Finite Group Databasebeta

← #46845591

Subgroups — #46845591

Order 34688.

994 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 994 conjugacy classes of subgroups — too many to draw. Only its 173 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×#22576186C212yesmaximal, complemented
2#31774408C211yesmaximal, characteristic, complemented
3#31774048C214yesmaximal, complemented
4#31774031C214yesmaximal, complemented
5#31773902C212yesmaximal, complemented
6#31773861C211yesmaximal, characteristic, complemented
7C23×D4×C271C211yesmaximal, characteristic, complemented
8C2×#16864654C2211yescharacteristic, complemented
9#22576186—28no—
10C2×#16864642C2214yescomplemented
11C2×#16864639C2214yescomplemented
12C2×#16864638C2212yescomplemented
13#22576143—21no—
14#22576139C2212yescomplemented
15#22576134C2212yescomplemented
16#22576129—21no—
17C4×C22⋊D271—22no—
18#22576119C2211yescharacteristic, complemented
19#22576081C2218yescomplemented
20C22⋊(C271⋊C8)C2214yescomplemented
21C25×C271—21no—
22C22×D4×C271—28no—
23C22×D4×C271—21no—
24C22×D4×C271—22no—
25C22×D4×C271C2214yes—
26C22×D4×C271C2212yes—
27C23×C4×C271C2211yescharacteristic
28#16864654—24no—
29C2×Dic542C2311yescharacteristic, complemented
30C2×D1084C2311yescharacteristic, complemented
31C2×C22⋊D271—22no—
32C2×C22⋊D271—21no—
33C2×C4×D271—21no—
34#16864642—28no—
35#16864639—28no—
36#16864638—24no—
37C2×C271⋊C8C2314yescomplemented
38C4⋊Dic271C2312yescomplemented
39Dic271⋊C4—22no—
40C22⋊Dic271—22no—
41C22⋊Dic271—21no—
42(C2×D271)⋊C4—22no—
43C4×Dic271—21no—
44C24×C271—28no—
45C24×C271—24no—
46C24×C271—22no—
47C24×C271—48no—
48C24×C271D411yescharacteristic
49C2×D4×C271—28no—
50C2×D4×C271—216no—
51C2×D4×C271—216no—
52C2×D4×C271—24no—
53C2×D4×C271—22no—
54C2×D4×C271—24no—
55C2×D4×C271D418yescomplemented
56C2×D4×C271C2314yes—
57C22×C4×C271—24no—
58C22×C4×C271D411yescharacteristic
59C22×C4×C271D412yes—
60C22×C4×C271C2312yes—
61C22×C4×C271C2311yescharacteristic
62C22×D271—21no—
63Dic542—22no—
64D1084—22no—
65C22⋊D271—42no—
66C22⋊D271—44no—
67C2×Dic271—22no—
68C2×Dic271—21no—
69C4×D271—42no—
70C271⋊C8—24no—
71C23×C271—22no—
72C23×C271—24no—
73C23×C271—216no—
74C23×C271—28no—
75C23×C271—24no—
76C23×C271—432no—
77C23×C271—48no—
78C23×C271—48no—
79C23×C271—48no—
80C23×C271C2×D411yescharacteristic
81C23×C271C2×D412yes—
82D4×C271—216no—
83D4×C271—28no—
84D4×C271—24no—
85D4×C271C2×D418yes—
86C2×C4×C271—22no—
87C2×C4×C271—24no—
88C2×C4×C271—24no—
89C2×C4×C271C2×D412yes—
90C2×C4×C271C2×D411yescharacteristic
91C2×C4×C271C2×D414yes—
92C2×C4×C271C2411yescharacteristic
93C22×C271—28no—
94C22×C271—22no—
95C22×C271—22no—
96C22×C271—22no—
97C22×C271—28no—
98C22×C271—416no—
99C22×C271—432no—
100C22×C271—48no—
101C22×C271#12012yes—
102C22×C271AGL(1,ℤ/8)14yescomplemented
103C22×C271C22×D411yescharacteristic
104C2×D271—41no—
105C2×D271—42no—
106C4×C271—22no—
107C4×C271#12012yes—
108C4×C271C22×D412yes—
109Dic271—41no—
110Dic271—42no—
111C2×C271—22no—
112C2×C271—24no—
113C2×C271—48no—
114C2×C271C2×#12011yescharacteristic
115C2×C271C2×AGL(1,ℤ/8)12yes—
116D271—81no—
117C271#300311yescharacteristic, complemented, Sylow
118#3003—2711nomaximal, Sylow
119C23×D4D27111yescharacteristic, complemented
120C2×AGL(1,ℤ/8)—2712no—
121#525—2711no—
122#450—2714no—
123#447—2714no—
124#418—2712no—
125#407—2711no—
126C25—21no—
127C2×#41—2711no—
128C22×D4—28no—
129C22×D4—22no—
130C22×D4—21no—
131C22×D4C2×D27114yes—
132C22×D4C2×D27112yes—
133C23×C4C2×D27111yescharacteristic
134AGL(1,ℤ/8)—5428no—
135C2×SD16—2714no—
136C2×D8—2714no—
137C2×M16—2712no—
138#123—5421no—
139C22⋊Q8—2712no—
140#121—2712no—
141#120—5421no—
142C4×D4—5422no—
143#117—2711no—
144#102—2718no—
145C22⋊C8—2714no—
146C24—28no—
147C24—24no—
148C24—22no—
149C24—48no—
150C24C22⋊D27111yescharacteristic
151#41—5424no—
152C2×Q8—2711no—
153C2×D4—28no—
154C2×D4—216no—
155C2×D4—216no—
156C2×D4—24no—
157C2×D4—22no—
158C2×D4—24no—
159C2×D4—2711no—
160C2×D4—5422no—
161C2×D4—5421no—
162C2×D4C22⋊D27118yescomplemented
163C2×D4C22×D27114yes—
164C22×C4—24no—
165C22×C4—5421no—
166C22×C4C22⋊D27112yes—
167C22×C4C22⋊D27111yescharacteristic
168C22×C4C22×D27112yes—
169C22×C4C22×D27111yescharacteristic
170SD16—5428no—
171D8—5428no—
172M16—5424no—
173C2×C8—2714no—
174C4⋊C4—2712no—
175C4⋊C4—5422no—
176C22⋊C4—5421no—
177C22⋊C4—5422no—
178C22⋊C4—5422no—
179C42—5421no—
180C23—22no—
181C23—24no—
182C23—216no—
183C23—28no—
184C23—24no—
185C23—432no—
186C23—48no—
187C23—48no—
188C23—48no—
189C23—5421no—
190C23C2×C22⋊D27111yescharacteristic
191C23C2×C22⋊D27112yes—
192Q8—5422no—
193D4—216no—
194D4—28no—
195D4—24no—
196D4—5422no—
197D4—10842no—
198D4—10844no—
199D4C2×C22⋊D27118yes—
200C2×C4—22no—
201C2×C4—24no—
202C2×C4—24no—
203C2×C4—5422no—
204C2×C4—5421no—
205C2×C4—10842no—
206C2×C4C2×C22⋊D27112yes—
207C2×C4C2×C22⋊D27111yescharacteristic
208C2×C4C2×C22⋊D27114yes—
209C2×C4C23×D27111yescharacteristic
210C8—5424no—
211C22—28no—
212C22—22no—
213C22—22no—
214C22—22no—
215C22—28no—
216C22—416no—
217C22—432no—
218C22—48no—
219C22—10841no—
220C22—10842no—
221C22#2257612912yes—
222C22#2257618614yescomplemented
223C22C22×C22⋊D27111yescharacteristic
224C4—22no—
225C4—10841no—
226C4—10842no—
227C4#2257612912yes—
228C4C22×C22⋊D27112yes—
229C2—22no—
230C2—24no—
231C2—48no—
232C2—21681no—
233C2C2×#2257612911yescharacteristic
234C2C2×#2257618612yes—
subgroupG/H
centerC22C22×C22⋊D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
FittingC23×D4×C271C2
socleC22×C271C22×D4
hypercenterC23×D4D271
hypocenterC271#3003
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. C2×#22576186C2semidirect
2. #31774408C2semidirect
3. #31774048C2semidirect
4. #31774031C2semidirect
5. #31773902C2semidirect
6. #31773861C2semidirect
7. C23×D4×C271C2semidirect
8. C2×#16864654C22semidirect
10. C2×#16864642C22semidirect
11. C2×#16864639C22semidirect
12. C2×#16864638C22semidirect
14. #22576139C22semidirect
15. #22576134C22semidirect
18. #22576119C22semidirect
19. #22576081C22semidirect
20. C22⋊(C271⋊C8)C22semidirect
29. C2×Dic542C23semidirect
30. C2×D1084C23semidirect
37. C2×C271⋊C8C23semidirect
38. C4⋊Dic271C23semidirect
55. C2×D4×C271D4semidirect
102. C22×C271AGL(1,ℤ/8)semidirect
117. C271#3003semidirect
119. C23×D4D271semidirect
162. C2×D4C22⋊D271semidirect
222. C22#22576186semidirect

Maximal subgroups

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

subgroupindex
C2×#225761862
#317744082
#317740482
#317740312
#317739022
#317738612
C23×D4×C2712
#3003271