Finite Group Databasebeta

← #6852765

Subgroups — #6852765

Order 512.

1455 conjugacy classes of subgroups.

The subgroup lattice has 617 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×C4⋊Q8C211yesmaximal, characteristic
2#34950C211yesmaximal, characteristic
3#33196C211yesmaximal, characteristic
4#32132C214yesmaximal
5#30949C214yesmaximal
6#29983C212yesmaximal
7#29287C211yesmaximal, characteristic
8#29276C212yesmaximal
9#29206C211yesmaximal, characteristic
10#28809C212yesmaximal
11#28687C212yesmaximal
12#27815C212yesmaximal
13#26280C212yesmaximal
14#25060C212yesmaximal
15#22821C211yesmaximal, characteristic
16#22112C212yesmaximal
17C4×#475C211yesmaximal, characteristic
18C2×Q82—21no—
19C2×#498C2214yes—
20C2×#475C2212yes—
21C2×#475C2211yescharacteristic
22#2868C2212yes—
23#2868C2211yescharacteristic
24#2586—22no—
25#2582C2211yescharacteristic
26#2567—21no—
27#2567—22no—
28#2567C2211yescharacteristic
29#2567C2211yescharacteristic
30#2551—22no—
31#2551—21no—
32#2551C2214yes—
33#2551C2211yescharacteristic
34#2551C2212yes—
35#2507—21no—
36#2507C2212yes—
37#2451—21no—
38#2451C2212yes—
39#2399—22no—
40Q8×C4⋊C4—21no—
41Q8×C4⋊C4C2212yes—
42C4×C4⋊Q8—21no—
43C4×C4⋊Q8C2212yes—
44C4×C4⋊Q8C2211yescharacteristic
45C4×C4⋊Q8C2212yes—
46C4×C4⋊Q8C2211yescharacteristic
47#2295—21no—
48#2295C2212yes—
49C42×Q8C2211yescharacteristic
50C42×Q8C2211yescharacteristic
51#2100C2214yes—
52#2099C2214yes—
53#2096C2214yes—
54#2088C2214yes—
55#2084C2214yes—
56#2084C2214yes—
57#2058C2214yes—
58#2058C2214yes—
59#2058C2214yes—
60#2024C2214yes—
61#1990C2214yes—
62#1990C2212yes—
63#1988C2211yescharacteristic
64#1988C2212yes—
65#1988C2212yes—
66#1987C2212yes—
67#1984C2212yes—
68#1948C2211yescharacteristic
69#1948C2212yes—
70#1948C2212yes—
71#1948C2212yes—
72#1948C2212yes—
73#1948C2212yes—
74#1948C2212yes—
75#1941C2214yes—
76#1933C2214yes—
77#1927C2214yes—
78#1920C2214yes—
79#1920C2214yes—
80#1920C2214yes—
81#1919C2212yes—
82#1916C2214yes—
83#1865C2212yes—
84#1844C2212yes—
85#1842C2211yescharacteristic
86#1831C2212yes—
87#1803C2211yescharacteristic
88#1803C2212yes—
89#1803C2212yes—
90C4×#106C2212yes—
91C8×C4⋊C4C2212yes—
92C4×C4⋊C8C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#103C2212yes—
95Q82—48no—
96C2×C4⋊Q8—21no—
97C2×C4⋊Q8—24no—
98C2×C4⋊Q8—22no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—22no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—22no—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2312yes—
105C2×C4⋊Q8C2311yescharacteristic
106C2×C4⋊Q8C2312yes—
107C2×C4⋊Q8C2314yes—
108C2×C4×Q8—22no—
109C2×C4×Q8—21no—
110C2×C4×Q8—22no—
111C2×C4×Q8—22no—
112C2×C4×Q8—21no—
113C2×C4×Q8D411yescharacteristic
114C2×C4×Q8D411yescharacteristic
115C2×C4×Q8C2311yescharacteristic
116C2×C4×Q8C2311yescharacteristic
117C2×C4×Q8C2311yescharacteristic
118C2×C4×Q8C2311yescharacteristic
119#498—48no—
120#475—44no—
121#475Q818yescomplemented
122C2×#106C2314yes—
123C2×#106C2312yes—
124C2×#106C2314yes—
125C2×#106C2314yes—
126C2×#106C2314yes—
127C2×#106C2312yes—
128C2×C4⋊C8C2311yescharacteristic
129C2×C4⋊C8C2312yes—
130C2×C4⋊C8C2312yes—
131C2×C4⋊C8C2312yes—
132C2×C4⋊C8C2311yescharacteristic
133C2×#103C2314yes—
134C2×#103C2312yes—
135C2×#103C2312yes—
136C2×C4×C8C2312yes—
137C2×C4×C8C2314yes—
138#398—22no—
139#398—22no—
140#398—22no—
141#398—22no—
142#395—22no—
143#395—22no—
144#395—24no—
145#395—24no—
146#395—24no—
147#395—22no—
148#395—22no—
149#391—21no—
150#391—21no—
151#391—22no—
152#391—21no—
153#391—21no—
154#391—22no—
155#391—22no—
156#391C2312yes—
157#391C2312yes—
158#389—21no—
159#389—21no—
160#389—24no—
161#389—22no—
162#389—22no—
163#389—24no—
164#389—21no—
165#389—21no—
166#389—22no—
167#389—21no—
168#389—21no—
169#389—22no—
170#389—22no—
171#389C2312yes—
172#389C2312yes—
173#389C2314yes—
174#389C2312yes—
175#389C2312yes—
176#389C2312yes—
177#389C2312yes—
178#387—22no—
179#387—22no—
180#387—22no—
181#387—22no—
182#384—21no—
183#384—22no—
184#384—21no—
185#384—21no—
186#384—22no—
187#384C2312yes—
188#384C2312yes—
189#384C2311yescharacteristic
190#384C2311yescharacteristic
191#384C2312yes—
192#384C2314yes—
193C4×C4⋊C4—21no—
194C4×C4⋊C4—22no—
195C4×C4⋊C4—21no—
196C4×C4⋊C4—22no—
197C4×C4⋊C4—22no—
198C4×C4⋊C4—21no—
199C4×C4⋊C4—21no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4D411yescharacteristic
202C4×C4⋊C4D412yes—
203C4×C4⋊C4D412yes—
204C4×C4⋊C4D411yescharacteristic
205C4×C4⋊C4C2311yescharacteristic
206C4×C4⋊C4C2312yes—
207C4×C4⋊C4C2314yes—
208C4×C4⋊C4C2312yes—
209C4×C4⋊C4C2311yescharacteristic
210C4×C4⋊C4C2311yescharacteristic
211C4×C4⋊C4C2311yescharacteristic
212C4×C4⋊C4C2312yes—
213#376—22no—
214C43—21no—
215C43C2312yes—
216C43C2311yescharacteristic
217#340C2312yes—
218#340C2312yes—
219#340C2314yes—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2312yes—
225#340C2314yes—
226#340C2314yes—
227#340C2312yes—
228#336C2314yes—
229#336C2312yes—
230#336C2314yes—
231#336C2314yes—
232C22×Q8—21no—
233C22×Q8—22no—
234C22×Q8—21no—
235C22×Q8—22no—
236C22×Q8—24no—
237C22×Q8C2×D411yescharacteristic
238C22×Q8C2411yescharacteristic
239C22×C8C2412yes—
240C22×C8C2414yes—
241C22×C8C2412yes—
242C4⋊Q8—24no—
243C4⋊Q8—22no—
244C4⋊Q8—22no—
245C4⋊Q8—22no—
246C4⋊Q8—42no—
247C4⋊Q8—42no—
248C4⋊Q8—44no—
249C4⋊Q8—42no—
250C4⋊Q8—44no—
251C4⋊Q8—416no—
252C4⋊Q8—48no—
253C4⋊Q8—44no—
254C4⋊Q8—48no—
255C4⋊Q8—44no—
256C4⋊Q8—82no—
257C4⋊Q8—84no—
258C4⋊Q8C2×Q814yes—
259C4×Q8—24no—
260C4×Q8—22no—
261C4×Q8—22no—
262C4×Q8—22no—
263C4×Q8—22no—
264C4×Q8—22no—
265C4×Q8—22no—
266C4×Q8—41no—
267C4×Q8—48no—
268C4×Q8—48no—
269C4×Q8—44no—
270C4×Q8—84no—
271C4×Q8—82no—
272C4×Q8SD1618yes—
273C4×Q8C2×Q814yes—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4C2×D412yes—
302C2×C4⋊C4C2×D411yescharacteristic
303C2×C4⋊C4C2×D412yes—
304C2×C4⋊C4C2×D412yes—
305C2×C4⋊C4C2×D411yescharacteristic
306C2×C4⋊C4C2×D411yescharacteristic
307C2×C4⋊C4C2411yescharacteristic
308C2×C4⋊C4C2414yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2411yescharacteristic
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2411yescharacteristic
315C2×C42—21no—
316C2×C42—24no—
317C2×C42—22no—
318C2×C42—21no—
319C2×C42—22no—
320C2×C42—21no—
321C2×C42—21no—
322C2×C42—21no—
323C2×C42—21no—
324C2×C42C2×D411yescharacteristic
325C2×C42C2×D412yes—
326C2×C42C2×D411yescharacteristic
327C2×C42C2×D412yes—
328C2×C42C2×D411yescharacteristic
329C2×C42C2×D412yes—
330C2×C42C2×D411yescharacteristic
331C2×C42C2×D412yes—
332C2×C42C2411yescharacteristic
333C2×C42C2412yes—
334C2×C42C2412yes—
335C2×C42C2411yescharacteristic
336C2×C42C2411yescharacteristic
337#106—28no—
338#106—24no—
339#106—44no—
340#106—44no—
341#106—44no—
342#106C2×Q818yes—
343C4⋊C8—24no—
344C4⋊C8—22no—
345C4⋊C8—24no—
346C4⋊C8—42no—
347C4⋊C8C2×Q814yes—
348#103—42no—
349#103—44no—
350#103C2×Q818yes—
351C4×C8—24no—
352C4×C8—44no—
353#95—22no—
354#95—22no—
355#95—22no—
356#95—22no—
357#95—21no—
358#95—21no—
359#95—21no—
360#95—22no—
361#95—22no—
362#95—22no—
363#95—22no—
364#95—22no—
365#95—22no—
366#95—24no—
367#95—22no—
368#95—24no—
369#95—22no—
370#95C2×D412yes—
371#95C2×D412yes—
372#95C2×D412yes—
373C2×Q8—22no—
374C2×Q8—24no—
375C2×Q8—22no—
376C2×Q8—21no—
377C2×Q8—21no—
378C2×Q8—21no—
379C2×Q8—21no—
380C2×Q8—21no—
381C2×Q8—44no—
382C2×Q8—42no—
383C2×Q8—42no—
384C2×Q8—44no—
385C2×Q8—48no—
386C2×Q8—44no—
387C2×Q8—42no—
388C2×Q8—82no—
389C2×Q8—81no—
390C2×Q8—81no—
391C2×Q8—82no—
392C2×Q8—84no—
393C2×Q8—82no—
394C2×Q8—84no—
395C2×Q8C2×SD1614yes—
396C2×Q8C22×Q812yes—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4C22×D412yes—
407C22×C4C22×D411yescharacteristic
408C22×C4C22×D411yescharacteristic
409C22×C4C22×D411yescharacteristic
410C22×C4C22×D412yes—
411C22×C4C22×D411yescharacteristic
412C22×C4C22×D412yes—
413C22×C4C22×D412yes—
414C22×C4C22×D411yescharacteristic
415C22×C4C22×D411yescharacteristic
416C22×C4C2511yescharacteristic
417C2×C8—22no—
418C2×C8—22no—
419C2×C8—24no—
420C2×C8—22no—
421C2×C8—24no—
422C2×C8—24no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8C22×Q814yes—
426C4⋊C4—24no—
427C4⋊C4—22no—
428C4⋊C4—24no—
429C4⋊C4—24no—
430C4⋊C4—24no—
431C4⋊C4—21no—
432C4⋊C4—22no—
433C4⋊C4—21no—
434C4⋊C4—24no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—24no—
444C4⋊C4—42no—
445C4⋊C4—44no—
446C4⋊C4—44no—
447C4⋊C4—42no—
448C4⋊C4—42no—
449C4⋊C4—42no—
450C4⋊C4—44no—
451C4⋊C4—48no—
452C4⋊C4—48no—
453C4⋊C4—44no—
454C4⋊C4—42no—
455C4⋊C4—44no—
456C4⋊C4—44no—
457C4⋊C4—42no—
458C4⋊C4—82no—
459C4⋊C4—82no—
460C4⋊C4—82no—
461C4⋊C4—81no—
462C4⋊C4—82no—
463C4⋊C4—82no—
464C4⋊C4—82no—
465C4⋊C4—84no—
466C4⋊C4—82no—
467C4⋊C4—82no—
468C4⋊C4—84no—
469C4⋊C4C2×SD1618yes—
470C4⋊C4C2×SD1614yes—
471C4⋊C4#13714yes—
472C4⋊C4C22×Q814yes—
473C4⋊C4C22×Q812yes—
474C4⋊C4H(5,𝔽2)12yes—
475C42—24no—
476C42—24no—
477C42—22no—
478C42—24no—
479C42—22no—
480C42—22no—
481C42—21no—
482C42—22no—
483C42—22no—
484C42—21no—
485C42—22no—
486C42—41no—
487C42—42no—
488C42—42no—
489C42—42no—
490C42—48no—
491C42—44no—
492C42—42no—
493C42—42no—
494C42—44no—
495C42—81no—
496C42C2×SD1618yes—
497C42C2×SD1614yes—
498C42C22×Q812yes—
499C422−1+412yes—
500C422−1+412yes—
501C422−1+412yes—
502C23C23×D411yescharacteristic
503Q8—28no—
504Q8—22no—
505Q8—22no—
506Q8—48no—
507Q8—41no—
508Q8—41no—
509Q8—88no—
510Q8—84no—
511Q8—164no—
512Q8—168no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—21no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—21no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—21no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—42no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—42no—
543C2×C4—41no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—41no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—44no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—41no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4#53612yes—
559C2×C4#53612yes—
560C2×C4D4×Q812yes—
561C2×C4D4×Q812yes—
562C2×C4C22×SD1614yes—
563C2×C4C22×SD1612yes—
564C2×C4C22×SD1614yes—
565C2×C4C22×SD1612yes—
566C2×C4C22×SD1612yes—
567C2×C4C2×#13714yes—
568C2×C4C2×#13712yes—
569C2×C4#57712yes—
570C2×C4C23×Q811yescharacteristic
571C2×C4C2×H(5,𝔽2)11yescharacteristic
572C2×C4C2×2−1+411yescharacteristic
573C2×C4C2×2−1+411yescharacteristic
574C2×C4C2×2−1+411yescharacteristic
575C2×C4#58511yescharacteristic
576C8—24no—
577C8—44no—
578C8—42no—
579C22C2×#53611yescharacteristic
580C22C2×D4×Q811yescharacteristic
581C22#348811yescharacteristic
582C22C23×SD1611yescharacteristic
583C22C22×#13711yescharacteristic
584C22C2×#57711yescharacteristic
585C22#358511yescharacteristic
586C4—24no—
587C4—24no—
588C4—22no—
589C4—22no—
590C4—22no—
591C4—24no—
592C4—41no—
593C4—42no—
594C4—44no—
595C4—42no—
596C4—42no—
597C4—42no—
598C4—82no—
599C4—81no—
600C4—82no—
601C4—84no—
602C4—82no—
603C4#322914yes—
604C4Q8×SD1614yes—
605C4#350211yescharacteristic
606C4#350311yescharacteristic
607C4#355011yescharacteristic
608C4#355511yescharacteristic
609C2#3959711yescharacteristic
610C2C2×#322911yescharacteristic
611C2#6089211yescharacteristic
612C2C2×Q8×SD1611yescharacteristic
613C2Q8×#13711yescharacteristic
614C2#6195111yescharacteristic
615C2#6198011yescharacteristic
subgroupG/H
center, socle, prime radicalC23C23×D4
commutator, Frattini, cosocleC22×C4C25
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
121. #475Q8semidirect

Maximal subgroups

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

subgroupindex
Q8×C4⋊Q82
#349502
#331962
#321322
#309492
#299832
#292872
#292762
#292062
#288092
#286872
#278152
#262802
#250602
#228212
#221122
C4×#4752