Finite Group Databasebeta

← #6683840

Subgroups — #6683840

Order 512.

1814 conjugacy classes of subgroups.

The subgroup lattice has 690 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3381C211yesmaximal, characteristic, complemented
2C2×#3361C211yesmaximal, characteristic
3#39746C211yesmaximal, characteristic, complemented
4#36180C211yesmaximal, characteristic, complemented
5#33024C211yesmaximal, characteristic
6#30951C214yesmaximal, complemented
7#30810C214yesmaximal, complemented
8#30277C211yesmaximal, characteristic, complemented
9#29869C211yesmaximal, characteristic, complemented
10#28677C211yesmaximal, characteristic, complemented
11#28640C211yesmaximal, characteristic, complemented
12#26382C214yesmaximal, complemented
13#26172C214yesmaximal, complemented
14#25930C211yesmaximal, characteristic
15#25836C211yesmaximal, characteristic, complemented
16#25835C211yesmaximal, characteristic, complemented
17#25648C211yesmaximal, characteristic, complemented
18#22238C211yesmaximal, characteristic, complemented
19#21117C211yesmaximal, characteristic, complemented
20C2×#559C2211yescharacteristic
21C2×#557C2211yescharacteristic
22C2×#554C2211yescharacteristic
23C2×#554C2211yescharacteristic
24#3381—28no—
25#3361—28no—
26C2×#499C2212yes—
27C2×#498C2211yescharacteristic
28C2×#498C2211yescharacteristic
29C2×#488C2212yes—
30C2×#486C2211yescharacteristic
31C2×#486C2211yescharacteristic
32C2×#479C2212yes—
33C2×#477C2211yescharacteristic
34C2×#477C2211yescharacteristic
35C2×#478C2212yes—
36C2×#478C2212yes—
37C2×#474C2211yescharacteristic
38C2×#463C2212yes—
39C2×#459C2211yescharacteristic
40C2×C8×Q8C2211yescharacteristic
41C2×C8×Q8C2211yescharacteristic
42C2×C4×SD16C2212yes—
43C2×C4×SD16C2211yescharacteristic
44C2×C4×D8C2212yes—
45#2868C2211yescharacteristic
46#2858C2211yescharacteristic
47#2857—21no—
48#2694—24no—
49#2658—21no—
50#2567C2211yescharacteristic
51#2567C2211yescharacteristic
52#2552C2214yes—
53#2548C2211yescharacteristic
54#2547—21no—
55#2546C2211yescharacteristic
56#2545—21no—
57#2502—21no—
58#2502—21no—
59#2454C2214yes—
60#2448C2214yes—
61#2440—24no—
62D4×C4⋊C4C2214yes—
63#2326—21no—
64C4×C4⋊Q8C2211yescharacteristic
65#2295C2211yescharacteristic
66C2×#384—21no—
67C42×D4C2211yescharacteristic
68#2094C2214yes—
69#2094C2214yes—
70#2088C2214yes—
71#2083C2214yes—
72#2072C2214yes—
73#2065C2214yes—
74#2059C2214yes—
75#2059C2214yes—
76#2042C2214yes—
77#2025C2214yes—
78#1988C2211yescharacteristic
79#1987C2211yescharacteristic
80#1987C2211yescharacteristic
81#1984C2211yescharacteristic
82#1984C2211yescharacteristic
83#1982C2211yescharacteristic
84#1941C2214yes—
85#1934C2214yes—
86#1927C2212yes—
87#1925C2214yes—
88#1920C2212yes—
89#1916C2214yes—
90#1916C2214yes—
91#1879C2214yes—
92#1879C2214yes—
93#1877C2212yes—
94#1865C2214yes—
95#1864C2212yes—
96#1864C2214yes—
97#1862C2214yes—
98#1831C2211yescharacteristic
99C4×#106C2211yescharacteristic
100C4×#103C2211yescharacteristic
101C4×#102C2211yescharacteristic
102#1744C2211yescharacteristic
103C22×SD16C2311yescharacteristic
104C22×D8C2312yes—
105#559—24no—
106#559—42no—
107#557—24no—
108#557—42no—
109#554—24no—
110#554—24no—
111#554—42no—
112#554—42no—
113C2×C4⋊Q8—21no—
114C2×C4⋊Q8—21no—
115C2×C4⋊Q8—22no—
116C2×C4⋊Q8D411yescharacteristic
117C2×C4⋊Q8C2311yescharacteristic
118C2×C4⋊Q8C2311yescharacteristic
119C2×C4⋊Q8C2311yescharacteristic
120C2×#127C2311yescharacteristic
121C2×#125—22no—
122C2×#125C2312yes—
123C2×#125C2312yes—
124C2×#125C2312yes—
125C2×#124—21no—
126C2×#124—21no—
127C2×#124—21no—
128C2×#124—21no—
129C2×#124C2312yes—
130C2×#123—22no—
131C2×C22⋊Q8—21no—
132C2×C22⋊Q8—21no—
133C2×C22⋊Q8—22no—
134C2×C22⋊Q8—22no—
135C2×C4×Q8—21no—
136C2×C4×Q8C2312yes—
137C2×C4×Q8C2311yescharacteristic
138C2×C4×Q8C2311yescharacteristic
139C2×C4×Q8C2311yescharacteristic
140C2×C4×D4—21no—
141C2×C4×D4C2312yes—
142C2×C4×D4C2312yes—
143C2×C4×D4C2312yes—
144C2×C4×D4C2312yes—
145C2×C4×D4C2311yescharacteristic
146C22×C4⋊C4—21no—
147C22×C4⋊C4—21no—
148C22×C4⋊C4—24no—
149C22×C42—21no—
150#499—28no—
151#498—24no—
152#498—42no—
153#488—28no—
154#486—24no—
155#486—42no—
156#479—28no—
157#478—28no—
158#478—28no—
159#477—24no—
160#477—24no—
161#474—24no—
162#463—28no—
163#459—24no—
164C8×Q8—24no—
165C8×Q8—24no—
166C4×SD16—28no—
167C4×SD16D418yes—
168C4×D8—28no—
169C2×#107C2312yes—
170C2×#107C2312yes—
171C2×#107C2312yes—
172C2×#106C2312yes—
173C2×#106C2312yes—
174C2×#106C2312yes—
175C2×#106C2314yes—
176C2×#106C2311yescharacteristic
177C2×C4⋊C8C2312yes—
178C2×C4⋊C8C2312yes—
179C2×C4⋊C8C2311yescharacteristic
180C2×C4⋊C8C2311yescharacteristic
181C2×#103C2312yes—
182C2×#103C2312yes—
183C2×#103C2312yes—
184C2×#103C2311yescharacteristic
185C2×#102C2312yes—
186C2×#102C2312yes—
187C2×#102C2312yes—
188C2×#102C2312yes—
189C2×#102C2314yes—
190C2×#102C2312yes—
191C2×#102C2314yes—
192C2×#102C2311yescharacteristic
193C2×C4×C8C2311yescharacteristic
194C2×C4×C8C2312yes—
195C2×C4×C8C2312yes—
196C2×C4×C8C2311yescharacteristic
197#399—24no—
198#399—24no—
199#398—22no—
200#398—24no—
201#398—24no—
202#397—24no—
203#396—24no—
204#395—22no—
205#395—22no—
206#394—22no—
207#391—21no—
208#391—21no—
209#391—21no—
210#391D411yescharacteristic
211#391D411yescharacteristic
212#391C2312yes—
213#390—21no—
214#390—22no—
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—24no—
232#384—21no—
233#384—21no—
234#384D411yescharacteristic
235#384C2314yes—
236#380—24no—
237#380—24no—
238C4×C4⋊C4—21no—
239C4×C4⋊C4C2311yescharacteristic
240C4×C4⋊C4C2312yes—
241C4×C4⋊C4C2311yescharacteristic
242C4×C4⋊C4C2314yes—
243C4×C22⋊C4—22no—
244C4×C22⋊C4—21no—
245#376—21no—
246C43C2311yescharacteristic
247#340C2312yes—
248#340C2312yes—
249#340C2314yes—
250#340C2314yes—
251#340C2314yes—
252#340C2312yes—
253#336C2314yes—
254#336C2314yes—
255#336C2314yes—
256#336C2312yes—
257#336C2312yes—
258C22×Q8—21no—
259C22×Q8—21no—
260C22×Q8—21no—
261C22×Q8C2×D411yescharacteristic
262C22×Q8C2×D411yescharacteristic
263C22×Q8C2411yescharacteristic
264C22×D4—21no—
265C22×D4—21no—
266C22×D4C2412yes—
267C22×D4C2411yescharacteristic
268C23×C4—21no—
269C23×C4—22no—
270C23×C4—22no—
271C23×C4—22no—
272C23×C4—22no—
273C2×SD16—24no—
274C2×SD16C2×D414yes—
275C2×D8—24no—
276C2×D8—28no—
277C22×C8C2412yes—
278C22×C8C2412yes—
279C22×C8C2412yes—
280C22×C8C2411yescharacteristic
281C22×C8C2411yescharacteristic
282C4⋊Q8—24no—
283C4⋊Q8—22no—
284C4⋊Q8—22no—
285C4⋊Q8—22no—
286C4⋊Q8—22no—
287C4⋊Q8—22no—
288C4⋊Q8—22no—
289C4⋊Q8—44no—
290C4⋊Q8—84no—
291C4⋊Q8—82no—
292#127—22no—
293#127—22no—
294#125—24no—
295#125—24no—
296#125—24no—
297#125—42no—
298#125—42no—
299#125—42no—
300#125—48no—
301#124—24no—
302#124—42no—
303#124—44no—
304#124—44no—
305#124—82no—
306#124—82no—
307#123—48no—
308C22⋊Q8—48no—
309C22⋊Q8—48no—
310C22⋊Q8—44no—
311C22⋊Q8—44no—
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—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—24no—
345C2×C4⋊C4—24no—
346C2×C4⋊C4—24no—
347C2×C4⋊C4—24no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—44no—
350C2×C4⋊C4—44no—
351C2×C4⋊C4—88no—
352C2×C4⋊C4C2×D411yescharacteristic
353C2×C4⋊C4C2×D411yescharacteristic
354C2×C4⋊C4C2×D412yes—
355C2×C4⋊C4C2×D412yes—
356C2×C4⋊C4C2×D414yes—
357C2×C4⋊C4C2×D414yes—
358C2×C4⋊C4C2412yes—
359C2×C4⋊C4C2412yes—
360C2×C4⋊C4C2412yes—
361C2×C4⋊C4C2412yes—
362C2×C4⋊C4C2411yescharacteristic
363C2×C4⋊C4C2411yescharacteristic
364C2×C4⋊C4C2412yes—
365C2×C22⋊C4—21no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—22no—
370C2×C22⋊C4—22no—
371C2×C22⋊C4—22no—
372C2×C22⋊C4—24no—
373C2×C22⋊C4—24no—
374C2×C22⋊C4—24no—
375C2×C22⋊C4—24no—
376C2×C42—21no—
377C2×C42—22no—
378C2×C42—21no—
379C2×C42—22no—
380C2×C42—21no—
381C2×C42—44no—
382C2×C42C2×D411yescharacteristic
383C2×C42C2×D412yes—
384C2×C42C2×D411yescharacteristic
385C2×C42C2412yes—
386C2×C42C2412yes—
387C2×C42C2412yes—
388C2×C42C2411yescharacteristic
389#107—24no—
390#107—24no—
391#107—24no—
392#106—24no—
393#106—24no—
394#106—24no—
395#106—44no—
396#106C2×D414yes—
397C4⋊C8—24no—
398C4⋊C8—24no—
399C4⋊C8—22no—
400C4⋊C8—22no—
401#103—24no—
402#103—24no—
403#103—24no—
404#103C2×D414yes—
405#102—24no—
406#102—24no—
407#102—24no—
408#102—24no—
409#102—24no—
410#102—28no—
411#102—44no—
412#102C2×D414yes—
413C4×C8—24no—
414C4×C8—24no—
415C4×C8—22no—
416C4×C8C2×D414yes—
417#95—22no—
418#95—22no—
419#95—21no—
420#95—22no—
421#95—22no—
422#95—21no—
423#95—24no—
424#95—24no—
425#95—24no—
426#95C2×D414yes—
427#95C2×D414yes—
428C24—21no—
429C24—22no—
430C2×Q8—22no—
431C2×Q8—22no—
432C2×Q8—22no—
433C2×Q8—22no—
434C2×Q8—22no—
435C2×Q8—22no—
436C2×Q8—21no—
437C2×Q8—22no—
438C2×Q8—22no—
439C2×Q8—42no—
440C2×Q8—42no—
441C2×Q8—42no—
442C2×Q8—42no—
443C2×Q8—42no—
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—44no—
475C22×C4—42no—
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—44no—
527C4⋊C4—48no—
528C4⋊C4—42no—
529C4⋊C4—42no—
530C4⋊C4—42no—
531C4⋊C4—82no—
532C4⋊C4—84no—
533C4⋊C4—84no—
534C4⋊C4—84no—
535C4⋊C4—84no—
536C4⋊C4C22×D412yes—
537C4⋊C4C22×D412yes—
538C4⋊C42−1+414yes—
539C22⋊C4—28no—
540C22⋊C4—24no—
541C22⋊C4—44no—
542C22⋊C4—44no—
543C22⋊C4—44no—
544C22⋊C4—44no—
545C22⋊C4—44no—
546C22⋊C4—48no—
547C22⋊C4—84no—
548C22⋊C4—84no—
549C22⋊C4—84no—
550C42—24no—
551C42—22no—
552C42—22no—
553C42—22no—
554C42—22no—
555C42—22no—
556C42—22no—
557C42—22no—
558C42—42no—
559C42—44no—
560C42—44no—
561C42—42no—
562C42—42no—
563C42#13514yes—
564C42C22×D412yes—
565C42H(5,𝔽2)12yes—
566C23—22no—
567C23—24no—
568C23—42no—
569C23—41no—
570C23—41no—
571C23—44no—
572C23—42no—
573C23—44no—
574C23—42no—
575C23—42no—
576C23C23×D411yescharacteristic
577Q8—22no—
578Q8—22no—
579Q8—44no—
580Q8—44no—
581Q8—42no—
582Q8—84no—
583Q8—162no—
584Q8—162no—
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—24no—
598C2×C4—24no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—24no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—416no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—41no—
618C2×C4—42no—
619C2×C4—41no—
620C2×C4—48no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—164no—
632C2×C4—164no—
633C2×C4—164no—
634C2×C4D4212yes—
635C2×C4D4212yes—
636C2×C4C2×#13514yes—
637C2×C4C2×#13512yes—
638C2×C4#57712yes—
639C2×C4C23×D411yescharacteristic
640C2×C4C2×H(5,𝔽2)11yescharacteristic
641C2×C4C2×2−1+412yes—
642C2×C4C2×2−1+412yes—
643C8—22no—
644C8—22no—
645C8—42no—
646C8—42no—
647C8—42no—
648C22—24no—
649C22—42no—
650C22—44no—
651C22—42no—
652C22—44no—
653C22—82no—
654C22—82no—
655C22—84no—
656C22—82no—
657C22—84no—
658C22C2×D4211yescharacteristic
659C22#348612yes—
660C22C22×#13511yescharacteristic
661C22C2×#57711yescharacteristic
662C22#358512yes—
663C4—24no—
664C4—22no—
665C4—42no—
666C4—42no—
667C4—42no—
668C4—42no—
669C4—44no—
670C4—42no—
671C4—42no—
672C4—42no—
673C4—42no—
674C4—82no—
675C4—82no—
676C4—82no—
677C4—82no—
678C4#328414yes—
679C4C2×#53611yescharacteristic
680C4#349911yescharacteristic
681C4#355212yes—
682C2—44no—
683C2—162no—
684C2#3767511yescharacteristic
685C2C2×#328411yescharacteristic
686C2#6108211yescharacteristic
687C2#6181912yes—
688C2#6186012yes—
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×#3381C2semidirect
3. #39746C2semidirect
4. #36180C2semidirect
6. #30951C2semidirect
7. #30810C2semidirect
8. #30277C2semidirect
9. #29869C2semidirect
10. #28677C2semidirect
11. #28640C2semidirect
12. #26382C2semidirect
13. #26172C2semidirect
15. #25836C2semidirect
16. #25835C2semidirect
17. #25648C2semidirect
18. #22238C2semidirect
19. #21117C2semidirect

Maximal subgroups

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

subgroupindex
C2×#33812
C2×#33612
#397462
#361802
#330242
#309512
#308102
#302772
#298692
#286772
#286402
#263822
#261722
#259302
#258362
#258352
#256482
#222382
#211172