Finite Group Databasebeta

← #6556237

Subgroups — #6556237

Order 512.

1560 conjugacy classes of subgroups.

The subgroup lattice has 641 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36276C211yesmaximal, characteristic
2#36194C211yesmaximal, characteristic
3#30205C211yesmaximal, characteristic, complemented
4#30199C212yesmaximal, complemented
5#29347C211yesmaximal, characteristic, complemented
6#29111C212yesmaximal, complemented
7#28717C211yesmaximal, characteristic, complemented
8#27980C212yesmaximal, complemented
9#27757C212yesmaximal, complemented
10#27738C212yesmaximal, complemented
11#27721C214yesmaximal, complemented
12#27277C214yesmaximal, complemented
13#23816C211yesmaximal, characteristic, complemented
14#23629C212yesmaximal, complemented
15#22502C212yesmaximal
16#22430C212yesmaximal
17#20568C211yesmaximal, characteristic
18C2×#554C2211yescharacteristic
19C2×D4×Q8C2211yescharacteristic
20C2×#487C2212yes—
21C2×#436C2212yes—
22C2×#419C2214yes—
23#2868C2211yescharacteristic
24#2862C2211yescharacteristic
25#2857C2212yes—
26#2857C2212yes—
27#2571C2212yes—
28#2567C2211yescharacteristic
29#2563—21no—
30#2550—21no—
31#2550—21no—
32#2549—22no—
33#2549—22no—
34#2546C2211yescharacteristic
35#2538—21no—
36#2502C2212yes—
37#2497—22no—
38#2492—22no—
39#2439C2214yes—
40#2434C2212yes—
41#2369—21no—
42#2355—21no—
43#2354C2212yes—
44#2349—21no—
45#2349—21no—
46C4×C4⋊Q8C2211yescharacteristic
47C4×C4⋊Q8C2211yescharacteristic
48C2×#391C2212yes—
49#2285C2212yes—
50C42×D4C2211yescharacteristic
51#2100C2212yes—
52#2099C2214yes—
53#2074C2214yes—
54#2074C2214yes—
55#2072C2211yescharacteristic
56#2072C2211yescharacteristic
57#2072C2212yes—
58#2072C2212yes—
59#2072C2212yes—
60#2072C2212yes—
61#2072C2212yes—
62#2043C2212yes—
63#2043C2214yes—
64#2024C2211yescharacteristic
65#2024C2211yescharacteristic
66#2024C2214yes—
67#2024C2212yes—
68#1990C2212yes—
69#1990C2212yes—
70#1987C2212yes—
71#1981C2212yes—
72#1960C2214yes—
73#1956C2212yes—
74#1935C2214yes—
75#1935C2214yes—
76#1933C2214yes—
77#1894C2214yes—
78#1894C2214yes—
79#1873C2214yes—
80#1873C2214yes—
81#1836C2214yes—
82#1823C2214yes—
83#1803C2212yes—
84#1803C2212yes—
85#1787C2214yes—
86#1770C2212yes—
87C4×C4⋊C8C2211yescharacteristic
88C4×#103C2212yes—
89C4×#103C2212yes—
90#1753C2214yes—
91#1749C2212yes—
92#1744C2211yescharacteristic
93#1736C2214yes—
94#1728C2212yes—
95C23×Q8C2312yes—
96C22×M16C2314yes—
97#554—28no—
98D4×Q8—28no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2312yes—
105C2×C4⋊Q8C2311yescharacteristic
106C2×C4⋊Q8C2311yescharacteristic
107C2×C4⋊Q8C2312yes—
108C2×#125—21no—
109C2×C22⋊Q8—22no—
110C2×C22⋊Q8—22no—
111C2×C22⋊Q8—22no—
112C2×C22⋊Q8C2312yes—
113C2×C22⋊Q8C2312yes—
114C2×C4×Q8—21no—
115C2×C4×Q8C2311yescharacteristic
116C2×C4×Q8C2311yescharacteristic
117C2×C4×D4—21no—
118C2×C4×D4—21no—
119C2×C4×D4D411yescharacteristic
120C2×C4×D4D411yescharacteristic
121C2×C4×D4C2311yescharacteristic
122C2×#117C2314yes—
123C22×C4⋊C4C2312yes—
124C22×C42C2312yes—
125#487—44no—
126#436—28no—
127C2×C4⋊C8C2312yes—
128#419—48no—
129C2×#103C2314yes—
130C2×#103C2312yes—
131C2×#103C2314yes—
132C2×#103C2314yes—
133C2×#103C2314yes—
134C2×#103C2312yes—
135C2×#103C2314yes—
136C2×#103C2312yes—
137C2×#103C2314yes—
138C2×#103C2312yes—
139C2×C22⋊C8C2314yes—
140C2×#97C2314yes—
141C2×C4×C8C2312yes—
142C2×C4×C8C2312yes—
143#400—24no—
144#398—22no—
145#398—22no—
146#398—22no—
147#396—24no—
148#395—22no—
149#395—22no—
150#395—22no—
151#393—24no—
152#391—21no—
153#391—22no—
154#391—21no—
155#391—21no—
156#391—22no—
157#391—22no—
158#391—24no—
159#391C2311yescharacteristic
160#391C2311yescharacteristic
161#391C2312yes—
162#391C2312yes—
163#391C2312yes—
164#391C2312yes—
165#390—22no—
166#389—21no—
167#389—21no—
168#389—22no—
169#389—21no—
170#389—21no—
171#389—21no—
172#389—21no—
173#389—21no—
174#389—22no—
175#389—22no—
176#389—22no—
177#389C2312yes—
178#389C2314yes—
179#388—24no—
180#387—24no—
181#385—22no—
182#385—22no—
183#385—22no—
184#385—22no—
185#384—21no—
186#384—21no—
187#384C2311yescharacteristic
188#384C2311yescharacteristic
189#384C2312yes—
190#384C2312yes—
191#384C2312yes—
192#382—22no—
193#382—21no—
194#382—21no—
195#382C2314yes—
196#380—22no—
197#380—22no—
198#380—22no—
199#380—22no—
200#380—22no—
201#380C2314yes—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4D411yescharacteristic
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4C2312yes—
211C4×C4⋊C4C2311yescharacteristic
212C4×C22⋊C4D412yes—
213C4×C22⋊C4D412yes—
214C4×C22⋊C4C2312yes—
215C2×#95—22no—
216C2×#95—22no—
217C43C2311yescharacteristic
218#340C2314yes—
219#340C2314yes—
220#340C2312yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2312yes—
225#340C2312yes—
226#340C2314yes—
227#340C2312yes—
228#336C2311yescharacteristic
229#336C2312yes—
230#336C2311yescharacteristic
231#336C2314yes—
232#336C2312yes—
233C22×Q8—21no—
234C22×Q8—22no—
235C22×Q8—21no—
236C22×Q8—22no—
237C22×Q8—24no—
238C22×Q8—28no—
239C22×Q8—42no—
240C22×Q8—44no—
241C22×Q8C2411yescharacteristic
242C22×Q8C2412yes—
243C22×Q8C2411yescharacteristic
244C22×D4C2×D411yescharacteristic
245C23×C4—22no—
246C23×C4—22no—
247C23×C4C2×D412yes—
248C23×C4C2×D412yes—
249C23×C4C2412yes—
250C2×M16—28no—
251C2×M16—44no—
252C2×M16—44no—
253C22×C8C2412yes—
254C22×C8C2414yes—
255C22×C8C2412yes—
256C4⋊Q8—22no—
257C4⋊Q8—24no—
258C4⋊Q8—24no—
259C4⋊Q8—42no—
260C4⋊Q8—42no—
261C4⋊Q8—42no—
262C4⋊Q8—41no—
263C4⋊Q8—44no—
264C4⋊Q8—44no—
265C4⋊Q8—44no—
266C4⋊Q8—82no—
267#125—44no—
268C22⋊Q8—28no—
269C22⋊Q8—28no—
270C22⋊Q8—48no—
271C22⋊Q8—48no—
272C22⋊Q8—84no—
273C4×Q8—24no—
274C4×Q8—24no—
275C4×Q8—44no—
276C4×D4—24no—
277C4×D4—24no—
278C4×D4—44no—
279C4×D4—44no—
280C4×D4#4118yes—
281#117—44no—
282#117—44no—
283C2×C4⋊C4—28no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4C2×D411yescharacteristic
311C2×C4⋊C4C2×D411yescharacteristic
312C2×C4⋊C4C2×D411yescharacteristic
313C2×C4⋊C4C2411yescharacteristic
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2414yes—
317C2×C4⋊C4C2411yescharacteristic
318C2×C4⋊C4C2412yes—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4C2×D412yes—
324C2×C22⋊C4C2×D412yes—
325C2×C22⋊C4C2×D412yes—
326C2×C42—21no—
327C2×C42—21no—
328C2×C42—22no—
329C2×C42—24no—
330C2×C42—44no—
331C2×C42C2×D411yescharacteristic
332C2×C42C2×D411yescharacteristic
333C2×C42C2×D412yes—
334C2×C42C2×D412yes—
335C2×C42C2×D411yescharacteristic
336C2×C42C2×D411yescharacteristic
337C2×C42C2411yescharacteristic
338C2×C42C2411yescharacteristic
339C2×C42C2412yes—
340C2×C42C2411yescharacteristic
341C4⋊C8—24no—
342#103—24no—
343#103—28no—
344#103—42no—
345#103—44no—
346#103—42no—
347#103—42no—
348#103—44no—
349#103—44no—
350#103—44no—
351#103—44no—
352C22⋊C8—28no—
353#97—44no—
354C4×C8—24no—
355C4×C8—42no—
356#95—21no—
357#95—21no—
358#95—22no—
359#95—22no—
360#95—22no—
361#95—22no—
362#95—22no—
363#95—22no—
364#95—21no—
365#95—21no—
366#95—24no—
367#95—24no—
368#95—24no—
369#95—22no—
370#95—22no—
371#95—24no—
372#95—22no—
373#95—22no—
374#95C2×D412yes—
375#95C2×D412yes—
376#95C2×D412yes—
377C24C22×D412yes—
378C2×Q8—21no—
379C2×Q8—22no—
380C2×Q8—22no—
381C2×Q8—24no—
382C2×Q8—21no—
383C2×Q8—22no—
384C2×Q8—22no—
385C2×Q8—21no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—24no—
389C2×Q8—44no—
390C2×Q8—42no—
391C2×Q8—42no—
392C2×Q8—416no—
393C2×Q8—44no—
394C2×Q8—42no—
395C2×Q8—42no—
396C2×Q8—44no—
397C2×Q8—82no—
398C2×Q8—82no—
399C2×Q8—82no—
400C2×Q8—81no—
401C2×Q8—81no—
402C2×Q8—88no—
403C2×Q8H(5,𝔽2)12yes—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4#12414yes—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—21no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—44no—
426C22×C4—44no—
427C22×C4—44no—
428C22×C4—82no—
429C22×C4—82no—
430C22×C4—82no—
431C22×C4C22×D412yes—
432C22×C4C22×D412yes—
433C22×C4C22×D412yes—
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D411yescharacteristic
440C22×C4C2×#4118yes—
441C22×C4C2511yescharacteristic
442M16—88no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—24no—
446C2×C8—24no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—24no—
450C2×C8—22no—
451C2×C8—22no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—24no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—24no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—24no—
463C4⋊C4—42no—
464C4⋊C4—42no—
465C4⋊C4—42no—
466C4⋊C4—44no—
467C4⋊C4—42no—
468C4⋊C4—44no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—42no—
472C4⋊C4—42no—
473C4⋊C4—44no—
474C4⋊C4—44no—
475C4⋊C4—42no—
476C4⋊C4—42no—
477C4⋊C4—44no—
478C4⋊C4—44no—
479C4⋊C4—44no—
480C4⋊C4—81no—
481C4⋊C4—81no—
482C4⋊C4—82no—
483C4⋊C4—82no—
484C4⋊C4—84no—
485C4⋊C4—82no—
486C4⋊C4—82no—
487C4⋊C4—82no—
488C4⋊C4—82no—
489C4⋊C4#12414yes—
490C22⋊C4—24no—
491C22⋊C4—24no—
492C22⋊C4—44no—
493C22⋊C4—44no—
494C22⋊C4—84no—
495C22⋊C4—84no—
496C22⋊C4#12418yes—
497C42—21no—
498C42—24no—
499C42—22no—
500C42—21no—
501C42—42no—
502C42—42no—
503C42—41no—
504C42—41no—
505C42—42no—
506C42—42no—
507C42—44no—
508C42—84no—
509C42#13714yes—
510C42#13714yes—
511C42C2×#4114yes—
512C23—22no—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23C2×#12414yes—
519C23C23×D411yescharacteristic
520Q8—24no—
521Q8—42no—
522Q8—42no—
523Q8—44no—
524Q8—42no—
525Q8—44no—
526Q8—88no—
527Q8—84no—
528Q8—164no—
529Q8—162no—
530D4—28no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—22no—
543C2×C4—28no—
544C2×C4—28no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—42no—
555C2×C4—44no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—42no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—84no—
579C2×C4—84no—
580C2×C4—88no—
581C2×C4—164no—
582C2×C4C2×#12412yes—
583C2×C4C2×#12414yes—
584C2×C4C2×#12412yes—
585C2×C4C2×#13712yes—
586C2×C4C2×#13712yes—
587C2×C4C2×#13712yes—
588C2×C4C2×#13712yes—
589C2×C4C2×#13712yes—
590C2×C4C2×#13712yes—
591C2×C4C22×#4112yes—
592C2×C4C2×H(5,𝔽2)11yescharacteristic
593C2×C4#58512yes—
594C2×C4#58511yescharacteristic
595C8—44no—
596C8—42no—
597C8—42no—
598C22—24no—
599C22—24no—
600C22—24no—
601C22—42no—
602C22—42no—
603C22—42no—
604C22—42no—
605C22C22×#12411yescharacteristic
606C22#348511yescharacteristic
607C22#348811yescharacteristic
608C22C22×#13711yescharacteristic
609C22C22×#13711yescharacteristic
610C22#358511yescharacteristic
611C22#358611yescharacteristic
612C4—24no—
613C4—22no—
614C4—22no—
615C4—22no—
616C4—22no—
617C4—22no—
618C4—42no—
619C4—42no—
620C4—42no—
621C4—82no—
622C4—81no—
623C4—84no—
624C4—82no—
625C4—82no—
626C4—81no—
627C4—81no—
628C4—82no—
629C4#313914yes—
630C4#351312yes—
631C4#352112yes—
632C2—44no—
633C2#3584711yescharacteristic
634C2C2×#313911yescharacteristic
635C2#6062511yescharacteristic
636C2#6172111yescharacteristic
637C2#6172311yescharacteristic
638C2#6196511yescharacteristic
639C2#6196611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #30205C2semidirect
4. #30199C2semidirect
5. #29347C2semidirect
6. #29111C2semidirect
7. #28717C2semidirect
8. #27980C2semidirect
9. #27757C2semidirect
10. #27738C2semidirect
11. #27721C2semidirect
12. #27277C2semidirect
13. #23816C2semidirect
14. #23629C2semidirect

Maximal subgroups

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

subgroupindex
#362762
#361942
#302052
#301992
#293472
#291112
#287172
#279802
#277572
#277382
#277212
#272772
#238162
#236292
#225022
#224302
#205682