Finite Group Databasebeta

← #499630

Subgroups — #499630

Order 512.

449 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 449 conjugacy classes of subgroups — too many to draw. Only its 136 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#21736C211yesmaximal, characteristic
2#21287C211yesmaximal, characteristic
3#13541C211yesmaximal, characteristic
4#13536C211yesmaximal, characteristic
5#13521C212yesmaximal
6#13479C212yesmaximal
7#13454C211yesmaximal, characteristic
8#13368C211yesmaximal, characteristic
9#12677C212yesmaximal
10#12659C211yesmaximal, characteristic
11#12648C212yesmaximal
12#3162C2211yescharacteristic
13#2961—21no—
14C2×#441C2211yescharacteristic
15#2295—21no—
16#2159C2212yes—
17C2×#365C2212yes—
18C2×#364C2212yes—
19C2×#362C2212yes—
20#2141C2212yes—
21C2×#347C2212yes—
22#1994—21no—
23#1971C2211yescharacteristic
24#1969C2211yescharacteristic
25#1946—21no—
26#1945C2212yes—
27#1944C2211yescharacteristic
28#1941C2211yescharacteristic
29#1935—21no—
30#1933—21no—
31#1849C2211yescharacteristic
32#1776C2211yescharacteristic
33#1773C2212yes—
34#1772—21no—
35#1763—21no—
36C4×#97C2211yescharacteristic
37#1387C2212yes—
38#1386C2212yes—
39#1386C2212yes—
40#1383C2212yes—
41#1377C2211yescharacteristic
42#1377C2211yescharacteristic
43C22×Q16C2311yescharacteristic
44#533—21no—
45C2×C4×Q8—21no—
46C2×M32C2312yes—
47C2×M32C2312yes—
48#501—22no—
49#501—22no—
50#446—42no—
51#446—42no—
52#441—22no—
53#441Q814yescomplemented
54C2×#108—21no—
55C2×#108C2312yes—
56#427—21no—
57#427C2312yes—
58C2×#107C2311yescharacteristic
59C2×#106C2311yescharacteristic
60C2×#106C2311yescharacteristic
61#424—22no—
62#423—21no—
63C2×#103—21no—
64C2×#103—21no—
65C2×#103—21no—
66#405—22no—
67C2×#97D411yescharacteristic
68C2×#97D411yescharacteristic
69C2×#97C2311yescharacteristic
70C2×#97C2311yescharacteristic
71C2×C4×C8D411yescharacteristic
72C2×C4×C8C2311yescharacteristic
73#391—21no—
74#391—21no—
75#389—21no—
76#389—21no—
77#387—41no—
78#387—41no—
79C4×C4⋊C4—21no—
80C4×C4⋊C4—41no—
81#376—41no—
82C43D411yescharacteristic
83#367—22no—
84#366—22no—
85#365—24no—
86#364—24no—
87#362—24no—
88#358—22no—
89#358—22no—
90#347—24no—
91#340—21no—
92#340—21no—
93#340—21no—
94#339—22no—
95#339—22no—
96C22×Q8—21no—
97C2×Q16—22no—
98C2×Q16—22no—
99C2×Q16—21no—
100C2×Q16C2×Q812yes—
101C2×M16—22no—
102C22×C8C2×D411yescharacteristic
103C22×C8C2×D411yescharacteristic
104C22×C8C2×D411yescharacteristic
105C22×C8C2411yescharacteristic
106C4⋊Q8—21no—
107C4⋊Q8—21no—
108#125—22no—
109C22⋊Q8—41no—
110C22⋊Q8—41no—
111C4×Q8—24no—
112C4×Q8—42no—
113C4×Q8—41no—
114C4×Q8—41no—
115#117—22no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—41no—
120C2×C4⋊C4—41no—
121C2×C4⋊C4—41no—
122C2×C42—21no—
123C2×C42—21no—
124C2×C42—41no—
125C2×C42—41no—
126C2×C42C2×D411yescharacteristic
127C2×C42C2×D411yescharacteristic
128C2×C42C2×D411yescharacteristic
129M32—22no—
130M32—22no—
131M32—22no—
132M32#4114yes—
133C2×C16—22no—
134C2×C16—22no—
135#108—22no—
136#108—22no—
137#108—42no—
138#107—21no—
139#107—21no—
140#107—22no—
141#107#4112yes—
142#107#4112yes—
143#106—22no—
144#106—21no—
145#106—22no—
146#106—22no—
147#106C2×Q812yes—
148C4⋊C8—41no—
149C4⋊C8—41no—
150C4⋊C8—44no—
151#103—24no—
152#103—42no—
153#103—42no—
154#97—22no—
155#97—22no—
156#97—41no—
157#97—42no—
158#97C2×Q812yes—
159#97#4112yes—
160C4×C8—22no—
161C4×C8—22no—
162C4×C8—42no—
163#95—41no—
164#95—41no—
165#95—41no—
166#95—41no—
167#95—41no—
168#95—41no—
169#95—41no—
170C2×Q8—22no—
171C2×Q8—41no—
172C2×Q8—41no—
173C2×Q8—41no—
174C2×Q8—41no—
175C2×Q8—41no—
176C2×Q8—41no—
177C2×Q8—41no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4AGL(1,ℤ/8)11yescharacteristic
185C22×C4AGL(1,ℤ/8)11yescharacteristic
186C22×C4C22×D411yescharacteristic
187Q16—24no—
188Q16—42no—
189M16—42no—
190M16—42no—
191C2×C8—21no—
192C2×C8—21no—
193C2×C8—21no—
194C2×C8—21no—
195C2×C8—21no—
196C2×C8—21no—
197C2×C8—42no—
198C2×C8C22⋊Q812yes—
199C2×C8AGL(1,ℤ/8)12yes—
200C2×C8#13712yes—
201C2×C8C22×Q811yescharacteristic
202C2×C8C2×#4111yescharacteristic
203C2×C8C2×#4112yes—
204C2×C8C2×#4112yes—
205C4⋊C4—22no—
206C4⋊C4—22no—
207C4⋊C4—22no—
208C4⋊C4—44no—
209C4⋊C4—42no—
210C4⋊C4—42no—
211C4⋊C4—41no—
212C4⋊C4—41no—
213C4⋊C4—41no—
214C4⋊C4—41no—
215C4⋊C4—41no—
216C4⋊C4—41no—
217C4⋊C4—41no—
218C4⋊C4—82no—
219C4⋊C4—82no—
220C22⋊C4—42no—
221C42—21no—
222C42—21no—
223C42—21no—
224C42—21no—
225C42—21no—
226C42—42no—
227C42—44no—
228C42—42no—
229C42—81no—
230C42—82no—
231C42C22⋊Q812yes—
232C16—24no—
233C16—42no—
234C23C2×AGL(1,ℤ/8)11yescharacteristic
235Q8—42no—
236Q8—42no—
237Q8—81no—
238Q8—81no—
239Q8—161no—
240Q8—161no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—81no—
256C2×C4—81no—
257C2×C4—81no—
258C2×C4—81no—
259C2×C4—81no—
260C2×C4—81no—
261C2×C4—81no—
262C2×C4—81no—
263C2×C4—81no—
264C2×C4—81no—
265C2×C4C2×C22⋊Q811yescharacteristic
266C2×C4#52511yescharacteristic
267C2×C4#53211yescharacteristic
268C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
269C2×C4C2×#13711yescharacteristic
270C2×C4#57511yescharacteristic
271C8—22no—
272C8—22no—
273C8—22no—
274C8—82no—
275C8#52912yes—
276C8#53312yes—
277C22—21no—
278C22—21no—
279C22#307711yescharacteristic
280C22#310911yescharacteristic
281C22#318211yescharacteristic
282C4—42no—
283C4—42no—
284C4—41no—
285C4—41no—
286C4—82no—
287C4—82no—
288C4—82no—
289C4—82no—
290C4—161no—
291C4—161no—
292C4#239611yescharacteristic
293C4#307911yescharacteristic
294C4#311711yescharacteristic
295C4#318011yescharacteristic
296C2—21no—
297C2—21no—
298C2#2378011yescharacteristic
299C2#3368911yescharacteristic
300C2#3369011yescharacteristic
subgroupG/H
center, socleC22#3109
commutator, Frattini, cosocleC22×C8C24
prime radicalC23C2×AGL(1,ℤ/8)
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C4×#97 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
53. #441Q8semidirect

Maximal subgroups

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

subgroupindex
#217362
#212872
#135412
#135362
#135212
#134792
#134542
#133682
#126772
#126592
#126482