Finite Group Databasebeta

← #6445713

Subgroups — #6445713

Order 512.

3094 conjugacy classes of subgroups.

The subgroup lattice has 685 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3503C211yesmaximal, characteristic
2C2×#3165C211yesmaximal, characteristic, complemented
3C2×#3160C211yesmaximal, characteristic, complemented
4#35280C211yesmaximal, characteristic
5#25822C218yesmaximal, complemented
6#25813C212yesmaximal, complemented
7#25812C212yesmaximal, complemented
8#25812C212yesmaximal, complemented
9#25811C212yesmaximal, complemented
10#23557C214yesmaximal, complemented
11#21322C211yesmaximal, characteristic, complemented
12C2×#1990C211yesmaximal, characteristic, complemented
13#20849C214yesmaximal
14C4×#408C211yesmaximal, 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×#533C2212yes—
26C2×#533C2211yescharacteristic
27C2×#533C2211yescharacteristic
28C22×C4⋊Q8C2212yes—
29C22×C4⋊Q8C2211yescharacteristic
30C2×#520C2212yes—
31C2×C4×#41C2211yescharacteristic
32#3165—28no—
33C2×#501C2214yes—
34#3160—28no—
35C2×#500C2211yescharacteristic
36C2×#500C2212yes—
37C2×#500C2211yescharacteristic
38C2×#499C2214yes—
39C2×#498C2211yescharacteristic
40C2×#498C2212yes—
41C2×#498C2211yescharacteristic
42C2×#428C2214yes—
43C2×#427C2214yes—
44C2×#408C2212yes—
45C2×C4×M16C2211yescharacteristic
46C22×C4×C8C2211yescharacteristic
47#2857C2212yes—
48#2857C2212yes—
49#2629C2214yes—
50#2582—21no—
51#2582—21no—
52#2538—21no—
53#2538—21no—
54#2417—22no—
55#2417—22no—
56#2316C2212yes—
57#2311C2212yes—
58C2×#384C2211yescharacteristic
59#2059C2218yes—
60#2059C2218yes—
61#2058C2218yes—
62#2057C2218yes—
63#1992C2214yes—
64#1990C2212yes—
65#1990C2212yes—
66#1990C2214yes—
67#1990C2212yes—
68#1990C2212yes—
69#1854C2218yes—
70#1854C2218yes—
71#1853C2218yes—
72#1852C2218yes—
73#1750C2214yes—
74C4×C22⋊C8C2212yes—
75C4×C22⋊C8C2212yes—
76#1736C2214yes—
77C2×#340C2214yes—
78#1729C2212yes—
79C22×#41C2311yescharacteristic
80C23×Q8—22no—
81C23×Q8—21no—
82C22×M16C2312yes—
83C23×C8C2312yes—
84Q82—216no—
85#557—216no—
86#554—216no—
87#554—232no—
88D4×Q8—232no—
89D4×Q8—216no—
90#544—216no—
91#544—216no—
92#544—28no—
93#544—28no—
94#533—28no—
95#533—28no—
96#533—24no—
97#533—24no—
98#533—24no—
99#533D418yes—
100C2×C4⋊Q8—28no—
101C2×C4⋊Q8—28no—
102C2×C4⋊Q8—22no—
103C2×C4⋊Q8—24no—
104C2×C4⋊Q8—22no—
105C2×C4⋊Q8—21no—
106C2×C4⋊Q8—22no—
107C2×C4⋊Q8—21no—
108C2×C4⋊Q8—24no—
109C2×C4⋊Q8D418yes—
110C2×C4⋊Q8C2314yes—
111C2×C4⋊Q8C2311yescharacteristic
112C2×C4⋊Q8C2311yescharacteristic
113C2×C4⋊Q8C2311yescharacteristic
114C2×C4⋊Q8C2312yes—
115C2×C4⋊Q8C2311yescharacteristic
116C2×C4⋊Q8C2312yes—
117C2×#125—22no—
118C2×#125—24no—
119C2×#125C2314yes—
120C2×#125C2312yes—
121C2×#125C2312yes—
122C2×C22⋊Q8—22no—
123C2×C22⋊Q8—21no—
124C2×C22⋊Q8—22no—
125C2×C22⋊Q8—21no—
126C2×C22⋊Q8—21no—
127C2×C22⋊Q8—21no—
128C2×C22⋊Q8—22no—
129C2×C22⋊Q8—22no—
130C2×C22⋊Q8—24no—
131C2×C22⋊Q8—24no—
132C2×C22⋊Q8—24no—
133#520—28no—
134#520—28no—
135C4×#41—24no—
136C4×#41—24no—
137C2×C4×Q8—22no—
138C2×C4×Q8—22no—
139C2×C4×Q8—21no—
140C2×C4×Q8—22no—
141C2×C4×Q8—21no—
142C2×C4×Q8—21no—
143C2×C4×Q8—21no—
144C2×C4×Q8D411yescharacteristic
145C2×C4×Q8D411yescharacteristic
146C2×C4×D4—24no—
147C2×C4×D4—22no—
148C2×C4×D4D412yes—
149C2×C4×D4D412yes—
150C2×C4×D4D411yescharacteristic
151C2×C4×D4D411yescharacteristic
152C2×#117C2312yes—
153C2×#117C2314yes—
154C2×#117C2312yes—
155C2×#117C2311yescharacteristic
156C2×#117C2312yes—
157C22×C4⋊C4—22no—
158C22×C4⋊C4—21no—
159C22×C4⋊C4—21no—
160C22×C4⋊C4—21no—
161C22×C4⋊C4C2312yes—
162C22×C4⋊C4C2314yes—
163C22×C4⋊C4C2312yes—
164C22×C42C2312yes—
165C22×C42C2311yescharacteristic
166#501—216no—
167#500—24no—
168#500—24no—
169#500—28no—
170#499—216no—
171#498—24no—
172#498—24no—
173#498—28no—
174#428—216no—
175#427—216no—
176C2×#107C2314yes—
177C2×#107C2318yes—
178C2×#107C2314yes—
179C2×#106C2314yes—
180C2×#106C2318yes—
181C2×#106C2314yes—
182#408Q8116yescomplemented
183C2×C22⋊C8C2314yes—
184C2×C22⋊C8C2314yes—
185C4×M16—24no—
186C2×#97C2312yes—
187C2×C4×C8—24no—
188C2×C4×C8C2311yescharacteristic
189C2×C4×C8C2312yes—
190C2×C4×C8C2311yescharacteristic
191C2×C4×C8C2312yes—
192#398—24no—
193#398—24no—
194#396—24no—
195#396—24no—
196#391—22no—
197#391—22no—
198#391—22no—
199#390—22no—
200#389—22no—
201#389—22no—
202#389—22no—
203#389—22no—
204#389C2314yes—
205#389C2314yes—
206#386—24no—
207#384—21no—
208#384—21no—
209#384C2312yes—
210#384C2312yes—
211#380—22no—
212#380—22no—
213#380—24no—
214#380C2314yes—
215#380C2314yes—
216#340C2318yes—
217#340C2318yes—
218#340C2318yes—
219#340C2318yes—
220#336C2314yes—
221#336C2314yes—
222C2×#41—22no—
223C2×#41—22no—
224C2×#41—24no—
225C2×#41C2×Q818yes—
226C22×Q8—216no—
227C22×Q8—28no—
228C22×Q8—21no—
229C22×Q8—22no—
230C22×Q8—21no—
231C22×Q8—24no—
232C22×Q8—22no—
233C22×Q8—21no—
234C22×Q8—21no—
235C22×Q8—82no—
236C22×Q8—82no—
237C22×Q8—84no—
238C22×Q8—84no—
239C22×Q8C2×D411yescharacteristic
240C22×D4C2×D412yes—
241C22×D4C2×D411yescharacteristic
242C23×C4—24no—
243C23×C4—22no—
244C23×C4—22no—
245C23×C4C2×D412yes—
246C23×C4C2×D412yes—
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2412yes—
250C23×C4C2411yescharacteristic
251C2×M16—24no—
252C2×M16—24no—
253C2×M16C2×Q818yescomplemented
254C22×C8—24no—
255C22×C8—24no—
256C22×C8C2×Q818yescomplemented
257C22×C8C2412yes—
258C22×C8C2412yes—
259C22×C8C2414yes—
260C4⋊Q8—216no—
261C4⋊Q8—216no—
262C4⋊Q8—232no—
263C4⋊Q8—216no—
264C4⋊Q8—28no—
265C4⋊Q8—28no—
266C4⋊Q8—28no—
267C4⋊Q8—28no—
268C4⋊Q8—22no—
269C4⋊Q8—24no—
270C4⋊Q8—24no—
271C4⋊Q8—22no—
272C4⋊Q8—22no—
273C4⋊Q8—22no—
274C4⋊Q8—22no—
275C4⋊Q8—22no—
276C4⋊Q8C2×D414yes—
277C4⋊Q8C2×D418yes—
278C4⋊Q8C2×D418yes—
279C4⋊Q8C2×D414yes—
280#125—232no—
281#125—216no—
282#125—28no—
283#125—28no—
284#125—24no—
285#125—24no—
286#125—24no—
287#125C2×D418yes—
288C22⋊Q8—216no—
289C22⋊Q8—216no—
290C22⋊Q8—232no—
291C22⋊Q8—232no—
292C22⋊Q8—232no—
293C22⋊Q8—28no—
294C22⋊Q8—28no—
295C22⋊Q8—44no—
296C22⋊Q8—44no—
297C22⋊Q8—84no—
298C22⋊Q8—84no—
299C4×Q8—216no—
300C4×Q8—216no—
301C4×Q8—216no—
302C4×Q8—28no—
303C4×Q8—28no—
304C4×Q8—24no—
305C4×Q8—24no—
306C4×Q8—44no—
307C4×Q8—44no—
308C4×D4—232no—
309C4×D4—216no—
310C4×D4—28no—
311C4×D4—28no—
312C4×D4—24no—
313C4×D4—24no—
314#117—22no—
315#117—22no—
316#117—24no—
317#117—28no—
318#117—28no—
319#117—24no—
320#117—24no—
321#117—24no—
322#117—24no—
323#117C2×D418yes—
324C2×C4⋊C4—216no—
325C2×C4⋊C4—28no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—24no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—24no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—24no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—24no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—24no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—28no—
339C2×C4⋊C4—28no—
340C2×C4⋊C4—28no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—24no—
351C2×C4⋊C4—24no—
352C2×C4⋊C4—82no—
353C2×C4⋊C4—82no—
354C2×C4⋊C4C2×D411yescharacteristic
355C2×C4⋊C4C2×D412yes—
356C2×C4⋊C4C2×D411yescharacteristic
357C2×C4⋊C4C2×D412yes—
358C2×C4⋊C4C2×D418yes—
359C2×C4⋊C4C2414yes—
360C2×C4⋊C4C2412yes—
361C2×C4⋊C4C2412yes—
362C2×C4⋊C4C2412yes—
363C2×C4⋊C4C2414yes—
364C2×C4⋊C4C2412yes—
365C2×C22⋊C4—24no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—24no—
369C2×C22⋊C4—24no—
370C2×C22⋊C4C2×D412yes—
371C2×C22⋊C4C2×D412yes—
372C2×C22⋊C4C2×D412yes—
373C2×C42—24no—
374C2×C42—22no—
375C2×C42—24no—
376C2×C42—24no—
377C2×C42—22no—
378C2×C42C2×D412yes—
379C2×C42C2×D412yes—
380C2×C42C2×D411yescharacteristic
381C2×C42C2×D411yescharacteristic
382C2×C42C2×D414yes—
383C2×C42C2×D414yes—
384C2×C42C2412yes—
385C2×C42C2411yescharacteristic
386C2×C42C2411yescharacteristic
387#107—28no—
388#107—28no—
389#107—216no—
390#106—28no—
391#106—28no—
392#106—216no—
393C22⋊C8C2×Q8116yescomplemented
394C22⋊C8C2×Q8116yescomplemented
395#97—24no—
396C4×C8—22no—
397C4×C8—24no—
398C4×C8—22no—
399C4×C8—24no—
400#95—24no—
401#95—24no—
402#95—24no—
403#95—24no—
404C24C22×D412yes—
405C24C22×D411yescharacteristic
406#41—216no—
407#41—216no—
408C2×Q8—216no—
409C2×Q8—22no—
410C2×Q8—24no—
411C2×Q8—28no—
412C2×Q8—24no—
413C2×Q8—28no—
414C2×Q8—22no—
415C2×Q8—42no—
416C2×Q8—42no—
417C2×Q8—44no—
418C2×Q8—42no—
419C2×Q8—42no—
420C2×Q8—42no—
421C2×Q8—44no—
422C2×Q8—42no—
423C2×Q8—42no—
424C2×Q8—42no—
425C2×Q8—42no—
426C2×Q8—42no—
427C2×Q8—82no—
428C2×Q8—82no—
429C2×Q8—84no—
430C2×Q8—84no—
431C2×Q8—816no—
432C2×Q8—832no—
433C2×Q8C4⋊Q814yes—
434C2×D4—24no—
435C2×D4—24no—
436C2×D4—22no—
437C2×D4—22no—
438C2×D4C4⋊Q818yes—
439C2×D4C4⋊Q814yes—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—216no—
451C22×C4—28no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—24no—
461C22×C4—44no—
462C22×C4—82no—
463C22×C4—84no—
464C22×C4—82no—
465C22×C4—84no—
466C22×C4—82no—
467C22×C4—82no—
468C22×C4#12014yes—
469C22×C4#12014yes—
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D412yes—
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D412yes—
479C22×C4C22×D412yes—
480C22×C4C22×Q818yes—
481C22×C4C22×Q814yes—
482C22×C42−1+414yes—
483C22×C4C2511yescharacteristic
484M16—216no—
485C2×C8—216no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—24no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—24no—
492C2×C8C22×Q814yes—
493C2×C8C22×Q814yes—
494C2×C8C22×Q818yes—
495C4⋊C4—216no—
496C4⋊C4—216no—
497C4⋊C4—216no—
498C4⋊C4—216no—
499C4⋊C4—24no—
500C4⋊C4—28no—
501C4⋊C4—28no—
502C4⋊C4—28no—
503C4⋊C4—24no—
504C4⋊C4—28no—
505C4⋊C4—28no—
506C4⋊C4—24no—
507C4⋊C4—22no—
508C4⋊C4—24no—
509C4⋊C4—22no—
510C4⋊C4—24no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—28no—
518C4⋊C4—44no—
519C4⋊C4—44no—
520C4⋊C4—42no—
521C4⋊C4—42no—
522C4⋊C4—44no—
523C4⋊C4—44no—
524C4⋊C4—44no—
525C4⋊C4—82no—
526C4⋊C4—82no—
527C4⋊C4—84no—
528C4⋊C4—84no—
529C4⋊C4C22×D414yes—
530C4⋊C4C22×D418yes—
531C4⋊C4C22×D414yes—
532C22⋊C4—216no—
533C22⋊C4—28no—
534C22⋊C4—24no—
535C22⋊C4—24no—
536C22⋊C4—24no—
537C22⋊C4—44no—
538C22⋊C4—84no—
539C22⋊C4—84no—
540C42—216no—
541C42—24no—
542C42—24no—
543C42—28no—
544C42—22no—
545C42—22no—
546C42—44no—
547C42#12014yes—
548C42#12014yes—
549C42C22×D412yes—
550C42C22×D412yes—
551C42C22×D412yes—
552C42C22×D412yes—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—22no—
558C23—22no—
559C23—22no—
560C23—22no—
561C23—22no—
562C23—22no—
563C23C2×#12012yes—
564C23C2×C4⋊Q814yes—
565C23C2×C4⋊Q812yes—
566C23#53612yes—
567C23C23×D411yescharacteristic
568Q8—28no—
569Q8—416no—
570Q8—48no—
571Q8—48no—
572Q8—88no—
573Q8—816no—
574Q8—84no—
575Q8—84no—
576D4—216no—
577D4—28no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—28no—
585C2×C4—28no—
586C2×C4—28no—
587C2×C4—28no—
588C2×C4—24no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—28no—
592C2×C4—24no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—48no—
604C2×C4—48no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—88no—
615C2×C4—816no—
616C2×C4—164no—
617C2×C4C2×#12012yes—
618C2×C4C2×#12012yes—
619C2×C4C2×#12012yes—
620C2×C4C2×#12012yes—
621C2×C4C2×#12012yes—
622C2×C4C2×C4⋊Q814yes—
623C2×C4C2×C4⋊Q812yes—
624C2×C4#53612yes—
625C2×C4D4×Q818yes—
626C2×C4D4×Q818yes—
627C2×C4#57612yes—
628C2×C4#57712yes—
629C2×C4#57712yes—
630C2×C4#57812yes—
631C2×C4C23×D411yescharacteristic
632C2×C4C23×D411yescharacteristic
633C2×C4C23×Q812yes—
634C2×C4C2×2−1+412yes—
635C8—24no—
636C8—24no—
637C8—28no—
638C22—22no—
639C22—22no—
640C22—22no—
641C22—22no—
642C22—22no—
643C22—22no—
644C22—24no—
645C22—24no—
646C22—24no—
647C22—42no—
648C22—42no—
649C22—42no—
650C22—42no—
651C22#243414yes—
652C22C22×#12011yescharacteristic
653C22C22×C4⋊Q811yescharacteristic
654C22C2×#53611yescharacteristic
655C22C2×#57611yescharacteristic
656C22C2×#57711yescharacteristic
657C22C2×#57711yescharacteristic
658C22C2×#57811yescharacteristic
659C4—24no—
660C4—22no—
661C4—22no—
662C4—42no—
663C4—48no—
664C4—44no—
665C4—42no—
666C4—44no—
667C4—82no—
668C4—82no—
669C4—84no—
670C4#243414yes—
671C4#301214yes—
672C4#301314yes—
673C4C2×D4×Q812yes—
674C4C2×D4×Q812yes—
675C2—24no—
676C2—44no—
677C2C2×#243411yescharacteristic
678C2C2×#301211yescharacteristic
679C2C2×#301311yescharacteristic
680C2#6066511yescharacteristic
681C2#6066711yescharacteristic
682C2#6087711yescharacteristic
683C2#6087811yescharacteristic
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×#3165C2semidirect
3. C2×#3160C2semidirect
5. #25822C2semidirect
6. #25813C2semidirect
7. #25812C2semidirect
8. #25812C2semidirect
9. #25811C2semidirect
10. #23557C2semidirect
11. #21322C2semidirect
12. C2×#1990C2semidirect
182. #408Q8semidirect
253. C2×M16C2×Q8semidirect
256. C22×C8C2×Q8semidirect
393. C22⋊C8C2×Q8semidirect
394. C22⋊C8C2×Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#35032
C2×#31652
C2×#31602
#352802
#258222
#258132
#258122
#258112
#235572
#213222
C2×#19902
#208492
C4×#4082