Finite Group Databasebeta

← #46835621

Subgroups — #46835621

Order 34688.

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

normal subgroups only

This group has 200 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#31774300C211yesmaximal, characteristic, complemented
3#31774183C211yesmaximal, characteristic, complemented
4#31774103C211yesmaximal, characteristic
5#31773627C211yesmaximal, characteristic, complemented
6#31773624C211yesmaximal, characteristic
7#332×C271C211yesmaximal, characteristic, complemented
8C2×#16864643—21no—
9#22576160C2211yescharacteristic
10#22576155—22no—
11#22576149—21no—
12#22576140—21no—
13Q8×Dic271—21no—
14C4×Dic542C2211yescharacteristic
15C4×D1084C2211yescharacteristic
16#22576118—21no—
17#22576093—22no—
18#22576092—21no—
19#22576085—21no—
20#22576085—21no—
21C4×C271⋊C8C2211yescharacteristic
22C4⋊Q8×C271C2211yescharacteristic
23C4⋊C8×C271C2211yescharacteristic
24C4⋊C8×C271C2211yescharacteristic
25C2×Dic542—21no—
26C2×Dic542D411yescharacteristic
27C2×D1084D411yescharacteristic
28C2×C4×D271—21no—
29#16864643—42no—
30C2×C271⋊C8—21no—
31C4⋊Dic271—22no—
32C4⋊Dic271—21no—
33C4⋊Dic271—21no—
34C4⋊Dic271D411yescharacteristic
35C4⋊Dic271D411yescharacteristic
36Dic271⋊C4—21no—
37(C2×D271)⋊C4—22no—
38(C2×D271)⋊C4—22no—
39(C2×D271)⋊C4—21no—
40C4×Dic271—22no—
41C4×Dic271—21no—
42C2×Q8×C271—21no—
43C2×C8×C271—21no—
44C2×C8×C271—21no—
45C4⋊C4×C271—21no—
46C4⋊C4×C271D412yes—
47C42×C271C2311yescharacteristic
48C22×D271—21no—
49Dic542—21no—
50Dic542—21no—
51Dic542—41no—
52Dic542—41no—
53D1084—21no—
54D1084—21no—
55C2×Dic271—21no—
56C2×Dic271—21no—
57C2×Dic271—21no—
58C2×Dic271—41no—
59C4×D271—41no—
60C4×D271—41no—
61C271⋊C8—22no—
62Q8×C271—41no—
63Q8×C271—41no—
64C2×C4×C271—22no—
65C2×C4×C271C2×D411yescharacteristic
66C2×C4×C271C2×D411yescharacteristic
67C2×C4×C271C2×D411yescharacteristic
68C8×C271—41no—
69C8×C271—41no—
70C22×C271#12011yescharacteristic
71C2×D271—41no—
72C2×D271—41no—
73C2×D271—41no—
74C4×C271—21no—
75C4×C271—42no—
76C4×C271#13511yescharacteristic
77C4×C271#13511yescharacteristic
78C4×C271AGL(1,ℤ/8)11yescharacteristic
79C4×C271#13711yescharacteristic
80Dic271—41no—
81Dic271—41no—
82Dic271—41no—
83Dic271—81no—
84C2×C271#44911yescharacteristic
85C2×C271#45211yescharacteristic
86C2×C271#45611yescharacteristic
87D271—81no—
88C271#165011yescharacteristic, complemented, Sylow
89#1650—2711nomaximal, Sylow
90#556—2711no—
91#500—2711no—
92#475—2711no—
93#461—2711no—
94#332D27111yescharacteristic, complemented
95Q8⋊C8—2711no—
96D4⋊C8—2711no—
97C2×SD16—5421no—
98C4⋊Q8—2711no—
99C4⋊Q8C2×D27111yescharacteristic
100#126—5422no—
101#124—5421no—
102C22⋊Q8—5421no—
103C4×Q8—2711no—
104C4×Q8—5421no—
105C4×D4—2711no—
106#117—5421no—
107#107—5422no—
108#106—5421no—
109C4⋊C8C2×D27111yescharacteristic
110C4⋊C8C2×D27111yescharacteristic
111#103—5421no—
112#103—5421no—
113C4×C8—2711no—
114C2×Q8—21no—
115C2×Q8—2711no—
116C2×Q8—5421no—
117C2×D4—2711no—
118C22×C4—5421no—
119SD16—10842no—
120C2×C8—21no—
121C2×C8—21no—
122C2×C8—5421no—
123C4⋊C4—21no—
124C4⋊C4—2711no—
125C4⋊C4—2711no—
126C4⋊C4—5422no—
127C4⋊C4—5421no—
128C4⋊C4—5421no—
129C4⋊C4—5421no—
130C4⋊C4D108412yes—
131C22⋊C4—5422no—
132C22⋊C4—5422no—
133C22⋊C4—5421no—
134C42—5422no—
135C42—5421no—
136C42C22×D27111yescharacteristic
137C23—5421no—
138Q8—41no—
139Q8—41no—
140Q8—5421no—
141Q8—5421no—
142Q8—10841no—
143Q8—10841no—
144D4—5421no—
145D4—5421no—
146C2×C4—22no—
147C2×C4—5421no—
148C2×C4—5421no—
149C2×C4—5421no—
150C2×C4—10841no—
151C2×C4—10841no—
152C2×C4—10841no—
153C2×C4D4×D27111yescharacteristic
154C2×C4D4×D27111yescharacteristic
155C2×C4C2×D108411yescharacteristic
156C8—41no—
157C8—41no—
158C8—5422no—
159C22—10841no—
160C22—10841no—
161C22—10841no—
162C22#2257613011yescharacteristic
163C4—21no—
164C4—42no—
165C4—10841no—
166C4—10841no—
167C4—10841no—
168C4—21681no—
169C4#2257618111yescharacteristic
170C4#2257618111yescharacteristic
171C4#2257618411yescharacteristic
172C4#2257619211yescharacteristic
173C2—21681no—
174C2#3177404311yescharacteristic
175C2#3177406211yescharacteristic
176C2#3177408211yescharacteristic
subgroupG/H
centerC22#22576130
commutator, cosocleC42×C271C23
FrattiniC42C22×D271
Fitting#332×C271C2
socleC22×C271#120
prime radicalC2×D1084D4
hypercenter#332D271
hypocenterC271#1650
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. #31774300C2semidirect
3. #31774183C2semidirect
5. #31773627C2semidirect
7. #332×C271C2semidirect
88. C271#1650semidirect
94. #332D271semidirect

Maximal subgroups

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

subgroupindex
#317746212
#317743002
#317741832
#317741032
#317736272
#317736242
#332×C2712
#1650271