Finite Group Databasebeta

← #188149

Subgroups — #188149

Order 512.

2552 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#554C211yesmaximal, characteristic
2#21753C212yesmaximal, complemented
3#17362C212yesmaximal
4C2×#1463C212yesmaximal, complemented
5#8555C212yesmaximal, complemented
6#8554C212yesmaximal, complemented
7#8551C212yesmaximal, complemented
8#8550C212yesmaximal, complemented
9C2×#554—28no—
10C2×#554—24no—
11C2×#554—24no—
12C2×#554—24no—
13C2×#554—24no—
14C22×C4⋊Q8C2211yescharacteristic
15C22×#125C411yescharacteristic
16C22×#125C2211yescharacteristic
17C22×C22⋊Q8C411yescharacteristic
18C22×C22⋊Q8C411yescharacteristic
19C22×C22⋊Q8C411yescharacteristic
20C22×C22⋊Q8C2211yescharacteristic
21C22×C22⋊Q8C2211yescharacteristic
22C22×C22⋊Q8C2211yescharacteristic
23C22×C4×Q8C411yescharacteristic
24C22×C4×D4C411yescharacteristic
25C22×C4×D4C411yescharacteristic
26C22×C4×D4C2211yescharacteristic
27C23×C4⋊C4C411yescharacteristic
28C23×C4⋊C4C2211yescharacteristic
29C2×#407C2214yes—
30#1901—28no—
31#1821C2214yes—
32#1784C2214yes—
33#1465—24no—
34#1463—24no—
35#1463C2214yes—
36#1463C2214yes—
37#1463C2214yes—
38#1462C2214yes—
39#1459—24no—
40C2×#324—24no—
41C23×Q8C2×C411yescharacteristic
42C23×Q8C2×C411yescharacteristic
43C23×Q8C2311yescharacteristic
44C23×D4C2×C411yescharacteristic
45C24×C4C2×C411yescharacteristic
46C24×C4C2×C411yescharacteristic
47C24×C4C2×C411yescharacteristic
48C24×C4C2×C411yescharacteristic
49C24×C4C2×C411yescharacteristic
50C24×C4C2311yescharacteristic
51C22×M16—24no—
52#554—232no—
53#554—232no—
54#554—232no—
55#554—232no—
56C2×C4⋊Q8—24no—
57C2×C4⋊Q8—24no—
58C2×C4⋊Q8—24no—
59C2×#125—24no—
60C2×#125—22no—
61C2×#125—22no—
62C2×#125—24no—
63C2×#125—24no—
64C2×#125—24no—
65C2×#125—24no—
66C2×C22⋊Q8—24no—
67C2×C22⋊Q8—24no—
68C2×C22⋊Q8—24no—
69C2×C22⋊Q8—24no—
70C2×C22⋊Q8—22no—
71C2×C22⋊Q8—22no—
72C2×C22⋊Q8—22no—
73C2×C22⋊Q8—22no—
74C2×C22⋊Q8—24no—
75C2×C22⋊Q8—22no—
76C2×C22⋊Q8—22no—
77C2×C22⋊Q8—24no—
78C2×C22⋊Q8—24no—
79C2×C22⋊Q8—24no—
80C2×C22⋊Q8—24no—
81C2×C22⋊Q8—24no—
82C2×C22⋊Q8—22no—
83C2×C22⋊Q8—22no—
84C2×C22⋊Q8—22no—
85C2×C22⋊Q8—22no—
86C2×C22⋊Q8—24no—
87C2×C22⋊Q8—24no—
88C2×C22⋊Q8D418yes—
89C2×C4×Q8—24no—
90C2×C4×Q8—22no—
91C2×C4×Q8—22no—
92C2×C4×Q8—24no—
93C2×C4×D4—22no—
94C2×C4×D4—22no—
95C2×C4×D4—22no—
96C2×C4×D4—22no—
97C2×C4×D4—24no—
98C2×C4×D4—24no—
99C2×C4×D4—24no—
100C2×C4×D4—22no—
101C2×C4×D4—22no—
102C2×C4×D4—24no—
103C2×C4×D4—24no—
104C2×C4×D4D414yes—
105C2×C4×D4D414yes—
106C22×C4⋊C4—24no—
107C22×C4⋊C4—24no—
108C22×C4⋊C4—22no—
109C22×C4⋊C4—22no—
110C22×C4⋊C4—24no—
111C22×C4⋊C4—22no—
112C22×C4⋊C4—22no—
113C22×C4⋊C4C2×C411yescharacteristic
114C22×C4⋊C4C2×C411yescharacteristic
115C22×C4⋊C4C2×C411yescharacteristic
116C22×C4⋊C4C2×C411yescharacteristic
117C22×C4⋊C4C2×C411yescharacteristic
118C22×C4⋊C4C2×C411yescharacteristic
119C22×C4⋊C4C2×C411yescharacteristic
120C22×C4⋊C4C2×C411yescharacteristic
121C22×C4⋊C4C2×C411yescharacteristic
122C22×C4⋊C4C2×C411yescharacteristic
123C22×C4⋊C4C2×C411yescharacteristic
124C22×C4⋊C4C2×C411yescharacteristic
125C22×C4⋊C4D414yes—
126C22×C4⋊C4D414yes—
127C22×C4⋊C4C2311yescharacteristic
128C22×C4⋊C4C2311yescharacteristic
129C22×C4⋊C4C2311yescharacteristic
130C22×C4⋊C4C2311yescharacteristic
131C22×C22⋊C4C2×C411yescharacteristic
132C22×C22⋊C4C2×C411yescharacteristic
133C22×C22⋊C4C2×C411yescharacteristic
134C22×C22⋊C4C2×C411yescharacteristic
135C22×C22⋊C4C2×C411yescharacteristic
136C22×C22⋊C4C2311yescharacteristic
137C22×C42C2×C411yescharacteristic
138C22×C42C2×C411yescharacteristic
139C22×C42C2×C411yescharacteristic
140C2×#101—28no—
141#407—24no—
142#407—24no—
143#407—24no—
144#407C2318yes—
145C2×C22⋊C8—24no—
146C2×C22⋊C8—24no—
147#343—28no—
148#324—28no—
149#324—28no—
150#324—28no—
151#323—28no—
152C25C22×C411yescharacteristic
153C25C22×C411yescharacteristic
154C22×Q8—22no—
155C22×Q8—22no—
156C22×Q8—22no—
157C22×Q8—22no—
158C22×Q8—22no—
159C22×Q8—22no—
160C22×Q8—22no—
161C22×Q8—22no—
162C22×Q8—44no—
163C22×Q8—82no—
164C22×Q8—82no—
165C22×Q8C2×D414yes—
166C22×D4—22no—
167C22×D4—22no—
168C22×D4—28no—
169C22×D4C22⋊C412yes—
170C22×D4C22⋊C412yes—
171C23×C4—22no—
172C23×C4—22no—
173C23×C4—21no—
174C23×C4—21no—
175C23×C4—22no—
176C23×C4—22no—
177C23×C4—21no—
178C23×C4—21no—
179C23×C4—22no—
180C23×C4—22no—
181C23×C4—22no—
182C23×C4—22no—
183C23×C4—22no—
184C23×C4—21no—
185C23×C4—21no—
186C23×C4—44no—
187C23×C4—44no—
188C23×C4—82no—
189C23×C4—82no—
190C23×C4—82no—
191C23×C4—82no—
192C23×C4C22⋊C414yes—
193C23×C4C22⋊C414yes—
194C23×C4C22⋊C414yes—
195C23×C4C22×C411yescharacteristic
196C23×C4C22×C411yescharacteristic
197C23×C4C22×C411yescharacteristic
198C23×C4C22×C411yescharacteristic
199C23×C4C22×C411yescharacteristic
200C23×C4C22×C411yescharacteristic
201C23×C4C22×C411yescharacteristic
202C23×C4C22×C411yescharacteristic
203C23×C4C22×C411yescharacteristic
204C23×C4C22×C411yescharacteristic
205C23×C4C22×C411yescharacteristic
206C23×C4C22×C411yescharacteristic
207C23×C4C2×D414yes—
208C23×C4C2×D412yes—
209C23×C4C2×D412yes—
210C23×C4C2×D412yes—
211C23×C4C2×D412yes—
212C23×C4C2411yescharacteristic
213C2×M16—28no—
214C2×M16—44no—
215C2×M16—88no—
216C22×C8—44no—
217C4⋊Q8—232no—
218#125—232no—
219#125—216no—
220#125—216no—
221C22⋊Q8—216no—
222C22⋊Q8—216no—
223C22⋊Q8—232no—
224C22⋊Q8—232no—
225C22⋊Q8—216no—
226C22⋊Q8—216no—
227C22⋊Q8—216no—
228C22⋊Q8—216no—
229C22⋊Q8—28no—
230C22⋊Q8—28no—
231C22⋊Q8—28no—
232C22⋊Q8—28no—
233C4×Q8—216no—
234C4×Q8—216no—
235C4×D4—216no—
236C4×D4—216no—
237C4×D4—216no—
238C4×D4—216no—
239C4×D4—28no—
240C4×D4—28no—
241C4×D4—28no—
242C4×D4—28no—
243C2×C4⋊C4—216no—
244C2×C4⋊C4—216no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—28no—
280C2×C4⋊C4—28no—
281C2×C4⋊C4—28no—
282C2×C4⋊C4—28no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4C22⋊C412yes—
292C2×C4⋊C4C22⋊C412yes—
293C2×C4⋊C4C22⋊C414yes—
294C2×C4⋊C4C22⋊C412yes—
295C2×C4⋊C4C22⋊C412yes—
296C2×C4⋊C4C22⋊C414yes—
297C2×C4⋊C4C2×D414yes—
298C2×C4⋊C4C2×D414yes—
299C2×C4⋊C4C2×D414yes—
300C2×C4⋊C4C2×D414yes—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—21no—
307C2×C22⋊C4—21no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—21no—
310C2×C22⋊C4—21no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4C22⋊C414yes—
318C2×C22⋊C4C22⋊C414yes—
319C2×C22⋊C4C22⋊C412yes—
320C2×C22⋊C4C22⋊C412yes—
321C2×C22⋊C4C2×D414yes—
322C2×C42—22no—
323C2×C42—22no—
324C2×C42—22no—
325C2×C42—22no—
326C2×C42—21no—
327C2×C42—21no—
328C2×C42—22no—
329C2×C42—22no—
330C2×C42—22no—
331C2×C42C22⋊C414yes—
332#101—88no—
333C22⋊C8—28no—
334C22⋊C8—28no—
335C22⋊C8—44no—
336C22⋊C8—44no—
337C24—21no—
338C24—21no—
339C24—24no—
340C24—24no—
341C24—42no—
342C24—42no—
343C24—42no—
344C24—42no—
345C24C2×C22⋊C412yes—
346C24C2×C22⋊C412yes—
347C24C2×C22⋊C411yescharacteristic
348C24C2×C22⋊C412yes—
349C24C2×C22⋊C411yescharacteristic
350C24C23×C411yescharacteristic
351C2×Q8—28no—
352C2×Q8—28no—
353C2×Q8—28no—
354C2×Q8—48no—
355C2×Q8—48no—
356C2×Q8—48no—
357C2×Q8—44no—
358C2×Q8—44no—
359C2×Q8—84no—
360C2×Q8—84no—
361C2×Q8—84no—
362C2×Q8—84no—
363C2×Q8—84no—
364C2×D4—216no—
365C2×D4—216no—
366C2×D4—24no—
367C2×D4—24no—
368C2×D4—24no—
369C2×D4—24no—
370C2×D4—24no—
371C2×D4#10118yescomplemented
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—28no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—28no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—28no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—48no—
406C22×C4—44no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—44no—
413C22×C4—48no—
414C22×C4—44no—
415C22×C4—48no—
416C22×C4—48no—
417C22×C4—44no—
418C22×C4—44no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—84no—
426C22×C4—84no—
427C22×C4—82no—
428C22×C4—82no—
429C22×C4—82no—
430C22×C4—82no—
431C22×C4—84no—
432C22×C4—82no—
433C22×C4—82no—
434C22×C4—82no—
435C22×C4—82no—
436C22×C4—84no—
437C22×C4—84no—
438C22×C4—84no—
439C22×C4—84no—
440C22×C4C2×C22⋊C412yes—
441C22×C4C2×C22⋊C412yes—
442C22×C4C2×C22⋊C412yes—
443C22×C4C2×C22⋊C412yes—
444C22×C4C2×C22⋊C412yes—
445C22×C4C2×C22⋊C412yes—
446C22×C4C2×C22⋊C412yes—
447C22×C4C2×C22⋊C412yes—
448C22×C4C2×C22⋊C412yes—
449C22×C4C2×C22⋊C412yes—
450C22×C4C2×C22⋊C412yes—
451C22×C4C2×C22⋊C412yes—
452C22×C4C2×C22⋊C411yescharacteristic
453C22×C4C2×C22⋊C411yescharacteristic
454C22×C4C2×C22⋊C412yes—
455C22×C4#12014yes—
456C22×C4#12014yes—
457C22×C4#12014yes—
458C22×C4#12014yes—
459C22×C4C22×D412yes—
460C22×C4C22×D412yes—
461C22×C4C22×D412yes—
462M16—416no—
463M16—88no—
464C2×C8—48no—
465C2×C8—88no—
466C4⋊C4—28no—
467C4⋊C4—28no—
468C4⋊C4—28no—
469C4⋊C4—28no—
470C4⋊C4—28no—
471C4⋊C4—28no—
472C4⋊C4—28no—
473C4⋊C4—28no—
474C4⋊C4—28no—
475C4⋊C4—28no—
476C4⋊C4—28no—
477C4⋊C4—28no—
478C4⋊C4—24no—
479C4⋊C4—24no—
480C4⋊C4—24no—
481C4⋊C4—24no—
482C4⋊C4—24no—
483C4⋊C4—24no—
484C4⋊C4—24no—
485C4⋊C4—24no—
486C22⋊C4—28no—
487C22⋊C4—28no—
488C22⋊C4—24no—
489C22⋊C4—24no—
490C22⋊C4—24no—
491C22⋊C4—24no—
492C22⋊C4—24no—
493C22⋊C4—24no—
494C22⋊C4—28no—
495C42—28no—
496C42—28no—
497C42—24no—
498C42—24no—
499C23—22no—
500C23—22no—
501C23—22no—
502C23—24no—
503C23—24no—
504C23—24no—
505C23—22no—
506C23—22no—
507C23—24no—
508C23—22no—
509C23—24no—
510C23—24no—
511C23—41no—
512C23—42no—
513C23—41no—
514C23—41no—
515C23—42no—
516C23—41no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—416no—
522C23—416no—
523C23—42no—
524C23—42no—
525C23#37912yes—
526C23#37912yes—
527C23C2×#10114yes—
528C23C2×#10114yes—
529C23C22×C22⋊C411yescharacteristic
530C23C22×C22⋊C411yescharacteristic
531C23C22×C22⋊C411yescharacteristic
532Q8—416no—
533Q8—88no—
534Q8—88no—
535D4—216no—
536D4—216no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—48no—
569C2×C4—48no—
570C2×C4—416no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—84no—
596C2×C4—84no—
597C2×C4—84no—
598C2×C4—84no—
599C2×C4—88no—
600C2×C4#37912yes—
601C2×C4#37912yes—
602C2×C4C2×#10114yes—
603C2×C4#41314yes—
604C2×C4C2×#12014yes—
605C8—88no—
606C22—28no—
607C22—21no—
608C22—21no—
609C22—22no—
610C22—21no—
611C22—21no—
612C22—21no—
613C22—21no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22—24no—
618C22—22no—
619C22—44no—
620C22—48no—
621C22—48no—
622C22—44no—
623C22—48no—
624C22—44no—
625C22—44no—
626C22—48no—
627C22—42no—
628C22—42no—
629C22#178414yes—
630C22C2×#37911yescharacteristic
631C22#288312yes—
632C22C22×#10112yes—
633C22C2×#41312yes—
634C4—24no—
635C4—22no—
636C4—22no—
637C4—24no—
638C4—24no—
639C4—42no—
640C4—42no—
641C4—42no—
642C4—42no—
643C4—42no—
644C4—42no—
645C4—42no—
646C4—42no—
647C4—42no—
648C2—22no—
649C2—22no—
650C2—22no—
651C2—24no—
652C2—42no—
653C2—44no—
654C2#2058511yescharacteristic
655C2C2×#178411yescharacteristic
656C2#2059111yescharacteristic
subgroupG/H
center, socleC22C2×#379
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C2×C4
LCSG ▷ C24 ▷ 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
2. #21753C2semidirect
4. C2×#1463C2semidirect
5. #8555C2semidirect
6. #8554C2semidirect
7. #8551C2semidirect
8. #8550C2semidirect
371. C2×D4#101semidirect

Maximal subgroups

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

subgroupindex
C22×#5542
#217532
#173622
C2×#14632
#85552
#85542
#85512
#85502