Finite Group Databasebeta

← #46835515

Subgroups — #46835515

Order 34688.

234 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 61 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#31774249C211yesmaximal, characteristic, complemented
2#31774187C211yesmaximal, characteristic, complemented
3#31774187C211yesmaximal, characteristic, complemented
4#31773643C211yesmaximal, characteristic, complemented
5D4⋊(C271⋊C8)C211yesmaximal, characteristic, complemented
6D4⋊(C271⋊C8)C211yesmaximal, characteristic, complemented
7#547×C271C211yesmaximal, characteristic
8#22576160C2211yescharacteristic
9#22576091—21no—
10#22576091—21no—
11C4⋊(C271⋊C8)C2211yescharacteristic
12C4⋊(C271⋊C8)C2211yescharacteristic
13#22576083—22no—
14#22576081—21no—
15#22576081—21no—
16C4×C271⋊C8C2211yescharacteristic
17C2×#41×C271—21no—
18C4⋊Q8×C271C2211yescharacteristic
19#123×C271—22no—
20C22⋊Q8×C271—21no—
21#121×C271—21no—
22C4×D4×C271C2211yescharacteristic
23C4×D4×C271C2211yescharacteristic
24C2×C4⋊C4×C271—21no—
25C2×Dic542—21no—
26C2×C271⋊C8—21no—
27C2×C271⋊C8—21no—
28C2×C271⋊C8—21no—
29C4⋊Dic271—21no—
30C4⋊Dic271—21no—
31#41×C271—42no—
32#41×C271—42no—
33C2×Q8×C271—21no—
34C2×D4×C271—22no—
35C2×D4×C271D411yescharacteristic
36C2×D4×C271D411yescharacteristic
37C22×C4×C271—22no—
38C22×C4×C271—21no—
39C22×C4×C271—21no—
40C4⋊C4×C271—22no—
41C4⋊C4×C271—21no—
42C4⋊C4×C271D411yescharacteristic
43C4⋊C4×C271D411yescharacteristic
44C4⋊C4×C271D412yes—
45C22⋊C4×C271—22no—
46C22⋊C4×C271—21no—
47C22⋊C4×C271—21no—
48C42×C271C2311yescharacteristic
49Dic542—42no—
50C2×Dic271—41no—
51C271⋊C8—41no—
52C271⋊C8—41no—
53C271⋊C8—41no—
54C23×C271—21no—
55C23×C271—21no—
56Q8×C271—41no—
57Q8×C271—41no—
58D4×C271—21no—
59D4×C271—21no—
60D4×C271—21no—
61D4×C271—44no—
62D4×C271SD1612yes—
63C2×C4×C271—21no—
64C2×C4×C271—21no—
65C2×C4×C271—22no—
66C2×C4×C271—44no—
67C2×C4×C271—41no—
68C2×C4×C271—41no—
69C2×C4×C271—41no—
70C2×C4×C271—41no—
71C2×C4×C271C2×D411yescharacteristic
72C2×C4×C271C2×D411yescharacteristic
73C2×C4×C271C2×D411yescharacteristic
74C22×C271—22no—
75C22×C271—41no—
76C22×C271—41no—
77C22×C271—41no—
78C22×C271—41no—
79C22×C271—41no—
80C22×C271#12011yescharacteristic
81C4×C271—21no—
82C4×C271—41no—
83C4×C271—42no—
84C4×C271—41no—
85C4×C271C2×SD1611yescharacteristic
86C4×C271#13511yescharacteristic
87C4×C271AGL(1,ℤ/8)11yescharacteristic
88C4×C271#13711yescharacteristic
89Dic271—81no—
90C2×C271—42no—
91C2×C271—81no—
92C2×C271#45011yescharacteristic
93C2×C271#45211yescharacteristic
94C2×C271#45411yescharacteristic
95C271#163511yescharacteristic, complemented, Sylow
96#1635—2711nomaximal, Sylow
97#547D27111yescharacteristic
98#487—2711no—
99#476—2711no—
100#476—2711no—
101#332—2711no—
102D4⋊C8—2711no—
103D4⋊C8—2711no—
104C2×#41—21no—
105C4⋊Q8—2711no—
106C4⋊Q8C2×D27111yescharacteristic
107#123—22no—
108C22⋊Q8—21no—
109#121—21no—
110C4×D4C2×D27111yescharacteristic
111C4×D4C2×D27111yescharacteristic
112C2×C4⋊C4—21no—
113#106—5421no—
114#106—5421no—
115C4⋊C8—2711no—
116C4⋊C8—2711no—
117#103—5422no—
118#102—5421no—
119#102—5421no—
120C4×C8—2711no—
121#41—42no—
122#41—42no—
123C2×Q8—21no—
124C2×Q8—5421no—
125C2×D4—22no—
126C2×D4C22⋊D27111yescharacteristic
127C2×D4C22⋊D27111yescharacteristic
128C22×C4—22no—
129C22×C4—21no—
130C22×C4—21no—
131C2×C8—5421no—
132C2×C8—5421no—
133C2×C8—5421no—
134C4⋊C4—22no—
135C4⋊C4—21no—
136C4⋊C4—5421no—
137C4⋊C4—5421no—
138C4⋊C4C22⋊D27112yes—
139C4⋊C4C22⋊D27111yescharacteristic
140C4⋊C4C22⋊D27111yescharacteristic
141C22⋊C4—22no—
142C22⋊C4—21no—
143C22⋊C4—21no—
144C42C22×D27111yescharacteristic
145C23—21no—
146C23—21no—
147Q8—41no—
148Q8—41no—
149Q8—10842no—
150D4—21no—
151D4—21no—
152D4—21no—
153D4—44no—
154D4#1686464212yes—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—22no—
158C2×C4—44no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—10841no—
164C2×C4C2×C22⋊D27111yescharacteristic
165C2×C4C2×C22⋊D27111yescharacteristic
166C2×C4C2×C22⋊D27111yescharacteristic
167C8—10841no—
168C8—10841no—
169C8—10841no—
170C22—22no—
171C22—41no—
172C22—41no—
173C22—41no—
174C22—41no—
175C22—41no—
176C22#2257612911yescharacteristic
177C4—21no—
178C4—41no—
179C4—42no—
180C4—41no—
181C4—21681no—
182C4C2×#1686464211yescharacteristic
183C4#2257618011yescharacteristic
184C4#2257618611yescharacteristic
185C4#2257619011yescharacteristic
186C2—42no—
187C2—81no—
188C2#3177404811yescharacteristic
189C2#3177405911yescharacteristic
190C2#3177407011yescharacteristic
subgroupG/H
centerC22#22576129
commutator, cosocleC42×C271C23
FrattiniC42C22×D271
Fitting, prime radical#547×C271C2
socleC22×C271#120
hypercenter#547D271
hypocenterC271#1635
LCSG ▷ C42×C271 ▷ C22×C271 ▷ C271
UCSC1 ◁ C22 ◁ C42 ◁ #547

Complements

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

NH ≅ G/Ntype
1. #31774249C2semidirect
2. #31774187C2semidirect
3. #31774187C2semidirect
4. #31773643C2semidirect
5. D4⋊(C271⋊C8)C2semidirect
6. D4⋊(C271⋊C8)C2semidirect
95. C271#1635semidirect

Maximal subgroups

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

subgroupindex
#317742492
#317741872
#317736432
D4⋊(C271⋊C8)2
#547×C2712
#1635271