Finite Group Databasebeta

← #6447229

Subgroups — #6447229

Order 512.

1816 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35180C211yesmaximal, characteristic
2#34206C211yesmaximal, characteristic
3#26313C214yesmaximal, complemented
4#26312C214yesmaximal, complemented
5#26311C214yesmaximal, complemented
6#26310C214yesmaximal, complemented
7#23643C212yesmaximal, complemented
8#23639C212yesmaximal, complemented
9#21321C211yesmaximal, characteristic, complemented
10#21312C211yesmaximal, characteristic, complemented
11#21188C211yesmaximal, characteristic
12#21170C211yesmaximal, characteristic, complemented
13#20848C212yesmaximal, complemented
14#20703C212yesmaximal, complemented
15C2×#1776C211yesmaximal, characteristic, complemented
16C2×#533C2211yescharacteristic
17C2×#532C2211yescharacteristic
18C2×#520C2212yes—
19C2×#428C2212yes—
20C22×#107C2211yescharacteristic
21C22×#106C2211yescharacteristic
22C2×#423C2211yescharacteristic
23C2×#408C2212yes—
24C2×#408C2212yes—
25C22×#97C2211yescharacteristic
26#2850—22no—
27#2639—22no—
28#2592—22no—
29#2568—21no—
30#2563C2212yes—
31#2562C2214yes—
32#2548—21no—
33#2547C2212yes—
34#2367—22no—
35#2366—21no—
36#2365—21no—
37#2354C2212yes—
38#2352C2214yes—
39D4×C4⋊C4C2212yes—
40#2325—22no—
41#2311C2212yes—
42#2311C2212yes—
43#2311C2212yes—
44C4×#121—22no—
45#2285C2211yescharacteristic
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×C4×C4⋊C4C2211yescharacteristic
48#2077C2214yes—
49#2077C2214yes—
50#2076C2214yes—
51#2076C2214yes—
52#2076C2214yes—
53#2075C2214yes—
54#2075C2214yes—
55#2075C2214yes—
56#1993C2212yes—
57#1993C2212yes—
58#1992C2212yes—
59#1992C2212yes—
60#1989C2214yes—
61#1988C2212yes—
62#1987C2212yes—
63#1955C2214yes—
64#1955C2214yes—
65#1949C2214yes—
66#1948C2212yes—
67#1947C2212yes—
68#1854C2214yes—
69#1854C2214yes—
70#1853C2214yes—
71#1853C2214yes—
72#1825C2214yes—
73#1825C2214yes—
74#1824C2214yes—
75#1823C2214yes—
76#1776C2214yes—
77#1776C2214yes—
78#1736C2212yes—
79#1736C2212yes—
80#1736C2212yes—
81#1735C2212yes—
82#1735C2212yes—
83C2×#340C2211yescharacteristic
84C2×#340C2211yescharacteristic
85C22×#41C2312yes—
86C22×M16C2312yes—
87C23×C8C2311yescharacteristic
88C23×C8C2311yescharacteristic
89#533—24no—
90#533—24no—
91#532—24no—
92#532—24no—
93C2×C4⋊Q8D411yescharacteristic
94C2×C4⋊Q8C2311yescharacteristic
95C2×C4⋊Q8C2311yescharacteristic
96C2×#127D411yescharacteristic
97C2×#125C2312yes—
98C2×#124—22no—
99C2×#124D412yes—
100C2×C22⋊Q8—21no—
101C2×C22⋊Q8—21no—
102C2×#121D412yes—
103C2×#121D412yes—
104#520—28no—
105#520—44no—
106C2×C4×Q8—22no—
107C2×C4×Q8—21no—
108C2×C4×Q8—21no—
109C2×C4×D4—24no—
110C2×C4×D4—22no—
111C2×C4×D4—22no—
112C2×C4×D4C2312yes—
113C2×C4×D4C2312yes—
114C2×#117C2312yes—
115C2×#117C2312yes—
116C2×#117C2314yes—
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2314yes—
121C22×C4⋊C4C2311yescharacteristic
122C22×C42—21no—
123C22×C42—21no—
124C22×C42C2311yescharacteristic
125#428—44no—
126C2×#107—24no—
127C2×#107C2314yes—
128C2×#107C2314yes—
129C2×#106—24no—
130C2×#106C2314yes—
131C2×#106C2314yes—
132#423—24no—
133C2×C4⋊C8C2312yes—
134C2×C4⋊C8C2312yes—
135#408—28no—
136#408—44no—
137C2×C22⋊C8C2314yes—
138C2×C22⋊C8C2314yes—
139C2×C22⋊C8C2314yes—
140C2×C22⋊C8C2314yes—
141C2×#97—24no—
142C2×#97C2312yes—
143C2×#97C2312yes—
144#400—24no—
145#397—24no—
146#394—24no—
147#393—24no—
148#391—22no—
149#391—22no—
150#391—22no—
151#391—22no—
152#390—22no—
153#390—22no—
154#390—22no—
155#390—22no—
156#390—22no—
157#389—22no—
158#388—24no—
159#388—24no—
160#388—24no—
161#388—24no—
162#387—24no—
163#386—24no—
164#386—24no—
165#386—24no—
166#386—24no—
167#385—24no—
168#384C2311yescharacteristic
169#384C2311yescharacteristic
170#382—22no—
171#382C2312yes—
172#380—21no—
173#380—21no—
174#380C2314yes—
175#380C2314yes—
176#380C2314yes—
177#380C2314yes—
178#380C2314yes—
179#380C2314yes—
180C4×C4⋊C4—22no—
181C4×C4⋊C4—21no—
182C4×C4⋊C4—21no—
183C4×C4⋊C4—21no—
184C4×C4⋊C4—21no—
185C4×C4⋊C4—21no—
186C4×C4⋊C4—21no—
187C4×C4⋊C4C2312yes—
188C4×C4⋊C4C2312yes—
189C4×C4⋊C4C2312yes—
190C4×C4⋊C4C2312yes—
191C4×C22⋊C4—24no—
192C2×#95—21no—
193C2×#95—21no—
194#340C2314yes—
195#340C2312yes—
196#340C2312yes—
197#340C2312yes—
198#340C2314yes—
199#340C2314yes—
200#340C2314yes—
201#340C2314yes—
202#340C2312yes—
203#340C2312yes—
204#340C2312yes—
205#340C2312yes—
206#340C2312yes—
207#340C2314yes—
208#340C2314yes—
209#340C2314yes—
210C2×#41—24no—
211C2×#41—24no—
212C2×#41—24no—
213C2×#41—24no—
214C2×#41—24no—
215C2×#41—24no—
216C22×Q8—21no—
217C22×Q8—21no—
218C22×Q8C2×D412yes—
219C22×D4C2×D414yes—
220C22×D4C2×D412yes—
221C22×D4C2×D412yes—
222C22×D4C2×D412yes—
223C23×C4—21no—
224C23×C4—24no—
225C23×C4—21no—
226C23×C4—21no—
227C23×C4—21no—
228C23×C4C2×D411yescharacteristic
229C23×C4C2×D411yescharacteristic
230C23×C4C2414yes—
231C23×C4C2411yescharacteristic
232C2×M16—24no—
233C2×M16—24no—
234C2×M16—24no—
235C22×C8—22no—
236C22×C8—22no—
237C22×C8—22no—
238C22×C8—22no—
239C22×C8—22no—
240C22×C8#4114yes—
241C22×C8C2412yes—
242C22×C8C2414yes—
243C22×C8C2412yes—
244C4⋊Q8—24no—
245C4⋊Q8—22no—
246C4⋊Q8—22no—
247C4⋊Q8—22no—
248C4⋊Q8—22no—
249#127—24no—
250#125—24no—
251#125—24no—
252#124—28no—
253#124—84no—
254C22⋊Q8—44no—
255C22⋊Q8—44no—
256#121—28no—
257#121—28no—
258C4×Q8—48no—
259C4×Q8—44no—
260C4×Q8—44no—
261C4×D4—24no—
262C4×D4—24no—
263C4×D4—24no—
264C4×D4—24no—
265C4×D4—48no—
266C4×D4—416no—
267C4×D4—84no—
268#117—24no—
269#117—24no—
270#117—24no—
271#117—24no—
272#117—28no—
273#117—44no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—28no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—44no—
296C2×C4⋊C4C2×D412yes—
297C2×C4⋊C4C2×D412yes—
298C2×C4⋊C4#4114yes—
299C2×C4⋊C4C2412yes—
300C2×C4⋊C4C2412yes—
301C2×C4⋊C4C2412yes—
302C2×C4⋊C4C2412yes—
303C2×C4⋊C4C2412yes—
304C2×C4⋊C4C2412yes—
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2412yes—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4C2×D414yes—
317C2×C22⋊C4C2×D414yes—
318C2×C42—24no—
319C2×C42—22no—
320C2×C42—22no—
321C2×C42—22no—
322C2×C42—22no—
323C2×C42—22no—
324C2×C42—22no—
325C2×C42—82no—
326C2×C42—82no—
327C2×C42C2×D411yescharacteristic
328C2×C42C2×D411yescharacteristic
329C2×C42#4114yes—
330C2×C42#4114yes—
331C2×C42C2411yescharacteristic
332C2×C42C2411yescharacteristic
333#107—28no—
334#107—44no—
335#106—28no—
336#106—44no—
337C4⋊C8—24no—
338C4⋊C8—24no—
339C22⋊C8—28no—
340C22⋊C8—28no—
341C22⋊C8—44no—
342C22⋊C8—44no—
343#97—24no—
344#97—24no—
345#95—22no—
346#95—22no—
347#95—22no—
348#95—22no—
349#95—22no—
350#95—24no—
351#95—22no—
352#95—24no—
353#95—24no—
354#95—24no—
355#95—24no—
356#95—24no—
357C24C22×D414yes—
358C24C22×D411yescharacteristic
359#41—48no—
360#41—48no—
361#41—48no—
362#41—48no—
363C2×Q8—24no—
364C2×Q8—24no—
365C2×Q8—24no—
366C2×Q8—42no—
367C2×Q8—42no—
368C2×Q8—42no—
369C2×Q8—42no—
370C2×Q8—81no—
371C2×Q8—81no—
372C2×D4—28no—
373C2×D4—28no—
374C2×D4—28no—
375C2×D4—24no—
376C2×D4—24no—
377C2×D4—24no—
378C2×D4—24no—
379C2×D4—24no—
380C2×D4—42no—
381C2×D4—42no—
382C2×D4—42no—
383C2×D4—42no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—48no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—81no—
408C22×C4—81no—
409C22×C4—81no—
410C22×C4—81no—
411C22×C4—81no—
412C22×C4—81no—
413C22×C4—81no—
414C22×C4—81no—
415C22×C4—84no—
416C22×C4—84no—
417C22×C4#12314yes—
418C22×C4#12314yes—
419C22×C4C22×D414yes—
420C22×C4C22×D411yescharacteristic
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D411yescharacteristic
423C22×C4C22×D411yescharacteristic
424C22×C4C22×D411yescharacteristic
425C22×C4C2×#4112yes—
426C22×C4C2×#4112yes—
427C22×C4C2×#4112yes—
428C22×C4C2×#4112yes—
429C22×C4H(5,𝔽2)12yes—
430C22×C42−1+412yes—
431C22×C4C2511yescharacteristic
432M16—48no—
433C2×C8—28no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—24no—
438C2×C8—24no—
439C2×C8—22no—
440C2×C8—24no—
441C2×C8—22no—
442C2×C8—44no—
443C2×C8C2×#4114yes—
444C4⋊C4—24no—
445C4⋊C4—24no—
446C4⋊C4—24no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—48no—
463C4⋊C4—42no—
464C4⋊C4—42no—
465C4⋊C4—44no—
466C4⋊C4—44no—
467C4⋊C4—42no—
468C4⋊C4—42no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—42no—
472C4⋊C4—81no—
473C4⋊C4—81no—
474C4⋊C4C2×#4114yes—
475C22⋊C4—28no—
476C22⋊C4—28no—
477C22⋊C4—48no—
478C22⋊C4—44no—
479C22⋊C4—44no—
480C22⋊C4—84no—
481C22⋊C4—84no—
482C22⋊C4—84no—
483C22⋊C4—84no—
484C22⋊C4—84no—
485C22⋊C4—84no—
486C42—48no—
487C42—44no—
488C42—44no—
489C42—44no—
490C42—44no—
491C42—82no—
492C42—82no—
493C42#12314yes—
494C42#12314yes—
495C42C2×#4112yes—
496C42C2×#4112yes—
497C42C2×#4112yes—
498C42C2×#4112yes—
499C23—21no—
500C23—21no—
501C23—21no—
502C23—21no—
503C23—24no—
504C23—24no—
505C23—24no—
506C23—24no—
507C23—24no—
508C23—24no—
509C23—24no—
510C23C2×#12312yes—
511C23#54612yes—
512C23#54712yes—
513C23C23×D411yescharacteristic
514Q8—48no—
515Q8—84no—
516Q8—84no—
517D4—48no—
518D4—416no—
519D4—48no—
520D4—84no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—84no—
563C2×C4—84no—
564C2×C4—84no—
565C2×C4—84no—
566C2×C4—168no—
567C2×C4C2×#12312yes—
568C2×C4C2×#12312yes—
569C2×C4C2×#12312yes—
570C2×C4C2×#12312yes—
571C2×C4C2×#12312yes—
572C2×C4#52914yes—
573C2×C4#54214yes—
574C2×C4#54612yes—
575C2×C4#54712yes—
576C2×C4#55614yes—
577C2×C4#55914yes—
578C2×C4C22×#4111yescharacteristic
579C2×C4C22×#4111yescharacteristic
580C2×C4C22×#4111yescharacteristic
581C2×C4C22×#4111yescharacteristic
582C2×C4C2×H(5,𝔽2)11yescharacteristic
583C2×C4C2×2−1+411yescharacteristic
584C8—24no—
585C8—42no—
586C8—44no—
587C22—22no—
588C22—22no—
589C22—22no—
590C22—22no—
591C22—22no—
592C22—22no—
593C22—44no—
594C22—44no—
595C22—44no—
596C22—44no—
597C22—44no—
598C22—44no—
599C22—44no—
600C22#244714yes—
601C22C22×#12311yescharacteristic
602C22C2×#54611yescharacteristic
603C22C2×#54711yescharacteristic
604C22#358511yescharacteristic
605C22#358511yescharacteristic
606C22#358611yescharacteristic
607C22#358611yescharacteristic
608C4—22no—
609C4—22no—
610C4—22no—
611C4—22no—
612C4—44no—
613C4—44no—
614C4—82no—
615C4—82no—
616C4—82no—
617C4—82no—
618C4—82no—
619C4—82no—
620C4—82no—
621C4#244714yes—
622C4C2×#52911yescharacteristic
623C4C2×#54211yescharacteristic
624C4C2×#55611yescharacteristic
625C4C2×#55911yescharacteristic
626C2—24no—
627C2—84no—
628C2C2×#244711yescharacteristic
629C2#6051711yescharacteristic
630C2#6052011yescharacteristic
631C2#6121911yescharacteristic
632C2#6122011yescharacteristic
633C2#6129211yescharacteristic
634C2#6129311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#532 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #26313C2semidirect
4. #26312C2semidirect
5. #26311C2semidirect
6. #26310C2semidirect
7. #23643C2semidirect
8. #23639C2semidirect
9. #21321C2semidirect
10. #21312C2semidirect
12. #21170C2semidirect
13. #20848C2semidirect
14. #20703C2semidirect
15. C2×#1776C2semidirect

Maximal subgroups

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

subgroupindex
#351802
#342062
#263132
#263122
#263112
#263102
#236432
#236392
#213212
#213122
#211882
#211702
#208482
#207032
C2×#17762