Finite Group Databasebeta

← #45665362

Subgroups — #45665362

Order 32896.

406 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 75 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
1C2×#22551617C212yesmaximal, complemented
2C2×#22551519C212yesmaximal, complemented
3#30656638C211yesmaximal, characteristic, complemented
4#30656506C211yesmaximal, characteristic, complemented
5#537×C257C211yesmaximal, characteristic, complemented
6C22×D1028C2211yescharacteristic
7#22551617—24no—
8#22551617—24no—
9C2×#16861605—22no—
10C2×#16861603—22no—
11C2×#16861602C2212yes—
12C2×#16861595—22no—
13C2×C4×Dic257C2211yescharacteristic
14#22551519—24no—
15C2×#41×C257C2212yes—
16C22⋊Q8×C257—22no—
17#121×C257—22no—
18#121×C257—22no—
19C2×C4⋊C4×C257C2211yescharacteristic
20C23×D257—21no—
21C2×D1028—22no—
22C2×D1028—22no—
23C2×D1028—21no—
24C2×D1028D412yescomplemented
25C22×Dic257—21no—
26#16861605—28no—
27#16861603—28no—
28#16861602—22no—
29#16861602D414yescomplemented
30C2×#14523339—22no—
31#16861595—44no—
32C4×Dic257—22no—
33#41×C257—22no—
34#41×C257—22no—
35#41×C257—22no—
36#41×C257—22no—
37C2×Q8×C257D412yes—
38C2×D4×C257—22no—
39C2×D4×C257—22no—
40C2×D4×C257D412yes—
41C22×C4×C257—22no—
42C22×C4×C257D412yes—
43C22×C4×C257C2311yescharacteristic
44C4⋊C4×C257—22no—
45C4⋊C4×C257—22no—
46C22⋊C4×C257—24no—
47C22×D257—22no—
48C22×D257—42no—
49C22×D257—42no—
50C22×D257—41no—
51C22×D257—41no—
52D1028—24no—
53D1028—42no—
54C2×Dic257—22no—
55C2×Dic257—41no—
56C2×Dic257—41no—
57#14523339—24no—
58C23×C257—22no—
59C23×C257C2×D411yescharacteristic
60Q8×C257—22no—
61Q8×C257—22no—
62D4×C257—22no—
63D4×C257—22no—
64D4×C257—42no—
65D4×C257—42no—
66D4×C257—82no—
67C2×C4×C257—22no—
68C2×C4×C257—22no—
69C2×C4×C257—42no—
70C2×C4×C257—42no—
71C2×C4×C257—42no—
72C2×C4×C257C2×D411yescharacteristic
73C2×C4×C257C2×D412yes—
74C2×C4×C257C2×D412yes—
75C2×C4×C257#4111yescharacteristic
76C22×C257—21no—
77C22×C257—21no—
78C22×C257—42no—
79C22×C257—42no—
80C22×C257—42no—
81C22×C257#12011yescharacteristic
82C22×C257#12111yescharacteristic
83C22×C257#12711yescharacteristic
84C2×D257—44no—
85C2×D257—42no—
86C2×D257—82no—
87C2×D257—81no—
88C2×D257—81no—
89C4×C257—42no—
90C4×C257—42no—
91C4×C257#12012yes—
92C4×C257#12112yes—
93Dic257—42no—
94C2×C257—21no—
95C2×C257—21no—
96C2×C257—82no—
97C2×C257#39211yescharacteristic
98C2×C257#45311yescharacteristic
99C2×C257#45411yescharacteristic
100D257—82no—
101C257#200611yescharacteristic, complemented, Sylow
102#2006—2571nomaximal, Sylow
103C2×AGL(1,ℤ/8)—2572no—
104#537D25711yescharacteristic, complemented
105C2×#104—2572no—
106#390—2571no—
107#343—2571no—
108C2×#41C2×D25712yes—
109C22×D4—2571no—
110AGL(1,ℤ/8)—5144no—
111AGL(1,ℤ/8)—5144no—
112C2×SD16—5142no—
113C2×D8—5142no—
114C2×M16—2572no—
115C22⋊Q8—22no—
116#121—22no—
117#121—22no—
118C2×C4⋊C4C2×D25711yescharacteristic
119C2×C22⋊C4—5142no—
120C2×C42—2571no—
121#104—5144no—
122C24—5141no—
123#41—22no—
124#41—22no—
125#41—22no—
126#41—22no—
127C2×Q8C22⋊D25712yes—
128C2×D4—2572no—
129C2×D4—22no—
130C2×D4—22no—
131C2×D4—5142no—
132C2×D4—5142no—
133C2×D4—5141no—
134C2×D4C22⋊D25712yes—
135C22×C4—22no—
136C22×C4—5141no—
137C22×C4C22⋊D25712yes—
138C22×C4C22×D25711yescharacteristic
139SD16—5148no—
140D8—5148no—
141M16—2574no—
142M16—5142no—
143C2×C8—5142no—
144C4⋊C4—22no—
145C4⋊C4—22no—
146C22⋊C4—24no—
147C22⋊C4—10284no—
148C42—5142no—
149C23—22no—
150C23—5142no—
151C23—10282no—
152C23—10282no—
153C23—10281no—
154C23—10281no—
155C23C2×C22⋊D25711yescharacteristic
156Q8—22no—
157Q8—22no—
158D4—22no—
159D4—22no—
160D4—5144no—
161D4—42no—
162D4—42no—
163D4—10282no—
164D4—82no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—5142no—
168C2×C4—42no—
169C2×C4—42no—
170C2×C4—42no—
171C2×C4—10281no—
172C2×C4—10281no—
173C2×C4D4×D25711yescharacteristic
174C2×C4D4×D25712yes—
175C2×C4C2×C22⋊D25712yes—
176C2×C4#1686161711yescharacteristic
177C8—5144no—
178C22—21no—
179C22—21no—
180C22—42no—
181C22—42no—
182C22—42no—
183C22—10284no—
184C22—10282no—
185C22—20562no—
186C22—20561no—
187C22—20561no—
188C22#2255156111yescharacteristic
189C22#2255156311yescharacteristic
190C22#2255158811yescharacteristic
191C4—42no—
192C4—42no—
193C4—10282no—
194C4#2255156012yes—
195C4#2255156412yes—
196C2—21no—
197C2—21no—
198C2—82no—
199C2—20562no—
200C2#3065664511yescharacteristic
201C2#3065690511yescharacteristic
202C2#3065691011yescharacteristic
subgroupG/H
centerC22#22551561
commutator, cosocleC22×C4×C257C23
FrattiniC22×C4C22×D257
Fitting#537×C257C2
socleC22×C257#120
hypercenter#537D257
hypocenterC257#2006
LCSG ▷ C22×C4×C257 ▷ C2×C257 ▷ C257
UCSC1 ◁ C22 ◁ C22×C4 ◁ #537

Complements

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

NH ≅ G/Ntype
1. C2×#22551617C2semidirect
2. C2×#22551519C2semidirect
3. #30656638C2semidirect
4. #30656506C2semidirect
5. #537×C257C2semidirect
24. C2×D1028D4semidirect
29. #16861602D4semidirect
101. C257#2006semidirect
104. #537D257semidirect

Maximal subgroups

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

subgroupindex
C2×#225516172
C2×#225515192
#306566382
#306565062
#537×C2572
#2006257