Finite Group Databasebeta

← #6570389

Subgroups — #6570389

Order 512.

1376 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36598C211yesmaximal, characteristic, complemented
2#36299C211yesmaximal, characteristic
3#30344C212yesmaximal, complemented
4#30305C211yesmaximal, characteristic, complemented
5#29554C212yesmaximal
6#29512C212yesmaximal, complemented
7#29148C211yesmaximal, characteristic
8#29132C211yesmaximal, characteristic, complemented
9#28886C214yesmaximal
10#28264C212yesmaximal
11#28187C212yesmaximal, complemented
12#28060C214yesmaximal, complemented
13#23797C211yesmaximal, characteristic
14#23781C212yesmaximal
15#22814C212yesmaximal, complemented
16#22762C212yesmaximal
17#20572C211yesmaximal, characteristic, complemented
18C2×C8×Q8C2212yes—
19C2×C4×Q16C2212yes—
20C2×C4×SD16C2212yes—
21#2868C2211yescharacteristic
22#2863C2212yes—
23#2862C2212yes—
24#2859C2211yescharacteristic
25#2582C2211yescharacteristic
26#2582C2212yes—
27#2571—21no—
28#2570—22no—
29#2568—21no—
30#2538C2211yescharacteristic
31#2515—22no—
32#2511C2214yes—
33#2509C2212yes—
34#2495—21no—
35#2475—22no—
36#2447C2212yes—
37#2446—22no—
38#2421—21no—
39#2365—21no—
40#2360—21no—
41#2355—21no—
42#2355C2211yescharacteristic
43#2355C2212yes—
44D4×C4⋊C4C2211yescharacteristic
45#2344—21no—
46C4×C4⋊Q8C2212yes—
47C4×C4⋊Q8C2211yescharacteristic
48C4×#124C2211yescharacteristic
49#2293C2212yes—
50C42×Q8C2211yescharacteristic
51#2099C2212yes—
52#2097C2212yes—
53#2096C2211yescharacteristic
54#2093C2211yescharacteristic
55#2087C2214yes—
56#2084C2214yes—
57#2066C2214yes—
58#2065C2214yes—
59#2058C2212yes—
60#2058C2212yes—
61#2056C2214yes—
62#2043C2212yes—
63#2024C2212yes—
64#2023C2212yes—
65#2016C2211yescharacteristic
66#2015C2212yes—
67#2013C2211yescharacteristic
68#1971C2214yes—
69#1968C2214yes—
70#1959C2212yes—
71#1941C2214yes—
72#1935C2214yes—
73#1935C2214yes—
74#1933C2214yes—
75#1933C2214yes—
76#1927C2214yes—
77#1927C2214yes—
78#1920C2214yes—
79#1877C2214yes—
80#1844C2212yes—
81#1836C2212yes—
82#1834C2212yes—
83#1803C2212yes—
84#1803C2212yes—
85#1803C2212yes—
86#1802C2212yes—
87#1776C2214yes—
88#1770C2212yes—
89#1770C2214yes—
90#1767C2212yes—
91#1767C2211yescharacteristic
92#1762C2214yes—
93#1744C2212yes—
94#1744C2211yescharacteristic
95C22×Q16C2312yes—
96C22×SD16C2312yes—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2312yes—
101C2×#125—21no—
102C2×#124C2311yescharacteristic
103C2×C22⋊Q8—21no—
104C2×C4×Q8—22no—
105C2×C4×Q8D411yescharacteristic
106C2×C4×Q8D411yescharacteristic
107C2×C4×Q8C2312yes—
108C2×C4×Q8C2311yescharacteristic
109C2×C4×Q8C2311yescharacteristic
110C2×C4×D4C2311yescharacteristic
111C2×C4×D4C2312yes—
112C2×#117—22no—
113C22×C4⋊C4—21no—
114C8×Q8—28no—
115C4×Q16—28no—
116C4×SD16—28no—
117C2×#107C2312yes—
118C2×#106C2312yes—
119C2×#106C2314yes—
120C2×#106C2314yes—
121C2×C4⋊C8C2314yes—
122C2×C4⋊C8C2312yes—
123C2×#103C2312yes—
124C2×#103C2312yes—
125C2×#103C2312yes—
126C2×#103C2314yes—
127C2×#103C2314yes—
128C2×#103C2314yes—
129C2×#102C2312yes—
130C2×#102C2314yes—
131C2×#97C2312yes—
132C2×#97C2314yes—
133C2×C4×C8C2314yes—
134C2×C4×C8C2312yes—
135#400—24no—
136#400—22no—
137#400—24no—
138#400—22no—
139#398—24no—
140#398—22no—
141#397—22no—
142#397—22no—
143#396—22no—
144#391—22no—
145#391—21no—
146#391—22no—
147#391—22no—
148#391—22no—
149#391C2312yes—
150#391C2312yes—
151#391C2314yes—
152#391C2314yes—
153#391C2311yescharacteristic
154#391C2312yes—
155#390C2312yes—
156#390C2311yescharacteristic
157#389—22no—
158#389—21no—
159#389—22no—
160#389—21no—
161#389—22no—
162#389—21no—
163#389—21no—
164#389—21no—
165#389C2314yes—
166#389C2312yes—
167#389C2312yes—
168#389C2312yes—
169#389C2312yes—
170#389C2312yes—
171#388—22no—
172#388—21no—
173#388—22no—
174#388—21no—
175#387—24no—
176#387—22no—
177#387—22no—
178#387—22no—
179#387—24no—
180#387—24no—
181#387—22no—
182#387—22no—
183#385—22no—
184#385—22no—
185#384—21no—
186#384—21no—
187#384C2311yescharacteristic
188#384C2312yes—
189#384C2311yescharacteristic
190#384C2311yescharacteristic
191#384C2312yes—
192#383—22no—
193#382—22no—
194#382—21no—
195#382—22no—
196#382—22no—
197#382—21no—
198#382C2311yescharacteristic
199#381—22no—
200#380—21no—
201#380—21no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4D412yes—
208C4×C4⋊C4D411yescharacteristic
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4D412yes—
211C4×C4⋊C4C2314yes—
212C4×C4⋊C4C2312yes—
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2312yes—
215C4×C22⋊C4—21no—
216C4×C22⋊C4—21no—
217C4×C22⋊C4—22no—
218C43C2311yescharacteristic
219C43C2312yes—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#336C2311yescharacteristic
227#336C2311yescharacteristic
228#336C2312yes—
229#336C2312yes—
230#336C2314yes—
231C22×Q8—22no—
232C22×Q8C2×D411yescharacteristic
233C22×Q8C2411yescharacteristic
234C22×Q8C2412yes—
235C22×D4C2411yescharacteristic
236C23×C4—21no—
237C23×C4—22no—
238C2×Q16—24no—
239C2×Q16—42no—
240C2×Q16—42no—
241C2×SD16—24no—
242C2×SD16—24no—
243C2×SD16—42no—
244C22×C8C2414yes—
245C22×C8C2412yes—
246C22×C8C2412yes—
247C4⋊Q8—44no—
248C4⋊Q8—42no—
249C4⋊Q8—44no—
250C4⋊Q8—82no—
251#125—82no—
252#124—22no—
253#124—41no—
254C22⋊Q8—44no—
255C4×Q8—24no—
256C4×Q8—24no—
257C4×Q8—24no—
258C4×Q8—24no—
259C4×Q8—24no—
260C4×Q8—42no—
261C4×Q8—84no—
262C4×D4—24no—
263C4×D4—24no—
264C4×D4—42no—
265#117—84no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—44no—
288C2×C4⋊C4C2×D411yescharacteristic
289C2×C4⋊C4C2×D411yescharacteristic
290C2×C4⋊C4C2×D412yes—
291C2×C4⋊C4C2×D411yescharacteristic
292C2×C4⋊C4C2×D412yes—
293C2×C4⋊C4C2×D412yes—
294C2×C4⋊C4C2412yes—
295C2×C4⋊C4C2411yescharacteristic
296C2×C4⋊C4C2411yescharacteristic
297C2×C4⋊C4C2412yes—
298C2×C4⋊C4C2414yes—
299C2×C4⋊C4C2412yes—
300C2×C22⋊C4—21no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—21no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—21no—
305C2×C22⋊C4—21no—
306C2×C22⋊C4—21no—
307C2×C22⋊C4—22no—
308C2×C42—21no—
309C2×C42—22no—
310C2×C42—22no—
311C2×C42—24no—
312C2×C42C2×D412yes—
313C2×C42C2×D412yes—
314C2×C42C2×D411yescharacteristic
315C2×C42C2×D411yescharacteristic
316C2×C42C2×D412yes—
317C2×C42C2×D411yescharacteristic
318C2×C42C2×D411yescharacteristic
319C2×C42C2×D412yes—
320C2×C42C2412yes—
321C2×C42C2412yes—
322C2×C42C2411yescharacteristic
323C2×C42C2411yescharacteristic
324C2×C42C2411yescharacteristic
325#107—24no—
326#106—24no—
327#106—44no—
328#106—44no—
329C4⋊C8—28no—
330C4⋊C8—24no—
331#103—24no—
332#103—24no—
333#103—24no—
334#103—44no—
335#103—44no—
336#103—44no—
337#102—24no—
338#102—44no—
339#97—44no—
340#97—42no—
341C4×C8—28no—
342C4×C8#4118yes—
343#95—22no—
344#95—21no—
345#95—21no—
346#95—21no—
347#95—21no—
348#95—22no—
349#95—22no—
350#95—24no—
351#95—22no—
352#95—22no—
353#95—24no—
354#95—22no—
355#95—24no—
356#95—22no—
357#95—22no—
358#95—22no—
359#95—22no—
360#95—24no—
361#95—22no—
362#95—24no—
363#95—24no—
364#95—22no—
365#95C2×D412yes—
366#95C2×D412yes—
367#95C2×D412yes—
368C24—21no—
369C2×Q8—22no—
370C2×Q8—22no—
371C2×Q8—22no—
372C2×Q8—22no—
373C2×Q8—22no—
374C2×Q8—21no—
375C2×Q8—22no—
376C2×Q8—22no—
377C2×Q8—22no—
378C2×Q8—22no—
379C2×Q8—21no—
380C2×Q8—22no—
381C2×Q8—44no—
382C2×Q8—44no—
383C2×Q8—44no—
384C2×Q8#13714yes—
385C2×D4—22no—
386C2×D4—22no—
387C2×D4—21no—
388C2×D42−1+412yes—
389C22×C4—21no—
390C22×C4—22no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—44no—
398C22×C4—44no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—82no—
403C22×C4C22×D412yes—
404C22×C4C22×D412yes—
405C22×C4C22×D411yescharacteristic
406C22×C4C22×D412yes—
407C22×C4C22×D411yescharacteristic
408C22×C4C22×D411yescharacteristic
409C22×C4C22×D412yes—
410C22×C4C22×D411yescharacteristic
411C22×C4C22×D411yescharacteristic
412C22×C4C22×D411yescharacteristic
413C22×C4C2511yescharacteristic
414Q16—48no—
415SD16—48no—
416C2×C8—22no—
417C2×C8—22no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—24no—
421C2×C8—24no—
422C2×C8—24no—
423C2×C8C2×#4114yes—
424C2×C8C2×#4114yes—
425C4⋊C4—24no—
426C4⋊C4—22no—
427C4⋊C4—24no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—24no—
436C4⋊C4—22no—
437C4⋊C4—24no—
438C4⋊C4—24no—
439C4⋊C4—42no—
440C4⋊C4—44no—
441C4⋊C4—44no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C4⋊C4—44no—
445C4⋊C4—42no—
446C4⋊C4—44no—
447C4⋊C4—42no—
448C4⋊C4—44no—
449C4⋊C4—42no—
450C4⋊C4—44no—
451C4⋊C4—42no—
452C4⋊C4—44no—
453C4⋊C4—42no—
454C4⋊C4—82no—
455C4⋊C4—84no—
456C4⋊C4—81no—
457C4⋊C4—81no—
458C4⋊C4—82no—
459C4⋊C4—82no—
460C4⋊C4—82no—
461C22⋊C4—44no—
462C22⋊C4—42no—
463C22⋊C4—42no—
464C22⋊C4—42no—
465C22⋊C4—44no—
466C22⋊C4—42no—
467C22⋊C4—42no—
468C22⋊C4—82no—
469C42—24no—
470C42—22no—
471C42—22no—
472C42—24no—
473C42—24no—
474C42—22no—
475C42—22no—
476C42—24no—
477C42—22no—
478C42—21no—
479C42—22no—
480C42—21no—
481C42—22no—
482C42—42no—
483C42—42no—
484C42—44no—
485C42—82no—
486C42—84no—
487C42C2×#4114yes—
488C23—22no—
489C23—41no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23C23×D411yescharacteristic
496Q8—28no—
497Q8—42no—
498Q8—42no—
499Q8—42no—
500Q8—42no—
501Q8—42no—
502Q8—42no—
503Q8—88no—
504D4—24no—
505D4—42no—
506C2×C4—22no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—22no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—22no—
515C2×C4—21no—
516C2×C4—22no—
517C2×C4—21no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—42no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—42no—
533C2×C4—41no—
534C2×C4—44no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—42no—
545C2×C4—44no—
546C2×C4—41no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—44no—
550C2×C4—41no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—88no—
554C2×C4—84no—
555C2×C4#53212yes—
556C2×C4#53212yes—
557C2×C4C2×#13714yes—
558C2×C4C2×#13712yes—
559C2×C4#57512yes—
560C2×C4#57712yes—
561C2×C4#57712yes—
562C2×C4C22×#4112yes—
563C2×C4C2×2−1+411yescharacteristic
564C2×C4#58511yescharacteristic
565C2×C4#58512yes—
566C8—24no—
567C8—24no—
568C8—44no—
569C22—42no—
570C22—42no—
571C22—42no—
572C22—81no—
573C22—81no—
574C22—81no—
575C22—81no—
576C22C2×#53211yescharacteristic
577C22#348711yescharacteristic
578C22#348911yescharacteristic
579C22C22×#13711yescharacteristic
580C22C2×#57511yescharacteristic
581C22C2×#57711yescharacteristic
582C22C2×#57711yescharacteristic
583C4—24no—
584C4—22no—
585C4—22no—
586C4—22no—
587C4—42no—
588C4—41no—
589C4—42no—
590C4—42no—
591C4—41no—
592C4—81no—
593C4—81no—
594C4—82no—
595C4—82no—
596C4—84no—
597C4—82no—
598C4—82no—
599C4—81no—
600C4#318114yes—
601C4#349312yes—
602C4#349512yes—
603C2—82no—
604C2#3606511yescharacteristic
605C2C2×#318111yescharacteristic
606C2#6071611yescharacteristic
607C2#6190411yescharacteristic
608C2#6190611yescharacteristic
609C2#6214411yescharacteristic
610C2#6214711yescharacteristic
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. #30344C2semidirect
4. #30305C2semidirect
6. #29512C2semidirect
8. #29132C2semidirect
11. #28187C2semidirect
12. #28060C2semidirect
15. #22814C2semidirect
17. #20572C2semidirect

Maximal subgroups

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

subgroupindex
#365982
#362992
#303442
#303052
#295542
#295122
#291482
#291322
#288862
#282642
#281872
#280602
#237972
#237812
#228142
#227622
#205722