Finite Group Databasebeta

← #370582

Subgroups — #370582

Order 512.

1316 conjugacy classes of subgroups.

The subgroup lattice has 685 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34938C211yesmaximal, characteristic
2C2×#1986C212yesmaximal
3C4⋊C4×M16C211yesmaximal, characteristic
4C4×#389C211yesmaximal, characteristic
5#11386C212yesmaximal
6#11226C212yesmaximal
7#10886C212yesmaximal
8#10852C211yesmaximal, characteristic
9#10851C211yesmaximal, characteristic
10#10839C211yesmaximal, characteristic
11C4×#339C211yesmaximal, characteristic
12C2×#533C2212yes—
13C2×#423C2211yescharacteristic
14C2×C4×M16C2212yes—
15C2×C4×M16C2211yescharacteristic
16#2552C411yescharacteristic, complemented
17#2552C411yescharacteristic, complemented
18#2551C411yescharacteristic, complemented
19#2551C411yescharacteristic, complemented
20#2550C412yescomplemented
21#2550C412yescomplemented
22#2549—22no—
23#2440—21no—
24#2440—21no—
25#2399—22no—
26C4×C4⋊Q8—21no—
27C4×C4⋊Q8—21no—
28#2296C2212yes—
29C2×#389—21no—
30#2285C2211yescharacteristic
31C4×#117C2211yescharacteristic
32C2×C4×C4⋊C4C2211yescharacteristic
33C2×C4×C4⋊C4C2212yes—
34C2×C4×C4⋊C4C2211yescharacteristic
35C2×C43C2211yescharacteristic
36#1986—28no—
37#1771—22no—
38#1770—21no—
39C8×C4⋊C4—21no—
40C2×#339C2212yes—
41C2×#339C2212yes—
42C2×#339C2212yes—
43C2×#339C2212yes—
44C2×#339C2211yescharacteristic
45C2×#339C2211yescharacteristic
46C2×#339C2212yes—
47C2×#339C2212yes—
48C2×#339C2211yescharacteristic
49C2×#339C2211yescharacteristic
50#1728C2212yes—
51#1436—22no—
52C4⋊#95—21no—
53C4×#95C2212yes—
54C22×M16C2312yes—
55C22×M16C2311yescharacteristic
56C22×M16C2311yescharacteristic
57#533—24no—
58#533—24no—
59#533—42no—
60#533—42no—
61C2×C4⋊Q8—21no—
62C2×C4⋊Q8—21no—
63C2×C4⋊Q8C2×C412yes—
64C2×C4⋊Q8C2×C412yes—
65C2×#125C2×C412yes—
66C2×#125C2×C412yes—
67C2×C22⋊Q8—22no—
68C2×C22⋊Q8—22no—
69C2×C4×Q8—22no—
70C2×C4×Q8—22no—
71C2×#117C2312yes—
72C2×#117C2311yescharacteristic
73C2×#117C2311yescharacteristic
74C22×C4⋊C4—21no—
75C22×C4⋊C4—22no—
76C22×C4⋊C4D411yescharacteristic
77C22×C4⋊C4D411yescharacteristic
78C22×C42—21no—
79C22×C42—21no—
80C22×C42C2312yes—
81C22×C42C2311yescharacteristic
82C22×C42C2311yescharacteristic
83C22×C42C2312yes—
84C22×C42C2311yescharacteristic
85#423—22no—
86#423Q814yescomplemented
87C2×C4⋊C8—21no—
88C2×C4⋊C8—21no—
89C4×M16—22no—
90C4×M16—24no—
91C4×M16—42no—
92C4×M16Q814yescomplemented
93C2×#97—21no—
94C2×#97—22no—
95C2×C4×C8—21no—
96C2×C4×C8—22no—
97#398—22no—
98#398—22no—
99#398—22no—
100#398—22no—
101#395—22no—
102#395—22no—
103#395—22no—
104#395—22no—
105#391—22no—
106#391—22no—
107#389—28no—
108#389—21no—
109#389—21no—
110#389—21no—
111#389—21no—
112#389—22no—
113#389—22no—
114#389C2×C412yes—
115#389C2×C412yes—
116#389C2×C412yes—
117#389C2×C412yes—
118#385—22no—
119#385—22no—
120#384—21no—
121#384—21no—
122#384C2×C411yescharacteristic
123#384C2×C411yescharacteristic
124#382C2×C412yes—
125#380—21no—
126#380—21no—
127C4×C4⋊C4—21no—
128C4×C4⋊C4—22no—
129C4×C4⋊C4—22no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4—24no—
132C4×C4⋊C4—22no—
133C4×C4⋊C4—22no—
134C4×C4⋊C4—42no—
135C4×C4⋊C4C2×C411yescharacteristic
136C4×C4⋊C4C2×C411yescharacteristic
137C4×C4⋊C4C2×C411yescharacteristic
138C4×C4⋊C4C2×C411yescharacteristic
139C4×C4⋊C4C2×C412yes—
140C4×C4⋊C4C2×C411yescharacteristic
141C4×C4⋊C4C2×C411yescharacteristic
142C4×C4⋊C4D411yescharacteristic
143C4×C4⋊C4D411yescharacteristic
144C4×C4⋊C4D412yes—
145C4×C22⋊C4—21no—
146C4×C22⋊C4—21no—
147#376—21no—
148C2×#95—22no—
149C2×#95—22no—
150C2×#95—22no—
151C2×#95—21no—
152C2×#95D412yes—
153C43—21no—
154C43—22no—
155C43—22no—
156#339—24no—
157#339—24no—
158#339—22no—
159#339—22no—
160#339—24no—
161#339—24no—
162#339—42no—
163#339—42no—
164#339D414yes—
165#339Q814yescomplemented
166#336—22no—
167#336—22no—
168C22×Q8—22no—
169C22×Q8—22no—
170C23×C4—21no—
171C23×C4—21no—
172C23×C4—22no—
173C23×C4C2×D412yes—
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D412yes—
176C23×C4C2×D411yescharacteristic
177C23×C4C2411yescharacteristic
178C2×M16—21no—
179C2×M16—21no—
180C2×M16—21no—
181C2×M16—21no—
182C2×M16—22no—
183C2×M16—22no—
184C2×M16—21no—
185C2×M16—21no—
186C2×M16—22no—
187C2×M16—21no—
188C2×M16—22no—
189C2×M16—21no—
190C2×M16—22no—
191C2×M16C2×D412yes—
192C2×M16C2×Q812yes—
193C2×M16C2×Q814yes—
194C2×M16C2×Q812yes—
195C2×M16C2×Q812yes—
196C22×C8—21no—
197C22×C8—21no—
198C22×C8—22no—
199C4⋊Q8—24no—
200C4⋊Q8—24no—
201C4⋊Q8—42no—
202C4⋊Q8—42no—
203C4⋊Q8—44no—
204C4⋊Q8—44no—
205#125—24no—
206#125—24no—
207#125—42no—
208#125—42no—
209C22⋊Q8—44no—
210C22⋊Q8—44no—
211C22⋊Q8—44no—
212C22⋊Q8—44no—
213C4×Q8—44no—
214C4×Q8—44no—
215C4×Q8—44no—
216C4×Q8—44no—
217#117—21no—
218#117—21no—
219#117—21no—
220#117—21no—
221#117—22no—
222#117—22no—
223#117—22no—
224#117—22no—
225#117C2×D412yes—
226#117C2×Q812yes—
227#117#4112yes—
228#117#4112yes—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4C22⋊C411yescharacteristic
252C2×C4⋊C4C22⋊C411yescharacteristic
253C2×C4⋊C4C22⋊C411yescharacteristic
254C2×C4⋊C4C22⋊C412yes—
255C2×C4⋊C4C22⋊C412yes—
256C2×C4⋊C4C22⋊C411yescharacteristic
257C2×C4⋊C4C22×C412yes—
258C2×C4⋊C4C22×C412yes—
259C2×C4⋊C4C22×C411yescharacteristic
260C2×C4⋊C4C22×C411yescharacteristic
261C2×C4⋊C4C22×C411yescharacteristic
262C2×C4⋊C4C22×C411yescharacteristic
263C2×C22⋊C4—21no—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—22no—
266C2×C42—22no—
267C2×C42—21no—
268C2×C42—21no—
269C2×C42—22no—
270C2×C42—22no—
271C2×C42—22no—
272C2×C42—21no—
273C2×C42—21no—
274C2×C42—21no—
275C2×C42—21no—
276C2×C42—21no—
277C2×C42—21no—
278C2×C42—21no—
279C2×C42—21no—
280C2×C42—24no—
281C2×C42—21no—
282C2×C42—21no—
283C2×C42—22no—
284C2×C42—21no—
285C2×C42—21no—
286C2×C42—22no—
287C2×C42—22no—
288C2×C42—22no—
289C2×C42—22no—
290C2×C42—22no—
291C2×C42—22no—
292C2×C42—21no—
293C2×C42—21no—
294C2×C42—42no—
295C2×C42—42no—
296C2×C42—42no—
297C2×C42—42no—
298C2×C42—42no—
299C2×C42C22×C412yes—
300C2×C42C22×C412yes—
301C2×C42C22×C411yescharacteristic
302C2×C42C22×C411yescharacteristic
303C2×C42C2×D412yes—
304C2×C42C2×D412yes—
305C2×C42C2×D412yes—
306C2×C42C2×D411yescharacteristic
307C2×C42C2×D411yescharacteristic
308C2×C42C2×Q812yes—
309C2×C42C2×Q812yes—
310C2×C42C2×Q812yes—
311C2×C42#4112yes—
312C2×C42#4112yes—
313C4⋊C8—24no—
314C4⋊C8—24no—
315#97—24no—
316#97—44no—
317C4×C8—24no—
318C4×C8—44no—
319#95—28no—
320#95—21no—
321#95—21no—
322#95—21no—
323#95—21no—
324#95—22no—
325#95—22no—
326#95—22no—
327#95—22no—
328#95—22no—
329#95—22no—
330#95—44no—
331#95—44no—
332#95—42no—
333#95C22⋊C412yes—
334#95C22⋊C412yes—
335#95C22⋊C412yes—
336#95C22⋊C412yes—
337C24C22×D411yescharacteristic
338C2×Q8—44no—
339C2×Q8—44no—
340C2×Q8—44no—
341C2×Q8—44no—
342C2×Q8—82no—
343C2×Q8—82no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4C2×C22⋊C412yes—
384C22×C4C2×C22⋊C412yes—
385C22×C4C2×C22⋊C412yes—
386C22×C4C2×C22⋊C411yescharacteristic
387C22×C4C2×C22⋊C411yescharacteristic
388C22×C4C2×C22⋊C411yescharacteristic
389C22×C4C2×C22⋊C412yes—
390C22×C4C2×C22⋊C411yescharacteristic
391C22×C4#12412yes—
392C22×C4#12412yes—
393C22×C4C4⋊Q812yes—
394C22×C4C4⋊Q812yes—
395C22×C4C23×C411yescharacteristic
396C22×C4C22×D411yescharacteristic
397C22×C4C22×D411yescharacteristic
398C22×C4C22×Q811yescharacteristic
399C22×C4C22×Q811yescharacteristic
400C22×C4C22×Q811yescharacteristic
401C22×C4C2×#4111yescharacteristic
402C22×C4C2×#4111yescharacteristic
403C22×C4C2×#4111yescharacteristic
404C22×C4C2×#4111yescharacteristic
405C22×C4H(5,𝔽2)11yescharacteristic
406C22×C4H(5,𝔽2)11yescharacteristic
407C22×C42−1+411yescharacteristic
408C22×C42−1+411yescharacteristic
409M16—22no—
410M16—22no—
411M16—22no—
412M16—22no—
413M16—22no—
414M16—24no—
415M16—24no—
416M16—42no—
417M16—42no—
418M16—41no—
419M16—41no—
420M16C4⋊Q814yescomplemented
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—24no—
426C2×C8—42no—
427C2×C8—41no—
428C2×C8—41no—
429C2×C8—42no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—21no—
434C4⋊C4—24no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—42no—
443C4⋊C4—44no—
444C4⋊C4—44no—
445C4⋊C4—44no—
446C4⋊C4—44no—
447C4⋊C4—44no—
448C4⋊C4—44no—
449C4⋊C4—48no—
450C4⋊C4—416no—
451C4⋊C4—44no—
452C4⋊C4—44no—
453C4⋊C4—42no—
454C4⋊C4—42no—
455C4⋊C4—81no—
456C4⋊C4—81no—
457C4⋊C4—82no—
458C4⋊C4—82no—
459C4⋊C4#10414yes—
460C4⋊C4C4×D412yes—
461C4⋊C4C4×D412yes—
462C4⋊C4C4×Q812yes—
463C4⋊C4C4×Q812yes—
464C22⋊C4—22no—
465C22⋊C4—22no—
466C22⋊C4—22no—
467C22⋊C4—22no—
468C22⋊C4—42no—
469C22⋊C4—42no—
470C42—22no—
471C42—22no—
472C42—22no—
473C42—22no—
474C42—22no—
475C42—22no—
476C42—22no—
477C42—22no—
478C42—22no—
479C42—22no—
480C42—22no—
481C42—22no—
482C42—22no—
483C42—22no—
484C42—22no—
485C42—22no—
486C42—48no—
487C42—48no—
488C42—41no—
489C42—41no—
490C42—42no—
491C42—42no—
492C42—41no—
493C42—41no—
494C42—42no—
495C42—42no—
496C42—42no—
497C42—42no—
498C42—42no—
499C42—42no—
500C42—81no—
501C42—81no—
502C42C4×Q812yes—
503C42C4×Q812yes—
504C42C4×Q812yes—
505C42C4×Q812yes—
506C42C22⋊Q812yes—
507C42C22⋊Q812yes—
508C42C22⋊Q812yes—
509C42C22⋊Q812yes—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23C22×C22⋊C411yescharacteristic
515C23C2×#12411yescharacteristic
516C23C2×C4⋊Q811yescharacteristic
517C23D4211yescharacteristic
518C23#54611yescharacteristic
519C23#54811yescharacteristic
520C23D4×Q811yescharacteristic
521Q8—88no—
522Q8—88no—
523C2×C4—22no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—22no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—22no—
536C2×C4—22no—
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—42no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
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—42no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—82no—
582C2×C4—82no—
583C2×C4—82no—
584C2×C4—82no—
585C2×C4—82no—
586C2×C4—82no—
587C2×C4#39112yes—
588C2×C4#39112yes—
589C2×C4#39112yes—
590C2×C4#39112yes—
591C2×C4C2×#10414yes—
592C2×C4C2×#10412yes—
593C2×C4#42112yes—
594C2×C4C2×C4×D411yescharacteristic
595C2×C4C2×C4×Q811yescharacteristic
596C2×C4C2×C4×Q811yescharacteristic
597C2×C4C2×C4×Q811yescharacteristic
598C2×C4#51811yescharacteristic
599C2×C4#51911yescharacteristic
600C2×C4C2×C22⋊Q811yescharacteristic
601C2×C4C2×C22⋊Q811yescharacteristic
602C2×C4C2×C4⋊Q811yescharacteristic
603C2×C4#53211yescharacteristic
604C2×C4#53311yescharacteristic
605C2×C4#53311yescharacteristic
606C2×C4#54211yescharacteristic
607C2×C4#54211yescharacteristic
608C2×C4#54411yescharacteristic
609C2×C4#54411yescharacteristic
610C2×C4D4211yescharacteristic
611C2×C4#54611yescharacteristic
612C2×C4#54711yescharacteristic
613C2×C4#54811yescharacteristic
614C2×C4D4×Q811yescharacteristic
615C2×C4D4×Q811yescharacteristic
616C2×C4#55411yescharacteristic
617C2×C4#55411yescharacteristic
618C2×C4#55511yescharacteristic
619C2×C4#55611yescharacteristic
620C2×C4#55711yescharacteristic
621C2×C4Q8211yescharacteristic
622C8—24no—
623C8—42no—
624C8—44no—
625C22—21no—
626C22—21no—
627C22—21no—
628C22—21no—
629C22—21no—
630C22—21no—
631C22—21no—
632C22—21no—
633C22—21no—
634C22—21no—
635C22—21no—
636C22—21no—
637C22C2×#39111yescharacteristic
638C22D4×C22⋊C411yescharacteristic
639C22Q8×C22⋊C411yescharacteristic
640C22#249111yescharacteristic
641C22#249211yescharacteristic
642C22#255011yescharacteristic
643C22#255111yescharacteristic
644C22C22×#10411yescharacteristic
645C22C2×#42111yescharacteristic
646C22#290111yescharacteristic
647C22#290111yescharacteristic
648C4—22no—
649C4—22no—
650C4—22no—
651C4—24no—
652C4—44no—
653C4—44no—
654C4—42no—
655C4—42no—
656C4—42no—
657C4—42no—
658C4—42no—
659C4—42no—
660C4—42no—
661C4—84no—
662C4—82no—
663C4—82no—
664C4#198614yescomplemented
665C4C4×C4⋊Q811yescharacteristic
666C4#233111yescharacteristic
667C4#236511yescharacteristic
668C4#236911yescharacteristic
669C4#243911yescharacteristic
670C4#244011yescharacteristic
671C4#245111yescharacteristic
672C4#245211yescharacteristic
673C2—21no—
674C2—21no—
675C2—21no—
676C2—21no—
677C2#1405311yescharacteristic
678C2C2×#198611yescharacteristic
679C2#2130711yescharacteristic
680C2D4×#10411yescharacteristic
681C2#2243511yescharacteristic
682C2Q8×#10411yescharacteristic
683C2#2252911yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C22×C22⋊C4
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
16. #2552C4semidirect
17. #2552C4semidirect
18. #2551C4semidirect
19. #2551C4semidirect
20. #2550C4semidirect
21. #2550C4semidirect
86. #423Q8semidirect
92. C4×M16Q8semidirect
165. #339Q8semidirect
420. M16C4⋊Q8semidirect
664. C4#1986semidirect

Maximal subgroups

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

subgroupindex
#349382
C2×#19862
C4⋊C4×M162
C4×#3892
#113862
#112262
#108862
#108522
#108512
#108392
C4×#3392