Finite Group Databasebeta

← #7503480

Subgroups — #7503480

Order 512.

1914 conjugacy classes of subgroups.

The subgroup lattice has 702 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3097C211yesmaximal, characteristic
2C22×#475C211yesmaximal, characteristic, complemented
3#35574C211yesmaximal, characteristic
4#35182C211yesmaximal, characteristic
5#31107C214yesmaximal, complemented
6#31103C214yesmaximal, complemented
7#31093C214yesmaximal, complemented
8#31089C214yesmaximal, complemented
9#23666C211yesmaximal, characteristic, complemented
10#23661C211yesmaximal, characteristic, complemented
11#23640C211yesmaximal, characteristic
12#23633C211yesmaximal, characteristic, complemented
13#23552C211yesmaximal, characteristic, complemented
14#23484C211yesmaximal, characteristic
15#21088C211yesmaximal, characteristic, complemented
16#21052C211yesmaximal, characteristic
17#20874C211yesmaximal, characteristic, complemented
18#20848C211yesmaximal, characteristic
19#20802C211yesmaximal, characteristic, complemented
20C2×#536C2211yescharacteristic
21C22×C4⋊Q8C2211yescharacteristic
22C2×#520C2211yescharacteristic
23C22×C4×Q8C2211yescharacteristic
24#3097—28no—
25C2×#484C2212yes—
26C2×#485C2212yes—
27C2×#485C2212yes—
28C2×#483C2212yes—
29C2×#475—28no—
30C2×#475C2214yes—
31C2×#475C2214yes—
32C2×#428C2211yescharacteristic
33C2×#428C2211yescharacteristic
34C2×#428C2211yescharacteristic
35C2×#427C2211yescharacteristic
36C22×#106C2211yescharacteristic
37C22×#106C2211yescharacteristic
38C22×C4⋊C8C2211yescharacteristic
39C2×#419C2211yescharacteristic
40C2×#419C2211yescharacteristic
41C22×#103C2211yescharacteristic
42C22×#103C2211yescharacteristic
43C2×#408C2211yescharacteristic
44C2×#408C2211yescharacteristic
45C2×#405C2211yescharacteristic
46#2853—22no—
47#2623—22no—
48#2579—22no—
49#2452—22no—
50#2451—22no—
51#2448—22no—
52#2446—22no—
53#2440C2214yes—
54#2439C2214yes—
55#2423C2214yes—
56#2421C2214yes—
57#2409—22no—
58#2317C2211yescharacteristic
59#2317C2211yescharacteristic
60#2316C2211yescharacteristic
61#2311C2211yescharacteristic
62#2311C2211yescharacteristic
63#2310C2211yescharacteristic
64#2297C2211yescharacteristic
65#2296C2211yescharacteristic
66#2296C2211yescharacteristic
67#2088C2212yes—
68#2088C2212yes—
69#2087C2212yes—
70#2086C2212yes—
71#2085C2214yes—
72#2084C2214yes—
73#2077C2214yes—
74#2077C2214yes—
75#2074C2214yes—
76#2072C2214yes—
77#2058C2214yes—
78#2057C2214yes—
79#2036C2214yes—
80#2035C2214yes—
81#1935C2214yes—
82#1933C2214yes—
83#1924C2214yes—
84#1923C2214yes—
85#1865C2214yes—
86#1863C2214yes—
87#1854C2214yes—
88#1853C2214yes—
89#1847C2212yes—
90#1847C2212yes—
91#1846C2212yes—
92#1844C2212yes—
93#1736C2211yescharacteristic
94#1736C2211yescharacteristic
95#1736C2211yescharacteristic
96#1735C2211yescharacteristic
97#1735C2211yescharacteristic
98#1735C2211yescharacteristic
99#1729C2211yescharacteristic
100#1728C2211yescharacteristic
101C22×#41C2311yescharacteristic
102C23×Q8—21no—
103C23×Q8C2311yescharacteristic
104C22×M16C2311yescharacteristic
105C22×M16C2311yescharacteristic
106C23×C8C2311yescharacteristic
107C23×C8C2311yescharacteristic
108#536—24no—
109#536—42no—
110C2×C4⋊Q8—21no—
111C2×C4⋊Q8—24no—
112C2×C4⋊Q8—22no—
113C2×C4⋊Q8—42no—
114C2×C4⋊Q8C2314yes—
115C2×#124—21no—
116C2×#123—22no—
117C2×C22⋊Q8—24no—
118C2×C22⋊Q8C2312yes—
119C2×C22⋊Q8C2312yes—
120#520—24no—
121#520—42no—
122C2×C4×Q8—21no—
123C2×C4×Q8—22no—
124C2×C4×Q8—24no—
125C2×C4×Q8—42no—
126C2×C4×Q8C2314yes—
127C2×C4×D4—22no—
128C2×C4×D4—21no—
129C2×#117C2311yescharacteristic
130C2×#117C2311yescharacteristic
131C2×#117C2311yescharacteristic
132C2×#117C2311yescharacteristic
133C2×#117C2312yes—
134C2×#117C2311yescharacteristic
135C22×C4⋊C4—21no—
136C22×C4⋊C4—21no—
137C22×C4⋊C4C2311yescharacteristic
138C22×C4⋊C4C2311yescharacteristic
139C22×C4⋊C4C2312yes—
140C22×C4⋊C4C2311yescharacteristic
141C22×C42—21no—
142C22×C42C2311yescharacteristic
143#485—28no—
144#485—28no—
145#484—28no—
146#483—28no—
147#475—216no—
148#475—216no—
149#428—24no—
150#428—42no—
151#428—42no—
152#427—24no—
153C2×#107C2312yes—
154C2×#107C2311yescharacteristic
155C2×#107C2311yescharacteristic
156C2×#107C2312yes—
157C2×#107C2312yes—
158C2×#106—24no—
159C2×#106—24no—
160C2×#106C2312yes—
161C2×#106C2312yes—
162C2×#106C2314yes—
163C2×#106C2312yes—
164C2×#106C2311yescharacteristic
165C2×#106C2311yescharacteristic
166C2×#106C2312yes—
167C2×#106C2312yes—
168C2×C4⋊C8—24no—
169C2×C4⋊C8C2312yes—
170C2×C4⋊C8C2312yes—
171#419—24no—
172#419—24no—
173C2×#103—24no—
174C2×#103D418yes—
175C2×#103C2314yes—
176C2×#103C2312yes—
177C2×#103C2312yes—
178C2×#103C2314yes—
179C2×#103C2312yes—
180C2×#103C2312yes—
181#408—24no—
182#408—24no—
183C2×C22⋊C8C2312yes—
184C2×C22⋊C8C2312yes—
185C2×C22⋊C8C2314yes—
186#405—24no—
187C2×#97C2312yes—
188C2×C4×C8C2312yes—
189#400—24no—
190#399—24no—
191#398—24no—
192#398—24no—
193#397—24no—
194#396—24no—
195#395—24no—
196#393—24no—
197#391—21no—
198#391—21no—
199#391—22no—
200#391—24no—
201#391D412yes—
202#391C2314yes—
203#390—21no—
204#390—21no—
205#390D412yes—
206#389—22no—
207#389—22no—
208#389—22no—
209#389C2312yes—
210#389C2312yes—
211#389C2314yes—
212#389C2312yes—
213#389C2312yes—
214#388—24no—
215#388—24no—
216#387—24no—
217#387—24no—
218#387—24no—
219#386—22no—
220#386—22no—
221#386D412yes—
222#386D412yes—
223#385—21no—
224#385—21no—
225#385—21no—
226#385—21no—
227#385—22no—
228#385—22no—
229#385—24no—
230#384C2312yes—
231#384C2312yes—
232#382C2312yes—
233#382C2312yes—
234#381—22no—
235#380—22no—
236#380C2314yes—
237#380C2314yes—
238C4×C4⋊C4—22no—
239C4×C4⋊C4—22no—
240C4×C22⋊C4C2314yes—
241#340C2314yes—
242#340C2314yes—
243#340C2312yes—
244#340C2312yes—
245#340C2314yes—
246#340C2312yes—
247#340C2312yes—
248#340C2312yes—
249#340C2312yes—
250#336C2314yes—
251#336C2314yes—
252C2×#41—22no—
253C2×#41—22no—
254C2×#41—22no—
255C2×#41—22no—
256C2×#41—22no—
257C2×#41—22no—
258C22×Q8—21no—
259C22×Q8—24no—
260C22×Q8—22no—
261C22×Q8—22no—
262C22×Q8—22no—
263C22×Q8—22no—
264C22×Q8—22no—
265C22×Q8—44no—
266C22×Q8—84no—
267C22×Q8C2×D414yes—
268C22×Q8C2×D411yescharacteristic
269C22×Q8C2412yes—
270C22×D4C2×D412yes—
271C22×D4C2×D411yescharacteristic
272C23×C4—21no—
273C23×C4—21no—
274C23×C4—21no—
275C23×C4—21no—
276C23×C4—22no—
277C23×C4C2×D411yescharacteristic
278C23×C4C2×D411yescharacteristic
279C23×C4C2412yes—
280C23×C4C2411yescharacteristic
281C2×M16—22no—
282C2×M16—22no—
283C2×M16—22no—
284C2×M16—22no—
285C2×M16—22no—
286C2×M16#4114yes—
287C22×C8—22no—
288C22×C8—22no—
289C22×C8—22no—
290C22×C8—22no—
291C22×C8—22no—
292C22×C8C2×D414yes—
293C22×C8C2412yes—
294C22×C8C2412yes—
295C22×C8C2411yescharacteristic
296C22×C8C2411yescharacteristic
297C22×C8C2412yes—
298C4⋊Q8—28no—
299C4⋊Q8—44no—
300C4⋊Q8—48no—
301C4⋊Q8—44no—
302#124—44no—
303#123—48no—
304C22⋊Q8—24no—
305C22⋊Q8—24no—
306C22⋊Q8—42no—
307C22⋊Q8—42no—
308C22⋊Q8—88no—
309C4×Q8—28no—
310C4×Q8—44no—
311C4×Q8—48no—
312C4×Q8—44no—
313C4×D4—44no—
314C4×D4—48no—
315#117—22no—
316#117—22no—
317#117—22no—
318#117—22no—
319#117—22no—
320#117—22no—
321#117—22no—
322#117—22no—
323#117—22no—
324#117—22no—
325#117—24no—
326#117—42no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—24no—
334C2×C4⋊C4—24no—
335C2×C4⋊C4—24no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—22no—
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—22no—
347C2×C4⋊C4—24no—
348C2×C4⋊C4—42no—
349C2×C4⋊C4—44no—
350C2×C4⋊C4—44no—
351C2×C4⋊C4C2×D412yes—
352C2×C4⋊C4C2×D414yes—
353C2×C4⋊C4C2411yescharacteristic
354C2×C4⋊C4C2412yes—
355C2×C4⋊C4C2412yes—
356C2×C4⋊C4C2411yescharacteristic
357C2×C4⋊C4C2411yescharacteristic
358C2×C4⋊C4C2411yescharacteristic
359C2×C4⋊C4C2412yes—
360C2×C4⋊C4C2412yes—
361C2×C4⋊C4C2412yes—
362C2×C22⋊C4—21no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—21no—
365C2×C22⋊C4—21no—
366C2×C22⋊C4—21no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—24no—
370C2×C22⋊C4—24no—
371C2×C22⋊C4C2×D412yes—
372C2×C22⋊C4C2×D414yes—
373C2×C22⋊C4C2×D414yes—
374C2×C42—22no—
375C2×C42—22no—
376C2×C42—21no—
377C2×C42—22no—
378C2×C42—21no—
379C2×C42—21no—
380C2×C42—21no—
381C2×C42—24no—
382C2×C42—44no—
383C2×C42C2×D412yes—
384C2×C42C2412yes—
385C2×C42C2412yes—
386#107—22no—
387#107—22no—
388#107—24no—
389#107—42no—
390#107—42no—
391#106—24no—
392#106—24no—
393#106—22no—
394#106—22no—
395#106—28no—
396#106—24no—
397#106—42no—
398#106—42no—
399C4⋊C8—24no—
400C4⋊C8—24no—
401#103—28no—
402#103—24no—
403#103—24no—
404#103—28no—
405#103C2×D418yes—
406#103C2×D418yes—
407C22⋊C8—24no—
408C22⋊C8—24no—
409C22⋊C8—28no—
410#97—24no—
411C4×C8—24no—
412#95—22no—
413#95—22no—
414#95—22no—
415#95—22no—
416#95—24no—
417#95—24no—
418#95—24no—
419#95—24no—
420#95—24no—
421#95—24no—
422#95—24no—
423#95C2×D414yes—
424#95C2×D414yes—
425C24C22×D412yes—
426C24C22×D411yescharacteristic
427#41—44no—
428#41—44no—
429#41—44no—
430#41—44no—
431C2×Q8—22no—
432C2×Q8—24no—
433C2×Q8—22no—
434C2×Q8—22no—
435C2×Q8—22no—
436C2×Q8—28no—
437C2×Q8—28no—
438C2×Q8—22no—
439C2×Q8—22no—
440C2×Q8—22no—
441C2×Q8—22no—
442C2×Q8—48no—
443C2×Q8—42no—
444C2×Q8—44no—
445C2×Q8—42no—
446C2×Q8—44no—
447C2×Q8—48no—
448C2×Q8—84no—
449C2×Q8—816no—
450C2×Q8C22×D414yes—
451C2×D4—24no—
452C2×D4—24no—
453C2×D4—24no—
454C2×D4—22no—
455C2×D4—22no—
456C2×D4—22no—
457C22×C4—22no—
458C22×C4—24no—
459C22×C4—22no—
460C22×C4—24no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—24no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—21no—
472C22×C4—22no—
473C22×C4—21no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—42no—
479C22×C4—44no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4—82no—
485C22×C4—82no—
486C22×C4—82no—
487C22×C4—84no—
488C22×C4C22×D414yes—
489C22×C4C22×D412yes—
490C22×C4C22×D412yes—
491C22×C4C22×D412yes—
492C22×C4C22×D411yescharacteristic
493C22×C4C22×D412yes—
494C22×C4C2×#4112yes—
495C22×C4H(5,𝔽2)12yes—
496C22×C42−1+412yes—
497C22×C42−1+412yes—
498C22×C42−1+412yes—
499C22×C4C2511yescharacteristic
500M16—28no—
501M16—44no—
502C2×C8—28no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—22no—
506C2×C8—21no—
507C2×C8—21no—
508C2×C8—22no—
509C2×C8—22no—
510C2×C8—22no—
511C2×C8—22no—
512C2×C8—22no—
513C2×C8—22no—
514C2×C8—22no—
515C2×C8—44no—
516C2×C8C22×D412yes—
517C2×C8C22×D412yes—
518C2×C8C2×#4114yes—
519C4⋊C4—24no—
520C4⋊C4—24no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—22no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—22no—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—22no—
532C4⋊C4—22no—
533C4⋊C4—21no—
534C4⋊C4—21no—
535C4⋊C4—22no—
536C4⋊C4—22no—
537C4⋊C4—22no—
538C4⋊C4—44no—
539C4⋊C4—48no—
540C4⋊C4—48no—
541C4⋊C4—44no—
542C4⋊C4—44no—
543C4⋊C4—44no—
544C4⋊C4—42no—
545C4⋊C4—42no—
546C4⋊C4—44no—
547C4⋊C4—44no—
548C4⋊C4—84no—
549C4⋊C4—82no—
550C4⋊C4—84no—
551C4⋊C4—84no—
552C4⋊C4C22×D414yes—
553C22⋊C4—24no—
554C22⋊C4—28no—
555C22⋊C4—44no—
556C22⋊C4—44no—
557C22⋊C4—44no—
558C22⋊C4—44no—
559C22⋊C4—42no—
560C22⋊C4—42no—
561C22⋊C4—42no—
562C22⋊C4—42no—
563C22⋊C4—84no—
564C22⋊C4—84no—
565C42—24no—
566C42—22no—
567C42—22no—
568C42—22no—
569C42—22no—
570C42—48no—
571C42—44no—
572C42—42no—
573C42—42no—
574C42—42no—
575C42—42no—
576C23—22no—
577C23—22no—
578C23—24no—
579C23—22no—
580C23—22no—
581C23—22no—
582C23—24no—
583C23#53612yes—
584C23D4212yes—
585C23#54712yes—
586C23C23×D411yescharacteristic
587Q8—24no—
588Q8—24no—
589Q8—48no—
590Q8—44no—
591Q8—42no—
592Q8—42no—
593Q8—816no—
594D4—48no—
595D4—44no—
596C2×C4—28no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—24no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—24no—
605C2×C4—22no—
606C2×C4—24no—
607C2×C4—24no—
608C2×C4—22no—
609C2×C4—24no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—48no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—41no—
626C2×C4—42no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—44no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—84no—
638C2×C4—84no—
639C2×C4—84no—
640C2×C4—164no—
641C2×C4#53614yes—
642C2×C4#53612yes—
643C2×C4#53914yes—
644C2×C4D4212yes—
645C2×C4#54712yes—
646C2×C4#54714yes—
647C2×C4#56314yes—
648C2×C4#57714yes—
649C2×C4C23×D411yescharacteristic
650C2×C4C22×#4111yescharacteristic
651C2×C4C2×H(5,𝔽2)11yescharacteristic
652C2×C4C2×2−1+411yescharacteristic
653C2×C4C2×2−1+411yescharacteristic
654C2×C4C2×2−1+411yescharacteristic
655C8—24no—
656C8—22no—
657C8—22no—
658C8—42no—
659C8—42no—
660C22—24no—
661C22—22no—
662C22—22no—
663C22—24no—
664C22—44no—
665C22—42no—
666C22—42no—
667C22—42no—
668C22—44no—
669C22#268014yes—
670C22C2×#53611yescharacteristic
671C22C2×D4211yescharacteristic
672C22C2×#54711yescharacteristic
673C22C2×#57712yes—
674C22#358612yes—
675C4—24no—
676C4—44no—
677C4—44no—
678C4—42no—
679C4—44no—
680C4—42no—
681C4—82no—
682C4—81no—
683C4—82no—
684C4—81no—
685C4—81no—
686C4—81no—
687C4—82no—
688C4—82no—
689C4#268014yes—
690C4C2×#53611yescharacteristic
691C4C2×#53911yescharacteristic
692C4C2×#54711yescharacteristic
693C4C2×#56311yescharacteristic
694C2—24no—
695C2—82no—
696C2C2×#268011yescharacteristic
697C2#6088912yes—
698C2#6105011yescharacteristic
699C2#6106111yescharacteristic
700C2#6128112yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2297 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×#475C2semidirect
5. #31107C2semidirect
6. #31103C2semidirect
7. #31093C2semidirect
8. #31089C2semidirect
9. #23666C2semidirect
10. #23661C2semidirect
12. #23633C2semidirect
13. #23552C2semidirect
15. #21088C2semidirect
17. #20874C2semidirect
19. #20802C2semidirect

Maximal subgroups

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

subgroupindex
C2×#30972
C22×#4752
#355742
#351822
#311072
#311032
#310932
#310892
#236662
#236612
#236402
#236332
#235522
#234842
#210882
#210522
#208742
#208482
#208022