Finite Group Databasebeta

← #45663547

Subgroups — #45663547

Order 32896.

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

normal subgroups only

This group has 212 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#30657452C211yesmaximal, characteristic
2#30657089C211yesmaximal, characteristic, complemented
3#30656939C211yesmaximal, characteristic, complemented
4#30656485C211yesmaximal, characteristic, complemented
5D4⋊#14523339C211yesmaximal, characteristic, complemented
6#30656464C211yesmaximal, characteristic, complemented
7#474×C257C211yesmaximal, characteristic, complemented
8C2×#16861605—21no—
9#22551592C2211yescharacteristic
10#22551578—22no—
11#22551566—21no—
12#22551563—21no—
13C4×D1028C2211yescharacteristic
14#22551551—21no—
15#22551550—21no—
16C4⋊#14523339C2211yescharacteristic
17#22551516—21no—
18#22551515—22no—
19C4×#14523339C2211yescharacteristic
20C4⋊Q8×C257C2211yescharacteristic
21C4×D4×C257C2211yescharacteristic
22#107×C257—21no—
23C4⋊C8×C257C2211yescharacteristic
24#102×C257—21no—
25C2×Dic514D412yes—
26C2×D1028D411yescharacteristic
27C2×C22⋊D257—22no—
28C2×C4×D257—21no—
29#16861605—42no—
30C2×#14523339—21no—
31C2×#14523339—21no—
32C4⋊Dic257—21no—
33C4⋊Dic257—21no—
34C4⋊Dic257D411yescharacteristic
35C22⋊Dic257—22no—
36#16861595—22no—
37#16861595—21no—
38C4×Dic257—22no—
39C2×Q8×C257—21no—
40C2×D4×C257D411yescharacteristic
41C22×C4×C257—21no—
42C2×C8×C257—21no—
43C4⋊C4×C257—21no—
44C4⋊C4×C257—21no—
45C4⋊C4×C257D411yescharacteristic
46C22⋊C4×C257—21no—
47C42×C257C2311yescharacteristic
48C22×D257—21no—
49Dic514—42no—
50D1028—21no—
51D1028—21no—
52C22⋊D257—82no—
53C2×Dic257—21no—
54C2×Dic257—22no—
55C4×D257—41no—
56C4×D257—41no—
57#14523339—41no—
58#14523339—41no—
59C23×C257—21no—
60Q8×C257—41no—
61Q8×C257—41no—
62D4×C257—21no—
63D4×C257—21no—
64C2×C4×C257—21no—
65C2×C4×C257—41no—
66C2×C4×C257—41no—
67C2×C4×C257—41no—
68C2×C4×C257C2×D411yescharacteristic
69C2×C4×C257C2×D411yescharacteristic
70C2×C4×C257C2×D411yescharacteristic
71C8×C257—41no—
72C22×C257—41no—
73C22×C257—41no—
74C22×C257—41no—
75C22×C257#12011yescharacteristic
76C2×D257—41no—
77C2×D257—41no—
78C2×D257—41no—
79C4×C257—21no—
80C4×C257—41no—
81C4×C257—81no—
82C4×C257#13511yescharacteristic
83C4×C257#13511yescharacteristic
84C4×C257AGL(1,ℤ/8)11yescharacteristic
85C4×C257#13711yescharacteristic
86Dic257—41no—
87Dic257—42no—
88C2×C257—81no—
89C2×C257#44911yescharacteristic
90C2×C257#45211yescharacteristic
91C2×C257#45511yescharacteristic
92D257—81no—
93C257#164611yescharacteristic, complemented, Sylow
94#1646—2571nomaximal, Sylow
95#555—2571no—
96#487—2571no—
97#474D25711yescharacteristic, complemented
98#461—2571no—
99#333—2571no—
100D4⋊C8—2571no—
101D4⋊C8—2571no—
102C2×SD16—5141no—
103C4⋊Q8—2571no—
104C4⋊Q8C2×D25711yescharacteristic
105#124—5142no—
106#121—5141no—
107#121—5141no—
108C4×D4—2571no—
109C4×D4C2×D25711yescharacteristic
110#117—5141no—
111#117—5141no—
112#107—21no—
113C4⋊C8—2571no—
114C4⋊C8C2×D25711yescharacteristic
115#103—5141no—
116#103—5142no—
117#102—21no—
118C4×C8—2571no—
119C2×Q8—2572no—
120C2×Q8—21no—
121C2×D4—2571no—
122C2×D4—5142no—
123C2×D4C22⋊D25711yescharacteristic
124C22×C4—21no—
125C22×C4—5141no—
126SD16—10282no—
127C2×C8—21no—
128C2×C8—5141no—
129C2×C8—5141no—
130C4⋊C4—2571no—
131C4⋊C4—21no—
132C4⋊C4—21no—
133C4⋊C4—5141no—
134C4⋊C4—5141no—
135C4⋊C4C22⋊D25711yescharacteristic
136C22⋊C4—21no—
137C22⋊C4—5142no—
138C22⋊C4—5142no—
139C22⋊C4—5141no—
140C42—5142no—
141C42C22×D25711yescharacteristic
142C23—21no—
143C23—5141no—
144Q8—41no—
145Q8—41no—
146Q8—10282no—
147D4—21no—
148D4—21no—
149D4—5141no—
150D4—5141no—
151D4—20562no—
152C2×C4—21no—
153C2×C4—5141no—
154C2×C4—5142no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—10281no—
159C2×C4—10281no—
160C2×C4D4×D25711yescharacteristic
161C2×C4D4×D25711yescharacteristic
162C2×C4C2×C22⋊D25711yescharacteristic
163C8—41no—
164C8—10281no—
165C8—10281no—
166C22—41no—
167C22—41no—
168C22—41no—
169C22—10281no—
170C22—10281no—
171C22—10281no—
172C22#2255156011yescharacteristic
173C4—21no—
174C4—41no—
175C4—10281no—
176C4—10282no—
177C4—81no—
178C4#2255161111yescharacteristic
179C4#2255161211yescharacteristic
180C4#2255161611yescharacteristic
181C4#2255162311yescharacteristic
182C2—81no—
183C2—20561no—
184C2#3065687911yescharacteristic
185C2#3065689711yescharacteristic
186C2#3065691311yescharacteristic
subgroupG/H
centerC22#22551560
commutator, cosocleC42×C257C23
FrattiniC42C22×D257
Fitting#474×C257C2
socleC22×C257#120
prime radical#30657452C2
hypercenter#474D257
hypocenterC257#1646
LCSG ▷ C42×C257 ▷ C22×C257 ▷ C257
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

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

NH ≅ G/Ntype
2. #30657089C2semidirect
3. #30656939C2semidirect
4. #30656485C2semidirect
5. D4⋊#14523339C2semidirect
6. #30656464C2semidirect
7. #474×C257C2semidirect
93. C257#1646semidirect
97. #474D257semidirect

Maximal subgroups

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

subgroupindex
#306574522
#306570892
#306569392
#306564852
D4⋊#145233392
#306564642
#474×C2572
#1646257