Finite Group Databasebeta

← #6439108

Subgroups — #6439108

Order 512.

2112 conjugacy classes of subgroups.

The subgroup lattice has 719 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#479C211yesmaximal, characteristic, complemented
2C2×#2947C211yesmaximal, characteristic
3#35639C211yesmaximal, characteristic
4#34249C211yesmaximal, characteristic, complemented
5#26399C214yesmaximal, complemented
6#26395C214yesmaximal, complemented
7#26383C214yesmaximal, complemented
8#26379C214yesmaximal, complemented
9#23737C211yesmaximal, characteristic
10#23661C211yesmaximal, characteristic, complemented
11#23609C211yesmaximal, characteristic
12#23553C211yesmaximal, characteristic, complemented
13#21156C211yesmaximal, characteristic, complemented
14#21024C211yesmaximal, characteristic, complemented
15#20930C211yesmaximal, characteristic
16#20926C211yesmaximal, characteristic
17#20882C211yesmaximal, characteristic
18#20880C211yesmaximal, characteristic, complemented
19#20877C211yesmaximal, characteristic, complemented
20C22×#137C2211yescharacteristic
21C22×#125C2211yescharacteristic
22C2×#520C2211yescharacteristic
23C2×#519C2211yescharacteristic
24C2×#519C2211yescharacteristic
25C22×C4×Q8C2211yescharacteristic
26C2×#479—28no—
27C2×#479C2214yes—
28C2×#479C2214yes—
29#2947—28no—
30C2×#441C2212yes—
31C2×#440C2212yes—
32C2×C4×Q16C2212yes—
33C2×C4×SD16C2212yes—
34C2×#428C2211yescharacteristic
35C2×#428C2211yescharacteristic
36C2×#427C2211yescharacteristic
37C22×#107C2211yescharacteristic
38C22×#106C2211yescharacteristic
39C22×C4⋊C8C2211yescharacteristic
40C2×#417C2211yescharacteristic
41C2×#419C2211yescharacteristic
42C2×#416C2211yescharacteristic
43C2×#416C2211yescharacteristic
44C22×#103C2211yescharacteristic
45C22×#103C2211yescharacteristic
46C2×#405C2211yescharacteristic
47#2579—22no—
48#2487—22no—
49#2454C2214yes—
50#2453C2214yes—
51#2442—22no—
52#2441—22no—
53#2351—22no—
54#2351C2214yes—
55Q8×C4⋊C4—22no—
56Q8×C4⋊C4C2214yes—
57#2317—22no—
58#2317C2211yescharacteristic
59#2317C2211yescharacteristic
60#2317C2211yescharacteristic
61#2316C2211yescharacteristic
62#2311C2211yescharacteristic
63#2297C2211yescharacteristic
64#2296—22no—
65#2296C2211yescharacteristic
66#2296C2211yescharacteristic
67#2097C2212yes—
68#2095C2212yes—
69#2093C2212yes—
70#2092C2212yes—
71#2085C2214yes—
72#2084C2214yes—
73#2068C2212yes—
74#2067C2212yes—
75#2066C2212yes—
76#2065C2212yes—
77#2059C2214yes—
78#2059C2214yes—
79#1944C2214yes—
80#1942C2214yes—
81#1917C2214yes—
82#1916C2214yes—
83#1880C2214yes—
84#1880C2214yes—
85#1878C2214yes—
86#1878C2214yes—
87#1865C2214yes—
88#1865C2214yes—
89#1865C2214yes—
90#1864C2214yes—
91#1863C2214yes—
92#1863C2214yes—
93#1736C2211yescharacteristic
94#1736C2211yescharacteristic
95#1736C2211yescharacteristic
96#1735C2211yescharacteristic
97#1729C2211yescharacteristic
98#1729C2211yescharacteristic
99#1729C2211yescharacteristic
100#1728C2211yescharacteristic
101C22×#41C2311yescharacteristic
102C23×Q8C2311yescharacteristic
103C2×#137—24no—
104C2×#137—24no—
105C2×#137Q818yes—
106C22×Q16C2312yes—
107C22×SD16C2312yes—
108C22×M16C2311yescharacteristic
109C22×M16C2311yescharacteristic
110C23×C8C2311yescharacteristic
111C23×C8C2311yescharacteristic
112C2×#125—24no—
113C2×#125—22no—
114C2×#125—42no—
115C2×#125C2314yes—
116#520—24no—
117#520—42no—
118#519—24no—
119#519—24no—
120#519—24no—
121#519—42no—
122C2×C4×Q8—22no—
123C2×C4×Q8—21no—
124C2×C4×Q8—21no—
125C2×C4×Q8—22no—
126C2×C4×Q8—24no—
127C2×C4×Q8—42no—
128C2×C4×Q8C2314yes—
129C2×C4×Q8C2314yes—
130C2×C4×Q8C2312yes—
131C2×C4×Q8C2312yes—
132C2×C4×Q8C2312yes—
133C2×C4×D4—22no—
134C2×C4×D4C2312yes—
135C2×#117—21no—
136C2×#117—22no—
137C2×#117—22no—
138C2×#117C2311yescharacteristic
139C2×#117C2311yescharacteristic
140C2×#117C2311yescharacteristic
141C2×#117C2311yescharacteristic
142C2×#117C2311yescharacteristic
143C22×C4⋊C4—21no—
144C22×C4⋊C4—21no—
145C22×C4⋊C4—21no—
146C22×C4⋊C4—21no—
147C22×C4⋊C4C2311yescharacteristic
148C22×C4⋊C4C2311yescharacteristic
149C22×C4⋊C4C2311yescharacteristic
150C22×C42—21no—
151C22×C42C2311yescharacteristic
152#479—216no—
153#479—216no—
154#441—28no—
155#440—28no—
156C4×Q16—28no—
157C4×SD16—28no—
158#428—24no—
159#428—42no—
160#427—24no—
161C2×#107—24no—
162C2×#107C2312yes—
163C2×#107C2312yes—
164C2×#107C2312yes—
165C2×#107C2312yes—
166C2×#107C2312yes—
167C2×#106—24no—
168C2×#106C2314yes—
169C2×#106C2312yes—
170C2×#106C2312yes—
171C2×#106C2312yes—
172C2×C4⋊C8—24no—
173C2×C4⋊C8C2312yes—
174C2×C4⋊C8C2312yes—
175#419—24no—
176#417—42no—
177#416—24no—
178#416—24no—
179C2×#103—24no—
180C2×#103D418yes—
181C2×#103C2314yes—
182C2×#103C2311yescharacteristic
183C2×#103C2312yes—
184C2×#103C2311yescharacteristic
185C2×#103C2314yes—
186C2×#103C2312yes—
187C2×#103C2312yes—
188C2×#103C2312yes—
189C2×#102C2311yescharacteristic
190C2×#102C2312yes—
191C2×#102C2311yescharacteristic
192C2×#102C2312yes—
193#405—24no—
194C2×#97C2312yes—
195C2×C4×C8C2312yes—
196#400—24no—
197#399—24no—
198#398—24no—
199#396—24no—
200#391C2312yes—
201#391C2312yes—
202#390C2312yes—
203#390C2312yes—
204#389—22no—
205#389—24no—
206#389—22no—
207#389—24no—
208#389—24no—
209#389—24no—
210#389—24no—
211#389D412yes—
212#389D412yes—
213#389C2312yes—
214#389C2312yes—
215#389C2314yes—
216#389C2314yes—
217#389C2314yes—
218#387—24no—
219#386—21no—
220#386—21no—
221#386—22no—
222#385—21no—
223#385—21no—
224#385—24no—
225#385—22no—
226#385—22no—
227#385—22no—
228#385D412yes—
229#385D412yes—
230#384—24no—
231#384—21no—
232#384—21no—
233#384C2312yes—
234#382—21no—
235#382—21no—
236#382—24no—
237#382C2312yes—
238#380—22no—
239#380—22no—
240#380—24no—
241C4×C4⋊C4—24no—
242C4×C4⋊C4—22no—
243C4×C4⋊C4—24no—
244C4×C4⋊C4C2314yes—
245C4×C22⋊C4—22no—
246#340C2314yes—
247#340C2314yes—
248#340C2312yes—
249#340C2312yes—
250#340C2314yes—
251#336C2314yes—
252#336C2314yes—
253#336C2312yes—
254#336C2312yes—
255#336C2314yes—
256C2×#41—22no—
257C2×#41—22no—
258C2×#41—22no—
259C2×#41—22no—
260C2×#41—22no—
261C2×#41C2×Q814yes—
262C22×Q8—21no—
263C22×Q8—22no—
264C22×Q8—22no—
265C22×Q8—22no—
266C22×Q8C2×D414yes—
267C22×Q8C2×Q814yes—
268C22×Q8#4114yes—
269C22×Q8C2411yescharacteristic
270C22×Q8C2412yes—
271C22×D4—21no—
272C22×D4C2411yescharacteristic
273C23×C4—21no—
274C23×C4—21no—
275C23×C4—21no—
276C23×C4—21no—
277C23×C4—21no—
278C23×C4—22no—
279C23×C4C2×D411yescharacteristic
280C23×C4C2×D411yescharacteristic
281C23×C4C2411yescharacteristic
282#137—232no—
283C2×Q16—24no—
284C2×Q16—24no—
285C2×Q16C2×Q818yes—
286C2×SD16—24no—
287C2×SD16—24no—
288C2×SD16C2×Q818yes—
289C2×M16—22no—
290C2×M16—22no—
291C2×M16—22no—
292C2×M16—22no—
293C2×M16—22no—
294C2×M16C2×Q814yes—
295C22×C8—22no—
296C22×C8—22no—
297C22×C8—22no—
298C22×C8—22no—
299C22×C8—22no—
300C22×C8C2×D414yes—
301C22×C8C2412yes—
302C22×C8C2412yes—
303C22×C8C2411yescharacteristic
304C22×C8C2411yescharacteristic
305C22×C8C2412yes—
306#125—28no—
307#125—44no—
308#125—48no—
309C4×Q8—216no—
310C4×Q8—28no—
311C4×Q8—28no—
312C4×Q8—28no—
313C4×Q8—24no—
314C4×Q8—24no—
315C4×Q8—24no—
316C4×Q8—44no—
317C4×Q8—42no—
318C4×Q8—42no—
319C4×Q8—42no—
320C4×Q8—48no—
321C4×Q8—44no—
322C4×Q8—44no—
323C4×D4—24no—
324C4×D4—42no—
325C4×D4—48no—
326#117—216no—
327#117—22no—
328#117—22no—
329#117—22no—
330#117—22no—
331#117—22no—
332#117—22no—
333#117—22no—
334#117—22no—
335#117—22no—
336#117—48no—
337#117—44no—
338#117#4114yes—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—24no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—24no—
348C2×C4⋊C4—24no—
349C2×C4⋊C4—24no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—24no—
356C2×C4⋊C4—22no—
357C2×C4⋊C4—24no—
358C2×C4⋊C4—24no—
359C2×C4⋊C4—24no—
360C2×C4⋊C4—44no—
361C2×C4⋊C4—44no—
362C2×C4⋊C4—44no—
363C2×C4⋊C4—44no—
364C2×C4⋊C4C2×D411yescharacteristic
365C2×C4⋊C4C2×D411yescharacteristic
366C2×C4⋊C4C2×D412yes—
367C2×C4⋊C4C2×D414yes—
368C2×C4⋊C4C2×D414yes—
369C2×C4⋊C4C2×D414yes—
370C2×C4⋊C4C2412yes—
371C2×C4⋊C4C2412yes—
372C2×C4⋊C4C2411yescharacteristic
373C2×C4⋊C4C2411yescharacteristic
374C2×C4⋊C4C2412yes—
375C2×C4⋊C4C2412yes—
376C2×C4⋊C4C2412yes—
377C2×C22⋊C4—21no—
378C2×C22⋊C4—21no—
379C2×C22⋊C4—21no—
380C2×C22⋊C4—22no—
381C2×C22⋊C4—22no—
382C2×C22⋊C4—22no—
383C2×C22⋊C4—24no—
384C2×C22⋊C4C2×D412yes—
385C2×C22⋊C4C2×D412yes—
386C2×C42—24no—
387C2×C42—21no—
388C2×C42—22no—
389C2×C42—22no—
390C2×C42—22no—
391C2×C42—22no—
392C2×C42—22no—
393C2×C42—21no—
394C2×C42—21no—
395C2×C42—24no—
396C2×C42—44no—
397C2×C42C2×D412yes—
398C2×C42C2412yes—
399C2×C42C2412yes—
400C2×C42C2412yes—
401#107—24no—
402#107—24no—
403#107—24no—
404#107—24no—
405#107—42no—
406#106—28no—
407#106—24no—
408#106—24no—
409#106—42no—
410C4⋊C8—24no—
411C4⋊C8—24no—
412#103—28no—
413#103—28no—
414#103—22no—
415#103—22no—
416#103—24no—
417#103—42no—
418#103C2×D418yes—
419#103C2×D418yes—
420#102—22no—
421#102—22no—
422#102—24no—
423#102—42no—
424#97—24no—
425C4×C8—24no—
426#95—22no—
427#95—22no—
428#95—22no—
429#95—22no—
430#95—24no—
431#95—24no—
432#95—24no—
433#95—24no—
434#95—24no—
435#95C2×D414yes—
436#95C2×D414yes—
437C24—21no—
438C24C22×D411yescharacteristic
439#41—28no—
440#41—28no—
441#41—44no—
442#41—44no—
443C2×Q8—24no—
444C2×Q8—28no—
445C2×Q8—24no—
446C2×Q8—28no—
447C2×Q8—24no—
448C2×Q8—28no—
449C2×Q8—22no—
450C2×Q8—22no—
451C2×Q8—22no—
452C2×Q8—21no—
453C2×Q8—21no—
454C2×Q8—21no—
455C2×Q8—21no—
456C2×Q8—21no—
457C2×Q8C22⋊Q8116yes—
458C2×Q8C22×D414yes—
459C2×Q8C22×Q812yes—
460C2×Q8C22×Q814yes—
461C2×Q8C2×#4114yes—
462C2×D4—24no—
463C2×D4—21no—
464C2×D4—21no—
465C2×D4—21no—
466C2×D4—21no—
467C2×D4—21no—
468C2×D4—42no—
469C2×D4—42no—
470C2×D4C22×Q812yes—
471C22×C4—24no—
472C22×C4—24no—
473C22×C4—24no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—21no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—24no—
481C22×C4—24no—
482C22×C4—21no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—21no—
487C22×C4—42no—
488C22×C4—42no—
489C22×C4—44no—
490C22×C4—42no—
491C22×C4—42no—
492C22×C4—42no—
493C22×C4—42no—
494C22×C4—84no—
495C22×C4—82no—
496C22×C4—82no—
497C22×C4—82no—
498C22×C4C22×D412yes—
499C22×C4C22×D414yes—
500C22×C4C22×D412yes—
501C22×C4C22×D412yes—
502C22×C4C22×D412yes—
503C22×C4C22×D412yes—
504C22×C4C22×D411yescharacteristic
505C22×C4C22×Q812yes—
506C22×C4C2×#4112yes—
507C22×C4C2×#4112yes—
508C22×C4H(5,𝔽2)12yes—
509C22×C42−1+412yes—
510C22×C4C2511yescharacteristic
511Q16—216no—
512SD16—216no—
513M16—28no—
514M16—44no—
515C2×C8—28no—
516C2×C8—22no—
517C2×C8—21no—
518C2×C8—21no—
519C2×C8—22no—
520C2×C8—21no—
521C2×C8—21no—
522C2×C8—22no—
523C2×C8—22no—
524C2×C8—22no—
525C2×C8—22no—
526C2×C8—22no—
527C2×C8—22no—
528C2×C8—44no—
529C2×C8C22×D412yes—
530C2×C8C22×D412yes—
531C2×C8C22×Q814yes—
532C4⋊C4—216no—
533C4⋊C4—24no—
534C4⋊C4—24no—
535C4⋊C4—22no—
536C4⋊C4—22no—
537C4⋊C4—22no—
538C4⋊C4—22no—
539C4⋊C4—21no—
540C4⋊C4—21no—
541C4⋊C4—22no—
542C4⋊C4—22no—
543C4⋊C4—22no—
544C4⋊C4—22no—
545C4⋊C4—22no—
546C4⋊C4—21no—
547C4⋊C4—22no—
548C4⋊C4—21no—
549C4⋊C4—44no—
550C4⋊C4—44no—
551C4⋊C4—44no—
552C4⋊C4—48no—
553C4⋊C4—48no—
554C4⋊C4—48no—
555C4⋊C4—44no—
556C4⋊C4—44no—
557C4⋊C4—44no—
558C4⋊C4—42no—
559C4⋊C4—42no—
560C4⋊C4—84no—
561C4⋊C4—82no—
562C4⋊C4—84no—
563C4⋊C4—84no—
564C4⋊C4—84no—
565C4⋊C4C22×D414yes—
566C4⋊C4C2×#4114yes—
567C22⋊C4—24no—
568C22⋊C4—24no—
569C22⋊C4—24no—
570C22⋊C4—44no—
571C22⋊C4—42no—
572C22⋊C4—42no—
573C22⋊C4—84no—
574C22⋊C4—82no—
575C22⋊C4—82no—
576C42—216no—
577C42—24no—
578C42—24no—
579C42—22no—
580C42—22no—
581C42—22no—
582C42—22no—
583C42—22no—
584C42—22no—
585C42—48no—
586C42—44no—
587C42—44no—
588C42—44no—
589C42—42no—
590C42—42no—
591C23—22no—
592C23—22no—
593C23—22no—
594C23—42no—
595C23—42no—
596C23—41no—
597C23—41no—
598C23#52512yes—
599C23D4212yes—
600C23D4×Q812yes—
601C23C23×D411yescharacteristic
602Q8—24no—
603Q8—24no—
604Q8—24no—
605Q8—216no—
606Q8—22no—
607Q8—22no—
608Q8—41no—
609Q8—41no—
610Q8C2×C22⋊Q818yes—
611D4—22no—
612D4—22no—
613D4—48no—
614D4—41no—
615D4—41no—
616C2×C4—24no—
617C2×C4—24no—
618C2×C4—24no—
619C2×C4—216no—
620C2×C4—24no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—24no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—24no—
627C2×C4—24no—
628C2×C4—24no—
629C2×C4—24no—
630C2×C4—22no—
631C2×C4—22no—
632C2×C4—22no—
633C2×C4—22no—
634C2×C4—24no—
635C2×C4—22no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—44no—
641C2×C4—41no—
642C2×C4—41no—
643C2×C4—48no—
644C2×C4—42no—
645C2×C4—44no—
646C2×C4—42no—
647C2×C4—44no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—41no—
652C2×C4—41no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—84no—
656C2×C4—84no—
657C2×C4—84no—
658C2×C4—164no—
659C2×C4C2×C22⋊Q814yes—
660C2×C4#52512yes—
661C2×C4#53914yes—
662C2×C4D4212yes—
663C2×C4#54714yes—
664C2×C4D4×Q812yes—
665C2×C4#55414yes—
666C2×C4C23×D411yescharacteristic
667C2×C4C23×Q811yescharacteristic
668C2×C4C22×#4111yescharacteristic
669C2×C4C22×#4111yescharacteristic
670C2×C4C2×H(5,𝔽2)11yescharacteristic
671C2×C4C2×2−1+411yescharacteristic
672C8—24no—
673C8—22no—
674C8—22no—
675C8—42no—
676C8—42no—
677C22—22no—
678C22—22no—
679C22—24no—
680C22—24no—
681C22—42no—
682C22—44no—
683C22—81no—
684C22—81no—
685C22—82no—
686C22#244914yes—
687C22C2×#52511yescharacteristic
688C22C2×D4211yescharacteristic
689C22C2×D4×Q811yescharacteristic
690C22#358412yes—
691C22#358612yes—
692C4—28no—
693C4—42no—
694C4—42no—
695C4—42no—
696C4—42no—
697C4—42no—
698C4—44no—
699C4—44no—
700C4—44no—
701C4—81no—
702C4—82no—
703C4—82no—
704C4—81no—
705C4—82no—
706C4#244914yes—
707C4C22×C22⋊Q811yescharacteristic
708C4C2×#53911yescharacteristic
709C4C2×#54711yescharacteristic
710C4C2×#55411yescharacteristic
711C2—24no—
712C2—82no—
713C2C2×#244911yescharacteristic
714C2#6059812yes—
715C2#6105411yescharacteristic
716C2#6106011yescharacteristic
717C2#6138612yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2296 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#479C2semidirect
4. #34249C2semidirect
5. #26399C2semidirect
6. #26395C2semidirect
7. #26383C2semidirect
8. #26379C2semidirect
10. #23661C2semidirect
12. #23553C2semidirect
13. #21156C2semidirect
14. #21024C2semidirect
18. #20880C2semidirect
19. #20877C2semidirect

Maximal subgroups

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

subgroupindex
C22×#4792
C2×#29472
#356392
#342492
#263992
#263952
#263832
#263792
#237372
#236612
#236092
#235532
#211562
#210242
#209302
#209262
#208822
#208802
#208772