Finite Group Databasebeta

← #44400017

Subgroups — #44400017

Order 29056.

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

normal subgroups only

This group has 374 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#28409507C211yesmaximal, characteristic
2#28408912C211yesmaximal, characteristic
3#28408907C212yesmaximal, complemented
4#28408798C212yesmaximal, complemented
5C2×#99×C227C211yesmaximal, characteristic
6C23×Dic227C2211yescharacteristic
7D4×Dic227—22no—
8D4×Dic227—22no—
9#20300247—21no—
10C2×C4⋊Dic227—22no—
11C2×C22⋊Dic227—21no—
12C2×C22⋊Dic227—21no—
13C2×C22⋊Dic227C2211yescharacteristic
14C2×C22⋊Dic227C2212yes—
15C227⋊#95—22no—
16C227⋊#95—21no—
17C22×D4×C227C2211yescharacteristic
18C2×C22⋊C4×C227C2212yes—
19#99×C227C414yescomplemented
20C22×Dic227—21no—
21C22×Dic227—21no—
22C22×Dic227—21no—
23C22×Dic227—22no—
24C22×Dic227—21no—
25C22×Dic227—42no—
26C22×Dic227D412yes—
27C22×Dic227D412yes—
28C4⋊Dic227—21no—
29C4⋊Dic227—21no—
30C4⋊Dic227—44no—
31C22⋊Dic227—22no—
32C22⋊Dic227—22no—
33C22⋊Dic227—22no—
34C22⋊Dic227—22no—
35C22⋊Dic227—42no—
36C4×Dic227—22no—
37C24×C227D411yescharacteristic
38C24×C227C2311yescharacteristic
39C2×D4×C227—22no—
40C2×D4×C227—22no—
41C2×D4×C227C2×C412yes—
42C2×D4×C227Q812yes—
43C22×C4×C227—22no—
44C22×C4×C227D411yescharacteristic
45C22⋊C4×C227—22no—
46C22⋊C4×C227C2×C414yescomplemented
47C2×Dic227—22no—
48C2×Dic227—24no—
49C2×Dic227—22no—
50C2×Dic227—22no—
51C2×Dic227—42no—
52C2×Dic227—41no—
53C2×Dic227—41no—
54C2×Dic227—42no—
55C2×Dic227—42no—
56C2×Dic227—41no—
57C2×Dic227—41no—
58C2×Dic227—41no—
59C2×Dic227—81no—
60C2×Dic227—81no—
61C23×C227—21no—
62C23×C227—21no—
63C23×C227—41no—
64C23×C227—41no—
65C23×C227—41no—
66C23×C227—41no—
67C23×C227C4⋊C412yescomplemented
68C23×C227C22×C411yescharacteristic
69C23×C227C2×D411yescharacteristic
70C23×C227C2×D412yes—
71C23×C227C2×Q811yescharacteristic
72C23×C227#4112yes—
73D4×C227—24no—
74D4×C227—24no—
75C2×C4×C227—21no—
76C2×C4×C227—21no—
77C2×C4×C227—24no—
78C2×C4×C227—42no—
79C2×C4×C227—42no—
80C2×C4×C227C4⋊C412yescomplemented
81C22×C227—22no—
82C22×C227—21no—
83C22×C227—22no—
84C22×C227—21no—
85C22×C227—22no—
86C22×C227—41no—
87C22×C227—41no—
88C22×C227—42no—
89C22×C227—42no—
90C22×C227—41no—
91C22×C227—41no—
92C22×C227—81no—
93C22×C227C2×C4⋊C411yescharacteristic
94C22×C227C4×D412yes—
95C22×C227#12011yescharacteristic
96C22×C227C22⋊Q812yes—
97C22×C227#12311yescharacteristic
98C4×C227—22no—
99C4×C227—44no—
100Dic227—42no—
101Dic227—42no—
102Dic227—42no—
103Dic227—81no—
104C2×C227—21no—
105C2×C227—22no—
106C2×C227—21no—
107C2×C227—22no—
108C2×C227—42no—
109C2×C227#38511yescharacteristic
110C2×C227B(4,𝔽2)11yescharacteristic
111C2×C227#45811yescharacteristic
112C227#186711yescharacteristic, complemented, Sylow
113#1867—2271nomaximal, Sylow
114#518—2271no—
115C2×#99D22711yescharacteristic
116#386—2271no—
117#385—2272no—
118#342—2272no—
119C22×D4C2×D22711yescharacteristic
120C23×C4—2271no—
121C4×D4—4542no—
122C4×D4—4542no—
123#117—4541no—
124C2×C4⋊C4—4542no—
125C2×C22⋊C4—4541no—
126C2×C22⋊C4—4541no—
127C2×C22⋊C4—2271no—
128C2×C22⋊C4—2272no—
129C2×C22⋊C4C2×D22712yes—
130#99Dic22714yescomplemented
131#95—4542no—
132#95—4541no—
133C24D90811yescharacteristic
134C24C22×D22711yescharacteristic
135C2×D4—22no—
136C2×D4—22no—
137C2×D4C2×Dic22712yes—
138C2×D4Dic45412yes—
139C22×C4—22no—
140C22×C4—9082no—
141C22×C4—4541no—
142C22×C4—4541no—
143C22×C4—4541no—
144C22×C4—4542no—
145C22×C4—4541no—
146C22×C4—2272no—
147C22×C4—2272no—
148C22×C4D90811yescharacteristic
149C4⋊C4—9084no—
150C4⋊C4—4541no—
151C4⋊C4—4541no—
152C22⋊C4—22no—
153C22⋊C4—9082no—
154C22⋊C4—4542no—
155C22⋊C4—4542no—
156C22⋊C4—4542no—
157C22⋊C4—4542no—
158C22⋊C4C2×Dic22714yescomplemented
159C42—4542no—
160C23—21no—
161C23—21no—
162C23—41no—
163C23—41no—
164C23—41no—
165C23—41no—
166C23C4⋊Dic22712yescomplemented
167C23C22×Dic22711yescharacteristic
168C23D4×D22712yes—
169C23C2×D90811yescharacteristic
170C23C2×Dic45411yescharacteristic
171C23#1681388312yes—
172D4—24no—
173D4—24no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—24no—
177C2×C4—42no—
178C2×C4—42no—
179C2×C4—18161no—
180C2×C4—18161no—
181C2×C4—9081no—
182C2×C4—9081no—
183C2×C4—9081no—
184C2×C4—9082no—
185C2×C4—9081no—
186C2×C4—9081no—
187C2×C4—9082no—
188C2×C4—9082no—
189C2×C4—4542no—
190C2×C4—4544no—
191C2×C4—4542no—
192C2×C4—4542no—
193C2×C4C4⋊Dic22712yescomplemented
194C22—22no—
195C22—21no—
196C22—22no—
197C22—21no—
198C22—22no—
199C22—41no—
200C22—41no—
201C22—42no—
202C22—42no—
203C22—41no—
204C22—41no—
205C22—81no—
206C22C2×C4⋊Dic22711yescharacteristic
207C22D4×Dic22712yes—
208C22#2030025811yescharacteristic
209C22#2030026512yes—
210C22#2030027211yescharacteristic
211C4—22no—
212C4—44no—
213C4—18161no—
214C4—9082no—
215C4—9082no—
216C4—9082no—
217C2—22no—
218C2—21no—
219C2—21no—
220C2—22no—
221C2—42no—
222C2#2840890711yescharacteristic
223C2#2840922011yescharacteristic
224C2#2840922311yescharacteristic
subgroupG/H
centerC22#20300258
commutatorC23×C227C22×C4
FrattiniC24C22×D227
FittingC2×#99×C227C2
socleC22×C227#120
prime radicalC22×D4×C227C22
cosocleC24×C227C23
hypercenterC2×#99D227
hypocenterC227#1867
LCSG ▷ C23×C227 ▷ C2×C227 ▷ C227
UCSC1 ◁ C22 ◁ C24 ◁ C2×#99

Complements

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

NH ≅ G/Ntype
3. #28408907C2semidirect
4. #28408798C2semidirect
19. #99×C227C4semidirect
46. C22⋊C4×C227C2×C4semidirect
67. C23×C227C4⋊C4semidirect
80. C2×C4×C227C4⋊C4semidirect
112. C227#1867semidirect
130. #99Dic227semidirect
158. C22⋊C4C2×Dic227semidirect
166. C23C4⋊Dic227semidirect
193. C2×C4C4⋊Dic227semidirect

Maximal subgroups

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

subgroupindex
#284095072
#284089122
#284089072
#284087982
C2×#99×C2272
#1867227