Finite Group Databasebeta

← #280920

Subgroups — #280920

Order 512.

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

normal subgroups only

This group has 494 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#22816C211yesmaximal, characteristic, complemented
3#10642C212yesmaximal, complemented
4#10465C212yesmaximal, complemented
5#9668C211yesmaximal, characteristic, complemented
6#9631C211yesmaximal, characteristic, complemented
7#9181C212yesmaximal, complemented
8#8790C211yesmaximal, characteristic, complemented
9#8771C212yesmaximal, complemented
10#8674C211yesmaximal, characteristic, complemented
11#8255C211yesmaximal, characteristic, complemented
12C2×#472—21no—
13#3036C2212yes—
14C2×C4×D8C2211yescharacteristic
15#2931C2211yescharacteristic
16#2357C2212yes—
17#2051—22no—
18#1972—21no—
19#1925—22no—
20#1848C2211yescharacteristic
21#1846C2211yescharacteristic
22#1816—21no—
23#1804C2212yes—
24#1800C2212yes—
25#1776—21no—
26#1770C2211yescharacteristic
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×#97—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—22no—
92#327—42no—
93D4⋊C8—42no—
94#323—42no—
95#322—21no—
96C82—21no—
97C2×#41—21no—
98C22×D4C2×D412yes—
99C23×C4—22no—
100AGL(1,ℤ/8)—44no—
101AGL(1,ℤ/8)—82no—
102C2×SD16—42no—
103C2×D8—21no—
104C2×D8—21no—
105C2×D8—22no—
106C2×D8—22no—
107C2×D8—42no—
108#131—42no—
109#131—42no—
110C2×M16—21no—
111C2×M16—22no—
112C2×M16—41no—
113C22×C8—21no—
114C22×C8—41no—
115C22×C8C2×D411yescharacteristic
116C22×C8C2×D411yescharacteristic
117C4⋊Q8#4111yescharacteristic
118#127#4111yescharacteristic
119#124—21no—
120#121—21no—
121#121—21no—
122C4×D4—22no—
123C4×D4—22no—
124C4×D4—22no—
125C4×D4—21no—
126C4×D4—21no—
127C4×D4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4C2×D412yes—
131C2×C22⋊C4—22no—
132C2×C22⋊C4—42no—
133C2×C22⋊C4—42no—
134C2×C42—21no—
135C2×C42—42no—
136C2×C42C2411yescharacteristic
137#108—42no—
138#108—81no—
139#107—21no—
140#107—21no—
141#107—81no—
142#106—42no—
143C4⋊C8—21no—
144C4⋊C8—21no—
145C4⋊C8—22no—
146C4⋊C8—21no—
147C4⋊C8—21no—
148C4⋊C8—22no—
149C4⋊C8—21no—
150C4⋊C8—21no—
151C4⋊C8—22no—
152#103—42no—
153#102—22no—
154#102—22no—
155#102—42no—
156#102—82no—
157#100—44no—
158C22⋊C8—21no—
159C22⋊C8—21no—
160C22⋊C8—22no—
161C22⋊C8—41no—
162C22⋊C8—41no—
163#97—42no—
164C4×C8—21no—
165C4×C8—21no—
166C4×C8—22no—
167C4×C8#4111yescharacteristic
168C4×C8#4111yescharacteristic
169C4×C8#4111yescharacteristic
170C4×C8#4111yescharacteristic
171#95—22no—
172#95—22no—
173#95—42no—
174#95#4112yes—
175C24—22no—
176#41—41no—
177#41—41no—
178#41—41no—
179#41—41no—
180C2×Q8—21no—
181C2×D4—22no—
182C2×D4—21no—
183C2×D4—21no—
184C2×D4—22no—
185C2×D4—21no—
186C2×D4—22no—
187C2×D4—22no—
188C2×D4—22no—
189C2×D4—22no—
190C2×D4—41no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—41no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—162no—
198C22×C4C22×D411yescharacteristic
199C22×C4C2×#4111yescharacteristic
200C22×C4H(5,𝔽2)11yescharacteristic
201SD16—84no—
202D8—42no—
203D8—42no—
204D8—84no—
205M16—41no—
206M16—41no—
207M16—42no—
208M16—42no—
209M16—82no—
210C2×C8—21no—
211C2×C8—21no—
212C2×C8—41no—
213C2×C8—42no—
214C2×C8—42no—
215C2×C8—41no—
216C2×C8—41no—
217C2×C8—41no—
218C2×C8—41no—
219C2×C8—41no—
220C2×C8—82no—
221C2×C8#12311yescharacteristic
222C2×C8#12311yescharacteristic
223C2×C8#12311yescharacteristic
224C2×C8#12311yescharacteristic
225C2×C8#13512yes—
226C2×C8AGL(1,ℤ/8)12yes—
227C4⋊C4—22no—
228C4⋊C4—21no—
229C4⋊C4—22no—
230C4⋊C4—21no—
231C4⋊C4—42no—
232C4⋊C4—42no—
233C4⋊C4—81no—
234C22⋊C4—42no—
235C22⋊C4—41no—
236C22⋊C4—42no—
237C22⋊C4—41no—
238C22⋊C4—84no—
239C22⋊C4—162no—
240C42—41no—
241C42—41no—
242C42—162no—
243C42C2×#4111yescharacteristic
244C42C2×#4111yescharacteristic
245C42C2×#4111yescharacteristic
246C42H(5,𝔽2)11yescharacteristic
247C23—42no—
248C23—41no—
249C23—42no—
250C23—42no—
251C23—82no—
252C23—82no—
253C23#54611yescharacteristic
254Q8—41no—
255Q8—41no—
256D4—42no—
257D4—42no—
258D4—42no—
259D4—41no—
260D4—41no—
261D4—42no—
262D4—81no—
263D4—81no—
264D4—81no—
265D4—81no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—41no—
272C2×C4—42no—
273C2×C4—81no—
274C2×C4—81no—
275C2×C4—82no—
276C2×C4—82no—
277C2×C4—82no—
278C2×C4—82no—
279C2×C4—162no—
280C2×C4—162no—
281C2×C4C2×#12311yescharacteristic
282C2×C4#52911yescharacteristic
283C2×C4#54011yescharacteristic
284C2×C4#54611yescharacteristic
285C2×C4#55311yescharacteristic
286C2×C4#55511yescharacteristic
287C2×C4C2×#13511yescharacteristic
288C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
289C2×C4#57711yescharacteristic
290C2×C4#57711yescharacteristic
291C8—21no—
292C8—21no—
293C8—82no—
294C8—81no—
295C8—81no—
296C8—81no—
297C8—81no—
298C8—82no—
299C8#48212yes—
300C8#48212yes—
301C22—21no—
302C22—21no—
303C22—81no—
304C22—82no—
305C22—81no—
306C22—82no—
307C22—82no—
308C22—81no—
309C22—162no—
310C22—162no—
311C22#242011yescharacteristic
312C22#330311yescharacteristic
313C22#333211yescharacteristic
314C4—21no—
315C4—21no—
316C4—21no—
317C4—21no—
318C4—82no—
319C4—81no—
320C4—82no—
321C4—162no—
322C4C2×#48211yescharacteristic
323C4#309911yescharacteristic
324C4#330311yescharacteristic
325C4#332411yescharacteristic
326C2—21no—
327C2—21no—
328C2—161no—
329C2—162no—
330C2#2496611yescharacteristic
331C2#2498611yescharacteristic
332C2#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. #22816C2semidirect
3. #10642C2semidirect
4. #10465C2semidirect
5. #9668C2semidirect
6. #9631C2semidirect
7. #9181C2semidirect
8. #8790C2semidirect
9. #8771C2semidirect
10. #8674C2semidirect
11. #8255C2semidirect

Maximal subgroups

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

subgroupindex
#282402
#228162
#106422
#104652
#96682
#96312
#91812
#87902
#87712
#86742
#82552