Finite Group Databasebeta

← #362750

Subgroups — #362750

Order 512.

2582 conjugacy classes of subgroups.

The subgroup lattice has 698 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#544C211yesmaximal, characteristic
2C2×#1989C211yesmaximal, characteristic, complemented
3C2×#1989C211yesmaximal, characteristic, complemented
4C2×#1988C211yesmaximal, characteristic, complemented
5C2×#1987C211yesmaximal, characteristic, complemented
6C2×#1798C211yesmaximal, characteristic
7#13777C211yesmaximal, characteristic
8#11385C212yesmaximal, complemented
9#11384C212yesmaximal, complemented
10#10731C212yesmaximal
11#10730C212yesmaximal
12C2×#544—24no—
13C2×#544—24no—
14C2×#544—24no—
15C2×#544—24no—
16C2×#544—24no—
17C2×#544C418yescomplemented
18C22×C4⋊Q8—21no—
19C22×C4⋊Q8C2211yescharacteristic
20C22×C4⋊Q8C2211yescharacteristic
21C22×#125—21no—
22C22×#125C2211yescharacteristic
23C22×#125C2211yescharacteristic
24C22×C22⋊Q8—21no—
25C22×C22⋊Q8—21no—
26C22×#117C2211yescharacteristic
27C22×#117C2211yescharacteristic
28C23×C4⋊C4C2211yescharacteristic
29C22×C4⋊C8C2211yescharacteristic
30C22×C4⋊C8C2211yescharacteristic
31C22×C22⋊C8C2212yes—
32C2×#380C2212yes—
33C2×C4×C4⋊C4C2211yescharacteristic
34C2×C4×C4⋊C4C2211yescharacteristic
35#1989—24no—
36#1989—24no—
37#1988—24no—
38#1987—24no—
39#1798—24no—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#340C2212yes—
44C2×#340C2212yes—
45C2×#340C2212yes—
46C2×#340C2212yes—
47C2×#340C2212yes—
48C2×#336C2212yes—
49#1437—21no—
50#1437—21no—
51#1436—21no—
52#1436—21no—
53C4⋊#95C2212yes—
54C23×Q8—21no—
55C23×Q8—21no—
56C24×C4—21no—
57C24×C4C2311yescharacteristic
58C23×C8C2312yes—
59C23×C8C2312yes—
60#544—232no—
61#544—232no—
62#544—232no—
63#544—232no—
64C2×C4⋊Q8—28no—
65C2×C4⋊Q8—28no—
66C2×C4⋊Q8—28no—
67C2×C4⋊Q8—22no—
68C2×C4⋊Q8—22no—
69C2×C4⋊Q8—22no—
70C2×C4⋊Q8—22no—
71C2×C4⋊Q8—22no—
72C2×C4⋊Q8—22no—
73C2×C4⋊Q8—22no—
74C2×C4⋊Q8—22no—
75C2×C4⋊Q8C2×C414yes—
76C2×C4⋊Q8C2×C414yes—
77C2×C4⋊Q8D414yes—
78C2×C4⋊Q8D414yes—
79C2×#125—28no—
80C2×#125—28no—
81C2×#125—28no—
82C2×#125—22no—
83C2×#125—22no—
84C2×#125—22no—
85C2×#125—22no—
86C2×#125—22no—
87C2×#125—22no—
88C2×#125—22no—
89C2×#125—22no—
90C2×#125C2×C414yes—
91C2×#125C2×C414yes—
92C2×#125D414yes—
93C2×#125D414yes—
94C2×C22⋊Q8—28no—
95C2×C22⋊Q8—28no—
96C2×C22⋊Q8—28no—
97C2×C22⋊Q8—28no—
98C2×C22⋊Q8—28no—
99C2×C22⋊Q8—28no—
100C2×#117—22no—
101C2×#117—22no—
102C2×#117—22no—
103C2×#117—22no—
104C2×#117—22no—
105C2×#117—22no—
106C2×#117—22no—
107C2×#117—22no—
108C2×#117—22no—
109C2×#117C2×C414yes—
110C2×#117C2×C414yes—
111C2×#117D414yes—
112C22×C4⋊C4—21no—
113C22×C4⋊C4—22no—
114C22×C4⋊C4—21no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4—21no—
117C22×C4⋊C4—21no—
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⋊C4D411yescharacteristic
126C22×C4⋊C4D411yescharacteristic
127C22×C4⋊C4C2312yes—
128C22×C4⋊C4C2312yes—
129C22×C4⋊C4C2311yescharacteristic
130C22×C4⋊C4C2311yescharacteristic
131C22×C4⋊C4C2311yescharacteristic
132C22×C4⋊C4C2311yescharacteristic
133C22×C22⋊C4—21no—
134C22×C22⋊C4—22no—
135C22×C22⋊C4D411yescharacteristic
136C22×C22⋊C4D411yescharacteristic
137C22×C42—21no—
138C22×C42—21no—
139C22×C42—21no—
140C22×C42C2311yescharacteristic
141C22×C42C2311yescharacteristic
142C2×C4⋊C8—22no—
143C2×C4⋊C8—22no—
144C2×C4⋊C8—22no—
145C2×C4⋊C8—22no—
146C2×C4⋊C8—22no—
147C2×C4⋊C8—22no—
148C2×C22⋊C8—24no—
149C2×C22⋊C8—28no—
150#380—24no—
151#380Q818yescomplemented
152C4×C4⋊C4—22no—
153C4×C4⋊C4—22no—
154C4×C4⋊C4—22no—
155C4×C4⋊C4—22no—
156C2×#95—22no—
157C2×#95—21no—
158C2×#95—21no—
159C2×#95—22no—
160C2×#95—22no—
161#340—24no—
162#340—24no—
163#340—24no—
164#340—24no—
165#340—24no—
166#340—24no—
167#340—24no—
168#340—24no—
169#336—24no—
170C25C2×D411yescharacteristic
171C22×Q8—24no—
172C22×Q8—24no—
173C22×Q8—24no—
174C22×Q8—24no—
175C22×Q8—24no—
176C22×Q8—24no—
177C22×Q8—82no—
178C22×Q8—82no—
179C22×Q8—82no—
180C22×Q8—82no—
181C23×C4—21no—
182C23×C4—21no—
183C23×C4—21no—
184C23×C4—21no—
185C23×C4—22no—
186C23×C4—22no—
187C23×C4—21no—
188C23×C4—21no—
189C23×C4—21no—
190C23×C4—21no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—22no—
195C23×C4—22no—
196C23×C4—82no—
197C23×C4—82no—
198C23×C4C22×C412yes—
199C23×C4C2×D411yescharacteristic
200C23×C4C2×D411yescharacteristic
201C23×C4C2×D411yescharacteristic
202C23×C4C2×D411yescharacteristic
203C23×C4C2×D411yescharacteristic
204C23×C4C2×D412yes—
205C23×C4C2×Q814yes—
206C23×C4#4114yes—
207C23×C4C2411yescharacteristic
208C22×C8—22no—
209C22×C8—22no—
210C22×C8—22no—
211C22×C8—22no—
212C22×C8—22no—
213C22×C8—22no—
214C22×C8—24no—
215C22×C8—24no—
216C22×C8—24no—
217C22×C8—24no—
218C4⋊Q8—264no—
219C4⋊Q8—28no—
220C4⋊Q8—28no—
221C4⋊Q8—28no—
222C4⋊Q8—28no—
223C4⋊Q8—28no—
224C4⋊Q8—28no—
225C4⋊Q8—28no—
226C4⋊Q8C22⋊C4116yescomplemented
227#125—264no—
228#125—28no—
229#125—28no—
230#125—28no—
231#125—28no—
232#125—28no—
233#125—28no—
234#125—28no—
235#125C22⋊C4116yescomplemented
236C22⋊Q8—264no—
237C22⋊Q8—264no—
238#117—28no—
239#117—28no—
240#117—28no—
241#117—28no—
242#117—28no—
243#117—28no—
244#117—28no—
245#117—28no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
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—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
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—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—28no—
286C2×C4⋊C4—28no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4C22⋊C414yes—
298C2×C4⋊C4C22⋊C414yes—
299C2×C4⋊C4C22×C412yes—
300C2×C4⋊C4C22×C412yes—
301C2×C4⋊C4C22×C412yes—
302C2×C4⋊C4C22×C412yes—
303C2×C4⋊C4C2×D412yes—
304C2×C4⋊C4C2×D412yes—
305C2×C4⋊C4C2×D412yes—
306C2×C4⋊C4C2×D412yes—
307C2×C4⋊C4C2×D412yes—
308C2×C4⋊C4C2×D412yes—
309C2×C4⋊C4C2×D412yes—
310C2×C4⋊C4C2×D412yes—
311C2×C4⋊C4C2×Q814yes—
312C2×C4⋊C4C2×Q814yes—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—28no—
321C2×C22⋊C4—82no—
322C2×C22⋊C4—82no—
323C2×C22⋊C4C22⋊C414yes—
324C2×C22⋊C4C22⋊C414yes—
325C2×C42—24no—
326C2×C42—24no—
327C2×C42—24no—
328C2×C42—21no—
329C2×C42—21no—
330C2×C42—21no—
331C2×C42—21no—
332C2×C42—42no—
333C2×C42—42no—
334C2×C42—82no—
335C2×C42—82no—
336C2×C42C22×C412yes—
337C2×C42C22×C412yes—
338C2×C42C2×D412yes—
339C2×C42C2×D412yes—
340C2×C42C2×D412yes—
341C2×C42C2×D412yes—
342C2×C42C2×D412yes—
343C2×C42C2×D412yes—
344C4⋊C8—44no—
345C4⋊C8—44no—
346C22⋊C8—48no—
347#95—28no—
348#95—42no—
349#95—42no—
350#95—82no—
351#95—82no—
352C24—21no—
353C24—21no—
354C24—22no—
355C24—22no—
356C24—22no—
357C24—22no—
358C24—22no—
359C24C2×C22⋊C412yes—
360C24C22⋊Q814yes—
361C24C22×D411yescharacteristic
362C2×Q8—216no—
363C2×Q8—216no—
364C2×Q8—88no—
365C2×Q8—84no—
366C2×Q8—84no—
367C2×Q8—88no—
368C2×Q8—88no—
369C2×Q8—84no—
370C2×Q8—84no—
371C2×Q8—88no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—216no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
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—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—88no—
438C22×C4—84no—
439C22×C4—84no—
440C22×C4—88no—
441C22×C4C2×C22⋊C412yes—
442C22×C4C2×C22⋊C412yes—
443C22×C4C2×C22⋊C412yes—
444C22×C4C2×C22⋊C412yes—
445C22×C4C2×C22⋊C412yes—
446C22×C4C4×Q818yes—
447C22×C4#12012yes—
448C22×C4#12012yes—
449C22×C4#12012yes—
450C22×C4#12012yes—
451C22×C4C22⋊Q814yes—
452C22×C4#12414yes—
453C22×C4C4⋊Q814yes—
454C22×C4C23×C411yescharacteristic
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D411yescharacteristic
460C22×C4C22×Q812yes—
461C22×C4C2×#4112yes—
462C22×C42−1+412yes—
463C22×C42−1+412yes—
464C2×C8—42no—
465C2×C8—42no—
466C2×C8—42no—
467C2×C8—42no—
468C2×C8—42no—
469C2×C8—42no—
470C2×C8—44no—
471C2×C8—44no—
472C2×C8—44no—
473C2×C8—44no—
474C4⋊C4—216no—
475C4⋊C4—216no—
476C4⋊C4—216no—
477C4⋊C4—216no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—216no—
491C4⋊C4—216no—
492C4⋊C4—216no—
493C4⋊C4—28no—
494C4⋊C4—28no—
495C4⋊C4—24no—
496C4⋊C4—24no—
497C4⋊C4—24no—
498C4⋊C4—24no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4C2×C22⋊C414yes—
504C4⋊C4C2×C22⋊C414yes—
505C4⋊C4C2×C22⋊C414yes—
506C4⋊C4C2×C22⋊C414yes—
507C22⋊C4—216no—
508C22⋊C4—28no—
509C22⋊C4—28no—
510C22⋊C4—88no—
511C42—216no—
512C42—22no—
513C42—22no—
514C42—22no—
515C42—22no—
516C42—84no—
517C42—84no—
518C42C2×C22⋊C414yes—
519C42C2×C22⋊C414yes—
520C42#12014yes—
521C42#12014yes—
522C23—22no—
523C23—24no—
524C23—24no—
525C23—24no—
526C23—24no—
527C23—24no—
528C23—24no—
529C23—24no—
530C23—24no—
531C23—22no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—42no—
541C23—42no—
542C23#39114yes—
543C23C22×C22⋊C411yescharacteristic
544C23C2×#12011yescharacteristic
545C23C2×#12011yescharacteristic
546C23C2×C22⋊Q812yes—
547C23#53612yes—
548C23#57612yes—
549C23#57712yes—
550C23#57712yes—
551C23#57812yes—
552Q8—816no—
553Q8—816no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—22no—
566C2×C4—28no—
567C2×C4—28no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—22no—
575C2×C4—24no—
576C2×C4—22no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—816no—
609C2×C4—82no—
610C2×C4—82no—
611C2×C4—82no—
612C2×C4—82no—
613C2×C4—82no—
614C2×C4—82no—
615C2×C4—82no—
616C2×C4—82no—
617C2×C4#37914yes—
618C2×C4#37914yes—
619C2×C4#39114yes—
620C2×C4C22×C22⋊C411yescharacteristic
621C2×C4C22×C22⋊C411yescharacteristic
622C2×C4C2×C4×Q812yes—
623C2×C4#51912yes—
624C2×C4C2×#12011yescharacteristic
625C2×C4C2×#12011yescharacteristic
626C2×C4C2×#12411yescharacteristic
627C2×C4C2×C4⋊Q811yescharacteristic
628C2×C4#53611yescharacteristic
629C2×C4#53611yescharacteristic
630C2×C4#54812yes—
631C2×C4#54812yes—
632C2×C4#54812yes—
633C2×C4#54812yes—
634C2×C4D4×Q812yes—
635C2×C4D4×Q812yes—
636C2×C4D4×Q812yes—
637C2×C4D4×Q812yes—
638C8—82no—
639C8—82no—
640C22—28no—
641C22—24no—
642C22—44no—
643C22—42no—
644C22—42no—
645C22—44no—
646C22—44no—
647C22—44no—
648C22—44no—
649C22C2×#37911yescharacteristic
650C22C2×#39111yescharacteristic
651C22#231011yescharacteristic
652C22#240912yes—
653C22#240912yes—
654C22#289712yes—
655C22#289812yes—
656C22#300811yescharacteristic
657C22#300911yescharacteristic
658C22#301011yescharacteristic
659C22#301111yescharacteristic
660C22#301212yes—
661C22#301312yes—
662C22#308112yes—
663C22#308212yes—
664C22#308312yes—
665C22#308412yes—
666C22#324112yes—
667C22#324212yes—
668C22#324312yes—
669C22#324412yes—
670C4—24no—
671C4—24no—
672C4—24no—
673C4—42no—
674C4—44no—
675C4—44no—
676C4—44no—
677C4—42no—
678C4—44no—
679C4—84no—
680C4—84no—
681C4Q8×C22⋊C412yes—
682C4Q8×C22⋊C412yes—
683C4#243411yescharacteristic
684C4#243411yescharacteristic
685C4#243611yescharacteristic
686C4#243611yescharacteristic
687C2—44no—
688C2#1392211yescharacteristic
689C2#2060211yescharacteristic
690C2#2060311yescharacteristic
691C2#2131512yes—
692C2#2184312yes—
693C2#2430812yes—
694C2#2430912yes—
695C2#2431012yes—
696C2#2431112yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1989C2semidirect
3. C2×#1989C2semidirect
4. C2×#1988C2semidirect
5. C2×#1987C2semidirect
8. #11385C2semidirect
9. #11384C2semidirect
17. C2×#544C4semidirect
151. #380Q8semidirect
226. C4⋊Q8C22⋊C4semidirect
235. #125C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#5442
C2×#19892
C2×#19882
C2×#19872
C2×#17982
#137772
#113852
#113842
#107312
#107302