Finite Group Databasebeta

← #46835614

Subgroups — #46835614

Order 34688.

234 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 57 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#31774562C211yesmaximal, characteristic
2#31774300C211yesmaximal, characteristic, complemented
3#31774178C211yesmaximal, characteristic
4#31774108C211yesmaximal, characteristic, complemented
5#31773627C211yesmaximal, characteristic, complemented
6#31773624C211yesmaximal, characteristic
7#332×C271C211yesmaximal, characteristic, complemented
8C2×Q8×D271—21no—
9C2×Dic1084—21no—
10#22576160C2211yescharacteristic
11#22576159—21no—
12#22576138—22no—
13#22576136—21no—
14C4×Dic542C2211yescharacteristic
15C4×D1084C2211yescharacteristic
16#22576122—21no—
17#22576094—21no—
18#22576093—22no—
19#22576085—21no—
20#22576082—21no—
21C4×C271⋊C8C2211yescharacteristic
22C4⋊Q8×C271C2211yescharacteristic
23C4⋊C8×C271C2211yescharacteristic
24C4⋊C8×C271C2211yescharacteristic
25C2×Dic542—22no—
26C2×Dic542—21no—
27C2×Dic542D411yescharacteristic
28Q8×D271—42no—
29Q8×D271—42no—
30C2×D1084D411yescharacteristic
31C2×C4×D271—22no—
32C2×C4×D271—21no—
33Dic1084—42no—
34C2×C271⋊C8—21no—
35C4⋊Dic271—21no—
36C4⋊Dic271—21no—
37C4⋊Dic271D411yescharacteristic
38C4⋊Dic271D411yescharacteristic
39Dic271⋊C4—22no—
40Dic271⋊C4—21no—
41(C2×D271)⋊C4—22no—
42(C2×D271)⋊C4—21no—
43C4×Dic271—21no—
44C2×Q8×C271—21no—
45C2×C8×C271—21no—
46C2×C8×C271—21no—
47C4⋊C4×C271—21no—
48C4⋊C4×C271D412yes—
49C42×C271C2311yescharacteristic
50C22×D271—21no—
51Dic542—21no—
52Dic542—44no—
53Dic542—41no—
54Dic542—41no—
55Dic542D812yes—
56D1084—21no—
57D1084Q1612yescomplemented
58C2×Dic271—21no—
59C2×Dic271—21no—
60C2×Dic271—21no—
61C2×Dic271—41no—
62C4×D271—44no—
63C4×D271—41no—
64C4×D271—41no—
65C271⋊C8—22no—
66Q8×C271—41no—
67Q8×C271—41no—
68C2×C4×C271—22no—
69C2×C4×C271C2×D411yescharacteristic
70C2×C4×C271C2×D411yescharacteristic
71C2×C4×C271C2×D411yescharacteristic
72C8×C271—41no—
73C8×C271—41no—
74C22×C271#12011yescharacteristic
75C2×D271—22no—
76C2×D271—41no—
77C2×D271—41no—
78C4×C271—21no—
79C4×C271—42no—
80C4×C271C2×D811yescharacteristic
81C4×C271C2×Q1611yescharacteristic
82C4×C271AGL(1,ℤ/8)11yescharacteristic
83C4×C271#13711yescharacteristic
84Dic271—22no—
85Dic271—41no—
86Dic271—41no—
87Dic271—81no—
88C2×C271#44711yescharacteristic
89C2×C271#45111yescharacteristic
90C2×C271#45611yescharacteristic
91D271—42no—
92C271#164911yescharacteristic, complemented, Sylow
93#1649—2711nomaximal, Sylow
94D4×Q8—2711no—
95#500—2711no—
96#474—2711no—
97#462—2711no—
98#332D27111yescharacteristic, complemented
99Q8⋊C8—2711no—
100D4⋊C8—2711no—
101C22×Q8—5421no—
102C2×Q16—5421no—
103C4⋊Q8—2711no—
104C4⋊Q8—5421no—
105C4⋊Q8C2×D27111yescharacteristic
106C22⋊Q8—5421no—
107C22⋊Q8—5422no—
108C4×Q8—2711no—
109C4×D4—2711no—
110C4×D4—5421no—
111#107—5421no—
112#107—5422no—
113C4⋊C8C2×D27111yescharacteristic
114C4⋊C8C2×D27111yescharacteristic
115#103—5421no—
116#102—5421no—
117C4×C8—2711no—
118C2×Q8—21no—
119C2×Q8—2711no—
120C2×Q8—5422no—
121C2×Q8—5421no—
122C2×Q8—10842no—
123C2×Q8—10842no—
124C2×D4—2711no—
125C22×C4—5422no—
126C22×C4—5421no—
127Q16—10842no—
128C2×C8—21no—
129C2×C8—21no—
130C2×C8—5421no—
131C4⋊C4—21no—
132C4⋊C4—2711no—
133C4⋊C4—2711no—
134C4⋊C4—5422no—
135C4⋊C4—5421no—
136C4⋊C4—5421no—
137C4⋊C4—5421no—
138C4⋊C4D108412yes—
139C22⋊C4—5422no—
140C22⋊C4—5421no—
141C42—5421no—
142C42C22×D27111yescharacteristic
143C23—5421no—
144Q8—41no—
145Q8—41no—
146Q8—2712no—
147Q8—5421no—
148Q8—10844no—
149Q8—10841no—
150Q8—10841no—
151D4—2712no—
152D4—5421no—
153C2×C4—22no—
154C2×C4—5421no—
155C2×C4—5421no—
156C2×C4—5421no—
157C2×C4—10844no—
158C2×C4—10841no—
159C2×C4—10841no—
160C2×C4—10841no—
161C2×C4D4×D27111yescharacteristic
162C2×C4D4×D27111yescharacteristic
163C2×C4C2×D108411yescharacteristic
164C8—41no—
165C8—41no—
166C8—5422no—
167C22—5422no—
168C22—10841no—
169C22—10841no—
170C22#2257613011yescharacteristic
171C4—21no—
172C4—42no—
173C4—5422no—
174C4—10841no—
175C4—10841no—
176C4—21681no—
177C4D8×D27111yescharacteristic
178C4Q16×D27111yescharacteristic
179C4#2257618711yescharacteristic
180C4#2257618911yescharacteristic
181C2—10842no—
182C2#3177403211yescharacteristic
183C2#3177405411yescharacteristic
184C2#3177408211yescharacteristic
subgroupG/H
centerC22#22576130
commutator, cosocleC42×C271C23
FrattiniC42C22×D271
Fitting#332×C271C2
socleC22×C271#120
prime radicalC2×D1084D4
hypercenter#332D271
hypocenterC271#1649
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
4. #31774108C2semidirect
5. #31773627C2semidirect
7. #332×C271C2semidirect
57. D1084Q16semidirect
92. C271#1649semidirect
98. #332D271semidirect

Maximal subgroups

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

subgroupindex
#317745622
#317743002
#317741782
#317741082
#317736272
#317736242
#332×C2712
#1649271