Finite Group Databasebeta

← #44549498

Subgroups — #44549498

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#30632397C211yesmaximal, characteristic
2#30632066C211yesmaximal, characteristic, complemented
3#30631974C211yesmaximal, characteristic
4#30631435C211yesmaximal, characteristic, complemented
5Q8⋊(C251⋊C8)C211yesmaximal, characteristic, complemented
6#30631414C211yesmaximal, characteristic
7#477×C251C211yesmaximal, characteristic, complemented
8C2×#16860800—22no—
9#22548715C2211yescharacteristic
10#22548707—21no—
11#22548693—22no—
12#22548691—21no—
13C4×D1004—21no—
14C4×D1004C2211yescharacteristic
15C4⋊C4×D251—21no—
16#22548646—21no—
17C4⋊(C251⋊C8)C2211yescharacteristic
18#22548635—21no—
19C4×C251⋊C8C2211yescharacteristic
20C4⋊Q8×C251C2211yescharacteristic
21C4×Q8×C251C2211yescharacteristic
22#107×C251—21no—
23C4⋊C8×C251C2211yescharacteristic
24#103×C251—21no—
25C2×Dic502D412yes—
26C2×D1004D411yescharacteristic
27C2×C4×D251—22no—
28C2×C4×D251—21no—
29#16860800—44no—
30C2×C251⋊C8—21no—
31C2×C251⋊C8—21no—
32C4⋊Dic251—21no—
33C4⋊Dic251—21no—
34C4⋊Dic251D411yescharacteristic
35Dic251⋊C4—22no—
36Dic251⋊C4—22no—
37(C2×D251)⋊C4—22no—
38(C2×D251)⋊C4—21no—
39C2×Q8×C251—21no—
40C2×Q8×C251D411yescharacteristic
41C2×C8×C251—21no—
42C4⋊C4×C251—21no—
43C4⋊C4×C251—21no—
44C4⋊C4×C251—21no—
45C4⋊C4×C251D411yescharacteristic
46C42×C251—21no—
47C42×C251C2311yescharacteristic
48C22×D251—21no—
49Dic502—42no—
50D1004—21no—
51D1004Q1612yescomplemented
52C2×Dic251—21no—
53C2×Dic251—22no—
54C4×D251—41no—
55C4×D251—41no—
56C4×D251—82no—
57C251⋊C8—22no—
58C251⋊C8—41no—
59Q8×C251—21no—
60Q8×C251—21no—
61Q8×C251—42no—
62C2×C4×C251—21no—
63C2×C4×C251—21no—
64C2×C4×C251—41no—
65C2×C4×C251C2×D411yescharacteristic
66C2×C4×C251C2×D411yescharacteristic
67C2×C4×C251C2×D411yescharacteristic
68C8×C251—41no—
69C22×C251#12011yescharacteristic
70C2×D251—22no—
71C2×D251—41no—
72C2×D251—41no—
73C4×C251—21no—
74C4×C251—41no—
75C4×C251—41no—
76C4×C251—81no—
77C4×C251C2×Q1611yescharacteristic
78C4×C251#13511yescharacteristic
79C4×C251#13711yescharacteristic
80C4×C251#13711yescharacteristic
81Dic251—41no—
82Dic251—42no—
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#477D25111yescharacteristic, complemented
92#476—2511no—
93#333—2511no—
94Q8⋊C8—2511no—
95D4⋊C8—2511no—
96C2×Q16—5022no—
97C4⋊Q8—2511no—
98C4⋊Q8C2×D25111yescharacteristic
99#125—5021no—
100C22⋊Q8—5022no—
101C22⋊Q8—5021no—
102C4×Q8C2×D25111yescharacteristic
103C4×D4—5021no—
104C4×D4—2511no—
105C2×C4⋊C4—5021no—
106#107—21no—
107#106—5021no—
108C4⋊C8—2511no—
109C4⋊C8C2×D25111yescharacteristic
110#103—21no—
111#102—5021no—
112C4×C8—2511no—
113C2×Q8—21no—
114C2×Q8—2512no—
115C2×Q8C22⋊D25111yescharacteristic
116C2×D4—2511no—
117C22×C4—5022no—
118C22×C4—5021no—
119Q16—10044no—
120C2×C8—21no—
121C2×C8—5021no—
122C2×C8—5021no—
123C4⋊C4—21no—
124C4⋊C4—21no—
125C4⋊C4—21no—
126C4⋊C4—5022no—
127C4⋊C4—5022no—
128C4⋊C4—5021no—
129C4⋊C4—5021no—
130C4⋊C4—2511no—
131C4⋊C4C22⋊D25111yescharacteristic
132C22⋊C4—5022no—
133C22⋊C4—5021no—
134C42—21no—
135C42C22×D25111yescharacteristic
136C23—5021no—
137Q8—21no—
138Q8—21no—
139Q8—42no—
140Q8—10042no—
141D4—5021no—
142D4—2512no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—41no—
146C2×C4—20082no—
147C2×C4—10041no—
148C2×C4—10041no—
149C2×C4—5021no—
150C2×C4—5022no—
151C2×C4D4×D25111yescharacteristic
152C2×C4D4×D25111yescharacteristic
153C2×C4C2×C22⋊D25111yescharacteristic
154C8—41no—
155C8—10041no—
156C8—5022no—
157C22—10041no—
158C22—10041no—
159C22—5022no—
160C22#2254868311yescharacteristic
161C4—21no—
162C4—41no—
163C4—41no—
164C4—81no—
165C4—10041no—
166C4—10042no—
167C4Q16×D25111yescharacteristic
168C4#2254873511yescharacteristic
169C4#2254874611yescharacteristic
170C4#2254874711yescharacteristic
171C2—10042no—
172C2#3063184111yescharacteristic
173C2#3063184711yescharacteristic
174C2#3063185311yescharacteristic
subgroupG/H
centerC22#22548683
commutator, cosocleC42×C251C23
FrattiniC42C22×D251
Fitting#477×C251C2
socleC22×C251#120
prime radicalC2×D1004D4
hypercenter#477D251
hypocenterC251#1632
LCSG ▷ C42×C251 ▷ C22×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ #477

Complements

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

NH ≅ G/Ntype
2. #30632066C2semidirect
4. #30631435C2semidirect
5. Q8⋊(C251⋊C8)C2semidirect
7. #477×C251C2semidirect
51. D1004Q16semidirect
87. C251#1632semidirect
91. #477D251semidirect

Maximal subgroups

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

subgroupindex
#306323972
#306320662
#306319742
#306314352
Q8⋊(C251⋊C8)2
#306314142
#477×C2512
#1632251