Finite Group Databasebeta

← #6556851

Subgroups — #6556851

Order 512.

1415 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36598C211yesmaximal, characteristic, complemented
2#36508C211yesmaximal, characteristic
3#30208C212yesmaximal
4#30205C211yesmaximal, characteristic, complemented
5#29355C211yesmaximal, characteristic, complemented
6#29352C212yesmaximal
7#29148C211yesmaximal, characteristic
8#29147C212yesmaximal, complemented
9#27757C212yesmaximal, complemented
10#27754C214yesmaximal
11#27736C212yesmaximal
12#27734C214yesmaximal, complemented
13#23817C211yesmaximal, characteristic
14#23816C212yesmaximal, complemented
15#22541C212yesmaximal
16#22521C212yesmaximal, complemented
17#20572C211yesmaximal, characteristic, complemented
18C2×#557C2211yescharacteristic
19C2×#556C2211yescharacteristic
20C2×C8×Q8C2212yes—
21#2868C2211yescharacteristic
22#2862C2211yescharacteristic
23#2862C2212yes—
24#2861C2212yes—
25#2582—21no—
26#2571C2212yes—
27#2570C2214yes—
28#2568—21no—
29#2565C2212yes—
30#2546C2211yescharacteristic
31#2545C2212yes—
32#2542C2211yescharacteristic
33#2519—22no—
34#2519—22no—
35#2511—22no—
36#2446—22no—
37#2429—21no—
38#2429—21no—
39#2365—21no—
40#2362—21no—
41#2362—21no—
42#2355—21no—
43#2355C2212yes—
44C4×C4⋊Q8C2212yes—
45C4×C4⋊Q8C2211yescharacteristic
46C4×C4⋊Q8C2212yes—
47C4×#124C2211yescharacteristic
48C42×Q8C2211yescharacteristic
49#2101C2212yes—
50#2100C2214yes—
51#2074C2214yes—
52#2073C2214yes—
53#2072C2211yescharacteristic
54#2072C2212yes—
55#2072C2214yes—
56#2071C2211yescharacteristic
57#2043C2214yes—
58#2042C2212yes—
59#2024C2212yes—
60#2024C2212yes—
61#2024C2212yes—
62#2024C2211yescharacteristic
63#2024C2212yes—
64#2023C2211yescharacteristic
65#2023C2212yes—
66#1991C2214yes—
67#1990C2212yes—
68#1989C2214yes—
69#1988C2212yes—
70#1983C2214yes—
71#1981C2212yes—
72#1960C2214yes—
73#1959C2212yes—
74#1958C2214yes—
75#1956C2212yes—
76#1936C2214yes—
77#1935C2214yes—
78#1935C2214yes—
79#1836C2212yes—
80#1836C2214yes—
81#1805C2214yes—
82#1805C2214yes—
83#1803C2212yes—
84#1802C2212yes—
85#1770C2214yes—
86#1770C2212yes—
87#1767C2212yes—
88#1767C2211yescharacteristic
89C4×#103C2214yes—
90C4×#103C2214yes—
91C4×#103C2212yes—
92C4×#102C2212yes—
93#1744C2212yes—
94#1744C2211yescharacteristic
95#557—28no—
96#556—28no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2312yes—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2312yes—
105C2×#126—22no—
106C2×#125—22no—
107C2×#125C2312yes—
108C2×#124C2312yes—
109C2×#124C2311yescharacteristic
110C2×C22⋊Q8—21no—
111C2×C22⋊Q8—21no—
112C2×C22⋊Q8—21no—
113C2×C4×Q8—21no—
114C2×C4×Q8—21no—
115C2×C4×Q8—22no—
116C2×C4×Q8D411yescharacteristic
117C2×C4×Q8D411yescharacteristic
118C2×C4×Q8C2312yes—
119C2×C4×Q8C2312yes—
120C2×C4×Q8C2311yescharacteristic
121C2×C4×Q8C2311yescharacteristic
122C2×C4×D4C2311yescharacteristic
123C2×#117—21no—
124C8×Q8—28no—
125C2×C4⋊C8C2312yes—
126C2×C4⋊C8C2314yes—
127C2×#103C2312yes—
128C2×#103C2314yes—
129C2×#103C2312yes—
130C2×#103C2314yes—
131C2×#103C2314yes—
132C2×#103C2314yes—
133C2×#103C2314yes—
134C2×#102C2314yes—
135C2×#102C2312yes—
136C2×#102C2312yes—
137C2×#97C2312yes—
138C2×#97C2314yes—
139C2×C4×C8C2312yes—
140C2×C4×C8C2314yes—
141#400—24no—
142#400—24no—
143#399—22no—
144#398—22no—
145#398—22no—
146#398—24no—
147#398—22no—
148#396—22no—
149#396—22no—
150#391—22no—
151#391—22no—
152#391—21no—
153#391—22no—
154#391—22no—
155#391—22no—
156#391—21no—
157#391—21no—
158#391—22no—
159#391C2311yescharacteristic
160#391C2312yes—
161#391C2312yes—
162#391C2312yes—
163#390C2311yescharacteristic
164#390C2312yes—
165#389—22no—
166#389—21no—
167#389—21no—
168#389—22no—
169#389—22no—
170#389—22no—
171#389—21no—
172#389C2312yes—
173#389C2314yes—
174#388—22no—
175#388—22no—
176#388—22no—
177#388—22no—
178#388—21no—
179#388—21no—
180#388—22no—
181#388—22no—
182#387—24no—
183#387—22no—
184#387—22no—
185#387—24no—
186#387—24no—
187#384—21no—
188#384—21no—
189#384C2311yescharacteristic
190#384C2312yes—
191#384C2312yes—
192#384C2312yes—
193#382—22no—
194#382—21no—
195#382—21no—
196#382C2311yescharacteristic
197#382C2314yes—
198#382C2314yes—
199C4×C4⋊C4—22no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4D411yescharacteristic
205C4×C4⋊C4D411yescharacteristic
206C4×C4⋊C4D412yes—
207C4×C4⋊C4D412yes—
208C4×C4⋊C4C2314yes—
209C4×C4⋊C4C2312yes—
210C4×C4⋊C4C2312yes—
211C4×C4⋊C4C2311yescharacteristic
212C4×C22⋊C4—21no—
213C4×C22⋊C4—21no—
214C43C2312yes—
215C43C2311yescharacteristic
216#340C2314yes—
217#340C2314yes—
218#340C2312yes—
219#340C2314yes—
220#340C2314yes—
221#340C2312yes—
222#340C2314yes—
223#340C2312yes—
224#340C2314yes—
225#340C2312yes—
226#336C2311yescharacteristic
227#336C2312yes—
228#336C2311yescharacteristic
229#336C2312yes—
230#336C2314yes—
231C22×Q8—21no—
232C22×Q8—22no—
233C22×Q8—21no—
234C22×Q8C2×D411yescharacteristic
235C22×Q8C2412yes—
236C22×Q8C2411yescharacteristic
237C22×D4C2411yescharacteristic
238C23×C4—21no—
239C22×C8C2412yes—
240C22×C8C2412yes—
241C22×C8C2414yes—
242C4⋊Q8—24no—
243C4⋊Q8—24no—
244C4⋊Q8—24no—
245C4⋊Q8—42no—
246C4⋊Q8—44no—
247C4⋊Q8—42no—
248C4⋊Q8—44no—
249C4⋊Q8—44no—
250C4⋊Q8—82no—
251#126—48no—
252#125—28no—
253#125—48no—
254#124—22no—
255#124—28no—
256#124—41no—
257C22⋊Q8—44no—
258C22⋊Q8—44no—
259C22⋊Q8—44no—
260C4×Q8—28no—
261C4×Q8—28no—
262C4×Q8—24no—
263C4×Q8—42no—
264C4×Q8—42no—
265C4×Q8—44no—
266C4×Q8—44no—
267C4×Q8—84no—
268C4×Q8#4118yes—
269C4×D4—24no—
270#117—44no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4C2×D411yescharacteristic
294C2×C4⋊C4C2×D411yescharacteristic
295C2×C4⋊C4C2×D412yes—
296C2×C4⋊C4C2×D412yes—
297C2×C4⋊C4C2×D411yescharacteristic
298C2×C4⋊C4C2×D412yes—
299C2×C4⋊C4C2411yescharacteristic
300C2×C4⋊C4C2411yescharacteristic
301C2×C4⋊C4C2412yes—
302C2×C4⋊C4C2412yes—
303C2×C4⋊C4C2414yes—
304C2×C4⋊C4C2412yes—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—21no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—21no—
311C2×C22⋊C4—21no—
312C2×C22⋊C4—21no—
313C2×C22⋊C4—21no—
314C2×C42—22no—
315C2×C42—22no—
316C2×C42—21no—
317C2×C42—22no—
318C2×C42—24no—
319C2×C42C2×D411yescharacteristic
320C2×C42C2×D412yes—
321C2×C42C2×D411yescharacteristic
322C2×C42C2×D411yescharacteristic
323C2×C42C2×D412yes—
324C2×C42C2×D411yescharacteristic
325C2×C42C2×D412yes—
326C2×C42C2×D412yes—
327C2×C42C2411yescharacteristic
328C2×C42C2411yescharacteristic
329C2×C42C2412yes—
330C2×C42C2411yescharacteristic
331C2×C42C2412yes—
332C4⋊C8—24no—
333C4⋊C8—28no—
334#103—24no—
335#103—44no—
336#103—42no—
337#103—44no—
338#103—44no—
339#103—44no—
340#103—44no—
341#102—42no—
342#102—44no—
343#102—42no—
344#97—42no—
345#97—44no—
346C4×C8—24no—
347C4×C8—28no—
348#95—21no—
349#95—21no—
350#95—21no—
351#95—21no—
352#95—24no—
353#95—22no—
354#95—22no—
355#95—24no—
356#95—22no—
357#95—22no—
358#95—24no—
359#95—24no—
360#95—22no—
361#95—22no—
362#95—24no—
363#95—24no—
364#95—22no—
365#95—22no—
366#95C2×D412yes—
367#95C2×D412yes—
368#95C2×D412yes—
369C24—21no—
370C2×Q8—22no—
371C2×Q8—22no—
372C2×Q8—21no—
373C2×Q8—22no—
374C2×Q8—24no—
375C2×Q8—22no—
376C2×Q8—22no—
377C2×Q8—24no—
378C2×Q8—22no—
379C2×Q8—42no—
380C2×Q8—42no—
381C2×Q8—44no—
382C2×Q8—44no—
383C2×Q8—42no—
384C2×Q8—42no—
385C2×Q8—42no—
386C2×Q8—44no—
387C2×Q8—42no—
388C2×Q8#12414yes—
389C2×Q82−1+412yes—
390C2×D4—22no—
391C2×D4—21no—
392C2×D4—22no—
393C2×D4—21no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4C22×D411yescharacteristic
406C22×C4C22×D412yes—
407C22×C4C22×D411yescharacteristic
408C22×C4C22×D412yes—
409C22×C4C22×D412yes—
410C22×C4C22×D412yes—
411C22×C4C22×D411yescharacteristic
412C22×C4C22×D411yescharacteristic
413C22×C4C22×D411yescharacteristic
414C22×C4C22×D411yescharacteristic
415C22×C4C2511yescharacteristic
416C2×C8—22no—
417C2×C8—22no—
418C2×C8—24no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—24no—
422C2×C8—22no—
423C2×C8—24no—
424C2×C8—22no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—24no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—24no—
434C4⋊C4—22no—
435C4⋊C4—24no—
436C4⋊C4—42no—
437C4⋊C4—42no—
438C4⋊C4—44no—
439C4⋊C4—44no—
440C4⋊C4—42no—
441C4⋊C4—44no—
442C4⋊C4—44no—
443C4⋊C4—44no—
444C4⋊C4—44no—
445C4⋊C4—42no—
446C4⋊C4—44no—
447C4⋊C4—42no—
448C4⋊C4—44no—
449C4⋊C4—42no—
450C4⋊C4—44no—
451C4⋊C4—44no—
452C4⋊C4—44no—
453C4⋊C4—42no—
454C4⋊C4—42no—
455C4⋊C4—44no—
456C4⋊C4—42no—
457C4⋊C4—84no—
458C4⋊C4—82no—
459C4⋊C4—82no—
460C4⋊C4#12414yes—
461C4⋊C4#12418yes—
462C22⋊C4—42no—
463C22⋊C4—42no—
464C22⋊C4—44no—
465C22⋊C4—44no—
466C22⋊C4—44no—
467C22⋊C4—42no—
468C22⋊C4—44no—
469C22⋊C4—42no—
470C22⋊C4—42no—
471C42—22no—
472C42—22no—
473C42—24no—
474C42—22no—
475C42—24no—
476C42—22no—
477C42—21no—
478C42—22no—
479C42—21no—
480C42—22no—
481C42—22no—
482C42—42no—
483C42—42no—
484C42—44no—
485C42—44no—
486C42—42no—
487C42—44no—
488C42—84no—
489C42C2×#4114yes—
490C42C2×#4118yes—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23C23×D411yescharacteristic
499Q8—28no—
500Q8—24no—
501Q8—42no—
502Q8—44no—
503Q8—44no—
504Q8—88no—
505Q8—84no—
506Q8—84no—
507D4—42no—
508D4—42no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—21no—
517C2×C4—22no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—42no—
536C2×C4—41no—
537C2×C4—42no—
538C2×C4—44no—
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—41no—
548C2×C4—41no—
549C2×C4—44no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—42no—
555C2×C4—84no—
556C2×C4C2×#12414yes—
557C2×C4C2×#12412yes—
558C2×C4C2×#12414yes—
559C2×C4C2×#12412yes—
560C2×C4#57512yes—
561C2×C4#57512yes—
562C2×C4#57712yes—
563C2×C4#57812yes—
564C2×C4C22×#4112yes—
565C2×C4C2×2−1+411yescharacteristic
566C2×C4#58512yes—
567C2×C4#58511yescharacteristic
568C8—42no—
569C8—42no—
570C8—44no—
571C22—81no—
572C22—81no—
573C22—81no—
574C22—81no—
575C22—81no—
576C22—81no—
577C22—81no—
578C22C22×#12411yescharacteristic
579C22#348711yescharacteristic
580C22#348911yescharacteristic
581C22C2×#57511yescharacteristic
582C22C2×#57511yescharacteristic
583C22C2×#57711yescharacteristic
584C22C2×#57811yescharacteristic
585C4—24no—
586C4—24no—
587C4—22no—
588C4—22no—
589C4—22no—
590C4—42no—
591C4—41no—
592C4—42no—
593C4—41no—
594C4—42no—
595C4—82no—
596C4—81no—
597C4—82no—
598C4—82no—
599C4—81no—
600C4—84no—
601C4—82no—
602C4#313514yes—
603C4#351612yes—
604C4#352412yes—
605C2—161no—
606C2#3585411yescharacteristic
607C2C2×#313511yescharacteristic
608C2#6062311yescharacteristic
609C2#6190411yescharacteristic
610C2#6190511yescharacteristic
611C2#6214511yescharacteristic
612C2#6214611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36598C2semidirect
4. #30205C2semidirect
5. #29355C2semidirect
8. #29147C2semidirect
9. #27757C2semidirect
12. #27734C2semidirect
14. #23816C2semidirect
16. #22521C2semidirect
17. #20572C2semidirect

Maximal subgroups

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

subgroupindex
#365982
#365082
#302082
#302052
#293552
#293522
#291482
#291472
#277572
#277542
#277362
#277342
#238172
#238162
#225412
#225212
#205722