Finite Group Databasebeta

← #7541680

Subgroups — #7541680

Order 512.

1521 conjugacy classes of subgroups.

The subgroup lattice has 625 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59863C211yesmaximal, characteristic, complemented
2#36134C211yesmaximal, characteristic, complemented
3#32446C214yesmaximal, complemented
4#29206C211yesmaximal, characteristic, complemented
5#29139C211yesmaximal, characteristic, complemented
6#27735C212yesmaximal, complemented
7#27144C214yesmaximal, complemented
8#26318C212yesmaximal, complemented
9#26090C212yesmaximal, complemented
10#25300C212yesmaximal
11#24991C212yesmaximal, complemented
12#24677C212yesmaximal, complemented
13#23153C211yesmaximal, characteristic, complemented
14#22813C212yesmaximal, complemented
15#22742C212yesmaximal, complemented
16C4×#476C211yesmaximal, characteristic, complemented
17C4×#459C211yesmaximal, characteristic
18C2×#533—21no—
19C2×#476C2211yescharacteristic
20C2×#459C2211yescharacteristic
21C2×#446C2212yes—
22C2×C4×D8C2212yes—
23#2868C2211yescharacteristic
24#2857—21no—
25#2808—22no—
26#2592—22no—
27#2563C2212yes—
28#2552C2212yes—
29#2548C2212yes—
30#2547—21no—
31#2542C2212yes—
32#2538C2212yes—
33#2513C2214yes—
34#2485—22no—
35#2441—22no—
36#2429—22no—
37#2423—22no—
38#2365C2212yes—
39#2355C2212yes—
40#2354—21no—
41#2354C2212yes—
42#2354C2212yes—
43Q8×C4⋊C4C2212yes—
44C4×C4⋊Q8C2211yescharacteristic
45C4×C4⋊Q8C2211yescharacteristic
46C4×#127C2211yescharacteristic
47C42×D4C2211yescharacteristic
48#2101C2214yes—
49#2094C2214yes—
50#2073C2214yes—
51#2071C2214yes—
52#2065C2214yes—
53#2058C2214yes—
54#2054C2214yes—
55#2023C2214yes—
56#1990C2212yes—
57#1987C2211yescharacteristic
58#1984C2212yes—
59#1982C2211yescharacteristic
60#1972C2214yes—
61#1957C2212yes—
62#1949C2214yes—
63#1948C2212yes—
64#1948C2212yes—
65#1948C2211yescharacteristic
66#1948C2214yes—
67#1947C2211yescharacteristic
68#1939C2212yes—
69#1936C2214yes—
70#1926C2214yes—
71#1926C2214yes—
72#1926C2214yes—
73#1925C2212yes—
74#1920C2212yes—
75#1920C2214yes—
76#1917C2214yes—
77#1879C2212yes—
78#1864C2212yes—
79#1844C2212yes—
80#1836C2214yes—
81#1834C2212yes—
82#1802C2211yescharacteristic
83#1802C2212yes—
84#1802C2212yes—
85#1802C2211yescharacteristic
86#1776C2214yes—
87C4×#106C2212yes—
88#1770C2212yes—
89C4×C4⋊C8C2211yescharacteristic
90C4×C4⋊C8C2211yescharacteristic
91C4×#102C2212yes—
92C4×#102C2212yes—
93C22×D8C2312yes—
94#533—48no—
95C2×C4⋊Q8—21no—
96C2×C4⋊Q8—21no—
97C2×C4⋊Q8D411yescharacteristic
98C2×C4⋊Q8C2311yescharacteristic
99C2×C4⋊Q8C2312yes—
100C2×#127C2311yescharacteristic
101C2×#125—22no—
102C2×#124—22no—
103C2×C22⋊Q8—22no—
104C2×C22⋊Q8—22no—
105C2×C22⋊Q8—22no—
106C2×C4×Q8—22no—
107C2×C4×Q8—22no—
108C2×C4×Q8—21no—
109C2×C4×Q8C2312yes—
110C2×C4×D4—21no—
111C2×C4×D4—21no—
112C2×C4×D4C2312yes—
113C2×C4×D4C2311yescharacteristic
114C2×C4×D4C2311yescharacteristic
115C2×C4×D4C2311yescharacteristic
116C2×#117—22no—
117C2×#117—22no—
118C22×C42—21no—
119#476—24no—
120#459—24no—
121#446—28no—
122C4×D8—28no—
123C2×#107C2312yes—
124C2×#106C2312yes—
125C2×#106C2314yes—
126C2×#106C2314yes—
127C2×#106C2312yes—
128C2×C4⋊C8C2314yes—
129C2×C4⋊C8C2311yescharacteristic
130C2×C4⋊C8C2311yescharacteristic
131C2×C4⋊C8C2311yescharacteristic
132C2×C4⋊C8C2311yescharacteristic
133C2×#102C2312yes—
134C2×#102C2312yes—
135C2×#102C2312yes—
136C2×#102C2312yes—
137C2×#102C2314yes—
138C2×#102C2314yes—
139C2×#102C2312yes—
140C2×#102C2312yes—
141C2×#97C2314yes—
142C2×C4×C8C2312yes—
143C2×C4×C8C2312yes—
144#399—24no—
145#398—24no—
146#397—24no—
147#397—24no—
148#396—24no—
149#394—24no—
150#393—24no—
151#391—22no—
152#391—22no—
153#391—21no—
154#391D412yes—
155#390—22no—
156#390—21no—
157#390—21no—
158#390C2312yes—
159#390C2314yes—
160#390C2314yes—
161#390C2314yes—
162#389—22no—
163#389—22no—
164#389—24no—
165#389—21no—
166#389D412yes—
167#389D412yes—
168#389C2312yes—
169#389C2314yes—
170#388—24no—
171#388—24no—
172#388—24no—
173#388—24no—
174#387—24no—
175#387—24no—
176#384—21no—
177#384—21no—
178#384D411yescharacteristic
179#384C2311yescharacteristic
180#384C2311yescharacteristic
181#384C2312yes—
182#384C2314yes—
183#384C2312yes—
184#382C2314yes—
185#380—22no—
186#380—22no—
187#380—22no—
188#380—22no—
189#380—22no—
190C4×C4⋊C4—22no—
191C4×C4⋊C4—21no—
192C4×C4⋊C4C2312yes—
193C4×C4⋊C4C2311yescharacteristic
194C4×C4⋊C4C2314yes—
195C4×C4⋊C4C2314yes—
196C4×C4⋊C4C2311yescharacteristic
197C4×C4⋊C4C2311yescharacteristic
198C4×C4⋊C4C2312yes—
199C4×C4⋊C4C2312yes—
200C4×C22⋊C4—21no—
201C4×C22⋊C4—21no—
202C4×C22⋊C4—21no—
203C43C2311yescharacteristic
204#340C2314yes—
205#340C2314yes—
206#340C2314yes—
207#340C2312yes—
208#340C2312yes—
209#340C2314yes—
210#340C2312yes—
211#340C2312yes—
212#340C2312yes—
213#340C2312yes—
214#336C2312yes—
215#336C2314yes—
216#336C2314yes—
217#336C2312yes—
218C22×Q8—22no—
219C22×Q8—21no—
220C22×Q8C2×D412yes—
221C22×D4—21no—
222C22×D4—21no—
223C22×D4C2412yes—
224C22×D4C2411yescharacteristic
225C23×C4—21no—
226C23×C4—22no—
227C23×C4—21no—
228C23×C4—21no—
229C2×D8—24no—
230C2×D8—24no—
231C2×D8—42no—
232C22×C8C2412yes—
233C22×C8C2412yes—
234C22×C8C2412yes—
235C22×C8C2412yes—
236C4⋊Q8—24no—
237C4⋊Q8—22no—
238C4⋊Q8—22no—
239C4⋊Q8—42no—
240C4⋊Q8—42no—
241C4⋊Q8—44no—
242C4⋊Q8—82no—
243#127—22no—
244#127—22no—
245#125—48no—
246#124—84no—
247C22⋊Q8—48no—
248C22⋊Q8—48no—
249C22⋊Q8—84no—
250C4×Q8—24no—
251C4×Q8—24no—
252C4×Q8—48no—
253C4×Q8—48no—
254C4×Q8—82no—
255C4×D4—24no—
256C4×D4—22no—
257C4×D4—22no—
258C4×D4—22no—
259C4×D4—24no—
260C4×D4—41no—
261C4×D4—41no—
262C4×D4—44no—
263C4×D4—82no—
264C4×D4#4114yes—
265#117—48no—
266#117—84no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4C2×D412yes—
282C2×C4⋊C4C2×D412yes—
283C2×C4⋊C4C2×D412yes—
284C2×C4⋊C4C2×D412yes—
285C2×C4⋊C4C2×D414yes—
286C2×C4⋊C4C2×D412yes—
287C2×C4⋊C4C2412yes—
288C2×C4⋊C4C2411yescharacteristic
289C2×C4⋊C4C2411yescharacteristic
290C2×C4⋊C4C2412yes—
291C2×C4⋊C4C2414yes—
292C2×C4⋊C4C2412yes—
293C2×C4⋊C4C2411yescharacteristic
294C2×C22⋊C4—21no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—21no—
298C2×C22⋊C4—21no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C42—24no—
307C2×C42—21no—
308C2×C42—21no—
309C2×C42—22no—
310C2×C42—21no—
311C2×C42—24no—
312C2×C42—44no—
313C2×C42C2×D411yescharacteristic
314C2×C42C2×D411yescharacteristic
315C2×C42C2×D411yescharacteristic
316C2×C42C2×D411yescharacteristic
317C2×C42C2411yescharacteristic
318C2×C42C2414yes—
319C2×C42C2411yescharacteristic
320C2×C42C2411yescharacteristic
321#107—24no—
322#106—24no—
323#106—44no—
324#106—44no—
325#106—42no—
326C4⋊C8—28no—
327C4⋊C8—22no—
328C4⋊C8—22no—
329C4⋊C8—22no—
330C4⋊C8#4114yes—
331#102—24no—
332#102—24no—
333#102—24no—
334#102—24no—
335#102—42no—
336#102—42no—
337#102—44no—
338#102—44no—
339#97—28no—
340C4×C8—24no—
341C4×C8—24no—
342#95—21no—
343#95—21no—
344#95—21no—
345#95—24no—
346#95—22no—
347#95—24no—
348#95—24no—
349#95—22no—
350#95—24no—
351#95—24no—
352#95C2×D414yes—
353#95C2×D414yes—
354C24—21no—
355C24—22no—
356C2×Q8—24no—
357C2×Q8—24no—
358C2×Q8—24no—
359C2×Q8—44no—
360C2×Q8—44no—
361C2×Q8—42no—
362C2×Q8—44no—
363C2×Q8—81no—
364C2×Q8—81no—
365C2×D4—21no—
366C2×D4—22no—
367C2×D4—22no—
368C2×D4—21no—
369C2×D4—21no—
370C2×D4—22no—
371C2×D4—22no—
372C2×D4—21no—
373C2×D4—21no—
374C2×D4—22no—
375C2×D4—22no—
376C2×D4—42no—
377C2×D4—42no—
378C2×D4—42no—
379C2×D4—42no—
380C2×D4—81no—
381C2×D4—81no—
382C2×D4C2×#4112yes—
383C22×C4—21no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—21no—
390C22×C4—24no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—44no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—44no—
400C22×C4—44no—
401C22×C4—42no—
402C22×C4C22×D414yes—
403C22×C4C22×D414yes—
404C22×C4C22×D411yescharacteristic
405C22×C4C22×D411yescharacteristic
406C22×C4C22×D411yescharacteristic
407C22×C4C22×D411yescharacteristic
408C22×C4C22×D411yescharacteristic
409C22×C4C22×D411yescharacteristic
410C22×C4C2511yescharacteristic
411D8—48no—
412C2×C8—22no—
413C2×C8—22no—
414C2×C8—22no—
415C2×C8—22no—
416C2×C8—22no—
417C2×C8—22no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8C2×#4114yes—
424C4⋊C4—24no—
425C4⋊C4—28no—
426C4⋊C4—24no—
427C4⋊C4—24no—
428C4⋊C4—24no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—21no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—21no—
437C4⋊C4—22no—
438C4⋊C4—24no—
439C4⋊C4—24no—
440C4⋊C4—22no—
441C4⋊C4—21no—
442C4⋊C4—42no—
443C4⋊C4—48no—
444C4⋊C4—48no—
445C4⋊C4—44no—
446C4⋊C4—42no—
447C4⋊C4—44no—
448C4⋊C4—42no—
449C4⋊C4—44no—
450C4⋊C4—42no—
451C4⋊C4—81no—
452C4⋊C4—81no—
453C4⋊C4—82no—
454C4⋊C4—84no—
455C4⋊C4—82no—
456C4⋊C4—82no—
457C4⋊C4C2×#4112yes—
458C22⋊C4—24no—
459C22⋊C4—48no—
460C22⋊C4—42no—
461C22⋊C4—42no—
462C22⋊C4—44no—
463C22⋊C4—44no—
464C22⋊C4—84no—
465C22⋊C4—84no—
466C22⋊C4—82no—
467C22⋊C4—84no—
468C22⋊C4—84no—
469C22⋊C4—84no—
470C42—22no—
471C42—22no—
472C42—24no—
473C42—24no—
474C42—21no—
475C42—21no—
476C42—41no—
477C42—48no—
478C42—42no—
479C42—42no—
480C42—42no—
481C42—44no—
482C42—84no—
483C42AGL(1,ℤ/8)14yes—
484C42C2×#4112yes—
485C42C2×#4112yes—
486C422−1+412yes—
487C422−1+412yes—
488C23—22no—
489C23—41no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23C23×D411yescharacteristic
503Q8—48no—
504Q8—88no—
505Q8—84no—
506D4—22no—
507D4—22no—
508D4—42no—
509D4—42no—
510D4—41no—
511D4—42no—
512D4—41no—
513D4—42no—
514D4—84no—
515D4—84no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—22no—
526C2×C4—21no—
527C2×C4—24no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—48no—
535C2×C4—42no—
536C2×C4—44no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—41no—
542C2×C4—44no—
543C2×C4—42no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—44no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—41no—
553C2×C4—84no—
554C2×C4—84no—
555C2×C4—88no—
556C2×C4#54712yes—
557C2×C4#54712yes—
558C2×C4#54812yes—
559C2×C4#54812yes—
560C2×C4C2×AGL(1,ℤ/8)12yes—
561C2×C4C2×AGL(1,ℤ/8)12yes—
562C2×C4C2×AGL(1,ℤ/8)12yes—
563C2×C4#57712yes—
564C2×C4#57712yes—
565C2×C4C22×#4111yescharacteristic
566C2×C4C22×#4111yescharacteristic
567C2×C4C2×2−1+411yescharacteristic
568C2×C4C2×2−1+411yescharacteristic
569C2×C4#58511yescharacteristic
570C2×C4#58511yescharacteristic
571C8—24no—
572C8—42no—
573C8—42no—
574C8—42no—
575C22—42no—
576C22—42no—
577C22—42no—
578C22—81no—
579C22—81no—
580C22—81no—
581C22—81no—
582C22—82no—
583C22—82no—
584C22—82no—
585C22—82no—
586C22—82no—
587C22—82no—
588C22—82no—
589C22C2×#54711yescharacteristic
590C22C2×#54811yescharacteristic
591C22#348711yescharacteristic
592C22C22×AGL(1,ℤ/8)11yescharacteristic
593C22C2×#57711yescharacteristic
594C22C2×#57711yescharacteristic
595C22#358511yescharacteristic
596C4—22no—
597C4—22no—
598C4—22no—
599C4—44no—
600C4—44no—
601C4—42no—
602C4—41no—
603C4—81no—
604C4—82no—
605C4—82no—
606C4—81no—
607C4—84no—
608C4—82no—
609C4#335414yes—
610C4#337314yes—
611C4#347511yescharacteristic
612C4#350711yescharacteristic
613C4#351811yescharacteristic
614C4#355511yescharacteristic
615C2—82no—
616C2—162no—
617C2#5906511yescharacteristic
618C2C2×#335411yescharacteristic
619C2#6127911yescharacteristic
620C2C2×#337311yescharacteristic
621C2#6134711yescharacteristic
622C2#6189011yescharacteristic
623C2#6193511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#459C22
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
1. #59863C2semidirect
2. #36134C2semidirect
3. #32446C2semidirect
4. #29206C2semidirect
5. #29139C2semidirect
6. #27735C2semidirect
7. #27144C2semidirect
8. #26318C2semidirect
9. #26090C2semidirect
11. #24991C2semidirect
12. #24677C2semidirect
13. #23153C2semidirect
14. #22813C2semidirect
15. #22742C2semidirect
16. C4×#476C2semidirect

Maximal subgroups

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

subgroupindex
#598632
#361342
#324462
#292062
#291392
#277352
#271442
#263182
#260902
#253002
#249912
#246772
#231532
#228132
#227422
C4×#4762
C4×#4592