, characteristic, complemented
10#26039991C211yesmaximal, characteristic, complemented
11#26039990C211yesmaximal, characteristic, complemented
12#26039989C211yesmaximal, characteristic, complemented
13#26039988C211yesmaximal, characteristic, complemented
14#26039988C211yesmaximal, characteristic, complemented
15C22⋊Q8×C359C211yesmaximal, characteristic, complemented
16C2×Dic718C2211yescharacteristic
17C2×D1436C2211yescharacteristic
18C2×D1436C2211yescharacteristic
19C2×D1436C2211yescharacteristic
20C2×C22⋊D359C2211yescharacteristic
21C2×C22⋊D359C2211yescharacteristic
22C2×C22⋊D359C2211yescharacteristic
23C2×C4×D359C2211yescharacteristic
24C2×C4×D359C2211yescharacteristic
25C2×C4×D359C2211yescharacteristic
26C4⋊Dic359C2211yescharacteristic
27C4⋊Dic359C2211yescharacteristic
28Dic359⋊C4C2211yescharacteristic
29Dic359⋊C4C2211yescharacteristic
30Dic359⋊C4C2211yescharacteristic
31Dic359⋊C4C2211yescharacteristic
32Dic359⋊C4C2211yescharacteristic
33C22⋊Dic359C2211yescharacteristic
34(C2×D359)⋊C4C2211yescharacteristic
35(C2×D359)⋊C4C2211yescharacteristic
36(C2×D359)⋊C4C2211yescharacteristic
37(C2×D359)⋊C4C2211yescharacteristic
38(C2×D359)⋊C4C2211yescharacteristic
39(C2×D359)⋊C4C2211yescharacteristic
40(C2×D359)⋊C4C2211yescharacteristic
41(C2×D359)⋊C4C2211yescharacteristic
42(C2×D359)⋊C4C2211yescharacteristic
43C4×Dic359C2211yescharacteristic
44C2×Q8×C359C2211yescharacteristic
45C22×C4×C359C2211yescharacteristic
46C4⋊C4×C359C2211yescharacteristic
47C4⋊C4×C359C2211yescharacteristic
48C4⋊C4×C359C2211yescharacteristic
49C22⋊C4×C359C2211yescharacteristic
50C22⋊C4×C359C2211yescharacteristic
51C22×D359C2311yescharacteristic
52C22×D359C2311yescharacteristic
53C22×D359C2311yescharacteristic
54Dic718—41no—
55D1436—41no—
56D1436—41no—
57D1436—41no—
58C22⋊D359—41no—
59C22⋊D359—41no—
60C22⋊D359—41no—
61C2×Dic359C2311yescharacteristic
62C2×Dic359C2311yescharacteristic
63C2×Dic359C2311yescharacteristic
64C2×Dic359C2311yescharacteristic
65C2×Dic359C2311yescharacteristic
66C4×D359—41no—
67C4×D359—41no—
68C4×D359—41no—
69C23×C359C2311yescharacteristic
70Q8×C359—41no—
71C2×C4×C359—41no—
72C2×C4×C359C2311yescharacteristic
73C2×C4×C359C2311yescharacteristic
74C2×C4×C359C2311yescharacteristic
75C2×C4×C359C2311yescharacteristic
76C2×C4×C359C2311yescharacteristic
77C2×C4×C359C2311yescharacteristic
78C22×C359—21no—
79C22×C359—21no—
80C22×C359—21no—
81C22×C359C2411yescharacteristic
82C2×D359—21no—
83C2×D359—21no—
84C2×D359—21no—
85C2×D359—21no—
86C2×D359—21no—
87C2×D359—21no—
88C2×D359—21no—
89C2×D359—21no—
90C2×D359—21no—
91C4×C359—21no—
92C4×C359—21no—
93C4×C359—21no—
94C4×C359—21no—
95C4×C359—21no—
96C4×C359—21no—
97Dic359—21no—
98Dic359—21no—
99Dic359—21no—
100Dic359—21no—
101Dic359—21no—
102C2×C359—41no—
103C2×C359H(5,𝔽2)11yescharacteristic
104C2×C359H(5,𝔽2)11yescharacteristic
105C2×C3592−1+411yescharacteristic
106D359—41no—
107D359—41no—
108D359—41no—
109C359#56211yescharacteristic, complemented, Sylow
110#562—3591nomaximal, Sylow
111#125—3591no—
112#124—3591no—
113#124—3591no—
114#123—3591no—
115#123—3591no—
116#123—3591no—
117#123—3591no—
118C22⋊Q8—3591no—
119C22⋊Q8—3591no—
120C22⋊Q8—3591no—
121C22⋊Q8D35911yescharacteristic, complemented
122#121—3591no—
123#121—3591no—
124#121—3591no—
125#121—3591no—
126C2×Q8—3591no—
127C2×Q8C2×D35911yescharacteristic
128C2×D4—3591no—
129C2×D4—3591no—
130C2×D4—3591no—
131C2×D4—3591no—
132C2×D4—3591no—
133C2×D4—3591no—
134C22×C4—3591no—
135C22×C4—3591no—
136C22×C4—3591no—
137C22×C4C2×D35911yescharacteristic
138C4⋊C4—3591no—
139C4⋊C4—3591no—
140C4⋊C4—3591no—
141C4⋊C4—3591no—
142C4⋊C4—3591no—
143C4⋊C4—3591no—
144C4⋊C4—3591no—
145C4⋊C4C2×D35911yescharacteristic
146C4⋊C4C2×D35911yescharacteristic
147C4⋊C4C2×D35911yescharacteristic
148C22⋊C4—3591no—
149C22⋊C4—3591no—
150C22⋊C4—3591no—
151C22⋊C4—3591no—
152C22⋊C4—3591no—
153C22⋊C4—3591no—
154C22⋊C4—3591no—
155C22⋊C4—3591no—
156C22⋊C4—3591no—
157C22⋊C4—3591no—
158C22⋊C4C2×D35911yescharacteristic
159C22⋊C4C2×D35911yescharacteristic
160C42—3591no—
161C23—3591no—
162C23—3591no—
163C23—3591no—
164C23C22×D35911yescharacteristic
165Q8—41no—
166Q8—14361no—
167D4—14361no—
168D4—14361no—
169D4—14361no—
170D4—14361no—
171D4—14361no—
172D4—14361no—
173C2×C4—41no—
174C2×C4—3591no—
175C2×C4—3591no—
176C2×C4—3591no—
177C2×C4—3591no—
178C2×C4—3591no—
179C2×C4—14361no—
180C2×C4—14361no—
181C2×C4—14361no—
182C2×C4C22×D35911yescharacteristic
183C2×C4C22×D35911yescharacteristic
184C2×C4C22×D35911yescharacteristic
185C2×C4C22×D35911yescharacteristic
186C2×C4C22×D35911yescharacteristic
187C2×C4C22×D35911yescharacteristic
188C22—21no—
189C22—21no—
190C22—21no—
191C22—7181no—
192C22—7181no—
193C22—7181no—
194C22—7181no—
195C22—7181no—
196C22—7181no—
197C22—7181no—
198C22—7181no—
199C22—7181no—
200C22C23×D35911yescharacteristic
201C4—21no—
202C4—21no—
203C4—21no—
204C4—21no—
205C4—21no—
206C4—21no—
207C4—7181no—
208C4—7181no—
209C4—7181no—
210C4—7181no—
211C4—7181no—
212C2—41no—
213C2—14361no—
214C2—14361no—
215C2—14361no—
216C2#2604006011yescharacteristic
217C2#2604006111yescharacteristic
218C2#2604006311yescharacteristic
subgroupG/H
center, FrattiniC22C23×D359
commutator, socle, cosocleC22×C359C24
FittingC22⋊Q8×C359C2
hypercenterC22⋊Q8D359
hypocenterC359#562
LCSG ▷ C22×C359 ▷ C359
UCSC1 ◁ C22 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. #26040005C2semidirect
2. #26040004C2semidirect
3. #26040001C2semidirect
4. #26040000C2semidirect
5. #26040000C2semidirect
6. #26039998C2semidirect
7. #26039997C2semidirect
8. #26039995C2semidirect
9. #26039993C2semidirect
10. #26039991C2semidirect
11. #26039990C2semidirect
12. #26039989C2semidirect
13. #26039988C2semidirect
14. #26039988C2semidirect
15. C22⋊Q8×C359C2semidirect
109. C359#562semidirect
121. C22⋊Q8D359semidirect

Maximal subgroups

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

subgroupindex
#260400052
#260400042
#260400012
#260400002
#260399982
#260399972
#260399952
#260399932
#260399912
#260399902
#260399892
#260399882
C22⋊Q8×C3592
#562359