Finite Group Databasebeta

← #6851986

Subgroups — #6851986

Order 512.

1701 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36551C211yesmaximal, characteristic
2C2×#2551C211yesmaximal, characteristic, complemented
3#33149C211yesmaximal, characteristic, complemented
4#32146C214yesmaximal, complemented
5#30962C214yesmaximal, complemented
6#29973C212yesmaximal, complemented
7#28816C212yesmaximal, complemented
8#27950C211yesmaximal, characteristic, complemented
9#27821C212yesmaximal, complemented
10#27678C211yesmaximal, characteristic
11#26442C212yesmaximal, complemented
12#26040C212yesmaximal, complemented
13#25816C212yesmaximal, complemented
14#24298C212yesmaximal
15#22629C211yesmaximal, characteristic, complemented
16#22082C212yesmaximal
17C4×#481C211yesmaximal, characteristic
18C22×C4⋊Q8—21no—
19C22×C4⋊Q8C2212yes—
20C2×#501C2214yes—
21C2×#481C2211yescharacteristic
22C2×#474C2214yes—
23#2857C2211yescharacteristic
24#2851C2212yes—
25#2589—22no—
26#2563C2211yescharacteristic
27#2552C2211yescharacteristic
28#2551—22no—
29#2551—21no—
30#2551—21no—
31#2551C2212yes—
32#2551C2212yes—
33#2551C2214yes—
34#2509C2212yes—
35#2497—21no—
36#2487—21no—
37#2454C2212yes—
38#2450—22no—
39C2×#395—22no—
40#2369C2211yescharacteristic
41D4×C4⋊C4—21no—
42D4×C4⋊C4C2211yescharacteristic
43#2345—21no—
44C4×#121C2211yescharacteristic
45C4×#121C2212yes—
46C2×#389—21no—
47C2×#389C2212yes—
48C2×#384C2211yescharacteristic
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×C4×C4⋊C4C2211yescharacteristic
51#2101C2214yes—
52#2100C2212yes—
53#2094C2212yes—
54#2088C2214yes—
55#2086C2214yes—
56#2083C2214yes—
57#2059C2214yes—
58#2058C2214yes—
59#2057C2214yes—
60#2057C2214yes—
61#2023C2214yes—
62#1993C2214yes—
63#1990C2212yes—
64#1988C2214yes—
65#1987C2212yes—
66#1948C2212yes—
67#1947C2212yes—
68#1944C2214yes—
69#1939C2212yes—
70#1936C2214yes—
71#1926C2212yes—
72#1921C2214yes—
73#1919C2214yes—
74#1919C2212yes—
75#1917C2214yes—
76#1874C2212yes—
77#1874C2211yescharacteristic
78#1862C2214yes—
79#1853C2212yes—
80#1853C2211yescharacteristic
81#1852C2212yes—
82#1852C2212yes—
83#1852C2212yes—
84#1845C2212yes—
85#1824C2211yescharacteristic
86#1824C2212yes—
87#1802C2212yes—
88#1797C2212yes—
89#1797C2211yescharacteristic
90C4×#106C2212yes—
91#1770C2212yes—
92C4×#102C2212yes—
93#1749C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8—22no—
96C2×C4⋊Q8—22no—
97C2×C4⋊Q8—24no—
98C2×C4⋊Q8—22no—
99C2×C4⋊Q8—22no—
100C2×C4⋊Q8—22no—
101C2×C4⋊Q8—48no—
102C2×C4⋊Q8—48no—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2314yes—
105C2×C4⋊Q8C2314yes—
106C2×#125C2312yes—
107C2×C22⋊Q8—21no—
108C2×#121C2311yescharacteristic
109C2×C4×D4—21no—
110C2×C4×D4—22no—
111C2×C4×D4C2311yescharacteristic
112C2×C4×D4C2312yes—
113C22×C4⋊C4—22no—
114C22×C4⋊C4—22no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4—21no—
117C22×C4⋊C4—21no—
118C22×C4⋊C4D411yescharacteristic
119C22×C4⋊C4D411yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2312yes—
123C22×C42—21no—
124C22×C42C2311yescharacteristic
125C22×C42C2312yes—
126#501—48no—
127#481Q818yescomplemented
128#474—48no—
129C2×#107C2314yes—
130C2×#107C2314yes—
131C2×#107C2314yes—
132C2×#106C2312yes—
133C2×#106C2314yes—
134C2×#106C2312yes—
135C2×C4⋊C8C2312yes—
136C2×C4⋊C8C2314yes—
137C2×#102C2312yes—
138C2×#102C2314yes—
139C2×#102C2312yes—
140C2×C22⋊C8C2311yescharacteristic
141C2×C22⋊C8C2312yes—
142C2×C22⋊C8C2311yescharacteristic
143C2×#97C2314yes—
144C2×C4×C8C2312yes—
145#400—22no—
146#400—22no—
147#399—22no—
148#398—22no—
149#396—22no—
150#395—24no—
151#395—24no—
152#395—24no—
153#395—24no—
154#393—22no—
155#391—21no—
156#391—22no—
157#390C2312yes—
158#390C2311yescharacteristic
159#389—22no—
160#389—21no—
161#389—21no—
162#389—22no—
163#389—22no—
164#389—24no—
165#389—21no—
166#389—22no—
167#389—22no—
168#389C2312yes—
169#389C2312yes—
170#389C2312yes—
171#389C2314yes—
172#389C2312yes—
173#389C2314yes—
174#389C2311yescharacteristic
175#388—21no—
176#388—22no—
177#387—22no—
178#387—22no—
179#387—22no—
180#386—21no—
181#386—22no—
182#385—22no—
183#385—22no—
184#385—22no—
185#385—22no—
186#384—21no—
187#384—21no—
188#384—21no—
189#384C2312yes—
190#384C2312yes—
191#384C2311yescharacteristic
192#382C2312yes—
193#380—21no—
194#380—21no—
195#380—21no—
196#380—21no—
197#380—22no—
198#380C2311yescharacteristic
199#380C2312yes—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4D412yes—
204C4×C4⋊C4D411yescharacteristic
205C4×C4⋊C4D411yescharacteristic
206C4×C4⋊C4C2312yes—
207C4×C4⋊C4C2311yescharacteristic
208C4×C4⋊C4C2312yes—
209C4×C4⋊C4C2312yes—
210C4×C4⋊C4C2311yescharacteristic
211C4×C4⋊C4C2311yescharacteristic
212C4×C4⋊C4C2312yes—
213C4×C22⋊C4—22no—
214C4×C22⋊C4—21no—
215C2×#95—21no—
216C2×#95—22no—
217C2×#95D412yes—
218#340C2312yes—
219#340C2312yes—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2312yes—
225#340C2314yes—
226#340C2312yes—
227#340C2314yes—
228#340C2314yes—
229#336C2314yes—
230#336C2312yes—
231#336C2314yes—
232#336C2312yes—
233C22×Q8—24no—
234C22×Q8—24no—
235C22×Q8—21no—
236C22×Q8—48no—
237C22×Q8—48no—
238C22×Q8—84no—
239C22×D4—21no—
240C22×D4C2411yescharacteristic
241C23×C4—21no—
242C23×C4—21no—
243C23×C4—22no—
244C23×C4—21no—
245C23×C4—22no—
246C23×C4C2×D412yes—
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D412yes—
250C23×C4C2411yescharacteristic
251C22×C8C2412yes—
252C22×C8C2414yes—
253C22×C8C2412yes—
254C4⋊Q8—28no—
255C4⋊Q8—28no—
256C4⋊Q8—24no—
257C4⋊Q8—44no—
258C4⋊Q8—44no—
259C4⋊Q8—48no—
260C4⋊Q8—48no—
261C4⋊Q8—416no—
262C4⋊Q8—48no—
263C4⋊Q8—48no—
264#125—44no—
265C22⋊Q8—82no—
266#121—22no—
267#121C2×Q814yes—
268C4×D4—24no—
269C4×D4—44no—
270C4×D4—44no—
271C4×D4—84no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
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—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—44no—
298C2×C4⋊C4—48no—
299C2×C4⋊C4—44no—
300C2×C4⋊C4—48no—
301C2×C4⋊C4—44no—
302C2×C4⋊C4—82no—
303C2×C4⋊C4C2×D411yescharacteristic
304C2×C4⋊C4C2×D412yes—
305C2×C4⋊C4C2×D411yescharacteristic
306C2×C4⋊C4C2×D411yescharacteristic
307C2×C4⋊C4C2×D412yes—
308C2×C4⋊C4C2×D411yescharacteristic
309C2×C4⋊C4C2×Q814yes—
310C2×C4⋊C4C2414yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2411yescharacteristic
313C2×C4⋊C4C2411yescharacteristic
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2411yescharacteristic
317C2×C4⋊C4C2412yes—
318C2×C22⋊C4—21no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—21no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—21no—
327C2×C42—24no—
328C2×C42—21no—
329C2×C42—22no—
330C2×C42—22no—
331C2×C42—22no—
332C2×C42—44no—
333C2×C42—44no—
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D411yescharacteristic
336C2×C42C2×D412yes—
337C2×C42C2×D412yes—
338C2×C42C2412yes—
339C2×C42C2412yes—
340C2×C42C2411yescharacteristic
341C2×C42C2411yescharacteristic
342#107—28no—
343#107—44no—
344#107—44no—
345#106—24no—
346#106—44no—
347#106C2×Q818yescomplemented
348C4⋊C8—24no—
349C4⋊C8—44no—
350#102—24no—
351#102—44no—
352#102C2×Q818yescomplemented
353C22⋊C8—22no—
354C22⋊C8—24no—
355C22⋊C8C2×Q814yes—
356#97—44no—
357C4×C8—24no—
358#95—22no—
359#95—22no—
360#95—24no—
361#95—24no—
362#95—21no—
363#95—22no—
364#95—21no—
365#95—22no—
366#95—22no—
367#95—22no—
368#95—21no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95C2×D412yes—
374#95C2×D412yes—
375#95C2×D412yes—
376#95C2×D412yes—
377C24—21no—
378C24C22×D411yescharacteristic
379C2×Q8—48no—
380C2×Q8—48no—
381C2×Q8—48no—
382C2×Q8—42no—
383C2×Q8—48no—
384C2×Q8—84no—
385C2×Q8—84no—
386C2×Q8—81no—
387C2×Q8—81no—
388C2×Q8—832no—
389C2×D4—24no—
390C2×D4—22no—
391C2×D4—22no—
392C2×D4—21no—
393C2×D4—42no—
394C2×D4—42no—
395C2×D4C22×Q812yes—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—41no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—44no—
424C22×C4—44no—
425C22×C4—44no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—44no—
430C22×C4—82no—
431C22×C4—82no—
432C22×C4#13714yes—
433C22×C4C22×D412yes—
434C22×C4C22×D412yes—
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D412yes—
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D412yes—
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D411yescharacteristic
442C22×C4C22×Q812yes—
443C22×C42−1+412yes—
444C22×C4C2511yescharacteristic
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—24no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—24no—
452C2×C8—24no—
453C2×C8C22×Q814yes—
454C4⋊C4—24no—
455C4⋊C4—24no—
456C4⋊C4—24no—
457C4⋊C4—22no—
458C4⋊C4—24no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—21no—
464C4⋊C4—24no—
465C4⋊C4—22no—
466C4⋊C4—24no—
467C4⋊C4—24no—
468C4⋊C4—22no—
469C4⋊C4—24no—
470C4⋊C4—48no—
471C4⋊C4—48no—
472C4⋊C4—44no—
473C4⋊C4—44no—
474C4⋊C4—48no—
475C4⋊C4—42no—
476C4⋊C4—44no—
477C4⋊C4—42no—
478C4⋊C4—48no—
479C4⋊C4—44no—
480C4⋊C4—42no—
481C4⋊C4—82no—
482C4⋊C4—82no—
483C4⋊C4—82no—
484C4⋊C4—82no—
485C4⋊C4—81no—
486C4⋊C4—82no—
487C4⋊C4—82no—
488C4⋊C4—82no—
489C4⋊C4AGL(1,ℤ/8)14yes—
490C4⋊C4C22×Q814yes—
491C4⋊C4C22×Q812yes—
492C4⋊C4H(5,𝔽2)14yes—
493C22⋊C4—24no—
494C22⋊C4—42no—
495C22⋊C4—44no—
496C22⋊C4—42no—
497C22⋊C4—42no—
498C22⋊C4—81no—
499C22⋊C4—82no—
500C22⋊C4—82no—
501C22⋊C4—82no—
502C42—22no—
503C42—22no—
504C42—24no—
505C42—24no—
506C42—22no—
507C42—42no—
508C42—42no—
509C42—44no—
510C42—44no—
511C42—42no—
512C42—82no—
513C422−1+414yes—
514C23—21no—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—22no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23D4×Q812yes—
526C23C2×#13712yes—
527C23#57612yes—
528C23C23×D411yescharacteristic
529Q8—816no—
530Q8—816no—
531Q8—162no—
532D4—24no—
533D4—48no—
534D4—42no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—22no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—22no—
542C2×C4—21no—
543C2×C4—28no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—22no—
547C2×C4—21no—
548C2×C4—22no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—21no—
561C2×C4—44no—
562C2×C4—48no—
563C2×C4—48no—
564C2×C4—48no—
565C2×C4—48no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—41no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—88no—
588C2×C4—84no—
589C2×C4—164no—
590C2×C4#53612yes—
591C2×C4#53612yes—
592C2×C4D4×Q812yes—
593C2×C4C2×AGL(1,ℤ/8)14yes—
594C2×C4C2×AGL(1,ℤ/8)12yes—
595C2×C4C2×#13712yes—
596C2×C4C2×#13712yes—
597C2×C4C23×Q811yescharacteristic
598C2×C4C2×H(5,𝔽2)12yes—
599C2×C4C2×2−1+412yes—
600C2×C4C2×2−1+411yescharacteristic
601C8—24no—
602C8—44no—
603C8—42no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—42no—
611C22—42no—
612C22—42no—
613C22—81no—
614C22—81no—
615C22—81no—
616C22—81no—
617C22#338514yes—
618C22C2×#53611yescharacteristic
619C22C2×D4×Q811yescharacteristic
620C22#348411yescharacteristic
621C22C22×AGL(1,ℤ/8)11yescharacteristic
622C22C22×#13711yescharacteristic
623C22C2×#57611yescharacteristic
624C22#358611yescharacteristic
625C4—24no—
626C4—22no—
627C4—22no—
628C4—22no—
629C4—42no—
630C4—42no—
631C4—42no—
632C4—44no—
633C4—42no—
634C4—44no—
635C4—41no—
636C4—81no—
637C4—84no—
638C4—82no—
639C4—81no—
640C4—82no—
641C4#323914yes—
642C4#350312yes—
643C4#355212yes—
644C2—24no—
645C2—82no—
646C2#3959211yescharacteristic
647C2C2×#323911yescharacteristic
648C2#6087111yescharacteristic
649C2C2×#338511yescharacteristic
650C2Q8×AGL(1,ℤ/8)11yescharacteristic
651C2#6158411yescharacteristic
652C2#6159111yescharacteristic
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×#2551C2semidirect
3. #33149C2semidirect
4. #32146C2semidirect
5. #30962C2semidirect
6. #29973C2semidirect
7. #28816C2semidirect
8. #27950C2semidirect
9. #27821C2semidirect
11. #26442C2semidirect
12. #26040C2semidirect
13. #25816C2semidirect
15. #22629C2semidirect
127. #481Q8semidirect
347. #106C2×Q8semidirect
352. #102C2×Q8semidirect

Maximal subgroups

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

subgroupindex
#365512
C2×#25512
#331492
#321462
#309622
#299732
#288162
#279502
#278212
#276782
#264422
#260402
#258162
#242982
#226292
#220822
C4×#4812