Finite Group Databasebeta

← #6458337

Subgroups — #6458337

Order 512.

1804 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35172C211yesmaximal, characteristic
2#35167C211yesmaximal, characteristic, complemented
3#27822C212yesmaximal, complemented
4#27821C212yesmaximal, complemented
5#27819C214yesmaximal, complemented
6#27817C214yesmaximal, complemented
7#27814C212yesmaximal, complemented
8#27813C212yesmaximal, complemented
9#23749C212yesmaximal, complemented
10#23456C212yesmaximal
11#21326C212yesmaximal, complemented
12#21318C211yesmaximal, characteristic, complemented
13#21312C211yesmaximal, characteristic, complemented
14#21089C212yesmaximal
15#20646C211yesmaximal, characteristic
16#20437C211yesmaximal, characteristic, complemented
17C2×C4×#102C211yesmaximal, characteristic
18C2×#544C2212yes—
19C2×#533C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21C2×#423C2211yescharacteristic
22C2×#418C2212yes—
23C22×#102C2211yescharacteristic
24C22×#102C2211yescharacteristic
25C2×#405C2212yes—
26C22×C4×C8C2211yescharacteristic
27#2857—22no—
28#2848—22no—
29#2659—22no—
30#2626—22no—
31#2580—22no—
32#2566C2214yes—
33#2563C2212yes—
34#2552—21no—
35#2552C2212yescomplemented
36#2551—21no—
37#2551C2212yescomplemented
38#2550—22no—
39#2550C2214yescomplemented
40#2542C2212yes—
41#2538—21no—
42#2538—21no—
43#2317C2212yes—
44#2316C2212yes—
45#2309C2212yes—
46#2285C2211yescharacteristic
47#2285C2211yescharacteristic
48C2×C4×C4⋊C4C2211yescharacteristic
49#2101C2214yescomplemented
50#2101C2214yescomplemented
51#2100C2214yescomplemented
52#2099C2214yescomplemented
53#2025C2214yes—
54#2025C2214yes—
55#2023C2214yes—
56#2023C2214yes—
57#1993C2214yescomplemented
58#1992C2214yescomplemented
59#1991C2212yescomplemented
60#1991C2212yescomplemented
61#1991C2214yescomplemented
62#1990C2211yescharacteristic, complemented
63#1990C2211yescharacteristic, complemented
64#1990C2211yescharacteristic, complemented
65#1990C2211yescharacteristic, complemented
66#1990C2214yescomplemented
67#1989C2214yescomplemented
68#1988C2212yescomplemented
69#1987C2212yescomplemented
70#1936C2214yes—
71#1936C2214yes—
72#1934C2214yes—
73#1934C2214yes—
74#1804C2214yes—
75#1802C2212yes—
76#1802C2212yes—
77#1770C2212yescomplemented
78#1770C2212yescomplemented
79C8×C4⋊C4C2212yescomplemented
80C8×C4⋊C4C2212yescomplemented
81C4×#102C2212yes—
82C4×#102C2212yes—
83C4×#102C2212yes—
84C4×#102C2212yes—
85#1736C2212yes—
86#1736C2212yes—
87#1736C2212yes—
88#1735C2212yes—
89#1735C2212yes—
90C2×#340C2211yescharacteristic
91C2×#340C2211yescharacteristic
92#1729C2212yes—
93C23×D4C2311yescharacteristic
94C24×C4—21no—
95C22×M16C2312yes—
96C23×C8C2311yescharacteristic
97C23×C8C2311yescharacteristic
98#544—28no—
99#544—44no—
100#533—24no—
101#533—24no—
102C2×C4⋊Q8—22no—
103C2×C4⋊Q8—22no—
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2311yescharacteristic
106C2×C4⋊Q8C2314yes—
107C2×#125—22no—
108C2×#125C2312yes—
109C2×#125C2314yes—
110C2×C22⋊Q8—21no—
111C2×C22⋊Q8—21no—
112C2×C22⋊Q8—22no—
113C2×C22⋊Q8—22no—
114C2×C22⋊Q8—24no—
115C2×C4×Q8—21no—
116C2×C4×Q8—21no—
117C2×C4×D4—21no—
118C2×C4×D4—21no—
119C2×C4×D4—24no—
120C2×C4×D4—24no—
121C2×C4×D4C2312yes—
122C2×C4×D4C2312yes—
123C2×#117C2312yes—
124C2×#117C2312yes—
125C2×#117C2312yes—
126C22×C4⋊C4C2311yescharacteristic
127C22×C4⋊C4C2311yescharacteristic
128C22×C4⋊C4C2311yescharacteristic
129C22×C22⋊C4—21no—
130C22×C42—21no—
131C22×C42C2311yescharacteristic
132#423—24no—
133C2×C4⋊C8C2312yes—
134C2×C4⋊C8C2312yes—
135#418—44no—
136C2×#102—24no—
137C2×#102Q818yescomplemented
138C2×#102C2312yes—
139C2×#102C2312yes—
140C2×#102C2314yes—
141C2×#102C2314yes—
142C2×#102C2312yes—
143C2×#102C2312yes—
144#405—44no—
145C2×#97C2314yes—
146C2×C4×C8—24no—
147C2×C4×C8C2311yescharacteristic
148C2×C4×C8C2311yescharacteristic
149C2×C4×C8C2311yescharacteristic
150C2×C4×C8C2311yescharacteristic
151C2×C4×C8C2314yes—
152#399—24no—
153#398—24no—
154#398—24no—
155#396—24no—
156#396—24no—
157#395—24no—
158#395—24no—
159#393—24no—
160#391—24no—
161#390—22no—
162#390—22no—
163#390C2314yes—
164#390C2314yes—
165#389—22no—
166#389—22no—
167#389—22no—
168#389—22no—
169#389—22no—
170#389—22no—
171#389—24no—
172#389C2314yes—
173#389C2314yes—
174#385—24no—
175#385—24no—
176#385—24no—
177#384—22no—
178#384D411yescharacteristic
179#384D411yescharacteristic
180#384C2314yes—
181#384C2311yescharacteristic
182#384C2311yescharacteristic
183#382D412yes—
184#382C2314yes—
185#382C2312yes—
186#380—21no—
187#380—21no—
188#380—22no—
189#380—22no—
190#380—24no—
191#380D412yes—
192#380D412yes—
193#379—24no—
194C4×C4⋊C4—21no—
195C4×C4⋊C4—21no—
196C4×C4⋊C4—21no—
197C4×C4⋊C4—21no—
198C4×C4⋊C4C2312yes—
199C4×C4⋊C4C2312yes—
200C4×C4⋊C4C2312yes—
201C4×C4⋊C4C2312yes—
202C2×#95—21no—
203#340C2314yes—
204#340C2314yes—
205#340C2312yes—
206#340C2312yes—
207#340C2314yes—
208#340C2312yes—
209#340C2312yes—
210#340C2314yes—
211#340C2312yes—
212#340C2312yes—
213#340C2312yes—
214#340C2312yes—
215#340C2312yes—
216#340C2312yes—
217#340C2314yes—
218#340C2314yes—
219#340C2312yes—
220#340C2312yes—
221#336C2314yes—
222#336C2314yes—
223C25—21no—
224C22×Q8—21no—
225C22×Q8—21no—
226C22×Q8—24no—
227C22×D4—21no—
228C22×D4—22no—
229C22×D4—22no—
230C22×D4—22no—
231C22×D4—22no—
232C22×D4—22no—
233C22×D4C2×Q814yes—
234C22×D4C2412yes—
235C23×C4—21no—
236C23×C4—21no—
237C23×C4—21no—
238C23×C4—22no—
239C23×C4—22no—
240C23×C4—44no—
241C23×C4—82no—
242C23×C4—82no—
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D411yescharacteristic
245C23×C4C2411yescharacteristic
246C2×M16—24no—
247C2×M16—24no—
248C2×M16—24no—
249C22×C8—22no—
250C22×C8—22no—
251C22×C8—22no—
252C22×C8—22no—
253C22×C8—22no—
254C22×C8C2×Q814yes—
255C22×C8C2411yescharacteristic
256C22×C8C2411yescharacteristic
257C22×C8C2414yes—
258C22×C8C2411yescharacteristic
259C22×C8C2411yescharacteristic
260C4⋊Q8—28no—
261C4⋊Q8—22no—
262C4⋊Q8—22no—
263C4⋊Q8—22no—
264C4⋊Q8—22no—
265C4⋊Q8—44no—
266C4⋊Q8—48no—
267C4⋊Q8—84no—
268#125—28no—
269#125—24no—
270#125—24no—
271#125—44no—
272#125—48no—
273C22⋊Q8—416no—
274C22⋊Q8—44no—
275C22⋊Q8—44no—
276C22⋊Q8—84no—
277C22⋊Q8—84no—
278C4×Q8—44no—
279C4×Q8—44no—
280C4×D4—24no—
281C4×D4—24no—
282C4×D4—24no—
283C4×D4—24no—
284C4×D4—44no—
285C4×D4—44no—
286#117—24no—
287#117—24no—
288#117—24no—
289#117—24no—
290#117—24no—
291#117—24no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4C2×D412yes—
315C2×C4⋊C4C2×D412yes—
316C2×C4⋊C4C2×D412yes—
317C2×C4⋊C4C2×D412yes—
318C2×C4⋊C4C2×D412yes—
319C2×C4⋊C4C2×D412yes—
320C2×C4⋊C4C2×Q814yes—
321C2×C4⋊C4#4114yes—
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2412yes—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—44no—
338C2×C22⋊C4C2×D414yes—
339C2×C42—22no—
340C2×C42—22no—
341C2×C42—22no—
342C2×C42—22no—
343C2×C42—82no—
344C2×C42C2×D411yescharacteristic
345C2×C42C2×D411yescharacteristic
346C2×C42#4114yes—
347C2×C42#4114yes—
348C2×C42C2414yes—
349C2×C42C2411yescharacteristic
350C2×C42C2411yescharacteristic
351C4⋊C8—24no—
352C4⋊C8—24no—
353#102—24no—
354#102—24no—
355#102—44no—
356#102—44no—
357#102C2×Q818yes—
358#102C2×Q818yes—
359#97—44no—
360C4×C8—22no—
361C4×C8—22no—
362C4×C8—22no—
363C4×C8—22no—
364C4×C8—44no—
365#95—22no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—24no—
370#95—24no—
371#95—24no—
372#95—24no—
373#95C2×D414yes—
374#95C2×D414yes—
375C24—22no—
376C24—24no—
377C24—42no—
378C24—42no—
379C24—81no—
380C24—81no—
381C24—81no—
382C24—81no—
383C24C22×D411yescharacteristic
384C2×Q8—48no—
385C2×Q8—42no—
386C2×Q8—42no—
387C2×Q8—42no—
388C2×Q8—42no—
389C2×Q8—81no—
390C2×Q8—81no—
391C2×Q8—84no—
392C2×Q8—84no—
393C2×D4—28no—
394C2×D4—24no—
395C2×D4—28no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—42no—
402C2×D4—44no—
403C2×D4—42no—
404C2×D4—44no—
405C2×D4C22×Q814yes—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—41no—
418C22×C4—41no—
419C22×C4—42no—
420C22×C4—44no—
421C22×C4—44no—
422C22×C4—44no—
423C22×C4—44no—
424C22×C4—82no—
425C22×C4—82no—
426C22×C4—816no—
427C22×C4—81no—
428C22×C4—81no—
429C22×C4—81no—
430C22×C4—81no—
431C22×C4—81no—
432C22×C4—81no—
433C22×C4#12414yes—
434C22×C4#12414yes—
435C22×C4C22×D414yes—
436C22×C4C22×D414yes—
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D411yescharacteristic
442C22×C4C22×Q812yes—
443C22×C4C2×#4112yes—
444C22×C4C2×#4112yes—
445C22×C4C2×#4112yes—
446C22×C4H(5,𝔽2)12yes—
447C22×C42−1+412yes—
448C22×C4C2511yescharacteristic
449M16—48no—
450C2×C8—28no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—24no—
460C2×C8—24no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—24no—
464C2×C8—44no—
465C2×C8C22×Q812yes—
466C2×C8C22×Q812yes—
467C4⋊C4—24no—
468C4⋊C4—24no—
469C4⋊C4—24no—
470C4⋊C4—24no—
471C4⋊C4—24no—
472C4⋊C4—24no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—48no—
488C4⋊C4—48no—
489C4⋊C4—48no—
490C4⋊C4—42no—
491C4⋊C4—44no—
492C4⋊C4—44no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—81no—
498C4⋊C4—81no—
499C4⋊C4—84no—
500C4⋊C4—84no—
501C4⋊C4—84no—
502C4⋊C4—84no—
503C4⋊C4C22×Q814yes—
504C4⋊C4C2×#4114yes—
505C22⋊C4—28no—
506C22⋊C4—44no—
507C22⋊C4—44no—
508C22⋊C4—44no—
509C22⋊C4—44no—
510C22⋊C4—84no—
511C22⋊C4—84no—
512C22⋊C4—84no—
513C22⋊C4—84no—
514C42—24no—
515C42—24no—
516C42—44no—
517C42—44no—
518C42—44no—
519C42—82no—
520C42#12414yes—
521C42#12414yes—
522C42C2×#4112yes—
523C42C2×#4112yes—
524C42C2×#4112yes—
525C42C2×#4112yes—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—24no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—48no—
536C23—42no—
537C23—42no—
538C23—84no—
539C23—84no—
540C23—84no—
541C23—84no—
542C23—84no—
543C23—84no—
544C23C2×#12412yes—
545C23#54612yes—
546C23D4×Q812yes—
547C23C23×D411yescharacteristic
548Q8—84no—
549Q8—84no—
550Q8—168no—
551D4—24no—
552D4—24no—
553D4—42no—
554D4—42no—
555D4—48no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—28no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—88no—
592C2×C4—84no—
593C2×C4—84no—
594C2×C4—84no—
595C2×C4—88no—
596C2×C4C2×#12412yes—
597C2×C4C2×#12412yes—
598C2×C4C2×#12412yes—
599C2×C4C2×#12412yes—
600C2×C4C2×#12412yes—
601C2×C4#53314yes—
602C2×C4#54214yes—
603C2×C4#54612yes—
604C2×C4D4×Q812yes—
605C2×C4#55414yes—
606C2×C4#55614yes—
607C2×C4#57612yes—
608C2×C4#57712yes—
609C2×C4C23×Q811yescharacteristic
610C2×C4C22×#4111yescharacteristic
611C2×C4C22×#4111yescharacteristic
612C2×C4C22×#4111yescharacteristic
613C2×C4C2×H(5,𝔽2)11yescharacteristic
614C2×C4C2×2−1+411yescharacteristic
615C8—22no—
616C8—22no—
617C8—44no—
618C8—41no—
619C8—41no—
620C22—22no—
621C22—22no—
622C22—22no—
623C22—22no—
624C22—22no—
625C22—22no—
626C22—44no—
627C22—44no—
628C22—44no—
629C22—816no—
630C22—82no—
631C22—82no—
632C22—82no—
633C22—82no—
634C22#249214yes—
635C22C22×#12411yescharacteristic
636C22C2×#54611yescharacteristic
637C22C2×D4×Q811yescharacteristic
638C22C2×#57611yescharacteristic
639C22C2×#57711yescharacteristic
640C22#358411yescharacteristic
641C22#358411yescharacteristic
642C4—22no—
643C4—22no—
644C4—22no—
645C4—44no—
646C4—44no—
647C4—44no—
648C4—44no—
649C4—82no—
650C4—82no—
651C4—82no—
652C4—82no—
653C4—82no—
654C4#249214yes—
655C4#313614yes—
656C4C2×#53311yescharacteristic
657C4C2×#54211yescharacteristic
658C4C2×#55411yescharacteristic
659C4C2×#55611yescharacteristic
660C2—24no—
661C2—84no—
662C2C2×#249211yescharacteristic
663C2C2×#313611yescharacteristic
664C2#6061211yescharacteristic
665C2#6121111yescharacteristic
666C2#6121311yescharacteristic
667C2#6137011yescharacteristic
668C2#6137111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2285 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #35167C2semidirect
3. #27822C2semidirect
4. #27821C2semidirect
5. #27819C2semidirect
6. #27817C2semidirect
7. #27814C2semidirect
8. #27813C2semidirect
9. #23749C2semidirect
11. #21326C2semidirect
12. #21318C2semidirect
13. #21312C2semidirect
16. #20437C2semidirect
35. #2552C22semidirect
37. #2551C22semidirect
39. #2550C22semidirect
49. #2101C22semidirect
50. #2101C22semidirect
51. #2100C22semidirect
52. #2099C22semidirect
57. #1993C22semidirect
58. #1992C22semidirect
59. #1991C22semidirect
60. #1991C22semidirect
61. #1991C22semidirect
62. #1990C22semidirect
63. #1990C22semidirect
64. #1990C22semidirect
65. #1990C22semidirect
66. #1990C22semidirect
67. #1989C22semidirect
68. #1988C22semidirect
69. #1987C22semidirect
77. #1770C22semidirect
78. #1770C22semidirect
79. C8×C4⋊C4C22semidirect
80. C8×C4⋊C4C22semidirect
137. C2×#102Q8semidirect

Maximal subgroups

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

subgroupindex
#351722
#351672
#278222
#278212
#278192
#278172
#278142
#278132
#237492
#234562
#213262
#213182
#213122
#210892
#206462
#204372
C2×C4×#1022