Finite Group Databasebeta

← #244173

Subgroups — #244173

Order 512.

585 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 585 conjugacy classes of subgroups — too many to draw. Only its 150 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#21716C211yesmaximal, characteristic, complemented
2#21661C211yesmaximal, characteristic
3#17931C212yesmaximal, complemented
4#17929C212yesmaximal
5#10670C211yesmaximal, characteristic, complemented
6#10667C211yesmaximal, characteristic
7#10664C212yesmaximal, complemented
8#10035C212yesmaximal
9#10017C211yesmaximal, characteristic, complemented
10#10014C211yesmaximal, characteristic
11#8138C211yesmaximal, characteristic, complemented
12#3135C2211yescharacteristic
13#3104C2212yes—
14#3103C2212yes—
15#2960C2211yescharacteristic
16C2×#446C2212yes—
17#2931C2211yescharacteristic
18#2399C2211yescharacteristic
19#1995C2211yescharacteristic
20#1985C2212yes—
21#1963C2211yescharacteristic
22#1940—22no—
23#1938—22no—
24#1843C2211yescharacteristic
25#1838C2211yescharacteristic
26#1770—22no—
27#1764—22no—
28#1726C2211yescharacteristic
29#1591—24no—
30#1590—24no—
31#1584—24no—
32#1582—24no—
33#1571C2212yes—
34#1570C2212yes—
35#1487C2212yes—
36#1487C2212yes—
37#1481C2212yes—
38#1480C2212yes—
39#1450C2212yes—
40#1275C2212yes—
41#1274C2211yescharacteristic
42#1274C2211yescharacteristic
43C2×#131—21no—
44#533C2311yescharacteristic
45#532C2311yescharacteristic
46C2×C4×Q8C2312yes—
47#491—21no—
48#490—21no—
49#489—22no—
50#484—22no—
51#481—22no—
52#477—22no—
53#475—22no—
54#462—22no—
55#460—22no—
56#451—22no—
57#448—22no—
58#446—22no—
59#446—22no—
60#446—22no—
61#446—22no—
62#446—22no—
63C8×Q8—22no—
64C4×Q16—24no—
65C4×SD16—24no—
66#436—22no—
67C8×D4—22no—
68#431C2312yes—
69#431C2312yes—
70#424—21no—
71#423C2311yescharacteristic
72#423C2311yescharacteristic
73C2×C4⋊C8—22no—
74C2×C4⋊C8—22no—
75C2×C4⋊C8C2312yes—
76C2×#104—22no—
77#416—22no—
78#408—21no—
79#405—21no—
80C2×#97—22no—
81C2×#97D412yes—
82C2×#97C2311yescharacteristic
83C2×C4×C8D412yes—
84C2×C4×C8C2311yescharacteristic
85#398—22no—
86#395—22no—
87C4×C4⋊C4—22no—
88#376C2311yescharacteristic
89#341—22no—
90#339—22no—
91#336—24no—
92#335—24no—
93#334—24no—
94Q8⋊C8—24no—
95Q8⋊C8—24no—
96Q8⋊C8—24no—
97D4⋊C8—24no—
98#322D414yes—
99C82D414yes—
100C2×#41—21no—
101C22×Q8#4112yes—
102C2×Q16—24no—
103C2×SD16—24no—
104#131—44no—
105C2×M16—21no—
106C2×M16—21no—
107C2×M16—41no—
108C22×C8—22no—
109C22×C8—41no—
110C22×C8C2×D412yes—
111C22×C8C2×D411yescharacteristic
112C22×C8C2×D411yescharacteristic
113C4⋊Q8#4111yescharacteristic
114C4⋊Q8#4112yes—
115#127#4111yescharacteristic
116#125—21no—
117#124—21no—
118C22⋊Q8—22no—
119#121—22no—
120C4×Q8—22no—
121C4×Q8—22no—
122C4×Q8—22no—
123C4×Q8—22no—
124C4×Q8—22no—
125C4×D4—22no—
126#117—21no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—24no—
129C2×C4⋊C4#4112yes—
130C2×C42—22no—
131C2×C42C2×D412yes—
132C2×C42C2411yescharacteristic
133#107—24no—
134#106—24no—
135C4⋊C8—22no—
136C4⋊C8—22no—
137C4⋊C8—22no—
138C4⋊C8—22no—
139C4⋊C8—22no—
140C4⋊C8—22no—
141C4⋊C8—42no—
142C4⋊C8—42no—
143C4⋊C8—42no—
144C4⋊C8—44no—
145#104—44no—
146#103—24no—
147#103—24no—
148#103—24no—
149#103—24no—
150#102—24no—
151#102—24no—
152C22⋊C8—22no—
153C22⋊C8—22no—
154C22⋊C8—42no—
155#97—24no—
156#97—21no—
157#97—21no—
158#97—42no—
159#97—42no—
160#97—41no—
161C4×C8—24no—
162C4×C8—41no—
163C4×C8C2×D414yes—
164C4×C8C2×D414yes—
165C4×C8C2×D412yes—
166C4×C8C2×D412yes—
167#95—22no—
168#95—22no—
169#95—22no—
170#41—42no—
171#41—42no—
172C2×Q8—21no—
173C2×Q8—22no—
174C2×Q8—22no—
175C2×Q8—22no—
176C2×Q8—82no—
177C2×D4—21no—
178C2×D4—22no—
179C2×D4—41no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—41no—
184C22×C4C22×D411yescharacteristic
185C22×C4C2×#4112yes—
186Q16—84no—
187SD16—84no—
188M16—42no—
189M16—42no—
190M16—82no—
191C2×C8—22no—
192C2×C8—24no—
193C2×C8—24no—
194C2×C8—22no—
195C2×C8—22no—
196C2×C8—22no—
197C2×C8—41no—
198C2×C8—42no—
199C2×C8—41no—
200C2×C8—44no—
201C2×C8—82no—
202C2×C8#12712yes—
203C2×C8#12711yescharacteristic
204C2×C8#12711yescharacteristic
205C2×C8#13714yes—
206C4⋊C4—22no—
207C4⋊C4—22no—
208C4⋊C4—22no—
209C4⋊C4—22no—
210C4⋊C4—22no—
211C4⋊C4—42no—
212C4⋊C4—42no—
213C4⋊C4—42no—
214C4⋊C4—84no—
215C22⋊C4—41no—
216C22⋊C4—42no—
217C42—22no—
218C42—42no—
219C42—41no—
220C42—42no—
221C42#13514yes—
222C42C22×D412yes—
223C42C2×#4111yescharacteristic
224C42C2×#4111yescharacteristic
225C23—41no—
226C23#52511yescharacteristic
227Q8—42no—
228Q8—42no—
229Q8—44no—
230Q8—42no—
231Q8—42no—
232Q8—42no—
233D4—44no—
234D4—42no—
235D4—82no—
236C2×C4—24no—
237C2×C4—42no—
238C2×C4—44no—
239C2×C4—42no—
240C2×C4—44no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—41no—
246C2×C4—81no—
247C2×C4—81no—
248C2×C4#52512yes—
249C2×C4#52512yes—
250C2×C4#52511yescharacteristic
251C2×C4C2×#12711yescharacteristic
252C2×C4C2×#13512yes—
253C2×C4C2×#13712yes—
254C8—22no—
255C8—44no—
256C8—44no—
257C8—42no—
258C8—82no—
259C8—81no—
260C8—81no—
261C8#49712yes—
262C8#49712yes—
263C22—22no—
264C22—82no—
265C22—81no—
266C22#239711yescharacteristic
267C22#310812yes—
268C4—22no—
269C4—22no—
270C4—44no—
271C4—82no—
272C4—82no—
273C4—82no—
274C4—81no—
275C4#310712yes—
276C4#314711yescharacteristic
277C4C2×#49711yescharacteristic
278C2—22no—
279C2—161no—
280C2#2379512yes—
281C2#2379711yescharacteristic
subgroupG/H
centerC2×C4C2×#127
commutator, Frattini, cosocleC2×C42C24
socleC22#2397
prime radical#532C23
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #21716C2semidirect
3. #17931C2semidirect
5. #10670C2semidirect
7. #10664C2semidirect
9. #10017C2semidirect
11. #8138C2semidirect

Maximal subgroups

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

subgroupindex
#217162
#216612
#179312
#179292
#106702
#106672
#106642
#100352
#100172
#100142
#81382