Finite Group Databasebeta

← #6439107

Subgroups — #6439107

Order 512.

2186 conjugacy classes of subgroups.

The subgroup lattice has 725 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#477C211yesmaximal, characteristic, complemented
2C2×#2947C211yesmaximal, characteristic
3#35574C211yesmaximal, characteristic
4#34249C211yesmaximal, characteristic, complemented
5#26397C214yesmaximal, complemented
6#26393C214yesmaximal, complemented
7#26381C214yesmaximal, complemented
8#26377C214yesmaximal, complemented
9#23737C211yesmaximal, characteristic
10#23660C211yesmaximal, characteristic, complemented
11#23609C211yesmaximal, characteristic
12#23552C211yesmaximal, characteristic, complemented
13#21156C211yesmaximal, characteristic, complemented
14#21024C211yesmaximal, characteristic, complemented
15#20930C211yesmaximal, characteristic
16#20926C211yesmaximal, characteristic
17#20881C211yesmaximal, characteristic
18#20880C211yesmaximal, characteristic, complemented
19#20877C211yesmaximal, characteristic, complemented
20C22×#137C2211yescharacteristic
21C22×C4⋊Q8C2211yescharacteristic
22C2×#520C2211yescharacteristic
23C2×#519C2211yescharacteristic
24C2×#519C2211yescharacteristic
25C22×C4×Q8C2211yescharacteristic
26C2×#477—28no—
27C2×#477C2214yes—
28C2×#477C2214yes—
29#2947—28no—
30C2×#441C2212yes—
31C2×#440C2212yes—
32C2×C4×Q16C2212yes—
33C2×C4×SD16C2212yes—
34C2×#428C2211yescharacteristic
35C2×#428C2211yescharacteristic
36C2×#427C2211yescharacteristic
37C22×#107C2211yescharacteristic
38C22×#107C2211yescharacteristic
39C22×C4⋊C8C2211yescharacteristic
40C2×#417C2211yescharacteristic
41C2×#419C2211yescharacteristic
42C2×#416C2211yescharacteristic
43C2×#416C2211yescharacteristic
44C22×#103C2211yescharacteristic
45C22×#103C2211yescharacteristic
46C2×#405C2211yescharacteristic
47#2579—22no—
48#2452C2214yes—
49#2451C2214yes—
50#2440—22no—
51#2439—22no—
52#2409—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#2083C2214yes—
73#2068C2212yes—
74#2067C2212yes—
75#2066C2212yes—
76#2065C2212yes—
77#2058C2214yes—
78#2057C2214yes—
79#1944C2214yes—
80#1942C2214yes—
81#1917C2214yes—
82#1916C2214yes—
83#1880C2214yes—
84#1880C2214yes—
85#1878C2214yes—
86#1878C2214yes—
87#1865C2214yes—
88#1865C2214yes—
89#1863C2214yes—
90#1863C2214yes—
91#1863C2214yes—
92#1862C2214yes—
93#1736C2211yescharacteristic
94#1736C2211yescharacteristic
95#1736C2211yescharacteristic
96#1735C2211yescharacteristic
97#1729C2211yescharacteristic
98#1729C2211yescharacteristic
99#1729C2211yescharacteristic
100#1728C2211yescharacteristic
101C22×#41C2311yescharacteristic
102C23×Q8—21no—
103C23×Q8C2311yescharacteristic
104C2×#137—24no—
105C2×#137—24no—
106C2×#137Q818yescomplemented
107C22×Q16C2312yes—
108C22×SD16C2312yes—
109C22×M16C2311yescharacteristic
110C22×M16C2311yescharacteristic
111C23×C8C2311yescharacteristic
112C23×C8C2311yescharacteristic
113C2×C4⋊Q8—24no—
114C2×C4⋊Q8—22no—
115C2×C4⋊Q8—42no—
116C2×C4⋊Q8C2314yes—
117C2×C22⋊Q8—24no—
118#520—24no—
119#520—42no—
120#519—24no—
121#519—24no—
122#519—24no—
123#519—42no—
124C2×C4×Q8—22no—
125C2×C4×Q8—21no—
126C2×C4×Q8—21no—
127C2×C4×Q8—22no—
128C2×C4×Q8—24no—
129C2×C4×Q8—42no—
130C2×C4×Q8C2314yes—
131C2×C4×Q8C2314yes—
132C2×C4×Q8C2312yes—
133C2×C4×Q8C2312yes—
134C2×C4×Q8C2312yes—
135C2×C4×D4—22no—
136C2×C4×D4C2312yes—
137C2×#117—21no—
138C2×#117—22no—
139C2×#117—22no—
140C2×#117C2311yescharacteristic
141C2×#117C2311yescharacteristic
142C2×#117C2311yescharacteristic
143C2×#117C2311yescharacteristic
144C2×#117C2311yescharacteristic
145C22×C4⋊C4—21no—
146C22×C4⋊C4—21no—
147C22×C4⋊C4—21no—
148C22×C4⋊C4C2311yescharacteristic
149C22×C4⋊C4C2311yescharacteristic
150C22×C4⋊C4C2311yescharacteristic
151C22×C42—21no—
152C22×C42C2311yescharacteristic
153#477—216no—
154#477—216no—
155#441—28no—
156#440—28no—
157C4×Q16—28no—
158C4×SD16—28no—
159#428—24no—
160#428—42no—
161#427—24no—
162C2×#107—24no—
163C2×#107—24no—
164C2×#107C2314yes—
165C2×#107C2312yes—
166C2×#107C2312yes—
167C2×#107C2312yes—
168C2×#107C2312yes—
169C2×#107C2312yes—
170C2×#106C2312yes—
171C2×#106C2312yes—
172C2×#106C2312yes—
173C2×C4⋊C8—24no—
174C2×C4⋊C8C2312yes—
175C2×C4⋊C8C2312yes—
176#419—24no—
177#417—42no—
178#416—24no—
179#416—24no—
180C2×#103—24no—
181C2×#103D418yes—
182C2×#103C2314yes—
183C2×#103C2311yescharacteristic
184C2×#103C2312yes—
185C2×#103C2311yescharacteristic
186C2×#103C2314yes—
187C2×#103C2312yes—
188C2×#103C2312yes—
189C2×#103C2312yes—
190C2×#102C2311yescharacteristic
191C2×#102C2312yes—
192C2×#102C2311yescharacteristic
193C2×#102C2312yes—
194#405—24no—
195C2×#97C2312yes—
196C2×C4×C8C2312yes—
197#398—24no—
198#396—24no—
199#395—24no—
200#393—24no—
201#391—24no—
202#391C2312yes—
203#391C2312yes—
204#390C2312yes—
205#390C2312yes—
206#389—22no—
207#389—24no—
208#389—22no—
209#389—24no—
210#389—24no—
211#389—24no—
212#389D412yes—
213#389D412yes—
214#389C2312yes—
215#389C2312yes—
216#389C2314yes—
217#389C2314yes—
218#389C2314yes—
219#387—24no—
220#386—21no—
221#386—21no—
222#386—22no—
223#385—21no—
224#385—21no—
225#385—22no—
226#385—22no—
227#385—22no—
228#385—24no—
229#385D412yes—
230#385D412yes—
231#384—24no—
232#384—21no—
233#384—21no—
234#384C2312yes—
235#382—21no—
236#382—21no—
237#382—24no—
238#382C2312yes—
239#380—22no—
240#380—22no—
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—24no—
264C22×Q8—22no—
265C22×Q8—22no—
266C22×Q8—22no—
267C22×Q8—44no—
268C22×Q8—84no—
269C22×Q8C2×D414yes—
270C22×Q8C2×Q814yes—
271C22×Q8#4114yes—
272C22×Q8C2411yescharacteristic
273C22×Q8C2412yes—
274C22×D4—21no—
275C22×D4C2411yescharacteristic
276C23×C4—21no—
277C23×C4—21no—
278C23×C4—21no—
279C23×C4—21no—
280C23×C4—21no—
281C23×C4—22no—
282C23×C4C2×D411yescharacteristic
283C23×C4C2×D411yescharacteristic
284C23×C4C2411yescharacteristic
285#137—232no—
286C2×Q16—24no—
287C2×Q16—24no—
288C2×Q16C2×Q818yescomplemented
289C2×SD16—24no—
290C2×SD16—24no—
291C2×SD16C2×Q818yescomplemented
292C2×M16—22no—
293C2×M16—22no—
294C2×M16—22no—
295C2×M16—22no—
296C2×M16—22no—
297C2×M16C2×Q814yes—
298C22×C8—22no—
299C22×C8—22no—
300C22×C8—22no—
301C22×C8—22no—
302C22×C8—22no—
303C22×C8C2×D414yes—
304C22×C8C2412yes—
305C22×C8C2412yes—
306C22×C8C2411yescharacteristic
307C22×C8C2411yescharacteristic
308C22×C8C2412yes—
309C4⋊Q8—28no—
310C4⋊Q8—44no—
311C4⋊Q8—48no—
312C22⋊Q8—88no—
313C4×Q8—216no—
314C4×Q8—28no—
315C4×Q8—28no—
316C4×Q8—28no—
317C4×Q8—24no—
318C4×Q8—24no—
319C4×Q8—24no—
320C4×Q8—44no—
321C4×Q8—42no—
322C4×Q8—42no—
323C4×Q8—42no—
324C4×Q8—48no—
325C4×Q8—44no—
326C4×Q8—44no—
327C4×D4—24no—
328C4×D4—42no—
329C4×D4—48no—
330#117—216no—
331#117—22no—
332#117—22no—
333#117—22no—
334#117—22no—
335#117—22no—
336#117—22no—
337#117—22no—
338#117—22no—
339#117—22no—
340#117—48no—
341#117—44no—
342#117#4114yes—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—24no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—24no—
352C2×C4⋊C4—24no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—22no—
357C2×C4⋊C4—22no—
358C2×C4⋊C4—24no—
359C2×C4⋊C4—24no—
360C2×C4⋊C4—24no—
361C2×C4⋊C4—22no—
362C2×C4⋊C4—24no—
363C2×C4⋊C4—44no—
364C2×C4⋊C4—44no—
365C2×C4⋊C4—44no—
366C2×C4⋊C4C2×D411yescharacteristic
367C2×C4⋊C4C2×D411yescharacteristic
368C2×C4⋊C4C2×D412yes—
369C2×C4⋊C4C2×D414yes—
370C2×C4⋊C4C2×D414yes—
371C2×C4⋊C4C2×D414yes—
372C2×C4⋊C4C2412yes—
373C2×C4⋊C4C2412yes—
374C2×C4⋊C4C2411yescharacteristic
375C2×C4⋊C4C2411yescharacteristic
376C2×C4⋊C4C2412yes—
377C2×C4⋊C4C2412yes—
378C2×C4⋊C4C2412yes—
379C2×C22⋊C4—21no—
380C2×C22⋊C4—21no—
381C2×C22⋊C4—21no—
382C2×C22⋊C4—22no—
383C2×C22⋊C4—22no—
384C2×C22⋊C4—22no—
385C2×C22⋊C4—24no—
386C2×C22⋊C4C2×D412yes—
387C2×C22⋊C4C2×D412yes—
388C2×C42—24no—
389C2×C42—21no—
390C2×C42—22no—
391C2×C42—22no—
392C2×C42—22no—
393C2×C42—22no—
394C2×C42—22no—
395C2×C42—21no—
396C2×C42—21no—
397C2×C42—24no—
398C2×C42—44no—
399C2×C42C2×D412yes—
400C2×C42C2412yes—
401C2×C42C2412yes—
402C2×C42C2412yes—
403#107—24no—
404#107—24no—
405#107—28no—
406#107—24no—
407#107—24no—
408#107—42no—
409#106—24no—
410#106—24no—
411#106—42no—
412C4⋊C8—24no—
413C4⋊C8—24no—
414#103—28no—
415#103—28no—
416#103—22no—
417#103—22no—
418#103—24no—
419#103—42no—
420#103C2×D418yes—
421#103C2×D418yes—
422#102—22no—
423#102—22no—
424#102—24no—
425#102—42no—
426#97—24no—
427C4×C8—24no—
428#95—22no—
429#95—22no—
430#95—22no—
431#95—22no—
432#95—24no—
433#95—24no—
434#95—24no—
435#95—24no—
436#95—24no—
437#95C2×D414yes—
438#95C2×D414yes—
439C24—21no—
440C24C22×D411yescharacteristic
441#41—28no—
442#41—28no—
443#41—44no—
444#41—44no—
445C2×Q8—24no—
446C2×Q8—28no—
447C2×Q8—24no—
448C2×Q8—28no—
449C2×Q8—24no—
450C2×Q8—28no—
451C2×Q8—22no—
452C2×Q8—22no—
453C2×Q8—22no—
454C2×Q8—21no—
455C2×Q8—21no—
456C2×Q8—21no—
457C2×Q8—21no—
458C2×Q8—21no—
459C2×Q8—48no—
460C2×Q8—48no—
461C2×Q8—84no—
462C2×Q8—816no—
463C2×Q8C22⋊Q8116yescomplemented
464C2×Q8C22×D414yes—
465C2×Q8C22×Q812yes—
466C2×Q8C22×Q814yes—
467C2×Q8C2×#4114yes—
468C2×D4—24no—
469C2×D4—21no—
470C2×D4—21no—
471C2×D4—21no—
472C2×D4—21no—
473C2×D4—21no—
474C2×D4—42no—
475C2×D4—42no—
476C2×D4C22×Q812yes—
477C22×C4—24no—
478C22×C4—22no—
479C22×C4—24no—
480C22×C4—24no—
481C22×C4—22no—
482C22×C4—24no—
483C22×C4—22no—
484C22×C4—21no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—24no—
488C22×C4—24no—
489C22×C4—21no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—21no—
494C22×C4—42no—
495C22×C4—44no—
496C22×C4—42no—
497C22×C4—42no—
498C22×C4—42no—
499C22×C4—42no—
500C22×C4—84no—
501C22×C4—82no—
502C22×C4—82no—
503C22×C4—82no—
504C22×C4C22×D412yes—
505C22×C4C22×D414yes—
506C22×C4C22×D412yes—
507C22×C4C22×D412yes—
508C22×C4C22×D412yes—
509C22×C4C22×D412yes—
510C22×C4C22×D411yescharacteristic
511C22×C4C22×Q812yes—
512C22×C4C2×#4112yes—
513C22×C4C2×#4112yes—
514C22×C4H(5,𝔽2)12yes—
515C22×C42−1+412yes—
516C22×C4C2511yescharacteristic
517Q16—216no—
518SD16—216no—
519M16—28no—
520M16—44no—
521C2×C8—28no—
522C2×C8—22no—
523C2×C8—21no—
524C2×C8—21no—
525C2×C8—22no—
526C2×C8—21no—
527C2×C8—21no—
528C2×C8—22no—
529C2×C8—22no—
530C2×C8—22no—
531C2×C8—22no—
532C2×C8—22no—
533C2×C8—22no—
534C2×C8—44no—
535C2×C8C22×D412yes—
536C2×C8C22×D412yes—
537C2×C8C22×Q814yes—
538C4⋊C4—216no—
539C4⋊C4—24no—
540C4⋊C4—24no—
541C4⋊C4—22no—
542C4⋊C4—22no—
543C4⋊C4—28no—
544C4⋊C4—22no—
545C4⋊C4—22no—
546C4⋊C4—21no—
547C4⋊C4—21no—
548C4⋊C4—22no—
549C4⋊C4—22no—
550C4⋊C4—22no—
551C4⋊C4—22no—
552C4⋊C4—22no—
553C4⋊C4—21no—
554C4⋊C4—22no—
555C4⋊C4—21no—
556C4⋊C4—44no—
557C4⋊C4—44no—
558C4⋊C4—48no—
559C4⋊C4—48no—
560C4⋊C4—44no—
561C4⋊C4—44no—
562C4⋊C4—42no—
563C4⋊C4—42no—
564C4⋊C4—44no—
565C4⋊C4—84no—
566C4⋊C4—84no—
567C4⋊C4—82no—
568C4⋊C4—84no—
569C4⋊C4—84no—
570C4⋊C4C22×D414yes—
571C4⋊C4C2×#4114yes—
572C22⋊C4—24no—
573C22⋊C4—24no—
574C22⋊C4—24no—
575C22⋊C4—44no—
576C22⋊C4—44no—
577C22⋊C4—42no—
578C22⋊C4—42no—
579C22⋊C4—44no—
580C22⋊C4—84no—
581C42—216no—
582C42—24no—
583C42—24no—
584C42—22no—
585C42—22no—
586C42—22no—
587C42—22no—
588C42—22no—
589C42—22no—
590C42—48no—
591C42—44no—
592C42—44no—
593C42—44no—
594C42—42no—
595C42—42no—
596C23—22no—
597C23—22no—
598C23—22no—
599C23—42no—
600C23—41no—
601C23—41no—
602C23—42no—
603C23#52512yes—
604C23D4212yes—
605C23D4×Q812yes—
606C23C23×D411yescharacteristic
607Q8—24no—
608Q8—24no—
609Q8—24no—
610Q8—216no—
611Q8—22no—
612Q8—22no—
613Q8—41no—
614Q8—41no—
615Q8—816no—
616Q8C2×C22⋊Q818yes—
617D4—22no—
618D4—22no—
619D4—48no—
620D4—41no—
621D4—41no—
622C2×C4—28no—
623C2×C4—24no—
624C2×C4—22no—
625C2×C4—24no—
626C2×C4—24no—
627C2×C4—216no—
628C2×C4—22no—
629C2×C4—22no—
630C2×C4—24no—
631C2×C4—24no—
632C2×C4—22no—
633C2×C4—24no—
634C2×C4—24no—
635C2×C4—22no—
636C2×C4—22no—
637C2×C4—22no—
638C2×C4—24no—
639C2×C4—22no—
640C2×C4—24no—
641C2×C4—24no—
642C2×C4—44no—
643C2×C4—42no—
644C2×C4—44no—
645C2×C4—41no—
646C2×C4—41no—
647C2×C4—42no—
648C2×C4—48no—
649C2×C4—42no—
650C2×C4—44no—
651C2×C4—42no—
652C2×C4—44no—
653C2×C4—42no—
654C2×C4—41no—
655C2×C4—41no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—84no—
661C2×C4—84no—
662C2×C4—84no—
663C2×C4—164no—
664C2×C4C2×C22⋊Q814yes—
665C2×C4#52512yes—
666C2×C4#53914yes—
667C2×C4D4212yes—
668C2×C4#54714yes—
669C2×C4D4×Q812yes—
670C2×C4#55414yes—
671C2×C4#57814yes—
672C2×C4C23×D411yescharacteristic
673C2×C4C23×Q811yescharacteristic
674C2×C4C22×#4111yescharacteristic
675C2×C4C22×#4111yescharacteristic
676C2×C4C2×H(5,𝔽2)11yescharacteristic
677C2×C4C2×2−1+411yescharacteristic
678C8—24no—
679C8—22no—
680C8—22no—
681C8—42no—
682C8—42no—
683C22—22no—
684C22—22no—
685C22—24no—
686C22—24no—
687C22—42no—
688C22—44no—
689C22—81no—
690C22—81no—
691C22—82no—
692C22#244914yes—
693C22C2×#52511yescharacteristic
694C22C2×D4211yescharacteristic
695C22C2×D4×Q811yescharacteristic
696C22C2×#57812yes—
697C22#358512yes—
698C4—28no—
699C4—24no—
700C4—42no—
701C4—42no—
702C4—42no—
703C4—42no—
704C4—44no—
705C4—44no—
706C4—44no—
707C4—81no—
708C4—81no—
709C4—82no—
710C4—82no—
711C4—82no—
712C4#244914yes—
713C4C22×C22⋊Q811yescharacteristic
714C4C2×#53911yescharacteristic
715C4C2×#54711yescharacteristic
716C4C2×#55411yescharacteristic
717C2—24no—
718C2—82no—
719C2C2×#244911yescharacteristic
720C2#6060012yes—
721C2#6105311yescharacteristic
722C2#6105911yescharacteristic
723C2#6138412yes—
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×#477C2semidirect
4. #34249C2semidirect
5. #26397C2semidirect
6. #26393C2semidirect
7. #26381C2semidirect
8. #26377C2semidirect
10. #23660C2semidirect
12. #23552C2semidirect
13. #21156C2semidirect
14. #21024C2semidirect
18. #20880C2semidirect
19. #20877C2semidirect
106. C2×#137Q8semidirect
288. C2×Q16C2×Q8semidirect
291. C2×SD16C2×Q8semidirect
463. C2×Q8C22⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C22×#4772
C2×#29472
#355742
#342492
#263972
#263932
#263812
#263772
#237372
#236602
#236092
#235522
#211562
#210242
#209302
#209262
#208812
#208802
#208772