Finite Group Databasebeta

← #46840002

Subgroups — #46840002

Order 34688.

1348 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 1348 conjugacy classes of subgroups — too many to draw. Only its 197 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
1C23×D1084C211yesmaximal, characteristic, complemented
2C2×#22576156C212yesmaximal, complemented
3C2×#22576128C214yesmaximal, complemented
4#31773806C214yesmaximal, complemented
5#31773798C212yesmaximal, complemented
6C4×C22⋊Dic271C211yesmaximal, characteristic, complemented
7C2×#121×C271C211yesmaximal, characteristic, complemented
8C24×D271C2212yes—
9C22×D1084—28no—
10C22×D1084—24no—
11C22×D1084C2212yes—
12C22×D1084C2212yes—
13C22×C22⋊D271C2218yescomplemented
14#22576156—28no—
15#22576128—216no—
16C2×C22⋊Dic271C2212yes—
17C2×(C2×D271)⋊C4C2218yescomplemented
18C2×C4×Dic271C2212yescomplemented
19C271⋊#95C2212yescomplemented
20C22×D4×C271C2212yes—
21C22×D4×C271C2211yescharacteristic, complemented
22C23×C4×C271C2211yescharacteristic
23#121×C271—24no—
24C2×C4⋊C4×C271C2211yescharacteristic, complemented
25C2×C22⋊C4×C271C2212yescomplemented
26C23×D271—28no—
27C23×D271—24no—
28C23×D271—28no—
29C23×D271—44no—
30C23×D271C2314yes—
31C2×D1084—232no—
32C2×D1084—28no—
33C2×D1084—24no—
34C2×D1084—24no—
35C2×D1084—24no—
36C2×D1084D418yescomplemented
37C2×C22⋊D271—216no—
38C2×C22⋊D271—216no—
39C2×C22⋊D271—48no—
40C22×Dic271C2314yes—
41C22⋊Dic271D418yescomplemented
42(C2×D271)⋊C4—216no—
43C4×Dic271—24no—
44C24×C271C2312yes—
45C24×C271C2311yescharacteristic
46C2×D4×C271—28no—
47C2×D4×C271—24no—
48C2×D4×C271—24no—
49C2×D4×C271—22no—
50C22×C4×C271—24no—
51C22×C4×C271D414yescomplemented
52C22×C4×C271C2312yes—
53C22×C4×C271C2311yescharacteristic
54C22×C4×C271C2311yescharacteristic
55C4⋊C4×C271—22no—
56C22⋊C4×C271—24no—
57C22×D271—216no—
58C22×D271—28no—
59C22×D271—24no—
60C22×D271—24no—
61C22×D271—28no—
62C22×D271—432no—
63C22×D271—48no—
64C22×D271—48no—
65C22×D271C2×D418yes—
66D1084—216no—
67D1084—216no—
68C22⋊D271—432no—
69C2×Dic271—24no—
70C2×Dic271—24no—
71C2×Dic271C2×D418yescomplemented
72C23×C271—22no—
73C23×C271—22no—
74C23×C271—22no—
75C23×C271—24no—
76C23×C271—24no—
77C23×C271—24no—
78C23×C271C2×D414yes—
79C23×C271C2×D412yes—
80C23×C271C2411yescharacteristic
81D4×C271—48no—
82D4×C271—44no—
83C2×C4×C271—22no—
84C2×C4×C271—28no—
85C2×C4×C271—22no—
86C2×C4×C271—24no—
87C2×C4×C271C2×D414yes—
88C2×C4×C271C2×D412yes—
89C2×C4×C271C2×D412yes—
90C22×C271—22no—
91C22×C271—22no—
92C22×C271—28no—
93C22×C271—42no—
94C22×C271—42no—
95C22×C271—42no—
96C22×C271—48no—
97C22×C271#12714yescomplemented
98C22×C271C22×D412yes—
99C22×C271C22×D412yes—
100C22×C271C22×D411yescharacteristic
101C22×C271H(5,𝔽2)12yes—
102C2×D271—28no—
103C2×D271—216no—
104C2×D271—432no—
105C2×D271—44no—
106C2×D271—44no—
107C2×D271—48no—
108C4×C271—22no—
109C4×C271—42no—
110C4×C271#12014yescomplemented
111Dic271—28no—
112C2×C271—24no—
113C2×C271—82no—
114C2×C271C2×#12011yescharacteristic
115C2×C271C2×#12711yescharacteristic
116C2×C271#53511yescharacteristic
117C2×C271D4214yes—
118D271—48no—
119C271#243311yescharacteristic, complemented, Sylow
120#2433—2711nomaximal, Sylow
121C23×D4—2711no—
122C2×#127—2712no—
123C2×#121D27111yescharacteristic, complemented
124C2×#120—2714no—
125#392—2714no—
126#390—2712no—
127C4×C22⋊C4—2711no—
128C25—2712no—
129C22×D4—2718no—
130C22×D4—2712no—
131C22×D4—2712no—
132C22×D4—5428no—
133C22×D4—5424no—
134C22×D4C2×D27112yes—
135C22×D4C2×D27111yescharacteristic, complemented
136C23×C4C2×D27111yescharacteristic
137#127—5428no—
138#121—24no—
139#120—54216no—
140C2×C4⋊C4C2×D27111yescharacteristic, complemented
141C2×C22⋊C4—2712no—
142C2×C22⋊C4—2718no—
143C2×C22⋊C4C2×D27112yescomplemented
144C2×C42—2712no—
145#95—2712no—
146C24—2714no—
147C24—5428no—
148C24—5424no—
149C24—5428no—
150C24—10844no—
151C24C22×D27112yes—
152C24C22×D27111yescharacteristic
153C2×D4—28no—
154C2×D4—24no—
155C2×D4—24no—
156C2×D4—22no—
157C2×D4—2718no—
158C2×D4—5428no—
159C2×D4—54232no—
160C2×D4—54216no—
161C2×D4—54216no—
162C2×D4—5424no—
163C2×D4—5424no—
164C2×D4—5424no—
165C2×D4—10848no—
166C22×C4—24no—
167C22×C4—2714no—
168C22×C4C22⋊D27114yescomplemented
169C22×C4C22×D27112yes—
170C22×C4C22×D27111yescharacteristic
171C22×C4C22×D27111yescharacteristic
172C4⋊C4—22no—
173C22⋊C4—24no—
174C22⋊C4—2718no—
175C22⋊C4—54216no—
176C42—5424no—
177C23—22no—
178C23—22no—
179C23—24no—
180C23—24no—
181C23—22no—
182C23—24no—
183C23—2718no—
184C23—54216no—
185C23—5428no—
186C23—5424no—
187C23—5424no—
188C23—5428no—
189C23—108432no—
190C23—10848no—
191C23—10848no—
192C23D4×D27114yes—
193C23C2×C22⋊D27112yes—
194C23C23×D27111yescharacteristic
195D4—48no—
196D4—44no—
197D4—54216no—
198D4—54216no—
199D4—108432no—
200C2×C4—22no—
201C2×C4—28no—
202C2×C4—24no—
203C2×C4—22no—
204C2×C4—2718no—
205C2×C4—5424no—
206C2×C4—5424no—
207C2×C4D4×D27114yes—
208C2×C4C2×C22⋊D27112yes—
209C2×C4C2×C22⋊D27112yes—
210C22—28no—
211C22—22no—
212C22—22no—
213C22—48no—
214C22—42no—
215C22—42no—
216C22—42no—
217C22—54216no—
218C22—5428no—
219C22—108432no—
220C22—10848no—
221C22—10844no—
222C22—10844no—
223C22#2257615614yescomplemented
224C22C2×D4×D27112yes—
225C22C2×D4×D27112yes—
226C22C22×C22⋊D27111yescharacteristic
227C22#2257620612yes—
228C4—22no—
229C4—42no—
230C4—5428no—
231C4#2257612814yescomplemented
232C2—24no—
233C2—82no—
234C2—10848no—
235C2C2×#2257612811yescharacteristic
236C2C2×#2257615611yescharacteristic
237C2#3177446111yescharacteristic
238C2#3177453114yes—
subgroupG/H
center, FrattiniC23C23×D271
commutator, socle, cosocleC23×C271C24
FittingC2×#121×C271C2
hypercenterC2×#121D271
hypocenterC271#2433
LCSG ▷ C23×C271 ▷ C271
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C23×D1084C2semidirect
2. C2×#22576156C2semidirect
3. C2×#22576128C2semidirect
4. #31773806C2semidirect
5. #31773798C2semidirect
6. C4×C22⋊Dic271C2semidirect
7. C2×#121×C271C2semidirect
13. C22×C22⋊D271C22semidirect
17. C2×(C2×D271)⋊C4C22semidirect
18. C2×C4×Dic271C22semidirect
19. C271⋊#95C22semidirect
21. C22×D4×C271C22semidirect
24. C2×C4⋊C4×C271C22semidirect
25. C2×C22⋊C4×C271C22semidirect
36. C2×D1084D4semidirect
41. C22⋊Dic271D4semidirect
51. C22×C4×C271D4semidirect
71. C2×Dic271C2×D4semidirect
97. C22×C271#127semidirect
110. C4×C271#120semidirect
119. C271#2433semidirect
123. C2×#121D271semidirect
135. C22×D4C2×D271semidirect
140. C2×C4⋊C4C2×D271semidirect
143. C2×C22⋊C4C2×D271semidirect
168. C22×C4C22⋊D271semidirect
223. C22#22576156semidirect
231. C4#22576128semidirect

Maximal subgroups

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

subgroupindex
C23×D10842
C2×#225761562
C2×#225761282
#317738062
#317737982
C4×C22⋊Dic2712
C2×#121×C2712
#2433271