Finite Group Databasebeta

← #6541182

Subgroups — #6541182

Order 512.

1603 conjugacy classes of subgroups.

The subgroup lattice has 633 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36559C211yesmaximal, characteristic
2#34335C211yesmaximal, characteristic, complemented
3#30094C212yesmaximal, complemented
4#29983C211yesmaximal, characteristic, complemented
5#29575C212yesmaximal, complemented
6#29366C212yesmaximal
7#29219C211yesmaximal, characteristic, complemented
8#28202C211yesmaximal, characteristic
9#27426C214yesmaximal, complemented
10#27002C214yesmaximal, complemented
11#24992C212yesmaximal, complemented
12#24329C212yesmaximal, complemented
13#23815C212yesmaximal
14#23678C211yesmaximal, characteristic
15#22963C212yesmaximal, complemented
16#22828C211yesmaximal, characteristic, complemented
17#22770C212yesmaximal, complemented
18C2×Q82C2211yescharacteristic
19C2×#554—21no—
20C2×#476C2212yes—
21C2×C4×SD16C2212yes—
22#2868C2211yescharacteristic
23#2859C2212yes—
24#2566—22no—
25#2563—21no—
26#2538—21no—
27#2538C2212yes—
28#2452—22no—
29#2452C2214yes—
30#2369C2212yes—
31#2369C2211yescharacteristic
32#2365—21no—
33#2365—22no—
34#2355C2212yes—
35#2355C2211yescharacteristic
36#2351—21no—
37Q8×C4⋊C4C2211yescharacteristic
38D4×C4⋊C4—21no—
39D4×C4⋊C4C2212yes—
40#2331—21no—
41#2331—22no—
42C4×C4⋊Q8C2212yes—
43C4×C4⋊Q8C2211yescharacteristic
44C4×#127C2212yes—
45C4×#127C2211yescharacteristic
46#2295C2211yescharacteristic
47#2295C2212yes—
48#2276—21no—
49C42×Q8C2211yescharacteristic
50#2100C2214yes—
51#2099C2212yes—
52#2099C2212yes—
53#2099C2212yes—
54#2096C2212yes—
55#2084C2212yes—
56#2084C2211yescharacteristic
57#2072C2211yescharacteristic
58#2071C2214yes—
59#2065C2212yes—
60#2058C2214yes—
61#2058C2212yes—
62#2055C2212yes—
63#2041C2211yescharacteristic
64#2041C2212yes—
65#2024C2212yes—
66#2023C2214yes—
67#2014C2211yescharacteristic
68#1990C2212yes—
69#1988C2214yes—
70#1984C2212yes—
71#1948C2212yes—
72#1941C2214yes—
73#1933C2214yes—
74#1926C2214yes—
75#1926C2214yes—
76#1919C2214yes—
77#1919C2214yes—
78#1919C2214yes—
79#1916C2214yes—
80#1880C2214yes—
81#1877C2214yes—
82#1865C2214yes—
83#1864C2214yes—
84#1839C2212yes—
85#1834C2212yes—
86#1831C2211yescharacteristic
87#1803C2212yes—
88#1802C2212yes—
89C8×C4⋊C4C2212yes—
90C4×C4⋊C8C2211yescharacteristic
91C4×#102C2214yes—
92C4×#102C2214yes—
93C4×#102C2212yes—
94#1744C2212yes—
95C22×SD16C2312yes—
96Q82—24no—
97Q82—24no—
98#554—48no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—22no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8C2312yes—
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2312yes—
106C2×#127C2311yescharacteristic
107C2×#125—22no—
108C2×C22⋊Q8—22no—
109C2×C22⋊Q8—24no—
110C2×C4×Q8—21no—
111C2×C4×Q8—21no—
112C2×C4×Q8—22no—
113C2×C4×Q8—21no—
114C2×C4×Q8D411yescharacteristic
115C2×C4×Q8D411yescharacteristic
116C2×C4×Q8C2312yes—
117C2×C4×Q8C2311yescharacteristic
118C2×C4×Q8C2311yescharacteristic
119C2×C4×D4—21no—
120C2×C4×D4—22no—
121C2×C4×D4—21no—
122C2×C4×D4—22no—
123C2×C4×D4C2312yes—
124C2×C4×D4C2314yes—
125C22×C4⋊C4—22no—
126#476—28no—
127C4×SD16—28no—
128C2×#106C2312yes—
129C2×#106C2314yes—
130C2×#106C2314yes—
131C2×C4⋊C8C2312yes—
132C2×C4⋊C8C2312yes—
133C2×#103C2312yes—
134C2×#103C2314yes—
135C2×#102C2312yes—
136C2×#102C2314yes—
137C2×#102C2312yes—
138C2×#102C2314yes—
139C2×#102C2312yes—
140C2×C4×C8C2312yes—
141C2×C4×C8C2314yes—
142#396—24no—
143#396—24no—
144#393—24no—
145#391—22no—
146#391—21no—
147#391—22no—
148#391—22no—
149#391—22no—
150#391C2311yescharacteristic
151#391C2311yescharacteristic
152#391C2312yes—
153#390—22no—
154#390—21no—
155#390—21no—
156#390—22no—
157#390C2312yes—
158#390C2314yes—
159#389—21no—
160#389—22no—
161#389—21no—
162#389—22no—
163#389—21no—
164#389—22no—
165#389—22no—
166#389—22no—
167#389C2311yescharacteristic
168#389C2312yes—
169#389C2311yescharacteristic
170#389C2314yes—
171#389C2314yes—
172#389C2312yes—
173#387—22no—
174#387—21no—
175#387—22no—
176#387—21no—
177#387—22no—
178#387—22no—
179#387—22no—
180#385—24no—
181#385—24no—
182#384—21no—
183#384—21no—
184#384C2311yescharacteristic
185#384C2312yes—
186#384C2311yescharacteristic
187#384C2312yes—
188#384C2312yes—
189#382—21no—
190#382—22no—
191#382—22no—
192#380—22no—
193#380—22no—
194C4×C4⋊C4—21no—
195C4×C4⋊C4—22no—
196C4×C4⋊C4—21no—
197C4×C4⋊C4—22no—
198C4×C4⋊C4—22no—
199C4×C4⋊C4—21no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4D411yescharacteristic
203C4×C4⋊C4D412yes—
204C4×C4⋊C4D411yescharacteristic
205C4×C4⋊C4D412yes—
206C4×C4⋊C4C2312yes—
207C4×C4⋊C4C2312yes—
208C4×C4⋊C4C2312yes—
209C4×C4⋊C4C2312yes—
210C4×C4⋊C4C2311yescharacteristic
211C4×C4⋊C4C2311yescharacteristic
212C4×C4⋊C4C2312yes—
213#376—22no—
214#376—21no—
215C43C2311yescharacteristic
216C43C2312yes—
217#340C2312yes—
218#340C2314yes—
219#340C2314yes—
220#340C2314yes—
221#340C2312yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#336C2312yes—
227#336C2314yes—
228#336C2314yes—
229#336C2311yescharacteristic
230#336C2312yes—
231#336C2311yescharacteristic
232#336C2314yes—
233C22×Q8—22no—
234C22×Q8—21no—
235C22×Q8—22no—
236C22×Q8C2×D411yescharacteristic
237C22×Q8C2411yescharacteristic
238C22×D4—21no—
239C22×D4—21no—
240C22×D4C2412yes—
241C23×C4—22no—
242C23×C4—24no—
243C2×SD16—24no—
244C2×SD16—24no—
245C2×SD16—42no—
246C22×C8C2412yes—
247C22×C8C2414yes—
248C22×C8C2412yes—
249C4⋊Q8—216no—
250C4⋊Q8—28no—
251C4⋊Q8—24no—
252C4⋊Q8—24no—
253C4⋊Q8—22no—
254C4⋊Q8—44no—
255C4⋊Q8—44no—
256C4⋊Q8—82no—
257C4⋊Q8#4114yes—
258#127—24no—
259#125—48no—
260C22⋊Q8—48no—
261C22⋊Q8—416no—
262C4×Q8—28no—
263C4×Q8—24no—
264C4×Q8—24no—
265C4×Q8—22no—
266C4×Q8—22no—
267C4×Q8—22no—
268C4×Q8—42no—
269C4×Q8—44no—
270C4×Q8—84no—
271C4×Q8—82no—
272C4×Q8SD1618yescomplemented
273C4×Q8#4114yes—
274C4×D4—24no—
275C4×D4—24no—
276C4×D4—44no—
277C4×D4—44no—
278C4×D4—44no—
279C4×D4—48no—
280C4×D4—44no—
281C4×D4—84no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—48no—
304C2×C4⋊C4C2×D411yescharacteristic
305C2×C4⋊C4C2×D411yescharacteristic
306C2×C4⋊C4C2×D412yes—
307C2×C4⋊C4C2×D412yes—
308C2×C4⋊C4C2×D411yescharacteristic
309C2×C4⋊C4C2×D412yes—
310C2×C4⋊C4C2414yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2411yescharacteristic
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2411yescharacteristic
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2411yescharacteristic
317C2×C22⋊C4—22no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—24no—
323C2×C42—22no—
324C2×C42—21no—
325C2×C42—22no—
326C2×C42—21no—
327C2×C42—22no—
328C2×C42—22no—
329C2×C42—22no—
330C2×C42C2×D412yes—
331C2×C42C2×D411yescharacteristic
332C2×C42C2×D411yescharacteristic
333C2×C42C2×D412yes—
334C2×C42C2×D412yes—
335C2×C42C2×D411yescharacteristic
336C2×C42C2×D412yes—
337C2×C42C2×D411yescharacteristic
338C2×C42C2412yes—
339C2×C42C2412yes—
340C2×C42C2411yescharacteristic
341C2×C42C2411yescharacteristic
342C2×C42C2411yescharacteristic
343#106—24no—
344#106—28no—
345#106—44no—
346C4⋊C8—24no—
347C4⋊C8—42no—
348#103—24no—
349#103—44no—
350#102—24no—
351#102—28no—
352#102—24no—
353#102—28no—
354#102—42no—
355C4×C8—24no—
356C4×C8—44no—
357#95—21no—
358#95—21no—
359#95—22no—
360#95—21no—
361#95—22no—
362#95—22no—
363#95—21no—
364#95—21no—
365#95—22no—
366#95—21no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95—21no—
372#95—24no—
373#95—24no—
374#95C2×D412yes—
375#95C2×D412yes—
376#95C2×D412yes—
377C24—22no—
378C2×Q8—28no—
379C2×Q8—22no—
380C2×Q8—24no—
381C2×Q8—22no—
382C2×Q8—21no—
383C2×Q8—22no—
384C2×Q8—22no—
385C2×Q8—44no—
386C2×Q8—81no—
387C2×Q8—82no—
388C2×Q8—81no—
389C2×Q8—82no—
390C2×Q8—82no—
391C2×Q8—82no—
392C2×Q8C2×SD1614yes—
393C2×Q8C2×#4112yes—
394C2×D4—24no—
395C2×D4—24no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—42no—
402C2×D4—42no—
403C2×D4—42no—
404C2×D4—42no—
405C2×D42−1+414yes—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—44no—
415C22×C4—44no—
416C22×C4—44no—
417C22×C4—48no—
418C22×C4—84no—
419C22×C4—84no—
420C22×C4C22×D412yes—
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D412yes—
423C22×C4C22×D411yescharacteristic
424C22×C4C22×D411yescharacteristic
425C22×C4C22×D411yescharacteristic
426C22×C4C22×D412yes—
427C22×C4C22×D412yes—
428C22×C4C22×D411yescharacteristic
429C22×C4C22×D411yescharacteristic
430C22×C4C2511yescharacteristic
431SD16—48no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—24no—
435C2×C8—22no—
436C2×C8—24no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—24no—
440C2×C8—22no—
441C4⋊C4—28no—
442C4⋊C4—28no—
443C4⋊C4—24no—
444C4⋊C4—24no—
445C4⋊C4—24no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—21no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—24no—
454C4⋊C4—22no—
455C4⋊C4—24no—
456C4⋊C4—24no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C4⋊C4—44no—
460C4⋊C4—48no—
461C4⋊C4—44no—
462C4⋊C4—42no—
463C4⋊C4—44no—
464C4⋊C4—48no—
465C4⋊C4—42no—
466C4⋊C4—44no—
467C4⋊C4—42no—
468C4⋊C4—44no—
469C4⋊C4—82no—
470C4⋊C4—81no—
471C4⋊C4—81no—
472C4⋊C4—82no—
473C4⋊C4—82no—
474C4⋊C4—82no—
475C4⋊C4C2×SD1614yes—
476C4⋊C4C2×SD1618yes—
477C4⋊C4C2×#4114yes—
478C4⋊C4C2×#4112yes—
479C22⋊C4—44no—
480C22⋊C4—44no—
481C22⋊C4—44no—
482C22⋊C4—48no—
483C22⋊C4—84no—
484C22⋊C4—84no—
485C42—28no—
486C42—24no—
487C42—22no—
488C42—24no—
489C42—22no—
490C42—22no—
491C42—22no—
492C42—22no—
493C42—24no—
494C42—22no—
495C42—41no—
496C42—42no—
497C42—42no—
498C42—44no—
499C42—42no—
500C42—82no—
501C42—82no—
502C42—82no—
503C42C2×SD1618yes—
504C42C2×SD1614yes—
505C42C2×#4112yes—
506C42C2×#4112yes—
507C23—24no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23C23×D411yescharacteristic
515Q8—28no—
516Q8—24no—
517Q8—42no—
518Q8—84no—
519Q8—88no—
520Q8—164no—
521D4—24no—
522D4—24no—
523D4—48no—
524D4—48no—
525D4—42no—
526D4—42no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—22no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—21no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—42no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—42no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—41no—
561C2×C4—44no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—41no—
566C2×C4—42no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—88no—
573C2×C4—168no—
574C2×C4#52512yes—
575C2×C4#52512yes—
576C2×C4C22×SD1612yes—
577C2×C4C22×SD1614yes—
578C2×C4C22×SD1612yes—
579C2×C4C22×SD1614yes—
580C2×C4C22×SD1612yes—
581C2×C4#57512yes—
582C2×C4#57612yes—
583C2×C4C22×#4111yescharacteristic
584C2×C4C22×#4111yescharacteristic
585C2×C4C2×2−1+412yes—
586C2×C4#58512yes—
587C8—42no—
588C8—42no—
589C8—44no—
590C22—44no—
591C22—44no—
592C22—44no—
593C22—82no—
594C22—82no—
595C22—82no—
596C22—82no—
597C22C2×#52511yescharacteristic
598C22#348611yescharacteristic
599C22#350011yescharacteristic
600C22C23×SD1611yescharacteristic
601C22C2×#57511yescharacteristic
602C22C2×#57611yescharacteristic
603C22#358611yescharacteristic
604C4—24no—
605C4—22no—
606C4—24no—
607C4—22no—
608C4—22no—
609C4—22no—
610C4—42no—
611C4—41no—
612C4—42no—
613C4—42no—
614C4—44no—
615C4—42no—
616C4—84no—
617C4—82no—
618C4—81no—
619C4—82no—
620C4—81no—
621C4#310314yes—
622C4#349512yes—
623C4#351612yes—
624C2—84no—
625C2#3573811yescharacteristic
626C2C2×#310311yescharacteristic
627C2#6059611yescharacteristic
628C2#6181811yescharacteristic
629C2#6183511yescharacteristic
630C2#6252511yescharacteristic
631C2#6258111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#127C23
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
2. #34335C2semidirect
3. #30094C2semidirect
4. #29983C2semidirect
5. #29575C2semidirect
7. #29219C2semidirect
9. #27426C2semidirect
10. #27002C2semidirect
11. #24992C2semidirect
12. #24329C2semidirect
15. #22963C2semidirect
16. #22828C2semidirect
17. #22770C2semidirect
272. C4×Q8SD16semidirect

Maximal subgroups

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

subgroupindex
#365592
#343352
#300942
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