Finite Group Databasebeta

← #45663548

Subgroups — #45663548

Order 32896.

212 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 57 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#30657087C211yesmaximal, characteristic, complemented
2#30657017C211yesmaximal, characteristic, complemented
3#30656942C211yesmaximal, characteristic, complemented
4#30656486C211yesmaximal, characteristic, complemented
5D4⋊#14523339C211yesmaximal, characteristic, complemented
6D4⋊#14523339C211yesmaximal, characteristic, complemented
7#555×C257C211yesmaximal, characteristic
8C2×#16861606—21no—
9#22551592C2211yescharacteristic
10#22551525—21no—
11C4⋊#14523339C2211yescharacteristic
12C4⋊#14523339C2211yescharacteristic
13#22551516—22no—
14#22551515—21no—
15#22551513—21no—
16C4×#14523339C2211yescharacteristic
17C4⋊Q8×C257C2211yescharacteristic
18#124×C257—22no—
19#121×C257—21no—
20#121×C257—21no—
21C4×D4×C257C2211yescharacteristic
22C4×D4×C257C2211yescharacteristic
23#117×C257—21no—
24#117×C257—21no—
25C2×Dic514—21no—
26#16861606—42no—
27C2×#14523339—21no—
28C2×#14523339—21no—
29C2×#14523339—21no—
30C4⋊Dic257—21no—
31C4⋊Dic257—21no—
32C2×Q8×C257D412yes—
33C2×D4×C257—22no—
34C2×D4×C257D411yescharacteristic
35C2×D4×C257D411yescharacteristic
36C22×C4×C257—21no—
37C22×C4×C257—21no—
38C4⋊C4×C257—21no—
39C4⋊C4×C257—21no—
40C4⋊C4×C257D411yescharacteristic
41C4⋊C4×C257D411yescharacteristic
42C22⋊C4×C257—22no—
43C22⋊C4×C257—22no—
44C22⋊C4×C257—21no—
45C22⋊C4×C257—21no—
46C42×C257—22no—
47C42×C257C2311yescharacteristic
48Dic514—41no—
49Dic514—41no—
50C2×Dic257—41no—
51#14523339—41no—
52#14523339—41no—
53#14523339—41no—
54C23×C257—21no—
55C23×C257—21no—
56Q8×C257—42no—
57D4×C257—21no—
58D4×C257—21no—
59D4×C257—21no—
60D4×C257—21no—
61D4×C257—82no—
62C2×C4×C257—21no—
63C2×C4×C257—21no—
64C2×C4×C257—22no—
65C2×C4×C257—41no—
66C2×C4×C257—41no—
67C2×C4×C257—41no—
68C2×C4×C257—41no—
69C2×C4×C257C2×D411yescharacteristic
70C2×C4×C257C2×D411yescharacteristic
71C2×C4×C257C2×D411yescharacteristic
72C22×C257—41no—
73C22×C257—41no—
74C22×C257—41no—
75C22×C257—41no—
76C22×C257—41no—
77C22×C257—41no—
78C22×C257#12011yescharacteristic
79C4×C257—21no—
80C4×C257—41no—
81C4×C257—41no—
82C4×C257—42no—
83C4×C257#13511yescharacteristic
84C4×C257#13511yescharacteristic
85C4×C257AGL(1,ℤ/8)11yescharacteristic
86C4×C257#13711yescharacteristic
87Dic257—81no—
88C2×C257—81no—
89C2×C257—81no—
90C2×C257#44911yescharacteristic
91C2×C257#45211yescharacteristic
92C2×C257#45511yescharacteristic
93C257#164611yescharacteristic, complemented, Sylow
94#1646—2571nomaximal, Sylow
95#555D25711yescharacteristic
96#487—2571no—
97#474—2571no—
98#461—2571no—
99#333—2571no—
100D4⋊C8—2571no—
101D4⋊C8—2571no—
102C2×SD16—5141no—
103C4⋊Q8—2571no—
104C4⋊Q8C2×D25711yescharacteristic
105#124—22no—
106#121—21no—
107#121—21no—
108C4×D4C2×D25711yescharacteristic
109C4×D4C2×D25711yescharacteristic
110#117—21no—
111#117—21no—
112#107—5141no—
113C4⋊C8—2571no—
114C4⋊C8—2571no—
115#103—5142no—
116#103—5141no—
117#102—5141no—
118C4×C8—2571no—
119C2×Q8—5141no—
120C2×Q8C22⋊D25712yes—
121C2×D4—22no—
122C2×D4C22⋊D25711yescharacteristic
123C2×D4C22⋊D25711yescharacteristic
124C22×C4—21no—
125C22×C4—21no—
126SD16—10282no—
127C2×C8—5141no—
128C2×C8—5141no—
129C2×C8—5141no—
130C4⋊C4—21no—
131C4⋊C4—21no—
132C4⋊C4—5141no—
133C4⋊C4—5141no—
134C4⋊C4C22⋊D25711yescharacteristic
135C4⋊C4C22⋊D25711yescharacteristic
136C22⋊C4—22no—
137C22⋊C4—22no—
138C22⋊C4—21no—
139C22⋊C4—21no—
140C42—22no—
141C42C22×D25711yescharacteristic
142C23—21no—
143C23—21no—
144Q8—42no—
145Q8—10281no—
146Q8—10281no—
147D4—21no—
148D4—21no—
149D4—21no—
150D4—21no—
151D4—82no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—22no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—10281no—
160C2×C4C2×C22⋊D25711yescharacteristic
161C2×C4C2×C22⋊D25711yescharacteristic
162C2×C4C2×C22⋊D25711yescharacteristic
163C8—10281no—
164C8—10281no—
165C8—10281no—
166C22—41no—
167C22—41no—
168C22—41no—
169C22—41no—
170C22—41no—
171C22—41no—
172C22#2255156111yescharacteristic
173C4—21no—
174C4—41no—
175C4—41no—
176C4—42no—
177C4—20561no—
178C4#2255161211yescharacteristic
179C4#2255161211yescharacteristic
180C4#2255161711yescharacteristic
181C4#2255162211yescharacteristic
182C2—81no—
183C2—81no—
184C2#3065688211yescharacteristic
185C2#3065689911yescharacteristic
186C2#3065691511yescharacteristic
subgroupG/H
centerC22#22551561
commutator, cosocleC42×C257C23
FrattiniC42C22×D257
Fitting, prime radical#555×C257C2
socleC22×C257#120
hypercenter#555D257
hypocenterC257#1646
LCSG ▷ C42×C257 ▷ C22×C257 ▷ C257
UCSC1 ◁ C22 ◁ C42 ◁ #555

Complements

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

NH ≅ G/Ntype
1. #30657087C2semidirect
2. #30657017C2semidirect
3. #30656942C2semidirect
4. #30656486C2semidirect
5. D4⋊#14523339C2semidirect
6. D4⋊#14523339C2semidirect
93. C257#1646semidirect

Maximal subgroups

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

subgroupindex
#306570872
#306570172
#306569422
#306564862
D4⋊#145233392
#555×C2572
#1646257