Finite Group Databasebeta

← #366940

Subgroups — #366940

Order 512.

1552 conjugacy classes of subgroups.

The subgroup lattice has 735 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#476C211yesmaximal, characteristic
2C2×#1993C211yesmaximal, characteristic, complemented
3C2×#1949C211yesmaximal, characteristic, complemented
4C2×#1771C211yesmaximal, characteristic, complemented
5C2×#1762C211yesmaximal, characteristic
6#14155C211yesmaximal, characteristic, complemented
7#13941C211yesmaximal, characteristic
8#11378C211yesmaximal, characteristic, complemented
9#11378C211yesmaximal, characteristic, complemented
10#11123C211yesmaximal, characteristic
11#11121C211yesmaximal, characteristic
12#10927C211yesmaximal, characteristic
13#10925C211yesmaximal, characteristic
14#10863C211yesmaximal, characteristic, complemented
15#10861C211yesmaximal, characteristic, complemented
16C22×C4⋊Q8C2211yescharacteristic
17C22×C4×D4C2211yescharacteristic
18C2×#476—24no—
19C2×#476—24no—
20C2×#476C418yescomplemented
21C22×#107C2211yescharacteristic
22C22×#106C2211yescharacteristic
23C22×#106C2211yescharacteristic
24C22×#106C2211yescharacteristic
25C22×C4⋊C8C2211yescharacteristic
26C22×#102C2211yescharacteristic
27C22×#102C2211yescharacteristic
28C22×#102C2211yescharacteristic
29C22×#102C2211yescharacteristic
30C22×#97C2211yescharacteristic
31C22×#97C2211yescharacteristic
32C22×C4×C8C2211yescharacteristic
33C2×#391—22no—
34C2×#390C2211yescharacteristic
35C2×#390C2211yescharacteristic
36C2×#382C2211yescharacteristic
37C2×C4×C4⋊C4C2211yescharacteristic
38C2×C4×C4⋊C4C2211yescharacteristic
39#1993—24no—
40#1949—24no—
41#1771—24no—
42#1762—24no—
43C2×#340C2211yescharacteristic
44C2×#340C2211yescharacteristic
45C2×#340C2211yescharacteristic
46C2×#340C2211yescharacteristic
47C2×#340C2211yescharacteristic
48C2×#340C2211yescharacteristic
49C2×#340C2211yescharacteristic
50C2×#340C2211yescharacteristic
51C2×#336C2211yescharacteristic
52C2×#336C2211yescharacteristic
53#1445—22no—
54#1439—22no—
55C4⋊#95C2211yescharacteristic
56C4⋊#95C2211yescharacteristic
57C4⋊#95C2211yescharacteristic
58C4⋊#95C2211yescharacteristic
59C4⋊#95C2211yescharacteristic
60C4⋊#95C2211yescharacteristic
61C22⋊#95—22no—
62C23×Q8—21no—
63C23×D4C2311yescharacteristic
64C24×C4—21no—
65C23×C8C2311yescharacteristic
66C23×C8C2311yescharacteristic
67C23×C8C2311yescharacteristic
68C23×C8C2311yescharacteristic
69C2×C4⋊Q8—22no—
70C2×C4⋊Q8—22no—
71C2×C4⋊Q8—22no—
72C2×C4⋊Q8—22no—
73C2×C4⋊Q8—22no—
74C2×C4⋊Q8C2×C414yes—
75C2×C4×D4—22no—
76C2×C4×D4—22no—
77C2×C4×D4—22no—
78C2×C4×D4—22no—
79C2×C4×D4—22no—
80C2×C4×D4C2×C414yes—
81C22×C4⋊C4—21no—
82C22×C4⋊C4D412yes—
83C22×C4⋊C4C2311yescharacteristic
84C22×C4⋊C4C2311yescharacteristic
85C22×C4⋊C4C2311yescharacteristic
86C22×C4⋊C4C2311yescharacteristic
87C22×C4⋊C4C2311yescharacteristic
88C22×C4⋊C4C2311yescharacteristic
89C22×C4⋊C4C2311yescharacteristic
90C22×C22⋊C4—21no—
91C22×C22⋊C4—22no—
92C22×C22⋊C4—22no—
93C22×C42—21no—
94C22×C42C2311yescharacteristic
95C22×C42C2311yescharacteristic
96C22×C42C2311yescharacteristic
97#476—232no—
98C2×#107—22no—
99C2×#107—22no—
100C2×#107—41no—
101C2×#106—22no—
102C2×#106—22no—
103C2×#106—22no—
104C2×#106—22no—
105C2×#106—22no—
106C2×#106—22no—
107C2×#106—41no—
108C2×#106C2×C414yes—
109C2×#106C2×C414yes—
110C2×C4⋊C8C2×C414yes—
111C2×C4⋊C8D414yes—
112C2×C4⋊C8D414yes—
113C2×#102—22no—
114C2×#102—22no—
115C2×#102—22no—
116C2×#102—22no—
117C2×#102—22no—
118C2×#102—22no—
119C2×#102—22no—
120C2×#102—22no—
121C2×#102—22no—
122C2×#102C2×C414yes—
123C2×#102C2×C414yes—
124C2×#102Q814yescomplemented
125C2×#97—22no—
126C2×#97—22no—
127C2×#97—22no—
128C2×#97—22no—
129C2×#97—41no—
130C2×#97D414yes—
131C2×C4×C8—22no—
132C2×C4×C8—22no—
133C2×C4×C8—41no—
134#391—48no—
135#390—22no—
136#390—22no—
137#390—41no—
138#390—41no—
139#382—22no—
140#382—22no—
141C4×C4⋊C4—22no—
142C4×C4⋊C4—22no—
143C4×C4⋊C4—22no—
144C4×C4⋊C4—22no—
145C2×#95—21no—
146C2×#95—22no—
147C2×#95—22no—
148C2×#95—22no—
149C2×#95—22no—
150C2×#95—21no—
151C2×#95—21no—
152C2×#95—22no—
153C2×#95—22no—
154C2×#95D412yes—
155#340—22no—
156#340—22no—
157#340—22no—
158#340—22no—
159#340—22no—
160#340—22no—
161#340—22no—
162#340—22no—
163#336—22no—
164#336—22no—
165C25—21no—
166C22×Q8—24no—
167C22×Q8—44no—
168C22×Q8—44no—
169C22×Q8—42no—
170C22×Q8—42no—
171C22×D4—21no—
172C22×D4—21no—
173C22×D4—21no—
174C22×D4—21no—
175C22×D4—21no—
176C22×D4—21no—
177C22×D4—21no—
178C22×D4—21no—
179C22×D4—21no—
180C22×D4—21no—
181C22×D4—21no—
182C22×D4C22×C412yes—
183C22×D4C2×Q812yes—
184C22×D4#4112yes—
185C23×C4—21no—
186C23×C4—21no—
187C23×C4—21no—
188C23×C4—21no—
189C23×C4—21no—
190C23×C4—24no—
191C23×C4—21no—
192C23×C4—21no—
193C23×C4—42no—
194C23×C4—42no—
195C23×C4—82no—
196C23×C4—82no—
197C23×C4C2×D412yes—
198C23×C4C2×D412yes—
199C23×C4C2×D411yescharacteristic
200C23×C4C2×D411yescharacteristic
201C23×C4C2411yescharacteristic
202C22×C8—21no—
203C22×C8—21no—
204C22×C8—21no—
205C22×C8—21no—
206C22×C8—21no—
207C22×C8—21no—
208C22×C8—21no—
209C22×C8—21no—
210C22×C8—21no—
211C22×C8—21no—
212C22×C8—21no—
213C22×C8—21no—
214C22×C8—21no—
215C22×C8—21no—
216C22×C8—21no—
217C22×C8—21no—
218C22×C8—21no—
219C22×C8—21no—
220C22×C8C22×C412yes—
221C22×C8C22×C412yes—
222C22×C8C2×D412yes—
223C22×C8C2×D412yes—
224C22×C8C2×D412yes—
225C22×C8C2×D412yes—
226C22×C8C2×D412yes—
227C22×C8C2×D412yes—
228C22×C8C2×Q812yes—
229C22×C8#4112yes—
230C4⋊Q8—28no—
231C4⋊Q8—28no—
232C4⋊Q8—44no—
233C4⋊Q8—44no—
234C4×D4—28no—
235C4×D4—28no—
236C4×D4—44no—
237C4×D4—44no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—42no—
279C2×C4⋊C4—42no—
280C2×C4⋊C4—42no—
281C2×C4⋊C4C22×C412yes—
282C2×C4⋊C4C22×C412yes—
283C2×C4⋊C4C22×C412yes—
284C2×C4⋊C4C2×Q812yes—
285C2×C4⋊C4#4112yes—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—42no—
288C2×C22⋊C4—42no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—82no—
294C2×C22⋊C4—82no—
295C2×C42—21no—
296C2×C42—21no—
297C2×C42—21no—
298C2×C42—21no—
299C2×C42—21no—
300C2×C42—21no—
301C2×C42—21no—
302C2×C42—21no—
303C2×C42—21no—
304C2×C42—21no—
305C2×C42—21no—
306C2×C42—21no—
307C2×C42—44no—
308C2×C42—42no—
309C2×C42C22×C412yes—
310C2×C42C2×D412yes—
311C2×C42C2×D412yes—
312C2×C42C2×D412yes—
313C2×C42#4112yes—
314C2×C42#4112yes—
315#107—44no—
316#106—28no—
317#106—28no—
318#106—44no—
319C4⋊C8C22⋊C4116yescomplemented
320#102—28no—
321#102—28no—
322#102—28no—
323#102—28no—
324#97—28no—
325#97—44no—
326C4×C8—44no—
327#95—24no—
328#95—44no—
329#95—44no—
330#95—42no—
331#95—42no—
332#95—42no—
333#95—44no—
334#95—44no—
335#95—82no—
336#95—82no—
337C24—22no—
338C24—22no—
339C24—22no—
340C24—41no—
341C24—41no—
342C24—41no—
343C24—41no—
344C24—42no—
345C24—42no—
346C24—42no—
347C24—42no—
348C24C22×D411yescharacteristic
349C2×Q8—416no—
350C2×Q8—48no—
351C2×Q8—84no—
352C2×Q8—84no—
353C2×Q8—88no—
354C2×D4—22no—
355C2×D4—22no—
356C2×D4—22no—
357C2×D4—22no—
358C2×D4—22no—
359C2×D4—22no—
360C2×D4—22no—
361C2×D4—22no—
362C2×D4—22no—
363C2×D4—22no—
364C2×D4—22no—
365C2×D4—22no—
366C2×D4—22no—
367C2×D4—22no—
368C2×D4—22no—
369C2×D4—41no—
370C2×D4—41no—
371C2×D4—41no—
372C2×D4—41no—
373C2×D4—41no—
374C2×D4—41no—
375C2×D4—41no—
376C2×D4—41no—
377C2×D4—41no—
378C2×D4—41no—
379C2×D4—41no—
380C2×D4—41no—
381C2×D4C4×Q814yes—
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—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
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—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—48no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—88no—
436C22×C4—84no—
437C22×C4—84no—
438C22×C4—88no—
439C22×C4C4×D412yes—
440C22×C4C4×D412yes—
441C22×C4#12112yes—
442C22×C4#12112yes—
443C22×C4#12712yes—
444C22×C4#12712yes—
445C22×C4AGL(1,ℤ/8)14yes—
446C22×C4C23×C411yescharacteristic
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D411yescharacteristic
450C22×C4C22×Q811yescharacteristic
451C22×C4C2×#4111yescharacteristic
452C22×C4C2×#4111yescharacteristic
453C22×C4C2×#4111yescharacteristic
454C22×C4C2×#4111yescharacteristic
455C22×C4C2×#4111yescharacteristic
456C22×C4H(5,𝔽2)11yescharacteristic
457C22×C42−1+411yescharacteristic
458C22×C42−1+411yescharacteristic
459C22×C42−1+411yescharacteristic
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—41no—
482C2×C8—41no—
483C2×C8—41no—
484C2×C8—41no—
485C2×C8C2×C22⋊C414yes—
486C2×C8C2×C22⋊C414yes—
487C2×C8C22⋊Q814yes—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—48no—
499C4⋊C4—41no—
500C4⋊C4—41no—
501C4⋊C4—41no—
502C4⋊C4—41no—
503C4⋊C4—41no—
504C4⋊C4—41no—
505C4⋊C4—41no—
506C4⋊C4—41no—
507C4⋊C4—41no—
508C4⋊C4—41no—
509C4⋊C4—48no—
510C4⋊C4—41no—
511C4⋊C4—41no—
512C4⋊C4—41no—
513C4⋊C4—41no—
514C4⋊C4—41no—
515C4⋊C4—41no—
516C4⋊C4—41no—
517C4⋊C4—41no—
518C22⋊C4—48no—
519C22⋊C4—88no—
520C22⋊C4—88no—
521C42—41no—
522C42—41no—
523C42—41no—
524C42—41no—
525C42—41no—
526C42—41no—
527C42—41no—
528C42—41no—
529C42—84no—
530C42C2×C22⋊C414yes—
531C42C4×D414yes—
532C42#12114yes—
533C42#12414yes—
534C23—24no—
535C23—44no—
536C23—42no—
537C23—44no—
538C23—42no—
539C23—42no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—42no—
544C23—42no—
545C23—44no—
546C23—42no—
547C23—82no—
548C23—82no—
549C23—82no—
550C23—82no—
551C23—81no—
552C23—81no—
553C23—81no—
554C23—81no—
555C23—82no—
556C23—82no—
557C23C2×C4×D411yescharacteristic
558C23C2×#12111yescharacteristic
559C23C2×#12711yescharacteristic
560C23#54611yescharacteristic
561C23#54711yescharacteristic
562C23#54811yescharacteristic
563C23D4×Q811yescharacteristic
564C23C2×AGL(1,ℤ/8)12yes—
565C23#57512yes—
566C23#57712yes—
567C23#57712yes—
568Q8—816no—
569D4—24no—
570D4—24no—
571D4—42no—
572D4—42no—
573D4—42no—
574D4—42no—
575D4—42no—
576D4—42no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—816no—
610C2×C4—82no—
611C2×C4—81no—
612C2×C4—81no—
613C2×C4—81no—
614C2×C4—81no—
615C2×C4—81no—
616C2×C4—81no—
617C2×C4—81no—
618C2×C4—81no—
619C2×C4—81no—
620C2×C4—81no—
621C2×C4—82no—
622C2×C4—82no—
623C2×C4—82no—
624C2×C4—82no—
625C2×C4—82no—
626C2×C4—82no—
627C2×C4—81no—
628C2×C4—81no—
629C2×C4—81no—
630C2×C4—81no—
631C2×C4—82no—
632C2×C4—81no—
633C2×C4—81no—
634C2×C4—82no—
635C2×C4#39014yes—
636C2×C4#39014yes—
637C2×C4C22×C22⋊C411yescharacteristic
638C2×C4C2×C4×D411yescharacteristic
639C2×C4C2×C4×Q811yescharacteristic
640C2×C4C4×#4111yescharacteristic
641C2×C4#51911yescharacteristic
642C2×C4#52011yescharacteristic
643C2×C4C2×#12111yescharacteristic
644C2×C4C2×C22⋊Q811yescharacteristic
645C2×C4#52511yescharacteristic
646C2×C4C2×#12411yescharacteristic
647C2×C4#52911yescharacteristic
648C2×C4#53311yescharacteristic
649C2×C4#53611yescharacteristic
650C2×C4#53711yescharacteristic
651C2×C4#54111yescharacteristic
652C2×C4#54211yescharacteristic
653C2×C4#54611yescharacteristic
654C2×C4#54611yescharacteristic
655C2×C4#54711yescharacteristic
656C2×C4#54711yescharacteristic
657C2×C4#54811yescharacteristic
658C2×C4#54811yescharacteristic
659C2×C4D4×Q811yescharacteristic
660C2×C4D4×Q811yescharacteristic
661C2×C4#55211yescharacteristic
662C2×C4#55211yescharacteristic
663C2×C4#55411yescharacteristic
664C2×C4#55611yescharacteristic
665C8—24no—
666C8—24no—
667C8—24no—
668C8—42no—
669C22—44no—
670C22—48no—
671C22—84no—
672C22—82no—
673C22—82no—
674C22—84no—
675C22—82no—
676C22—82no—
677C22—84no—
678C22—82no—
679C22—82no—
680C22C2×#39011yescharacteristic
681C22#236011yescharacteristic
682C22#236211yescharacteristic
683C22#242711yescharacteristic
684C22#242911yescharacteristic
685C22#254011yescharacteristic
686C22#254211yescharacteristic
687C22C4×AGL(1,ℤ/8)12yes—
688C22#294912yes—
689C22#305512yes—
690C22#305712yes—
691C22#315111yescharacteristic
692C22#315311yescharacteristic
693C22#315411yescharacteristic
694C22#315411yescharacteristic
695C22#330612yes—
696C22#331112yes—
697C22#334512yes—
698C22#335012yes—
699C22#337112yes—
700C22#337312yes—
701C22#338412yes—
702C22#338812yes—
703C4—24no—
704C4—24no—
705C4—44no—
706C4—82no—
707C4—82no—
708C4—82no—
709C4—82no—
710C4—82no—
711C4—82no—
712C4—161no—
713C4—161no—
714C4C4×C22⋊Q811yescharacteristic
715C4#232311yescharacteristic
716C4#233911yescharacteristic
717C4Q8×C22⋊C411yescharacteristic
718C4#241511yescharacteristic
719C4#241711yescharacteristic
720C4#248911yescharacteristic
721C4#249211yescharacteristic
722C2—84no—
723C2#1399711yescharacteristic
724C2#2123111yescharacteristic
725C2#2123611yescharacteristic
726C2#2297512yes—
727C2#2307512yes—
728C2#2535412yes—
729C2#2548512yes—
730C2#2843011yescharacteristic
731C2#2843611yescharacteristic
732C2#2854711yescharacteristic
733C2#2854811yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
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
2. C2×#1993C2semidirect
3. C2×#1949C2semidirect
4. C2×#1771C2semidirect
6. #14155C2semidirect
8. #11378C2semidirect
9. #11378C2semidirect
14. #10863C2semidirect
15. #10861C2semidirect
20. C2×#476C4semidirect
124. C2×#102Q8semidirect
319. C4⋊C8C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4762
C2×#19932
C2×#19492
C2×#17712
C2×#17622
#141552
#139412
#113782
#111232
#111212
#109272
#109252
#108632
#108612