Finite Group Databasebeta

← #362769

Subgroups — #362769

Order 512.

2526 conjugacy classes of subgroups.

The subgroup lattice has 708 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#557C211yesmaximal, characteristic
2C2×#1992C211yesmaximal, characteristic
3C2×#1989C211yesmaximal, characteristic
4C2×#1985C211yesmaximal, characteristic
5C2×#1803C211yesmaximal, characteristic
6C2×#1763C211yesmaximal, characteristic
7#13926C211yesmaximal, characteristic
8#11385C212yesmaximal
9#11384C212yesmaximal
10#10789C212yesmaximal
11#10789C212yesmaximal
12C2×#557—24no—
13C2×#557—24no—
14C2×#557—24no—
15C2×#557—24no—
16C2×#557—24no—
17C2×#557C418yescomplemented
18C22×C4⋊Q8—21no—
19C22×C4⋊Q8C2211yescharacteristic
20C22×#125—21no—
21C22×#125—21no—
22C22×#125C2211yescharacteristic
23C22×#125C2211yescharacteristic
24C22×C4×Q8—21no—
25C22×C4×Q8C2211yescharacteristic
26C22×C4×Q8C2211yescharacteristic
27C22×C4×Q8C2211yescharacteristic
28C22×C4×Q8C2211yescharacteristic
29C22×C4⋊C8C2211yescharacteristic
30C22×#103C2212yes—
31C22×#103C2212yes—
32C22×#97C2211yescharacteristic
33C2×#391—21no—
34C2×#384C2211yescharacteristic
35C2×C4×C4⋊C4C2211yescharacteristic
36#1992—24no—
37#1989—24no—
38#1985—24no—
39#1803—24no—
40#1763—24no—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#340C2212yes—
44C2×#340C2212yes—
45C2×#340C2212yes—
46C2×#340C2212yes—
47C2×#336C2212yes—
48C2×#336C2212yes—
49#1441—21no—
50#1441—21no—
51C4⋊#95—21no—
52C4⋊#95C2212yes—
53C4⋊#95C2212yes—
54C23×Q8—21no—
55C23×Q8D411yescharacteristic
56C23×Q8C2311yescharacteristic
57C23×C8C2312yes—
58C23×C8C2312yes—
59#557—232no—
60#557—232no—
61#557—232no—
62#557—232no—
63C2×C4⋊Q8—28no—
64C2×C4⋊Q8—28no—
65C2×C4⋊Q8—28no—
66C2×C4⋊Q8—22no—
67C2×C4⋊Q8—22no—
68C2×C4⋊Q8—22no—
69C2×C4⋊Q8—22no—
70C2×C4⋊Q8—22no—
71C2×C4⋊Q8C2×C414yes—
72C2×#125—28no—
73C2×#125—28no—
74C2×#125—28no—
75C2×#125—28no—
76C2×#125—28no—
77C2×#125—28no—
78C2×#125—22no—
79C2×#125—22no—
80C2×#125—22no—
81C2×#125—22no—
82C2×#125—22no—
83C2×#125—22no—
84C2×#125—22no—
85C2×#125—22no—
86C2×#125C2×C414yes—
87C2×#125C2×C414yes—
88C2×#125D414yes—
89C2×#125D414yes—
90C2×C4×Q8—28no—
91C2×C4×Q8—28no—
92C2×C4×Q8—22no—
93C2×C4×Q8—22no—
94C2×C4×Q8—22no—
95C2×C4×Q8—22no—
96C2×C4×Q8—28no—
97C2×C4×Q8—22no—
98C2×C4×Q8—22no—
99C2×C4×Q8—22no—
100C2×C4×Q8—22no—
101C2×C4×Q8—22no—
102C2×C4×Q8—22no—
103C2×C4×Q8—22no—
104C2×C4×Q8—22no—
105C2×C4×Q8—22no—
106C2×C4×Q8—22no—
107C2×C4×Q8—22no—
108C2×C4×Q8—22no—
109C2×C4×Q8—22no—
110C2×C4×Q8C2×C414yes—
111C2×C4×Q8C2×C414yes—
112C2×C4×Q8C2×C414yes—
113C2×C4×Q8C2×C414yes—
114C2×C4×Q8D414yes—
115C2×C4×Q8D414yes—
116C2×C4×Q8D414yes—
117C22×C4⋊C4—21no—
118C22×C4⋊C4—21no—
119C22×C4⋊C4—21no—
120C22×C4⋊C4—21no—
121C22×C4⋊C4—21no—
122C22×C4⋊C4—21no—
123C22×C4⋊C4—21no—
124C22×C4⋊C4—21no—
125C22×C4⋊C4—21no—
126C22×C4⋊C4—21no—
127C22×C4⋊C4D411yescharacteristic
128C22×C4⋊C4D411yescharacteristic
129C22×C4⋊C4D411yescharacteristic
130C22×C4⋊C4C2312yes—
131C22×C4⋊C4C2312yes—
132C22×C4⋊C4C2311yescharacteristic
133C22×C4⋊C4C2311yescharacteristic
134C22×C4⋊C4C2311yescharacteristic
135C22×C42—21no—
136C22×C42—21no—
137C22×C42—21no—
138C22×C42—21no—
139C22×C42C2311yescharacteristic
140C22×C42C2311yescharacteristic
141C22×C42C2311yescharacteristic
142C2×C4⋊C8—22no—
143C2×C4⋊C8—22no—
144C2×C4⋊C8—22no—
145C2×#103—24no—
146C2×#103—24no—
147C2×#103—24no—
148C2×#103—24no—
149C2×#103—24no—
150C2×#103Q818yescomplemented
151C2×#97—22no—
152C2×#97—22no—
153C2×#97—22no—
154#391—44no—
155#384—22no—
156#384—22no—
157C4×C4⋊C4—22no—
158C4×C4⋊C4—22no—
159C2×#95—22no—
160C2×#95—22no—
161C2×#95—21no—
162C2×#95—22no—
163C2×#95—22no—
164#340—24no—
165#340—24no—
166#340—24no—
167#340—24no—
168#340—24no—
169#340—24no—
170#336—24no—
171#336—24no—
172C22×Q8—24no—
173C22×Q8—24no—
174C22×Q8—24no—
175C22×Q8—22no—
176C22×Q8—22no—
177C22×Q8—22no—
178C22×Q8—22no—
179C22×Q8—21no—
180C22×Q8—21no—
181C22×Q8—21no—
182C22×Q8—22no—
183C22×Q8—22no—
184C22×Q8—41no—
185C22×Q8—41no—
186C22×Q8—42no—
187C22×Q8—42no—
188C22×Q8—42no—
189C22×Q8—42no—
190C22×Q8C22⋊C414yes—
191C22×Q8C22×C412yes—
192C22×Q8C2×D412yes—
193C22×Q8C2×D412yes—
194C22×Q8C2×Q814yes—
195C22×Q8#4114yes—
196C23×C4—21no—
197C23×C4—21no—
198C23×C4—21no—
199C23×C4—21no—
200C23×C4—22no—
201C23×C4—22no—
202C23×C4C2×D411yescharacteristic
203C23×C4C2×D411yescharacteristic
204C23×C4C2×D411yescharacteristic
205C23×C4C2×D411yescharacteristic
206C23×C4C2×D411yescharacteristic
207C23×C4C2×D411yescharacteristic
208C23×C4C2411yescharacteristic
209C22×C8—22no—
210C22×C8—22no—
211C22×C8—22no—
212C22×C8—22no—
213C22×C8—22no—
214C22×C8—22no—
215C22×C8—22no—
216C22×C8—22no—
217C22×C8—22no—
218C22×C8—22no—
219C22×C8—22no—
220C22×C8—22no—
221C22×C8—22no—
222C22×C8C2×Q814yes—
223C4⋊Q8—264no—
224C4⋊Q8—28no—
225C4⋊Q8—28no—
226C4⋊Q8—28no—
227C4⋊Q8—28no—
228#125—264no—
229#125—28no—
230#125—28no—
231#125—264no—
232#125—28no—
233#125—28no—
234#125—28no—
235#125—28no—
236#125—28no—
237#125C22⋊C4116yes—
238C4×Q8—28no—
239C4×Q8—28no—
240C4×Q8—264no—
241C4×Q8—28no—
242C4×Q8—28no—
243C4×Q8—28no—
244C4×Q8—28no—
245C4×Q8—28no—
246C4×Q8—28no—
247C4×Q8—28no—
248C4×Q8—28no—
249C4×Q8—28no—
250C4×Q8—28no—
251C4×Q8—28no—
252C4×Q8—28no—
253C4×Q8—28no—
254C4×Q8C22⋊C4116yes—
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—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—42no—
306C2×C4⋊C4—42no—
307C2×C4⋊C4—42no—
308C2×C4⋊C4—42no—
309C2×C4⋊C4—82no—
310C2×C4⋊C4C22⋊C414yes—
311C2×C4⋊C4C22⋊C414yes—
312C2×C4⋊C4C22⋊C414yes—
313C2×C4⋊C4C22×C412yes—
314C2×C4⋊C4C22×C412yes—
315C2×C4⋊C4C22×C412yes—
316C2×C4⋊C4C2×D412yes—
317C2×C4⋊C4C2×D412yes—
318C2×C4⋊C4C2×D412yes—
319C2×C4⋊C4C2×D412yes—
320C2×C4⋊C4C2×D412yes—
321C2×C4⋊C4C2×D412yes—
322C2×C4⋊C4C2×Q814yes—
323C2×C42—24no—
324C2×C42—24no—
325C2×C42—24no—
326C2×C42—24no—
327C2×C42—24no—
328C2×C42—24no—
329C2×C42—24no—
330C2×C42—24no—
331C2×C42—24no—
332C2×C42—21no—
333C2×C42—21no—
334C2×C42—21no—
335C2×C42—21no—
336C2×C42—21no—
337C2×C42—21no—
338C2×C42—21no—
339C2×C42—21no—
340C2×C42—42no—
341C2×C42—42no—
342C2×C42—42no—
343C2×C42C22×C412yes—
344C2×C42C22×C412yes—
345C2×C42C22×C412yes—
346C2×C42C2×D412yes—
347C2×C42C2×D412yes—
348C2×C42C2×D412yes—
349C2×C42C2×D412yes—
350C2×C42C2×D412yes—
351C2×C42C2×D412yes—
352C2×C42C2×D412yes—
353C4⋊C8—44no—
354#103—216no—
355#103—48no—
356#97—44no—
357#95—44no—
358#95—42no—
359#95—44no—
360#95—82no—
361#95—82no—
362C24C22×D411yescharacteristic
363C2×Q8—216no—
364C2×Q8—22no—
365C2×Q8—24no—
366C2×Q8—22no—
367C2×Q8—22no—
368C2×Q8—24no—
369C2×Q8—22no—
370C2×Q8—22no—
371C2×Q8—24no—
372C2×Q8—24no—
373C2×Q8—24no—
374C2×Q8—24no—
375C2×Q8—24no—
376C2×Q8—24no—
377C2×Q8—28no—
378C2×Q8—28no—
379C2×Q8—44no—
380C2×Q8—44no—
381C2×Q8—44no—
382C2×Q8—44no—
383C2×Q8—44no—
384C2×Q8—48no—
385C2×Q8—48no—
386C2×Q8—48no—
387C2×Q8—48no—
388C2×Q8—48no—
389C2×Q8—48no—
390C2×Q8C2×C22⋊C414yes—
391C2×Q8C4×Q818yes—
392C2×Q8C22⋊Q818yes—
393C2×Q8#12414yes—
394C2×Q8C4⋊Q814yes—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—41no—
435C22×C4—41no—
436C22×C4—41no—
437C22×C4—41no—
438C22×C4—41no—
439C22×C4—41no—
440C22×C4—41no—
441C22×C4—41no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4C2×C22⋊C412yes—
458C22×C4C2×C22⋊C412yes—
459C22×C4C2×C22⋊C412yes—
460C22×C4C2×C22⋊C412yes—
461C22×C4C2×C22⋊C412yes—
462C22×C4C2×C22⋊C412yes—
463C22×C4#12012yes—
464C22×C4#12012yes—
465C22×C4#12012yes—
466C22×C4#12012yes—
467C22×C4AGL(1,ℤ/8)12yes—
468C22×C4#13712yes—
469C22×C4C23×C411yescharacteristic
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C4C22×Q812yes—
476C22×C4C2×#4112yes—
477C22×C42−1+412yes—
478C22×C42−1+412yes—
479C2×C8—24no—
480C2×C8—24no—
481C2×C8—24no—
482C2×C8—42no—
483C2×C8—42no—
484C2×C8—42no—
485C2×C8—42no—
486C2×C8—42no—
487C2×C8—42no—
488C2×C8—42no—
489C2×C8—42no—
490C2×C8—42no—
491C2×C8—42no—
492C2×C8—42no—
493C4⋊C4—216no—
494C4⋊C4—216no—
495C4⋊C4—216no—
496C4⋊C4—216no—
497C4⋊C4—216no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—216no—
508C4⋊C4—216no—
509C4⋊C4—216no—
510C4⋊C4—28no—
511C4⋊C4—28no—
512C4⋊C4—28no—
513C4⋊C4—24no—
514C4⋊C4—24no—
515C4⋊C4—42no—
516C4⋊C4—42no—
517C4⋊C4—42no—
518C4⋊C4—42no—
519C4⋊C4—42no—
520C4⋊C4—42no—
521C4⋊C4—84no—
522C4⋊C4—84no—
523C4⋊C4C2×C22⋊C414yes—
524C4⋊C4C2×C22⋊C414yes—
525C4⋊C4C2×C22⋊C414yes—
526C42—216no—
527C42—216no—
528C42—22no—
529C42—22no—
530C42—22no—
531C42—22no—
532C42—22no—
533C42—22no—
534C42—22no—
535C42—22no—
536C42—216no—
537C42—84no—
538C42C2×C22⋊C414yes—
539C42C2×C22⋊C414yes—
540C42#12014yes—
541C42#12014yes—
542C23C22×C22⋊C411yescharacteristic
543C23C2×#12011yescharacteristic
544C23C2×#12011yescharacteristic
545C23#54812yes—
546C23D4×Q812yes—
547C23C2×AGL(1,ℤ/8)11yescharacteristic
548C23C2×#13711yescharacteristic
549C23#57511yescharacteristic
550C23#57511yescharacteristic
551C23#57712yes—
552C23#57812yes—
553Q8—24no—
554Q8—28no—
555Q8—28no—
556Q8—24no—
557Q8—24no—
558Q8—416no—
559Q8—432no—
560Q8#39118yescomplemented
561C2×C4—28no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—28no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—24no—
589C2×C4—22no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—82no—
622C2×C4—82no—
623C2×C4—82no—
624C2×C4—82no—
625C2×C4—82no—
626C2×C4—82no—
627C2×C4—82no—
628C2×C4—82no—
629C2×C4—82no—
630C2×C4—82no—
631C2×C4—82no—
632C2×C4#37914yes—
633C2×C4#37914yes—
634C2×C4#41714yes—
635C2×C4C22×C22⋊C411yescharacteristic
636C2×C4C22×C22⋊C411yescharacteristic
637C2×C4C2×C4×Q812yes—
638C2×C4#51912yes—
639C2×C4C2×#12011yescharacteristic
640C2×C4C2×#12011yescharacteristic
641C2×C4C2×C22⋊Q812yes—
642C2×C4C2×#12411yescharacteristic
643C2×C4C2×C4⋊Q811yescharacteristic
644C2×C4#53612yes—
645C2×C4#53611yescharacteristic
646C2×C4#53611yescharacteristic
647C2×C4#54812yes—
648C2×C4#54812yes—
649C2×C4#54812yes—
650C2×C4D4×Q812yes—
651C2×C4D4×Q812yes—
652C2×C4D4×Q812yes—
653C8—44no—
654C8—82no—
655C22C2×#37911yescharacteristic
656C22Q8×C22⋊C412yes—
657C22#240912yes—
658C22#243411yescharacteristic
659C22#243611yescharacteristic
660C22C2×#41711yescharacteristic
661C22#289611yescharacteristic
662C22#289812yes—
663C22#300611yescharacteristic
664C22#300611yescharacteristic
665C22#300711yescharacteristic
666C22#300711yescharacteristic
667C22#301011yescharacteristic
668C22#301111yescharacteristic
669C22#301312yes—
670C22#336812yes—
671C22#337112yes—
672C22#337412yes—
673C22#337712yes—
674C22#338612yes—
675C22#338712yes—
676C22#338812yes—
677C22#339012yes—
678C4—28no—
679C4—24no—
680C4—24no—
681C4—24no—
682C4—42no—
683C4—44no—
684C4—44no—
685C4—42no—
686C4—42no—
687C4—44no—
688C4—84no—
689C4—162no—
690C4C2×#39111yescharacteristic
691C4#231011yescharacteristic
692C4Q8×C22⋊C412yes—
693C4#240912yes—
694C4#243411yescharacteristic
695C4#243611yescharacteristic
696C2#1392211yescharacteristic
697C2#2060111yescharacteristic
698C2#2060311yescharacteristic
699C2#2252412yes—
700C2#2252712yes—
701C2#2434612yes—
702C2#2434812yes—
703C2#2584811yescharacteristic
704C2#2584911yescharacteristic
705C2#2594311yescharacteristic
706C2#2594411yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
17. C2×#557C4semidirect
150. C2×#103Q8semidirect
560. Q8#391semidirect

Maximal subgroups

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

subgroupindex
C22×#5572
C2×#19922
C2×#19892
C2×#19852
C2×#18032
C2×#17632
#139262
#113852
#113842
#107892