Finite Group Databasebeta

← #188188

Subgroups — #188188

Order 512.

2422 conjugacy classes of subgroups.

The subgroup lattice has 659 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#552C211yesmaximal, characteristic
2#20717C212yesmaximal, complemented
3C2×#1470C212yesmaximal, complemented
4C2×#1464C211yesmaximal, characteristic, complemented
5C2×#1458C211yesmaximal, characteristic, complemented
6#8309C212yesmaximal, complemented
7#8304C212yesmaximal
8#8303C212yesmaximal, complemented
9#8299C212yesmaximal, complemented
10C2×#552—28no—
11C2×#552—24no—
12C2×#552—24no—
13C2×#552—24no—
14C2×#552—24no—
15C22×#126C411yescharacteristic
16C22×#126C2211yescharacteristic
17C22×#125C411yescharacteristic
18C22×#125C411yescharacteristic
19C22×#125C2211yescharacteristic
20C22×#123C411yescharacteristic
21C22×#123C2211yescharacteristic
22C22×C22⋊Q8C411yescharacteristic
23C22×C22⋊Q8C411yescharacteristic
24C22×C4×Q8C2211yescharacteristic
25C22×C4×D4C411yescharacteristic
26C22×#117C411yescharacteristic
27C22×#117C2211yescharacteristic
28C22×#117C2211yescharacteristic
29C23×C4⋊C4C2211yescharacteristic
30C2×#432C2212yes—
31C2×#431C2212yes—
32#1817C2214yes—
33#1798—24no—
34#1779C2214yes—
35#1750—24no—
36#1470—28no—
37#1464—24no—
38#1458—24no—
39C2×#324C2212yes—
40C2×#324C2212yes—
41C2×#324C2214yes—
42C2×#324C2214yes—
43C2×#323C2212yes—
44C2×#323C2212yes—
45C23×Q8C2×C411yescharacteristic
46C23×D4C2×C411yescharacteristic
47C24×C4C2×C411yescharacteristic
48C24×C4C2×C411yescharacteristic
49C24×C4C2×C411yescharacteristic
50C24×C4C2311yescharacteristic
51#552—232no—
52#552—232no—
53#552—232no—
54#552—232no—
55C2×#126—24no—
56C2×#126—24no—
57C2×#126—24no—
58C2×#126—24no—
59C2×#126—22no—
60C2×#126—22no—
61C2×#126—24no—
62C2×#125—24no—
63C2×#125—22no—
64C2×#125—22no—
65C2×#125—24no—
66C2×#125—24no—
67C2×#125—24no—
68C2×#125—24no—
69C2×#125—22no—
70C2×#125—22no—
71C2×#125—22no—
72C2×#125—22no—
73C2×#125D418yes—
74C2×#123—24no—
75C2×#123—24no—
76C2×#123—24no—
77C2×#123—24no—
78C2×#123—24no—
79C2×#123—22no—
80C2×#123—22no—
81C2×C22⋊Q8—24no—
82C2×C22⋊Q8—24no—
83C2×C22⋊Q8—24no—
84C2×C22⋊Q8—24no—
85C2×C22⋊Q8—22no—
86C2×C22⋊Q8—22no—
87C2×C22⋊Q8—24no—
88C2×C4×Q8—24no—
89C2×C4×Q8—22no—
90C2×C4×Q8—22no—
91C2×C4×Q8—24no—
92C2×C4×D4—24no—
93C2×C4×D4—24no—
94C2×C4×D4—22no—
95C2×C4×D4—22no—
96C2×#117—24no—
97C2×#117—22no—
98C2×#117—22no—
99C2×#117—24no—
100C2×#117—24no—
101C2×#117—22no—
102C2×#117—22no—
103C2×#117—24no—
104C2×#117—22no—
105C2×#117—22no—
106C2×#117D414yes—
107C2×#117D414yes—
108C2×#117D414yes—
109C2×#117D414yes—
110C22×C4⋊C4—24no—
111C22×C4⋊C4—24no—
112C22×C4⋊C4—22no—
113C22×C4⋊C4—22no—
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⋊C4C2×C411yescharacteristic
126C22×C4⋊C4C2×C411yescharacteristic
127C22×C4⋊C4C2311yescharacteristic
128C22×C4⋊C4C2311yescharacteristic
129C22×C4⋊C4C2311yescharacteristic
130C22×C22⋊C4C2×C411yescharacteristic
131C22×C22⋊C4C2×C411yescharacteristic
132C22×C22⋊C4C2×C411yescharacteristic
133C22×C22⋊C4C2×C411yescharacteristic
134C22×C22⋊C4C2×C411yescharacteristic
135C22×C22⋊C4C2×C411yescharacteristic
136C22×C22⋊C4C2×C411yescharacteristic
137C22×C22⋊C4C2311yescharacteristic
138C22×C42C2×C411yescharacteristic
139C22×C42C2×C411yescharacteristic
140C22×C42C2×C411yescharacteristic
141C22×C42C2311yescharacteristic
142C22×C42C2311yescharacteristic
143#432—24no—
144#432—24no—
145#431—24no—
146#431—24no—
147C2×C4⋊C8—22no—
148C2×C4⋊C8—22no—
149C2×C22⋊C8—28no—
150C2×C22⋊C8C2314yes—
151C2×C22⋊C8C2314yes—
152C2×#97—22no—
153C2×C4×C8—22no—
154#336—28no—
155#324—24no—
156#324—24no—
157#324—28no—
158#324—28no—
159#323—24no—
160#323—24no—
161C25C22×C411yescharacteristic
162C25C22×C411yescharacteristic
163C22×Q8—22no—
164C22×Q8—22no—
165C22×Q8—22no—
166C22×Q8—24no—
167C22×Q8—42no—
168C22×D4—22no—
169C22×D4—22no—
170C22×D4—22no—
171C22×D4—44no—
172C23×C4—22no—
173C23×C4—22no—
174C23×C4—22no—
175C23×C4—22no—
176C23×C4—22no—
177C23×C4—22no—
178C23×C4—22no—
179C23×C4—21no—
180C23×C4—21no—
181C23×C4—24no—
182C23×C4—22no—
183C23×C4—21no—
184C23×C4—21no—
185C23×C4—24no—
186C23×C4—42no—
187C23×C4—42no—
188C23×C4—44no—
189C23×C4—44no—
190C23×C4C22⋊C414yes—
191C23×C4C22×C411yescharacteristic
192C23×C4C22×C411yescharacteristic
193C23×C4C22×C411yescharacteristic
194C23×C4C22×C411yescharacteristic
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×C4C2×D414yes—
204C23×C4C2411yescharacteristic
205C22×C8—24no—
206C22×C8—24no—
207#126—232no—
208#126—216no—
209#126—216no—
210#125—232no—
211#125—216no—
212#125—216no—
213#125—28no—
214#125—28no—
215#125—28no—
216#125—28no—
217#123—216no—
218#123—216no—
219#123—232no—
220C22⋊Q8—232no—
221C22⋊Q8—216no—
222C22⋊Q8—216no—
223C4×Q8—216no—
224C4×Q8—216no—
225C4×D4—216no—
226C4×D4—216no—
227#117—216no—
228#117—216no—
229#117—28no—
230#117—28no—
231#117—28no—
232#117—28no—
233#117—28no—
234#117—28no—
235#117—28no—
236#117—28no—
237C2×C4⋊C4—216no—
238C2×C4⋊C4—216no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
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—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—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—22no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4C22⋊C414yes—
282C2×C4⋊C4C22⋊C412yes—
283C2×C4⋊C4C22⋊C412yes—
284C2×C4⋊C4C22⋊C414yes—
285C2×C4⋊C4C22⋊C414yes—
286C2×C4⋊C4C22⋊C414yes—
287C2×C4⋊C4C22⋊C414yes—
288C2×C4⋊C4C22⋊C414yes—
289C2×C4⋊C4C2×D412yes—
290C2×C4⋊C4C2×D412yes—
291C2×C4⋊C4C2×D414yes—
292C2×C4⋊C4C2×D414yes—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4C22⋊C412yes—
314C2×C22⋊C4C22⋊C412yes—
315C2×C22⋊C4C22⋊C414yes—
316C2×C22⋊C4C22⋊C414yes—
317C2×C22⋊C4C2×D412yes—
318C2×C22⋊C4C2×D412yes—
319C2×C42—22no—
320C2×C42—21no—
321C2×C42—21no—
322C2×C42—22no—
323C2×C42—22no—
324C2×C42—22no—
325C2×C42—22no—
326C2×C42—22no—
327C2×C42—22no—
328C2×C42—22no—
329C2×C42—21no—
330C2×C42—21no—
331C2×C42C22⋊C414yes—
332C2×C42C2×D414yes—
333C2×C42C2×D414yes—
334C2×C42C2×D412yes—
335C2×C42C2×D412yes—
336C2×C42C2×D412yes—
337C2×C42C2×D412yes—
338C4⋊C8—44no—
339C4⋊C8—44no—
340C22⋊C8—24no—
341C22⋊C8—24no—
342C22⋊C8—24no—
343C22⋊C8—24no—
344C22⋊C8—416no—
345#97—44no—
346C4×C8—44no—
347C24—21no—
348C24—21no—
349C24—24no—
350C24—21no—
351C24—21no—
352C24—22no—
353C24—22no—
354C24—22no—
355C24—22no—
356C24—42no—
357C24—42no—
358C24—42no—
359C24C2×C22⋊C412yes—
360C24C2×C22⋊C412yes—
361C24C23×C411yescharacteristic
362C2×Q8—28no—
363C2×Q8—48no—
364C2×Q8—48no—
365C2×Q8—44no—
366C2×Q8—44no—
367C2×D4—28no—
368C2×D4—48no—
369C2×D4—44no—
370C2×D4—48no—
371C2×D4—44no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—28no—
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—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—28no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—28no—
409C22×C4—44no—
410C22×C4—42no—
411C22×C4—44no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—48no—
416C22×C4—44no—
417C22×C4—44no—
418C22×C4—48no—
419C22×C4—48no—
420C22×C4—44no—
421C22×C4—48no—
422C22×C4—44no—
423C22×C4—44no—
424C22×C4—48no—
425C22×C4—48no—
426C22×C4—44no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4C2×C22⋊C412yes—
434C22×C4C2×C22⋊C412yes—
435C22×C4C2×C22⋊C412yes—
436C22×C4C2×C22⋊C412yes—
437C22×C4C2×C22⋊C412yes—
438C22×C4C2×C22⋊C412yes—
439C22×C4C2×C22⋊C412yes—
440C22×C4C2×C22⋊C411yescharacteristic
441C22×C4C2×C22⋊C411yescharacteristic
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×C4#13114yes—
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D411yescharacteristic
455C2×C8—48no—
456C2×C8—48no—
457C2×C8—44no—
458C2×C8—44no—
459C4⋊C4—28no—
460C4⋊C4—28no—
461C4⋊C4—28no—
462C4⋊C4—28no—
463C4⋊C4—28no—
464C4⋊C4—28no—
465C4⋊C4—24no—
466C4⋊C4—24no—
467C4⋊C4—24no—
468C4⋊C4—24no—
469C4⋊C4—24no—
470C4⋊C4—24no—
471C4⋊C4—28no—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—24no—
475C4⋊C4—24no—
476C4⋊C4—28no—
477C4⋊C4—24no—
478C4⋊C4—24no—
479C4⋊C4—28no—
480C4⋊C4—28no—
481C22⋊C4—28no—
482C22⋊C4—28no—
483C22⋊C4—28no—
484C22⋊C4—28no—
485C22⋊C4—24no—
486C22⋊C4—24no—
487C22⋊C4—24no—
488C22⋊C4—24no—
489C22⋊C4—28no—
490C22⋊C4—28no—
491C42—28no—
492C42—28no—
493C42—24no—
494C42—24no—
495C42—24no—
496C42—24no—
497C42#12014yes—
498C42#12014yes—
499C42#12014yes—
500C42#12014yes—
501C23—22no—
502C23—22no—
503C23—24no—
504C23—24no—
505C23—24no—
506C23—22no—
507C23—24no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—48no—
513C23—48no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—41no—
521C23—41no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—84no—
528C23—84no—
529C23#41014yes—
530C23C22×C22⋊C411yescharacteristic
531C23C22×C22⋊C411yescharacteristic
532C23C22×C22⋊C411yescharacteristic
533C23C2×#13112yes—
534C23#56812yes—
535Q8—416no—
536D4—416no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—42no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—48no—
568C2×C4—48no—
569C2×C4—48no—
570C2×C4—48no—
571C2×C4—416no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—416no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
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—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4#37912yes—
599C2×C4#37912yes—
600C2×C4#37914yes—
601C2×C4#40814yes—
602C2×C4C2×#12012yes—
603C2×C4C2×#12012yes—
604C8—84no—
605C8—84no—
606C22—24no—
607C22—24no—
608C22—21no—
609C22—21no—
610C22—21no—
611C22—21no—
612C22—28no—
613C22—24no—
614C22—44no—
615C22—44no—
616C22—48no—
617C22—44no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—42no—
622C22—42no—
623C22—82no—
624C22—82no—
625C22—84no—
626C22—84no—
627C22C2×#37911yescharacteristic
628C22C2×#40811yescharacteristic
629C22#287911yescharacteristic
630C22#288012yes—
631C22#288012yes—
632C22C2×#41012yes—
633C22#288912yes—
634C4—24no—
635C4—22no—
636C4—22no—
637C4—22no—
638C4—22no—
639C4—24no—
640C4—42no—
641C4—42no—
642C4—42no—
643C4—42no—
644C4—42no—
645C4—42no—
646C4—42no—
647C4—42no—
648C4—42no—
649C2—21no—
650C2—21no—
651C2—22no—
652C2—44no—
653C2—82no—
654C2#1741714yes—
655C2#2058111yescharacteristic
656C2#2058411yescharacteristic
657C2#2059311yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C2×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20717C2semidirect
3. C2×#1470C2semidirect
4. C2×#1464C2semidirect
5. C2×#1458C2semidirect
6. #8309C2semidirect
8. #8303C2semidirect
9. #8299C2semidirect

Maximal subgroups

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

subgroupindex
C22×#5522
#207172
C2×#14702
C2×#14642
C2×#14582
#83092
#83042
#83032
#82992