Finite Group Databasebeta

← #319593

Subgroups — #319593

Order 512.

1106 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has 1106 conjugacy classes of subgroups — too many to draw. Only its 178 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#33210C211yesmaximal, characteristic, complemented
2#27878C212yesmaximal, complemented
3#25717C212yesmaximal
4#21203C211yesmaximal, characteristic
5#10673C211yesmaximal, characteristic, complemented
6#10671C211yesmaximal, characteristic, complemented
7#10315C211yesmaximal, characteristic, complemented
8#10315C211yesmaximal, characteristic, complemented
9#9842C212yesmaximal, complemented
10#9829C212yesmaximal, complemented
11#8147C211yesmaximal, characteristic, complemented
12C22×#127C2211yescharacteristic
13C2×#498C2211yescharacteristic, complemented
14C2×#492C2212yes—
15C2×#492C2212yes—
16C2×#450—24no—
17C2×#418—22no—
18C2×C4×M16C2211yescharacteristic
19#2857C2212yes—
20#2285C2211yescharacteristic
21#2076—24no—
22#2014—24no—
23#1990C2212yescomplemented
24#1990C2212yescomplemented
25#1990C2212yescomplemented
26#1961—22no—
27#1957C2211yescharacteristic
28#1957C2211yescharacteristic
29#1957C2211yescharacteristic
30#1957C2212yes—
31#1957C2211yescharacteristic
32#1957C2212yes—
33#1943—22no—
34#1921—24no—
35#1845C2212yescomplemented
36#1845C2211yescharacteristic, complemented
37#1834C2212yescomplemented
38#1809—22no—
39C4×C22⋊C8C2212yes—
40C2×#322C2212yescomplemented
41#1273C2211yescharacteristic, complemented
42#1273C2211yescharacteristic, complemented
43C23×D4—21no—
44C23×D4—22no—
45C22×SD16—24no—
46C22×SD16—24no—
47C22×M16D412yes—
48C2×C4⋊Q8C2312yes—
49C2×#127—22no—
50C2×#127—22no—
51C2×#127—21no—
52C2×#127—22no—
53C2×#127—42no—
54C2×#127C2311yescharacteristic
55C2×#127C2311yescharacteristic
56C2×C22⋊Q8—22no—
57C2×#117—21no—
58C22×C42C2311yescharacteristic
59#498—42no—
60#492—24no—
61#492—24no—
62#492—42no—
63#492—42no—
64#450—416no—
65C2×#107—22no—
66C2×#107—24no—
67C2×#106—22no—
68#418—48no—
69C2×#102—22no—
70C2×#102—22no—
71C2×#102—22no—
72C2×#102—22no—
73C2×#102—24no—
74C2×#102—24no—
75C2×C22⋊C8D414yes—
76C4×M16—42no—
77C2×#97D412yes—
78C2×C4×C8C2312yes—
79C2×C4×C8C2311yescharacteristic
80C2×C4×C8C2312yes—
81C2×C4×C8C2311yescharacteristic
82#391—22no—
83#389—22no—
84#389—22no—
85#384C2312yes—
86#384C2311yescharacteristic
87#384C2311yescharacteristic
88#382—21no—
89#380—22no—
90#380—22no—
91#380—22no—
92C4×C4⋊C4—22no—
93#340—24no—
94#340—24no—
95#340—24no—
96#340—24no—
97#336D414yes—
98#322—42no—
99C25—41no—
100C22×Q8—22no—
101C22×D4—24no—
102C22×D4—22no—
103C22×D4—22no—
104C22×D4—21no—
105C22×D4—21no—
106C22×D4—44no—
107C22×D4—44no—
108C22×D4—44no—
109C22×D4—42no—
110C22×D4—48no—
111C22×D4—42no—
112C22×D4—41no—
113C22×D4—44no—
114C22×D4—84no—
115C23×C4C2×D412yes—
116C23×C4C2×D411yescharacteristic
117C2×SD16—48no—
118C2×SD16—48no—
119C2×SD16—48no—
120C2×SD16—48no—
121C2×SD16—84no—
122C2×SD16—84no—
123C2×M16—24no—
124C2×M16—24no—
125C2×M16—42no—
126C22×C8C2×D412yes—
127C22×C8C2×D414yes—
128C22×C8C2×D412yes—
129C22×C8C2×D414yes—
130C4⋊Q8—22no—
131C4⋊Q8—22no—
132C4⋊Q8—22no—
133C4⋊Q8—22no—
134#127—24no—
135#127—21no—
136#127—21no—
137#127—22no—
138#127—22no—
139#127—44no—
140#127—44no—
141#127—42no—
142#127#4112yes—
143#127#4112yes—
144C22⋊Q8—48no—
145#117—44no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—42no—
156C2×C22⋊C4—41no—
157C2×C22⋊C4—42no—
158C2×C42—21no—
159C2×C42—21no—
160C2×C42—22no—
161C2×C42—41no—
162C2×C42C2×D412yes—
163C2×C42C2×D411yescharacteristic
164C2×C42C2411yescharacteristic
165#107—48no—
166#107—44no—
167#106—44no—
168#102—44no—
169#102—44no—
170#102—48no—
171#102—44no—
172#102—44no—
173#102—48no—
174C22⋊C8—28no—
175#97—42no—
176C4×C8—22no—
177C4×C8—22no—
178C4×C8—22no—
179C4×C8—22no—
180C4×C8—22no—
181C4×C8—22no—
182#95—42no—
183#95—42no—
184#95—42no—
185C24—44no—
186C24—41no—
187C24—41no—
188C24—82no—
189C24—84no—
190C24—84no—
191C24—82no—
192C24#12711yescharacteristic
193C2×Q8—42no—
194C2×Q8—42no—
195C2×Q8—44no—
196C2×Q8—44no—
197C2×D4—22no—
198C2×D4—22no—
199C2×D4—24no—
200C2×D4—24no—
201C2×D4—416no—
202C2×D4—48no—
203C2×D4—44no—
204C2×D4—41no—
205C2×D4—41no—
206C2×D4—42no—
207C2×D4—42no—
208C2×D4—42no—
209C2×D4—42no—
210C2×D4—44no—
211C2×D4—44no—
212C2×D4—42no—
213C2×D4—42no—
214C2×D4—44no—
215C2×D4—44no—
216C2×D4—82no—
217C2×D4—88no—
218C2×D4—84no—
219C2×D4—84no—
220C2×D4—816no—
221C2×D4—816no—
222C2×D4—82no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—24no—
229C22×C4—24no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4#12712yes—
235C22×C4#12711yescharacteristic
236C22×C4#13714yes—
237C22×C4C22×D411yescharacteristic
238C22×C4C22×D412yes—
239SD16—816no—
240SD16—816no—
241M16—48no—
242C2×C8—22no—
243C2×C8—24no—
244C2×C8—22no—
245C2×C8—24no—
246C2×C8—22no—
247C2×C8—24no—
248C2×C8—24no—
249C2×C8—22no—
250C2×C8—22no—
251C2×C8—24no—
252C2×C8AGL(1,ℤ/8)18yes—
253C2×C8AGL(1,ℤ/8)14yes—
254C4⋊C4—42no—
255C4⋊C4—42no—
256C4⋊C4—42no—
257C4⋊C4—42no—
258C4⋊C4—42no—
259C4⋊C4—42no—
260C4⋊C4—44no—
261C4⋊C4—44no—
262C4⋊C4—44no—
263C4⋊C4—44no—
264C4⋊C4—44no—
265C4⋊C4—84no—
266C22⋊C4—82no—
267C22⋊C4—84no—
268C42—21no—
269C42—21no—
270C42—22no—
271C42—22no—
272C42—82no—
273C42C2×#4111yescharacteristic
274C42C2×#4111yescharacteristic
275C42H(5,𝔽2)12yes—
276C23—21no—
277C23—22no—
278C23—22no—
279C23—21no—
280C23—42no—
281C23—42no—
282C23—88no—
283C23—816no—
284C23—81no—
285C23—81no—
286C23—81no—
287C23—81no—
288C23—82no—
289C23—82no—
290C23—82no—
291C23—82no—
292C23—168no—
293C23—164no—
294C23—164no—
295C23#49712yes—
296C23C2×#12711yescharacteristic
297Q8—84no—
298Q8—84no—
299D4—44no—
300D4—44no—
301D4—48no—
302D4—48no—
303D4—88no—
304D4—816no—
305D4—82no—
306D4—82no—
307D4—84no—
308D4—84no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—24no—
312C2×C4—24no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—44no—
317C2×C4—44no—
318C2×C4—81no—
319C2×C4—81no—
320C2×C4—82no—
321C2×C4—82no—
322C2×C4—82no—
323C2×C4—82no—
324C2×C4—82no—
325C2×C4—82no—
326C2×C4—82no—
327C2×C4—82no—
328C2×C4#49712yes—
329C2×C4#52511yescharacteristic
330C2×C4#53411yescharacteristic
331C2×C4#54612yes—
332C2×C4#54612yes—
333C2×C4C2×AGL(1,ℤ/8)12yes—
334C2×C4C2×AGL(1,ℤ/8)14yes—
335C2×C4C2×#13712yes—
336C2×C4#57714yes—
337C8—24no—
338C8—28no—
339C8—42no—
340C8—44no—
341C22—22no—
342C22—24no—
343C22—41no—
344C22—41no—
345C22—42no—
346C22—82no—
347C22—82no—
348C22—84no—
349C22—84no—
350C22—1616no—
351C22—161no—
352C22—161no—
353C22—161no—
354C22—161no—
355C22—162no—
356C22—162no—
357C22#253911yescharacteristic
358C22C2×#49711yescharacteristic
359C22#315111yescharacteristic
360C22#315112yes—
361C22#315412yes—
362C4—22no—
363C4—42no—
364C4—161no—
365C4—161no—
366C4—162no—
367C4—162no—
368C4#311012yes—
369C4#319312yes—
370C4#331014yes—
371C4#332414yes—
372C2—42no—
373C2—162no—
374C2—162no—
375C2#1967912yes—
376C2#1971311yescharacteristic
377C2#1974511yescharacteristic
378C2#2831111yescharacteristic
379C2#2836512yes—
subgroupG/H
center, socleC23C2×#127
commutator, Frattini, cosocleC2×C42C24
prime radicalC22×#127C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33210C2semidirect
2. #27878C2semidirect
5. #10673C2semidirect
6. #10671C2semidirect
7. #10315C2semidirect
8. #10315C2semidirect
9. #9842C2semidirect
10. #9829C2semidirect
11. #8147C2semidirect
13. C2×#498C22semidirect
23. #1990C22semidirect
24. #1990C22semidirect
25. #1990C22semidirect
35. #1845C22semidirect
36. #1845C22semidirect
37. #1834C22semidirect
40. C2×#322C22semidirect
41. #1273C22semidirect
42. #1273C22semidirect

Maximal subgroups

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

subgroupindex
#332102
#278782
#257172
#212032
#106732
#106712
#103152
#98422
#98292
#81472