Finite Group Databasebeta

← #7503478

Subgroups — #7503478

Order 512.

1992 conjugacy classes of subgroups.

The subgroup lattice has 722 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3096C211yesmaximal, characteristic
2C22×#478C211yesmaximal, characteristic, complemented
3#35639C211yesmaximal, characteristic, complemented
4#35152C211yesmaximal, characteristic
5#31106C214yesmaximal, complemented
6#31102C214yesmaximal, complemented
7#31092C214yesmaximal, complemented
8#31088C214yesmaximal, complemented
9#23666C211yesmaximal, characteristic, complemented
10#23660C211yesmaximal, characteristic, complemented
11#23640C211yesmaximal, characteristic, complemented
12#23632C211yesmaximal, characteristic, complemented
13#23553C211yesmaximal, characteristic, complemented
14#23483C211yesmaximal, characteristic
15#21089C211yesmaximal, characteristic, complemented
16#21052C211yesmaximal, characteristic, complemented
17#20873C211yesmaximal, characteristic, complemented
18#20847C211yesmaximal, characteristic, complemented
19#20802C211yesmaximal, characteristic, complemented
20C2×#535C2211yescharacteristic
21C22×#125C2211yescharacteristic
22C2×#520C2211yescharacteristic
23C22×C4×D4C2211yescharacteristic
24#3096—28no—
25C2×#481C2212yes—
26C2×#482C2212yes—
27C2×#482C2212yes—
28C2×#480C2212yes—
29C2×#478—28no—
30C2×#478C2214yes—
31C2×#478C2214yes—
32C2×#428C2211yescharacteristic
33C2×#428C2211yescharacteristic
34C2×#428C2211yescharacteristic
35C2×#427C2211yescharacteristic
36C22×#107C2211yescharacteristic
37C22×#106C2211yescharacteristic
38C22×C4⋊C8C2211yescharacteristic
39C2×#418C2211yescharacteristic
40C2×#418C2211yescharacteristic
41C22×#102C2211yescharacteristic
42C22×#102C2211yescharacteristic
43C2×#408C2211yescharacteristic
44C2×#408C2211yescharacteristic
45C2×#405C2211yescharacteristic
46#2849—22no—
47#2622—22no—
48#2579—22no—
49#2487—22no—
50#2454—22no—
51#2453—22no—
52#2447—22no—
53#2445—22no—
54#2442C2214yescomplemented
55#2441C2214yescomplemented
56#2422C2214yes—
57#2420C2214yes—
58#2317C2211yescharacteristic
59#2317C2211yescharacteristic
60#2316C2211yescharacteristic
61#2311C2211yescharacteristic
62#2311C2211yescharacteristic
63#2309C2211yescharacteristic
64#2297C2211yescharacteristic
65#2296C2211yescharacteristic
66#2296C2211yescharacteristic
67#2088C2212yescomplemented
68#2088C2212yescomplemented
69#2087C2212yescomplemented
70#2086C2212yescomplemented
71#2085C2214yescomplemented
72#2083C2214yescomplemented
73#2077C2214yescomplemented
74#2077C2214yescomplemented
75#2073C2214yes—
76#2071C2214yes—
77#2059C2214yescomplemented
78#2059C2214yescomplemented
79#2034C2214yes—
80#2033C2214yes—
81#1936C2214yes—
82#1934C2214yes—
83#1924C2214yescomplemented
84#1923C2214yescomplemented
85#1864C2214yes—
86#1862C2214yes—
87#1854C2214yescomplemented
88#1852C2214yescomplemented
89#1847C2212yescomplemented
90#1847C2212yescomplemented
91#1846C2212yescomplemented
92#1844C2212yescomplemented
93#1736C2211yescharacteristic
94#1736C2211yescharacteristic
95#1736C2211yescharacteristic
96#1735C2211yescharacteristic
97#1735C2211yescharacteristic
98#1735C2211yescharacteristic
99#1729C2211yescharacteristic
100#1728C2211yescharacteristic
101C22×#41C2311yescharacteristic
102C23×D4C2311yescharacteristic
103C24×C4—21no—
104C22×M16C2311yescharacteristic
105C22×M16C2311yescharacteristic
106C23×C8C2311yescharacteristic
107C23×C8C2311yescharacteristic
108#535—24no—
109#535—42no—
110C2×#127—21no—
111C2×#125—24no—
112C2×#125—22no—
113C2×#125—42no—
114C2×#125C2314yes—
115C2×#124—21no—
116C2×#123—24no—
117C2×#121C2312yes—
118C2×#121C2312yes—
119C2×#120—22no—
120#520—24no—
121#520—42no—
122C2×C4×Q8—21no—
123C2×C4×D4—22no—
124C2×C4×D4—21no—
125C2×C4×D4—24no—
126C2×C4×D4—22no—
127C2×C4×D4—42no—
128C2×C4×D4C2314yes—
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×C22⋊C4—21no—
142C22×C42C2311yescharacteristic
143#482—28no—
144#482—28no—
145#481—28no—
146#480—28no—
147#478—216no—
148#478—216no—
149#428—24no—
150#428—42no—
151#428—42no—
152#427—24no—
153C2×#107—24no—
154C2×#107C2314yes—
155C2×#107C2312yes—
156C2×#107C2311yescharacteristic
157C2×#107C2311yescharacteristic
158C2×#107C2312yes—
159C2×#107C2312yes—
160C2×#106—24no—
161C2×#106C2312yes—
162C2×#106C2312yes—
163C2×#106C2312yes—
164C2×#106C2311yescharacteristic
165C2×#106C2311yescharacteristic
166C2×#106C2312yes—
167C2×#106C2312yes—
168C2×C4⋊C8—24no—
169C2×C4⋊C8C2312yes—
170C2×C4⋊C8C2312yes—
171#418—24no—
172#418—24no—
173C2×#102—24no—
174C2×#102D418yes—
175C2×#102C2314yes—
176C2×#102C2312yes—
177C2×#102C2312yes—
178C2×#102C2314yes—
179C2×#102C2312yes—
180C2×#102C2312yes—
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#397—24no—
193#397—24no—
194#396—24no—
195#394—24no—
196#394—24no—
197#391—21no—
198#391—21no—
199#391D412yes—
200#390—21no—
201#390—21no—
202#390—22no—
203#390D412yes—
204#390C2314yes—
205#389—22no—
206#389—22no—
207#389—22no—
208#389—24no—
209#389C2312yes—
210#389C2312yes—
211#389C2314yes—
212#389C2312yes—
213#389C2312yes—
214#387—24no—
215#386—22no—
216#386—22no—
217#386—24no—
218#386D412yes—
219#386D412yes—
220#385—21no—
221#385—21no—
222#385—21no—
223#385—21no—
224#385—22no—
225#385—22no—
226#385—24no—
227#385—24no—
228#384C2312yes—
229#384C2312yes—
230#382C2312yes—
231#382C2312yes—
232#381—24no—
233#380—22no—
234#380—24no—
235#380C2314yes—
236#380C2314yes—
237#379—22no—
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—
252C25—21no—
253C2×#41—22no—
254C2×#41—22no—
255C2×#41—22no—
256C2×#41—22no—
257C2×#41—22no—
258C2×#41—22no—
259C22×Q8C2×D411yescharacteristic
260C22×D4—22no—
261C22×D4—22no—
262C22×D4—21no—
263C22×D4—22no—
264C22×D4—22no—
265C22×D4—22no—
266C22×D4—22no—
267C22×D4—22no—
268C22×D4C2×D412yes—
269C22×D4C2×D411yescharacteristic
270C22×D4C2×D414yes—
271C22×D4C2412yes—
272C23×C4—21no—
273C23×C4—21no—
274C23×C4—21no—
275C23×C4—22no—
276C23×C4—24no—
277C23×C4—44no—
278C23×C4—84no—
279C23×C4C2×D411yescharacteristic
280C23×C4C2×D411yescharacteristic
281C23×C4C2412yes—
282C23×C4C2411yescharacteristic
283C2×M16—22no—
284C2×M16—22no—
285C2×M16—22no—
286C2×M16—22no—
287C2×M16—22no—
288C2×M16#4114yes—
289C22×C8—22no—
290C22×C8—22no—
291C22×C8—22no—
292C22×C8—22no—
293C22×C8—22no—
294C22×C8C2×D414yes—
295C22×C8C2412yes—
296C22×C8C2412yes—
297C22×C8C2411yescharacteristic
298C22×C8C2411yescharacteristic
299C22×C8C2412yes—
300#127—44no—
301#125—28no—
302#125—44no—
303#125—48no—
304#124—44no—
305#123—88no—
306#121—24no—
307#121—24no—
308#121—42no—
309#121—42no—
310#120—48no—
311C4×Q8—44no—
312C4×D4—28no—
313C4×D4—44no—
314C4×D4—48no—
315C4×D4—44no—
316C4×D4—48no—
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—22no—
326#117—22no—
327#117—24no—
328#117—42no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—24no—
336C2×C4⋊C4—24no—
337C2×C4⋊C4—24no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—24no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—24no—
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—24no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—22no—
370C2×C22⋊C4—24no—
371C2×C22⋊C4—24no—
372C2×C22⋊C4—24no—
373C2×C22⋊C4—24no—
374C2×C22⋊C4—44no—
375C2×C22⋊C4C2×D412yes—
376C2×C22⋊C4C2×D414yes—
377C2×C22⋊C4C2×D414yes—
378C2×C42—22no—
379C2×C42—22no—
380C2×C42—21no—
381C2×C42—22no—
382C2×C42—21no—
383C2×C42—21no—
384C2×C42—21no—
385C2×C42C2×D412yes—
386C2×C42C2412yes—
387C2×C42C2412yes—
388#107—22no—
389#107—22no—
390#107—28no—
391#107—24no—
392#107—42no—
393#107—42no—
394#106—24no—
395#106—24no—
396#106—22no—
397#106—22no—
398#106—24no—
399#106—42no—
400#106—42no—
401C4⋊C8—24no—
402C4⋊C8—24no—
403#102—28no—
404#102—24no—
405#102—24no—
406#102—28no—
407#102C2×D418yes—
408#102C2×D418yes—
409C22⋊C8—24no—
410C22⋊C8—24no—
411C22⋊C8—28no—
412#97—24no—
413C4×C8—24no—
414#95—22no—
415#95—22no—
416#95—22no—
417#95—22no—
418#95—24no—
419#95—24no—
420#95—24no—
421#95—24no—
422#95—24no—
423#95C2×D414yes—
424#95C2×D414yes—
425C24—22no—
426C24—24no—
427C24—42no—
428C24—42no—
429C24—82no—
430C24—82no—
431C24C22×D412yes—
432C24C22×D411yescharacteristic
433#41—44no—
434#41—44no—
435#41—44no—
436#41—44no—
437C2×Q8—22no—
438C2×Q8—22no—
439C2×Q8—22no—
440C2×D4—24no—
441C2×D4—24no—
442C2×D4—24no—
443C2×D4—28no—
444C2×D4—22no—
445C2×D4—24no—
446C2×D4—22no—
447C2×D4—28no—
448C2×D4—22no—
449C2×D4—22no—
450C2×D4—22no—
451C2×D4—22no—
452C2×D4—22no—
453C2×D4—22no—
454C2×D4—44no—
455C2×D4—44no—
456C2×D4—42no—
457C2×D4—44no—
458C2×D4—42no—
459C2×D4—44no—
460C2×D4—84no—
461C2×D4—84no—
462C2×D4C22×D414yes—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—24no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—42no—
482C22×C4—42no—
483C22×C4—48no—
484C22×C4—48no—
485C22×C4—44no—
486C22×C4—42no—
487C22×C4—42no—
488C22×C4—84no—
489C22×C4—82no—
490C22×C4—82no—
491C22×C4—82no—
492C22×C4—84no—
493C22×C4—816no—
494C22×C4C22×D412yes—
495C22×C4C22×D414yes—
496C22×C4C22×D412yes—
497C22×C4C22×D412yes—
498C22×C4C22×D411yescharacteristic
499C22×C4C22×D412yes—
500C22×C4C2×#4112yes—
501C22×C4H(5,𝔽2)12yes—
502C22×C42−1+412yes—
503C22×C42−1+412yes—
504C22×C42−1+412yes—
505C22×C4C2511yescharacteristic
506M16—28no—
507M16—44no—
508C2×C8—28no—
509C2×C8—21no—
510C2×C8—21no—
511C2×C8—22no—
512C2×C8—21no—
513C2×C8—21no—
514C2×C8—22no—
515C2×C8—22no—
516C2×C8—22no—
517C2×C8—22no—
518C2×C8—22no—
519C2×C8—22no—
520C2×C8—22no—
521C2×C8—44no—
522C2×C8C22×D412yes—
523C2×C8C22×D412yes—
524C2×C8C2×#4114yes—
525C4⋊C4—24no—
526C4⋊C4—24no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—21no—
532C4⋊C4—22no—
533C4⋊C4—21no—
534C4⋊C4—22no—
535C4⋊C4—21no—
536C4⋊C4—21no—
537C4⋊C4—22no—
538C4⋊C4—22no—
539C4⋊C4—21no—
540C4⋊C4—21no—
541C4⋊C4—22no—
542C4⋊C4—22no—
543C4⋊C4—22no—
544C4⋊C4—44no—
545C4⋊C4—48no—
546C4⋊C4—48no—
547C4⋊C4—44no—
548C4⋊C4—42no—
549C4⋊C4—42no—
550C4⋊C4—84no—
551C4⋊C4—82no—
552C4⋊C4—82no—
553C4⋊C4—84no—
554C4⋊C4—82no—
555C4⋊C4—84no—
556C4⋊C4C22×D414yes—
557C22⋊C4—24no—
558C22⋊C4—44no—
559C22⋊C4—44no—
560C22⋊C4—48no—
561C22⋊C4—44no—
562C22⋊C4—44no—
563C22⋊C4—44no—
564C22⋊C4—42no—
565C22⋊C4—42no—
566C22⋊C4—42no—
567C22⋊C4—42no—
568C22⋊C4—84no—
569C22⋊C4—84no—
570C22⋊C4—84no—
571C22⋊C4—84no—
572C42—24no—
573C42—22no—
574C42—22no—
575C42—22no—
576C42—22no—
577C42—44no—
578C42—42no—
579C42—42no—
580C42—42no—
581C42—42no—
582C23—22no—
583C23—22no—
584C23—24no—
585C23—24no—
586C23—22no—
587C23—24no—
588C23—22no—
589C23—22no—
590C23—44no—
591C23—44no—
592C23—42no—
593C23—48no—
594C23—42no—
595C23—84no—
596C23—84no—
597C23—88no—
598C23—88no—
599C23#53612yes—
600C23D4212yes—
601C23#54712yes—
602C23C23×D411yescharacteristic
603Q8—44no—
604D4—24no—
605D4—24no—
606D4—48no—
607D4—48no—
608D4—44no—
609D4—42no—
610D4—42no—
611D4—164no—
612D4—164no—
613C2×C4—22no—
614C2×C4—24no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—24no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—24no—
624C2×C4—24no—
625C2×C4—22no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—44no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—44no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—41no—
645C2×C4—41no—
646C2×C4—41no—
647C2×C4—41no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—84no—
652C2×C4—816no—
653C2×C4—84no—
654C2×C4—84no—
655C2×C4—164no—
656C2×C4#53614yes—
657C2×C4#53612yes—
658C2×C4#53914yes—
659C2×C4D4212yes—
660C2×C4#54712yes—
661C2×C4#54714yes—
662C2×C4#56314yes—
663C2×C4C23×D411yescharacteristic
664C2×C4C22×#4111yescharacteristic
665C2×C4C2×H(5,𝔽2)11yescharacteristic
666C2×C4C2×2−1+411yescharacteristic
667C2×C4C2×2−1+411yescharacteristic
668C2×C4C2×2−1+411yescharacteristic
669C8—24no—
670C8—22no—
671C8—22no—
672C8—42no—
673C8—42no—
674C22—24no—
675C22—22no—
676C22—22no—
677C22—24no—
678C22—44no—
679C22—42no—
680C22—44no—
681C22—44no—
682C22—42no—
683C22—42no—
684C22—44no—
685C22—44no—
686C22—816no—
687C22—84no—
688C22—84no—
689C22#268014yes—
690C22C2×#53611yescharacteristic
691C22C2×D4211yescharacteristic
692C22C2×#54711yescharacteristic
693C22#358412yes—
694C22#358512yes—
695C4—44no—
696C4—42no—
697C4—42no—
698C4—44no—
699C4—42no—
700C4—82no—
701C4—81no—
702C4—82no—
703C4—81no—
704C4—82no—
705C4—81no—
706C4—81no—
707C4—82no—
708C4#268014yes—
709C4C2×#53611yescharacteristic
710C4C2×#53911yescharacteristic
711C4C2×#54711yescharacteristic
712C4C2×#56311yescharacteristic
713C2—24no—
714C2—82no—
715C2—84no—
716C2C2×#268011yescharacteristic
717C2#6089012yes—
718C2#6105211yescharacteristic
719C2#6105811yescharacteristic
720C2#6128312yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#535C22
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×#478C2semidirect
3. #35639C2semidirect
5. #31106C2semidirect
6. #31102C2semidirect
7. #31092C2semidirect
8. #31088C2semidirect
9. #23666C2semidirect
10. #23660C2semidirect
11. #23640C2semidirect
12. #23632C2semidirect
13. #23553C2semidirect
15. #21089C2semidirect
16. #21052C2semidirect
17. #20873C2semidirect
18. #20847C2semidirect
19. #20802C2semidirect
54. #2442C22semidirect
55. #2441C22semidirect
67. #2088C22semidirect
68. #2088C22semidirect
69. #2087C22semidirect
70. #2086C22semidirect
71. #2085C22semidirect
72. #2083C22semidirect
73. #2077C22semidirect
74. #2077C22semidirect
77. #2059C22semidirect
78. #2059C22semidirect
83. #1924C22semidirect
84. #1923C22semidirect
87. #1854C22semidirect
88. #1852C22semidirect
89. #1847C22semidirect
90. #1847C22semidirect
91. #1846C22semidirect
92. #1844C22semidirect

Maximal subgroups

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

subgroupindex
C2×#30962
C22×#4782
#356392
#351522
#311062
#311022
#310922
#310882
#236662
#236602
#236402
#236322
#235532
#234832
#210892
#210522
#208732
#208472
#208022