Finite Group Databasebeta

← #364892

Subgroups — #364892

Order 512.

1816 conjugacy classes of subgroups.

The subgroup lattice has 707 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#462C211yesmaximal, characteristic
2C2×#2365C211yesmaximal, characteristic
3C2×#1956C211yesmaximal, characteristic
4C2×#1878C212yesmaximal
5C2×C4×#103C211yesmaximal, characteristic
6#13926C211yesmaximal, characteristic
7#11190C212yesmaximal
8#11024C212yesmaximal
9#10849C211yesmaximal, characteristic
10#10834C211yesmaximal, characteristic
11#10789C212yesmaximal
12C23×Q16C2212yes—
13C22×C4⋊Q8C2211yescharacteristic
14C22×C4×Q8C2211yescharacteristic
15C22×C4×Q8C2212yes—
16C2×#462—24no—
17C2×#462—24no—
18C2×#462C418yescomplemented
19C22×C4⋊C8C2211yescharacteristic
20C22×#103C2211yescharacteristic
21C22×#103C2212yes—
22C22×#103C2212yes—
23C22×#103C2212yes—
24C22×#103C2211yescharacteristic
25C22×C4×C8C2211yescharacteristic
26#2365—24no—
27#2365—24no—
28C2×#391C2212yes—
29C2×#391C2212yes—
30C2×#389—21no—
31C2×#387—22no—
32C2×#384C2211yescharacteristic
33C2×C4×C4⋊C4—21no—
34C2×C4×C4⋊C4C2211yescharacteristic
35C2×C4×C4⋊C4C2211yescharacteristic
36#1956—24no—
37#1878—28no—
38C4×#103—24no—
39C2×#340C2211yescharacteristic
40C2×#340C2212yes—
41C2×#340C2211yescharacteristic
42C2×#340C2212yes—
43C2×#336C2212yes—
44C2×#336C2212yes—
45#1441—22no—
46C4⋊#95—21no—
47C4⋊#95—21no—
48C4⋊#95C2212yes—
49C23×Q8C2312yes—
50C23×Q8C2311yescharacteristic
51C22×Q16—24no—
52C22×Q16—24no—
53C22×Q16—24no—
54C22×Q16—24no—
55C22×Q16—42no—
56C22×Q16—42no—
57C22×Q16C2×C418yes—
58C23×C8C2312yes—
59C23×C8C2311yescharacteristic
60C23×C8C2311yescharacteristic
61C2×C4⋊Q8—22no—
62C2×C4⋊Q8—22no—
63C2×C4⋊Q8—22no—
64C2×C4⋊Q8C2×C414yes—
65C2×C4⋊Q8D414yes—
66C2×C4⋊Q8D414yes—
67C2×C4×Q8—24no—
68C2×C4×Q8—24no—
69C2×C4×Q8—22no—
70C2×C4×Q8—24no—
71C2×C4×Q8—24no—
72C2×C4×Q8—22no—
73C2×C4×Q8—22no—
74C2×C4×Q8—22no—
75C2×C4×Q8—22no—
76C2×C4×Q8—42no—
77C2×C4×Q8—42no—
78C2×C4×Q8C2×C414yes—
79C22×C4⋊C4—21no—
80C22×C4⋊C4—21no—
81C22×C4⋊C4—21no—
82C22×C4⋊C4—22no—
83C22×C4⋊C4—21no—
84C22×C4⋊C4—21no—
85C22×C4⋊C4D411yescharacteristic
86C22×C4⋊C4D411yescharacteristic
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2311yescharacteristic
89C22×C4⋊C4C2311yescharacteristic
90C22×C4⋊C4C2311yescharacteristic
91C22×C42—21no—
92C22×C42—22no—
93C22×C42—21no—
94C22×C42C2312yes—
95C22×C42C2311yescharacteristic
96#462—232no—
97C2×C4⋊C8—22no—
98C2×C4⋊C8—22no—
99C2×C4⋊C8C2×C414yes—
100C2×#103—24no—
101C2×#103—24no—
102C2×#103—22no—
103C2×#103—22no—
104C2×#103—22no—
105C2×#103—22no—
106C2×#103—22no—
107C2×#103—24no—
108C2×#103—24no—
109C2×#103—24no—
110C2×#103—24no—
111C2×#103—42no—
112C2×#103—42no—
113C2×#103C2×C418yes—
114C2×#103D414yes—
115C2×C4×C8—22no—
116C2×C4×C8—22no—
117C2×C4×C8—22no—
118#391—24no—
119#391—24no—
120#391—24no—
121#391—42no—
122#389—28no—
123#387—216no—
124#384—22no—
125#384—22no—
126C4×C4⋊C4—28no—
127C4×C4⋊C4—22no—
128C4×C4⋊C4—22no—
129C4×C4⋊C4—22no—
130C4×C4⋊C4D414yes—
131C2×#95—22no—
132C2×#95—21no—
133C2×#95—22no—
134C2×#95—22no—
135C2×#95—22no—
136C2×#95—22no—
137C2×#95—22no—
138C2×#95D412yes—
139#340—24no—
140#340—22no—
141#340—22no—
142#340—42no—
143#336—24no—
144#336—42no—
145C22×Q8—22no—
146C22×Q8—21no—
147C22×Q8—22no—
148C22×Q8—22no—
149C22×Q8—21no—
150C22×Q8—21no—
151C22×Q8—21no—
152C22×Q8—21no—
153C22×Q8—22no—
154C22×Q8—21no—
155C22×Q8—21no—
156C22×Q8—22no—
157C22×Q8—22no—
158C22×Q8—21no—
159C22×Q8—21no—
160C22×Q8—22no—
161C22×Q8—22no—
162C22×Q8—22no—
163C22×Q8—21no—
164C22×Q8—21no—
165C22×Q8—22no—
166C22×Q8—22no—
167C22×Q8C22×C414yes—
168C22×Q8C22×C412yes—
169C22×Q8C2×D414yes—
170C22×Q8C2×D414yes—
171C22×Q8C2×D412yes—
172C22×Q8#4112yes—
173C23×C4—22no—
174C23×C4—21no—
175C23×C4—21no—
176C23×C4—21no—
177C23×C4—21no—
178C23×C4—22no—
179C23×C4C2×D412yes—
180C23×C4C2×D412yes—
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2411yescharacteristic
184C2×Q16—216no—
185C2×Q16—216no—
186C2×Q16—48no—
187C2×Q16—48no—
188C2×Q16—48no—
189C2×Q16—48no—
190C2×Q16—48no—
191C22×C8—22no—
192C22×C8—22no—
193C22×C8—22no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—21no—
198C22×C8—22no—
199C22×C8—21no—
200C22×C8—21no—
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—22no—
204C22×C8—22no—
205C22×C8—21no—
206C22×C8—21no—
207C22×C8—21no—
208C22×C8—21no—
209C22×C8C22×C414yes—
210C22×C8C2×D412yes—
211C22×C8#4112yes—
212C4⋊Q8—28no—
213C4⋊Q8—28no—
214C4⋊Q8—28no—
215C4⋊Q8C22⋊C4116yes—
216C4×Q8—28no—
217C4×Q8—28no—
218C4×Q8—48no—
219C4×Q8—48no—
220C4×Q8—48no—
221C4×Q8—48no—
222C4×Q8—44no—
223C4×Q8—44no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4—42no—
263C2×C4⋊C4—42no—
264C2×C4⋊C4—81no—
265C2×C4⋊C4—81no—
266C2×C4⋊C4Q1614yescomplemented
267C2×C4⋊C4C22×C412yes—
268C2×C4⋊C4C2×D412yes—
269C2×C4⋊C4C2×D412yes—
270C2×C4⋊C4C2×D412yes—
271C2×C4⋊C4#4112yes—
272C2×C42—28no—
273C2×C42—24no—
274C2×C42—24no—
275C2×C42—22no—
276C2×C42—22no—
277C2×C42—22no—
278C2×C42—22no—
279C2×C42—22no—
280C2×C42—22no—
281C2×C42—44no—
282C2×C42—44no—
283C2×C42—42no—
284C2×C42—42no—
285C2×C42—42no—
286C2×C42—42no—
287C2×C42C22×C412yes—
288C2×C42C2×D412yes—
289C2×C42C2×D412yes—
290C2×C42C2×D412yes—
291C2×C42#4112yes—
292C2×C42#4112yes—
293C4⋊C8—28no—
294#103—28no—
295#103—216no—
296#103—28no—
297#103—48no—
298#103—48no—
299C4×C8—44no—
300#95—28no—
301#95—28no—
302#95—24no—
303#95—24no—
304#95—44no—
305#95—44no—
306#95—82no—
307#95—82no—
308C24C22×D411yescharacteristic
309C2×Q8—24no—
310C2×Q8—22no—
311C2×Q8—24no—
312C2×Q8—24no—
313C2×Q8—22no—
314C2×Q8—22no—
315C2×Q8—22no—
316C2×Q8—22no—
317C2×Q8—24no—
318C2×Q8—24no—
319C2×Q8—22no—
320C2×Q8—22no—
321C2×Q8—24no—
322C2×Q8—24no—
323C2×Q8—22no—
324C2×Q8—22no—
325C2×Q8—22no—
326C2×Q8—22no—
327C2×Q8—24no—
328C2×Q8—24no—
329C2×Q8—22no—
330C2×Q8—22no—
331C2×Q8—24no—
332C2×Q8—24no—
333C2×Q8—22no—
334C2×Q8—22no—
335C2×Q8—24no—
336C2×Q8—24no—
337C2×Q8—24no—
338C2×Q8—24no—
339C2×Q8—42no—
340C2×Q8—41no—
341C2×Q8—41no—
342C2×Q8—42no—
343C2×Q8—42no—
344C2×Q8—42no—
345C2×Q8—42no—
346C2×Q8—41no—
347C2×Q8—42no—
348C2×Q8—41no—
349C2×Q8—42no—
350C2×Q8—41no—
351C2×Q8—42no—
352C2×Q8—41no—
353C2×Q8—41no—
354C2×Q8—41no—
355C2×Q8—42no—
356C2×Q8—41no—
357C2×Q8—41no—
358C2×Q8—42no—
359C2×Q8—41no—
360C2×Q8—42no—
361C2×Q8—41no—
362C2×Q8—42no—
363C2×Q8C2×C22⋊C418yes—
364C2×Q8C4×D414yes—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—24no—
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—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—41no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—41no—
414C22×C4—41no—
415C22×C4—41no—
416C22×C4—41no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4C4×D412yes—
425C22×C4C4×D412yes—
426C22×C4#12012yes—
427C22×C4#12012yes—
428C22×C4#12112yes—
429C22×C4#12112yes—
430C22×C4C2×Q1614yes—
431C22×C4C2×Q1612yes—
432C22×C4#13512yes—
433C22×C4#13712yes—
434C22×C4#13712yes—
435C22×C4C23×C411yescharacteristic
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D411yescharacteristic
440C22×C4C2×#4111yescharacteristic
441C22×C4C2×#4111yescharacteristic
442C22×C4C2×#4111yescharacteristic
443C22×C4C2×#4111yescharacteristic
444C22×C4C2×#4111yescharacteristic
445C22×C4H(5,𝔽2)11yescharacteristic
446C22×C4H(5,𝔽2)11yescharacteristic
447C22×C4H(5,𝔽2)11yescharacteristic
448C22×C42−1+411yescharacteristic
449Q16—432no—
450C2×C8—24no—
451C2×C8—24no—
452C2×C8—24no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—41no—
460C2×C8—41no—
461C2×C8—41no—
462C2×C8—41no—
463C2×C8—42no—
464C2×C8—42no—
465C2×C8—41no—
466C2×C8—42no—
467C2×C8—42no—
468C2×C8—41no—
469C2×C8—41no—
470C2×C8—41no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—216no—
475C4⋊C4—216no—
476C4⋊C4—28no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—416no—
482C4⋊C4—41no—
483C4⋊C4—41no—
484C4⋊C4—41no—
485C4⋊C4—41no—
486C4⋊C4—48no—
487C4⋊C4—44no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C4⋊C4—84no—
493C4⋊C4—84no—
494C4⋊C4#12114yes—
495C42—416no—
496C42—42no—
497C42—42no—
498C42—42no—
499C42—42no—
500C42—48no—
501C42—48no—
502C42C2×C22⋊C414yes—
503C42C4×D414yes—
504C42#12114yes—
505C42#12314yes—
506C23C2×C4×D411yescharacteristic
507C23C2×#12011yescharacteristic
508C23C2×#12111yescharacteristic
509C23D4211yescharacteristic
510C23#54611yescharacteristic
511C23#54811yescharacteristic
512C23#55011yescharacteristic
513C23C22×Q1611yescharacteristic
514C23C2×#13511yescharacteristic
515C23C2×#13711yescharacteristic
516C23C2×#13711yescharacteristic
517C23#57711yescharacteristic
518C23#57711yescharacteristic
519C23#57811yescharacteristic
520C23#57811yescharacteristic
521Q8—24no—
522Q8—28no—
523Q8—28no—
524Q8—24no—
525Q8—44no—
526Q8—42no—
527Q8—42no—
528Q8—44no—
529Q8—42no—
530Q8—44no—
531Q8—42no—
532Q8—44no—
533Q8—42no—
534Q8—44no—
535Q8—42no—
536Q8—44no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—28no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—42no—
559C2×C4—44no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—81no—
585C2×C4—81no—
586C2×C4—81no—
587C2×C4—81no—
588C2×C4—81no—
589C2×C4—81no—
590C2×C4—81no—
591C2×C4—81no—
592C2×C4—82no—
593C2×C4—82no—
594C2×C4—82no—
595C2×C4—82no—
596C2×C4—82no—
597C2×C4—82no—
598C2×C4—82no—
599C2×C4#38614yes—
600C2×C4#38614yes—
601C2×C4C4×Q1614yes—
602C2×C4#44114yes—
603C2×C4#45114yes—
604C2×C4#45214yes—
605C2×C4#46214yes—
606C2×C4#46414yes—
607C2×C4C22×C22⋊C411yescharacteristic
608C2×C4C2×C4×D411yescharacteristic
609C2×C4C2×C4×D411yescharacteristic
610C2×C4C4×#4111yescharacteristic
611C2×C4#51811yescharacteristic
612C2×C4#52011yescharacteristic
613C2×C4C2×#12111yescharacteristic
614C2×C4C2×#12111yescharacteristic
615C2×C4C2×#12311yescharacteristic
616C2×C4#52511yescharacteristic
617C2×C4#52911yescharacteristic
618C2×C4#53211yescharacteristic
619C2×C4#53511yescharacteristic
620C2×C4#53711yescharacteristic
621C2×C4#54011yescharacteristic
622C2×C4#54211yescharacteristic
623C2×C4D4211yescharacteristic
624C2×C4D4211yescharacteristic
625C2×C4#54611yescharacteristic
626C2×C4#54611yescharacteristic
627C2×C4#54611yescharacteristic
628C2×C4#54611yescharacteristic
629C2×C4#54711yescharacteristic
630C2×C4#54711yescharacteristic
631C2×C4#54711yescharacteristic
632C2×C4#55311yescharacteristic
633C2×C4#55311yescharacteristic
634C2×C4#55511yescharacteristic
635C8—44no—
636C8—42no—
637C8—42no—
638C22C2×#38611yescharacteristic
639C22#236111yescharacteristic
640C22#236311yescharacteristic
641C22#240711yescharacteristic
642C22#240811yescharacteristic
643C22#242711yescharacteristic
644C22#242811yescharacteristic
645C22C2×C4×Q1611yescharacteristic
646C22C2×#44111yescharacteristic
647C22#294711yescharacteristic
648C22#294711yescharacteristic
649C22C2×#45111yescharacteristic
650C22C2×#45211yescharacteristic
651C22#301311yescharacteristic
652C22#301311yescharacteristic
653C22C2×#46211yescharacteristic
654C22C2×#46411yescharacteristic
655C22#304311yescharacteristic
656C22#304411yescharacteristic
657C22#327611yescharacteristic
658C22#327811yescharacteristic
659C22#328411yescharacteristic
660C22D4×Q1611yescharacteristic
661C22#329911yescharacteristic
662C22#330211yescharacteristic
663C22#332811yescharacteristic
664C22#333111yescharacteristic
665C22#336311yescharacteristic
666C22#336411yescharacteristic
667C22#336911yescharacteristic
668C22#337411yescharacteristic
669C22#339411yescharacteristic
670C22#339511yescharacteristic
671C22#339811yescharacteristic
672C22#340011yescharacteristic
673C4—24no—
674C4—44no—
675C4—48no—
676C4—44no—
677C4—44no—
678C4—82no—
679C4—82no—
680C4—82no—
681C4—162no—
682C4#187818yescomplemented
683C4C4×#12111yescharacteristic
684C4#232411yescharacteristic
685C4D4×C22⋊C411yescharacteristic
686C4#233911yescharacteristic
687C4#242011yescharacteristic
688C4#242211yescharacteristic
689C4#253511yescharacteristic
690C4#253711yescharacteristic
691C2#1397011yescharacteristic
692C2C2×#187811yescharacteristic
693C2#2093011yescharacteristic
694C2#2299811yescharacteristic
695C2#2300411yescharacteristic
696C2#2309611yescharacteristic
697C2#2310211yescharacteristic
698C2#2418911yescharacteristic
699C2#2419111yescharacteristic
700C2#2424111yescharacteristic
701C2#2424311yescharacteristic
702C2#2532311yescharacteristic
703C2#2532411yescharacteristic
704C2#2536911yescharacteristic
705C2#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
18. C2×#462C4semidirect
266. C2×C4⋊C4Q16semidirect
682. C4#1878semidirect

Maximal subgroups

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

subgroupindex
C22×#4622
C2×#23652
C2×#19562
C2×#18782
C2×C4×#1032
#139262
#111902
#110242
#108492
#108342
#107892