Finite Group Databasebeta

← #46835451

Subgroups — #46835451

Order 34688.

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

normal subgroups only

This group has 256 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#31774551C211yesmaximal, characteristic, complemented
2#31774243C211yesmaximal, characteristic, complemented
3#31774089C211yesmaximal, characteristic, complemented
4#31773644C211yesmaximal, characteristic, complemented
5D4⋊(C271⋊C8)C211yesmaximal, characteristic, complemented
6#31773624C211yesmaximal, characteristic, complemented
7#459×C271C211yesmaximal, characteristic, complemented
8C2×#16864653—21no—
9C2×#16864639—21no—
10#22576160C2211yescharacteristic
11#22576144—22no—
12#22576140—21no—
13#22576134—21no—
14C4×D1084C2211yescharacteristic
15C2×C4⋊Dic271—21no—
16C4⋊(C271⋊C8)C2211yescharacteristic
17#22576081—22no—
18#22576081—21no—
19C4×C271⋊C8C2211yescharacteristic
20C2×D8×C271—21no—
21#127×C271C2211yescharacteristic
22C4×D4×C271C2211yescharacteristic
23C4⋊C8×C271C2211yescharacteristic
24#102×C271—21no—
25#16864653—42no—
26#16864653—42no—
27C2×Dic542—21no—
28C2×D1084D411yescharacteristic
29C2×C22⋊D271—22no—
30C22×Dic271—22no—
31C2×C4×D271—21no—
32#16864639—42no—
33C2×C271⋊C8—21no—
34C2×C271⋊C8—21no—
35C4⋊Dic271—22no—
36C4⋊Dic271—21no—
37C4⋊Dic271D412yes—
38C4⋊Dic271D411yescharacteristic
39(C2×D271)⋊C4—22no—
40(C2×D271)⋊C4—21no—
41C2×D4×C271—21no—
42C2×D4×C271—21no—
43C2×D4×C271—21no—
44C2×D4×C271D411yescharacteristic
45C22×C4×C271—21no—
46D8×C271—42no—
47C2×C8×C271—21no—
48C4⋊C4×C271D411yescharacteristic
49C22⋊C4×C271—21no—
50C42×C271C2311yescharacteristic
51C22×D271—21no—
52Dic542—41no—
53Dic542—41no—
54D1084—21no—
55D1084—21no—
56C22⋊D271—44no—
57C2×Dic271—21no—
58C2×Dic271—22no—
59C2×Dic271—44no—
60C4×D271—41no—
61C4×D271—41no—
62C271⋊C8—41no—
63C271⋊C8—41no—
64C23×C271—21no—
65C23×C271—41no—
66D4×C271—21no—
67D4×C271—42no—
68D4×C271—41no—
69D4×C271—41no—
70D4×C271—41no—
71D4×C271—41no—
72D4×C271D812yescomplemented
73C2×C4×C271—21no—
74C2×C4×C271—41no—
75C2×C4×C271—41no—
76C2×C4×C271C2×D411yescharacteristic
77C2×C4×C271C2×D411yescharacteristic
78C2×C4×C271C2×D411yescharacteristic
79C8×C271—41no—
80C22×C271—22no—
81C22×C271—41no—
82C22×C271—41no—
83C22×C271—81no—
84C22×C271—81no—
85C22×C271—81no—
86C22×C271#12011yescharacteristic
87C2×D271—41no—
88C2×D271—41no—
89C2×D271—41no—
90C4×C271—21no—
91C4×C271—41no—
92C4×C271C2×D811yescharacteristic
93C4×C271#13511yescharacteristic
94C4×C271AGL(1,ℤ/8)11yescharacteristic
95C4×C271AGL(1,ℤ/8)11yescharacteristic
96Dic271—41no—
97Dic271—42no—
98C2×C271—42no—
99C2×C271—161no—
100C2×C271#44711yescharacteristic
101C2×C271#44911yescharacteristic
102C2×C271#45411yescharacteristic
103D271—81no—
104C271#162511yescharacteristic, complemented, Sylow
105#1625—2711nomaximal, Sylow
106#547—2711no—
107#486—2711no—
108#459—2711no—
109#459D27111yescharacteristic, complemented
110#332—2711no—
111D4⋊C8—2711no—
112D4⋊C8—2711no—
113C2×#41—5421no—
114C2×D8—21no—
115C2×D8—5421no—
116C4⋊Q8—2711no—
117#127C2×D27111yescharacteristic
118#123—5422no—
119C22⋊Q8—5421no—
120#121—5421no—
121C4×D4—2711no—
122C4×D4C2×D27111yescharacteristic
123C2×C4⋊C4—5421no—
124C4⋊C8—2711no—
125C4⋊C8C2×D27111yescharacteristic
126#102—21no—
127#102—5421no—
128#102—5422no—
129C4×C8—2711no—
130#41—10842no—
131#41—10842no—
132C2×Q8—5421no—
133C2×D4—21no—
134C2×D4—21no—
135C2×D4—21no—
136C2×D4—2711no—
137C2×D4—5422no—
138C2×D4C22⋊D27111yescharacteristic
139C22×C4—21no—
140C22×C4—5422no—
141C22×C4—5421no—
142D8—42no—
143D8—10842no—
144C2×C8—21no—
145C2×C8—5421no—
146C2×C8—5421no—
147C4⋊C4—2711no—
148C4⋊C4—2712no—
149C4⋊C4—5422no—
150C4⋊C4—5421no—
151C4⋊C4C22⋊D27111yescharacteristic
152C22⋊C4—21no—
153C22⋊C4—5422no—
154C22⋊C4—5421no—
155C42C22×D27111yescharacteristic
156C23—21no—
157C23—41no—
158C23—5421no—
159Q8—10841no—
160Q8—10841no—
161D4—21no—
162D4—42no—
163D4—41no—
164D4—41no—
165D4—41no—
166D4—41no—
167D4—5421no—
168D4—5421no—
169D4—10844no—
170D4#1686463912yescomplemented
171C2×C4—21no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—5421no—
175C2×C4—5422no—
176C2×C4—10844no—
177C2×C4—10841no—
178C2×C4—10841no—
179C2×C4D4×D27111yescharacteristic
180C2×C4D4×D27111yescharacteristic
181C2×C4C2×C22⋊D27111yescharacteristic
182C8—41no—
183C8—10841no—
184C8—10841no—
185C22—22no—
186C22—41no—
187C22—41no—
188C22—81no—
189C22—81no—
190C22—81no—
191C22—10841no—
192C22—10841no—
193C22—10841no—
194C22#2257612811yescharacteristic
195C4—21no—
196C4—41no—
197C4—10841no—
198C4—10842no—
199C4C2×#1686463911yescharacteristic
200C4#2257617911yescharacteristic
201C4#2257618411yescharacteristic
202C4#2257618511yescharacteristic
203C2—42no—
204C2—161no—
205C2—21681no—
206C2#3177403011yescharacteristic
207C2#3177403911yescharacteristic
208C2#3177406811yescharacteristic
subgroupG/H
centerC22#22576128
commutator, cosocleC42×C271C23
FrattiniC42C22×D271
Fitting#459×C271C2
socleC22×C271#120
hypercenter#459D271
hypocenterC271#1625
LCSG ▷ C42×C271 ▷ C22×C271 ▷ C271
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. #31774551C2semidirect
2. #31774243C2semidirect
3. #31774089C2semidirect
4. #31773644C2semidirect
5. D4⋊(C271⋊C8)C2semidirect
6. #31773624C2semidirect
7. #459×C271C2semidirect
72. D4×C271D8semidirect
104. C271#1625semidirect
109. #459D271semidirect
170. D4#16864639semidirect

Maximal subgroups

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

subgroupindex
#317745512
#317742432
#317740892
#317736442
D4⋊(C271⋊C8)2
#317736242
#459×C2712
#1625271