Finite Group Databasebeta

← #145246

Subgroups — #145246

Order 512.

280 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 62 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×#1693C211yesmaximal, characteristic
2#12028C211yesmaximal, characteristic, complemented
3#11975C211yesmaximal, characteristic, complemented
4#11975C211yesmaximal, characteristic, complemented
5#11636C211yesmaximal, characteristic
6#8624C211yesmaximal, characteristic
7#7294C211yesmaximal, characteristic, complemented
8C2×#500C2211yescharacteristic
9C2×#461—21no—
10C2×#436C2211yescharacteristic
11#2163C2211yescharacteristic
12#2163C2211yescharacteristic
13#1693—22no—
14#1693—22no—
15#1568C2211yescharacteristic
16#1568C2211yescharacteristic
17C2×#334C2211yescharacteristic
18C2×#332—21no—
19#1466—21no—
20#1368—22no—
21#1365—22no—
22#1365—22no—
23#1330—22no—
24#1290—21no—
25C22×SD16—41no—
26C22×M16—21no—
27C2×C4⋊Q8—21no—
28C2×C4×D4D411yescharacteristic
29#500—21no—
30#500—21no—
31#500—21no—
32#500D412yes—
33#461—24no—
34#461—42no—
35#436—21no—
36#436—21no—
37#436—21no—
38#436—21no—
39#432—21no—
40#432—21no—
41C2×#107—21no—
42C2×#107—21no—
43C2×C4⋊C8—21no—
44C2×C4⋊C8D411yescharacteristic
45C2×#103—41no—
46C2×C22⋊C8—21no—
47C2×C4×C8C2311yescharacteristic
48C4⋊C16—21no—
49C4⋊C16D412yes—
50C22⋊C16—21no—
51C22⋊C16—21no—
52#346—21no—
53#346—21no—
54#335—21no—
55#335—21no—
56#334—21no—
57#334—21no—
58#334—21no—
59#334D412yes—
60#332—24no—
61#323—42no—
62C22×Q8—41no—
63C22×D4SD1611yescharacteristic
64C23×C4—21no—
65C2×SD16—44no—
66C2×SD16—84no—
67C2×M16—41no—
68C2×M16—41no—
69C2×M16—82no—
70C22×C8—21no—
71C22×C8—41no—
72C22×C8#4111yescharacteristic
73C22×C8#4111yescharacteristic
74C4⋊Q8—22no—
75C4⋊Q8—41no—
76C4⋊Q8—41no—
77C4⋊Q8—41no—
78C4×D4—21no—
79C4×D4—21no—
80C4×D4—21no—
81C4×D4—21no—
82C2×C4⋊C4—41no—
83C2×C4⋊C4—41no—
84C2×C4⋊C4SD1611yescharacteristic
85C2×C22⋊C4—21no—
86C2×C42C2×D411yescharacteristic
87C2×C16—22no—
88C2×C16—41no—
89#107—22no—
90#107—22no—
91#107—41no—
92#107—41no—
93C4⋊C8—21no—
94C4⋊C8—22no—
95C4⋊C8—21no—
96C4⋊C8—41no—
97C4⋊C8—41no—
98C4⋊C8—41no—
99#103—44no—
100C22⋊C8—41no—
101C22⋊C8—41no—
102C22⋊C8—41no—
103C22⋊C8—41no—
104#97—41no—
105#97—41no—
106C4×C8C2×D411yescharacteristic
107C4×C8C2×D411yescharacteristic
108C4×C8C2×D411yescharacteristic
109C4×C8#4111yescharacteristic
110C24—21no—
111C2×Q8—42no—
112C2×Q8—82no—
113C2×Q8—82no—
114C2×Q8—81no—
115C2×D4—21no—
116C2×D4—21no—
117C2×D4—42no—
118C22×C4—21no—
119C22×C4—41no—
120C22×C4—41no—
121C22×C4—81no—
122C22×C4—81no—
123C22×C4—81no—
124C22×C4#12311yescharacteristic
125C22×C4C2×SD1611yescharacteristic
126C22×C4AGL(1,ℤ/8)11yescharacteristic
127SD16—88no—
128M16—82no—
129M16—82no—
130C2×C8—22no—
131C2×C8—41no—
132C2×C8—41no—
133C2×C8—42no—
134C2×C8—82no—
135C2×C8—81no—
136C2×C8—81no—
137C2×C8—81no—
138C2×C8#12111yescharacteristic
139C2×C8#12111yescharacteristic
140C2×C8#12311yescharacteristic
141C2×C8#12411yescharacteristic
142C4⋊C4—21no—
143C4⋊C4—21no—
144C4⋊C4—42no—
145C4⋊C4—42no—
146C4⋊C4—81no—
147C4⋊C4—81no—
148C22⋊C4—41no—
149C22⋊C4—41no—
150C42#12011yescharacteristic
151C42#12111yescharacteristic
152C42#13511yescharacteristic
153C42AGL(1,ℤ/8)11yescharacteristic
154C16—42no—
155C16—81no—
156C23—41no—
157C23—42no—
158C23—81no—
159C23—81no—
160C23#48111yescharacteristic
161Q8—82no—
162Q8—82no—
163Q8—162no—
164D4—42no—
165D4—81no—
166C2×C4—22no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—82no—
172C2×C4—81no—
173C2×C4—81no—
174C2×C4—81no—
175C2×C4—82no—
176C2×C4—162no—
177C2×C4#39411yescharacteristic
178C2×C4#44911yescharacteristic
179C2×C4#45011yescharacteristic
180C2×C4#46511yescharacteristic
181C2×C4#46911yescharacteristic
182C2×C4#48211yescharacteristic
183C8—21no—
184C8—21no—
185C8—21no—
186C8—82no—
187C8—81no—
188C8—161no—
189C8—161no—
190C8#47112yes—
191C22—21no—
192C22—21no—
193C22—81no—
194C22—81no—
195C22—81no—
196C22—162no—
197C22#203411yescharacteristic
198C22#223812yes—
199C4—21no—
200C4—21no—
201C4—21no—
202C4—21no—
203C4—21no—
204C4—81no—
205C4—162no—
206C4#204611yescharacteristic
207C4#204911yescharacteristic
208C2—22no—
209C2—161no—
210C2#1044511yescharacteristic
211C2#1339811yescharacteristic
212C2#1340511yescharacteristic
subgroupG/H
center, socleC22#2034
commutator, Frattini, cosocleC2×C4×C8C23
prime radicalC22×D4SD16
LCSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#436 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #12028C2semidirect
3. #11975C2semidirect
4. #11975C2semidirect
7. #7294C2semidirect

Maximal subgroups

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

subgroupindex
C2×#16932
#120282
#119752
#116362
#86242
#72942