Finite Group Databasebeta

← #6439119

Subgroups — #6439119

Order 512.

2174 conjugacy classes of subgroups.

The subgroup lattice has 756 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3084C211yesmaximal, characteristic
2C2×#2947C211yesmaximal, characteristic
3#35668C211yesmaximal, characteristic, complemented
4#34183C211yesmaximal, characteristic
5#26398C214yesmaximal, complemented
6#26389C214yesmaximal, complemented
7#26376C214yesmaximal, complemented
8#26371C214yesmaximal, complemented
9C2×#2096C211yesmaximal, characteristic, complemented
10#23658C211yesmaximal, characteristic, complemented
11#23613C211yesmaximal, characteristic, complemented
12#23556C211yesmaximal, characteristic, complemented
13#21155C211yesmaximal, characteristic, complemented
14#21025C211yesmaximal, characteristic, complemented
15#20931C211yesmaximal, characteristic
16#20927C211yesmaximal, characteristic
17#20882C211yesmaximal, characteristic
18#20877C211yesmaximal, characteristic, complemented
19#20872C211yesmaximal, characteristic
20C22×#137C2211yescharacteristic
21C2×#544C2211yescharacteristic
22C2×#520C2211yescharacteristic
23C2×#520C2211yescharacteristic
24C2×#519C2211yescharacteristic
25C2×C4×#41C2211yescharacteristic
26#3084—28no—
27C2×#479C2212yes—
28C2×#475C2212yes—
29C2×#478C2212yes—
30C2×#476C2212yes—
31#2947—28no—
32C2×#441C2212yes—
33C2×#440C2212yes—
34C2×C4×Q16C2212yes—
35C2×C4×SD16C2212yes—
36C2×#428C2211yescharacteristic
37C2×#428C2211yescharacteristic
38C2×#427C2211yescharacteristic
39C22×#106C2211yescharacteristic
40C22×#106C2211yescharacteristic
41C2×#424C2211yescharacteristic
42C2×#417C2211yescharacteristic
43C2×#419C2211yescharacteristic
44C2×#416C2211yescharacteristic
45C2×#416C2211yescharacteristic
46C22×#103C2211yescharacteristic
47C22×#103C2211yescharacteristic
48C2×#405C2211yescharacteristic
49#2568—21no—
50#2567—21no—
51#2565—21no—
52#2565—21no—
53#2453C2214yes—
54#2451C2214yes—
55#2438—24no—
56Q8×C4⋊C4C2214yes—
57D4×C4⋊C4—24no—
58D4×C4⋊C4C2214yes—
59#2317—21no—
60#2317—21no—
61#2317C2211yescharacteristic
62#2317C2211yescharacteristic
63#2317C2211yescharacteristic
64#2310C2211yescharacteristic
65#2296C2211yescharacteristic
66C2×#391C2211yescharacteristic
67#2285—21no—
68#2285C2211yescharacteristic
69C2×#384—21no—
70C2×#384C2211yescharacteristic
71#2096C2214yes—
72#2096C2214yes—
73#2084C2214yes—
74#2083C2214yes—
75#2068C2214yes—
76#2066C2214yes—
77#2059C2214yes—
78#2058C2214yes—
79#1943C2214yes—
80#1941C2214yes—
81#1917C2214yes—
82#1916C2214yes—
83#1878C2214yes—
84#1878C2214yes—
85#1877C2214yes—
86#1877C2214yes—
87#1865C2212yes—
88#1865C2214yes—
89#1865C2214yes—
90#1864C2212yes—
91#1864C2214yes—
92#1863C2214yes—
93#1863C2212yes—
94#1862C2212yes—
95#1736C2211yescharacteristic
96#1736C2211yescharacteristic
97#1736C2211yescharacteristic
98#1735C2211yescharacteristic
99#1729C2211yescharacteristic
100#1729C2211yescharacteristic
101#1728C2211yescharacteristic
102C2×#336C2211yescharacteristic
103C22×#41C2311yescharacteristic
104C23×Q8C2311yescharacteristic
105C2×#137—24no—
106C2×#137—24no—
107C2×#137Q818yescomplemented
108C22×Q16C2312yes—
109C22×SD16C2312yes—
110C22×M16C2311yescharacteristic
111C22×M16C2311yescharacteristic
112C23×C8C2311yescharacteristic
113C23×C8C2311yescharacteristic
114#544—24no—
115#544—42no—
116C2×C4⋊Q8—21no—
117C2×C4⋊Q8C2312yes—
118C2×#125—21no—
119C2×#125C2312yes—
120C2×C22⋊Q8—22no—
121#520—24no—
122#520—24no—
123#520—42no—
124#520—42no—
125#519—24no—
126#519—42no—
127C4×#41—24no—
128C4×#41—24no—
129C2×C4×Q8—21no—
130C2×C4×Q8—21no—
131C2×C4×Q8C2312yes—
132C2×C4×Q8C2312yes—
133C2×C4×Q8C2312yes—
134C2×C4×Q8C2312yes—
135C2×C4×Q8C2312yes—
136C2×C4×D4—22no—
137C2×C4×D4—22no—
138C2×C4×D4—22no—
139C2×C4×D4C2312yes—
140C2×C4×D4C2312yes—
141C2×C4×D4C2312yes—
142C2×#117—21no—
143C2×#117—21no—
144C2×#117—22no—
145C2×#117—21no—
146C2×#117C2311yescharacteristic
147C2×#117C2311yescharacteristic
148C2×#117C2311yescharacteristic
149C2×#117C2311yescharacteristic
150C2×#117C2311yescharacteristic
151C22×C4⋊C4—22no—
152C22×C4⋊C4—21no—
153C22×C4⋊C4—21no—
154C22×C4⋊C4—24no—
155C22×C4⋊C4C2311yescharacteristic
156C22×C4⋊C4C2311yescharacteristic
157C22×C4⋊C4C2311yescharacteristic
158C22×C42—21no—
159C22×C42C2311yescharacteristic
160#479—28no—
161#478—28no—
162#476—28no—
163#475—28no—
164#441—28no—
165#440—28no—
166C4×Q16—28no—
167C4×SD16—28no—
168#428—24no—
169#428—24no—
170#427—24no—
171C2×#107C2312yes—
172C2×#107C2312yes—
173C2×#107C2312yes—
174C2×#106—24no—
175C2×#106—42no—
176C2×#106C2312yes—
177C2×#106C2312yes—
178C2×#106C2312yes—
179C2×#106C2312yes—
180C2×#106C2312yes—
181C2×#106C2314yes—
182#424—24no—
183C2×C4⋊C8C2312yes—
184C2×C4⋊C8C2312yes—
185#419—24no—
186#417—24no—
187#416—24no—
188#416D418yes—
189C2×#103—24no—
190C2×#103—42no—
191C2×#103C2312yes—
192C2×#103C2312yes—
193C2×#103C2314yes—
194C2×#103C2312yes—
195C2×#103C2312yes—
196C2×#103C2314yes—
197C2×#103C2311yescharacteristic
198C2×#103C2311yescharacteristic
199C2×#102C2312yes—
200C2×#102C2312yes—
201C2×#102C2311yescharacteristic
202C2×#102C2311yescharacteristic
203#405—24no—
204C2×#97C2312yes—
205C2×C4×C8C2312yes—
206#400—24no—
207#398—24no—
208#398—24no—
209#395—24no—
210#391—21no—
211#391—21no—
212#391—21no—
213#391—21no—
214#391—24no—
215#391C2314yes—
216#391C2314yes—
217#389—24no—
218#389—24no—
219#389—24no—
220#389C2314yes—
221#389C2314yes—
222#389C2314yes—
223#389C2312yes—
224#389C2312yes—
225#387—24no—
226#387—24no—
227#385—22no—
228#385—22no—
229#385—22no—
230#385—22no—
231#385—24no—
232#385—24no—
233#384—21no—
234#384—21no—
235#384—21no—
236#384—24no—
237#384—21no—
238#384—21no—
239#384D411yescharacteristic
240#384D411yescharacteristic
241#384C2314yes—
242#382—21no—
243#382—21no—
244#382—21no—
245#382D411yescharacteristic
246#382D411yescharacteristic
247#381—22no—
248#380—24no—
249#380D414yes—
250C4×C4⋊C4—24no—
251C4×C4⋊C4—21no—
252C4×C4⋊C4—21no—
253C4×C4⋊C4C2314yes—
254C2×#95—22no—
255#340C2312yes—
256#340C2314yes—
257#340C2312yes—
258#340C2314yes—
259#340C2314yes—
260#336C2314yes—
261#336C2314yes—
262#336C2312yes—
263#336C2312yes—
264#336C2314yes—
265C2×#41—22no—
266C2×#41—22no—
267C2×#41—22no—
268C2×#41C2×D414yes—
269C2×#41C2×Q814yes—
270C2×#41#4114yes—
271C22×Q8—21no—
272C22×Q8—22no—
273C22×Q8—22no—
274C22×Q8—22no—
275C22×Q8—22no—
276C22×Q8—22no—
277C22×Q8—22no—
278C22×Q8C2×Q814yes—
279C22×Q8C2412yes—
280C22×Q8C2411yescharacteristic
281C22×D4—21no—
282C22×D4C2411yescharacteristic
283C23×C4—21no—
284C23×C4—21no—
285C23×C4—22no—
286C23×C4—21no—
287C23×C4—21no—
288C23×C4—21no—
289C23×C4—22no—
290C23×C4—22no—
291C23×C4C2×D412yes—
292C23×C4C2411yescharacteristic
293#137—232no—
294C2×Q16—24no—
295C2×Q16—24no—
296C2×Q16C2×Q818yes—
297C2×SD16—24no—
298C2×SD16—24no—
299C2×SD16C2×Q818yes—
300C2×M16—22no—
301C2×M16—22no—
302C2×M16—22no—
303C2×M16—22no—
304C2×M16—22no—
305C2×M16C2×Q814yes—
306C22×C8—22no—
307C22×C8—22no—
308C22×C8—22no—
309C22×C8—22no—
310C22×C8—22no—
311C22×C8C2×D414yes—
312C22×C8C2412yes—
313C22×C8C2412yes—
314C22×C8C2411yescharacteristic
315C22×C8C2411yescharacteristic
316C22×C8C2412yes—
317C4⋊Q8—24no—
318C4⋊Q8—42no—
319C4⋊Q8—44no—
320#125—24no—
321#125—42no—
322#125—44no—
323C22⋊Q8—48no—
324C4×Q8—24no—
325C4×Q8—24no—
326C4×Q8—24no—
327C4×Q8—24no—
328C4×Q8—24no—
329C4×Q8—24no—
330C4×Q8—42no—
331C4×Q8—42no—
332C4×Q8—42no—
333C4×Q8—42no—
334C4×Q8—44no—
335C4×Q8—44no—
336C4×D4—216no—
337C4×D4—24no—
338C4×D4—24no—
339C4×D4—24no—
340C4×D4—24no—
341C4×D4—42no—
342C4×D4—42no—
343C4×D4—48no—
344C4×D4—48no—
345#117—28no—
346#117—22no—
347#117—22no—
348#117—22no—
349#117—22no—
350#117—22no—
351#117—22no—
352#117—22no—
353#117—22no—
354#117—48no—
355#117—44no—
356#117—44no—
357#117C2×D414yes—
358#117#4114yes—
359C2×C4⋊C4—22no—
360C2×C4⋊C4—22no—
361C2×C4⋊C4—22no—
362C2×C4⋊C4—22no—
363C2×C4⋊C4—22no—
364C2×C4⋊C4—22no—
365C2×C4⋊C4—22no—
366C2×C4⋊C4—24no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—22no—
370C2×C4⋊C4—22no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—22no—
373C2×C4⋊C4—22no—
374C2×C4⋊C4—22no—
375C2×C4⋊C4—22no—
376C2×C4⋊C4—22no—
377C2×C4⋊C4—22no—
378C2×C4⋊C4—24no—
379C2×C4⋊C4—24no—
380C2×C4⋊C4—24no—
381C2×C4⋊C4—22no—
382C2×C4⋊C4—24no—
383C2×C4⋊C4—44no—
384C2×C4⋊C4—44no—
385C2×C4⋊C4—88no—
386C2×C4⋊C4—84no—
387C2×C4⋊C4C2×D411yescharacteristic
388C2×C4⋊C4C2×D411yescharacteristic
389C2×C4⋊C4C2×D414yes—
390C2×C4⋊C4C2×D411yescharacteristic
391C2×C4⋊C4C2×D411yescharacteristic
392C2×C4⋊C4C2×D414yes—
393C2×C4⋊C4C2412yes—
394C2×C4⋊C4C2412yes—
395C2×C4⋊C4C2412yes—
396C2×C4⋊C4C2412yes—
397C2×C4⋊C4C2411yescharacteristic
398C2×C4⋊C4C2411yescharacteristic
399C2×C4⋊C4C2412yes—
400C2×C22⋊C4—21no—
401C2×C22⋊C4—21no—
402C2×C22⋊C4—22no—
403C2×C22⋊C4—21no—
404C2×C22⋊C4—22no—
405C2×C22⋊C4—22no—
406C2×C22⋊C4C2×D412yes—
407C2×C22⋊C4C2×D412yes—
408C2×C42—28no—
409C2×C42—21no—
410C2×C42—21no—
411C2×C42—22no—
412C2×C42—22no—
413C2×C42—22no—
414C2×C42—22no—
415C2×C42—22no—
416C2×C42—22no—
417C2×C42—22no—
418C2×C42—21no—
419C2×C42C2×D411yescharacteristic
420C2×C42C2×D411yescharacteristic
421C2×C42C2412yes—
422C2×C42C2412yes—
423C2×C42C2412yes—
424#107—24no—
425#107—24no—
426#107—24no—
427#106—24no—
428#106—24no—
429#106—24no—
430#106—24no—
431#106—24no—
432#106—44no—
433C4⋊C8—24no—
434C4⋊C8—24no—
435#103—24no—
436#103—24no—
437#103—24no—
438#103—24no—
439#103—28no—
440#103—44no—
441#103C2×D414yes—
442#103C2×D414yes—
443#102—24no—
444#102—24no—
445#102C2×D414yes—
446#102C2×D414yes—
447#97—24no—
448C4×C8—24no—
449#95—24no—
450#95—24no—
451#95—24no—
452#95—24no—
453#95—22no—
454#95—22no—
455#95—24no—
456#95—24no—
457#95—24no—
458#95C2×D414yes—
459#95C2×D414yes—
460C24—21no—
461C24C22×D411yescharacteristic
462#41—28no—
463#41—28no—
464#41—28no—
465#41C22⋊Q8116yes—
466C2×Q8—24no—
467C2×Q8—21no—
468C2×Q8—21no—
469C2×Q8—21no—
470C2×Q8—22no—
471C2×Q8—28no—
472C2×Q8—28no—
473C2×Q8—22no—
474C2×Q8—22no—
475C2×Q8—22no—
476C2×Q8—22no—
477C2×Q8—44no—
478C2×Q8—42no—
479C2×Q8—44no—
480C2×Q8—42no—
481C2×Q8—44no—
482C2×Q8—84no—
483C2×Q8C22×D412yes—
484C2×Q8C22×Q814yes—
485C2×Q8C22×Q812yes—
486C2×Q8C2×#4112yes—
487C2×D4—24no—
488C2×D4—24no—
489C2×D4—24no—
490C2×D4—21no—
491C2×D4—21no—
492C2×D4—21no—
493C2×D4C22×D412yes—
494C2×D4C22×Q812yes—
495C2×D4C2×#4112yes—
496C22×C4—21no—
497C22×C4—24no—
498C22×C4—24no—
499C22×C4—24no—
500C22×C4—28no—
501C22×C4—24no—
502C22×C4—24no—
503C22×C4—22no—
504C22×C4—22no—
505C22×C4—21no—
506C22×C4—21no—
507C22×C4—24no—
508C22×C4—22no—
509C22×C4—22no—
510C22×C4—22no—
511C22×C4—22no—
512C22×C4—42no—
513C22×C4—44no—
514C22×C4—44no—
515C22×C4—44no—
516C22×C4—42no—
517C22×C4—42no—
518C22×C4—44no—
519C22×C4—42no—
520C22×C4—42no—
521C22×C4—42no—
522C22×C4—84no—
523C22×C4—84no—
524C22×C4—82no—
525C22×C4—82no—
526C22×C4—82no—
527C22×C4C22×D412yes—
528C22×C4C22×D412yes—
529C22×C4C22×D412yes—
530C22×C4C22×D412yes—
531C22×C4C22×D412yes—
532C22×C4C22×D412yes—
533C22×C4C22×D412yes—
534C22×C4C22×D411yescharacteristic
535C22×C4C22×Q812yes—
536C22×C4C2×#4112yes—
537C22×C4C2×#4112yes—
538C22×C4H(5,𝔽2)12yes—
539C22×C42−1+412yes—
540C22×C4C2511yescharacteristic
541Q16—216no—
542SD16—216no—
543M16—28no—
544M16—44no—
545C2×C8—28no—
546C2×C8—22no—
547C2×C8—21no—
548C2×C8—21no—
549C2×C8—22no—
550C2×C8—21no—
551C2×C8—21no—
552C2×C8—22no—
553C2×C8—22no—
554C2×C8—22no—
555C2×C8—22no—
556C2×C8—22no—
557C2×C8—22no—
558C2×C8—44no—
559C2×C8C22×D412yes—
560C2×C8C22×D412yes—
561C2×C8C22×Q814yes—
562C4⋊C4—28no—
563C4⋊C4—22no—
564C4⋊C4—22no—
565C4⋊C4—22no—
566C4⋊C4—22no—
567C4⋊C4—28no—
568C4⋊C4—22no—
569C4⋊C4—21no—
570C4⋊C4—21no—
571C4⋊C4—22no—
572C4⋊C4—22no—
573C4⋊C4—22no—
574C4⋊C4—22no—
575C4⋊C4—22no—
576C4⋊C4—22no—
577C4⋊C4—22no—
578C4⋊C4—22no—
579C4⋊C4—44no—
580C4⋊C4—44no—
581C4⋊C4—44no—
582C4⋊C4—44no—
583C4⋊C4—44no—
584C4⋊C4—44no—
585C4⋊C4—44no—
586C4⋊C4—44no—
587C4⋊C4—42no—
588C4⋊C4—42no—
589C4⋊C4—42no—
590C4⋊C4—44no—
591C4⋊C4—48no—
592C4⋊C4—84no—
593C4⋊C4—82no—
594C4⋊C4—84no—
595C4⋊C4—84no—
596C4⋊C4—84no—
597C4⋊C4C22×D412yes—
598C4⋊C4C22×D412yes—
599C4⋊C4C2×#4114yes—
600C22⋊C4—28no—
601C22⋊C4—24no—
602C22⋊C4—24no—
603C22⋊C4—24no—
604C22⋊C4—24no—
605C22⋊C4—44no—
606C22⋊C4—44no—
607C22⋊C4—42no—
608C42—28no—
609C42—22no—
610C42—24no—
611C42—22no—
612C42—24no—
613C42—22no—
614C42—22no—
615C42—22no—
616C42—22no—
617C42—22no—
618C42—22no—
619C42—44no—
620C42—44no—
621C42—44no—
622C42—44no—
623C42—42no—
624C23—22no—
625C23—22no—
626C23—22no—
627C23—22no—
628C23—22no—
629C23—42no—
630C23—42no—
631C23#52512yes—
632C23D4212yes—
633C23D4×Q812yes—
634C23C23×D411yescharacteristic
635Q8—22no—
636Q8—22no—
637Q8—22no—
638Q8—24no—
639Q8—24no—
640Q8—42no—
641Q8—42no—
642Q8—48no—
643Q8—164no—
644Q8C2×C22⋊Q814yes—
645D4—216no—
646D4—22no—
647D4—22no—
648D4—22no—
649D4C2×C22⋊Q814yes—
650C2×C4—22no—
651C2×C4—22no—
652C2×C4—22no—
653C2×C4—22no—
654C2×C4—22no—
655C2×C4—216no—
656C2×C4—24no—
657C2×C4—24no—
658C2×C4—22no—
659C2×C4—22no—
660C2×C4—24no—
661C2×C4—22no—
662C2×C4—22no—
663C2×C4—22no—
664C2×C4—22no—
665C2×C4—22no—
666C2×C4—22no—
667C2×C4—24no—
668C2×C4—24no—
669C2×C4—24no—
670C2×C4—22no—
671C2×C4—48no—
672C2×C4—416no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—42no—
676C2×C4—42no—
677C2×C4—48no—
678C2×C4—42no—
679C2×C4—42no—
680C2×C4—42no—
681C2×C4—44no—
682C2×C4—42no—
683C2×C4—42no—
684C2×C4—42no—
685C2×C4—42no—
686C2×C4—44no—
687C2×C4—44no—
688C2×C4—42no—
689C2×C4—164no—
690C2×C4—164no—
691C2×C4—162no—
692C2×C4—162no—
693C2×C4—162no—
694C2×C4C2×C22⋊Q814yes—
695C2×C4#52512yes—
696C2×C4#53914yes—
697C2×C4D4212yes—
698C2×C4#54714yes—
699C2×C4D4×Q812yes—
700C2×C4#55414yes—
701C2×C4#57714yes—
702C2×C4C23×D411yescharacteristic
703C2×C4C23×Q811yescharacteristic
704C2×C4C22×#4111yescharacteristic
705C2×C4C22×#4111yescharacteristic
706C2×C4C2×H(5,𝔽2)11yescharacteristic
707C2×C4C2×2−1+411yescharacteristic
708C8—24no—
709C8—22no—
710C8—22no—
711C8—42no—
712C8—42no—
713C22—22no—
714C22—22no—
715C22—24no—
716C22—24no—
717C22—24no—
718C22—42no—
719C22—42no—
720C22—44no—
721C22—44no—
722C22#244914yes—
723C22C2×#52511yescharacteristic
724C22C2×D4211yescharacteristic
725C22C2×D4×Q811yescharacteristic
726C22C2×#57712yes—
727C22#358612yes—
728C4—24no—
729C4—24no—
730C4—44no—
731C4—42no—
732C4—42no—
733C4—42no—
734C4—42no—
735C4—42no—
736C4—42no—
737C4—42no—
738C4—44no—
739C4—82no—
740C4—82no—
741C4—82no—
742C4—82no—
743C4#244914yes—
744C4C22×C22⋊Q811yescharacteristic
745C4C2×#53911yescharacteristic
746C4C2×#54711yescharacteristic
747C4C2×#55411yescharacteristic
748C2—24no—
749C2—44no—
750C2C2×#244911yescharacteristic
751C2#6058512yes—
752C2#6108311yescharacteristic
753C2#6108711yescharacteristic
754C2#6138312yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2285 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #35668C2semidirect
5. #26398C2semidirect
6. #26389C2semidirect
7. #26376C2semidirect
8. #26371C2semidirect
9. C2×#2096C2semidirect
10. #23658C2semidirect
11. #23613C2semidirect
12. #23556C2semidirect
13. #21155C2semidirect
14. #21025C2semidirect
18. #20877C2semidirect
107. C2×#137Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#30842
C2×#29472
#356682
#341832
#263982
#263892
#263762
#263712
C2×#20962
#236582
#236132
#235562
#211552
#210252
#209312
#209272
#208822
#208772
#208722