Finite Group Databasebeta

← #6970958

Subgroups — #6970958

Order 512.

1688 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36276C211yesmaximal, characteristic
2#36134C211yesmaximal, characteristic
3#32774C212yesmaximal, complemented
4#31936C214yesmaximal, complemented
5#30990C214yesmaximal, complemented
6#30205C211yesmaximal, characteristic, complemented
7#29953C212yesmaximal, complemented
8#29664C212yesmaximal, complemented
9#29141C211yesmaximal, characteristic, complemented
10#28373C211yesmaximal, characteristic, complemented
11#24817C212yesmaximal, complemented
12#24732C212yesmaximal, complemented
13#24243C212yesmaximal
14#24122C212yesmaximal
15#23934C212yesmaximal, complemented
16#23801C211yesmaximal, characteristic, complemented
17#22008C211yesmaximal, characteristic
18C2×#554C2211yescharacteristic
19C2×D4×Q8C2211yescharacteristic
20C2×#533C2212yes—
21C2×#477C2212yes—
22C2×#462C2214yes—
23C2×#462C2212yes—
24C2×#451C2214yes—
25C2×C4×Q16C2212yes—
26C2×C4×Q16C2212yes—
27#2868C2211yescharacteristic
28#2862C2211yescharacteristic
29#2857C2212yes—
30#2857C2212yes—
31#2567C2211yescharacteristic
32#2552—21no—
33#2550—21no—
34#2550—21no—
35#2549—22no—
36#2549—22no—
37#2546C2211yescharacteristic
38#2542—21no—
39#2538C2212yes—
40#2502C2212yes—
41#2441—22no—
42#2434C2212yes—
43#2429—22no—
44#2423C2214yes—
45#2365—21no—
46#2354—21no—
47Q8×C4⋊C4C2212yes—
48#2349—21no—
49#2349—21no—
50C4×C4⋊Q8C2211yescharacteristic
51C4×C4⋊Q8C2211yescharacteristic
52C2×#391C2212yes—
53C42×D4C2211yescharacteristic
54#2095C2211yescharacteristic
55#2095C2212yes—
56#2095C2211yescharacteristic
57#2095C2212yes—
58#2083C2214yes—
59#2072C2214yes—
60#2068C2214yes—
61#2068C2214yes—
62#2043C2214yes—
63#2036C2214yes—
64#2036C2214yes—
65#2024C2212yes—
66#2024C2212yes—
67#2024C2212yes—
68#2024C2212yes—
69#2016C2212yes—
70#2016C2211yescharacteristic
71#2016C2212yes—
72#2016C2211yescharacteristic
73#2001C2214yes—
74#2001C2214yes—
75#1990C2212yes—
76#1969C2214yes—
77#1956C2212yes—
78#1956C2212yes—
79#1954C2212yes—
80#1939C2212yes—
81#1923C2214yes—
82#1878C2214yes—
83#1878C2214yes—
84#1872C2214yes—
85#1852C2214yes—
86#1834C2212yes—
87#1817C2212yes—
88#1803C2212yes—
89#1803C2212yes—
90#1803C2214yes—
91#1797C2212yes—
92#1787C2214yes—
93C4×C4⋊C8C2211yescharacteristic
94#1744C2211yescharacteristic
95C23×Q8C2312yes—
96C22×Q16C2312yes—
97C22×Q16C2314yes—
98C22×Q16C2312yes—
99#554—44no—
100D4×Q8—44no—
101#533—48no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8—21no—
105C2×C4⋊Q8C2311yescharacteristic
106C2×C4⋊Q8C2311yescharacteristic
107C2×C4⋊Q8C2311yescharacteristic
108C2×C4⋊Q8C2312yes—
109C2×C4⋊Q8C2311yescharacteristic
110C2×C4⋊Q8C2312yes—
111C2×#125—21no—
112C2×#125—22no—
113C2×C22⋊Q8—22no—
114C2×C22⋊Q8—22no—
115C2×C22⋊Q8—22no—
116C2×C22⋊Q8—22no—
117C2×C22⋊Q8C2314yes—
118C2×C22⋊Q8C2312yes—
119C2×C22⋊Q8C2312yes—
120C2×C4×Q8—22no—
121C2×C4×Q8C2312yes—
122C2×C4×Q8C2311yescharacteristic
123C2×C4×Q8C2311yescharacteristic
124C2×C4×Q8C2314yes—
125C2×C4×D4—21no—
126C2×C4×D4—21no—
127C2×C4×D4D411yescharacteristic
128C2×C4×D4D411yescharacteristic
129C2×C4×D4C2311yescharacteristic
130C2×#117—22no—
131C22×C4⋊C4C2312yes—
132C22×C42C2312yes—
133#477—44no—
134#462—44no—
135#462—48no—
136#451—48no—
137C4×Q16—44no—
138C4×Q16—44no—
139C2×#107C2314yes—
140C2×#107C2312yes—
141C2×#107C2312yes—
142C2×C4⋊C8C2312yes—
143C2×C4⋊C8C2314yes—
144C2×#103C2314yes—
145C2×#103C2312yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2312yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#103C2312yes—
153C2×#103C2312yes—
154C2×C22⋊C8C2314yes—
155C2×C22⋊C8C2314yes—
156C2×C4×C8C2312yes—
157C2×C4×C8C2312yes—
158#399—24no—
159#398—24no—
160#398—22no—
161#398—22no—
162#398—22no—
163#396—24no—
164#395—22no—
165#395—22no—
166#395—22no—
167#391—22no—
168#391—21no—
169#391—21no—
170#391—22no—
171#391—22no—
172#391C2311yescharacteristic
173#391C2312yes—
174#391C2311yescharacteristic
175#391C2312yes—
176#391C2312yes—
177#391C2312yes—
178#391C2312yes—
179#390—22no—
180#389—21no—
181#389—21no—
182#389—21no—
183#389—21no—
184#389—22no—
185#389—21no—
186#389—21no—
187#389—21no—
188#389—21no—
189#389—22no—
190#389—22no—
191#389—22no—
192#389—24no—
193#389C2314yes—
194#388—24no—
195#387—24no—
196#385—22no—
197#385—22no—
198#385—22no—
199#385—22no—
200#384—22no—
201#384—21no—
202#384—21no—
203#384C2311yescharacteristic
204#384C2311yescharacteristic
205#384C2312yes—
206#384C2312yes—
207#384C2312yes—
208#382—21no—
209#382—21no—
210#380—22no—
211#380—22no—
212#380—22no—
213#380C2314yes—
214C4×C4⋊C4—22no—
215C4×C4⋊C4—21no—
216C4×C4⋊C4—21no—
217C4×C4⋊C4—21no—
218C4×C4⋊C4—21no—
219C4×C4⋊C4D411yescharacteristic
220C4×C4⋊C4D411yescharacteristic
221C4×C4⋊C4C2311yescharacteristic
222C4×C22⋊C4D412yes—
223C4×C22⋊C4D412yes—
224C4×C22⋊C4C2312yes—
225C2×#95—22no—
226C2×#95—22no—
227C43C2311yescharacteristic
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#340C2314yes—
232#336C2311yescharacteristic
233#336C2312yes—
234#336C2312yes—
235#336C2311yescharacteristic
236#336C2314yes—
237C22×Q8—22no—
238C22×Q8—22no—
239C22×Q8—21no—
240C22×Q8—24no—
241C22×Q8—44no—
242C22×Q8—42no—
243C22×Q8—44no—
244C22×Q8C2411yescharacteristic
245C22×Q8C2412yes—
246C22×Q8C2412yes—
247C22×Q8C2411yescharacteristic
248C22×D4C2×D411yescharacteristic
249C23×C4—22no—
250C23×C4—22no—
251C23×C4C2×D412yes—
252C23×C4C2×D412yes—
253C23×C4C2412yes—
254C2×Q16—24no—
255C2×Q16—24no—
256C2×Q16—24no—
257C2×Q16—28no—
258C2×Q16—42no—
259C2×Q16—42no—
260C2×Q16—42no—
261C2×Q16—44no—
262C2×Q16—44no—
263C22×C8C2412yes—
264C22×C8C2412yes—
265C22×C8C2414yes—
266C4⋊Q8—24no—
267C4⋊Q8—22no—
268C4⋊Q8—24no—
269C4⋊Q8—24no—
270C4⋊Q8—44no—
271C4⋊Q8—41no—
272C4⋊Q8—44no—
273C4⋊Q8—44no—
274C4⋊Q8—44no—
275C4⋊Q8—82no—
276#125—48no—
277#125—44no—
278C22⋊Q8—28no—
279C22⋊Q8—48no—
280C22⋊Q8—48no—
281C22⋊Q8—44no—
282C22⋊Q8—44no—
283C22⋊Q8—44no—
284C22⋊Q8—48no—
285C22⋊Q8—84no—
286C4×Q8—28no—
287C4×Q8—24no—
288C4×Q8—24no—
289C4×Q8—24no—
290C4×Q8—24no—
291C4×Q8—44no—
292C4×Q8—48no—
293C4×D4—24no—
294C4×D4—24no—
295C4×D4—44no—
296C4×D4—44no—
297C4×D4Q1618yescomplemented
298#117—48no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—24no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—44no—
329C2×C4⋊C4C2×D411yescharacteristic
330C2×C4⋊C4C2×D411yescharacteristic
331C2×C4⋊C4C2×D411yescharacteristic
332C2×C4⋊C4C2411yescharacteristic
333C2×C4⋊C4C2412yes—
334C2×C4⋊C4C2411yescharacteristic
335C2×C4⋊C4C2414yes—
336C2×C4⋊C4C2412yes—
337C2×C22⋊C4—24no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—24no—
341C2×C22⋊C4C2×D412yes—
342C2×C22⋊C4C2×D412yes—
343C2×C22⋊C4C2×D412yes—
344C2×C42—24no—
345C2×C42—21no—
346C2×C42—21no—
347C2×C42—22no—
348C2×C42—44no—
349C2×C42C2×D412yes—
350C2×C42C2×D412yes—
351C2×C42C2×D411yescharacteristic
352C2×C42C2×D411yescharacteristic
353C2×C42C2×D411yescharacteristic
354C2×C42C2×D411yescharacteristic
355C2×C42C2412yes—
356C2×C42C2411yescharacteristic
357C2×C42C2411yescharacteristic
358C2×C42C2411yescharacteristic
359#107—42no—
360#107—42no—
361#107—44no—
362C4⋊C8—24no—
363C4⋊C8—44no—
364#103—42no—
365#103—42no—
366#103—42no—
367#103—42no—
368#103—44no—
369#103—44no—
370#103—44no—
371#103—44no—
372#103—44no—
373#103—44no—
374C22⋊C8—44no—
375C22⋊C8—44no—
376C4×C8—24no—
377C4×C8—42no—
378#95—21no—
379#95—21no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—21no—
387#95—21no—
388#95—24no—
389#95—22no—
390#95—22no—
391#95—24no—
392#95—22no—
393#95—22no—
394#95C2×D412yes—
395#95C2×D412yes—
396#95C2×D412yes—
397C24C22×D412yes—
398C2×Q8—22no—
399C2×Q8—21no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—24no—
406C2×Q8—22no—
407C2×Q8—24no—
408C2×Q8—22no—
409C2×Q8—21no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8—21no—
413C2×Q8—22no—
414C2×Q8—44no—
415C2×Q8—44no—
416C2×Q8—44no—
417C2×Q8—44no—
418C2×Q8—42no—
419C2×Q8—44no—
420C2×Q8—82no—
421C2×Q8—81no—
422C2×Q8—81no—
423C2×Q8—88no—
424C2×Q8—88no—
425C2×Q8H(5,𝔽2)14yes—
426C2×Q82−1+412yes—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4C2×Q1614yes—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—24no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4—82no—
452C22×C4—82no—
453C22×C4—82no—
454C22×C4C2×Q1618yes—
455C22×C4C22×D412yes—
456C22×C4C22×D412yes—
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D412yes—
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D411yescharacteristic
464C22×C4C2511yescharacteristic
465Q16—48no—
466Q16—48no—
467Q16—88no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—24no—
472C2×C8—24no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—24no—
476C2×C8—22no—
477C4⋊C4—24no—
478C4⋊C4—24no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—24no—
483C4⋊C4—24no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—24no—
487C4⋊C4—22no—
488C4⋊C4—48no—
489C4⋊C4—48no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—44no—
494C4⋊C4—44no—
495C4⋊C4—44no—
496C4⋊C4—44no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—44no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—44no—
505C4⋊C4—44no—
506C4⋊C4—81no—
507C4⋊C4—81no—
508C4⋊C4—82no—
509C4⋊C4—82no—
510C4⋊C4—84no—
511C4⋊C4—82no—
512C4⋊C4—82no—
513C4⋊C4—82no—
514C4⋊C4—82no—
515C4⋊C4C2×Q1614yes—
516C22⋊C4—24no—
517C22⋊C4—24no—
518C22⋊C4—48no—
519C22⋊C4—44no—
520C22⋊C4—44no—
521C22⋊C4—84no—
522C22⋊C4C2×Q1618yes—
523C42—24no—
524C42—22no—
525C42—21no—
526C42—21no—
527C42—42no—
528C42—42no—
529C42—41no—
530C42—41no—
531C42—48no—
532C42—42no—
533C42—42no—
534C42—44no—
535C42C2×Q1614yes—
536C42#13714yes—
537C42H(5,𝔽2)14yes—
538C23—22no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—22no—
543C23—22no—
544C23C22×Q1614yes—
545C23C23×D411yescharacteristic
546Q8—24no—
547Q8—24no—
548Q8—24no—
549Q8—42no—
550Q8—42no—
551Q8—42no—
552Q8—42no—
553Q8—44no—
554Q8—42no—
555Q8—44no—
556Q8—88no—
557Q8—88no—
558Q8—162no—
559D4—28no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—24no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—48no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—41no—
604C2×C4—42no—
605C2×C4—41no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—84no—
609C2×C4—88no—
610C2×C4—164no—
611C2×C4#53512yes—
612C2×C4#53512yes—
613C2×C4C22×Q1612yes—
614C2×C4C22×Q1612yes—
615C2×C4C22×Q1614yes—
616C2×C4C22×Q1612yes—
617C2×C4C2×#13712yes—
618C2×C4C2×#13712yes—
619C2×C4C2×#13712yes—
620C2×C4C2×H(5,𝔽2)12yes—
621C2×C4C2×H(5,𝔽2)12yes—
622C2×C4C2×2−1+411yescharacteristic
623C2×C4#58511yescharacteristic
624C8—42no—
625C8—42no—
626C8—44no—
627C22—24no—
628C22—24no—
629C22—24no—
630C22—42no—
631C22—42no—
632C22—42no—
633C22—42no—
634C22C2×#53511yescharacteristic
635C22#348411yescharacteristic
636C22#348911yescharacteristic
637C22C23×Q1611yescharacteristic
638C22C22×#13711yescharacteristic
639C22#358511yescharacteristic
640C22#358611yescharacteristic
641C4—22no—
642C4—24no—
643C4—22no—
644C4—22no—
645C4—22no—
646C4—22no—
647C4—42no—
648C4—42no—
649C4—42no—
650C4—44no—
651C4—84no—
652C4—82no—
653C4—81no—
654C4—81no—
655C4—82no—
656C4—82no—
657C4—81no—
658C4#320914yes—
659C4#353612yes—
660C4#354212yes—
661C2—44no—
662C2#4247811yescharacteristic
663C2C2×#320911yescharacteristic
664C2#6082911yescharacteristic
665C2#6156311yescharacteristic
666C2#6158811yescharacteristic
667C2#6210011yescharacteristic
668C2#6211311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #32774C2semidirect
4. #31936C2semidirect
5. #30990C2semidirect
6. #30205C2semidirect
7. #29953C2semidirect
8. #29664C2semidirect
9. #29141C2semidirect
10. #28373C2semidirect
11. #24817C2semidirect
12. #24732C2semidirect
15. #23934C2semidirect
16. #23801C2semidirect
297. C4×D4Q16semidirect

Maximal subgroups

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

subgroupindex
#362762
#361342
#327742
#319362
#309902
#302052
#299532
#296642
#291412
#283732
#248172
#247322
#242432
#241222
#239342
#238012
#220082