Finite Group Databasebeta

← #46835537

Subgroups — #46835537

Order 34688.

194 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 194 conjugacy classes of subgroups — too many to draw. Only its 53 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#31774621C211yesmaximal, characteristic
2#31774249C211yesmaximal, characteristic, complemented
3#31774193C211yesmaximal, characteristic, complemented
4#31774188C211yesmaximal, characteristic
5#31773627C211yesmaximal, characteristic, complemented
6#31773624C211yesmaximal, characteristic
7#332×C271C211yesmaximal, characteristic, complemented
8#22576160C2211yescharacteristic
9#22576155—22no—
10#22576149—21no—
11#22576140—21no—
12Q8×Dic271—21no—
13C4×Dic542C2211yescharacteristic
14C4×D1084C2211yescharacteristic
15#22576118—21no—
16#22576094—21no—
17#22576092—21no—
18#22576085—21no—
19#22576083—22no—
20#22576082—21no—
21C4×C271⋊C8C2211yescharacteristic
22C4⋊Q8×C271C2211yescharacteristic
23C4⋊C8×C271C2211yescharacteristic
24C4⋊C8×C271C2211yescharacteristic
25C2×Dic542—21no—
26C2×Dic542D411yescharacteristic
27C2×D1084D411yescharacteristic
28C2×C4×D271—21no—
29C2×C271⋊C8—21no—
30C4⋊Dic271—22no—
31C4⋊Dic271—21no—
32C4⋊Dic271—21no—
33C4⋊Dic271D411yescharacteristic
34C4⋊Dic271D411yescharacteristic
35Dic271⋊C4—21no—
36(C2×D271)⋊C4—22no—
37(C2×D271)⋊C4—22no—
38(C2×D271)⋊C4—21no—
39C4×Dic271—22no—
40C4×Dic271—21no—
41C2×Q8×C271—21no—
42C2×C8×C271—21no—
43C2×C8×C271—21no—
44C4⋊C4×C271—21no—
45C4⋊C4×C271D412yes—
46C42×C271C2311yescharacteristic
47C22×D271—21no—
48Dic542—21no—
49Dic542—21no—
50Dic542—42no—
51D1084—21no—
52D1084—21no—
53C2×Dic271—21no—
54C2×Dic271—21no—
55C2×Dic271—21no—
56C2×Dic271—41no—
57C4×D271—41no—
58C4×D271—41no—
59C271⋊C8—41no—
60Q8×C271—41no—
61Q8×C271—41no—
62C2×C4×C271—22no—
63C2×C4×C271C2×D411yescharacteristic
64C2×C4×C271C2×D411yescharacteristic
65C2×C4×C271C2×D411yescharacteristic
66C8×C271—41no—
67C8×C271—41no—
68C22×C271#12011yescharacteristic
69C2×D271—41no—
70C2×D271—41no—
71C2×D271—41no—
72C4×C271—21no—
73C4×C271—42no—
74C4×C271#13511yescharacteristic
75C4×C271#13511yescharacteristic
76C4×C271#13711yescharacteristic
77C4×C271#13711yescharacteristic
78Dic271—41no—
79Dic271—41no—
80Dic271—41no—
81Dic271—81no—
82C2×C271#45211yescharacteristic
83C2×C271#45211yescharacteristic
84C2×C271#45411yescharacteristic
85D271—81no—
86C271#163811yescharacteristic, complemented, Sylow
87#1638—2711nomaximal, Sylow
88#556—2711no—
89#487—2711no—
90#477—2711no—
91#476—2711no—
92#332D27111yescharacteristic, complemented
93Q8⋊C8—2711no—
94D4⋊C8—2711no—
95C4⋊Q8—2711no—
96C4⋊Q8C2×D27111yescharacteristic
97#126—5422no—
98#124—5421no—
99C22⋊Q8—5421no—
100C4×Q8—2711no—
101C4×Q8—5421no—
102C4×D4—2711no—
103#117—5421no—
104#107—5421no—
105#106—5421no—
106C4⋊C8C2×D27111yescharacteristic
107C4⋊C8C2×D27111yescharacteristic
108#103—5421no—
109#103—5422no—
110#102—5421no—
111C4×C8—2711no—
112C2×Q8—21no—
113C2×Q8—2711no—
114C2×Q8—5421no—
115C2×D4—2711no—
116C22×C4—5421no—
117C2×C8—21no—
118C2×C8—21no—
119C2×C8—5421no—
120C4⋊C4—21no—
121C4⋊C4—2711no—
122C4⋊C4—2711no—
123C4⋊C4—5422no—
124C4⋊C4—5421no—
125C4⋊C4—5421no—
126C4⋊C4—5421no—
127C4⋊C4D108412yes—
128C22⋊C4—5422no—
129C22⋊C4—5422no—
130C22⋊C4—5421no—
131C42—5422no—
132C42—5421no—
133C42C22×D27111yescharacteristic
134C23—5421no—
135Q8—41no—
136Q8—41no—
137Q8—5421no—
138Q8—5421no—
139Q8—10842no—
140D4—5421no—
141D4—5421no—
142C2×C4—22no—
143C2×C4—5421no—
144C2×C4—5421no—
145C2×C4—5421no—
146C2×C4—10841no—
147C2×C4—10841no—
148C2×C4—10841no—
149C2×C4D4×D27111yescharacteristic
150C2×C4D4×D27111yescharacteristic
151C2×C4C2×D108411yescharacteristic
152C8—41no—
153C8—41no—
154C8—10841no—
155C22—10841no—
156C22—10841no—
157C22—10841no—
158C22#2257613011yescharacteristic
159C4—21no—
160C4—42no—
161C4—10841no—
162C4—10841no—
163C4—10841no—
164C4—21681no—
165C4#2257617911yescharacteristic
166C4#2257618111yescharacteristic
167C4#2257618911yescharacteristic
168C4#2257619211yescharacteristic
169C2—21681no—
170C2#3177406011yescharacteristic
171C2#3177406211yescharacteristic
172C2#3177407211yescharacteristic
subgroupG/H
centerC22#22576130
commutator, cosocleC42×C271C23
FrattiniC42C22×D271
Fitting#332×C271C2
socleC22×C271#120
prime radicalC2×D1084D4
hypercenter#332D271
hypocenterC271#1638
LCSG ▷ C42×C271 ▷ C22×C271 ▷ C271
UCSC1 ◁ C22 ◁ C42 ◁ #332

Complements

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

NH ≅ G/Ntype
2. #31774249C2semidirect
3. #31774193C2semidirect
5. #31773627C2semidirect
7. #332×C271C2semidirect
86. C271#1638semidirect
92. #332D271semidirect

Maximal subgroups

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

subgroupindex
#317746212
#317742492
#317741932
#317741882
#317736272
#317736242
#332×C2712
#1638271