Finite Group Databasebeta

← #208073

Subgroups — #208073

Order 512.

725 conjugacy classes of subgroups.

The subgroup lattice has 643 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35142C211yesmaximal, characteristic
2#21435C211yesmaximal, characteristic
3#20467C211yesmaximal, characteristic
4#9893C211yesmaximal, characteristic, complemented
5#9893C211yesmaximal, characteristic, complemented
6#9690C211yesmaximal, characteristic, complemented
7#9690C211yesmaximal, characteristic, complemented
8#9126C211yesmaximal, characteristic
9#9126C211yesmaximal, characteristic
10#9125C211yesmaximal, characteristic
11#9125C211yesmaximal, characteristic
12#8227C211yesmaximal, characteristic, complemented
13#8227C211yesmaximal, characteristic, complemented
14C4×#330C211yesmaximal, characteristic, complemented
15C4×#330C211yesmaximal, characteristic, complemented
16C2×#432C2211yescharacteristic
17C2×#432C2211yescharacteristic
18#2864C2211yescharacteristic
19#2859C2211yescharacteristic
20#2851C2211yescharacteristic
21#2851C2211yescharacteristic
22#2655—21no—
23#2652—21no—
24#2533—21no—
25#2528—21no—
26#2509C411yescharacteristic
27#2508C411yescharacteristic
28#2507C411yescharacteristic
29#2506C411yescharacteristic
30#2497C411yescharacteristic
31#2496C411yescharacteristic
32#2495C411yescharacteristic
33#2494C411yescharacteristic
34#2478—21no—
35#2475—21no—
36#2336—21no—
37#2334—21no—
38C4×#125C2211yescharacteristic
39C4×#125C2211yescharacteristic
40C4×#117C2211yescharacteristic
41#1924—21no—
42#1923—21no—
43#1917—21no—
44#1916—21no—
45#1768—21no—
46#1768—21no—
47#1767C2211yescharacteristic
48#1767C2211yescharacteristic
49#1750—21no—
50#1750—21no—
51#1749C2211yescharacteristic
52#1749C2211yescharacteristic
53#1745C2211yescharacteristic
54#1745C2211yescharacteristic
55#1745C2211yescharacteristic
56C4×#97C2211yescharacteristic
57C4×#97C2211yescharacteristic
58C4×#97C2211yescharacteristic
59C2×#330C2211yescharacteristic
60C2×#330C2211yescharacteristic
61C2×#330C2211yescharacteristic
62C2×#330C2211yescharacteristic
63C2×#330C2211yescharacteristic
64C2×#330C2211yescharacteristic
65C2×#330C2211yescharacteristic
66C2×#330C2211yescharacteristic
67#1295C2211yescharacteristic
68#1295C2211yescharacteristic
69#1295C2211yescharacteristic
70#1295C2211yescharacteristic
71#1295C2211yescharacteristic
72#1295C2211yescharacteristic
73#1295C2211yescharacteristic
74#1295C2211yescharacteristic
75C2×#126—21no—
76C2×#125—21no—
77C2×#125C2311yescharacteristic
78C2×#125C2311yescharacteristic
79C2×#117D411yescharacteristic
80C2×#117D411yescharacteristic
81C22×C42C2311yescharacteristic
82#432—22no—
83#432—22no—
84#432—22no—
85#432—22no—
86C2×C4⋊C8—21no—
87C2×C4⋊C8—21no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8—21no—
90C2×C22⋊C8—21no—
91C2×C22⋊C8—21no—
92C2×C22⋊C8—21no—
93C2×C22⋊C8—21no—
94C2×#97—21no—
95C2×#97—21no—
96C2×#97—21no—
97C2×#97C2311yescharacteristic
98C2×#97C2311yescharacteristic
99C2×#97C2311yescharacteristic
100C2×#97C2311yescharacteristic
101C2×#97C2311yescharacteristic
102C2×#97C2311yescharacteristic
103C2×#97C2311yescharacteristic
104C2×#97C2311yescharacteristic
105C2×C4×C8—21no—
106C2×C4×C8—21no—
107C2×C4×C8—21no—
108#401—21no—
109#400—21no—
110#400—21no—
111#400—21no—
112#400—21no—
113#400—21no—
114#400—21no—
115#400—21no—
116#398—21no—
117#398—21no—
118#398—21no—
119#398—21no—
120#397—21no—
121#397—21no—
122#397—21no—
123#396—21no—
124#389—21no—
125#389—21no—
126#389—21no—
127#389—21no—
128#389—21no—
129#389—21no—
130#389C2×C411yescharacteristic
131#389C2×C411yescharacteristic
132#389C2×C411yescharacteristic
133#389C2×C411yescharacteristic
134#389C2×C411yescharacteristic
135#389C2×C411yescharacteristic
136#389C2×C411yescharacteristic
137#389C2×C411yescharacteristic
138#389C2×C411yescharacteristic
139#389C2×C411yescharacteristic
140#389C2×C411yescharacteristic
141#389C2×C411yescharacteristic
142#388—21no—
143#388—21no—
144#388—21no—
145#387—21no—
146#387—21no—
147#387—21no—
148#387—21no—
149#387—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#387—21no—
161#387—21no—
162#387—21no—
163#387—21no—
164#385—21no—
165#385—21no—
166#385—21no—
167#385—21no—
168#384C2×C411yescharacteristic
169#384C2×C411yescharacteristic
170#383—21no—
171#382—21no—
172#382C2×C411yescharacteristic
173#382C2×C411yescharacteristic
174#382C2311yescharacteristic
175#382C2311yescharacteristic
176#380C2×C411yescharacteristic
177#380C2×C411yescharacteristic
178#380C2×C411yescharacteristic
179#380C2×C411yescharacteristic
180C4×C4⋊C4—21no—
181C4×C4⋊C4—21no—
182C4×C4⋊C4—21no—
183C4×C4⋊C4—21no—
184C4×C4⋊C4C2×C411yescharacteristic
185C4×C4⋊C4C2×C411yescharacteristic
186C4×C4⋊C4C2×C411yescharacteristic
187C4×C4⋊C4C2×C411yescharacteristic
188C4×C4⋊C4C2×C411yescharacteristic
189C4×C4⋊C4C2×C411yescharacteristic
190C4×C4⋊C4D411yescharacteristic
191C4×C4⋊C4D411yescharacteristic
192C4×C22⋊C4C2×C411yescharacteristic
193C4×C22⋊C4C2×C411yescharacteristic
194C4×C22⋊C4D411yescharacteristic
195C4×C22⋊C4D411yescharacteristic
196#376D411yescharacteristic
197#376D411yescharacteristic
198C43C2311yescharacteristic
199C43C2311yescharacteristic
200#336—21no—
201#336—21no—
202#336—21no—
203#336—21no—
204#336—21no—
205#336—21no—
206#336—21no—
207#336—21no—
208#336—21no—
209#336—21no—
210#330—22no—
211#330—22no—
212#330—22no—
213#330—22no—
214#330—22no—
215#330—22no—
216#330—22no—
217#330—22no—
218C23×C4C22×C411yescharacteristic
219C23×C4C2×D411yescharacteristic
220C23×C4C2×D411yescharacteristic
221C22×C8—21no—
222C22×C8—21no—
223C22×C8—21no—
224C22×C8—21no—
225C22×C8—21no—
226C22×C8—21no—
227C22×C8—21no—
228C22×C8—21no—
229#126—44no—
230#125—21no—
231#125—21no—
232#125—21no—
233#125—21no—
234#125—21no—
235#125—21no—
236#125—21no—
237#125—21no—
238#125—44no—
239#117—42no—
240#117—42no—
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⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
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×C4⋊C4C22×C411yescharacteristic
268C2×C4⋊C4C22×C411yescharacteristic
269C2×C4⋊C4C22×C411yescharacteristic
270C2×C4⋊C4C22×C411yescharacteristic
271C2×C4⋊C4C22×C411yescharacteristic
272C2×C4⋊C4C22×C411yescharacteristic
273C2×C22⋊C4—21no—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4C22⋊C411yescharacteristic
278C2×C22⋊C4C22⋊C411yescharacteristic
279C2×C22⋊C4C22⋊C411yescharacteristic
280C2×C22⋊C4C22⋊C411yescharacteristic
281C2×C42—21no—
282C2×C42—21no—
283C2×C42—21no—
284C2×C42—21no—
285C2×C42—21no—
286C2×C42—21no—
287C2×C42—21no—
288C2×C42—21no—
289C2×C42C22×C411yescharacteristic
290C2×C42C22×C411yescharacteristic
291C2×C42C22×C411yescharacteristic
292C2×C42C22×C411yescharacteristic
293C2×C42C22×C411yescharacteristic
294C2×C42C2×D411yescharacteristic
295C2×C42C2×D411yescharacteristic
296C2×C42C2×D411yescharacteristic
297C2×C42C2×D411yescharacteristic
298C2×C42C2×D411yescharacteristic
299C2×C42C2×D411yescharacteristic
300C2×C42C2×D411yescharacteristic
301C2×C42C2×D411yescharacteristic
302C2×C42C2×D411yescharacteristic
303C2×C42C2×D411yescharacteristic
304C2×C42C2411yescharacteristic
305C4⋊C8—24no—
306C4⋊C8—24no—
307C4⋊C8—42no—
308C4⋊C8—42no—
309C22⋊C8—24no—
310C22⋊C8—24no—
311C22⋊C8—42no—
312C22⋊C8—42no—
313#97—21no—
314#97—21no—
315#97—21no—
316#97—21no—
317#97—21no—
318#97—21no—
319#97—21no—
320#97—21no—
321#97—21no—
322#97—21no—
323#97—21no—
324#97—21no—
325#97—42no—
326#97—42no—
327#97—42no—
328#97#4112yes—
329#97#4112yes—
330#97#4112yes—
331#97#4112yes—
332C4×C8—42no—
333C4×C8—42no—
334C4×C8—42no—
335#95—21no—
336#95—21no—
337#95—21no—
338#95—21no—
339#95—21no—
340#95—21no—
341#95—21no—
342#95—21no—
343#95—21no—
344#95—21no—
345#95—21no—
346#95—21no—
347#95—21no—
348#95—21no—
349#95—21no—
350#95—21no—
351#95—21no—
352#95—21no—
353#95—21no—
354#95—21no—
355#95—21no—
356#95—21no—
357#95—21no—
358#95—21no—
359#95—21no—
360#95—21no—
361#95—21no—
362#95—21no—
363#95C22⋊C411yescharacteristic
364#95C22⋊C411yescharacteristic
365#95C22⋊C411yescharacteristic
366#95C22⋊C411yescharacteristic
367#95C22⋊C411yescharacteristic
368#95C22⋊C411yescharacteristic
369#95C22⋊C411yescharacteristic
370#95C22⋊C411yescharacteristic
371C24C2×C22⋊C411yescharacteristic
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4C2×C22⋊C411yescharacteristic
393C22×C4C2×C22⋊C411yescharacteristic
394C22×C4C2×C22⋊C411yescharacteristic
395C22×C4C2×C22⋊C411yescharacteristic
396C22×C4C2×C22⋊C411yescharacteristic
397C22×C4C2×C22⋊C411yescharacteristic
398C22×C4C2×C22⋊C411yescharacteristic
399C22×C4C2×C22⋊C411yescharacteristic
400C22×C4C2×C22⋊C411yescharacteristic
401C22×C4C2×C22⋊C411yescharacteristic
402C22×C4C2×C22⋊C411yescharacteristic
403C22×C4C23×C411yescharacteristic
404C22×C4C22×D411yescharacteristic
405C22×C4C22×D411yescharacteristic
406C2×C8—22no—
407C2×C8—22no—
408C2×C8—22no—
409C2×C8—22no—
410C2×C8—22no—
411C2×C8—22no—
412C2×C8—41no—
413C2×C8—41no—
414C2×C8—41no—
415C2×C8—41no—
416C2×C8—41no—
417C2×C8—41no—
418C2×C8—41no—
419C2×C8—41no—
420C2×C8—41no—
421C2×C8—41no—
422C2×C8—41no—
423C2×C8—41no—
424C2×C8—41no—
425C2×C8—41no—
426C2×C8—41no—
427C2×C8—41no—
428C2×C8—41no—
429C2×C8—41no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—21no—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—41no—
441C4⋊C4—41no—
442C4⋊C4—41no—
443C4⋊C4—41no—
444C4⋊C4—41no—
445C4⋊C4—41no—
446C4⋊C4—42no—
447C4⋊C4—42no—
448C4⋊C4—42no—
449C4⋊C4—42no—
450C4⋊C4—42no—
451C4⋊C4—42no—
452C4⋊C4—41no—
453C4⋊C4—41no—
454C4⋊C4—41no—
455C4⋊C4—41no—
456C4⋊C4—41no—
457C4⋊C4—41no—
458C4⋊C4—41no—
459C4⋊C4—41no—
460C4⋊C4—42no—
461C4⋊C4—42no—
462C4⋊C4—81no—
463C4⋊C4—81no—
464C4⋊C4—81no—
465C4⋊C4—81no—
466C4⋊C4—81no—
467C4⋊C4—81no—
468C4⋊C4—81no—
469C4⋊C4—81no—
470C22⋊C4—22no—
471C22⋊C4—22no—
472C22⋊C4—41no—
473C22⋊C4—41no—
474C22⋊C4—42no—
475C22⋊C4—42no—
476C22⋊C4—81no—
477C22⋊C4—81no—
478C42—21no—
479C42—21no—
480C42—21no—
481C42—21no—
482C42—21no—
483C42—21no—
484C42—22no—
485C42—22no—
486C42—41no—
487C42—41no—
488C42—41no—
489C42—41no—
490C42—41no—
491C42—41no—
492C42—41no—
493C42—41no—
494C42—41no—
495C42—41no—
496C42—42no—
497C42—42no—
498C42—81no—
499C42—81no—
500C42C2×#4111yescharacteristic
501C42C2×#4111yescharacteristic
502C42C2×#4111yescharacteristic
503C42C2×#4111yescharacteristic
504C23—21no—
505C23—21no—
506C23—21no—
507C23—21no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23C22×C22⋊C411yescharacteristic
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—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—81no—
576C2×C4—81no—
577C2×C4—81no—
578C2×C4—81no—
579C2×C4#42112yes—
580C2×C4#42112yes—
581C2×C4C4×#4111yescharacteristic
582C2×C4C4×#4111yescharacteristic
583C2×C4#52511yescharacteristic
584C2×C4#52511yescharacteristic
585C2×C4#53211yescharacteristic
586C2×C4#53211yescharacteristic
587C8—42no—
588C8—42no—
589C8—42no—
590C8—42no—
591C8—42no—
592C8—42no—
593C8—81no—
594C8—81no—
595C22—41no—
596C22—41no—
597C22—41no—
598C22—41no—
599C22—41no—
600C22—41no—
601C22—41no—
602C22#229711yescharacteristic
603C22#288911yescharacteristic
604C22#288911yescharacteristic
605C22C2×#42111yescharacteristic
606C22C2×#42111yescharacteristic
607C22#290111yescharacteristic
608C22#290111yescharacteristic
609C4—22no—
610C4—22no—
611C4—21no—
612C4—21no—
613C4—21no—
614C4—21no—
615C4—21no—
616C4—21no—
617C4—41no—
618C4—41no—
619C4—41no—
620C4—41no—
621C4—41no—
622C4—41no—
623C4—41no—
624C4—41no—
625C4—41no—
626C4—81no—
627C4—81no—
628C4—81no—
629C4—81no—
630C4—81no—
631C4—81no—
632C4—81no—
633C4—81no—
634C2—81no—
635C2#1768011yescharacteristic
636C2#1768011yescharacteristic
637C2#1768211yescharacteristic
638C2#1768311yescharacteristic
639C2#2138411yescharacteristic
640C2#2141611yescharacteristic
641C2#2142911yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9893C2semidirect
5. #9893C2semidirect
6. #9690C2semidirect
7. #9690C2semidirect
12. #8227C2semidirect
13. #8227C2semidirect
14. C4×#330C2semidirect
15. C4×#330C2semidirect

Maximal subgroups

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

subgroupindex
#351422
#214352
#204672
#98932
#96902
#91262
#91252
#82272
C4×#3302