Finite Group Databasebeta

← #6489559

Subgroups — #6489559

Order 512.

2812 conjugacy classes of subgroups.

The subgroup lattice has 699 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3458C211yesmaximal, characteristic
2C2×#3237C211yesmaximal, characteristic, complemented
3C2×#3231C211yesmaximal, characteristic, complemented
4#35325C211yesmaximal, characteristic
5#32985C214yesmaximal
6#29645C214yesmaximal, complemented
7#25914C212yesmaximal, complemented
8#25914C212yesmaximal, complemented
9#25821C212yesmaximal, complemented
10#25820C212yesmaximal, complemented
11#24311C214yesmaximal, complemented
12#24310C214yesmaximal, complemented
13#21842C211yesmaximal, characteristic, complemented
14#21837C211yesmaximal, characteristic, complemented
15#20717C211yesmaximal, characteristic
16#3458—28no—
17#3458—28no—
18C2×#544—21no—
19C2×#544—21no—
20C2×#544C2212yes—
21C2×#544C2212yes—
22C2×#544C2212yes—
23C2×#544C2212yes—
24C2×#543—21no—
25C2×#543—21no—
26C2×#533—22no—
27C2×#533—22no—
28C2×#533C2211yescharacteristic
29C2×#533C2212yes—
30C2×#533C2211yescharacteristic
31C2×#533C2211yescharacteristic
32C22×#117C2212yes—
33C22×#117C2211yescharacteristic
34#3237—28no—
35#3231—28no—
36C2×#501C2212yescomplemented
37C2×#500C2211yescharacteristic, complemented
38C2×#498C2211yescharacteristic, complemented
39C2×#491C2212yescomplemented
40C2×#487C2211yescharacteristic, complemented
41C2×#487C2211yescharacteristic, complemented
42C2×#485C2214yes—
43C2×#485C2214yes—
44C2×#479C2214yescomplemented
45C2×#477C2212yescomplemented
46C2×#475C2212yescomplemented
47C2×#432C2211yescharacteristic
48C2×#431C2211yescharacteristic
49#2855C2212yes—
50#2848C2212yes—
51#2628C2212yes—
52#2621C2212yes—
53#2611—22no—
54#2599—22no—
55#2583—21no—
56#2561—21no—
57#2484—21no—
58#2482—21no—
59#2323C2214yes—
60#2283C2211yescharacteristic
61#2059C2214yescomplemented
62#2059C2214yescomplemented
63#2058C2214yescomplemented
64#2057C2214yescomplemented
65#2026C2214yescomplemented
66#2026C2214yescomplemented
67#2024C2214yescomplemented
68#2024C2214yescomplemented
69#1993C2212yescomplemented
70#1993C2212yescomplemented
71#1991C2212yescomplemented
72#1991C2212yescomplemented
73#1989C2214yescomplemented
74#1988C2212yescomplemented
75#1987C2212yescomplemented
76#1873C2214yes—
77#1873C2214yes—
78#1872C2214yes—
79#1872C2214yes—
80#1854C2214yes—
81#1854C2214yes—
82#1853C2214yes—
83#1852C2214yes—
84#1825C2214yes—
85#1825C2214yes—
86#1817C2212yes—
87#1798C2214yes—
88#1779C2212yes—
89#1750C2212yes—
90#1750C2212yes—
91C24×C4C2311yescharacteristic
92#544—216no—
93#544—216no—
94#544—28no—
95#544—28no—
96#544—28no—
97#544—28no—
98#544—28no—
99#544—28no—
100#544—28no—
101#544—28no—
102#543—216no—
103#543—216no—
104#533—232no—
105#533—232no—
106#533—28no—
107#533—28no—
108#533—24no—
109#533—24no—
110#533—24no—
111#533—24no—
112#533D418yes—
113#533D418yes—
114C2×C4⋊Q8—22no—
115C2×C4⋊Q8—22no—
116C2×C4⋊Q8—22no—
117C2×C4⋊Q8—22no—
118C2×C4⋊Q8C2311yescharacteristic
119C2×C4⋊Q8C2312yes—
120C2×C4⋊Q8C2312yes—
121C2×C4⋊Q8C2311yescharacteristic
122C2×C4⋊Q8C2311yescharacteristic
123C2×C4⋊Q8C2311yescharacteristic
124C2×C4⋊Q8C2311yescharacteristic
125C2×C4⋊Q8C2311yescharacteristic
126C2×#125—22no—
127C2×#125—22no—
128C2×#125—22no—
129C2×#125—22no—
130C2×#125—21no—
131C2×#125C2312yes—
132C2×#125C2312yes—
133C2×#125C2312yes—
134C2×#125C2312yes—
135C2×C22⋊Q8—22no—
136C2×C22⋊Q8—22no—
137C2×C22⋊Q8—21no—
138C2×C22⋊Q8—21no—
139C2×C22⋊Q8—21no—
140C2×C22⋊Q8—21no—
141C2×C22⋊Q8—21no—
142C2×C22⋊Q8—22no—
143C2×C22⋊Q8—22no—
144C2×C22⋊Q8—22no—
145C2×C22⋊Q8—22no—
146C2×C22⋊Q8—22no—
147C2×C22⋊Q8—22no—
148C2×C22⋊Q8—22no—
149C2×C22⋊Q8—22no—
150C2×C22⋊Q8C2312yes—
151C2×C22⋊Q8C2314yes—
152C2×C4×Q8—22no—
153C2×C4×Q8—22no—
154C2×C4×Q8—21no—
155C2×C4×Q8—21no—
156C2×C4×Q8—21no—
157C2×C4×Q8—21no—
158C2×C4×Q8—21no—
159C2×C4×Q8C2312yes—
160C2×#117—22no—
161C2×#117—22no—
162C2×#117—28no—
163C2×#117—28no—
164C2×#117—21no—
165C2×#117—24no—
166C2×#117—24no—
167C2×#117D412yes—
168C2×#117D412yes—
169C2×#117C2312yes—
170C2×#117C2312yes—
171C2×#117C2312yes—
172C2×#117C2312yes—
173C2×#117C2312yes—
174C2×#117C2312yes—
175C22×C4⋊C4D411yescharacteristic
176C22×C4⋊C4D411yescharacteristic
177C22×C4⋊C4C2312yes—
178C22×C4⋊C4C2312yes—
179C22×C22⋊C4—22no—
180C22×C22⋊C4D411yescharacteristic
181C22×C22⋊C4D411yescharacteristic
182C22×C42—22no—
183C22×C42C2311yescharacteristic
184C22×C42C2311yescharacteristic
185#501—28no—
186#500—24no—
187#498—24no—
188#491—28no—
189#487—24no—
190#487—24no—
191#485—216no—
192#485—216no—
193#479—216no—
194#477—28no—
195#475—28no—
196#432—24no—
197#431—24no—
198C2×#107C2314yes—
199C2×#107C2314yes—
200C2×#106C2314yes—
201C2×#106C2314yes—
202C2×C4⋊C8C2312yes—
203C2×C4⋊C8C2312yes—
204C2×#103C2314yes—
205C2×#103C2314yes—
206C2×#103C2314yes—
207C2×#103C2314yes—
208C2×C22⋊C8C2314yes—
209C2×C22⋊C8C2314yes—
210C2×C22⋊C8C2312yes—
211C2×C22⋊C8C2312yes—
212C2×#97C2312yes—
213C2×C4×C8C2311yescharacteristic
214C2×C4×C8C2311yescharacteristic
215#400—22no—
216#399—22no—
217#398—22no—
218#398—22no—
219#397—22no—
220#397—22no—
221#395—22no—
222#393—22no—
223#391C2314yes—
224#389—24no—
225#389C2314yes—
226#388—24no—
227#388—24no—
228#387—24no—
229#387—24no—
230#385—22no—
231#385—22no—
232#385—24no—
233#383—21no—
234#382C2312yes—
235#381—22no—
236#381—21no—
237#380C2314yes—
238#380C2312yes—
239#379—22no—
240C4×C4⋊C4C2312yes—
241C4×C22⋊C4—21no—
242#376—21no—
243#340C2314yes—
244#340C2314yes—
245#340C2314yes—
246#340C2314yes—
247#340C2314yes—
248#340C2314yes—
249#340C2314yes—
250#340C2314yes—
251#336C2314yes—
252#336C2314yes—
253C25C2×D411yescharacteristic
254C22×Q8—22no—
255C22×Q8—22no—
256C22×Q8—21no—
257C22×Q8—21no—
258C22×Q8—21no—
259C22×Q8—21no—
260C22×Q8—21no—
261C22×Q8C2412yes—
262C23×C4—22no—
263C23×C4—22no—
264C23×C4—28no—
265C23×C4—22no—
266C23×C4—22no—
267C23×C4C2×D411yescharacteristic
268C23×C4C2×D412yes—
269C23×C4C2×D411yescharacteristic
270C23×C4C2×D411yescharacteristic
271C23×C4C2×D411yescharacteristic
272C23×C4C2×D411yescharacteristic
273C23×C4C2411yescharacteristic
274C23×C4C2412yes—
275C22×C8C2412yes—
276C22×C8C2412yes—
277C22×C8C2414yes—
278C4⋊Q8—216no—
279C4⋊Q8—216no—
280C4⋊Q8—216no—
281C4⋊Q8—216no—
282C4⋊Q8—24no—
283C4⋊Q8—24no—
284C4⋊Q8—24no—
285C4⋊Q8—24no—
286C4⋊Q8—22no—
287C4⋊Q8—22no—
288C4⋊Q8—22no—
289C4⋊Q8—22no—
290C4⋊Q8—22no—
291C4⋊Q8—22no—
292C4⋊Q8—22no—
293C4⋊Q8—22no—
294C4⋊Q8C2×D414yes—
295C4⋊Q8C2×D414yes—
296C4⋊Q8C2×D414yes—
297C4⋊Q8C2×D414yes—
298#125—216no—
299#125—216no—
300#125—216no—
301#125—216no—
302#125—24no—
303#125—24no—
304#125—24no—
305#125—24no—
306#125—24no—
307#125—24no—
308#125—44no—
309#125C2×D418yes—
310#125C2×D418yes—
311C22⋊Q8—216no—
312C22⋊Q8—216no—
313C22⋊Q8—216no—
314C22⋊Q8—216no—
315C22⋊Q8—216no—
316C22⋊Q8—216no—
317C22⋊Q8—216no—
318C22⋊Q8—216no—
319C22⋊Q8—216no—
320C22⋊Q8—216no—
321C22⋊Q8—28no—
322C22⋊Q8—28no—
323C22⋊Q8—28no—
324C22⋊Q8—28no—
325C22⋊Q8—24no—
326C22⋊Q8—24no—
327C22⋊Q8—216no—
328C22⋊Q8—44no—
329C4×Q8—216no—
330C4×Q8—216no—
331C4×Q8—28no—
332C4×Q8—28no—
333C4×Q8—28no—
334C4×Q8—28no—
335C4×Q8—24no—
336C4×Q8—24no—
337C4×Q8—44no—
338#117—216no—
339#117—216no—
340#117—24no—
341#117—24no—
342#117—24no—
343#117—24no—
344#117—24no—
345#117—24no—
346#117—28no—
347#117—28no—
348#117—24no—
349#117—24no—
350#117—24no—
351#117—24no—
352#117—44no—
353#117C2×D418yes—
354#117C2×D418yes—
355C2×C4⋊C4—24no—
356C2×C4⋊C4—24no—
357C2×C4⋊C4—22no—
358C2×C4⋊C4—22no—
359C2×C4⋊C4—22no—
360C2×C4⋊C4—22no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—22no—
366C2×C4⋊C4—22no—
367C2×C4⋊C4—22no—
368C2×C4⋊C4—22no—
369C2×C4⋊C4—22no—
370C2×C4⋊C4—22no—
371C2×C4⋊C4—22no—
372C2×C4⋊C4—22no—
373C2×C4⋊C4—24no—
374C2×C4⋊C4—24no—
375C2×C4⋊C4—24no—
376C2×C4⋊C4—24no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—22no—
381C2×C4⋊C4—22no—
382C2×C4⋊C4—24no—
383C2×C4⋊C4—24no—
384C2×C4⋊C4C2×D412yes—
385C2×C4⋊C4C2×D412yes—
386C2×C4⋊C4C2×D412yes—
387C2×C4⋊C4C2×D412yes—
388C2×C4⋊C4C2412yes—
389C2×C4⋊C4C2414yes—
390C2×C4⋊C4C2412yes—
391C2×C4⋊C4C2412yes—
392C2×C4⋊C4C2412yes—
393C2×C4⋊C4C2412yes—
394C2×C22⋊C4—216no—
395C2×C22⋊C4—22no—
396C2×C22⋊C4—22no—
397C2×C22⋊C4—22no—
398C2×C22⋊C4—22no—
399C2×C22⋊C4—24no—
400C2×C22⋊C4—24no—
401C2×C22⋊C4—22no—
402C2×C22⋊C4—24no—
403C2×C22⋊C4—22no—
404C2×C22⋊C4—24no—
405C2×C22⋊C4C2×D412yes—
406C2×C22⋊C4C2×D412yes—
407C2×C22⋊C4C2×D412yes—
408C2×C22⋊C4C2×D412yes—
409C2×C42—216no—
410C2×C42—22no—
411C2×C42—22no—
412C2×C42—22no—
413C2×C42—22no—
414C2×C42—22no—
415C2×C42—22no—
416C2×C42—22no—
417C2×C42—22no—
418C2×C42C2×D412yes—
419C2×C42C2×D411yescharacteristic
420C2×C42C2×D411yescharacteristic
421C2×C42C2×D414yes—
422C2×C42C2×D414yes—
423C2×C42C2412yes—
424C2×C42C2411yescharacteristic
425C2×C42C2411yescharacteristic
426#107—28no—
427#107—28no—
428#106—28no—
429#106—28no—
430C4⋊C8—24no—
431C4⋊C8—24no—
432#103—28no—
433#103—28no—
434#103—28no—
435#103—28no—
436C22⋊C8—24no—
437C22⋊C8—24no—
438C22⋊C8—28no—
439C22⋊C8—28no—
440#97—24no—
441C4×C8—22no—
442C4×C8—22no—
443#95—22no—
444#95—22no—
445#95—22no—
446#95—22no—
447#95—24no—
448#95—24no—
449#95—24no—
450#95—24no—
451C24—22no—
452C24—24no—
453C24—42no—
454C24—42no—
455C24C22×D411yescharacteristic
456C24C22×D412yes—
457C2×Q8—28no—
458C2×Q8—28no—
459C2×Q8—24no—
460C2×Q8—24no—
461C2×Q8—24no—
462C2×Q8—24no—
463C2×Q8—22no—
464C2×Q8—22no—
465C2×Q8—28no—
466C2×Q8—42no—
467C2×Q8—82no—
468C2×Q8—82no—
469C2×Q8—82no—
470C2×Q8—82no—
471C2×Q8—82no—
472C2×Q8—84no—
473C2×Q8—84no—
474C22×C4—216no—
475C22×C4—216no—
476C22×C4—22no—
477C22×C4—24no—
478C22×C4—24no—
479C22×C4—24no—
480C22×C4—24no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—24no—
484C22×C4—24no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—28no—
490C22×C4—28no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—24no—
494C22×C4—22no—
495C22×C4—22no—
496C22×C4—42no—
497C22×C4—42no—
498C22×C4—42no—
499C22×C4—42no—
500C22×C4—84no—
501C22×C4—84no—
502C22×C4#12014yes—
503C22×C4#12014yes—
504C22×C4C22×D412yes—
505C22×C4C22×D412yes—
506C22×C4C22×D411yescharacteristic
507C22×C4C22×D411yescharacteristic
508C22×C4C22×D411yescharacteristic
509C22×C4C22×D411yescharacteristic
510C22×C4C22×D411yescharacteristic
511C22×C4C22×D412yes—
512C22×C4C22×D412yes—
513C22×C4C22×D412yes—
514C22×C42−1+418yes—
515C22×C4C2511yescharacteristic
516C2×C8—22no—
517C2×C8—22no—
518C2×C8—24no—
519C2×C8—24no—
520C2×C8—24no—
521C2×C8—24no—
522C2×C8—24no—
523C4⋊C4—28no—
524C4⋊C4—28no—
525C4⋊C4—28no—
526C4⋊C4—28no—
527C4⋊C4—28no—
528C4⋊C4—28no—
529C4⋊C4—28no—
530C4⋊C4—28no—
531C4⋊C4—24no—
532C4⋊C4—24no—
533C4⋊C4—24no—
534C4⋊C4—24no—
535C4⋊C4—24no—
536C4⋊C4—24no—
537C4⋊C4—24no—
538C4⋊C4—24no—
539C4⋊C4—28no—
540C4⋊C4—28no—
541C4⋊C4—28no—
542C4⋊C4—28no—
543C4⋊C4—22no—
544C4⋊C4—22no—
545C4⋊C4—22no—
546C4⋊C4—22no—
547C4⋊C4—22no—
548C4⋊C4—22no—
549C4⋊C4—28no—
550C4⋊C4—42no—
551C4⋊C4—42no—
552C4⋊C4—44no—
553C4⋊C4—44no—
554C4⋊C4—42no—
555C4⋊C4—48no—
556C4⋊C4—84no—
557C4⋊C4C22×D414yes—
558C4⋊C4C22×D414yes—
559C4⋊C4C22×D414yes—
560C4⋊C4C22×D414yes—
561C22⋊C4—28no—
562C22⋊C4—28no—
563C22⋊C4—28no—
564C22⋊C4—28no—
565C22⋊C4—24no—
566C22⋊C4—24no—
567C22⋊C4—24no—
568C22⋊C4—24no—
569C22⋊C4—44no—
570C22⋊C4—44no—
571C22⋊C4—48no—
572C22⋊C4—84no—
573C42—28no—
574C42—28no—
575C42—28no—
576C42—28no—
577C42—24no—
578C42—22no—
579C42—22no—
580C42—44no—
581C42—82no—
582C42#12014yes—
583C42#12014yes—
584C42C22×D412yes—
585C42C22×D412yes—
586C42C22×D412yes—
587C42C22×D412yes—
588C23—28no—
589C23—24no—
590C23—22no—
591C23—22no—
592C23—24no—
593C23—22no—
594C23—22no—
595C23—22no—
596C23—88no—
597C23—82no—
598C23—82no—
599C23C2×#12012yes—
600C23#53614yes—
601C23C23×D411yescharacteristic
602Q8—44no—
603Q8—44no—
604Q8—84no—
605Q8—84no—
606Q8—84no—
607Q8—84no—
608Q8—84no—
609Q8—88no—
610Q8—88no—
611C2×C4—216no—
612C2×C4—28no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—24no—
621C2×C4—24no—
622C2×C4—28no—
623C2×C4—28no—
624C2×C4—24no—
625C2×C4—24no—
626C2×C4—22no—
627C2×C4—22no—
628C2×C4—24no—
629C2×C4—24no—
630C2×C4—24no—
631C2×C4—24no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—82no—
646C2×C4—82no—
647C2×C4—88no—
648C2×C4—88no—
649C2×C4C2×#12012yes—
650C2×C4C2×#12012yes—
651C2×C4C2×#12012yes—
652C2×C4C2×#12012yes—
653C2×C4C2×#12012yes—
654C2×C4#53614yes—
655C2×C4C23×D411yescharacteristic
656C2×C4C23×D411yescharacteristic
657C2×C4C2×2−1+414yes—
658C8—42no—
659C8—42no—
660C8—44no—
661C22—22no—
662C22—24no—
663C22—28no—
664C22—22no—
665C22—22no—
666C22—44no—
667C22—44no—
668C22—42no—
669C22—42no—
670C22—88no—
671C22C22×#12011yescharacteristic
672C22C2×#53612yes—
673C22#358412yes—
674C22#358612yes—
675C4—22no—
676C4—24no—
677C4—22no—
678C4—22no—
679C4—22no—
680C4—22no—
681C4—42no—
682C4—44no—
683C4—44no—
684C4—42no—
685C4—44no—
686C4—44no—
687C4—82no—
688C4—84no—
689C4—82no—
690C4#348612yes—
691C4#348612yes—
692C2—42no—
693C2—44no—
694C2#3524211yescharacteristic
695C2#6036611yescharacteristic
696C2#6037011yescharacteristic
697C2#6087614yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
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×#3237C2semidirect
3. C2×#3231C2semidirect
6. #29645C2semidirect
7. #25914C2semidirect
8. #25914C2semidirect
9. #25821C2semidirect
10. #25820C2semidirect
11. #24311C2semidirect
12. #24310C2semidirect
13. #21842C2semidirect
14. #21837C2semidirect
36. C2×#501C22semidirect
37. C2×#500C22semidirect
38. C2×#498C22semidirect
39. C2×#491C22semidirect
40. C2×#487C22semidirect
41. C2×#487C22semidirect
44. C2×#479C22semidirect
45. C2×#477C22semidirect
46. C2×#475C22semidirect
61. #2059C22semidirect
62. #2059C22semidirect
63. #2058C22semidirect
64. #2057C22semidirect
65. #2026C22semidirect
66. #2026C22semidirect
67. #2024C22semidirect
68. #2024C22semidirect
69. #1993C22semidirect
70. #1993C22semidirect
71. #1991C22semidirect
72. #1991C22semidirect
73. #1989C22semidirect
74. #1988C22semidirect
75. #1987C22semidirect

Maximal subgroups

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

subgroupindex
C2×#34582
C2×#32372
C2×#32312
#353252
#329852
#296452
#259142
#258212
#258202
#243112
#243102
#218422
#218372
#207172