Finite Group Databasebeta

← #45663455

Subgroups — #45663455

Order 32896.

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

normal subgroups only

This group has 214 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#30657448C211yesmaximal, characteristic
2#30657116C211yesmaximal, characteristic, complemented
3#30657033C211yesmaximal, characteristic, complemented
4#30657028C211yesmaximal, characteristic
5#30656467C211yesmaximal, characteristic, complemented
6#30656464C211yesmaximal, characteristic
7#333×C257C211yesmaximal, characteristic, complemented
8C2×#16861608—22no—
9#22551592C2211yescharacteristic
10#22551583—21no—
11#22551572—21no—
12#22551568—22no—
13C4×Dic514C2211yescharacteristic
14C4×D1028C2211yescharacteristic
15#22551554—21no—
16C4⋊C4×D257—21no—
17#22551526—21no—
18#22551524—21no—
19#22551517—21no—
20#22551514—21no—
21C4×#14523339C2211yescharacteristic
22C4⋊Q8×C257C2211yescharacteristic
23C4⋊C8×C257C2211yescharacteristic
24C4⋊C8×C257C2211yescharacteristic
25C2×Dic514—21no—
26C2×Dic514D411yescharacteristic
27C2×D1028D411yescharacteristic
28C2×C4×D257—22no—
29C2×C4×D257—21no—
30#16861608—44no—
31C2×#14523339—21no—
32C4⋊Dic257—22no—
33C4⋊Dic257—21no—
34C4⋊Dic257—21no—
35C4⋊Dic257D411yescharacteristic
36C4⋊Dic257D411yescharacteristic
37#16861597—22no—
38#16861597—21no—
39#16861595—22no—
40#16861595—21no—
41C4×Dic257—21no—
42C2×Q8×C257D412yes—
43C2×C8×C257—21no—
44C2×C8×C257—21no—
45C4⋊C4×C257—21no—
46C4⋊C4×C257—21no—
47C42×C257C2311yescharacteristic
48C22×D257—21no—
49Dic514—21no—
50Dic514—21no—
51Dic514—42no—
52D1028—21no—
53D1028Q1612yescomplemented
54C2×Dic257—21no—
55C2×Dic257—21no—
56C2×Dic257—21no—
57C2×Dic257—41no—
58C4×D257—41no—
59C4×D257—41no—
60C4×D257—82no—
61#14523339—22no—
62Q8×C257—42no—
63C2×C4×C257—22no—
64C2×C4×C257C2×D411yescharacteristic
65C2×C4×C257C2×D411yescharacteristic
66C2×C4×C257C2×D411yescharacteristic
67C8×C257—41no—
68C8×C257—41no—
69C22×C257#12011yescharacteristic
70C2×D257—22no—
71C2×D257—41no—
72C2×D257—41no—
73C4×C257—21no—
74C4×C257—42no—
75C4×C257C2×Q1611yescharacteristic
76C4×C257#13511yescharacteristic
77C4×C257#13711yescharacteristic
78C4×C257#13711yescharacteristic
79Dic257—41no—
80Dic257—41no—
81Dic257—41no—
82Dic257—81no—
83C2×C257#45111yescharacteristic
84C2×C257#45211yescharacteristic
85C2×C257#45311yescharacteristic
86D257—42no—
87C257#163211yescharacteristic, complemented, Sylow
88#1632—2571nomaximal, Sylow
89#554—2571no—
90#494—2571no—
91#477—2571no—
92#476—2571no—
93#333D25711yescharacteristic, complemented
94Q8⋊C8—2571no—
95D4⋊C8—2571no—
96C2×Q16—5142no—
97C4⋊Q8—2571no—
98C4⋊Q8C2×D25711yescharacteristic
99#125—5141no—
100C22⋊Q8—5141no—
101C22⋊Q8—5142no—
102C4×Q8—2571no—
103C4×D4—2571no—
104C4×D4—5141no—
105C2×C4⋊C4—5141no—
106#107—5141no—
107#106—5141no—
108C4⋊C8C2×D25711yescharacteristic
109C4⋊C8C2×D25711yescharacteristic
110#103—5141no—
111#102—5141no—
112C4×C8—2571no—
113C2×Q8—2571no—
114C2×Q8—5141no—
115C2×Q8D102812yes—
116C2×D4—2571no—
117C22×C4—5142no—
118C22×C4—5141no—
119Q16—10284no—
120C2×C8—21no—
121C2×C8—21no—
122C2×C8—5141no—
123C4⋊C4—2571no—
124C4⋊C4—2571no—
125C4⋊C4—21no—
126C4⋊C4—21no—
127C4⋊C4—5142no—
128C4⋊C4—5142no—
129C4⋊C4—5141no—
130C4⋊C4—5141no—
131C4⋊C4—5141no—
132C22⋊C4—5142no—
133C22⋊C4—5141no—
134C42—5141no—
135C42C22×D25711yescharacteristic
136C23—5141no—
137Q8—5141no—
138Q8—5141no—
139Q8—42no—
140Q8—10282no—
141D4—2572no—
142D4—5141no—
143C2×C4—22no—
144C2×C4—5141no—
145C2×C4—5141no—
146C2×C4—5141no—
147C2×C4—10281no—
148C2×C4—10281no—
149C2×C4—10281no—
150C2×C4—20562no—
151C2×C4D4×D25711yescharacteristic
152C2×C4D4×D25711yescharacteristic
153C2×C4C2×D102811yescharacteristic
154C8—5142no—
155C8—41no—
156C8—41no—
157C22—5142no—
158C22—10281no—
159C22—10281no—
160C22#2255156211yescharacteristic
161C4—21no—
162C4—42no—
163C4—10281no—
164C4—10281no—
165C4—10281no—
166C4—20561no—
167C4Q16×D25711yescharacteristic
168C4#2255161311yescharacteristic
169C4#2255162111yescharacteristic
170C4#2255162411yescharacteristic
171C2—10282no—
172C2#3065689411yescharacteristic
173C2#3065690211yescharacteristic
174C2#3065690711yescharacteristic
subgroupG/H
centerC22#22551562
commutator, cosocleC42×C257C23
FrattiniC42C22×D257
Fitting#333×C257C2
socleC22×C257#120
prime radicalC2×D1028D4
hypercenter#333D257
hypocenterC257#1632
LCSG ▷ C42×C257 ▷ C22×C257 ▷ C257
UCSC1 ◁ C22 ◁ C42 ◁ #333

Complements

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

NH ≅ G/Ntype
2. #30657116C2semidirect
3. #30657033C2semidirect
5. #30656467C2semidirect
7. #333×C257C2semidirect
53. D1028Q16semidirect
87. C257#1632semidirect
93. #333D257semidirect

Maximal subgroups

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

subgroupindex
#306574482
#306571162
#306570332
#306570282
#306564672
#306564642
#333×C2572
#1632257