Finite Group Databasebeta

← #237806

Subgroups — #237806

Order 512.

1098 conjugacy classes of subgroups; only the 198 normal subgroups are shown.

normal subgroups only

This group has 1098 conjugacy classes of subgroups — too many to draw. Only its 198 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#35052C211yesmaximal, characteristic
2#20699C212yesmaximal
3#13958C211yesmaximal, characteristic
4#10347C212yesmaximal, complemented
5#10346C212yesmaximal, complemented
6#10343C212yesmaximal, complemented
7#10342C212yesmaximal, complemented
8#8024C212yesmaximal
9C2×#1280C211yesmaximal, characteristic
10C22×#117C2211yescharacteristic
11C22×C22⋊C8C2212yes—
12#2848—22no—
13#2659C2212yes—
14#2646C2212yes—
15#2569—22no—
16#2549—22no—
17#2438—24no—
18#2329—22no—
19#2280C2212yes—
20C2×#381C2211yescharacteristic
21C22×#95C2211yescharacteristic
22#1825—22no—
23#1825—22no—
24#1824C2214yes—
25#1823C2214yes—
26#1735—24no—
27#1445C2211yescharacteristic
28#1445C2211yescharacteristic
29#1445C2211yescharacteristic
30#1445C2211yescharacteristic
31#1434—22no—
32C22⋊#95—22no—
33#1294C2214yes—
34#1294C2214yes—
35#1280C2212yes—
36#1280C2212yes—
37C24×C4D412yes—
38C24×C4C2311yescharacteristic
39C23×C8—22no—
40C2×C22⋊Q8—22no—
41C2×C22⋊Q8—22no—
42C2×C22⋊Q8—22no—
43C2×#117—22no—
44C2×#117—22no—
45C2×#117—22no—
46C2×#117—24no—
47C2×#117D414yes—
48C2×#117Q814yescomplemented
49C22×C4⋊C4C2311yescharacteristic
50C22×C4⋊C4C2311yescharacteristic
51C22×C22⋊C4—21no—
52C22×C22⋊C4—21no—
53C22×C42—21no—
54C2×C22⋊C8—24no—
55C2×C22⋊C8—24no—
56C2×C22⋊C8—22no—
57C2×C22⋊C8C2314yes—
58#400—24no—
59#398—24no—
60#398—24no—
61#395—24no—
62#391—24no—
63#389—24no—
64#387—24no—
65#387—24no—
66#385—24no—
67#385—24no—
68#385—24no—
69#381—22no—
70#381—24no—
71#380—22no—
72#380—22no—
73#380—22no—
74#380D412yes—
75#380D412yes—
76#380C2314yes—
77#379—22no—
78C4×C22⋊C4—22no—
79C4×C22⋊C4—22no—
80C2×#95—22no—
81C2×#95—22no—
82C2×#95—22no—
83C2×#95—22no—
84C2×#95—21no—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95D412yes—
88C2×#95C2311yescharacteristic
89C2×#95C2311yescharacteristic
90C2×#95C2311yescharacteristic
91C2×#95C2311yescharacteristic
92#340—24no—
93#340—24no—
94#340—24no—
95#340—24no—
96C25C2×D411yescharacteristic
97C22×Q8—22no—
98C23×C4—21no—
99C23×C4—21no—
100C23×C4—22no—
101C23×C4—22no—
102C23×C4—24no—
103C23×C4—21no—
104C23×C4—21no—
105C23×C4—44no—
106C23×C4—42no—
107C23×C4—44no—
108C23×C4—82no—
109C23×C4C2×D412yes—
110C23×C4C2×D412yes—
111C23×C4C2×D412yes—
112C23×C4C2×D411yescharacteristic
113C23×C4C2×D412yes—
114C23×C4C2×Q812yes—
115C23×C4#4114yes—
116C23×C4#4112yes—
117C23×C4C2411yescharacteristic
118C22×C8—24no—
119C22×C8—48no—
120C22×C8—44no—
121C22⋊Q8—44no—
122C22⋊Q8—44no—
123C22⋊Q8—84no—
124C22⋊Q8—84no—
125#117—28no—
126#117—28no—
127#117—44no—
128#117—44no—
129#117—48no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—24no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—24no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4C2×D414yes—
143C2×C4⋊C4C2×D414yes—
144C2×C4⋊C4C2×Q812yes—
145C2×C4⋊C4C2×Q812yes—
146C2×C22⋊C4—24no—
147C2×C22⋊C4—22no—
148C2×C22⋊C4—22no—
149C2×C22⋊C4—24no—
150C2×C22⋊C4—24no—
151C2×C22⋊C4—41no—
152C2×C22⋊C4—41no—
153C2×C22⋊C4—42no—
154C2×C22⋊C4—44no—
155C2×C22⋊C4—42no—
156C2×C42—24no—
157C2×C42—24no—
158C2×C42—42no—
159C2×C42—42no—
160C22⋊C8—24no—
161C22⋊C8—24no—
162C22⋊C8—44no—
163#95—24no—
164#95—24no—
165#95—24no—
166#95—24no—
167#95—22no—
168#95—22no—
169#95—22no—
170#95—22no—
171#95—24no—
172#95—22no—
173#95—22no—
174#95—44no—
175#95—42no—
176#95—44no—
177#95—82no—
178#95—82no—
179C24—22no—
180C24—22no—
181C24—24no—
182C24—41no—
183C24—41no—
184C24#12011yescharacteristic
185C24#12011yescharacteristic
186C24C22⋊Q812yes—
187C24#12412yes—
188C24C22×D411yescharacteristic
189C2×Q8—44no—
190C2×Q8—82no—
191C2×Q8—82no—
192C22×C4—28no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—24no—
197C22×C4—24no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—42no—
203C22×C4—44no—
204C22×C4—44no—
205C22×C4—44no—
206C22×C4—48no—
207C22×C4—42no—
208C22×C4—44no—
209C22×C4—44no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—44no—
216C22×C4—42no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—44no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—82no—
235C22×C4—82no—
236C22×C4—88no—
237C22×C4—88no—
238C22×C4#12012yes—
239C22×C4C22⋊Q814yes—
240C22×C4C22⋊Q814yes—
241C22×C4C22⋊Q812yes—
242C22×C4#12414yes—
243C22×C4#12414yes—
244C22×C4#12412yes—
245C22×C4#13514yes—
246C22×C4C22×D412yes—
247C22×C4C22×Q811yescharacteristic
248C22×C4C2×#4112yes—
249C22×C4C2×#4111yescharacteristic
250C2×C8—48no—
251C2×C8—44no—
252C2×C8—88no—
253C4⋊C4—24no—
254C4⋊C4—24no—
255C4⋊C4—44no—
256C4⋊C4—44no—
257C4⋊C4—44no—
258C4⋊C4—48no—
259C4⋊C4—42no—
260C4⋊C4—42no—
261C4⋊C4—82no—
262C4⋊C4—82no—
263C4⋊C4—84no—
264C4⋊C4—84no—
265C22⋊C4—44no—
266C22⋊C4—44no—
267C22⋊C4—44no—
268C22⋊C4—44no—
269C22⋊C4—44no—
270C22⋊C4—44no—
271C22⋊C4—48no—
272C22⋊C4—84no—
273C42—48no—
274C42—84no—
275C23—22no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—22no—
281C23—24no—
282C23—22no—
283C23—22no—
284C23—22no—
285C23—21no—
286C23—21no—
287C23—22no—
288C23—24no—
289C23—24no—
290C23—42no—
291C23—42no—
292C23—44no—
293C23—44no—
294C23—44no—
295C23—44no—
296C23#39312yes—
297C23#39312yes—
298C23#45814yes—
299C23C2×#12011yescharacteristic
300C23C2×C22⋊Q811yescharacteristic
301C23C2×#12411yescharacteristic
302C23C2×#13512yes—
303C23#57512yes—
304Q8—164no—
305C2×C4—24no—
306C2×C4—24no—
307C2×C4—28no—
308C2×C4—28no—
309C2×C4—24no—
310C2×C4—44no—
311C2×C4—42no—
312C2×C4—44no—
313C2×C4—42no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—44no—
320C2×C4—44no—
321C2×C4—44no—
322C2×C4—44no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—44no—
326C2×C4—44no—
327C2×C4—42no—
328C2×C4—44no—
329C2×C4—42no—
330C2×C4—44no—
331C2×C4—44no—
332C2×C4—44no—
333C2×C4—42no—
334C2×C4—82no—
335C2×C4—88no—
336C2×C4—88no—
337C2×C4—82no—
338C2×C4—81no—
339C2×C4—81no—
340C2×C4—82no—
341C2×C4—81no—
342C2×C4—81no—
343C2×C4—82no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4—82no—
347C2×C4—82no—
348C2×C4—82no—
349C2×C4—82no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4#39314yes—
355C2×C4C2×C22⋊Q812yes—
356C2×C4C2×#12412yes—
357C8—84no—
358C22—22no—
359C22—22no—
360C22—24no—
361C22—22no—
362C22—22no—
363C22—24no—
364C22—22no—
365C22—21no—
366C22—21no—
367C22—22no—
368C22—21no—
369C22—21no—
370C22—22no—
371C22—22no—
372C22—22no—
373C22—44no—
374C22—44no—
375C22—48no—
376C22—42no—
377C22—42no—
378C22#203014yes—
379C22C2×#39311yescharacteristic
380C22#300212yes—
381C22#301911yescharacteristic
382C22#302011yescharacteristic
383C22C2×#45812yes—
384C22#307612yes—
385C22#313111yescharacteristic
386C22#313511yescharacteristic
387C4—22no—
388C4—22no—
389C4—24no—
390C4—44no—
391C4—82no—
392C4—84no—
393C4—84no—
394C4—82no—
395C4—82no—
396C4—161no—
397C4—161no—
398C2—21no—
399C2—21no—
400C2—22no—
401C2—24no—
402C2—42no—
403C2#1820012yes—
404C2#1820112yes—
405C2#2345111yescharacteristic
406C2#2346211yescharacteristic
407C2C2×#203011yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #10347C2semidirect
5. #10346C2semidirect
6. #10343C2semidirect
7. #10342C2semidirect
48. C2×#117Q8semidirect

Maximal subgroups

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

subgroupindex
#350522
#206992
#139582
#103472
#103462
#103432
#103422
#80242
C2×#12802