Finite Group Databasebeta

← #6583849

Subgroups — #6583849

Order 512.

1828 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3369C211yesmaximal, characteristic
2C2×#3364C211yesmaximal, characteristic
3#36474C211yesmaximal, characteristic, complemented
4#36188C211yesmaximal, characteristic
5#33236C211yesmaximal, characteristic, complemented
6#33201C211yesmaximal, characteristic
7#33163C211yesmaximal, characteristic, complemented
8#28646C211yesmaximal, characteristic, complemented
9#28638C211yesmaximal, characteristic, complemented
10#27784C214yesmaximal
11#26473C214yesmaximal, complemented
12#25936C211yesmaximal, characteristic, complemented
13#25931C211yesmaximal, characteristic, complemented
14#25650C211yesmaximal, characteristic
15#25646C211yesmaximal, characteristic
16#24743C214yesmaximal
17#23014C214yesmaximal, complemented
18#21714C211yesmaximal, characteristic, complemented
19C4×#440C211yesmaximal, characteristic
20C2×Q82C2211yescharacteristic
21C2×#557C2211yescharacteristic
22C2×#556C2211yescharacteristic
23C2×#556C2211yescharacteristic
24#3369—28no—
25#3364—28no—
26C2×#489C2212yes—
27C2×#488C2212yes—
28C2×#487C2211yescharacteristic
29C2×#487C2211yescharacteristic
30C2×#487C2211yescharacteristic
31C2×#487C2211yescharacteristic
32C2×#479C2212yes—
33C2×#475C2212yes—
34C2×#476C2211yescharacteristic
35C2×#474C2211yescharacteristic
36C2×#464C2212yes—
37C2×#464C2212yes—
38C2×#462C2211yescharacteristic
39C2×#462C2211yescharacteristic
40C2×#462C2214yes—
41C2×#446C2211yescharacteristic
42C2×C8×Q8C2211yescharacteristic
43C2×#441C2212yes—
44C2×#440C2212yes—
45C2×C4×Q16C2212yes—
46C2×C4×SD16C2211yescharacteristic
47#2868C2211yescharacteristic
48#2862—21no—
49#2862C2211yescharacteristic
50#2857—21no—
51#2568C2211yescharacteristic
52#2568C2211yescharacteristic
53#2567—21no—
54#2567C2211yescharacteristic
55#2550—24no—
56#2550C2214yes—
57#2547C2214yes—
58#2546—21no—
59#2546C2211yescharacteristic
60#2499—21no—
61#2499—21no—
62#2496—24no—
63#2365C2214yes—
64#2351C2214yes—
65C4×C4⋊Q8—21no—
66C4×C4⋊Q8C2211yescharacteristic
67#2285—21no—
68C42×Q8C2211yescharacteristic
69C42×D4C2211yescharacteristic
70#2068C2214yes—
71#2068C2214yes—
72#2042C2214yes—
73#2042C2214yes—
74#2026C2214yes—
75#2024C2214yes—
76#1988C2211yescharacteristic
77#1987C2211yescharacteristic
78#1984C2211yescharacteristic
79#1984C2211yescharacteristic
80#1981C2211yescharacteristic
81#1981C2211yescharacteristic
82#1981C2214yes—
83#1960C2214yes—
84#1960C2214yes—
85#1949C2214yes—
86#1940C2214yes—
87#1935C2214yes—
88#1933C2214yes—
89#1878C2214yes—
90#1878C2214yes—
91#1865C2214yes—
92#1863C2214yes—
93#1842C2211yescharacteristic
94#1804C2214yes—
95#1776C2212yes—
96C4×#107C2211yescharacteristic
97#1771C2214yes—
98#1763C2212yes—
99#1762C2212yes—
100C4×#103C2211yescharacteristic
101C4×#102C2211yescharacteristic
102C4×#97C2211yescharacteristic
103C22×Q16C2312yes—
104C22×Q16C2312yes—
105C22×SD16C2311yescharacteristic
106Q82—24no—
107Q82—42no—
108#557—24no—
109#557—42no—
110#556—24no—
111#556—24no—
112#556—42no—
113#556—42no—
114C2×C4⋊Q8—21no—
115C2×C4⋊Q8—21no—
116C2×C4⋊Q8—21no—
117C2×C4⋊Q8—22no—
118C2×C4⋊Q8D411yescharacteristic
119C2×C4⋊Q8C2311yescharacteristic
120C2×C4⋊Q8C2311yescharacteristic
121C2×C4⋊Q8C2314yes—
122C2×C4⋊Q8C2311yescharacteristic
123C2×C4⋊Q8C2311yescharacteristic
124C2×C4⋊Q8C2312yes—
125C2×#126—22no—
126C2×#126—22no—
127C2×#125—22no—
128C2×#125C2312yes—
129C2×#124C2312yes—
130C2×#124C2312yes—
131C2×C22⋊Q8—21no—
132C2×C22⋊Q8—21no—
133C2×C4×Q8—21no—
134C2×C4×Q8—21no—
135C2×C4×Q8C2312yes—
136C2×C4×Q8C2312yes—
137C2×C4×Q8C2312yes—
138C2×C4×Q8C2312yes—
139C2×C4×Q8C2311yescharacteristic
140C2×C4×Q8C2312yes—
141C2×C4×Q8C2311yescharacteristic
142C2×C4×Q8C2311yescharacteristic
143C2×C4×D4C2312yes—
144C2×C4×D4C2311yescharacteristic
145C2×#117—21no—
146C2×#117—21no—
147C22×C42—21no—
148#489—28no—
149#488—28no—
150#487—24no—
151#487—24no—
152#487—42no—
153#487—42no—
154#479—28no—
155#476—24no—
156#475—28no—
157#474—24no—
158#464—28no—
159#464—28no—
160#462—24no—
161#462—24no—
162#462—216no—
163#446—24no—
164C8×Q8—24no—
165#441—28no—
166#440—28no—
167C4×Q16—28no—
168C4×SD16D418yes—
169C2×#107C2312yes—
170C2×#107C2312yes—
171C2×#106C2312yes—
172C2×#106C2311yescharacteristic
173C2×C4⋊C8C2312yes—
174C2×C4⋊C8C2312yes—
175C2×C4⋊C8C2314yes—
176C2×C4⋊C8C2311yescharacteristic
177C2×C4⋊C8C2311yescharacteristic
178C2×#103C2312yes—
179C2×#103C2312yes—
180C2×#103C2312yes—
181C2×#103C2312yes—
182C2×#103C2314yes—
183C2×#103C2314yes—
184C2×#103C2312yes—
185C2×#103C2312yes—
186C2×#103C2312yes—
187C2×#103C2314yes—
188C2×#103C2314yes—
189C2×#103C2311yescharacteristic
190C2×#102C2312yes—
191C2×#102C2312yes—
192C2×#102C2312yes—
193C2×#102C2311yescharacteristic
194C2×#97C2312yes—
195C2×#97C2312yes—
196C2×#97C2312yes—
197C2×C4×C8C2311yescharacteristic
198C2×C4×C8C2312yes—
199C2×C4×C8C2311yescharacteristic
200#400—22no—
201#400—22no—
202#398—24no—
203#398—22no—
204#397—24no—
205#397—24no—
206#395—24no—
207#395—22no—
208#391—21no—
209#391—21no—
210#391—21no—
211#391—21no—
212#391—21no—
213#391—21no—
214#391—22no—
215#391—24no—
216#391D411yescharacteristic
217#391D411yescharacteristic
218#391C2314yes—
219#391C2314yes—
220#389—24no—
221#389—24no—
222#389—24no—
223#389—24no—
224#389D414yes—
225#389C2314yes—
226#389C2314yes—
227#387—24no—
228#387—24no—
229#384—21no—
230#384—21no—
231#384—22no—
232#384—21no—
233#384D411yescharacteristic
234#382—21no—
235#382—21no—
236#382—24no—
237#382C2314yes—
238#380—24no—
239#380—24no—
240C4×C4⋊C4—24no—
241C4×C4⋊C4—21no—
242C4×C4⋊C4—22no—
243C4×C4⋊C4—24no—
244C4×C4⋊C4C2311yescharacteristic
245C4×C4⋊C4C2311yescharacteristic
246C4×C4⋊C4C2312yes—
247C4×C4⋊C4C2312yes—
248C4×C4⋊C4C2314yes—
249C4×C22⋊C4—22no—
250C4×C22⋊C4—21no—
251C43—21no—
252C43C2311yescharacteristic
253#340C2314yes—
254#340C2314yes—
255#340C2312yes—
256#340C2312yes—
257#340C2312yes—
258#336C2314yes—
259#336C2314yes—
260C22×Q8—21no—
261C22×Q8—22no—
262C22×Q8—21no—
263C22×Q8C2×D411yescharacteristic
264C22×Q8C2×D411yescharacteristic
265C22×Q8C2412yes—
266C22×Q8C2412yes—
267C22×Q8C2411yescharacteristic
268C22×D4C2411yescharacteristic
269C23×C4—21no—
270C23×C4—22no—
271C2×Q16—24no—
272C2×Q16—24no—
273C2×Q16—28no—
274C2×Q16—28no—
275C2×SD16—24no—
276C2×SD16C2×D414yes—
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—28no—
288C4⋊Q8—22no—
289C4⋊Q8—22no—
290C4⋊Q8—22no—
291C4⋊Q8—22no—
292C4⋊Q8—28no—
293C4⋊Q8—42no—
294C4⋊Q8—44no—
295C4⋊Q8—42no—
296C4⋊Q8—44no—
297C4⋊Q8—48no—
298C4⋊Q8—44no—
299#126—48no—
300#126—48no—
301#125—24no—
302#125—42no—
303#125—48no—
304#124—24no—
305#124—24no—
306#124—42no—
307#124—42no—
308C22⋊Q8—44no—
309C22⋊Q8—44no—
310C4×Q8—22no—
311C4×Q8—22no—
312C4×Q8—22no—
313C4×Q8—22no—
314C4×Q8—22no—
315C4×Q8—24no—
316C4×Q8—24no—
317C4×Q8—24no—
318C4×Q8—24no—
319C4×Q8—24no—
320C4×Q8—42no—
321C4×Q8—42no—
322C4×Q8—42no—
323C4×Q8—42no—
324C4×Q8—44no—
325C4×Q8—44no—
326C4×Q8C2×D414yes—
327C4×Q8#4118yes—
328C4×D4—24no—
329C4×D4—22no—
330C4×D4—24no—
331C4×D4C2×D414yes—
332#117—44no—
333#117—44no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—21no—
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—22no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—24no—
350C2×C4⋊C4—24no—
351C2×C4⋊C4—24no—
352C2×C4⋊C4—24no—
353C2×C4⋊C4C2×D411yescharacteristic
354C2×C4⋊C4C2×D411yescharacteristic
355C2×C4⋊C4C2×D412yes—
356C2×C4⋊C4C2×D412yes—
357C2×C4⋊C4C2×D414yes—
358C2×C4⋊C4C2×D414yes—
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×C42—21no—
372C2×C42—22no—
373C2×C42—22no—
374C2×C42—22no—
375C2×C42—22no—
376C2×C42—22no—
377C2×C42—21no—
378C2×C42—22no—
379C2×C42—21no—
380C2×C42—22no—
381C2×C42—44no—
382C2×C42C2×D411yescharacteristic
383C2×C42C2×D411yescharacteristic
384C2×C42C2×D412yes—
385C2×C42C2412yes—
386C2×C42C2412yes—
387C2×C42C2411yescharacteristic
388C2×C42C2412yes—
389#107—24no—
390#107—24no—
391#106—24no—
392#106C2×D414yes—
393C4⋊C8—24no—
394C4⋊C8—24no—
395C4⋊C8—22no—
396C4⋊C8—22no—
397C4⋊C8—28no—
398#103—24no—
399#103—24no—
400#103—24no—
401#103—24no—
402#103—24no—
403#103—24no—
404#103—24no—
405#103—28no—
406#103—28no—
407#103—44no—
408#103—44no—
409#103C2×D414yes—
410#102—24no—
411#102—24no—
412#102—24no—
413#102C2×D414yes—
414#97—24no—
415#97—24no—
416#97—24no—
417C4×C8—24no—
418C4×C8—22no—
419C4×C8C2×D414yes—
420#95—22no—
421#95—21no—
422#95—21no—
423#95—22no—
424#95—24no—
425#95—24no—
426#95—24no—
427#95—24no—
428#95—24no—
429#95—24no—
430#95—24no—
431#95—24no—
432#95C2×D414yes—
433#95C2×D414yes—
434C24—21no—
435C2×Q8—22no—
436C2×Q8—22no—
437C2×Q8—22no—
438C2×Q8—22no—
439C2×Q8—22no—
440C2×Q8—22no—
441C2×Q8—21no—
442C2×Q8—22no—
443C2×Q8—22no—
444C2×Q8—22no—
445C2×Q8—24no—
446C2×Q8—24no—
447C2×Q8—22no—
448C2×Q8—22no—
449C2×Q8—24no—
450C2×Q8—24no—
451C2×Q8—44no—
452C2×Q8—42no—
453C2×Q8—42no—
454C2×Q8—42no—
455C2×Q8—42no—
456C2×Q8—82no—
457C2×Q8—84no—
458C2×Q8C22×D412yes—
459C2×Q8C2×#4114yes—
460C2×D4—22no—
461C2×D4—21no—
462C2×D4—22no—
463C2×D4C22×D412yes—
464C22×C4—21no—
465C22×C4—22no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—24no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—44no—
475C22×C4—44no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—44no—
479C22×C4C22×D412yes—
480C22×C4C22×D412yes—
481C22×C4C22×D412yes—
482C22×C4C22×D412yes—
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D412yes—
485C22×C4C22×D412yes—
486C22×C4C22×D411yescharacteristic
487C22×C4C2511yescharacteristic
488Q16—48no—
489Q16—48no—
490SD16—28no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—24no—
497C2×C8—24no—
498C2×C8—22no—
499C2×C8—22no—
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—44no—
518C4⋊C4—44no—
519C4⋊C4—44no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C4⋊C4—44no—
523C4⋊C4—44no—
524C4⋊C4—44no—
525C4⋊C4—44no—
526C4⋊C4—48no—
527C4⋊C4—42no—
528C4⋊C4—42no—
529C4⋊C4—42no—
530C4⋊C4—42no—
531C4⋊C4—42no—
532C4⋊C4—42no—
533C4⋊C4—48no—
534C4⋊C4—48no—
535C4⋊C4—84no—
536C4⋊C4C22×D412yes—
537C4⋊C4C22×D412yes—
538C4⋊C4C2×#4114yes—
539C22⋊C4—24no—
540C22⋊C4—44no—
541C22⋊C4—44no—
542C22⋊C4—44no—
543C22⋊C4—44no—
544C22⋊C4—44no—
545C42—24no—
546C42—28no—
547C42—22no—
548C42—22no—
549C42—22no—
550C42—22no—
551C42—22no—
552C42—22no—
553C42—42no—
554C42—44no—
555C42—44no—
556C42—44no—
557C42—44no—
558C42—44no—
559C42—42no—
560C42—42no—
561C42—82no—
562C42C22×D412yes—
563C42C2×#4114yes—
564C42H(5,𝔽2)12yes—
565C422−1+414yes—
566C23—22no—
567C23—42no—
568C23—42no—
569C23—41no—
570C23—41no—
571C23C23×D411yescharacteristic
572Q8—22no—
573Q8—24no—
574Q8—44no—
575Q8—44no—
576Q8—42no—
577Q8—42no—
578Q8—42no—
579Q8—42no—
580Q8—44no—
581Q8—44no—
582Q8—84no—
583Q8—162no—
584Q8—164no—
585Q8#53214yes—
586D4—22no—
587D4—22no—
588D4—42no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—24no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—24no—
597C2×C4—24no—
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—22no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—41no—
612C2×C4—42no—
613C2×C4—41no—
614C2×C4—48no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—88no—
628C2×C4D4212yes—
629C2×C4D4212yes—
630C2×C4#54814yes—
631C2×C4#54814yes—
632C2×C4#57512yes—
633C2×C4#57812yes—
634C2×C4C23×D411yescharacteristic
635C2×C4C22×#4112yes—
636C2×C4C2×H(5,𝔽2)11yescharacteristic
637C2×C4C2×2−1+412yes—
638C8—22no—
639C8—22no—
640C8—42no—
641C8—42no—
642C8—42no—
643C22—42no—
644C22—42no—
645C22—42no—
646C22—82no—
647C22—82no—
648C22C2×D4211yescharacteristic
649C22C2×#54812yes—
650C22C2×#57511yescharacteristic
651C22C2×#57811yescharacteristic
652C22#358612yes—
653C4—24no—
654C4—24no—
655C4—22no—
656C4—24no—
657C4—42no—
658C4—42no—
659C4—42no—
660C4—44no—
661C4—42no—
662C4—42no—
663C4—42no—
664C4—82no—
665C4—82no—
666C4—82no—
667C4—82no—
668C4—82no—
669C4#326914yes—
670C4C2×#53211yescharacteristic
671C4#349911yescharacteristic
672C4#351112yes—
673C2—82no—
674C2#3612211yescharacteristic
675C2C2×#326911yescharacteristic
676C2#6108311yescharacteristic
677C2#6132312yes—
678C2#6134212yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C24
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
3. #36474C2semidirect
5. #33236C2semidirect
7. #33163C2semidirect
8. #28646C2semidirect
9. #28638C2semidirect
11. #26473C2semidirect
12. #25936C2semidirect
13. #25931C2semidirect
17. #23014C2semidirect
18. #21714C2semidirect

Maximal subgroups

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

subgroupindex
C2×#33692
C2×#33642
#364742
#361882
#332362
#332012
#331632
#286462
#286382
#277842
#264732
#259362
#259312
#256502
#256462
#247432
#230142
#217142
C4×#4402