Finite Group Databasebeta

← #44549452

Subgroups — #44549452

Order 32128.

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#30632341C211yesmaximal, characteristic, complemented
2#30632033C211yesmaximal, characteristic, complemented
3#30631879C211yesmaximal, characteristic, complemented
4#30631434C211yesmaximal, characteristic, complemented
5D4⋊(C251⋊C8)C211yesmaximal, characteristic, complemented
6#30631414C211yesmaximal, characteristic, complemented
7#459×C251C211yesmaximal, characteristic, complemented
8C2×#16860809—21no—
9C2×#16860795—21no—
10#22548715C2211yescharacteristic
11#22548699—22no—
12#22548695—21no—
13#22548689—21no—
14C4×D1004C2211yescharacteristic
15C2×C4⋊Dic251—21no—
16C4⋊(C251⋊C8)C2211yescharacteristic
17#22548636—22no—
18#22548636—21no—
19C4×C251⋊C8C2211yescharacteristic
20C2×D8×C251—21no—
21#127×C251C2211yescharacteristic
22C4×D4×C251C2211yescharacteristic
23C4⋊C8×C251C2211yescharacteristic
24#102×C251—21no—
25#16860809—42no—
26#16860809—42no—
27C2×Dic502—21no—
28C2×D1004D411yescharacteristic
29C2×C22⋊D251—22no—
30C22×Dic251—22no—
31C2×C4×D251—21no—
32#16860795—42no—
33C2×C251⋊C8—21no—
34C2×C251⋊C8—21no—
35C4⋊Dic251—22no—
36C4⋊Dic251—21no—
37C4⋊Dic251D411yescharacteristic
38C4⋊Dic251D412yes—
39(C2×D251)⋊C4—22no—
40(C2×D251)⋊C4—21no—
41C2×D4×C251—21no—
42C2×D4×C251—21no—
43C2×D4×C251—21no—
44C2×D4×C251D411yescharacteristic
45C22×C4×C251—21no—
46D8×C251—42no—
47C2×C8×C251—21no—
48C4⋊C4×C251D411yescharacteristic
49C22⋊C4×C251—21no—
50C42×C251C2311yescharacteristic
51C22×D251—21no—
52Dic502—41no—
53Dic502—41no—
54D1004—21no—
55D1004—21no—
56C22⋊D251—44no—
57C2×Dic251—21no—
58C2×Dic251—22no—
59C2×Dic251—44no—
60C4×D251—41no—
61C4×D251—41no—
62C251⋊C8—41no—
63C251⋊C8—41no—
64C23×C251—21no—
65C23×C251—41no—
66D4×C251—21no—
67D4×C251—42no—
68D4×C251—41no—
69D4×C251—41no—
70D4×C251—41no—
71D4×C251—41no—
72D4×C251D812yescomplemented
73C2×C4×C251—21no—
74C2×C4×C251—41no—
75C2×C4×C251—41no—
76C2×C4×C251C2×D411yescharacteristic
77C2×C4×C251C2×D411yescharacteristic
78C2×C4×C251C2×D411yescharacteristic
79C8×C251—41no—
80C22×C251—22no—
81C22×C251—41no—
82C22×C251—41no—
83C22×C251—81no—
84C22×C251—81no—
85C22×C251—81no—
86C22×C251#12011yescharacteristic
87C2×D251—41no—
88C2×D251—41no—
89C2×D251—41no—
90C4×C251—21no—
91C4×C251—41no—
92C4×C251C2×D811yescharacteristic
93C4×C251#13511yescharacteristic
94C4×C251AGL(1,ℤ/8)11yescharacteristic
95C4×C251AGL(1,ℤ/8)11yescharacteristic
96Dic251—41no—
97Dic251—42no—
98C2×C251—42no—
99C2×C251—161no—
100C2×C251#44711yescharacteristic
101C2×C251#44911yescharacteristic
102C2×C251#45411yescharacteristic
103D251—81no—
104C251#162511yescharacteristic, complemented, Sylow
105#1625—2511nomaximal, Sylow
106#547—2511no—
107#486—2511no—
108#459—2511no—
109#459D25111yescharacteristic, complemented
110#332—2511no—
111D4⋊C8—2511no—
112D4⋊C8—2511no—
113C2×#41—5021no—
114C2×D8—21no—
115C2×D8—5021no—
116C4⋊Q8—2511no—
117#127C2×D25111yescharacteristic
118#123—5022no—
119C22⋊Q8—5021no—
120#121—5021no—
121C4×D4—2511no—
122C4×D4C2×D25111yescharacteristic
123C2×C4⋊C4—5021no—
124C4⋊C8—2511no—
125C4⋊C8C2×D25111yescharacteristic
126#102—21no—
127#102—5021no—
128#102—5022no—
129C4×C8—2511no—
130#41—10042no—
131#41—10042no—
132C2×Q8—5021no—
133C2×D4—21no—
134C2×D4—21no—
135C2×D4—21no—
136C2×D4—5022no—
137C2×D4—2511no—
138C2×D4C22⋊D25111yescharacteristic
139C22×C4—21no—
140C22×C4—5022no—
141C22×C4—5021no—
142D8—42no—
143D8—10042no—
144C2×C8—21no—
145C2×C8—5021no—
146C2×C8—5021no—
147C4⋊C4—5022no—
148C4⋊C4—5021no—
149C4⋊C4—2511no—
150C4⋊C4—2512no—
151C4⋊C4C22⋊D25111yescharacteristic
152C22⋊C4—21no—
153C22⋊C4—5022no—
154C22⋊C4—5021no—
155C42C22×D25111yescharacteristic
156C23—21no—
157C23—41no—
158C23—5021no—
159Q8—10041no—
160Q8—10041no—
161D4—21no—
162D4—42no—
163D4—41no—
164D4—41no—
165D4—41no—
166D4—41no—
167D4—10044no—
168D4—5021no—
169D4—5021no—
170D4#1686079512yescomplemented
171C2×C4—21no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—10044no—
175C2×C4—10041no—
176C2×C4—10041no—
177C2×C4—5021no—
178C2×C4—5022no—
179C2×C4D4×D25111yescharacteristic
180C2×C4D4×D25111yescharacteristic
181C2×C4C2×C22⋊D25111yescharacteristic
182C8—41no—
183C8—10041no—
184C8—10041no—
185C22—22no—
186C22—41no—
187C22—41no—
188C22—81no—
189C22—81no—
190C22—81no—
191C22—10041no—
192C22—10041no—
193C22—10041no—
194C22#2254868311yescharacteristic
195C4—21no—
196C4—41no—
197C4—10041no—
198C4—10042no—
199C4C2×#1686079511yescharacteristic
200C4#2254873411yescharacteristic
201C4#2254873911yescharacteristic
202C4#2254874011yescharacteristic
203C2—42no—
204C2—161no—
205C2—20081no—
206C2#3063182011yescharacteristic
207C2#3063182911yescharacteristic
208C2#3063185811yescharacteristic
subgroupG/H
centerC22#22548683
commutator, cosocleC42×C251C23
FrattiniC42C22×D251
Fitting#459×C251C2
socleC22×C251#120
hypercenter#459D251
hypocenterC251#1625
LCSG ▷ C42×C251 ▷ C22×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. #30632341C2semidirect
2. #30632033C2semidirect
3. #30631879C2semidirect
4. #30631434C2semidirect
5. D4⋊(C251⋊C8)C2semidirect
6. #30631414C2semidirect
7. #459×C251C2semidirect
72. D4×C251D8semidirect
104. C251#1625semidirect
109. #459D251semidirect
170. D4#16860795semidirect

Maximal subgroups

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

subgroupindex
#306323412
#306320332
#306318792
#306314342
D4⋊(C251⋊C8)2
#306314142
#459×C2512
#1625251