Finite Group Databasebeta

← #6557749

Subgroups — #6557749

Order 512.

1726 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×#127C211yesmaximal, characteristic
2#36497C211yesmaximal, characteristic, complemented
3#30233C212yesmaximal, complemented
4#30230C211yesmaximal, characteristic, complemented
5#29225C211yesmaximal, characteristic, complemented
6#29218C212yesmaximal, complemented
7#29130C211yesmaximal, characteristic
8#29124C212yesmaximal
9#27830C214yesmaximal, complemented
10#27813C212yesmaximal, complemented
11#27744C214yesmaximal, complemented
12#27727C212yesmaximal, complemented
13#23822C212yesmaximal
14#23819C211yesmaximal, characteristic
15#22523C212yesmaximal, complemented
16#22523C212yesmaximal, complemented
17#20573C211yesmaximal, characteristic, complemented
18C2×D4×Q8—21no—
19C2×D4×Q8—21no—
20C2×D4×Q8C2212yes—
21C2×#446C2212yes—
22#2868C2211yescharacteristic
23#2868C2212yes—
24#2864C2212yes—
25#2859C2211yescharacteristic
26#2570C2212yes—
27#2552C2211yescharacteristic
28#2551C2211yescharacteristic
29#2550C2212yes—
30#2542—21no—
31#2542—21no—
32#2542—22no—
33#2542—22no—
34#2542C2212yes—
35#2542C2214yes—
36#2520—22no—
37#2462—22no—
38#2453—21no—
39#2451—21no—
40#2370—21no—
41#2365—21no—
42#2355C2211yescharacteristic
43#2355C2211yescharacteristic
44C4×C4⋊Q8C2212yes—
45C4×#127C2211yescharacteristic
46C4×#127C2212yes—
47C4×#125C2211yescharacteristic
48C42×Q8C2211yescharacteristic
49#2100C2211yescharacteristic
50#2100C2211yescharacteristic
51#2100C2212yes—
52#2100C2212yes—
53#2100C2212yes—
54#2100C2214yes—
55#2100C2212yes—
56#2100C2212yes—
57#2099C2214yes—
58#2099C2214yes—
59#2073C2212yes—
60#2071C2214yes—
61#2041C2211yescharacteristic
62#2041C2211yescharacteristic
63#2041C2212yes—
64#2025C2212yes—
65#2023C2214yes—
66#1993C2214yes—
67#1993C2214yes—
68#1992C2212yes—
69#1990C2212yes—
70#1990C2212yes—
71#1990C2214yes—
72#1988C2212yes—
73#1988C2212yes—
74#1987C2214yes—
75#1987C2214yes—
76#1936C2214yes—
77#1934C2214yes—
78#1934C2214yes—
79#1836C2212yes—
80#1834C2214yes—
81#1804C2214yes—
82#1804C2214yes—
83#1802C2212yes—
84#1802C2212yes—
85#1771C2214yes—
86C8×C4⋊C4C2212yes—
87#1767C2211yescharacteristic
88C4×C4⋊C8C2212yes—
89#1762C2214yes—
90#1762C2214yes—
91C4×#102C2212yes—
92C4×#102C2212yes—
93#1745C2212yes—
94#1744C2211yescharacteristic
95C23×Q8—22no—
96D4×Q8—28no—
97D4×Q8—28no—
98D4×Q8—48no—
99D4×Q8—48no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—22no—
104C2×C4⋊Q8—22no—
105C2×C4⋊Q8C2314yes—
106C2×C4⋊Q8C2312yes—
107C2×C4⋊Q8C2312yes—
108C2×#127C2311yescharacteristic
109C2×#125—21no—
110C2×#125—21no—
111C2×#125C2311yescharacteristic
112C2×C22⋊Q8—24no—
113C2×C22⋊Q8—24no—
114C2×C22⋊Q8—22no—
115C2×C22⋊Q8—22no—
116C2×C22⋊Q8—22no—
117C2×C22⋊Q8—24no—
118C2×C4×Q8—22no—
119C2×C4×Q8D411yescharacteristic
120C2×C4×Q8D411yescharacteristic
121C2×C4×Q8C2311yescharacteristic
122C2×C4×D4—21no—
123C2×C4×D4—21no—
124C2×C4×D4—22no—
125C2×C4×D4—22no—
126C2×C4×D4C2312yes—
127C2×C4×D4C2314yes—
128#446—28no—
129C2×C4⋊C8C2312yes—
130C2×C4⋊C8C2314yes—
131C2×#102C2314yes—
132C2×#102C2312yes—
133C2×#102C2312yes—
134C2×#102C2312yes—
135C2×#102C2314yes—
136C2×#102C2312yes—
137C2×#97C2314yes—
138C2×#97C2312yes—
139C2×C4×C8C2312yes—
140C2×C4×C8C2314yes—
141#400—22no—
142#399—24no—
143#399—24no—
144#399—24no—
145#398—22no—
146#398—22no—
147#398—22no—
148#395—22no—
149#395—22no—
150#391—21no—
151#391—21no—
152#391—22no—
153#391—22no—
154#390—22no—
155#390—21no—
156#390—22no—
157#390—21no—
158#390C2314yes—
159#390C2312yes—
160#389—21no—
161#389—21no—
162#389—22no—
163#389—21no—
164#389—21no—
165#389—22no—
166#389C2311yescharacteristic
167#389C2311yescharacteristic
168#389C2312yes—
169#389C2312yes—
170#389C2312yes—
171#389C2312yes—
172#387—22no—
173#387—22no—
174#387—21no—
175#387—21no—
176#387—22no—
177#387—22no—
178#387—22no—
179#387—22no—
180#387—22no—
181#387—22no—
182#384—22no—
183#384—21no—
184#384—22no—
185#384—21no—
186#384C2311yescharacteristic
187#384C2312yes—
188#384C2312yes—
189#384C2311yescharacteristic
190#384C2311yescharacteristic
191#384C2314yes—
192#384C2312yes—
193#382—21no—
194#382C2311yescharacteristic
195#382C2312yes—
196#382C2312yes—
197C4×C4⋊C4—21no—
198C4×C4⋊C4—21no—
199C4×C4⋊C4—21no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4D411yescharacteristic
203C4×C4⋊C4D411yescharacteristic
204C4×C4⋊C4D412yes—
205C4×C4⋊C4D412yes—
206C4×C4⋊C4C2311yescharacteristic
207C4×C4⋊C4C2311yescharacteristic
208C4×C4⋊C4C2312yes—
209C4×C4⋊C4C2312yes—
210C4×C4⋊C4C2312yes—
211C4×C4⋊C4C2311yescharacteristic
212C43C2311yescharacteristic
213C43C2312yes—
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#340C2314yes—
218#340C2314yes—
219#340C2314yes—
220#340C2314yes—
221#340C2314yes—
222#340C2312yes—
223#340C2312yes—
224#340C2312yes—
225#340C2314yes—
226#340C2312yes—
227#340C2314yes—
228#336C2314yes—
229#336C2312yes—
230#336C2311yescharacteristic
231#336C2311yescharacteristic
232#336C2312yes—
233C22×Q8—24no—
234C22×Q8—22no—
235C22×Q8—22no—
236C22×Q8—48no—
237C22×Q8—48no—
238C22×Q8—48no—
239C22×Q8C2×D411yescharacteristic
240C22×D4—21no—
241C22×D4—21no—
242C22×D4C2412yes—
243C23×C4—24no—
244C23×C4—22no—
245C22×C8C2412yes—
246C22×C8C2412yes—
247C22×C8C2414yes—
248C4⋊Q8—24no—
249C4⋊Q8—24no—
250C4⋊Q8—24no—
251C4⋊Q8—28no—
252C4⋊Q8—28no—
253C4⋊Q8—42no—
254C4⋊Q8—44no—
255C4⋊Q8—44no—
256C4⋊Q8—48no—
257C4⋊Q8—48no—
258C4⋊Q8—44no—
259#127—24no—
260#125—22no—
261#125—41no—
262#125—44no—
263#125—82no—
264C22⋊Q8—416no—
265C22⋊Q8—416no—
266C22⋊Q8—416no—
267C22⋊Q8—48no—
268C22⋊Q8—48no—
269C22⋊Q8—48no—
270C4×Q8—24no—
271C4×Q8—24no—
272C4×Q8—84no—
273C4×Q8#4118yes—
274C4×D4—24no—
275C4×D4—24no—
276C4×D4—28no—
277C4×D4—28no—
278C4×D4—44no—
279C4×D4—44no—
280C4×D4—48no—
281C4×D4—48no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4C2×D412yes—
308C2×C4⋊C4C2×D412yes—
309C2×C4⋊C4C2×D411yescharacteristic
310C2×C4⋊C4C2×D412yes—
311C2×C4⋊C4C2×D411yescharacteristic
312C2×C4⋊C4C2×D411yescharacteristic
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2414yes—
315C2×C4⋊C4C2411yescharacteristic
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2411yescharacteristic
318C2×C4⋊C4C2411yescharacteristic
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2411yescharacteristic
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—22no—
327C2×C42—21no—
328C2×C42—21no—
329C2×C42—22no—
330C2×C42—22no—
331C2×C42C2×D411yescharacteristic
332C2×C42C2×D411yescharacteristic
333C2×C42C2×D412yes—
334C2×C42C2×D412yes—
335C2×C42C2×D412yes—
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D412yes—
338C2×C42C2×D411yescharacteristic
339C2×C42C2411yescharacteristic
340C2×C42C2411yescharacteristic
341C2×C42C2412yes—
342C2×C42C2412yes—
343C2×C42C2411yescharacteristic
344C4⋊C8—24no—
345C4⋊C8—28no—
346#102—28no—
347#102—24no—
348#102—24no—
349#102—24no—
350#102—42no—
351#102—44no—
352#97—28no—
353#97—42no—
354C4×C8—24no—
355C4×C8—44no—
356#95—21no—
357#95—22no—
358#95—22no—
359#95—22no—
360#95—21no—
361#95—21no—
362#95—21no—
363#95—21no—
364#95—21no—
365#95—22no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95C2×D412yes—
373#95C2×D412yes—
374#95C2×D412yes—
375C24—22no—
376C2×Q8—22no—
377C2×Q8—22no—
378C2×Q8—44no—
379C2×Q8—48no—
380C2×Q8—48no—
381C2×Q8—48no—
382C2×Q8—44no—
383C2×Q8—44no—
384C2×Q8—44no—
385C2×Q8—44no—
386C2×Q8—82no—
387C2×Q8—832no—
388C2×Q8#12414yes—
389C2×D4—24no—
390C2×D4—24no—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—22no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—42no—
397C2×D4—42no—
398C2×D4—42no—
399C2×D4—42no—
400C2×D42−1+414yes—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—48no—
410C22×C4—48no—
411C22×C4—48no—
412C22×C4—44no—
413C22×C4—44no—
414C22×C4—44no—
415C22×C4C22×D411yescharacteristic
416C22×C4C22×D411yescharacteristic
417C22×C4C22×D412yes—
418C22×C4C22×D412yes—
419C22×C4C22×D411yescharacteristic
420C22×C4C22×D412yes—
421C22×C4C22×D412yes—
422C22×C4C22×D411yescharacteristic
423C22×C4C22×D411yescharacteristic
424C22×C4C22×D411yescharacteristic
425C22×C4C2511yescharacteristic
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—24no—
429C2×C8—22no—
430C2×C8—24no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—24no—
435C4⋊C4—24no—
436C4⋊C4—24no—
437C4⋊C4—21no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—24no—
441C4⋊C4—24no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—24no—
445C4⋊C4—21no—
446C4⋊C4—22no—
447C4⋊C4—21no—
448C4⋊C4—22no—
449C4⋊C4—42no—
450C4⋊C4—42no—
451C4⋊C4—44no—
452C4⋊C4—42no—
453C4⋊C4—44no—
454C4⋊C4—42no—
455C4⋊C4—44no—
456C4⋊C4—44no—
457C4⋊C4—44no—
458C4⋊C4—44no—
459C4⋊C4—48no—
460C4⋊C4—48no—
461C4⋊C4—48no—
462C4⋊C4—44no—
463C4⋊C4—44no—
464C4⋊C4—42no—
465C4⋊C4—44no—
466C4⋊C4—42no—
467C4⋊C4—42no—
468C4⋊C4—44no—
469C4⋊C4—42no—
470C4⋊C4—82no—
471C4⋊C4—81no—
472C4⋊C4—81no—
473C4⋊C4—82no—
474C4⋊C4#12418yes—
475C4⋊C4#12414yes—
476C4⋊C4AGL(1,ℤ/8)14yes—
477C4⋊C4AGL(1,ℤ/8)14yes—
478C4⋊C42−1+412yes—
479C22⋊C4—44no—
480C22⋊C4—44no—
481C22⋊C4—44no—
482C22⋊C4—48no—
483C22⋊C4—48no—
484C22⋊C4—48no—
485C42—24no—
486C42—24no—
487C42—22no—
488C42—22no—
489C42—24no—
490C42—22no—
491C42—24no—
492C42—22no—
493C42—21no—
494C42—22no—
495C42—22no—
496C42—22no—
497C42—21no—
498C42—42no—
499C42—42no—
500C42—82no—
501C42—82no—
502C42C2×#4114yes—
503C42C2×#4118yes—
504C23—24no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23C23×D411yescharacteristic
512Q8—28no—
513Q8—88no—
514Q8—816no—
515Q8—88no—
516D4—24no—
517D4—24no—
518D4—48no—
519D4—48no—
520D4—42no—
521D4—42no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—22no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—44no—
546C2×C4—42no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—42no—
550C2×C4—41no—
551C2×C4—42no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—41no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—816no—
567C2×C4—88no—
568C2×C4C2×#12412yes—
569C2×C4C2×#12414yes—
570C2×C4C2×#12414yes—
571C2×C4C2×#12412yes—
572C2×C4C2×AGL(1,ℤ/8)14yes—
573C2×C4C2×AGL(1,ℤ/8)14yes—
574C2×C4C2×AGL(1,ℤ/8)12yes—
575C2×C4C2×AGL(1,ℤ/8)12yes—
576C2×C4#57712yes—
577C2×C4#57812yes—
578C2×C4C22×#4112yes—
579C2×C4C2×2−1+412yes—
580C2×C4C2×2−1+411yescharacteristic
581C2×C4#58511yescharacteristic
582C8—42no—
583C8—44no—
584C8—42no—
585C22—44no—
586C22—44no—
587C22—44no—
588C22—82no—
589C22—82no—
590C22—82no—
591C22—82no—
592C22C22×#12411yescharacteristic
593C22#348611yescharacteristic
594C22#348911yescharacteristic
595C22C22×AGL(1,ℤ/8)11yescharacteristic
596C22C22×AGL(1,ℤ/8)11yescharacteristic
597C22C2×#57711yescharacteristic
598C22C2×#57811yescharacteristic
599C4—22no—
600C4—22no—
601C4—24no—
602C4—24no—
603C4—24no—
604C4—24no—
605C4—22no—
606C4—41no—
607C4—42no—
608C4—41no—
609C4—81no—
610C4—81no—
611C4—82no—
612C4—82no—
613C4—81no—
614C4—84no—
615C4—82no—
616C4#313814yes—
617C4#351612yes—
618C4#352312yes—
619C2—84no—
620C2#3591011yescharacteristic
621C2C2×#313811yescharacteristic
622C2#6062311yescharacteristic
623C2#6183711yescharacteristic
624C2#6184311yescharacteristic
625C2#6212811yescharacteristic
626C2#6213211yescharacteristic
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. #36497C2semidirect
3. #30233C2semidirect
4. #30230C2semidirect
5. #29225C2semidirect
6. #29218C2semidirect
9. #27830C2semidirect
10. #27813C2semidirect
11. #27744C2semidirect
12. #27727C2semidirect
15. #22523C2semidirect
16. #22523C2semidirect
17. #20573C2semidirect

Maximal subgroups

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

subgroupindex
Q8×#1272
#364972
#302332
#302302
#292252
#292182
#291302
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