Finite Group Databasebeta

← #318591

Subgroups — #318591

Order 512.

643 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 643 conjugacy classes of subgroups — too many to draw. Only its 148 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#28240C212yesmaximal, complemented
2#19280C212yesmaximal, complemented
3#17975C211yesmaximal, characteristic, complemented
4C2×#1594C211yesmaximal, characteristic, complemented
5#10457C214yesmaximal, complemented
6#8825C211yesmaximal, characteristic, complemented
7#8682C212yesmaximal, complemented
8#8618C212yesmaximal, complemented
9#3169C2211yescharacteristic
10C2×#472—22no—
11#3036C2214yes—
12C2×C4×D8C2211yescharacteristic
13#2931C2212yes—
14C2×#436C2212yes—
15#2051—24no—
16#1848C2212yes—
17#1816—22no—
18#1800C2214yes—
19#1701—24no—
20#1693—22no—
21#1687—22no—
22#1594—24no—
23#1581—24no—
24#1567C2211yescharacteristic
25#1566C2212yes—
26#1566C2212yes—
27C2×#334C2211yescharacteristic
28#1547C2214yes—
29#1492C2212yes—
30C2×D4⋊C8C2212yes—
31C2×#322C2211yescharacteristic
32#1292C2214yes—
33C2×AGL(1,ℤ/8)—24no—
34C22×D8D411yescharacteristic
35C2×#131—22no—
36C22×M16—22no—
37#532C2312yes—
38C2×C4×D4C2312yes—
39#500D411yescharacteristic
40#493D411yescharacteristic
41#486D412yes—
42#472—48no—
43#466—22no—
44#463—44no—
45#461—24no—
46#459—24no—
47C4×D8—22no—
48C4×D8—22no—
49C4×D8—22no—
50#436—22no—
51#436—22no—
52#436—22no—
53#436—22no—
54#436—22no—
55#436—22no—
56C8×D4—22no—
57C8×D4—22no—
58C2×#108—22no—
59C2×#108—22no—
60C2×#107D411yescharacteristic
61#423C2312yes—
62#423C2314yes—
63C2×C4⋊C8C2312yes—
64#417—24no—
65C2×#102D412yes—
66C2×#100—24no—
67#408—22no—
68C2×C22⋊C8—22no—
69C2×C4×C8C2311yescharacteristic
70C2×C4×C8C2311yescharacteristic
71C2×C4×C8C2311yescharacteristic
72#343—24no—
73#335—22no—
74#334—22no—
75#334—22no—
76#334—22no—
77#334—42no—
78#332—24no—
79#331—24no—
80D4⋊C8—24no—
81D4⋊C8—24no—
82#322D414yes—
83C2×#41—22no—
84C22×D4C2×D412yes—
85C23×C4—22no—
86AGL(1,ℤ/8)—48no—
87AGL(1,ℤ/8)—84no—
88C2×SD16—44no—
89C2×D8—22no—
90C2×D8—22no—
91C2×D8—24no—
92C2×D8—44no—
93#131—44no—
94#131—44no—
95C2×M16—22no—
96C2×M16—24no—
97C2×M16—42no—
98C2×M16—42no—
99C2×M16—42no—
100C2×M16—84no—
101C22×C8—21no—
102C22×C8—22no—
103C22×C8—42no—
104C22×C8C2×D411yescharacteristic
105C22×C8C2×D411yescharacteristic
106C22×C8#4112yes—
107C4⋊Q8C2×D412yes—
108#127C2×D412yes—
109#124—22no—
110#121—22no—
111#121—22no—
112C4×D4—22no—
113C4×D4—22no—
114C4×D4—22no—
115C4×D4—22no—
116C4×D4—22no—
117C4×D4—22no—
118C2×C4⋊C4C2×D412yes—
119C2×C22⋊C4—22no—
120C2×C42C2411yescharacteristic
121#108—44no—
122#108—82no—
123#107—22no—
124#107—22no—
125C4⋊C8—22no—
126C4⋊C8—22no—
127C4⋊C8—22no—
128C4⋊C8—24no—
129C4⋊C8—22no—
130C4⋊C8—24no—
131#103—44no—
132#102—24no—
133#102—24no—
134#102—44no—
135#100—48no—
136C22⋊C8—42no—
137C22⋊C8—42no—
138C22⋊C8—42no—
139C22⋊C8—42no—
140C4×C8—21no—
141C4×C8C2×D412yes—
142C4×C8C2×D412yes—
143C4×C8C2×D412yes—
144C4×C8C2×D411yescharacteristic
145C4×C8C2×D411yescharacteristic
146C4×C8#4112yes—
147C24—22no—
148#41—42no—
149#41—42no—
150#41—42no—
151#41—42no—
152C2×Q8—22no—
153C2×D4—22no—
154C2×D4—22no—
155C2×D4—22no—
156C2×D4—22no—
157C2×D4—22no—
158C2×D4—24no—
159C2×D4—24no—
160C2×D4—42no—
161C22×C4—22no—
162C22×C4—42no—
163C22×C4—42no—
164C22×C4—42no—
165C22×C4—82no—
166C22×C4—82no—
167C22×C4C22×D411yescharacteristic
168C22×C4C2×#4111yescharacteristic
169C22×C42−1+411yescharacteristic
170SD16—88no—
171D8—44no—
172D8—82no—
173D8—88no—
174M16—42no—
175M16—44no—
176M16—42no—
177M16—44no—
178M16—84no—
179M16—84no—
180M16—84no—
181C2×C8—22no—
182C2×C8—22no—
183C2×C8—42no—
184C2×C8—42no—
185C2×C8—42no—
186C2×C8—44no—
187C2×C8—81no—
188C2×C8—84no—
189C2×C8—82no—
190C2×C8#12112yes—
191C2×C8#12112yes—
192C2×C8#12112yes—
193C2×C8#12112yes—
194C2×C8AGL(1,ℤ/8)14yes—
195C4⋊C4—22no—
196C4⋊C4—22no—
197C4⋊C4—22no—
198C4⋊C4—22no—
199C22⋊C4—42no—
200C22⋊C4—42no—
201C22⋊C4—42no—
202C22⋊C4—42no—
203C42C22×D411yescharacteristic
204C42C22×D411yescharacteristic
205C42C2×#4111yescharacteristic
206C42H(5,𝔽2)11yescharacteristic
207C23—42no—
208C23—42no—
209C23—44no—
210C23—82no—
211C23—82no—
212C23#54711yescharacteristic
213Q8—42no—
214Q8—42no—
215D4—42no—
216D4—44no—
217D4—42no—
218D4—42no—
219D4—42no—
220D4—44no—
221D4—82no—
222D4—82no—
223D4—82no—
224C2×C4—22no—
225C2×C4—41no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—84no—
229C2×C4—82no—
230C2×C4—82no—
231C2×C4—82no—
232C2×C4—82no—
233C2×C4—82no—
234C2×C4C2×#12111yescharacteristic
235C2×C4C2×#12111yescharacteristic
236C2×C4#53711yescharacteristic
237C2×C4D4211yescharacteristic
238C2×C4#54711yescharacteristic
239C2×C4#55511yescharacteristic
240C2×C4C2×AGL(1,ℤ/8)12yes—
241C2×C4#57712yes—
242C8—22no—
243C8—22no—
244C8—22no—
245C8—42no—
246C8—84no—
247C8—82no—
248C8—82no—
249C8#46914yes—
250C22—21no—
251C22—21no—
252C22—82no—
253C22—82no—
254C22—82no—
255C22—82no—
256C22—82no—
257C22—82no—
258C22—164no—
259C22#253711yescharacteristic
260C22#335412yes—
261C4—21no—
262C4—21no—
263C4—21no—
264C4—82no—
265C4—82no—
266C4C2×#46911yescharacteristic
267C4#306011yescharacteristic
268C4#306812yes—
269C4#327212yes—
270C2—22no—
271C2—162no—
272C2—162no—
273C2#2823111yescharacteristic
274C2#2824311yescharacteristic
275C2#2825011yescharacteristic
subgroupG/H
center, socleC22#2537
commutator, Frattini, cosocleC2×C42C24
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #28240C2semidirect
2. #19280C2semidirect
3. #17975C2semidirect
4. C2×#1594C2semidirect
5. #10457C2semidirect
6. #8825C2semidirect
7. #8682C2semidirect
8. #8618C2semidirect

Maximal subgroups

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

subgroupindex
#282402
#192802
#179752
C2×#15942
#104572
#88252
#86822
#86182