Finite Group Databasebeta

← #6443403

Subgroups — #6443403

Order 512.

2224 conjugacy classes of subgroups.

The subgroup lattice has 703 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3081C211yesmaximal, characteristic
2C22×#474C211yesmaximal, characteristic
3#35576C211yesmaximal, characteristic, complemented
4#34248C211yesmaximal, characteristic
5#26058C214yesmaximal, complemented
6#26052C214yesmaximal, complemented
7#26049C214yesmaximal, complemented
8#26040C214yesmaximal, complemented
9#23667C211yesmaximal, characteristic, complemented
10#23666C211yesmaximal, characteristic, complemented
11#23555C211yesmaximal, characteristic, complemented
12#23552C211yesmaximal, characteristic, complemented
13#21173C211yesmaximal, characteristic, complemented
14#21147C211yesmaximal, characteristic, complemented
15#20876C211yesmaximal, characteristic
16#20873C211yesmaximal, characteristic
17#20803C211yesmaximal, characteristic, complemented
18#20650C211yesmaximal, characteristic
19#20416C211yesmaximal, characteristic
20C2×#544C2211yescharacteristic
21C22×C4⋊Q8C2211yescharacteristic
22C2×#518C2211yescharacteristic
23C2×#518C2211yescharacteristic
24C22×C4×D4C2211yescharacteristic
25#3081—28no—
26C2×#478C2212yes—
27C2×#478C2212yes—
28C2×#476C2212yes—
29C2×#474—28no—
30C2×#474C2214yes—
31C2×#474C2214yes—
32C2×#474C2212yes—
33C2×#428C2211yescharacteristic
34C2×#428C2211yescharacteristic
35C2×#428C2211yescharacteristic
36C2×#427C2211yescharacteristic
37C22×#107C2211yescharacteristic
38C22×#107C2211yescharacteristic
39C2×#424C2211yescharacteristic
40C2×#424C2211yescharacteristic
41C22×C4⋊C8C2211yescharacteristic
42C2×#418C2211yescharacteristic
43C2×#418C2211yescharacteristic
44C22×#102C2211yescharacteristic
45C22×#102C2211yescharacteristic
46C2×#405C2211yescharacteristic
47#2583—22no—
48#2551—22no—
49#2551C2214yescomplemented
50#2550—22no—
51#2550C2214yescomplemented
52#2434—22no—
53#2349—22no—
54#2349C2214yes—
55D4×C4⋊C4—22no—
56D4×C4⋊C4C2214yes—
57#2317C2211yescharacteristic
58#2317C2211yescharacteristic
59#2317C2211yescharacteristic
60#2317C2211yescharacteristic
61#2315—22no—
62#2296C2211yescharacteristic
63#2296C2211yescharacteristic
64#2296C2211yescharacteristic
65C2×#385—22no—
66#2274C2211yescharacteristic
67#2088C2212yescomplemented
68#2088C2212yescomplemented
69#2087C2212yescomplemented
70#2087C2214yescomplemented
71#2086C2212yescomplemented
72#2086C2214yescomplemented
73#2059C2214yescomplemented
74#2058C2214yescomplemented
75#2057C2214yescomplemented
76#2057C2214yescomplemented
77#1949C2214yescomplemented
78#1947C2214yescomplemented
79#1940C2214yescomplemented
80#1939C2214yescomplemented
81#1864C2214yes—
82#1864C2214yes—
83#1862C2214yes—
84#1862C2214yes—
85#1847C2212yescomplemented
86#1847C2212yescomplemented
87#1846C2212yescomplemented
88#1845C2212yescomplemented
89#1804C2214yes—
90#1802C2214yes—
91#1762C2214yes—
92C4×#102C2214yes—
93#1736C2211yescharacteristic
94#1736C2211yescharacteristic
95#1736C2211yescharacteristic
96#1735C2211yescharacteristic
97#1735C2211yescharacteristic
98#1735C2211yescharacteristic
99#1729C2211yescharacteristic
100#1728C2211yescharacteristic
101C23×Q8—21no—
102C23×D4C2311yescharacteristic
103C24×C4—21no—
104C22×M16C2311yescharacteristic
105C22×M16C2311yescharacteristic
106C23×C8C2311yescharacteristic
107C23×C8C2311yescharacteristic
108#544—24no—
109#544—42no—
110C2×C4⋊Q8—21no—
111C2×C4⋊Q8—24no—
112C2×C4⋊Q8—24no—
113C2×C4⋊Q8—22no—
114C2×C4⋊Q8—42no—
115C2×C4⋊Q8C2312yes—
116C2×C4⋊Q8C2314yes—
117C2×#125—21no—
118C2×#125C2312yes—
119C2×C22⋊Q8—22no—
120#518—24no—
121#518—24no—
122#518—42no—
123#518Q818yescomplemented
124C2×C4×D4—22no—
125C2×C4×D4—21no—
126C2×C4×D4—21no—
127C2×C4×D4—24no—
128C2×C4×D4—22no—
129C2×C4×D4—42no—
130C2×C4×D4C2312yes—
131C2×C4×D4C2312yes—
132C2×C4×D4C2314yes—
133C2×C4×D4C2314yes—
134C2×#117C2311yescharacteristic
135C2×#117C2311yescharacteristic
136C2×#117C2311yescharacteristic
137C2×#117C2311yescharacteristic
138C2×#117C2311yescharacteristic
139C2×#117C2311yescharacteristic
140C22×C4⋊C4—21no—
141C22×C4⋊C4—24no—
142C22×C4⋊C4C2311yescharacteristic
143C22×C4⋊C4C2311yescharacteristic
144C22×C4⋊C4C2311yescharacteristic
145C22×C22⋊C4—22no—
146C22×C22⋊C4—21no—
147C22×C22⋊C4—22no—
148C22×C42C2311yescharacteristic
149#478—28no—
150#478—28no—
151#476—28no—
152#474—28no—
153#474—216no—
154#474—216no—
155#428—24no—
156#428—42no—
157#428—42no—
158#427—24no—
159C2×#107—24no—
160C2×#107—24no—
161C2×#107C2311yescharacteristic
162C2×#107C2312yes—
163C2×#107C2314yes—
164C2×#107C2311yescharacteristic
165C2×#107C2312yes—
166C2×#107C2312yes—
167C2×#107C2312yes—
168C2×#107C2312yes—
169C2×#106C2311yescharacteristic
170C2×#106C2312yes—
171C2×#106C2311yescharacteristic
172C2×#106C2312yes—
173C2×#106C2312yes—
174#424—24no—
175#424—24no—
176C2×C4⋊C8—24no—
177C2×C4⋊C8C2312yes—
178C2×C4⋊C8C2312yes—
179C2×C4⋊C8C2314yes—
180C2×C4⋊C8C2312yes—
181C2×C4⋊C8C2312yes—
182#418—24no—
183#418Q818yescomplemented
184C2×#102—24no—
185C2×#102D418yes—
186C2×#102C2312yes—
187C2×#102C2312yes—
188C2×#102C2314yes—
189C2×#102C2312yes—
190C2×#102C2312yes—
191C2×#102C2314yes—
192#405—24no—
193C2×#97C2312yes—
194C2×C4×C8C2312yes—
195#398—24no—
196#395—24no—
197#395—24no—
198#395—24no—
199#391—24no—
200#389—22no—
201#389—22no—
202#389—24no—
203#389—24no—
204#389D412yes—
205#389D412yes—
206#389C2312yes—
207#389C2314yes—
208#389C2312yes—
209#389C2314yes—
210#389C2312yes—
211#389C2312yes—
212#385—21no—
213#385—21no—
214#385—21no—
215#385—21no—
216#385—24no—
217#385—24no—
218#385—22no—
219#385—22no—
220#385—22no—
221#385—22no—
222#385—24no—
223#385—24no—
224#385D412yes—
225#385D412yes—
226#384—21no—
227#384—21no—
228#384C2312yes—
229#384C2312yes—
230#382—21no—
231#382—21no—
232#382C2312yes—
233#382C2312yes—
234#380—24no—
235#380—24no—
236C4×C4⋊C4—22no—
237C4×C4⋊C4—22no—
238C4×C4⋊C4—22no—
239C4×C4⋊C4C2314yes—
240C4×C4⋊C4C2314yes—
241C4×C22⋊C4—22no—
242C2×#95—24no—
243#340C2312yes—
244#340C2314yes—
245#340C2314yes—
246#340C2312yes—
247#340C2312yes—
248#340C2314yes—
249#340C2312yes—
250#340C2312yes—
251#340C2312yes—
252#336C2314yes—
253#336C2314yes—
254C25—21no—
255C22×Q8—24no—
256C22×Q8—22no—
257C22×Q8—44no—
258C22×Q8—84no—
259C22×D4—21no—
260C22×D4—22no—
261C22×D4—22no—
262C22×D4—22no—
263C22×D4C2×D414yes—
264C22×D4C2×Q814yes—
265C22×D4C2×Q814yes—
266C22×D4C2412yes—
267C23×C4—21no—
268C23×C4—24no—
269C23×C4—21no—
270C23×C4—21no—
271C23×C4—24no—
272C23×C4—22no—
273C23×C4—22no—
274C23×C4—44no—
275C23×C4—84no—
276C23×C4C2×D411yescharacteristic
277C23×C4C2×D411yescharacteristic
278C23×C4C2411yescharacteristic
279C2×M16—22no—
280C2×M16—22no—
281C2×M16—22no—
282C2×M16—22no—
283C2×M16—22no—
284C2×M16C2×Q814yes—
285C22×C8—22no—
286C22×C8—22no—
287C22×C8—22no—
288C22×C8—22no—
289C22×C8—22no—
290C22×C8C2×D414yes—
291C22×C8C2412yes—
292C22×C8C2412yes—
293C22×C8C2411yescharacteristic
294C22×C8C2411yescharacteristic
295C22×C8C2412yes—
296C4⋊Q8—28no—
297C4⋊Q8—24no—
298C4⋊Q8—44no—
299C4⋊Q8—42no—
300C4⋊Q8—48no—
301C4⋊Q8—44no—
302C4⋊Q8—88no—
303#125—24no—
304#125—42no—
305#125—44no—
306C22⋊Q8—48no—
307C4×D4—216no—
308C4×D4—28no—
309C4×D4—24no—
310C4×D4—24no—
311C4×D4—28no—
312C4×D4—44no—
313C4×D4—42no—
314C4×D4—42no—
315C4×D4—48no—
316C4×D4—44no—
317C4×D4—44no—
318C4×D4C2×Q8116yescomplemented
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#117C2×Q814yes—
328#117C2×Q814yes—
329#117C2×Q814yes—
330#117#4114yes—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—24no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—24no—
347C2×C4⋊C4—24no—
348C2×C4⋊C4—24no—
349C2×C4⋊C4—24no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—44no—
352C2×C4⋊C4—88no—
353C2×C4⋊C4C2×D414yes—
354C2×C4⋊C4C2×D411yescharacteristic
355C2×C4⋊C4C2×D411yescharacteristic
356C2×C4⋊C4C2×D414yes—
357C2×C4⋊C4C2×D414yes—
358C2×C4⋊C4C2×D412yes—
359C2×C4⋊C4C2412yes—
360C2×C4⋊C4C2412yes—
361C2×C4⋊C4C2411yescharacteristic
362C2×C4⋊C4C2411yescharacteristic
363C2×C4⋊C4C2411yescharacteristic
364C2×C4⋊C4C2411yescharacteristic
365C2×C4⋊C4C2412yes—
366C2×C4⋊C4C2412yes—
367C2×C4⋊C4C2412yes—
368C2×C22⋊C4—21no—
369C2×C22⋊C4—21no—
370C2×C22⋊C4—24no—
371C2×C22⋊C4—216no—
372C2×C22⋊C4—21no—
373C2×C22⋊C4—21no—
374C2×C22⋊C4—22no—
375C2×C22⋊C4—24no—
376C2×C22⋊C4—22no—
377C2×C22⋊C4—48no—
378C2×C22⋊C4—44no—
379C2×C22⋊C4C2×D412yes—
380C2×C22⋊C4C2×D412yes—
381C2×C42—21no—
382C2×C42—21no—
383C2×C42—22no—
384C2×C42—22no—
385C2×C42—21no—
386C2×C42—21no—
387C2×C42—24no—
388C2×C42C2×D412yes—
389C2×C42C2412yes—
390C2×C42C2412yes—
391C2×C42C2412yes—
392#107—24no—
393#107—24no—
394#107—22no—
395#107—22no—
396#107—28no—
397#107—24no—
398#107—42no—
399#107—42no—
400#106—22no—
401#106—22no—
402#106—24no—
403#106—42no—
404#106—42no—
405C4⋊C8—24no—
406C4⋊C8—24no—
407C4⋊C8—24no—
408C4⋊C8—24no—
409C4⋊C8—28no—
410#102—28no—
411#102—24no—
412#102—24no—
413#102C2×D418yes—
414#102C2×D418yes—
415#102C2×Q8116yescomplemented
416#97—24no—
417C4×C8—24no—
418#95—22no—
419#95—24no—
420#95—24no—
421#95—24no—
422#95—22no—
423#95—22no—
424#95—22no—
425#95—24no—
426#95—24no—
427#95C2×D414yes—
428#95C2×D414yes—
429C24—22no—
430C24—24no—
431C24—24no—
432C24—24no—
433C24—82no—
434C24—82no—
435C24C22×D411yescharacteristic
436C2×Q8—48no—
437C2×Q8—44no—
438C2×Q8—48no—
439C2×Q8—84no—
440C2×Q8—84no—
441C2×Q8—816no—
442C2×D4—24no—
443C2×D4—28no—
444C2×D4—24no—
445C2×D4—28no—
446C2×D4—24no—
447C2×D4—28no—
448C2×D4—22no—
449C2×D4—22no—
450C2×D4—22no—
451C2×D4C4⋊Q8116yescomplemented
452C2×D4C22×D414yes—
453C2×D4C22×Q814yes—
454C2×D4C22×Q814yes—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—24no—
462C22×C4—216no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—42no—
470C22×C4—48no—
471C22×C4—44no—
472C22×C4—44no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—48no—
476C22×C4—84no—
477C22×C4—84no—
478C22×C4—84no—
479C22×C4—84no—
480C22×C4—84no—
481C22×C4—816no—
482C22×C4—82no—
483C22×C4—82no—
484C22×C4—82no—
485C22×C4C22×D412yes—
486C22×C4C22×D412yes—
487C22×C4C22×D412yes—
488C22×C4C22×D414yes—
489C22×C4C22×D412yes—
490C22×C4C22×D411yescharacteristic
491C22×C4C22×D412yes—
492C22×C4C22×Q812yes—
493C22×C4C22×Q812yes—
494C22×C4C2×#4112yes—
495C22×C4H(5,𝔽2)12yes—
496C22×C42−1+412yes—
497C22×C4C2511yescharacteristic
498M16—28no—
499M16—44no—
500C2×C8—28no—
501C2×C8—21no—
502C2×C8—22no—
503C2×C8—21no—
504C2×C8—22no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—22no—
508C2×C8—22no—
509C2×C8—22no—
510C2×C8—22no—
511C2×C8—22no—
512C2×C8—22no—
513C2×C8—44no—
514C2×C8C22×D412yes—
515C2×C8C22×D412yes—
516C2×C8C22×Q814yes—
517C4⋊C4—24no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—28no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—22no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—22no—
530C4⋊C4—21no—
531C4⋊C4—21no—
532C4⋊C4—22no—
533C4⋊C4—22no—
534C4⋊C4—21no—
535C4⋊C4—21no—
536C4⋊C4—44no—
537C4⋊C4—48no—
538C4⋊C4—44no—
539C4⋊C4—44no—
540C4⋊C4—44no—
541C4⋊C4—44no—
542C4⋊C4—42no—
543C4⋊C4—42no—
544C4⋊C4—44no—
545C4⋊C4—44no—
546C4⋊C4—82no—
547C4⋊C4—84no—
548C4⋊C4—84no—
549C4⋊C4—84no—
550C4⋊C4—84no—
551C4⋊C4C22⋊Q818yes—
552C4⋊C4C22×D414yes—
553C4⋊C4C22×Q814yes—
554C4⋊C4C22×Q814yes—
555C22⋊C4—24no—
556C22⋊C4—24no—
557C22⋊C4—24no—
558C22⋊C4—216no—
559C22⋊C4—48no—
560C22⋊C4—44no—
561C22⋊C4—44no—
562C22⋊C4—42no—
563C22⋊C4—42no—
564C22⋊C4C22⋊Q818yes—
565C42—24no—
566C42—24no—
567C42—24no—
568C42—22no—
569C42—22no—
570C42—22no—
571C42—22no—
572C42—42no—
573C42—42no—
574C42—84no—
575C42C22×Q814yes—
576C42C2×#4114yes—
577C23—22no—
578C23—22no—
579C23—24no—
580C23—24no—
581C23—24no—
582C23—216no—
583C23—44no—
584C23—44no—
585C23—84no—
586C23—88no—
587C23—84no—
588C23—88no—
589C23C2×C22⋊Q812yes—
590C23D4212yes—
591C23D4×Q812yes—
592C23C23×D411yescharacteristic
593Q8—816no—
594Q8—164no—
595D4—24no—
596D4—216no—
597D4—24no—
598D4—24no—
599D4C2×C4⋊Q818yes—
600C2×C4—28no—
601C2×C4—24no—
602C2×C4—22no—
603C2×C4—24no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—24no—
607C2×C4—22no—
608C2×C4—24no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—24no—
613C2×C4—24no—
614C2×C4—24no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—816no—
634C2×C4—816no—
635C2×C4—84no—
636C2×C4—84no—
637C2×C4—84no—
638C2×C4—164no—
639C2×C4—164no—
640C2×C4C2×C22⋊Q818yes—
641C2×C4C2×C22⋊Q812yes—
642C2×C4C2×C4⋊Q814yes—
643C2×C4#54414yes—
644C2×C4D4212yes—
645C2×C4#54714yes—
646C2×C4D4×Q812yes—
647C2×C4#55414yes—
648C2×C4#57614yes—
649C2×C4C23×D411yescharacteristic
650C2×C4C23×Q811yescharacteristic
651C2×C4C23×Q811yescharacteristic
652C2×C4C22×#4111yescharacteristic
653C2×C4C2×H(5,𝔽2)11yescharacteristic
654C2×C4C2×2−1+411yescharacteristic
655C8—22no—
656C8—22no—
657C8—24no—
658C8—42no—
659C8—42no—
660C22—24no—
661C22—22no—
662C22—22no—
663C22—24no—
664C22—28no—
665C22—48no—
666C22—44no—
667C22—44no—
668C22—48no—
669C22—816no—
670C22#244014yes—
671C22C22×C22⋊Q811yescharacteristic
672C22C2×D4211yescharacteristic
673C22C2×D4×Q811yescharacteristic
674C22C2×#57612yes—
675C22#358612yes—
676C4—28no—
677C4—24no—
678C4—44no—
679C4—42no—
680C4—44no—
681C4—42no—
682C4—42no—
683C4—44no—
684C4—82no—
685C4—82no—
686C4—81no—
687C4—81no—
688C4—81no—
689C4—81no—
690C4#244014yes—
691C4C22×C4⋊Q811yescharacteristic
692C4C2×#54411yescharacteristic
693C4C2×#54711yescharacteristic
694C4C2×#55411yescharacteristic
695C2—24no—
696C2—48no—
697C2C2×#244011yescharacteristic
698C2#6049412yes—
699C2#6104911yescharacteristic
700C2#6106011yescharacteristic
701C2#6138112yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
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
3. #35576C2semidirect
5. #26058C2semidirect
6. #26052C2semidirect
7. #26049C2semidirect
8. #26040C2semidirect
9. #23667C2semidirect
10. #23666C2semidirect
11. #23555C2semidirect
12. #23552C2semidirect
13. #21173C2semidirect
14. #21147C2semidirect
17. #20803C2semidirect
49. #2551C22semidirect
51. #2550C22semidirect
67. #2088C22semidirect
68. #2088C22semidirect
69. #2087C22semidirect
70. #2087C22semidirect
71. #2086C22semidirect
72. #2086C22semidirect
73. #2059C22semidirect
74. #2058C22semidirect
75. #2057C22semidirect
76. #2057C22semidirect
77. #1949C22semidirect
78. #1947C22semidirect
79. #1940C22semidirect
80. #1939C22semidirect
85. #1847C22semidirect
86. #1847C22semidirect
87. #1846C22semidirect
88. #1845C22semidirect
123. #518Q8semidirect
183. #418Q8semidirect
318. C4×D4C2×Q8semidirect
415. #102C2×Q8semidirect
451. C2×D4C4⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#30812
C22×#4742
#355762
#342482
#260582
#260522
#260492
#260402
#236672
#236662
#235552
#235522
#211732
#211472
#208762
#208732
#208032
#206502
#204162