Finite Group Databasebeta

← #46840008

Subgroups — #46840008

Order 34688.

800 conjugacy classes of subgroups; only the 213 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 213 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×Dic542C211yesmaximal, characteristic
2C2×#22576160C212yesmaximal, complemented
3C2×#22576139C211yesmaximal, characteristic
4#31773818C214yesmaximal, complemented
5#31773803C212yesmaximal, complemented
6#31773772C214yesmaximal
7C4×C22⋊C4×C271C211yesmaximal, characteristic
8C22×Dic542—28no—
9C22×Dic542—24no—
10C22×Dic542C2212yes—
11C22×Dic542C2211yescharacteristic
12C22×Dic542C2212yes—
13C23×Dic271C2212yes—
14#22576160—28no—
15#22576139—24no—
16C2×C4⋊Dic271C2218yes—
17C2×C4⋊Dic271C2211yescharacteristic
18C2×Dic271⋊C4C2212yes—
19C2×C22⋊Dic271C2212yes—
20C271⋊#95C2218yes—
21C23×C4×C271C2211yescharacteristic
22C2×C22⋊C4×C271C2212yes—
23C2×C42×C271C2212yes—
24#95×C271C2212yes—
25C2×Dic542—232no—
26C2×Dic542—28no—
27C2×Dic542—24no—
28C2×Dic542—24no—
29C2×Dic542—24no—
30C2×Dic542—22no—
31C2×Dic542—24no—
32C2×Dic542D418yes—
33C22×Dic271—28no—
34C22×Dic271—24no—
35C22×Dic271C2312yes—
36C22×Dic271C2312yes—
37C22×Dic271C2314yes—
38C4⋊Dic271—216no—
39C4⋊Dic271—22no—
40Dic271⋊C4—24no—
41C22⋊Dic271—24no—
42C24×C271C2311yescharacteristic
43C22×C4×C271—24no—
44C22×C4×C271D414yes—
45C22×C4×C271C2311yescharacteristic
46C22×C4×C271C2314yes—
47C22×C4×C271C2311yescharacteristic
48C22⋊C4×C271Q818yescomplemented
49C42×C271—24no—
50Dic542—216no—
51Dic542—216no—
52Dic542—44no—
53C2×Dic271—216no—
54C2×Dic271—24no—
55C2×Dic271—24no—
56C2×Dic271—22no—
57C2×Dic271—22no—
58C2×Dic271—24no—
59C2×Dic271—24no—
60C2×Dic271C2×D418yes—
61C23×C271—22no—
62C23×C271—22no—
63C23×C271C2×D412yes—
64C23×C271C2×Q814yes—
65C23×C271C2411yescharacteristic
66C2×C4×C271—24no—
67C2×C4×C271—22no—
68C2×C4×C271—24no—
69C2×C4×C271—28no—
70C2×C4×C271C2×D414yes—
71C2×C4×C271C2×D412yes—
72C2×C4×C271C2×D412yes—
73C2×C4×C271C2×Q818yescomplemented
74C22×C271—22no—
75C22×C271—22no—
76C22×C271—28no—
77C22×C271C4⋊Q814yescomplemented
78C22×C271C22×D412yes—
79C22×C271C22×D411yescharacteristic
80C22×C271C22×Q812yes—
81C22×C2712−1+412yes—
82C4×C271—22no—
83C4×C271—28no—
84C4×C271#12014yes—
85Dic271—28no—
86Dic271—42no—
87Dic271—42no—
88C2×C271—24no—
89C2×C271C2×#12011yescharacteristic
90C2×C271C2×C4⋊Q811yescharacteristic
91C2×C271#53611yescharacteristic
92C2×C271D4×Q814yes—
93C271#243411yescharacteristic, complemented, Sylow
94#2434—2711nomaximal, Sylow
95C23×Q8—2711no—
96C2×C4⋊Q8—2712no—
97C2×C22⋊Q8—2711no—
98#395—2714no—
99#391—2712no—
100#385—2714no—
101C4×C22⋊C4D27111yescharacteristic
102C22×Q8—2712no—
103C22×Q8—2711no—
104C22×Q8—2712no—
105C22×Q8—5428no—
106C22×Q8—5424no—
107C23×C4—2712no—
108C23×C4C2×D27111yescharacteristic
109C4⋊Q8—5428no—
110C22⋊Q8—5424no—
111C2×C4⋊C4—2711no—
112C2×C4⋊C4—2712no—
113C2×C4⋊C4—2718no—
114C2×C22⋊C4—2712no—
115C2×C22⋊C4C2×D27112yes—
116C2×C42C2×D27112yes—
117#95—2718no—
118#95C2×D27112yes—
119C24C22×D27111yescharacteristic
120C2×Q8—2718no—
121C2×Q8—5428no—
122C2×Q8—54232no—
123C2×Q8—5424no—
124C2×Q8—5424no—
125C2×Q8—5424no—
126C2×Q8—5424no—
127C2×Q8—5422no—
128C22×C4—24no—
129C22×C4—2714no—
130C22×C4—2712no—
131C22×C4—2712no—
132C22×C4—5428no—
133C22×C4—5424no—
134C22×C4D108414yes—
135C22×C4C22×D27111yescharacteristic
136C22×C4C22×D27114yes—
137C22×C4C22×D27111yescharacteristic
138C4⋊C4—54216no—
139C4⋊C4—5424no—
140C4⋊C4—5422no—
141C22⋊C4—5424no—
142C22⋊C4Dic54218yescomplemented
143C42—24no—
144C23—22no—
145C23—22no—
146C23C2×D108412yes—
147C23C2×Dic54214yes—
148C23C23×D27111yescharacteristic
149Q8—54216no—
150Q8—54216no—
151Q8—10844no—
152C2×C4—24no—
153C2×C4—22no—
154C2×C4—24no—
155C2×C4—28no—
156C2×C4—2718no—
157C2×C4—54216no—
158C2×C4—5424no—
159C2×C4—5424no—
160C2×C4—5424no—
161C2×C4—5422no—
162C2×C4—5422no—
163C2×C4—5424no—
164C2×C4D4×D27114yes—
165C2×C4C2×D108412yes—
166C2×C4C2×D108412yes—
167C2×C4C2×Dic54218yescomplemented
168C22—22no—
169C22—28no—
170C22—22no—
171C22#2257616014yescomplemented
172C22C2×D4×D27112yes—
173C22C22×D108411yescharacteristic
174C22C22×Dic54212yes—
175C22#2257620712yes—
176C4—28no—
177C4—22no—
178C4—5428no—
179C4—10842no—
180C4—10842no—
181C4#2257613014yes—
182C2—24no—
183C2C2×#2257613011yescharacteristic
184C2C2×#2257616011yescharacteristic
185C2#3177447311yescharacteristic
186C2D4×Dic54214yes—
subgroupG/H
center, FrattiniC23C23×D271
commutator, socle, cosocleC23×C271C24
FittingC4×C22⋊C4×C271C2
prime radicalC24×C271C23
hypercenterC4×C22⋊C4D271
hypocenterC271#2434
LCSG ▷ C23×C271 ▷ C271
UCSC1 ◁ C23 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#22576160C2semidirect
4. #31773818C2semidirect
5. #31773803C2semidirect
48. C22⋊C4×C271Q8semidirect
73. C2×C4×C271C2×Q8semidirect
77. C22×C271C4⋊Q8semidirect
93. C271#2434semidirect
142. C22⋊C4Dic542semidirect
167. C2×C4C2×Dic542semidirect
171. C22#22576160semidirect

Maximal subgroups

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

subgroupindex
C23×Dic5422
C2×#225761602
C2×#225761392
#317738182
#317738032
#317737722
C4×C22⋊C4×C2712
#2434271