Finite Group Databasebeta

← #315865

Subgroups — #315865

Order 512.

767 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 767 conjugacy classes of subgroups — too many to draw. Only its 166 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#27974C212yesmaximal, complemented
3#27891C212yesmaximal, complemented
4#20824C211yesmaximal, characteristic, complemented
5#10677C211yesmaximal, characteristic, complemented
6#10677C211yesmaximal, characteristic, complemented
7#10673C211yesmaximal, characteristic, complemented
8#10671C211yesmaximal, characteristic, complemented
9#9832C212yesmaximal, complemented
10#8821C212yesmaximal, complemented
11#8147C211yesmaximal, characteristic, complemented
12C2×#493C2211yescharacteristic
13C2×#469—22no—
14C2×C4×M16C2211yescharacteristic
15#2856C2211yescharacteristic
16#2856C2211yescharacteristic
17#2285C2212yes—
18#2073—24no—
19#2033—24no—
20#1972—22no—
21#1957C2212yes—
22#1957C2212yes—
23#1957C2212yes—
24#1957C2212yes—
25#1957C2211yescharacteristic
26#1957C2211yescharacteristic
27#1925—24no—
28#1875—24no—
29#1845C2211yescharacteristic
30#1845C2212yes—
31#1844C2211yescharacteristic
32#1844C2211yescharacteristic
33#1834C2212yes—
34#1834C2212yes—
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#469—48no—
55C2×#107—22no—
56C2×#107—24no—
57C2×#106—22no—
58C2×#106—22no—
59C2×#102—24no—
60C2×#102—22no—
61C2×#102—22no—
62C2×#102—24no—
63C2×#102—24no—
64C2×#102—24no—
65C2×C22⋊C8D414yes—
66C4×M16—24no—
67C2×#97D412yes—
68C2×C4×C8C2312yes—
69C2×C4×C8C2312yes—
70C2×C4×C8C2311yescharacteristic
71C2×C4×C8C2311yescharacteristic
72#390—22no—
73#390—22no—
74#390—21no—
75#390—21no—
76#384C2311yescharacteristic
77#384C2311yescharacteristic
78#384C2312yes—
79#384C2312yes—
80#382—22no—
81#380—22no—
82#380—22no—
83C4×C4⋊C4—22no—
84C4×C4⋊C4—22no—
85#340—24no—
86#340—24no—
87#340—24no—
88#340—24no—
89#336D414yes—
90#322—24no—
91C22×D4—21no—
92C22×D4—21no—
93C22×D4—22no—
94C22×D4—22no—
95C22×D4—41no—
96C22×D4—41no—
97C23×C4C2×D412yes—
98C23×C4C2×D411yescharacteristic
99C2×D8—44no—
100C2×D8—44no—
101C2×D8—44no—
102C2×M16—24no—
103C2×M16—24no—
104C2×M16—42no—
105C22×C8C2×D414yes—
106C22×C8C2×D414yes—
107C22×C8C2×D412yes—
108C22×C8C2×D412yes—
109#127—22no—
110#127—22no—
111#127—21no—
112#127—21no—
113#127—21no—
114#127—21no—
115#121—48no—
116#121—44no—
117#121—44no—
118#121—84no—
119#117—84no—
120C2×C4⋊C4—22no—
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×C22⋊C4—42no—
131C2×C22⋊C4—41no—
132C2×C22⋊C4—41no—
133C2×C22⋊C4—42no—
134C2×C22⋊C4—42no—
135C2×C42—22no—
136C2×C42—21no—
137C2×C42—21no—
138C2×C42—42no—
139C2×C42C2×D412yes—
140C2×C42C2×D411yescharacteristic
141C2×C42C2411yescharacteristic
142#107—44no—
143#107—84no—
144#106—44no—
145#106—44no—
146#102—48no—
147#102—44no—
148#102—44no—
149#102—84no—
150#102—84no—
151#102—84no—
152C22⋊C8—44no—
153#97—24no—
154C4×C8—22no—
155C4×C8—22no—
156C4×C8—22no—
157C4×C8—22no—
158C4×C8—22no—
159C4×C8#4114yes—
160#95—42no—
161#95—42no—
162C24—41no—
163C24—41no—
164C24#12711yescharacteristic
165C2×D4—41no—
166C2×D4—41no—
167C2×D4—41no—
168C2×D4—41no—
169C2×D4—42no—
170C2×D4—42no—
171C2×D4—42no—
172C2×D4—42no—
173C2×D4—42no—
174C2×D4—42no—
175C2×D4—44no—
176C2×D4—44no—
177C2×D4—42no—
178C2×D4—42no—
179C2×D4—44no—
180C2×D4—44no—
181C2×D4—82no—
182C2×D4—82no—
183C2×D4—84no—
184C2×D4—84no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—24no—
191C22×C4—22no—
192C22×C4—24no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—41no—
196C22×C4—41no—
197C22×C4#12711yescharacteristic
198C22×C4#12712yes—
199C22×C4C22×D411yescharacteristic
200C22×C4C22×D412yes—
201D8—88no—
202M16—48no—
203C2×C8—22no—
204C2×C8—22no—
205C2×C8—22no—
206C2×C8—22no—
207C2×C8—24no—
208C2×C8—24no—
209C2×C8—24no—
210C2×C8—24no—
211C2×C8—24no—
212C2×C8—24no—
213C2×C8#12314yes—
214C2×C8AGL(1,ℤ/8)14yes—
215C4⋊C4—42no—
216C4⋊C4—42no—
217C4⋊C4—44no—
218C4⋊C4—44no—
219C4⋊C4—44no—
220C4⋊C4—44no—
221C4⋊C4—44no—
222C4⋊C4—44no—
223C4⋊C4—44no—
224C4⋊C4—44no—
225C22⋊C4—82no—
226C22⋊C4—82no—
227C22⋊C4—84no—
228C22⋊C4—84no—
229C22⋊C4—84no—
230C42—21no—
231C42—21no—
232C42—22no—
233C42—22no—
234C42—84no—
235C42C2×#4112yes—
236C422−1+411yescharacteristic
237C422−1+411yescharacteristic
238C23—21no—
239C23—21no—
240C23—22no—
241C23—22no—
242C23—21no—
243C23—81no—
244C23—81no—
245C23—81no—
246C23—81no—
247C23—82no—
248C23—82no—
249C23—82no—
250C23—82no—
251C23—81no—
252C23—81no—
253C23C2×#12711yescharacteristic
254D4—84no—
255D4—84no—
256D4—82no—
257D4—82no—
258D4—84no—
259D4—84no—
260D4—82no—
261D4—82no—
262D4—164no—
263D4—164no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—21no—
269C2×C4—22no—
270C2×C4—44no—
271C2×C4—42no—
272C2×C4—44no—
273C2×C4—42no—
274C2×C4—82no—
275C2×C4—81no—
276C2×C4—81no—
277C2×C4—82no—
278C2×C4—82no—
279C2×C4—82no—
280C2×C4—82no—
281C2×C4—82no—
282C2×C4—82no—
283C2×C4—82no—
284C2×C4C2×#12311yescharacteristic
285C2×C4#53611yescharacteristic
286C2×C4#54712yes—
287C2×C4#54712yes—
288C2×C4C2×AGL(1,ℤ/8)12yes—
289C2×C4#57514yes—
290C2×C4#57512yes—
291C2×C4#57614yes—
292C8—24no—
293C8—24no—
294C8—44no—
295C8—44no—
296C22—41no—
297C22—41no—
298C22—41no—
299C22—42no—
300C22—42no—
301C22—161no—
302C22—161no—
303C22—161no—
304C22—161no—
305C22—161no—
306C22—161no—
307C22—162no—
308C22—162no—
309C22—162no—
310C22—162no—
311C22#250111yescharacteristic
312C22#315011yescharacteristic
313C22#315111yescharacteristic
314C22#315312yes—
315C22#315512yes—
316C4—41no—
317C4—42no—
318C4—162no—
319C4—161no—
320C4—161no—
321C4—162no—
322C4#309412yes—
323C4#323512yes—
324C4#334714yes—
325C4#335514yes—
326C2—81no—
327C2—321no—
328C2—321no—
329C2#1969512yes—
330C2#1972711yescharacteristic
331C2#1974911yescharacteristic
332C2#2791711yescharacteristic
333C2#2794212yes—
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. #27974C2semidirect
3. #27891C2semidirect
4. #20824C2semidirect
5. #10677C2semidirect
6. #10677C2semidirect
7. #10673C2semidirect
8. #10671C2semidirect
9. #9832C2semidirect
10. #8821C2semidirect
11. #8147C2semidirect

Maximal subgroups

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

subgroupindex
#331752
#279742
#278912
#208242
#106772
#106732
#106712
#98322
#88212
#81472