Finite Group Databasebeta

← #6445392

Subgroups — #6445392

Order 512.

3172 conjugacy classes of subgroups.

The subgroup lattice has 692 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3503C211yesmaximal, characteristic
2C2×#3078C211yesmaximal, characteristic, complemented
3C22×#475C211yesmaximal, characteristic, complemented
4C4×#536C211yesmaximal, characteristic
5#24310C218yesmaximal, complemented
6#24303C214yesmaximal, complemented
7#24301C214yesmaximal, complemented
8#23556C212yesmaximal, complemented
9#23457C212yesmaximal, complemented
10#21313C211yesmaximal, characteristic, complemented
11#21311C211yesmaximal, characteristic, complemented
12#20913C212yesmaximal
13#20848C212yesmaximal
14#20629C211yesmaximal, characteristic
15#3503—28no—
16#3503—28no—
17C2×Q82—21no—
18C2×#557—21no—
19C2×#554—22no—
20C2×#554—21no—
21C2×D4×Q8—22no—
22C2×D4×Q8—21no—
23C2×#544C2212yes—
24C2×#544C2214yes—
25C2×#536C2211yescharacteristic
26C2×#533C2212yes—
27C2×#533C2211yescharacteristic
28C22×C4⋊Q8C2212yes—
29C22×C4⋊Q8C2211yescharacteristic
30C2×#520C2212yes—
31C2×C4×#41C2211yescharacteristic
32C22×C4×Q8C2211yescharacteristic
33#3078—28no—
34C2×#479C2214yes—
35C2×#477C2212yes—
36C2×#475—28no—
37C2×#475C2214yes—
38C2×#475C2214yes—
39C2×#475C2212yes—
40C2×#428C2212yes—
41C22×#106C2212yes—
42C2×#423C2211yescharacteristic
43C22×C4⋊C8C2211yescharacteristic
44C2×#419C2212yes—
45C22×#103C2212yes—
46C2×#408C2212yes—
47#2327—21no—
48#2326—21no—
49#2323C2214yes—
50#2322—22no—
51#2317C2211yescharacteristic
52#2311C2211yescharacteristic
53#2310C2211yescharacteristic
54C4×C4⋊Q8—21no—
55C4×#124—21no—
56C4×C22⋊Q8C2214yes—
57C4×#123—22no—
58C4×#117C2211yescharacteristic
59#2059C2218yes—
60#2058C2218yes—
61#2026C2218yes—
62#2024C2218yes—
63#1989C2214yes—
64#1988C2212yes—
65#1988C2214yes—
66#1987C2212yes—
67#1987C2214yes—
68#1873C2218yes—
69#1872C2218yes—
70#1854C2218yes—
71#1853C2218yes—
72#1798C2214yes—
73#1797C2214yes—
74#1736C2212yes—
75#1736C2212yes—
76#1735C2212yes—
77#1735C2212yes—
78#1729C2212yes—
79C22×#41C2311yescharacteristic
80C23×Q8—22no—
81C23×Q8—21no—
82C23×Q8C2311yescharacteristic
83C22×M16C2312yes—
84C23×C8C2312yes—
85Q82—216no—
86#557—216no—
87#554—216no—
88#554—232no—
89D4×Q8—232no—
90D4×Q8—216no—
91#544—216no—
92#544—216no—
93#544—28no—
94#544—28no—
95#536—24no—
96#536Q818yescomplemented
97#533—28no—
98#533—28no—
99#533—24no—
100#533D418yes—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—28no—
103C2×C4⋊Q8—28no—
104C2×C4⋊Q8—22no—
105C2×C4⋊Q8—24no—
106C2×C4⋊Q8—22no—
107C2×C4⋊Q8—21no—
108C2×C4⋊Q8—22no—
109C2×C4⋊Q8—21no—
110C2×C4⋊Q8—24no—
111C2×C4⋊Q8D418yes—
112C2×C4⋊Q8C2314yes—
113C2×C4⋊Q8C2311yescharacteristic
114C2×C4⋊Q8C2312yes—
115C2×C4⋊Q8C2312yes—
116C2×C4⋊Q8C2311yescharacteristic
117C2×#125—22no—
118C2×#125—24no—
119C2×#125C2314yes—
120C2×#125C2312yes—
121C2×#124—21no—
122C2×#123—22no—
123C2×C22⋊Q8—22no—
124C2×C22⋊Q8—21no—
125C2×C22⋊Q8—22no—
126C2×C22⋊Q8—21no—
127C2×C22⋊Q8—24no—
128C2×C22⋊Q8—24no—
129C2×C22⋊Q8—24no—
130C2×C22⋊Q8C2314yes—
131#520—28no—
132#520—28no—
133C4×#41—24no—
134C4×#41—24no—
135C2×C4×Q8—22no—
136C2×C4×Q8—22no—
137C2×C4×Q8—21no—
138C2×C4×Q8—22no—
139C2×C4×Q8—21no—
140C2×C4×Q8—21no—
141C2×C4×Q8—21no—
142C2×C4×Q8—24no—
143C2×C4×Q8—24no—
144C2×C4×Q8D411yescharacteristic
145C2×C4×Q8D411yescharacteristic
146C2×C4×Q8C2312yes—
147C2×C4×Q8C2312yes—
148C2×C4×D4—24no—
149C2×C4×D4—22no—
150C2×C4×D4D412yes—
151C2×C4×D4D411yescharacteristic
152C2×C4×D4D412yes—
153C2×C4×D4D411yescharacteristic
154C2×#117C2311yescharacteristic
155C2×#117C2314yes—
156C2×#117C2312yes—
157C2×#117C2311yescharacteristic
158C2×#117C2312yes—
159C2×#117C2311yescharacteristic
160C22×C4⋊C4—22no—
161C22×C4⋊C4—21no—
162C22×C4⋊C4—21no—
163C22×C4⋊C4C2311yescharacteristic
164C22×C4⋊C4C2314yes—
165C22×C4⋊C4C2312yes—
166C22×C42—21no—
167C22×C42C2312yes—
168C22×C42C2311yescharacteristic
169#479—216no—
170#477—28no—
171#475—216no—
172#475—216no—
173#475—28no—
174#428—28no—
175C2×#107C2314yes—
176C2×#106—28no—
177C2×#106C2318yes—
178C2×#106C2314yes—
179#423—24no—
180C2×C4⋊C8—24no—
181C2×C4⋊C8C2312yes—
182C2×C4⋊C8C2312yes—
183C2×C4⋊C8C2312yes—
184C2×C4⋊C8C2312yes—
185#419—28no—
186C2×#103—28no—
187C2×#103C2318yes—
188C2×#103C2314yes—
189C2×#103C2314yes—
190#408—28no—
191C2×C22⋊C8C2318yes—
192#391—21no—
193#391—21no—
194#391—21no—
195#391C2314yes—
196#390—21no—
197#389—24no—
198#389C2314yes—
199#388—24no—
200#388—24no—
201#387—24no—
202#387—24no—
203#386—22no—
204#385—24no—
205#385—22no—
206#384—21no—
207#382—21no—
208#381—22no—
209#380C2314yes—
210C4×C4⋊C4—24no—
211C4×C4⋊C4C2312yes—
212C4×C4⋊C4C2312yes—
213C4×C22⋊C4—21no—
214C4×C22⋊C4—21no—
215C4×C22⋊C4—24no—
216#376—21no—
217C43—21no—
218#340C2314yes—
219#340C2318yes—
220#340C2314yes—
221#340C2314yes—
222#340C2318yes—
223#340C2314yes—
224#336C2318yes—
225C2×#41—22no—
226C2×#41—22no—
227C2×#41—22no—
228C2×#41—22no—
229C2×#41C2×Q814yes—
230C2×#41#4114yes—
231C22×Q8—216no—
232C22×Q8—28no—
233C22×Q8—21no—
234C22×Q8—21no—
235C22×Q8—22no—
236C22×Q8—21no—
237C22×Q8—24no—
238C22×Q8—22no—
239C22×Q8—22no—
240C22×Q8—22no—
241C22×Q8—22no—
242C22×Q8—22no—
243C22×Q8—22no—
244C22×Q8—84no—
245C22×Q8—88no—
246C22×Q8C2×D411yescharacteristic
247C22×Q8C2×Q814yescomplemented
248C22×Q8C2412yes—
249C22×D4C2×D412yes—
250C22×D4C2×D411yescharacteristic
251C23×C4—21no—
252C23×C4—24no—
253C23×C4—22no—
254C23×C4—21no—
255C23×C4C2×D412yes—
256C23×C4C2×D411yescharacteristic
257C23×C4C2×D412yes—
258C23×C4C2×D411yescharacteristic
259C23×C4C2412yes—
260C23×C4C2411yescharacteristic
261C2×M16—24no—
262C2×M16—24no—
263C2×M16—24no—
264C22×C8—24no—
265C22×C8—24no—
266C22×C8—24no—
267C22×C8C2414yes—
268C22×C8C2414yes—
269C4⋊Q8—216no—
270C4⋊Q8—216no—
271C4⋊Q8—232no—
272C4⋊Q8—216no—
273C4⋊Q8—28no—
274C4⋊Q8—28no—
275C4⋊Q8—28no—
276C4⋊Q8—28no—
277C4⋊Q8—28no—
278C4⋊Q8—22no—
279C4⋊Q8—24no—
280C4⋊Q8—24no—
281C4⋊Q8—22no—
282C4⋊Q8C2×D414yes—
283C4⋊Q8C2×D418yes—
284C4⋊Q8C2×D418yes—
285C4⋊Q8C2×D414yes—
286#125—232no—
287#125—216no—
288#125—28no—
289#125—28no—
290#125—24no—
291#125C2×D418yes—
292#124—28no—
293#123—216no—
294C22⋊Q8—216no—
295C22⋊Q8—216no—
296C22⋊Q8—232no—
297C22⋊Q8—232no—
298C22⋊Q8—232no—
299C22⋊Q8—28no—
300C22⋊Q8—28no—
301C22⋊Q8—28no—
302C22⋊Q8C2×Q8116yescomplemented
303C4×Q8—216no—
304C4×Q8—216no—
305C4×Q8—216no—
306C4×Q8—28no—
307C4×Q8—28no—
308C4×Q8—24no—
309C4×Q8—24no—
310C4×Q8—24no—
311C4×Q8—24no—
312C4×Q8—24no—
313C4×Q8—24no—
314C4×Q8—44no—
315C4×Q8—44no—
316C4×D4—232no—
317C4×D4—216no—
318C4×D4—28no—
319C4×D4—28no—
320C4×D4—24no—
321C4×D4—24no—
322#117—22no—
323#117—22no—
324#117—24no—
325#117—22no—
326#117—28no—
327#117—28no—
328#117—24no—
329#117—24no—
330#117—22no—
331#117—22no—
332#117C2×D418yes—
333#117C2×Q814yescomplemented
334C2×C4⋊C4—216no—
335C2×C4⋊C4—28no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—24no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—24no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—24no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—24no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—24no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—24no—
349C2×C4⋊C4—28no—
350C2×C4⋊C4—28no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—24no—
357C2×C4⋊C4—44no—
358C2×C4⋊C4C2×D411yescharacteristic
359C2×C4⋊C4C2×D412yes—
360C2×C4⋊C4C2×D411yescharacteristic
361C2×C4⋊C4C2×D412yes—
362C2×C4⋊C4C2×D418yes—
363C2×C4⋊C4C2412yes—
364C2×C4⋊C4C2412yes—
365C2×C4⋊C4C2412yes—
366C2×C4⋊C4C2414yes—
367C2×C4⋊C4C2412yes—
368C2×C4⋊C4C2412yes—
369C2×C22⋊C4—24no—
370C2×C22⋊C4—22no—
371C2×C22⋊C4—21no—
372C2×C22⋊C4—24no—
373C2×C22⋊C4—21no—
374C2×C22⋊C4—24no—
375C2×C22⋊C4C2×D412yes—
376C2×C22⋊C4C2×D412yes—
377C2×C22⋊C4C2×D412yes—
378C2×C42—24no—
379C2×C42—22no—
380C2×C42—21no—
381C2×C42—24no—
382C2×C42—24no—
383C2×C42—21no—
384C2×C42—22no—
385C2×C42—22no—
386C2×C42—22no—
387C2×C42—22no—
388C2×C42—44no—
389C2×C42C2×D412yes—
390C2×C42C2×D412yes—
391C2×C42C2×D411yescharacteristic
392C2×C42C2×D411yescharacteristic
393C2×C42C2×D414yes—
394C2×C42C2×D414yes—
395C2×C42C2412yes—
396C2×C42C2411yescharacteristic
397C2×C42C2411yescharacteristic
398#107—28no—
399#106—216no—
400#106—28no—
401C4⋊C8—24no—
402C4⋊C8—24no—
403C4⋊C8—24no—
404C4⋊C8—24no—
405#103—216no—
406#103—28no—
407#103—28no—
408C22⋊C8—216no—
409#95—22no—
410#95—22no—
411#95—24no—
412#95—24no—
413#95—24no—
414#95—24no—
415C24C22×D412yes—
416C24C22×D411yescharacteristic
417#41—28no—
418#41—28no—
419#41—28no—
420#41—28no—
421C2×Q8—216no—
422C2×Q8—22no—
423C2×Q8—24no—
424C2×Q8—28no—
425C2×Q8—24no—
426C2×Q8—28no—
427C2×Q8—24no—
428C2×Q8—22no—
429C2×Q8—22no—
430C2×Q8—28no—
431C2×Q8—28no—
432C2×Q8—22no—
433C2×Q8—22no—
434C2×Q8—22no—
435C2×Q8—22no—
436C2×Q8—44no—
437C2×Q8—42no—
438C2×Q8—42no—
439C2×Q8—44no—
440C2×Q8—44no—
441C2×Q8—48no—
442C2×Q8—44no—
443C2×Q8—84no—
444C2×Q8—88no—
445C2×Q8—816no—
446C2×Q8—832no—
447C2×Q8C22⋊Q814yes—
448C2×Q8C22×Q814yes—
449C2×D4—24no—
450C2×D4—24no—
451C2×D4—22no—
452C2×D4—22no—
453C2×D4C22⋊Q818yes—
454C2×D4C22⋊Q814yes—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—24no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—24no—
465C22×C4—22no—
466C22×C4—216no—
467C22×C4—28no—
468C22×C4—24no—
469C22×C4—24no—
470C22×C4—28no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—42no—
474C22×C4—44no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—82no—
479C22×C4—84no—
480C22×C4—88no—
481C22×C4#12014yes—
482C22×C4#12014yes—
483C22×C4C22×D412yes—
484C22×C4C22×D412yes—
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D412yes—
489C22×C4C22×D411yescharacteristic
490C22×C4C22×D411yescharacteristic
491C22×C4C22×D412yes—
492C22×C4C22×D412yes—
493C22×C4C22×Q814yes—
494C22×C4C22×Q812yes—
495C22×C4C2×#4114yes—
496C22×C4C2×#4112yes—
497C22×C42−1+412yes—
498C22×C42−1+412yes—
499C22×C4C2511yescharacteristic
500M16—48no—
501C2×C8—24no—
502C2×C8—24no—
503C2×C8—24no—
504C2×C8—24no—
505C2×C8—24no—
506C2×C8—24no—
507C2×C8—48no—
508C4⋊C4—216no—
509C4⋊C4—216no—
510C4⋊C4—216no—
511C4⋊C4—216no—
512C4⋊C4—24no—
513C4⋊C4—28no—
514C4⋊C4—28no—
515C4⋊C4—28no—
516C4⋊C4—24no—
517C4⋊C4—28no—
518C4⋊C4—28no—
519C4⋊C4—24no—
520C4⋊C4—22no—
521C4⋊C4—24no—
522C4⋊C4—22no—
523C4⋊C4—216no—
524C4⋊C4—24no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—42no—
533C4⋊C4—44no—
534C4⋊C4—44no—
535C4⋊C4—84no—
536C4⋊C4C22×D414yes—
537C4⋊C4C22×D418yes—
538C4⋊C4C22×D414yes—
539C4⋊C4C22×Q814yes—
540C22⋊C4—216no—
541C22⋊C4—28no—
542C22⋊C4—24no—
543C22⋊C4—24no—
544C22⋊C4—216no—
545C22⋊C4—24no—
546C22⋊C4—24no—
547C22⋊C4—42no—
548C22⋊C4—84no—
549C42—216no—
550C42—24no—
551C42—24no—
552C42—28no—
553C42—24no—
554C42—22no—
555C42—22no—
556C42—44no—
557C42—44no—
558C42—42no—
559C42—44no—
560C42—82no—
561C42#12014yes—
562C42#12014yes—
563C42C22×D412yes—
564C42C22×D412yes—
565C42C22×D412yes—
566C42C22×D412yes—
567C23—22no—
568C23—22no—
569C23—24no—
570C23—22no—
571C23—22no—
572C23—24no—
573C23C2×#12012yes—
574C23C2×C22⋊Q814yes—
575C23C2×C22⋊Q812yes—
576C23#53612yes—
577C23C23×D411yescharacteristic
578Q8—28no—
579Q8—24no—
580Q8—24no—
581Q8—48no—
582Q8—48no—
583Q8—42no—
584Q8—42no—
585Q8—416no—
586Q8—48no—
587Q8—88no—
588Q8—816no—
589D4—216no—
590D4—28no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—28no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—28no—
606C2×C4—24no—
607C2×C4—24no—
608C2×C4—24no—
609C2×C4—24no—
610C2×C4—44no—
611C2×C4—48no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—48no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—48no—
619C2×C4—44no—
620C2×C4—48no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—88no—
627C2×C4—816no—
628C2×C4—84no—
629C2×C4C2×#12012yes—
630C2×C4C2×#12012yes—
631C2×C4C2×#12012yes—
632C2×C4C2×#12012yes—
633C2×C4C2×#12012yes—
634C2×C4C2×C22⋊Q814yes—
635C2×C4C2×C22⋊Q812yes—
636C2×C4#53612yes—
637C2×C4#54814yes—
638C2×C4#54814yes—
639C2×C4D4×Q814yes—
640C2×C4D4×Q814yes—
641C2×C4#57714yes—
642C2×C4C23×D411yescharacteristic
643C2×C4C23×D411yescharacteristic
644C2×C4C23×Q811yescharacteristic
645C2×C4C22×#4111yescharacteristic
646C2×C4C2×2−1+411yescharacteristic
647C2×C4C2×2−1+411yescharacteristic
648C8—44no—
649C8—44no—
650C22—24no—
651C22—22no—
652C22—22no—
653C22—24no—
654C22—28no—
655C22—24no—
656C22—42no—
657C22—42no—
658C22—44no—
659C22#240914yes—
660C22C22×#12011yescharacteristic
661C22C22×C22⋊Q811yescharacteristic
662C22C2×#53611yescharacteristic
663C22C2×#57712yes—
664C22#358612yes—
665C4—24no—
666C4—22no—
667C4—22no—
668C4—22no—
669C4—44no—
670C4—44no—
671C4—42no—
672C4—48no—
673C4—44no—
674C4—42no—
675C4—44no—
676C4—82no—
677C4—82no—
678C4#240914yes—
679C4#300914yes—
680C4C2×#54811yescharacteristic
681C4C2×#54811yescharacteristic
682C4C2×D4×Q811yescharacteristic
683C4C2×D4×Q811yescharacteristic
684C2—24no—
685C2—44no—
686C2C2×#240911yescharacteristic
687C2C2×#300911yescharacteristic
688C2#6036811yescharacteristic
689C2#6049712yes—
690C2#6088912yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#3078C2semidirect
3. C22×#475C2semidirect
5. #24310C2semidirect
6. #24303C2semidirect
7. #24301C2semidirect
8. #23556C2semidirect
9. #23457C2semidirect
10. #21313C2semidirect
11. #21311C2semidirect
96. #536Q8semidirect
247. C22×Q8C2×Q8semidirect
302. C22⋊Q8C2×Q8semidirect
333. #117C2×Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#35032
C2×#30782
C22×#4752
C4×#5362
#243102
#243032
#243012
#235562
#234572
#213132
#213112
#209132
#208482
#206292