Finite Group Databasebeta

← #315760

Subgroups — #315760

Order 512.

829 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 829 conjugacy classes of subgroups — too many to draw. Only its 186 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#10677C212yesmaximal, complemented
6#10672C212yesmaximal, complemented
7#9837C211yesmaximal, characteristic, complemented
8#9824C211yesmaximal, characteristic, complemented
9#8820C211yesmaximal, characteristic, complemented
10#8813C211yesmaximal, characteristic, complemented
11#8146C211yesmaximal, 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×#322C2211yescharacteristic
41C2×C82C2211yescharacteristic
42#1273C2212yes—
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—22no—
54#493—22no—
55#482—416no—
56#469—48no—
57C2×#107—24no—
58C2×#107—22no—
59C2×#106—24no—
60C2×#106—22no—
61C2×#106—22no—
62C2×#106—24no—
63C2×#102—24no—
64C2×#102—24no—
65C2×#102—24no—
66C2×#102—22no—
67C2×#102—22no—
68C2×#102—24no—
69C2×C22⋊C8D414yes—
70C4×M16—22no—
71C4×M16—22no—
72C2×#97D412yes—
73C2×C4×C8C2311yescharacteristic
74C2×C4×C8C2312yes—
75C2×C4×C8C2312yes—
76C2×C4×C8C2311yescharacteristic
77#390—22no—
78#390—22no—
79#390—21no—
80#390—21no—
81#384C2311yescharacteristic
82#384C2311yescharacteristic
83#384C2312yes—
84#384C2312yes—
85#382—22no—
86#380—22no—
87#380—22no—
88C4×C4⋊C4—22no—
89C4×C4⋊C4—22no—
90#340—24no—
91#340—24no—
92#336D414yes—
93#322—22no—
94C82—22no—
95C22×D4—21no—
96C22×D4—21no—
97C22×D4—22no—
98C22×D4—22no—
99C22×D4—41no—
100C22×D4—41no—
101C23×C4C2×D412yes—
102C23×C4C2×D411yescharacteristic
103C2×D8—44no—
104C2×D8—44no—
105C2×D8—44no—
106C2×M16—24no—
107C2×M16—24no—
108C2×M16—24no—
109C22×C8C2×D412yes—
110C22×C8C2×D414yes—
111C22×C8C2×D414yes—
112C22×C8C2×D412yes—
113#127—21no—
114#127—21no—
115#127—21no—
116#127—21no—
117#127—22no—
118#127—22no—
119#121—44no—
120#121—44no—
121#121—48no—
122#121—84no—
123#117—48no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C22⋊C4—42no—
135C2×C22⋊C4—41no—
136C2×C22⋊C4—41no—
137C2×C22⋊C4—42no—
138C2×C22⋊C4—42no—
139C2×C42—21no—
140C2×C42—21no—
141C2×C42—22no—
142C2×C42—42no—
143C2×C42C2×D412yes—
144C2×C42C2×D411yescharacteristic
145C2×C42C2411yescharacteristic
146#107—48no—
147#107—44no—
148#106—44no—
149#106—44no—
150#106—48no—
151#106—84no—
152#102—44no—
153#102—44no—
154#102—48no—
155#102—48no—
156#102—48no—
157#102—84no—
158C22⋊C8—28no—
159#97—24no—
160C4×C8—22no—
161C4×C8—22no—
162C4×C8—21no—
163C4×C8—21no—
164C4×C8—22no—
165C4×C8—22no—
166C4×C8#4112yes—
167C4×C8#4112yes—
168#95—42no—
169#95—42no—
170C24—41no—
171C24—41no—
172C24#12711yescharacteristic
173C2×D4—41no—
174C2×D4—41no—
175C2×D4—41no—
176C2×D4—41no—
177C2×D4—42no—
178C2×D4—42no—
179C2×D4—42no—
180C2×D4—42no—
181C2×D4—42no—
182C2×D4—42no—
183C2×D4—44no—
184C2×D4—42no—
185C2×D4—42no—
186C2×D4—44no—
187C2×D4—44no—
188C2×D4—44no—
189C2×D4—84no—
190C2×D4—82no—
191C2×D4—82no—
192C2×D4—84no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—24no—
199C22×C4—22no—
200C22×C4—24no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—41no—
204C22×C4—41no—
205C22×C4#12711yescharacteristic
206C22×C4#12712yes—
207C22×C4C22×D411yescharacteristic
208C22×C4C22×D412yes—
209D8—88no—
210M16—48no—
211C2×C8—24no—
212C2×C8—24no—
213C2×C8—22no—
214C2×C8—24no—
215C2×C8—22no—
216C2×C8—24no—
217C2×C8—22no—
218C2×C8—24no—
219C2×C8#12314yes—
220C2×C8#13514yes—
221C2×C8AGL(1,ℤ/8)18yes—
222C2×C8AGL(1,ℤ/8)14yes—
223C4⋊C4—44no—
224C4⋊C4—44no—
225C4⋊C4—42no—
226C4⋊C4—42no—
227C4⋊C4—44no—
228C4⋊C4—44no—
229C4⋊C4—44no—
230C4⋊C4—44no—
231C4⋊C4—44no—
232C4⋊C4—44no—
233C22⋊C4—84no—
234C22⋊C4—82no—
235C22⋊C4—82no—
236C22⋊C4—84no—
237C22⋊C4—84no—
238C42—21no—
239C42—21no—
240C42—22no—
241C42—22no—
242C42—84no—
243C42C2×#4111yescharacteristic
244C42C2×#4111yescharacteristic
245C422−1+412yes—
246C23—21no—
247C23—22no—
248C23—21no—
249C23—22no—
250C23—21no—
251C23—81no—
252C23—81no—
253C23—81no—
254C23—81no—
255C23—82no—
256C23—82no—
257C23—82no—
258C23—82no—
259C23—81no—
260C23—81no—
261C23C2×#12711yescharacteristic
262D4—82no—
263D4—82no—
264D4—84no—
265D4—84no—
266D4—84no—
267D4—84no—
268D4—82no—
269D4—82no—
270D4—164no—
271D4—164no—
272C2×C4—21no—
273C2×C4—22no—
274C2×C4—21no—
275C2×C4—22no—
276C2×C4—21no—
277C2×C4—22no—
278C2×C4—42no—
279C2×C4—44no—
280C2×C4—44no—
281C2×C4—42no—
282C2×C4—82no—
283C2×C4—82no—
284C2×C4—82no—
285C2×C4—81no—
286C2×C4—81no—
287C2×C4—82no—
288C2×C4—82no—
289C2×C4—82no—
290C2×C4—82no—
291C2×C4—82no—
292C2×C4C2×#12311yescharacteristic
293C2×C4#53611yescharacteristic
294C2×C4#54712yes—
295C2×C4#54712yes—
296C2×C4C2×#13512yes—
297C2×C4C2×AGL(1,ℤ/8)12yes—
298C2×C4C2×AGL(1,ℤ/8)14yes—
299C2×C4#57714yes—
300C8—28no—
301C8—42no—
302C8—44no—
303C8#48218yescomplemented
304C22—41no—
305C22—41no—
306C22—41no—
307C22—42no—
308C22—42no—
309C22—161no—
310C22—161no—
311C22—161no—
312C22—161no—
313C22—161no—
314C22—161no—
315C22—162no—
316C22—162no—
317C22—162no—
318C22—162no—
319C22#250111yescharacteristic
320C22#314711yescharacteristic
321C22#315111yescharacteristic
322C22#315112yes—
323C22#315412yes—
324C4—41no—
325C4—42no—
326C4—161no—
327C4—162no—
328C4—161no—
329C4—162no—
330C4C2×#48212yes—
331C4#322812yes—
332C4#334514yes—
333C4#335414yes—
334C2—81no—
335C2—321no—
336C2—321no—
337C2#1964112yes—
338C2#1971311yescharacteristic
339C2#1974511yescharacteristic
340C2#2790711yescharacteristic
341C2#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. #10677C2semidirect
6. #10672C2semidirect
7. #9837C2semidirect
8. #9824C2semidirect
9. #8820C2semidirect
10. #8813C2semidirect
11. #8146C2semidirect
303. C8#482semidirect

Maximal subgroups

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

subgroupindex
#331752
#279072
#278902
#208242
#106772
#106722
#98372
#98242
#88202
#88132
#81462