Finite Group Databasebeta

← #44399766

Subgroups — #44399766

Order 29056.

316 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 101 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
1C22×#16813867C211yesmaximal, characteristic
2C2×#20300221C211yesmaximal, characteristic, complemented
3C2×#20300219C211yesmaximal, characteristic, complemented
4#28408889C211yesmaximal, characteristic
5#28408889C211yesmaximal, characteristic
6#28408795C211yesmaximal, characteristic, complemented
7#340×C227C211yesmaximal, characteristic, complemented
8C22×C4×D227C2211yescharacteristic
9C2×#16813867—22no—
10C2×#16813867—22no—
11C2×#16813867C414yescomplemented
12C22×C227⋊C8C2211yescharacteristic
13C2×C4⋊Dic227C2211yescharacteristic
14C2×C4⋊Dic227C2211yescharacteristic
15C2×(C2×D227)⋊C4—21no—
16C2×(C2×D227)⋊C4—21no—
17#20300221—22no—
18#20300219—22no—
19C227⋊#95—21no—
20C227⋊#95—21no—
21C22×C8×C227C2211yescharacteristic
22C2×C4⋊C4×C227C2211yescharacteristic
23C2×C4⋊C4×C227C2211yescharacteristic
24C23×D227D411yescharacteristic
25C22×Dic227—21no—
26C22×Dic227—21no—
27C22×Dic227D411yescharacteristic
28C2×C4×D227—21no—
29C2×C4×D227—21no—
30C2×C4×D227—21no—
31C2×C4×D227—21no—
32C2×C4×D227C2×C412yes—
33C2×C4×D227Q812yes—
34#16813867—28no—
35C2×C227⋊C8C2×C412yes—
36C2×C227⋊C8D412yes—
37C2×C227⋊C8D412yes—
38C4⋊Dic227—21no—
39C4⋊Dic227—21no—
40C4⋊Dic227—21no—
41C4⋊Dic227—21no—
42(C2×D227)⋊C4—42no—
43(C2×D227)⋊C4—42no—
44C22×C4×C227—21no—
45C22×C4×C227—21no—
46C22×C4×C227C2311yescharacteristic
47C2×C8×C227—21no—
48C2×C8×C227—21no—
49C2×C8×C227C2×C412yes—
50C4⋊C4×C227—21no—
51C4⋊C4×C227—21no—
52C4⋊C4×C227—21no—
53C4⋊C4×C227—21no—
54C22×D227—21no—
55C22×D227—21no—
56C22×D227—21no—
57C22×D227—21no—
58C22×D227—21no—
59C22×D227—21no—
60C22×D227C4⋊C412yes—
61C2×Dic227—21no—
62C2×Dic227—21no—
63C2×Dic227—41no—
64C2×Dic227—41no—
65C2×Dic227—41no—
66C2×Dic227—41no—
67C2×Dic227—41no—
68C2×Dic227—41no—
69C2×Dic227C4⋊C412yes—
70C4×D227—22no—
71C4×D227—22no—
72C4×D227—22no—
73C4×D227—22no—
74C227⋊C8C22⋊C414yescomplemented
75C23×C227C2×D411yescharacteristic
76C2×C4×C227—41no—
77C2×C4×C227—41no—
78C2×C4×C227—41no—
79C2×C4×C227—41no—
80C2×C4×C227—41no—
81C2×C4×C227—41no—
82C2×C4×C227C22×C411yescharacteristic
83C2×C4×C227C2×D411yescharacteristic
84C2×C4×C227C2×D411yescharacteristic
85C2×C4×C227C2×Q811yescharacteristic
86C2×C4×C227#4111yescharacteristic
87C2×C4×C227#4111yescharacteristic
88C8×C227—22no—
89C22×C227C2×C4⋊C411yescharacteristic
90C22×C227#12111yescharacteristic
91C22×C227#12111yescharacteristic
92C22×C227AGL(1,ℤ/8)11yescharacteristic
93C22×C227AGL(1,ℤ/8)11yescharacteristic
94C22×C227#13711yescharacteristic
95C22×C227#13711yescharacteristic
96C2×D227—22no—
97C2×D227—22no—
98C2×D227—22no—
99C2×D227—41no—
100C2×D227—41no—
101C2×D227—41no—
102C2×D227—41no—
103C4×C227—81no—
104C4×C227—81no—
105C4×C227C2×C22⋊C411yescharacteristic
106C4×C227#11711yescharacteristic
107C4×C227C22⋊Q811yescharacteristic
108C4×C227C22⋊Q811yescharacteristic
109Dic227—22no—
110Dic227—81no—
111Dic227—81no—
112C2×C227#38011yescharacteristic
113C2×C227#42811yescharacteristic
114C2×C227#42811yescharacteristic
115C2×C227#46811yescharacteristic
116C2×C227#46811yescharacteristic
117C2×C227#46911yescharacteristic
118C2×C227#47011yescharacteristic
119D227—42no—
120C227#180911yescharacteristic, complemented, Sylow
121#1809—2271nomaximal, Sylow
122C22×M16—2271no—
123C2×#107—2271no—
124C2×#106—2271no—
125#380—2271no—
126#380—2271no—
127#340—2271no—
128#340D22711yescharacteristic, complemented
129C23×C4—2271no—
130C2×M16—4542no—
131C2×M16—4542no—
132C2×M16—2274no—
133C22×C8—2271no—
134C22×C8C2×D22711yescharacteristic
135C2×C4⋊C4—2271no—
136C2×C4⋊C4—2271no—
137C2×C4⋊C4C2×D22711yescharacteristic
138C2×C4⋊C4C2×D22711yescharacteristic
139C2×C22⋊C4—4541no—
140C2×C22⋊C4—4541no—
141#107—4542no—
142#106—4542no—
143#95—4541no—
144#95—4541no—
145C24—2271no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—4541no—
149C22×C4—4541no—
150C22×C4—4541no—
151C22×C4—4541no—
152C22×C4—4541no—
153C22×C4—4541no—
154C22×C4—2271no—
155C22×C4—2272no—
156C22×C4—2272no—
157C22×C4C22×D22711yescharacteristic
158M16—4548no—
159C2×C8—21no—
160C2×C8—21no—
161C2×C8—2272no—
162C2×C8—2272no—
163C2×C8—2272no—
164C2×C8C4×D22712yes—
165C4⋊C4—21no—
166C4⋊C4—21no—
167C4⋊C4—21no—
168C4⋊C4—21no—
169C4⋊C4—4541no—
170C4⋊C4—4541no—
171C4⋊C4—4541no—
172C4⋊C4—4541no—
173C22⋊C4—9082no—
174C22⋊C4—9082no—
175C23—4541no—
176C23—4541no—
177C23—4541no—
178C23—4541no—
179C23—4541no—
180C23—4541no—
181C23—2272no—
182C23D4×D22711yescharacteristic
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—9081no—
190C2×C4—9081no—
191C2×C4—9081no—
192C2×C4—9081no—
193C2×C4—9081no—
194C2×C4—9081no—
195C2×C4—4541no—
196C2×C4—4541no—
197C2×C4—4542no—
198C2×C4—4542no—
199C2×C4—4542no—
200C2×C4—4542no—
201C2×C4—2272no—
202C2×C4C2×C4×D22711yescharacteristic
203C2×C4D4×D22711yescharacteristic
204C2×C4D4×D22711yescharacteristic
205C2×C4Q8×D22711yescharacteristic
206C2×C4#1681388411yescharacteristic
207C2×C4#1681388411yescharacteristic
208C8—22no—
209C8—2274no—
210C22—9081no—
211C22—9081no—
212C22—9081no—
213C22—9081no—
214C22—4542no—
215C22—4542no—
216C22—4542no—
217C22C4⋊C4×D22711yescharacteristic
218C22#2030026111yescharacteristic
219C22#2030026111yescharacteristic
220C22#2030031211yescharacteristic
221C22#2030031511yescharacteristic
222C22#2030031711yescharacteristic
223C22#2030032011yescharacteristic
224C4—81no—
225C4—81no—
226C4—18161no—
227C4—18161no—
228C4—4542no—
229C4C22⋊C4×D22711yescharacteristic
230C4#2030024511yescharacteristic
231C4#2030026611yescharacteristic
232C4#2030026611yescharacteristic
233C2—9082no—
234C2#2840889111yescharacteristic
235C2#2840907111yescharacteristic
236C2#2840907311yescharacteristic
237C2#2840927711yescharacteristic
238C2#2840928111yescharacteristic
239C2#2840928411yescharacteristic
240C2#2840928911yescharacteristic
subgroupG/H
centerC23D4×D227
commutatorC2×C4×C227C22×C4
FrattiniC22×C4C22×D227
Fitting#340×C227C2
socleC23×C227C2×D4
prime radicalC23×D227D4
cosocleC22×C4×C227C23
hypercenter#340D227
hypocenterC227#1809
LCSG ▷ C2×C4×C227 ▷ C2×C227 ▷ C227
UCSC1 ◁ C23 ◁ C22×C4 ◁ #340

Complements

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

NH ≅ G/Ntype
2. C2×#20300221C2semidirect
3. C2×#20300219C2semidirect
6. #28408795C2semidirect
7. #340×C227C2semidirect
11. C2×#16813867C4semidirect
74. C227⋊C8C22⋊C4semidirect
120. C227#1809semidirect
128. #340D227semidirect

Maximal subgroups

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

subgroupindex
C22×#168138672
C2×#203002212
C2×#203002192
#284088892
#284087952
#340×C2272
#1809227