Finite Group Databasebeta

← #6800019

Subgroups — #6800019

Order 512.

1767 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36564C211yesmaximal, characteristic
2C2×#2546C211yesmaximal, characteristic
3#32957C211yesmaximal, characteristic
4#32084C214yesmaximal, complemented
5#30880C214yesmaximal, complemented
6#29953C212yesmaximal
7#28401C212yesmaximal, complemented
8#27769C212yesmaximal, complemented
9#25856C212yesmaximal
10#25856C211yesmaximal, characteristic
11#25278C212yesmaximal, complemented
12#24732C212yesmaximal, complemented
13#24212C212yesmaximal
14#24212C211yesmaximal, characteristic
15#22625C211yesmaximal, characteristic
16#22044C212yesmaximal
17C4×#451C211yesmaximal, characteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×C4⋊Q8C2212yes—
20C22×C4×Q8C2211yescharacteristic
21C2×#487C2214yes—
22C2×#487C2212yes—
23C2×#477C2212yes—
24C2×#462C2214yes—
25C2×#462C2212yes—
26C2×#451C2212yes—
27C2×#451C2211yescharacteristic
28C2×C4×Q16C2212yes—
29#2857C2212yes—
30#2857C2211yescharacteristic
31#2586—22no—
32#2567C2211yescharacteristic
33#2551C2211yescharacteristic
34#2549—22no—
35#2546—22no—
36#2546—21no—
37#2546—21no—
38#2546C2214yes—
39#2546C2212yes—
40#2546C2212yes—
41#2507—21no—
42#2507C2212yes—
43#2451C2212yes—
44#2440—21no—
45C2×#398—22no—
46#2369—21no—
47#2369C2211yescharacteristic
48Q8×C4⋊C4C2211yescharacteristic
49#2345—21no—
50C4×C22⋊Q8C2212yes—
51C4×C22⋊Q8C2211yescharacteristic
52C2×#391C2212yes—
53C2×#389—21no—
54C2×#389—21no—
55C2×C4×C4⋊C4C2211yescharacteristic
56#2095C2214yes—
57#2068C2214yes—
58#2068C2214yes—
59#2043C2214yes—
60#2043C2214yes—
61#2043C2214yes—
62#2043C2212yes—
63#2043C2214yes—
64#2024C2214yes—
65#2016C2212yes—
66#2016C2214yes—
67#1981C2214yes—
68#1981C2212yes—
69#1969C2214yes—
70#1956C2214yes—
71#1956C2212yes—
72#1939C2212yes—
73#1933C2214yes—
74#1928C2214yes—
75#1916C2214yes—
76#1878C2214yes—
77#1878C2212yes—
78#1872C2211yescharacteristic
79#1863C2212yes—
80#1863C2214yes—
81#1852C2211yescharacteristic
82#1852C2212yes—
83#1803C2212yes—
84#1797C2212yes—
85#1797C2211yescharacteristic
86#1787C2212yes—
87#1787C2211yescharacteristic
88#1787C2212yes—
89#1787C2212yes—
90#1787C2212yes—
91C8×C4⋊C4C2212yes—
92C4×#103C2212yes—
93C4×C22⋊C8C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8C2311yescharacteristic
96C23×Q8C2312yes—
97C22×Q16C2312yes—
98C22×Q16C2314yes—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—22no—
103C2×C4⋊Q8—22no—
104C2×C4⋊Q8—44no—
105C2×C4⋊Q8—48no—
106C2×C4⋊Q8C2312yes—
107C2×C4⋊Q8C2314yes—
108C2×C4⋊Q8C2314yes—
109C2×C4⋊Q8C2312yes—
110C2×C4⋊Q8C2311yescharacteristic
111C2×C4⋊Q8C2312yes—
112C2×C22⋊Q8—22no—
113C2×C22⋊Q8—21no—
114C2×C22⋊Q8C2311yescharacteristic
115C2×C4×Q8—21no—
116C2×C4×Q8—21no—
117C2×C4×Q8—44no—
118C2×C4×Q8C2312yes—
119C2×C4×Q8C2311yescharacteristic
120C2×C4×Q8C2312yes—
121C2×C4×Q8C2312yes—
122C22×C4⋊C4—21no—
123C22×C4⋊C4—21no—
124C22×C4⋊C4—22no—
125C22×C4⋊C4—22no—
126C22×C4⋊C4—21no—
127C22×C4⋊C4D411yescharacteristic
128C22×C4⋊C4D411yescharacteristic
129C22×C4⋊C4C2311yescharacteristic
130C22×C42—21no—
131C22×C42C2311yescharacteristic
132C22×C42C2312yes—
133#487—44no—
134#487—48no—
135#477—44no—
136#462—48no—
137#462—44no—
138#451—24no—
139#451—28no—
140C4×Q16—44no—
141C2×#107C2314yes—
142C2×#107C2312yes—
143C2×C4⋊C8C2312yes—
144C2×C4⋊C8C2314yes—
145C2×#103C2314yes—
146C2×#103C2312yes—
147C2×#103C2312yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×#103C2314yes—
154C2×#103C2314yes—
155C2×#103C2314yes—
156C2×#103C2312yes—
157C2×#103C2312yes—
158C2×C22⋊C8C2311yescharacteristic
159C2×C22⋊C8C2312yes—
160C2×C22⋊C8C2311yescharacteristic
161C2×C4×C8C2312yes—
162C2×C4×C8C2314yes—
163#398—22no—
164#398—24no—
165#398—24no—
166#398—24no—
167#398—24no—
168#398—22no—
169#398—22no—
170#395—22no—
171#395—22no—
172#395—22no—
173#391—21no—
174#391—22no—
175#391—22no—
176#391—22no—
177#391—22no—
178#391—22no—
179#391C2312yes—
180#391C2314yes—
181#391C2314yes—
182#391C2311yescharacteristic
183#389—21no—
184#389—21no—
185#389—21no—
186#389—21no—
187#389—22no—
188#389—24no—
189#389—22no—
190#389—24no—
191#389—22no—
192#389—22no—
193#389—22no—
194#389—21no—
195#389—22no—
196#389—22no—
197#389C2312yes—
198#389C2312yes—
199#389C2312yes—
200#389C2311yescharacteristic
201#389C2312yes—
202#387—22no—
203#387—22no—
204#387—21no—
205#387—22no—
206#385—22no—
207#385—21no—
208#384—21no—
209#384C2312yes—
210#384C2311yescharacteristic
211#380—21no—
212#380—22no—
213#380C2312yes—
214#380C2311yescharacteristic
215C4×C4⋊C4—21no—
216C4×C4⋊C4—21no—
217C4×C4⋊C4—22no—
218C4×C4⋊C4—21no—
219C4×C4⋊C4D412yes—
220C4×C4⋊C4D411yescharacteristic
221C4×C4⋊C4D411yescharacteristic
222C4×C4⋊C4C2312yes—
223C4×C4⋊C4C2311yescharacteristic
224C4×C4⋊C4C2311yescharacteristic
225C4×C4⋊C4C2311yescharacteristic
226C4×C4⋊C4C2312yes—
227C4×C22⋊C4—21no—
228C4×C22⋊C4—22no—
229C2×#95—22no—
230C2×#95—22no—
231C2×#95D412yes—
232#340C2314yes—
233#340C2312yes—
234#340C2312yes—
235#336C2314yes—
236#336C2312yes—
237#336C2312yes—
238#336C2314yes—
239C22×Q8—24no—
240C22×Q8—21no—
241C22×Q8—22no—
242C22×Q8—22no—
243C22×Q8—21no—
244C22×Q8—22no—
245C22×Q8—24no—
246C22×Q8—22no—
247C22×Q8—24no—
248C22×Q8—42no—
249C22×Q8—44no—
250C22×Q8—44no—
251C22×Q8C2414yes—
252C22×Q8C2411yescharacteristic
253C22×Q8C2412yes—
254C23×C4—21no—
255C23×C4—22no—
256C23×C4—21no—
257C23×C4C2×D411yescharacteristic
258C23×C4C2×D412yes—
259C23×C4C2×D411yescharacteristic
260C23×C4C2×D412yes—
261C23×C4C2411yescharacteristic
262C2×Q16—24no—
263C2×Q16—24no—
264C2×Q16—28no—
265C2×Q16—42no—
266C2×Q16—44no—
267C2×Q16—44no—
268C22×C8C2412yes—
269C22×C8C2414yes—
270C22×C8C2412yes—
271C4⋊Q8—28no—
272C4⋊Q8—28no—
273C4⋊Q8—24no—
274C4⋊Q8—24no—
275C4⋊Q8—24no—
276C4⋊Q8—44no—
277C4⋊Q8—44no—
278C4⋊Q8—44no—
279C4⋊Q8—44no—
280C4⋊Q8—44no—
281C4⋊Q8—44no—
282C4⋊Q8—48no—
283C4⋊Q8—48no—
284C4⋊Q8—44no—
285C22⋊Q8—22no—
286C22⋊Q8—84no—
287C22⋊Q8—82no—
288C22⋊Q8#4114yes—
289C4×Q8—24no—
290C4×Q8—24no—
291C4×Q8—24no—
292C4×Q8—42no—
293C4×Q8—42no—
294C4×Q8—44no—
295C4×Q8—44no—
296C4×Q8—44no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—42no—
324C2×C4⋊C4—44no—
325C2×C4⋊C4—48no—
326C2×C4⋊C4—48no—
327C2×C4⋊C4—44no—
328C2×C4⋊C4—44no—
329C2×C4⋊C4Q1618yescomplemented
330C2×C4⋊C4C2×D411yescharacteristic
331C2×C4⋊C4C2×D412yes—
332C2×C4⋊C4C2×D411yescharacteristic
333C2×C4⋊C4C2×D412yes—
334C2×C4⋊C4C2×D411yescharacteristic
335C2×C4⋊C4C2×D411yescharacteristic
336C2×C4⋊C4C2412yes—
337C2×C4⋊C4C2412yes—
338C2×C4⋊C4C2411yescharacteristic
339C2×C4⋊C4C2411yescharacteristic
340C2×C4⋊C4C2412yes—
341C2×C4⋊C4C2411yescharacteristic
342C2×C22⋊C4—21no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—21no—
345C2×C42—24no—
346C2×C42—22no—
347C2×C42—22no—
348C2×C42—21no—
349C2×C42—21no—
350C2×C42—22no—
351C2×C42—22no—
352C2×C42—44no—
353C2×C42—44no—
354C2×C42C2×D411yescharacteristic
355C2×C42C2×D411yescharacteristic
356C2×C42C2×D412yes—
357C2×C42C2×D412yes—
358C2×C42C2411yescharacteristic
359C2×C42C2412yes—
360C2×C42C2412yes—
361C2×C42C2411yescharacteristic
362#107—42no—
363#107—44no—
364C4⋊C8—24no—
365C4⋊C8—44no—
366#103—28no—
367#103—24no—
368#103—24no—
369#103—28no—
370#103—42no—
371#103—42no—
372#103—44no—
373#103—44no—
374#103—44no—
375#103—44no—
376#103—44no—
377#103—44no—
378#103—44no—
379C22⋊C8—22no—
380C22⋊C8—24no—
381C22⋊C8—22no—
382C4×C8—24no—
383C4×C8—44no—
384#95—24no—
385#95—24no—
386#95—24no—
387#95—24no—
388#95—21no—
389#95—21no—
390#95—22no—
391#95—22no—
392#95—22no—
393#95—22no—
394#95—22no—
395#95—21no—
396#95—22no—
397#95—21no—
398#95—22no—
399#95—21no—
400#95—22no—
401#95C2×D412yes—
402#95C2×D412yes—
403#95C2×D412yes—
404#95C2×D412yes—
405C24C22×D411yescharacteristic
406C2×Q8—24no—
407C2×Q8—22no—
408C2×Q8—21no—
409C2×Q8—22no—
410C2×Q8—24no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8—22no—
415C2×Q8—24no—
416C2×Q8—24no—
417C2×Q8—24no—
418C2×Q8—24no—
419C2×Q8—22no—
420C2×Q8—24no—
421C2×Q8—48no—
422C2×Q8—48no—
423C2×Q8—42no—
424C2×Q8—44no—
425C2×Q8—44no—
426C2×Q8—42no—
427C2×Q8—42no—
428C2×Q8—44no—
429C2×Q8—44no—
430C2×Q8—81no—
431C2×Q8—82no—
432C2×Q8—82no—
433C2×Q8—81no—
434C2×Q8—88no—
435C2×Q8—88no—
436C2×Q8C2×#4112yes—
437C2×Q8H(5,𝔽2)14yes—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—41no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—42no—
466C22×C4—44no—
467C22×C4—42no—
468C22×C4C2×Q1614yes—
469C22×C4C2×Q1618yes—
470C22×C4C22×D412yes—
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D412yes—
475C22×C4C22×D412yes—
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D412yes—
478C22×C4C22×D411yescharacteristic
479C22×C4C2×#4112yes—
480C22×C42−1+412yes—
481C22×C4C2511yescharacteristic
482Q16—48no—
483Q16—88no—
484C2×C8—22no—
485C2×C8—24no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—24no—
490C2×C8—24no—
491C2×C8—22no—
492C2×C8—22no—
493C4⋊C4—24no—
494C4⋊C4—24no—
495C4⋊C4—22no—
496C4⋊C4—21no—
497C4⋊C4—24no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—24no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—48no—
504C4⋊C4—48no—
505C4⋊C4—48no—
506C4⋊C4—48no—
507C4⋊C4—42no—
508C4⋊C4—44no—
509C4⋊C4—44no—
510C4⋊C4—44no—
511C4⋊C4—44no—
512C4⋊C4—44no—
513C4⋊C4—44no—
514C4⋊C4—42no—
515C4⋊C4—42no—
516C4⋊C4—44no—
517C4⋊C4—42no—
518C4⋊C4—82no—
519C4⋊C4—81no—
520C4⋊C4—82no—
521C4⋊C4—82no—
522C4⋊C4—81no—
523C4⋊C4—82no—
524C4⋊C4—82no—
525C4⋊C4—82no—
526C4⋊C4—82no—
527C4⋊C4C2×Q1618yes—
528C4⋊C4C2×Q1614yes—
529C4⋊C4C2×Q1614yes—
530C4⋊C4#13714yes—
531C4⋊C4C2×#4112yes—
532C22⋊C4—24no—
533C22⋊C4—42no—
534C22⋊C4—44no—
535C42—22no—
536C42—22no—
537C42—24no—
538C42—24no—
539C42—22no—
540C42—42no—
541C42—42no—
542C42—44no—
543C42—44no—
544C42—42no—
545C42—42no—
546C42—82no—
547C42—82no—
548C42H(5,𝔽2)14yes—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23#54812yes—
554C23C22×Q1612yes—
555C23#57712yes—
556C23C23×D411yescharacteristic
557Q8—24no—
558Q8—24no—
559Q8—24no—
560Q8—48no—
561Q8—42no—
562Q8—42no—
563Q8—42no—
564Q8—44no—
565Q8—44no—
566Q8—44no—
567Q8—44no—
568Q8—88no—
569Q8—162no—
570Q8—164no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—28no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—24no—
592C2×C4—22no—
593C2×C4—21no—
594C2×C4—22no—
595C2×C4—48no—
596C2×C4—44no—
597C2×C4—48no—
598C2×C4—48no—
599C2×C4—41no—
600C2×C4—42no—
601C2×C4—41no—
602C2×C4—42no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—42no—
606C2×C4—41no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—41no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—41no—
615C2×C4—44no—
616C2×C4—41no—
617C2×C4—42no—
618C2×C4—41no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—41no—
622C2×C4—42no—
623C2×C4—84no—
624C2×C4—88no—
625C2×C4#53512yes—
626C2×C4#53512yes—
627C2×C4#54812yes—
628C2×C4C22×Q1612yes—
629C2×C4C22×Q1614yes—
630C2×C4C22×Q1614yes—
631C2×C4C22×Q1612yes—
632C2×C4C2×#13714yes—
633C2×C4C2×#13712yes—
634C2×C4C22×#4111yescharacteristic
635C2×C4C2×H(5,𝔽2)12yes—
636C2×C4C2×H(5,𝔽2)12yes—
637C2×C4C2×2−1+411yescharacteristic
638C8—42no—
639C8—44no—
640C8—42no—
641C22—22no—
642C22—22no—
643C22—22no—
644C22—22no—
645C22—22no—
646C22—22no—
647C22#336314yes—
648C22C2×#53511yescharacteristic
649C22C2×#54811yescharacteristic
650C22#348411yescharacteristic
651C22C23×Q1611yescharacteristic
652C22C22×#13711yescharacteristic
653C22C2×#57711yescharacteristic
654C22#358611yescharacteristic
655C4—22no—
656C4—24no—
657C4—24no—
658C4—22no—
659C4—24no—
660C4—22no—
661C4—42no—
662C4—41no—
663C4—42no—
664C4—44no—
665C4—42no—
666C4—81no—
667C4—82no—
668C4—82no—
669C4—82no—
670C4—81no—
671C4—84no—
672C4#320914yes—
673C4#351912yes—
674C4#353612yes—
675C2—24no—
676C2#3912011yescharacteristic
677C2C2×#320911yescharacteristic
678C2#6081311yescharacteristic
679C2C2×#336311yescharacteristic
680C2#6132211yescharacteristic
681C2#6156311yescharacteristic
682C2#6159311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #32084C2semidirect
5. #30880C2semidirect
7. #28401C2semidirect
8. #27769C2semidirect
11. #25278C2semidirect
12. #24732C2semidirect
329. C2×C4⋊C4Q16semidirect

Maximal subgroups

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

subgroupindex
#365642
C2×#25462
#329572
#320842
#308802
#299532
#284012
#277692
#258562
#252782
#247322
#242122
#226252
#220442
C4×#4512