Finite Group Databasebeta

← #315862

Subgroups — #315862

Order 512.

823 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 823 conjugacy classes of subgroups — too many to draw. Only its 174 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#33175C211yesmaximal, characteristic
2#27907C212yesmaximal, complemented
3#27890C212yesmaximal, complemented
4#20824C211yesmaximal, characteristic, complemented
5#10683C211yesmaximal, characteristic, complemented
6#10683C211yesmaximal, characteristic, complemented
7#10673C211yesmaximal, characteristic, complemented
8#10671C211yesmaximal, characteristic, complemented
9#9832C212yesmaximal, complemented
10#8819C212yesmaximal, complemented
11#8147C211yesmaximal, characteristic, complemented
12C2×#493C2211yescharacteristic
13C2×#482—24no—
14C2×#469—22no—
15C2×C4×M16C2211yescharacteristic
16#2856C2211yescharacteristic
17#2856C2211yescharacteristic
18#2285C2212yes—
19#2092—24no—
20#2034—24no—
21#1972—22no—
22#1957C2212yes—
23#1957C2212yes—
24#1957C2212yes—
25#1957C2212yes—
26#1957C2211yescharacteristic
27#1957C2211yescharacteristic
28#1926—24no—
29#1845C2212yes—
30#1845C2211yescharacteristic
31#1844C2212yes—
32#1844C2211yescharacteristic
33#1844C2211yescharacteristic
34#1844C2212yes—
35#1834C2212yes—
36#1834C2212yes—
37#1809—22no—
38#1776—22no—
39C4×C22⋊C8C2212yes—
40C2×#322C2212yes—
41#1273C2211yescharacteristic
42#1273C2211yescharacteristic
43C22×D8—22no—
44C22×M16D412yes—
45C2×#127C2311yescharacteristic
46C2×#127C2311yescharacteristic
47C2×#121—22no—
48C2×#121—21no—
49C2×#121—21no—
50C2×#121—22no—
51C2×#117—22no—
52C22×C42C2311yescharacteristic
53#493—24no—
54#482—416no—
55#469—48no—
56C2×#107—24no—
57C2×#107—22no—
58C2×#106—24no—
59C2×#106—22no—
60C2×#106—22no—
61C2×#106—24no—
62C2×#102—24no—
63C2×#102—24no—
64C2×#102—24no—
65C2×#102—22no—
66C2×#102—22no—
67C2×#102—24no—
68C2×C22⋊C8D414yes—
69C4×M16—24no—
70C2×#97D412yes—
71C2×C4×C8C2312yes—
72C2×C4×C8C2311yescharacteristic
73C2×C4×C8C2312yes—
74C2×C4×C8C2311yescharacteristic
75#390—22no—
76#390—22no—
77#390—21no—
78#390—21no—
79#384C2311yescharacteristic
80#384C2311yescharacteristic
81#384C2312yes—
82#384C2312yes—
83#382—22no—
84#380—22no—
85#380—22no—
86C4×C4⋊C4—22no—
87C4×C4⋊C4—22no—
88#340—24no—
89#340—24no—
90#336D414yes—
91#322—24no—
92C22×D4—21no—
93C22×D4—21no—
94C22×D4—22no—
95C22×D4—22no—
96C22×D4—41no—
97C22×D4—41no—
98C23×C4C2×D412yes—
99C23×C4C2×D411yescharacteristic
100C2×D8—44no—
101C2×D8—44no—
102C2×D8—44no—
103C2×M16—24no—
104C2×M16—24no—
105C2×M16—42no—
106C22×C8C2×D414yes—
107C22×C8C2×D412yes—
108C22×C8C2×D412yes—
109C22×C8C2×D414yes—
110#127—22no—
111#127—22no—
112#127—21no—
113#127—21no—
114#127—21no—
115#127—21no—
116#121—44no—
117#121—44no—
118#121—48no—
119#121—84no—
120#117—48no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C22⋊C4—42no—
132C2×C22⋊C4—41no—
133C2×C22⋊C4—41no—
134C2×C22⋊C4—42no—
135C2×C22⋊C4—42no—
136C2×C42—22no—
137C2×C42—21no—
138C2×C42—21no—
139C2×C42—42no—
140C2×C42C2×D412yes—
141C2×C42C2×D411yescharacteristic
142C2×C42C2411yescharacteristic
143#107—44no—
144#107—48no—
145#106—44no—
146#106—44no—
147#106—48no—
148#106—84no—
149#102—48no—
150#102—48no—
151#102—44no—
152#102—44no—
153#102—48no—
154#102—84no—
155C22⋊C8—28no—
156#97—24no—
157C4×C8—22no—
158C4×C8—22no—
159C4×C8—22no—
160C4×C8—22no—
161C4×C8—22no—
162C4×C8#4114yes—
163#95—42no—
164#95—42no—
165C24—41no—
166C24—41no—
167C24#12711yescharacteristic
168C2×D4—41no—
169C2×D4—41no—
170C2×D4—41no—
171C2×D4—41no—
172C2×D4—42no—
173C2×D4—42no—
174C2×D4—42no—
175C2×D4—42no—
176C2×D4—44no—
177C2×D4—42no—
178C2×D4—42no—
179C2×D4—44no—
180C2×D4—42no—
181C2×D4—42no—
182C2×D4—44no—
183C2×D4—44no—
184C2×D4—84no—
185C2×D4—82no—
186C2×D4—82no—
187C2×D4—84no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—24no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—24no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4#12711yescharacteristic
201C22×C4#12712yes—
202C22×C4C22×D411yescharacteristic
203C22×C4C22×D412yes—
204D8—88no—
205M16—48no—
206C2×C8—22no—
207C2×C8—24no—
208C2×C8—22no—
209C2×C8—22no—
210C2×C8—22no—
211C2×C8—24no—
212C2×C8—24no—
213C2×C8—24no—
214C2×C8—24no—
215C2×C8#12314yes—
216C2×C8AGL(1,ℤ/8)18yes—
217C2×C8AGL(1,ℤ/8)14yes—
218C4⋊C4—44no—
219C4⋊C4—42no—
220C4⋊C4—42no—
221C4⋊C4—44no—
222C4⋊C4—44no—
223C4⋊C4—44no—
224C4⋊C4—44no—
225C4⋊C4—44no—
226C4⋊C4—44no—
227C4⋊C4—44no—
228C22⋊C4—84no—
229C22⋊C4—82no—
230C22⋊C4—82no—
231C22⋊C4—84no—
232C22⋊C4—84no—
233C42—21no—
234C42—21no—
235C42—22no—
236C42—22no—
237C42—84no—
238C42C2×#4112yes—
239C422−1+411yescharacteristic
240C422−1+411yescharacteristic
241C23—21no—
242C23—22no—
243C23—21no—
244C23—22no—
245C23—21no—
246C23—81no—
247C23—81no—
248C23—81no—
249C23—81no—
250C23—82no—
251C23—82no—
252C23—82no—
253C23—82no—
254C23—81no—
255C23—81no—
256C23C2×#12711yescharacteristic
257D4—84no—
258D4—84no—
259D4—82no—
260D4—82no—
261D4—84no—
262D4—84no—
263D4—82no—
264D4—82no—
265D4—164no—
266D4—164no—
267C2×C4—21no—
268C2×C4—22no—
269C2×C4—21no—
270C2×C4—22no—
271C2×C4—21no—
272C2×C4—22no—
273C2×C4—44no—
274C2×C4—42no—
275C2×C4—44no—
276C2×C4—42no—
277C2×C4—82no—
278C2×C4—82no—
279C2×C4—82no—
280C2×C4—81no—
281C2×C4—81no—
282C2×C4—82no—
283C2×C4—82no—
284C2×C4—82no—
285C2×C4—82no—
286C2×C4—82no—
287C2×C4C2×#12311yescharacteristic
288C2×C4#53611yescharacteristic
289C2×C4#54712yes—
290C2×C4#54712yes—
291C2×C4C2×AGL(1,ℤ/8)14yes—
292C2×C4C2×AGL(1,ℤ/8)12yes—
293C2×C4#57512yes—
294C2×C4#57714yes—
295C8—28no—
296C8—24no—
297C8—24no—
298C8—44no—
299C22—41no—
300C22—41no—
301C22—41no—
302C22—42no—
303C22—42no—
304C22—161no—
305C22—161no—
306C22—161no—
307C22—161no—
308C22—161no—
309C22—161no—
310C22—162no—
311C22—162no—
312C22—162no—
313C22—162no—
314C22#250111yescharacteristic
315C22#315011yescharacteristic
316C22#315111yescharacteristic
317C22#315112yes—
318C22#315412yes—
319C4—41no—
320C4—42no—
321C4—161no—
322C4—162no—
323C4—161no—
324C4—162no—
325C4#309412yes—
326C4#323512yes—
327C4#334514yes—
328C4#335414yes—
329C2—81no—
330C2—321no—
331C2—321no—
332C2#1969412yes—
333C2#1971311yescharacteristic
334C2#1974511yescharacteristic
335C2#2791711yescharacteristic
336C2#2794112yes—
subgroupG/H
center, socleC23C2×#127
commutator, Frattini, cosocleC2×C42C24
prime radical#33175C2
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #27907C2semidirect
3. #27890C2semidirect
4. #20824C2semidirect
5. #10683C2semidirect
6. #10683C2semidirect
7. #10673C2semidirect
8. #10671C2semidirect
9. #9832C2semidirect
10. #8819C2semidirect
11. #8147C2semidirect

Maximal subgroups

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

subgroupindex
#331752
#279072
#278902
#208242
#106832
#106732
#106712
#98322
#88192
#81472