Finite Group Databasebeta

← #6532190

Subgroups — #6532190

Order 512.

1564 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2567C211yesmaximal, characteristic
2#34196C211yesmaximal, characteristic
3#27616C212yesmaximal
4#27616C212yesmaximal
5#27613C212yesmaximal
6#27610C212yesmaximal
7#26295C214yesmaximal, complemented
8#26293C214yesmaximal, complemented
9#26282C212yesmaximal, complemented
10#26280C212yesmaximal, complemented
11#26279C212yesmaximal, complemented
12#26277C212yesmaximal, complemented
13#21840C211yesmaximal, characteristic
14#21833C211yesmaximal, characteristic
15#21820C211yesmaximal, characteristic
16#21365C211yesmaximal, characteristic
17#21361C211yesmaximal, characteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C2×#432C2211yescharacteristic
21C2×#431C2211yescharacteristic
22#2567—22no—
23#2567—21no—
24#2567—21no—
25#2567C2212yes—
26#2567C2212yes—
27#2567C2214yes—
28C2×#395—22no—
29#2355C2211yescharacteristic
30#2355C2211yescharacteristic
31#2355C2212yes—
32#2355C2212yes—
33Q8×C4⋊C4C2211yescharacteristic
34Q8×C4⋊C4C2211yescharacteristic
35#2344—22no—
36#2344—21no—
37Q8×C22⋊C4—21no—
38#2320—22no—
39#2320—21no—
40C2×#391—21no—
41C2×#389C2212yes—
42C2×#389C2212yes—
43#2285C2211yescharacteristic
44#2285C2212yes—
45C2×#384—21no—
46#2283C2211yescharacteristic
47#2283C2212yes—
48#2276—21no—
49C2×C4×C4⋊C4C2211yescharacteristic
50#2101C2214yes—
51#2101C2214yes—
52#2100C2214yes—
53#2099C2214yes—
54#2085C2214yes—
55#2085C2214yes—
56#2084C2214yes—
57#2083C2214yes—
58#1993C2212yes—
59#1993C2212yes—
60#1990C2212yes—
61#1990C2212yes—
62#1988C2214yes—
63#1987C2212yes—
64#1987C2212yes—
65#1985C2214yes—
66#1984C2212yes—
67#1984C2212yes—
68#1949C2212yes—
69#1949C2212yes—
70#1948C2212yes—
71#1947C2212yes—
72#1921C2214yes—
73#1921C2214yes—
74#1920C2214yes—
75#1920C2214yes—
76#1919C2214yes—
77#1919C2214yes—
78#1918C2214yes—
79#1918C2214yes—
80#1853C2211yescharacteristic
81#1853C2211yescharacteristic
82#1853C2212yes—
83#1852C2211yescharacteristic
84#1852C2211yescharacteristic
85#1852C2212yes—
86#1823C2212yes—
87#1823C2212yes—
88#1823C2212yes—
89#1823C2212yes—
90#1817C2212yes—
91#1817C2212yes—
92#1776C2214yes—
93C4×#107C2212yes—
94C4×#106C2212yes—
95C23×Q8—21no—
96C23×Q8D411yescharacteristic
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—24no—
100C2×C4⋊Q8—24no—
101C2×C4⋊Q8C2312yes—
102C2×C4⋊Q8C2312yes—
103C2×C4×Q8—21no—
104C2×C4×Q8—21no—
105C2×C4×Q8—24no—
106C2×C4×Q8—24no—
107C2×C4×Q8D412yes—
108C2×C4×Q8D411yescharacteristic
109C2×C4×Q8D411yescharacteristic
110C2×C4×Q8C2311yescharacteristic
111C2×C4×Q8C2311yescharacteristic
112C2×C4×Q8C2312yes—
113C2×#117—22no—
114C2×#117—21no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4—22no—
117C22×C4⋊C4—22no—
118C22×C4⋊C4D412yes—
119C22×C4⋊C4D411yescharacteristic
120C22×C4⋊C4C2312yes—
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C42—21no—
124C22×C42C2312yes—
125C22×C42C2311yescharacteristic
126#432—24no—
127#431—24no—
128C2×#107C2312yes—
129C2×#107C2314yes—
130C2×#107C2312yes—
131C2×#106C2312yes—
132C2×#106C2314yes—
133C2×#106C2312yes—
134C2×C4⋊C8C2312yes—
135C2×C4⋊C8C2312yes—
136C2×C22⋊C8C2311yescharacteristic
137C2×C22⋊C8C2312yes—
138C2×C22⋊C8C2311yescharacteristic
139C2×#97C2312yes—
140C2×C4×C8C2312yes—
141#395—24no—
142#395—24no—
143#395—24no—
144#395—24no—
145#391—22no—
146#391—24no—
147#391—22no—
148#391—21no—
149#391—21no—
150#391—22no—
151#391—22no—
152#389—21no—
153#389—21no—
154#389—22no—
155#389—22no—
156#389—22no—
157#389—22no—
158#389—22no—
159#389—22no—
160#389C2314yes—
161#389C2314yes—
162#389C2314yes—
163#389C2314yes—
164#387—22no—
165#387—22no—
166#387—22no—
167#387—22no—
168#387—22no—
169#387—22no—
170#384—22no—
171#384—24no—
172#384—22no—
173#384C2312yes—
174#384C2312yes—
175#384C2311yescharacteristic
176#384C2311yescharacteristic
177#383—22no—
178#383—21no—
179#382—22no—
180#382—21no—
181#382C2312yes—
182#382C2312yes—
183#382C2311yescharacteristic
184#382C2311yescharacteristic
185#381—22no—
186#381—21no—
187#380—22no—
188#380—21no—
189#380C2311yescharacteristic
190#380C2312yes—
191#380C2311yescharacteristic
192C4×C4⋊C4—21no—
193C4×C4⋊C4—21no—
194C4×C4⋊C4—22no—
195C4×C4⋊C4—22no—
196C4×C4⋊C4—21no—
197C4×C4⋊C4—21no—
198C4×C4⋊C4—22no—
199C4×C4⋊C4—21no—
200C4×C4⋊C4C2312yes—
201C4×C4⋊C4C2312yes—
202C4×C4⋊C4C2312yes—
203C4×C4⋊C4C2312yes—
204C4×C4⋊C4C2311yescharacteristic
205C4×C4⋊C4C2311yescharacteristic
206C4×C4⋊C4C2312yes—
207C4×C4⋊C4C2312yes—
208C4×C4⋊C4C2312yes—
209C4×C22⋊C4—22no—
210C4×C22⋊C4—21no—
211#376—22no—
212#376—21no—
213C2×#95—21no—
214C2×#95—22no—
215C2×#95—22no—
216#340C2314yes—
217#340C2314yes—
218#340C2314yes—
219#340C2314yes—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2312yes—
225#340C2314yes—
226#340C2312yes—
227#340C2314yes—
228#340C2312yes—
229#340C2312yes—
230#336C2314yes—
231#336C2314yes—
232#336C2314yes—
233#336C2314yes—
234C22×Q8—24no—
235C22×Q8—24no—
236C22×Q8—22no—
237C22×Q8—22no—
238C22×Q8—42no—
239C22×Q8—44no—
240C22×Q8—82no—
241C22×Q8—82no—
242C22×Q8C2×D411yescharacteristic
243C22×Q8C2×D412yes—
244C22×Q8C2×D411yescharacteristic
245C23×C4—21no—
246C23×C4—21no—
247C23×C4—22no—
248C23×C4C2×D412yes—
249C23×C4C2×D412yes—
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D411yescharacteristic
252C23×C4C2411yescharacteristic
253C22×C8C2412yes—
254C22×C8C2412yes—
255C22×C8C2414yes—
256C4⋊Q8—24no—
257C4⋊Q8—24no—
258C4⋊Q8—24no—
259C4⋊Q8—24no—
260C4⋊Q8—44no—
261C4⋊Q8—44no—
262C4×Q8—24no—
263C4×Q8—28no—
264C4×Q8—24no—
265C4×Q8—22no—
266C4×Q8—22no—
267C4×Q8—24no—
268C4×Q8—22no—
269C4×Q8—22no—
270C4×Q8—24no—
271C4×Q8—44no—
272C4×Q8—44no—
273#117—82no—
274#117—84no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—28no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—44no—
299C2×C4⋊C4—84no—
300C2×C4⋊C4—84no—
301C2×C4⋊C4C2×D411yescharacteristic
302C2×C4⋊C4C2×D412yes—
303C2×C4⋊C4C2×D412yes—
304C2×C4⋊C4C2×D411yescharacteristic
305C2×C4⋊C4C2×D412yes—
306C2×C4⋊C4C2×D412yes—
307C2×C4⋊C4C2×D412yes—
308C2×C4⋊C4C2414yes—
309C2×C4⋊C4C2411yescharacteristic
310C2×C4⋊C4C2411yescharacteristic
311C2×C4⋊C4C2411yescharacteristic
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2411yescharacteristic
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C22⋊C4—21no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—21no—
320C2×C42—24no—
321C2×C42—24no—
322C2×C42—22no—
323C2×C42—22no—
324C2×C42—21no—
325C2×C42—21no—
326C2×C42—22no—
327C2×C42—22no—
328C2×C42—22no—
329C2×C42—22no—
330C2×C42—22no—
331C2×C42—44no—
332C2×C42C2×D412yes—
333C2×C42C2×D412yes—
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D411yescharacteristic
336C2×C42#4114yes—
337C2×C42#4114yes—
338C2×C42C2412yes—
339C2×C42C2412yes—
340C2×C42C2411yescharacteristic
341C2×C42C2411yescharacteristic
342#107—24no—
343#107—24no—
344#107—28no—
345#106—24no—
346#106—28no—
347#106—24no—
348C4⋊C8—24no—
349C4⋊C8—24no—
350C22⋊C8—22no—
351C22⋊C8—22no—
352C22⋊C8—24no—
353#97—24no—
354C4×C8—24no—
355#95—22no—
356#95—22no—
357#95—24no—
358#95—24no—
359#95—24no—
360#95—24no—
361#95—21no—
362#95—22no—
363#95—21no—
364#95—22no—
365#95—22no—
366#95—21no—
367#95—21no—
368#95—22no—
369#95—22no—
370#95—21no—
371#95—21no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—21no—
378#95—21no—
379C24C22×D411yescharacteristic
380C2×Q8—22no—
381C2×Q8—24no—
382C2×Q8—22no—
383C2×Q8—24no—
384C2×Q8—22no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—42no—
388C2×Q8—416no—
389C2×Q8—44no—
390C2×Q8—44no—
391C2×Q8—82no—
392C2×Q8—82no—
393C2×Q8—82no—
394C2×Q8—82no—
395C2×Q8—816no—
396C2×Q8#13714yes—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—24no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—44no—
425C22×C4—44no—
426C22×C4—44no—
427C22×C4#12314yes—
428C22×C4#12314yes—
429C22×C4C22×D412yes—
430C22×C4C22×D412yes—
431C22×C4C22×D412yes—
432C22×C4C22×D411yescharacteristic
433C22×C4C22×D411yescharacteristic
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D412yes—
437C22×C4C22×D411yescharacteristic
438C22×C4C2×#4112yes—
439C22×C4C2×#4112yes—
440C22×C4C2511yescharacteristic
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—24no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—24no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—24no—
450C4⋊C4—24no—
451C4⋊C4—24no—
452C4⋊C4—24no—
453C4⋊C4—24no—
454C4⋊C4—24no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—24no—
458C4⋊C4—24no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—24no—
462C4⋊C4—24no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—24no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—24no—
469C4⋊C4—44no—
470C4⋊C4—44no—
471C4⋊C4—44no—
472C4⋊C4—44no—
473C4⋊C4—44no—
474C4⋊C4—42no—
475C4⋊C4—42no—
476C4⋊C4—84no—
477C4⋊C4—84no—
478C4⋊C4—84no—
479C4⋊C4—84no—
480C4⋊C4—81no—
481C4⋊C4—81no—
482C4⋊C4—82no—
483C4⋊C4—82no—
484C4⋊C42−1+414yes—
485C4⋊C42−1+414yes—
486C22⋊C4—44no—
487C22⋊C4—42no—
488C22⋊C4—42no—
489C42—24no—
490C42—24no—
491C42—22no—
492C42—22no—
493C42—22no—
494C42—22no—
495C42—44no—
496C42—44no—
497C42—44no—
498C42—42no—
499C42—44no—
500C42—42no—
501C42—82no—
502C42—82no—
503C42—82no—
504C42#12314yes—
505C42#12314yes—
506C42C2×#4112yes—
507C42C2×#4112yes—
508C42C2×#4112yes—
509C42C2×#4112yes—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23C2×#12312yes—
515C23#57612yes—
516C23#57712yes—
517C23C23×D411yescharacteristic
518Q8—28no—
519Q8—48no—
520Q8—44no—
521Q8—88no—
522Q8—88no—
523C2×C4—28no—
524C2×C4—22no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—22no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—28no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—48no—
549C2×C4—48no—
550C2×C4—48no—
551C2×C4—48no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—42no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—44no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—44no—
577C2×C4—88no—
578C2×C4C2×#12312yes—
579C2×C4C2×#12312yes—
580C2×C4C2×#12312yes—
581C2×C4C2×#12312yes—
582C2×C4C2×#12312yes—
583C2×C4C2×#13714yes—
584C2×C4C2×#13712yes—
585C2×C4#57512yes—
586C2×C4C22×#4111yescharacteristic
587C2×C4C22×#4111yescharacteristic
588C2×C4C2×2−1+412yes—
589C2×C4C2×2−1+412yes—
590C8—42no—
591C8—42no—
592C8—44no—
593C22—22no—
594C22—22no—
595C22—22no—
596C22—22no—
597C22—22no—
598C22—22no—
599C22#309614yes—
600C22C22×#12311yescharacteristic
601C22#348611yescharacteristic
602C22#348611yescharacteristic
603C22C22×#13711yescharacteristic
604C22C2×#57511yescharacteristic
605C22C2×#57611yescharacteristic
606C22C2×#57711yescharacteristic
607C4—22no—
608C4—22no—
609C4—24no—
610C4—22no—
611C4—22no—
612C4—42no—
613C4—42no—
614C4—42no—
615C4—44no—
616C4—44no—
617C4—84no—
618C4—82no—
619C4—81no—
620C4—81no—
621C4—81no—
622C4—82no—
623C4—81no—
624C4#309514yes—
625C4#350912yes—
626C4#350912yes—
627C2—24no—
628C2#3567411yescharacteristic
629C2C2×#309511yescharacteristic
630C2C2×#309611yescharacteristic
631C2#6183511yescharacteristic
632C2#6183811yescharacteristic
633C2#6184111yescharacteristic
634C2#6184411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
7. #26295C2semidirect
8. #26293C2semidirect
9. #26282C2semidirect
10. #26280C2semidirect
11. #26279C2semidirect
12. #26277C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25672
#341962
#276162
#276132
#276102
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