Finite Group Databasebeta

← #44400985

Subgroups — #44400985

Order 29056.

382 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 69 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#28409594C211yesmaximal, characteristic
2C2×#20300202C212yesmaximal, complemented
3#28408911C212yesmaximal, complemented
4#28408799C211yesmaximal, characteristic, complemented
5#385×C227C211yesmaximal, characteristic, complemented
6C22×C22⋊D227C2212yes—
7#20300272—22no—
8#20300262—21no—
9#20300262—21no—
10#20300258—22no—
11C2×C22⋊Dic227C2212yes—
12C2×(C2×D227)⋊C4—22no—
13#20300202—24no—
14C227⋊#95—22no—
15C23×C4×C227C2211yescharacteristic
16C2×C4⋊C4×C227—21no—
17C2×C22⋊C4×C227C2211yescharacteristic
18C2×C22⋊C4×C227C2211yescharacteristic
19#95×C227—21no—
20C23×D227D412yes—
21C2×D908—21no—
22C2×D908—21no—
23C2×C22⋊D227—22no—
24C2×C22⋊D227—22no—
25C2×C22⋊D227—22no—
26C2×C22⋊D227—22no—
27C2×C22⋊D227—22no—
28C2×C22⋊D227—22no—
29C22×Dic227—22no—
30C22×Dic227D412yes—
31C4⋊Dic227—21no—
32C4⋊Dic227—21no—
33C22⋊Dic227—22no—
34C22⋊Dic227—22no—
35(C2×D227)⋊C4—22no—
36(C2×D227)⋊C4—22no—
37(C2×D227)⋊C4—44no—
38C24×C227C2311yescharacteristic
39C22×C4×C227—21no—
40C22×C4×C227—21no—
41C22×C4×C227—22no—
42C22×C4×C227—42no—
43C22×C4×C227D411yescharacteristic
44C22×C4×C227D411yescharacteristic
45C4⋊C4×C227—42no—
46C22⋊C4×C227—22no—
47C22⋊C4×C227—21no—
48C22⋊C4×C227D412yescomplemented
49C22×D227—22no—
50C22×D227—22no—
51C22×D227—22no—
52C22×D227—42no—
53C22×D227—42no—
54D908—81no—
55D908—81no—
56C22⋊D227—42no—
57C22⋊D227—42no—
58C22⋊D227—42no—
59C22⋊D227—42no—
60C2×Dic227—22no—
61C2×Dic227—22no—
62C2×Dic227—22no—
63C2×Dic227—42no—
64C2×Dic227—42no—
65C2×Dic227—42no—
66C23×C227—41no—
67C23×C227—41no—
68C23×C227C2×D412yes—
69C23×C227C2×D411yescharacteristic
70C23×C227C2×D411yescharacteristic
71C23×C227#4111yescharacteristic
72C23×C227#4112yes—
73C2×C4×C227—22no—
74C2×C4×C227—24no—
75C2×C4×C227—21no—
76C2×C4×C227—21no—
77C2×C4×C227—22no—
78C2×C4×C227—41no—
79C2×C4×C227—41no—
80C2×C4×C227—41no—
81C2×C4×C227—42no—
82C2×C4×C227—41no—
83C2×C4×C227—41no—
84C2×C4×C227—41no—
85C22×C227—21no—
86C22×C227—22no—
87C22×C227—21no—
88C22×C227—22no—
89C22×C227—41no—
90C22×C227—41no—
91C22×C227—42no—
92C22×C227#12011yescharacteristic
93C22×C227#12111yescharacteristic
94C22×C227#12112yes—
95C22×C227#12312yes—
96C22×C227#12411yescharacteristic
97C2×D227—44no—
98C2×D227—42no—
99C2×D227—42no—
100C2×D227—42no—
101C2×D227—82no—
102C4×C227—42no—
103C4×C227—42no—
104C4×C227—41no—
105C4×C227—41no—
106Dic227—42no—
107Dic227—82no—
108C2×C227—22no—
109C2×C227—21no—
110C2×C227—22no—
111C2×C227—21no—
112C2×C227#39411yescharacteristic
113C2×C227B(4,𝔽2)11yescharacteristic
114C2×C227#45811yescharacteristic
115D227—82no—
116C227#204411yescharacteristic, complemented, Sylow
117#2044—2271nomaximal, Sylow
118#534—2271no—
119C2×#99—2272no—
120#386—2272no—
121#385D22711yescharacteristic, complemented
122#342—2271no—
123C22×D4—2272no—
124C23×C4C2×D22711yescharacteristic
125#123—4542no—
126#121—4541no—
127#121—4541no—
128#120—4542no—
129C2×C4⋊C4—21no—
130C2×C22⋊C4—4542no—
131C2×C22⋊C4—2272no—
132C2×C22⋊C4C2×D22711yescharacteristic
133C2×C22⋊C4C2×D22711yescharacteristic
134#99—4544no—
135#95—21no—
136#95—4542no—
137C24—2272no—
138C24C22×D22711yescharacteristic
139C2×D4—4541no—
140C2×D4—4541no—
141C2×D4—4542no—
142C2×D4—4542no—
143C2×D4—4542no—
144C2×D4—4542no—
145C2×D4—4542no—
146C2×D4—4542no—
147C22×C4—22no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—42no—
151C22×C4—4542no—
152C22×C4—2272no—
153C22×C4C22⋊D22711yescharacteristic
154C22×C4C22⋊D22711yescharacteristic
155C4⋊C4—42no—
156C4⋊C4—4541no—
157C4⋊C4—4541no—
158C22⋊C4—22no—
159C22⋊C4—21no—
160C22⋊C4—9084no—
161C22⋊C4—4542no—
162C22⋊C4—4542no—
163C22⋊C4—4542no—
164C22⋊C4—4542no—
165C22⋊C4D90812yescomplemented
166C23—41no—
167C23—41no—
168C23—9082no—
169C23—9082no—
170C23—4542no—
171C23—4542no—
172C23—4542no—
173C23D4×D22712yes—
174C23C2×C22⋊D22711yescharacteristic
175C23C2×D90811yescharacteristic
176C23#1681388411yescharacteristic
177C23#1681388512yes—
178D4—18161no—
179D4—18161no—
180D4—9082no—
181D4—9082no—
182D4—9082no—
183D4—9082no—
184C2×C4—24no—
185C2×C4—22no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—22no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—42no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—9082no—
197C2×C4—9082no—
198C2×C4—9082no—
199C2×C4—4542no—
200C2×C4—4542no—
201C2×C4—4542no—
202C22—21no—
203C22—22no—
204C22—21no—
205C22—22no—
206C22—41no—
207C22—41no—
208C22—42no—
209C22—18162no—
210C22—9084no—
211C22—9082no—
212C22—9082no—
213C22—9082no—
214C22#2030025611yescharacteristic
215C22#2030026211yescharacteristic
216C22#2030026312yes—
217C22#2030027312yes—
218C22#2030027711yescharacteristic
219C4—42no—
220C4—42no—
221C4—41no—
222C4—41no—
223C4—18162no—
224C4—9082no—
225C2—21no—
226C2—22no—
227C2—22no—
228C2—21no—
229C2—18162no—
230C2#2840894911yescharacteristic
231C2#2840921811yescharacteristic
232C2#2840922111yescharacteristic
subgroupG/H
centerC22#20300256
commutator, cosocleC24×C227C23
FrattiniC24C22×D227
Fitting#385×C227C2
socleC22×C227#120
prime radical#28409594C2
hypercenter#385D227
hypocenterC227#2044
LCSG ▷ C24×C227 ▷ C2×C227 ▷ C227
UCSC1 ◁ C22 ◁ C24 ◁ #385

Complements

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

NH ≅ G/Ntype
2. C2×#20300202C2semidirect
3. #28408911C2semidirect
4. #28408799C2semidirect
5. #385×C227C2semidirect
48. C22⋊C4×C227D4semidirect
116. C227#2044semidirect
121. #385D227semidirect
165. C22⋊C4D908semidirect

Maximal subgroups

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

subgroupindex
#284095942
C2×#203002022
#284089112
#284087992
#385×C2272
#2044227