Finite Group Databasebeta

← #1632×C251 #44546219

Subgroups — #1632×C251 #44546219

Order 32128.

214 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 72 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#554×C251C211yesmaximal, characteristic
2#494×C251C211yesmaximal, characteristic, complemented
3#477×C251C211yesmaximal, characteristic, complemented
4#476×C251C211yesmaximal, characteristic
5#333×C251C211yesmaximal, characteristic, complemented
6Q8⋊C8×C251C211yesmaximal, characteristic, complemented
7D4⋊C8×C251C211yesmaximal, characteristic
8C2×Q16×C251—22no—
9C4⋊Q8×C251C2211yescharacteristic
10C4⋊Q8×C251C2211yescharacteristic
11#125×C251—21no—
12C22⋊Q8×C251—21no—
13C22⋊Q8×C251—22no—
14C4×Q8×C251C2211yescharacteristic
15C4×D4×C251—21no—
16C4×D4×C251C2211yescharacteristic
17C2×C4⋊C4×C251—21no—
18#107×C251—21no—
19#106×C251—21no—
20C4⋊C8×C251C2211yescharacteristic
21C4⋊C8×C251C2211yescharacteristic
22#103×C251—21no—
23#102×C251—21no—
24C4×C8×C251C2211yescharacteristic
25C2×Q8×C251—21no—
26C2×Q8×C251D411yescharacteristic
27C2×Q8×C251D412yes—
28C2×D4×C251D411yescharacteristic
29C22×C4×C251—22no—
30C22×C4×C251—21no—
31Q16×C251—44no—
32C2×C8×C251—21no—
33C2×C8×C251—21no—
34C2×C8×C251—21no—
35C4⋊C4×C251—22no—
36C4⋊C4×C251—22no—
37C4⋊C4×C251—21no—
38C4⋊C4×C251—21no—
39C4⋊C4×C251—21no—
40C4⋊C4×C251—21no—
41C4⋊C4×C251—21no—
42C4⋊C4×C251D411yescharacteristic
43C4⋊C4×C251D411yescharacteristic
44C22⋊C4×C251—22no—
45C22⋊C4×C251—21no—
46C42×C251—21no—
47C42×C251C2311yescharacteristic
48C23×C251—21no—
49Q8×C251—21no—
50Q8×C251—21no—
51Q8×C251—42no—
52Q8×C251—42no—
53D4×C251—21no—
54D4×C251Q1612yescomplemented
55C2×C4×C251—21no—
56C2×C4×C251—21no—
57C2×C4×C251—21no—
58C2×C4×C251—22no—
59C2×C4×C251—41no—
60C2×C4×C251—41no—
61C2×C4×C251—41no—
62C2×C4×C251—82no—
63C2×C4×C251C2×D411yescharacteristic
64C2×C4×C251C2×D411yescharacteristic
65C2×C4×C251C2×D411yescharacteristic
66C8×C251—22no—
67C8×C251—41no—
68C8×C251—41no—
69C22×C251—22no—
70C22×C251—41no—
71C22×C251—41no—
72C22×C251#12011yescharacteristic
73C4×C251—21no—
74C4×C251—41no—
75C4×C251—41no—
76C4×C251—41no—
77C4×C251—42no—
78C4×C251—81no—
79C4×C251C2×Q1611yescharacteristic
80C4×C251#13511yescharacteristic
81C4×C251#13711yescharacteristic
82C4×C251#13711yescharacteristic
83C2×C251—42no—
84C2×C251#45111yescharacteristic
85C2×C251#45211yescharacteristic
86C2×C251#45311yescharacteristic
87C251#163211yescharacteristic, direct product factor, Sylow
88#1632C25111yesmaximal, characteristic, direct product factor, Sylow
89#554C2×C25111yescharacteristic
90#494C2×C25111yescharacteristic, complemented
91#477C2×C25111yescharacteristic, complemented
92#476C2×C25111yescharacteristic
93#333C2×C25111yescharacteristic, complemented
94Q8⋊C8C2×C25111yescharacteristic, complemented
95D4⋊C8C2×C25111yescharacteristic
96C2×Q16—22no—
97C4⋊Q8C22×C25111yescharacteristic
98C4⋊Q8C22×C25111yescharacteristic
99#125—21no—
100C22⋊Q8—21no—
101C22⋊Q8—22no—
102C4×Q8C22×C25111yescharacteristic
103C4×D4—21no—
104C4×D4C22×C25111yescharacteristic
105C2×C4⋊C4—21no—
106#107—21no—
107#106—21no—
108C4⋊C8C22×C25111yescharacteristic
109C4⋊C8C22×C25111yescharacteristic
110#103—21no—
111#102—21no—
112C4×C8C22×C25111yescharacteristic
113C2×Q8—21no—
114C2×Q8D4×C25111yescharacteristic
115C2×Q8D4×C25112yes—
116C2×D4D4×C25111yescharacteristic
117C22×C4—22no—
118C22×C4—21no—
119Q16—44no—
120C2×C8—21no—
121C2×C8—21no—
122C2×C8—21no—
123C4⋊C4—22no—
124C4⋊C4—22no—
125C4⋊C4—21no—
126C4⋊C4—21no—
127C4⋊C4—21no—
128C4⋊C4—21no—
129C4⋊C4—21no—
130C4⋊C4D4×C25111yescharacteristic
131C4⋊C4D4×C25111yescharacteristic
132C22⋊C4—22no—
133C22⋊C4—21no—
134C42—21no—
135C42C23×C25111yescharacteristic
136C23—21no—
137Q8—21no—
138Q8—21no—
139Q8—42no—
140Q8—42no—
141D4—21no—
142D4Q16×C25112yescomplemented
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—21no—
146C2×C4—22no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—41no—
150C2×C4—82no—
151C2×C4C2×D4×C25111yescharacteristic
152C2×C4C2×D4×C25111yescharacteristic
153C2×C4C2×D4×C25111yescharacteristic
154C8—22no—
155C8—41no—
156C8—41no—
157C22—22no—
158C22—41no—
159C22—41no—
160C22#120×C25111yescharacteristic
161C4—21no—
162C4—41no—
163C4—41no—
164C4—41no—
165C4—42no—
166C4—81no—
167C4C2×Q16×C25111yescharacteristic
168C4#135×C25111yescharacteristic
169C4#137×C25111yescharacteristic
170C4#137×C25111yescharacteristic
171C2—42no—
172C2#451×C25111yescharacteristic
173C2#452×C25111yescharacteristic
174C2#453×C25111yescharacteristic
subgroupG/H
center, socleC22×C251#120
commutator, Frattini, cosocleC42C23×C251
prime radicalC2×D4×C251D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C251 ◁ C42×C251 ◁ G

Complements

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

NH ≅ G/Ntype
2. #494×C251C2semidirect
3. #477×C251C2semidirect
5. #333×C251C2semidirect
6. Q8⋊C8×C251C2semidirect
54. D4×C251Q16semidirect
87. C251#1632direct (N × H)
88. #1632C251direct (N × H)
90. #494C2×C251semidirect
91. #477C2×C251semidirect
93. #333C2×C251semidirect
94. Q8⋊C8C2×C251semidirect
142. D4Q16×C251semidirect

Maximal subgroups

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

subgroupindex
#554×C2512
#494×C2512
#477×C2512
#476×C2512
#333×C2512
Q8⋊C8×C2512
D4⋊C8×C2512
#1632251