Finite Group Databasebeta

← #392882

Subgroups — #392882

Order 512.

1710 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2274C211yesmaximal, characteristic
2C2×#1948C211yesmaximal, characteristic, complemented
3C2×#1947C211yesmaximal, characteristic, complemented
4C2×#1854C212yesmaximal
5C2×#1798C211yesmaximal, characteristic
6#14340C211yesmaximal, characteristic
7#10908C212yesmaximal, complemented
8#10907C212yesmaximal, complemented
9#10850C211yesmaximal, characteristic, complemented
10#10850C211yesmaximal, characteristic, complemented
11#10730C212yesmaximal
12C22×#117C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C22×#107C2212yes—
16C22×#106C2212yes—
17C22×C4⋊C8C2211yescharacteristic
18C22×C4⋊C8C2211yescharacteristic
19C22×C22⋊C8C2212yes—
20C2×#384C2211yescharacteristic
21C2×#382C2211yescharacteristic
22C2×#380C2212yes—
23#2274—24no—
24#2274—24no—
25C2×C4×C4⋊C4—21no—
26C2×C4×C4⋊C4—21no—
27C2×C4×C4⋊C4C2211yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29C2×C4×C4⋊C4C2211yescharacteristic
30C2×C4×C4⋊C4C2211yescharacteristic
31C2×C4×C22⋊C4—21no—
32C2×C4×C22⋊C4—21no—
33#1948—24no—
34#1947—24no—
35#1854—28no—
36#1798C418yescomplemented
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#336C2212yes—
44#1441—21no—
45#1441—21no—
46#1439—21no—
47#1439—21no—
48C4⋊#95C2212yes—
49C24×C4—21no—
50C24×C4C2311yescharacteristic
51C23×C8C2312yes—
52C23×C8C2312yes—
53C2×#117—22no—
54C2×#117—22no—
55C2×#117—22no—
56C2×#117—22no—
57C2×#117—22no—
58C2×#117—22no—
59C2×#117—22no—
60C2×#117—22no—
61C2×#117—22no—
62C2×#117C2×C414yes—
63C2×#117D414yes—
64C2×#117Q814yes—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4—22no—
70C22×C4⋊C4—22no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4D411yescharacteristic
74C22×C4⋊C4D411yescharacteristic
75C22×C4⋊C4Q814yes—
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2311yescharacteristic
79C22×C4⋊C4C2311yescharacteristic
80C22×C4⋊C4C2311yescharacteristic
81C22×C4⋊C4C2311yescharacteristic
82C22×C22⋊C4—21no—
83C22×C22⋊C4—22no—
84C22×C22⋊C4D411yescharacteristic
85C22×C22⋊C4D411yescharacteristic
86C22×C42—21no—
87C22×C42—21no—
88C22×C42—21no—
89C22×C42—21no—
90C22×C42—21no—
91C22×C42C2311yescharacteristic
92C22×C42C2311yescharacteristic
93C2×#107—24no—
94C2×#107—24no—
95C2×#107—24no—
96C2×#106—24no—
97C2×#106—24no—
98C2×#106—24no—
99C2×C4⋊C8—22no—
100C2×C4⋊C8—22no—
101C2×C4⋊C8—22no—
102C2×C4⋊C8—22no—
103C2×C4⋊C8C2×C414yescomplemented
104C2×C4⋊C8C2×C414yescomplemented
105C2×C22⋊C8—24no—
106C2×C22⋊C8—24no—
107C2×C22⋊C8C2×C418yes—
108#384—22no—
109#384—22no—
110#382—22no—
111#382—22no—
112#380—24no—
113#380—24no—
114C4×C4⋊C4—28no—
115C4×C4⋊C4—28no—
116C4×C4⋊C4—22no—
117C4×C4⋊C4—22no—
118C4×C4⋊C4—22no—
119C4×C4⋊C4—22no—
120C4×C4⋊C4—22no—
121C4×C4⋊C4—22no—
122C4×C4⋊C4D414yes—
123C4×C4⋊C4D414yes—
124C4×C22⋊C4—28no—
125C4×C22⋊C4—28no—
126C2×#95—21no—
127C2×#95—21no—
128C2×#95—21no—
129C2×#95—21no—
130C2×#95—21no—
131C2×#95—21no—
132C2×#95—22no—
133C2×#95—22no—
134C2×#95—22no—
135#340—24no—
136#340—24no—
137#340—24no—
138#340—24no—
139#340—24no—
140#340—24no—
141#336C2×C418yescomplemented
142C25C2×D411yescharacteristic
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—24no—
156C23×C4—24no—
157C23×C4—24no—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4—82no—
161C23×C4—82no—
162C23×C4C22×C412yes—
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D412yes—
169C23×C4C2×Q812yes—
170C23×C4C2×Q812yes—
171C23×C4#4112yes—
172C23×C4#4112yes—
173C23×C4C2411yescharacteristic
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8C22×C414yes—
187C22×C8C22×C414yes—
188#117—28no—
189#117—28no—
190#117—28no—
191#117—28no—
192#117—28no—
193#117—28no—
194#117—28no—
195#117—28no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—28no—
198C2×C4⋊C4—28no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—28no—
218C2×C4⋊C4—28no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—82no—
234C2×C4⋊C4—82no—
235C2×C4⋊C4C4⋊C414yes—
236C2×C4⋊C4C4⋊C414yes—
237C2×C4⋊C4C2×D412yes—
238C2×C4⋊C4C2×D412yes—
239C2×C4⋊C4C2×D412yes—
240C2×C4⋊C4C2×D412yes—
241C2×C4⋊C4C2×D412yes—
242C2×C4⋊C4C2×D412yes—
243C2×C4⋊C4C2×Q812yes—
244C2×C4⋊C4C2×Q812yes—
245C2×C4⋊C4#4112yes—
246C2×C4⋊C4#4112yes—
247C2×C4⋊C4#4112yes—
248C2×C4⋊C4#4112yes—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—82no—
258C2×C22⋊C4—82no—
259C2×C22⋊C4C4⋊C414yes—
260C2×C22⋊C4C4⋊C414yes—
261C2×C42—24no—
262C2×C42—24no—
263C2×C42—24no—
264C2×C42—21no—
265C2×C42—21no—
266C2×C42—21no—
267C2×C42—21no—
268C2×C42—24no—
269C2×C42—24no—
270C2×C42—24no—
271C2×C42—24no—
272C2×C42—82no—
273C2×C42—82no—
274C2×C42—82no—
275C2×C42—82no—
276C2×C42C22×C412yes—
277C2×C42C22×C412yes—
278C2×C42C2×D412yes—
279C2×C42C2×D412yes—
280C2×C42C2×Q812yes—
281C2×C42C2×Q812yes—
282C2×C42#4112yes—
283C2×C42#4112yes—
284#107—48no—
285#106—48no—
286C4⋊C8—28no—
287C4⋊C8—28no—
288C22⋊C8—216no—
289#95—24no—
290#95—24no—
291#95—24no—
292#95—24no—
293#95—44no—
294#95—42no—
295#95—42no—
296#95—82no—
297#95—82no—
298C24—21no—
299C24—21no—
300C24—21no—
301C24—21no—
302C24—22no—
303C24—22no—
304C24—22no—
305C24—22no—
306C24C2×C4⋊C412yes—
307C24#12112yes—
308C24C22⋊Q812yes—
309C24C22×D411yescharacteristic
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—21no—
343C22×C4—21no—
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—21no—
353C22×C4—21no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—216no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
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—88no—
386C22×C4—88no—
387C22×C4—84no—
388C22×C4—84no—
389C22×C4C2×C4⋊C412yes—
390C22×C4C2×C4⋊C412yes—
391C22×C4C2×C4⋊C412yes—
392C22×C4C2×C4⋊C412yes—
393C22×C4C2×C4⋊C412yes—
394C22×C4C4×D414yes—
395C22×C4C4×Q814yes—
396C22×C4#12012yes—
397C22×C4#12012yes—
398C22×C4#12112yes—
399C22×C4C22⋊Q812yes—
400C22×C4#12312yes—
401C22×C4#12312yes—
402C22×C4#12514yes—
403C22×C4C4⋊Q814yes—
404C22×C4C23×C411yescharacteristic
405C22×C4C22×D411yescharacteristic
406C22×C4C22×D411yescharacteristic
407C22×C4C22×D411yescharacteristic
408C22×C4C22×Q811yescharacteristic
409C22×C4C22×Q811yescharacteristic
410C22×C4C2×#4111yescharacteristic
411C22×C4C2×#4111yescharacteristic
412C22×C4C2×#4111yescharacteristic
413C22×C4C2×#4111yescharacteristic
414C22×C4C2×#4111yescharacteristic
415C22×C4C2×#4111yescharacteristic
416C22×C4C2×#4111yescharacteristic
417C22×C4C2×#4111yescharacteristic
418C2×C8—24no—
419C2×C8—24no—
420C2×C8—24no—
421C2×C8—24no—
422C2×C8—24no—
423C2×C8—24no—
424C2×C8—42no—
425C2×C8—42no—
426C2×C8—42no—
427C2×C8—42no—
428C2×C8—42no—
429C2×C8—42no—
430C2×C8—42no—
431C2×C8—42no—
432C4⋊C4—216no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—28no—
446C4⋊C4—28no—
447C4⋊C4—42no—
448C4⋊C4—42no—
449C4⋊C4—42no—
450C4⋊C4—42no—
451C4⋊C4—42no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C4⋊C4—42no—
455C4⋊C4—84no—
456C4⋊C4—84no—
457C4⋊C4#12114yes—
458C4⋊C4#12114yes—
459C4⋊C4C22⋊Q814yes—
460C4⋊C4C22⋊Q814yes—
461C22⋊C4—216no—
462C22⋊C4—28no—
463C22⋊C4—28no—
464C22⋊C4—88no—
465C42—216no—
466C42—22no—
467C42—22no—
468C42—22no—
469C42—22no—
470C42—84no—
471C42—84no—
472C42—84no—
473C42—84no—
474C42C4×D414yes—
475C42C4×D414yes—
476C42C22⋊Q814yes—
477C42C22⋊Q814yes—
478C23—24no—
479C23—24no—
480C23—22no—
481C23—22no—
482C23—22no—
483C23—22no—
484C23—24no—
485C23—24no—
486C23—24no—
487C23—24no—
488C23—22no—
489C23—22no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—41no—
497C23—41no—
498C23—41no—
499C23—41no—
500C23#38914yes—
501C23C22×C4⋊C411yescharacteristic
502C23C2×#12011yescharacteristic
503C23C2×#12111yescharacteristic
504C23C2×C22⋊Q811yescharacteristic
505C23C2×#12311yescharacteristic
506C23#52511yescharacteristic
507C23#52511yescharacteristic
508C23#57612yes—
509C23#57712yes—
510C23#57712yes—
511C23#57812yes—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—41no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—41no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—41no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—816no—
565C2×C4—82no—
566C2×C4—82no—
567C2×C4—82no—
568C2×C4—82no—
569C2×C4—82no—
570C2×C4—82no—
571C2×C4—82no—
572C2×C4—82no—
573C2×C4—82no—
574C2×C4—82no—
575C2×C4—82no—
576C2×C4—82no—
577C2×C4—82no—
578C2×C4—82no—
579C2×C4#38514yes—
580C2×C4#38514yes—
581C2×C4#38914yes—
582C2×C4C2×C4×D411yescharacteristic
583C2×C4C2×C4×D411yescharacteristic
584C2×C4C2×C4×D411yescharacteristic
585C2×C4C2×C4×Q811yescharacteristic
586C2×C4C4×#4111yescharacteristic
587C2×C4C4×#4111yescharacteristic
588C2×C4C2×#12111yescharacteristic
589C2×C4C2×#12111yescharacteristic
590C2×C4C2×C22⋊Q811yescharacteristic
591C2×C4C2×C22⋊Q811yescharacteristic
592C2×C4C2×C22⋊Q811yescharacteristic
593C2×C4C2×C22⋊Q811yescharacteristic
594C2×C4#52511yescharacteristic
595C2×C4#52511yescharacteristic
596C2×C4#52511yescharacteristic
597C2×C4#52511yescharacteristic
598C2×C4C2×#12511yescharacteristic
599C2×C4#52911yescharacteristic
600C2×C4#52911yescharacteristic
601C2×C4#52911yescharacteristic
602C2×C4#52911yescharacteristic
603C2×C4C2×C4⋊Q811yescharacteristic
604C2×C4#53211yescharacteristic
605C2×C4#53211yescharacteristic
606C2×C4#53311yescharacteristic
607C2×C4#53311yescharacteristic
608C2×C4#53311yescharacteristic
609C2×C4#53311yescharacteristic
610C8—44no—
611C8—44no—
612C22—24no—
613C22—28no—
614C22—44no—
615C22—42no—
616C22—42no—
617C22—44no—
618C22—44no—
619C22—42no—
620C22—42no—
621C22—42no—
622C22—42no—
623C22C2×#38511yescharacteristic
624C22C2×#38911yescharacteristic
625C22#229611yescharacteristic
626C22#284711yescharacteristic
627C22#284811yescharacteristic
628C22#285411yescharacteristic
629C22#285511yescharacteristic
630C22#291212yes—
631C22#291312yes—
632C22#300811yescharacteristic
633C22#300911yescharacteristic
634C22#301011yescharacteristic
635C22#301111yescharacteristic
636C22#304512yes—
637C22#304612yes—
638C22#308112yes—
639C22#308212yes—
640C22#309812yes—
641C22#309912yes—
642C22#311812yes—
643C22#311912yes—
644C22#312012yes—
645C22#312112yes—
646C4—24no—
647C4—24no—
648C4—24no—
649C4—42no—
650C4—42no—
651C4—44no—
652C4—44no—
653C4—44no—
654C4—44no—
655C4—162no—
656C4—162no—
657C4C4×#12111yescharacteristic
658C4C4×#12111yescharacteristic
659C4C4×C22⋊Q811yescharacteristic
660C4C4×C22⋊Q811yescharacteristic
661C4#285111yescharacteristic
662C4#285111yescharacteristic
663C4#285711yescharacteristic
664C4#285711yescharacteristic
665C2—44no—
666C2#1469711yescharacteristic
667C2#2084711yescharacteristic
668C2#2084811yescharacteristic
669C2#2117512yes—
670C2#2136712yes—
671C2#3291311yescharacteristic
672C2#3291611yescharacteristic
673C2#3296211yescharacteristic
674C2#3296411yescharacteristic
675C2#3311512yes—
676C2#3315812yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1948C2semidirect
3. C2×#1947C2semidirect
7. #10908C2semidirect
8. #10907C2semidirect
9. #10850C2semidirect
10. #10850C2semidirect
36. #1798C4semidirect
103. C2×C4⋊C8C2×C4semidirect
104. C2×C4⋊C8C2×C4semidirect
141. #336C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#22742
C2×#19482
C2×#19472
C2×#18542
C2×#17982
#143402
#109082
#109072
#108502
#107302