Finite Group Databasebeta

← #512664

Subgroups — #512664

Order 512.

843 conjugacy classes of subgroups; only the 184 normal subgroups are shown.

normal subgroups only

This group has 843 conjugacy classes of subgroups — too many to draw. Only its 184 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#60808C211yesmaximal, characteristic
2#33358C212yesmaximal, complemented
3#13609C218yesmaximal, complemented
4#13596C212yesmaximal, complemented
5#13585C212yesmaximal, complemented
6C2×#540C411yescharacteristic, complemented
7#3449C2211yescharacteristic
8#3221—22no—
9#3221C414yescomplemented
10#3220—22no—
11#3220C2214yes—
12#3151C411yescharacteristic, complemented
13#3151C2211yescharacteristic
14#3142C411yescharacteristic, complemented
15#3142C2211yescharacteristic
16#3036—22no—
17#3006—22no—
18#2921C411yescharacteristic, complemented
19C2×#410—22no—
20#2129—22no—
21#2128C2214yescomplemented
22#2127—22no—
23#2127C2214yescomplemented
24#2126—22no—
25#2126C2214yes—
26#1412C2218yescomplemented
27#1408C2218yescomplemented
28C22×#41D412yes—
29C2×AGL(1,ℤ/8)—22no—
30C2×AGL(1,ℤ/8)—22no—
31#540—22no—
32#540—42no—
33#540C2×C414yescomplemented
34#537—41no—
35#536C2×C412yes—
36#536D412yes—
37#535D412yes—
38#535C2312yes—
39#534—41no—
40#532C2×C411yescharacteristic
41#532D412yes—
42#532C2311yescharacteristic
43C2×#127C2×C411yescharacteristic, complemented
44C2×#125C2×C411yescharacteristic, complemented
45#525—42no—
46C2×#123—41no—
47C2×#121—21no—
48C2×#121—41no—
49C2×C4×D4—21no—
50C2×#117—42no—
51C2×#117C2×C411yescharacteristic
52#496—22no—
53#496C2×C414yescomplemented
54#493—21no—
55#493C2312yes—
56#492—21no—
57#492C2×C412yescomplemented
58#491—21no—
59#491C2×C412yescomplemented
60#490—21no—
61#490C2312yes—
62#488—22no—
63#488C2×C414yescomplemented
64#463—24no—
65#463—24no—
66#461—24no—
67#459—24no—
68#449—24no—
69#449—24no—
70#448—24no—
71#447—24no—
72#433—22no—
73#433C2×C414yescomplemented
74#423—21no—
75#417—22no—
76#417—22no—
77#410—24no—
78#410—24no—
79#410—42no—
80#410—42no—
81C2×#99—24no—
82C2×#99—42no—
83#407—21no—
84C4×M16C2×C411yescharacteristic, complemented
85C4×M16C2×C411yescharacteristic, complemented
86#354—24no—
87#354—24no—
88#353—24no—
89#353C2318yes—
90C2×#41—22no—
91C2×#41—22no—
92C2×#41—82no—
93C2×#41—82no—
94C2×#41C22⋊C412yes—
95C2×#41C22⋊C412yes—
96C2×#41C2×D412yes—
97C2×#41C2×D412yes—
98C22×Q8C22⋊C412yes—
99C22×D4—22no—
100C22×D4—21no—
101C22×D4—41no—
102C22×D4—41no—
103C22×D4C2×D412yes—
104C23×C4—22no—
105C23×C4—41no—
106C23×C4C22⋊C411yescharacteristic
107AGL(1,ℤ/8)—82no—
108AGL(1,ℤ/8)—82no—
109AGL(1,ℤ/8)—82no—
110AGL(1,ℤ/8)—82no—
111C2×SD16—24no—
112C2×SD16—24no—
113C2×D8—24no—
114C2×D8—24no—
115C2×M16—21no—
116C2×M16—21no—
117C4⋊Q8C22⋊C411yescharacteristic
118C4⋊Q8C22⋊C412yes—
119C4⋊Q8C22×C411yescharacteristic
120#127—21no—
121#127—41no—
122#127C22⋊C411yescharacteristic
123#127C22×C412yes—
124#127C2×D412yes—
125#127C2411yescharacteristic
126#125—21no—
127#125—41no—
128#125C22×C412yes—
129#124C22⋊C412yes—
130#124C22×C412yes—
131#124C2×D412yes—
132#124C2×D412yes—
133#123—42no—
134#123—42no—
135#123—44no—
136#123—44no—
137C22⋊Q8—42no—
138C22⋊Q8—42no—
139#121—24no—
140#121—44no—
141#121—44no—
142#121—42no—
143#121—42no—
144#121—41no—
145#121—42no—
146#121—41no—
147#121—42no—
148#120—42no—
149#120—42no—
150C4×D4—24no—
151C4×D4—41no—
152C4×D4—42no—
153C4×D4—41no—
154C4×D4—42no—
155#117—22no—
156#117—44no—
157#117—44no—
158#117—82no—
159#117—82no—
160#117C22⋊C412yes—
161#117C22×C412yes—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—82no—
166C2×C22⋊C4—21no—
167C2×C22⋊C4—41no—
168C2×C22⋊C4—41no—
169C2×C22⋊C4—44no—
170C2×C42—82no—
171C2×C42C22⋊C411yescharacteristic
172C2×C42C22×C411yescharacteristic
173C4⋊C8—24no—
174#103—24no—
175#103—24no—
176#102—24no—
177#102—24no—
178#99—28no—
179#99—44no—
180#99—44no—
181#99—44no—
182C22⋊C8—24no—
183#97—21no—
184#97C22×C412yes—
185C4×C8—21no—
186C4×C8C22×C412yes—
187C24—22no—
188C24—41no—
189C24#9911yescharacteristic
190#41—42no—
191#41—42no—
192#41—42no—
193#41—42no—
194#41—82no—
195#41—82no—
196#41—82no—
197#41—82no—
198#41—82no—
199#41—82no—
200C2×Q8—22no—
201C2×Q8—82no—
202C2×Q8C2×C22⋊C412yes—
203C2×Q8C2×C22⋊C412yes—
204C2×D4—22no—
205C2×D4—22no—
206C2×D4—22no—
207C2×D4—22no—
208C2×D4—22no—
209C2×D4—22no—
210C2×D4—42no—
211C2×D4—42no—
212C2×D4—41no—
213C2×D4—42no—
214C2×D4—41no—
215C2×D4—42no—
216C2×D4—42no—
217C2×D4—44no—
218C2×D4—41no—
219C2×D4—82no—
220C2×D4—82no—
221C2×D4—82no—
222C2×D4—162no—
223C2×D4—162no—
224C2×D4C2×C22⋊C412yes—
225C2×D4C22×D412yes—
226C22×C4—22no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—44no—
239C22×C4—41no—
240C22×C4—84no—
241C22×C4—162no—
242C22×C4#9912yes—
243C22×C4#9911yescharacteristic
244C22×C4C2×C22⋊C411yescharacteristic
245C22×C4C2×C22⋊C412yes—
246SD16—84no—
247SD16—84no—
248D8—84no—
249D8—84no—
250M16—42no—
251M16—42no—
252C2×C8—22no—
253C2×C8—22no—
254C4⋊C4—22no—
255C4⋊C4—22no—
256C4⋊C4—22no—
257C4⋊C4—42no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4—42no—
261C4⋊C4—42no—
262C4⋊C4—41no—
263C4⋊C4—84no—
264C4⋊C4—84no—
265C22⋊C4—22no—
266C22⋊C4—22no—
267C22⋊C4—42no—
268C22⋊C4—42no—
269C22⋊C4—42no—
270C22⋊C4—42no—
271C22⋊C4—44no—
272C22⋊C4—44no—
273C22⋊C4—48no—
274C22⋊C4—84no—
275C42—21no—
276C42—21no—
277C42—84no—
278C42—84no—
279C42C2×C22⋊C411yescharacteristic
280C42C2×C22⋊C412yes—
281C42C23×C411yescharacteristic
282C23—22no—
283C23—22no—
284C23—41no—
285C23—42no—
286C23—42no—
287C23—42no—
288C23—81no—
289C23—81no—
290C23—81no—
291C23—82no—
292C23—82no—
293C23—162no—
294C23C2×#9912yes—
295C23C2×#9911yescharacteristic
296Q8—42no—
297Q8—42no—
298Q8—82no—
299D4—42no—
300D4—42no—
301D4—44no—
302D4—82no—
303D4—82no—
304D4—82no—
305D4—82no—
306D4—82no—
307D4—82no—
308D4—162no—
309D4—161no—
310D4—161no—
311D4—162no—
312D4—162no—
313D4—164no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—41no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4—82no—
338C2×C4—88no—
339C2×C4—164no—
340C2×C4—168no—
341C2×C4C2×#9912yes—
342C2×C4C2×#9911yescharacteristic
343C2×C4C22×C22⋊C411yescharacteristic
344C8—42no—
345C8—42no—
346C22—22no—
347C22—21no—
348C22—41no—
349C22—42no—
350C22—42no—
351C22—42no—
352C22—82no—
353C22—82no—
354C22—82no—
355C22—82no—
356C22—84no—
357C22—82no—
358C22—161no—
359C22—162no—
360C22—162no—
361C22—162no—
362C22C22×#9911yescharacteristic
363C4—21no—
364C4—22no—
365C4—42no—
366C4—41no—
367C4—41no—
368C4—42no—
369C4—42no—
370C4—82no—
371C4—168no—
372C4#212712yes—
373C2—21no—
374C2—42no—
375C2—41no—
376C2—82no—
377C2—82no—
378C2—162no—
379C2C2×#212711yescharacteristic
subgroupG/H
center, socleC2C2×#2127
commutatorC42C23×C4
Frattini, cosocle#127C24
prime radical#60808C2
LCSG ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C2×C4 ◁ C23×C4 ◁ #3449 ◁ G
LECSG ▷ #127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #33358C2semidirect
3. #13609C2semidirect
4. #13596C2semidirect
5. #13585C2semidirect
6. C2×#540C4semidirect
9. #3221C4semidirect
12. #3151C4semidirect
14. #3142C4semidirect
18. #2921C4semidirect
21. #2128C22semidirect
23. #2127C22semidirect
26. #1412C22semidirect
27. #1408C22semidirect
33. #540C2×C4semidirect
43. C2×#127C2×C4semidirect
44. C2×#125C2×C4semidirect
53. #496C2×C4semidirect
57. #492C2×C4semidirect
59. #491C2×C4semidirect
63. #488C2×C4semidirect
73. #433C2×C4semidirect
84. C4×M16C2×C4semidirect
85. C4×M16C2×C4semidirect

Maximal subgroups

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

subgroupindex
#608082
#333582
#136092
#135962
#135852