Finite Group Databasebeta

← #244408

Subgroups — #244408

Order 512.

792 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 792 conjugacy classes of subgroups — too many to draw. Only its 170 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#21727C211yesmaximal, characteristic, complemented
2#21692C211yesmaximal, characteristic
3#21647C212yesmaximal, complemented
4#21646C212yesmaximal
5#10679C212yesmaximal, complemented
6#10676C211yesmaximal, characteristic, complemented
7#10671C211yesmaximal, characteristic
8#10317C211yesmaximal, characteristic, complemented
9#10312C211yesmaximal, characteristic
10#10310C212yesmaximal
11#8136C211yesmaximal, characteristic, complemented
12C2×#498C2211yescharacteristic
13C2×#492C2211yescharacteristic
14C2×#487C2212yes—
15C2×#486C2212yes—
16C2×#440—22no—
17C4×C4⋊Q8C2212yes—
18C4×C4⋊Q8C2211yescharacteristic
19C4×#127C2211yescharacteristic
20#1990C2212yes—
21#1990C2212yes—
22#1990C2212yes—
23#1973—22no—
24#1957C2212yes—
25#1957C2211yescharacteristic
26#1956C2212yes—
27#1956C2211yescharacteristic
28#1944—22no—
29#1936—24no—
30#1935—24no—
31#1845C2211yescharacteristic
32#1845C2211yescharacteristic
33#1834C2212yes—
34#1834C2212yes—
35#1776—22no—
36#1763—24no—
37#1762—24no—
38C4×#97C2212yes—
39C4×#97C2211yescharacteristic
40#1273C2212yes—
41#1273C2211yescharacteristic
42#1273C2211yescharacteristic
43C22×SD16—22no—
44C2×C4⋊Q8C2312yes—
45C2×C4⋊Q8C2311yescharacteristic
46C2×#127C2311yescharacteristic
47C2×C4×Q8—21no—
48C2×C4×Q8—22no—
49C2×C4×D4—21no—
50C2×C4×D4—22no—
51#498—24no—
52#492—22no—
53#492—22no—
54#487—24no—
55#487—24no—
56#486—24no—
57#486—24no—
58#440—48no—
59C2×#107—22no—
60C2×#107—22no—
61C2×#106—22no—
62C2×#103—24no—
63C2×#103—22no—
64C2×#103—24no—
65C2×#102—24no—
66C2×#102—22no—
67C2×#102—24no—
68C2×#97D412yes—
69C2×#97D412yes—
70C2×#97D414yes—
71C2×#97D414yes—
72C2×C4×C8C2312yes—
73C2×C4×C8C2312yes—
74C2×C4×C8C2311yescharacteristic
75C2×C4×C8C2311yescharacteristic
76#391—22no—
77#391—21no—
78#390—22no—
79#390—21no—
80#389—22no—
81#389—22no—
82#389—22no—
83#384C2311yescharacteristic
84#384C2311yescharacteristic
85#384C2312yes—
86C4×C4⋊C4—22no—
87C4×C4⋊C4—22no—
88C4×C4⋊C4—22no—
89C43C2311yescharacteristic
90#340—24no—
91#340—24no—
92#340—24no—
93#340—24no—
94#340—24no—
95C22×Q8—21no—
96C22×Q8—22no—
97C22×D4—21no—
98C22×D4—22no—
99C23×C4—41no—
100C2×SD16—28no—
101C2×SD16—44no—
102C2×SD16—44no—
103C22×C8C2×D414yes—
104C22×C8C2×D414yes—
105C22×C8C2×D412yes—
106C22×C8C2×D412yes—
107C4⋊Q8—22no—
108C4⋊Q8—22no—
109C4⋊Q8—22no—
110C4⋊Q8—22no—
111C4⋊Q8—21no—
112C4⋊Q8#4114yes—
113C4⋊Q8#4112yes—
114#127—22no—
115#127—21no—
116#127#4112yes—
117C4×Q8—48no—
118C4×Q8—44no—
119C4×D4—48no—
120C4×D4—44no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—41no—
132C2×C4⋊C4—42no—
133C2×C22⋊C4—42no—
134C2×C22⋊C4—41no—
135C2×C42—42no—
136C2×C42—41no—
137C2×C42C2×D411yescharacteristic
138C2×C42C2×D412yes—
139C2×C42C2×D412yes—
140C2×C42C2×D411yescharacteristic
141C2×C42C2411yescharacteristic
142#107—44no—
143#107—44no—
144#106—28no—
145#103—216no—
146#103—48no—
147#103—44no—
148#102—216no—
149#102—48no—
150#102—44no—
151#97—24no—
152#97—24no—
153#97—44no—
154#97—44no—
155C4×C8—22no—
156C4×C8—22no—
157C4×C8—22no—
158C4×C8—22no—
159C4×C8—21no—
160C4×C8—21no—
161C4×C8—22no—
162#95—42no—
163#95—42no—
164#95—42no—
165C24—41no—
166C2×Q8—24no—
167C2×Q8—22no—
168C2×Q8—42no—
169C2×Q8—41no—
170C2×Q8—42no—
171C2×Q8—42no—
172C2×Q8—44no—
173C2×Q8—44no—
174C2×D4—24no—
175C2×D4—22no—
176C2×D4—42no—
177C2×D4—41no—
178C2×D4—42no—
179C2×D4—42no—
180C2×D4—44no—
181C2×D4—44no—
182C22×C4—41no—
183C22×C4—42no—
184C22×C4—41no—
185C22×C4—42no—
186C22×C4—41no—
187C22×C4—82no—
188C22×C4—84no—
189C22×C4#12712yes—
190C22×C4#12711yescharacteristic
191C22×C4#12711yescharacteristic
192C22×C4C22×D411yescharacteristic
193C22×C4C22×D412yes—
194SD16—416no—
195C2×C8—22no—
196C2×C8—22no—
197C2×C8—22no—
198C2×C8—22no—
199C2×C8—24no—
200C2×C8—24no—
201C2×C8—24no—
202C2×C8—24no—
203C2×C8—24no—
204C2×C8—24no—
205C2×C8AGL(1,ℤ/8)14yes—
206C2×C8#13714yes—
207C4⋊C4—24no—
208C4⋊C4—24no—
209C4⋊C4—24no—
210C4⋊C4—42no—
211C4⋊C4—42no—
212C4⋊C4—42no—
213C4⋊C4—42no—
214C4⋊C4—42no—
215C4⋊C4—44no—
216C4⋊C4—44no—
217C4⋊C4—44no—
218C4⋊C4—44no—
219C4⋊C4—44no—
220C4⋊C4—82no—
221C4⋊C4—84no—
222C22⋊C4—82no—
223C22⋊C4—84no—
224C42—22no—
225C42—22no—
226C42—24no—
227C42—24no—
228C42—82no—
229C42—84no—
230C42C2×#4111yescharacteristic
231C42C2×#4112yes—
232C42C2×#4111yescharacteristic
233C23—42no—
234C23—81no—
235C23—81no—
236C23—82no—
237C23—82no—
238C23C2×#12711yescharacteristic
239Q8—44no—
240Q8—48no—
241Q8—84no—
242Q8—82no—
243D4—44no—
244D4—48no—
245D4—84no—
246D4—82no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—42no—
255C2×C4—44no—
256C2×C4—81no—
257C2×C4—81no—
258C2×C4—82no—
259C2×C4—82no—
260C2×C4—81no—
261C2×C4—81no—
262C2×C4—82no—
263C2×C4—82no—
264C2×C4—82no—
265C2×C4—82no—
266C2×C4—82no—
267C2×C4—164no—
268C2×C4#52511yescharacteristic
269C2×C4#52512yes—
270C2×C4#52512yes—
271C2×C4#52511yescharacteristic
272C2×C4C2×AGL(1,ℤ/8)12yes—
273C2×C4C2×#13712yes—
274C2×C4#57514yes—
275C2×C4#57514yes—
276C8—24no—
277C8—24no—
278C8—44no—
279C8—44no—
280C22—82no—
281C22—84no—
282C22—161no—
283C22—161no—
284C22—162no—
285C22#239711yescharacteristic
286C22#315111yescharacteristic
287C22#315211yescharacteristic
288C22#315312yes—
289C22#315312yes—
290C4—41no—
291C4—41no—
292C4—42no—
293C4—82no—
294C4—84no—
295C4—161no—
296C4—161no—
297C4—162no—
298C4#311012yes—
299C4#311114yes—
300C4#311114yes—
301C4#311212yes—
302C2—162no—
303C2#1974911yescharacteristic
304C2#1975011yescharacteristic
305C2#1975212yes—
306C2#2380311yescharacteristic
307C2#2380412yes—
subgroupG/H
center, socleC23C2×#127
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#127C23
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #21727C2semidirect
3. #21647C2semidirect
5. #10679C2semidirect
6. #10676C2semidirect
8. #10317C2semidirect
11. #8136C2semidirect

Maximal subgroups

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

subgroupindex
#217272
#216922
#216472
#216462
#106792
#106762
#106712
#103172
#103122
#103102
#81362