Finite Group Databasebeta

← #6854496

Subgroups — #6854496

Order 512.

1488 conjugacy classes of subgroups.

The subgroup lattice has 633 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×C4⋊Q8C211yesmaximal, characteristic, complemented
2#36555C211yesmaximal, characteristic
3#33294C211yesmaximal, characteristic, complemented
4#31536C214yesmaximal, complemented
5#31534C214yesmaximal
6#30309C212yesmaximal, complemented
7#29196C212yesmaximal, complemented
8#29150C211yesmaximal, characteristic, complemented
9#29148C212yesmaximal
10#28519C211yesmaximal, characteristic
11#24817C212yesmaximal, complemented
12#24815C212yesmaximal
13#24333C212yesmaximal, complemented
14#24329C212yesmaximal
15#22273C212yesmaximal
16C4×#497C211yesmaximal, characteristic
17#21345C211yesmaximal, characteristic, complemented
18C2×Q82C2212yes—
19C2×#497C2211yescharacteristic
20C2×#477C2212yes—
21C2×#476C2212yes—
22C2×C4×Q16C2212yes—
23C2×C4×SD16C2212yes—
24#2868C2211yescharacteristic
25#2868C2212yes—
26#2864C2212yes—
27#2859C2211yescharacteristic
28#2587—22no—
29#2582C2211yescharacteristic
30#2567C2214yes—
31#2567C2212yes—
32#2551—22no—
33#2551—21no—
34#2551—21no—
35#2551—22no—
36#2538C2211yescharacteristic
37#2522—22no—
38#2497—21no—
39#2454C2212yes—
40#2453C2212yes—
41#2423—21no—
42#2360—21no—
43Q8×C4⋊C4—21no—
44Q8×C4⋊C4—21no—
45Q8×C4⋊C4C2211yescharacteristic
46D4×C4⋊C4C2211yescharacteristic
47#2345—21no—
48C4×C4⋊Q8C2212yes—
49C4×C4⋊Q8C2211yescharacteristic
50C4×#124C2212yes—
51C4×#124C2211yescharacteristic
52C42×Q8C2211yescharacteristic
53#2096C2212yes—
54#2093C2212yes—
55#2084C2214yes—
56#2083C2214yes—
57#2068C2214yes—
58#2066C2214yes—
59#2058C2214yes—
60#2057C2214yes—
61#2026C2214yes—
62#2024C2214yes—
63#2023C2214yes—
64#1992C2214yes—
65#1990C2212yes—
66#1990C2212yes—
67#1984C2212yes—
68#1973C2212yes—
69#1971C2212yes—
70#1960C2212yes—
71#1960C2211yescharacteristic
72#1959C2212yes—
73#1959C2211yescharacteristic
74#1939C2212yes—
75#1917C2214yes—
76#1880C2214yes—
77#1878C2214yes—
78#1878C2214yes—
79#1877C2214yes—
80#1865C2214yes—
81#1863C2214yes—
82#1847C2212yes—
83#1847C2211yescharacteristic
84#1836C2212yes—
85#1836C2212yes—
86#1836C2212yes—
87#1803C2212yes—
88#1803C2212yes—
89#1803C2214yes—
90#1802C2214yes—
91#1767C2212yes—
92#1767C2211yescharacteristic
93#1745C2212yes—
94C4×#97C2211yescharacteristic
95C22×Q16C2312yes—
96C22×SD16C2312yes—
97Q82—44no—
98Q82—44no—
99C2×C4⋊Q8—22no—
100C2×C4⋊Q8—24no—
101C2×C4⋊Q8—22no—
102C2×C4⋊Q8—22no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8—21no—
105C2×C4⋊Q8C2311yescharacteristic
106C2×C4⋊Q8C2312yes—
107C2×C4⋊Q8C2314yes—
108C2×#125C2312yes—
109C2×#124C2311yescharacteristic
110C2×C22⋊Q8—21no—
111C2×C4×Q8—22no—
112C2×C4×Q8D411yescharacteristic
113C2×C4×Q8D411yescharacteristic
114C2×C4×Q8C2311yescharacteristic
115C2×C4×Q8C2312yes—
116C2×C4×Q8C2314yes—
117C2×C4×Q8C2312yes—
118C2×C4×Q8C2311yescharacteristic
119C2×C4×D4C2311yescharacteristic
120C2×C4×D4C2312yes—
121C22×C4⋊C4—21no—
122#497Q818yescomplemented
123#477—44no—
124#476—44no—
125C4×Q16—44no—
126C4×SD16—44no—
127C2×#107C2314yes—
128C2×#107C2312yes—
129C2×#106C2312yes—
130C2×#106C2314yes—
131C2×C4⋊C8C2312yes—
132C2×C4⋊C8C2314yes—
133C2×#103C2314yes—
134C2×#103C2312yes—
135C2×#103C2314yes—
136C2×#103C2314yes—
137C2×#103C2312yes—
138C2×#103C2312yes—
139C2×#102C2312yes—
140C2×#102C2314yes—
141C2×#97C2311yescharacteristic
142C2×#97C2312yes—
143C2×#97C2312yes—
144C2×#97C2311yescharacteristic
145C2×C4×C8C2312yes—
146#400—22no—
147#399—22no—
148#398—22no—
149#398—22no—
150#396—22no—
151#396—22no—
152#395—24no—
153#395—24no—
154#395—24no—
155#391—21no—
156#391C2314yes—
157#391C2312yes—
158#391C2312yes—
159#391C2311yescharacteristic
160#390C2311yescharacteristic
161#390C2312yes—
162#389—22no—
163#389—24no—
164#389—24no—
165#389—22no—
166#389—21no—
167#389—22no—
168#389—22no—
169#389—22no—
170#389—21no—
171#389—21no—
172#389—21no—
173#389—21no—
174#389—21no—
175#389—21no—
176#389—22no—
177#389C2312yes—
178#389C2312yes—
179#389C2312yes—
180#389C2312yes—
181#388—21no—
182#388—22no—
183#388—22no—
184#388—21no—
185#387—22no—
186#387—22no—
187#387—22no—
188#387—22no—
189#387—22no—
190#385—22no—
191#385—22no—
192#384—21no—
193#384—22no—
194#384—21no—
195#384—21no—
196#384—22no—
197#384C2312yes—
198#384C2311yescharacteristic
199#384C2314yes—
200#384C2312yes—
201#384C2311yescharacteristic
202#384C2311yescharacteristic
203#382C2312yes—
204#382C2311yescharacteristic
205#380—21no—
206#380—21no—
207#380—22no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4D412yes—
214C4×C4⋊C4D411yescharacteristic
215C4×C4⋊C4D411yescharacteristic
216C4×C4⋊C4D412yes—
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2312yes—
219C4×C22⋊C4—22no—
220C4×C22⋊C4—21no—
221C4×C22⋊C4—21no—
222C4×C22⋊C4—22no—
223C43C2311yescharacteristic
224C43C2312yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#336C2314yes—
232#336C2314yes—
233#336C2312yes—
234#336C2314yes—
235C22×Q8—24no—
236C22×Q8—22no—
237C22×Q8C2×D411yescharacteristic
238C22×Q8C2411yescharacteristic
239C22×Q8C2412yes—
240C22×D4C2411yescharacteristic
241C23×C4—21no—
242C23×C4—22no—
243C2×Q16—42no—
244C2×Q16—42no—
245C2×Q16C2×Q818yes—
246C2×SD16—42no—
247C2×SD16—42no—
248C2×SD16C2×Q818yes—
249C22×C8C2414yes—
250C22×C8C2412yes—
251C22×C8C2412yes—
252C4⋊Q8—24no—
253C4⋊Q8—24no—
254C4⋊Q8—22no—
255C4⋊Q8—44no—
256C4⋊Q8—44no—
257C4⋊Q8—416no—
258C4⋊Q8—48no—
259C4⋊Q8—84no—
260C4⋊Q8—82no—
261C4⋊Q8—82no—
262C4⋊Q8—84no—
263C4⋊Q8C2×Q814yes—
264#125—44no—
265#124—22no—
266#124C2×Q814yes—
267C22⋊Q8—44no—
268C4×Q8—24no—
269C4×Q8—24no—
270C4×Q8—24no—
271C4×Q8—24no—
272C4×Q8—24no—
273C4×Q8—24no—
274C4×Q8—44no—
275C4×Q8—44no—
276C4×Q8—44no—
277C4×Q8—48no—
278C4×D4—24no—
279C4×D4—44no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—44no—
307C2×C4⋊C4C2×D411yescharacteristic
308C2×C4⋊C4C2×D412yes—
309C2×C4⋊C4C2×D412yes—
310C2×C4⋊C4C2×D411yescharacteristic
311C2×C4⋊C4C2×D412yes—
312C2×C4⋊C4C2×D411yescharacteristic
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2411yescharacteristic
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2414yes—
318C2×C4⋊C4C2411yescharacteristic
319C2×C22⋊C4—21no—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—21no—
326C2×C22⋊C4—21no—
327C2×C42—24no—
328C2×C42—22no—
329C2×C42—21no—
330C2×C42—22no—
331C2×C42C2×D412yes—
332C2×C42C2×D411yescharacteristic
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D412yes—
335C2×C42C2×D412yes—
336C2×C42C2×D412yes—
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D411yescharacteristic
339C2×C42C2412yes—
340C2×C42C2412yes—
341C2×C42C2411yescharacteristic
342C2×C42C2411yescharacteristic
343C2×C42C2411yescharacteristic
344#107—42no—
345#107—44no—
346#106—42no—
347#106—44no—
348C4⋊C8—24no—
349C4⋊C8—44no—
350#103—44no—
351#103—44no—
352#103—42no—
353#103—42no—
354#103—42no—
355#103—44no—
356#102—42no—
357#102—44no—
358#97—22no—
359#97—24no—
360#97—42no—
361#97C2×Q814yes—
362C4×C8—24no—
363#95—21no—
364#95—21no—
365#95—22no—
366#95—22no—
367#95—21no—
368#95—22no—
369#95—21no—
370#95—22no—
371#95—22no—
372#95—24no—
373#95—22no—
374#95—22no—
375#95—24no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95C2×D412yes—
380#95C2×D412yes—
381#95C2×D412yes—
382C24—21no—
383C2×Q8—22no—
384C2×Q8—22no—
385C2×Q8—21no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—48no—
394C2×Q8—44no—
395C2×Q8—82no—
396C2×Q8—82no—
397C2×Q8—84no—
398C2×Q8—84no—
399C2×Q8#13714yes—
400C2×Q8C22×Q812yes—
401C2×Q8C22×Q814yes—
402C2×D4—22no—
403C2×D4—21no—
404C2×D4—22no—
405C2×D4C22×Q812yes—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—44no—
418C22×C4—82no—
419C22×C4—82no—
420C22×C4C22×D411yescharacteristic
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D412yes—
423C22×C4C22×D412yes—
424C22×C4C22×D411yescharacteristic
425C22×C4C22×D411yescharacteristic
426C22×C4C22×D411yescharacteristic
427C22×C4C22×D412yes—
428C22×C4C22×D412yes—
429C22×C4C22×D411yescharacteristic
430C22×C4C2511yescharacteristic
431Q16—48no—
432SD16—48no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—24no—
437C2×C8—24no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—24no—
441C2×C8C22×Q814yes—
442C4⋊C4—24no—
443C4⋊C4—22no—
444C4⋊C4—24no—
445C4⋊C4—24no—
446C4⋊C4—24no—
447C4⋊C4—24no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—24no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—24no—
455C4⋊C4—24no—
456C4⋊C4—24no—
457C4⋊C4—48no—
458C4⋊C4—48no—
459C4⋊C4—44no—
460C4⋊C4—42no—
461C4⋊C4—44no—
462C4⋊C4—42no—
463C4⋊C4—42no—
464C4⋊C4—44no—
465C4⋊C4—44no—
466C4⋊C4—44no—
467C4⋊C4—42no—
468C4⋊C4—42no—
469C4⋊C4—44no—
470C4⋊C4—81no—
471C4⋊C4—81no—
472C4⋊C4—82no—
473C4⋊C4—82no—
474C4⋊C4—82no—
475C4⋊C4—82no—
476C4⋊C4—82no—
477C4⋊C4—84no—
478C4⋊C4—82no—
479C4⋊C4—82no—
480C4⋊C4—84no—
481C4⋊C4#13714yes—
482C22⋊C4—24no—
483C22⋊C4—24no—
484C22⋊C4—42no—
485C22⋊C4—44no—
486C22⋊C4—81no—
487C22⋊C4—81no—
488C22⋊C4—82no—
489C22⋊C4—82no—
490C42—24no—
491C42—24no—
492C42—22no—
493C42—22no—
494C42—24no—
495C42—22no—
496C42—24no—
497C42—42no—
498C42—41no—
499C42—41no—
500C42—42no—
501C42—48no—
502C42—42no—
503C42—44no—
504C42—82no—
505C42C22×Q812yes—
506C42H(5,𝔽2)14yes—
507C422−1+412yes—
508C23—22no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23C23×D411yescharacteristic
516Q8—28no—
517Q8—24no—
518Q8—24no—
519Q8—24no—
520Q8—42no—
521Q8—42no—
522Q8—42no—
523Q8—164no—
524Q8—168no—
525D4—24no—
526D4—42no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—22no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—22no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—42no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—42no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—84no—
574C2×C4—164no—
575C2×C4#53512yes—
576C2×C4#53512yes—
577C2×C4D4×Q812yes—
578C2×C4D4×Q812yes—
579C2×C4C2×#13714yes—
580C2×C4C2×#13714yes—
581C2×C4C2×#13712yes—
582C2×C4C2×#13712yes—
583C2×C4C23×Q811yescharacteristic
584C2×C4C2×H(5,𝔽2)12yes—
585C2×C4C2×2−1+411yescharacteristic
586C2×C4#58512yes—
587C8—24no—
588C8—42no—
589C8—44no—
590C22—42no—
591C22—42no—
592C22—42no—
593C22—81no—
594C22—81no—
595C22—81no—
596C22—81no—
597C22C2×#53511yescharacteristic
598C22C2×D4×Q811yescharacteristic
599C22#350011yescharacteristic
600C22C22×#13711yescharacteristic
601C22C22×#13711yescharacteristic
602C22#358511yescharacteristic
603C22#358511yescharacteristic
604C4—24no—
605C4—24no—
606C4—22no—
607C4—22no—
608C4—22no—
609C4—41no—
610C4—41no—
611C4—42no—
612C4—42no—
613C4—42no—
614C4—44no—
615C4—81no—
616C4—82no—
617C4—82no—
618C4—81no—
619C4—84no—
620C4—82no—
621C4#321814yes—
622C4#350112yes—
623C4#354712yes—
624C2—82no—
625C2#3961811yescharacteristic
626C2C2×#321811yescharacteristic
627C2#6083011yescharacteristic
628C2Q8×#13711yescharacteristic
629C2Q8×#13711yescharacteristic
630C2#6258211yescharacteristic
631C2#6259411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C24
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
1. Q8×C4⋊Q8C2semidirect
3. #33294C2semidirect
4. #31536C2semidirect
6. #30309C2semidirect
7. #29196C2semidirect
8. #29150C2semidirect
11. #24817C2semidirect
13. #24333C2semidirect
17. #21345C2semidirect
122. #497Q8semidirect

Maximal subgroups

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

subgroupindex
Q8×C4⋊Q82
#365552
#332942
#315362
#315342
#303092
#291962
#291502
#291482
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