Finite Group Databasebeta

← #6556869

Subgroups — #6556869

Order 512.

1450 conjugacy classes of subgroups.

The subgroup lattice has 617 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36598C211yesmaximal, characteristic, complemented
2#36446C211yesmaximal, characteristic
3#30215C212yesmaximal, complemented
4#30205C211yesmaximal, characteristic, complemented
5#29355C211yesmaximal, characteristic, complemented
6#29347C212yesmaximal, complemented
7#29148C211yesmaximal, characteristic
8#29142C212yesmaximal
9#27764C214yesmaximal
10#27748C212yesmaximal
11#27742C214yesmaximal, complemented
12#27730C212yesmaximal, complemented
13#23821C212yesmaximal
14#23817C211yesmaximal, characteristic
15#22547C212yesmaximal, complemented
16#22523C212yesmaximal
17#20573C211yesmaximal, characteristic, complemented
18C2×#557C2211yescharacteristic
19C2×D4×Q8C2211yescharacteristic
20C2×#446C2212yes—
21#2868C2211yescharacteristic
22#2862C2212yes—
23#2862C2211yescharacteristic
24#2862C2212yes—
25#2582—21no—
26#2571C2212yes—
27#2570C2214yes—
28#2568—21no—
29#2568C2211yescharacteristic
30#2568C2212yes—
31#2547C2211yescharacteristic
32#2547C2212yes—
33#2511—22no—
34#2489—21no—
35#2489—21no—
36#2489—22no—
37#2489—22no—
38#2446—22no—
39#2365—21no—
40#2355—21no—
41#2355C2212yes—
42Q8×C22⋊C4—21no—
43Q8×C22⋊C4—21no—
44C4×C4⋊Q8C2211yescharacteristic
45C4×C4⋊Q8C2212yes—
46C4×#124C2211yescharacteristic
47C4×#124C2212yes—
48C42×Q8C2211yescharacteristic
49#2101C2212yes—
50#2101C2214yes—
51#2074C2214yes—
52#2074C2214yes—
53#2072C2211yescharacteristic
54#2072C2212yes—
55#2072C2214yes—
56#2071C2211yescharacteristic
57#2042C2212yes—
58#2042C2214yes—
59#2024C2211yescharacteristic
60#2024C2212yes—
61#2024C2212yes—
62#2024C2212yes—
63#2024C2212yes—
64#2023C2211yescharacteristic
65#2023C2212yes—
66#1993C2214yes—
67#1990C2212yes—
68#1988C2212yes—
69#1987C2214yes—
70#1981C2212yes—
71#1981C2214yes—
72#1960C2214yes—
73#1959C2212yes—
74#1958C2214yes—
75#1956C2212yes—
76#1935C2214yes—
77#1933C2214yes—
78#1933C2214yes—
79#1836C2212yes—
80#1835C2214yes—
81#1805C2214yes—
82#1804C2214yes—
83#1803C2212yes—
84#1802C2212yes—
85#1771C2214yes—
86C8×C4⋊C4C2212yes—
87#1767C2211yescharacteristic
88C4×C4⋊C8C2212yes—
89#1763C2214yes—
90#1762C2214yes—
91C4×#103C2212yes—
92C4×#102C2212yes—
93#1745C2212yes—
94#1744C2211yescharacteristic
95C23×Q8—21no—
96#557—28no—
97D4×Q8—28no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2312yes—
102C2×C4⋊Q8C2312yes—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2312yes—
106C2×#125—21no—
107C2×#125—22no—
108C2×#124C2312yes—
109C2×#124C2311yescharacteristic
110C2×C22⋊Q8—21no—
111C2×C22⋊Q8—22no—
112C2×C4×Q8—21no—
113C2×C4×Q8—21no—
114C2×C4×Q8—22no—
115C2×C4×Q8D411yescharacteristic
116C2×C4×Q8D411yescharacteristic
117C2×C4×Q8C2311yescharacteristic
118C2×C4×Q8C2312yes—
119C2×C4×Q8C2311yescharacteristic
120C2×C4×D4C2311yescharacteristic
121C2×C4×D4C2312yes—
122#446—28no—
123C2×C4⋊C8C2312yes—
124C2×C4⋊C8C2314yes—
125C2×#103C2314yes—
126C2×#103C2314yes—
127C2×#103C2314yes—
128C2×#103C2314yes—
129C2×#103C2312yes—
130C2×#103C2314yes—
131C2×#103C2312yes—
132C2×#102C2312yes—
133C2×#102C2314yes—
134C2×#102C2312yes—
135C2×#97C2314yes—
136C2×#97C2312yes—
137C2×C4×C8C2312yes—
138C2×C4×C8C2314yes—
139#400—24no—
140#400—22no—
141#400—22no—
142#400—22no—
143#400—24no—
144#398—24no—
145#397—22no—
146#397—22no—
147#397—22no—
148#391—22no—
149#391—21no—
150#391—21no—
151#391—22no—
152#391—22no—
153#391—22no—
154#391—21no—
155#391—22no—
156#391—22no—
157#391—21no—
158#391—21no—
159#391—22no—
160#391—22no—
161#391—22no—
162#391—22no—
163#391C2312yes—
164#391C2312yes—
165#391C2312yes—
166#391C2311yescharacteristic
167#391C2312yes—
168#390C2311yescharacteristic
169#389—22no—
170#389—22no—
171#389—21no—
172#389C2312yes—
173#389C2314yes—
174#388—22no—
175#388—21no—
176#388—22no—
177#388—21no—
178#387—24no—
179#387—24no—
180#387—24no—
181#384—21no—
182#384—21no—
183#384C2311yescharacteristic
184#384C2312yes—
185#384C2312yes—
186#382—22no—
187#382—22no—
188#382—21no—
189#382—22no—
190#382—21no—
191#382C2312yes—
192#382C2311yescharacteristic
193#382C2314yes—
194#382C2314yes—
195#380—21no—
196#380—21no—
197#380—22no—
198#380—22no—
199C4×C4⋊C4—22no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4D411yescharacteristic
203C4×C4⋊C4D411yescharacteristic
204C4×C4⋊C4D412yes—
205C4×C4⋊C4D412yes—
206C4×C4⋊C4C2314yes—
207C4×C4⋊C4C2312yes—
208C4×C4⋊C4C2312yes—
209C4×C4⋊C4C2311yescharacteristic
210C4×C22⋊C4—21no—
211C4×C22⋊C4—21no—
212C4×C22⋊C4—22no—
213C4×C22⋊C4—22no—
214C43C2312yes—
215C43C2311yescharacteristic
216#340C2314yes—
217#340C2314yes—
218#340C2314yes—
219#340C2314yes—
220#340C2312yes—
221#340C2312yes—
222#340C2314yes—
223#340C2312yes—
224#340C2314yes—
225#340C2312yes—
226#336C2311yescharacteristic
227#336C2312yes—
228#336C2311yescharacteristic
229#336C2312yes—
230#336C2314yes—
231C22×Q8—22no—
232C22×Q8—21no—
233C22×Q8—22no—
234C22×Q8—21no—
235C22×Q8—44no—
236C22×Q8—44no—
237C22×Q8—44no—
238C22×Q8C2×D411yescharacteristic
239C22×Q8C2411yescharacteristic
240C22×Q8C2412yes—
241C22×D4C2411yescharacteristic
242C23×C4—22no—
243C23×C4—21no—
244C22×C8C2412yes—
245C22×C8C2412yes—
246C22×C8C2414yes—
247C4⋊Q8—24no—
248C4⋊Q8—28no—
249C4⋊Q8—24no—
250C4⋊Q8—28no—
251C4⋊Q8—42no—
252C4⋊Q8—44no—
253C4⋊Q8—44no—
254C4⋊Q8—82no—
255#125—48no—
256#125—44no—
257#124—24no—
258#124—22no—
259#124—42no—
260#124—41no—
261C22⋊Q8—48no—
262C22⋊Q8—44no—
263C4×Q8—24no—
264C4×Q8—28no—
265C4×Q8—42no—
266C4×Q8—42no—
267C4×Q8—44no—
268C4×Q8—44no—
269C4×Q8—84no—
270C4×Q8#4118yes—
271C4×D4—24no—
272C4×D4—28no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—21no—
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—21no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4C2×D411yescharacteristic
292C2×C4⋊C4C2×D411yescharacteristic
293C2×C4⋊C4C2×D412yes—
294C2×C4⋊C4C2×D412yes—
295C2×C4⋊C4C2×D411yescharacteristic
296C2×C4⋊C4C2×D412yes—
297C2×C4⋊C4C2411yescharacteristic
298C2×C4⋊C4C2412yes—
299C2×C4⋊C4C2411yescharacteristic
300C2×C4⋊C4C2412yes—
301C2×C4⋊C4C2414yes—
302C2×C4⋊C4C2412yes—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—21no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—21no—
308C2×C22⋊C4—21no—
309C2×C22⋊C4—21no—
310C2×C22⋊C4—21no—
311C2×C42—22no—
312C2×C42—21no—
313C2×C42—22no—
314C2×C42—24no—
315C2×C42C2×D411yescharacteristic
316C2×C42C2×D412yes—
317C2×C42C2×D411yescharacteristic
318C2×C42C2×D411yescharacteristic
319C2×C42C2×D412yes—
320C2×C42C2×D411yescharacteristic
321C2×C42C2×D412yes—
322C2×C42C2×D412yes—
323C2×C42C2412yes—
324C2×C42C2411yescharacteristic
325C2×C42C2411yescharacteristic
326C2×C42C2412yes—
327C2×C42C2411yescharacteristic
328C4⋊C8—24no—
329C4⋊C8—28no—
330#103—44no—
331#103—42no—
332#103—42no—
333#103—44no—
334#103—44no—
335#103—44no—
336#103—44no—
337#102—24no—
338#102—44no—
339#102—42no—
340#97—28no—
341#97—42no—
342C4×C8—24no—
343C4×C8—44no—
344#95—21no—
345#95—22no—
346#95—21no—
347#95—22no—
348#95—22no—
349#95—22no—
350#95—21no—
351#95—21no—
352#95—24no—
353#95—24no—
354#95—22no—
355#95—22no—
356#95—24no—
357#95—22no—
358#95—22no—
359#95—22no—
360#95—22no—
361#95—22no—
362#95—22no—
363#95—24no—
364#95—24no—
365#95—24no—
366#95C2×D412yes—
367#95C2×D412yes—
368#95C2×D412yes—
369C24—21no—
370C2×Q8—22no—
371C2×Q8—22no—
372C2×Q8—22no—
373C2×Q8—24no—
374C2×Q8—22no—
375C2×Q8—21no—
376C2×Q8—24no—
377C2×Q8—22no—
378C2×Q8—21no—
379C2×Q8—22no—
380C2×Q8—42no—
381C2×Q8—42no—
382C2×Q8—44no—
383C2×Q8—44no—
384C2×Q8—44no—
385C2×Q8—44no—
386C2×Q8—44no—
387C2×Q8—42no—
388C2×Q8—42no—
389C2×Q8—42no—
390C2×Q8—44no—
391C2×Q8—42no—
392C2×Q8—816no—
393C2×Q8#12414yes—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—21no—
397C2×D42−1+412yes—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—44no—
407C22×C4—44no—
408C22×C4—44no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4C22×D412yes—
413C22×C4C22×D411yescharacteristic
414C22×C4C22×D412yes—
415C22×C4C22×D411yescharacteristic
416C22×C4C22×D412yes—
417C22×C4C22×D412yes—
418C22×C4C22×D411yescharacteristic
419C22×C4C22×D411yescharacteristic
420C22×C4C22×D411yescharacteristic
421C22×C4C22×D411yescharacteristic
422C22×C4C2511yescharacteristic
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—24no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—24no—
429C2×C8—22no—
430C2×C8—24no—
431C2×C8—22no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—24no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—24no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—24no—
443C4⋊C4—42no—
444C4⋊C4—42no—
445C4⋊C4—42no—
446C4⋊C4—44no—
447C4⋊C4—44no—
448C4⋊C4—42no—
449C4⋊C4—44no—
450C4⋊C4—42no—
451C4⋊C4—44no—
452C4⋊C4—44no—
453C4⋊C4—42no—
454C4⋊C4—42no—
455C4⋊C4—44no—
456C4⋊C4—42no—
457C4⋊C4—44no—
458C4⋊C4—44no—
459C4⋊C4—42no—
460C4⋊C4—84no—
461C4⋊C4—82no—
462C4⋊C4—82no—
463C4⋊C4#12418yes—
464C4⋊C4#12414yes—
465C22⋊C4—44no—
466C22⋊C4—42no—
467C22⋊C4—44no—
468C22⋊C4—42no—
469C22⋊C4—42no—
470C22⋊C4—44no—
471C22⋊C4—42no—
472C22⋊C4—42no—
473C42—22no—
474C42—22no—
475C42—24no—
476C42—22no—
477C42—24no—
478C42—22no—
479C42—22no—
480C42—21no—
481C42—22no—
482C42—21no—
483C42—22no—
484C42—42no—
485C42—42no—
486C42—44no—
487C42—42no—
488C42—44no—
489C42—84no—
490C42C2×#4114yes—
491C42C2×#4118yes—
492C23—22no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23—41no—
499C23C23×D411yescharacteristic
500Q8—28no—
501Q8—42no—
502Q8—44no—
503Q8—42no—
504Q8—44no—
505Q8—88no—
506Q8—84no—
507Q8—88no—
508Q8—84no—
509D4—24no—
510D4—42no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—21no—
519C2×C4—22no—
520C2×C4—21no—
521C2×C4—22no—
522C2×C4—21no—
523C2×C4—22no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—21no—
533C2×C4—41no—
534C2×C4—42no—
535C2×C4—41no—
536C2×C4—42no—
537C2×C4—41no—
538C2×C4—44no—
539C2×C4—42no—
540C2×C4—41no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—41no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—41no—
550C2×C4—42no—
551C2×C4—41no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—42no—
557C2×C4—88no—
558C2×C4—84no—
559C2×C4C2×#12414yes—
560C2×C4C2×#12412yes—
561C2×C4C2×#12414yes—
562C2×C4C2×#12412yes—
563C2×C4#57512yes—
564C2×C4#57512yes—
565C2×C4#57712yes—
566C2×C4#57812yes—
567C2×C4C22×#4112yes—
568C2×C4C2×2−1+411yescharacteristic
569C2×C4#58512yes—
570C2×C4#58511yescharacteristic
571C8—42no—
572C8—42no—
573C8—44no—
574C22—42no—
575C22—42no—
576C22—42no—
577C22—81no—
578C22—81no—
579C22—81no—
580C22—81no—
581C22C22×#12411yescharacteristic
582C22#348711yescharacteristic
583C22#348911yescharacteristic
584C22C2×#57511yescharacteristic
585C22C2×#57511yescharacteristic
586C22C2×#57711yescharacteristic
587C22C2×#57811yescharacteristic
588C4—24no—
589C4—24no—
590C4—22no—
591C4—22no—
592C4—22no—
593C4—41no—
594C4—42no—
595C4—41no—
596C4—42no—
597C4—82no—
598C4—81no—
599C4—82no—
600C4—82no—
601C4—81no—
602C4—81no—
603C4—84no—
604C4—82no—
605C4#314014yes—
606C4#351612yes—
607C4#352412yes—
608C2—82no—
609C2#3585411yescharacteristic
610C2C2×#314011yescharacteristic
611C2#6062311yescharacteristic
612C2#6190411yescharacteristic
613C2#6190511yescharacteristic
614C2#6213311yescharacteristic
615C2#6213411yescharacteristic
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
3. #30215C2semidirect
4. #30205C2semidirect
5. #29355C2semidirect
6. #29347C2semidirect
11. #27742C2semidirect
12. #27730C2semidirect
15. #22547C2semidirect
17. #20573C2semidirect

Maximal subgroups

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

subgroupindex
#365982
#364462
#302152
#302052
#293552
#293472
#291482
#291422
#277642
#277482
#277422
#277302
#238212
#238172
#225472
#225232
#205732