Finite Group Databasebeta

← #45663653

Subgroups — #45663653

Order 32896.

184 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 45 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#30657487C211yesmaximal, characteristic
2#30657143C211yesmaximal, characteristic, complemented
3#30657105C211yesmaximal, characteristic, complemented
4#30656478C211yesmaximal, characteristic, complemented
5#30656478C211yesmaximal, characteristic, complemented
6#30656477C211yesmaximal, characteristic
7#489×C257C211yesmaximal, characteristic
8#22551590—21no—
9#22551585—21no—
10#22551585—21no—
11#22551584C2211yescharacteristic
12#22551584C2211yescharacteristic
13#22551573—21no—
14#22551569—21no—
15#22551569—21no—
16#22551525—21no—
17#22551523—21no—
18#22551516—21no—
19#22551516—21no—
20#22551503C2211yescharacteristic
21#22551503C2211yescharacteristic
22C4⋊Q8×C257C2211yescharacteristic
23#124×C257C2211yescharacteristic
24#103×C257—21no—
25#102×C257—21no—
26#97×C257C2211yescharacteristic
27C2×Dic514—21no—
28C22×Dic257—21no—
29C2×#14523339—21no—
30C2×#14523339—21no—
31C4⋊Dic257D411yescharacteristic
32C4⋊Dic257D411yescharacteristic
33C4⋊Dic257D411yescharacteristic
34C4⋊Dic257D411yescharacteristic
35#16861597—21no—
36#16861597—21no—
37#16861597—21no—
38#16861597—21no—
39#16861597—21no—
40#16861597—21no—
41C22⋊Dic257—21no—
42C22⋊Dic257—21no—
43C22⋊Dic257—21no—
44C4×Dic257—21no—
45C2×Q8×C257—21no—
46C2×Q8×C257D411yescharacteristic
47C2×D4×C257D411yescharacteristic
48C2×C8×C257—21no—
49C4⋊C4×C257—21no—
50C4⋊C4×C257—21no—
51C22⋊C4×C257—21no—
52C22⋊C4×C257—21no—
53C42×C257C2311yescharacteristic
54Dic514—81no—
55C2×Dic257—21no—
56C2×Dic257—21no—
57C2×Dic257—21no—
58C2×Dic257—21no—
59C2×Dic257—81no—
60#14523339—22no—
61#14523339—41no—
62C23×C257—21no—
63Q8×C257—41no—
64Q8×C257—42no—
65D4×C257—41no—
66C2×C4×C257—21no—
67C2×C4×C257—41no—
68C2×C4×C257C2×D411yescharacteristic
69C2×C4×C257C2×D411yescharacteristic
70C2×C4×C257C2×D411yescharacteristic
71C8×C257—41no—
72C22×C257—41no—
73C22×C257—41no—
74C22×C257—41no—
75C22×C257#12011yescharacteristic
76C4×C257—21no—
77C4×C257—21no—
78C4×C257—21no—
79C4×C257—41no—
80C4×C257—81no—
81Dic257—41no—
82Dic257—41no—
83Dic257—41no—
84Dic257—41no—
85C2×C257—81no—
86C2×C257#45411yescharacteristic
87C2×C257#45511yescharacteristic
88C2×C257#45611yescharacteristic
89C257#166411yescharacteristic, complemented, Sylow
90#1664—2571nomaximal, Sylow
91#563—2571no—
92#501—2571no—
93#491—2571no—
94#489D25711yescharacteristic
95#330—2571no—
96#330—2571no—
97#329—2571no—
98C4⋊Q8—5141no—
99C4⋊Q8C2×D25711yescharacteristic
100#126—5141no—
101#126—5141no—
102#125—2571no—
103#125—2571no—
104#124C2×D25711yescharacteristic
105#123—5141no—
106C22⋊Q8—5141no—
107C22⋊Q8—5141no—
108#107—5141no—
109#106—5141no—
110#103—21no—
111#103—5141no—
112#103—5141no—
113#102—21no—
114#97—2571no—
115#97—2571no—
116#97C2×D25711yescharacteristic
117C2×Q8—21no—
118C2×Q8—5141no—
119C2×Q8C22⋊D25711yescharacteristic
120C2×D4C22⋊D25711yescharacteristic
121C22×C4—5141no—
122C2×C8—21no—
123C2×C8—5141no—
124C2×C8—5141no—
125C4⋊C4—2571no—
126C4⋊C4—2571no—
127C4⋊C4—2571no—
128C4⋊C4—2571no—
129C4⋊C4—21no—
130C4⋊C4—21no—
131C4⋊C4—5141no—
132C4⋊C4—5141no—
133C4⋊C4—5141no—
134C4⋊C4—5141no—
135C4⋊C4—5141no—
136C4⋊C4—5141no—
137C22⋊C4—21no—
138C22⋊C4—21no—
139C22⋊C4—5141no—
140C22⋊C4—5141no—
141C22⋊C4—5141no—
142C42—5141no—
143C42C22×D25711yescharacteristic
144C23—21no—
145Q8—41no—
146Q8—42no—
147Q8—20561no—
148D4—41no—
149C2×C4—21no—
150C2×C4—5141no—
151C2×C4—5141no—
152C2×C4—5141no—
153C2×C4—5141no—
154C2×C4—41no—
155C2×C4—20561no—
156C2×C4D4×D25711yescharacteristic
157C2×C4D4×D25711yescharacteristic
158C2×C4C2×C22⋊D25711yescharacteristic
159C8—5142no—
160C8—41no—
161C8—10281no—
162C22—41no—
163C22—41no—
164C22—41no—
165C22#2255156011yescharacteristic
166C4—21no—
167C4—21no—
168C4—21no—
169C4—41no—
170C4—10281no—
171C4—10281no—
172C4—10281no—
173C4—10281no—
174C4—81no—
175C2—81no—
176C2#3065690811yescharacteristic
177C2#3065691411yescharacteristic
178C2#3065691811yescharacteristic
subgroupG/H
centerC22#22551560
commutator, cosocleC42×C257C23
FrattiniC42C22×D257
Fitting#489×C257C2
socleC22×C257#120
prime radicalC2×D4×C257D4
hypercenter#489D257
hypocenterC257#1664
LCSG ▷ C42×C257 ▷ C22×C257 ▷ C257
UCSC1 ◁ C22 ◁ C42 ◁ #489

Complements

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

NH ≅ G/Ntype
2. #30657143C2semidirect
3. #30657105C2semidirect
4. #30656478C2semidirect
5. #30656478C2semidirect
89. C257#1664semidirect

Maximal subgroups

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

subgroupindex
#306574872
#306571432
#306571052
#306564782
#306564772
#489×C2572
#1664257