Finite Group Databasebeta

← #44400770

Subgroups — #44400770

Order 29056.

362 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 75 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
1C2×#20300319C211yesmaximal, characteristic
2#28409606C211yesmaximal, characteristic
3C2×#20300216C211yesmaximal, characteristic
4#28409041C211yesmaximal, characteristic, complemented
5C2×#20300207C211yesmaximal, characteristic, complemented
6#28408792C211yesmaximal, characteristic, complemented
7C2×C4⋊Q8×C227C211yesmaximal, characteristic, complemented
8C2×#16813884C2211yescharacteristic
9#20300319—22no—
10#20300319—22no—
11C2×#16813874—21no—
12C2×#16813871—21no—
13C2×#16813868C2211yescharacteristic
14C2×#16813868C2211yescharacteristic
15#20300286—21no—
16#20300277—21no—
17#20300271—22no—
18#20300264—22no—
19#20300247C2211yescharacteristic
20#20300216—22no—
21#20300212—22no—
22#20300207—22no—
23C22×Q8×C227C2211yescharacteristic
24C22×Q8×C227C2211yescharacteristic
25C4⋊Q8×C227—22no—
26C4⋊Q8×C227—22no—
27C2×C4⋊C4×C227—22no—
28C2×C42×C227C2211yescharacteristic
29#16813884—21no—
30#16813884—21no—
31#16813884—21no—
32#16813884—21no—
33C2×Dic454D411yescharacteristic
34C2×D908D411yescharacteristic
35C2×C22⋊D227—21no—
36C2×C4×D227—21no—
37#16813874—24no—
38#16813871—24no—
39#16813868—21no—
40#16813868—21no—
41#16813868—21no—
42#16813868D412yes—
43C2×C227⋊C8—21no—
44C2×C227⋊C8—21no—
45C4⋊Dic227D412yes—
46Dic227⋊C4—24no—
47C22⋊Dic227—21no—
48(C2×D227)⋊C4—24no—
49C4×Dic227—21no—
50C2×Q8×C227—21no—
51C2×Q8×C227—21no—
52C2×Q8×C227—21no—
53C2×Q8×C227—21no—
54C2×Q8×C227—21no—
55C2×Q8×C227—21no—
56C2×Q8×C227—21no—
57C2×Q8×C227—21no—
58C2×Q8×C227—21no—
59C2×Q8×C227—21no—
60C2×Q8×C227—21no—
61C2×Q8×C227D412yes—
62C22×C4×C227—21no—
63C22×C4×C227—21no—
64C22×C4×C227D412yes—
65C22×C4×C227C2311yescharacteristic
66C4⋊C4×C227—28no—
67C42×C227—21no—
68C42×C227D412yes—
69C22×D227—21no—
70Dic454—21no—
71Dic454—21no—
72D908—21no—
73D908—21no—
74C22⋊D227—41no—
75C22⋊D227—41no—
76C2×Dic227—22no—
77C2×Dic227—21no—
78C4×D227—41no—
79C4×D227—41no—
80C227⋊C8—22no—
81C227⋊C8—41no—
82C23×C227C2×D411yescharacteristic
83Q8×C227—22no—
84Q8×C227—22no—
85Q8×C227—41no—
86Q8×C227—41no—
87Q8×C227—41no—
88Q8×C227—41no—
89Q8×C227—41no—
90Q8×C227—41no—
91C2×C4×C227—24no—
92C2×C4×C227—22no—
93C2×C4×C227—22no—
94C2×C4×C227—22no—
95C2×C4×C227—22no—
96C2×C4×C227—22no—
97C2×C4×C227—41no—
98C2×C4×C227—41no—
99C2×C4×C227C2×D411yescharacteristic
100C2×C4×C227C2×D411yescharacteristic
101C2×C4×C227C2×D411yescharacteristic
102C2×C4×C227C2×D411yescharacteristic
103C2×C4×C227C2×D411yescharacteristic
104C2×C4×C227#4111yescharacteristic
105C22×C227—22no—
106C22×C227#12011yescharacteristic
107C22×C227#12011yescharacteristic
108C22×C227#12111yescharacteristic
109C2×D227—41no—
110C2×D227—42no—
111C4×C227—24no—
112C4×C227—42no—
113C4×C227—42no—
114C4×C227#12011yescharacteristic
115C4×C227#12111yescharacteristic
116C4×C227#12111yescharacteristic
117C4×C227#12711yescharacteristic
118Dic227—41no—
119Dic227—42no—
120C2×C227—21no—
121C2×C227—21no—
122C2×C227#39211yescharacteristic
123C2×C227#45612yes—
124D227—81no—
125C227#200511yescharacteristic, complemented, Sylow
126#2005—2271nomaximal, Sylow
127C2×#137—2271no—
128#536—2271no—
129C2×C4⋊Q8D22711yescharacteristic, complemented
130C2×#104—2271no—
131#419—2271no—
132C2×#101—2271no—
133#339—2271no—
134C2×#41—2271no—
135C22×Q8C2×D22711yescharacteristic
136C22×Q8C2×D22711yescharacteristic
137#137—4542no—
138#137—4542no—
139C2×Q16—4541no—
140C2×SD16—4541no—
141C2×M16—2271no—
142C2×M16—2271no—
143C4⋊Q8—22no—
144C4⋊Q8—22no—
145C4⋊Q8—4541no—
146#124—4541no—
147#123—4542no—
148C22⋊Q8—4542no—
149#117—2271no—
150C2×C4⋊C4—22no—
151C2×C42C2×D22711yescharacteristic
152#104—4542no—
153#103—4542no—
154#101—4542no—
155#41—4541no—
156#41—4541no—
157#41—4541no—
158#41—4541no—
159C2×Q8—21no—
160C2×Q8—21no—
161C2×Q8—21no—
162C2×Q8—21no—
163C2×Q8—21no—
164C2×Q8—21no—
165C2×Q8—21no—
166C2×Q8—21no—
167C2×Q8—21no—
168C2×Q8—21no—
169C2×Q8—21no—
170C2×Q8—2271no—
171C2×Q8C22⋊D22712yes—
172C2×D4—4541no—
173C2×D4—2271no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—4541no—
177C22×C4C22⋊D22712yes—
178C22×C4C22×D22711yescharacteristic
179Q16—4544no—
180SD16—4544no—
181M16—4541no—
182M16—4541no—
183M16—4541no—
184M16—2272no—
185C2×C8—4541no—
186C2×C8—4541no—
187C4⋊C4—28no—
188C4⋊C4—4544no—
189C4⋊C4—2272no—
190C22⋊C4—4544no—
191C22⋊C4—4541no—
192C42—21no—
193C42—4541no—
194C42C22⋊D22712yes—
195C23—4541no—
196C23C2×C22⋊D22711yescharacteristic
197Q8—22no—
198Q8—22no—
199Q8—41no—
200Q8—41no—
201Q8—41no—
202Q8—41no—
203Q8—41no—
204Q8—41no—
205Q8—4541no—
206Q8—4541no—
207D4—9081no—
208D4—9081no—
209D4—4541no—
210D4—4541no—
211C2×C4—24no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—9081no—
220C2×C4—9081no—
221C2×C4—4542no—
222C2×C4—4541no—
223C2×C4D4×D22711yescharacteristic
224C2×C4D4×D22711yescharacteristic
225C2×C4D4×D22711yescharacteristic
226C2×C4C2×C22⋊D22711yescharacteristic
227C2×C4C2×C22⋊D22711yescharacteristic
228C2×C4#1681388311yescharacteristic
229C8—9081no—
230C8—4542no—
231C22—22no—
232C22—9081no—
233C22—9082no—
234C22#2030025611yescharacteristic
235C22#2030025711yescharacteristic
236C22#2030026011yescharacteristic
237C4—24no—
238C4—42no—
239C4—42no—
240C4—9081no—
241C4—9082no—
242C4#2030025611yescharacteristic
243C4#2030025911yescharacteristic
244C4#2030026011yescharacteristic
245C4#2030028411yescharacteristic
246C2—21no—
247C2—21no—
248C2—18161no—
249C2#2840894011yescharacteristic
250C2#2840921412yes—
subgroupG/H
centerC22#20300256
commutator, cosocleC22×C4×C227C23
FrattiniC22×C4C22×D227
FittingC2×C4⋊Q8×C227C2
socleC22×C227#120
prime radicalC2×#16813884C22
hypercenterC2×C4⋊Q8D227
hypocenterC227#2005
LCSG ▷ C22×C4×C227 ▷ C2×C227 ▷ C227
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
4. #28409041C2semidirect
5. C2×#20300207C2semidirect
6. #28408792C2semidirect
7. C2×C4⋊Q8×C227C2semidirect
125. C227#2005semidirect
129. C2×C4⋊Q8D227semidirect

Maximal subgroups

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

subgroupindex
C2×#203003192
#284096062
C2×#203002162
#284090412
C2×#203002072
#284087922
C2×C4⋊Q8×C2272
#2005227