Finite Group Databasebeta

← #46846287

Subgroups — #46846287

Order 34688.

458 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 141 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#31774408C211yesmaximal, characteristic
2#31774232C212yesmaximal, complemented
3#31774227C212yesmaximal, complemented
4#31774215C212yesmaximal, complemented
5#31774210C212yesmaximal, complemented
6#31773937C212yesmaximal, complemented
7#31773896C212yesmaximal, complemented
8#31773861C211yesmaximal, characteristic, complemented
9C22×C4⋊C4×C271C211yesmaximal, characteristic, complemented
10C2×#16864654C2211yescharacteristic
11C2×#16864638C2212yes—
12#22576143—21no—
13#22576139C2211yescharacteristic
14#22576139C2211yescharacteristic
15#22576134C2211yescharacteristic
16#22576134C2211yescharacteristic
17#22576129—21no—
18C4×C22⋊D271—21no—
19C4×C22⋊D271—21no—
20#22576119C2211yescharacteristic
21#22576093C2214yescomplemented
22#22576091C2214yescomplemented
23#22576083C2214yescomplemented
24#22576080C2214yescomplemented
25C22⋊(C271⋊C8)C2212yes—
26C22⋊(C271⋊C8)C2212yes—
27C23×C4×C271—21no—
28C23×C4×C271C2211yescharacteristic
29C2×C4⋊C4×C271—21no—
30C2×C4⋊C4×C271—21no—
31C2×C4⋊C4×C271—21no—
32C2×C4⋊C4×C271C2212yes—
33C2×C4⋊C4×C271C2212yes—
34C2×C4⋊C4×C271C2212yes—
35#16864654—24no—
36C2×Dic542C2311yescharacteristic
37C2×D1084C2311yescharacteristic
38C2×C22⋊D271—21no—
39C2×C22⋊D271—21no—
40C2×C22⋊D271—21no—
41C2×C4×D271—21no—
42#16864638—24no—
43C2×C271⋊C8C2314yes—
44C4⋊Dic271C2311yescharacteristic
45C4⋊Dic271C2311yescharacteristic
46Dic271⋊C4—21no—
47Dic271⋊C4—21no—
48C22⋊Dic271—21no—
49C22⋊Dic271—21no—
50C22⋊Dic271—21no—
51(C2×D271)⋊C4—21no—
52(C2×D271)⋊C4—21no—
53C4×Dic271—21no—
54C24×C271D411yescharacteristic
55C22×C4×C271—24no—
56C22×C4×C271—22no—
57C22×C4×C271—22no—
58C22×C4×C271—22no—
59C22×C4×C271—44no—
60C22×C4×C271D411yescharacteristic
61C22×C4×C271D411yescharacteristic
62C22×C4×C271D411yescharacteristic
63C22×C4×C271C2311yescharacteristic
64C22×C4×C271C2311yescharacteristic
65C22×C4×C271C2311yescharacteristic
66C4⋊C4×C271—22no—
67C4⋊C4×C271—22no—
68C4⋊C4×C271—22no—
69C4⋊C4×C271C2314yes—
70C22×D271—21no—
71Dic542—22no—
72D1084—22no—
73C22⋊D271—42no—
74C22⋊D271—42no—
75C22⋊D271—42no—
76C2×Dic271—21no—
77C2×Dic271—21no—
78C2×Dic271—21no—
79C4×D271—42no—
80C271⋊C8—24no—
81C23×C271—22no—
82C23×C271—22no—
83C23×C271—22no—
84C23×C271C2×D411yescharacteristic
85C23×C271C2×D411yescharacteristic
86C23×C271C2×D411yescharacteristic
87C2×C4×C271—22no—
88C2×C4×C271—22no—
89C2×C4×C271—22no—
90C2×C4×C271—22no—
91C2×C4×C271—22no—
92C2×C4×C271—24no—
93C2×C4×C271—44no—
94C2×C4×C271—44no—
95C2×C4×C271—44no—
96C2×C4×C271C2×D411yescharacteristic
97C2×C4×C271C2×D411yescharacteristic
98C2×C4×C271C2×D411yescharacteristic
99C2×C4×C271#4114yes—
100C2×C4×C271C2411yescharacteristic
101C22×C271—28no—
102C22×C271—21no—
103C22×C271—21no—
104C22×C271—21no—
105C22×C271—21no—
106C22×C271—21no—
107C22×C271—21no—
108C22×C271#12312yes—
109C22×C271AGL(1,ℤ/8)12yes—
110C22×C271#13712yes—
111C22×C271C22×D411yescharacteristic
112C2×D271—41no—
113C2×D271—41no—
114C2×D271—41no—
115C4×C271—21no—
116C4×C271—21no—
117C4×C271—44no—
118C4×C271#12312yes—
119C4×C271C2×#4112yes—
120Dic271—41no—
121Dic271—41no—
122Dic271—41no—
123C2×C271—22no—
124C2×C271—22no—
125C2×C271—22no—
126C2×C271C2×#12311yescharacteristic
127C2×C271C2×AGL(1,ℤ/8)11yescharacteristic
128C2×C271C2×#13711yescharacteristic
129D271—81no—
130C271#309211yescharacteristic, complemented, Sylow
131#3092—2711nomaximal, Sylow
132#525—2711no—
133C22×C4⋊C4D27111yescharacteristic, complemented
134#484—2712no—
135#483—2712no—
136#481—2712no—
137#480—2712no—
138#428—2712no—
139#417—2712no—
140#407—2711no—
141C2×#41—2711no—
142C23×C4—21no—
143C23×C4C2×D27111yescharacteristic
144C2×M16—2712no—
145#123—5421no—
146C22⋊Q8—2711no—
147C22⋊Q8—2711no—
148#121—2711no—
149#121—2711no—
150#120—5421no—
151C4×D4—5421no—
152C4×D4—5421no—
153#117—2711no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4C2×D27112yes—
158C2×C4⋊C4C2×D27112yes—
159C2×C4⋊C4C2×D27112yes—
160#107—2714no—
161#106—2714no—
162#103—2714no—
163#102—2714no—
164C22⋊C8—2712no—
165C22⋊C8—2712no—
166C24C22⋊D27111yescharacteristic
167#41—5424no—
168C2×Q8—2711no—
169C2×D4—2711no—
170C2×D4—5421no—
171C2×D4—5421no—
172C2×D4—5421no—
173C22×C4—24no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—44no—
178C22×C4—5421no—
179C22×C4C22⋊D27111yescharacteristic
180C22×C4C22⋊D27111yescharacteristic
181C22×C4C22⋊D27111yescharacteristic
182C22×C4C22×D27111yescharacteristic
183C22×C4C22×D27111yescharacteristic
184C22×C4C22×D27111yescharacteristic
185M16—5424no—
186C2×C8—2714no—
187C4⋊C4—22no—
188C4⋊C4—22no—
189C4⋊C4—22no—
190C4⋊C4—2711no—
191C4⋊C4—2711no—
192C4⋊C4—5421no—
193C4⋊C4—5421no—
194C4⋊C4C22×D27114yes—
195C22⋊C4—5421no—
196C22⋊C4—5421no—
197C22⋊C4—5421no—
198C22⋊C4—5421no—
199C22⋊C4—5421no—
200C42—5421no—
201C23—22no—
202C23—22no—
203C23—22no—
204C23—5421no—
205C23C2×C22⋊D27111yescharacteristic
206C23C2×C22⋊D27111yescharacteristic
207C23C2×C22⋊D27111yescharacteristic
208Q8—5422no—
209D4—5422no—
210D4—10842no—
211D4—10842no—
212D4—10842no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—24no—
219C2×C4—44no—
220C2×C4—44no—
221C2×C4—44no—
222C2×C4—5421no—
223C2×C4—5421no—
224C2×C4—5421no—
225C2×C4—10842no—
226C2×C4C2×C22⋊D27111yescharacteristic
227C2×C4C2×C22⋊D27111yescharacteristic
228C2×C4C2×C22⋊D27111yescharacteristic
229C2×C4#1686465414yes—
230C2×C4C23×D27111yescharacteristic
231C8—5424no—
232C22—28no—
233C22—21no—
234C22—21no—
235C22—21no—
236C22—21no—
237C22—21no—
238C22—21no—
239C22—10841no—
240C22—10841no—
241C22—10841no—
242C22#2257614312yes—
243C22#2257618612yes—
244C22#2257619112yes—
245C22C22×C22⋊D27111yescharacteristic
246C4—21no—
247C4—21no—
248C4—44no—
249C4—10841no—
250C4—10841no—
251C4—10841no—
252C4#2257614312yes—
253C4C2×#1686465412yes—
254C2—22no—
255C2—22no—
256C2—22no—
257C2—21681no—
258C2C2×#2257614311yescharacteristic
259C2C2×#2257618611yescharacteristic
260C2C2×#2257619111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
FittingC22×C4⋊C4×C271C2
socleC22×C271C22×D4
prime radical#31774408C2
hypercenterC22×C4⋊C4D271
hypocenterC271#3092
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #31774232C2semidirect
3. #31774227C2semidirect
4. #31774215C2semidirect
5. #31774210C2semidirect
6. #31773937C2semidirect
7. #31773896C2semidirect
8. #31773861C2semidirect
9. C22×C4⋊C4×C271C2semidirect
21. #22576093C22semidirect
22. #22576091C22semidirect
23. #22576083C22semidirect
24. #22576080C22semidirect
130. C271#3092semidirect
133. C22×C4⋊C4D271semidirect

Maximal subgroups

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

subgroupindex
#317744082
#317742322
#317742272
#317742152
#317742102
#317739372
#317738962
#317738612
C22×C4⋊C4×C2712
#3092271