Finite Group Databasebeta

← #185647

Subgroups — #185647

Order 512.

2568 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#554C211yesmaximal, characteristic
2#20722C211yesmaximal, characteristic
3#20700C212yesmaximal, complemented
4#17578C212yesmaximal, complemented
5#17362C211yesmaximal, characteristic
6#8380C218yesmaximal, complemented
7C2×#554—24no—
8C2×#554—24no—
9C2×#554—22no—
10C2×#554—22no—
11C2×#554—24no—
12C2×#554—24no—
13C2×#554C414yescomplemented
14C2×#554C2214yes—
15C22×C4⋊Q8C411yescharacteristic, complemented
16C22×#125—21no—
17C22×C22⋊Q8—21no—
18C22×C22⋊Q8—21no—
19C22×C22⋊Q8C411yescharacteristic, complemented
20C22×C22⋊Q8C2211yescharacteristic
21C22×C4×Q8C411yescharacteristic, complemented
22C22×C4×D4—21no—
23C22×C4×D4C2211yescharacteristic
24C23×C4⋊C4C411yescharacteristic, complemented
25C23×C4⋊C4C2211yescharacteristic
26C2×#407C2212yes—
27#2280C2212yes—
28#1824C2218yescomplemented
29#1823—24no—
30#1819—21no—
31#1799C2214yes—
32#1781—21no—
33C4×#99—22no—
34#1736—24no—
35#1520C2218yescomplemented
36#1511—24no—
37#1508—24no—
38#1465—22no—
39#1463—21no—
40#1462—21no—
41#1459C2214yes—
42C23×Q8—21no—
43C23×Q8C2×C411yescharacteristic
44C23×D4D411yescharacteristic
45C24×C4—21no—
46C24×C4—21no—
47C24×C4C2×C411yescharacteristic
48C24×C4C2311yescharacteristic
49C22×M16—22no—
50#554—216no—
51#554—216no—
52#554—216no—
53#554—216no—
54#554—216no—
55#554—216no—
56#554—216no—
57#554—216no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8—22no—
60C2×C4⋊Q8—22no—
61C2×C4⋊Q8—22no—
62C2×C4⋊Q8—22no—
63C2×C4⋊Q8C2×C414yes—
64C2×#125—28no—
65C2×#125—28no—
66C2×#125—24no—
67C2×#125—24no—
68C2×C22⋊Q8—28no—
69C2×C22⋊Q8—28no—
70C2×C22⋊Q8—28no—
71C2×C22⋊Q8—28no—
72C2×C22⋊Q8—22no—
73C2×C22⋊Q8—22no—
74C2×C22⋊Q8—22no—
75C2×C22⋊Q8—21no—
76C2×C22⋊Q8—22no—
77C2×C22⋊Q8—22no—
78C2×C22⋊Q8—21no—
79C2×C22⋊Q8—22no—
80C2×C22⋊Q8—22no—
81C2×C22⋊Q8—24no—
82C2×C22⋊Q8—24no—
83C2×C22⋊Q8—24no—
84C2×C22⋊Q8—24no—
85C2×C22⋊Q8C2×C414yes—
86C2×C22⋊Q8C2×C412yes—
87C2×C22⋊Q8D414yes—
88C2×C22⋊Q8D414yes—
89C2×C22⋊Q8C2312yes—
90C2×C4×Q8—22no—
91C2×C4×Q8—22no—
92C2×C4×Q8—22no—
93C2×C4×Q8—22no—
94C2×C4×Q8—22no—
95C2×C4×Q8C2×C414yes—
96C2×C4×D4—28no—
97C2×C4×D4—28no—
98C2×C4×D4—24no—
99C2×C4×D4—24no—
100C2×C4×D4—22no—
101C2×C4×D4—22no—
102C2×C4×D4—22no—
103C2×C4×D4—22no—
104C2×C4×D4C2×C412yes—
105C2×C4×D4D412yes—
106C2×C4×D4D412yes—
107C2×C4×D4C2312yes—
108C2×#117—22no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—22no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4—21no—
114C22×C4⋊C4—21no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4—22no—
117C22×C4⋊C4—22no—
118C22×C4⋊C4—21no—
119C22×C4⋊C4—21no—
120C22×C4⋊C4—22no—
121C22×C4⋊C4—22no—
122C22×C4⋊C4—22no—
123C22×C4⋊C4—22no—
124C22×C4⋊C4C2×C414yes—
125C22×C4⋊C4C2×C411yescharacteristic
126C22×C4⋊C4C2×C411yescharacteristic
127C22×C4⋊C4C2×C411yescharacteristic
128C22×C4⋊C4C2×C412yes—
129C22×C4⋊C4D411yescharacteristic
130C22×C4⋊C4D414yes—
131C22×C4⋊C4D414yes—
132C22×C4⋊C4C2312yes—
133C22×C22⋊C4—21no—
134C22×C22⋊C4—21no—
135C22×C22⋊C4D411yescharacteristic
136C22×C22⋊C4D411yescharacteristic
137C22×C42—21no—
138C22×C42C2×C411yescharacteristic
139C2×#99—24no—
140#407—22no—
141#407—22no—
142#407—22no—
143#407—22no—
144C2×C22⋊C8—22no—
145C2×C22⋊C8C2314yes—
146#381—22no—
147#380—22no—
148#380C2314yes—
149#379—22no—
150C4×C22⋊C4—22no—
151C4×C22⋊C4—22no—
152#342—24no—
153#340—216no—
154#328—216no—
155#324—24no—
156#323—24no—
157C25C22⋊C411yescharacteristic
158C25C2×D411yescharacteristic
159C22×Q8—24no—
160C22×Q8—24no—
161C22×Q8—21no—
162C22×Q8—21no—
163C22×Q8—21no—
164C22×Q8—24no—
165C22×Q8—44no—
166C22×Q8—84no—
167C22×Q8C22⋊C412yes—
168C22×Q8C22⋊C412yes—
169C22×Q8C22×C412yes—
170C22×D4—22no—
171C22×D4—22no—
172C22×D4—28no—
173C22×D4C22⋊C412yes—
174C22×D4C2×D412yes—
175C23×C4—21no—
176C23×C4—21no—
177C23×C4—21no—
178C23×C4—21no—
179C23×C4—24no—
180C23×C4—24no—
181C23×C4—24no—
182C23×C4—22no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—22no—
186C23×C4—21no—
187C23×C4—21no—
188C23×C4—21no—
189C23×C4—21no—
190C23×C4—44no—
191C23×C4—44no—
192C23×C4—84no—
193C23×C4—84no—
194C23×C4C22⋊C411yescharacteristic
195C23×C4C22⋊C411yescharacteristic
196C23×C4C22⋊C411yescharacteristic
197C23×C4C22⋊C412yes—
198C23×C4C22×C411yescharacteristic
199C23×C4C22×C412yes—
200C23×C4C22×C411yescharacteristic
201C23×C4C2×D412yes—
202C23×C4C2×D411yescharacteristic
203C23×C4C2×D412yes—
204C23×C4C2×D412yes—
205C23×C4C2×D412yes—
206C23×C4C2×D411yescharacteristic
207C23×C4C2×D411yescharacteristic
208C23×C4C2411yescharacteristic
209C2×M16—48no—
210C2×M16—42no—
211C2×M16—42no—
212C22×C8—24no—
213C4⋊Q8—28no—
214C4⋊Q8—28no—
215C4⋊Q8—28no—
216C4⋊Q8—28no—
217#125—232no—
218#125—232no—
219C22⋊Q8—28no—
220C22⋊Q8—28no—
221C22⋊Q8—232no—
222C22⋊Q8—232no—
223C22⋊Q8—232no—
224C22⋊Q8—232no—
225C22⋊Q8—28no—
226C22⋊Q8—24no—
227C22⋊Q8—24no—
228C22⋊Q8—24no—
229C22⋊Q8—24no—
230C22⋊Q8—24no—
231C22⋊Q8—24no—
232C22⋊Q8—28no—
233C22⋊Q8C22⋊C418yes—
234C22⋊Q8C2×D418yes—
235C4×Q8—28no—
236C4×Q8—28no—
237C4×Q8—28no—
238C4×Q8—28no—
239C4×D4—232no—
240C4×D4—232no—
241C4×D4—24no—
242C4×D4—24no—
243C4×D4—24no—
244C4×D4—24no—
245C4×D4—24no—
246C4×D4—24no—
247C4×D4—24no—
248C4×D4—24no—
249#117—48no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—22no—
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—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—28no—
290C2×C4⋊C4—28no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4C22⋊C412yes—
293C2×C4⋊C4C22⋊C412yes—
294C2×C4⋊C4C22⋊C412yes—
295C2×C4⋊C4C22⋊C412yes—
296C2×C4⋊C4C22⋊C412yes—
297C2×C4⋊C4C22⋊C412yes—
298C2×C4⋊C4C22⋊C412yes—
299C2×C4⋊C4C22⋊C418yes—
300C2×C4⋊C4C22×C412yes—
301C2×C4⋊C4C22×C412yes—
302C2×C4⋊C4C22×C412yes—
303C2×C4⋊C4C2×D412yes—
304C2×C4⋊C4C2×D418yes—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—42no—
318C2×C22⋊C4—42no—
319C2×C22⋊C4C22⋊C412yes—
320C2×C22⋊C4C22⋊C412yes—
321C2×C22⋊C4C2×D412yes—
322C2×C22⋊C4C2×D412yes—
323C2×C42—24no—
324C2×C42—24no—
325C2×C42—21no—
326C2×C42—21no—
327C2×C42—21no—
328C2×C42—21no—
329C2×C42—24no—
330C2×C42—42no—
331C2×C42C22⋊C412yes—
332C2×C42C22×C412yes—
333#99—48no—
334C22⋊C8—24no—
335C22⋊C8—24no—
336C22⋊C8—42no—
337C22⋊C8—42no—
338#95—24no—
339#95—42no—
340C24—24no—
341C24—21no—
342C24—21no—
343C24—24no—
344C24—42no—
345C24—42no—
346C24—42no—
347C24—42no—
348C24C2×C22⋊C411yescharacteristic
349C24C2×C22⋊C411yescharacteristic
350C24C2×C22⋊C412yes—
351C24#12012yes—
352C24#12012yes—
353C24C22×D411yescharacteristic
354C2×Q8—216no—
355C2×Q8—22no—
356C2×Q8—22no—
357C2×Q8—22no—
358C2×Q8—44no—
359C2×Q8—44no—
360C2×Q8—44no—
361C2×Q8—44no—
362C2×Q8—44no—
363C2×Q8—44no—
364C2×Q8—88no—
365C2×Q8—88no—
366C2×Q8—88no—
367C2×Q8C2×C22⋊C414yes—
368C2×D4—216no—
369C2×D4—216no—
370C2×D4—24no—
371C2×D4—24no—
372C2×D4—24no—
373C2×D4—24no—
374C2×D4—24no—
375C2×D4#13718yes—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—28no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—216no—
398C22×C4—28no—
399C22×C4—28no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—42no—
405C22×C4—44no—
406C22×C4—42no—
407C22×C4—44no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—44no—
412C22×C4—44no—
413C22×C4—44no—
414C22×C4—44no—
415C22×C4—44no—
416C22×C4—44no—
417C22×C4—44no—
418C22×C4—44no—
419C22×C4—44no—
420C22×C4—44no—
421C22×C4—44no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—44no—
425C22×C4—88no—
426C22×C4—84no—
427C22×C4—88no—
428C22×C4—84no—
429C22×C4—88no—
430C22×C4—88no—
431C22×C4—88no—
432C22×C4C2×C22⋊C411yescharacteristic
433C22×C4C2×C22⋊C411yescharacteristic
434C22×C4C2×C22⋊C411yescharacteristic
435C22×C4C2×C22⋊C412yes—
436C22×C4C2×C22⋊C411yescharacteristic
437C22×C4C2×C22⋊C412yes—
438C22×C4C2×C22⋊C412yes—
439C22×C4C2×C22⋊C412yes—
440C22×C4C2×C22⋊C412yes—
441C22×C4C2×C22⋊C411yescharacteristic
442C22×C4C2×C22⋊C411yescharacteristic
443C22×C4#12012yes—
444C22×C4#12012yes—
445C22×C4#12012yes—
446C22×C4#12012yes—
447C22×C4#12012yes—
448C22×C4#12012yes—
449C22×C4C23×C411yescharacteristic
450C22×C4C22×D412yes—
451C22×C4C22×D412yes—
452C22×C4C22×D411yescharacteristic
453M16—84no—
454M16—84no—
455C2×C8—48no—
456C2×C8—44no—
457C4⋊C4—216no—
458C4⋊C4—216no—
459C4⋊C4—216no—
460C4⋊C4—216no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—28no—
465C4⋊C4—28no—
466C4⋊C4—24no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—28no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—28no—
475C4⋊C4—24no—
476C4⋊C4—48no—
477C4⋊C4C2×C22⋊C414yes—
478C4⋊C4C2×C22⋊C414yes—
479C4⋊C4C2×C22⋊C414yes—
480C22⋊C4—216no—
481C22⋊C4—28no—
482C22⋊C4—24no—
483C22⋊C4—28no—
484C22⋊C4—24no—
485C22⋊C4—24no—
486C22⋊C4—24no—
487C22⋊C4—48no—
488C22⋊C4—84no—
489C22⋊C4—84no—
490C42—216no—
491C42—22no—
492C42—22no—
493C42—22no—
494C42—22no—
495C42—84no—
496C23—22no—
497C23—22no—
498C23—22no—
499C23—24no—
500C23—24no—
501C23—24no—
502C23—22no—
503C23—22no—
504C23—24no—
505C23—24no—
506C23—22no—
507C23—24no—
508C23—41no—
509C23—42no—
510C23—41no—
511C23—41no—
512C23—42no—
513C23—41no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—416no—
519C23—416no—
520C23—42no—
521C23—42no—
522C23#37912yes—
523C23#37911yescharacteristic
524C23#37911yescharacteristic
525C23#41914yes—
526C23C22×C22⋊C411yescharacteristic
527C23C2×#12012yes—
528C23C2×#13714yes—
529Q8—44no—
530Q8—44no—
531Q8—44no—
532Q8—44no—
533Q8—816no—
534D4—216no—
535D4—216no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—24no—
541C2×C4—22no—
542C2×C4—24no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—24no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—22no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—42no—
557C2×C4—41no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—41no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—88no—
586C2×C4—88no—
587C2×C4—816no—
588C2×C4—84no—
589C2×C4—84no—
590C2×C4—84no—
591C2×C4#37911yescharacteristic
592C2×C4#37912yes—
593C2×C4#37911yescharacteristic
594C2×C4#41014yes—
595C2×C4#41914yes—
596C2×C4C22×C22⋊C411yescharacteristic
597C2×C4C22×C22⋊C411yescharacteristic
598C2×C4C2×#12012yes—
599C8—84no—
600C22—28no—
601C22—21no—
602C22—21no—
603C22—22no—
604C22—21no—
605C22—21no—
606C22—21no—
607C22—21no—
608C22—22no—
609C22—22no—
610C22—22no—
611C22—24no—
612C22—22no—
613C22—44no—
614C22—48no—
615C22—48no—
616C22—44no—
617C22—48no—
618C22—44no—
619C22—44no—
620C22—48no—
621C22—42no—
622C22—42no—
623C22#178212yes—
624C22#179512yes—
625C22C2×#37911yescharacteristic
626C22C2×#41912yes—
627C22#300414yes—
628C4—22no—
629C4—21no—
630C4—21no—
631C4—22no—
632C4—24no—
633C4—42no—
634C4—44no—
635C4—44no—
636C4—44no—
637C4—44no—
638C4—42no—
639C4—164no—
640C4#178212yes—
641C4C2×#41012yes—
642C2—24no—
643C2—22no—
644C2—22no—
645C2—22no—
646C2—42no—
647C2—44no—
648C2C2×#178211yescharacteristic
649C2#2059911yescharacteristic
650C2C2×#179511yescharacteristic
subgroupG/H
center, socleC22C2×#379
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #20700C2semidirect
4. #17578C2semidirect
6. #8380C2semidirect
13. C2×#554C4semidirect
15. C22×C4⋊Q8C4semidirect
19. C22×C22⋊Q8C4semidirect
21. C22×C4×Q8C4semidirect
24. C23×C4⋊C4C4semidirect
28. #1824C22semidirect
35. #1520C22semidirect

Maximal subgroups

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

subgroupindex
C22×#5542
#207222
#207002
#175782
#173622
#83802