Finite Group Databasebeta

← #44549501

Subgroups — #44549501

Order 32128.

214 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 214 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#30632394C211yesmaximal, characteristic
2#30632067C211yesmaximal, characteristic, complemented
3#30631983C211yesmaximal, characteristic, complemented
4#30631977C211yesmaximal, characteristic
5#30631435C211yesmaximal, characteristic, complemented
6#30631419C211yesmaximal, characteristic, complemented
7D4⋊C8×C251C211yesmaximal, characteristic
8C2×Dic1004—22no—
9#22548715C2211yescharacteristic
10#22548715C2211yescharacteristic
11#22548706—21no—
12#22548694—21no—
13#22548692—22no—
14C4×Dic502C2211yescharacteristic
15D4×Dic251—21no—
16C2×C4⋊Dic251—21no—
17#22548649—21no—
18#22548646—21no—
19C4⋊(C251⋊C8)C2211yescharacteristic
20#22548640—21no—
21#22548636—21no—
22C4×D4×C251C2211yescharacteristic
23C4⋊C8×C251C2211yescharacteristic
24C4×C8×C251C2211yescharacteristic
25C2×Dic502—21no—
26C2×Dic502D411yescharacteristic
27C2×Dic502D412yes—
28C22×Dic251—22no—
29Dic1004—44no—
30C2×C251⋊C8—21no—
31C4⋊Dic251—22no—
32C4⋊Dic251—21no—
33C4⋊Dic251—21no—
34C4⋊Dic251—21no—
35C4⋊Dic251—21no—
36C4⋊Dic251D411yescharacteristic
37Dic251⋊C4—22no—
38Dic251⋊C4—21no—
39C22⋊Dic251—22no—
40C4×Dic251—21no—
41C2×D4×C251D411yescharacteristic
42C22×C4×C251—21no—
43C2×C8×C251—21no—
44C2×C8×C251—21no—
45C4⋊C4×C251D411yescharacteristic
46C22⋊C4×C251—21no—
47C42×C251C2311yescharacteristic
48Dic502—21no—
49Dic502—21no—
50Dic502—42no—
51Dic502—42no—
52C2×Dic251—21no—
53C2×Dic251—21no—
54C2×Dic251—22no—
55C2×Dic251—41no—
56C2×Dic251—82no—
57C251⋊C8—41no—
58C23×C251—21no—
59D4×C251—21no—
60D4×C251Q1612yescomplemented
61C2×C4×C251—21no—
62C2×C4×C251—41no—
63C2×C4×C251—41no—
64C2×C4×C251C2×D411yescharacteristic
65C2×C4×C251C2×D411yescharacteristic
66C2×C4×C251C2×D411yescharacteristic
67C8×C251—22no—
68C8×C251—41no—
69C22×C251—22no—
70C22×C251—41no—
71C22×C251—41no—
72C22×C251#12011yescharacteristic
73C4×C251—21no—
74C4×C251—41no—
75C4×C251C2×Q1611yescharacteristic
76C4×C251#13511yescharacteristic
77C4×C251#13711yescharacteristic
78C4×C251#13711yescharacteristic
79Dic251—41no—
80Dic251—41no—
81Dic251—42no—
82Dic251—81no—
83C2×C251—42no—
84C2×C251#45111yescharacteristic
85C2×C251#45211yescharacteristic
86C2×C251#45311yescharacteristic
87C251#163211yescharacteristic, complemented, Sylow
88#1632—2511nomaximal, Sylow
89#554—2511no—
90#494—2511no—
91#477—2511no—
92#476—2511no—
93#333—2511no—
94Q8⋊C8—2511no—
95D4⋊C8D25111yescharacteristic
96C2×Q16—5022no—
97C4⋊Q8—2511no—
98C4⋊Q8—2511no—
99#125—5021no—
100C22⋊Q8—5022no—
101C22⋊Q8—5021no—
102C4×Q8—2511no—
103C4×D4—5021no—
104C4×D4C2×D25111yescharacteristic
105C2×C4⋊C4—5021no—
106#107—5021no—
107#106—5021no—
108C4⋊C8—2511no—
109C4⋊C8C2×D25111yescharacteristic
110#103—5021no—
111#102—5021no—
112C4×C8C2×D25111yescharacteristic
113C2×Q8—5021no—
114C2×Q8—2512no—
115C2×Q8—2511no—
116C2×D4D100411yescharacteristic
117C22×C4—21no—
118C22×C4—5022no—
119Q16—10044no—
120C2×C8—21no—
121C2×C8—21no—
122C2×C8—5021no—
123C4⋊C4—5022no—
124C4⋊C4—5022no—
125C4⋊C4—5021no—
126C4⋊C4—5021no—
127C4⋊C4—5021no—
128C4⋊C4—5021no—
129C4⋊C4—5021no—
130C4⋊C4—2511no—
131C4⋊C4D100411yescharacteristic
132C22⋊C4—21no—
133C22⋊C4—5022no—
134C42—5021no—
135C42C22×D25111yescharacteristic
136C23—21no—
137Q8—10042no—
138Q8—10042no—
139Q8—5021no—
140Q8—5021no—
141D4—21no—
142D4Dic100412yescomplemented
143C2×C4—21no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—20082no—
147C2×C4—10041no—
148C2×C4—5021no—
149C2×C4—5021no—
150C2×C4—5022no—
151C2×C4D4×D25111yescharacteristic
152C2×C4D4×D25111yescharacteristic
153C2×C4C2×D100411yescharacteristic
154C8—22no—
155C8—41no—
156C8—10041no—
157C22—22no—
158C22—41no—
159C22—41no—
160C22#2254868511yescharacteristic
161C4—21no—
162C4—41no—
163C4—20081no—
164C4—10041no—
165C4—10041no—
166C4—10042no—
167C4C2×Dic100411yescharacteristic
168C4#2254873411yescharacteristic
169C4#2254874411yescharacteristic
170C4#2254874811yescharacteristic
171C2—42no—
172C2#3063184511yescharacteristic
173C2#3063185011yescharacteristic
174C2#3063185611yescharacteristic
subgroupG/H
centerC22#22548685
commutator, cosocleC42×C251C23
FrattiniC42C22×D251
FittingD4⋊C8×C251C2
socleC22×C251#120
prime radicalC2×D4×C251D4
hypercenterD4⋊C8D251
hypocenterC251#1632
LCSG ▷ C42×C251 ▷ C22×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ D4⋊C8

Complements

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

NH ≅ G/Ntype
2. #30632067C2semidirect
3. #30631983C2semidirect
5. #30631435C2semidirect
6. #30631419C2semidirect
60. D4×C251Q16semidirect
87. C251#1632semidirect
142. D4Dic1004semidirect

Maximal subgroups

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

subgroupindex
#306323942
#306320672
#306319832
#306319772
#306314352
#306314192
D4⋊C8×C2512
#1632251