Finite Group Databasebeta

← #44549538

Subgroups — #44549538

Order 32128.

194 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 194 conjugacy classes of subgroups — too many to draw. Only its 53 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#30632411C211yesmaximal, characteristic
2#30632039C211yesmaximal, characteristic, complemented
3#30631983C211yesmaximal, characteristic, complemented
4#30631978C211yesmaximal, characteristic
5#30631417C211yesmaximal, characteristic, complemented
6#30631414C211yesmaximal, characteristic
7#332×C251C211yesmaximal, characteristic, complemented
8#22548715C2211yescharacteristic
9#22548710—22no—
10#22548704—21no—
11#22548695—21no—
12Q8×Dic251—21no—
13C4×Dic502C2211yescharacteristic
14C4×D1004C2211yescharacteristic
15#22548673—21no—
16#22548649—21no—
17#22548647—21no—
18#22548640—21no—
19#22548638—22no—
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—21no—
29C2×C251⋊C8—21no—
30C4⋊Dic251—22no—
31C4⋊Dic251—21no—
32C4⋊Dic251—21no—
33C4⋊Dic251D411yescharacteristic
34C4⋊Dic251D411yescharacteristic
35Dic251⋊C4—21no—
36(C2×D251)⋊C4—22no—
37(C2×D251)⋊C4—22no—
38(C2×D251)⋊C4—21no—
39C4×Dic251—22no—
40C4×Dic251—21no—
41C2×Q8×C251—21no—
42C2×C8×C251—21no—
43C2×C8×C251—21no—
44C4⋊C4×C251—21no—
45C4⋊C4×C251D412yes—
46C42×C251C2311yescharacteristic
47C22×D251—21no—
48Dic502—21no—
49Dic502—21no—
50Dic502—42no—
51D1004—21no—
52D1004—21no—
53C2×Dic251—21no—
54C2×Dic251—21no—
55C2×Dic251—21no—
56C2×Dic251—41no—
57C4×D251—41no—
58C4×D251—41no—
59C251⋊C8—41no—
60Q8×C251—41no—
61Q8×C251—41no—
62C2×C4×C251—22no—
63C2×C4×C251C2×D411yescharacteristic
64C2×C4×C251C2×D411yescharacteristic
65C2×C4×C251C2×D411yescharacteristic
66C8×C251—41no—
67C8×C251—41no—
68C22×C251#12011yescharacteristic
69C2×D251—41no—
70C2×D251—41no—
71C2×D251—41no—
72C4×C251—21no—
73C4×C251—42no—
74C4×C251#13511yescharacteristic
75C4×C251#13511yescharacteristic
76C4×C251#13711yescharacteristic
77C4×C251#13711yescharacteristic
78Dic251—41no—
79Dic251—41no—
80Dic251—41no—
81Dic251—81no—
82C2×C251#45211yescharacteristic
83C2×C251#45211yescharacteristic
84C2×C251#45411yescharacteristic
85D251—81no—
86C251#163811yescharacteristic, complemented, Sylow
87#1638—2511nomaximal, Sylow
88#556—2511no—
89#487—2511no—
90#477—2511no—
91#476—2511no—
92#332D25111yescharacteristic, complemented
93Q8⋊C8—2511no—
94D4⋊C8—2511no—
95C4⋊Q8—2511no—
96C4⋊Q8C2×D25111yescharacteristic
97#126—5022no—
98#124—5021no—
99C22⋊Q8—5021no—
100C4×Q8—5021no—
101C4×Q8—2511no—
102C4×D4—2511no—
103#117—5021no—
104#107—5021no—
105#106—5021no—
106C4⋊C8C2×D25111yescharacteristic
107C4⋊C8C2×D25111yescharacteristic
108#103—5021no—
109#103—5022no—
110#102—5021no—
111C4×C8—2511no—
112C2×Q8—21no—
113C2×Q8—5021no—
114C2×Q8—2511no—
115C2×D4—2511no—
116C22×C4—5021no—
117C2×C8—21no—
118C2×C8—21no—
119C2×C8—5021no—
120C4⋊C4—21no—
121C4⋊C4—5022no—
122C4⋊C4—5021no—
123C4⋊C4—5021no—
124C4⋊C4—5021no—
125C4⋊C4—2511no—
126C4⋊C4—2511no—
127C4⋊C4D100412yes—
128C22⋊C4—5022no—
129C22⋊C4—5022no—
130C22⋊C4—5021no—
131C42—5022no—
132C42—5021no—
133C42C22×D25111yescharacteristic
134C23—5021no—
135Q8—41no—
136Q8—41no—
137Q8—10042no—
138Q8—5021no—
139Q8—5021no—
140D4—5021no—
141D4—5021no—
142C2×C4—22no—
143C2×C4—10041no—
144C2×C4—10041no—
145C2×C4—10041no—
146C2×C4—5021no—
147C2×C4—5021no—
148C2×C4—5021no—
149C2×C4D4×D25111yescharacteristic
150C2×C4D4×D25111yescharacteristic
151C2×C4C2×D100411yescharacteristic
152C8—41no—
153C8—41no—
154C8—10041no—
155C22—10041no—
156C22—10041no—
157C22—10041no—
158C22#2254868511yescharacteristic
159C4—21no—
160C4—42no—
161C4—20081no—
162C4—10041no—
163C4—10041no—
164C4—10041no—
165C4#2254873411yescharacteristic
166C4#2254873611yescharacteristic
167C4#2254874411yescharacteristic
168C4#2254874711yescharacteristic
169C2—20081no—
170C2#3063185011yescharacteristic
171C2#3063185211yescharacteristic
172C2#3063186211yescharacteristic
subgroupG/H
centerC22#22548685
commutator, cosocleC42×C251C23
FrattiniC42C22×D251
Fitting#332×C251C2
socleC22×C251#120
prime radicalC2×D1004D4
hypercenter#332D251
hypocenterC251#1638
LCSG ▷ C42×C251 ▷ C22×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ #332

Complements

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

NH ≅ G/Ntype
2. #30632039C2semidirect
3. #30631983C2semidirect
5. #30631417C2semidirect
7. #332×C251C2semidirect
86. C251#1638semidirect
92. #332D251semidirect

Maximal subgroups

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

subgroupindex
#306324112
#306320392
#306319832
#306319782
#306314172
#306314142
#332×C2512
#1638251