Finite Group Databasebeta

← #364343

Subgroups — #364343

Order 512.

1462 conjugacy classes of subgroups.

The subgroup lattice has 736 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1854C211yesmaximal, characteristic
2C2×#1853C211yesmaximal, characteristic
3C2×#1809C211yesmaximal, characteristic
4C2×#1753C211yesmaximal, characteristic
5C2×#1736C211yesmaximal, characteristic
6#13881C211yesmaximal, characteristic
7#13879C211yesmaximal, characteristic
8#11325C211yesmaximal, characteristic, complemented
9#11325C211yesmaximal, characteristic, complemented
10#11325C211yesmaximal, characteristic, complemented
11#11323C211yesmaximal, characteristic, complemented
12#10863C211yesmaximal, characteristic, complemented
13#10861C211yesmaximal, characteristic, complemented
14#10843C211yesmaximal, characteristic, complemented
15#10842C211yesmaximal, characteristic, complemented
16C23×M16C2211yescharacteristic
17C22×#117C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C22×#107C2211yescharacteristic
20C22×#107C2211yescharacteristic
21C22×#106C2211yescharacteristic
22C22×#106C2211yescharacteristic
23C22×#106C2211yescharacteristic
24C22×#106C2211yescharacteristic
25C22×C22⋊C8C2211yescharacteristic
26C22×#97C2211yescharacteristic
27C22×C4×C8C2211yescharacteristic
28C2×#384C2211yescharacteristic
29C2×#382C2211yescharacteristic
30C2×#382C2211yescharacteristic
31C2×#382C2211yescharacteristic
32C2×#380C2211yescharacteristic
33C2×#380C2211yescharacteristic
34C2×C4×C22⋊C4C2211yescharacteristic
35#1854—24no—
36#1853—24no—
37#1809—24no—
38#1753C418yescomplemented
39#1736—24no—
40C2×#340C2211yescharacteristic
41C2×#340C2211yescharacteristic
42C2×#340C2211yescharacteristic
43C2×#340C2211yescharacteristic
44C2×#340C2211yescharacteristic
45C2×#340C2211yescharacteristic
46C2×#340C2211yescharacteristic
47C2×#340C2211yescharacteristic
48C2×#340C2211yescharacteristic
49C2×#340C2211yescharacteristic
50C2×#336C2211yescharacteristic
51C2×#336C2211yescharacteristic
52#1442—22no—
53#1439—22no—
54#1436—22no—
55C4⋊#95C2211yescharacteristic
56C4⋊#95C2211yescharacteristic
57C4⋊#95C2211yescharacteristic
58C4⋊#95C2211yescharacteristic
59C22⋊#95—22no—
60C24×C4—21no—
61C24×C4C2311yescharacteristic
62C22×M16—41no—
63C22×M16—41no—
64C22×M16—41no—
65C22×M16—41no—
66C22×M16C2×C414yes—
67C22×M16D414yes—
68C22×M16Q814yes—
69C23×C8C2311yescharacteristic
70C23×C8C2311yescharacteristic
71C23×C8C2311yescharacteristic
72C23×C8C2311yescharacteristic
73C2×#117—22no—
74C2×#117—22no—
75C2×#117—22no—
76C2×#117—22no—
77C2×#117—41no—
78C2×#117—41no—
79C22×C4⋊C4—22no—
80C22×C4⋊C4—22no—
81C22×C4⋊C4—22no—
82C22×C4⋊C4—22no—
83C22×C4⋊C4—21no—
84C22×C4⋊C4—22no—
85C22×C4⋊C4—41no—
86C22×C4⋊C4—41no—
87C22×C4⋊C4C2311yescharacteristic
88C22×C4⋊C4C2311yescharacteristic
89C22×C4⋊C4C2311yescharacteristic
90C22×C4⋊C4C2311yescharacteristic
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2311yescharacteristic
95C22×C22⋊C4—21no—
96C22×C22⋊C4—21no—
97C22×C22⋊C4—21no—
98C22×C22⋊C4D412yes—
99C22×C42—21no—
100C22×C42C2311yescharacteristic
101C22×C42C2311yescharacteristic
102C2×#107—22no—
103C2×#107—22no—
104C2×#107—22no—
105C2×#107—41no—
106C2×#107—41no—
107C2×#107—41no—
108C2×#106—22no—
109C2×#106—22no—
110C2×#106—22no—
111C2×#106—22no—
112C2×#106—22no—
113C2×#106—22no—
114C2×#106—22no—
115C2×#106—41no—
116C2×#106—41no—
117C2×#106—41no—
118C2×#106—41no—
119C2×#106—41no—
120C2×C22⋊C8C2×C414yes—
121C2×C22⋊C8D414yes—
122C2×C22⋊C8Q814yes—
123C2×#97—41no—
124C2×#97—41no—
125C2×#97C2×C414yescomplemented
126C2×C4×C8—41no—
127C2×C4×C8—41no—
128C2×C4×C8C2×C414yescomplemented
129#384—22no—
130#384—41no—
131#382—22no—
132#382—22no—
133#382—22no—
134#382—41no—
135#382—41no—
136#382—41no—
137#380—22no—
138#380—22no—
139#380—22no—
140#380—22no—
141C4×C22⋊C4C2×C414yes—
142C4×C22⋊C4Q814yes—
143C2×#95—22no—
144C2×#95—22no—
145C2×#95—22no—
146C2×#95—21no—
147C2×#95—22no—
148C2×#95—21no—
149C2×#95—22no—
150C2×#95—22no—
151C2×#95—22no—
152C2×#95D412yes—
153#340—22no—
154#340—22no—
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#336C2×C414yescomplemented
164#336C2×C414yescomplemented
165C25C2×D411yescharacteristic
166C23×C4—21no—
167C23×C4—21no—
168C23×C4—21no—
169C23×C4—21no—
170C23×C4—21no—
171C23×C4—21no—
172C23×C4—21no—
173C23×C4—21no—
174C23×C4—21no—
175C23×C4—21no—
176C23×C4—21no—
177C23×C4—21no—
178C23×C4—21no—
179C23×C4—21no—
180C23×C4—21no—
181C23×C4—21no—
182C23×C4—42no—
183C23×C4—42no—
184C23×C4—42no—
185C23×C4—44no—
186C23×C4—82no—
187C23×C4C22×C412yes—
188C23×C4C2×D412yes—
189C23×C4C2×D412yes—
190C23×C4C2×D411yescharacteristic
191C23×C4C2×D412yes—
192C23×C4C2×D412yes—
193C23×C4C2×Q812yes—
194C23×C4C2×Q812yes—
195C23×C4C2×Q812yes—
196C23×C4C2411yescharacteristic
197C2×M16—44no—
198C2×M16—44no—
199C2×M16—44no—
200C2×M16—44no—
201C2×M16—44no—
202C2×M16—44no—
203C2×M16C4⋊C4116yescomplemented
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×C8C22×C412yes—
223C22×C8C22×C412yes—
224C22×C8C2×D412yes—
225C22×C8C2×D412yes—
226C22×C8C2×D412yes—
227C22×C8C2×D412yes—
228C22×C8C2×Q812yes—
229C22×C8C2×Q812yes—
230C22×C8C2×Q812yes—
231C22×C8C2×Q812yes—
232#117—28no—
233#117—44no—
234#117—44no—
235#117—44no—
236C2×C4⋊C4—28no—
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—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—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
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—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4—42no—
284C2×C4⋊C4—42no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4—42no—
287C2×C4⋊C4—44no—
288C2×C4⋊C4—44no—
289C2×C4⋊C4—82no—
290C2×C4⋊C4#4112yes—
291C2×C4⋊C4#4112yes—
292C2×C4⋊C4#4112yes—
293C2×C4⋊C4#4112yes—
294C2×C22⋊C4—42no—
295C2×C22⋊C4—42no—
296C2×C22⋊C4—42no—
297C2×C22⋊C4—42no—
298C2×C22⋊C4—42no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—42no—
301C2×C22⋊C4—42no—
302C2×C22⋊C4—82no—
303C2×C22⋊C4C4⋊C418yes—
304C2×C42—21no—
305C2×C42—21no—
306C2×C42—21no—
307C2×C42—21no—
308C2×C42—21no—
309C2×C42—21no—
310C2×C42—21no—
311C2×C42—21no—
312C2×C42—42no—
313C2×C42—42no—
314C2×C42—42no—
315C2×C42C22×C412yes—
316C2×C42C22×C412yes—
317C2×C42C2×Q812yes—
318C2×C42C2×Q812yes—
319#107—44no—
320#107—44no—
321#106—44no—
322#106—44no—
323#106—44no—
324#106—44no—
325C22⋊C8C4⋊C4116yescomplemented
326#97—44no—
327C4×C8—44no—
328#95—44no—
329#95—44no—
330#95—44no—
331#95—44no—
332#95—42no—
333#95—42no—
334#95—82no—
335#95—82no—
336#95—82no—
337#95C4⋊C418yes—
338C24—21no—
339C24—21no—
340C24—21no—
341C24—21no—
342C24—22no—
343C24—22no—
344C24—22no—
345C24—22no—
346C24C2×C4⋊C412yes—
347C24#12712yes—
348C24C4⋊Q812yes—
349C24C22×D411yescharacteristic
350C22×C4—21no—
351C22×C4—21no—
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—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
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—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—48no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—41no—
414C22×C4—41no—
415C22×C4—41no—
416C22×C4—41no—
417C22×C4—41no—
418C22×C4—41no—
419C22×C4—41no—
420C22×C4—41no—
421C22×C4—41no—
422C22×C4—41no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—88no—
432C22×C4—88no—
433C22×C4—84no—
434C22×C4—84no—
435C22×C4C2×C4⋊C414yes—
436C22×C4C2×C4⋊C414yes—
437C22×C4C2×C4⋊C414yes—
438C22×C4C2×C4⋊C414yes—
439C22×C4C2×C4⋊C412yes—
440C22×C4#12712yes—
441C22×C4C4⋊Q814yes—
442C22×C4C4⋊Q814yes—
443C22×C4C4⋊Q812yes—
444C22×C4C23×C411yescharacteristic
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D411yescharacteristic
447C22×C4C22×Q811yescharacteristic
448C22×C4C22×Q811yescharacteristic
449C22×C4C22×Q811yescharacteristic
450C22×C4C2×#4111yescharacteristic
451C22×C4C2×#4111yescharacteristic
452C22×C4C2×#4111yescharacteristic
453C22×C4C2×#4111yescharacteristic
454C22×C4H(5,𝔽2)11yescharacteristic
455C22×C4H(5,𝔽2)11yescharacteristic
456C22×C4H(5,𝔽2)11yescharacteristic
457C22×C4H(5,𝔽2)11yescharacteristic
458M16—416no—
459C2×C8—22no—
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—22no—
482C2×C8—22no—
483C2×C8C2×C4⋊C414yes—
484C2×C8C2×C4⋊C414yes—
485C2×C8C2×C4⋊C414yes—
486C2×C8C2×C4⋊C414yes—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—48no—
496C4⋊C4—41no—
497C4⋊C4—41no—
498C4⋊C4—41no—
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—41no—
510C4⋊C4—41no—
511C4⋊C4—41no—
512C4⋊C4—41no—
513C4⋊C4—41no—
514C4⋊C4—41no—
515C4⋊C4—41no—
516C4⋊C4—41no—
517C4⋊C4—41no—
518C4⋊C4—41no—
519C4⋊C4—41no—
520C4⋊C4—88no—
521C22⋊C4—48no—
522C22⋊C4—48no—
523C22⋊C4—84no—
524C22⋊C4—84no—
525C42—22no—
526C42—22no—
527C42—22no—
528C42—22no—
529C42—22no—
530C42—22no—
531C42—22no—
532C42—22no—
533C42—48no—
534C23—22no—
535C23—22no—
536C23—22no—
537C23—22no—
538C23—22no—
539C23—22no—
540C23—24no—
541C23—24no—
542C23—24no—
543C23—24no—
544C23—22no—
545C23—22no—
546C23—22no—
547C23—22no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23#38414yes—
557C23C22×C4⋊C411yescharacteristic
558C23C2×#12711yescharacteristic
559C23C2×C4⋊Q811yescharacteristic
560C23#54611yescharacteristic
561C23#54611yescharacteristic
562C23#54611yescharacteristic
563C23#54611yescharacteristic
564C23#57512yes—
565C23#57512yes—
566C23#57712yes—
567C23#57712yes—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
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—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—816no—
605C2×C4—81no—
606C2×C4—81no—
607C2×C4—81no—
608C2×C4—81no—
609C2×C4—81no—
610C2×C4—81no—
611C2×C4—81no—
612C2×C4—81no—
613C2×C4—82no—
614C2×C4—82no—
615C2×C4—82no—
616C2×C4—82no—
617C2×C4—81no—
618C2×C4—81no—
619C2×C4—81no—
620C2×C4—81no—
621C2×C4—81no—
622C2×C4—81no—
623C2×C4—81no—
624C2×C4—81no—
625C2×C4—81no—
626C2×C4—81no—
627C2×C4—81no—
628C2×C4—81no—
629C2×C4—82no—
630C2×C4—82no—
631C2×C4—82no—
632C2×C4—82no—
633C2×C4—82no—
634C2×C4—82no—
635C2×C4—82no—
636C2×C4—82no—
637C2×C4#38414yes—
638C2×C4C22×C4⋊C411yescharacteristic
639C2×C4C22×C4⋊C411yescharacteristic
640C2×C4C4×#4111yescharacteristic
641C2×C4C4×#4111yescharacteristic
642C2×C4#51811yescharacteristic
643C2×C4#51811yescharacteristic
644C2×C4#52511yescharacteristic
645C2×C4#52511yescharacteristic
646C2×C4C2×C4⋊Q811yescharacteristic
647C2×C4C2×C4⋊Q811yescharacteristic
648C2×C4#53311yescharacteristic
649C2×C4#53311yescharacteristic
650C2×C4#53411yescharacteristic
651C2×C4#53411yescharacteristic
652C2×C4#54311yescharacteristic
653C2×C4#54311yescharacteristic
654C2×C4#54611yescharacteristic
655C2×C4#54611yescharacteristic
656C2×C4#54611yescharacteristic
657C2×C4#54611yescharacteristic
658C2×C4#55411yescharacteristic
659C2×C4#55411yescharacteristic
660C2×C4#55411yescharacteristic
661C2×C4#55411yescharacteristic
662C2×C4#55411yescharacteristic
663C2×C4#55411yescharacteristic
664C2×C4#55411yescharacteristic
665C2×C4#55411yescharacteristic
666C8—24no—
667C8—24no—
668C8—24no—
669C8—24no—
670C22—24no—
671C22—24no—
672C22—24no—
673C22—44no—
674C22—44no—
675C22—44no—
676C22—44no—
677C22—42no—
678C22—42no—
679C22—42no—
680C22—42no—
681C22C2×#38411yescharacteristic
682C22#234911yescharacteristic
683C22#234911yescharacteristic
684C22#253911yescharacteristic
685C22#253911yescharacteristic
686C22#254911yescharacteristic
687C22#254911yescharacteristic
688C22#291112yes—
689C22#291212yes—
690C22#315311yescharacteristic
691C22#315311yescharacteristic
692C22#315411yescharacteristic
693C22#315411yescharacteristic
694C22#316312yes—
695C22#316612yes—
696C22#330712yes—
697C22#331112yes—
698C22#331712yes—
699C22#332112yes—
700C22#332312yes—
701C22#332312yes—
702C22#332612yes—
703C22#332612yes—
704C4—24no—
705C4—44no—
706C4—44no—
707C4—82no—
708C4—82no—
709C4—82no—
710C4—82no—
711C4—161no—
712C4—161no—
713C4—161no—
714C4—161no—
715C4#229611yescharacteristic
716C4#231511yescharacteristic
717C4#234911yescharacteristic
718C4#234911yescharacteristic
719C4#254911yescharacteristic
720C4#254911yescharacteristic
721C4#258311yescharacteristic
722C4#258311yescharacteristic
723C2—44no—
724C2#1396511yescharacteristic
725C2#2080111yescharacteristic
726C2#2080211yescharacteristic
727C2#2271012yes—
728C2#2271412yes—
729C2#2836111yescharacteristic
730C2#2836311yescharacteristic
731C2#2836611yescharacteristic
732C2#2836811yescharacteristic
733C2#2878512yes—
734C2#2878812yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
8. #11325C2semidirect
9. #11325C2semidirect
10. #11325C2semidirect
11. #11323C2semidirect
12. #10863C2semidirect
13. #10861C2semidirect
14. #10843C2semidirect
15. #10842C2semidirect
38. #1753C4semidirect
125. C2×#97C2×C4semidirect
128. C2×C4×C8C2×C4semidirect
163. #336C2×C4semidirect
164. #336C2×C4semidirect
203. C2×M16C4⋊C4semidirect
325. C22⋊C8C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#18542
C2×#18532
C2×#18092
C2×#17532
C2×#17362
#138812
#138792
#113252
#113232
#108632
#108612
#108432
#108422