Finite Group Databasebeta

← #367037

Subgroups — #367037

Order 512.

1694 conjugacy classes of subgroups.

The subgroup lattice has 705 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2546C211yesmaximal, characteristic
2C2×#1981C211yesmaximal, characteristic
3C2×#1981C212yesmaximal
4C2×#1956C211yesmaximal, characteristic
5C2×#1939C211yesmaximal, characteristic
6C2×#1803C211yesmaximal, characteristic
7C2×C4×#103C211yesmaximal, characteristic
8C4⋊C42C211yesmaximal, characteristic
9#11077C212yesmaximal
10#11010C212yesmaximal
11#10932C212yesmaximal
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4⋊Q8C2212yes—
14C22×C4×Q8C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16C22×C4⋊C8C2212yes—
17C22×#103C2211yescharacteristic
18C22×#103C2212yes—
19C22×#103C2212yes—
20C22×#103C2212yes—
21C22×#103C2212yes—
22C22×#103C2211yescharacteristic
23C22×#103C2212yes—
24C22×C4×C8C2211yescharacteristic
25#2546—24no—
26#2546C418yescomplemented
27C2×#398—22no—
28C2×#391C2212yes—
29C2×#389—21no—
30C2×#389—21no—
31C2×#384C2211yescharacteristic
32C2×C4×C4⋊C4—21no—
33C2×C4×C4⋊C4C2212yes—
34C2×C4×C4⋊C4C2211yescharacteristic
35C2×C4×C4⋊C4C2211yescharacteristic
36#1981—28no—
37#1981—24no—
38#1956—24no—
39#1939—24no—
40#1803—24no—
41C4×#103—24no—
42C2×#340C2211yescharacteristic
43C2×#340C2212yes—
44C2×#340C2211yescharacteristic
45C2×#336C2212yes—
46#1436—22no—
47C4⋊#95—21no—
48C4⋊#95C2212yes—
49C23×Q8C2312yes—
50C23×Q8C2311yescharacteristic
51C23×C8C2312yes—
52C23×C8C2311yescharacteristic
53C23×C8C2311yescharacteristic
54C2×C4⋊Q8—24no—
55C2×C4⋊Q8—24no—
56C2×C4⋊Q8—24no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8—22no—
60C2×C4⋊Q8—22no—
61C2×C4⋊Q8—22no—
62C2×C4⋊Q8—42no—
63C2×C4⋊Q8—42no—
64C2×C4⋊Q8C2×C414yes—
65C2×C4⋊Q8C2×C418yes—
66C2×C4×Q8—22no—
67C2×C4×Q8—22no—
68C2×C4×Q8—22no—
69C2×C4×Q8—22no—
70C2×C4×Q8—22no—
71C2×C4×Q8C2×C414yes—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—22no—
75C22×C4⋊C4—22no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4—21no—
78C22×C4⋊C4—21no—
79C22×C4⋊C4D411yescharacteristic
80C22×C4⋊C4D411yescharacteristic
81C22×C4⋊C4C2311yescharacteristic
82C22×C4⋊C4C2311yescharacteristic
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2311yescharacteristic
85C22×C42—21no—
86C22×C42—21no—
87C22×C42—22no—
88C22×C42C2312yes—
89C22×C42C2311yescharacteristic
90C2×C4⋊C8—24no—
91C2×C4⋊C8—22no—
92C2×C4⋊C8—24no—
93C2×C4⋊C8—22no—
94C2×C4⋊C8—22no—
95C2×C4⋊C8—42no—
96C2×#103—22no—
97C2×#103—22no—
98C2×#103—22no—
99C2×#103—22no—
100C2×#103—22no—
101C2×#103—24no—
102C2×#103—24no—
103C2×#103—24no—
104C2×#103—24no—
105C2×#103—24no—
106C2×#103—24no—
107C2×#103—24no—
108C2×#103—24no—
109C2×#103—24no—
110C2×#103—24no—
111C2×#103—42no—
112C2×#103—42no—
113C2×#103—42no—
114C2×#103—42no—
115C2×#103—42no—
116C2×#103D414yes—
117C2×C4×C8—22no—
118C2×C4×C8—22no—
119C2×C4×C8—22no—
120#398—216no—
121#391—24no—
122#391C2×C418yes—
123#389—28no—
124#389—28no—
125#384—22no—
126#384D414yes—
127C4×C4⋊C4—22no—
128C4×C4⋊C4—24no—
129C4×C4⋊C4—42no—
130C4×C4⋊C4—44no—
131C4×C4⋊C4C2×C414yes—
132C4×C4⋊C4D414yes—
133C4×C4⋊C4D414yes—
134C2×#95—22no—
135C2×#95—22no—
136C2×#95—21no—
137C2×#95—22no—
138C2×#95—22no—
139C2×#95—22no—
140C2×#95D412yes—
141#340—24no—
142#340—22no—
143#340—22no—
144#336—24no—
145C22×Q8—21no—
146C22×Q8—22no—
147C22×Q8—21no—
148C22×Q8—21no—
149C22×Q8—21no—
150C22×Q8—21no—
151C22×Q8—22no—
152C22×Q8—22no—
153C22×Q8—22no—
154C22×Q8—21no—
155C22×Q8—21no—
156C22×Q8—21no—
157C22×Q8—21no—
158C22×Q8—22no—
159C22×Q8—22no—
160C22×Q8—22no—
161C22×Q8—22no—
162C22×Q8—22no—
163C22×Q8—21no—
164C22×Q8—22no—
165C22×Q8—22no—
166C22×Q8—22no—
167C22×Q8—21no—
168C22×Q8—22no—
169C22×Q8C22×C414yes—
170C22×Q8C22×C412yes—
171C22×Q8C2×D412yes—
172C22×Q8#4112yes—
173C23×C4—21no—
174C23×C4—21no—
175C23×C4—22no—
176C23×C4—22no—
177C23×C4—21no—
178C23×C4—21no—
179C23×C4C2×D412yes—
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D412yes—
182C23×C4C2×D411yescharacteristic
183C23×C4C2411yescharacteristic
184C22×C8—21no—
185C22×C8—21no—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—21no—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—22no—
197C22×C8—22no—
198C22×C8—22no—
199C22×C8—22no—
200C22×C8—22no—
201C22×C8—22no—
202C22×C8—22no—
203C22×C8C2×D412yes—
204C22×C8#4112yes—
205C4⋊Q8—216no—
206C4⋊Q8—28no—
207C4⋊Q8—28no—
208C4⋊Q8—48no—
209C4⋊Q8—48no—
210C4⋊Q8—48no—
211C4⋊Q8—44no—
212C4⋊Q8—44no—
213C4×Q8—28no—
214C4×Q8—28no—
215C4×Q8—44no—
216C4×Q8—44no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—81no—
257C2×C4⋊C4—81no—
258C2×C4⋊C4C22⋊C414yes—
259C2×C4⋊C4C22⋊C414yes—
260C2×C4⋊C4SD1614yes—
261C2×C4⋊C4Q1614yes—
262C2×C4⋊C4C22×C412yes—
263C2×C4⋊C4C2×D412yes—
264C2×C4⋊C4C2×D414yes—
265C2×C4⋊C4C2×D412yes—
266C2×C4⋊C4C2×D412yes—
267C2×C4⋊C4#4112yes—
268C2×C42—22no—
269C2×C42—22no—
270C2×C42—22no—
271C2×C42—22no—
272C2×C42—24no—
273C2×C42—24no—
274C2×C42—42no—
275C2×C42—42no—
276C2×C42—42no—
277C2×C42—42no—
278C2×C42—44no—
279C2×C42—44no—
280C2×C42—82no—
281C2×C42C22×C414yes—
282C2×C42C22×C412yes—
283C2×C42C2×D414yes—
284C2×C42C2×D412yes—
285C2×C42C2×D412yes—
286C2×C42C2×D412yes—
287C2×C42#4112yes—
288C2×C42#4112yes—
289C4⋊C8—48no—
290C4⋊C8—44no—
291#103—28no—
292#103—28no—
293#103—48no—
294#103—48no—
295#103—48no—
296#103—48no—
297#103—48no—
298C4×C8—44no—
299#95—28no—
300#95—28no—
301#95—24no—
302#95—44no—
303#95—44no—
304#95—82no—
305#95C22⋊C418yes—
306C24C22×D411yescharacteristic
307C2×Q8—22no—
308C2×Q8—22no—
309C2×Q8—22no—
310C2×Q8—22no—
311C2×Q8—22no—
312C2×Q8—24no—
313C2×Q8—24no—
314C2×Q8—24no—
315C2×Q8—22no—
316C2×Q8—24no—
317C2×Q8—24no—
318C2×Q8—22no—
319C2×Q8—24no—
320C2×Q8—22no—
321C2×Q8—22no—
322C2×Q8—22no—
323C2×Q8—22no—
324C2×Q8—22no—
325C2×Q8—22no—
326C2×Q8—22no—
327C2×Q8—22no—
328C2×Q8—41no—
329C2×Q8—42no—
330C2×Q8—41no—
331C2×Q8—42no—
332C2×Q8—41no—
333C2×Q8—41no—
334C2×Q8—41no—
335C2×Q8—41no—
336C2×Q8—42no—
337C2×Q8—41no—
338C2×Q8—42no—
339C2×Q8—41no—
340C2×Q8—42no—
341C2×Q8—41no—
342C2×Q8—41no—
343C2×Q8—42no—
344C2×Q8—41no—
345C2×Q8—41no—
346C2×Q8—42no—
347C2×Q8—42no—
348C2×Q8—42no—
349C2×Q8—42no—
350C2×Q8—42no—
351C2×Q8—42no—
352C2×Q8—42no—
353C2×Q8—42no—
354C2×Q8—42no—
355C2×Q8—42no—
356C2×Q8—42no—
357C2×Q8—42no—
358C2×Q8—42no—
359C2×Q8—42no—
360C2×Q8—42no—
361C2×Q8—42no—
362C2×Q8C4×D414yes—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—22no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—41no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—41no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—41no—
408C22×C4—42no—
409C22×C4—41no—
410C22×C4—42no—
411C22×C4—41no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—41no—
415C22×C4—41no—
416C22×C4—41no—
417C22×C4—41no—
418C22×C4—41no—
419C22×C4C2×C22⋊C414yes—
420C22×C4C2×C22⋊C412yes—
421C22×C4C2×C22⋊C414yes—
422C22×C4C2×C22⋊C412yes—
423C22×C4#12112yes—
424C22×C4#12112yes—
425C22×C4#12112yes—
426C22×C4#12112yes—
427C22×C4C2×SD1614yes—
428C22×C4C2×SD1612yes—
429C22×C4C2×Q1614yes—
430C22×C4C2×Q1612yes—
431C22×C4#13712yes—
432C22×C4#13712yes—
433C22×C4C23×C411yescharacteristic
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D411yescharacteristic
438C22×C4C2×#4111yescharacteristic
439C22×C4C2×#4111yescharacteristic
440C22×C4C2×#4111yescharacteristic
441C22×C4C2×#4111yescharacteristic
442C22×C4C2×#4111yescharacteristic
443C22×C4H(5,𝔽2)11yescharacteristic
444C22×C4H(5,𝔽2)11yescharacteristic
445C22×C4H(5,𝔽2)11yescharacteristic
446C22×C42−1+411yescharacteristic
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—41no—
454C2×C8—41no—
455C2×C8—41no—
456C2×C8—41no—
457C2×C8—41no—
458C2×C8—41no—
459C2×C8—41no—
460C2×C8—41no—
461C2×C8—42no—
462C2×C8—42no—
463C2×C8—42no—
464C2×C8—42no—
465C2×C8—42no—
466C2×C8—42no—
467C2×C8—42no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—28no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—24no—
477C4⋊C4—24no—
478C4⋊C4—416no—
479C4⋊C4—416no—
480C4⋊C4—41no—
481C4⋊C4—41no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—41no—
485C4⋊C4—41no—
486C4⋊C4—48no—
487C4⋊C4—48no—
488C4⋊C4—48no—
489C4⋊C4—48no—
490C4⋊C4—84no—
491C4⋊C4#103116yescomplemented
492C4⋊C4#12114yes—
493C42—24no—
494C42—24no—
495C42—24no—
496C42—24no—
497C42—48no—
498C42—48no—
499C42—88no—
500C42C4×D414yes—
501C42C4×D414yes—
502C42#12414yes—
503C42#12714yes—
504C23C22×C22⋊C411yescharacteristic
505C23C2×#12111yescharacteristic
506C23C2×#12111yescharacteristic
507C23D4211yescharacteristic
508C23#54611yescharacteristic
509C23#54811yescharacteristic
510C23#55011yescharacteristic
511C23C22×SD1611yescharacteristic
512C23C22×Q1611yescharacteristic
513C23C2×#13711yescharacteristic
514C23C2×#13711yescharacteristic
515C23#57711yescharacteristic
516C23#57711yescharacteristic
517C23#57811yescharacteristic
518C23#57811yescharacteristic
519Q8—24no—
520Q8—24no—
521Q8—42no—
522Q8—42no—
523Q8—42no—
524Q8—42no—
525Q8—42no—
526Q8—42no—
527Q8—44no—
528Q8—44no—
529Q8—44no—
530Q8—44no—
531Q8—82no—
532Q8—82no—
533Q8—82no—
534Q8—82no—
535C2×C4—24no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—22no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—81no—
581C2×C4—82no—
582C2×C4—81no—
583C2×C4—81no—
584C2×C4—82no—
585C2×C4—82no—
586C2×C4—82no—
587C2×C4—82no—
588C2×C4—82no—
589C2×C4—82no—
590C2×C4—82no—
591C2×C4—81no—
592C2×C4—81no—
593C2×C4—81no—
594C2×C4—81no—
595C2×C4—81no—
596C2×C4#39014yes—
597C2×C4#39014yes—
598C2×C4C2×#10318yes—
599C2×C4C2×#10314yes—
600C2×C4#41914yes—
601C2×C4#46114yes—
602C2×C4#46114yes—
603C2×C4#46214yes—
604C2×C4#46214yes—
605C2×C4C2×C4×D411yescharacteristic
606C2×C4C2×C4×D411yescharacteristic
607C2×C4C2×C4×D411yescharacteristic
608C2×C4C4×#4111yescharacteristic
609C2×C4#51811yescharacteristic
610C2×C4#52011yescharacteristic
611C2×C4C2×#12111yescharacteristic
612C2×C4#52511yescharacteristic
613C2×C4C2×#12411yescharacteristic
614C2×C4#52911yescharacteristic
615C2×C4C2×#12711yescharacteristic
616C2×C4#53211yescharacteristic
617C2×C4#53511yescharacteristic
618C2×C4#53711yescharacteristic
619C2×C4#54011yescharacteristic
620C2×C4#54211yescharacteristic
621C2×C4D4211yescharacteristic
622C2×C4D4211yescharacteristic
623C2×C4#54611yescharacteristic
624C2×C4#54611yescharacteristic
625C2×C4#54711yescharacteristic
626C2×C4#54811yescharacteristic
627C2×C4#54811yescharacteristic
628C2×C4#55011yescharacteristic
629C2×C4#55011yescharacteristic
630C2×C4#55311yescharacteristic
631C2×C4#55311yescharacteristic
632C2×C4#55511yescharacteristic
633C8—42no—
634C8—42no—
635C8—82no—
636C22C2×#39011yescharacteristic
637C22D4×C22⋊C411yescharacteristic
638C22#234111yescharacteristic
639C22#241411yescharacteristic
640C22#241611yescharacteristic
641C22#242611yescharacteristic
642C22#242811yescharacteristic
643C22C22×#10311yescharacteristic
644C22C2×#41911yescharacteristic
645C22#289811yescharacteristic
646C22#289811yescharacteristic
647C22C2×#46111yescharacteristic
648C22C2×#46111yescharacteristic
649C22C2×#46211yescharacteristic
650C22C2×#46211yescharacteristic
651C22#304211yescharacteristic
652C22#304411yescharacteristic
653C22#304611yescharacteristic
654C22#304611yescharacteristic
655C22#327611yescharacteristic
656C22#327811yescharacteristic
657C22D4×SD1611yescharacteristic
658C22D4×Q1611yescharacteristic
659C22#329811yescharacteristic
660C22#329911yescharacteristic
661C22#332811yescharacteristic
662C22#332911yescharacteristic
663C22#335911yescharacteristic
664C22#336311yescharacteristic
665C22#337211yescharacteristic
666C22#337411yescharacteristic
667C22#339211yescharacteristic
668C22#339511yescharacteristic
669C22#339811yescharacteristic
670C22#340011yescharacteristic
671C4—24no—
672C4—28no—
673C4—44no—
674C4—44no—
675C4—44no—
676C4—82no—
677C4—84no—
678C4—82no—
679C4—84no—
680C4#198118yescomplemented
681C4C4×#12111yescharacteristic
682C4#232411yescharacteristic
683C4#236111yescharacteristic
684C4#236311yescharacteristic
685C4#249011yescharacteristic
686C4#249111yescharacteristic
687C4#254011yescharacteristic
688C4#254111yescharacteristic
689C2#1399811yescharacteristic
690C2C2×#198111yescharacteristic
691C2#2126211yescharacteristic
692C2D4×#10311yescharacteristic
693C2#2243011yescharacteristic
694C2#2254211yescharacteristic
695C2#2254711yescharacteristic
696C2#2463011yescharacteristic
697C2#2463211yescharacteristic
698C2#2473011yescharacteristic
699C2#2473211yescharacteristic
700C2#2527611yescharacteristic
701C2#2527811yescharacteristic
702C2#2536811yescharacteristic
703C2#2537011yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
26. #2546C4semidirect
491. C4⋊C4#103semidirect
680. C4#1981semidirect

Maximal subgroups

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

subgroupindex
C2×#25462
C2×#19812
C2×#19562
C2×#19392
C2×#18032
C2×C4×#1032
C4⋊C422
#110772
#110102
#109322