Finite Group Databasebeta

← #316203

Subgroups — #316203

Order 512.

499 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 499 conjugacy classes of subgroups — too many to draw. Only its 122 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#29810C211yesmaximal, characteristic
2#21995C211yesmaximal, characteristic, complemented
3#10634C212yesmaximal, complemented
4#10631C212yesmaximal, complemented
5#9780C211yesmaximal, characteristic, complemented
6#9773C211yesmaximal, characteristic, complemented
7#9198C212yesmaximal, complemented
8#9197C212yesmaximal, complemented
9#8794C211yesmaximal, characteristic, complemented
10#8791C211yesmaximal, characteristic, complemented
11#8158C211yesmaximal, characteristic, complemented
12C2×#493C2211yescharacteristic
13#2921C2211yescharacteristic
14C2×#433C2211yescharacteristic
15#2905C2211yescharacteristic
16#2619C2212yes—
17#2054—22no—
18#1999—22no—
19#1961C2212yes—
20#1950—22no—
21#1855—22no—
22#1849C2211yescharacteristic
23#1848C2211yescharacteristic
24#1838C2211yescharacteristic
25#1837C2211yescharacteristic
26#1835C2211yescharacteristic
27#1545C2212yes—
28#1545C2212yes—
29#1545C2212yes—
30#1545C2212yes—
31#1458C2212yes—
32#1452C2211yescharacteristic
33#1450C2211yescharacteristic
34#1286C2212yes—
35#1286C2212yes—
36#1286C2212yes—
37#1286C2212yes—
38#1281C2212yes—
39C22×D8—22no—
40C22×M16—21no—
41C2×#127C2312yes—
42C2×#121—22no—
43C2×#117C2311yescharacteristic
44#493—21no—
45#493—21no—
46#493—22no—
47#433—21no—
48#433—21no—
49#433—22no—
50#433—42no—
51#431C2312yes—
52#431C2312yes—
53C2×#108—22no—
54C2×#108—22no—
55#424—42no—
56#423—21no—
57#423C2311yescharacteristic
58#423C2311yescharacteristic
59#423C2311yescharacteristic
60#423C2311yescharacteristic
61C2×C4⋊C8D412yes—
62C2×#102—22no—
63C2×#102—22no—
64#407—21no—
65C2×C22⋊C8D412yes—
66C4×M16—21no—
67C2×#97C2311yescharacteristic
68C2×C4×C8C2311yescharacteristic
69#394—22no—
70#394—22no—
71#392—22no—
72#392—22no—
73#390—22no—
74#382C2312yes—
75#341—22no—
76#341—22no—
77#340—22no—
78#331—42no—
79#331—42no—
80#331—42no—
81#331—42no—
82#324—22no—
83#323—22no—
84#322—21no—
85#322—22no—
86C82—21no—
87C22×D4—22no—
88C22×D4—22no—
89C22×D4—22no—
90C22×D4—42no—
91C23×C4C2×D411yescharacteristic
92C2×D8—42no—
93C2×D8—42no—
94C2×D8—82no—
95C2×D8—82no—
96C2×M16—21no—
97C2×M16—21no—
98C2×M16—21no—
99C2×M16—21no—
100C2×M16—41no—
101C2×M16—82no—
102C22×C8—41no—
103C22×C8C2×D412yes—
104C22×C8C2×D412yes—
105#127—22no—
106#127—22no—
107#127—22no—
108#127—22no—
109#121—82no—
110#121—82no—
111#117—21no—
112#117—21no—
113#117—22no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4#4112yes—
116C2×C22⋊C4—42no—
117C2×C22⋊C4—42no—
118C2×C22⋊C4—42no—
119C2×C22⋊C4#4112yes—
120C2×C42C2×D411yescharacteristic
121C2×C42C2411yescharacteristic
122#108—82no—
123#108—82no—
124C4⋊C8—22no—
125C4⋊C8—22no—
126C4⋊C8—22no—
127C4⋊C8—22no—
128C4⋊C8—22no—
129C4⋊C8—22no—
130C4⋊C8—22no—
131C4⋊C8—22no—
132C4⋊C8—42no—
133#102—82no—
134#102—82no—
135C22⋊C8—22no—
136C22⋊C8—22no—
137C22⋊C8—22no—
138C22⋊C8—22no—
139C22⋊C8—42no—
140#97—21no—
141#97—21no—
142#97—41no—
143C4×C8—22no—
144C4×C8—22no—
145C4×C8—41no—
146C4×C8#4111yescharacteristic
147C4×C8#4111yescharacteristic
148C4×C8#4112yes—
149#95—22no—
150#95—22no—
151C24—42no—
152C24#12411yescharacteristic
153C2×D4—42no—
154C2×D4—42no—
155C2×D4—42no—
156C2×D4—42no—
157C2×D4—42no—
158C2×D4—42no—
159C2×D4—42no—
160C2×D4—42no—
161C2×D4—42no—
162C2×D4—42no—
163C2×D4—84no—
164C2×D4—162no—
165C2×D4—162no—
166C2×D4—162no—
167C2×D4—162no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—42no—
173C22×C4#12411yescharacteristic
174C22×C4#12412yes—
175C22×C4C22×D411yescharacteristic
176C22×C4C2×#4112yes—
177D8—84no—
178D8—84no—
179M16—42no—
180M16—42no—
181M16—42no—
182M16—42no—
183M16—82no—
184C2×C8—22no—
185C2×C8—22no—
186C2×C8—22no—
187C2×C8—22no—
188C2×C8—22no—
189C2×C8—41no—
190C2×C8—41no—
191C2×C8—41no—
192C2×C8—41no—
193C2×C8—42no—
194C2×C8—42no—
195C2×C8—82no—
196C2×C8#12312yes—
197C2×C8#12312yes—
198C4⋊C4—22no—
199C4⋊C4—22no—
200C4⋊C4—42no—
201C4⋊C4—42no—
202C22⋊C4—22no—
203C22⋊C4—22no—
204C22⋊C4—84no—
205C22⋊C4—162no—
206C22⋊C4—162no—
207C42—21no—
208C42—21no—
209C42C2×#4111yescharacteristic
210C42C2×#4111yescharacteristic
211C42H(5,𝔽2)12yes—
212C23—21no—
213C23—22no—
214C23—42no—
215C23—82no—
216C23—82no—
217C23—82no—
218C23—82no—
219C23—82no—
220C23—82no—
221C23—82no—
222C23C2×#12411yescharacteristic
223D4—82no—
224D4—82no—
225D4—82no—
226D4—82no—
227D4—162no—
228D4—162no—
229D4—162no—
230D4—162no—
231D4—168no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—21no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—42no—
240C2×C4—82no—
241C2×C4—82no—
242C2×C4—82no—
243C2×C4C2×#12311yescharacteristic
244C2×C4#53511yescharacteristic
245C2×C4#55312yes—
246C2×C4#55312yes—
247C2×C4#57612yes—
248C2×C4#57712yes—
249C8—22no—
250C8—22no—
251C8—42no—
252C8—82no—
253C8—82no—
254C8—81no—
255C8—81no—
256C8—81no—
257C8—81no—
258C22—22no—
259C22—41no—
260C22—42no—
261C22—82no—
262C22—162no—
263C22—162no—
264C22—162no—
265C22—162no—
266C22—162no—
267C22—162no—
268C22—162no—
269C22#250411yescharacteristic
270C22#313612yes—
271C4—22no—
272C4—22no—
273C4—41no—
274C4—42no—
275C4—162no—
276C4#309612yes—
277C4#322011yescharacteristic
278C4#322111yescharacteristic
279C2—22no—
280C2—81no—
281C2—322no—
282C2#2798611yescharacteristic
283C2#2798912yes—
subgroupG/H
center, socleC22#2504
commutator, Frattini, cosocleC2×C42C24
prime radical#29810C2
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
2. #21995C2semidirect
3. #10634C2semidirect
4. #10631C2semidirect
5. #9780C2semidirect
6. #9773C2semidirect
7. #9198C2semidirect
8. #9197C2semidirect
9. #8794C2semidirect
10. #8791C2semidirect
11. #8158C2semidirect

Maximal subgroups

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

subgroupindex
#298102
#219952
#106342
#106312
#97802
#97732
#91982
#91972
#87942
#87912
#81582