Finite Group Databasebeta

← #46835646

Subgroups — #46835646

Order 34688.

240 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 55 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#31774552C211yesmaximal, characteristic
2#31774293C211yesmaximal, characteristic, complemented
3#31774174C211yesmaximal, characteristic, complemented
4#31773642C211yesmaximal, characteristic, complemented
5D4⋊(C271⋊C8)C211yesmaximal, characteristic, complemented
6#31773624C211yesmaximal, characteristic, complemented
7#461×C271C211yesmaximal, characteristic, complemented
8C2×#16864654—21no—
9#22576160C2211yescharacteristic
10#22576142—22no—
11#22576139—21no—
12#22576131—21no—
13C4×D1084C2211yescharacteristic
14C4⋊C4×D271—21no—
15#22576093—21no—
16#22576091—22no—
17C4⋊(C271⋊C8)C2211yescharacteristic
18#22576080—21no—
19C4×C271⋊C8C2211yescharacteristic
20C2×SD16×C271—21no—
21C4⋊Q8×C271C2211yescharacteristic
22C4×D4×C271C2211yescharacteristic
23C4⋊C8×C271C2211yescharacteristic
24#103×C271—21no—
25#16864654—42no—
26#16864654—42no—
27C2×Dic542—21no—
28C2×D1084D411yescharacteristic
29C2×C22⋊D271—22no—
30C2×C4×D271—22no—
31C2×C4×D271—21no—
32C2×C271⋊C8—21no—
33C2×C271⋊C8—21no—
34C4⋊Dic271—21no—
35C4⋊Dic271D411yescharacteristic
36C4⋊Dic271D412yes—
37Dic271⋊C4—22no—
38C22⋊Dic271—22no—
39(C2×D271)⋊C4—21no—
40C2×Q8×C271—21no—
41C2×D4×C271D411yescharacteristic
42C22×C4×C271—21no—
43SD16×C271—42no—
44C2×C8×C271—21no—
45C4⋊C4×C271—21no—
46C4⋊C4×C271—21no—
47C4⋊C4×C271D411yescharacteristic
48C22⋊C4×C271—21no—
49C42×C271C2311yescharacteristic
50C22×D271—21no—
51Dic542—41no—
52Dic542—41no—
53D1084—21no—
54D1084SD1612yescomplemented
55C22⋊D271—44no—
56C2×Dic271—21no—
57C2×Dic271—22no—
58C4×D271—44no—
59C4×D271—41no—
60C4×D271—41no—
61C271⋊C8—22no—
62C271⋊C8—41no—
63C23×C271—21no—
64Q8×C271—41no—
65Q8×C271—41no—
66D4×C271—21no—
67D4×C271—21no—
68C2×C4×C271—21no—
69C2×C4×C271—41no—
70C2×C4×C271—41no—
71C2×C4×C271—41no—
72C2×C4×C271C2×D411yescharacteristic
73C2×C4×C271C2×D411yescharacteristic
74C2×C4×C271C2×D411yescharacteristic
75C8×C271—41no—
76C22×C271—41no—
77C22×C271—41no—
78C22×C271—41no—
79C22×C271#12011yescharacteristic
80C2×D271—22no—
81C2×D271—41no—
82C2×D271—41no—
83C4×C271—21no—
84C4×C271—41no—
85C4×C271—81no—
86C4×C271C2×SD1611yescharacteristic
87C4×C271#13511yescharacteristic
88C4×C271AGL(1,ℤ/8)11yescharacteristic
89C4×C271AGL(1,ℤ/8)11yescharacteristic
90Dic271—41no—
91Dic271—42no—
92C2×C271—81no—
93C2×C271#44911yescharacteristic
94C2×C271#45011yescharacteristic
95C2×C271#45611yescharacteristic
96D271—42no—
97C271#165411yescharacteristic, complemented, Sylow
98#1654—2711nomaximal, Sylow
99#547—2711no—
100#498—2711no—
101#474—2711no—
102#461D27111yescharacteristic, complemented
103#332—2711no—
104D4⋊C8—2711no—
105D4⋊C8—2711no—
106C2×#41—5421no—
107C2×SD16—21no—
108C4⋊Q8—2711no—
109C4⋊Q8C2×D27111yescharacteristic
110#123—5422no—
111C22⋊Q8—5421no—
112#121—5421no—
113C4×D4—2711no—
114C4×D4C2×D27111yescharacteristic
115C2×C4⋊C4—5421no—
116#107—5421no—
117#106—5422no—
118C4⋊C8—2711no—
119C4⋊C8C2×D27111yescharacteristic
120#103—21no—
121#102—5421no—
122C4×C8—2711no—
123#41—10842no—
124#41—10842no—
125C2×Q8—21no—
126C2×Q8—5421no—
127C2×D4—2711no—
128C2×D4—5422no—
129C2×D4C22⋊D27111yescharacteristic
130C22×C4—21no—
131C22×C4—5422no—
132C22×C4—5421no—
133SD16—42no—
134C2×C8—21no—
135C2×C8—5421no—
136C2×C8—5421no—
137C4⋊C4—21no—
138C4⋊C4—21no—
139C4⋊C4—2712no—
140C4⋊C4—2711no—
141C4⋊C4—5422no—
142C4⋊C4—5421no—
143C4⋊C4C22⋊D27111yescharacteristic
144C22⋊C4—21no—
145C22⋊C4—5422no—
146C22⋊C4—5421no—
147C42C22×D27111yescharacteristic
148C23—21no—
149C23—5421no—
150Q8—41no—
151Q8—41no—
152Q8—10841no—
153Q8—10841no—
154D4—21no—
155D4—21no—
156D4—2712no—
157D4—5421no—
158D4—10844no—
159C2×C4—21no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—5421no—
164C2×C4—5422no—
165C2×C4—10844no—
166C2×C4—10841no—
167C2×C4—10841no—
168C2×C4D4×D27111yescharacteristic
169C2×C4D4×D27111yescharacteristic
170C2×C4C2×C22⋊D27111yescharacteristic
171C8—41no—
172C8—5422no—
173C8—10841no—
174C22—41no—
175C22—41no—
176C22—41no—
177C22—5422no—
178C22—10841no—
179C22—10841no—
180C22#2257612811yescharacteristic
181C4—21no—
182C4—41no—
183C4—81no—
184C4—10841no—
185C4—10842no—
186C4SD16×D27111yescharacteristic
187C4#2257618011yescharacteristic
188C4#2257618611yescharacteristic
189C4#2257618711yescharacteristic
190C2—81no—
191C2—10842no—
192C2#3177404011yescharacteristic
193C2#3177404611yescharacteristic
194C2#3177407811yescharacteristic
subgroupG/H
centerC22#22576128
commutator, cosocleC42×C271C23
FrattiniC42C22×D271
Fitting#461×C271C2
socleC22×C271#120
prime radical#31774552C2
hypercenter#461D271
hypocenterC271#1654
LCSG ▷ C42×C271 ▷ C22×C271 ▷ C271
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
2. #31774293C2semidirect
3. #31774174C2semidirect
4. #31773642C2semidirect
5. D4⋊(C271⋊C8)C2semidirect
6. #31773624C2semidirect
7. #461×C271C2semidirect
54. D1084SD16semidirect
97. C271#1654semidirect
102. #461D271semidirect

Maximal subgroups

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

subgroupindex
#317745522
#317742932
#317741742
#317736422
D4⋊(C271⋊C8)2
#317736242
#461×C2712
#1654271