Finite Group Databasebeta

← #6498027

Subgroups — #6498027

Order 512.

1344 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35337C211yesmaximal, characteristic
2#35275C211yesmaximal, characteristic
3#33299C211yesmaximal, characteristic, complemented
4#33296C211yesmaximal, characteristic, complemented
5#33294C211yesmaximal, characteristic, complemented
6#33293C211yesmaximal, characteristic, complemented
7#30232C214yesmaximal, complemented
8#29229C214yesmaximal, complemented
9#29227C211yesmaximal, characteristic, complemented
10#29227C211yesmaximal, characteristic, complemented
11#29226C211yesmaximal, characteristic, complemented
12#29225C211yesmaximal, characteristic, complemented
13#27971C214yesmaximal
14#27967C211yesmaximal, characteristic
15#27967C211yesmaximal, characteristic
16#27966C211yesmaximal, characteristic
17#27965C211yesmaximal, characteristic
18#21843C214yesmaximal
19#20465C211yesmaximal, characteristic
20C2×#544C2212yes—
21C2×#433C2212yes—
22#2868C2211yescharacteristic
23#2868C2211yescharacteristic
24#2862C2211yescharacteristic
25#2859C2211yescharacteristic
26#2859C2211yescharacteristic
27#2859C2211yescharacteristic
28#2851C2211yescharacteristic
29#2851C2211yescharacteristic
30#2660C2212yes—
31#2660C2212yes—
32#2647C2212yes—
33#2628C2212yes—
34#2589—22no—
35#2589—22no—
36#2588—22no—
37#2588—22no—
38#2552—22no—
39#2551—22no—
40#2440—22no—
41#2440—22no—
42#2331C2212yes—
43#2317C2212yes—
44#2314C2212yes—
45C4×#125C2211yescharacteristic
46C4×#125C2211yescharacteristic
47#2285C2211yescharacteristic
48C2×#384C2211yescharacteristic
49C4×#117C2211yescharacteristic
50#2101C2214yes—
51#2101C2214yes—
52#2101C2214yes—
53#2101C2214yes—
54#2100C2214yes—
55#2100C2214yes—
56#2099C2214yes—
57#2099C2214yes—
58#1993C2214yes—
59#1993C2214yes—
60#1992C2211yescharacteristic
61#1992C2211yescharacteristic
62#1992C2211yescharacteristic
63#1992C2211yescharacteristic
64#1992C2214yes—
65#1992C2211yescharacteristic
66#1992C2211yescharacteristic
67#1992C2211yescharacteristic
68#1992C2211yescharacteristic
69#1991C2214yes—
70#1990C2214yes—
71#1990C2214yes—
72#1989C2214yes—
73#1989C2214yes—
74#1988C2214yes—
75#1987C2214yes—
76#1836C2211yescharacteristic
77#1836C2211yescharacteristic
78#1836C2211yescharacteristic
79#1836C2211yescharacteristic
80#1836C2214yes—
81#1836C2211yescharacteristic
82#1836C2211yescharacteristic
83#1836C2211yescharacteristic
84#1836C2211yescharacteristic
85#1835C2214yes—
86#1825C2214yes—
87#1825C2214yes—
88#1824C2214yes—
89#1823C2214yes—
90#1768C2212yes—
91#1767C2211yescharacteristic
92#1767C2211yescharacteristic
93#1750C2212yes—
94#1749C2211yescharacteristic
95#1749C2211yescharacteristic
96#1746C2212yes—
97C4×#97C2211yescharacteristic
98C4×#97C2211yescharacteristic
99#544—216no—
100C2×C4⋊Q8—22no—
101C2×C4⋊Q8—22no—
102C2×C4⋊Q8—22no—
103C2×C4⋊Q8C2312yes—
104C2×C4⋊Q8C2312yes—
105C2×C4⋊Q8C2312yes—
106C2×#125—22no—
107C2×#125C2311yescharacteristic
108C2×#125C2311yescharacteristic
109C2×#125C2312yes—
110C2×#125C2312yes—
111C2×C22⋊Q8—22no—
112C2×C22⋊Q8—22no—
113C2×#117C2312yes—
114C2×#117C2311yescharacteristic
115C2×#117C2311yescharacteristic
116C22×C4⋊C4—22no—
117C22×C4⋊C4C2312yes—
118C22×C42C2311yescharacteristic
119#433—28no—
120C2×C4⋊C8C2312yes—
121C2×C4⋊C8C2312yes—
122C2×C22⋊C8C2312yes—
123C2×C22⋊C8C2312yes—
124C2×#97C2311yescharacteristic
125C2×#97C2311yescharacteristic
126C2×#97C2311yescharacteristic
127C2×#97C2311yescharacteristic
128C2×#97C2312yes—
129C2×#97C2312yes—
130C2×C4×C8C2312yes—
131C2×C4×C8C2312yes—
132#400—22no—
133#400—22no—
134#400—22no—
135#399—22no—
136#398—22no—
137#398—22no—
138#398—24no—
139#398—24no—
140#395—22no—
141#395—24no—
142#395—24no—
143#393—22no—
144#391—22no—
145#391—22no—
146#391—22no—
147#391—22no—
148#389—24no—
149#389—22no—
150#389—22no—
151#389—22no—
152#389—22no—
153#389—22no—
154#389—22no—
155#389C2312yes—
156#389C2312yes—
157#389C2312yes—
158#389C2312yes—
159#388—22no—
160#388—22no—
161#387—22no—
162#387—22no—
163#387—22no—
164#387—22no—
165#387—22no—
166#387—22no—
167#387—22no—
168#387—22no—
169#385—22no—
170#385—22no—
171#385—24no—
172#384—22no—
173#384—22no—
174#384—22no—
175#384C2312yes—
176#384C2312yes—
177#384C2312yes—
178#384C2311yescharacteristic
179#384C2311yescharacteristic
180#384C2311yescharacteristic
181#384C2311yescharacteristic
182#384C2312yes—
183#382C2311yescharacteristic
184#382C2311yescharacteristic
185#382C2312yes—
186#382C2312yes—
187#382C2312yes—
188#382C2312yes—
189#382C2312yes—
190#380—22no—
191#380C2312yes—
192#380C2312yes—
193C4×C4⋊C4—22no—
194C4×C4⋊C4—22no—
195C4×C4⋊C4—22no—
196C4×C4⋊C4D412yes—
197C4×C4⋊C4D412yes—
198C4×C4⋊C4C2312yes—
199C4×C4⋊C4C2312yes—
200C4×C22⋊C4D412yes—
201C4×C22⋊C4D412yes—
202#376C2311yescharacteristic
203#376C2311yescharacteristic
204C43C2311yescharacteristic
205C43C2311yescharacteristic
206#340C2314yes—
207#340C2314yes—
208#340C2314yes—
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#340C2314yes—
218#340C2314yes—
219#340C2314yes—
220#340C2314yes—
221#340C2314yes—
222#336C2312yes—
223#336C2312yes—
224#336C2312yes—
225#336C2312yes—
226C22×Q8—22no—
227C22×Q8—22no—
228C22×Q8—22no—
229C23×C4—22no—
230C23×C4C2×D412yes—
231C23×C4C2411yescharacteristic
232C22×C8C2412yes—
233C22×C8C2412yes—
234C22×C8C2412yes—
235C22×C8C2412yes—
236C4⋊Q8—28no—
237C4⋊Q8—28no—
238C4⋊Q8—28no—
239C4⋊Q8—48no—
240C4⋊Q8—84no—
241C4⋊Q8—84no—
242#125—28no—
243#125—28no—
244#125—22no—
245#125—22no—
246#125—22no—
247#125—22no—
248#125—48no—
249C22⋊Q8—48no—
250C22⋊Q8—48no—
251#117—24no—
252#117—28no—
253#117#4118yes—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—84no—
279C2×C4⋊C4C2×D412yes—
280C2×C4⋊C4C2×D411yescharacteristic
281C2×C4⋊C4C2×D411yescharacteristic
282C2×C4⋊C4C2412yes—
283C2×C4⋊C4C2412yes—
284C2×C4⋊C4C2412yes—
285C2×C4⋊C4C2412yes—
286C2×C4⋊C4C2412yes—
287C2×C4⋊C4C2412yes—
288C2×C4⋊C4C2412yes—
289C2×C4⋊C4C2412yes—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4C2×D412yes—
294C2×C22⋊C4C2×D411yescharacteristic
295C2×C22⋊C4C2×D411yescharacteristic
296C2×C42—22no—
297C2×C42—22no—
298C2×C42—22no—
299C2×C42—22no—
300C2×C42—22no—
301C2×C42C2×D412yes—
302C2×C42C2×D412yes—
303C2×C42C2×D412yes—
304C2×C42C2×D412yes—
305C2×C42C2×D412yes—
306C2×C42C2411yescharacteristic
307C2×C42C2411yescharacteristic
308C2×C42C2411yescharacteristic
309C2×C42C2411yescharacteristic
310C2×C42C2411yescharacteristic
311C2×C42C2411yescharacteristic
312C4⋊C8—24no—
313C4⋊C8—24no—
314C22⋊C8—24no—
315C22⋊C8—24no—
316#97—24no—
317#97—24no—
318#97—22no—
319#97—22no—
320#97—22no—
321#97—22no—
322C4×C8—24no—
323C4×C8—24no—
324#95—24no—
325#95—22no—
326#95—22no—
327#95—22no—
328#95—22no—
329#95—22no—
330#95—22no—
331#95—22no—
332#95—22no—
333#95—24no—
334#95—24no—
335#95—22no—
336#95—22no—
337#95—22no—
338#95—22no—
339#95—22no—
340#95—22no—
341#95C2×D412yes—
342#95C2×D412yes—
343#95C2×D412yes—
344#95C2×D412yes—
345C24C22×D411yescharacteristic
346C2×Q8—44no—
347C2×Q8—44no—
348C2×Q8—44no—
349C2×Q8—84no—
350C2×Q8—84no—
351C2×Q8—82no—
352C2×Q8—82no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—24no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—44no—
368C22×C4—84no—
369C22×C4C22×D412yes—
370C22×C4C22×D412yes—
371C22×C4C22×D411yescharacteristic
372C22×C4C22×D411yescharacteristic
373C22×C4C22×D411yescharacteristic
374C22×C4C22×D411yescharacteristic
375C22×C4C22×D411yescharacteristic
376C22×C4C22×D412yes—
377C22×C4C22×D412yes—
378C22×C4C2×#4114yes—
379C22×C4C2511yescharacteristic
380C2×C8—22no—
381C2×C8—22no—
382C2×C8—22no—
383C2×C8—22no—
384C2×C8—22no—
385C2×C8—22no—
386C2×C8—22no—
387C2×C8—22no—
388C2×C8—22no—
389C2×C8—22no—
390C2×C8—22no—
391C2×C8—22no—
392C4⋊C4—24no—
393C4⋊C4—24no—
394C4⋊C4—24no—
395C4⋊C4—24no—
396C4⋊C4—24no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—24no—
402C4⋊C4—24no—
403C4⋊C4—44no—
404C4⋊C4—44no—
405C4⋊C4—44no—
406C4⋊C4—44no—
407C4⋊C4—44no—
408C4⋊C4—44no—
409C4⋊C4—44no—
410C4⋊C4—44no—
411C4⋊C4—44no—
412C4⋊C4—44no—
413C4⋊C4—44no—
414C4⋊C4—44no—
415C4⋊C4—44no—
416C4⋊C4—44no—
417C4⋊C4—82no—
418C4⋊C4—82no—
419C4⋊C4—82no—
420C4⋊C4—82no—
421C4⋊C4—82no—
422C4⋊C4—82no—
423C4⋊C4—84no—
424C4⋊C4—84no—
425C4⋊C4—84no—
426C4⋊C4#12414yes—
427C4⋊C4#12414yes—
428C22⋊C4—24no—
429C22⋊C4—44no—
430C22⋊C4—82no—
431C22⋊C4—82no—
432C22⋊C4#12414yes—
433C22⋊C4#12414yes—
434C42—24no—
435C42—21no—
436C42—21no—
437C42—22no—
438C42—42no—
439C42—42no—
440C42—42no—
441C42—42no—
442C42—44no—
443C42—82no—
444C42—82no—
445C42C2×#4114yes—
446C42C2×#4114yes—
447C422−1+414yes—
448C422−1+412yes—
449C422−1+412yes—
450C23—22no—
451C23—22no—
452C23—21no—
453C23—21no—
454C23C2×#12412yes—
455C23C23×D411yescharacteristic
456Q8—164no—
457Q8—164no—
458Q8—164no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—48no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—44no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—164no—
499C2×C4C2×#12412yes—
500C2×C4C2×#12412yes—
501C2×C4C2×#12412yes—
502C2×C4C2×#12412yes—
503C2×C4C2×#12412yes—
504C2×C4#53614yes—
505C2×C4#53614yes—
506C2×C4C22×#4112yes—
507C2×C4C2×2−1+411yescharacteristic
508C2×C4C2×2−1+411yescharacteristic
509C2×C4C2×2−1+412yes—
510C8—42no—
511C8—42no—
512C8—42no—
513C8—42no—
514C22—22no—
515C22—22no—
516C22—22no—
517C22—42no—
518C22—42no—
519C22C22×#12411yescharacteristic
520C22C2×#53611yescharacteristic
521C22C2×#53611yescharacteristic
522C22#358512yes—
523C22#358612yes—
524C4—24no—
525C4—22no—
526C4—22no—
527C4—22no—
528C4—22no—
529C4—24no—
530C4—22no—
531C4—42no—
532C4—42no—
533C4—82no—
534C4—82no—
535C4—82no—
536C4—82no—
537C4—82no—
538C4—82no—
539C4—82no—
540C4—82no—
541C4#350912yes—
542C4#350912yes—
543C2—42no—
544C2#3532011yescharacteristic
545C2#6062611yescharacteristic
546C2#6062711yescharacteristic
547C2#6086712yes—
548C2#6088512yes—
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
3. #33299C2semidirect
4. #33296C2semidirect
5. #33294C2semidirect
6. #33293C2semidirect
7. #30232C2semidirect
8. #29229C2semidirect
9. #29227C2semidirect
10. #29227C2semidirect
11. #29226C2semidirect
12. #29225C2semidirect

Maximal subgroups

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

subgroupindex
#353372
#352752
#332992
#332962
#332942
#332932
#302322
#292292
#292272
#292262
#292252
#279712
#279672
#279662
#279652
#218432
#204652