Finite Group Databasebeta

← #363136

Subgroups — #363136

Order 512.

1292 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34145C211yesmaximal, characteristic
2C2×#1994C211yesmaximal, characteristic
3C2×#1986C211yesmaximal, characteristic
4#21146C211yesmaximal, characteristic
5C4×#389C211yesmaximal, characteristic
6#11226C212yesmaximal
7#11209C212yesmaximal
8#10852C211yesmaximal, characteristic
9#10852C212yesmaximal
10#10808C212yesmaximal
11C4×#339C211yesmaximal, characteristic
12C2×#533C2211yescharacteristic
13C2×#423C2211yescharacteristic
14C2×#423C2212yes—
15C2×C4×M16C2211yescharacteristic
16#2370C412yescomplemented
17#2370C412yescomplemented
18#2369C411yescharacteristic, complemented
19#2369C411yescharacteristic, complemented
20#2368—22no—
21#2365C411yescharacteristic, complemented
22#2365C411yescharacteristic, complemented
23#2360—21no—
24#2360—21no—
25#2296C2212yes—
26#2295—21no—
27#2295—22no—
28C2×#389—21no—
29C42×Q8—21no—
30C4×#117C2212yes—
31C4×#117C2211yescharacteristic
32C2×C4×C4⋊C4C2211yescharacteristic
33C2×C4×C4⋊C4C2212yes—
34C2×C4×C4⋊C4C2211yescharacteristic
35C2×C43C2211yescharacteristic
36#1994—24no—
37#1986—24no—
38#1940—22no—
39#1939—21no—
40#1939—21no—
41C2×#339C2212yes—
42C2×#339C2212yes—
43C2×#339C2212yes—
44C2×#339C2212yes—
45C2×#339C2212yes—
46C2×#339C2211yescharacteristic
47C2×#339C2211yescharacteristic
48C2×#339C2211yescharacteristic
49C2×#339C2212yes—
50C2×#339C2211yescharacteristic
51#1728C2212yes—
52#1436—22no—
53C4⋊#95—21no—
54C4×#95C2212yes—
55C22×M16C2312yes—
56C22×M16C2311yescharacteristic
57C22×M16C2311yescharacteristic
58#533—22no—
59#533—22no—
60#533D414yes—
61#533Q814yes—
62C2×C4⋊Q8C2×C411yescharacteristic
63C2×C4⋊Q8C2×C411yescharacteristic
64C2×#125C2×C412yes—
65C2×C22⋊Q8—21no—
66C2×C22⋊Q8—21no—
67C2×C4×Q8—21no—
68C2×C4×Q8—21no—
69C2×C4×Q8—21no—
70C2×C4×Q8—21no—
71C2×#117C2311yescharacteristic
72C2×#117C2312yes—
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—24no—
86#423—22no—
87#423—22no—
88#423—42no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8—22no—
91C2×C4⋊C8—22no—
92C2×C4⋊C8—21no—
93C4×M16—22no—
94C4×M16—22no—
95C2×#97—21no—
96C2×C4×C8—21no—
97#391—21no—
98#391—21no—
99#391—22no—
100#391—22no—
101#389—28no—
102#389—21no—
103#389—21no—
104#389—22no—
105#389—22no—
106#389C2×C412yes—
107#389C2×C412yes—
108#389C2×C412yes—
109#389C2×C412yes—
110#387—22no—
111#387—22no—
112#387—22no—
113#387—22no—
114#387—22no—
115#387—22no—
116#385—22no—
117#385—22no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4—22no—
121C4×C4⋊C4—22no—
122C4×C4⋊C4—22no—
123C4×C4⋊C4—22no—
124C4×C4⋊C4—22no—
125C4×C4⋊C4—24no—
126C4×C4⋊C4—22no—
127C4×C4⋊C4—42no—
128C4×C4⋊C4C2×C411yescharacteristic
129C4×C4⋊C4C2×C411yescharacteristic
130C4×C4⋊C4C2×C412yes—
131C4×C4⋊C4C2×C412yes—
132C4×C4⋊C4C2×C412yes—
133C4×C4⋊C4C2×C412yes—
134C4×C4⋊C4C2×C411yescharacteristic
135C4×C4⋊C4C2×C411yescharacteristic
136C4×C4⋊C4C2×C411yescharacteristic
137C4×C4⋊C4C2×C411yescharacteristic
138C4×C4⋊C4C2×C412yes—
139C4×C4⋊C4D411yescharacteristic
140C4×C4⋊C4D411yescharacteristic
141C4×C4⋊C4D412yes—
142C4×C22⋊C4—21no—
143C4×C22⋊C4—22no—
144C4×C22⋊C4—22no—
145C4×C22⋊C4—21no—
146#376—21no—
147#376—22no—
148C2×#95—22no—
149C2×#95—22no—
150C2×#95—22no—
151C2×#95—21no—
152C2×#95D412yes—
153C43—22no—
154C43—21no—
155C43—22no—
156C43—22no—
157#339—22no—
158#339—24no—
159#339—24no—
160#339—24no—
161#339—24no—
162#339—22no—
163#339—24no—
164#339—22no—
165#339—42no—
166#339Q814yescomplemented
167#336—22no—
168#336—22no—
169C22×Q8—21no—
170C22×Q8—21no—
171C23×C4—21no—
172C23×C4—21no—
173C23×C4—22no—
174C23×C4C2×D412yes—
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D412yes—
177C23×C4C2×D411yescharacteristic
178C23×C4C2411yescharacteristic
179C2×M16—21no—
180C2×M16—21no—
181C2×M16—21no—
182C2×M16—21no—
183C2×M16—21no—
184C2×M16—21no—
185C2×M16—21no—
186C2×M16—21no—
187C2×M16—21no—
188C2×M16—21no—
189C2×M16—22no—
190C2×M16—22no—
191C2×M16—22no—
192C2×M16—22no—
193C2×M16—22no—
194C2×M16—22no—
195C2×M16C2×Q812yes—
196C2×M16#4112yes—
197C22×C8—21no—
198C22×C8—21no—
199C22×C8—22no—
200C4⋊Q8—22no—
201C4⋊Q8—22no—
202C4⋊Q8C4⋊C414yescomplemented
203C4⋊Q8C4⋊C414yescomplemented
204#125—24no—
205#125C4⋊C418yescomplemented
206C22⋊Q8—24no—
207C22⋊Q8—24no—
208C22⋊Q8—24no—
209C22⋊Q8—24no—
210C4×Q8—24no—
211C4×Q8—24no—
212C4×Q8—24no—
213C4×Q8—24no—
214C4×Q8—44no—
215C4×Q8—44no—
216#117—22no—
217#117—22no—
218#117—21no—
219#117—21no—
220#117—21no—
221#117—21no—
222#117—21no—
223#117—21no—
224#117C2×D414yes—
225#117C2×Q814yes—
226#117C2×Q812yes—
227#117#4112yes—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4C22⋊C411yescharacteristic
245C2×C4⋊C4C22⋊C412yes—
246C2×C4⋊C4C22⋊C411yescharacteristic
247C2×C4⋊C4C22⋊C411yescharacteristic
248C2×C4⋊C4C22⋊C412yes—
249C2×C4⋊C4C22⋊C411yescharacteristic
250C2×C4⋊C4C22×C411yescharacteristic
251C2×C4⋊C4C22×C411yescharacteristic
252C2×C4⋊C4C22×C412yes—
253C2×C4⋊C4C22×C412yes—
254C2×C4⋊C4C22×C411yescharacteristic
255C2×C4⋊C4C22×C411yescharacteristic
256C2×C22⋊C4—22no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—21no—
259C2×C42—22no—
260C2×C42—21no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—21no—
266C2×C42—21no—
267C2×C42—21no—
268C2×C42—21no—
269C2×C42—21no—
270C2×C42—21no—
271C2×C42—21no—
272C2×C42—21no—
273C2×C42—24no—
274C2×C42—22no—
275C2×C42—22no—
276C2×C42—21no—
277C2×C42—21no—
278C2×C42—21no—
279C2×C42—21no—
280C2×C42—21no—
281C2×C42—22no—
282C2×C42—22no—
283C2×C42—22no—
284C2×C42—22no—
285C2×C42—22no—
286C2×C42—22no—
287C2×C42—21no—
288C2×C42—21no—
289C2×C42—22no—
290C2×C42—22no—
291C2×C42—42no—
292C2×C42—42no—
293C2×C42—42no—
294C2×C42—42no—
295C2×C42—42no—
296C2×C42C22×C412yes—
297C2×C42C22×C412yes—
298C2×C42C22×C411yescharacteristic
299C2×C42C22×C411yescharacteristic
300C2×C42C2×D412yes—
301C2×C42C2×D412yes—
302C2×C42C2×D412yes—
303C2×C42C2×D411yescharacteristic
304C2×C42C2×D411yescharacteristic
305C2×C42C2×Q812yes—
306C2×C42C2×Q812yes—
307C2×C42#4112yes—
308C2×C42#4112yes—
309C2×C42#4112yes—
310C4⋊C8—44no—
311C4⋊C8—44no—
312C4⋊C8—42no—
313C4⋊C8—42no—
314#97—42no—
315C4×C8—42no—
316#95—28no—
317#95—21no—
318#95—21no—
319#95—22no—
320#95—22no—
321#95—22no—
322#95—22no—
323#95—21no—
324#95—21no—
325#95—22no—
326#95—22no—
327#95—22no—
328#95—22no—
329#95—44no—
330#95—44no—
331#95—42no—
332#95C22⋊C412yes—
333#95C22⋊C412yes—
334#95C22⋊C412yes—
335#95C22⋊C412yes—
336C24C22×D411yescharacteristic
337C2×Q8—24no—
338C2×Q8—24no—
339C2×Q8—42no—
340C2×Q8—42no—
341C2×Q8—81no—
342C2×Q8—81no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—21no—
350C22×C4—21no—
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—24no—
360C22×C4—24no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4C2×C22⋊C412yes—
383C22×C4C2×C22⋊C412yes—
384C22×C4C2×C22⋊C412yes—
385C22×C4C2×C22⋊C411yescharacteristic
386C22×C4C2×C22⋊C411yescharacteristic
387C22×C4C2×C22⋊C411yescharacteristic
388C22×C4C2×C22⋊C412yes—
389C22×C4C2×C22⋊C411yescharacteristic
390C22×C4#12112yes—
391C22×C4#12112yes—
392C22×C4C22⋊Q812yes—
393C22×C4C22⋊Q812yes—
394C22×C4C23×C411yescharacteristic
395C22×C4C22×D411yescharacteristic
396C22×C4C22×D411yescharacteristic
397C22×C4C22×Q811yescharacteristic
398C22×C4C22×Q811yescharacteristic
399C22×C4C2×#4111yescharacteristic
400C22×C4C2×#4111yescharacteristic
401C22×C4C2×#4111yescharacteristic
402C22×C4C2×#4111yescharacteristic
403C22×C4C2×#4111yescharacteristic
404C22×C4H(5,𝔽2)11yescharacteristic
405C22×C42−1+411yescharacteristic
406C22×C42−1+411yescharacteristic
407C22×C42−1+411yescharacteristic
408M16—22no—
409M16—22no—
410M16—22no—
411M16—22no—
412M16—41no—
413M16—41no—
414M16—41no—
415M16—41no—
416M16—42no—
417M16—42no—
418M16—42no—
419M16—42no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—41no—
423C2×C8—41no—
424C2×C8—41no—
425C2×C8—41no—
426C2×C8—42no—
427C2×C8—42no—
428C2×C8—42no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—28no—
434C4⋊C4—28no—
435C4⋊C4—24no—
436C4⋊C4—24no—
437C4⋊C4—24no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—42no—
446C4⋊C4—48no—
447C4⋊C4—416no—
448C4⋊C4—44no—
449C4⋊C4—44no—
450C4⋊C4—82no—
451C4⋊C4—82no—
452C4⋊C4#10414yes—
453C4⋊C4C2×C4⋊C414yes—
454C4⋊C4C2×C4⋊C414yes—
455C4⋊C4C4×Q812yes—
456C4⋊C4C4×Q812yes—
457C22⋊C4—24no—
458C22⋊C4—24no—
459C22⋊C4—22no—
460C22⋊C4—22no—
461C22⋊C4—41no—
462C22⋊C4—41no—
463C42—22no—
464C42—22no—
465C42—22no—
466C42—22no—
467C42—24no—
468C42—24no—
469C42—22no—
470C42—22no—
471C42—22no—
472C42—22no—
473C42—22no—
474C42—22no—
475C42—22no—
476C42—22no—
477C42—22no—
478C42—22no—
479C42—48no—
480C42—44no—
481C42—48no—
482C42—41no—
483C42—41no—
484C42—41no—
485C42—41no—
486C42—41no—
487C42—41no—
488C42—44no—
489C42—42no—
490C42—42no—
491C42—42no—
492C42—42no—
493C42—42no—
494C42—42no—
495C42—82no—
496C42—81no—
497C42—81no—
498C42—82no—
499C42C2×C4⋊C412yes—
500C42C2×C4⋊C412yes—
501C42#11712yes—
502C42#11712yes—
503C42#12112yes—
504C42#12112yes—
505C42C22⋊Q812yes—
506C42C22⋊Q812yes—
507C23—21no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23C22×C22⋊C411yescharacteristic
512C23C2×#12111yescharacteristic
513C23C2×C22⋊Q811yescharacteristic
514C23#54611yescharacteristic
515C23#54711yescharacteristic
516C23#54811yescharacteristic
517C23D4×Q811yescharacteristic
518Q8—48no—
519Q8—84no—
520C2×C4—22no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—22no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
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—24no—
544C2×C4—24no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—41no—
560C2×C4—42no—
561C2×C4—41no—
562C2×C4—42no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—82no—
579C2×C4—82no—
580C2×C4—82no—
581C2×C4—82no—
582C2×C4—82no—
583C2×C4—82no—
584C2×C4#38012yes—
585C2×C4#38012yes—
586C2×C4#38012yes—
587C2×C4#38012yes—
588C2×C4C2×#10414yes—
589C2×C4C2×#10412yes—
590C2×C4#42112yes—
591C2×C4C22×C4⋊C411yescharacteristic
592C2×C4C2×#11711yescharacteristic
593C2×C4C2×C4×Q811yescharacteristic
594C2×C4C4×#4111yescharacteristic
595C2×C4#51911yescharacteristic
596C2×C4#52011yescharacteristic
597C2×C4C2×#12111yescharacteristic
598C2×C4C2×C22⋊Q811yescharacteristic
599C2×C4#52911yescharacteristic
600C2×C4#52911yescharacteristic
601C2×C4#53311yescharacteristic
602C2×C4#53311yescharacteristic
603C2×C4#54111yescharacteristic
604C2×C4#54111yescharacteristic
605C2×C4#54211yescharacteristic
606C2×C4#54211yescharacteristic
607C2×C4#54611yescharacteristic
608C2×C4#54711yescharacteristic
609C2×C4#54811yescharacteristic
610C2×C4#54811yescharacteristic
611C2×C4#54811yescharacteristic
612C2×C4D4×Q811yescharacteristic
613C2×C4D4×Q811yescharacteristic
614C2×C4#55011yescharacteristic
615C2×C4#55711yescharacteristic
616C2×C4#55711yescharacteristic
617C2×C4#55711yescharacteristic
618C2×C4Q8211yescharacteristic
619C8—42no—
620C8—42no—
621C8—82no—
622C22—21no—
623C22—21no—
624C22—21no—
625C22—21no—
626C22—21no—
627C22—21no—
628C22—21no—
629C22—21no—
630C22—21no—
631C22—21no—
632C22—21no—
633C22—21no—
634C22C2×#38011yescharacteristic
635C22#233911yescharacteristic
636C22Q8×C22⋊C411yescharacteristic
637C22#242611yescharacteristic
638C22#242911yescharacteristic
639C22#245111yescharacteristic
640C22#245311yescharacteristic
641C22C22×#10411yescharacteristic
642C22C2×#42111yescharacteristic
643C22#290111yescharacteristic
644C22#290111yescharacteristic
645C4—22no—
646C4—22no—
647C4—22no—
648C4—24no—
649C4—44no—
650C4—44no—
651C4—42no—
652C4—42no—
653C4—44no—
654C4—44no—
655C4—42no—
656C4—42no—
657C4—42no—
658C4—84no—
659C4—81no—
660C4—81no—
661C4#180614yes—
662C4#229511yescharacteristic
663C4#232011yescharacteristic
664C4Q8×C4⋊C411yescharacteristic
665C4#235111yescharacteristic
666C4#242611yescharacteristic
667C4#242911yescharacteristic
668C4#245111yescharacteristic
669C4#245311yescharacteristic
670C2—21no—
671C2—21no—
672C2—21no—
673C2—21no—
674C2#1393511yescharacteristic
675C2C2×#180611yescharacteristic
676C2#2065511yescharacteristic
677C2#2250511yescharacteristic
678C2#2250711yescharacteristic
679C2Q8×#10411yescharacteristic
680C2#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. #2370C4semidirect
17. #2370C4semidirect
18. #2369C4semidirect
19. #2369C4semidirect
21. #2365C4semidirect
22. #2365C4semidirect
166. #339Q8semidirect
202. C4⋊Q8C4⋊C4semidirect
203. C4⋊Q8C4⋊C4semidirect
205. #125C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#341452
C2×#19942
C2×#19862
#211462
C4×#3892
#112262
#112092
#108522
#108082
C4×#3392