Finite Group Databasebeta

← #44549500

Subgroups — #44549500

Order 32128.

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#30632398C211yesmaximal, characteristic
2#30632066C211yesmaximal, characteristic, complemented
3#30631983C211yesmaximal, characteristic, complemented
4#30631978C211yesmaximal, characteristic
5#30631417C211yesmaximal, characteristic, complemented
6#30631414C211yesmaximal, characteristic
7#333×C251C211yesmaximal, characteristic, complemented
8C2×#16860800—22no—
9#22548715C2211yescharacteristic
10#22548706—21no—
11#22548695—21no—
12#22548691—22no—
13C4×Dic502C2211yescharacteristic
14C4×D1004C2211yescharacteristic
15#22548677—21no—
16C4⋊C4×D251—21no—
17#22548649—21no—
18#22548647—21no—
19#22548640—21no—
20#22548637—21no—
21C4×C251⋊C8C2211yescharacteristic
22C4⋊Q8×C251C2211yescharacteristic
23C4⋊C8×C251C2211yescharacteristic
24C4⋊C8×C251C2211yescharacteristic
25C2×Dic502—21no—
26C2×Dic502D411yescharacteristic
27C2×D1004D411yescharacteristic
28C2×C4×D251—22no—
29C2×C4×D251—21no—
30#16860800—44no—
31C2×C251⋊C8—21no—
32C4⋊Dic251—22no—
33C4⋊Dic251—21no—
34C4⋊Dic251—21no—
35C4⋊Dic251D411yescharacteristic
36C4⋊Dic251D411yescharacteristic
37Dic251⋊C4—22no—
38Dic251⋊C4—21no—
39(C2×D251)⋊C4—22no—
40(C2×D251)⋊C4—21no—
41C4×Dic251—21no—
42C2×Q8×C251D412yes—
43C2×C8×C251—21no—
44C2×C8×C251—21no—
45C4⋊C4×C251—21no—
46C4⋊C4×C251—21no—
47C42×C251C2311yescharacteristic
48C22×D251—21no—
49Dic502—21no—
50Dic502—21no—
51Dic502—42no—
52D1004—21no—
53D1004Q1612yescomplemented
54C2×Dic251—21no—
55C2×Dic251—21no—
56C2×Dic251—21no—
57C2×Dic251—41no—
58C4×D251—41no—
59C4×D251—41no—
60C4×D251—82no—
61C251⋊C8—22no—
62Q8×C251—42no—
63C2×C4×C251—22no—
64C2×C4×C251C2×D411yescharacteristic
65C2×C4×C251C2×D411yescharacteristic
66C2×C4×C251C2×D411yescharacteristic
67C8×C251—41no—
68C8×C251—41no—
69C22×C251#12011yescharacteristic
70C2×D251—22no—
71C2×D251—41no—
72C2×D251—41no—
73C4×C251—21no—
74C4×C251—42no—
75C4×C251C2×Q1611yescharacteristic
76C4×C251#13511yescharacteristic
77C4×C251#13711yescharacteristic
78C4×C251#13711yescharacteristic
79Dic251—41no—
80Dic251—41no—
81Dic251—41no—
82Dic251—81no—
83C2×C251#45111yescharacteristic
84C2×C251#45211yescharacteristic
85C2×C251#45311yescharacteristic
86D251—42no—
87C251#163211yescharacteristic, complemented, Sylow
88#1632—2511nomaximal, Sylow
89#554—2511no—
90#494—2511no—
91#477—2511no—
92#476—2511no—
93#333D25111yescharacteristic, complemented
94Q8⋊C8—2511no—
95D4⋊C8—2511no—
96C2×Q16—5022no—
97C4⋊Q8—2511no—
98C4⋊Q8C2×D25111yescharacteristic
99#125—5021no—
100C22⋊Q8—5021no—
101C22⋊Q8—5022no—
102C4×Q8—2511no—
103C4×D4—5021no—
104C4×D4—2511no—
105C2×C4⋊C4—5021no—
106#107—5021no—
107#106—5021no—
108C4⋊C8C2×D25111yescharacteristic
109C4⋊C8C2×D25111yescharacteristic
110#103—5021no—
111#102—5021no—
112C4×C8—2511no—
113C2×Q8—5021no—
114C2×Q8—2511no—
115C2×Q8D100412yes—
116C2×D4—2511no—
117C22×C4—5022no—
118C22×C4—5021no—
119Q16—10044no—
120C2×C8—21no—
121C2×C8—21no—
122C2×C8—5021no—
123C4⋊C4—21no—
124C4⋊C4—21no—
125C4⋊C4—5022no—
126C4⋊C4—5022no—
127C4⋊C4—5021no—
128C4⋊C4—5021no—
129C4⋊C4—5021no—
130C4⋊C4—2511no—
131C4⋊C4—2511no—
132C22⋊C4—5022no—
133C22⋊C4—5021no—
134C42—5021no—
135C42C22×D25111yescharacteristic
136C23—5021no—
137Q8—42no—
138Q8—10042no—
139Q8—5021no—
140Q8—5021no—
141D4—5021no—
142D4—2512no—
143C2×C4—22no—
144C2×C4—20082no—
145C2×C4—10041no—
146C2×C4—10041no—
147C2×C4—10041no—
148C2×C4—5021no—
149C2×C4—5021no—
150C2×C4—5021no—
151C2×C4D4×D25111yescharacteristic
152C2×C4D4×D25111yescharacteristic
153C2×C4C2×D100411yescharacteristic
154C8—41no—
155C8—41no—
156C8—5022no—
157C22—10041no—
158C22—10041no—
159C22—5022no—
160C22#2254868511yescharacteristic
161C4—21no—
162C4—42no—
163C4—20081no—
164C4—10041no—
165C4—10041no—
166C4—10041no—
167C4Q16×D25111yescharacteristic
168C4#2254873611yescharacteristic
169C4#2254874411yescharacteristic
170C4#2254874711yescharacteristic
171C2—10042no—
172C2#3063184411yescharacteristic
173C2#3063185211yescharacteristic
174C2#3063185711yescharacteristic
subgroupG/H
centerC22#22548685
commutator, cosocleC42×C251C23
FrattiniC42C22×D251
Fitting#333×C251C2
socleC22×C251#120
prime radicalC2×D1004D4
hypercenter#333D251
hypocenterC251#1632
LCSG ▷ C42×C251 ▷ C22×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ #333

Complements

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

NH ≅ G/Ntype
2. #30632066C2semidirect
3. #30631983C2semidirect
5. #30631417C2semidirect
7. #333×C251C2semidirect
53. D1004Q16semidirect
87. C251#1632semidirect
93. #333D251semidirect

Maximal subgroups

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

subgroupindex
#306323982
#306320662
#306319832
#306319782
#306314172
#306314142
#333×C2512
#1632251