Finite Group Databasebeta

← #208167

Subgroups — #208167

Order 512.

988 conjugacy classes of subgroups.

The subgroup lattice has 669 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35277C211yesmaximal, characteristic
2#21922C211yesmaximal, characteristic, complemented
3#20431C211yesmaximal, characteristic, complemented
4#9905C211yesmaximal, characteristic, complemented
5#9898C211yesmaximal, characteristic, complemented
6#9704C211yesmaximal, characteristic, complemented
7#9696C211yesmaximal, characteristic, complemented
8#9141C211yesmaximal, characteristic
9#9139C211yesmaximal, characteristic, complemented
10#9133C211yesmaximal, characteristic
11#9131C211yesmaximal, characteristic, complemented
12#8236C211yesmaximal, characteristic, complemented
13#8234C211yesmaximal, characteristic, complemented
14C4×#332C211yesmaximal, characteristic, complemented
15C4×#331C211yesmaximal, characteristic, complemented
16C2×#542—21no—
17C2×#423C2211yescharacteristic
18C22×C4⋊C8C2211yescharacteristic
19C2×#405—21no—
20#2868C2211yescharacteristic
21#2858C2211yescharacteristic
22#2857C2211yescharacteristic
23#2856C2211yescharacteristic
24#2640—21no—
25#2620—21no—
26#2512—21no—
27#2512C412yes—
28#2491—21no—
29#2491C412yes—
30#2452—21no—
31#2452C412yes—
32#2422—21no—
33#2422C412yes—
34#2326—21no—
35C4×C4⋊Q8C2211yescharacteristic
36C4×#127C2211yescharacteristic
37C4×#117C2211yescharacteristic
38#1923—21no—
39#1923—21no—
40#1916—21no—
41#1916—21no—
42#1842C2211yescharacteristic
43#1831C2211yescharacteristic
44#1797C2211yescharacteristic
45#1797C2211yescharacteristic
46#1768—21no—
47#1768—21no—
48#1767C2211yescharacteristic
49#1767C2211yescharacteristic
50C4×C4⋊C8C2211yescharacteristic
51C4×C4⋊C8C2211yescharacteristic
52C4×C4⋊C8C2211yescharacteristic
53C4×C4⋊C8C2211yescharacteristic
54#1729—21no—
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#423—22no—
98#423—22no—
99C2×C4⋊C8—21no—
100C2×C4⋊C8—21no—
101C2×C4⋊C8—22no—
102C2×C4⋊C8—22no—
103C2×C4⋊C8—21no—
104C2×C4⋊C8—21no—
105C2×C4⋊C8C2311yescharacteristic
106C2×C4⋊C8C2311yescharacteristic
107C2×C4⋊C8C2311yescharacteristic
108C2×C4⋊C8C2311yescharacteristic
109C2×C4⋊C8C2311yescharacteristic
110C2×C4⋊C8C2311yescharacteristic
111C2×C4⋊C8C2311yescharacteristic
112C2×C4⋊C8C2311yescharacteristic
113C2×C22⋊C8—21no—
114C2×C22⋊C8—21no—
115#405—44no—
116C2×#97—21no—
117C2×#97—21no—
118C2×C4×C8—21no—
119C2×C4×C8—21no—
120C2×C4×C8—21no—
121C2×C4×C8—21no—
122#397—22no—
123#397—22no—
124#396—22no—
125#396—22no—
126#394—22no—
127#394—22no—
128#393—22no—
129#393—22no—
130#391—21no—
131#391—21no—
132#390—21no—
133#390—21no—
134#390—21no—
135#390—21no—
136#390—21no—
137#390—21no—
138#390C2×C412yes—
139#390C2×C412yes—
140#390C2×C412yes—
141#390C2×C412yes—
142#389—21no—
143#389—21no—
144#389C2×C412yes—
145#389C2×C412yes—
146#388—22no—
147#388—22no—
148#385—22no—
149#385—22no—
150#384—21no—
151#384—21no—
152#384C2×C412yes—
153#384C2311yescharacteristic
154#384C2311yescharacteristic
155#380—21no—
156#380C2×C412yes—
157C4×C4⋊C4—21no—
158C4×C4⋊C4C2×C412yes—
159C4×C4⋊C4C2×C411yescharacteristic
160C4×C4⋊C4C2×C411yescharacteristic
161C4×C4⋊C4D411yescharacteristic
162C4×C4⋊C4D411yescharacteristic
163C4×C22⋊C4C2×C411yescharacteristic
164C4×C22⋊C4C2×C411yescharacteristic
165C4×C22⋊C4D411yescharacteristic
166C4×C22⋊C4D411yescharacteristic
167#376D411yescharacteristic
168#376D411yescharacteristic
169C43C2311yescharacteristic
170C43C2311yescharacteristic
171#336—21no—
172#336—21no—
173#336—21no—
174#336—21no—
175#336—21no—
176#336—21no—
177#336—21no—
178#336—21no—
179#336—21no—
180#336—21no—
181#332—22no—
182#332—22no—
183#332—22no—
184#332—22no—
185#331—22no—
186#331—22no—
187#331—22no—
188#331—22no—
189C22×Q8—22no—
190C22×Q8—21no—
191C22×D4—22no—
192C22×D4—21no—
193C22×D4—21no—
194C22×D4C22×C412yes—
195C23×C4—22no—
196C23×C4C22×C411yescharacteristic
197C23×C4C2×D411yescharacteristic
198C23×C4C2×D411yescharacteristic
199C2×M16—24no—
200C2×M16—42no—
201C2×M16—42no—
202C22×C8—24no—
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—
213C4⋊Q8—21no—
214C4⋊Q8—21no—
215C4⋊Q8—21no—
216C4⋊Q8—21no—
217C4⋊Q8—42no—
218C4⋊Q8—42no—
219C4⋊Q8—81no—
220C4⋊Q8—81no—
221C4⋊Q8—82no—
222#127—21no—
223#127—21no—
224#127—21no—
225#127—21no—
226#126—48no—
227#124—42no—
228#124—42no—
229#124—48no—
230#124—42no—
231#124—42no—
232#123—48no—
233C22⋊Q8—42no—
234C22⋊Q8—42no—
235C22⋊Q8—48no—
236#121—42no—
237#121—42no—
238#121—81no—
239#121—81no—
240#121—82no—
241C4×Q8—42no—
242C4×Q8—42no—
243C4×D4—42no—
244C4×D4—42no—
245C4×D4—81no—
246C4×D4—81no—
247C4×D4—82no—
248#117—41no—
249#117—41no—
250#117D814yes—
251#117SD1614yes—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4C22⋊C411yescharacteristic
263C2×C4⋊C4C22⋊C411yescharacteristic
264C2×C4⋊C4C22⋊C411yescharacteristic
265C2×C4⋊C4C22⋊C411yescharacteristic
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⋊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⋊C4C22⋊C411yescharacteristic
281C2×C22⋊C4C22⋊C411yescharacteristic
282C2×C22⋊C4C22⋊C411yescharacteristic
283C2×C22⋊C4C22⋊C411yescharacteristic
284C2×C42—21no—
285C2×C42—21no—
286C2×C42—21no—
287C2×C42—21no—
288C2×C42—22no—
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—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—81no—
320C4⋊C8—81no—
321C4⋊C8#4112yes—
322C4⋊C8#4112yes—
323C4⋊C8#4112yes—
324C4⋊C8#4112yes—
325C22⋊C8—81no—
326C22⋊C8—81no—
327#97—42no—
328#97—42no—
329C4×C8—42no—
330C4×C8—42no—
331C4×C8—81no—
332C4×C8—81no—
333#95—22no—
334#95—22no—
335#95—22no—
336#95—22no—
337#95C22⋊C411yescharacteristic
338#95C22⋊C411yescharacteristic
339#95C22⋊C411yescharacteristic
340#95C22⋊C411yescharacteristic
341#95C22⋊C411yescharacteristic
342#95C22⋊C411yescharacteristic
343#95C22⋊C411yescharacteristic
344#95C22⋊C411yescharacteristic
345C24—22no—
346C24C2×C22⋊C411yescharacteristic
347C2×Q8—41no—
348C2×Q8—41no—
349C2×Q8—41no—
350C2×Q8—41no—
351C2×Q8—41no—
352C2×Q8—41no—
353C2×Q8—44no—
354C2×Q8—82no—
355C2×Q8—82no—
356C2×D4—22no—
357C2×D4—22no—
358C2×D4—42no—
359C2×D4—42no—
360C2×D4—44no—
361C2×D4—41no—
362C2×D4—41no—
363C2×D4—41no—
364C2×D4—41no—
365C2×D4—41no—
366C2×D4—41no—
367C2×D4—41no—
368C2×D4—41no—
369C2×D4—41no—
370C2×D4—41no—
371C2×D4—41no—
372C2×D4—41no—
373C2×D4—82no—
374C2×D4—82no—
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—21no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—44no—
393C22×C4—82no—
394C22×C4—82no—
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×C4C2×C22⋊C411yescharacteristic
404C22×C4C2×C22⋊C411yescharacteristic
405C22×C4C2×C22⋊C411yescharacteristic
406C22×C4C2×D812yes—
407C22×C4C2×SD1612yes—
408C22×C4C23×C411yescharacteristic
409C22×C4C22×D411yescharacteristic
410C22×C4C22×D411yescharacteristic
411M16—48no—
412C2×C8—22no—
413C2×C8—22no—
414C2×C8—22no—
415C2×C8—22no—
416C2×C8—48no—
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—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—41no—
440C4⋊C4—41no—
441C4⋊C4—42no—
442C4⋊C4—42no—
443C4⋊C4—41no—
444C4⋊C4—41no—
445C4⋊C4—41no—
446C4⋊C4—41no—
447C4⋊C4—41no—
448C4⋊C4—41no—
449C4⋊C4—41no—
450C4⋊C4—41no—
451C4⋊C4—41no—
452C4⋊C4—41no—
453C4⋊C4—44no—
454C4⋊C4—44no—
455C4⋊C4—44no—
456C4⋊C4—82no—
457C4⋊C4—82no—
458C4⋊C4#10214yes—
459C4⋊C4#10214yes—
460C22⋊C4—41no—
461C22⋊C4—41no—
462C22⋊C4—44no—
463C22⋊C4—44no—
464C22⋊C4—44no—
465C22⋊C4—44no—
466C22⋊C4—44no—
467C22⋊C4—44no—
468C22⋊C4—44no—
469C22⋊C4—82no—
470C22⋊C4—82no—
471C22⋊C4—82no—
472C22⋊C4—82no—
473C22⋊C4#10214yes—
474C22⋊C4#10214yes—
475C42—21no—
476C42—21no—
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—41no—
494C42—41no—
495C42—41no—
496C42—41no—
497C42—44no—
498C42C2×D812yes—
499C42C2×D812yes—
500C42C2×SD1612yes—
501C42C2×SD1612yes—
502C42C2×#4111yescharacteristic
503C42C2×#4111yescharacteristic
504C42C2×#4111yescharacteristic
505C42C2×#4111yescharacteristic
506C23—21no—
507C23—21no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23C2×#10212yes—
520C23C22×C22⋊C411yescharacteristic
521Q8—81no—
522Q8—81no—
523Q8—81no—
524Q8—81no—
525Q8—164no—
526D4—82no—
527D4—82no—
528D4—81no—
529D4—81no—
530D4—81no—
531D4—81no—
532D4—81no—
533D4—81no—
534D4—81no—
535D4—81no—
536D4—164no—
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—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
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—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—81no—
585C2×C4—81no—
586C2×C4—164no—
587C2×C4C2×#10212yes—
588C2×C4C2×#10212yes—
589C2×C4C2×#10212yes—
590C2×C4C2×#10212yes—
591C2×C4C2×#10212yes—
592C2×C4C4×#4111yescharacteristic
593C2×C4C4×#4111yescharacteristic
594C2×C4#52511yescharacteristic
595C2×C4#52511yescharacteristic
596C2×C4#53211yescharacteristic
597C2×C4#53211yescharacteristic
598C2×C4C22×D811yescharacteristic
599C2×C4C22×SD1611yescharacteristic
600C2×C4#57511yescharacteristic
601C2×C4#57511yescharacteristic
602C2×C4#57611yescharacteristic
603C2×C4#57711yescharacteristic
604C2×C4#57711yescharacteristic
605C2×C4#57811yescharacteristic
606C8—42no—
607C8—42no—
608C8—42no—
609C8—42no—
610C8—81no—
611C8—81no—
612C8—81no—
613C8—81no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22—41no—
618C22—41no—
619C22—41no—
620C22—41no—
621C22—82no—
622C22—82no—
623C22—82no—
624C22—82no—
625C22—82no—
626C22—82no—
627C22—82no—
628C22#229711yescharacteristic
629C22#288811yescharacteristic
630C22#288911yescharacteristic
631C22C22×#10211yescharacteristic
632C22#289611yescharacteristic
633C22#289711yescharacteristic
634C22#289811yescharacteristic
635C4—22no—
636C4—22no—
637C4—22no—
638C4—22no—
639C4—22no—
640C4—21no—
641C4—21no—
642C4—41no—
643C4—41no—
644C4—41no—
645C4—41no—
646C4—41no—
647C4—41no—
648C4—82no—
649C4—82no—
650C4—82no—
651C4#310011yescharacteristic
652C4#310011yescharacteristic
653C4#310311yescharacteristic
654C4#310311yescharacteristic
655C4#317311yescharacteristic
656C4#317511yescharacteristic
657C4#317811yescharacteristic
658C4#317911yescharacteristic
659C2—42no—
660C2—162no—
661C2#1772011yescharacteristic
662C2#1772111yescharacteristic
663C2#1776811yescharacteristic
664C2#1776911yescharacteristic
665C2#2138611yescharacteristic
666C2#2141211yescharacteristic
667C2#2142711yescharacteristic
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. #20431C2semidirect
4. #9905C2semidirect
5. #9898C2semidirect
6. #9704C2semidirect
7. #9696C2semidirect
9. #9139C2semidirect
11. #9131C2semidirect
12. #8236C2semidirect
13. #8234C2semidirect
14. C4×#332C2semidirect
15. C4×#331C2semidirect

Maximal subgroups

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

subgroupindex
#352772
#219222
#204312
#99052
#98982
#97042
#96962
#91412
#91392
#91332
#91312
#82362
#82342
C4×#3322
C4×#3312