Finite Group Databasebeta

← #6854499

Subgroups — #6854499

Order 512.

1464 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36563C211yesmaximal, characteristic, complemented
2#36562C211yesmaximal, characteristic
3#33293C211yesmaximal, characteristic, complemented
4#31548C214yesmaximal, complemented
5#31544C214yesmaximal
6#30312C212yesmaximal, complemented
7#29197C212yesmaximal
8#29150C212yesmaximal, complemented
9#29148C211yesmaximal, characteristic
10#28522C211yesmaximal, characteristic, complemented
11#24817C212yesmaximal, complemented
12#24813C212yesmaximal
13#24332C212yesmaximal, complemented
14#24331C212yesmaximal
15#22273C212yesmaximal
16C4×#497C211yesmaximal, characteristic
17#21345C211yesmaximal, characteristic, complemented
18C2×#557C2211yescharacteristic
19C2×#554C2211yescharacteristic
20C2×#497C2211yescharacteristic
21C2×#477C2212yes—
22C2×#475C2212yes—
23C2×C4×Q16C2212yes—
24C2×C4×SD16C2212yes—
25#2868C2211yescharacteristic
26#2862C2211yescharacteristic
27#2859C2212yes—
28#2859C2212yes—
29#2582C2212yes—
30#2567—21no—
31#2567C2211yescharacteristic
32#2566C2212yes—
33#2565C2212yes—
34#2556—22no—
35#2556—22no—
36#2552—21no—
37#2550—22no—
38#2542C2211yescharacteristic
39#2507—22no—
40#2451C2214yes—
41#2429—21no—
42#2429—21no—
43#2369—21no—
44#2360—21no—
45#2360—21no—
46#2355—21no—
47Q8×C4⋊C4C2212yes—
48C4×C4⋊Q8C2212yes—
49C4×C4⋊Q8C2211yescharacteristic
50C4×#124C2212yes—
51C4×#124C2211yescharacteristic
52C42×Q8C2211yescharacteristic
53#2097C2214yes—
54#2084C2214yes—
55#2083C2214yes—
56#2068C2214yes—
57#2065C2214yes—
58#2059C2214yes—
59#2059C2214yes—
60#2026C2214yes—
61#2026C2214yes—
62#2024C2212yes—
63#2023C2212yes—
64#1993C2214yes—
65#1990C2212yes—
66#1990C2212yes—
67#1984C2212yes—
68#1973C2212yes—
69#1971C2212yes—
70#1960C2212yes—
71#1960C2211yescharacteristic
72#1959C2212yes—
73#1959C2211yescharacteristic
74#1939C2212yes—
75#1917C2214yes—
76#1880C2214yes—
77#1878C2214yes—
78#1878C2214yes—
79#1877C2214yes—
80#1865C2214yes—
81#1862C2214yes—
82#1847C2212yes—
83#1847C2211yescharacteristic
84#1836C2212yes—
85#1836C2212yes—
86#1836C2212yes—
87#1803C2212yes—
88#1803C2214yes—
89#1803C2214yes—
90#1802C2212yes—
91#1767C2212yes—
92#1767C2211yescharacteristic
93#1745C2212yes—
94C4×#97C2211yescharacteristic
95C22×Q16C2312yes—
96C22×SD16C2312yes—
97#557—44no—
98#554—44no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2312yes—
106C2×#125—22no—
107C2×#125—22no—
108C2×#125C2312yes—
109C2×#125C2312yes—
110C2×#124C2311yescharacteristic
111C2×C22⋊Q8—21no—
112C2×C22⋊Q8—21no—
113C2×C22⋊Q8—21no—
114C2×C22⋊Q8—22no—
115C2×C4×Q8—21no—
116C2×C4×Q8D411yescharacteristic
117C2×C4×Q8D411yescharacteristic
118C2×C4×Q8C2312yes—
119C2×C4×Q8C2311yescharacteristic
120C2×C4×Q8C2314yes—
121C2×C4×Q8C2312yes—
122C2×C4×Q8C2311yescharacteristic
123C2×C4×D4C2311yescharacteristic
124C2×C4×D4C2312yes—
125C22×C4⋊C4—21no—
126#497Q818yescomplemented
127#477—44no—
128#475—44no—
129C4×Q16—44no—
130C4×SD16—44no—
131C2×#107C2314yes—
132C2×#107C2312yes—
133C2×#106C2314yes—
134C2×#106C2312yes—
135C2×C4⋊C8C2312yes—
136C2×C4⋊C8C2314yes—
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×#103C2312yes—
140C2×#103C2312yes—
141C2×#103C2314yes—
142C2×#103C2312yes—
143C2×#102C2314yes—
144C2×#102C2312yes—
145C2×#97C2311yescharacteristic
146C2×#97C2312yes—
147C2×#97C2312yes—
148C2×#97C2311yescharacteristic
149C2×C4×C8C2312yes—
150#399—22no—
151#398—22no—
152#398—22no—
153#398—24no—
154#398—24no—
155#396—22no—
156#396—22no—
157#395—24no—
158#395—22no—
159#391—22no—
160#391—21no—
161#391—21no—
162#391—21no—
163#391—21no—
164#391—24no—
165#391C2314yes—
166#391C2312yes—
167#391C2311yescharacteristic
168#391C2312yes—
169#390C2311yescharacteristic
170#390C2312yes—
171#389—21no—
172#389—21no—
173#389—22no—
174#389—22no—
175#389—21no—
176#389—21no—
177#389—22no—
178#389—22no—
179#389—22no—
180#389—22no—
181#389—22no—
182#389—22no—
183#389—22no—
184#389—24no—
185#389C2314yes—
186#389C2314yes—
187#388—22no—
188#388—22no—
189#388—21no—
190#388—21no—
191#387—24no—
192#387—22no—
193#387—22no—
194#385—22no—
195#385—22no—
196#384—22no—
197#384—21no—
198#384—21no—
199#384C2311yescharacteristic
200#384C2312yes—
201#384C2312yes—
202#384C2312yes—
203#384C2312yes—
204#382—22no—
205#382C2312yes—
206#382C2311yescharacteristic
207#382C2312yes—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4D411yescharacteristic
214C4×C4⋊C4D411yescharacteristic
215C4×C4⋊C4D412yes—
216C4×C4⋊C4D412yes—
217C4×C4⋊C4C2312yes—
218C4×C4⋊C4C2311yescharacteristic
219C4×C22⋊C4—21no—
220C4×C22⋊C4—21no—
221C4×C22⋊C4—22no—
222C4×C22⋊C4—22no—
223C43C2312yes—
224C43C2311yescharacteristic
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#336C2314yes—
232#336C2314yes—
233#336C2312yes—
234#336C2314yes—
235C22×Q8—21no—
236C22×Q8—21no—
237C22×Q8—22no—
238C22×Q8C2×D411yescharacteristic
239C22×Q8C2411yescharacteristic
240C22×Q8C2412yes—
241C22×D4C2411yescharacteristic
242C23×C4—21no—
243C23×C4—22no—
244C2×Q16—42no—
245C2×Q16—42no—
246C2×Q16C2×Q818yes—
247C2×SD16—42no—
248C2×SD16—42no—
249C2×SD16C2×Q818yes—
250C22×C8C2414yes—
251C22×C8C2412yes—
252C22×C8C2412yes—
253C4⋊Q8—24no—
254C4⋊Q8—24no—
255C4⋊Q8—22no—
256C4⋊Q8—44no—
257C4⋊Q8—44no—
258C4⋊Q8—44no—
259C4⋊Q8—82no—
260C4⋊Q8C2×Q814yes—
261#125—44no—
262#125—44no—
263#125—48no—
264#125—84no—
265#124—22no—
266#124C2×Q814yes—
267C22⋊Q8—44no—
268C22⋊Q8—48no—
269C22⋊Q8—44no—
270C22⋊Q8—44no—
271C4×Q8—24no—
272C4×Q8—24no—
273C4×Q8—24no—
274C4×Q8—24no—
275C4×Q8—24no—
276C4×Q8—42no—
277C4×Q8—44no—
278C4×Q8—44no—
279C4×Q8—44no—
280C4×Q8—44no—
281C4×D4—24no—
282C4×D4—44no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—44no—
309C2×C4⋊C4C2×D411yescharacteristic
310C2×C4⋊C4C2×D411yescharacteristic
311C2×C4⋊C4C2×D412yes—
312C2×C4⋊C4C2×D412yes—
313C2×C4⋊C4C2×D412yes—
314C2×C4⋊C4C2×D411yescharacteristic
315C2×C4⋊C4C2411yescharacteristic
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2411yescharacteristic
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2414yes—
320C2×C4⋊C4C2412yes—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—21no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—21no—
328C2×C22⋊C4—21no—
329C2×C42—22no—
330C2×C42—21no—
331C2×C42—22no—
332C2×C42—24no—
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D412yes—
336C2×C42C2×D412yes—
337C2×C42C2×D412yes—
338C2×C42C2×D411yescharacteristic
339C2×C42C2×D412yes—
340C2×C42C2×D411yescharacteristic
341C2×C42C2412yes—
342C2×C42C2412yes—
343C2×C42C2411yescharacteristic
344C2×C42C2411yescharacteristic
345C2×C42C2411yescharacteristic
346#107—42no—
347#107—44no—
348#106—42no—
349#106—44no—
350C4⋊C8—24no—
351C4⋊C8—44no—
352#103—44no—
353#103—42no—
354#103—42no—
355#103—42no—
356#103—44no—
357#103—44no—
358#102—44no—
359#102—42no—
360#97—22no—
361#97—24no—
362#97—42no—
363#97C2×Q814yes—
364C4×C8—24no—
365#95—21no—
366#95—21no—
367#95—22no—
368#95—22no—
369#95—21no—
370#95—21no—
371#95—22no—
372#95—22no—
373#95—24no—
374#95—22no—
375#95—24no—
376#95—24no—
377#95—22no—
378#95—24no—
379#95—22no—
380#95—22no—
381#95C2×D412yes—
382#95C2×D412yes—
383#95C2×D412yes—
384C24—21no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—21no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—44no—
397C2×Q8—42no—
398C2×Q8—42no—
399C2×Q8—42no—
400C2×Q8—42no—
401C2×Q8—82no—
402C2×Q8—82no—
403C2×Q8—81no—
404C2×Q8—81no—
405C2×Q8C22×Q812yes—
406C2×Q8C22×Q814yes—
407C2×D4—22no—
408C2×D4—21no—
409C2×D4—22no—
410C2×D4C22×Q812yes—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—44no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—82no—
424C22×C4—82no—
425C22×C4C22×D411yescharacteristic
426C22×C4C22×D411yescharacteristic
427C22×C4C22×D412yes—
428C22×C4C22×D412yes—
429C22×C4C22×D412yes—
430C22×C4C22×D411yescharacteristic
431C22×C4C22×D411yescharacteristic
432C22×C4C22×D411yescharacteristic
433C22×C4C22×D412yes—
434C22×C4C22×D411yescharacteristic
435C22×C4C2511yescharacteristic
436Q16—48no—
437SD16—48no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—24no—
442C2×C8—24no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—24no—
446C2×C8C22×Q814yes—
447C4⋊C4—22no—
448C4⋊C4—24no—
449C4⋊C4—24no—
450C4⋊C4—24no—
451C4⋊C4—24no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—24no—
456C4⋊C4—22no—
457C4⋊C4—24no—
458C4⋊C4—22no—
459C4⋊C4—24no—
460C4⋊C4—24no—
461C4⋊C4—42no—
462C4⋊C4—42no—
463C4⋊C4—44no—
464C4⋊C4—44no—
465C4⋊C4—44no—
466C4⋊C4—44no—
467C4⋊C4—44no—
468C4⋊C4—44no—
469C4⋊C4—42no—
470C4⋊C4—44no—
471C4⋊C4—44no—
472C4⋊C4—42no—
473C4⋊C4—44no—
474C4⋊C4—42no—
475C4⋊C4—42no—
476C4⋊C4—44no—
477C4⋊C4—44no—
478C4⋊C4—82no—
479C4⋊C4—82no—
480C4⋊C4—84no—
481C4⋊C4—84no—
482C4⋊C4—84no—
483C4⋊C4—82no—
484C4⋊C4—82no—
485C4⋊C4#13714yes—
486C22⋊C4—24no—
487C22⋊C4—24no—
488C22⋊C4—44no—
489C22⋊C4—44no—
490C22⋊C4—42no—
491C22⋊C4—44no—
492C22⋊C4—81no—
493C22⋊C4—81no—
494C42—22no—
495C42—24no—
496C42—22no—
497C42—24no—
498C42—22no—
499C42—24no—
500C42—42no—
501C42—41no—
502C42—42no—
503C42—42no—
504C42—41no—
505C42—44no—
506C42—44no—
507C42—42no—
508C42—84no—
509C42C22×Q812yes—
510C42H(5,𝔽2)14yes—
511C422−1+412yes—
512C23—22no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23C23×D411yescharacteristic
520Q8—24no—
521Q8—24no—
522Q8—24no—
523Q8—44no—
524Q8—42no—
525Q8—42no—
526Q8—42no—
527Q8—84no—
528Q8—162no—
529Q8—164no—
530D4—24no—
531D4—42no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—22no—
536C2×C4—21no—
537C2×C4—22no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—21no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—22no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—41no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—42no—
565C2×C4—41no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—44no—
579C2×C4—84no—
580C2×C4—164no—
581C2×C4#53512yes—
582C2×C4#53512yes—
583C2×C4D4×Q812yes—
584C2×C4D4×Q812yes—
585C2×C4C2×#13714yes—
586C2×C4C2×#13712yes—
587C2×C4#57512yes—
588C2×C4C23×Q811yescharacteristic
589C2×C4C2×H(5,𝔽2)12yes—
590C2×C4C2×2−1+411yescharacteristic
591C2×C4#58512yes—
592C8—24no—
593C8—42no—
594C8—44no—
595C22—42no—
596C22—42no—
597C22—42no—
598C22—81no—
599C22—81no—
600C22—81no—
601C22—81no—
602C22C2×#53511yescharacteristic
603C22C2×D4×Q811yescharacteristic
604C22#350011yescharacteristic
605C22C22×#13711yescharacteristic
606C22C2×#57511yescharacteristic
607C22#358511yescharacteristic
608C22#358611yescharacteristic
609C4—24no—
610C4—22no—
611C4—22no—
612C4—22no—
613C4—41no—
614C4—42no—
615C4—42no—
616C4—41no—
617C4—42no—
618C4—42no—
619C4—44no—
620C4—81no—
621C4—82no—
622C4—84no—
623C4—81no—
624C4—82no—
625C4—82no—
626C4#321614yes—
627C4#350112yes—
628C4#354712yes—
629C2—82no—
630C2#3961811yescharacteristic
631C2C2×#321611yescharacteristic
632C2#6082911yescharacteristic
633C2Q8×#13711yescharacteristic
634C2#6136911yescharacteristic
635C2#6257711yescharacteristic
636C2#6259011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36563C2semidirect
3. #33293C2semidirect
4. #31548C2semidirect
6. #30312C2semidirect
8. #29150C2semidirect
10. #28522C2semidirect
11. #24817C2semidirect
13. #24332C2semidirect
17. #21345C2semidirect
126. #497Q8semidirect

Maximal subgroups

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

subgroupindex
#365632
#365622
#332932
#315482
#315442
#303122
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