Finite Group Databasebeta

← #6447644

Subgroups — #6447644

Order 512.

3101 conjugacy classes of subgroups.

The subgroup lattice has 738 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3458C211yesmaximal, characteristic, complemented
2C2×#3137C212yesmaximal, complemented
3#35306C211yesmaximal, characteristic
4#25916C214yesmaximal, complemented
5#25914C214yesmaximal, complemented
6#25910C212yesmaximal, complemented
7#25909C212yesmaximal, complemented
8#25906C212yesmaximal, complemented
9#25904C212yesmaximal, complemented
10#23458C212yesmaximal, complemented
11#23457C212yesmaximal, complemented
12#21326C212yesmaximal, complemented
13#20911C211yesmaximal, characteristic, complemented
14#20910C211yesmaximal, characteristic
15#20908C211yesmaximal, characteristic
16C2×#1873C211yesmaximal, characteristic, complemented
17#20379C211yesmaximal, characteristic, complemented
18#3458—216no—
19C2×#544—22no—
20C2×#544C2214yes—
21C2×#544C2212yes—
22C2×#544C2212yes—
23C2×#543—22no—
24C2×#536C2211yescharacteristic
25C2×#533—22no—
26C2×#533—22no—
27C2×#533C2212yes—
28C2×#533C2212yes—
29C2×#525C2211yescharacteristic
30C22×C22⋊Q8C2211yescharacteristic
31C22×#117C2211yescharacteristic
32C22×#117C2211yescharacteristic
33C22×#117C2211yescharacteristic
34#3137—216no—
35C2×#491C2214yescomplemented
36C2×#489C2214yescomplemented
37C2×#488C2214yescomplemented
38C2×#487C2214yescomplemented
39C2×#417C2212yes—
40C2×#419C2212yes—
41C2×#416C2212yes—
42C22×#103C2212yes—
43C2×#407C2211yescharacteristic
44C22×C22⋊C8C2211yescharacteristic
45C2×#405C2212yes—
46#2857C2212yes—
47#2850C2212yes—
48#2621C2212yes—
49#2620C2212yes—
50#2417—22no—
51#2415—22no—
52#2413—22no—
53#2412—22no—
54#2317C2211yescharacteristic
55#2311C2211yescharacteristic
56#2310C2211yescharacteristic
57#2026C2214yescomplemented
58#2026C2214yescomplemented
59#2026C2214yescomplemented
60#2025C2214yescomplemented
61#2024C2214yescomplemented
62#2024C2214yescomplemented
63#2024C2214yescomplemented
64#2023C2214yescomplemented
65#1993C2214yescomplemented
66#1992C2214yescomplemented
67#1991C2214yescomplemented
68#1990C2214yescomplemented
69#1875C2211yescharacteristic
70#1875C2212yes—
71#1875C2211yescharacteristic
72#1875C2212yes—
73#1874C2212yes—
74#1873C2211yescharacteristic
75#1873C2212yes—
76#1873C2212yes—
77#1873C2212yes—
78#1873C2212yes—
79#1873C2212yes—
80#1873C2211yescharacteristic
81#1873C2212yes—
82#1873C2214yes—
83#1872C2212yes—
84#1872C2214yes—
85#1750C2214yes—
86#1749C2212yes—
87C4×C22⋊C8C2212yes—
88#1736C2212yes—
89#1736C2212yes—
90#1735C2212yes—
91C2×#340C2212yes—
92#1729C2212yes—
93C22×#41C2311yescharacteristic
94C23×Q8C2311yescharacteristic
95C24×C4C2311yescharacteristic
96C22×M16C2312yes—
97C23×C8C2312yes—
98#544—232no—
99#544—216no—
100#544—216no—
101#544—28no—
102#544—28no—
103#544—28no—
104#544D4116yescomplemented
105#543—232no—
106#536—28no—
107#533—232no—
108#533—232no—
109#533—216no—
110#533—216no—
111C2×C4⋊Q8—22no—
112C2×C4⋊Q8—24no—
113C2×C4⋊Q8—22no—
114C2×C4⋊Q8C2312yes—
115C2×C4⋊Q8C2314yes—
116C2×C4⋊Q8C2314yes—
117C2×#125—22no—
118C2×#125—24no—
119C2×#125—22no—
120C2×#125C2314yes—
121C2×#125C2314yes—
122C2×#124C2312yes—
123#525—24no—
124#525—24no—
125C2×#123—21no—
126C2×#123—22no—
127C2×C22⋊Q8—24no—
128C2×C22⋊Q8—24no—
129C2×C22⋊Q8—21no—
130C2×C22⋊Q8—21no—
131C2×C22⋊Q8—22no—
132C2×C22⋊Q8—24no—
133C2×C22⋊Q8—24no—
134C2×C22⋊Q8—24no—
135C2×C22⋊Q8—24no—
136C2×C22⋊Q8—24no—
137C2×C22⋊Q8—24no—
138C2×C22⋊Q8C2312yes—
139C2×C22⋊Q8C2311yescharacteristic
140C2×C22⋊Q8C2312yes—
141C2×C22⋊Q8C2311yescharacteristic
142C2×#121C2311yescharacteristic
143C2×#121C2311yescharacteristic
144C2×#120—21no—
145C2×C4×Q8—24no—
146C2×C4×Q8—24no—
147C2×C4×D4—21no—
148C2×C4×D4—21no—
149C2×#117—21no—
150C2×#117—21no—
151C2×#117—21no—
152C2×#117—21no—
153C2×#117—28no—
154C2×#117—28no—
155C2×#117—28no—
156C2×#117D412yes—
157C2×#117D412yes—
158C2×#117C2311yescharacteristic
159C2×#117C2311yescharacteristic
160C2×#117C2312yes—
161C2×#117C2311yescharacteristic
162C2×#117C2312yes—
163C2×#117C2312yes—
164C2×#117C2311yescharacteristic
165C2×#117C2312yes—
166C2×#117C2311yescharacteristic
167C22×C4⋊C4—21no—
168C22×C4⋊C4D412yes—
169C22×C4⋊C4C2311yescharacteristic
170C22×C4⋊C4C2312yes—
171C22×C4⋊C4C2312yes—
172C22×C22⋊C4—21no—
173C22×C22⋊C4—21no—
174C22×C22⋊C4—21no—
175C22×C22⋊C4D412yes—
176C22×C42—21no—
177C22×C42—21no—
178C22×C42C2312yes—
179#491—216no—
180#489—216no—
181#488—216no—
182#487—216no—
183#419—28no—
184#417—28no—
185#416—28no—
186C2×#103—28no—
187C2×#103C2312yes—
188C2×#103C2314yes—
189C2×#103C2314yes—
190C2×#103C2314yes—
191C2×#103C2314yes—
192C2×#103C2314yes—
193C2×#103C2312yes—
194C2×#102C2312yes—
195C2×#102C2314yes—
196C2×#102C2312yes—
197#407—24no—
198C2×C22⋊C8—24no—
199C2×C22⋊C8C2312yes—
200C2×C22⋊C8C2311yescharacteristic
201C2×C22⋊C8C2311yescharacteristic
202C2×C22⋊C8C2312yes—
203C2×C22⋊C8C2312yes—
204#405—28no—
205C2×#97C2314yes—
206C2×C4×C8C2314yes—
207#398—24no—
208#397—24no—
209#396—24no—
210#394—24no—
211#391—22no—
212#391C2312yes—
213#391C2312yes—
214#391C2312yes—
215#390C2312yes—
216#389—22no—
217#389—24no—
218#388—24no—
219#386—24no—
220#386—22no—
221#385—22no—
222#384C2312yes—
223#382C2312yes—
224#381—22no—
225#380—24no—
226#380—22no—
227#340C2314yes—
228#340C2314yes—
229#340C2312yes—
230#340C2312yes—
231#340C2314yes—
232#340C2314yes—
233#340C2312yes—
234#340C2312yes—
235#340C2314yes—
236#340C2314yes—
237#336C2314yes—
238#336C2314yes—
239C25C2×D411yescharacteristic
240C2×#41—24no—
241C2×#41—22no—
242C2×#41—22no—
243C2×#41—24no—
244C22×Q8—24no—
245C22×Q8—24no—
246C22×Q8—21no—
247C22×Q8—24no—
248C22×Q8—22no—
249C22×Q8—22no—
250C22×Q8—24no—
251C22×Q8C2412yes—
252C22×Q8C2411yescharacteristic
253C22×D4—21no—
254C22×D4—21no—
255C22×D4—21no—
256C22×D4C2411yescharacteristic
257C23×C4—24no—
258C23×C4—24no—
259C23×C4—22no—
260C23×C4—22no—
261C23×C4—21no—
262C23×C4—21no—
263C23×C4—21no—
264C23×C4C2×D411yescharacteristic
265C23×C4C2×D411yescharacteristic
266C23×C4C2×D411yescharacteristic
267C23×C4C2×D414yes—
268C23×C4#4114yes—
269C23×C4#4114yes—
270C23×C4C2411yescharacteristic
271C23×C4C2411yescharacteristic
272C23×C4C2411yescharacteristic
273C2×M16—24no—
274C2×M16—24no—
275C2×M16—24no—
276C22×C8—24no—
277C22×C8—24no—
278C22×C8—24no—
279C22×C8C2412yes—
280C22×C8C2412yes—
281C22×C8C2414yes—
282C4⋊Q8—232no—
283C4⋊Q8—216no—
284C4⋊Q8—216no—
285C4⋊Q8—28no—
286C4⋊Q8—28no—
287C4⋊Q8—28no—
288C4⋊Q8—28no—
289C4⋊Q8C2×D4116yes—
290#125—232no—
291#125—216no—
292#125—216no—
293#125—28no—
294#125—28no—
295#125—28no—
296#125C2×D4116yes—
297#124—28no—
298#123—44no—
299#123—48no—
300C22⋊Q8—232no—
301C22⋊Q8—232no—
302C22⋊Q8—232no—
303C22⋊Q8—232no—
304C22⋊Q8—232no—
305C22⋊Q8—232no—
306C22⋊Q8—24no—
307C22⋊Q8—24no—
308C22⋊Q8—24no—
309C22⋊Q8—24no—
310C22⋊Q8—22no—
311C22⋊Q8—22no—
312C22⋊Q8—22no—
313C22⋊Q8—22no—
314C22⋊Q8—44no—
315C22⋊Q8—44no—
316C22⋊Q8—48no—
317#121—22no—
318#121—22no—
319#121—22no—
320#121—22no—
321#120—44no—
322C4×Q8—232no—
323C4×Q8—232no—
324C4×D4—44no—
325C4×D4—44no—
326#117—28no—
327#117—28no—
328#117—28no—
329#117—28no—
330#117—24no—
331#117—24no—
332#117—28no—
333#117—24no—
334#117—24no—
335#117—28no—
336#117—28no—
337#117—28no—
338#117—28no—
339#117—22no—
340#117—22no—
341#117C2×D418yes—
342#117C2×D418yes—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—24no—
346C2×C4⋊C4—24no—
347C2×C4⋊C4—24no—
348C2×C4⋊C4—24no—
349C2×C4⋊C4—24no—
350C2×C4⋊C4—24no—
351C2×C4⋊C4—24no—
352C2×C4⋊C4—28no—
353C2×C4⋊C4—24no—
354C2×C4⋊C4—24no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—22no—
357C2×C4⋊C4—22no—
358C2×C4⋊C4—22no—
359C2×C4⋊C4—21no—
360C2×C4⋊C4—21no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—24no—
363C2×C4⋊C4—24no—
364C2×C4⋊C4—44no—
365C2×C4⋊C4C2×D412yes—
366C2×C4⋊C4C2×D414yes—
367C2×C4⋊C4C2×D418yes—
368C2×C4⋊C4C2×D412yes—
369C2×C4⋊C4C2412yes—
370C2×C4⋊C4C2411yescharacteristic
371C2×C4⋊C4C2411yescharacteristic
372C2×C4⋊C4C2412yes—
373C2×C4⋊C4C2412yes—
374C2×C4⋊C4C2414yes—
375C2×C22⋊C4—28no—
376C2×C22⋊C4—28no—
377C2×C22⋊C4—22no—
378C2×C22⋊C4—21no—
379C2×C22⋊C4—21no—
380C2×C22⋊C4—22no—
381C2×C22⋊C4—22no—
382C2×C22⋊C4—28no—
383C2×C22⋊C4—22no—
384C2×C22⋊C4—22no—
385C2×C22⋊C4—21no—
386C2×C22⋊C4—21no—
387C2×C22⋊C4—21no—
388C2×C22⋊C4—21no—
389C2×C22⋊C4—21no—
390C2×C22⋊C4—24no—
391C2×C22⋊C4—24no—
392C2×C22⋊C4—44no—
393C2×C22⋊C4C2×D412yes—
394C2×C22⋊C4C2×D414yes—
395C2×C22⋊C4C2×D412yes—
396C2×C42—28no—
397C2×C42—28no—
398C2×C42—21no—
399C2×C42—21no—
400C2×C42—22no—
401C2×C42—22no—
402C2×C42—22no—
403C2×C42—28no—
404C2×C42—21no—
405C2×C42C2×D414yes—
406C2×C42C2412yes—
407C2×C42C2412yes—
408#103—28no—
409#103—28no—
410#103—24no—
411#103—24no—
412#103—28no—
413#103—28no—
414#103—28no—
415#102—24no—
416#102—24no—
417#102—28no—
418C22⋊C8—24no—
419C22⋊C8—22no—
420C22⋊C8—22no—
421C22⋊C8—24no—
422C22⋊C8—24no—
423#97—28no—
424C4×C8—28no—
425#95—24no—
426#95—24no—
427#95—24no—
428#95—24no—
429C24—22no—
430C24—22no—
431C24—21no—
432C24—42no—
433C24—42no—
434C24#12414yes—
435C24C22×D411yescharacteristic
436C24C22×D411yescharacteristic
437C24C22×D411yescharacteristic
438#41—48no—
439#41—48no—
440C2×Q8—216no—
441C2×Q8—216no—
442C2×Q8—24no—
443C2×Q8—22no—
444C2×Q8—22no—
445C2×Q8—22no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—24no—
449C2×Q8—22no—
450C2×Q8—42no—
451C2×Q8—42no—
452C2×Q8—42no—
453C2×Q8—48no—
454C2×Q8—48no—
455C2×Q8—84no—
456C2×Q8—84no—
457C2×Q8—84no—
458C2×Q8—84no—
459C2×D4—22no—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4—22no—
463C2×D4—44no—
464C2×D4—42no—
465C2×D4—42no—
466C2×D4—42no—
467C2×D4—42no—
468C2×D4—42no—
469C2×D4—82no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—216no—
477C22×C4—216no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—24no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—28no—
486C22×C4—28no—
487C22×C4—28no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—21no—
493C22×C4—44no—
494C22×C4—44no—
495C22×C4—42no—
496C22×C4—42no—
497C22×C4—42no—
498C22×C4—42no—
499C22×C4—42no—
500C22×C4—42no—
501C22×C4—42no—
502C22×C4—82no—
503C22×C4—82no—
504C22×C4—82no—
505C22×C4—82no—
506C22×C4—82no—
507C22×C4#12414yes—
508C22×C4#12414yes—
509C22×C4#12414yes—
510C22×C4C22×D414yes—
511C22×C4C22×D414yes—
512C22×C4C22×D411yescharacteristic
513C22×C4C22×D411yescharacteristic
514C22×C4C22×D411yescharacteristic
515C22×C4C22×D414yes—
516C22×C4C2×#4112yes—
517C22×C4C2×#4112yes—
518C22×C4C2×#4112yes—
519C22×C4C2×#4112yes—
520C22×C4C2×#4112yes—
521C22×C4C2×#4112yes—
522C22×C42−1+414yes—
523C22×C4C2511yescharacteristic
524M16—48no—
525C2×C8—22no—
526C2×C8—24no—
527C2×C8—22no—
528C2×C8—22no—
529C2×C8—22no—
530C2×C8—24no—
531C2×C8—22no—
532C2×C8—22no—
533C2×C8—24no—
534C2×C8—48no—
535C4⋊C4—28no—
536C4⋊C4—216no—
537C4⋊C4—216no—
538C4⋊C4—216no—
539C4⋊C4—28no—
540C4⋊C4—24no—
541C4⋊C4—28no—
542C4⋊C4—216no—
543C4⋊C4—216no—
544C4⋊C4—216no—
545C4⋊C4—24no—
546C4⋊C4—22no—
547C4⋊C4—22no—
548C4⋊C4—22no—
549C4⋊C4—22no—
550C4⋊C4—21no—
551C4⋊C4—21no—
552C4⋊C4—21no—
553C4⋊C4—21no—
554C4⋊C4—48no—
555C4⋊C4—42no—
556C4⋊C4—44no—
557C4⋊C4—44no—
558C4⋊C4—42no—
559C4⋊C4—42no—
560C4⋊C4—84no—
561C4⋊C4#12018yes—
562C4⋊C4C22×D414yes—
563C4⋊C4C22×D414yes—
564C4⋊C4C22×D418yes—
565C22⋊C4—28no—
566C22⋊C4—28no—
567C22⋊C4—24no—
568C22⋊C4—24no—
569C22⋊C4—28no—
570C22⋊C4—24no—
571C22⋊C4—28no—
572C22⋊C4—44no—
573C22⋊C4—44no—
574C22⋊C4—42no—
575C22⋊C4—42no—
576C22⋊C4—42no—
577C22⋊C4—48no—
578C22⋊C4—42no—
579C22⋊C4—42no—
580C22⋊C4—84no—
581C22⋊C4#12018yes—
582C42—28no—
583C42—24no—
584C42—24no—
585C42—28no—
586C42—28no—
587C42—28no—
588C42—24no—
589C42—42no—
590C42C22×D418yes—
591C23—22no—
592C23—22no—
593C23—22no—
594C23—22no—
595C23—22no—
596C23—22no—
597C23—21no—
598C23—21no—
599C23—21no—
600C23—21no—
601C23—28no—
602C23—44no—
603C23—44no—
604C23—42no—
605C23—42no—
606C23—41no—
607C23—41no—
608C23—41no—
609C23—84no—
610C23—84no—
611C23C2×#12012yes—
612C23C2×#12412yes—
613C23C2×#12412yes—
614C23C2×#12412yes—
615C23#53612yes—
616C23C23×D411yescharacteristic
617Q8—44no—
618Q8—44no—
619Q8—42no—
620Q8—42no—
621Q8—816no—
622Q8—816no—
623Q8—84no—
624D4—42no—
625D4—42no—
626D4—84no—
627D4—84no—
628D4—84no—
629C2×C4—24no—
630C2×C4—24no—
631C2×C4—21no—
632C2×C4—21no—
633C2×C4—21no—
634C2×C4—21no—
635C2×C4—28no—
636C2×C4—28no—
637C2×C4—28no—
638C2×C4—24no—
639C2×C4—24no—
640C2×C4—24no—
641C2×C4—24no—
642C2×C4—24no—
643C2×C4—24no—
644C2×C4—24no—
645C2×C4—28no—
646C2×C4—24no—
647C2×C4—24no—
648C2×C4—44no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—44no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—44no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—41no—
659C2×C4—41no—
660C2×C4—41no—
661C2×C4—42no—
662C2×C4—41no—
663C2×C4—42no—
664C2×C4—41no—
665C2×C4—42no—
666C2×C4—88no—
667C2×C4—88no—
668C2×C4—88no—
669C2×C4—84no—
670C2×C4—84no—
671C2×C4—84no—
672C2×C4C2×#12018yes—
673C2×C4C2×#12012yes—
674C2×C4C2×#12412yes—
675C2×C4C2×#12412yes—
676C2×C4C2×#12412yes—
677C2×C4#53612yes—
678C2×C4#54818yes—
679C2×C4#54818yes—
680C2×C4C23×D412yes—
681C2×C4C22×#4111yescharacteristic
682C2×C4C22×#4111yescharacteristic
683C2×C4C2×2−1+412yes—
684C8—42no—
685C8—42no—
686C8—44no—
687C22—22no—
688C22—22no—
689C22—24no—
690C22—24no—
691C22—22no—
692C22—22no—
693C22—22no—
694C22—22no—
695C22—22no—
696C22—22no—
697C22—42no—
698C22—42no—
699C22—42no—
700C22—42no—
701C22—88no—
702C22—81no—
703C22—81no—
704C22—81no—
705C22—82no—
706C22—82no—
707C22#243614yes—
708C22C22×#12011yescharacteristic
709C22C22×#12411yescharacteristic
710C22C2×#53611yescharacteristic
711C22#358412yes—
712C22#358612yes—
713C4—22no—
714C4—22no—
715C4—28no—
716C4—44no—
717C4—48no—
718C4—44no—
719C4—44no—
720C4—81no—
721C4—82no—
722C4—82no—
723C4—81no—
724C4—81no—
725C4#243614yes—
726C4C2×#54812yes—
727C4C2×#54812yes—
728C2—24no—
729C2—42no—
730C2—42no—
731C2—161no—
732C2C2×#243611yescharacteristic
733C2#6037712yes—
734C2#6061211yescharacteristic
735C2#6062011yescharacteristic
736C2#6087612yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#525C22
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
1. C2×#3458C2semidirect
2. C2×#3137C2semidirect
4. #25916C2semidirect
5. #25914C2semidirect
6. #25910C2semidirect
7. #25909C2semidirect
8. #25906C2semidirect
9. #25904C2semidirect
10. #23458C2semidirect
11. #23457C2semidirect
12. #21326C2semidirect
13. #20911C2semidirect
16. C2×#1873C2semidirect
17. #20379C2semidirect
35. C2×#491C22semidirect
36. C2×#489C22semidirect
37. C2×#488C22semidirect
38. C2×#487C22semidirect
57. #2026C22semidirect
58. #2026C22semidirect
59. #2026C22semidirect
60. #2025C22semidirect
61. #2024C22semidirect
62. #2024C22semidirect
63. #2024C22semidirect
64. #2023C22semidirect
65. #1993C22semidirect
66. #1992C22semidirect
67. #1991C22semidirect
68. #1990C22semidirect
104. #544D4semidirect

Maximal subgroups

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

subgroupindex
C2×#34582
C2×#31372
#353062
#259162
#259142
#259102
#259092
#259062
#259042
#234582
#234572
#213262
#209112
#209102
#209082
C2×#18732
#203792