Finite Group Databasebeta

← #46835696

Subgroups — #46835696

Order 34688.

184 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 45 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#31774651C211yesmaximal, characteristic
2#31774265C211yesmaximal, characteristic, complemented
3#31774259C211yesmaximal, characteristic
4#31773639C211yesmaximal, characteristic, complemented
5#31773638C211yesmaximal, characteristic, complemented
6#31773636C211yesmaximal, characteristic
7#501×C271C211yesmaximal, characteristic, complemented
8#22576159—21no—
9#22576155—21no—
10#22576155—21no—
11#22576152C2211yescharacteristic
12#22576148C2211yescharacteristic
13#22576145—21no—
14#22576140—21no—
15#22576138—21no—
16#22576084—21no—
17#22576084—21no—
18#22576083—21no—
19#22576080—21no—
20#22576071C2211yescharacteristic
21#22576071C2211yescharacteristic
22C4⋊Q8×C271C2211yescharacteristic
23#125×C271C2211yescharacteristic
24#107×C271—21no—
25#106×C271—21no—
26#97×C271C2211yescharacteristic
27C2×Dic542—21no—
28C2×Dic542D411yescharacteristic
29C2×D1084D411yescharacteristic
30C2×C4×D271—21no—
31C2×C271⋊C8—21no—
32C2×C271⋊C8—21no—
33C4⋊Dic271—21no—
34C4⋊Dic271D411yescharacteristic
35C4⋊Dic271D411yescharacteristic
36Dic271⋊C4—21no—
37Dic271⋊C4—21no—
38Dic271⋊C4—21no—
39(C2×D271)⋊C4—21no—
40(C2×D271)⋊C4—21no—
41(C2×D271)⋊C4—21no—
42(C2×D271)⋊C4—21no—
43(C2×D271)⋊C4—21no—
44C4×Dic271—21no—
45C2×Q8×C271—21no—
46C2×C8×C271—21no—
47C4⋊C4×C271—21no—
48C4⋊C4×C271—21no—
49C4⋊C4×C271—21no—
50C4⋊C4×C271—21no—
51C4⋊C4×C271D411yescharacteristic
52C4⋊C4×C271D411yescharacteristic
53C42×C271C2311yescharacteristic
54C22×D271—21no—
55Dic542—41no—
56Dic542—81no—
57D1084—41no—
58C2×Dic271—21no—
59C2×Dic271—21no—
60C2×Dic271—21no—
61C4×D271—81no—
62C271⋊C8—41no—
63C271⋊C8—41no—
64Q8×C271—42no—
65C2×C4×C271—21no—
66C2×C4×C271—21no—
67C2×C4×C271—41no—
68C2×C4×C271C2×D411yescharacteristic
69C2×C4×C271C2×D411yescharacteristic
70C2×C4×C271C2×D411yescharacteristic
71C8×C271—22no—
72C22×C271#12011yescharacteristic
73C2×D271—41no—
74C2×D271—41no—
75C2×D271—41no—
76C4×C271—21no—
77C4×C271—21no—
78C4×C271—21no—
79C4×C271—41no—
80C4×C271—41no—
81C4×C271—81no—
82Dic271—41no—
83Dic271—41no—
84Dic271—41no—
85C2×C271#45411yescharacteristic
86C2×C271#45511yescharacteristic
87C2×C271#45611yescharacteristic
88D271—81no—
89C271#166411yescharacteristic, complemented, Sylow
90#1664—2711nomaximal, Sylow
91#563—2711no—
92#501D27111yescharacteristic, complemented
93#491—2711no—
94#489—2711no—
95#330—2711no—
96#330—2711no—
97#329—2711no—
98C4⋊Q8—5421no—
99C4⋊Q8C2×D27111yescharacteristic
100#126—5421no—
101#126—5421no—
102#125—2711no—
103#125C2×D27111yescharacteristic
104#124—2711no—
105#123—5421no—
106C22⋊Q8—5421no—
107C22⋊Q8—5421no—
108#107—21no—
109#106—21no—
110#103—5421no—
111#103—5421no—
112#103—5421no—
113#102—5421no—
114#97—2711no—
115#97—2711no—
116#97C2×D27111yescharacteristic
117C2×Q8—21no—
118C2×Q8—2711no—
119C2×Q8—5421no—
120C2×D4—2711no—
121C22×C4—5421no—
122C2×C8—21no—
123C2×C8—5421no—
124C2×C8—5421no—
125C4⋊C4—21no—
126C4⋊C4—21no—
127C4⋊C4—21no—
128C4⋊C4—21no—
129C4⋊C4—2711no—
130C4⋊C4—2711no—
131C4⋊C4—5421no—
132C4⋊C4—5421no—
133C4⋊C4—5421no—
134C4⋊C4—5421no—
135C4⋊C4C22⋊D27111yescharacteristic
136C4⋊C4C22⋊D27111yescharacteristic
137C22⋊C4—5421no—
138C22⋊C4—5421no—
139C22⋊C4—5421no—
140C22⋊C4—5421no—
141C22⋊C4—5421no—
142C42—5421no—
143C42C22×D27111yescharacteristic
144C23—5421no—
145Q8—42no—
146Q8—10841no—
147Q8—21681no—
148D4—10841no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—41no—
152C2×C4—5421no—
153C2×C4—5421no—
154C2×C4—5421no—
155C2×C4—21681no—
156C2×C4D4×D27111yescharacteristic
157C2×C4D4×D27111yescharacteristic
158C2×C4C2×C22⋊D27111yescharacteristic
159C8—22no—
160C8—10841no—
161C8—10841no—
162C22—10841no—
163C22—10841no—
164C22—10841no—
165C22#2257612811yescharacteristic
166C4—21no—
167C4—21no—
168C4—21no—
169C4—41no—
170C4—41no—
171C4—81no—
172C4—10841no—
173C4—10841no—
174C4—10841no—
175C2—21681no—
176C2#3177406811yescharacteristic
177C2#3177407311yescharacteristic
178C2#3177407911yescharacteristic
subgroupG/H
centerC22#22576128
commutator, cosocleC42×C271C23
FrattiniC42C22×D271
Fitting#501×C271C2
socleC22×C271#120
prime radicalC2×D1084D4
hypercenter#501D271
hypocenterC271#1664
LCSG ▷ C42×C271 ▷ C22×C271 ▷ C271
UCSC1 ◁ C22 ◁ C42 ◁ #501

Complements

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

NH ≅ G/Ntype
2. #31774265C2semidirect
4. #31773639C2semidirect
5. #31773638C2semidirect
7. #501×C271C2semidirect
89. C271#1664semidirect
92. #501D271semidirect

Maximal subgroups

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

subgroupindex
#317746512
#317742652
#317742592
#317736392
#317736382
#317736362
#501×C2712
#1664271