Finite Group Databasebeta

← #358776

Subgroups — #358776

Order 512.

2970 conjugacy classes of subgroups.

The subgroup lattice has 665 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#544C211yesmaximal, characteristic
2C2×#1736C211yesmaximal, characteristic
3C2×#1736C211yesmaximal, characteristic
4C2×#1735C211yesmaximal, characteristic
5C22×#340C211yesmaximal, characteristic
6C2×#1729C211yesmaximal, characteristic
7#13795C211yesmaximal, characteristic
8#11385C214yesmaximal, complemented
9#11384C214yesmaximal, complemented
10C23×M16C2211yescharacteristic
11C24×C8C2211yescharacteristic
12C2×#544—24no—
13C2×#544—24no—
14C2×#544—24no—
15C2×#544—24no—
16C2×#544—24no—
17C2×#544C418yescomplemented
18C22×C4⋊Q8—21no—
19C22×C4⋊Q8C2212yes—
20C22×#125—21no—
21C22×#125C2212yes—
22C22×C22⋊Q8—22no—
23C22×#117C2211yescharacteristic
24C22×#117C2211yescharacteristic
25C22×#117C2211yescharacteristic
26C23×C4⋊C4C2211yescharacteristic
27C23×C4⋊C4C2211yescharacteristic
28#1736—24no—
29#1736—24no—
30#1735—24no—
31C2×#340—24no—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2214yes—
35C2×#340C2214yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38#1729—24no—
39C2×#336C2214yes—
40C4⋊#95—21no—
41C4⋊#95—21no—
42C4⋊#95C2214yes—
43C22⋊#95—22no—
44C23×Q8—22no—
45C24×C4—21no—
46C24×C4—21no—
47C24×C4C2311yescharacteristic
48C22×M16—22no—
49C22×M16—22no—
50C22×M16—22no—
51C22×M16—22no—
52C22×M16—22no—
53C22×M16—22no—
54C22×M16Q814yescomplemented
55C23×C8—22no—
56C23×C8—22no—
57C23×C8—22no—
58C23×C8—22no—
59C23×C8—22no—
60C23×C8—22no—
61C23×C8Q814yescomplemented
62C23×C8C2312yes—
63C23×C8C2312yes—
64#544—232no—
65#544—232no—
66#544—232no—
67#544—232no—
68C2×C4⋊Q8—28no—
69C2×C4⋊Q8—28no—
70C2×C4⋊Q8—28no—
71C2×C4⋊Q8—24no—
72C2×C4⋊Q8—24no—
73C2×C4⋊Q8—24no—
74C2×C4⋊Q8—24no—
75C2×C4⋊Q8—24no—
76C2×C4⋊Q8C2×C418yescomplemented
77C2×#125—28no—
78C2×#125—28no—
79C2×#125—28no—
80C2×#125—24no—
81C2×#125—24no—
82C2×#125—24no—
83C2×#125—24no—
84C2×#125—24no—
85C2×#125C2×C418yescomplemented
86C2×C22⋊Q8—216no—
87C2×C22⋊Q8—216no—
88C2×C22⋊Q8—216no—
89C2×#117—22no—
90C2×#117—22no—
91C2×#117—22no—
92C2×#117—22no—
93C2×#117—22no—
94C2×#117—22no—
95C2×#117—22no—
96C2×#117—22no—
97C2×#117—22no—
98C2×#117—22no—
99C2×#117—22no—
100C2×#117—22no—
101C2×#117C2×C414yes—
102C2×#117C2×C414yes—
103C2×#117D414yes—
104C2×#117D414yes—
105C2×#117D414yes—
106C2×#117Q814yescomplemented
107C22×C4⋊C4—21no—
108C22×C4⋊C4—22no—
109C22×C4⋊C4—22no—
110C22×C4⋊C4—22no—
111C22×C4⋊C4—22no—
112C22×C4⋊C4—22no—
113C22×C4⋊C4—22no—
114C22×C4⋊C4—22no—
115C22×C4⋊C4—22no—
116C22×C4⋊C4—22no—
117C22×C4⋊C4—22no—
118C22×C4⋊C4—21no—
119C22×C4⋊C4—22no—
120C22×C4⋊C4C2×C414yes—
121C22×C4⋊C4D411yescharacteristic
122C22×C4⋊C4D411yescharacteristic
123C22×C4⋊C4D414yes—
124C22×C4⋊C4D414yes—
125C22×C4⋊C4Q814yescomplemented
126C22×C4⋊C4C2312yes—
127C22×C4⋊C4C2312yes—
128C22×C4⋊C4C2312yes—
129C22×C4⋊C4C2311yescharacteristic
130C22×C4⋊C4C2311yescharacteristic
131C22×C22⋊C4—21no—
132C22×C22⋊C4—21no—
133C22×C22⋊C4D412yes—
134C22×C42—21no—
135C22×C42—21no—
136C22×C42C2312yes—
137C2×#95—24no—
138C2×#95—24no—
139#340—28no—
140#340—24no—
141#340—24no—
142#340—24no—
143#340—24no—
144#340—28no—
145#336—28no—
146C25C2×D411yescharacteristic
147C22×Q8—28no—
148C22×Q8—28no—
149C22×Q8—28no—
150C22×Q8—84no—
151C22×Q8—84no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—22no—
155C23×C4—24no—
156C23×C4—24no—
157C23×C4—24no—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4—24no—
161C23×C4—42no—
162C23×C4—42no—
163C23×C4—82no—
164C23×C4—82no—
165C23×C4—82no—
166C23×C4—82no—
167C23×C4C22×C412yes—
168C23×C4C2×D412yes—
169C23×C4C2×D412yes—
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D412yes—
172C23×C4C2×D412yes—
173C23×C4C2×D412yes—
174C23×C4C2×D412yes—
175C23×C4C2×D412yes—
176C23×C4C2×Q812yes—
177C23×C4C2×Q812yes—
178C23×C4C2×Q812yes—
179C23×C4C2×Q812yes—
180C23×C4#4112yes—
181C23×C4#4112yes—
182C23×C4#4112yes—
183C23×C4#4112yes—
184C23×C4C2411yescharacteristic
185C2×M16—28no—
186C2×M16—28no—
187C2×M16—28no—
188C2×M16—44no—
189C2×M16—44no—
190C2×M16—44no—
191C2×M16—44no—
192C22×C8—28no—
193C22×C8—28no—
194C22×C8—28no—
195C22×C8—22no—
196C22×C8—22no—
197C22×C8—22no—
198C22×C8—22no—
199C22×C8—22no—
200C22×C8—22no—
201C22×C8—22no—
202C22×C8—22no—
203C22×C8—22no—
204C22×C8—22no—
205C22×C8—22no—
206C22×C8—22no—
207C22×C8—44no—
208C22×C8—44no—
209C22×C8—44no—
210C22×C8—44no—
211C22×C8C2×Q814yes—
212C22×C8C2×Q814yes—
213C4⋊Q8—216no—
214C4⋊Q8—216no—
215C4⋊Q8—264no—
216C4⋊Q8—216no—
217C4⋊Q8—216no—
218#125—216no—
219#125—216no—
220#125—264no—
221#125—216no—
222#125—216no—
223C22⋊Q8—2128no—
224#117—28no—
225#117—28no—
226#117—28no—
227#117—28no—
228#117—28no—
229#117—28no—
230#117—28no—
231#117—28no—
232#117—28no—
233#117—44no—
234#117—44no—
235#117C22⋊C4116yes—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
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—28no—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—42no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—42no—
281C2×C4⋊C4C22⋊C414yes—
282C2×C4⋊C4C22⋊C414yes—
283C2×C4⋊C4C22⋊C4116yes—
284C2×C4⋊C4C22×C414yes—
285C2×C4⋊C4C22×C412yes—
286C2×C4⋊C4C22×C412yes—
287C2×C4⋊C4C2×D414yes—
288C2×C4⋊C4C2×D414yes—
289C2×C4⋊C4C2×D412yes—
290C2×C4⋊C4C2×D412yes—
291C2×C4⋊C4C2×D412yes—
292C2×C4⋊C4C2×D412yes—
293C2×C4⋊C4C2×Q814yes—
294C2×C4⋊C4C2×Q814yes—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—42no—
302C2×C22⋊C4—42no—
303C2×C22⋊C4C22⋊C418yes—
304C2×C42—24no—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—22no—
308C2×C42—22no—
309C2×C42—22no—
310C2×C42—22no—
311C2×C42—24no—
312C2×C42—42no—
313C2×C42—42no—
314C2×C42C22×C414yes—
315C2×C42C2×D414yes—
316#95—84no—
317#95—84no—
318C24—22no—
319C24—22no—
320C24—22no—
321C24—22no—
322C24C2×C22⋊C412yes—
323C24#12012yes—
324C24#12012yes—
325C24C22⋊Q812yes—
326C24C22⋊Q812yes—
327C24#12412yes—
328C24C4⋊Q812yes—
329C24C22×D411yescharacteristic
330C2×Q8—232no—
331C2×Q8—816no—
332C2×Q8—88no—
333C2×Q8—88no—
334C2×Q8—816no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—24no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—216no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—44no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—44no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—44no—
367C22×C4—42no—
368C22×C4—44no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—48no—
372C22×C4—88no—
373C22×C4—84no—
374C22×C4—84no—
375C22×C4—88no—
376C22×C4—88no—
377C22×C4—88no—
378C22×C4—84no—
379C22×C4—84no—
380C22×C4C2×C22⋊C414yes—
381C22×C4C2×C22⋊C414yes—
382C22×C4C2×C22⋊C414yes—
383C22×C4C2×C22⋊C414yes—
384C22×C4C2×C22⋊C412yes—
385C22×C4C4×Q814yes—
386C22×C4C4×Q814yes—
387C22×C4C4×Q814yes—
388C22×C4C4×Q814yes—
389C22×C4#12014yes—
390C22×C4#12014yes—
391C22×C4#12012yes—
392C22×C4#12012yes—
393C22×C4C22⋊Q814yes—
394C22×C4C22⋊Q814yes—
395C22×C4C22⋊Q814yes—
396C22×C4C22⋊Q814yes—
397C22×C4C22⋊Q814yes—
398C22×C4C22⋊Q814yes—
399C22×C4C22⋊Q814yes—
400C22×C4C22⋊Q814yes—
401C22×C4C22⋊Q814yes—
402C22×C4C22⋊Q814yes—
403C22×C4C22⋊Q812yes—
404C22×C4C22⋊Q812yes—
405C22×C4#12414yes—
406C22×C4#12414yes—
407C22×C4#12414yes—
408C22×C4#12414yes—
409C22×C4#12414yes—
410C22×C4#12412yes—
411C22×C4C4⋊Q814yes—
412C22×C4C4⋊Q814yes—
413C22×C4C4⋊Q814yes—
414C22×C4C4⋊Q814yes—
415C22×C4C4⋊Q814yes—
416C22×C4C4⋊Q812yes—
417C22×C4C23×C411yescharacteristic
418C22×C4C22×D411yescharacteristic
419C22×C4C22×D411yescharacteristic
420C22×C4C22×D411yescharacteristic
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D411yescharacteristic
423C22×C4C22×Q811yescharacteristic
424C22×C4C22×Q811yescharacteristic
425C22×C4C22×Q811yescharacteristic
426C22×C4C22×Q811yescharacteristic
427C22×C4C2×#4111yescharacteristic
428C22×C4C2×#4111yescharacteristic
429C22×C4C2×#4111yescharacteristic
430C22×C4C2×#4111yescharacteristic
431M16—416no—
432C2×C8—24no—
433C2×C8—24no—
434C2×C8—24no—
435C2×C8—24no—
436C2×C8—24no—
437C2×C8—24no—
438C2×C8—42no—
439C2×C8—42no—
440C2×C8—42no—
441C2×C8—42no—
442C2×C8—42no—
443C2×C8—42no—
444C2×C8—42no—
445C2×C8—42no—
446C2×C8—416no—
447C4⋊C4—232no—
448C4⋊C4—232no—
449C4⋊C4—232no—
450C4⋊C4—24no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—24no—
454C4⋊C4—24no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—216no—
460C4⋊C4—28no—
461C4⋊C4—28no—
462C4⋊C4—24no—
463C4⋊C4—24no—
464C4⋊C4—24no—
465C4⋊C4—24no—
466C4⋊C4—42no—
467C4⋊C4—42no—
468C4⋊C4—42no—
469C4⋊C4—42no—
470C4⋊C4—48no—
471C4⋊C4C2×C22⋊C418yes—
472C4⋊C4C2×C22⋊C414yes—
473C4⋊C4C2×C22⋊C414yes—
474C22⋊C4—216no—
475C22⋊C4—216no—
476C22⋊C4—48no—
477C42—24no—
478C42—24no—
479C42—24no—
480C42—24no—
481C42—216no—
482C42—48no—
483C23—24no—
484C23—24no—
485C23—24no—
486C23—24no—
487C23—24no—
488C23—24no—
489C23—24no—
490C23—24no—
491C23—24no—
492C23—24no—
493C23—22no—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—24no—
498C23—24no—
499C23#37914yes—
500C23#39114yes—
501C23#39114yes—
502C23#39314yes—
503C23#39314yes—
504C23#39514yes—
505C23#39514yes—
506C23C22×C22⋊C411yescharacteristic
507C23C2×#12011yescharacteristic
508C23C2×#12011yescharacteristic
509C23C2×C22⋊Q811yescharacteristic
510C23C2×C22⋊Q811yescharacteristic
511C23C2×#12411yescharacteristic
512C23C2×C4⋊Q811yescharacteristic
513C23#57612yes—
514C23#57712yes—
515C23#57712yes—
516C23#57812yes—
517Q8—832no—
518C2×C4—28no—
519C2×C4—28no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—28no—
525C2×C4—48no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—42no—
529C2×C4—44no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—44no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—44no—
539C2×C4—48no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—48no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—816no—
549C2×C4—816no—
550C2×C4—82no—
551C2×C4—84no—
552C2×C4—82no—
553C2×C4—82no—
554C2×C4—82no—
555C2×C4—82no—
556C2×C4—82no—
557C2×C4#37914yes—
558C2×C4#39114yes—
559C2×C4#39114yes—
560C2×C4#39118yes—
561C2×C4#39118yes—
562C2×C4#39118yes—
563C2×C4#39118yes—
564C2×C4#39318yes—
565C2×C4#39318yes—
566C2×C4#39314yes—
567C2×C4#39314yes—
568C2×C4#39518yes—
569C2×C4#39518yes—
570C2×C4#39514yes—
571C2×C4#39514yes—
572C2×C4C22×C22⋊C411yescharacteristic
573C2×C4C22×C22⋊C411yescharacteristic
574C2×C4C2×C4×Q811yescharacteristic
575C2×C4C2×C4×Q811yescharacteristic
576C2×C4C2×C4×Q811yescharacteristic
577C2×C4C2×C4×Q811yescharacteristic
578C2×C4C2×#12011yescharacteristic
579C2×C4C2×#12011yescharacteristic
580C2×C4C2×C22⋊Q811yescharacteristic
581C2×C4C2×C22⋊Q811yescharacteristic
582C2×C4C2×C22⋊Q811yescharacteristic
583C2×C4C2×C22⋊Q811yescharacteristic
584C2×C4C2×C22⋊Q811yescharacteristic
585C2×C4C2×C22⋊Q811yescharacteristic
586C2×C4C2×C22⋊Q811yescharacteristic
587C2×C4C2×C22⋊Q811yescharacteristic
588C2×C4C2×C22⋊Q811yescharacteristic
589C2×C4C2×C22⋊Q811yescharacteristic
590C2×C4C2×#12411yescharacteristic
591C2×C4C2×#12411yescharacteristic
592C2×C4C2×#12411yescharacteristic
593C2×C4C2×#12411yescharacteristic
594C2×C4C2×#12411yescharacteristic
595C2×C4C2×C4⋊Q811yescharacteristic
596C2×C4C2×C4⋊Q811yescharacteristic
597C2×C4C2×C4⋊Q811yescharacteristic
598C2×C4C2×C4⋊Q811yescharacteristic
599C2×C4C2×C4⋊Q811yescharacteristic
600C8—44no—
601C8—44no—
602C22—24no—
603C22—24no—
604C22—28no—
605C22—28no—
606C22—28no—
607C22—24no—
608C22—24no—
609C22—24no—
610C22—24no—
611C22—28no—
612C22#144618yes—
613C22C2×#37911yescharacteristic
614C22C2×#39111yescharacteristic
615C22C2×#39111yescharacteristic
616C22C2×#39311yescharacteristic
617C22C2×#39311yescharacteristic
618C22C2×#39511yescharacteristic
619C22C2×#39511yescharacteristic
620C22#289712yes—
621C22#289812yes—
622C22#300811yescharacteristic
623C22#300911yescharacteristic
624C22#301011yescharacteristic
625C22#301111yescharacteristic
626C22#301212yes—
627C22#301312yes—
628C22#308112yes—
629C22#308212yes—
630C22#308312yes—
631C22#308412yes—
632C22#313612yes—
633C22#313712yes—
634C22#316312yes—
635C22#316412yes—
636C4—44no—
637C4—48no—
638C4—44no—
639C4—44no—
640C4—44no—
641C4—84no—
642C4—84no—
643C4#144618yes—
644C4C2×#39111yescharacteristic
645C4C2×#39111yescharacteristic
646C4C2×#39111yescharacteristic
647C4C2×#39111yescharacteristic
648C4C2×#39311yescharacteristic
649C4C2×#39311yescharacteristic
650C4C2×#39511yescharacteristic
651C4C2×#39511yescharacteristic
652C2—28no—
653C2C2×#144611yescharacteristic
654C2#2060211yescharacteristic
655C2#2060311yescharacteristic
656C2#2131512yes—
657C2#2132612yes—
658C2#2345611yescharacteristic
659C2#2345711yescharacteristic
660C2#2345812yes—
661C2#2355511yescharacteristic
662C2#2355611yescharacteristic
663C2#2355712yes—
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
8. #11385C2semidirect
9. #11384C2semidirect
17. C2×#544C4semidirect
54. C22×M16Q8semidirect
61. C23×C8Q8semidirect
76. C2×C4⋊Q8C2×C4semidirect
85. C2×#125C2×C4semidirect
106. C2×#117Q8semidirect
125. C22×C4⋊C4Q8semidirect

Maximal subgroups

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

subgroupindex
C22×#5442
C2×#17362
C2×#17352
C22×#3402
C2×#17292
#137952
#113852
#113842