Finite Group Databasebeta

← #6816548

Subgroups — #6816548

Order 512.

1746 conjugacy classes of subgroups.

The subgroup lattice has 683 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3385C211yesmaximal, characteristic, complemented
2C2×#3353C211yesmaximal, characteristic
3#39172C211yesmaximal, characteristic, complemented
4#38955C211yesmaximal, characteristic
5#33313C211yesmaximal, characteristic, complemented
6#31082C214yesmaximal, complemented
7#31079C214yesmaximal, complemented
8#29867C211yesmaximal, characteristic, complemented
9#29865C211yesmaximal, characteristic, complemented
10#28711C211yesmaximal, characteristic, complemented
11#28683C211yesmaximal, characteristic, complemented
12#26076C214yesmaximal, complemented
13#25837C211yesmaximal, characteristic, complemented
14#25835C211yesmaximal, characteristic, complemented
15#25701C211yesmaximal, characteristic, complemented
16#25699C211yesmaximal, characteristic
17#25044C214yesmaximal, complemented
18#21631C211yesmaximal, characteristic, complemented
19#21274C211yesmaximal, characteristic, complemented
20C2×Q82C2211yescharacteristic
21C2×#556C2211yescharacteristic
22C2×#554C2211yescharacteristic
23C2×#547C2211yescharacteristic
24#3385—28no—
25#3353—28no—
26C2×#501C2212yes—
27C2×#500C2211yescharacteristic
28C2×#500C2211yescharacteristic
29C2×#498C2211yescharacteristic
30C2×#498C2211yescharacteristic
31C2×#484C2212yes—
32C2×#482C2212yes—
33C2×#479C2212yes—
34C2×#477C2211yescharacteristic
35C2×#475C2211yescharacteristic
36C2×#476C2211yescharacteristic
37C2×#474C2211yescharacteristic
38C2×#474C2212yes—
39C2×#468C2212yes—
40C2×#446C2211yescharacteristic
41C2×#440C2212yes—
42C2×C4×SD16C2211yescharacteristic
43C2×#436C2211yescharacteristic
44C2×#428C2212yes—
45#2862C2211yescharacteristic
46#2862C2211yescharacteristic
47#2716—24no—
48#2687—24no—
49#2640—21no—
50#2621—21no—
51#2568—21no—
52#2563C2211yescharacteristic
53#2563C2211yescharacteristic
54#2546C2211yescharacteristic
55#2546C2211yescharacteristic
56#2545—21no—
57#2503—21no—
58#2442C2214yes—
59#2439C2214yes—
60#2432—21no—
61#2421C2214yes—
62#2420C2214yes—
63#2327—21no—
64#2311—21no—
65C4×#124C2211yescharacteristic
66#2295C2211yescharacteristic
67#2293C2211yescharacteristic
68#2087C2214yes—
69#2085C2214yes—
70#2085C2214yes—
71#2075C2214yes—
72#2074C2214yes—
73#2073C2214yes—
74#2059C2214yes—
75#2058C2214yes—
76#2035C2214yes—
77#2034C2214yes—
78#1988C2211yescharacteristic
79#1988C2211yescharacteristic
80#1987C2211yescharacteristic
81#1987C2211yescharacteristic
82#1984C2211yescharacteristic
83#1968C2212yes—
84#1954C2211yescharacteristic
85#1944C2214yes—
86#1941C2212yes—
87#1934C2212yes—
88#1933C2212yes—
89#1923C2214yes—
90#1865C2214yes—
91#1864C2214yes—
92#1854C2214yes—
93#1853C2214yes—
94#1846C2214yes—
95#1846C2214yes—
96#1831C2211yescharacteristic
97#1809C2214yes—
98C4×#106C2211yescharacteristic
99C4×#103C2211yescharacteristic
100C4×#102C2211yescharacteristic
101#1753C2214yes—
102C42×C8C2211yescharacteristic
103C22×#41—21no—
104C22×SD16C2311yescharacteristic
105C22×M16C2312yes—
106Q82—24no—
107Q82—42no—
108#556—24no—
109#556—42no—
110#554—24no—
111#554—42no—
112#547—24no—
113#547—42no—
114C2×C4⋊Q8—21no—
115C2×C4⋊Q8—21no—
116C2×C4⋊Q8—21no—
117C2×C4⋊Q8—22no—
118C2×C4⋊Q8C2311yescharacteristic
119C2×C4⋊Q8C2311yescharacteristic
120C2×C4⋊Q8C2311yescharacteristic
121C2×C4⋊Q8C2311yescharacteristic
122C2×C4⋊Q8C2312yes—
123C2×#126—22no—
124C2×#125C2312yes—
125C2×#124—21no—
126C2×#124D411yescharacteristic
127C2×#123—22no—
128C2×C22⋊Q8—21no—
129C2×C22⋊Q8—21no—
130C2×C22⋊Q8—22no—
131C2×C22⋊Q8C2312yes—
132C2×#121C2312yes—
133C2×C4×Q8—21no—
134C2×C4×Q8—21no—
135C2×C4×Q8C2312yes—
136C2×C4×Q8C2311yescharacteristic
137C2×C4×Q8C2311yescharacteristic
138C2×C4×D4C2312yes—
139C2×C4×D4C2311yescharacteristic
140C2×C4×D4C2311yescharacteristic
141C2×#117—21no—
142C2×#117C2312yes—
143C22×C4⋊C4—21no—
144C22×C4⋊C4C2312yes—
145#501—28no—
146#500—24no—
147#500—24no—
148#498—42no—
149#498—42no—
150#484—28no—
151#482—28no—
152#479—28no—
153#477—24no—
154#476—24no—
155#475—24no—
156#474—24no—
157#474—28no—
158#468—28no—
159#446—24no—
160#440—28no—
161C4×SD16D418yes—
162#436—24no—
163#428—28no—
164C2×#107C2312yes—
165C2×#107C2312yes—
166C2×#107C2312yes—
167C2×#107C2312yes—
168C2×#107C2312yes—
169C2×#107C2312yes—
170C2×#106C2312yes—
171C2×#106C2312yes—
172C2×#106C2312yes—
173C2×#106C2314yes—
174C2×#106C2314yes—
175C2×#106C2311yescharacteristic
176C2×C4⋊C8C2311yescharacteristic
177C2×C4⋊C8C2311yescharacteristic
178C2×C4⋊C8C2312yes—
179C2×#103C2312yes—
180C2×#103C2312yes—
181C2×#103C2312yes—
182C2×#103C2311yescharacteristic
183C2×#102C2312yes—
184C2×#102C2312yes—
185C2×#102C2312yes—
186C2×#102C2311yescharacteristic
187C2×C22⋊C8C2312yes—
188C2×C22⋊C8C2314yes—
189C2×#97C2312yes—
190C2×C4×C8C2312yes—
191C2×C4×C8C2312yes—
192C2×C4×C8C2311yescharacteristic
193C2×C4×C8C2311yescharacteristic
194#400—24no—
195#400—24no—
196#398—22no—
197#398—22no—
198#398—24no—
199#397—24no—
200#397—24no—
201#396—24no—
202#394—24no—
203#393—22no—
204#393—22no—
205#393—24no—
206#391—21no—
207#391—21no—
208#391—21no—
209#391—21no—
210#391—21no—
211#391—22no—
212#391—21no—
213#391—22no—
214#391—24no—
215#391D411yescharacteristic
216#391C2312yes—
217#390—21no—
218#390—21no—
219#390D411yescharacteristic
220#390C2312yes—
221#389C2314yes—
222#389C2314yes—
223#389C2312yes—
224#389C2312yes—
225#388—24no—
226#388—22no—
227#388D414yes—
228#387—22no—
229#387—24no—
230#387—24no—
231#386—24no—
232#385—24no—
233#382—21no—
234#382—21no—
235#382D411yescharacteristic
236#382C2314yes—
237#382C2314yes—
238#381—24no—
239#380—24no—
240#380C2314yes—
241#380C2314yes—
242C4×C4⋊C4—21no—
243C4×C4⋊C4C2311yescharacteristic
244C4×C4⋊C4C2311yescharacteristic
245C4×C22⋊C4—21no—
246C4×C22⋊C4C2314yes—
247#376—21no—
248C43C2311yescharacteristic
249#340C2312yes—
250#340C2314yes—
251#340C2312yes—
252#340C2314yes—
253#340C2314yes—
254#340C2314yes—
255#340C2312yes—
256#336C2314yes—
257#336C2314yes—
258C2×#41—44no—
259C2×#41—44no—
260C2×#41—82no—
261C22×Q8—21no—
262C22×Q8—21no—
263C22×Q8—21no—
264C22×Q8—22no—
265C22×Q8—21no—
266C22×Q8C2×D411yescharacteristic
267C22×Q8C2411yescharacteristic
268C22×D4—22no—
269C22×D4C2×D411yescharacteristic
270C22×D4C2411yescharacteristic
271C23×C4—21no—
272C23×C4—22no—
273C23×C4—22no—
274C23×C4C2412yes—
275C2×SD16—24no—
276C2×SD16C2×D414yes—
277C2×M16—24no—
278C2×M16—28no—
279C22×C8C2412yes—
280C22×C8C2412yes—
281C22×C8C2411yescharacteristic
282C22×C8C2412yes—
283C22×C8C2411yescharacteristic
284C4⋊Q8—24no—
285C4⋊Q8—22no—
286C4⋊Q8—22no—
287C4⋊Q8—22no—
288C4⋊Q8—22no—
289C4⋊Q8—22no—
290C4⋊Q8—22no—
291C4⋊Q8—22no—
292C4⋊Q8—22no—
293C4⋊Q8—42no—
294C4⋊Q8—44no—
295C4⋊Q8—48no—
296C4⋊Q8—44no—
297C4⋊Q8—82no—
298#126—48no—
299#125—24no—
300#125—42no—
301#124—42no—
302#124—44no—
303#123—48no—
304C22⋊Q8—24no—
305C22⋊Q8—48no—
306C22⋊Q8—42no—
307C22⋊Q8—44no—
308C22⋊Q8—44no—
309#121—24no—
310#121—42no—
311C4×Q8—22no—
312C4×Q8—22no—
313C4×Q8—22no—
314C4×Q8—24no—
315C4×Q8—42no—
316C4×Q8—44no—
317C4×Q8—44no—
318C4×Q8C2×D414yes—
319C4×D4—24no—
320C4×D4—22no—
321C4×D4—22no—
322C4×D4—22no—
323C4×D4—42no—
324C4×D4C2×D414yes—
325#117—24no—
326#117—42no—
327#117—82no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—24no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—24no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—24no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—42no—
345C2×C4⋊C4—44no—
346C2×C4⋊C4C2×D411yescharacteristic
347C2×C4⋊C4C2×D411yescharacteristic
348C2×C4⋊C4C2×D412yes—
349C2×C4⋊C4C2412yes—
350C2×C4⋊C4C2412yes—
351C2×C4⋊C4C2412yes—
352C2×C4⋊C4C2412yes—
353C2×C4⋊C4C2412yes—
354C2×C4⋊C4C2412yes—
355C2×C4⋊C4C2411yescharacteristic
356C2×C4⋊C4C2411yescharacteristic
357C2×C22⋊C4—21no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—24no—
363C2×C22⋊C4—24no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—22no—
366C2×C22⋊C4C2×D412yes—
367C2×C22⋊C4C2×D414yes—
368C2×C22⋊C4C2×D414yes—
369C2×C42—21no—
370C2×C42—21no—
371C2×C42—22no—
372C2×C42—22no—
373C2×C42—21no—
374C2×C42C2×D411yescharacteristic
375C2×C42C2×D411yescharacteristic
376C2×C42C2×D412yes—
377C2×C42C2412yes—
378C2×C42C2412yes—
379C2×C42C2411yescharacteristic
380#107—24no—
381#107—24no—
382#107—24no—
383#107—24no—
384#107—24no—
385#107—24no—
386#106—24no—
387#106—24no—
388#106—24no—
389#106—44no—
390#106—44no—
391#106C2×D414yes—
392C4⋊C8—24no—
393C4⋊C8—22no—
394C4⋊C8—22no—
395#103—24no—
396#103—24no—
397#103—24no—
398#103C2×D414yes—
399#102—24no—
400#102—24no—
401#102—24no—
402#102C2×D414yes—
403C22⋊C8—24no—
404C22⋊C8—28no—
405#97—24no—
406C4×C8—24no—
407C4×C8—24no—
408C4×C8—22no—
409C4×C8C2×D414yes—
410#95—22no—
411#95—22no—
412#95—22no—
413#95—22no—
414#95—21no—
415#95—21no—
416#95—24no—
417#95—24no—
418#95—24no—
419#95—24no—
420#95—24no—
421#95—24no—
422#95—24no—
423#95—24no—
424#95C2×D414yes—
425#95C2×D414yes—
426C24—21no—
427C24C22×D412yes—
428#41—816no—
429C2×Q8—22no—
430C2×Q8—22no—
431C2×Q8—22no—
432C2×Q8—21no—
433C2×Q8—22no—
434C2×Q8—22no—
435C2×Q8—44no—
436C2×Q8—42no—
437C2×Q8—42no—
438C2×Q8—42no—
439C2×Q8—42no—
440C2×Q8—42no—
441C2×Q8—42no—
442C2×Q8—42no—
443C2×Q8—42no—
444C2×Q8—82no—
445C2×Q8—82no—
446C2×Q8—84no—
447C2×Q8C22×D412yes—
448C2×D4—22no—
449C2×D4—22no—
450C2×D4—22no—
451C2×D4—21no—
452C2×D4—22no—
453C2×D4—22no—
454C2×D4—44no—
455C2×D4—44no—
456C2×D4—82no—
457C2×D4C22×D412yes—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—24no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—44no—
473C22×C4—44no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—44no—
477C22×C4—84no—
478C22×C4—82no—
479C22×C4C22×D412yes—
480C22×C4C22×D412yes—
481C22×C4C22×D412yes—
482C22×C4C22×D412yes—
483C22×C4C22×D412yes—
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D411yescharacteristic
486C22×C4C2511yescharacteristic
487SD16—28no—
488M16—48no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—24no—
495C2×C8—24no—
496C2×C8C22×D412yes—
497C2×C8C22×D412yes—
498C2×C82−1+414yes—
499C2×C82−1+414yes—
500C4⋊C4—24no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—24no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
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—44no—
519C4⋊C4—44no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C4⋊C4—44no—
523C4⋊C4—44no—
524C4⋊C4—48no—
525C4⋊C4—42no—
526C4⋊C4—42no—
527C4⋊C4—42no—
528C4⋊C4—42no—
529C4⋊C4—82no—
530C4⋊C4—82no—
531C4⋊C4—84no—
532C4⋊C4C22×D412yes—
533C4⋊C4C22×D412yes—
534C22⋊C4—24no—
535C22⋊C4—24no—
536C22⋊C4—44no—
537C22⋊C4—44no—
538C22⋊C4—44no—
539C22⋊C4—44no—
540C22⋊C4—44no—
541C22⋊C4—44no—
542C22⋊C4—82no—
543C22⋊C4—82no—
544C22⋊C4—84no—
545C22⋊C4—84no—
546C42—24no—
547C42—22no—
548C42—22no—
549C42—22no—
550C42—22no—
551C42—22no—
552C42—22no—
553C42—42no—
554C42—44no—
555C42—44no—
556C42—42no—
557C42—42no—
558C42C22×D412yes—
559C42H(5,𝔽2)12yes—
560C23—24no—
561C23—22no—
562C23—22no—
563C23—24no—
564C23—22no—
565C23—22no—
566C23—42no—
567C23—42no—
568C23—41no—
569C23—41no—
570C23C23×D411yescharacteristic
571Q8—22no—
572Q8—22no—
573Q8—44no—
574Q8—42no—
575Q8—84no—
576Q8—84no—
577Q8—162no—
578Q8—162no—
579Q8—164no—
580D4—22no—
581D4—22no—
582D4—44no—
583D4—42no—
584D4—88no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—24no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—24no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—48no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—88no—
627C2×C4—164no—
628C2×C4D4212yes—
629C2×C4D4212yes—
630C2×C4#57512yes—
631C2×C4#57712yes—
632C2×C4C23×D411yescharacteristic
633C2×C4C2×H(5,𝔽2)11yescharacteristic
634C2×C4C2×2−1+412yes—
635C2×C4C2×2−1+412yes—
636C8—22no—
637C8—24no—
638C8—42no—
639C8—42no—
640C8#53614yes—
641C22—44no—
642C22—44no—
643C22—42no—
644C22—42no—
645C22—42no—
646C22—42no—
647C22—42no—
648C22—42no—
649C22—82no—
650C22—82no—
651C22C2×D4211yescharacteristic
652C22#349912yes—
653C22C2×#57511yescharacteristic
654C22C2×#57711yescharacteristic
655C22#358412yes—
656C4—22no—
657C4—42no—
658C4—42no—
659C4—42no—
660C4—42no—
661C4—42no—
662C4—42no—
663C4—42no—
664C4—42no—
665C4—82no—
666C4—82no—
667C4—82no—
668C4—82no—
669C4—82no—
670C4—82no—
671C4#327514yes—
672C4C2×#53611yescharacteristic
673C4#349911yescharacteristic
674C4#355212yes—
675C2—82no—
676C2—82no—
677C2#3936011yescharacteristic
678C2C2×#327511yescharacteristic
679C2#6109011yescharacteristic
680C2#6248512yes—
681C2#6249412yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#547C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#124 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3385C2semidirect
3. #39172C2semidirect
5. #33313C2semidirect
6. #31082C2semidirect
7. #31079C2semidirect
8. #29867C2semidirect
9. #29865C2semidirect
10. #28711C2semidirect
11. #28683C2semidirect
12. #26076C2semidirect
13. #25837C2semidirect
14. #25835C2semidirect
15. #25701C2semidirect
17. #25044C2semidirect
18. #21631C2semidirect
19. #21274C2semidirect

Maximal subgroups

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

subgroupindex
C2×#33852
C2×#33532
#391722
#389552
#333132
#310822
#310792
#298672
#298652
#287112
#286832
#260762
#258372
#258352
#257012
#256992
#250442
#216312
#212742