Finite Group Databasebeta

← #46837588

Subgroups — #46837588

Order 34688.

306 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 69 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#31774478C211yesmaximal, characteristic
2C2×#22576088C212yesmaximal, complemented
3#31773896C212yesmaximal, complemented
4#31773666C211yesmaximal, characteristic, complemented
5#384×C271C211yesmaximal, characteristic, complemented
6C2×#16864654C2212yes—
7C2×#16864638C2212yes—
8#22576140—22no—
9#22576135—22no—
10#22576134—22no—
11#22576088—24no—
12#22576083—22no—
13#22576080—22no—
14C2×C4⋊C4×C271—22no—
15C2×C4⋊C4×C271C2211yescharacteristic
16C2×C4⋊C4×C271C2211yescharacteristic
17C2×C42×C271C2211yescharacteristic
18#16864654—22no—
19#16864654—22no—
20#16864654—22no—
21#16864654—22no—
22C2×Dic542D412yes—
23C2×D1084—22no—
24C2×D1084D412yes—
25C2×C22⋊D271—22no—
26C2×C4×D271—22no—
27#16864638—22no—
28#16864638—22no—
29C2×C271⋊C8—22no—
30C4⋊Dic271—22no—
31(C2×D271)⋊C4—24no—
32C22×C4×C271—21no—
33C22×C4×C271—21no—
34C22×C4×C271D412yes—
35C22×C4×C271C2311yescharacteristic
36C4⋊C4×C271—22no—
37C4⋊C4×C271—22no—
38C4⋊C4×C271—44no—
39C42×C271—21no—
40C42×C271D412yescomplemented
41C22×D271—22no—
42Dic542—22no—
43Dic542—22no—
44D1084—22no—
45D1084—22no—
46D1084—82no—
47C22⋊D271—42no—
48C22⋊D271—42no—
49C2×Dic271—22no—
50C4×D271—42no—
51C4×D271—42no—
52C271⋊C8—42no—
53C23×C271C2×D411yescharacteristic
54C2×C4×C271—22no—
55C2×C4×C271—22no—
56C2×C4×C271—22no—
57C2×C4×C271—22no—
58C2×C4×C271—42no—
59C2×C4×C271—42no—
60C2×C4×C271—41no—
61C2×C4×C271C2×D412yes—
62C2×C4×C271C2×D411yescharacteristic
63C2×C4×C271#4111yescharacteristic
64C2×C4×C271#4112yes—
65C22×C271—21no—
66C22×C271—21no—
67C22×C271#12011yescharacteristic
68C22×C271#12111yescharacteristic
69C22×C271#12411yescharacteristic
70C2×D271—42no—
71C2×D271—42no—
72C2×D271—42no—
73C4×C271—24no—
74C4×C271—42no—
75C4×C271—81no—
76C4×C271#12112yes—
77C4×C271#12312yes—
78Dic271—42no—
79C2×C271—21no—
80C2×C271—21no—
81C2×C271#39411yescharacteristic
82C2×C271#45311yescharacteristic
83C2×C271#45611yescharacteristic
84D271—82no—
85C271#203911yescharacteristic, complemented, Sylow
86#2039—2711nomaximal, Sylow
87#537—2711no—
88C2×#104—2712no—
89#417—2712no—
90#384D27111yescharacteristic, complemented
91#343—2711no—
92C2×#41—2712no—
93C2×M16—2712no—
94C22⋊Q8—5422no—
95#121—5422no—
96#121—5422no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4C2×D27111yescharacteristic
99C2×C4⋊C4C2×D27111yescharacteristic
100C2×C42C2×D27111yescharacteristic
101#104—5424no—
102#103—5422no—
103#102—5422no—
104#41—5422no—
105#41—5422no—
106#41—5422no—
107#41—5422no—
108C2×Q8—2712no—
109C2×D4—2712no—
110C2×D4—5422no—
111C2×D4—5422no—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—5422no—
115C22×C4C22⋊D27112yes—
116C22×C4C22×D27111yescharacteristic
117M16—5422no—
118M16—5422no—
119C2×C8—5422no—
120C4⋊C4—22no—
121C4⋊C4—22no—
122C4⋊C4—44no—
123C4⋊C4—5422no—
124C22⋊C4—5424no—
125C42—21no—
126C42D108412yescomplemented
127C23—5422no—
128C23C2×C22⋊D27111yescharacteristic
129Q8—5422no—
130Q8—5422no—
131D4—5422no—
132D4—5422no—
133D4—10842no—
134D4—10842no—
135D4—21682no—
136C2×C4—22no—
137C2×C4—22no—
138C2×C4—22no—
139C2×C4—22no—
140C2×C4—42no—
141C2×C4—42no—
142C2×C4—41no—
143C2×C4—5422no—
144C2×C4—10842no—
145C2×C4—10842no—
146C2×C4D4×D27112yes—
147C2×C4C2×D108411yescharacteristic
148C2×C4#1686465411yescharacteristic
149C2×C4#1686465512yes—
150C8—10842no—
151C22—21no—
152C22—21no—
153C22—10842no—
154C22—10842no—
155C22—10842no—
156C22#2257612811yescharacteristic
157C22#2257613411yescharacteristic
158C22#2257614911yescharacteristic
159C4—24no—
160C4—42no—
161C4—81no—
162C4—10842no—
163C4#2257613512yes—
164C4#2257614512yes—
165C2—21no—
166C2—21no—
167C2—21682no—
168C2#3177381411yescharacteristic
169C2#3177406411yescharacteristic
170C2#3177407911yescharacteristic
subgroupG/H
centerC22#22576128
commutator, cosocleC22×C4×C271C23
FrattiniC22×C4C22×D271
Fitting#384×C271C2
socleC22×C271#120
prime radical#31774478C2
hypercenter#384D271
hypocenterC271#2039
LCSG ▷ C22×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #384

Complements

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

NH ≅ G/Ntype
2. C2×#22576088C2semidirect
3. #31773896C2semidirect
4. #31773666C2semidirect
5. #384×C271C2semidirect
40. C42×C271D4semidirect
85. C271#2039semidirect
90. #384D271semidirect
126. C42D1084semidirect

Maximal subgroups

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

subgroupindex
#317744782
C2×#225760882
#317738962
#317736662
#384×C2712
#2039271