Finite Group Databasebeta

← #280919

Subgroups — #280919

Order 512.

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

normal subgroups only

This group has 470 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#28241C211yesmaximal, characteristic
2#22815C211yesmaximal, characteristic, complemented
3#10644C212yesmaximal, complemented
4#10467C212yesmaximal
5#9669C211yesmaximal, characteristic, complemented
6#9631C211yesmaximal, characteristic, complemented
7#9182C212yesmaximal
8#8790C211yesmaximal, characteristic, complemented
9#8769C212yesmaximal, complemented
10#8674C211yesmaximal, characteristic
11#8255C211yesmaximal, characteristic
12C2×#473—21no—
13#3037C2212yes—
14C2×C4×Q16C2211yescharacteristic
15#2931C2211yescharacteristic
16#2356C2212yes—
17#2052—22no—
18#1971—21no—
19#1928—22no—
20#1848C2211yescharacteristic
21#1846C2211yescharacteristic
22#1816—21no—
23#1805C2212yes—
24#1801C2212yes—
25#1776—21no—
26#1770C2211yescharacteristic
27#1570C2212yes—
28#1564C2211yescharacteristic
29#1540C2211yescharacteristic
30#1517C2212yes—
31#1502C2211yescharacteristic
32#1479C2212yes—
33#1461C2212yes—
34#1450C2211yescharacteristic
35#1303C2211yescharacteristic
36#1303C2212yes—
37#1301C2211yescharacteristic
38#1286C2212yes—
39#1286C2212yes—
40#1286C2212yes—
41C2×#137—22no—
42C22×Q16D411yescharacteristic
43C2×#131—21no—
44#532C2311yescharacteristic
45C2×C4×Q8C2312yes—
46#473—44no—
47#464—42no—
48#462—22no—
49#460—22no—
50C4×Q16—21no—
51C4×Q16—21no—
52C4×Q16—21no—
53C4×Q16—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×#103—22no—
71C2×#103D412yes—
72C2×#101—22no—
73#408—21no—
74C2×#97—21no—
75C2×C4×C8C2311yescharacteristic
76#391—22no—
77#387—42no—
78#383—22no—
79#382C2312yes—
80C4×C4⋊C4—22no—
81C4×C4⋊C4C2311yescharacteristic
82#376—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—
93Q8⋊C8—42no—
94#324—42no—
95#322—21no—
96C82—21no—
97C2×#41—21no—
98C22×Q8C2×D412yes—
99#137—44no—
100#137—82no—
101C2×Q16—21no—
102C2×Q16—21no—
103C2×Q16—22no—
104C2×Q16—22no—
105C2×Q16—42no—
106C2×SD16—42no—
107#131—42no—
108#131—42no—
109C2×M16—21no—
110C2×M16—22no—
111C2×M16—41no—
112C22×C8—21no—
113C22×C8—41no—
114C22×C8C2×D411yescharacteristic
115C22×C8C2×D411yescharacteristic
116C4⋊Q8#4111yescharacteristic
117#127#4111yescharacteristic
118#124—21no—
119#121—21no—
120#121—21no—
121C4×Q8—22no—
122C4×Q8—22no—
123C4×Q8—22no—
124C4×Q8—22no—
125C4×D4—21no—
126C4×D4—21no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4C2×D412yes—
131C2×C42—21no—
132C2×C42—22no—
133C2×C42—42no—
134C2×C42C2411yescharacteristic
135#108—42no—
136#108—81no—
137#107—21no—
138#107—21no—
139#107—81no—
140#106—42no—
141C4⋊C8—21no—
142C4⋊C8—21no—
143C4⋊C8—22no—
144C4⋊C8—21no—
145C4⋊C8—21no—
146C4⋊C8—22no—
147C4⋊C8—21no—
148C4⋊C8—21no—
149C4⋊C8—22no—
150#103—22no—
151#103—22no—
152#103—42no—
153#103—82no—
154#102—42no—
155#101—44no—
156C22⋊C8—21no—
157C22⋊C8—21no—
158C22⋊C8—22no—
159C22⋊C8—41no—
160C22⋊C8—41no—
161#97—42no—
162C4×C8—21no—
163C4×C8—21no—
164C4×C8—22no—
165C4×C8#4111yescharacteristic
166C4×C8#4111yescharacteristic
167C4×C8#4111yescharacteristic
168C4×C8#4111yescharacteristic
169#95—22no—
170#95—22no—
171#95—42no—
172#95—42no—
173#95—42no—
174#95#4112yes—
175#41—41no—
176#41—41no—
177#41—41no—
178#41—41no—
179C2×Q8—22no—
180C2×Q8—21no—
181C2×Q8—22no—
182C2×Q8—22no—
183C2×Q8—22no—
184C2×Q8—22no—
185C2×Q8—22no—
186C2×D4—21no—
187C2×D4—21no—
188C2×D4—21no—
189C2×D4—41no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—41no—
194C22×C4—42no—
195C22×C4C22×D411yescharacteristic
196C22×C4C2×#4111yescharacteristic
197C22×C4H(5,𝔽2)11yescharacteristic
198Q16—42no—
199Q16—42no—
200Q16—84no—
201SD16—84no—
202M16—41no—
203M16—41no—
204M16—42no—
205M16—42no—
206M16—82no—
207C2×C8—21no—
208C2×C8—21no—
209C2×C8—41no—
210C2×C8—42no—
211C2×C8—42no—
212C2×C8—41no—
213C2×C8—41no—
214C2×C8—41no—
215C2×C8—41no—
216C2×C8—41no—
217C2×C8—82no—
218C2×C8#12311yescharacteristic
219C2×C8#12311yescharacteristic
220C2×C8#12311yescharacteristic
221C2×C8#12311yescharacteristic
222C2×C8#13512yes—
223C2×C8AGL(1,ℤ/8)12yes—
224C4⋊C4—22no—
225C4⋊C4—21no—
226C4⋊C4—22no—
227C4⋊C4—21no—
228C4⋊C4—42no—
229C4⋊C4—42no—
230C4⋊C4—42no—
231C4⋊C4—42no—
232C4⋊C4—81no—
233C22⋊C4—41no—
234C22⋊C4—41no—
235C42—42no—
236C42—42no—
237C42—41no—
238C42—41no—
239C42—84no—
240C42C2×#4111yescharacteristic
241C42C2×#4111yescharacteristic
242C42C2×#4111yescharacteristic
243C42H(5,𝔽2)11yescharacteristic
244C23—41no—
245C23#54611yescharacteristic
246Q8—42no—
247Q8—42no—
248Q8—42no—
249Q8—41no—
250Q8—42no—
251Q8—41no—
252D4—41no—
253D4—41no—
254D4—81no—
255D4—81no—
256D4—81no—
257D4—81no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—41no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—81no—
269C2×C4—81no—
270C2×C4—82no—
271C2×C4—82no—
272C2×C4—82no—
273C2×C4—82no—
274C2×C4C2×#12311yescharacteristic
275C2×C4#52911yescharacteristic
276C2×C4#54011yescharacteristic
277C2×C4#54611yescharacteristic
278C2×C4#55311yescharacteristic
279C2×C4#55511yescharacteristic
280C2×C4C2×#13511yescharacteristic
281C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
282C2×C4#57711yescharacteristic
283C2×C4#57711yescharacteristic
284C8—21no—
285C8—21no—
286C8—82no—
287C8—81no—
288C8—81no—
289C8—81no—
290C8—81no—
291C8—82no—
292C8#48212yes—
293C8#48212yes—
294C22—21no—
295C22—21no—
296C22—81no—
297C22—81no—
298C22—81no—
299C22#242011yescharacteristic
300C22#330311yescharacteristic
301C22#333211yescharacteristic
302C4—21no—
303C4—21no—
304C4—21no—
305C4—21no—
306C4—82no—
307C4—82no—
308C4—81no—
309C4—82no—
310C4—162no—
311C4C2×#48211yescharacteristic
312C4#309911yescharacteristic
313C4#330311yescharacteristic
314C4#332411yescharacteristic
315C2—21no—
316C2—21no—
317C2—161no—
318C2#2496611yescharacteristic
319C2#2498611yescharacteristic
320C2#2498911yescharacteristic
subgroupG/H
center, socleC22#2420
commutator, Frattini, cosocleC2×C42C24
prime radical#532C23
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. #22815C2semidirect
3. #10644C2semidirect
5. #9669C2semidirect
6. #9631C2semidirect
8. #8790C2semidirect
9. #8769C2semidirect

Maximal subgroups

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

subgroupindex
#282412
#228152
#106442
#104672
#96692
#96312
#91822
#87902
#87692
#86742
#82552