Finite Group Databasebeta

← #44549367

Subgroups — #44549367

Order 32128.

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

normal subgroups only

This group has 254 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#30632000C212yesmaximal, complemented
3#30631420C212yesmaximal, complemented
4#30631413C211yesmaximal, characteristic, complemented
5#398×C251C211yesmaximal, characteristic, complemented
6C2×#16860810—21no—
7#22548695—22no—
8#22548690—22no—
9#22548689—21no—
10#22548689C2212yes—
11#22548648—22no—
12#22548635—22no—
13C22⋊(C251⋊C8)C2212yes—
14C2×C4⋊C4×C251—21no—
15C2×C4⋊C4×C251C2211yescharacteristic
16C2×C4⋊C4×C251C2211yescharacteristic
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—
39C4⋊C4×C251—42no—
40C22×D251—22no—
41Dic502—42no—
42D1004—42no—
43D1004—42no—
44D1004—81no—
45C22⋊D251—42no—
46C22⋊D251—42no—
47C2×Dic251—22no—
48C4×D251—44no—
49C251⋊C8—42no—
50C23×C251C2×D411yescharacteristic
51C2×C4×C251—21no—
52C2×C4×C251—21no—
53C2×C4×C251—22no—
54C2×C4×C251—21no—
55C2×C4×C251—41no—
56C2×C4×C251—41no—
57C2×C4×C251—41no—
58C2×C4×C251—41no—
59C2×C4×C251—41no—
60C2×C4×C251—41no—
61C2×C4×C251—41no—
62C2×C4×C251D812yes—
63C2×C4×C251C2×D412yes—
64C22×C251—21no—
65C22×C251—21no—
66C22×C251#12011yescharacteristic
67C22×C251C2×D811yescharacteristic
68C22×C251AGL(1,ℤ/8)11yescharacteristic
69C2×D251—42no—
70C2×D251—42no—
71C2×D251—42no—
72C4×C251—22no—
73C4×C251—22no—
74C4×C251—42no—
75C4×C251—41no—
76C4×C251—81no—
77Dic251—42no—
78C2×C251—21no—
79C2×C251—21no—
80C2×C251#44711yescharacteristic
81C2×C251#45611yescharacteristic
82C2×C251#45811yescharacteristic
83D251—82no—
84C251#161011yescharacteristic, complemented, Sylow
85#1610—2511nomaximal, Sylow
86#537—2511no—
87#480—2512no—
88#398D25111yescharacteristic, complemented
89#327—2512no—
90#324—2511no—
91C2×#41—5021no—
92C22⋊Q8—5022no—
93#121—5022no—
94#121—5021no—
95#121—2512no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4C2×D25111yescharacteristic
98C2×C4⋊C4C2×D25111yescharacteristic
99#107—5022no—
100#102—5022no—
101C22⋊C8—2512no—
102#95—21no—
103#95C2×D25111yescharacteristic
104#41—10042no—
105#41—10042no—
106C2×Q8—5021no—
107C2×D4—5021no—
108C2×D4—5021no—
109C2×D4—5022no—
110C2×D4—2512no—
111C22×C4—21no—
112C22×C4—21no—
113C22×C4—5022no—
114C22×C4C22⋊D25111yescharacteristic
115C22×C4C22⋊D25111yescharacteristic
116C22×C4C22×D25111yescharacteristic
117C2×C8—5022no—
118C4⋊C4—22no—
119C4⋊C4—22no—
120C4⋊C4—42no—
121C4⋊C4—5021no—
122C4⋊C4—2512no—
123C22⋊C4—5022no—
124C22⋊C4—5022no—
125C23—5022no—
126C23C2×C22⋊D25111yescharacteristic
127Q8—10042no—
128D4—20081no—
129D4—10042no—
130D4—10042no—
131D4—10042no—
132D4—10042no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—22no—
136C2×C4—21no—
137C2×C4—41no—
138C2×C4—41no—
139C2×C4—41no—
140C2×C4—41no—
141C2×C4—41no—
142C2×C4—41no—
143C2×C4—41no—
144C2×C4—10044no—
145C2×C4—5022no—
146C2×C4#1686079512yes—
147C2×C4D4×D25112yes—
148C8—10042no—
149C22—21no—
150C22—21no—
151C22—10042no—
152C22—10042no—
153C22—10042no—
154C22#2254868311yescharacteristic
155C22C2×#1686079511yescharacteristic
156C22#2254874011yescharacteristic
157C4—22no—
158C4—22no—
159C4—42no—
160C4—41no—
161C4—81no—
162C4—10042no—
163C2—21no—
164C2—21no—
165C2—20082no—
166C2#3063182011yescharacteristic
167C2#3063186911yescharacteristic
168C2#3063187611yescharacteristic
subgroupG/H
centerC22#22548683
commutator, cosocleC22×C4×C251C23
FrattiniC22×C4C22×D251
Fitting#398×C251C2
socleC22×C251#120
prime radical#30632268C2
hypercenter#398D251
hypocenterC251#1610
LCSG ▷ C22×C4×C251 ▷ C22×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #398

Complements

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

NH ≅ G/Ntype
2. #30632000C2semidirect
3. #30631420C2semidirect
4. #30631413C2semidirect
5. #398×C251C2semidirect
84. C251#1610semidirect
88. #398D251semidirect

Maximal subgroups

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

subgroupindex
#306322682
#306320002
#306314202
#306314132
#398×C2512
#1610251