Finite Group Databasebeta

← #45665424

Subgroups — #45665424

Order 32896.

350 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 69 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#30657537C211yesmaximal, characteristic, complemented
2#30657535C212yesmaximal, complemented
3#30656703C212yesmaximal, complemented
4#30656495C211yesmaximal, characteristic, complemented
5#408×C257C211yesmaximal, characteristic, complemented
6C2×#16861618C2212yes—
7#22551625—22no—
8#22551623—22no—
9#22551622—22no—
10#22551620—22no—
11#22551618—22no—
12#22551617—22no—
13#22551614—22no—
14#22551612—24no—
15C2×#16861602C2212yes—
16#22551504—24no—
17C2×#41×C257C2211yescharacteristic
18C2×M16×C257C2211yescharacteristic
19C22×C8×C257C2211yescharacteristic
20C22⋊C8×C257—22no—
21#16861618—22no—
22#16861618—22no—
23#16861618—22no—
24#16861618—22no—
25C2×Dic514D412yes—
26C2×D1028D412yes—
27C2×C22⋊D257—22no—
28C2×C4×D257—22no—
29Dic1028—22no—
30#16861608—24no—
31#16861607—24no—
32#16861606—24no—
33#16861605—24no—
34D2056—22no—
35#16861603—24no—
36#16861602—22no—
37#16861602—22no—
38C2×#14523339D414yes—
39#41×C257—22no—
40#41×C257—22no—
41C2×Q8×C257D411yescharacteristic
42C2×D4×C257—21no—
43C2×D4×C257D411yescharacteristic
44C22×C4×C257—21no—
45C22×C4×C257C2311yescharacteristic
46M16×C257—22no—
47C2×C8×C257—21no—
48C2×C8×C257—22no—
49C2×C8×C257D412yes—
50C22×D257—22no—
51Dic514—22no—
52Dic514—22no—
53D1028—22no—
54D1028—22no—
55C22⋊D257—42no—
56C22⋊D257—42no—
57C2×Dic257—22no—
58C4×D257—42no—
59C4×D257—42no—
60#14523339—24no—
61C23×C257—21no—
62C23×C257#4111yescharacteristic
63Q8×C257—22no—
64D4×C257—22no—
65D4×C257—42no—
66C2×C4×C257—21no—
67C2×C4×C257—41no—
68C2×C4×C257—41no—
69C2×C4×C257C2×D412yes—
70C2×C4×C257C2×D412yes—
71C2×C4×C257C2×D411yescharacteristic
72C2×C4×C257C2×D411yescharacteristic
73C8×C257—22no—
74C8×C257—41no—
75C22×C257—41no—
76C22×C257—42no—
77C22×C257—41no—
78C22×C257#12112yes—
79C22×C257#12111yescharacteristic
80C2×D257—44no—
81C2×D257—42no—
82C4×C257—41no—
83C4×C257#12011yescharacteristic
84C4×C257#12012yes—
85C4×C257#12711yescharacteristic
86Dic257—42no—
87C2×C257—22no—
88C2×C257—21no—
89C2×C257—81no—
90C2×C257#39211yescharacteristic
91D257—82no—
92C257#201711yescharacteristic, complemented, Sylow
93#2017—2571nomaximal, Sylow
94#575—2572no—
95#575—2571no—
96#408—2572no—
97#408D25711yescharacteristic, complemented
98#338—2571no—
99C2×#41—2572no—
100C2×#41C2×D25711yescharacteristic
101#137—5142no—
102#137—5142no—
103#137—5142no—
104AGL(1,ℤ/8)—5142no—
105AGL(1,ℤ/8)—5142no—
106AGL(1,ℤ/8)—5142no—
107#135—5144no—
108#135—5142no—
109C2×M16—2572no—
110C2×M16C2×D25711yescharacteristic
111C22×C8C2×D25711yescharacteristic
112C22⋊C8—22no—
113C22⋊C8—5144no—
114#41—22no—
115#41—22no—
116#41—5142no—
117#41—5142no—
118#41—5142no—
119#41—5142no—
120C2×Q8—2572no—
121C2×Q8D102811yescharacteristic
122C2×D4—2572no—
123C2×D4—21no—
124C2×D4—5142no—
125C2×D4D102811yescharacteristic
126C22×C4—21no—
127C22×C4—5142no—
128C22×C4C22×D25711yescharacteristic
129Q16—5142no—
130Q16—5144no—
131SD16—5144no—
132SD16—5144no—
133SD16—5144no—
134D8—5142no—
135D8—5144no—
136M16—22no—
137M16—5142no—
138M16—5142no—
139C2×C8—2574no—
140C2×C8—22no—
141C2×C8—21no—
142C2×C8C22⋊D25712yes—
143C23—21no—
144C23—5142no—
145C23#1686161811yescharacteristic
146Q8—22no—
147Q8—5142no—
148Q8—5142no—
149D4—22no—
150D4—5142no—
151D4—5142no—
152D4—42no—
153D4—10282no—
154D4—10282no—
155C2×C4—21no—
156C2×C4—5142no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—10282no—
160C2×C4—10282no—
161C2×C4D4×D25712yes—
162C2×C4D4×D25712yes—
163C2×C4C2×C22⋊D25711yescharacteristic
164C2×C4C2×D102811yescharacteristic
165C8—22no—
166C8—5144no—
167C8—41no—
168C22—41no—
169C22—42no—
170C22—41no—
171C22—10284no—
172C22—10282no—
173C22#2255156512yes—
174C22#2255156611yescharacteristic
175C4—41no—
176C4—10282no—
177C4#2255156011yescharacteristic
178C4#2255156212yes—
179C4#2255158811yescharacteristic
180C2—21no—
181C2—22no—
182C2—81no—
183C2—20562no—
184C2#3065664611yescharacteristic
subgroupG/H
centerC4#22551588
commutator, cosocleC22×C4×C257C23
FrattiniC22×C4C22×D257
Fitting#408×C257C2
socleC2×C257#392
hypercenter#408D257
hypocenterC257#2017
LCSG ▷ C22×C4×C257 ▷ C2×C257 ▷ C257
UCSC1 ◁ C4 ◁ C22×C4 ◁ #408

Complements

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

NH ≅ G/Ntype
1. #30657537C2semidirect
2. #30657535C2semidirect
3. #30656703C2semidirect
4. #30656495C2semidirect
5. #408×C257C2semidirect
92. C257#2017semidirect
97. #408D257semidirect

Maximal subgroups

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

subgroupindex
#306575372
#306575352
#306567032
#306564952
#408×C2572
#2017257