Finite Group Databasebeta

← #205040

Subgroups — #205040

Order 512.

1465 conjugacy classes of subgroups.

The subgroup lattice has 720 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35306C211yesmaximal, characteristic
2#20628C211yesmaximal, characteristic, complemented
3C2×#1516C211yesmaximal, characteristic
4C2×#1511C211yesmaximal, characteristic, complemented
5C2×#1504C211yesmaximal, characteristic
6C22×#328C211yesmaximal, characteristic, complemented
7#13799C211yesmaximal, characteristic, complemented
8#9716C212yesmaximal, complemented
9#9714C212yesmaximal, complemented
10#9713C212yesmaximal, complemented
11#9711C212yesmaximal, complemented
12C2×#536C411yescharacteristic, complemented
13C2×#525C2211yescharacteristic
14C22×C22⋊Q8C2211yescharacteristic
15C22×#117C2211yescharacteristic
16C2×#424—21no—
17C2×#407C2211yescharacteristic
18C22×C22⋊C8C2211yescharacteristic
19#2857C2212yes—
20#2850C2212yes—
21#2621C412yescomplemented
22#2620C412yescomplemented
23#2417—22no—
24#2415—22no—
25#2413—21no—
26#2413—21no—
27#2412—21no—
28#2412—21no—
29#2317C411yescharacteristic, complemented
30#2311C411yescharacteristic, complemented
31#2310C411yescharacteristic, complemented
32C2×#381C2211yescharacteristic
33C2×C4×C22⋊C4—21no—
34C22×#95C2211yescharacteristic
35#1798—22no—
36#1797C2212yes—
37#1797C2212yes—
38#1729—21no—
39#1516—24no—
40#1511—24no—
41#1504—24no—
42C2×#328—24no—
43C2×#328C2212yes—
44C2×#328C2212yes—
45C2×#328C2211yescharacteristic
46C2×#328C2211yescharacteristic
47C2×#328C2212yes—
48C2×#328C2212yes—
49C2×#328C2212yes—
50C2×#327C2211yescharacteristic
51C2×#327C2211yescharacteristic
52C2×#327C2212yes—
53#1436—22no—
54#1434C2212yes—
55C22⋊#95—21no—
56C4×#95C2212yes—
57C22×#41C2×C411yescharacteristic
58C23×Q8C2×C411yescharacteristic
59C24×C4—21no—
60C24×C4C2311yescharacteristic
61C22×M16—21no—
62C23×C8—21no—
63#536—44no—
64C2×C4⋊Q8C2×C412yescomplemented
65C2×#124C2×C412yescomplemented
66#525—22no—
67#525—22no—
68#525—22no—
69#525—22no—
70C2×#123—21no—
71C2×#123—21no—
72C2×#123—21no—
73C2×C22⋊Q8—21no—
74C2×C22⋊Q8—21no—
75C2×C22⋊Q8—22no—
76C2×C22⋊Q8—22no—
77C2×C22⋊Q8—22no—
78C2×C22⋊Q8—22no—
79C2×C22⋊Q8—22no—
80C2×C22⋊Q8C2×C412yes—
81C2×C22⋊Q8C2×C411yescharacteristic
82C2×C22⋊Q8C2312yes—
83C2×C22⋊Q8C2311yescharacteristic
84C2×#121C2×C411yescharacteristic
85C2×#121C2311yescharacteristic
86C2×#120—21no—
87C2×C4×D4—21no—
88C2×C4×D4—21no—
89C2×#117—44no—
90C2×#117C2×C411yescharacteristic
91C2×#117C2×C411yescharacteristic
92C2×#117C2×C411yescharacteristic
93C2×#117C2×C412yes—
94C2×#117D411yescharacteristic
95C2×#117D412yes—
96C2×#117D411yescharacteristic
97C22×C4⋊C4—21no—
98C22×C4⋊C4C2×C411yescharacteristic
99C22×C4⋊C4D412yes—
100C22×C22⋊C4—21no—
101C22×C22⋊C4—21no—
102C22×C22⋊C4D411yescharacteristic
103C22×C22⋊C4D411yescharacteristic
104C22×C42—22no—
105C22×C42C2312yes—
106#424—44no—
107C2×C4⋊C8—22no—
108C2×C4⋊C8—22no—
109#407—22no—
110#407—22no—
111C2×C22⋊C8—22no—
112C2×C22⋊C8—22no—
113C2×C22⋊C8—21no—
114C2×C22⋊C8—21no—
115C2×C22⋊C8C2312yes—
116C2×C22⋊C8C2311yescharacteristic
117C2×C22⋊C8C2311yescharacteristic
118#398—22no—
119#398—22no—
120#397—22no—
121#397—22no—
122#396—22no—
123#396—22no—
124#394—22no—
125#394—22no—
126#391—22no—
127#391C2×C412yescomplemented
128#391C2×C412yescomplemented
129#391C2×C412yescomplemented
130#390C2×C412yescomplemented
131#389—22no—
132#389—22no—
133#389—22no—
134#388—22no—
135#388—22no—
136#386—22no—
137#386—22no—
138#386—21no—
139#386—21no—
140#385—21no—
141#385—21no—
142#384C2×C412yescomplemented
143#382C2×C412yescomplemented
144#381—22no—
145#381—21no—
146#381—21no—
147#381D414yes—
148#380—22no—
149#380—22no—
150#380—22no—
151C4×C22⋊C4—44no—
152C2×#95—21no—
153C2×#95—21no—
154C2×#95—22no—
155C2×#95—22no—
156C2×#95—22no—
157C2×#95—22no—
158C2×#95D414yes—
159C2×#95C2311yescharacteristic
160C2×#95C2312yes—
161C2×#95C2311yescharacteristic
162#336—22no—
163#336—22no—
164#328—24no—
165#328—24no—
166#328—22no—
167#328—22no—
168#328—24no—
169#328—24no—
170#328—24no—
171#327—22no—
172#327—22no—
173#327—24no—
174C25C2×D411yescharacteristic
175C2×#41—22no—
176C2×#41—22no—
177C2×#41—42no—
178C2×#41—42no—
179C22×Q8—21no—
180C22×Q8—22no—
181C22×Q8—22no—
182C22×Q8—42no—
183C22×Q8—42no—
184C22×Q8C22×C412yes—
185C22×Q8C22×C411yescharacteristic
186C22×D4—21no—
187C22×D4—21no—
188C22×D4—21no—
189C22×D4C22×C411yescharacteristic
190C23×C4—21no—
191C23×C4—21no—
192C23×C4—22no—
193C23×C4—21no—
194C23×C4—24no—
195C23×C4—21no—
196C23×C4—21no—
197C23×C4—42no—
198C23×C4—42no—
199C23×C4—44no—
200C23×C4—42no—
201C23×C4—42no—
202C23×C4—82no—
203C23×C4C22×C411yescharacteristic
204C23×C4C22×C411yescharacteristic
205C23×C4C2×D412yes—
206C23×C4C2×D411yescharacteristic
207C23×C4C2×D412yes—
208C23×C4C2×D411yescharacteristic
209C23×C4C2×D411yescharacteristic
210C23×C4C2×D412yes—
211C23×C4C2×D412yes—
212C23×C4#4112yes—
213C23×C4#4112yes—
214C23×C4C2411yescharacteristic
215C2×M16—42no—
216C2×M16—42no—
217C2×M16—42no—
218C22×C8—22no—
219C22×C8—21no—
220C22×C8—21no—
221C22×C8—42no—
222C22×C8—42no—
223C22×C8—42no—
224C4⋊Q8—44no—
225#124—44no—
226#123—44no—
227#123—42no—
228#123—42no—
229#123—44no—
230C22⋊Q8—24no—
231C22⋊Q8—24no—
232C22⋊Q8—22no—
233C22⋊Q8—22no—
234C22⋊Q8—22no—
235C22⋊Q8—22no—
236C22⋊Q8—21no—
237C22⋊Q8—21no—
238C22⋊Q8—22no—
239C22⋊Q8—22no—
240C22⋊Q8—21no—
241C22⋊Q8—21no—
242C22⋊Q8—48no—
243C22⋊Q8—42no—
244C22⋊Q8—42no—
245C22⋊Q8—42no—
246C22⋊Q8—42no—
247#121—22no—
248#121—22no—
249#121—21no—
250#121—21no—
251#121—21no—
252#121—21no—
253#120—42no—
254#120—42no—
255C4×D4—42no—
256C4×D4—42no—
257C4×D4—42no—
258C4×D4—42no—
259#117—24no—
260#117—24no—
261#117—22no—
262#117—22no—
263#117—44no—
264#117—44no—
265#117—42no—
266#117—42no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—42no—
280C2×C4⋊C4—42no—
281C2×C4⋊C4C22⋊C412yes—
282C2×C4⋊C4C22⋊C412yes—
283C2×C4⋊C4C22⋊C412yes—
284C2×C4⋊C4C22⋊C412yes—
285C2×C4⋊C4C22×C412yes—
286C2×C4⋊C4C22×C411yescharacteristic
287C2×C4⋊C4C22×C411yescharacteristic
288C2×C22⋊C4—24no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—21no—
291C2×C22⋊C4—21no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—21no—
295C2×C22⋊C4—21no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—21no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—42no—
302C2×C22⋊C4—42no—
303C2×C22⋊C4—42no—
304C2×C22⋊C4—42no—
305C2×C22⋊C4—42no—
306C2×C22⋊C4C22⋊C411yescharacteristic
307C2×C22⋊C4C22⋊C411yescharacteristic
308C2×C22⋊C4C22⋊C411yescharacteristic
309C2×C22⋊C4C22⋊C412yes—
310C2×C22⋊C4C22⋊C412yes—
311C2×C22⋊C4C22⋊C411yescharacteristic
312C2×C42—21no—
313C2×C42—44no—
314C2×C42—44no—
315C2×C42—44no—
316C2×C42—82no—
317C2×C42C22×C412yes—
318C2×C42C22×C412yes—
319C2×C42C2×D412yes—
320C2×C42C2×D412yes—
321C4⋊C8—44no—
322C4⋊C8—44no—
323C22⋊C8—22no—
324C22⋊C8—21no—
325C22⋊C8—21no—
326C22⋊C8—22no—
327C22⋊C8—21no—
328C22⋊C8—21no—
329C22⋊C8—42no—
330C22⋊C8—42no—
331#95—22no—
332#95—22no—
333#95—22no—
334#95—22no—
335#95—22no—
336#95—22no—
337#95—22no—
338#95—22no—
339#95—24no—
340#95—24no—
341#95—21no—
342#95—21no—
343#95—22no—
344#95—44no—
345#95—44no—
346#95—82no—
347#95C2×D412yes—
348#95C2×D412yes—
349#95C2×D414yes—
350C24—22no—
351C24—22no—
352C24—22no—
353C24—22no—
354C24—21no—
355C24—42no—
356C24C2×C22⋊C411yescharacteristic
357C24C2×C22⋊C411yescharacteristic
358C24#12112yes—
359C24#12112yes—
360C24C22×D411yescharacteristic
361#41—84no—
362#41—84no—
363C2×Q8—22no—
364C2×Q8—22no—
365C2×Q8—21no—
366C2×Q8—21no—
367C2×Q8—42no—
368C2×Q8—42no—
369C2×Q8—42no—
370C2×Q8—41no—
371C2×Q8—42no—
372C2×Q8—41no—
373C2×Q8—42no—
374C2×Q8—84no—
375C2×Q8—84no—
376C2×D4—21no—
377C2×D4—21no—
378C2×D4—41no—
379C2×D4—41no—
380C2×D4—41no—
381C2×D4—41no—
382C2×D4—42no—
383C2×D4—42no—
384C2×D4—42no—
385C2×D4—42no—
386C2×D4—42no—
387C2×D4—42no—
388C2×D4—82no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—41no—
408C22×C4—42no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—41no—
414C22×C4—41no—
415C22×C4—44no—
416C22×C4—44no—
417C22×C4—44no—
418C22×C4—44no—
419C22×C4—41no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—48no—
424C22×C4—42no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—42no—
428C22×C4—41no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—48no—
434C22×C4—41no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—82no—
445C22×C4—84no—
446C22×C4—88no—
447C22×C4—88no—
448C22×C4C2×C22⋊C412yes—
449C22×C4C2×C22⋊C412yes—
450C22×C4C2×C22⋊C412yes—
451C22×C4C2×C22⋊C411yescharacteristic
452C22×C4C2×C22⋊C411yescharacteristic
453C22×C4C2×C22⋊C412yes—
454C22×C4C4×D412yes—
455C22×C4C4×D412yes—
456C22×C4C4×D412yes—
457C22×C4C4×D412yes—
458C22×C4#12112yes—
459C22×C4#12112yes—
460C22×C4#12412yes—
461C22×C4#12412yes—
462C22×C4#12712yes—
463C22×C4#12712yes—
464C22×C4C23×C411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468C22×C4C2×#4111yescharacteristic
469C22×C4C2×#4111yescharacteristic
470M16—84no—
471C2×C8—42no—
472C2×C8—41no—
473C2×C8—41no—
474C2×C8—42no—
475C2×C8—41no—
476C2×C8—41no—
477C2×C8—42no—
478C2×C8—41no—
479C2×C8—41no—
480C2×C8—84no—
481C4⋊C4—24no—
482C4⋊C4—24no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—44no—
492C4⋊C4—42no—
493C4⋊C4—41no—
494C4⋊C4—42no—
495C4⋊C4—41no—
496C4⋊C4—42no—
497C4⋊C4—44no—
498C4⋊C4—42no—
499C4⋊C4—44no—
500C4⋊C4—82no—
501C4⋊C4—82no—
502C22⋊C4—22no—
503C22⋊C4—22no—
504C22⋊C4—22no—
505C22⋊C4—24no—
506C22⋊C4—42no—
507C22⋊C4—44no—
508C22⋊C4—48no—
509C22⋊C4—42no—
510C22⋊C4—42no—
511C22⋊C4—41no—
512C22⋊C4—41no—
513C22⋊C4—44no—
514C22⋊C4—42no—
515C22⋊C4—42no—
516C22⋊C4—41no—
517C22⋊C4—41no—
518C22⋊C4—42no—
519C22⋊C4—44no—
520C22⋊C4—82no—
521C22⋊C4—82no—
522C22⋊C4#9914yes—
523C42—24no—
524C42—24no—
525C42—42no—
526C42—42no—
527C42—42no—
528C42—88no—
529C23—21no—
530C23—21no—
531C23—22no—
532C23—22no—
533C23—22no—
534C23—21no—
535C23—21no—
536C23—22no—
537C23—22no—
538C23—24no—
539C23—24no—
540C23—22no—
541C23—22no—
542C23—24no—
543C23—42no—
544C23—42no—
545C23—44no—
546C23—44no—
547C23—44no—
548C23—44no—
549C23—41no—
550C23—41no—
551C23—42no—
552C23—42no—
553C23—41no—
554C23#39012yes—
555C23#39012yes—
556C23C2×#9912yes—
557C23#41012yes—
558C23C22×C22⋊C411yescharacteristic
559C23C2×#12111yescharacteristic
560C23C2×#12111yescharacteristic
561C23#57712yes—
562C23#57812yes—
563Q8—82no—
564Q8—82no—
565Q8—81no—
566Q8—81no—
567Q8—84no—
568D4—81no—
569D4—81no—
570D4—84no—
571D4—84no—
572D4—81no—
573D4—81no—
574D4—82no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—44no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—42no—
632C2×C4—41no—
633C2×C4—42no—
634C2×C4—84no—
635C2×C4—82no—
636C2×C4—82no—
637C2×C4—816no—
638C2×C4—82no—
639C2×C4—82no—
640C2×C4—82no—
641C2×C4—82no—
642C2×C4—82no—
643C2×C4—82no—
644C2×C4—84no—
645C2×C4—84no—
646C2×C4#39012yes—
647C2×C4#39012yes—
648C2×C4C2×#9914yes—
649C2×C4C2×C4×D411yescharacteristic
650C2×C4C2×C4×D411yescharacteristic
651C2×C4C2×#12411yescharacteristic
652C2×C4C2×#12711yescharacteristic
653C8—82no—
654C8—81no—
655C8—81no—
656C22—22no—
657C22—22no—
658C22—22no—
659C22—24no—
660C22—22no—
661C22—22no—
662C22—24no—
663C22—21no—
664C22—21no—
665C22—22no—
666C22—21no—
667C22—21no—
668C22—22no—
669C22—22no—
670C22—22no—
671C22—44no—
672C22—48no—
673C22—44no—
674C22—42no—
675C22—42no—
676C22—81no—
677C22—82no—
678C22—82no—
679C22—81no—
680C22—81no—
681C22#196414yes—
682C22C2×#39011yescharacteristic
683C22C22×#9911yescharacteristic
684C22C2×#41011yescharacteristic
685C22#289812yes—
686C22#290112yes—
687C22#304211yescharacteristic
688C22#304411yescharacteristic
689C22#304612yes—
690C4—24no—
691C4—21no—
692C4—22no—
693C4—21no—
694C4—21no—
695C4—21no—
696C4—22no—
697C4—44no—
698C4—42no—
699C4—42no—
700C4—42no—
701C4—84no—
702C4—82no—
703C4—81no—
704C4—82no—
705C4—81no—
706C4—81no—
707C4—82no—
708C2—21no—
709C2—22no—
710C2—24no—
711C2—21no—
712C2—42no—
713C2—161no—
714C2#1760812yes—
715C2#1760912yes—
716C2#2121011yescharacteristic
717C2C2×#196411yescharacteristic
718C2#2126211yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#525C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20628C2semidirect
4. C2×#1511C2semidirect
6. C22×#328C2semidirect
7. #13799C2semidirect
8. #9716C2semidirect
9. #9714C2semidirect
10. #9713C2semidirect
11. #9711C2semidirect
12. C2×#536C4semidirect
21. #2621C4semidirect
22. #2620C4semidirect
29. #2317C4semidirect
30. #2311C4semidirect
31. #2310C4semidirect
64. C2×C4⋊Q8C2×C4semidirect
65. C2×#124C2×C4semidirect
127. #391C2×C4semidirect
128. #391C2×C4semidirect
129. #391C2×C4semidirect
130. #390C2×C4semidirect
142. #384C2×C4semidirect
143. #382C2×C4semidirect

Maximal subgroups

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

subgroupindex
#353062
#206282
C2×#15162
C2×#15112
C2×#15042
C22×#3282
#137992
#97162
#97142
#97132
#97112