Finite Group Databasebeta

← #280848

Subgroups — #280848

Order 512.

497 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 497 conjugacy classes of subgroups — too many to draw. Only its 126 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#28240C211yesmaximal, characteristic
2#22810C211yesmaximal, characteristic, complemented
3#10642C212yesmaximal, complemented
4#10465C212yesmaximal, complemented
5#9668C211yesmaximal, characteristic, complemented
6#9628C211yesmaximal, characteristic, complemented
7#9181C212yesmaximal, complemented
8#8793C211yesmaximal, characteristic, complemented
9#8771C212yesmaximal, complemented
10#8676C211yesmaximal, characteristic, complemented
11#8251C211yesmaximal, characteristic, complemented
12C2×#472—21no—
13#3036C2212yes—
14C2×C4×D8C2211yescharacteristic
15#2931C2211yescharacteristic
16#2357C2212yes—
17#2051—22no—
18#1970—21no—
19#1926—22no—
20#1848C2211yescharacteristic
21#1846C2211yescharacteristic
22#1816—21no—
23#1804C2212yes—
24#1800C2212yes—
25C4×#107—21no—
26C8×C4⋊C4C2211yescharacteristic
27#1570C2212yes—
28#1564C2211yescharacteristic
29#1540C2211yescharacteristic
30#1517C2212yes—
31#1502C2211yescharacteristic
32#1478C2212yes—
33#1460C2212yes—
34#1450C2211yescharacteristic
35#1303C2211yescharacteristic
36#1303C2212yes—
37#1301C2211yescharacteristic
38#1286C2212yes—
39#1286C2212yes—
40#1286C2212yes—
41C2×AGL(1,ℤ/8)—22no—
42C22×D8D411yescharacteristic
43C2×#131—21no—
44#532C2311yescharacteristic
45C2×C4×D4C2312yes—
46#472—44no—
47#463—42no—
48#461—22no—
49#459—22no—
50C4×D8—21no—
51C4×D8—21no—
52C4×D8—21no—
53C4×D8—21no—
54#436—21no—
55#436—21no—
56C8×D4—21no—
57C8×D4—21no—
58#431C2311yescharacteristic
59#431C2311yescharacteristic
60#431C2312yes—
61C2×#108—21no—
62C2×#108—21no—
63C2×#107—21no—
64C2×#107D411yescharacteristic
65C2×#106—21no—
66#423C2311yescharacteristic
67#423C2312yes—
68C2×C4⋊C8C2311yescharacteristic
69#417—22no—
70C2×#102—22no—
71C2×#102D412yes—
72C2×#100—22no—
73#408—21no—
74C2×C4×C8—21no—
75C2×C4×C8C2311yescharacteristic
76#390—22no—
77#388—42no—
78#382C2312yes—
79#381—22no—
80C4×C4⋊C4—22no—
81C4×C4⋊C4C2311yescharacteristic
82C4×C22⋊C4—22no—
83#343—22no—
84#340—22no—
85#336—21no—
86#335—21no—
87#334—21no—
88#334—42no—
89#332—21no—
90#331—21no—
91#327—21no—
92#327—21no—
93#327—42no—
94D4⋊C8—42no—
95#323—42no—
96#322—21no—
97C82—21no—
98C2×#41—21no—
99C22×D4C2×D412yes—
100C23×C4—22no—
101AGL(1,ℤ/8)—44no—
102AGL(1,ℤ/8)—82no—
103C2×SD16—42no—
104C2×D8—21no—
105C2×D8—21no—
106C2×D8—22no—
107C2×D8—22no—
108C2×D8—42no—
109#131—42no—
110#131—42no—
111C2×M16—21no—
112C2×M16—22no—
113C2×M16—41no—
114C22×C8—21no—
115C22×C8—41no—
116C22×C8C2×D411yescharacteristic
117C22×C8C2×D411yescharacteristic
118C4⋊Q8#4111yescharacteristic
119#127#4111yescharacteristic
120#124—21no—
121#121—21no—
122#121—21no—
123C4×D4—22no—
124C4×D4—22no—
125C4×D4—22no—
126C4×D4—21no—
127C4×D4—21no—
128C4×D4—22no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4C2×D412yes—
132C2×C22⋊C4—22no—
133C2×C22⋊C4—42no—
134C2×C22⋊C4—42no—
135C2×C42—21no—
136C2×C42—42no—
137C2×C42C2411yescharacteristic
138#108—42no—
139#108—81no—
140#107—21no—
141#107—21no—
142#107—42no—
143#106—81no—
144C4⋊C8—21no—
145C4⋊C8—21no—
146C4⋊C8—22no—
147C4⋊C8—21no—
148C4⋊C8—21no—
149C4⋊C8—22no—
150C4⋊C8—21no—
151C4⋊C8—21no—
152C4⋊C8—22no—
153#103—42no—
154#102—22no—
155#102—22no—
156#102—42no—
157#102—82no—
158#100—44no—
159C22⋊C8—21no—
160C22⋊C8—21no—
161C22⋊C8—22no—
162C22⋊C8—41no—
163C22⋊C8—41no—
164C4×C8—21no—
165C4×C8—21no—
166C4×C8—22no—
167C4×C8—42no—
168C4×C8#4111yescharacteristic
169C4×C8#4111yescharacteristic
170C4×C8#4111yescharacteristic
171C4×C8#4111yescharacteristic
172#95—22no—
173#95—22no—
174#95—42no—
175#95#4111yescharacteristic
176#95#4111yescharacteristic
177C24—22no—
178#41—41no—
179#41—41no—
180#41—41no—
181#41—41no—
182C2×Q8—21no—
183C2×D4—22no—
184C2×D4—21no—
185C2×D4—21no—
186C2×D4—22no—
187C2×D4—21no—
188C2×D4—22no—
189C2×D4—22no—
190C2×D4—22no—
191C2×D4—22no—
192C2×D4—41no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—22no—
196C22×C4—41no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—162no—
201C22×C4C22×D411yescharacteristic
202C22×C4C2×#4111yescharacteristic
203C22×C4H(5,𝔽2)11yescharacteristic
204SD16—84no—
205D8—42no—
206D8—42no—
207D8—84no—
208M16—41no—
209M16—41no—
210M16—42no—
211M16—42no—
212M16—82no—
213C2×C8—21no—
214C2×C8—21no—
215C2×C8—41no—
216C2×C8—42no—
217C2×C8—42no—
218C2×C8—41no—
219C2×C8—41no—
220C2×C8—41no—
221C2×C8—41no—
222C2×C8—41no—
223C2×C8—82no—
224C2×C8#12311yescharacteristic
225C2×C8#12311yescharacteristic
226C2×C8#12311yescharacteristic
227C2×C8#12311yescharacteristic
228C2×C8#13512yes—
229C2×C8AGL(1,ℤ/8)12yes—
230C4⋊C4—22no—
231C4⋊C4—21no—
232C4⋊C4—22no—
233C4⋊C4—21no—
234C4⋊C4—42no—
235C4⋊C4—42no—
236C4⋊C4—42no—
237C22⋊C4—42no—
238C22⋊C4—41no—
239C22⋊C4—42no—
240C22⋊C4—41no—
241C22⋊C4—84no—
242C22⋊C4—162no—
243C42—41no—
244C42—41no—
245C42—162no—
246C42C2×#4111yescharacteristic
247C42C2×#4111yescharacteristic
248C42C2×#4111yescharacteristic
249C42H(5,𝔽2)11yescharacteristic
250C23—42no—
251C23—41no—
252C23—42no—
253C23—42no—
254C23—82no—
255C23—82no—
256C23#54611yescharacteristic
257Q8—41no—
258Q8—41no—
259D4—42no—
260D4—42no—
261D4—42no—
262D4—41no—
263D4—41no—
264D4—42no—
265D4—81no—
266D4—81no—
267D4—81no—
268D4—81no—
269C2×C4—22no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—42no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—81no—
280C2×C4—81no—
281C2×C4—82no—
282C2×C4—82no—
283C2×C4—82no—
284C2×C4—82no—
285C2×C4—162no—
286C2×C4—162no—
287C2×C4C2×#12311yescharacteristic
288C2×C4#52911yescharacteristic
289C2×C4#54011yescharacteristic
290C2×C4#54611yescharacteristic
291C2×C4#55311yescharacteristic
292C2×C4#55511yescharacteristic
293C2×C4C2×#13511yescharacteristic
294C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
295C2×C4#57711yescharacteristic
296C2×C4#57711yescharacteristic
297C8—21no—
298C8—21no—
299C8—82no—
300C8—81no—
301C8—81no—
302C8—81no—
303C8—81no—
304C8—82no—
305C8#48212yes—
306C8#48212yes—
307C22—21no—
308C22—21no—
309C22—81no—
310C22—82no—
311C22—81no—
312C22—82no—
313C22—82no—
314C22—81no—
315C22—162no—
316C22—162no—
317C22#242011yescharacteristic
318C22#330311yescharacteristic
319C22#333211yescharacteristic
320C4—21no—
321C4—21no—
322C4—21no—
323C4—21no—
324C4—42no—
325C4—81no—
326C4—81no—
327C4—82no—
328C4—162no—
329C4C2×#48211yescharacteristic
330C4#309911yescharacteristic
331C4#330311yescharacteristic
332C4#332411yescharacteristic
333C2—21no—
334C2—21no—
335C2—161no—
336C2—162no—
337C2#2496611yescharacteristic
338C2#2498511yescharacteristic
339C2#2498811yescharacteristic
subgroupG/H
center, socleC22#2420
commutator, Frattini, cosocleC2×C42C24
prime radical#28240C2
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22810C2semidirect
3. #10642C2semidirect
4. #10465C2semidirect
5. #9668C2semidirect
6. #9628C2semidirect
7. #9181C2semidirect
8. #8793C2semidirect
9. #8771C2semidirect
10. #8676C2semidirect
11. #8251C2semidirect

Maximal subgroups

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

subgroupindex
#282402
#228102
#106422
#104652
#96682
#96282
#91812
#87932
#87712
#86762
#82512