Finite Group Databasebeta

← #44549499

Subgroups — #44549499

Order 32128.

214 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 61 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#30632066C211yesmaximal, characteristic, complemented
2#30631982C211yesmaximal, characteristic, complemented
3#30631977C211yesmaximal, characteristic
4#30631436C211yesmaximal, characteristic, complemented
5Q8⋊(C251⋊C8)C211yesmaximal, characteristic, complemented
6D4⋊(C251⋊C8)C211yesmaximal, characteristic
7#554×C251C211yesmaximal, characteristic
8C2×#16860800—22no—
9#22548715C2211yescharacteristic
10#22548648—21no—
11#22548646—21no—
12C4⋊(C251⋊C8)C2211yescharacteristic
13C4⋊(C251⋊C8)C2211yescharacteristic
14#22548639—21no—
15#22548636—21no—
16C4×C251⋊C8C2211yescharacteristic
17C4⋊Q8×C251C2211yescharacteristic
18#125×C251—21no—
19C22⋊Q8×C251—21no—
20C22⋊Q8×C251—22no—
21C4×Q8×C251C2211yescharacteristic
22C4×D4×C251—21no—
23C4×D4×C251C2211yescharacteristic
24C2×C4⋊C4×C251—21no—
25C2×Dic502—21no—
26#16860800—44no—
27C2×C251⋊C8—21no—
28C2×C251⋊C8—21no—
29C2×C251⋊C8—21no—
30C4⋊Dic251—21no—
31C4⋊Dic251—21no—
32C2×Q8×C251D411yescharacteristic
33C2×Q8×C251D412yes—
34C2×D4×C251D411yescharacteristic
35C22×C4×C251—22no—
36C22×C4×C251—21no—
37C4⋊C4×C251—22no—
38C4⋊C4×C251—22no—
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
48Dic502—42no—
49C2×Dic251—41no—
50C251⋊C8—22no—
51C251⋊C8—41no—
52C251⋊C8—41no—
53C23×C251—21no—
54Q8×C251—21no—
55Q8×C251—21no—
56Q8×C251—42no—
57D4×C251—21no—
58D4×C251Q1612yescomplemented
59C2×C4×C251—21no—
60C2×C4×C251—21no—
61C2×C4×C251—21no—
62C2×C4×C251—22no—
63C2×C4×C251—41no—
64C2×C4×C251—41no—
65C2×C4×C251—82no—
66C2×C4×C251C2×D411yescharacteristic
67C2×C4×C251C2×D411yescharacteristic
68C2×C4×C251C2×D411yescharacteristic
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×C251C2×Q1611yescharacteristic
79C4×C251#13511yescharacteristic
80C4×C251#13711yescharacteristic
81C4×C251#13711yescharacteristic
82Dic251—81no—
83C2×C251—42no—
84C2×C251#45111yescharacteristic
85C2×C251#45211yescharacteristic
86C2×C251#45311yescharacteristic
87C251#163211yescharacteristic, complemented, Sylow
88#1632—2511nomaximal, Sylow
89#554D25111yescharacteristic
90#494—2511no—
91#477—2511no—
92#476—2511no—
93#333—2511no—
94Q8⋊C8—2511no—
95D4⋊C8—2511no—
96C2×Q16—5022no—
97C4⋊Q8—2511no—
98C4⋊Q8C2×D25111yescharacteristic
99#125—21no—
100C22⋊Q8—21no—
101C22⋊Q8—22no—
102C4×Q8C2×D25111yescharacteristic
103C4×D4—21no—
104C4×D4C2×D25111yescharacteristic
105C2×C4⋊C4—21no—
106#107—5021no—
107#106—5021no—
108C4⋊C8—2511no—
109C4⋊C8—2511no—
110#103—5021no—
111#102—5021no—
112C4×C8—2511no—
113C2×Q8—5021no—
114C2×Q8C22⋊D25111yescharacteristic
115C2×Q8C22⋊D25112yes—
116C2×D4C22⋊D25111yescharacteristic
117C22×C4—22no—
118C22×C4—21no—
119Q16—10044no—
120C2×C8—5021no—
121C2×C8—5021no—
122C2×C8—5021no—
123C4⋊C4—22no—
124C4⋊C4—22no—
125C4⋊C4—21no—
126C4⋊C4—21no—
127C4⋊C4—21no—
128C4⋊C4—5021no—
129C4⋊C4—5021no—
130C4⋊C4C22⋊D25111yescharacteristic
131C4⋊C4C22⋊D25111yescharacteristic
132C22⋊C4—22no—
133C22⋊C4—21no—
134C42—21no—
135C42C22×D25111yescharacteristic
136C23—21no—
137Q8—21no—
138Q8—21no—
139Q8—42no—
140Q8—10042no—
141D4—21no—
142D4#1686080012yescomplemented
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—21no—
146C2×C4—22no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—82no—
150C2×C4—10041no—
151C2×C4C2×C22⋊D25111yescharacteristic
152C2×C4C2×C22⋊D25111yescharacteristic
153C2×C4C2×C22⋊D25111yescharacteristic
154C8—10041no—
155C8—10041no—
156C8—5022no—
157C22—22no—
158C22—41no—
159C22—41no—
160C22#2254868411yescharacteristic
161C4—21no—
162C4—41no—
163C4—41no—
164C4—41no—
165C4—42no—
166C4—20081no—
167C4C2×#1686080011yescharacteristic
168C4#2254873511yescharacteristic
169C4#2254874511yescharacteristic
170C4#2254874611yescharacteristic
171C2—42no—
172C2#3063184311yescharacteristic
173C2#3063184911yescharacteristic
174C2#3063185511yescharacteristic
subgroupG/H
centerC22#22548684
commutator, cosocleC42×C251C23
FrattiniC42C22×D251
Fitting#554×C251C2
socleC22×C251#120
prime radicalC2×D4×C251D4
hypercenter#554D251
hypocenterC251#1632
LCSG ▷ C42×C251 ▷ C22×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ #554

Complements

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

NH ≅ G/Ntype
1. #30632066C2semidirect
2. #30631982C2semidirect
4. #30631436C2semidirect
5. Q8⋊(C251⋊C8)C2semidirect
58. D4×C251Q16semidirect
87. C251#1632semidirect
142. D4#16860800semidirect

Maximal subgroups

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

subgroupindex
#306320662
#306319822
#306319772
#306314362
Q8⋊(C251⋊C8)2
D4⋊(C251⋊C8)2
#554×C2512
#1632251