Finite Group Databasebeta

← #6626453

Subgroups — #6626453

Order 512.

1691 conjugacy classes of subgroups.

The subgroup lattice has 659 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36561C211yesmaximal, characteristic
2C2×#2369C211yesmaximal, characteristic, complemented
3#33148C212yesmaximal, complemented
4#33146C211yesmaximal, characteristic, complemented
5#29302C212yesmaximal, complemented
6#29237C212yesmaximal
7#27019C214yesmaximal, complemented
8#26877C214yesmaximal, complemented
9#26446C212yesmaximal, complemented
10#25811C211yesmaximal, characteristic, complemented
11#25451C212yesmaximal, complemented
12#24299C211yesmaximal, characteristic
13#23163C212yesmaximal, complemented
14#22969C212yesmaximal, complemented
15#22629C211yesmaximal, characteristic, complemented
16#21578C212yesmaximal
17C4×#480C211yesmaximal, characteristic
18C2×#554—21no—
19C22×C4⋊Q8C2211yescharacteristic
20C2×#498C2212yes—
21C2×#480C2211yescharacteristic
22C2×#476C2212yes—
23#2857C2212yes—
24#2857C2211yescharacteristic
25#2564—22no—
26#2556—22no—
27#2551C2211yescharacteristic
28#2542C2211yescharacteristic
29#2506C2212yes—
30#2492—21no—
31#2452C2212yes—
32#2417—21no—
33#2369—21no—
34#2369—21no—
35#2369—22no—
36#2369C2212yes—
37#2369C2212yes—
38#2369C2214yes—
39#2365C2211yescharacteristic
40#2360—21no—
41D4×C4⋊C4C2211yescharacteristic
42C4×#121C2212yes—
43C4×#121C2211yescharacteristic
44C2×#391—22no—
45C2×#389—21no—
46C2×#389C2212yes—
47C2×C4×C4⋊C4—21no—
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C4⋊C4C2212yes—
50C2×C4×C4⋊C4C2211yescharacteristic
51#2087C2214yes—
52#2084C2214yes—
53#2065C2214yes—
54#2058C2214yes—
55#2058C2212yes—
56#2057C2214yes—
57#2023C2212yes—
58#1990C2212yes—
59#1988C2212yes—
60#1987C2212yes—
61#1987C2214yes—
62#1948C2214yes—
63#1947C2212yes—
64#1944C2214yes—
65#1944C2214yes—
66#1939C2212yes—
67#1926C2214yes—
68#1925C2212yes—
69#1921C2214yes—
70#1920C2214yes—
71#1920C2214yes—
72#1919C2214yes—
73#1918C2212yes—
74#1918C2214yes—
75#1917C2214yes—
76#1874C2211yescharacteristic
77#1874C2212yes—
78#1862C2214yes—
79#1853C2212yes—
80#1853C2212yes—
81#1853C2212yes—
82#1852C2212yes—
83#1852C2211yescharacteristic
84#1823C2211yescharacteristic
85#1823C2212yes—
86#1797C2212yes—
87#1797C2211yescharacteristic
88#1776C2214yes—
89C4×#107C2212yes—
90#1770C2212yes—
91#1762C2214yes—
92C4×#102C2212yes—
93#1749C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8—21no—
96#554—48no—
97C2×C4⋊Q8—28no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8C2312yes—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2312yes—
104C2×C4⋊Q8C2312yes—
105C2×#125—22no—
106C2×C22⋊Q8—22no—
107C2×C22⋊Q8—21no—
108C2×C22⋊Q8—21no—
109C2×C22⋊Q8—22no—
110C2×C22⋊Q8—21no—
111C2×#121C2311yescharacteristic
112C2×C4×Q8—21no—
113C2×C4×D4—22no—
114C2×C4×D4—21no—
115C2×C4×D4C2311yescharacteristic
116C2×C4×D4C2312yes—
117C22×C4⋊C4—21no—
118C22×C4⋊C4—22no—
119C22×C4⋊C4—21no—
120C22×C4⋊C4D411yescharacteristic
121C22×C4⋊C4D411yescharacteristic
122C22×C4⋊C4C2312yes—
123C22×C4⋊C4C2311yescharacteristic
124C22×C4⋊C4C2311yescharacteristic
125C22×C42—21no—
126C22×C42—22no—
127C22×C42C2311yescharacteristic
128C22×C42C2312yes—
129#498—28no—
130#480Q818yescomplemented
131#476—28no—
132C2×#107C2312yes—
133C2×#107C2314yes—
134C2×#107C2312yes—
135C2×#106C2314yes—
136C2×#106C2314yes—
137C2×#106C2314yes—
138C2×C4⋊C8C2312yes—
139C2×C4⋊C8C2314yes—
140C2×#102C2314yes—
141C2×#102C2312yes—
142C2×#102C2312yes—
143C2×C22⋊C8C2312yes—
144C2×C22⋊C8C2311yescharacteristic
145C2×C22⋊C8C2311yescharacteristic
146C2×#97C2314yes—
147C2×C4×C8C2312yes—
148#400—22no—
149#399—22no—
150#398—22no—
151#396—22no—
152#395—22no—
153#393—22no—
154#391—21no—
155#391—21no—
156#391—22no—
157#391—24no—
158#391—24no—
159#391—24no—
160#391—24no—
161#390C2312yes—
162#390C2311yescharacteristic
163#389—21no—
164#389—22no—
165#389—24no—
166#389—22no—
167#389—22no—
168#389—21no—
169#389—22no—
170#389—22no—
171#389C2314yes—
172#389C2312yes—
173#389C2311yescharacteristic
174#389C2312yes—
175#389C2312yes—
176#389C2312yes—
177#389C2314yes—
178#388—22no—
179#388—21no—
180#387—22no—
181#387—22no—
182#386—21no—
183#386—22no—
184#385—22no—
185#385—22no—
186#384C2312yes—
187#384C2311yescharacteristic
188#380—21no—
189#380—22no—
190#380—21no—
191#380C2312yes—
192#380C2311yescharacteristic
193C4×C4⋊C4—22no—
194C4×C4⋊C4—24no—
195C4×C4⋊C4—22no—
196C4×C4⋊C4—22no—
197C4×C4⋊C4—22no—
198C4×C4⋊C4—21no—
199C4×C4⋊C4—21no—
200C4×C4⋊C4D411yescharacteristic
201C4×C4⋊C4D411yescharacteristic
202C4×C4⋊C4D412yes—
203C4×C4⋊C4C2312yes—
204C4×C4⋊C4C2311yescharacteristic
205C4×C4⋊C4C2312yes—
206C4×C4⋊C4C2314yes—
207C4×C4⋊C4C2314yes—
208C4×C4⋊C4C2312yes—
209C4×C4⋊C4C2311yescharacteristic
210C4×C4⋊C4C2312yes—
211C4×C4⋊C4C2311yescharacteristic
212C4×C22⋊C4—21no—
213C4×C22⋊C4—22no—
214C2×#95—22no—
215C2×#95—22no—
216C2×#95—21no—
217C2×#95D412yes—
218#340C2314yes—
219#340C2314yes—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2312yes—
224#340C2314yes—
225#340C2314yes—
226#340C2312yes—
227#340C2312yes—
228#340C2312yes—
229#336C2312yes—
230#336C2314yes—
231#336C2314yes—
232#336C2312yes—
233C22×Q8—28no—
234C22×Q8—22no—
235C22×Q8—22no—
236C22×Q8—21no—
237C22×Q8—22no—
238C22×Q8—82no—
239C22×Q8—82no—
240C22×D4—21no—
241C22×D4C2411yescharacteristic
242C23×C4—22no—
243C23×C4—21no—
244C23×C4—21no—
245C23×C4—21no—
246C23×C4—22no—
247C23×C4C2×D412yes—
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D412yes—
251C23×C4C2411yescharacteristic
252C22×C8C2412yes—
253C22×C8C2414yes—
254C22×C8C2412yes—
255C4⋊Q8—28no—
256C4⋊Q8—28no—
257C4⋊Q8—28no—
258C4⋊Q8—24no—
259C4⋊Q8—24no—
260C4⋊Q8—44no—
261C4⋊Q8—44no—
262C4⋊Q8#4118yes—
263#125—48no—
264C22⋊Q8—44no—
265C22⋊Q8—48no—
266C22⋊Q8—48no—
267C22⋊Q8—44no—
268C22⋊Q8—44no—
269#121—22no—
270#121C2×Q814yes—
271C4×Q8—44no—
272C4×D4—24no—
273C4×D4—44no—
274C4×D4—48no—
275C4×D4—44no—
276C2×C4⋊C4—28no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—28no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
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—24no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—42no—
300C2×C4⋊C4—44no—
301C2×C4⋊C4—84no—
302C2×C4⋊C4C2×D411yescharacteristic
303C2×C4⋊C4C2×D412yes—
304C2×C4⋊C4C2×D411yescharacteristic
305C2×C4⋊C4C2×D411yescharacteristic
306C2×C4⋊C4C2×D412yes—
307C2×C4⋊C4C2×D411yescharacteristic
308C2×C4⋊C4C2×Q814yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2411yescharacteristic
312C2×C4⋊C4C2411yescharacteristic
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2411yescharacteristic
316C2×C4⋊C4C2414yes—
317C2×C22⋊C4—21no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—21no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—21no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—21no—
326C2×C42—24no—
327C2×C42—22no—
328C2×C42—21no—
329C2×C42—22no—
330C2×C42—22no—
331C2×C42—24no—
332C2×C42—24no—
333C2×C42—44no—
334C2×C42—44no—
335C2×C42—84no—
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D412yes—
339C2×C42C2×D412yes—
340C2×C42C2412yes—
341C2×C42C2411yescharacteristic
342C2×C42C2412yes—
343C2×C42C2411yescharacteristic
344#107—42no—
345#107—44no—
346#107C2×Q818yescomplemented
347#106—28no—
348#106—28no—
349#106—28no—
350C4⋊C8—24no—
351C4⋊C8—44no—
352#102—28no—
353#102—42no—
354#102C2×Q818yescomplemented
355C22⋊C8—24no—
356C22⋊C8—22no—
357C22⋊C8C2×Q814yes—
358#97—44no—
359C4×C8—24no—
360#95—24no—
361#95—24no—
362#95—24no—
363#95—24no—
364#95—22no—
365#95—22no—
366#95—22no—
367#95—21no—
368#95—21no—
369#95—22no—
370#95—21no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95C2×D412yes—
376#95C2×D412yes—
377#95C2×D412yes—
378#95C2×D412yes—
379C24—21no—
380C24C22×D411yescharacteristic
381C2×Q8—28no—
382C2×Q8—28no—
383C2×Q8—44no—
384C2×Q8—42no—
385C2×Q8—42no—
386C2×Q8—42no—
387C2×Q8—82no—
388C2×Q8—82no—
389C2×Q8—82no—
390C2×Q8—82no—
391C2×Q8—82no—
392C2×Q8—82no—
393C2×Q8—816no—
394C2×D4—24no—
395C2×D4—22no—
396C2×D4—21no—
397C2×D4—22no—
398C2×D4—42no—
399C2×D4—42no—
400C2×D4C22×Q812yes—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—28no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—21no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—42no—
421C22×C4—41no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—44no—
425C22×C4—44no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—44no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—82no—
436C22×C4—82no—
437C22×C4AGL(1,ℤ/8)14yes—
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D412yes—
440C22×C4C22×D412yes—
441C22×C4C22×D412yes—
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D412yes—
446C22×C4C22×D411yescharacteristic
447C22×C4C22×Q812yes—
448C22×C42−1+412yes—
449C22×C4C2511yescharacteristic
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—24no—
454C2×C8—22no—
455C2×C8—24no—
456C2×C8—24no—
457C2×C8—22no—
458C2×C8C22×Q814yes—
459C4⋊C4—28no—
460C4⋊C4—28no—
461C4⋊C4—28no—
462C4⋊C4—24no—
463C4⋊C4—22no—
464C4⋊C4—24no—
465C4⋊C4—22no—
466C4⋊C4—24no—
467C4⋊C4—22no—
468C4⋊C4—24no—
469C4⋊C4—22no—
470C4⋊C4—21no—
471C4⋊C4—22no—
472C4⋊C4—24no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—24no—
476C4⋊C4—22no—
477C4⋊C4—42no—
478C4⋊C4—44no—
479C4⋊C4—44no—
480C4⋊C4—44no—
481C4⋊C4—44no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4—44no—
486C4⋊C4—44no—
487C4⋊C4—81no—
488C4⋊C4—84no—
489C4⋊C4—84no—
490C4⋊C4—82no—
491C4⋊C4#13714yes—
492C4⋊C4C22×Q814yes—
493C4⋊C4C22×Q812yes—
494C4⋊C4C2×#4118yes—
495C22⋊C4—24no—
496C22⋊C4—44no—
497C22⋊C4—44no—
498C22⋊C4—42no—
499C22⋊C4—42no—
500C22⋊C4—42no—
501C22⋊C4—44no—
502C22⋊C4—81no—
503C22⋊C4—82no—
504C42—22no—
505C42—22no—
506C42—22no—
507C42—24no—
508C42—24no—
509C42—42no—
510C42—42no—
511C42—44no—
512C42—44no—
513C42—44no—
514C42—42no—
515C42—84no—
516C42—84no—
517C42C2×#4114yes—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—22no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23D4×Q812yes—
530C23C2×AGL(1,ℤ/8)12yes—
531C23#57712yes—
532C23C23×D411yescharacteristic
533Q8—88no—
534Q8—88no—
535Q8—84no—
536Q8—164no—
537D4—24no—
538D4—48no—
539D4—42no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—22no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—22no—
547C2×C4—21no—
548C2×C4—28no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—21no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—21no—
555C2×C4—28no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—48no—
567C2×C4—44no—
568C2×C4—416no—
569C2×C4—44no—
570C2×C4—48no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—84no—
592C2×C4—84no—
593C2×C4—84no—
594C2×C4—164no—
595C2×C4#52512yes—
596C2×C4#52512yes—
597C2×C4D4×Q812yes—
598C2×C4C2×AGL(1,ℤ/8)12yes—
599C2×C4C2×AGL(1,ℤ/8)12yes—
600C2×C4C2×#13714yes—
601C2×C4C2×#13712yes—
602C2×C4C23×Q811yescharacteristic
603C2×C4C22×#4112yes—
604C2×C4C2×2−1+411yescharacteristic
605C2×C4#58512yes—
606C8—24no—
607C8—42no—
608C8—44no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—22no—
614C22—22no—
615C22—42no—
616C22—42no—
617C22—42no—
618C22—81no—
619C22—81no—
620C22—81no—
621C22—81no—
622C22#338414yes—
623C22C2×#52511yescharacteristic
624C22C2×D4×Q811yescharacteristic
625C22#348511yescharacteristic
626C22C22×AGL(1,ℤ/8)11yescharacteristic
627C22C22×#13711yescharacteristic
628C22C2×#57711yescharacteristic
629C22#358511yescharacteristic
630C4—24no—
631C4—22no—
632C4—22no—
633C4—22no—
634C4—42no—
635C4—42no—
636C4—42no—
637C4—41no—
638C4—48no—
639C4—44no—
640C4—42no—
641C4—82no—
642C4—82no—
643C4—81no—
644C4—81no—
645C4—82no—
646C4#311214yes—
647C4Q8×#4112yes—
648C4#351612yes—
649C2—24no—
650C2—82no—
651C2#3654711yescharacteristic
652C2C2×#311211yescharacteristic
653C2#6057911yescharacteristic
654C2C2×#338411yescharacteristic
655C2Q8×#13711yescharacteristic
656C2#6172011yescharacteristic
657C2#6173511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#121C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2369C2semidirect
3. #33148C2semidirect
4. #33146C2semidirect
5. #29302C2semidirect
7. #27019C2semidirect
8. #26877C2semidirect
9. #26446C2semidirect
10. #25811C2semidirect
11. #25451C2semidirect
13. #23163C2semidirect
14. #22969C2semidirect
15. #22629C2semidirect
130. #480Q8semidirect
346. #107C2×Q8semidirect
354. #102C2×Q8semidirect

Maximal subgroups

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

subgroupindex
#365612
C2×#23692
#331482
#331462
#293022
#292372
#270192
#268772
#264462
#258112
#254512
#242992
#231632
#229692
#226292
#215782
C4×#4802