Finite Group Databasebeta

← #6683880

Subgroups — #6683880

Order 512.

1828 conjugacy classes of subgroups.

The subgroup lattice has 689 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3385C211yesmaximal, characteristic, complemented
2C2×#3361C211yesmaximal, characteristic
3#39168C211yesmaximal, characteristic, complemented
4#36180C211yesmaximal, characteristic, complemented
5#33030C211yesmaximal, characteristic
6#30967C214yesmaximal, complemented
7#30818C214yesmaximal, complemented
8#30276C211yesmaximal, characteristic, complemented
9#29869C211yesmaximal, characteristic, complemented
10#28676C211yesmaximal, characteristic, complemented
11#28645C211yesmaximal, characteristic, complemented
12#26368C214yesmaximal, complemented
13#26188C214yesmaximal, complemented
14#25930C211yesmaximal, characteristic
15#25841C211yesmaximal, characteristic, complemented
16#25839C211yesmaximal, characteristic, complemented
17#25648C211yesmaximal, characteristic, complemented
18#22240C211yesmaximal, characteristic, complemented
19#21114C211yesmaximal, characteristic, complemented
20C2×#559C2211yescharacteristic
21C2×Q82C2211yescharacteristic
22C2×#554C2211yescharacteristic
23C2×#554C2211yescharacteristic
24#3385—28no—
25#3361—28no—
26C2×#501C2212yes—
27C2×#500C2211yescharacteristic
28C2×#500C2211yescharacteristic
29C2×#488C2212yes—
30C2×#486C2211yescharacteristic
31C2×#486C2211yescharacteristic
32C2×#479C2212yes—
33C2×#477C2211yescharacteristic
34C2×#477C2211yescharacteristic
35C2×#478C2212yes—
36C2×#476C2211yescharacteristic
37C2×#474C2212yes—
38C2×#463C2212yes—
39C2×#459C2211yescharacteristic
40C2×#446C2211yescharacteristic
41C2×C8×Q8C2211yescharacteristic
42C2×#440C2212yes—
43C2×C4×SD16C2211yescharacteristic
44C2×C4×D8C2212yes—
45#2868C2211yescharacteristic
46#2858C2211yescharacteristic
47#2857—21no—
48#2674—24no—
49#2640—21no—
50#2567C2211yescharacteristic
51#2567C2211yescharacteristic
52#2567C2211yescharacteristic
53#2552C2214yes—
54#2548C2211yescharacteristic
55#2545—21no—
56#2542—21no—
57#2502—21no—
58#2502—21no—
59#2452C2214yes—
60#2448C2214yes—
61#2440—24no—
62D4×C4⋊C4C2214yes—
63#2326—21no—
64C4×C4⋊Q8C2211yescharacteristic
65#2295C2211yescharacteristic
66C2×#384—21no—
67C42×D4C2211yescharacteristic
68#2093C2214yes—
69#2093C2214yes—
70#2087C2214yes—
71#2083C2214yes—
72#2072C2214yes—
73#2067C2214yes—
74#2059C2214yes—
75#2057C2214yes—
76#2042C2214yes—
77#2025C2214yes—
78#1987C2211yescharacteristic
79#1987C2211yescharacteristic
80#1987C2211yescharacteristic
81#1984C2211yescharacteristic
82#1984C2211yescharacteristic
83#1982C2211yescharacteristic
84#1942C2214yes—
85#1936C2214yes—
86#1928C2212yes—
87#1926C2214yes—
88#1919C2212yes—
89#1917C2214yes—
90#1916C2214yes—
91#1879C2214yes—
92#1879C2214yes—
93#1877C2212yes—
94#1863C2214yes—
95#1862C2212yes—
96#1862C2214yes—
97#1862C2214yes—
98#1831C2211yescharacteristic
99C4×#107C2211yescharacteristic
100C4×#103C2211yescharacteristic
101C4×#102C2211yescharacteristic
102#1744C2211yescharacteristic
103C22×SD16C2311yescharacteristic
104C22×D8C2312yes—
105#559—24no—
106#559—42no—
107Q82—24no—
108Q82—42no—
109#554—24no—
110#554—24no—
111#554—42no—
112#554—42no—
113C2×C4⋊Q8—21no—
114C2×C4⋊Q8—22no—
115C2×C4⋊Q8—22no—
116C2×C4⋊Q8D411yescharacteristic
117C2×C4⋊Q8C2311yescharacteristic
118C2×C4⋊Q8C2311yescharacteristic
119C2×C4⋊Q8C2311yescharacteristic
120C2×C4⋊Q8C2312yes—
121C2×#127C2311yescharacteristic
122C2×#125C2312yes—
123C2×#125C2312yes—
124C2×#124—21no—
125C2×#124—21no—
126C2×#124—22no—
127C2×#124C2312yes—
128C2×#123—22no—
129C2×C22⋊Q8—21no—
130C2×C22⋊Q8—21no—
131C2×C22⋊Q8—22no—
132C2×C22⋊Q8—22no—
133C2×C22⋊Q8—24no—
134C2×C4×Q8—21no—
135C2×C4×Q8C2312yes—
136C2×C4×Q8C2311yescharacteristic
137C2×C4×Q8C2311yescharacteristic
138C2×C4×Q8C2311yescharacteristic
139C2×C4×D4—21no—
140C2×C4×D4C2312yes—
141C2×C4×D4C2312yes—
142C2×C4×D4C2312yes—
143C2×C4×D4C2312yes—
144C2×C4×D4C2311yescharacteristic
145C22×C4⋊C4—21no—
146C22×C4⋊C4—21no—
147C22×C4⋊C4—24no—
148C22×C42—21no—
149#501—28no—
150#500—24no—
151#500—42no—
152#488—28no—
153#486—24no—
154#486—42no—
155#479—28no—
156#478—28no—
157#477—24no—
158#477—24no—
159#476—24no—
160#474—28no—
161#463—28no—
162#459—24no—
163#446—24no—
164C8×Q8—24no—
165#440—28no—
166C4×SD16D418yes—
167C4×D8—28no—
168C2×#107C2312yes—
169C2×#107C2312yes—
170C2×#107C2312yes—
171C2×#107C2312yes—
172C2×#107C2312yes—
173C2×#107C2314yes—
174C2×#106C2312yes—
175C2×#106C2311yescharacteristic
176C2×C4⋊C8C2312yes—
177C2×C4⋊C8C2311yescharacteristic
178C2×C4⋊C8C2311yescharacteristic
179C2×C4⋊C8C2312yes—
180C2×#103C2312yes—
181C2×#103C2312yes—
182C2×#103C2312yes—
183C2×#103C2311yescharacteristic
184C2×#102C2312yes—
185C2×#102C2312yes—
186C2×#102C2312yes—
187C2×#102C2314yes—
188C2×#102C2312yes—
189C2×#102C2312yes—
190C2×#102C2314yes—
191C2×#102C2311yescharacteristic
192C2×#97C2312yes—
193C2×C4×C8C2311yescharacteristic
194C2×C4×C8C2312yes—
195C2×C4×C8C2311yescharacteristic
196#399—24no—
197#398—24no—
198#398—24no—
199#397—24no—
200#396—24no—
201#395—22no—
202#395—22no—
203#395—22no—
204#394—22no—
205#393—24no—
206#391—21no—
207#391—21no—
208#391—21no—
209#391D411yescharacteristic
210#391D411yescharacteristic
211#391C2312yes—
212#390—21no—
213#390—21no—
214#390—21no—
215#390C2314yes—
216#390C2314yes—
217#389—24no—
218#389D414yes—
219#389C2314yes—
220#389C2314yes—
221#389C2314yes—
222#389C2312yes—
223#388—24no—
224#388—24no—
225#387—22no—
226#385—24no—
227#385—24no—
228#385—24no—
229#384—21no—
230#384—21no—
231#384—21no—
232#384—24no—
233#384—21no—
234#384—21no—
235#384D411yescharacteristic
236#384C2314yes—
237#380—24no—
238C4×C4⋊C4—21no—
239C4×C4⋊C4C2311yescharacteristic
240C4×C4⋊C4C2311yescharacteristic
241C4×C4⋊C4C2312yes—
242C4×C4⋊C4C2314yes—
243C4×C22⋊C4—22no—
244C4×C22⋊C4—21no—
245#376—21no—
246C43C2311yescharacteristic
247#340C2312yes—
248#340C2314yes—
249#340C2312yes—
250#340C2314yes—
251#340C2314yes—
252#340C2312yes—
253#336C2314yes—
254#336C2314yes—
255#336C2314yes—
256#336C2312yes—
257#336C2312yes—
258C22×Q8—22no—
259C22×Q8—21no—
260C22×Q8—21no—
261C22×Q8—21no—
262C22×Q8C2×D411yescharacteristic
263C22×Q8C2×D411yescharacteristic
264C22×Q8C2411yescharacteristic
265C22×D4—21no—
266C22×D4—21no—
267C22×D4C2412yes—
268C22×D4C2411yescharacteristic
269C23×C4—21no—
270C23×C4—22no—
271C23×C4—22no—
272C23×C4—22no—
273C23×C4—22no—
274C2×SD16—24no—
275C2×SD16C2×D414yes—
276C2×D8—24no—
277C2×D8—28no—
278C22×C8C2412yes—
279C22×C8C2412yes—
280C22×C8C2412yes—
281C22×C8C2411yescharacteristic
282C22×C8C2411yescharacteristic
283C4⋊Q8—24no—
284C4⋊Q8—22no—
285C4⋊Q8—22no—
286C4⋊Q8—22no—
287C4⋊Q8—22no—
288C4⋊Q8—22no—
289C4⋊Q8—22no—
290C4⋊Q8—42no—
291C4⋊Q8—44no—
292C4⋊Q8—42no—
293C4⋊Q8—48no—
294C4⋊Q8—84no—
295#127—22no—
296#127—22no—
297#125—24no—
298#125—24no—
299#125—42no—
300#125—42no—
301#124—24no—
302#124—42no—
303#124—44no—
304#124—44no—
305#124—84no—
306#123—48no—
307C22⋊Q8—48no—
308C22⋊Q8—48no—
309C22⋊Q8—44no—
310C22⋊Q8—44no—
311C22⋊Q8—88no—
312C4×Q8—22no—
313C4×Q8—22no—
314C4×Q8—22no—
315C4×Q8—22no—
316C4×Q8—22no—
317C4×Q8—24no—
318C4×Q8—42no—
319C4×Q8—44no—
320C4×Q8C2×D414yes—
321C4×D4—24no—
322C4×D4—24no—
323C4×D4—24no—
324C4×D4—24no—
325C4×D4—24no—
326C4×D4—22no—
327C4×D4—42no—
328C4×D4—42no—
329C4×D4—42no—
330C4×D4—44no—
331C4×D4C2×D414yes—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—24no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—24no—
345C2×C4⋊C4—24no—
346C2×C4⋊C4—24no—
347C2×C4⋊C4—24no—
348C2×C4⋊C4—44no—
349C2×C4⋊C4—44no—
350C2×C4⋊C4—88no—
351C2×C4⋊C4C2×D411yescharacteristic
352C2×C4⋊C4C2×D411yescharacteristic
353C2×C4⋊C4C2×D414yes—
354C2×C4⋊C4C2×D412yes—
355C2×C4⋊C4C2×D412yes—
356C2×C4⋊C4C2×D414yes—
357C2×C4⋊C4C2412yes—
358C2×C4⋊C4C2412yes—
359C2×C4⋊C4C2412yes—
360C2×C4⋊C4C2412yes—
361C2×C4⋊C4C2411yescharacteristic
362C2×C4⋊C4C2411yescharacteristic
363C2×C4⋊C4C2412yes—
364C2×C22⋊C4—21no—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—22no—
370C2×C22⋊C4—24no—
371C2×C22⋊C4—22no—
372C2×C22⋊C4—24no—
373C2×C22⋊C4—24no—
374C2×C22⋊C4—24no—
375C2×C42—21no—
376C2×C42—22no—
377C2×C42—21no—
378C2×C42—22no—
379C2×C42—21no—
380C2×C42—44no—
381C2×C42C2×D412yes—
382C2×C42C2×D411yescharacteristic
383C2×C42C2×D411yescharacteristic
384C2×C42C2412yes—
385C2×C42C2412yes—
386C2×C42C2412yes—
387C2×C42C2411yescharacteristic
388#107—24no—
389#107—24no—
390#107—24no—
391#107—24no—
392#107—24no—
393#107—44no—
394#106—24no—
395#106C2×D414yes—
396C4⋊C8—24no—
397C4⋊C8—24no—
398C4⋊C8—22no—
399C4⋊C8—22no—
400#103—24no—
401#103—24no—
402#103—24no—
403#103C2×D414yes—
404#102—24no—
405#102—24no—
406#102—24no—
407#102—24no—
408#102—24no—
409#102—28no—
410#102—44no—
411#102C2×D414yes—
412#97—24no—
413C4×C8—24no—
414C4×C8—22no—
415C4×C8C2×D414yes—
416#95—22no—
417#95—22no—
418#95—21no—
419#95—22no—
420#95—22no—
421#95—21no—
422#95—24no—
423#95—24no—
424#95—24no—
425#95C2×D414yes—
426#95C2×D414yes—
427C24—21no—
428C24—22no—
429C2×Q8—22no—
430C2×Q8—22no—
431C2×Q8—22no—
432C2×Q8—22no—
433C2×Q8—22no—
434C2×Q8—22no—
435C2×Q8—21no—
436C2×Q8—22no—
437C2×Q8—22no—
438C2×Q8—44no—
439C2×Q8—42no—
440C2×Q8—42no—
441C2×Q8—42no—
442C2×Q8—84no—
443C2×Q8—82no—
444C2×Q8—82no—
445C2×Q8—82no—
446C2×Q8C22×D412yes—
447C2×D4—22no—
448C2×D4—21no—
449C2×D4—22no—
450C2×D4—22no—
451C2×D4—24no—
452C2×D4—24no—
453C2×D4—42no—
454C2×D4—42no—
455C2×D4—42no—
456C2×D4—42no—
457C2×D4—82no—
458C2×D4C22×D412yes—
459C2×D42−1+414yes—
460C22×C4—28no—
461C22×C4—22no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—24no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—44no—
471C22×C4—44no—
472C22×C4—44no—
473C22×C4—44no—
474C22×C4—42no—
475C22×C4—44no—
476C22×C4—42no—
477C22×C4—84no—
478C22×C4—84no—
479C22×C4—84no—
480C22×C4C22×D412yes—
481C22×C4C22×D412yes—
482C22×C4C22×D412yes—
483C22×C4C22×D412yes—
484C22×C4C22×D412yes—
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D412yes—
487C22×C4C22×D411yescharacteristic
488C22×C4C2511yescharacteristic
489SD16—28no—
490D8—48no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—24no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—24no—
500C2×C8C22×D412yes—
501C2×C8C22×D412yes—
502C4⋊C4—24no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—24no—
506C4⋊C4—24no—
507C4⋊C4—28no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—44no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C4⋊C4—44no—
523C4⋊C4—44no—
524C4⋊C4—44no—
525C4⋊C4—44no—
526C4⋊C4—42no—
527C4⋊C4—42no—
528C4⋊C4—42no—
529C4⋊C4—48no—
530C4⋊C4—44no—
531C4⋊C4—84no—
532C4⋊C4—84no—
533C4⋊C4—84no—
534C4⋊C4—84no—
535C4⋊C4C22×D412yes—
536C4⋊C4C22×D412yes—
537C4⋊C42−1+414yes—
538C22⋊C4—28no—
539C22⋊C4—24no—
540C22⋊C4—44no—
541C22⋊C4—44no—
542C22⋊C4—44no—
543C22⋊C4—44no—
544C22⋊C4—44no—
545C22⋊C4—48no—
546C22⋊C4—48no—
547C22⋊C4—84no—
548C22⋊C4—84no—
549C42—24no—
550C42—22no—
551C42—24no—
552C42—22no—
553C42—22no—
554C42—22no—
555C42—22no—
556C42—22no—
557C42—22no—
558C42—22no—
559C42—44no—
560C42—44no—
561C42—42no—
562C42—42no—
563C42C22×D412yes—
564C42H(5,𝔽2)12yes—
565C23—22no—
566C23—24no—
567C23—41no—
568C23—42no—
569C23—41no—
570C23—42no—
571C23—44no—
572C23—42no—
573C23—44no—
574C23—42no—
575C23C23×D411yescharacteristic
576Q8—22no—
577Q8—22no—
578Q8—44no—
579Q8—44no—
580Q8—42no—
581Q8—162no—
582Q8—162no—
583Q8—162no—
584Q8—164no—
585D4—22no—
586D4—24no—
587D4—42no—
588D4—42no—
589D4—44no—
590D4—84no—
591D4—162no—
592D4#53614yes—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—24no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—24no—
609C2×C4—24no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—416no—
614C2×C4—42no—
615C2×C4—41no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—41no—
620C2×C4—48no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—44no—
627C2×C4—42no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—164no—
631C2×C4—164no—
632C2×C4—164no—
633C2×C4D4212yes—
634C2×C4D4212yes—
635C2×C4#57512yes—
636C2×C4#57614yes—
637C2×C4#57812yes—
638C2×C4C23×D411yescharacteristic
639C2×C4C2×H(5,𝔽2)11yescharacteristic
640C2×C4C2×2−1+412yes—
641C2×C4C2×2−1+412yes—
642C8—22no—
643C8—22no—
644C8—42no—
645C8—42no—
646C8—42no—
647C22—24no—
648C22—42no—
649C22—44no—
650C22—42no—
651C22—44no—
652C22—82no—
653C22—82no—
654C22—84no—
655C22—82no—
656C22—84no—
657C22C2×D4211yescharacteristic
658C22#348612yes—
659C22C2×#57511yescharacteristic
660C22C2×#57612yes—
661C22C2×#57811yescharacteristic
662C4—24no—
663C4—22no—
664C4—42no—
665C4—42no—
666C4—42no—
667C4—42no—
668C4—44no—
669C4—42no—
670C4—42no—
671C4—42no—
672C4—42no—
673C4—82no—
674C4—82no—
675C4—82no—
676C4—82no—
677C4#326914yes—
678C4C2×#53611yescharacteristic
679C4#349911yescharacteristic
680C4#355212yes—
681C2—44no—
682C2—162no—
683C2#3767511yescharacteristic
684C2C2×#326911yescharacteristic
685C2#6108911yescharacteristic
686C2#6183512yes—
687C2#6187012yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#459C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3385C2semidirect
3. #39168C2semidirect
4. #36180C2semidirect
6. #30967C2semidirect
7. #30818C2semidirect
8. #30276C2semidirect
9. #29869C2semidirect
10. #28676C2semidirect
11. #28645C2semidirect
12. #26368C2semidirect
13. #26188C2semidirect
15. #25841C2semidirect
16. #25839C2semidirect
17. #25648C2semidirect
18. #22240C2semidirect
19. #21114C2semidirect

Maximal subgroups

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

subgroupindex
C2×#33852
C2×#33612
#391682
#361802
#330302
#309672
#308182
#302762
#298692
#286762
#286452
#263682
#261882
#259302
#258412
#258392
#256482
#222402
#211142