Finite Group Databasebeta

← #223876

Subgroups — #223876

Order 512.

995 conjugacy classes of subgroups.

The subgroup lattice has 667 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36181C211yesmaximal, characteristic
2#22648C211yesmaximal, characteristic
3D4×C4⋊C8C211yesmaximal, characteristic
4#9904C211yesmaximal, characteristic, complemented
5#9900C211yesmaximal, characteristic, complemented
6#9139C211yesmaximal, characteristic, complemented
7#9131C211yesmaximal, characteristic, complemented
8#9129C211yesmaximal, characteristic, complemented
9#9129C211yesmaximal, characteristic, complemented
10#8726C211yesmaximal, characteristic, complemented
11#8723C211yesmaximal, characteristic, complemented
12#8586C211yesmaximal, characteristic, complemented
13#8236C211yesmaximal, characteristic, complemented
14#8045C211yesmaximal, characteristic, complemented
15C2×#1296C211yesmaximal, characteristic, complemented
16C2×#552—21no—
17C2×#435—21no—
18C2×C8×D4—21no—
19C2×#423C2211yescharacteristic
20C22×C4⋊C8C2211yescharacteristic
21C22×C4⋊C8C2211yescharacteristic
22C22×C4⋊C8C2211yescharacteristic
23#2859C2211yescharacteristic
24#2857C2211yescharacteristic
25#2857C2211yescharacteristic
26#2565—21no—
27#2506—21no—
28#2506C412yes—
29#2503—21no—
30#2499—21no—
31#2496—21no—
32#2496C412yes—
33#2486—21no—
34#2486C412yes—
35#2344—21no—
36#2344C412yes—
37C4×C4⋊Q8C2211yescharacteristic
38#2285C2211yescharacteristic
39C2×#384C2211yescharacteristic
40C42×D4C2211yescharacteristic
41#1940—21no—
42#1939—21no—
43#1851—21no—
44#1850—21no—
45#1850—21no—
46#1850—21no—
47#1842C2211yescharacteristic
48#1842C2211yescharacteristic
49#1797C2211yescharacteristic
50#1797C2211yescharacteristic
51#1797C2211yescharacteristic
52#1797C2211yescharacteristic
53#1767C2211yescharacteristic
54C4×C4⋊C8C2211yescharacteristic
55C2×#332C2211yescharacteristic
56C2×#332C2211yescharacteristic
57C2×#332C2211yescharacteristic
58C2×#332C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1296C2211yescharacteristic
66#1296C2211yescharacteristic
67#1296C2211yescharacteristic
68#1296C2211yescharacteristic
69#1296C2211yescharacteristic
70#1296C2211yescharacteristic
71C22×M16—21no—
72C23×C8—21no—
73C23×C8—21no—
74C23×C8—21no—
75#552—44no—
76#552—44no—
77C2×C4⋊Q8C2311yescharacteristic
78C2×#126—22no—
79C2×#125—22no—
80C2×#125—21no—
81C2×#123—22no—
82C2×C22⋊Q8—21no—
83C2×C22⋊Q8—21no—
84C2×C4×Q8—21no—
85C2×C4×D4C2×C411yescharacteristic
86C2×C4×D4D411yescharacteristic
87C2×C4×D4D411yescharacteristic
88C2×#117—21no—
89C2×#117—22no—
90C22×C4⋊C4—21no—
91C22×C4⋊C4—21no—
92C22×C4⋊C4C2×C412yes—
93C22×C42C2311yescharacteristic
94C22×C42C2311yescharacteristic
95#435—44no—
96C8×D4—44no—
97#423—22no—
98#423—41no—
99C2×C4⋊C8—22no—
100C2×C4⋊C8—22no—
101C2×C4⋊C8—22no—
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—41no—
111C2×C4⋊C8—41no—
112C2×C4⋊C8—41no—
113C2×C4⋊C8C2311yescharacteristic
114C2×C4⋊C8C2311yescharacteristic
115C2×C4⋊C8C2311yescharacteristic
116C2×C4⋊C8C2311yescharacteristic
117C2×C4⋊C8C2311yescharacteristic
118C2×C4⋊C8C2311yescharacteristic
119C2×C4⋊C8C2311yescharacteristic
120C2×C4⋊C8C2311yescharacteristic
121C2×C22⋊C8—21no—
122C2×C22⋊C8—21no—
123C2×C22⋊C8—21no—
124C2×C22⋊C8—21no—
125C2×#97—21no—
126C2×C4×C8—21no—
127#400—22no—
128#400—22no—
129#398—22no—
130#398—22no—
131#397—22no—
132#397—22no—
133#391—21no—
134#391—21no—
135#391—21no—
136#389—21no—
137#389—21no—
138#389—21no—
139#389—21no—
140#389—21no—
141#389C2×C412yes—
142#389C2×C412yes—
143#389C2×C412yes—
144#389C2×C412yes—
145#389C2×C412yes—
146#387—22no—
147#387—22no—
148#387—22no—
149#387—22no—
150#384—21no—
151#384—21no—
152#384C2×C412yes—
153#384C2311yescharacteristic
154#384C2311yescharacteristic
155#384C2311yescharacteristic
156#382—21no—
157#382—21no—
158#381—22no—
159#381—22no—
160#380—21no—
161#380—21no—
162#380—21no—
163#380C2×C412yes—
164#380C2×C412yes—
165#380C2×C412yes—
166C4×C4⋊C4—21no—
167C4×C4⋊C4—21no—
168C4×C4⋊C4C2×C412yes—
169C4×C4⋊C4C2×C412yes—
170C4×C4⋊C4C2×C411yescharacteristic
171C4×C4⋊C4D411yescharacteristic
172C4×C4⋊C4D411yescharacteristic
173C4×C22⋊C4C2×C411yescharacteristic
174C4×C22⋊C4C2×C411yescharacteristic
175C4×C22⋊C4D411yescharacteristic
176C4×C22⋊C4D411yescharacteristic
177C4×C22⋊C4D411yescharacteristic
178C4×C22⋊C4D411yescharacteristic
179C43C2311yescharacteristic
180#336—21no—
181#336—21no—
182#336—21no—
183#336—21no—
184#336—21no—
185#336—21no—
186#332—22no—
187#332—22no—
188#332—41no—
189#332—41no—
190C22×Q8—21no—
191C22×D4C22⋊C411yescharacteristic
192C23×C4—22no—
193C23×C4C22×C411yescharacteristic
194C23×C4C22×C411yescharacteristic
195C23×C4C2×D411yescharacteristic
196C23×C4C2×D411yescharacteristic
197C23×C4C2×D411yescharacteristic
198C23×C4C2×D411yescharacteristic
199C2×M16—42no—
200C2×M16—42no—
201C2×M16—81no—
202C22×C8—21no—
203C22×C8—21no—
204C22×C8—21no—
205C22×C8—21no—
206C22×C8—21no—
207C22×C8—21no—
208C22×C8—21no—
209C22×C8—21no—
210C22×C8—42no—
211C22×C8—42no—
212C22×C8—42no—
213C22×C8—42no—
214C22×C8—42no—
215C22×C8—42no—
216C22×C8—81no—
217C22×C8—81no—
218C22×C8—81no—
219C4⋊Q8—21no—
220C4⋊Q8—21no—
221C4⋊Q8—21no—
222C4⋊Q8—21no—
223#126—48no—
224#125—48no—
225#125—42no—
226#125—42no—
227#123—48no—
228C22⋊Q8—42no—
229C22⋊Q8—42no—
230C22⋊Q8—42no—
231C22⋊Q8—42no—
232C4×Q8—42no—
233C4×Q8—42no—
234C4×D4—22no—
235C4×D4—22no—
236C4×D4—42no—
237C4×D4D814yes—
238C4×D4SD1614yes—
239#117—42no—
240#117—42no—
241#117—48no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
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—44no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4C22⋊C411yescharacteristic
263C2×C4⋊C4C22⋊C411yescharacteristic
264C2×C4⋊C4C22⋊C411yescharacteristic
265C2×C4⋊C4C22×C412yes—
266C2×C4⋊C4C22×C412yes—
267C2×C4⋊C4C22×C412yes—
268C2×C4⋊C4C22×C412yes—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4C22⋊C411yescharacteristic
273C2×C22⋊C4C22⋊C411yescharacteristic
274C2×C22⋊C4C22⋊C411yescharacteristic
275C2×C22⋊C4C22⋊C411yescharacteristic
276C2×C22⋊C4C22⋊C411yescharacteristic
277C2×C22⋊C4C22⋊C411yescharacteristic
278C2×C42—21no—
279C2×C42—21no—
280C2×C42—21no—
281C2×C42—21no—
282C2×C42—21no—
283C2×C42—21no—
284C2×C42—21no—
285C2×C42—21no—
286C2×C42—22no—
287C2×C42—22no—
288C2×C42C22×C411yescharacteristic
289C2×C42C22×C411yescharacteristic
290C2×C42C22×C411yescharacteristic
291C2×C42C22×C411yescharacteristic
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×C42C2411yescharacteristic
301C4⋊C8—21no—
302C4⋊C8—21no—
303C4⋊C8—21no—
304C4⋊C8—21no—
305C4⋊C8—21no—
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—42no—
316C4⋊C8—42no—
317C4⋊C8—42no—
318C4⋊C8—42no—
319C4⋊C8—42no—
320C4⋊C8—42no—
321C4⋊C8—81no—
322C4⋊C8—81no—
323C4⋊C8#4112yes—
324C4⋊C8#4112yes—
325C22⋊C8—42no—
326C22⋊C8—42no—
327C22⋊C8—42no—
328C22⋊C8—42no—
329#97—42no—
330C4×C8—42no—
331#95—22no—
332#95—22no—
333#95—22no—
334#95—22no—
335#95—22no—
336#95—22no—
337#95—22no—
338#95—22no—
339#95—22no—
340#95—22no—
341#95C22⋊C411yescharacteristic
342#95C22⋊C411yescharacteristic
343#95C22⋊C411yescharacteristic
344#95C22⋊C411yescharacteristic
345#95C22⋊C411yescharacteristic
346#95C22⋊C411yescharacteristic
347C24C2×C22⋊C411yescharacteristic
348C24C2×C22⋊C411yescharacteristic
349C2×Q8—41no—
350C2×Q8—41no—
351C2×Q8—41no—
352C2×Q8—41no—
353C2×Q8—41no—
354C2×Q8—41no—
355C2×D4—22no—
356C2×D4—22no—
357C2×D4#10214yes—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—22no—
371C22×C4—21no—
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—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—44no—
389C22×C4—44no—
390C22×C4—44no—
391C22×C4C2×C22⋊C411yescharacteristic
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×D812yes—
402C22×C4C2×D812yes—
403C22×C4C2×SD1612yes—
404C22×C4C2×SD1612yes—
405C22×C4AGL(1,ℤ/8)12yes—
406C22×C4#13712yes—
407C22×C4C23×C411yescharacteristic
408C22×C4C22×D411yescharacteristic
409C22×C4C22×D411yescharacteristic
410M16—84no—
411C2×C8—22no—
412C2×C8—22no—
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—
430C2×C8—41no—
431C2×C8—41no—
432C2×C8—41no—
433C2×C8—41no—
434C2×C8—41no—
435C2×C8—84no—
436C2×C8—84no—
437C2×C8—84no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—41no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C4⋊C4—42no—
445C4⋊C4—44no—
446C4⋊C4—41no—
447C4⋊C4—41no—
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—41no—
459C4⋊C4—41no—
460C4⋊C4—44no—
461C4⋊C4—44no—
462C4⋊C4—44no—
463C4⋊C4—44no—
464C4⋊C4—44no—
465C4⋊C4—44no—
466C4⋊C4—44no—
467C4⋊C4#10214yes—
468C22⋊C4—22no—
469C22⋊C4—22no—
470C22⋊C4—22no—
471C22⋊C4—41no—
472C22⋊C4—44no—
473C22⋊C4—44no—
474C22⋊C4—44no—
475C22⋊C4#10214yes—
476C22⋊C4#10214yes—
477C42—21no—
478C42—21no—
479C42—21no—
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—41no—
496C42—41no—
497C42—44no—
498C42—44no—
499C42C2×D812yes—
500C42C2×SD1612yes—
501C42C2×#4111yescharacteristic
502C42C2×#4111yescharacteristic
503C42H(5,𝔽2)11yescharacteristic
504C422−1+411yescharacteristic
505C23—21no—
506C23—21no—
507C23—21no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23#41312yes—
516C23C2×#10212yes—
517C23C2×#10212yes—
518C23#41712yes—
519C23C22×C22⋊C411yescharacteristic
520Q8—81no—
521Q8—81no—
522Q8—81no—
523Q8—81no—
524D4—28no—
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—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
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—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—81no—
583C2×C4—81no—
584C2×C4—88no—
585C2×C4C2×#10212yes—
586C2×C4C2×#10212yes—
587C2×C4C2×#10212yes—
588C2×C4C2×#10212yes—
589C2×C4#41712yes—
590C2×C4C4×#4111yescharacteristic
591C2×C4#52011yescharacteristic
592C2×C4#54611yescharacteristic
593C2×C4#54611yescharacteristic
594C2×C4#54711yescharacteristic
595C2×C4#54711yescharacteristic
596C2×C4C22×D811yescharacteristic
597C2×C4C22×SD1611yescharacteristic
598C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
599C2×C4C2×#13711yescharacteristic
600C2×C4#57711yescharacteristic
601C2×C4#57711yescharacteristic
602C2×C4#57811yescharacteristic
603C2×C4#57811yescharacteristic
604C8—42no—
605C8—42no—
606C8—81no—
607C8—81no—
608C8—81no—
609C8—81no—
610C8—81no—
611C8—81no—
612C22—22no—
613C22—22no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—22no—
621C22—41no—
622C22—41no—
623C22—41no—
624C22—41no—
625C22#154714yes—
626C22#233911yescharacteristic
627C22C2×#41311yescharacteristic
628C22#288911yescharacteristic
629C22C22×#10211yescharacteristic
630C22C2×#41711yescharacteristic
631C22#289811yescharacteristic
632C22#289811yescharacteristic
633C4—22no—
634C4—22no—
635C4—21no—
636C4—21no—
637C4—22no—
638C4—22no—
639C4—22no—
640C4—41no—
641C4—41no—
642C4—41no—
643C4—41no—
644C4—41no—
645C4—82no—
646C4—82no—
647C4—82no—
648C4—82no—
649C4#329411yescharacteristic
650C4#329811yescharacteristic
651C4#331011yescharacteristic
652C4#331211yescharacteristic
653C4#333411yescharacteristic
654C4#333511yescharacteristic
655C4#334311yescharacteristic
656C4#334411yescharacteristic
657C2—24no—
658C2—42no—
659C2C2×#154711yescharacteristic
660C2#1772111yescharacteristic
661C2#1776311yescharacteristic
662C2#1778011yescharacteristic
663C2#2248011yescharacteristic
664C2#2248511yescharacteristic
665C2#2249111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D4C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9904C2semidirect
5. #9900C2semidirect
6. #9139C2semidirect
7. #9131C2semidirect
8. #9129C2semidirect
9. #9129C2semidirect
10. #8726C2semidirect
11. #8723C2semidirect
12. #8586C2semidirect
13. #8236C2semidirect
14. #8045C2semidirect
15. C2×#1296C2semidirect

Maximal subgroups

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

subgroupindex
#361812
#226482
D4×C4⋊C82
#99042
#99002
#91392
#91312
#91292
#87262
#87232
#85862
#82362
#80452
C2×#12962