Finite Group Databasebeta

← #188209

Subgroups — #188209

Order 512.

2376 conjugacy classes of subgroups.

The subgroup lattice has 655 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#556C211yesmaximal, characteristic
2#21820C212yesmaximal, complemented
3C2×#1473C212yesmaximal, complemented
4C2×#1464C211yesmaximal, characteristic, complemented
5C2×#1458C211yesmaximal, characteristic, complemented
6#8309C212yesmaximal, complemented
7#8305C212yesmaximal
8#8303C212yesmaximal, complemented
9#8300C212yesmaximal, complemented
10C2×#556—28no—
11C2×#556—24no—
12C2×#556—24no—
13C2×#556—24no—
14C2×#556—24no—
15C22×C4⋊Q8C411yescharacteristic
16C22×#126C411yescharacteristic
17C22×#126C411yescharacteristic
18C22×#126C2211yescharacteristic
19C22×#126C2211yescharacteristic
20C22×#124C411yescharacteristic
21C22×#124C2211yescharacteristic
22C22×C22⋊Q8C411yescharacteristic
23C22×C22⋊Q8C411yescharacteristic
24C22×C4×Q8C411yescharacteristic
25C22×C4×Q8C2211yescharacteristic
26C22×C4×Q8C2211yescharacteristic
27C22×C4×D4C411yescharacteristic
28C22×#117C2211yescharacteristic
29C22×#117C2211yescharacteristic
30C2×#432C2212yes—
31C2×#431C2212yes—
32#1985—24no—
33#1984—24no—
34#1817C2214yes—
35#1817C2214yes—
36#1473—28no—
37#1464—24no—
38#1458—24no—
39C2×#324C2212yes—
40C2×#324C2212yes—
41C2×#324C2214yes—
42C2×#323C2214yes—
43C2×#323C2212yes—
44C2×#323C2212yes—
45C23×Q8C2×C411yescharacteristic
46C23×Q8C2×C411yescharacteristic
47C23×Q8C2×C411yescharacteristic
48C23×D4C2×C411yescharacteristic
49C24×C4C2×C411yescharacteristic
50C24×C4C2×C411yescharacteristic
51#556—232no—
52#556—232no—
53#556—232no—
54#556—232no—
55C2×C4⋊Q8—24no—
56C2×C4⋊Q8—24no—
57C2×C4⋊Q8—24no—
58C2×#126—24no—
59C2×#126—24no—
60C2×#126—22no—
61C2×#126—22no—
62C2×#126—24no—
63C2×#126—24no—
64C2×#126—24no—
65C2×#126—22no—
66C2×#126—22no—
67C2×#126—24no—
68C2×#126—24no—
69C2×#126—22no—
70C2×#126—22no—
71C2×#126—22no—
72C2×#126—22no—
73C2×#126D418yes—
74C2×#124—24no—
75C2×#124—24no—
76C2×#124—24no—
77C2×#124—24no—
78C2×#124—22no—
79C2×#124—22no—
80C2×#124—24no—
81C2×C22⋊Q8—24no—
82C2×C22⋊Q8—24no—
83C2×C22⋊Q8—22no—
84C2×C22⋊Q8—22no—
85C2×C22⋊Q8—24no—
86C2×C22⋊Q8—24no—
87C2×C22⋊Q8—24no—
88C2×C4×Q8—24no—
89C2×C4×Q8—22no—
90C2×C4×Q8—22no—
91C2×C4×Q8—24no—
92C2×C4×Q8—22no—
93C2×C4×Q8—22no—
94C2×C4×Q8—24no—
95C2×C4×Q8—24no—
96C2×C4×Q8—24no—
97C2×C4×Q8—22no—
98C2×C4×Q8—22no—
99C2×C4×Q8D414yes—
100C2×C4×Q8D414yes—
101C2×C4×D4—24no—
102C2×C4×D4—22no—
103C2×C4×D4—22no—
104C2×C4×D4—24no—
105C2×#117—22no—
106C2×#117—22no—
107C2×#117—24no—
108C2×#117—24no—
109C2×#117—24no—
110C2×#117—22no—
111C2×#117—22no—
112C2×#117D414yes—
113C2×#117D414yes—
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⋊C4C2311yescharacteristic
125C22×C4⋊C4C2311yescharacteristic
126C22×C22⋊C4C2×C411yescharacteristic
127C22×C22⋊C4C2×C411yescharacteristic
128C22×C22⋊C4C2×C411yescharacteristic
129C22×C22⋊C4C2×C411yescharacteristic
130C22×C22⋊C4C2×C411yescharacteristic
131C22×C22⋊C4C2×C411yescharacteristic
132C22×C22⋊C4C2×C411yescharacteristic
133C22×C22⋊C4C2×C411yescharacteristic
134C22×C22⋊C4C2311yescharacteristic
135C22×C22⋊C4C2311yescharacteristic
136C22×C42C2×C411yescharacteristic
137C22×C42C2×C411yescharacteristic
138C22×C42C2×C411yescharacteristic
139C22×C42C2×C411yescharacteristic
140C22×C42C2311yescharacteristic
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⋊C8C2314yes—
150C2×C22⋊C8C2314yes—
151C2×#97—22no—
152C2×C4×C8—22no—
153#336—28no—
154#336—28no—
155#324—24no—
156#324—24no—
157#324—28no—
158#323—24no—
159#323—24no—
160#323—28no—
161C25C22×C411yescharacteristic
162C25C22×C411yescharacteristic
163C22×Q8—22no—
164C22×Q8—22no—
165C22×Q8—22no—
166C22×Q8—22no—
167C22×Q8—22no—
168C22×Q8—22no—
169C22×Q8—22no—
170C22×Q8—22no—
171C22×Q8—24no—
172C22×Q8—42no—
173C22×Q8—44no—
174C22×Q8—44no—
175C22×Q8C22⋊C414yes—
176C22×D4—22no—
177C22×D4—22no—
178C22×D4—22no—
179C22×D4—44no—
180C23×C4—22no—
181C23×C4—22no—
182C23×C4—22no—
183C23×C4—21no—
184C23×C4—21no—
185C23×C4—22no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—42no—
189C23×C4—42no—
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×C4C2411yescharacteristic
204C22×C8—24no—
205C22×C8—24no—
206C4⋊Q8—232no—
207#126—232no—
208#126—216no—
209#126—216no—
210#126—216no—
211#126—216no—
212#126—28no—
213#126—28no—
214#126—28no—
215#126—28no—
216#124—232no—
217#124—216no—
218#124—216no—
219C22⋊Q8—216no—
220C22⋊Q8—216no—
221C22⋊Q8—232no—
222C4×Q8—216no—
223C4×Q8—216no—
224C4×Q8—216no—
225C4×Q8—216no—
226C4×Q8—28no—
227C4×Q8—28no—
228C4×Q8—28no—
229C4×Q8—28no—
230C4×D4—216no—
231C4×D4—216no—
232#117—216no—
233#117—216no—
234#117—28no—
235#117—28no—
236#117—28no—
237#117—28no—
238C2×C4⋊C4—22no—
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—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
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—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4C22⋊C414yes—
269C2×C4⋊C4C22⋊C412yes—
270C2×C4⋊C4C22⋊C412yes—
271C2×C4⋊C4C22⋊C414yes—
272C2×C4⋊C4C22⋊C414yes—
273C2×C4⋊C4C22⋊C412yes—
274C2×C4⋊C4C22⋊C412yes—
275C2×C4⋊C4C22⋊C414yes—
276C2×C4⋊C4C2×D412yes—
277C2×C4⋊C4C2×D412yes—
278C2×C4⋊C4C2×D414yes—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—21no—
292C2×C22⋊C4—21no—
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—21no—
302C2×C22⋊C4—21no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4C22⋊C414yes—
307C2×C22⋊C4C22⋊C414yes—
308C2×C22⋊C4C22⋊C414yes—
309C2×C22⋊C4C2×D414yes—
310C2×C22⋊C4C2×D412yes—
311C2×C22⋊C4C2×D412yes—
312C2×C42—22no—
313C2×C42—22no—
314C2×C42—22no—
315C2×C42—22no—
316C2×C42—21no—
317C2×C42—21no—
318C2×C42—22no—
319C2×C42—22no—
320C2×C42—22no—
321C2×C42—22no—
322C2×C42—22no—
323C2×C42—22no—
324C2×C42—21no—
325C2×C42—21no—
326C2×C42—22no—
327C2×C42—21no—
328C2×C42—21no—
329C2×C42—22no—
330C2×C42C22⋊C414yes—
331C2×C42C2×D414yes—
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—
344#97—44no—
345C4×C8—44no—
346C24—21no—
347C24—21no—
348C24—21no—
349C24—21no—
350C24—22no—
351C24—22no—
352C24—22no—
353C24—22no—
354C24—42no—
355C24—42no—
356C24—42no—
357C24—42no—
358C24C2×C22⋊C412yes—
359C24C2×C22⋊C412yes—
360C24C23×C411yescharacteristic
361C2×Q8—28no—
362C2×Q8—28no—
363C2×Q8—28no—
364C2×Q8—28no—
365C2×Q8—44no—
366C2×Q8—48no—
367C2×Q8—48no—
368C2×Q8—48no—
369C2×Q8—48no—
370C2×Q8—48no—
371C2×Q8—48no—
372C2×Q8—44no—
373C2×Q8—44no—
374C2×Q8—44no—
375C2×Q8—44no—
376C2×D4—28no—
377C2×D4—48no—
378C2×D4—48no—
379C2×D4—44no—
380C2×D4—44no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—28no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—28no—
415C22×C4—44no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—48no—
420C22×C4—44no—
421C22×C4—44no—
422C22×C4—48no—
423C22×C4—48no—
424C22×C4—44no—
425C22×C4—44no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4C2×C22⋊C412yes—
431C22×C4C2×C22⋊C412yes—
432C22×C4C2×C22⋊C412yes—
433C22×C4C2×C22⋊C412yes—
434C22×C4C2×C22⋊C412yes—
435C22×C4C2×C22⋊C412yes—
436C22×C4C2×C22⋊C411yescharacteristic
437C22×C4C2×C22⋊C411yescharacteristic
438C22×C4C2×C22⋊C412yes—
439C22×C4C2×C22⋊C412yes—
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×C4#13114yes—
448C22×C4C22×D412yes—
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D412yes—
452C2×C8—48no—
453C2×C8—48no—
454C2×C8—44no—
455C2×C8—44no—
456C4⋊C4—28no—
457C4⋊C4—28no—
458C4⋊C4—28no—
459C4⋊C4—28no—
460C4⋊C4—24no—
461C4⋊C4—24no—
462C4⋊C4—24no—
463C4⋊C4—24no—
464C4⋊C4—28no—
465C4⋊C4—24no—
466C4⋊C4—24no—
467C4⋊C4—28no—
468C4⋊C4—24no—
469C4⋊C4—24no—
470C4⋊C4—24no—
471C4⋊C4—24no—
472C4⋊C4—28no—
473C22⋊C4—28no—
474C22⋊C4—28no—
475C22⋊C4—28no—
476C22⋊C4—28no—
477C22⋊C4—28no—
478C22⋊C4—24no—
479C22⋊C4—24no—
480C22⋊C4—24no—
481C22⋊C4—24no—
482C22⋊C4—28no—
483C22⋊C4—24no—
484C22⋊C4—24no—
485C22⋊C4—28no—
486C42—28no—
487C42—28no—
488C42—28no—
489C42—24no—
490C42—24no—
491C42—24no—
492C42—24no—
493C42—24no—
494C42—24no—
495C42#12014yes—
496C42#12014yes—
497C42#12014yes—
498C42#12014yes—
499C23—22no—
500C23—22no—
501C23—22no—
502C23—22no—
503C23—22no—
504C23—22no—
505C23—24no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—41no—
515C23—41no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—84no—
524C23—84no—
525C23—84no—
526C23—84no—
527C23C22×C22⋊C411yescharacteristic
528C23C22×C22⋊C411yescharacteristic
529C23C22×C22⋊C411yescharacteristic
530C23C2×#13112yes—
531C23#56812yes—
532Q8—416no—
533Q8—416no—
534Q8—416no—
535D4—416no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—24no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—41no—
566C2×C4—41no—
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—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
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#37912yes—
595C2×C4#37912yes—
596C2×C4#37914yes—
597C2×C4#40814yes—
598C2×C4#41314yes—
599C2×C4C2×#12012yes—
600C2×C4C2×#12012yes—
601C8—84no—
602C8—84no—
603C22—24no—
604C22—24no—
605C22—21no—
606C22—21no—
607C22—21no—
608C22—21no—
609C22—42no—
610C22—42no—
611C22—42no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22—82no—
616C22—82no—
617C22—84no—
618C22—82no—
619C22—82no—
620C22—84no—
621C22—84no—
622C22—84no—
623C22C2×#37911yescharacteristic
624C22C2×#40811yescharacteristic
625C22#287911yescharacteristic
626C22#288012yes—
627C22#288012yes—
628C22#288412yes—
629C22C2×#41312yes—
630C4—24no—
631C4—22no—
632C4—22no—
633C4—22no—
634C4—22no—
635C4—24no—
636C4—24no—
637C4—42no—
638C4—42no—
639C4—42no—
640C4—42no—
641C4—42no—
642C4—42no—
643C4—42no—
644C4—42no—
645C2—21no—
646C2—21no—
647C2—22no—
648C2—82no—
649C2—82no—
650C2#1742814yes—
651C2#2058111yescharacteristic
652C2#2058711yescharacteristic
653C2#2059111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C2×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21820C2semidirect
3. C2×#1473C2semidirect
4. C2×#1464C2semidirect
5. C2×#1458C2semidirect
6. #8309C2semidirect
8. #8303C2semidirect
9. #8300C2semidirect

Maximal subgroups

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

subgroupindex
C22×#5562
#218202
C2×#14732
C2×#14642
C2×#14582
#83092
#83052
#83032
#83002