Finite Group Databasebeta

← #216944

Subgroups — #216944

Order 512.

972 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35277C211yesmaximal, characteristic
2#21922C211yesmaximal, characteristic, complemented
3#20793C211yesmaximal, characteristic, complemented
4#9898C211yesmaximal, characteristic, complemented
5#9896C211yesmaximal, characteristic, complemented
6#9698C211yesmaximal, characteristic, complemented
7#9698C211yesmaximal, characteristic, complemented
8#9141C211yesmaximal, characteristic
9#9139C211yesmaximal, characteristic, complemented
10#9133C211yesmaximal, characteristic
11#9131C211yesmaximal, characteristic, complemented
12#8589C211yesmaximal, characteristic, complemented
13#8588C211yesmaximal, characteristic, complemented
14#8586C211yesmaximal, characteristic, complemented
15#8585C211yesmaximal, characteristic, complemented
16C2×#542—21no—
17C2×#424—21no—
18C2×#424—21no—
19C2×#423C2211yescharacteristic
20C22×C4⋊C8C2211yescharacteristic
21#2868C2211yescharacteristic
22#2858C2211yescharacteristic
23#2857C2211yescharacteristic
24#2856C2211yescharacteristic
25#2640—21no—
26#2620—21no—
27#2512—21no—
28#2512C412yes—
29#2491—21no—
30#2491C412yes—
31#2452—21no—
32#2452C412yes—
33#2422—21no—
34#2422C412yes—
35#2326—21no—
36C4×C4⋊Q8C2211yescharacteristic
37C4×#127C2211yescharacteristic
38C4×#117C2211yescharacteristic
39#1923—21no—
40#1923—21no—
41#1916—21no—
42#1916—21no—
43#1843—21no—
44#1843—21no—
45#1842C2211yescharacteristic
46#1842C2211yescharacteristic
47#1842C2211yescharacteristic
48#1842C2211yescharacteristic
49#1842C2211yescharacteristic
50#1831C2211yescharacteristic
51#1797C2211yescharacteristic
52#1797C2211yescharacteristic
53C4×C4⋊C8C2211yescharacteristic
54C4×C4⋊C8C2211yescharacteristic
55C2×#332C2211yescharacteristic
56C2×#332C2211yescharacteristic
57C2×#332C2211yescharacteristic
58C2×#332C2211yescharacteristic
59C2×#331C2211yescharacteristic
60C2×#331C2211yescharacteristic
61C2×#331C2211yescharacteristic
62C2×#331C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1296C2211yescharacteristic
66#1296C2211yescharacteristic
67#1296C2211yescharacteristic
68#1296C2211yescharacteristic
69#1296C2211yescharacteristic
70#1296C2211yescharacteristic
71C22×M16—21no—
72C23×C8—21no—
73#542—44no—
74#542—44no—
75C2×C4⋊Q8—21no—
76C2×C4⋊Q8—21no—
77C2×C4⋊Q8C2×C412yes—
78C2×C4⋊Q8C2311yescharacteristic
79C2×#127C2311yescharacteristic
80C2×#126—22no—
81C2×#124—22no—
82C2×#124—21no—
83C2×#124—21no—
84C2×#123—22no—
85C2×C22⋊Q8—21no—
86C2×C22⋊Q8—22no—
87C2×#121—21no—
88C2×#121—21no—
89C2×#121C2×C412yes—
90C2×C4×Q8—21no—
91C2×C4×D4—21no—
92C2×C4×D4—21no—
93C2×C4×D4C2×C412yes—
94C2×#117D411yescharacteristic
95C2×#117D411yescharacteristic
96C22×C42C2311yescharacteristic
97#424—44no—
98#424—44no—
99#423—41no—
100#423—41no—
101C2×C4⋊C8—21no—
102C2×C4⋊C8—21no—
103C2×C4⋊C8—21no—
104C2×C4⋊C8—21no—
105C2×C4⋊C8—21no—
106C2×C4⋊C8—21no—
107C2×C4⋊C8—21no—
108C2×C4⋊C8—21no—
109C2×C4⋊C8—21no—
110C2×C4⋊C8—21no—
111C2×C4⋊C8—21no—
112C2×C4⋊C8—21no—
113C2×C4⋊C8—41no—
114C2×C4⋊C8—41no—
115C2×C4⋊C8C2311yescharacteristic
116C2×C4⋊C8C2311yescharacteristic
117C2×C4⋊C8C2311yescharacteristic
118C2×C4⋊C8C2311yescharacteristic
119C2×C4⋊C8C2311yescharacteristic
120C2×C4⋊C8C2311yescharacteristic
121C2×C4⋊C8C2311yescharacteristic
122C2×C4⋊C8C2311yescharacteristic
123C2×C22⋊C8—21no—
124C2×C22⋊C8—21no—
125C2×C4×C8—21no—
126C2×C4×C8—21no—
127#397—22no—
128#397—22no—
129#396—22no—
130#396—22no—
131#394—22no—
132#394—22no—
133#393—22no—
134#393—22no—
135#391—21no—
136#391—21no—
137#390—21no—
138#390—21no—
139#390—21no—
140#390—21no—
141#390—21no—
142#390—21no—
143#390C2×C412yes—
144#390C2×C412yes—
145#390C2×C412yes—
146#390C2×C412yes—
147#389—21no—
148#389—21no—
149#389C2×C412yes—
150#389C2×C412yes—
151#388—22no—
152#388—22no—
153#385—22no—
154#385—22no—
155#384—21no—
156#384—21no—
157#384C2×C412yes—
158#384C2311yescharacteristic
159#384C2311yescharacteristic
160#380—21no—
161#380C2×C412yes—
162C4×C4⋊C4—21no—
163C4×C4⋊C4C2×C412yes—
164C4×C4⋊C4C2×C411yescharacteristic
165C4×C4⋊C4C2×C411yescharacteristic
166C4×C4⋊C4D411yescharacteristic
167C4×C4⋊C4D411yescharacteristic
168C4×C22⋊C4C2×C411yescharacteristic
169C4×C22⋊C4C2×C411yescharacteristic
170C4×C22⋊C4D411yescharacteristic
171C4×C22⋊C4D411yescharacteristic
172#376D411yescharacteristic
173#376D411yescharacteristic
174C43C2311yescharacteristic
175C43C2311yescharacteristic
176#336—21no—
177#336—21no—
178#336—21no—
179#336—21no—
180#336—21no—
181#336—21no—
182#332—41no—
183#332—41no—
184#332—41no—
185#332—41no—
186#331—41no—
187#331—41no—
188#331—41no—
189#331—41no—
190C22×Q8—22no—
191C22×Q8—21no—
192C22×D4—22no—
193C22×D4—21no—
194C22×D4—21no—
195C22×D4C22×C412yes—
196C23×C4—22no—
197C23×C4C22×C411yescharacteristic
198C23×C4C2×D411yescharacteristic
199C23×C4C2×D411yescharacteristic
200C2×M16—42no—
201C2×M16—42no—
202C2×M16—42no—
203C22×C8—21no—
204C22×C8—21no—
205C22×C8—21no—
206C22×C8—21no—
207C22×C8—21no—
208C22×C8—21no—
209C22×C8—21no—
210C22×C8—21no—
211C22×C8—42no—
212C22×C8—42no—
213C22×C8—42no—
214C4⋊Q8—21no—
215C4⋊Q8—21no—
216C4⋊Q8—21no—
217C4⋊Q8—21no—
218C4⋊Q8—42no—
219C4⋊Q8—42no—
220C4⋊Q8—81no—
221C4⋊Q8—81no—
222C4⋊Q8—82no—
223#127—21no—
224#127—21no—
225#127—21no—
226#127—21no—
227#126—48no—
228#124—48no—
229#124—42no—
230#124—42no—
231#124—42no—
232#124—42no—
233#123—48no—
234C22⋊Q8—42no—
235C22⋊Q8—42no—
236C22⋊Q8—48no—
237#121—42no—
238#121—42no—
239#121—81no—
240#121—81no—
241#121—82no—
242C4×Q8—42no—
243C4×Q8—42no—
244C4×D4—42no—
245C4×D4—42no—
246C4×D4—81no—
247C4×D4—81no—
248C4×D4—82no—
249#117—22no—
250#117—22no—
251#117D814yes—
252#117SD1614yes—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4C22⋊C411yescharacteristic
264C2×C4⋊C4C22⋊C411yescharacteristic
265C2×C4⋊C4C22⋊C411yescharacteristic
266C2×C4⋊C4C22⋊C411yescharacteristic
267C2×C4⋊C4C22×C412yes—
268C2×C4⋊C4C22×C412yes—
269C2×C4⋊C4C22×C412yes—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4C22⋊C411yescharacteristic
282C2×C22⋊C4C22⋊C411yescharacteristic
283C2×C22⋊C4C22⋊C411yescharacteristic
284C2×C22⋊C4C22⋊C411yescharacteristic
285C2×C42—21no—
286C2×C42—21no—
287C2×C42—21no—
288C2×C42—21no—
289C2×C42—22no—
290C2×C42C22×C411yescharacteristic
291C2×C42C22×C411yescharacteristic
292C2×C42C22×C411yescharacteristic
293C2×C42C22×C411yescharacteristic
294C2×C42C22×C411yescharacteristic
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×C42C2×D411yescharacteristic
305C2×C42C2411yescharacteristic
306C4⋊C8—21no—
307C4⋊C8—21no—
308C4⋊C8—21no—
309C4⋊C8—21no—
310C4⋊C8—21no—
311C4⋊C8—21no—
312C4⋊C8—21no—
313C4⋊C8—21no—
314C4⋊C8—21no—
315C4⋊C8—21no—
316C4⋊C8—21no—
317C4⋊C8—21no—
318C4⋊C8—21no—
319C4⋊C8—21no—
320C4⋊C8—21no—
321C4⋊C8—21no—
322C4⋊C8—42no—
323C4⋊C8—42no—
324C4⋊C8—42no—
325C4⋊C8—42no—
326C4⋊C8—42no—
327C4⋊C8—42no—
328C4⋊C8—42no—
329C4⋊C8—42no—
330C4⋊C8—42no—
331C4⋊C8—42no—
332C4⋊C8—81no—
333C4⋊C8—81no—
334C22⋊C8—81no—
335C22⋊C8—81no—
336C4×C8—81no—
337C4×C8—81no—
338#95—22no—
339#95—22no—
340#95—22no—
341#95—22no—
342#95C22⋊C411yescharacteristic
343#95C22⋊C411yescharacteristic
344#95C22⋊C411yescharacteristic
345#95C22⋊C411yescharacteristic
346#95C22⋊C411yescharacteristic
347#95C22⋊C411yescharacteristic
348#95C22⋊C411yescharacteristic
349#95C22⋊C411yescharacteristic
350C24—22no—
351C24C2×C22⋊C411yescharacteristic
352C2×Q8—41no—
353C2×Q8—41no—
354C2×Q8—41no—
355C2×Q8—41no—
356C2×Q8—41no—
357C2×Q8—41no—
358C2×Q8—44no—
359C2×Q8—82no—
360C2×Q8—82no—
361C2×D4—22no—
362C2×D4—22no—
363C2×D4—42no—
364C2×D4—42no—
365C2×D4—44no—
366C2×D4—41no—
367C2×D4—41no—
368C2×D4—41no—
369C2×D4—41no—
370C2×D4—41no—
371C2×D4—41no—
372C2×D4—41no—
373C2×D4—41no—
374C2×D4—41no—
375C2×D4—41no—
376C2×D4—41no—
377C2×D4—41no—
378C2×D4—82no—
379C2×D4—82no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—44no—
398C22×C4—82no—
399C22×C4—82no—
400C22×C4C2×C22⋊C411yescharacteristic
401C22×C4C2×C22⋊C411yescharacteristic
402C22×C4C2×C22⋊C411yescharacteristic
403C22×C4C2×C22⋊C411yescharacteristic
404C22×C4C2×C22⋊C411yescharacteristic
405C22×C4C2×C22⋊C411yescharacteristic
406C22×C4C2×C22⋊C411yescharacteristic
407C22×C4C2×C22⋊C411yescharacteristic
408C22×C4C2×C22⋊C411yescharacteristic
409C22×C4C2×C22⋊C411yescharacteristic
410C22×C4C2×C22⋊C411yescharacteristic
411C22×C4C2×D812yes—
412C22×C4C2×SD1612yes—
413C22×C4C23×C411yescharacteristic
414C22×C4C22×D411yescharacteristic
415C22×C4C22×D411yescharacteristic
416M16—84no—
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—
430C2×C8—41no—
431C2×C8—41no—
432C2×C8—41no—
433C2×C8—41no—
434C2×C8—41no—
435C2×C8—41no—
436C2×C8—41no—
437C2×C8—41no—
438C2×C8—41no—
439C2×C8—41no—
440C2×C8—41no—
441C2×C8—84no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—42no—
447C4⋊C4—42no—
448C4⋊C4—41no—
449C4⋊C4—41no—
450C4⋊C4—41no—
451C4⋊C4—41no—
452C4⋊C4—41no—
453C4⋊C4—41no—
454C4⋊C4—41no—
455C4⋊C4—41no—
456C4⋊C4—41no—
457C4⋊C4—41no—
458C4⋊C4—44no—
459C4⋊C4—44no—
460C4⋊C4—44no—
461C4⋊C4—82no—
462C4⋊C4—82no—
463C4⋊C4#10214yes—
464C4⋊C4#10214yes—
465C22⋊C4—22no—
466C22⋊C4—22no—
467C22⋊C4—44no—
468C22⋊C4—44no—
469C22⋊C4—44no—
470C22⋊C4—44no—
471C22⋊C4—44no—
472C22⋊C4—44no—
473C22⋊C4—44no—
474C22⋊C4—82no—
475C22⋊C4—82no—
476C22⋊C4—82no—
477C22⋊C4—82no—
478C22⋊C4#10214yes—
479C22⋊C4#10214yes—
480C42—21no—
481C42—21no—
482C42—21no—
483C42—21no—
484C42—21no—
485C42—21no—
486C42—21no—
487C42—21no—
488C42—21no—
489C42—21no—
490C42—21no—
491C42—21no—
492C42—21no—
493C42—21no—
494C42—21no—
495C42—21no—
496C42—41no—
497C42—41no—
498C42—41no—
499C42—41no—
500C42—44no—
501C42C2×D812yes—
502C42C2×D812yes—
503C42C2×SD1612yes—
504C42C2×SD1612yes—
505C42AGL(1,ℤ/8)12yes—
506C42#13712yes—
507C42H(5,𝔽2)11yescharacteristic
508C42H(5,𝔽2)11yescharacteristic
509C422−1+411yescharacteristic
510C422−1+411yescharacteristic
511C23—21no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23C2×#10212yes—
525C23C22×C22⋊C411yescharacteristic
526Q8—81no—
527Q8—81no—
528Q8—81no—
529Q8—81no—
530Q8—164no—
531D4—82no—
532D4—82no—
533D4—81no—
534D4—81no—
535D4—81no—
536D4—81no—
537D4—81no—
538D4—81no—
539D4—81no—
540D4—81no—
541D4—164no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—81no—
588C2×C4—81no—
589C2×C4—164no—
590C2×C4C2×#10212yes—
591C2×C4C2×#10212yes—
592C2×C4C2×#10212yes—
593C2×C4C2×#10212yes—
594C2×C4C2×#10212yes—
595C2×C4#41712yes—
596C2×C4#41712yes—
597C2×C4#52011yescharacteristic
598C2×C4#52011yescharacteristic
599C2×C4#53511yescharacteristic
600C2×C4#53611yescharacteristic
601C2×C4#53711yescharacteristic
602C2×C4#53711yescharacteristic
603C2×C4C22×D811yescharacteristic
604C2×C4C22×SD1611yescharacteristic
605C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
606C2×C4C2×#13711yescharacteristic
607C2×C4#57611yescharacteristic
608C2×C4#57711yescharacteristic
609C2×C4#57711yescharacteristic
610C2×C4#57811yescharacteristic
611C8—81no—
612C8—81no—
613C8—81no—
614C8—81no—
615C8—81no—
616C8—81no—
617C8—81no—
618C8—81no—
619C22—22no—
620C22—22no—
621C22—22no—
622C22—41no—
623C22—41no—
624C22—41no—
625C22—41no—
626C22—82no—
627C22—82no—
628C22—82no—
629C22—82no—
630C22—82no—
631C22—82no—
632C22—82no—
633C22#231111yescharacteristic
634C22#288811yescharacteristic
635C22#288911yescharacteristic
636C22C22×#10211yescharacteristic
637C22C2×#41711yescharacteristic
638C22#289711yescharacteristic
639C22#289811yescharacteristic
640C4—22no—
641C4—22no—
642C4—22no—
643C4—22no—
644C4—22no—
645C4—22no—
646C4—22no—
647C4—21no—
648C4—21no—
649C4—41no—
650C4—41no—
651C4—41no—
652C4—41no—
653C4—82no—
654C4—82no—
655C4—82no—
656C4#320511yescharacteristic
657C4#320611yescharacteristic
658C4#322611yescharacteristic
659C4#322711yescharacteristic
660C4#325711yescharacteristic
661C4#325811yescharacteristic
662C4#326211yescharacteristic
663C4#326311yescharacteristic
664C2—42no—
665C2—162no—
666C2#1772011yescharacteristic
667C2#1772111yescharacteristic
668C2#1776411yescharacteristic
669C2#1776511yescharacteristic
670C2#2185911yescharacteristic
671C2#2188111yescharacteristic
672C2#2189611yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#2856C22
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
2. #21922C2semidirect
3. #20793C2semidirect
4. #9898C2semidirect
5. #9896C2semidirect
6. #9698C2semidirect
7. #9698C2semidirect
9. #9139C2semidirect
11. #9131C2semidirect
12. #8589C2semidirect
13. #8588C2semidirect
14. #8586C2semidirect
15. #8585C2semidirect

Maximal subgroups

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

subgroupindex
#352772
#219222
#207932
#98982
#98962
#96982
#91412
#91392
#91332
#91312
#85892
#85882
#85862
#85852