Finite Group Databasebeta

← #368363

Subgroups — #368363

Order 512.

1736 conjugacy classes of subgroups.

The subgroup lattice has 716 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2354C211yesmaximal, characteristic
2C2×#1954C211yesmaximal, characteristic
3C2×#1948C211yesmaximal, characteristic, complemented
4C2×#1854C212yesmaximal, complemented
5C2×C4×#107C211yesmaximal, characteristic, complemented
6#13906C211yesmaximal, characteristic
7#11096C212yesmaximal, complemented
8#11095C212yesmaximal, complemented
9#10849C211yesmaximal, characteristic, complemented
10#10841C211yesmaximal, characteristic, complemented
11#10731C212yesmaximal, complemented
12C22×C4×D4C2211yescharacteristic
13C22×#117C2212yes—
14C22×#107C2211yescharacteristic
15C22×#107C2211yescharacteristic
16C22×#106C2212yes—
17C22×C4⋊C8C2211yescharacteristic
18C22×C22⋊C8C2212yes—
19C22×C22⋊C8C2212yes—
20C22×C4×C8C2211yescharacteristic
21#2354—24no—
22#2354—24no—
23C2×#390—21no—
24C2×#390—21no—
25C2×#388—22no—
26C2×#384C2211yescharacteristic
27C2×#384C2211yescharacteristic
28C2×#380C2212yes—
29C2×#380C2212yes—
30C2×#380C2212yes—
31C2×C4×C4⋊C4—21no—
32C2×C4×C4⋊C4C2211yescharacteristic
33C2×C4×C4⋊C4C2211yescharacteristic
34#1954C418yescomplemented
35#1948—24no—
36#1854—28no—
37C4×#107—24no—
38C2×#340C2211yescharacteristic
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2211yescharacteristic
43C2×#340C2212yes—
44C2×#340C2212yes—
45#1439—22no—
46C4⋊#95—21no—
47C23×D4D411yescharacteristic
48C24×C4C2312yes—
49C23×C8C2311yescharacteristic
50C23×C8C2312yes—
51C23×C8C2311yescharacteristic
52C2×C4×D4—22no—
53C2×C4×D4—22no—
54C2×C4×D4—22no—
55C2×C4×D4—22no—
56C2×C4×D4C2×C414yes—
57C2×C4×D4Q814yes—
58C2×#117—24no—
59C2×#117—24no—
60C2×#117—24no—
61C2×#117—24no—
62C2×#117—42no—
63C2×#117—42no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4D411yescharacteristic
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2311yescharacteristic
75C22×C22⋊C4—22no—
76C22×C22⋊C4—22no—
77C22×C22⋊C4—22no—
78C22×C22⋊C4—22no—
79C22×C22⋊C4D412yes—
80C22×C42—22no—
81C22×C42—21no—
82C22×C42—21no—
83C22×C42C2311yescharacteristic
84C2×#107—22no—
85C2×#107—22no—
86C2×#107—22no—
87C2×#107—22no—
88C2×#107—22no—
89C2×#107D414yescomplemented
90C2×#106—24no—
91C2×#106—24no—
92C2×#106—42no—
93C2×C4⋊C8—22no—
94C2×C4⋊C8—22no—
95C2×C4⋊C8C2×C414yes—
96C2×C22⋊C8—24no—
97C2×C22⋊C8—24no—
98C2×C22⋊C8—42no—
99C2×C22⋊C8—42no—
100C2×C22⋊C8C2×C418yes—
101C2×C22⋊C8C2×C418yes—
102C2×C4×C8—22no—
103C2×C4×C8—22no—
104C2×C4×C8C2×C414yes—
105#390—28no—
106#390—44no—
107#388—216no—
108#384—22no—
109#384—22no—
110#384—22no—
111#384D414yes—
112#380—24no—
113#380—24no—
114#380—24no—
115#380—24no—
116#380—42no—
117#380—42no—
118C4×C4⋊C4—28no—
119C4×C4⋊C4—22no—
120C4×C4⋊C4—22no—
121C4×C4⋊C4—22no—
122C4×C4⋊C4D414yes—
123C2×#95—22no—
124C2×#95—21no—
125C2×#95—22no—
126C2×#95—22no—
127C2×#95—21no—
128#340—24no—
129#340—24no—
130#340—24no—
131#340—22no—
132#340—22no—
133#340—42no—
134#340—42no—
135C25C2×D412yes—
136C22×D4—22no—
137C22×D4—22no—
138C22×D4—22no—
139C22×D4—22no—
140C22×D4—22no—
141C22×D4—22no—
142C22×D4C4⋊C414yes—
143C23×C4—21no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—21no—
160C23×C4—22no—
161C23×C4C22×C414yes—
162C23×C4C2×D412yes—
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2×Q814yes—
166C23×C4C2411yescharacteristic
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—21no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—21no—
178C22×C8—22no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—22no—
182C22×C8—21no—
183C22×C8C22×C412yes—
184C22×C8C22×C414yes—
185C22×C8C22×C412yes—
186C22×C8C2×D412yes—
187C22×C8#4112yes—
188C4×D4—28no—
189C4×D4—28no—
190C4×D4—28no—
191C4×D4—28no—
192#117—48no—
193#117—48no—
194#117—48no—
195#117—48no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
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—24no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—81no—
236C2×C4⋊C4—81no—
237C2×C4⋊C4—81no—
238C2×C4⋊C4—81no—
239C2×C4⋊C4C4⋊C414yes—
240C2×C4⋊C4C2×D414yes—
241C2×C4⋊C4C2×D412yes—
242C2×C4⋊C4C2×D412yes—
243C2×C4⋊C4C2×D412yes—
244C2×C4⋊C4C2×D412yes—
245C2×C4⋊C4#4112yes—
246C2×C4⋊C4#4112yes—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—82no—
258C2×C22⋊C4—82no—
259C2×C22⋊C4—82no—
260C2×C22⋊C4—82no—
261C2×C22⋊C4C4⋊C418yes—
262C2×C42—28no—
263C2×C42—24no—
264C2×C42—44no—
265C2×C42—44no—
266C2×C42—42no—
267C2×C42—42no—
268C2×C42—42no—
269C2×C42—42no—
270C2×C42—42no—
271C2×C42C22×C412yes—
272C2×C42C2×D412yes—
273C2×C42C2×D412yes—
274C2×C42C2×Q812yes—
275C2×C42#4112yes—
276C2×C42#4112yes—
277#107—28no—
278#107—28no—
279#106—48no—
280C4⋊C8—28no—
281C22⋊C8—216no—
282C22⋊C8—48no—
283C4×C8—28no—
284#95—28no—
285#95—24no—
286#95—44no—
287#95—44no—
288#95—42no—
289C24—22no—
290C24—22no—
291C24—22no—
292C24—22no—
293C24—22no—
294C24—22no—
295C24—22no—
296C24—22no—
297C24—22no—
298C24—22no—
299C24—22no—
300C24—22no—
301C24—22no—
302C24—22no—
303C24C2×C4⋊C414yes—
304C24C22×D411yescharacteristic
305C2×D4—28no—
306C2×D4—28no—
307C2×D4—28no—
308C2×D4—44no—
309C2×D4—44no—
310C2×D4—44no—
311C2×D4—44no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
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—24no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—22no—
338C22×C4—24no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—42no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—42no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—42no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—42no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4C2×C4⋊C414yes—
393C22×C4C2×C4⋊C412yes—
394C22×C4C2×C4⋊C412yes—
395C22×C4#12012yes—
396C22×C4#12012yes—
397C22×C4#12312yes—
398C22×C4#12312yes—
399C22×C4#13512yes—
400C22×C4#13512yes—
401C22×C4AGL(1,ℤ/8)12yes—
402C22×C4#13712yes—
403C22×C4C23×C411yescharacteristic
404C22×C4C22×D411yescharacteristic
405C22×C4C22×D411yescharacteristic
406C22×C4C22×D411yescharacteristic
407C22×C4C22×Q811yescharacteristic
408C22×C4C2×#4111yescharacteristic
409C22×C4C2×#4111yescharacteristic
410C22×C4C2×#4111yescharacteristic
411C22×C4C2×#4111yescharacteristic
412C22×C4C2×#4111yescharacteristic
413C22×C4H(5,𝔽2)11yescharacteristic
414C22×C4H(5,𝔽2)11yescharacteristic
415C22×C4H(5,𝔽2)11yescharacteristic
416C22×C42−1+411yescharacteristic
417C2×C8—22no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—24no—
425C2×C8—24no—
426C2×C8—22no—
427C2×C8—24no—
428C2×C8—22no—
429C2×C8—41no—
430C2×C8—41no—
431C2×C8—41no—
432C2×C8—41no—
433C2×C8—42no—
434C2×C8—42no—
435C2×C8—42no—
436C2×C8—42no—
437C2×C8C4×D414yes—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—28no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—24no—
448C4⋊C4—24no—
449C4⋊C4—22no—
450C4⋊C4—42no—
451C4⋊C4—41no—
452C4⋊C4—42no—
453C4⋊C4—41no—
454C4⋊C4—41no—
455C4⋊C4—41no—
456C4⋊C4—41no—
457C4⋊C4—41no—
458C4⋊C4—48no—
459C4⋊C4—42no—
460C4⋊C4—42no—
461C4⋊C4—42no—
462C4⋊C4—42no—
463C4⋊C4—84no—
464C4⋊C4—84no—
465C4⋊C4—84no—
466C4⋊C4#12114yes—
467C22⋊C4—216no—
468C22⋊C4—416no—
469C22⋊C4—88no—
470C22⋊C4—88no—
471C22⋊C4—88no—
472C42—416no—
473C42—48no—
474C42—84no—
475C42C4×D414yes—
476C42C4×D414yes—
477C42C22⋊Q814yes—
478C42C22⋊Q814yes—
479C23—24no—
480C23—24no—
481C23—24no—
482C23—24no—
483C23—24no—
484C23—24no—
485C23—24no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23C22×C4⋊C411yescharacteristic
515C23C2×#12011yescharacteristic
516C23C2×#12311yescharacteristic
517C23D4211yescharacteristic
518C23#54611yescharacteristic
519C23#54611yescharacteristic
520C23#54711yescharacteristic
521C23C2×#13511yescharacteristic
522C23C2×#13511yescharacteristic
523C23C2×AGL(1,ℤ/8)11yescharacteristic
524C23C2×#13711yescharacteristic
525C23#57611yescharacteristic
526C23#57711yescharacteristic
527C23#57711yescharacteristic
528C23#57811yescharacteristic
529D4—416no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—81no—
571C2×C4—82no—
572C2×C4—81no—
573C2×C4—81no—
574C2×C4—81no—
575C2×C4—81no—
576C2×C4—81no—
577C2×C4—81no—
578C2×C4—81no—
579C2×C4—81no—
580C2×C4—82no—
581C2×C4—82no—
582C2×C4—81no—
583C2×C4—81no—
584C2×C4—82no—
585C2×C4—81no—
586C2×C4—82no—
587C2×C4—82no—
588C2×C4—82no—
589C2×C4—82no—
590C2×C4—82no—
591C2×C4—82no—
592C2×C4—82no—
593C2×C4#38514yes—
594C2×C4#38514yes—
595C2×C4#42714yes—
596C2×C4#42814yes—
597C2×C4#44914yes—
598C2×C4#45214yes—
599C2×C4#48214yes—
600C2×C4#48514yes—
601C2×C4C2×C4×D411yescharacteristic
602C2×C4C2×C4×D411yescharacteristic
603C2×C4C2×C4×D411yescharacteristic
604C2×C4C4×#4111yescharacteristic
605C2×C4#51811yescharacteristic
606C2×C4#52011yescharacteristic
607C2×C4C2×#12111yescharacteristic
608C2×C4C2×C22⋊Q811yescharacteristic
609C2×C4C2×C22⋊Q811yescharacteristic
610C2×C4#52511yescharacteristic
611C2×C4#52911yescharacteristic
612C2×C4#53211yescharacteristic
613C2×C4#53511yescharacteristic
614C2×C4#53711yescharacteristic
615C2×C4#54011yescharacteristic
616C2×C4#54211yescharacteristic
617C2×C4D4211yescharacteristic
618C2×C4#54611yescharacteristic
619C2×C4#54811yescharacteristic
620C2×C4D4×Q811yescharacteristic
621C2×C4#55011yescharacteristic
622C2×C4#55011yescharacteristic
623C2×C4#55411yescharacteristic
624C2×C4#55411yescharacteristic
625C2×C4#55411yescharacteristic
626C2×C4#55611yescharacteristic
627C2×C4#55911yescharacteristic
628C2×C4#55911yescharacteristic
629C8—24no—
630C8—42no—
631C8—44no—
632C22—44no—
633C22—44no—
634C22—44no—
635C22—44no—
636C22—44no—
637C22—44no—
638C22—44no—
639C22—82no—
640C22—82no—
641C22—82no—
642C22—82no—
643C22—82no—
644C22—82no—
645C22—82no—
646C22—82no—
647C22C2×#38511yescharacteristic
648C22D4×C4⋊C411yescharacteristic
649C22#234911yescharacteristic
650C22#240711yescharacteristic
651C22#240811yescharacteristic
652C22#244511yescharacteristic
653C22#244711yescharacteristic
654C22C2×#42711yescharacteristic
655C22C2×#42811yescharacteristic
656C22#291211yescharacteristic
657C22#291311yescharacteristic
658C22C2×#44911yescharacteristic
659C22C2×#45211yescharacteristic
660C22#301211yescharacteristic
661C22#301311yescharacteristic
662C22C2×#48211yescharacteristic
663C22C2×#48511yescharacteristic
664C22#309611yescharacteristic
665C22#309711yescharacteristic
666C22#327211yescharacteristic
667C22#327811yescharacteristic
668C22#328311yescharacteristic
669C22#328711yescharacteristic
670C22#330311yescharacteristic
671C22#330311yescharacteristic
672C22#330411yescharacteristic
673C22#330411yescharacteristic
674C22#332511yescharacteristic
675C22#332811yescharacteristic
676C22#333011yescharacteristic
677C22#333311yescharacteristic
678C22#334111yescharacteristic
679C22#334211yescharacteristic
680C22#335411yescharacteristic
681C22#335611yescharacteristic
682C4—24no—
683C4—44no—
684C4—44no—
685C4—82no—
686C4—82no—
687C4—82no—
688C4—84no—
689C4—162no—
690C4#185418yescomplemented
691C4C4×#12111yescharacteristic
692C4#232411yescharacteristic
693C4#236611yescharacteristic
694C4#236711yescharacteristic
695C4#245211yescharacteristic
696C4#245411yescharacteristic
697C4#253811yescharacteristic
698C4#258011yescharacteristic
699C2—84no—
700C2#1401811yescharacteristic
701C2C2×#185411yescharacteristic
702C2#2084911yescharacteristic
703C2#2265811yescharacteristic
704C2#2266311yescharacteristic
705C2#2270811yescharacteristic
706C2#2271311yescharacteristic
707C2#2411711yescharacteristic
708C2#2412311yescharacteristic
709C2#2423711yescharacteristic
710C2#2424211yescharacteristic
711C2#2623111yescharacteristic
712C2#2623711yescharacteristic
713C2#2629711yescharacteristic
714C2#2630311yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1948C2semidirect
4. C2×#1854C2semidirect
5. C2×C4×#107C2semidirect
7. #11096C2semidirect
8. #11095C2semidirect
9. #10849C2semidirect
10. #10841C2semidirect
11. #10731C2semidirect
34. #1954C4semidirect
89. C2×#107D4semidirect
690. C4#1854semidirect

Maximal subgroups

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

subgroupindex
C2×#23542
C2×#19542
C2×#19482
C2×#18542
C2×C4×#1072
#139062
#110962
#110952
#108492
#108412
#107312