Finite Group Databasebeta

← #6683906

Subgroups — #6683906

Order 512.

1750 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3381C211yesmaximal, characteristic
2C2×#3372C211yesmaximal, characteristic
3#39170C211yesmaximal, characteristic, complemented
4#36180C211yesmaximal, characteristic
5#33032C211yesmaximal, characteristic
6#30960C214yesmaximal, complemented
7#30826C214yesmaximal, complemented
8#30280C211yesmaximal, characteristic, complemented
9#29869C211yesmaximal, characteristic, complemented
10#28684C211yesmaximal, characteristic, complemented
11#28638C211yesmaximal, characteristic, complemented
12#26398C214yesmaximal
13#26166C214yesmaximal
14#25934C211yesmaximal, characteristic
15#25836C211yesmaximal, characteristic, complemented
16#25835C211yesmaximal, characteristic
17#25651C211yesmaximal, characteristic, complemented
18#22240C211yesmaximal, characteristic, complemented
19#21114C211yesmaximal, characteristic
20C2×#557C2211yescharacteristic
21C2×#557C2211yescharacteristic
22C2×#554C2211yescharacteristic
23C2×#554C2211yescharacteristic
24#3381—28no—
25#3372—28no—
26C2×#499C2212yes—
27C2×#498C2211yescharacteristic
28C2×#498C2211yescharacteristic
29C2×#491C2212yes—
30C2×#487C2211yescharacteristic
31C2×#487C2211yescharacteristic
32C2×#479C2212yes—
33C2×#479C2212yes—
34C2×#477C2211yescharacteristic
35C2×#477C2211yescharacteristic
36C2×#478C2212yes—
37C2×#474C2211yescharacteristic
38C2×#461C2211yescharacteristic
39C2×#461C2212yes—
40C2×#446C2211yescharacteristic
41C2×C8×Q8C2211yescharacteristic
42C2×#440C2212yes—
43C2×C4×SD16C2212yes—
44C2×C4×SD16C2211yescharacteristic
45#2868C2211yescharacteristic
46#2862C2211yescharacteristic
47#2857—21no—
48#2719—24no—
49#2642—21no—
50#2568C2211yescharacteristic
51#2567C2211yescharacteristic
52#2567C2211yescharacteristic
53#2565—21no—
54#2552C2214yes—
55#2546—21no—
56#2546C2211yescharacteristic
57#2502—21no—
58#2502—21no—
59#2453C2214yes—
60#2446C2214yes—
61#2440—24no—
62D4×C4⋊C4C2214yes—
63#2327—21no—
64C4×C4⋊Q8C2211yescharacteristic
65#2295C2211yescharacteristic
66C2×#384—21no—
67C42×D4C2211yescharacteristic
68#2096C2214yes—
69#2096C2214yes—
70#2088C2214yes—
71#2083C2214yes—
72#2072C2214yes—
73#2068C2214yes—
74#2059C2214yes—
75#2059C2214yes—
76#2043C2214yes—
77#2026C2214yes—
78#1988C2211yescharacteristic
79#1987C2211yescharacteristic
80#1987C2211yescharacteristic
81#1984C2211yescharacteristic
82#1984C2211yescharacteristic
83#1981C2211yescharacteristic
84#1943C2214yes—
85#1933C2214yes—
86#1928C2214yes—
87#1928C2212yes—
88#1919C2212yes—
89#1917C2214yes—
90#1916C2214yes—
91#1877C2214yes—
92#1877C2214yes—
93#1877C2212yes—
94#1865C2214yes—
95#1864C2214yes—
96#1863C2214yes—
97#1862C2212yes—
98#1831C2211yescharacteristic
99C4×#107C2211yescharacteristic
100C4×#103C2211yescharacteristic
101C4×#102C2211yescharacteristic
102#1744C2211yescharacteristic
103C22×SD16C2312yes—
104C22×SD16C2311yescharacteristic
105#557—24no—
106#557—24no—
107#557—42no—
108#557—42no—
109#554—24no—
110#554—24no—
111#554—42no—
112#554—42no—
113C2×C4⋊Q8—21no—
114C2×C4⋊Q8—21no—
115C2×C4⋊Q8—22no—
116C2×C4⋊Q8D411yescharacteristic
117C2×C4⋊Q8C2311yescharacteristic
118C2×C4⋊Q8C2312yes—
119C2×C4⋊Q8C2311yescharacteristic
120C2×C4⋊Q8C2311yescharacteristic
121C2×C4⋊Q8C2311yescharacteristic
122C2×#125—21no—
123C2×#125—22no—
124C2×#125—22no—
125C2×#125C2312yes—
126C2×#125C2312yes—
127C2×#125C2312yes—
128C2×C22⋊Q8—21no—
129C2×C22⋊Q8—21no—
130C2×C22⋊Q8—22no—
131C2×C22⋊Q8—22no—
132C2×C4×Q8—21no—
133C2×C4×Q8—21no—
134C2×C4×Q8C2312yes—
135C2×C4×Q8C2312yes—
136C2×C4×Q8C2311yescharacteristic
137C2×C4×Q8C2311yescharacteristic
138C2×C4×Q8C2311yescharacteristic
139C2×C4×D4C2312yes—
140C2×C4×D4C2312yes—
141C2×C4×D4C2312yes—
142C2×C4×D4C2311yescharacteristic
143C22×C4⋊C4—21no—
144C22×C4⋊C4—21no—
145C22×C4⋊C4—24no—
146C22×C42—21no—
147#499—28no—
148#498—24no—
149#498—42no—
150#491—28no—
151#487—24no—
152#487—42no—
153#479—28no—
154#479—28no—
155#478—28no—
156#477—24no—
157#477—24no—
158#474—24no—
159#461—28no—
160#461—24no—
161#446—24no—
162C8×Q8—24no—
163#440—28no—
164C4×SD16—28no—
165C4×SD16D418yes—
166C2×#107C2312yes—
167C2×#107C2312yes—
168C2×#107C2312yes—
169C2×#107C2312yes—
170C2×#106C2312yes—
171C2×#106C2312yes—
172C2×#106C2314yes—
173C2×#106C2311yescharacteristic
174C2×C4⋊C8C2312yes—
175C2×C4⋊C8C2311yescharacteristic
176C2×C4⋊C8C2311yescharacteristic
177C2×C4⋊C8C2312yes—
178C2×#103C2312yes—
179C2×#103C2312yes—
180C2×#103C2312yes—
181C2×#103C2312yes—
182C2×#103C2312yes—
183C2×#103C2314yes—
184C2×#103C2312yes—
185C2×#103C2314yes—
186C2×#103C2311yescharacteristic
187C2×#102C2312yes—
188C2×#102C2312yes—
189C2×#102C2311yescharacteristic
190C2×#97C2312yes—
191C2×C4×C8C2311yescharacteristic
192C2×C4×C8C2312yes—
193C2×C4×C8C2311yescharacteristic
194#400—22no—
195#400—24no—
196#400—24no—
197#398—24no—
198#398—24no—
199#398—22no—
200#398—24no—
201#398—24no—
202#395—22no—
203#395—22no—
204#391—21no—
205#391—21no—
206#391—21no—
207#391—21no—
208#391—21no—
209#391—21no—
210#391D411yescharacteristic
211#391D411yescharacteristic
212#391C2314yes—
213#391C2312yes—
214#391C2314yes—
215#389—24no—
216#389—24no—
217#389D414yes—
218#389C2314yes—
219#389C2314yes—
220#389C2314yes—
221#389C2312yes—
222#387—24no—
223#387—22no—
224#387—24no—
225#387—24no—
226#385—24no—
227#385—24no—
228#384—21no—
229#384—21no—
230#384—24no—
231#384—21no—
232#384—21no—
233#384D411yescharacteristic
234#384C2314yes—
235#382—21no—
236#382—22no—
237#380—24no—
238C4×C4⋊C4—21no—
239C4×C4⋊C4C2311yescharacteristic
240C4×C4⋊C4C2311yescharacteristic
241C4×C4⋊C4C2312yes—
242C4×C4⋊C4C2314yes—
243C4×C22⋊C4—22no—
244C4×C22⋊C4—21no—
245#376—21no—
246C43C2311yescharacteristic
247#340C2312yes—
248#340C2312yes—
249#340C2314yes—
250#340C2314yes—
251#340C2314yes—
252#340C2312yes—
253#336C2314yes—
254#336C2314yes—
255#336C2314yes—
256#336C2312yes—
257#336C2312yes—
258C22×Q8—21no—
259C22×Q8—21no—
260C22×Q8—21no—
261C22×Q8C2×D411yescharacteristic
262C22×Q8C2×D411yescharacteristic
263C22×Q8C2412yes—
264C22×Q8C2411yescharacteristic
265C22×D4C2411yescharacteristic
266C23×C4—21no—
267C23×C4—22no—
268C23×C4—22no—
269C23×C4—22no—
270C2×SD16—24no—
271C2×SD16—24no—
272C2×SD16—28no—
273C2×SD16C2×D414yes—
274C22×C8C2412yes—
275C22×C8C2412yes—
276C22×C8C2412yes—
277C22×C8C2411yescharacteristic
278C22×C8C2411yescharacteristic
279C4⋊Q8—24no—
280C4⋊Q8—22no—
281C4⋊Q8—22no—
282C4⋊Q8—22no—
283C4⋊Q8—22no—
284C4⋊Q8—22no—
285C4⋊Q8—22no—
286C4⋊Q8—22no—
287C4⋊Q8—22no—
288C4⋊Q8—42no—
289C4⋊Q8—42no—
290C4⋊Q8—44no—
291C4⋊Q8—44no—
292C4⋊Q8—84no—
293#125—24no—
294#125—24no—
295#125—24no—
296#125—42no—
297#125—42no—
298#125—42no—
299#125—44no—
300#125—48no—
301#125—48no—
302C22⋊Q8—48no—
303C22⋊Q8—48no—
304C22⋊Q8—44no—
305C22⋊Q8—44no—
306C4×Q8—22no—
307C4×Q8—22no—
308C4×Q8—22no—
309C4×Q8—22no—
310C4×Q8—22no—
311C4×Q8—24no—
312C4×Q8—24no—
313C4×Q8—42no—
314C4×Q8—42no—
315C4×Q8—44no—
316C4×Q8—44no—
317C4×Q8C2×D414yes—
318C4×D4—24no—
319C4×D4—24no—
320C4×D4—24no—
321C4×D4—24no—
322C4×D4—22no—
323C4×D4—42no—
324C4×D4—42no—
325C4×D4C2×D414yes—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—24no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—24no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—24no—
346C2×C4⋊C4—24no—
347C2×C4⋊C4—24no—
348C2×C4⋊C4—44no—
349C2×C4⋊C4—44no—
350C2×C4⋊C4—88no—
351C2×C4⋊C4C2×D411yescharacteristic
352C2×C4⋊C4C2×D411yescharacteristic
353C2×C4⋊C4C2×D414yes—
354C2×C4⋊C4C2×D412yes—
355C2×C4⋊C4C2×D412yes—
356C2×C4⋊C4C2×D414yes—
357C2×C4⋊C4C2412yes—
358C2×C4⋊C4C2412yes—
359C2×C4⋊C4C2412yes—
360C2×C4⋊C4C2412yes—
361C2×C4⋊C4C2411yescharacteristic
362C2×C4⋊C4C2411yescharacteristic
363C2×C4⋊C4C2412yes—
364C2×C22⋊C4—21no—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—22no—
368C2×C42—21no—
369C2×C42—22no—
370C2×C42—22no—
371C2×C42—21no—
372C2×C42—22no—
373C2×C42—21no—
374C2×C42—44no—
375C2×C42C2×D412yes—
376C2×C42C2×D411yescharacteristic
377C2×C42C2×D411yescharacteristic
378C2×C42C2412yes—
379C2×C42C2412yes—
380C2×C42C2412yes—
381C2×C42C2411yescharacteristic
382#107—24no—
383#107—24no—
384#107—24no—
385#107—24no—
386#106—24no—
387#106—24no—
388#106—44no—
389#106C2×D414yes—
390C4⋊C8—24no—
391C4⋊C8—24no—
392C4⋊C8—22no—
393C4⋊C8—22no—
394#103—24no—
395#103—24no—
396#103—24no—
397#103—24no—
398#103—24no—
399#103—24no—
400#103—28no—
401#103—44no—
402#103C2×D414yes—
403#102—24no—
404#102—24no—
405#102C2×D414yes—
406#97—24no—
407C4×C8—24no—
408C4×C8—22no—
409C4×C8C2×D414yes—
410#95—22no—
411#95—22no—
412#95—21no—
413#95—22no—
414#95—22no—
415#95—21no—
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—
425C24—21no—
426C2×Q8—22no—
427C2×Q8—22no—
428C2×Q8—22no—
429C2×Q8—22no—
430C2×Q8—22no—
431C2×Q8—22no—
432C2×Q8—21no—
433C2×Q8—22no—
434C2×Q8—22no—
435C2×Q8—22no—
436C2×Q8—22no—
437C2×Q8—24no—
438C2×Q8—24no—
439C2×Q8—42no—
440C2×Q8—42no—
441C2×Q8—42no—
442C2×Q8—42no—
443C2×Q8—42no—
444C2×Q8—82no—
445C2×Q8—82no—
446C2×Q8C22×D412yes—
447C2×D4—21no—
448C2×D4—22no—
449C2×D4C22×D412yes—
450C2×D42−1+414yes—
451C22×C4—28no—
452C22×C4—22no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—24no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—42no—
466C22×C4—44no—
467C22×C4—42no—
468C22×C4—84no—
469C22×C4—84no—
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C22×D412yes—
474C22×C4C22×D412yes—
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D412yes—
477C22×C4C22×D411yescharacteristic
478C22×C4C2511yescharacteristic
479SD16—28no—
480SD16—48no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—24no—
488C2×C8—22no—
489C2×C8—24no—
490C2×C8C22×D412yes—
491C2×C8C22×D412yes—
492C4⋊C4—24no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—24no—
496C4⋊C4—24no—
497C4⋊C4—28no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—44no—
510C4⋊C4—44no—
511C4⋊C4—44no—
512C4⋊C4—44no—
513C4⋊C4—44no—
514C4⋊C4—44no—
515C4⋊C4—44no—
516C4⋊C4—44no—
517C4⋊C4—44no—
518C4⋊C4—44no—
519C4⋊C4—44no—
520C4⋊C4—48no—
521C4⋊C4—42no—
522C4⋊C4—42no—
523C4⋊C4—42no—
524C4⋊C4—42no—
525C4⋊C4—42no—
526C4⋊C4—44no—
527C4⋊C4—48no—
528C4⋊C4—84no—
529C4⋊C4—84no—
530C4⋊C4—84no—
531C4⋊C4C22×D412yes—
532C4⋊C4C22×D412yes—
533C4⋊C42−1+414yes—
534C22⋊C4—28no—
535C22⋊C4—24no—
536C22⋊C4—44no—
537C22⋊C4—44no—
538C42—24no—
539C42—22no—
540C42—24no—
541C42—22no—
542C42—22no—
543C42—22no—
544C42—22no—
545C42—22no—
546C42—22no—
547C42—22no—
548C42—44no—
549C42—44no—
550C42—44no—
551C42—42no—
552C42—42no—
553C42C22×D412yes—
554C42H(5,𝔽2)12yes—
555C23—22no—
556C23—24no—
557C23—41no—
558C23—42no—
559C23—41no—
560C23C23×D411yescharacteristic
561Q8—22no—
562Q8—22no—
563Q8—44no—
564Q8—44no—
565Q8—42no—
566Q8—42no—
567Q8—42no—
568Q8—44no—
569Q8—84no—
570Q8—162no—
571Q8—162no—
572D4—22no—
573D4—24no—
574D4#53614yes—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—24no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—416no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—41no—
602C2×C4—48no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—164no—
616C2×C4—164no—
617C2×C4D4212yes—
618C2×C4D4212yes—
619C2×C4#57512yes—
620C2×C4#57714yes—
621C2×C4#57812yes—
622C2×C4C23×D411yescharacteristic
623C2×C4C2×H(5,𝔽2)11yescharacteristic
624C2×C4C2×2−1+412yes—
625C2×C4C2×2−1+412yes—
626C8—22no—
627C8—22no—
628C8—42no—
629C8—42no—
630C8—42no—
631C22—24no—
632C22—42no—
633C22—44no—
634C22—42no—
635C22—44no—
636C22C2×D4211yescharacteristic
637C22#348612yes—
638C22C2×#57511yescharacteristic
639C22C2×#57712yes—
640C22C2×#57811yescharacteristic
641C4—24no—
642C4—22no—
643C4—42no—
644C4—42no—
645C4—42no—
646C4—42no—
647C4—44no—
648C4—42no—
649C4—42no—
650C4—42no—
651C4—42no—
652C4—82no—
653C4—82no—
654C4—82no—
655C4—82no—
656C4—82no—
657C4#326914yes—
658C4C2×#53611yescharacteristic
659C4#349911yescharacteristic
660C4#355212yes—
661C2—44no—
662C2#3767511yescharacteristic
663C2C2×#326911yescharacteristic
664C2#6108711yescharacteristic
665C2#6184112yes—
666C2#6186912yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #39170C2semidirect
6. #30960C2semidirect
7. #30826C2semidirect
8. #30280C2semidirect
9. #29869C2semidirect
10. #28684C2semidirect
11. #28638C2semidirect
15. #25836C2semidirect
17. #25651C2semidirect
18. #22240C2semidirect

Maximal subgroups

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

subgroupindex
C2×#33812
C2×#33722
#391702
#361802
#330322
#309602
#308262
#302802
#298692
#286842
#286382
#263982
#261662
#259342
#258362
#258352
#256512
#222402
#211142