Finite Group Databasebeta

← #6585873

Subgroups — #6585873

Order 512.

1494 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36598C211yesmaximal, characteristic
2#36593C211yesmaximal, characteristic
3#33289C211yesmaximal, characteristic
4#33196C211yesmaximal, characteristic
5#33196C211yesmaximal, characteristic
6#28790C212yesmaximal
7#27816C212yesmaximal
8#26447C212yesmaximal
9#26283C212yesmaximal
10#26113C212yesmaximal
11#26027C212yesmaximal
12#25472C212yesmaximal
13#24818C212yesmaximal
14#24332C212yesmaximal
15#22964C212yesmaximal
16#22771C212yesmaximal
17#22521C212yesmaximal
18C8×C4⋊Q8C211yesmaximal, characteristic
19C42×Q16C211yesmaximal, characteristic
20C2×#557C2212yes—
21C2×#475C2212yes—
22C2×#475C2212yes—
23C2×C8×Q8C2212yes—
24C2×C4×Q16C2211yescharacteristic
25C2×C4×Q16C2211yescharacteristic
26C2×C4×Q16C2211yescharacteristic
27#2868—21no—
28#2868C2211yescharacteristic
29#2868C2211yescharacteristic
30#2862—21no—
31#2582C2212yes—
32#2571C2212yes—
33#2570—22no—
34#2568C2212yes—
35#2552—22no—
36#2552—22no—
37#2552C2212yes—
38#2552C2212yes—
39#2546—22no—
40#2546C2212yes—
41#2511—22no—
42#2446—22no—
43#2365C2212yes—
44#2355C2212yes—
45#2355C2212yes—
46Q8×C4⋊C4C2212yes—
47Q8×C4⋊C4C2212yes—
48C4×C4⋊Q8—21no—
49C4×C4⋊Q8—21no—
50C4×C4⋊Q8C2211yescharacteristic
51C42×Q8C2211yescharacteristic
52C42×Q8C2211yescharacteristic
53#2101C2214yes—
54#2088C2214yes—
55#2085C2214yes—
56#2066C2214yes—
57#2059C2214yes—
58#2026C2214yes—
59#1991C2214yes—
60#1990C2212yes—
61#1990C2212yes—
62#1990C2212yes—
63#1990C2212yes—
64#1989C2214yes—
65#1988C2212yes—
66#1988C2212yes—
67#1984C2212yes—
68#1948C2212yes—
69#1948C2212yes—
70#1948C2212yes—
71#1948C2212yes—
72#1941C2214yes—
73#1939C2212yes—
74#1939C2212yes—
75#1933C2214yes—
76#1921C2214yes—
77#1920C2214yes—
78#1878C2214yes—
79#1865C2214yes—
80#1842C2211yescharacteristic
81#1834C2212yes—
82#1834C2212yes—
83#1805C2214yes—
84#1803C2212yes—
85#1803C2212yes—
86#1803C2212yes—
87#1803C2212yes—
88C4×#107C2211yescharacteristic
89C4×#107C2211yescharacteristic
90C4×#107C2211yescharacteristic
91C8×C4⋊C4C2212yes—
92C8×C4⋊C4C2212yes—
93C4×#103C2211yescharacteristic
94C4×#103C2211yescharacteristic
95C4×#103C2211yescharacteristic
96C4×#103C2211yescharacteristic
97C4×#103C2211yescharacteristic
98C4×#103C2211yescharacteristic
99C4×#103C2214yes—
100C42×C8C2211yescharacteristic
101C22×Q16C2311yescharacteristic
102#557—28no—
103#557—44no—
104C2×C4⋊Q8—22no—
105C2×C4⋊Q8—22no—
106C2×C4⋊Q8—21no—
107C2×C4⋊Q8—21no—
108C2×C4⋊Q8—21no—
109C2×C4⋊Q8—21no—
110C2×C4⋊Q8D411yescharacteristic
111C2×C4⋊Q8C2312yes—
112C2×C4⋊Q8C2312yes—
113C2×#125—24no—
114C2×#125C2314yes—
115C2×C4×Q8—22no—
116C2×C4×Q8—22no—
117C2×C4×Q8C2311yescharacteristic
118C2×C4×Q8C2311yescharacteristic
119C2×C4×Q8C2314yes—
120C2×C4×Q8C2311yescharacteristic
121C2×C4×Q8C2311yescharacteristic
122C2×C4×Q8C2311yescharacteristic
123C2×C4×Q8C2311yescharacteristic
124C2×C4×Q8C2312yes—
125#475—28no—
126#475—44no—
127C8×Q8—44no—
128C4×Q16—24no—
129C4×Q16—24no—
130C4×Q16Q818yescomplemented
131C2×#107C2311yescharacteristic
132C2×#107C2311yescharacteristic
133C2×#107C2311yescharacteristic
134C2×#106C2314yes—
135C2×#106C2314yes—
136C2×C4⋊C8C2312yes—
137C2×C4⋊C8C2312yes—
138C2×C4⋊C8C2312yes—
139C2×C4⋊C8C2314yes—
140C2×#103C2314yes—
141C2×#103C2311yescharacteristic
142C2×#103C2311yescharacteristic
143C2×#103C2311yescharacteristic
144C2×#103C2311yescharacteristic
145C2×#103C2314yes—
146C2×#103C2311yescharacteristic
147C2×#103C2311yescharacteristic
148C2×C4×C8C2311yescharacteristic
149C2×C4×C8C2311yescharacteristic
150C2×C4×C8C2311yescharacteristic
151C2×C4×C8C2311yescharacteristic
152C2×C4×C8C2311yescharacteristic
153C2×C4×C8C2314yes—
154C2×C4×C8C2311yescharacteristic
155#400—24no—
156#400—24no—
157#398—24no—
158#398—24no—
159#398—24no—
160#398—24no—
161#391—22no—
162#391—22no—
163#391—22no—
164#391—22no—
165#391—22no—
166#391—22no—
167#391D412yes—
168#391C2314yes—
169#389—22no—
170#389—22no—
171#389—22no—
172#389—22no—
173#389—24no—
174#389—24no—
175#389D412yes—
176#389D412yes—
177#389C2314yes—
178#389C2314yes—
179#389C2314yes—
180#387—24no—
181#387—24no—
182#387—24no—
183#384—22no—
184#384—22no—
185#384—22no—
186#384—22no—
187#384—21no—
188#384—21no—
189#384—21no—
190#384—21no—
191#384D411yescharacteristic
192#384C2312yes—
193#384C2312yes—
194#384C2312yes—
195#384C2312yes—
196#382—22no—
197#382—22no—
198#382C2314yes—
199C4×C4⋊C4—22no—
200C4×C4⋊C4—22no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4C2311yescharacteristic
210C4×C4⋊C4C2314yes—
211C4×C4⋊C4C2311yescharacteristic
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2312yes—
218C43—21no—
219C43—21no—
220C43C2311yescharacteristic
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2311yescharacteristic
228#340C2314yes—
229#340C2314yes—
230#340C2311yescharacteristic
231#340C2311yescharacteristic
232#340C2311yescharacteristic
233#340C2311yescharacteristic
234#340C2311yescharacteristic
235#336C2314yes—
236#336C2314yes—
237C22×Q8—22no—
238C22×Q8—22no—
239C22×Q8C2×D412yes—
240C22×Q8C2411yescharacteristic
241C22×Q8C2411yescharacteristic
242C2×Q16—22no—
243C2×Q16—22no—
244C2×Q16C2×Q814yes—
245C22×C8C2411yescharacteristic
246C22×C8C2414yes—
247C22×C8C2411yescharacteristic
248C22×C8C2411yescharacteristic
249C22×C8C2411yescharacteristic
250C4⋊Q8—24no—
251C4⋊Q8—24no—
252C4⋊Q8—24no—
253C4⋊Q8—24no—
254C4⋊Q8—24no—
255C4⋊Q8—48no—
256C4⋊Q8—44no—
257C4⋊Q8—44no—
258C4⋊Q8—82no—
259C4⋊Q8—82no—
260C4⋊Q8—84no—
261#125—28no—
262#125—44no—
263#125—416no—
264C4×Q8—24no—
265C4×Q8—22no—
266C4×Q8—24no—
267C4×Q8—22no—
268C4×Q8—22no—
269C4×Q8—22no—
270C4×Q8—28no—
271C4×Q8—22no—
272C4×Q8—22no—
273C4×Q8—22no—
274C4×Q8—22no—
275C4×Q8—44no—
276C4×Q8—48no—
277C4×Q8—84no—
278C4×Q8C2×Q814yes—
279C4×Q8C2×Q814yes—
280C4×Q8#4114yes—
281C4×Q8#4114yes—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4C2×D412yes—
305C2×C4⋊C4C2×D412yes—
306C2×C4⋊C4C2×D414yes—
307C2×C4⋊C4C2×D412yes—
308C2×C4⋊C4C2×D412yes—
309C2×C4⋊C4C2×D412yes—
310C2×C4⋊C4C2414yes—
311C2×C4⋊C4C2411yescharacteristic
312C2×C4⋊C4C2411yescharacteristic
313C2×C4⋊C4C2414yes—
314C2×C4⋊C4C2411yescharacteristic
315C2×C4⋊C4C2411yescharacteristic
316C2×C4⋊C4C2411yescharacteristic
317C2×C4⋊C4C2411yescharacteristic
318C2×C42—21no—
319C2×C42—21no—
320C2×C42—21no—
321C2×C42—21no—
322C2×C42—21no—
323C2×C42—24no—
324C2×C42—21no—
325C2×C42—21no—
326C2×C42—21no—
327C2×C42—21no—
328C2×C42—21no—
329C2×C42—21no—
330C2×C42—21no—
331C2×C42—24no—
332C2×C42C2×D411yescharacteristic
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D411yescharacteristic
336C2×C42C2414yes—
337C2×C42C2411yescharacteristic
338C2×C42C2411yescharacteristic
339C2×C42C2411yescharacteristic
340#107—22no—
341#107—22no—
342#107C2×Q814yes—
343#106—28no—
344#106—44no—
345C4⋊C8—24no—
346C4⋊C8—24no—
347C4⋊C8—24no—
348C4⋊C8—44no—
349#103—22no—
350#103—22no—
351#103—28no—
352#103—22no—
353#103—22no—
354#103—44no—
355#103C2×Q814yes—
356#103C2×Q814yes—
357C4×C8—22no—
358C4×C8—22no—
359C4×C8—22no—
360C4×C8—22no—
361C4×C8—22no—
362C4×C8—44no—
363C4×C8C2×Q814yes—
364#95—21no—
365#95—21no—
366#95—21no—
367#95—21no—
368#95—21no—
369#95—21no—
370#95—24no—
371#95—24no—
372#95—24no—
373#95—24no—
374#95—24no—
375#95—24no—
376#95—24no—
377#95—24no—
378#95C2×D414yes—
379#95C2×D414yes—
380C2×Q8—24no—
381C2×Q8—24no—
382C2×Q8—24no—
383C2×Q8—21no—
384C2×Q8—21no—
385C2×Q8—21no—
386C2×Q8—21no—
387C2×Q8—21no—
388C2×Q8—21no—
389C2×Q8—21no—
390C2×Q8—21no—
391C2×Q8—44no—
392C2×Q8—44no—
393C2×Q8—44no—
394C2×Q8—44no—
395C2×Q8—44no—
396C2×Q8—44no—
397C2×Q8C22×Q812yes—
398C2×Q8C22×Q812yes—
399C2×Q8C2×#4112yes—
400C2×Q8C2×#4112yes—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4C22×D414yes—
412C22×C4C22×D414yes—
413C22×C4C22×D411yescharacteristic
414C22×C4C22×D411yescharacteristic
415C22×C4C22×D411yescharacteristic
416C22×C4C22×D411yescharacteristic
417C22×C4C22×D411yescharacteristic
418C22×C4C22×D411yescharacteristic
419C22×C4C2511yescharacteristic
420Q16—28no—
421C2×C8—21no—
422C2×C8—21no—
423C2×C8—21no—
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—21no—
427C2×C8—24no—
428C2×C8—21no—
429C2×C8—24no—
430C2×C8—21no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—24no—
434C2×C8C22×Q812yes—
435C2×C8C22×Q812yes—
436C4⋊C4—24no—
437C4⋊C4—24no—
438C4⋊C4—28no—
439C4⋊C4—24no—
440C4⋊C4—24no—
441C4⋊C4—24no—
442C4⋊C4—24no—
443C4⋊C4—24no—
444C4⋊C4—24no—
445C4⋊C4—24no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—24no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—24no—
454C4⋊C4—24no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—24no—
458C4⋊C4—48no—
459C4⋊C4—48no—
460C4⋊C4—48no—
461C4⋊C4—42no—
462C4⋊C4—44no—
463C4⋊C4—42no—
464C4⋊C4—44no—
465C4⋊C4—44no—
466C4⋊C4—44no—
467C4⋊C4—44no—
468C4⋊C4—44no—
469C4⋊C4—84no—
470C4⋊C4—82no—
471C4⋊C4—82no—
472C4⋊C4—82no—
473C4⋊C4—82no—
474C4⋊C4—84no—
475C4⋊C4—84no—
476C4⋊C4C22×Q812yes—
477C4⋊C4C22×Q812yes—
478C4⋊C4C2×#4112yes—
479C4⋊C4C2×#4112yes—
480C42—24no—
481C42—24no—
482C42—24no—
483C42—24no—
484C42—22no—
485C42—22no—
486C42—22no—
487C42—24no—
488C42—24no—
489C42—48no—
490C42—42no—
491C42—42no—
492C42—42no—
493C42—42no—
494C42—42no—
495C42—42no—
496C42—42no—
497C42—42no—
498C42—84no—
499C42#13514yes—
500C42C22×Q812yes—
501C42C2×#4112yes—
502C42C2×#4112yes—
503C42H(5,𝔽2)12yes—
504C422−1+412yes—
505C422−1+412yes—
506C23C23×D411yescharacteristic
507Q8—22no—
508Q8—22no—
509Q8—22no—
510Q8—22no—
511Q8—22no—
512Q8—41no—
513Q8—41no—
514Q8—48no—
515Q8—88no—
516Q8—88no—
517Q8#53314yes—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—24no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—44no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4#54612yes—
561C2×C4#54612yes—
562C2×C4#54812yes—
563C2×C4#54812yes—
564C2×C4D4×Q812yes—
565C2×C4D4×Q812yes—
566C2×C4C2×#13512yes—
567C2×C4C2×#13512yes—
568C2×C4C2×#13512yes—
569C2×C4#57712yes—
570C2×C4#57712yes—
571C2×C4#57812yes—
572C2×C4C23×Q811yescharacteristic
573C2×C4C22×#4111yescharacteristic
574C2×C4C22×#4111yescharacteristic
575C2×C4C2×H(5,𝔽2)11yescharacteristic
576C2×C4C2×2−1+411yescharacteristic
577C2×C4C2×2−1+411yescharacteristic
578C8—22no—
579C8—22no—
580C8—41no—
581C8—41no—
582C8—44no—
583C22C2×#54611yescharacteristic
584C22C2×#54811yescharacteristic
585C22C2×D4×Q811yescharacteristic
586C22C22×#13511yescharacteristic
587C22C2×#57711yescharacteristic
588C22C2×#57711yescharacteristic
589C22C2×#57811yescharacteristic
590C4—24no—
591C4—22no—
592C4—22no—
593C4—22no—
594C4—42no—
595C4—44no—
596C4—42no—
597C4—42no—
598C4—42no—
599C4—42no—
600C4—44no—
601C4—81no—
602C4—84no—
603C4—81no—
604C4—84no—
605C4#330414yes—
606C4#336514yes—
607C4#338314yes—
608C4C2×#53311yescharacteristic
609C4#350311yescharacteristic
610C4#350911yescharacteristic
611C4#351111yescharacteristic
612C2#3612911yescharacteristic
613C2C2×#330411yescharacteristic
614C2#6121711yescharacteristic
615C2C2×#336511yescharacteristic
616C2#6133011yescharacteristic
617C2C2×#338311yescharacteristic
618C2#6137411yescharacteristic
subgroupG/H
center, socle, prime radicalC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
130. C4×Q16Q8semidirect

Maximal subgroups

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

subgroupindex
#365982
#365932
#332892
#331962
#287902
#278162
#264472
#262832
#261132
#260272
#254722
#248182
#243322
#229642
#227712
#225212
C8×C4⋊Q82
C42×Q162