Finite Group Databasebeta

← #364268

Subgroups — #364268

Order 512.

1618 conjugacy classes of subgroups.

The subgroup lattice has 713 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34109C211yesmaximal, characteristic
2#20837C211yesmaximal, characteristic
3C2×#1826C211yesmaximal, characteristic
4C22×#339C211yesmaximal, characteristic
5C4×#385C211yesmaximal, characteristic
6#11226C212yesmaximal, complemented
7#11209C212yesmaximal, complemented
8#10852C212yesmaximal, complemented
9#10808C212yesmaximal, complemented
10#10732C211yesmaximal, characteristic
11#10719C211yesmaximal, characteristic
12C23×M16C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C23×C42C2211yescharacteristic
15C2×#423C2212yes—
16C2×#407C2211yescharacteristic
17#2347C412yescomplemented
18#2346C412yescomplemented
19#2345C411yescharacteristic, complemented
20#2345C411yescharacteristic, complemented
21#2344C411yescharacteristic, complemented
22#2344C411yescharacteristic, complemented
23#2343—22no—
24#2342—21no—
25#2342—21no—
26#2298—22no—
27#2296C2212yes—
28C2×#385—21no—
29#2283—21no—
30#2280C2211yescharacteristic
31C4×#117—21no—
32C4×#117C2212yes—
33C2×C4×C4⋊C4C2212yes—
34C2×C4×C22⋊C4C2211yescharacteristic
35C2×C4×C22⋊C4C2211yescharacteristic
36#1851—22no—
37#1850—21no—
38#1850—21no—
39#1826—24no—
40C2×#339—24no—
41C2×#339C2211yescharacteristic
42C2×#339C2212yes—
43C2×#339C2212yes—
44C2×#339C2212yes—
45C2×#339C2211yescharacteristic
46C2×#339C2212yes—
47C2×#339C2211yescharacteristic
48C2×#339C2211yescharacteristic
49C2×#339C2212yes—
50C2×#339C2212yes—
51#1728C2212yes—
52#1436—22no—
53C22⋊#95—21no—
54C4×#95C2212yes—
55C24×C4—21no—
56C24×C4C2311yescharacteristic
57C22×M16—21no—
58C22×M16—22no—
59C22×M16—24no—
60C22×M16—22no—
61C22×M16—42no—
62C22×M16C2311yescharacteristic
63C22×M16C2311yescharacteristic
64C22×M16C2312yes—
65C23×C8—21no—
66C2×#117—21no—
67C2×#117—21no—
68C2×#117—22no—
69C2×#117—22no—
70C2×#117—21no—
71C2×#117—21no—
72C2×#117C2×C411yescharacteristic
73C2×#117C2×C411yescharacteristic
74C2×#117D414yes—
75C2×#117Q814yes—
76C2×#117C2312yes—
77C2×#117C2311yescharacteristic
78C2×#117C2311yescharacteristic
79C22×C4⋊C4—22no—
80C22×C4⋊C4C2×C411yescharacteristic
81C22×C4⋊C4C2×C411yescharacteristic
82C22×C22⋊C4—21no—
83C22×C22⋊C4—21no—
84C22×C22⋊C4D411yescharacteristic
85C22×C22⋊C4D411yescharacteristic
86C22×C42—21no—
87C22×C42—22no—
88C22×C42—24no—
89C22×C42—22no—
90C22×C42—21no—
91C22×C42—42no—
92C22×C42C2311yescharacteristic
93C22×C42C2311yescharacteristic
94C22×C42C2312yes—
95C22×C42C2312yes—
96#423—24no—
97#423—42no—
98C2×C4⋊C8—22no—
99C2×C4⋊C8—22no—
100#407—22no—
101#407—22no—
102C2×C22⋊C8—21no—
103C2×C22⋊C8—21no—
104#389C2×C412yescomplemented
105#389C2×C412yescomplemented
106#389C2×C412yescomplemented
107#389C2×C412yescomplemented
108#388—22no—
109#388—22no—
110#388—22no—
111#388—22no—
112#387—22no—
113#387—22no—
114#387—22no—
115#387—22no—
116#385—28no—
117#385—22no—
118#385—22no—
119#383—22no—
120#382—22no—
121#381—21no—
122#381—21no—
123#381—21no—
124#380—21no—
125#380—21no—
126#380C2×C411yescharacteristic
127#380C2×C411yescharacteristic
128#380C2×C412yes—
129#379—21no—
130C4×C4⋊C4—22no—
131C4×C4⋊C4—22no—
132C4×C4⋊C4—22no—
133C4×C4⋊C4—24no—
134C4×C4⋊C4—42no—
135C4×C4⋊C4C2×C412yescomplemented
136C4×C4⋊C4C2×C412yescomplemented
137C4×C4⋊C4C2×C412yescomplemented
138C4×C4⋊C4C2×C412yescomplemented
139C4×C22⋊C4—22no—
140C4×C22⋊C4—22no—
141C4×C22⋊C4—21no—
142C4×C22⋊C4—21no—
143C4×C22⋊C4—21no—
144C4×C22⋊C4—21no—
145C4×C22⋊C4—22no—
146C4×C22⋊C4—22no—
147C4×C22⋊C4C2×C411yescharacteristic
148C4×C22⋊C4C2×C411yescharacteristic
149C4×C22⋊C4C2×C412yes—
150C4×C22⋊C4D411yescharacteristic
151C4×C22⋊C4D411yescharacteristic
152C4×C22⋊C4D412yes—
153#376—22no—
154#376—22no—
155C2×#95—22no—
156C2×#95—22no—
157C2×#95—22no—
158C2×#95—21no—
159C2×#95D412yes—
160C43—22no—
161#339—24no—
162#339—22no—
163#339—22no—
164#339—22no—
165#339—22no—
166#339—24no—
167#339—24no—
168#339—24no—
169#339—42no—
170#339—42no—
171#336—22no—
172#336—22no—
173C25C2×D411yescharacteristic
174C23×C4—22no—
175C23×C4—22no—
176C23×C4—22no—
177C23×C4—22no—
178C23×C4—21no—
179C23×C4—22no—
180C23×C4—21no—
181C23×C4—24no—
182C23×C4—21no—
183C23×C4—21no—
184C23×C4—44no—
185C23×C4—44no—
186C23×C4—42no—
187C23×C4—42no—
188C23×C4C22×C411yescharacteristic
189C23×C4C22×C411yescharacteristic
190C23×C4C2×D412yes—
191C23×C4C2×D411yescharacteristic
192C23×C4C2×D411yescharacteristic
193C23×C4C2×D411yescharacteristic
194C23×C4C2×D412yes—
195C23×C4C2×D412yes—
196C23×C4C2×Q812yes—
197C23×C4#4112yes—
198C23×C4#4112yes—
199C23×C4C2411yescharacteristic
200C2×M16—22no—
201C2×M16—28no—
202C2×M16—22no—
203C2×M16—22no—
204C2×M16—22no—
205C2×M16—22no—
206C2×M16—21no—
207C2×M16—21no—
208C2×M16—22no—
209C2×M16—21no—
210C2×M16—21no—
211C2×M16—22no—
212C2×M16—22no—
213C2×M16—22no—
214C2×M16—44no—
215C2×M16—48no—
216C2×M16—44no—
217C2×M16—42no—
218C2×M16#4114yes—
219C22×C8—22no—
220C22×C8—21no—
221C22×C8—21no—
222C22×C8—44no—
223C22×C8—42no—
224#117—22no—
225#117—22no—
226#117—24no—
227#117—24no—
228#117—24no—
229#117—24no—
230#117—21no—
231#117—21no—
232#117—21no—
233#117—21no—
234#117—22no—
235#117—22no—
236#117—22no—
237#117—22no—
238#117—44no—
239#117—44no—
240#117C4⋊C414yescomplemented
241#117C4⋊C414yescomplemented
242#117C2×D412yes—
243#117C2×D412yes—
244#117C2×Q812yes—
245#117C2×Q812yes—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4C4⋊C414yescomplemented
258C2×C4⋊C4C4⋊C414yescomplemented
259C2×C4⋊C4C22×C412yes—
260C2×C4⋊C4C22×C412yes—
261C2×C4⋊C4C22×C411yescharacteristic
262C2×C4⋊C4C22×C411yescharacteristic
263C2×C4⋊C4C22×C411yescharacteristic
264C2×C4⋊C4C22×C411yescharacteristic
265C2×C22⋊C4—21no—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—42no—
278C2×C22⋊C4—42no—
279C2×C22⋊C4—42no—
280C2×C22⋊C4C22⋊C411yescharacteristic
281C2×C22⋊C4C22⋊C411yescharacteristic
282C2×C22⋊C4C22⋊C411yescharacteristic
283C2×C22⋊C4C22⋊C412yes—
284C2×C22⋊C4C22⋊C412yes—
285C2×C22⋊C4C22⋊C411yescharacteristic
286C2×C42—21no—
287C2×C42—21no—
288C2×C42—24no—
289C2×C42—24no—
290C2×C42—21no—
291C2×C42—21no—
292C2×C42—22no—
293C2×C42—22no—
294C2×C42—22no—
295C2×C42—22no—
296C2×C42—28no—
297C2×C42—22no—
298C2×C42—22no—
299C2×C42—22no—
300C2×C42—22no—
301C2×C42—22no—
302C2×C42—22no—
303C2×C42—22no—
304C2×C42—22no—
305C2×C42—21no—
306C2×C42—21no—
307C2×C42—22no—
308C2×C42—21no—
309C2×C42—21no—
310C2×C42—22no—
311C2×C42—22no—
312C2×C42—22no—
313C2×C42—22no—
314C2×C42—22no—
315C2×C42—48no—
316C2×C42—44no—
317C2×C42—44no—
318C2×C42—42no—
319C2×C42C22×C412yes—
320C2×C42C22×C412yes—
321C2×C42C2×D412yes—
322C2×C42C2×D412yes—
323C2×C42#4114yes—
324C4⋊C8—44no—
325C4⋊C8—44no—
326C22⋊C8—42no—
327C22⋊C8—42no—
328#95—28no—
329#95—22no—
330#95—22no—
331#95—22no—
332#95—22no—
333#95—22no—
334#95—22no—
335#95—22no—
336#95—21no—
337#95—21no—
338#95—22no—
339#95—22no—
340#95—44no—
341#95—44no—
342#95—42no—
343#95C22⋊C412yes—
344#95C22⋊C412yes—
345#95C22⋊C412yes—
346#95C22⋊C412yes—
347C24—22no—
348C24—22no—
349C24—22no—
350C24—22no—
351C24—42no—
352C24C2×C22⋊C411yescharacteristic
353C24C2×C22⋊C411yescharacteristic
354C24#12112yes—
355C24C22⋊Q812yes—
356C24C22×D411yescharacteristic
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—416no—
399C22×C4—48no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—48no—
403C22×C4—48no—
404C22×C4—416no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—42no—
413C22×C4—41no—
414C22×C4—41no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—82no—
418C22×C4C2×C22⋊C412yes—
419C22×C4C2×C22⋊C412yes—
420C22×C4C2×C22⋊C412yes—
421C22×C4C2×C22⋊C411yescharacteristic
422C22×C4C2×C22⋊C411yescharacteristic
423C22×C4C2×C22⋊C412yes—
424C22×C4C2×C4⋊C412yes—
425C22×C4C2×C4⋊C412yes—
426C22×C4#11712yes—
427C22×C4#11712yes—
428C22×C4#12112yes—
429C22×C4#12112yes—
430C22×C4#12112yes—
431C22×C4C22⋊Q812yes—
432C22×C4C22⋊Q812yes—
433C22×C4C22⋊Q812yes—
434C22×C4C23×C411yescharacteristic
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D411yescharacteristic
437C22×C4C22×Q811yescharacteristic
438C22×C4C2×#4111yescharacteristic
439C22×C4C2×#4111yescharacteristic
440C22×C4C2×#4112yes—
441C22×C4C2×#4112yes—
442C22×C4H(5,𝔽2)12yes—
443C22×C42−1+412yes—
444M16—24no—
445M16—24no—
446M16—48no—
447M16—42no—
448M16—42no—
449M16—41no—
450M16—41no—
451M16—41no—
452M16—41no—
453M16—42no—
454M16—42no—
455C2×C8—24no—
456C2×C8—48no—
457C2×C8—42no—
458C2×C8—42no—
459C2×C8—41no—
460C2×C8—41no—
461C2×C8—42no—
462C2×C8—42no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—24no—
468C4⋊C4—24no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—416no—
474C4⋊C4—44no—
475C4⋊C4—44no—
476C4⋊C4—82no—
477C4⋊C4—82no—
478C4⋊C4C2×C4⋊C412yes—
479C4⋊C4C2×C4⋊C412yes—
480C4⋊C4C2×C4⋊C412yes—
481C4⋊C4C2×C4⋊C412yes—
482C22⋊C4—22no—
483C22⋊C4—22no—
484C22⋊C4—22no—
485C22⋊C4—22no—
486C22⋊C4—24no—
487C22⋊C4—24no—
488C22⋊C4—22no—
489C22⋊C4—22no—
490C22⋊C4—22no—
491C22⋊C4—24no—
492C22⋊C4—42no—
493C22⋊C4—48no—
494C22⋊C4—42no—
495C22⋊C4—42no—
496C22⋊C4—42no—
497C22⋊C4—42no—
498C22⋊C4—82no—
499C22⋊C4#10414yes—
500C42—22no—
501C42—22no—
502C42—22no—
503C42—22no—
504C42—22no—
505C42—22no—
506C42—22no—
507C42—22no—
508C42—22no—
509C42—22no—
510C42—22no—
511C42—22no—
512C42—24no—
513C42—24no—
514C42—24no—
515C42—24no—
516C42—48no—
517C42—44no—
518C42—42no—
519C42—42no—
520C42—42no—
521C42—42no—
522C42—44no—
523C42—41no—
524C42—41no—
525C42—41no—
526C42—41no—
527C42—42no—
528C42—42no—
529C42—82no—
530C42—82no—
531C23—22no—
532C23—21no—
533C23—21no—
534C23—22no—
535C23—22no—
536C23—24no—
537C23—24no—
538C23—22no—
539C23—21no—
540C23—21no—
541C23—22no—
542C23—22no—
543C23—22no—
544C23—24no—
545C23—42no—
546C23—42no—
547C23—44no—
548C23—44no—
549C23—44no—
550C23—44no—
551C23#38012yes—
552C23#38012yes—
553C23C2×#10412yes—
554C23#42112yes—
555C23C22×C22⋊C411yescharacteristic
556C23C2×#12111yescharacteristic
557C23C2×C22⋊Q811yescharacteristic
558C23#54612yes—
559C23#54712yes—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—28no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—432no—
601C2×C4—48no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—82no—
620C2×C4—82no—
621C2×C4—84no—
622C2×C4—81no—
623C2×C4—81no—
624C2×C4—82no—
625C2×C4—82no—
626C2×C4—81no—
627C2×C4—81no—
628C2×C4#38012yes—
629C2×C4#38012yes—
630C2×C4C2×#10414yes—
631C2×C4C22×C4⋊C411yescharacteristic
632C2×C4C2×#11711yescharacteristic
633C2×C4C4×#4112yes—
634C2×C4#52012yes—
635C2×C4C2×#12111yescharacteristic
636C2×C4C2×C22⋊Q811yescharacteristic
637C2×C4#52511yescharacteristic
638C2×C4#52511yescharacteristic
639C2×C4#53312yes—
640C2×C4#53511yescharacteristic
641C2×C4#53611yescharacteristic
642C2×C4#54412yes—
643C2×C4#54612yes—
644C2×C4#54712yes—
645C2×C4#55112yes—
646C2×C4#55112yes—
647C2×C4#55212yes—
648C2×C4#55212yes—
649C8—44no—
650C8—81no—
651C8—81no—
652C22—22no—
653C22—21no—
654C22—21no—
655C22—21no—
656C22—22no—
657C22—21no—
658C22—22no—
659C22—22no—
660C22—24no—
661C22—22no—
662C22—22no—
663C22—24no—
664C22—22no—
665C22—22no—
666C22—22no—
667C22—44no—
668C22—44no—
669C22—48no—
670C22—42no—
671C22—42no—
672C22#180614yes—
673C22C2×#38011yescharacteristic
674C22#233912yes—
675C22#241111yescharacteristic
676C22#241311yescharacteristic
677C22#243812yes—
678C22C22×#10411yescharacteristic
679C22C2×#42111yescharacteristic
680C22#290112yes—
681C4—22no—
682C4—22no—
683C4—24no—
684C4—48no—
685C4—42no—
686C4—42no—
687C4—42no—
688C4—42no—
689C4—42no—
690C4—42no—
691C4—42no—
692C4—82no—
693C4—82no—
694C4—82no—
695C4—82no—
696C4#229611yescharacteristic
697C4#231711yescharacteristic
698C4#234312yes—
699C4#241111yescharacteristic
700C4#241311yescharacteristic
701C4#243812yes—
702C2—21no—
703C2—21no—
704C2—24no—
705C2—22no—
706C2—42no—
707C2#1396211yescharacteristic
708C2C2×#180611yescharacteristic
709C2#2065511yescharacteristic
710C2#2250512yes—
711C2#2250612yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C22×C22⋊C4
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #11226C2semidirect
7. #11209C2semidirect
8. #10852C2semidirect
9. #10808C2semidirect
17. #2347C4semidirect
18. #2346C4semidirect
19. #2345C4semidirect
20. #2345C4semidirect
21. #2344C4semidirect
22. #2344C4semidirect
104. #389C2×C4semidirect
105. #389C2×C4semidirect
106. #389C2×C4semidirect
107. #389C2×C4semidirect
135. C4×C4⋊C4C2×C4semidirect
136. C4×C4⋊C4C2×C4semidirect
137. C4×C4⋊C4C2×C4semidirect
138. C4×C4⋊C4C2×C4semidirect
240. #117C4⋊C4semidirect
241. #117C4⋊C4semidirect
257. C2×C4⋊C4C4⋊C4semidirect
258. C2×C4⋊C4C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#341092
#208372
C2×#18262
C22×#3392
C4×#3852
#112262
#112092
#108522
#108082
#107322
#107192