Finite Group Databasebeta

← #6686303

Subgroups — #6686303

Order 512.

1778 conjugacy classes of subgroups.

The subgroup lattice has 687 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3372C211yesmaximal, characteristic, complemented
2C2×#3340C211yesmaximal, characteristic
3#39489C211yesmaximal, characteristic
4#36508C211yesmaximal, characteristic, complemented
5#33198C211yesmaximal, characteristic, complemented
6#30652C214yesmaximal, complemented
7#30597C214yesmaximal, complemented
8#30280C211yesmaximal, characteristic, complemented
9#29829C211yesmaximal, characteristic, complemented
10#28684C211yesmaximal, characteristic, complemented
11#28598C211yesmaximal, characteristic, complemented
12#26569C214yesmaximal, complemented
13#26351C214yesmaximal, complemented
14#25835C211yesmaximal, characteristic, complemented
15#25701C211yesmaximal, characteristic, complemented
16#25651C211yesmaximal, characteristic, complemented
17#25568C211yesmaximal, characteristic
18#22212C211yesmaximal, characteristic, complemented
19#21120C211yesmaximal, characteristic, complemented
20C2×#557C2211yescharacteristic
21C2×#556C2211yescharacteristic
22C2×#555C2211yescharacteristic
23C2×#554C2211yescharacteristic
24#3372—28no—
25#3340—28no—
26C2×#498C2211yescharacteristic
27C2×#498C2211yescharacteristic
28C2×#491C2212yes—
29C2×#487C2211yescharacteristic
30C2×#487C2211yescharacteristic
31C2×#485C2212yes—
32C2×#482C2212yes—
33C2×#479C2212yes—
34C2×#477C2211yescharacteristic
35C2×#477C2211yescharacteristic
36C2×#474C2211yescharacteristic
37C2×#465C2212yes—
38C2×#461C2211yescharacteristic
39C2×#461C2212yes—
40C2×#446C2211yescharacteristic
41C2×#440C2212yes—
42C2×C4×SD16C2211yescharacteristic
43C2×C8×D4C2211yescharacteristic
44C2×#427C2212yes—
45#2868C2211yescharacteristic
46#2862C2211yescharacteristic
47#2861—21no—
48#2674—24no—
49#2620—21no—
50#2567C2211yescharacteristic
51#2565—21no—
52#2563C2211yescharacteristic
53#2546C2211yescharacteristic
54#2545—21no—
55#2542C2211yescharacteristic
56#2519—24no—
57#2501—21no—
58#2499—21no—
59#2441C2214yes—
60#2429C2214yes—
61#2422C2214yes—
62#2362C2214yes—
63#2317—21no—
64C4×C4⋊Q8—21no—
65C4×#124C2211yescharacteristic
66#2293C2211yescharacteristic
67C42×Q8C2211yescharacteristic
68#2096C2214yes—
69#2096C2214yes—
70#2088C2214yes—
71#2068C2214yes—
72#2059C2214yes—
73#2055C2214yes—
74#2043C2214yes—
75#2036C2214yes—
76#2035C2214yes—
77#2000C2214yes—
78#1988C2211yescharacteristic
79#1987C2211yescharacteristic
80#1987C2211yescharacteristic
81#1984C2211yescharacteristic
82#1981C2211yescharacteristic
83#1954C2211yescharacteristic
84#1947C2214yes—
85#1943C2214yes—
86#1925C2212yes—
87#1924C2214yes—
88#1920C2212yes—
89#1892C2214yes—
90#1880C2212yes—
91#1873C2214yes—
92#1864C2214yes—
93#1863C2214yes—
94#1863C2212yes—
95#1854C2214yes—
96#1850C2214yes—
97#1842C2211yescharacteristic
98#1803C2214yes—
99C4×#107C2211yescharacteristic
100C4×#103C2211yescharacteristic
101C4×#102C2211yescharacteristic
102#1744C2211yescharacteristic
103C22×SD16C2312yes—
104C22×SD16C2311yescharacteristic
105C23×C8C2312yes—
106#557—24no—
107#557—42no—
108#556—24no—
109#556—42no—
110#555—24no—
111#555—42no—
112#554—24no—
113#554—42no—
114C2×C4⋊Q8—21no—
115C2×C4⋊Q8—22no—
116C2×C4⋊Q8C2311yescharacteristic
117C2×C4⋊Q8C2312yes—
118C2×C4⋊Q8C2311yescharacteristic
119C2×C4⋊Q8C2311yescharacteristic
120C2×C4⋊Q8C2311yescharacteristic
121C2×#126—22no—
122C2×#125—21no—
123C2×#125—22no—
124C2×#125C2312yes—
125C2×#124—21no—
126C2×#124—22no—
127C2×#124D411yescharacteristic
128C2×#123—24no—
129C2×C22⋊Q8—21no—
130C2×C22⋊Q8—22no—
131C2×C22⋊Q8C2312yes—
132C2×C22⋊Q8C2314yes—
133C2×#121—21no—
134C2×#121C2312yes—
135C2×C4×Q8—21no—
136C2×C4×Q8—21no—
137C2×C4×Q8C2312yes—
138C2×C4×Q8C2312yes—
139C2×C4×Q8C2311yescharacteristic
140C2×C4×Q8C2311yescharacteristic
141C2×C4×D4C2312yes—
142C2×C4×D4C2311yescharacteristic
143C2×C4×D4C2311yescharacteristic
144C2×#117—21no—
145C2×#117—21no—
146C2×#117C2312yes—
147C2×#117C2312yes—
148C22×C4⋊C4—21no—
149#498—24no—
150#498—42no—
151#491—28no—
152#487—24no—
153#487—42no—
154#485—28no—
155#482—28no—
156#479—28no—
157#477—24no—
158#477—24no—
159#474—24no—
160#465—28no—
161#461—28no—
162#461—24no—
163#446—24no—
164#440—28no—
165C4×SD16D418yes—
166C8×D4—24no—
167#427—28no—
168C2×#107C2312yes—
169C2×#107C2312yes—
170C2×#107C2312yes—
171C2×#107C2312yes—
172C2×#106C2312yes—
173C2×#106C2312yes—
174C2×#106C2314yes—
175C2×#106C2311yescharacteristic
176C2×C4⋊C8C2311yescharacteristic
177C2×C4⋊C8C2311yescharacteristic
178C2×C4⋊C8C2312yes—
179C2×C4⋊C8C2314yes—
180C2×#103C2312yes—
181C2×#103C2312yes—
182C2×#103C2312yes—
183C2×#103C2312yes—
184C2×#103C2312yes—
185C2×#103C2314yes—
186C2×#103C2314yes—
187C2×#103C2312yes—
188C2×#103C2311yescharacteristic
189C2×#102C2312yes—
190C2×#102C2312yes—
191C2×#102C2311yescharacteristic
192C2×C22⋊C8C2312yes—
193C2×C22⋊C8C2314yes—
194C2×#97C2312yes—
195C2×C4×C8C2311yescharacteristic
196C2×C4×C8C2311yescharacteristic
197#399—24no—
198#399—22no—
199#398—22no—
200#398—24no—
201#397—24no—
202#396—24no—
203#396—24no—
204#395—22no—
205#394—24no—
206#393—22no—
207#391—21no—
208#391—21no—
209#391—21no—
210#391—21no—
211#391D411yescharacteristic
212#391C2314yes—
213#390—21no—
214#390—21no—
215#390D411yescharacteristic
216#390C2312yes—
217#389—24no—
218#389—24no—
219#389C2314yes—
220#389C2314yes—
221#389C2312yes—
222#388—24no—
223#388—24no—
224#388—24no—
225#388—22no—
226#388D414yes—
227#387—24no—
228#387—24no—
229#385—24no—
230#384—21no—
231#384—21no—
232#384—21no—
233#384—21no—
234#384C2314yes—
235#382—21no—
236#382—22no—
237#382D411yescharacteristic
238#380C2314yes—
239C4×C4⋊C4—21no—
240C4×C4⋊C4—22no—
241C4×C4⋊C4—24no—
242C4×C4⋊C4C2311yescharacteristic
243C4×C4⋊C4C2312yes—
244C4×C4⋊C4C2311yescharacteristic
245C4×C22⋊C4—21no—
246C4×C22⋊C4C2314yes—
247C43—21no—
248C43C2311yescharacteristic
249#340C2314yes—
250#340C2314yes—
251#340C2314yes—
252#340C2312yes—
253#340C2312yes—
254#340C2312yes—
255#336C2314yes—
256#336C2312yes—
257#336C2312yes—
258C22×Q8—21no—
259C22×Q8—21no—
260C22×Q8—21no—
261C22×Q8C2×D411yescharacteristic
262C22×Q8C2412yes—
263C22×Q8C2411yescharacteristic
264C22×D4—22no—
265C22×D4C2×D411yescharacteristic
266C22×D4C2411yescharacteristic
267C23×C4—21no—
268C23×C4—22no—
269C23×C4C2412yes—
270C2×SD16—24no—
271C2×SD16—42no—
272C2×SD16—44no—
273C2×SD16C2×D414yes—
274C22×C8—24no—
275C22×C8—44no—
276C22×C8C2412yes—
277C22×C8C2412yes—
278C22×C8C2412yes—
279C22×C8C2411yescharacteristic
280C22×C8C2411yescharacteristic
281C4⋊Q8—24no—
282C4⋊Q8—22no—
283C4⋊Q8—22no—
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—84no—
293#126—48no—
294#125—24no—
295#125—42no—
296#125—44no—
297#125—48no—
298#124—42no—
299#124—44no—
300#124—48no—
301#123—88no—
302C22⋊Q8—24no—
303C22⋊Q8—28no—
304C22⋊Q8—48no—
305C22⋊Q8—44no—
306C22⋊Q8—42no—
307C22⋊Q8—44no—
308#121—24no—
309#121—42no—
310#121—44no—
311C4×Q8—22no—
312C4×Q8—22no—
313C4×Q8—22no—
314C4×Q8—24no—
315C4×Q8—24no—
316C4×Q8—42no—
317C4×Q8—42no—
318C4×Q8—44no—
319C4×Q8—44no—
320C4×Q8C2×D414yes—
321C4×D4—24no—
322C4×D4—22no—
323C4×D4—22no—
324C4×D4—22no—
325C4×D4—42no—
326C4×D4C2×D414yes—
327#117—24no—
328#117—24no—
329#117—44no—
330#117—42no—
331#117—42no—
332#117—44no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—21no—
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—21no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—24no—
350C2×C4⋊C4—24no—
351C2×C4⋊C4—24no—
352C2×C4⋊C4—44no—
353C2×C4⋊C4C2×D411yescharacteristic
354C2×C4⋊C4C2×D411yescharacteristic
355C2×C4⋊C4C2×D412yes—
356C2×C4⋊C4C2412yes—
357C2×C4⋊C4C2412yes—
358C2×C4⋊C4C2412yes—
359C2×C4⋊C4C2412yes—
360C2×C4⋊C4C2411yescharacteristic
361C2×C4⋊C4C2411yescharacteristic
362C2×C4⋊C4C2412yes—
363C2×C22⋊C4—21no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—24no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—22no—
370C2×C22⋊C4—22no—
371C2×C22⋊C4—22no—
372C2×C22⋊C4—24no—
373C2×C22⋊C4C2×D412yes—
374C2×C22⋊C4C2×D414yes—
375C2×C22⋊C4C2×D414yes—
376C2×C42—21no—
377C2×C42—22no—
378C2×C42—21no—
379C2×C42—22no—
380C2×C42—22no—
381C2×C42—22no—
382C2×C42—21no—
383C2×C42—22no—
384C2×C42C2×D411yescharacteristic
385C2×C42C2×D412yes—
386C2×C42C2×D411yescharacteristic
387C2×C42C2412yes—
388C2×C42C2412yes—
389C2×C42C2411yescharacteristic
390#107—24no—
391#107—24no—
392#107—24no—
393#107—24no—
394#106—24no—
395#106—24no—
396#106—44no—
397#106C2×D414yes—
398C4⋊C8—24no—
399C4⋊C8—22no—
400C4⋊C8—22no—
401C4⋊C8—44no—
402#103—24no—
403#103—24no—
404#103—24no—
405#103—24no—
406#103—24no—
407#103—42no—
408#103—44no—
409#103—44no—
410#103C2×D414yes—
411#102—24no—
412#102—24no—
413#102C2×D414yes—
414C22⋊C8—24no—
415C22⋊C8—44no—
416#97—24no—
417C4×C8—22no—
418C4×C8C2×D414yes—
419#95—22no—
420#95—22no—
421#95—21no—
422#95—24no—
423#95—21no—
424#95—24no—
425#95—24no—
426#95—24no—
427#95—24no—
428#95C2×D414yes—
429#95C2×D414yes—
430C24—21no—
431C24C22×D412yes—
432C2×Q8—22no—
433C2×Q8—22no—
434C2×Q8—22no—
435C2×Q8—21no—
436C2×Q8—22no—
437C2×Q8—22no—
438C2×Q8—22no—
439C2×Q8—24no—
440C2×Q8—22no—
441C2×Q8—24no—
442C2×Q8—42no—
443C2×Q8—42no—
444C2×Q8—42no—
445C2×Q8—42no—
446C2×Q8—42no—
447C2×Q8—82no—
448C2×Q8—82no—
449C2×Q8C22×D412yes—
450C2×D4—22no—
451C2×D4—22no—
452C2×D4—22no—
453C2×D4—21no—
454C2×D4—22no—
455C2×D4—22no—
456C2×D4—44no—
457C2×D4—84no—
458C2×D4C22×D412yes—
459C22×C4—22no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—24no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—44no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—84no—
477C22×C4C22×D412yes—
478C22×C4C22×D412yes—
479C22×C4C22×D412yes—
480C22×C4C22×D412yes—
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D412yes—
483C22×C4C22×D411yescharacteristic
484C22×C4C2511yescharacteristic
485SD16—44no—
486SD16—48no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—24no—
495C2×C8—24no—
496C2×C8—48no—
497C2×C8C22×D412yes—
498C2×C8C22×D412yes—
499C4⋊C4—24no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—24no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—22no—
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—44no—
515C4⋊C4—44no—
516C4⋊C4—44no—
517C4⋊C4—44no—
518C4⋊C4—44no—
519C4⋊C4—48no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C4⋊C4—44no—
523C4⋊C4—48no—
524C4⋊C4—42no—
525C4⋊C4—42no—
526C4⋊C4—42no—
527C4⋊C4—42no—
528C4⋊C4—42no—
529C4⋊C4—48no—
530C4⋊C4—84no—
531C4⋊C4—84no—
532C4⋊C4C22×D412yes—
533C4⋊C4C22×D412yes—
534C4⋊C42−1+414yes—
535C22⋊C4—24no—
536C22⋊C4—24no—
537C22⋊C4—28no—
538C22⋊C4—44no—
539C22⋊C4—44no—
540C22⋊C4—44no—
541C22⋊C4—44no—
542C22⋊C4—44no—
543C22⋊C4—48no—
544C22⋊C4—44no—
545C22⋊C4—44no—
546C22⋊C4—44no—
547C22⋊C4—84no—
548C42—24no—
549C42—22no—
550C42—24no—
551C42—22no—
552C42—22no—
553C42—22no—
554C42—22no—
555C42—22no—
556C42—44no—
557C42—44no—
558C42—44no—
559C42—44no—
560C42—42no—
561C42—42no—
562C42—44no—
563C42C22×D412yes—
564C42H(5,𝔽2)12yes—
565C23—22no—
566C23—22no—
567C23—24no—
568C23—22no—
569C23—24no—
570C23—22no—
571C23—42no—
572C23—41no—
573C23—42no—
574C23—41no—
575C23C23×D411yescharacteristic
576Q8—22no—
577Q8—22no—
578Q8—44no—
579Q8—42no—
580Q8—42no—
581Q8—42no—
582Q8—44no—
583Q8—84no—
584Q8—162no—
585Q8—162no—
586D4—22no—
587D4—22no—
588D4—44no—
589D4—42no—
590D4—164no—
591C2×C4—22no—
592C2×C4—22no—
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—24no—
602C2×C4—24no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—24no—
607C2×C4—22no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—41no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—41no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—48no—
617C2×C4—44no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—42no—
630C2×C4—164no—
631C2×C4D4212yes—
632C2×C4D4212yes—
633C2×C4#57512yes—
634C2×C4#57714yes—
635C2×C4#57812yes—
636C2×C4C23×D411yescharacteristic
637C2×C4C2×H(5,𝔽2)11yescharacteristic
638C2×C4C2×2−1+412yes—
639C2×C4#58512yes—
640C8—22no—
641C8—22no—
642C8—42no—
643C8—42no—
644C8—42no—
645C22—42no—
646C22—42no—
647C22—42no—
648C22—42no—
649C22—42no—
650C22—42no—
651C22—44no—
652C22—44no—
653C22—82no—
654C22—82no—
655C22C2×D4211yescharacteristic
656C22#348912yes—
657C22C2×#57511yescharacteristic
658C22C2×#57712yes—
659C22C2×#57811yescharacteristic
660C4—22no—
661C4—24no—
662C4—42no—
663C4—42no—
664C4—42no—
665C4—42no—
666C4—42no—
667C4—42no—
668C4—42no—
669C4—44no—
670C4—82no—
671C4—82no—
672C4—82no—
673C4—82no—
674C4—82no—
675C4#326914yes—
676C4#344911yescharacteristic
677C4#349911yescharacteristic
678C4#355412yes—
679C2—82no—
680C2—82no—
681C2#3768011yescharacteristic
682C2C2×#326911yescharacteristic
683C2#6108711yescharacteristic
684C2#6213412yes—
685C2#6220412yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#555C22
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×#3372C2semidirect
4. #36508C2semidirect
5. #33198C2semidirect
6. #30652C2semidirect
7. #30597C2semidirect
8. #30280C2semidirect
9. #29829C2semidirect
10. #28684C2semidirect
11. #28598C2semidirect
12. #26569C2semidirect
13. #26351C2semidirect
14. #25835C2semidirect
15. #25701C2semidirect
16. #25651C2semidirect
18. #22212C2semidirect
19. #21120C2semidirect

Maximal subgroups

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

subgroupindex
C2×#33722
C2×#33402
#394892
#365082
#331982
#306522
#305972
#302802
#298292
#286842
#285982
#265692
#263512
#258352
#257012
#256512
#255682
#222122
#211202