Finite Group Databasebeta

← #6798432

Subgroups — #6798432

Order 512.

1438 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36165C211yesmaximal, characteristic
2#36134C211yesmaximal, characteristic
3#33297C211yesmaximal, characteristic, complemented
4#31368C214yesmaximal, complemented
5#31366C214yesmaximal, complemented
6#29884C212yesmaximal, complemented
7#29150C211yesmaximal, characteristic, complemented
8#28694C211yesmaximal, characteristic, complemented
9#27969C212yesmaximal, complemented
10#26113C212yesmaximal, complemented
11#26109C212yesmaximal, complemented
12#25677C212yesmaximal, complemented
13#24971C212yesmaximal
14#24969C212yesmaximal
15#21866C212yesmaximal, complemented
16C4×#491C211yesmaximal, characteristic, complemented
17#21270C211yesmaximal, characteristic
18C2×#533C2212yes—
19C2×#501C2212yes—
20C2×#491C2211yescharacteristic
21C2×#432C2212yes—
22#2868C2211yescharacteristic
23#2859C2211yescharacteristic
24#2857C2212yes—
25#2851C2212yes—
26#2571C2211yescharacteristic
27#2571C2211yescharacteristic
28#2552—21no—
29#2542—21no—
30#2538C2212yes—
31#2508—21no—
32#2508—21no—
33#2505—22no—
34#2505—22no—
35#2442C2212yes—
36#2442C2212yes—
37#2441—22no—
38#2429—22no—
39#2423C2214yes—
40#2365—21no—
41#2354—21no—
42Q8×C4⋊C4C2212yes—
43#2349—21no—
44#2349—21no—
45D4×C4⋊C4C2211yescharacteristic
46D4×C4⋊C4C2211yescharacteristic
47C4×C4⋊Q8C2211yescharacteristic
48C4×#125C2211yescharacteristic
49C2×#382C2212yes—
50C42×D4C2211yescharacteristic
51#2101C2214yes—
52#2101C2214yes—
53#2088C2214yes—
54#2086C2214yes—
55#2084C2212yes—
56#2083C2212yes—
57#2077C2214yes—
58#2076C2214yes—
59#2036C2214yes—
60#2035C2214yes—
61#2026C2214yes—
62#1993C2212yes—
63#1993C2212yes—
64#1993C2212yes—
65#1992C2214yes—
66#1992C2211yescharacteristic
67#1960C2212yes—
68#1960C2211yescharacteristic
69#1954C2212yes—
70#1948C2212yes—
71#1947C2212yes—
72#1939C2212yes—
73#1935C2212yes—
74#1935C2212yes—
75#1924C2214yes—
76#1872C2214yes—
77#1872C2214yes—
78#1865C2214yes—
79#1863C2214yes—
80#1836C2211yescharacteristic
81#1836C2212yes—
82#1834C2212yes—
83#1834C2212yes—
84#1824C2214yes—
85#1823C2214yes—
86#1805C2214yes—
87#1805C2214yes—
88#1803C2212yes—
89#1803C2212yes—
90#1767C2211yescharacteristic
91C4×#103C2212yes—
92C4×#103C2212yes—
93#1749C2212yes—
94C4×#97C2211yescharacteristic
95#533—28no—
96#533—44no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2312yes—
101C2×#125—22no—
102C2×#125C2311yescharacteristic
103C2×#125C2312yes—
104C2×C22⋊Q8—22no—
105C2×C22⋊Q8—22no—
106C2×C22⋊Q8C2314yes—
107C2×C4×Q8—22no—
108C2×C4×Q8C2312yes—
109C2×C4×Q8C2314yes—
110C2×C4×D4—21no—
111C2×C4×D4—21no—
112C2×C4×D4D411yescharacteristic
113C2×C4×D4D411yescharacteristic
114C2×C4×D4C2311yescharacteristic
115C2×#117—22no—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2312yes—
118C22×C42C2312yes—
119#501—44no—
120#491—24no—
121#432—44no—
122C2×#107C2314yes—
123C2×#106C2314yes—
124C2×C4⋊C8C2312yes—
125C2×C4⋊C8C2314yes—
126C2×#103C2312yes—
127C2×#103C2312yes—
128C2×#103C2314yes—
129C2×#103C2314yes—
130C2×#103C2312yes—
131C2×#103C2312yes—
132C2×C22⋊C8C2314yes—
133C2×C22⋊C8C2314yes—
134C2×#97C2311yescharacteristic
135C2×#97C2312yes—
136C2×#97C2311yescharacteristic
137C2×#97C2312yes—
138C2×C4×C8C2312yes—
139#400—22no—
140#400—22no—
141#400—22no—
142#400—22no—
143#399—24no—
144#398—24no—
145#398—22no—
146#398—22no—
147#396—24no—
148#391—22no—
149#391C2312yes—
150#390—22no—
151#389—21no—
152#389—21no—
153#389—21no—
154#389—21no—
155#389—22no—
156#389—21no—
157#389—21no—
158#389—22no—
159#389—21no—
160#389—21no—
161#389—21no—
162#389—21no—
163#389—24no—
164#389C2311yescharacteristic
165#389C2311yescharacteristic
166#389C2312yes—
167#389C2312yes—
168#389C2314yes—
169#388—24no—
170#387—21no—
171#387—21no—
172#387—22no—
173#387—22no—
174#387—24no—
175#385—22no—
176#385—22no—
177#385—22no—
178#385—22no—
179#385—22no—
180#385—22no—
181#384—22no—
182#384—21no—
183#384—21no—
184#384C2311yescharacteristic
185#384C2311yescharacteristic
186#384C2311yescharacteristic
187#384C2312yes—
188#384C2312yes—
189#382—21no—
190#382—21no—
191#382—22no—
192#382—22no—
193#382C2311yescharacteristic
194#382C2312yes—
195#382C2312yes—
196#382C2312yes—
197#380—22no—
198#380—22no—
199#380—22no—
200#380C2312yes—
201#380C2314yes—
202#380C2312yes—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4D411yescharacteristic
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4C2311yescharacteristic
211C4×C4⋊C4C2311yescharacteristic
212C4×C4⋊C4C2311yescharacteristic
213C4×C22⋊C4D412yes—
214C4×C22⋊C4D412yes—
215C4×C22⋊C4C2312yes—
216C2×#95—22no—
217C2×#95—22no—
218C43C2311yescharacteristic
219#340C2314yes—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2312yes—
226#340C2312yes—
227#340C2312yes—
228#340C2314yes—
229#340C2312yes—
230#340C2314yes—
231#336C2312yes—
232#336C2314yes—
233C22×Q8—22no—
234C22×Q8C2412yes—
235C22×D4C2×D411yescharacteristic
236C23×C4—22no—
237C23×C4—22no—
238C23×C4C2×D412yes—
239C23×C4C2×D412yes—
240C23×C4C2412yes—
241C22×C8C2412yes—
242C22×C8C2414yes—
243C22×C8C2412yes—
244C4⋊Q8—24no—
245C4⋊Q8—22no—
246C4⋊Q8—24no—
247C4⋊Q8—44no—
248C4⋊Q8—82no—
249C4⋊Q8#4114yes—
250#125—22no—
251#125—22no—
252#125—44no—
253#125—48no—
254C22⋊Q8—28no—
255C22⋊Q8—44no—
256C22⋊Q8—48no—
257C22⋊Q8—84no—
258C4×Q8—28no—
259C4×Q8—24no—
260C4×Q8—24no—
261C4×Q8—44no—
262C4×Q8—48no—
263C4×D4—24no—
264C4×D4—24no—
265C4×D4—42no—
266C4×D4—44no—
267C4×D4—44no—
268#117—48no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—44no—
297C2×C4⋊C4—44no—
298C2×C4⋊C4C2×D411yescharacteristic
299C2×C4⋊C4C2×D411yescharacteristic
300C2×C4⋊C4C2×D411yescharacteristic
301C2×C4⋊C4C2411yescharacteristic
302C2×C4⋊C4C2411yescharacteristic
303C2×C4⋊C4C2412yes—
304C2×C4⋊C4C2412yes—
305C2×C4⋊C4C2414yes—
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2411yescharacteristic
308C2×C4⋊C4C2411yescharacteristic
309C2×C22⋊C4—24no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4C2×D412yes—
314C2×C22⋊C4C2×D412yes—
315C2×C22⋊C4C2×D412yes—
316C2×C42—24no—
317C2×C42—24no—
318C2×C42—24no—
319C2×C42—21no—
320C2×C42—21no—
321C2×C42—22no—
322C2×C42C2×D411yescharacteristic
323C2×C42C2×D411yescharacteristic
324C2×C42C2×D412yes—
325C2×C42C2×D412yes—
326C2×C42C2×D411yescharacteristic
327C2×C42C2×D411yescharacteristic
328C2×C42C2412yes—
329C2×C42C2411yescharacteristic
330C2×C42C2411yescharacteristic
331C2×C42C2411yescharacteristic
332#107—44no—
333#106—44no—
334C4⋊C8—24no—
335C4⋊C8—44no—
336#103—24no—
337#103—24no—
338#103—42no—
339#103—42no—
340#103—44no—
341#103—44no—
342C22⋊C8—44no—
343C22⋊C8—44no—
344#97—22no—
345#97—22no—
346#97—24no—
347#97—42no—
348C4×C8—24no—
349#95—21no—
350#95—21no—
351#95—22no—
352#95—22no—
353#95—22no—
354#95—22no—
355#95—22no—
356#95—22no—
357#95—21no—
358#95—21no—
359#95—24no—
360#95—22no—
361#95—22no—
362#95—21no—
363#95—21no—
364#95—22no—
365#95—22no—
366#95—24no—
367#95—22no—
368#95—22no—
369#95C2×D412yes—
370#95C2×D412yes—
371#95C2×D412yes—
372C24C22×D412yes—
373C2×Q8—22no—
374C2×Q8—22no—
375C2×Q8—22no—
376C2×Q8—22no—
377C2×Q8—22no—
378C2×Q8—44no—
379C2×Q8—44no—
380C2×Q8—44no—
381C2×Q8C2×#4114yes—
382C2×D4—22no—
383C2×D4—22no—
384C2×D4#13714yes—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—44no—
405C22×C4—44no—
406C22×C4—82no—
407C22×C4—82no—
408C22×C4—82no—
409C22×C4—82no—
410C22×C4C22×D412yes—
411C22×C4C22×D412yes—
412C22×C4C22×D411yescharacteristic
413C22×C4C22×D411yescharacteristic
414C22×C4C22×D412yes—
415C22×C4C22×D411yescharacteristic
416C22×C4C22×D411yescharacteristic
417C22×C4C22×D411yescharacteristic
418C22×C4C22×D411yescharacteristic
419C22×C4C2511yescharacteristic
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—24no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—24no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—24no—
429C4⋊C4—22no—
430C4⋊C4—24no—
431C4⋊C4—24no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—24no—
437C4⋊C4—21no—
438C4⋊C4—24no—
439C4⋊C4—21no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—24no—
445C4⋊C4—24no—
446C4⋊C4—48no—
447C4⋊C4—48no—
448C4⋊C4—42no—
449C4⋊C4—42no—
450C4⋊C4—44no—
451C4⋊C4—44no—
452C4⋊C4—44no—
453C4⋊C4—42no—
454C4⋊C4—42no—
455C4⋊C4—42no—
456C4⋊C4—42no—
457C4⋊C4—44no—
458C4⋊C4—81no—
459C4⋊C4—81no—
460C4⋊C4—81no—
461C4⋊C4—82no—
462C4⋊C4—81no—
463C4⋊C4—82no—
464C4⋊C4—82no—
465C4⋊C4—82no—
466C4⋊C4—84no—
467C4⋊C4—82no—
468C4⋊C4—82no—
469C4⋊C4—82no—
470C4⋊C4—82no—
471C22⋊C4—24no—
472C22⋊C4—24no—
473C22⋊C4—42no—
474C22⋊C4—48no—
475C22⋊C4—44no—
476C22⋊C4—44no—
477C22⋊C4—84no—
478C42—24no—
479C42—24no—
480C42—22no—
481C42—22no—
482C42—22no—
483C42—22no—
484C42—22no—
485C42—41no—
486C42—41no—
487C42—48no—
488C42—44no—
489C42—42no—
490C42—42no—
491C42C2×#4112yes—
492C42H(5,𝔽2)12yes—
493C422−1+414yes—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—22no—
498C23—22no—
499C23—22no—
500C23C2×#13714yes—
501C23C23×D411yescharacteristic
502Q8—24no—
503Q8—24no—
504Q8—42no—
505Q8—42no—
506Q8—88no—
507D4—28no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—22no—
516C2×C4—21no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—24no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—22no—
526C2×C4—21no—
527C2×C4—22no—
528C2×C4—21no—
529C2×C4—22no—
530C2×C4—48no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—42no—
536C2×C4—44no—
537C2×C4—41no—
538C2×C4—42no—
539C2×C4—41no—
540C2×C4—42no—
541C2×C4—41no—
542C2×C4—42no—
543C2×C4—41no—
544C2×C4—42no—
545C2×C4—44no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—42no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—84no—
560C2×C4—164no—
561C2×C4—164no—
562C2×C4#53612yes—
563C2×C4#53612yes—
564C2×C4#54612yes—
565C2×C4#54612yes—
566C2×C4C2×#13712yes—
567C2×C4#57512yes—
568C2×C4C22×#4111yescharacteristic
569C2×C4C2×H(5,𝔽2)11yescharacteristic
570C2×C4C2×2−1+412yes—
571C2×C4#58512yes—
572C8—42no—
573C8—42no—
574C8—44no—
575C22—24no—
576C22—24no—
577C22—24no—
578C22—42no—
579C22—42no—
580C22—42no—
581C22—42no—
582C22C2×#53611yescharacteristic
583C22C2×#54611yescharacteristic
584C22#350011yescharacteristic
585C22C22×#13711yescharacteristic
586C22C2×#57511yescharacteristic
587C22#358511yescharacteristic
588C22#358611yescharacteristic
589C4—22no—
590C4—22no—
591C4—22no—
592C4—42no—
593C4—44no—
594C4—41no—
595C4—41no—
596C4—42no—
597C4—42no—
598C4—82no—
599C4—81no—
600C4—82no—
601C4—81no—
602C4—82no—
603C4—81no—
604C4—81no—
605C4—84no—
606C4#323614yes—
607C4#351512yes—
608C4#354712yes—
609C2—44no—
610C2#3895511yescharacteristic
611C2C2×#323611yescharacteristic
612C2#6088611yescharacteristic
613C2#6115711yescharacteristic
614C2#6116011yescharacteristic
615C2#6258211yescharacteristic
616C2#6259311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #33297C2semidirect
4. #31368C2semidirect
5. #31366C2semidirect
6. #29884C2semidirect
7. #29150C2semidirect
8. #28694C2semidirect
9. #27969C2semidirect
10. #26113C2semidirect
11. #26109C2semidirect
12. #25677C2semidirect
15. #21866C2semidirect
16. C4×#491C2semidirect

Maximal subgroups

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

subgroupindex
#361652
#361342
#332972
#313682
#313662
#298842
#291502
#286942
#279692
#261132
#261092
#256772
#249712
#249692
#218662
C4×#4912
#212702