Finite Group Databasebeta

← #7533407

Subgroups — #7533407

Order 512.

1483 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#59828C211yesmaximal, characteristic, complemented
2#36194C211yesmaximal, characteristic
3#31663C214yesmaximal
4#29141C211yesmaximal, characteristic, complemented
5#29141C211yesmaximal, characteristic, complemented
6#28195C212yesmaximal
7#27757C212yesmaximal, complemented
8#26711C214yesmaximal, complemented
9#26113C212yesmaximal, complemented
10#24997C212yesmaximal, complemented
11#24630C212yesmaximal
12#24187C212yesmaximal
13#23153C211yesmaximal, characteristic, complemented
14#22759C212yesmaximal
15#22547C212yesmaximal, complemented
16C4×#461C211yesmaximal, characteristic
17C4×#461C211yesmaximal, characteristic
18C2×#557C2212yes—
19C2×#461C2211yescharacteristic
20C2×#461C2211yescharacteristic
21C2×#446C2212yes—
22C2×C4×SD16C2212yes—
23C2×C4×SD16C2212yes—
24#2868C2211yescharacteristic
25#2857—21no—
26#2726—22no—
27#2582C2212yes—
28#2571C2212yes—
29#2563C2212yes—
30#2552—21no—
31#2546C2212yes—
32#2538C2212yes—
33#2506—22no—
34#2497—22no—
35#2492—22no—
36#2453—22no—
37#2446C2214yes—
38#2439—22no—
39#2369C2212yes—
40#2365—21no—
41#2355C2212yes—
42#2354C2212yes—
43Q8×C4⋊C4C2212yes—
44C4×C4⋊Q8C2211yescharacteristic
45C4×C4⋊Q8C2211yescharacteristic
46C4×C4⋊Q8C2211yescharacteristic
47#2285—21no—
48C42×D4C2211yescharacteristic
49#2101C2214yes—
50#2088C2214yes—
51#2074C2214yes—
52#2055C2214yes—
53#2043C2214yes—
54#2024C2214yes—
55#2024C2214yes—
56#2013C2214yes—
57#1984C2212yes—
58#1981C2211yescharacteristic
59#1981C2214yes—
60#1981C2211yescharacteristic
61#1960C2214yes—
62#1956C2212yes—
63#1956C2212yes—
64#1948C2211yescharacteristic
65#1948C2211yescharacteristic
66#1939C2212yes—
67#1936C2214yes—
68#1935C2214yes—
69#1927C2214yes—
70#1927C2212yes—
71#1927C2212yes—
72#1919C2214yes—
73#1917C2214yes—
74#1880C2214yes—
75#1877C2212yes—
76#1877C2212yes—
77#1865C2214yes—
78#1834C2212yes—
79#1834C2212yes—
80#1805C2214yes—
81#1803C2212yes—
82#1803C2212yes—
83#1803C2212yes—
84#1803C2212yes—
85#1802C2211yescharacteristic
86#1802C2211yescharacteristic
87#1770C2212yes—
88C4×C4⋊C8C2211yescharacteristic
89C4×C4⋊C8C2211yescharacteristic
90#1763C2214yes—
91C4×#103C2212yes—
92C4×#103C2212yes—
93C22×SD16C2312yes—
94C22×SD16C2312yes—
95#557—28no—
96#557—44no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8D411yescharacteristic
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2314yes—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2312yes—
104C2×#125—22no—
105C2×#125—24no—
106C2×C22⋊Q8—22no—
107C2×C4×Q8—22no—
108C2×C4×Q8C2312yes—
109C2×C4×Q8C2312yes—
110C2×C4×Q8C2314yes—
111C2×C4×Q8C2312yes—
112C2×C4×D4C2311yescharacteristic
113C2×C4×D4C2311yescharacteristic
114C2×C4×D4C2311yescharacteristic
115C2×#117—22no—
116C22×C42—21no—
117#461—24no—
118#461—24no—
119#446—44no—
120C4×SD16—28no—
121C4×SD16—44no—
122C2×#106C2312yes—
123C2×#106C2312yes—
124C2×C4⋊C8C2311yescharacteristic
125C2×C4⋊C8C2311yescharacteristic
126C2×C4⋊C8C2311yescharacteristic
127C2×C4⋊C8C2311yescharacteristic
128C2×C4⋊C8C2314yes—
129C2×#103C2312yes—
130C2×#103C2312yes—
131C2×#103C2312yes—
132C2×#103C2314yes—
133C2×#103C2314yes—
134C2×#103C2314yes—
135C2×#103C2314yes—
136C2×#103C2312yes—
137C2×#103C2312yes—
138C2×#103C2312yes—
139C2×#102C2312yes—
140C2×#102C2312yes—
141C2×#97C2314yes—
142C2×C4×C8C2312yes—
143C2×C4×C8C2312yes—
144#400—24no—
145#400—24no—
146#398—24no—
147#398—24no—
148#398—24no—
149#396—24no—
150#393—24no—
151#391—22no—
152#391—22no—
153#391—24no—
154#391D412yes—
155#391C2312yes—
156#391C2312yes—
157#391C2314yes—
158#390C2314yes—
159#389—22no—
160#389—22no—
161#389—22no—
162#389—22no—
163#389—22no—
164#389—21no—
165#389—21no—
166#389—21no—
167#389—21no—
168#389—24no—
169#389D412yes—
170#389D412yes—
171#389C2314yes—
172#389C2314yes—
173#388—24no—
174#387—24no—
175#387—24no—
176#387—24no—
177#384—22no—
178#384—21no—
179#384—21no—
180#384D411yescharacteristic
181#384C2311yescharacteristic
182#384C2311yescharacteristic
183#384C2312yes—
184#384C2312yes—
185#382—22no—
186#382C2314yes—
187#380—22no—
188#380—22no—
189#380—22no—
190#380—22no—
191C4×C4⋊C4—22no—
192C4×C4⋊C4—22no—
193C4×C4⋊C4—22no—
194C4×C4⋊C4—21no—
195C4×C4⋊C4—21no—
196C4×C4⋊C4—21no—
197C4×C4⋊C4—21no—
198C4×C4⋊C4C2311yescharacteristic
199C4×C4⋊C4C2314yes—
200C4×C4⋊C4C2311yescharacteristic
201C4×C4⋊C4C2311yescharacteristic
202C4×C4⋊C4C2312yes—
203C4×C4⋊C4C2312yes—
204C4×C22⋊C4—21no—
205C4×C22⋊C4—21no—
206C4×C22⋊C4—21no—
207C43C2311yescharacteristic
208#340C2314yes—
209#340C2312yes—
210#340C2312yes—
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#340C2312yes—
215#340C2312yes—
216#336C2314yes—
217#336C2314yes—
218#336C2312yes—
219#336C2312yes—
220C22×Q8—22no—
221C22×Q8C2×D412yes—
222C22×Q8C2412yes—
223C22×Q8C2412yes—
224C22×D4C2411yescharacteristic
225C23×C4—21no—
226C23×C4—21no—
227C23×C4—21no—
228C2×SD16—24no—
229C2×SD16—24no—
230C2×SD16—24no—
231C2×SD16—42no—
232C2×SD16—42no—
233C2×SD16—42no—
234C22×C8C2412yes—
235C22×C8C2412yes—
236C22×C8C2412yes—
237C22×C8C2412yes—
238C4⋊Q8—28no—
239C4⋊Q8—24no—
240C4⋊Q8—22no—
241C4⋊Q8—22no—
242C4⋊Q8—22no—
243C4⋊Q8—42no—
244C4⋊Q8—42no—
245C4⋊Q8—44no—
246C4⋊Q8—82no—
247C4⋊Q8—82no—
248C4⋊Q8#4114yes—
249#125—416no—
250#125—48no—
251C22⋊Q8—84no—
252C4×Q8—24no—
253C4×Q8—24no—
254C4×Q8—28no—
255C4×Q8—24no—
256C4×Q8—24no—
257C4×Q8—24no—
258C4×Q8—24no—
259C4×Q8—44no—
260C4×Q8—48no—
261C4×D4—22no—
262C4×D4—22no—
263C4×D4—22no—
264C4×D4—41no—
265C4×D4—41no—
266C4×D4#4114yes—
267#117—84no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4C2×D412yes—
289C2×C4⋊C4C2×D412yes—
290C2×C4⋊C4C2×D414yes—
291C2×C4⋊C4C2×D412yes—
292C2×C4⋊C4C2×D412yes—
293C2×C4⋊C4C2×D412yes—
294C2×C4⋊C4C2411yescharacteristic
295C2×C4⋊C4C2411yescharacteristic
296C2×C4⋊C4C2412yes—
297C2×C4⋊C4C2412yes—
298C2×C4⋊C4C2414yes—
299C2×C4⋊C4C2411yescharacteristic
300C2×C22⋊C4—21no—
301C2×C22⋊C4—21no—
302C2×C22⋊C4—21no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C42—22no—
306C2×C42—24no—
307C2×C42—22no—
308C2×C42—21no—
309C2×C42—21no—
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#106—24no—
322#106—42no—
323C4⋊C8—22no—
324C4⋊C8—22no—
325C4⋊C8—22no—
326C4⋊C8—22no—
327C4⋊C8—44no—
328#103—24no—
329#103—24no—
330#103—24no—
331#103—42no—
332#103—42no—
333#103—44no—
334#103—44no—
335#103—44no—
336#103—44no—
337#103—42no—
338#102—24no—
339#102—42no—
340#97—44no—
341C4×C8—24no—
342C4×C8—24no—
343#95—21no—
344#95—21no—
345#95—21no—
346#95—24no—
347#95—24no—
348#95—22no—
349#95—22no—
350#95—22no—
351#95—22no—
352#95—24no—
353#95—24no—
354#95—24no—
355#95—24no—
356#95—24no—
357#95C2×D414yes—
358#95C2×D414yes—
359C24—21no—
360C2×Q8—24no—
361C2×Q8—24no—
362C2×Q8—24no—
363C2×Q8—22no—
364C2×Q8—22no—
365C2×Q8—22no—
366C2×Q8—22no—
367C2×Q8—22no—
368C2×Q8—22no—
369C2×Q8—22no—
370C2×Q8—22no—
371C2×Q8—22no—
372C2×Q8—22no—
373C2×Q8—22no—
374C2×Q8—44no—
375C2×Q8—82no—
376C2×Q8—82no—
377C2×Q8C2×#4114yes—
378C2×D4—21no—
379C2×D4—21no—
380C2×D4—21no—
381C2×D4—21no—
382C2×D4—21no—
383C2×D4C2×#4112yes—
384C22×C4—21no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—21no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4C22×D411yescharacteristic
402C22×C4C22×D414yes—
403C22×C4C22×D414yes—
404C22×C4C22×D411yescharacteristic
405C22×C4C22×D411yescharacteristic
406C22×C4C22×D411yescharacteristic
407C22×C4C22×D411yescharacteristic
408C22×C4C22×D411yescharacteristic
409C22×C4C2511yescharacteristic
410SD16—48no—
411SD16—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×C8—22no—
424C4⋊C4—24no—
425C4⋊C4—24no—
426C4⋊C4—28no—
427C4⋊C4—24no—
428C4⋊C4—24no—
429C4⋊C4—24no—
430C4⋊C4—24no—
431C4⋊C4—21no—
432C4⋊C4—22no—
433C4⋊C4—21no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—21no—
437C4⋊C4—22no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—24no—
441C4⋊C4—24no—
442C4⋊C4—42no—
443C4⋊C4—48no—
444C4⋊C4—48no—
445C4⋊C4—48no—
446C4⋊C4—44no—
447C4⋊C4—44no—
448C4⋊C4—44no—
449C4⋊C4—42no—
450C4⋊C4—42no—
451C4⋊C4—42no—
452C4⋊C4—42no—
453C4⋊C4—44no—
454C4⋊C4—44no—
455C4⋊C4—81no—
456C4⋊C4—81no—
457C4⋊C4—84no—
458C4⋊C4—84no—
459C4⋊C4—82no—
460C4⋊C4—82no—
461C4⋊C4C2×#4112yes—
462C22⋊C4—24no—
463C22⋊C4—42no—
464C22⋊C4—42no—
465C22⋊C4—84no—
466C22⋊C4—84no—
467C42—22no—
468C42—22no—
469C42—24no—
470C42—24no—
471C42—21no—
472C42—21no—
473C42—41no—
474C42—48no—
475C42—44no—
476C42—44no—
477C42—42no—
478C42—42no—
479C42—42no—
480C42—84no—
481C42#13714yes—
482C42C2×#4112yes—
483C42C2×#4112yes—
484C42H(5,𝔽2)12yes—
485C422−1+412yes—
486C23—22no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23C23×D411yescharacteristic
494Q8—24no—
495Q8—24no—
496Q8—48no—
497Q8—42no—
498Q8—42no—
499Q8—42no—
500Q8—42no—
501Q8—42no—
502Q8—42no—
503Q8—164no—
504D4—22no—
505D4—22no—
506D4—41no—
507D4—41no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—22no—
519C2×C4—21no—
520C2×C4—24no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—48no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—41no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—41no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—41no—
539C2×C4—44no—
540C2×C4—42no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—41no—
548C2×C4—84no—
549C2×C4—84no—
550C2×C4#54612yes—
551C2×C4#54612yes—
552C2×C4#54812yes—
553C2×C4#54812yes—
554C2×C4C2×#13712yes—
555C2×C4C2×#13712yes—
556C2×C4C2×#13712yes—
557C2×C4#57712yes—
558C2×C4#57712yes—
559C2×C4C22×#4111yescharacteristic
560C2×C4C22×#4111yescharacteristic
561C2×C4C2×H(5,𝔽2)11yescharacteristic
562C2×C4C2×2−1+411yescharacteristic
563C2×C4#58511yescharacteristic
564C2×C4#58511yescharacteristic
565C8—42no—
566C8—42no—
567C8—42no—
568C8—42no—
569C22—42no—
570C22—42no—
571C22—42no—
572C22—81no—
573C22—81no—
574C22—81no—
575C22—81no—
576C22C2×#54611yescharacteristic
577C22C2×#54811yescharacteristic
578C22#348711yescharacteristic
579C22C22×#13711yescharacteristic
580C22C2×#57711yescharacteristic
581C22C2×#57711yescharacteristic
582C22#358611yescharacteristic
583C4—22no—
584C4—22no—
585C4—22no—
586C4—44no—
587C4—42no—
588C4—44no—
589C4—44no—
590C4—41no—
591C4—82no—
592C4—82no—
593C4—82no—
594C4—81no—
595C4—81no—
596C4—84no—
597C4#332914yes—
598C4#337214yes—
599C4#347511yescharacteristic
600C4#350611yescharacteristic
601C4#351511yescharacteristic
602C4#354711yescharacteristic
603C2—82no—
604C2#5903411yescharacteristic
605C2C2×#332911yescharacteristic
606C2#6120211yescharacteristic
607C2C2×#337211yescharacteristic
608C2#6134611yescharacteristic
609C2#6188911yescharacteristic
610C2#6193311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C24
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. #59828C2semidirect
4. #29141C2semidirect
5. #29141C2semidirect
7. #27757C2semidirect
8. #26711C2semidirect
9. #26113C2semidirect
10. #24997C2semidirect
13. #23153C2semidirect
15. #22547C2semidirect

Maximal subgroups

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

subgroupindex
#598282
#361942
#316632
#291412
#281952
#277572
#267112
#261132
#249972
#246302
#241872
#231532
#227592
#225472
C4×#4612