Finite Group Databasebeta

← #145216

Subgroups — #145216

Order 512.

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

normal subgroups only

This group has 322 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×#1687C211yesmaximal, characteristic
2#12028C211yesmaximal, characteristic, complemented
3#11964C211yesmaximal, characteristic, complemented
4#11964C211yesmaximal, characteristic, complemented
5#11636C211yesmaximal, characteristic, complemented
6#8624C211yesmaximal, characteristic, complemented
7#7294C211yesmaximal, characteristic, complemented
8C2×#493C2211yescharacteristic
9C2×#459—21no—
10C2×#436C2211yescharacteristic
11#2163C2211yescharacteristic
12#2163C2211yescharacteristic
13#1687—22no—
14#1687—22no—
15#1568C2211yescharacteristic
16#1568C2211yescharacteristic
17C2×#334C2211yescharacteristic
18C2×#331—21no—
19#1466—21no—
20#1368—22no—
21#1361—22no—
22#1361—22no—
23#1330—22no—
24#1290—21no—
25C22×D8—21no—
26C22×D8—21no—
27C22×D8—41no—
28C22×M16—21no—
29C2×#127—21no—
30C2×C4×D4D411yescharacteristic
31#493—21no—
32#493—21no—
33#493—21no—
34#493D412yes—
35#459—24no—
36#459—42no—
37#436—21no—
38#436—21no—
39#436—21no—
40#436—21no—
41#432—21no—
42#432—21no—
43C2×C4⋊C8—21no—
44C2×C4⋊C8D411yescharacteristic
45C2×#102—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#331—24no—
61#323—42no—
62C22×D4—41no—
63C22×D4—41no—
64C22×D4—41no—
65C22×D4SD1611yescharacteristic
66C23×C4—21no—
67C2×D8—22no—
68C2×D8—22no—
69C2×D8—44no—
70C2×D8—41no—
71C2×D8—41no—
72C2×D8—82no—
73C2×D8—82no—
74C2×D8—84no—
75C2×M16—41no—
76C2×M16—41no—
77C2×M16—82no—
78C22×C8—21no—
79C22×C8—41no—
80C22×C8#4111yescharacteristic
81C22×C8#4111yescharacteristic
82#127—22no—
83#127—41no—
84#127—41no—
85#127—41no—
86C4×D4—21no—
87C4×D4—21no—
88C4×D4—21no—
89C4×D4—21no—
90C2×C4⋊C4SD1611yescharacteristic
91C2×C22⋊C4—21no—
92C2×C42C2×D411yescharacteristic
93C2×C16—22no—
94C2×C16—41no—
95C4⋊C8—21no—
96C4⋊C8—22no—
97C4⋊C8—21no—
98C4⋊C8—41no—
99C4⋊C8—41no—
100C4⋊C8—41no—
101#102—44no—
102C22⋊C8—41no—
103C22⋊C8—41no—
104C22⋊C8—41no—
105C22⋊C8—41no—
106#97—41no—
107#97—41no—
108C4×C8C2×D411yescharacteristic
109C4×C8C2×D411yescharacteristic
110C4×C8C2×D411yescharacteristic
111C4×C8#4111yescharacteristic
112C24—21no—
113C24—81no—
114C2×D4—21no—
115C2×D4—21no—
116C2×D4—42no—
117C2×D4—42no—
118C2×D4—42no—
119C2×D4—42no—
120C2×D4—82no—
121C2×D4—82no—
122C2×D4—82no—
123C2×D4—82no—
124C2×D4—81no—
125C2×D4—81no—
126C2×D4—81no—
127C2×D4—162no—
128C22×C4—21no—
129C22×C4—41no—
130C22×C4—41no—
131C22×C4—81no—
132C22×C4—81no—
133C22×C4#12311yescharacteristic
134C22×C4C2×SD1611yescharacteristic
135C22×C4AGL(1,ℤ/8)11yescharacteristic
136D8—82no—
137D8—82no—
138D8—82no—
139D8—88no—
140D8—82no—
141M16—82no—
142M16—82no—
143C2×C8—22no—
144C2×C8—41no—
145C2×C8—41no—
146C2×C8—42no—
147C2×C8—82no—
148C2×C8—81no—
149C2×C8—81no—
150C2×C8—81no—
151C2×C8#12111yescharacteristic
152C2×C8#12111yescharacteristic
153C2×C8#12311yescharacteristic
154C2×C8#12411yescharacteristic
155C4⋊C4—21no—
156C4⋊C4—21no—
157C22⋊C4—41no—
158C22⋊C4—41no—
159C42#12011yescharacteristic
160C42#12111yescharacteristic
161C42#13511yescharacteristic
162C42AGL(1,ℤ/8)11yescharacteristic
163C16—42no—
164C16—81no—
165C23—41no—
166C23—42no—
167C23—81no—
168C23—81no—
169C23—82no—
170C23—162no—
171C23—162no—
172C23—162no—
173C23#48111yescharacteristic
174D4—42no—
175D4—81no—
176D4—82no—
177D4—82no—
178D4—82no—
179D4—82no—
180D4—162no—
181D4—162no—
182D4—162no—
183D4—162no—
184C2×C4—22no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—82no—
190C2×C4—81no—
191C2×C4—81no—
192C2×C4—81no—
193C2×C4#39411yescharacteristic
194C2×C4#44911yescharacteristic
195C2×C4#45011yescharacteristic
196C2×C4#46511yescharacteristic
197C2×C4#46911yescharacteristic
198C2×C4#48211yescharacteristic
199C8—21no—
200C8—21no—
201C8—21no—
202C8—82no—
203C8—81no—
204C8—161no—
205C8—161no—
206C8#47212yes—
207C22—21no—
208C22—21no—
209C22—81no—
210C22—81no—
211C22—81no—
212C22—162no—
213C22—162no—
214C22—162no—
215C22—162no—
216C22—324no—
217C22#203411yescharacteristic
218C22#223812yes—
219C4—21no—
220C4—21no—
221C4—21no—
222C4—21no—
223C4—21no—
224C4—81no—
225C4#204511yescharacteristic
226C4#204811yescharacteristic
227C2—22no—
228C2—161no—
229C2—322no—
230C2#1044111yescharacteristic
231C2#1339811yescharacteristic
232C2#1340411yescharacteristic
subgroupG/H
center, socleC22#2034
commutator, Frattini, cosocleC2×C4×C8C23
prime radicalC2×#1687C2
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. #11964C2semidirect
4. #11964C2semidirect
5. #11636C2semidirect
6. #8624C2semidirect
7. #7294C2semidirect

Maximal subgroups

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

subgroupindex
C2×#16872
#120282
#119642
#116362
#86242
#72942