Finite Group Databasebeta

← #145231

Subgroups — #145231

Order 512.

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

normal subgroups only

This group has 303 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×#1688C211yesmaximal, characteristic
2#12028C211yesmaximal, characteristic, complemented
3#11964C211yesmaximal, characteristic
4#11964C211yesmaximal, characteristic
5#11644C211yesmaximal, characteristic, complemented
6#8625C211yesmaximal, characteristic, complemented
7#7294C211yesmaximal, characteristic, complemented
8C2×#493C2211yescharacteristic
9C2×#460—21no—
10C2×#446C2211yescharacteristic
11#2163C2211yescharacteristic
12#2163C2211yescharacteristic
13#1688—22no—
14#1688—22no—
15#1568C2211yescharacteristic
16#1568C2211yescharacteristic
17C2×#334C2211yescharacteristic
18C2×#331—21no—
19#1467—21no—
20#1368—22no—
21#1361—22no—
22#1361—22no—
23#1338—22no—
24#1290—21no—
25C22×SD16—41no—
26C22×D8—21no—
27C22×D8—21no—
28C2×#127—21no—
29C2×C4×Q8D411yescharacteristic
30#493—21no—
31#493—21no—
32#493—21no—
33#493D412yes—
34#460—24no—
35#460—42no—
36#446—21no—
37#446—21no—
38#446—21no—
39#446—21no—
40#432—21no—
41#432—21no—
42C2×C4⋊C8—21no—
43C2×C4⋊C8—21no—
44C2×C4⋊C8D411yescharacteristic
45C2×#102—41no—
46C2×#97—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#324—42no—
62C22×Q8SD1611yescharacteristic
63C22×D4—41no—
64C22×D4—41no—
65C22×D4—41no—
66C2×SD16—44no—
67C2×SD16—84no—
68C2×D8—22no—
69C2×D8—22no—
70C2×D8—41no—
71C2×D8—41no—
72C2×D8—82no—
73C2×D8—82no—
74C22×C8—21no—
75C22×C8—41no—
76C22×C8#4111yescharacteristic
77C22×C8#4111yescharacteristic
78#127—22no—
79#127—41no—
80#127—41no—
81#127—41no—
82C4×Q8—21no—
83C4×Q8—21no—
84C4×Q8—21no—
85C4×Q8—21no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4SD1611yescharacteristic
88C2×C42—21no—
89C2×C42C2×D411yescharacteristic
90C2×C16—22no—
91C2×C16—41no—
92C4⋊C8—21no—
93C4⋊C8—22no—
94C4⋊C8—21no—
95C4⋊C8—41no—
96C4⋊C8—41no—
97C4⋊C8—41no—
98C4⋊C8—41no—
99C4⋊C8—41no—
100#102—44no—
101C22⋊C8—41no—
102C22⋊C8—41no—
103#97—41no—
104#97—41no—
105#97—41no—
106#97—41no—
107C4×C8C2×D411yescharacteristic
108C4×C8C2×D411yescharacteristic
109C4×C8C2×D411yescharacteristic
110C4×C8#4111yescharacteristic
111C24—81no—
112C2×Q8—21no—
113C2×Q8—21no—
114C2×Q8—42no—
115C2×D4—42no—
116C2×D4—42no—
117C2×D4—42no—
118C2×D4—82no—
119C2×D4—82no—
120C2×D4—82no—
121C2×D4—82no—
122C2×D4—81no—
123C2×D4—81no—
124C2×D4—81no—
125C2×D4—162no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4#12311yescharacteristic
129C22×C4C2×SD1611yescharacteristic
130C22×C4AGL(1,ℤ/8)11yescharacteristic
131SD16—88no—
132D8—82no—
133D8—82no—
134D8—82no—
135D8—82no—
136C2×C8—22no—
137C2×C8—41no—
138C2×C8—41no—
139C2×C8—42no—
140C2×C8—82no—
141C2×C8—81no—
142C2×C8—81no—
143C2×C8—81no—
144C2×C8#12111yescharacteristic
145C2×C8#12111yescharacteristic
146C2×C8#12311yescharacteristic
147C2×C8#12411yescharacteristic
148C4⋊C4—21no—
149C4⋊C4—21no—
150C4⋊C4—41no—
151C4⋊C4—41no—
152C42—41no—
153C42—41no—
154C42#12011yescharacteristic
155C42#12111yescharacteristic
156C42#13511yescharacteristic
157C42AGL(1,ℤ/8)11yescharacteristic
158C16—42no—
159C16—81no—
160C23—82no—
161C23—162no—
162C23—162no—
163C23—162no—
164C23#48111yescharacteristic
165Q8—42no—
166Q8—81no—
167D4—82no—
168D4—82no—
169D4—82no—
170D4—82no—
171D4—162no—
172D4—162no—
173D4—162no—
174D4—162no—
175C2×C4—22no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—41no—
179C2×C4—42no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—81no—
183C2×C4#39411yescharacteristic
184C2×C4#44911yescharacteristic
185C2×C4#45011yescharacteristic
186C2×C4#46511yescharacteristic
187C2×C4#46911yescharacteristic
188C2×C4#48211yescharacteristic
189C8—21no—
190C8—21no—
191C8—21no—
192C8—82no—
193C8—81no—
194C8—161no—
195C8—161no—
196C8#47112yes—
197C22—21no—
198C22—21no—
199C22—162no—
200C22—162no—
201C22—162no—
202C22—324no—
203C22#203411yescharacteristic
204C22#223812yes—
205C4—21no—
206C4—21no—
207C4—21no—
208C4—21no—
209C4—21no—
210C4—81no—
211C4—81no—
212C4#204611yescharacteristic
213C4#204811yescharacteristic
214C2—22no—
215C2—322no—
216C2#1044311yescharacteristic
217C2#1339811yescharacteristic
218C2#1340411yescharacteristic
subgroupG/H
center, socleC22#2034
commutator, Frattini, cosocleC2×C4×C8C23
prime radicalC2×#493C22
LCSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#446 ◁ 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
5. #11644C2semidirect
6. #8625C2semidirect
7. #7294C2semidirect

Maximal subgroups

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

subgroupindex
C2×#16882
#120282
#119642
#116442
#86252
#72942