Finite Group Databasebeta

← #6854616

Subgroups — #6854616

Order 512.

1188 conjugacy classes of subgroups.

The subgroup lattice has 653 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#39637C211yesmaximal, characteristic
2#39621C211yesmaximal, characteristic
3#33290C211yesmaximal, characteristic
4#32136C212yesmaximal
5#32135C212yesmaximal
6#32133C212yesmaximal
7#32130C212yesmaximal
8#30313C212yesmaximal
9#29229C212yesmaximal
10#29223C211yesmaximal, characteristic
11#29211C211yesmaximal, characteristic
12#29189C212yesmaximal
13#26451C212yesmaximal
14#26448C212yesmaximal
15#26287C212yesmaximal
16#26284C212yesmaximal
17#22254C212yesmaximal
18C4×#500C211yesmaximal, characteristic
19#21341C211yesmaximal, characteristic
20C2×#500C2211yescharacteristic
21#2862C2211yescharacteristic
22#2859C2211yescharacteristic
23#2811—21no—
24#2809—21no—
25#2806—21no—
26#2806—21no—
27#2806—21no—
28#2805—21no—
29#2734—21no—
30#2726—21no—
31#2660C2212yes—
32#2647C2212yes—
33#2586—22no—
34#2585—22no—
35#2584C2212yes—
36#2551C2212yes—
37#2550C2212yes—
38#2550C2212yes—
39#2507C2212yes—
40#2506C2212yes—
41#2453—22no—
42#2451—22no—
43#2370C2212yes—
44#2355C2212yes—
45#2331C2212yes—
46#2331C2212yes—
47C4×C4⋊Q8C2211yescharacteristic
48C4×C4⋊Q8C2211yescharacteristic
49#2295C2211yescharacteristic
50#2101C2212yes—
51#2101C2212yes—
52#2100C2212yes—
53#2100C2212yes—
54#2099C2212yes—
55#2099C2212yes—
56#2088C2212yes—
57#2088C2212yes—
58#2087C2212yes—
59#2086C2212yes—
60#2085C2212yes—
61#2085C2212yes—
62#2083C2212yes—
63#2083C2212yes—
64#2059C2212yes—
65#2059C2212yes—
66#2058C2212yes—
67#2057C2212yes—
68#2057C2212yes—
69#2057C2212yes—
70#1993C2212yes—
71#1993C2212yes—
72#1991C2212yes—
73#1991C2212yes—
74#1990C2211yescharacteristic
75#1990C2211yescharacteristic
76#1990C2211yescharacteristic
77#1990C2211yescharacteristic
78#1990C2211yescharacteristic
79#1990C2211yescharacteristic
80#1989C2212yes—
81#1989C2212yes—
82#1989C2212yes—
83#1989C2212yes—
84#1988C2212yes—
85#1987C2212yes—
86#1984C2211yescharacteristic
87#1984C2211yescharacteristic
88#1949C2212yes—
89#1949C2212yes—
90#1942C2212yes—
91#1942C2212yes—
92#1939C2211yescharacteristic
93#1939C2211yescharacteristic
94#1921C2212yes—
95#1921C2212yes—
96#1921C2212yes—
97#1921C2212yes—
98#1921C2212yes—
99#1920C2212yes—
100#1919C2212yes—
101#1918C2212yes—
102#1918C2212yes—
103#1918C2212yes—
104#1918C2212yes—
105#1918C2212yes—
106#1916C2212yes—
107#1916C2212yes—
108#1847C2212yes—
109#1845C2211yescharacteristic
110#1836C2212yes—
111#1835C2212yes—
112#1835C2212yes—
113C4×#107C2212yes—
114C4×#107C2212yes—
115#1768C2212yes—
116#1767C2211yescharacteristic
117#1746C2212yes—
118C42×C8C2211yescharacteristic
119C2×C4⋊Q8—22no—
120C2×C4⋊Q8C2311yescharacteristic
121C2×C4⋊Q8C2311yescharacteristic
122C2×C4⋊Q8C2311yescharacteristic
123C2×C4⋊Q8C2312yes—
124C2×C4⋊Q8C2311yescharacteristic
125C2×#125—22no—
126C2×#125C2312yes—
127C2×#125C2312yes—
128C2×C4×Q8—21no—
129C2×C4×Q8—21no—
130C2×C4×Q8C2311yescharacteristic
131#500Q818yescomplemented
132C2×#107C2312yes—
133C2×#107C2312yes—
134C2×#107C2312yes—
135C2×#107C2312yes—
136C2×#107C2312yes—
137C2×#107C2312yes—
138C2×#106C2312yes—
139C2×#106C2312yes—
140C2×C4⋊C8C2311yescharacteristic
141C2×C4⋊C8C2311yescharacteristic
142C2×C4⋊C8C2312yes—
143C2×C4⋊C8C2312yes—
144C2×#97C2312yes—
145C2×C4×C8C2311yescharacteristic
146C2×C4×C8C2311yescharacteristic
147C2×C4×C8C2311yescharacteristic
148C2×C4×C8C2311yescharacteristic
149C2×C4×C8C2311yescharacteristic
150C2×C4×C8C2311yescharacteristic
151#400—22no—
152#400—22no—
153#400—22no—
154#398—22no—
155#398—22no—
156#398—22no—
157#398—22no—
158#398—22no—
159#398—22no—
160#398—22no—
161#398—22no—
162#398—22no—
163#398—22no—
164#395—22no—
165#395—22no—
166#395—22no—
167#395—22no—
168#395—22no—
169#395—22no—
170#395—22no—
171#391—22no—
172#391—21no—
173#391—21no—
174#391—21no—
175#391—21no—
176#391—21no—
177#391—21no—
178#391—22no—
179#391—22no—
180#391—22no—
181#391—22no—
182#391—22no—
183#391—22no—
184#391D411yescharacteristic
185#391D411yescharacteristic
186#389—22no—
187#389—22no—
188#389—22no—
189#389—22no—
190#389—21no—
191#389—21no—
192#389—21no—
193#389—21no—
194#389—21no—
195#389—21no—
196#389—22no—
197#389—22no—
198#389—22no—
199#389D411yescharacteristic
200#389D411yescharacteristic
201#389C2312yes—
202#389C2312yes—
203#389C2312yes—
204#389C2312yes—
205#389C2312yes—
206#389C2312yes—
207#389C2312yes—
208#389C2312yes—
209#389C2312yes—
210#389C2312yes—
211#387—22no—
212#387—22no—
213#387—22no—
214#387—22no—
215#387—22no—
216#387—22no—
217#387—22no—
218#387—22no—
219#387—22no—
220#387—22no—
221#387D412yes—
222#387D412yes—
223#384C2311yescharacteristic
224#384C2311yescharacteristic
225#384C2312yes—
226#384C2311yescharacteristic
227#384C2311yescharacteristic
228#382C2312yes—
229#382C2312yes—
230#382C2312yes—
231#382C2312yes—
232C4×C4⋊C4—22no—
233C4×C4⋊C4—21no—
234C4×C4⋊C4—21no—
235C4×C4⋊C4—22no—
236C4×C4⋊C4C2312yes—
237C4×C4⋊C4C2312yes—
238C4×C4⋊C4C2312yes—
239C4×C4⋊C4C2312yes—
240C4×C4⋊C4C2312yes—
241C4×C4⋊C4C2312yes—
242C4×C4⋊C4C2311yescharacteristic
243C4×C4⋊C4C2312yes—
244#376C2312yes—
245C43C2311yescharacteristic
246#340C2312yes—
247#340C2312yes—
248#340C2312yes—
249#340C2312yes—
250#340C2312yes—
251#340C2312yes—
252#340C2312yes—
253#340C2312yes—
254#340C2312yes—
255#340C2312yes—
256#340C2312yes—
257#340C2312yes—
258#340C2312yes—
259#340C2312yes—
260#340C2312yes—
261#340C2312yes—
262#340C2312yes—
263#340C2312yes—
264#340C2312yes—
265#340C2312yes—
266#340C2312yes—
267#340C2312yes—
268#340C2312yes—
269#340C2312yes—
270#336C2312yes—
271#336C2312yes—
272#336C2312yes—
273#336C2312yes—
274#336C2312yes—
275#336C2312yes—
276#336C2312yes—
277C22×Q8—21no—
278C22×Q8—21no—
279C22×Q8—21no—
280C22×Q8—21no—
281C22×Q8—22no—
282C22×Q8C2×D411yescharacteristic
283C22×C8C2412yes—
284C22×C8C2412yes—
285C22×C8C2411yescharacteristic
286C22×C8C2411yescharacteristic
287C22×C8C2411yescharacteristic
288C22×C8C2411yescharacteristic
289C4⋊Q8—22no—
290C4⋊Q8—22no—
291C4⋊Q8—41no—
292C4⋊Q8—41no—
293C4⋊Q8—41no—
294C4⋊Q8—41no—
295C4⋊Q8—42no—
296C4⋊Q8—42no—
297C4⋊Q8—84no—
298C4⋊Q8C2×Q814yes—
299C4⋊Q8C2×Q814yes—
300#125—24no—
301#125—24no—
302#125—42no—
303#125—42no—
304#125—84no—
305C4×Q8—22no—
306C4×Q8—22no—
307C4×Q8—82no—
308C4×Q8—82no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4C2×D412yes—
339C2×C4⋊C4C2×D411yescharacteristic
340C2×C4⋊C4C2×D412yes—
341C2×C4⋊C4C2×D412yes—
342C2×C4⋊C4C2×D411yescharacteristic
343C2×C4⋊C4C2×D411yescharacteristic
344C2×C4⋊C4C2412yes—
345C2×C4⋊C4C2412yes—
346C2×C4⋊C4C2412yes—
347C2×C4⋊C4C2412yes—
348C2×C4⋊C4C2412yes—
349C2×C4⋊C4C2412yes—
350C2×C4⋊C4C2412yes—
351C2×C4⋊C4C2412yes—
352C2×C42—22no—
353C2×C42—22no—
354C2×C42—22no—
355C2×C42—22no—
356C2×C42—22no—
357C2×C42—22no—
358C2×C42C2×D411yescharacteristic
359C2×C42C2×D411yescharacteristic
360C2×C42C2×D411yescharacteristic
361C2×C42C2×D411yescharacteristic
362C2×C42C2412yes—
363C2×C42C2412yes—
364C2×C42C2411yescharacteristic
365C2×C42C2411yescharacteristic
366C2×C42C2411yescharacteristic
367#107—42no—
368#107—42no—
369#107—42no—
370#107—42no—
371#107C2×Q818yes—
372#107C2×Q818yes—
373#106—42no—
374#106—42no—
375C4⋊C8—22no—
376C4⋊C8—22no—
377C4⋊C8—42no—
378C4⋊C8—42no—
379#97—24no—
380C4×C8—22no—
381C4×C8—22no—
382C4×C8—22no—
383C4×C8—41no—
384C4×C8—41no—
385C4×C8C2×Q814yes—
386#95—22no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95—22no—
392#95—22no—
393#95—22no—
394#95—22no—
395#95—22no—
396#95—22no—
397#95—22no—
398#95—22no—
399#95—22no—
400#95—22no—
401#95—22no—
402#95—22no—
403#95—22no—
404#95—22no—
405#95—22no—
406#95—22no—
407#95—22no—
408#95—22no—
409#95—22no—
410#95C2×D412yes—
411#95C2×D412yes—
412#95C2×D412yes—
413#95C2×D412yes—
414#95C2×D412yes—
415#95C2×D412yes—
416#95C2×D412yes—
417C2×Q8—22no—
418C2×Q8—24no—
419C2×Q8—24no—
420C2×Q8—41no—
421C2×Q8—41no—
422C2×Q8—42no—
423C2×Q8—42no—
424C2×Q8—81no—
425C2×Q8—81no—
426C2×Q8—81no—
427C2×Q8—81no—
428C2×Q8—81no—
429C2×Q8—81no—
430C2×Q8—82no—
431C2×Q8—82no—
432C2×Q8—81no—
433C2×Q8—81no—
434C2×Q8—82no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4C22×D412yes—
444C22×C4C22×D412yes—
445C22×C4C22×D412yes—
446C22×C4C22×D412yes—
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C2511yescharacteristic
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8C22×Q812yes—
471C2×C8C22×Q812yes—
472C4⋊C4—22no—
473C4⋊C4—24no—
474C4⋊C4—24no—
475C4⋊C4—24no—
476C4⋊C4—24no—
477C4⋊C4—24no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—41no—
493C4⋊C4—41no—
494C4⋊C4—44no—
495C4⋊C4—44no—
496C4⋊C4—44no—
497C4⋊C4—44no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—81no—
501C4⋊C4—81no—
502C4⋊C4—81no—
503C4⋊C4—81no—
504C4⋊C4—82no—
505C4⋊C4—82no—
506C4⋊C4—82no—
507C4⋊C4—82no—
508C4⋊C4—82no—
509C4⋊C4—81no—
510C4⋊C4—81no—
511C4⋊C4—82no—
512C4⋊C4—82no—
513C4⋊C4—82no—
514C4⋊C4—82no—
515C4⋊C4—82no—
516C4⋊C4—81no—
517C4⋊C4—81no—
518C4⋊C4—82no—
519C4⋊C4—82no—
520C4⋊C4—82no—
521C4⋊C4C22×Q814yes—
522C4⋊C4C22×Q814yes—
523C42—22no—
524C42—22no—
525C42—22no—
526C42—22no—
527C42—21no—
528C42—21no—
529C42—22no—
530C42—22no—
531C42—41no—
532C42—41no—
533C42—82no—
534C42—82no—
535C42—82no—
536C42—82no—
537C42—82no—
538C42—82no—
539C42C22×Q812yes—
540C422−1+412yes—
541C422−1+412yes—
542C422−1+412yes—
543C23C23×D411yescharacteristic
544Q8—82no—
545Q8—84no—
546Q8—84no—
547Q8—162no—
548Q8—162no—
549Q8—164no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4#53612yes—
599C2×C4#53612yes—
600C2×C4D4×Q812yes—
601C2×C4D4×Q812yes—
602C2×C4#57612yes—
603C2×C4#57812yes—
604C2×C4C23×Q811yescharacteristic
605C2×C4C2×2−1+411yescharacteristic
606C2×C4C2×2−1+411yescharacteristic
607C2×C4C2×2−1+411yescharacteristic
608C2×C4#58511yescharacteristic
609C2×C4#58511yescharacteristic
610C8—22no—
611C8—22no—
612C8—41no—
613C8—41no—
614C8—42no—
615C8—42no—
616C22C2×#53611yescharacteristic
617C22C2×D4×Q811yescharacteristic
618C22#350011yescharacteristic
619C22C2×#57611yescharacteristic
620C22C2×#57811yescharacteristic
621C22#358411yescharacteristic
622C22#358411yescharacteristic
623C4—22no—
624C4—22no—
625C4—22no—
626C4—22no—
627C4—22no—
628C4—22no—
629C4—42no—
630C4—42no—
631C4—44no—
632C4—44no—
633C4—42no—
634C4—42no—
635C4—82no—
636C4—82no—
637C4—82no—
638C4—82no—
639C4—82no—
640C4—82no—
641C4#350211yescharacteristic
642C4#350211yescharacteristic
643C4#355511yescharacteristic
644C4#355511yescharacteristic
645C2#3961911yescharacteristic
646C2#6085911yescharacteristic
647C2#6088011yescharacteristic
648C2#6137011yescharacteristic
649C2#6137211yescharacteristic
650C2#6255811yescharacteristic
651C2#6255911yescharacteristic
subgroupG/H
center, socle, prime radicalC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2295 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
131. #500Q8semidirect

Maximal subgroups

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

subgroupindex
#396372
#396212
#332902
#321362
#321352
#321332
#321302
#303132
#292292
#292232
#292112
#291892
#264512
#264482
#262872
#262842
#222542
C4×#5002
#213412