Finite Group Databasebeta

← #6435441

Subgroups — #6435441

Order 512.

3140 conjugacy classes of subgroups.

The subgroup lattice has 741 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#542C211yesmaximal, characteristic
2C2×#3078C211yesmaximal, characteristic, complemented
3C2×#3077C211yesmaximal, characteristic, complemented
4#34634C211yesmaximal, characteristic, complemented
5#29702C218yesmaximal, complemented
6#29697C214yesmaximal, complemented
7#29696C214yesmaximal, complemented
8#23552C212yesmaximal, complemented
9#23506C212yesmaximal, complemented
10#23481C212yesmaximal, complemented
11#23480C212yesmaximal
12#20647C211yesmaximal, characteristic, complemented
13#20646C211yesmaximal, characteristic, complemented
14#20627C211yesmaximal, characteristic, complemented
15C2×#542—28no—
16C2×#542—28no—
17C2×#542—22no—
18C2×#542—22no—
19C2×#542C2214yes—
20C2×#542C2214yes—
21C2×#533C2211yescharacteristic
22C22×C4⋊Q8C2211yescharacteristic
23C22×#126—21no—
24C22×#124—21no—
25C22×#124C2211yescharacteristic
26C22×#123—21no—
27C22×C22⋊Q8—21no—
28C22×C22⋊Q8C2211yescharacteristic
29C22×#121C2211yescharacteristic
30C22×C4×Q8C2211yescharacteristic
31C22×C4×D4C2211yescharacteristic
32C22×#117C2211yescharacteristic
33#3078—28no—
34#3077—28no—
35C2×#479C2214yescomplemented
36C2×#477C2212yescomplemented
37C2×#475C2212yescomplemented
38C2×#478C2214yescomplemented
39C2×#476C2212yescomplemented
40C2×#474C2212yescomplemented
41C2×#428C2212yes—
42C2×#428C2212yes—
43C2×#423C2211yescharacteristic
44C2×#423C2211yescharacteristic
45C2×#419C2212yes—
46C2×#419C2212yes—
47C2×#418C2212yes—
48C2×#418C2212yes—
49C2×#407C2212yes—
50#2629C2214yescomplemented
51#2628C2212yescomplemented
52#2628C2212yescomplemented
53#2627—21no—
54#2627—21no—
55#2626—21no—
56#2626—21no—
57#2396—22no—
58#2396—22no—
59#2296C2212yes—
60#2285C2211yescharacteristic
61#2280C2212yes—
62#2058C2218yescomplemented
63#2057C2218yescomplemented
64#2042C2218yescomplemented
65#2042C2218yescomplemented
66#2036C2218yescomplemented
67#2035C2218yescomplemented
68#2034C2218yes—
69#2033C2218yes—
70#1805C2214yescomplemented
71#1804C2214yescomplemented
72#1803C2214yescomplemented
73#1802C2214yescomplemented
74#1798C2214yescomplemented
75#1798C2214yescomplemented
76#1736C2212yes—
77#1736C2212yes—
78#1728C2212yes—
79C23×Q8—21no—
80C23×Q8C2311yescharacteristic
81C23×D4—21no—
82C23×D4C2311yescharacteristic
83C24×C4—21no—
84C24×C4C2311yescharacteristic
85C22×M16C2312yes—
86C22×M16C2312yes—
87#542—232no—
88#542—232no—
89#542—216no—
90#542—216no—
91#542—216no—
92#542—216no—
93#533—24no—
94#533—24no—
95C2×C4⋊Q8—24no—
96C2×C4⋊Q8—24no—
97C2×C4⋊Q8—22no—
98C2×C4⋊Q8C2311yescharacteristic
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2314yes—
101C2×#126—216no—
102C2×#126—24no—
103C2×#126—24no—
104C2×#125C2312yes—
105C2×#124—216no—
106C2×#124—24no—
107C2×#124—24no—
108C2×#124—24no—
109C2×#124—24no—
110C2×#124—22no—
111C2×#124C2314yes—
112C2×#123—216no—
113C2×#123—24no—
114C2×#123—24no—
115C2×C22⋊Q8—216no—
116C2×C22⋊Q8—24no—
117C2×C22⋊Q8—22no—
118C2×C22⋊Q8—21no—
119C2×C22⋊Q8—21no—
120C2×C22⋊Q8—24no—
121C2×C22⋊Q8—24no—
122C2×C22⋊Q8—24no—
123C2×C22⋊Q8C2314yes—
124C2×#121—24no—
125C2×#121—22no—
126C2×#121—24no—
127C2×#121C2314yes—
128C2×C4×Q8—24no—
129C2×C4×Q8—21no—
130C2×C4×Q8—21no—
131C2×C4×Q8—21no—
132C2×C4×Q8—21no—
133C2×C4×Q8D418yes—
134C2×C4×Q8C2312yes—
135C2×C4×Q8C2312yes—
136C2×C4×D4—21no—
137C2×C4×D4—21no—
138C2×C4×D4—24no—
139C2×C4×D4D418yes—
140C2×C4×D4C2312yes—
141C2×C4×D4C2312yes—
142C2×#117—24no—
143C2×#117—24no—
144C2×#117D412yes—
145C2×#117D412yes—
146C2×#117C2312yes—
147C2×#117C2312yes—
148C2×#117C2312yes—
149C2×#117C2312yes—
150C22×C4⋊C4—21no—
151C22×C4⋊C4—21no—
152C22×C4⋊C4—21no—
153C22×C4⋊C4D411yescharacteristic
154C22×C4⋊C4D411yescharacteristic
155C22×C4⋊C4C2311yescharacteristic
156C22×C4⋊C4C2311yescharacteristic
157C22×C22⋊C4—21no—
158C22×C22⋊C4—21no—
159C22×C22⋊C4—21no—
160C22×C22⋊C4—21no—
161C22×C22⋊C4—21no—
162C22×C22⋊C4D411yescharacteristic
163C22×C22⋊C4D411yescharacteristic
164C22×C42—21no—
165C22×C42C2311yescharacteristic
166C22×C42C2311yescharacteristic
167#479—216no—
168#478—216no—
169#477—28no—
170#476—28no—
171#475—28no—
172#474—28no—
173#428—28no—
174#428—28no—
175C2×#107C2314yes—
176C2×#107C2314yes—
177C2×#106C2314yes—
178C2×#106C2314yes—
179#423—24no—
180#423—24no—
181C2×C4⋊C8C2312yes—
182C2×C4⋊C8C2312yes—
183C2×C4⋊C8C2312yes—
184C2×C4⋊C8C2312yes—
185#419—28no—
186#419—44no—
187#418—28no—
188#418—44no—
189C2×#103C2314yes—
190C2×#103C2314yes—
191C2×#103C2318yes—
192C2×#102C2314yes—
193C2×#102C2314yes—
194C2×#102C2318yes—
195#407—44no—
196C2×C22⋊C8C2318yes—
197#400—22no—
198#400—22no—
199#399—22no—
200#399—22no—
201#398—24no—
202#398—24no—
203#396—24no—
204#396—24no—
205#395—22no—
206#395—22no—
207#393—22no—
208#393—22no—
209#389C2314yes—
210#389C2314yes—
211#385—22no—
212#385—22no—
213#384C2312yes—
214#382C2312yes—
215#381—22no—
216#380—21no—
217#380—21no—
218#380C2314yes—
219#380C2314yes—
220#379—22no—
221C4×C4⋊C4—21no—
222C4×C4⋊C4—21no—
223C4×C4⋊C4—24no—
224C4×C22⋊C4—24no—
225#340C2318yes—
226#340C2318yes—
227#336C2318yes—
228C25—21no—
229C25C2×D411yescharacteristic
230C22×Q8—28no—
231C22×Q8—22no—
232C22×Q8—21no—
233C22×Q8—24no—
234C22×Q8—21no—
235C22×Q8—21no—
236C22×Q8—22no—
237C22×Q8—22no—
238C22×Q8—22no—
239C22×Q8—22no—
240C22×Q8—84no—
241C22×Q8C2×D414yes—
242C22×Q8C2412yes—
243C22×D4—28no—
244C22×D4—24no—
245C22×D4—21no—
246C22×D4—22no—
247C22×D4—22no—
248C22×D4—22no—
249C22×D4—22no—
250C22×D4—22no—
251C22×D4—84no—
252C22×D4C2×D414yes—
253C22×D4C2412yes—
254C23×C4—21no—
255C23×C4—21no—
256C23×C4—21no—
257C23×C4—22no—
258C23×C4—22no—
259C23×C4—22no—
260C23×C4—22no—
261C23×C4—22no—
262C23×C4—22no—
263C23×C4—28no—
264C23×C4—22no—
265C23×C4—22no—
266C23×C4—84no—
267C23×C4C2×D411yescharacteristic
268C23×C4C2×D412yes—
269C23×C4C2×D411yescharacteristic
270C23×C4C2×D411yescharacteristic
271C23×C4C2×D411yescharacteristic
272C23×C4C2×D411yescharacteristic
273C23×C4C2412yes—
274C23×C4C2411yescharacteristic
275C2×M16—24no—
276C2×M16—24no—
277C2×M16—24no—
278C2×M16—24no—
279C2×M16—24no—
280C2×M16—24no—
281C22×C8C2414yes—
282C22×C8C2414yes—
283C4⋊Q8—216no—
284C4⋊Q8—28no—
285C4⋊Q8—28no—
286C4⋊Q8—22no—
287C4⋊Q8—22no—
288C4⋊Q8—22no—
289C4⋊Q8—22no—
290#126—232no—
291#126—232no—
292#125—24no—
293#125—24no—
294#124—232no—
295#124—232no—
296#124—216no—
297#124—28no—
298#124—28no—
299#123—232no—
300#123—232no—
301C22⋊Q8—232no—
302C22⋊Q8—232no—
303C22⋊Q8—216no—
304C22⋊Q8—28no—
305C22⋊Q8—28no—
306C22⋊Q8—44no—
307C22⋊Q8—44no—
308#121—216no—
309#121—28no—
310#121—28no—
311C4×Q8—28no—
312C4×Q8—28no—
313C4×Q8—24no—
314C4×Q8—24no—
315C4×Q8—44no—
316C4×Q8—44no—
317C4×Q8C2×D418yes—
318C4×Q8C2×D418yes—
319C4×D4—28no—
320C4×D4—28no—
321C4×D4—24no—
322C4×D4—24no—
323C4×D4C2×D418yes—
324C4×D4C2×D418yes—
325#117—28no—
326#117—28no—
327#117—24no—
328#117—24no—
329#117—24no—
330#117—24no—
331#117—24no—
332#117—24no—
333#117—24no—
334#117—24no—
335C2×C4⋊C4—28no—
336C2×C4⋊C4—28no—
337C2×C4⋊C4—28no—
338C2×C4⋊C4—24no—
339C2×C4⋊C4—24no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—24no—
343C2×C4⋊C4—24no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—24no—
355C2×C4⋊C4—24no—
356C2×C4⋊C4C2×D412yes—
357C2×C4⋊C4C2×D412yes—
358C2×C4⋊C4C2×D412yes—
359C2×C4⋊C4C2×D412yes—
360C2×C4⋊C4C2×D414yes—
361C2×C4⋊C4C2×D414yes—
362C2×C4⋊C4C2412yes—
363C2×C4⋊C4C2412yes—
364C2×C4⋊C4C2414yes—
365C2×C4⋊C4C2412yes—
366C2×C4⋊C4C2412yes—
367C2×C22⋊C4—28no—
368C2×C22⋊C4—28no—
369C2×C22⋊C4—28no—
370C2×C22⋊C4—28no—
371C2×C22⋊C4—24no—
372C2×C22⋊C4—28no—
373C2×C22⋊C4—24no—
374C2×C22⋊C4—24no—
375C2×C22⋊C4—22no—
376C2×C22⋊C4—22no—
377C2×C22⋊C4—24no—
378C2×C22⋊C4—24no—
379C2×C22⋊C4—24no—
380C2×C22⋊C4—22no—
381C2×C22⋊C4—22no—
382C2×C22⋊C4—24no—
383C2×C22⋊C4—24no—
384C2×C22⋊C4C2×D412yes—
385C2×C22⋊C4C2×D412yes—
386C2×C22⋊C4C2×D412yes—
387C2×C22⋊C4C2×D412yes—
388C2×C42—28no—
389C2×C42—22no—
390C2×C42—22no—
391C2×C42—22no—
392C2×C42—22no—
393C2×C42—22no—
394C2×C42—22no—
395C2×C42C2×D412yes—
396C2×C42C2×D411yescharacteristic
397C2×C42C2×D411yescharacteristic
398C2×C42C2×D414yes—
399C2×C42C2×D414yes—
400C2×C42C2412yes—
401C2×C42C2411yescharacteristic
402C2×C42C2411yescharacteristic
403#107—28no—
404#107—28no—
405#106—28no—
406#106—28no—
407C4⋊C8—24no—
408C4⋊C8—24no—
409C4⋊C8—24no—
410C4⋊C8—24no—
411#103—28no—
412#103—28no—
413#103—48no—
414#102—28no—
415#102—28no—
416#102—48no—
417C22⋊C8—48no—
418#95—24no—
419#95—24no—
420#95—22no—
421#95—22no—
422#95—24no—
423#95—24no—
424C24—22no—
425C24—24no—
426C24—22no—
427C24—22no—
428C24—22no—
429C24—42no—
430C24—42no—
431C24—44no—
432C24—44no—
433C24—42no—
434C24—42no—
435C24C22×D412yes—
436C24C22×D411yescharacteristic
437C2×Q8—216no—
438C2×Q8—24no—
439C2×Q8—22no—
440C2×Q8—28no—
441C2×Q8—28no—
442C2×Q8—22no—
443C2×Q8—22no—
444C2×Q8—22no—
445C2×Q8—22no—
446C2×Q8—42no—
447C2×Q8—42no—
448C2×Q8—44no—
449C2×Q8—42no—
450C2×Q8—44no—
451C2×Q8—42no—
452C2×Q8—82no—
453C2×Q8—82no—
454C2×Q8—88no—
455C2×Q8—816no—
456C2×Q8C22×D414yes—
457C2×D4—216no—
458C2×D4—24no—
459C2×D4—28no—
460C2×D4—28no—
461C2×D4—22no—
462C2×D4—22no—
463C2×D4—22no—
464C2×D4—22no—
465C2×D4—22no—
466C2×D4—42no—
467C2×D4—44no—
468C2×D4—42no—
469C2×D4—44no—
470C2×D4—88no—
471C2×D4—816no—
472C2×D4C22×D414yes—
473C22×C4—22no—
474C22×C4—24no—
475C22×C4—22no—
476C22×C4—24no—
477C22×C4—22no—
478C22×C4—24no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—24no—
485C22×C4—24no—
486C22×C4—24no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—28no—
494C22×C4—28no—
495C22×C4—24no—
496C22×C4—22no—
497C22×C4—22no—
498C22×C4—42no—
499C22×C4—42no—
500C22×C4—42no—
501C22×C4—42no—
502C22×C4—44no—
503C22×C4—44no—
504C22×C4—42no—
505C22×C4—42no—
506C22×C4—44no—
507C22×C4—44no—
508C22×C4—44no—
509C22×C4—44no—
510C22×C4—84no—
511C22×C4—84no—
512C22×C4—816no—
513C22×C4#12014yes—
514C22×C4#12014yes—
515C22×C4C22×D412yes—
516C22×C4C22×D412yes—
517C22×C4C22×D412yes—
518C22×C4C22×D411yescharacteristic
519C22×C4C22×D411yescharacteristic
520C22×C4C22×D411yescharacteristic
521C22×C4C22×D411yescharacteristic
522C22×C4C22×D411yescharacteristic
523C22×C4C22×D412yes—
524C22×C4C22×D412yes—
525C22×C4H(5,𝔽2)12yes—
526C22×C42−1+412yes—
527C22×C42−1+412yes—
528C22×C42−1+412yes—
529C22×C4C2511yescharacteristic
530M16—48no—
531M16—48no—
532C2×C8—24no—
533C2×C8—24no—
534C2×C8—24no—
535C2×C8—24no—
536C2×C8—24no—
537C2×C8—24no—
538C4⋊C4—216no—
539C4⋊C4—216no—
540C4⋊C4—28no—
541C4⋊C4—28no—
542C4⋊C4—24no—
543C4⋊C4—24no—
544C4⋊C4—24no—
545C4⋊C4—24no—
546C4⋊C4—24no—
547C4⋊C4—24no—
548C4⋊C4—22no—
549C4⋊C4—22no—
550C4⋊C4—22no—
551C4⋊C4—22no—
552C4⋊C4—22no—
553C4⋊C4—22no—
554C4⋊C4—24no—
555C4⋊C4—24no—
556C4⋊C4—44no—
557C4⋊C4—44no—
558C4⋊C4—42no—
559C4⋊C4—42no—
560C4⋊C4—82no—
561C4⋊C4—82no—
562C4⋊C4—84no—
563C4⋊C4—84no—
564C4⋊C4C22×D414yes—
565C4⋊C4C22×D414yes—
566C22⋊C4—216no—
567C22⋊C4—216no—
568C22⋊C4—216no—
569C22⋊C4—216no—
570C22⋊C4—28no—
571C22⋊C4—28no—
572C22⋊C4—24no—
573C22⋊C4—24no—
574C22⋊C4—24no—
575C22⋊C4—24no—
576C22⋊C4—44no—
577C22⋊C4—44no—
578C22⋊C4—84no—
579C22⋊C4—84no—
580C42—28no—
581C42—28no—
582C42—24no—
583C42—22no—
584C42—22no—
585C42—44no—
586C42—44no—
587C42#12014yes—
588C42#12014yes—
589C42C22×D412yes—
590C42C22×D412yes—
591C42C22×D412yes—
592C42C22×D412yes—
593C23—22no—
594C23—22no—
595C23—24no—
596C23—24no—
597C23—22no—
598C23—24no—
599C23—22no—
600C23—22no—
601C23—48no—
602C23—44no—
603C23—44no—
604C23—48no—
605C23—44no—
606C23—44no—
607C23—44no—
608C23—48no—
609C23—42no—
610C23—42no—
611C23—84no—
612C23—84no—
613C23—88no—
614C23—88no—
615C23C2×#12012yes—
616C23#53612yes—
617C23#53712yes—
618C23C23×D411yescharacteristic
619Q8—24no—
620Q8—24no—
621Q8—48no—
622Q8—42no—
623Q8—42no—
624Q8—84no—
625Q8—84no—
626Q8—816no—
627D4—24no—
628D4—24no—
629D4—48no—
630D4—42no—
631D4—42no—
632D4—816no—
633C2×C4—22no—
634C2×C4—22no—
635C2×C4—22no—
636C2×C4—22no—
637C2×C4—24no—
638C2×C4—24no—
639C2×C4—22no—
640C2×C4—24no—
641C2×C4—22no—
642C2×C4—24no—
643C2×C4—22no—
644C2×C4—24no—
645C2×C4—22no—
646C2×C4—22no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—48no—
651C2×C4—48no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4—44no—
656C2×C4—48no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—48no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—44no—
665C2×C4—44no—
666C2×C4—44no—
667C2×C4—44no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—44no—
671C2×C4—88no—
672C2×C4—88no—
673C2×C4C2×#12012yes—
674C2×C4C2×#12012yes—
675C2×C4C2×#12012yes—
676C2×C4C2×#12012yes—
677C2×C4C2×#12012yes—
678C2×C4#53612yes—
679C2×C4#53614yes—
680C2×C4#53614yes—
681C2×C4#53712yes—
682C2×C4#53714yes—
683C2×C4#53714yes—
684C2×C4C23×D411yescharacteristic
685C2×C4C23×D411yescharacteristic
686C2×C4C2×H(5,𝔽2)11yescharacteristic
687C2×C4C2×2−1+411yescharacteristic
688C2×C4C2×2−1+411yescharacteristic
689C2×C4C2×2−1+411yescharacteristic
690C8—44no—
691C8—44no—
692C22—24no—
693C22—22no—
694C22—22no—
695C22—24no—
696C22—42no—
697C22—44no—
698C22—44no—
699C22—44no—
700C22—42no—
701C22—42no—
702C22—44no—
703C22—44no—
704C22—88no—
705C22—816no—
706C22—84no—
707C22—84no—
708C22#261614yes—
709C22C22×#12011yescharacteristic
710C22C2×#53611yescharacteristic
711C22C2×#53711yescharacteristic
712C22#358512yes—
713C22#358612yes—
714C4—22no—
715C4—22no—
716C4—22no—
717C4—22no—
718C4—42no—
719C4—44no—
720C4—42no—
721C4—44no—
722C4—44no—
723C4—42no—
724C4—82no—
725C4—82no—
726C4—84no—
727C4#261614yes—
728C4C2×#53611yescharacteristic
729C4C2×#53611yescharacteristic
730C4C2×#53711yescharacteristic
731C4C2×#53711yescharacteristic
732C2—24no—
733C2—82no—
734C2—84no—
735C2C2×#261611yescharacteristic
736C2#6036711yescharacteristic
737C2#6036811yescharacteristic
738C2#6087012yes—
739C2#6091912yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#3078C2semidirect
3. C2×#3077C2semidirect
4. #34634C2semidirect
5. #29702C2semidirect
6. #29697C2semidirect
7. #29696C2semidirect
8. #23552C2semidirect
9. #23506C2semidirect
10. #23481C2semidirect
12. #20647C2semidirect
13. #20646C2semidirect
14. #20627C2semidirect
35. C2×#479C22semidirect
36. C2×#477C22semidirect
37. C2×#475C22semidirect
38. C2×#478C22semidirect
39. C2×#476C22semidirect
40. C2×#474C22semidirect
50. #2629C22semidirect
51. #2628C22semidirect
52. #2628C22semidirect
62. #2058C22semidirect
63. #2057C22semidirect
64. #2042C22semidirect
65. #2042C22semidirect
66. #2036C22semidirect
67. #2035C22semidirect
70. #1805C22semidirect
71. #1804C22semidirect
72. #1803C22semidirect
73. #1802C22semidirect
74. #1798C22semidirect
75. #1798C22semidirect

Maximal subgroups

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

subgroupindex
C22×#5422
C2×#30782
C2×#30772
#346342
#297022
#296972
#296962
#235522
#235062
#234812
#234802
#206472
#206462
#206272