Finite Group Databasebeta

← #44549401

Subgroups — #44549401

Order 32128.

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

normal subgroups only

This group has 246 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#30632268C211yesmaximal, characteristic
2#30632005C212yesmaximal, complemented
3#30631420C212yesmaximal, complemented
4#30631413C211yesmaximal, characteristic, complemented
5#400×C251C211yesmaximal, characteristic, complemented
6C2×#16860810—21no—
7#22548695—22no—
8#22548690—22no—
9#22548689—21no—
10#22548689C2212yes—
11#22548646—22no—
12#22548635—22no—
13C22⋊(C251⋊C8)C2212yes—
14C2×C4⋊C4×C251C2211yescharacteristic
15C2×C4⋊C4×C251C2211yescharacteristic
16#95×C251—21no—
17#95×C251—21no—
18#95×C251C2211yescharacteristic
19#16860810—42no—
20#16860810—42no—
21C2×Dic502—21no—
22C2×D1004—21no—
23C2×D1004—21no—
24C2×D1004D412yes—
25C2×C22⋊D251—22no—
26C2×C4×D251—22no—
27C2×C251⋊C8—22no—
28C4⋊Dic251—21no—
29C4⋊Dic251D412yes—
30(C2×D251)⋊C4—22no—
31(C2×D251)⋊C4—22no—
32C22×C4×C251—21no—
33C22×C4×C251—21no—
34C22×C4×C251D411yescharacteristic
35C22×C4×C251D411yescharacteristic
36C22×C4×C251C2311yescharacteristic
37C4⋊C4×C251—22no—
38C4⋊C4×C251—22no—
39C22×D251—22no—
40Dic502—42no—
41D1004—42no—
42D1004—42no—
43D1004—81no—
44C22⋊D251—42no—
45C22⋊D251—42no—
46C2×Dic251—22no—
47C4×D251—44no—
48C251⋊C8—42no—
49C23×C251C2×D411yescharacteristic
50C2×C4×C251—21no—
51C2×C4×C251—21no—
52C2×C4×C251—21no—
53C2×C4×C251—41no—
54C2×C4×C251—41no—
55C2×C4×C251—41no—
56C2×C4×C251—41no—
57C2×C4×C251—42no—
58C2×C4×C251—41no—
59C2×C4×C251—41no—
60C2×C4×C251SD1612yes—
61C2×C4×C251C2×D412yes—
62C22×C251—21no—
63C22×C251—21no—
64C22×C251#12011yescharacteristic
65C22×C251C2×SD1611yescharacteristic
66C22×C251AGL(1,ℤ/8)11yescharacteristic
67C2×D251—42no—
68C2×D251—42no—
69C2×D251—42no—
70C4×C251—22no—
71C4×C251—22no—
72C4×C251—41no—
73C4×C251—81no—
74C4×C251—81no—
75Dic251—42no—
76C2×C251—21no—
77C2×C251—21no—
78C2×C251#45011yescharacteristic
79C2×C251#45411yescharacteristic
80C2×C251#45811yescharacteristic
81D251—82no—
82C251#161611yescharacteristic, complemented, Sylow
83#1616—2511nomaximal, Sylow
84#537—2511no—
85#481—2512no—
86#400D25111yescharacteristic, complemented
87#327—2512no—
88#324—2511no—
89C2×#41—5021no—
90C22⋊Q8—5022no—
91#121—5022no—
92#121—5021no—
93#121—2512no—
94C2×C4⋊C4C2×D25111yescharacteristic
95C2×C4⋊C4C2×D25111yescharacteristic
96#106—5022no—
97#102—5022no—
98C22⋊C8—2512no—
99#95—21no—
100#95—21no—
101#95C2×D25111yescharacteristic
102#41—10042no—
103#41—10042no—
104C2×Q8—5021no—
105C2×D4—5021no—
106C2×D4—5021no—
107C2×D4—5022no—
108C2×D4—2512no—
109C22×C4—21no—
110C22×C4—21no—
111C22×C4—5022no—
112C22×C4C22⋊D25111yescharacteristic
113C22×C4C22⋊D25111yescharacteristic
114C22×C4C22×D25111yescharacteristic
115C2×C8—5022no—
116C4⋊C4—22no—
117C4⋊C4—22no—
118C4⋊C4—5021no—
119C4⋊C4—2512no—
120C22⋊C4—5022no—
121C22⋊C4—5022no—
122C23—5022no—
123C23C2×C22⋊D25111yescharacteristic
124Q8—10042no—
125D4—20081no—
126D4—10042no—
127D4—10042no—
128D4—10042no—
129D4—10042no—
130C2×C4—21no—
131C2×C4—21no—
132C2×C4—21no—
133C2×C4—41no—
134C2×C4—41no—
135C2×C4—41no—
136C2×C4—41no—
137C2×C4—42no—
138C2×C4—41no—
139C2×C4—41no—
140C2×C4—10044no—
141C2×C4—5022no—
142C2×C4#1686079712yes—
143C2×C4D4×D25112yes—
144C8—10042no—
145C22—21no—
146C22—21no—
147C22—10042no—
148C22—10042no—
149C22—10042no—
150C22#2254868311yescharacteristic
151C22C2×#1686079711yescharacteristic
152C22#2254874011yescharacteristic
153C4—22no—
154C4—22no—
155C4—41no—
156C4—81no—
157C4—81no—
158C4—10042no—
159C2—21no—
160C2—21no—
161C2—20082no—
162C2#3063183711yescharacteristic
163C2#3063185911yescharacteristic
164C2#3063187611yescharacteristic
subgroupG/H
centerC22#22548683
commutator, cosocleC22×C4×C251C23
FrattiniC22×C4C22×D251
Fitting#400×C251C2
socleC22×C251#120
prime radical#30632268C2
hypercenter#400D251
hypocenterC251#1616
LCSG ▷ C22×C4×C251 ▷ C22×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #400

Complements

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

NH ≅ G/Ntype
2. #30632005C2semidirect
3. #30631420C2semidirect
4. #30631413C2semidirect
5. #400×C251C2semidirect
82. C251#1616semidirect
86. #400D251semidirect

Maximal subgroups

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

subgroupindex
#306322682
#306320052
#306314202
#306314132
#400×C2512
#1616251