Finite Group Databasebeta

← #413262

Subgroups — #413262

Order 512.

1296 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#501C211yesmaximal, characteristic
2C2×#2059C211yesmaximal, characteristic
3C2×#2057C211yesmaximal, characteristic
4C2×#1943C211yesmaximal, characteristic
5C2×#1847C211yesmaximal, characteristic
6C2×#1846C211yesmaximal, characteristic
7#15230C211yesmaximal, characteristic
8#15228C211yesmaximal, characteristic
9#11325C211yesmaximal, characteristic
10#11325C211yesmaximal, characteristic
11#11325C211yesmaximal, characteristic
12#11324C211yesmaximal, characteristic
13#10863C211yesmaximal, characteristic
14#10862C211yesmaximal, characteristic
15#10696C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×#125C2211yescharacteristic
18C2×#501—24no—
19C2×#501—24no—
20C2×#501—24no—
21C22×#107C2211yescharacteristic
22C22×#107C2211yescharacteristic
23C22×#107C2211yescharacteristic
24C22×#107C2211yescharacteristic
25C22×#107C2211yescharacteristic
26C22×#107C2211yescharacteristic
27C22×#106C2211yescharacteristic
28C22×#106C2211yescharacteristic
29C22×#106C2211yescharacteristic
30C22×#106C2211yescharacteristic
31C22×#97C2211yescharacteristic
32C22×#97C2211yescharacteristic
33C22×C4×C8C2211yescharacteristic
34C2×#389C2211yescharacteristic
35C2×#389C2211yescharacteristic
36C2×#384C2211yescharacteristic
37C2×#382C2211yescharacteristic
38C2×#382C2211yescharacteristic
39C2×#382C2211yescharacteristic
40#2059—24no—
41#2057—24no—
42#1943—24no—
43#1847—24no—
44#1846—24no—
45C2×#340C2211yescharacteristic
46C2×#340C2211yescharacteristic
47C2×#340C2211yescharacteristic
48C2×#340C2211yescharacteristic
49C2×#340C2211yescharacteristic
50C2×#340C2211yescharacteristic
51C2×#336C2211yescharacteristic
52C2×#336C2211yescharacteristic
53C2×#336C2211yescharacteristic
54#1446—22no—
55#1444—22no—
56#1441—22no—
57#1441—22no—
58C4⋊#95C2211yescharacteristic
59C4⋊#95C2211yescharacteristic
60C4⋊#95C2211yescharacteristic
61C4⋊#95C2211yescharacteristic
62C4×#95C2211yescharacteristic
63C23×Q8—21no—
64C23×C8C2311yescharacteristic
65C23×C8C2311yescharacteristic
66C23×C8C2311yescharacteristic
67C23×C8C2311yescharacteristic
68C2×C4⋊Q8—22no—
69C2×C4⋊Q8—22no—
70C2×C4⋊Q8—22no—
71C2×C4⋊Q8—22no—
72C2×C4⋊Q8—41no—
73C2×C4⋊Q8—41no—
74C2×#125—22no—
75C2×#125—22no—
76C2×#125—22no—
77C2×#125—22no—
78C2×#125—41no—
79C2×#125—41no—
80C22×C4⋊C4—21no—
81C22×C4⋊C4—21no—
82C22×C4⋊C4—21no—
83C22×C4⋊C4—21no—
84C22×C4⋊C4—22no—
85C22×C4⋊C4—21no—
86C22×C4⋊C4C2311yescharacteristic
87C22×C4⋊C4C2311yescharacteristic
88C22×C4⋊C4C2311yescharacteristic
89C22×C4⋊C4C2311yescharacteristic
90C22×C4⋊C4C2311yescharacteristic
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2311yescharacteristic
94C22×C42C2311yescharacteristic
95C22×C42C2311yescharacteristic
96C22×C42C2311yescharacteristic
97#501—232no—
98C2×#107—22no—
99C2×#107—22no—
100C2×#107—22no—
101C2×#107—22no—
102C2×#107—22no—
103C2×#107—22no—
104C2×#107—22no—
105C2×#107—22no—
106C2×#107—22no—
107C2×#107—22no—
108C2×#107—22no—
109C2×#107—22no—
110C2×#107—22no—
111C2×#107—41no—
112C2×#107—41no—
113C2×#107—41no—
114C2×#107D414yes—
115C2×#107Q814yescomplemented
116C2×#106—22no—
117C2×#106—22no—
118C2×#106—22no—
119C2×#106—22no—
120C2×#106—22no—
121C2×#106—22no—
122C2×#106—22no—
123C2×#106—22no—
124C2×#106—22no—
125C2×#106—22no—
126C2×#106—41no—
127C2×#106D414yes—
128C2×#97—22no—
129C2×#97—22no—
130C2×#97—22no—
131C2×#97—22no—
132C2×#97D414yes—
133C2×#97Q814yescomplemented
134C2×C4×C8—22no—
135C2×C4×C8—22no—
136C2×C4×C8Q814yescomplemented
137#389—22no—
138#389—22no—
139#389—22no—
140#389—22no—
141#384—22no—
142#384—41no—
143#382—22no—
144#382—22no—
145#382—22no—
146#382—41no—
147#382—41no—
148#382—41no—
149C2×#95—22no—
150C2×#95—21no—
151C2×#95—22no—
152C2×#95—22no—
153C2×#95—22no—
154C2×#95—21no—
155C2×#95—22no—
156C2×#95—22no—
157C2×#95—22no—
158C2×#95D412yes—
159C2×#95D412yes—
160#340—22no—
161#340—22no—
162#340—22no—
163#340—22no—
164#340—22no—
165#340—22no—
166#336D414yes—
167#336Q814yescomplemented
168#336Q814yescomplemented
169C22×Q8—42no—
170C22×Q8—42no—
171C22×Q8—44no—
172C22×Q8—42no—
173C22×Q8—82no—
174C23×C4—21no—
175C23×C4—21no—
176C23×C4—21no—
177C23×C4—21no—
178C23×C4—21no—
179C23×C4—21no—
180C23×C4—21no—
181C23×C4—21no—
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D412yes—
185C23×C4C2×D412yes—
186C23×C4C2411yescharacteristic
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—21no—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—21no—
198C22×C8—21no—
199C22×C8C2×D412yes—
200C22×C8C2×D412yes—
201C22×C8C2×D412yes—
202C22×C8C2×D412yes—
203C22×C8C2×D412yes—
204C22×C8C2×D412yes—
205C22×C8C2×Q812yes—
206C22×C8C2×Q812yes—
207C22×C8C2×Q812yes—
208C22×C8C2×Q812yes—
209C22×C8C2×Q812yes—
210C22×C8C2×Q812yes—
211C22×C8C2×Q812yes—
212C22×C8C2×Q812yes—
213C22×C8C2×Q812yes—
214C22×C8#4112yes—
215C4⋊Q8—28no—
216C4⋊Q8—44no—
217C4⋊Q8—44no—
218C4⋊Q8—44no—
219#125—28no—
220#125—44no—
221#125—44no—
222#125—44no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
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⋊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⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—42no—
266C2×C4⋊C4—42no—
267C2×C4⋊C4—42no—
268C2×C4⋊C4—42no—
269C2×C4⋊C4—42no—
270C2×C4⋊C4—42no—
271C2×C4⋊C4—42no—
272C2×C4⋊C4—42no—
273C2×C4⋊C4—42no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—42no—
278C2×C4⋊C4—42no—
279C2×C4⋊C4—82no—
280C2×C4⋊C4—82no—
281C2×C4⋊C4C2×D412yes—
282C2×C4⋊C4C2×D412yes—
283C2×C4⋊C4C2×D412yes—
284C2×C4⋊C4C2×D412yes—
285C2×C4⋊C4C2×Q812yes—
286C2×C4⋊C4C2×Q812yes—
287C2×C42—21no—
288C2×C42—21no—
289C2×C42—21no—
290C2×C42—21no—
291C2×C42—21no—
292C2×C42—21no—
293C2×C42—21no—
294C2×C42—21no—
295C2×C42—21no—
296C2×C42—21no—
297C2×C42—21no—
298C2×C42—21no—
299C2×C42C2×D412yes—
300C2×C42C2×D412yes—
301C2×C42C2×Q812yes—
302C2×C42C2×Q812yes—
303C2×C42C2×Q812yes—
304C2×C42C2×Q812yes—
305#107—28no—
306#107—28no—
307#107—28no—
308#107—44no—
309#107—44no—
310#107—44no—
311#106—28no—
312#106—28no—
313#106—44no—
314#106—44no—
315#97—28no—
316#97—28no—
317C4×C8—28no—
318#95—42no—
319#95—42no—
320#95—44no—
321#95—44no—
322#95—44no—
323#95—44no—
324#95—42no—
325#95—42no—
326#95—82no—
327#95—82no—
328#95—82no—
329C24C22×D411yescharacteristic
330C2×Q8—48no—
331C2×Q8—84no—
332C2×Q8—84no—
333C2×Q8—88no—
334C2×Q8—88no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4#12712yes—
396C22×C4#12712yes—
397C22×C4C4⋊Q814yes—
398C22×C4C4⋊Q814yes—
399C22×C4C22×D411yescharacteristic
400C22×C4C22×D411yescharacteristic
401C22×C4C22×D411yescharacteristic
402C22×C4C22×D411yescharacteristic
403C22×C4C22×Q811yescharacteristic
404C22×C4C22×Q811yescharacteristic
405C22×C4C22×Q811yescharacteristic
406C22×C4C22×Q811yescharacteristic
407C22×C4C22×Q811yescharacteristic
408C22×C4C2×#4111yescharacteristic
409C22×C4H(5,𝔽2)11yescharacteristic
410C22×C4H(5,𝔽2)11yescharacteristic
411C22×C42−1+411yescharacteristic
412C22×C42−1+411yescharacteristic
413C2×C8—22no—
414C2×C8—22no—
415C2×C8—22no—
416C2×C8—22no—
417C2×C8—22no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8#12014yes—
432C2×C8C22⋊Q814yes—
433C2×C8C22⋊Q814yes—
434C2×C8C22⋊Q814yes—
435C2×C8C4⋊Q814yes—
436C2×C8C4⋊Q814yes—
437C2×C8C4⋊Q814yes—
438C2×C8C4⋊Q814yes—
439C2×C8C4⋊Q814yes—
440C2×C8C4⋊Q814yes—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—48no—
454C4⋊C4—48no—
455C4⋊C4—48no—
456C4⋊C4—41no—
457C4⋊C4—41no—
458C4⋊C4—41no—
459C4⋊C4—41no—
460C4⋊C4—41no—
461C4⋊C4—41no—
462C4⋊C4—41no—
463C4⋊C4—41no—
464C4⋊C4—41no—
465C4⋊C4—41no—
466C4⋊C4—41no—
467C4⋊C4—41no—
468C4⋊C4—41no—
469C4⋊C4—41no—
470C4⋊C4—41no—
471C4⋊C4—41no—
472C4⋊C4—41no—
473C4⋊C4—41no—
474C4⋊C4—41no—
475C4⋊C4—41no—
476C4⋊C4—84no—
477C4⋊C4—84no—
478C4⋊C4—88no—
479C42—22no—
480C42—22no—
481C42—22no—
482C42—22no—
483C42—22no—
484C42—22no—
485C42—22no—
486C42—22no—
487C42—22no—
488C42—22no—
489C42—22no—
490C42—22no—
491C23C2×#12711yescharacteristic
492C23C2×C4⋊Q811yescharacteristic
493C23C2×C4⋊Q811yescharacteristic
494C23D4211yescharacteristic
495C23D4211yescharacteristic
496C23#54711yescharacteristic
497C23D4×Q811yescharacteristic
498C23#57512yes—
499C23#57512yes—
500C23#57612yes—
501C23#57812yes—
502Q8—816no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—82no—
540C2×C4—82no—
541C2×C4—81no—
542C2×C4—81no—
543C2×C4—81no—
544C2×C4—81no—
545C2×C4—81no—
546C2×C4—81no—
547C2×C4—81no—
548C2×C4—81no—
549C2×C4—81no—
550C2×C4—81no—
551C2×C4—81no—
552C2×C4—81no—
553C2×C4—82no—
554C2×C4—82no—
555C2×C4—82no—
556C2×C4—82no—
557C2×C4—81no—
558C2×C4—81no—
559C2×C4—81no—
560C2×C4—81no—
561C2×C4—81no—
562C2×C4—81no—
563C2×C4—82no—
564C2×C4—82no—
565C2×C4—82no—
566C2×C4—82no—
567C2×C4C2×#12011yescharacteristic
568C2×C4C2×C22⋊Q811yescharacteristic
569C2×C4C2×C22⋊Q811yescharacteristic
570C2×C4C2×C22⋊Q811yescharacteristic
571C2×C4C2×C4⋊Q811yescharacteristic
572C2×C4C2×C4⋊Q811yescharacteristic
573C2×C4C2×C4⋊Q811yescharacteristic
574C2×C4C2×C4⋊Q811yescharacteristic
575C2×C4C2×C4⋊Q811yescharacteristic
576C2×C4C2×C4⋊Q811yescharacteristic
577C2×C4#53511yescharacteristic
578C2×C4#53611yescharacteristic
579C2×C4#54411yescharacteristic
580C2×C4#54411yescharacteristic
581C2×C4#54411yescharacteristic
582C2×C4#54411yescharacteristic
583C2×C4D4211yescharacteristic
584C2×C4D4211yescharacteristic
585C2×C4#54711yescharacteristic
586C2×C4D4×Q811yescharacteristic
587C2×C4D4×Q811yescharacteristic
588C2×C4D4×Q811yescharacteristic
589C2×C4D4×Q811yescharacteristic
590C2×C4D4×Q811yescharacteristic
591C2×C4#55411yescharacteristic
592C2×C4#55411yescharacteristic
593C2×C4Q8211yescharacteristic
594C2×C4Q8211yescharacteristic
595C8—24no—
596C8—24no—
597C8—24no—
598C8#39518yes—
599C22#243311yescharacteristic
600C22#244011yescharacteristic
601C22#244011yescharacteristic
602C22#254211yescharacteristic
603C22#255111yescharacteristic
604C22#255111yescharacteristic
605C22#286811yescharacteristic
606C22#315311yescharacteristic
607C22#315311yescharacteristic
608C22#315511yescharacteristic
609C22#315611yescharacteristic
610C22#316212yes—
611C22#316412yes—
612C22#316412yes—
613C22#316612yes—
614C22#326812yes—
615C22#326912yes—
616C22#327312yes—
617C22#327712yes—
618C22#334712yes—
619C22#334912yes—
620C22#338912yes—
621C22#339012yes—
622C4—42no—
623C4—42no—
624C4—44no—
625C4—44no—
626C4—82no—
627C4—82no—
628C4—82no—
629C4—82no—
630C4—82no—
631C4—82no—
632C4—161no—
633C4—161no—
634C4C2×#39511yescharacteristic
635C4#243411yescharacteristic
636C4#244011yescharacteristic
637C4#244011yescharacteristic
638C4#255111yescharacteristic
639C4#255111yescharacteristic
640C4#256711yescharacteristic
641C4#266011yescharacteristic
642C2#1514311yescharacteristic
643C2#2579211yescharacteristic
644C2#2579311yescharacteristic
645C2#2606912yes—
646C2#2607912yes—
647C2#2854411yescharacteristic
648C2#2854611yescharacteristic
649C2#2881812yes—
650C2#2882012yes—
651C2#3331711yescharacteristic
652C2#3331911yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C4×#95 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
115. C2×#107Q8semidirect
133. C2×#97Q8semidirect
136. C2×C4×C8Q8semidirect
167. #336Q8semidirect
168. #336Q8semidirect

Maximal subgroups

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

subgroupindex
C22×#5012
C2×#20592
C2×#20572
C2×#19432
C2×#18472
C2×#18462
#152302
#152282
#113252
#113242
#108632
#108622
#106962