Finite Group Databasebeta

← #6479899

Subgroups — #6479899

Order 512.

1374 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35321C211yesmaximal, characteristic
2#34214C211yesmaximal, characteristic
3#33157C212yesmaximal
4#33117C212yesmaximal
5#29908C212yesmaximal
6#29645C212yesmaximal
7#29520C212yesmaximal, complemented
8#29512C212yesmaximal, complemented
9#29405C214yesmaximal, complemented
10#29361C212yesmaximal, complemented
11#29327C212yesmaximal, complemented
12#29280C212yesmaximal, complemented
13#29255C212yesmaximal, complemented
14#22092C214yesmaximal
15#20780C211yesmaximal, characteristic
16C2×#533C2211yescharacteristic
17C2×#485C2214yes—
18#2863C2212yes—
19#2855C2212yes—
20#2647C2211yescharacteristic
21#2642C2211yescharacteristic
22#2629C2211yescharacteristic
23#2621C2211yescharacteristic
24#2585—22no—
25#2573—22no—
26#2524—22no—
27#2515—22no—
28#2360—22no—
29#2360—22no—
30#2355—22no—
31Q8×C4⋊C4—22no—
32#2330C2212yes—
33#2330C2212yes—
34#2323C2212yes—
35#2320C2212yes—
36#2317C2212yes—
37C4×C4⋊Q8C2212yes—
38C4×C4⋊Q8C2212yes—
39C4×C4⋊Q8C2211yescharacteristic
40C4×#125C2211yescharacteristic
41C2×#384C2211yescharacteristic
42#2274C2212yes—
43C4×#117C2211yescharacteristic
44#2099C2212yes—
45#2099C2212yes—
46#2096C2212yes—
47#2096C2212yes—
48#2084C2214yes—
49#2072C2214yes—
50#2059C2214yes—
51#2058C2212yes—
52#2057C2212yes—
53#2026C2214yes—
54#2024C2212yes—
55#2024C2212yes—
56#1989C2214yes—
57#1981C2214yes—
58#1949C2214yes—
59#1948C2214yes—
60#1943C2214yes—
61#1941C2214yes—
62#1935C2214yes—
63#1933C2214yes—
64#1928C2214yes—
65#1927C2214yes—
66#1921C2214yes—
67#1920C2214yes—
68#1920C2214yes—
69#1919C2214yes—
70#1872C2214yes—
71#1865C2214yes—
72#1865C2214yes—
73#1853C2214yes—
74#1840C2212yes—
75#1831C2212yes—
76#1825C2214yes—
77#1817C2212yes—
78#1798C2212yes—
79#1768C2212yes—
80#1750C2212yes—
81#1745C2211yescharacteristic
82#1744C2211yescharacteristic
83#533—28no—
84C2×C4⋊Q8C2311yescharacteristic
85C2×C4⋊Q8C2312yes—
86C2×#125C2311yescharacteristic
87C2×#125C2312yes—
88C2×C22⋊Q8—22no—
89C2×C22⋊Q8C2312yes—
90C2×C4×Q8—22no—
91C2×C4×Q8—22no—
92C2×C4×Q8—22no—
93C2×C4×Q8C2312yes—
94C2×#117C2312yes—
95C2×#117C2311yescharacteristic
96C2×#117C2311yescharacteristic
97C22×C4⋊C4—22no—
98C22×C4⋊C4C2312yes—
99C22×C42C2311yescharacteristic
100#485—216no—
101C2×#107C2314yes—
102C2×#106C2314yes—
103C2×#106C2314yes—
104C2×#106C2314yes—
105C2×C4⋊C8C2314yes—
106C2×#103C2314yes—
107C2×#103C2314yes—
108C2×#103C2314yes—
109C2×#103C2314yes—
110C2×C22⋊C8C2314yes—
111C2×#97C2312yes—
112C2×C4×C8C2312yes—
113#400—22no—
114#400—22no—
115#400—22no—
116#398—22no—
117#398—22no—
118#398—22no—
119#397—22no—
120#396—22no—
121#391—22no—
122#391—22no—
123#391C2312yes—
124#391C2312yes—
125#391C2312yes—
126#391C2312yes—
127#389—22no—
128#389—22no—
129#389—22no—
130#389—22no—
131#389—24no—
132#389—24no—
133#389C2312yes—
134#389C2314yes—
135#389C2314yes—
136#389C2312yes—
137#389C2312yes—
138#389C2312yes—
139#388—22no—
140#388—22no—
141#387—22no—
142#387—22no—
143#387—22no—
144#387—22no—
145#387—22no—
146#387—22no—
147#387—22no—
148#387—22no—
149#387—22no—
150#387—24no—
151#387—24no—
152#385—22no—
153#385—22no—
154#385—22no—
155#384—22no—
156#384—22no—
157#384C2312yes—
158#384C2311yescharacteristic
159#384C2312yes—
160#383—22no—
161#383—22no—
162#382—22no—
163#382—22no—
164#382C2311yescharacteristic
165#381—22no—
166#380C2312yes—
167C4×C4⋊C4—24no—
168C4×C4⋊C4—22no—
169C4×C4⋊C4—22no—
170C4×C4⋊C4—22no—
171C4×C4⋊C4—22no—
172C4×C4⋊C4D412yes—
173C4×C4⋊C4D412yes—
174C4×C4⋊C4C2312yes—
175C4×C4⋊C4C2314yes—
176C4×C4⋊C4C2314yes—
177C4×C22⋊C4—24no—
178C4×C22⋊C4D412yes—
179C4×C22⋊C4D412yes—
180#376C2311yescharacteristic
181#376C2311yescharacteristic
182C43C2311yescharacteristic
183C43C2311yescharacteristic
184#340C2314yes—
185#340C2314yes—
186#340C2314yes—
187#340C2314yes—
188#340C2314yes—
189#340C2314yes—
190#340C2314yes—
191#340C2314yes—
192#336C2312yes—
193#336C2312yes—
194#336C2312yes—
195#336C2312yes—
196#336C2314yes—
197#336C2314yes—
198C22×Q8—22no—
199C22×Q8C2412yes—
200C23×C4—22no—
201C23×C4C2×D412yes—
202C23×C4C2411yescharacteristic
203C22×C8C2414yes—
204C22×C8C2414yes—
205C4⋊Q8—28no—
206C4⋊Q8—24no—
207#125—28no—
208#125—24no—
209C22⋊Q8—216no—
210C22⋊Q8—28no—
211C4×Q8—216no—
212C4×Q8—44no—
213C4×Q8—84no—
214C4×Q8—84no—
215#117—24no—
216#117—24no—
217#117—42no—
218#117#4118yes—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—84no—
238C2×C4⋊C4C2×D412yes—
239C2×C4⋊C4C2×D411yescharacteristic
240C2×C4⋊C4C2×D411yescharacteristic
241C2×C4⋊C4C2414yes—
242C2×C4⋊C4C2412yes—
243C2×C4⋊C4C2412yes—
244C2×C4⋊C4C2412yes—
245C2×C4⋊C4C2414yes—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4C2×D412yes—
250C2×C22⋊C4C2×D411yescharacteristic
251C2×C22⋊C4C2×D411yescharacteristic
252C2×C42—22no—
253C2×C42—24no—
254C2×C42—22no—
255C2×C42—24no—
256C2×C42—24no—
257C2×C42—22no—
258C2×C42C2×D412yes—
259C2×C42C2×D412yes—
260C2×C42C2×D412yes—
261C2×C42C2×D412yes—
262C2×C42C2×D412yes—
263C2×C42C2411yescharacteristic
264C2×C42C2411yescharacteristic
265C2×C42C2411yescharacteristic
266C2×C42C2411yescharacteristic
267C2×C42C2411yescharacteristic
268C2×C42C2411yescharacteristic
269#107—28no—
270#106—28no—
271#106—44no—
272#106—44no—
273C4⋊C8—28no—
274#103—28no—
275#103—28no—
276#103—44no—
277#103—44no—
278C22⋊C8—28no—
279#97—42no—
280C4×C8—42no—
281#95—24no—
282#95—24no—
283#95—22no—
284#95—22no—
285#95—24no—
286#95—24no—
287#95—22no—
288#95—22no—
289#95—22no—
290#95—22no—
291#95—22no—
292#95—22no—
293#95—22no—
294#95—22no—
295#95—22no—
296#95—22no—
297#95—22no—
298#95—22no—
299#95—22no—
300#95C2×D412yes—
301#95C2×D412yes—
302#95C2×D412yes—
303#95C2×D412yes—
304C24C22×D411yescharacteristic
305C2×Q8—28no—
306C2×Q8—24no—
307C2×Q8—24no—
308C2×Q8—24no—
309C2×Q8—44no—
310C2×Q8—44no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—24no—
321C22×C4—42no—
322C22×C4—44no—
323C22×C4—84no—
324C22×C4C22×D411yescharacteristic
325C22×C4C22×D412yes—
326C22×C4C22×D411yescharacteristic
327C22×C4C22×D411yescharacteristic
328C22×C4C22×D412yes—
329C22×C4C22×D411yescharacteristic
330C22×C4C22×D411yescharacteristic
331C22×C4C22×D412yes—
332C22×C4C22×D412yes—
333C22×C4C2×#4114yes—
334C22×C4C2511yescharacteristic
335C2×C8—24no—
336C2×C8—24no—
337C2×C8—24no—
338C2×C8—24no—
339C2×C8—24no—
340C2×C8—24no—
341C4⋊C4—216no—
342C4⋊C4—28no—
343C4⋊C4—24no—
344C4⋊C4—24no—
345C4⋊C4—24no—
346C4⋊C4—24no—
347C4⋊C4—24no—
348C4⋊C4—24no—
349C4⋊C4—24no—
350C4⋊C4—41no—
351C4⋊C4—41no—
352C4⋊C4—44no—
353C4⋊C4—44no—
354C4⋊C4—44no—
355C4⋊C4—44no—
356C4⋊C4—44no—
357C4⋊C4—44no—
358C4⋊C4—44no—
359C4⋊C4—44no—
360C4⋊C4—82no—
361C4⋊C4—82no—
362C4⋊C4—82no—
363C4⋊C4—82no—
364C4⋊C4—84no—
365C4⋊C4—84no—
366C4⋊C4—84no—
367C4⋊C4C2×#4118yes—
368C22⋊C4—28no—
369C22⋊C4—24no—
370C22⋊C4—41no—
371C22⋊C4—41no—
372C22⋊C4—44no—
373C22⋊C4—82no—
374C42—24no—
375C42—28no—
376C42—24no—
377C42—24no—
378C42—22no—
379C42—22no—
380C42—22no—
381C42—22no—
382C42—22no—
383C42—42no—
384C42—42no—
385C42—44no—
386C42—82no—
387C42—84no—
388C42—84no—
389C422−1+414yes—
390C23—22no—
391C23—21no—
392C23—21no—
393C23—22no—
394C23#52512yes—
395C23C23×D411yescharacteristic
396Q8—416no—
397Q8—48no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—22no—
404C2×C4—28no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—28no—
411C2×C4—22no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—48no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—164no—
439C2×C4#52512yes—
440C2×C4#53614yes—
441C2×C4#57714yes—
442C2×C4C22×#4112yes—
443C2×C4C2×2−1+412yes—
444C2×C4#58512yes—
445C8—44no—
446C8—44no—
447C22—22no—
448C22—24no—
449C22—41no—
450C22—41no—
451C22—42no—
452C22C2×#52511yescharacteristic
453C22C2×#53611yescharacteristic
454C22#344911yescharacteristic
455C22C2×#57712yes—
456C22#358612yes—
457C4—22no—
458C4—24no—
459C4—24no—
460C4—41no—
461C4—41no—
462C4—41no—
463C4—41no—
464C4—42no—
465C4—48no—
466C4—42no—
467C4—82no—
468C4—84no—
469C4—82no—
470C4—82no—
471C4—82no—
472C4#350714yes—
473C2—42no—
474C2#3511111yescharacteristic
475C2#6058512yes—
476C2#6088912yes—
477C2#6101211yescharacteristic
478C2#6102011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
7. #29520C2semidirect
8. #29512C2semidirect
9. #29405C2semidirect
10. #29361C2semidirect
11. #29327C2semidirect
12. #29280C2semidirect
13. #29255C2semidirect

Maximal subgroups

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

subgroupindex
#353212
#342142
#331572
#331172
#299082
#296452
#295202
#295122
#294052
#293612
#293272
#292802
#292552
#220922
#207802