Finite Group Databasebeta

← #162995

Subgroups — #162995

Order 512.

709 conjugacy classes of subgroups.

The subgroup lattice has 613 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35116C211yesmaximal, characteristic
2#22830C211yesmaximal, characteristic, complemented
3#20570C211yesmaximal, characteristic, complemented
4#7712C211yesmaximal, characteristic
5#7700C211yesmaximal, characteristic, complemented
6#7692C211yesmaximal, characteristic
7#7692C211yesmaximal, characteristic
8#7680C211yesmaximal, characteristic, complemented
9#7677C211yesmaximal, characteristic, complemented
10#7672C211yesmaximal, characteristic
11#7657C211yesmaximal, characteristic, complemented
12#7636C211yesmaximal, characteristic, complemented
13#7628C211yesmaximal, characteristic
14#7619C211yesmaximal, characteristic, complemented
15#7614C211yesmaximal, characteristic
16C2×#446—21no—
17#2864C2211yescharacteristic
18#2860C2211yescharacteristic
19#2860C2211yescharacteristic
20#2859C2211yescharacteristic
21#2654—21no—
22#2637—21no—
23#2584C411yescharacteristic
24#2580C411yescharacteristic
25#2570C411yescharacteristic
26#2566C411yescharacteristic
27#2519—21no—
28#2513C411yescharacteristic
29#2512C411yescharacteristic
30#2508C411yescharacteristic
31#2507C411yescharacteristic
32#2478—21no—
33#2475—21no—
34#2469—21no—
35#2334—21no—
36#2334—21no—
37C4×#125C2211yescharacteristic
38#2293C2211yescharacteristic
39C42×Q8C2211yescharacteristic
40#1985—21no—
41#1917—21no—
42#1916—21no—
43#1839C2211yescharacteristic
44#1839C2211yescharacteristic
45#1831C2211yescharacteristic
46#1771—21no—
47#1771—21no—
48#1771—21no—
49#1770—21no—
50#1767C2211yescharacteristic
51#1767C2211yescharacteristic
52#1767C2211yescharacteristic
53C4×C4⋊C8C2211yescharacteristic
54#1745C2211yescharacteristic
55#1745C2211yescharacteristic
56#1745C2211yescharacteristic
57#1744C2211yescharacteristic
58#1744C2211yescharacteristic
59#1271C2211yescharacteristic
60#1271C2211yescharacteristic
61#1271C2211yescharacteristic
62#1271C2211yescharacteristic
63#1271C2211yescharacteristic
64#1271C2211yescharacteristic
65#1271C2211yescharacteristic
66#1271C2211yescharacteristic
67#1271C2211yescharacteristic
68#1271C2211yescharacteristic
69#1271C2211yescharacteristic
70#1271C2211yescharacteristic
71#1270C2211yescharacteristic
72#1270C2211yescharacteristic
73#1270C2211yescharacteristic
74#1270C2211yescharacteristic
75C2×#126—21no—
76C2×#126—21no—
77C2×#126—21no—
78C2×#125—21no—
79C2×#125C2×C411yescharacteristic
80C2×C4×Q8C2×C411yescharacteristic
81C2×C4×Q8D411yescharacteristic
82C2×C4×Q8D411yescharacteristic
83C2×C4×D4C2×C411yescharacteristic
84C2×#117—21no—
85#446—28no—
86C2×C4⋊C8—21no—
87C2×C4⋊C8—21no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8—21no—
91C2×#97—21no—
92C2×#97—21no—
93C2×#97—21no—
94C2×#97—21no—
95C2×#97—21no—
96C2×#97—21no—
97C2×C4×C8—21no—
98C2×C4×C8—21no—
99C2×C4×C8—21no—
100#400—21no—
101#400—21no—
102#399—21no—
103#398—21no—
104#398—21no—
105#398—21no—
106#398—21no—
107#398—21no—
108#397—21no—
109#397—21no—
110#397—21no—
111#396—21no—
112#396—21no—
113#396—21no—
114#395—21no—
115#393—21no—
116#390C2×C411yescharacteristic
117#390C2×C411yescharacteristic
118#390C2×C411yescharacteristic
119#390C2×C411yescharacteristic
120#390C2×C411yescharacteristic
121#390C2311yescharacteristic
122#389—21no—
123#389—21no—
124#389—21no—
125#389—21no—
126#389—21no—
127#389C2×C411yescharacteristic
128#389C2×C411yescharacteristic
129#389C2×C411yescharacteristic
130#389C2×C411yescharacteristic
131#389C2×C411yescharacteristic
132#389C2×C411yescharacteristic
133#389C2×C411yescharacteristic
134#389C2×C411yescharacteristic
135#389C2×C411yescharacteristic
136#389C2311yescharacteristic
137#389C2311yescharacteristic
138#389C2311yescharacteristic
139#388—21no—
140#388—21no—
141#388—21no—
142#388—21no—
143#388—21no—
144#388—21no—
145#388—21no—
146#388—21no—
147#388—21no—
148#388—21no—
149#388—21no—
150#387—21no—
151#387—21no—
152#387—21no—
153#387—21no—
154#387—21no—
155#387—21no—
156#387—21no—
157#387—21no—
158#387—21no—
159#387—21no—
160#385—21no—
161#385—21no—
162#385—21no—
163#385—21no—
164#384C2×C411yescharacteristic
165#384C2×C411yescharacteristic
166#382—21no—
167#382—21no—
168#382C2×C411yescharacteristic
169#382C2×C411yescharacteristic
170#382C2×C411yescharacteristic
171C4×C4⋊C4—21no—
172C4×C4⋊C4—21no—
173C4×C4⋊C4C2×C411yescharacteristic
174C4×C4⋊C4C2×C411yescharacteristic
175C4×C4⋊C4C2×C411yescharacteristic
176C4×C4⋊C4C2×C411yescharacteristic
177C4×C4⋊C4C2×C411yescharacteristic
178C4×C4⋊C4C2×C411yescharacteristic
179C4×C4⋊C4D411yescharacteristic
180C4×C4⋊C4D411yescharacteristic
181C4×C4⋊C4D411yescharacteristic
182C4×C4⋊C4D411yescharacteristic
183C4×C4⋊C4D411yescharacteristic
184C4×C4⋊C4D411yescharacteristic
185C4×C22⋊C4—21no—
186C43C2311yescharacteristic
187C43C2311yescharacteristic
188C43C2311yescharacteristic
189#336—21no—
190#336—21no—
191#336—21no—
192#336—21no—
193#336—21no—
194#336—21no—
195#336—21no—
196#336—21no—
197#336—21no—
198#336—21no—
199#336C2311yescharacteristic
200#336C2311yescharacteristic
201#336C2311yescharacteristic
202#336C2311yescharacteristic
203#336C2311yescharacteristic
204#336C2311yescharacteristic
205#336C2311yescharacteristic
206#336C2311yescharacteristic
207C22×Q8C22⋊C411yescharacteristic
208C22×D4C22×C411yescharacteristic
209C23×C4—21no—
210C22×C8—21no—
211C22×C8—21no—
212C22×C8—21no—
213C22×C8—21no—
214C22×C8—21no—
215C22×C8—21no—
216C22×C8—21no—
217C22×C8—21no—
218#126—42no—
219#126—42no—
220#126—44no—
221#126—81no—
222#126—81no—
223#125—42no—
224#125—81no—
225#125—81no—
226C4×Q8—24no—
227C4×Q8—42no—
228C4×Q8—41no—
229C4×Q8—41no—
230C4×D4—42no—
231#117—81no—
232#117—81no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4C22⋊C411yescharacteristic
252C2×C4⋊C4C22⋊C411yescharacteristic
253C2×C4⋊C4C22⋊C411yescharacteristic
254C2×C4⋊C4C22⋊C411yescharacteristic
255C2×C4⋊C4C22⋊C411yescharacteristic
256C2×C4⋊C4C22⋊C411yescharacteristic
257C2×C4⋊C4C22⋊C411yescharacteristic
258C2×C4⋊C4C22⋊C411yescharacteristic
259C2×C4⋊C4C22⋊C411yescharacteristic
260C2×C4⋊C4C22×C411yescharacteristic
261C2×C4⋊C4C22×C411yescharacteristic
262C2×C4⋊C4C22×C411yescharacteristic
263C2×C4⋊C4C22×C411yescharacteristic
264C2×C4⋊C4C22×C411yescharacteristic
265C2×C4⋊C4C22×C411yescharacteristic
266C2×C4⋊C4C22×C411yescharacteristic
267C2×C22⋊C4—21no—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—21no—
280C2×C42—21no—
281C2×C42—21no—
282C2×C42—21no—
283C2×C42C22×C411yescharacteristic
284C2×C42C22×C411yescharacteristic
285C2×C42C22×C411yescharacteristic
286C2×C42C22×C411yescharacteristic
287C2×C42C22×C411yescharacteristic
288C2×C42C22×C411yescharacteristic
289C2×C42C2×D411yescharacteristic
290C2×C42C2×D411yescharacteristic
291C2×C42C2×D411yescharacteristic
292C2×C42C2×D411yescharacteristic
293C2×C42C2×D411yescharacteristic
294C2×C42C2×D411yescharacteristic
295C2×C42C2×D411yescharacteristic
296C2×C42C2×D411yescharacteristic
297C2×C42C2×D411yescharacteristic
298C2×C42C2×D411yescharacteristic
299C2×C42C2×D411yescharacteristic
300C2×C42C2×D411yescharacteristic
301C2×C42C2411yescharacteristic
302C4⋊C8—24no—
303C4⋊C8—24no—
304C4⋊C8—24no—
305C4⋊C8—42no—
306C4⋊C8—42no—
307#97—24no—
308#97—24no—
309#97—42no—
310#97—42no—
311#97—81no—
312#97—81no—
313C4×C8—24no—
314C4×C8—42no—
315C4×C8—81no—
316#95—21no—
317#95—21no—
318#95—21no—
319#95—21no—
320#95—21no—
321#95—21no—
322#95—21no—
323#95—21no—
324#95—21no—
325#95—21no—
326#95—21no—
327#95—21no—
328#95—21no—
329#95—21no—
330#95—21no—
331#95—21no—
332#95—21no—
333#95—21no—
334#95—21no—
335#95—21no—
336#95—21no—
337#95C22⋊C411yescharacteristic
338#95C22⋊C411yescharacteristic
339#95C22⋊C411yescharacteristic
340#95C22⋊C411yescharacteristic
341#95C22⋊C411yescharacteristic
342#95C22⋊C411yescharacteristic
343C24—21no—
344C2×Q8—22no—
345C2×Q8—41no—
346C2×Q8—41no—
347C2×D4—22no—
348C2×D4—41no—
349C2×D4—41no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—42no—
358C22×C4—81no—
359C22×C4—81no—
360C22×C4C2×C22⋊C411yescharacteristic
361C22×C4C2×C22⋊C411yescharacteristic
362C22×C4C2×C22⋊C411yescharacteristic
363C22×C4C2×C22⋊C411yescharacteristic
364C22×C4C2×C22⋊C411yescharacteristic
365C22×C4C2×C22⋊C411yescharacteristic
366C22×C4C2×C22⋊C411yescharacteristic
367C22×C4C2×C22⋊C411yescharacteristic
368C22×C4C2×C22⋊C411yescharacteristic
369C22×C4C2×C22⋊C411yescharacteristic
370C22×C4C2×C22⋊C411yescharacteristic
371C22×C4C2×C22⋊C411yescharacteristic
372C22×C4C23×C411yescharacteristic
373C22×C4C22×D411yescharacteristic
374C22×C4C22×D411yescharacteristic
375C2×C8—22no—
376C2×C8—22no—
377C2×C8—22no—
378C2×C8—22no—
379C2×C8—22no—
380C2×C8—41no—
381C2×C8—41no—
382C2×C8—41no—
383C2×C8—41no—
384C2×C8—41no—
385C2×C8—41no—
386C2×C8—41no—
387C2×C8—41no—
388C2×C8—41no—
389C2×C8—41no—
390C2×C8—41no—
391C2×C8—41no—
392C2×C8—41no—
393C2×C8—41no—
394C2×C8—41no—
395C2×C8—41no—
396C2×C8—41no—
397C2×C8—41no—
398C2×C8—41no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—41no—
405C4⋊C4—41no—
406C4⋊C4—41no—
407C4⋊C4—41no—
408C4⋊C4—41no—
409C4⋊C4—41no—
410C4⋊C4—41no—
411C4⋊C4—41no—
412C4⋊C4—41no—
413C4⋊C4—41no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—41no—
419C4⋊C4—41no—
420C4⋊C4—41no—
421C4⋊C4—41no—
422C4⋊C4—41no—
423C4⋊C4—41no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—81no—
427C4⋊C4—81no—
428C4⋊C4—81no—
429C4⋊C4—81no—
430C4⋊C4—81no—
431C4⋊C4—81no—
432C4⋊C4—81no—
433C4⋊C4—81no—
434C4⋊C4—81no—
435C4⋊C4#10414yes—
436C22⋊C4—42no—
437C22⋊C4—42no—
438C22⋊C4—42no—
439C22⋊C4—41no—
440C22⋊C4—41no—
441C22⋊C4—41no—
442C22⋊C4—41no—
443C22⋊C4—42no—
444C22⋊C4—42no—
445C22⋊C4—42no—
446C22⋊C4—81no—
447C22⋊C4—81no—
448C22⋊C4—81no—
449C22⋊C4—81no—
450C22⋊C4—81no—
451C42—22no—
452C42—22no—
453C42—22no—
454C42—21no—
455C42—21no—
456C42—21no—
457C42—21no—
458C42—21no—
459C42—21no—
460C42—21no—
461C42—21no—
462C42—21no—
463C42—21no—
464C42—21no—
465C42—21no—
466C42—22no—
467C42—22no—
468C42—22no—
469C42—41no—
470C42—41no—
471C42—41no—
472C42—41no—
473C42—41no—
474C42—41no—
475C42—41no—
476C42—42no—
477C42—81no—
478C42—81no—
479C23—41no—
480C23—41no—
481C23—41no—
482C23—41no—
483C23—41no—
484C23—41no—
485C23—41no—
486C23C22×C22⋊C411yescharacteristic
487Q8—44no—
488D4—82no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—84no—
550C2×C4C2×#10412yes—
551C2×C4C2×#10412yes—
552C2×C4C2×#10412yes—
553C2×C4#56812yes—
554C2×C4#57511yescharacteristic
555C2×C4#57511yescharacteristic
556C2×C4#57611yescharacteristic
557C2×C4#57711yescharacteristic
558C8—42no—
559C8—42no—
560C8—42no—
561C8—42no—
562C8—42no—
563C8—81no—
564C8—81no—
565C8—81no—
566C22—81no—
567C22—81no—
568C22—81no—
569C22—81no—
570C22—81no—
571C22—81no—
572C22—81no—
573C22#287911yescharacteristic
574C22#288411yescharacteristic
575C22#288911yescharacteristic
576C22#289611yescharacteristic
577C22#289711yescharacteristic
578C22C22×#10411yescharacteristic
579C22#290111yescharacteristic
580C4—22no—
581C4—22no—
582C4—21no—
583C4—21no—
584C4—21no—
585C4—21no—
586C4—22no—
587C4—22no—
588C4—41no—
589C4—41no—
590C4—41no—
591C4—41no—
592C4—41no—
593C4—41no—
594C4—41no—
595C4—41no—
596C4—41no—
597C4—81no—
598C4—81no—
599C4—81no—
600C4—81no—
601C4—81no—
602C4—81no—
603C4—81no—
604C2—161no—
605C2#1741011yescharacteristic
606C2#1750311yescharacteristic
607C2#1750511yescharacteristic
608C2#1755011yescharacteristic
609C2#1762411yescharacteristic
610C2#1765311yescharacteristic
611C2#1776911yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D4C22×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22830C2semidirect
3. #20570C2semidirect
5. #7700C2semidirect
8. #7680C2semidirect
9. #7677C2semidirect
11. #7657C2semidirect
12. #7636C2semidirect
14. #7619C2semidirect

Maximal subgroups

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

subgroupindex
#351162
#228302
#205702
#77122
#77002
#76922
#76802
#76772
#76722
#76572
#76362
#76282
#76192
#76142