Finite Group Databasebeta

← #6457789

Subgroups — #6457789

Order 512.

1516 conjugacy classes of subgroups.

The subgroup lattice has 611 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35629C211yesmaximal, characteristic
2#35557C211yesmaximal, characteristic
3#27848C214yesmaximal, complemented
4#27847C214yesmaximal, complemented
5#27842C214yesmaximal, complemented
6#27841C214yesmaximal, complemented
7#23746C211yesmaximal, characteristic, complemented
8#23745C211yesmaximal, characteristic, complemented
9#23640C211yesmaximal, characteristic
10#23639C211yesmaximal, characteristic
11#21158C211yesmaximal, characteristic, complemented
12#21154C211yesmaximal, characteristic, complemented
13#21038C211yesmaximal, characteristic, complemented
14#21037C211yesmaximal, characteristic, complemented
15C2×#1847C211yesmaximal, characteristic, complemented
16#20663C211yesmaximal, characteristic
17#20662C211yesmaximal, characteristic
18#20454C211yesmaximal, characteristic, complemented
19#20384C211yesmaximal, characteristic
20C22×#131C2211yescharacteristic
21C2×#427C2211yescharacteristic
22C2×#427C2211yescharacteristic
23C22×#107C2211yescharacteristic
24C22×#107C2211yescharacteristic
25C22×#106C2211yescharacteristic
26C22×#106C2211yescharacteristic
27C2×#408C2211yescharacteristic
28C2×#405C2211yescharacteristic
29C22×#97C2211yescharacteristic
30#2514—22no—
31#2513—22no—
32#2512—22no—
33#2511—22no—
34#2510—24no—
35#2497C2214yes—
36#2496C2214yes—
37#2495C2214yes—
38#2494C2214yes—
39#2481—24no—
40#2316C2211yescharacteristic
41#2316C2211yescharacteristic
42#2311C2211yescharacteristic
43#2311C2211yescharacteristic
44#2311C2211yescharacteristic
45#2311C2211yescharacteristic
46#2297C2211yescharacteristic
47#2296C2211yescharacteristic
48#2296C2211yescharacteristic
49C2×#384C2211yescharacteristic
50C2×#382C2211yescharacteristic
51#2274C2211yescharacteristic
52#2274C2211yescharacteristic
53#2101C2214yes—
54#2101C2214yes—
55#2100C2214yes—
56#2099C2214yes—
57#2077C2214yes—
58#2077C2214yes—
59#2076C2214yes—
60#2075C2214yes—
61#1944C2212yes—
62#1944C2212yes—
63#1943C2212yes—
64#1943C2212yes—
65#1943C2214yes—
66#1942C2214yes—
67#1921C2214yes—
68#1920C2214yes—
69#1919C2214yes—
70#1918C2214yes—
71#1847C2212yes—
72#1847C2212yes—
73#1847C2212yes—
74#1847C2212yes—
75#1809C2214yes—
76#1809C2214yes—
77#1808C2214yes—
78#1807C2214yes—
79#1776C2214yes—
80C4×#106C2214yes—
81#1753C2212yes—
82#1753C2212yes—
83#1752C2212yes—
84#1752C2212yes—
85#1736C2211yescharacteristic
86#1736C2211yescharacteristic
87#1736C2211yescharacteristic
88#1736C2211yescharacteristic
89#1736C2211yescharacteristic
90#1736C2211yescharacteristic
91#1736C2211yescharacteristic
92#1736C2211yescharacteristic
93#1735C2211yescharacteristic
94#1735C2211yescharacteristic
95#1729C2211yescharacteristic
96#1728C2211yescharacteristic
97C22×#41C2311yescharacteristic
98C2×#131—24no—
99C2×#131—24no—
100C2×#131—24no—
101C22×M16C2312yes—
102C22×M16C2311yescharacteristic
103C22×M16C2311yescharacteristic
104C23×C8C2312yes—
105C23×C8C2311yescharacteristic
106C23×C8C2311yescharacteristic
107C2×#126—24no—
108C2×#117C2311yescharacteristic
109C2×#117C2311yescharacteristic
110C2×#117C2311yescharacteristic
111C2×#117C2311yescharacteristic
112C2×#117C2311yescharacteristic
113C22×C4⋊C4—22no—
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C42C2311yescharacteristic
119#427—24no—
120#427—24no—
121C2×#107—24no—
122C2×#107—24no—
123C2×#107C2312yes—
124C2×#107C2312yes—
125C2×#107C2312yes—
126C2×#107C2312yes—
127C2×#107C2312yes—
128C2×#107C2312yes—
129C2×#106—24no—
130C2×#106—24no—
131C2×#106C2312yes—
132C2×#106C2312yes—
133C2×#106C2312yes—
134C2×#106C2312yes—
135C2×#106C2312yes—
136C2×#106C2312yes—
137#408—24no—
138C2×C22⋊C8C2312yes—
139C2×C22⋊C8C2312yes—
140#405—24no—
141C2×#97—24no—
142C2×#97C2311yescharacteristic
143C2×#97C2311yescharacteristic
144C2×#97C2311yescharacteristic
145C2×#97C2311yescharacteristic
146C2×#97C2312yes—
147C2×C4×C8C2312yes—
148#401—24no—
149#400—24no—
150#400—24no—
151#400—24no—
152#397—24no—
153#397—24no—
154#397—24no—
155#396—24no—
156#391—21no—
157#391—21no—
158#391—21no—
159#391—21no—
160#391D412yes—
161#390—21no—
162#390—21no—
163#390—21no—
164#390—21no—
165#390D412yes—
166#389—22no—
167#389—22no—
168#389C2314yes—
169#389C2314yes—
170#389C2312yes—
171#389C2312yes—
172#389C2312yes—
173#389C2312yes—
174#388—24no—
175#388—24no—
176#388—24no—
177#387—24no—
178#387—24no—
179#387—24no—
180#386—22no—
181#386—22no—
182#386—22no—
183#386—22no—
184#386D414yes—
185#385—22no—
186#385—22no—
187#384—22no—
188#384C2312yes—
189#384C2312yes—
190#383—24no—
191#382—22no—
192#382C2312yes—
193#382C2312yes—
194#380—22no—
195#380—22no—
196#380C2314yes—
197#380C2314yes—
198#380C2314yes—
199#380C2314yes—
200C4×C4⋊C4—22no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4C2314yes—
205C4×C4⋊C4C2314yes—
206C4×C22⋊C4—22no—
207C4×C22⋊C4—22no—
208C4×C22⋊C4C2312yes—
209C4×C22⋊C4C2312yes—
210C2×#95—22no—
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#340C2314yes—
215#340C2312yes—
216#340C2314yes—
217#340C2314yes—
218#340C2312yes—
219#340C2314yes—
220#340C2314yes—
221#340C2312yes—
222#340C2312yes—
223#336C2312yes—
224#336C2312yes—
225#336C2312yes—
226#336C2312yes—
227C2×#41—22no—
228C2×#41—22no—
229C2×#41—22no—
230C2×#41—22no—
231C2×#41—22no—
232C2×#41—22no—
233C22×Q8C2×D411yescharacteristic
234C22×D4C2×D412yes—
235C22×D4C2×D411yescharacteristic
236C23×C4—21no—
237C23×C4—21no—
238C23×C4—21no—
239C23×C4—21no—
240C23×C4C2×D412yes—
241C23×C4C2412yes—
242C23×C4C2411yescharacteristic
243#131—416no—
244C2×M16—22no—
245C2×M16—22no—
246C2×M16—22no—
247C2×M16—22no—
248C2×M16—22no—
249C2×M16—22no—
250C2×M16—24no—
251C2×M16—24no—
252C2×M16—24no—
253C22×C8—22no—
254C22×C8—22no—
255C22×C8—24no—
256C22×C8—24no—
257C22×C8—24no—
258C22×C8#4114yes—
259C22×C8#4114yes—
260C22×C8#4114yes—
261C22×C8#4114yes—
262C22×C8C2412yes—
263C22×C8C2412yes—
264C22×C8C2411yescharacteristic
265C22×C8C2411yescharacteristic
266C22×C8C2411yescharacteristic
267C22×C8C2411yescharacteristic
268#126—88no—
269#117—22no—
270#117—22no—
271#117—22no—
272#117—22no—
273#117—22no—
274#117—22no—
275#117—22no—
276#117—22no—
277#117—22no—
278#117—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—84no—
300C2×C4⋊C4C2×D412yes—
301C2×C4⋊C4C2412yes—
302C2×C4⋊C4C2412yes—
303C2×C4⋊C4C2412yes—
304C2×C4⋊C4C2412yes—
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2412yes—
309C2×C22⋊C4—21no—
310C2×C22⋊C4—21no—
311C2×C22⋊C4—21no—
312C2×C22⋊C4—21no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4C2×D412yes—
318C2×C22⋊C4C2×D414yes—
319C2×C22⋊C4C2×D414yes—
320C2×C42—22no—
321C2×C42—22no—
322C2×C42—21no—
323C2×C42—21no—
324C2×C42—21no—
325C2×C42—21no—
326C2×C42—24no—
327C2×C42—24no—
328C2×C42C2×D412yes—
329C2×C42C2412yes—
330C2×C42C2411yescharacteristic
331C2×C42C2411yescharacteristic
332#107—24no—
333#107—24no—
334#107—24no—
335#107—24no—
336#107—24no—
337#107—24no—
338#106—24no—
339#106—24no—
340#106—24no—
341#106—24no—
342#106—24no—
343#106—24no—
344C22⋊C8—24no—
345C22⋊C8—24no—
346#97—22no—
347#97—22no—
348#97—22no—
349#97—22no—
350#97—24no—
351C4×C8—24no—
352#95—24no—
353#95—24no—
354#95—24no—
355#95—24no—
356#95—22no—
357#95—22no—
358#95—22no—
359#95—22no—
360#95—24no—
361#95—24no—
362#95—24no—
363#95—24no—
364#95—24no—
365#95—24no—
366#95C2×D414yes—
367#95C2×D414yes—
368C24C22×D412yes—
369C24C22×D411yescharacteristic
370#41—44no—
371#41—44no—
372#41—44no—
373#41—44no—
374C2×Q8—22no—
375C2×Q8—22no—
376C2×Q8—22no—
377C2×D4—24no—
378C2×D4—24no—
379C2×D4—24no—
380C2×D4—22no—
381C2×D4—22no—
382C2×D4—22no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—82no—
405C22×C4—82no—
406C22×C4—82no—
407C22×C4—82no—
408C22×C4C22×D412yes—
409C22×C4C22×D414yes—
410C22×C4C22×D412yes—
411C22×C4C22×D412yes—
412C22×C4C22×D411yescharacteristic
413C22×C4C2×#4112yes—
414C22×C4C2×#4112yes—
415C22×C4C2×#4112yes—
416C22×C4C2×#4112yes—
417C22×C42−1+412yes—
418C22×C42−1+412yes—
419C22×C4C2511yescharacteristic
420M16—44no—
421M16—44no—
422M16—48no—
423C2×C8—28no—
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—21no—
427C2×C8—21no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—48no—
435C2×C8#126116yescomplemented
436C2×C8C2×#4112yes—
437C2×C8C2×#4112yes—
438C2×C8C2×#4112yes—
439C2×C8C2×#4112yes—
440C2×C8C2×#4112yes—
441C2×C8C2×#4112yes—
442C2×C8C2×#4112yes—
443C2×C8C2×#4112yes—
444C4⋊C4—24no—
445C4⋊C4—22no—
446C4⋊C4—22no—
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—44no—
462C4⋊C4—48no—
463C4⋊C4—44no—
464C4⋊C4—42no—
465C4⋊C4—42no—
466C4⋊C4—42no—
467C4⋊C4—42no—
468C4⋊C4—84no—
469C4⋊C4—82no—
470C4⋊C4—82no—
471C4⋊C4—84no—
472C4⋊C4—84no—
473C22⋊C4—24no—
474C22⋊C4—44no—
475C22⋊C4—44no—
476C22⋊C4—42no—
477C22⋊C4—42no—
478C22⋊C4—42no—
479C22⋊C4—42no—
480C22⋊C4—84no—
481C22⋊C4—84no—
482C22⋊C4—84no—
483C22⋊C4—84no—
484C42—24no—
485C42—22no—
486C42—21no—
487C42—21no—
488C42—21no—
489C42—21no—
490C42—22no—
491C42—42no—
492C42—42no—
493C42—42no—
494C42—42no—
495C42—84no—
496C42—84no—
497C23—22no—
498C23—22no—
499C23—24no—
500C23—24no—
501C23—22no—
502C23—22no—
503C23—22no—
504C23#53212yes—
505C23#54712yes—
506C23#54712yes—
507C23C23×D411yescharacteristic
508Q8—44no—
509D4—48no—
510D4—44no—
511C2×C4—24no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—24no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—24no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—48no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—44no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—44no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—162no—
549C2×C4—162no—
550C2×C4—162no—
551C2×C4—162no—
552C2×C4C2×#12614yes—
553C2×C4#53212yes—
554C2×C4#54114yes—
555C2×C4#54712yes—
556C2×C4#54712yes—
557C2×C4#55214yes—
558C2×C4#55614yes—
559C2×C4C22×#4111yescharacteristic
560C2×C4C22×#4111yescharacteristic
561C2×C4C22×#4111yescharacteristic
562C2×C4C22×#4111yescharacteristic
563C2×C4C2×2−1+411yescharacteristic
564C2×C4C2×2−1+411yescharacteristic
565C8—22no—
566C8—22no—
567C8—42no—
568C8—42no—
569C8C2×#12614yes—
570C8C2×#12614yes—
571C22—22no—
572C22—22no—
573C22—24no—
574C22—24no—
575C22—42no—
576C22—42no—
577C22—44no—
578C22—44no—
579C22—42no—
580C22#249614yes—
581C22C2×#53211yescharacteristic
582C22C2×#54711yescharacteristic
583C22C2×#54711yescharacteristic
584C22#358512yes—
585C22#358612yes—
586C4—42no—
587C4—42no—
588C4—44no—
589C4—42no—
590C4—82no—
591C4—82no—
592C4—82no—
593C4—82no—
594C4—82no—
595C4—82no—
596C4—82no—
597C4—82no—
598C4#249614yes—
599C4C22×#12611yescharacteristic
600C4C2×#54111yescharacteristic
601C4C2×#55211yescharacteristic
602C4C2×#55611yescharacteristic
603C2—24no—
604C2—82no—
605C2C2×#249611yescharacteristic
606C2#6071911yescharacteristic
607C2#6072011yescharacteristic
608C2#6126512yes—
609C2#6129412yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2297 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #27848C2semidirect
4. #27847C2semidirect
5. #27842C2semidirect
6. #27841C2semidirect
7. #23746C2semidirect
8. #23745C2semidirect
11. #21158C2semidirect
12. #21154C2semidirect
13. #21038C2semidirect
14. #21037C2semidirect
15. C2×#1847C2semidirect
18. #20454C2semidirect
435. C2×C8#126semidirect

Maximal subgroups

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

subgroupindex
#356292
#355572
#278482
#278472
#278422
#278412
#237462
#237452
#236402
#236392
#211582
#211542
#210382
#210372
C2×#18472
#206632
#206622
#204542
#203842