Finite Group Databasebeta

← #7198560

Subgroups — #7198560

Order 512.

1960 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×SD16C212yesmaximal, complemented
2#59832C211yesmaximal, characteristic, complemented
3#59817C211yesmaximal, characteristic
4#30954C214yesmaximal, complemented
5#30949C214yesmaximal, complemented
6#28205C211yesmaximal, characteristic, complemented
7#28202C211yesmaximal, characteristic
8#28186C212yesmaximal, complemented
9#26902C214yesmaximal, complemented
10#26897C214yesmaximal
11#24330C212yesmaximal, complemented
12#24329C212yesmaximal, complemented
13#23799C211yesmaximal, characteristic
14#22246C211yesmaximal, characteristic, complemented
15#22238C211yesmaximal, characteristic, complemented
16C2×Q82C2212yes—
17C2×D4×Q8—21no—
18C2×D4×Q8C2212yes—
19Q8×SD16—216no—
20C2×#498C2214yes—
21C2×#498C2211yescharacteristic
22C2×#475C2212yes—
23C2×#475C2214yes—
24C2×#476C2212yes—
25C2×#476C2214yes—
26C2×C8×Q8C2212yes—
27C2×C4×SD16C2214yes—
28C2×C4×SD16C2212yes—
29C2×C4×SD16C2212yes—
30#2734—22no—
31#2683—22no—
32#2582C2212yes—
33#2551—21no—
34#2551C2212yes—
35#2551C2212yes—
36#2546—22no—
37#2538—21no—
38#2538C2212yes—
39#2452—22no—
40#2452C2214yes—
41#2451—22no—
42#2451C2214yes—
43#2409—22no—
44#2366C2212yes—
45#2365—21no—
46#2365C2212yes—
47C4×C4⋊Q8C2211yescharacteristic
48C4×C4⋊Q8C2211yescharacteristic
49C4×C4⋊Q8C2211yescharacteristic
50C4×C4⋊Q8C2211yescharacteristic
51C4×#127C2211yescharacteristic
52#2096C2212yes—
53#2094C2212yes—
54#2088C2214yes—
55#2084C2212yes—
56#2084C2212yes—
57#2058C2214yes—
58#2058C2214yes—
59#2055C2214yes—
60#2043C2212yes—
61#2041C2212yes—
62#2024C2214yes—
63#2023C2214yes—
64#2014C2212yes—
65#2013C2212yes—
66#1990C2211yescharacteristic
67#1990C2211yescharacteristic
68#1984C2212yes—
69#1934C2214yes—
70#1933C2214yes—
71#1927C2214yes—
72#1926C2214yes—
73#1920C2214yes—
74#1919C2214yes—
75#1916C2214yes—
76#1916C2214yes—
77#1880C2214yes—
78#1877C2214yes—
79#1865C2214yes—
80#1864C2214yes—
81#1834C2211yescharacteristic
82#1831C2211yescharacteristic
83#1803C2212yes—
84#1802C2212yes—
85#1744C2211yescharacteristic
86C23×Q8—22no—
87C22×SD16C2312yes—
88C22×SD16C2312yes—
89Q82—28no—
90Q82—44no—
91D4×Q8—28no—
92D4×Q8—44no—
93D4×Q8—48no—
94C2×C4⋊Q8—22no—
95C2×C4⋊Q8—22no—
96C2×C4⋊Q8—24no—
97C2×C4⋊Q8—22no—
98C2×C4⋊Q8D411yescharacteristic
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2314yes—
101C2×C4⋊Q8C2314yes—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2311yescharacteristic
104C2×#127C2311yescharacteristic
105C2×C22⋊Q8—22no—
106C2×C22⋊Q8—22no—
107C2×C22⋊Q8—24no—
108C2×C22⋊Q8—24no—
109C2×C4×Q8—22no—
110C2×C4×Q8—21no—
111C2×C4×Q8—21no—
112C2×C4×Q8C2312yes—
113C2×C4×Q8C2314yes—
114C2×C4×Q8C2312yes—
115C2×C4×D4—22no—
116C2×C4×D4—21no—
117C2×C4×D4—21no—
118C2×C4×D4C2312yes—
119C2×C4×D4C2314yes—
120#498—216no—
121#498D418yescomplemented
122#476—28no—
123#476—216no—
124#475—28no—
125#475—216no—
126C8×Q8—28no—
127C4×SD16—216no—
128C4×SD16—28no—
129C4×SD16—44no—
130C2×#106C2314yes—
131C2×#106C2314yes—
132C2×#106C2312yes—
133C2×#106C2312yes—
134C2×#106C2314yes—
135C2×C4⋊C8C2314yes—
136C2×C4⋊C8C2312yes—
137C2×#103C2312yes—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#103C2312yes—
141C2×#102C2314yes—
142C2×#102C2312yes—
143C2×#102C2314yes—
144C2×#102C2312yes—
145C2×C4×C8C2314yes—
146C2×C4×C8C2311yescharacteristic
147C2×C4×C8C2311yescharacteristic
148#398—24no—
149#398—24no—
150#396—24no—
151#396—24no—
152#395—24no—
153#395—24no—
154#395—24no—
155#393—24no—
156#391—22no—
157#391—21no—
158#391—21no—
159#391—24no—
160#391D412yes—
161#391C2312yes—
162#391C2314yes—
163#390—22no—
164#390—21no—
165#390—21no—
166#390C2312yes—
167#390C2314yes—
168#389—22no—
169#389—22no—
170#389—22no—
171#389—22no—
172#389—21no—
173#389—21no—
174#389—21no—
175#389—21no—
176#389D412yes—
177#389D412yes—
178#389C2312yes—
179#389C2312yes—
180#389C2314yes—
181#389C2314yes—
182#387—24no—
183#387—24no—
184#386—24no—
185#385—24no—
186#384D411yescharacteristic
187#384C2311yescharacteristic
188#384C2311yescharacteristic
189#384C2311yescharacteristic
190#384C2311yescharacteristic
191#380—22no—
192C4×C4⋊C4—22no—
193C4×C4⋊C4—21no—
194C4×C4⋊C4—21no—
195C4×C4⋊C4—21no—
196C4×C4⋊C4—21no—
197C4×C4⋊C4C2312yes—
198C4×C4⋊C4C2312yes—
199C4×C4⋊C4C2312yes—
200C4×C4⋊C4C2312yes—
201C43C2311yescharacteristic
202#340C2314yes—
203#340C2314yes—
204#340C2314yes—
205#340C2314yes—
206#340C2314yes—
207#336C2314yes—
208#336C2314yes—
209#336C2312yes—
210#336C2312yes—
211#336C2312yes—
212#336C2312yes—
213C22×Q8—21no—
214C22×Q8—21no—
215C22×Q8—24no—
216C22×Q8—24no—
217C22×Q8—48no—
218C22×Q8—48no—
219C22×Q8—84no—
220C22×Q8C2×D412yes—
221C22×Q8C2412yes—
222C22×D4—21no—
223C22×D4—21no—
224C22×D4C2412yes—
225C23×C4—22no—
226C23×C4—24no—
227C2×SD16—24no—
228C2×SD16—24no—
229C2×SD16—24no—
230C2×SD16—24no—
231C2×SD16—24no—
232C2×SD16—42no—
233C22×C8C2412yes—
234C22×C8C2414yes—
235C22×C8C2412yes—
236C4⋊Q8—24no—
237C4⋊Q8—28no—
238C4⋊Q8—28no—
239C4⋊Q8—22no—
240C4⋊Q8—22no—
241C4⋊Q8—44no—
242C4⋊Q8—44no—
243C4⋊Q8—48no—
244C4⋊Q8—416no—
245C4⋊Q8—48no—
246C4⋊Q8—84no—
247C4⋊Q8SD1618yescomplemented
248C4⋊Q8C2×D414yes—
249C4⋊Q8C2×D414yes—
250#127—24no—
251C22⋊Q8—416no—
252C22⋊Q8—48no—
253C22⋊Q8—48no—
254C22⋊Q8—416no—
255C4×Q8—24no—
256C4×Q8—24no—
257C4×Q8—28no—
258C4×Q8—24no—
259C4×Q8—24no—
260C4×Q8—44no—
261C4×Q8—48no—
262C4×Q8—44no—
263C4×Q8—82no—
264C4×D4—28no—
265C4×D4—24no—
266C4×D4—24no—
267C4×D4—44no—
268C4×D4—48no—
269C4×D4—44no—
270C4×D4—82no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4C2×D412yes—
292C2×C4⋊C4C2×D414yes—
293C2×C4⋊C4C2×D412yes—
294C2×C4⋊C4C2×D412yes—
295C2×C4⋊C4C2×D412yes—
296C2×C4⋊C4C2×D412yes—
297C2×C4⋊C4C2412yes—
298C2×C4⋊C4C2412yes—
299C2×C4⋊C4C2412yes—
300C2×C4⋊C4C2412yes—
301C2×C4⋊C4C2412yes—
302C2×C4⋊C4C2412yes—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—24no—
309C2×C42—22no—
310C2×C42—22no—
311C2×C42—24no—
312C2×C42—22no—
313C2×C42C2×D411yescharacteristic
314C2×C42C2×D411yescharacteristic
315C2×C42C2×D411yescharacteristic
316C2×C42C2×D411yescharacteristic
317C2×C42C2414yes—
318C2×C42C2411yescharacteristic
319C2×C42C2411yescharacteristic
320C2×C42C2411yescharacteristic
321#106—24no—
322#106—28no—
323#106—28no—
324#106—42no—
325#106C2×D4116yes—
326C4⋊C8—28no—
327C4⋊C8—24no—
328#103—28no—
329#103—24no—
330#103—28no—
331#103—42no—
332#102—28no—
333#102—24no—
334#102—28no—
335#102—42no—
336C4×C8—28no—
337C4×C8—22no—
338C4×C8C2×D414yes—
339#95—22no—
340#95—22no—
341#95—22no—
342#95—24no—
343#95—22no—
344#95—22no—
345#95—22no—
346#95—24no—
347#95—24no—
348#95—24no—
349#95C2×D414yes—
350#95C2×D414yes—
351C24—22no—
352C2×Q8—24no—
353C2×Q8—24no—
354C2×Q8—24no—
355C2×Q8—24no—
356C2×Q8—22no—
357C2×Q8—22no—
358C2×Q8—22no—
359C2×Q8—22no—
360C2×Q8—22no—
361C2×Q8—48no—
362C2×Q8—48no—
363C2×Q8—42no—
364C2×Q8—48no—
365C2×Q8—42no—
366C2×Q8—81no—
367C2×Q8—81no—
368C2×Q8—84no—
369C2×Q8—84no—
370C2×Q8—84no—
371C2×Q8—832no—
372C2×Q8C2×SD1618yes—
373C2×Q82−1+414yes—
374C2×D4—24no—
375C2×D4—24no—
376C2×D4—22no—
377C2×D4—22no—
378C2×D4—22no—
379C2×D4—22no—
380C2×D4—22no—
381C2×D4—42no—
382C2×D4—42no—
383C2×D4—81no—
384C2×D4—81no—
385C2×D42−1+414yes—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—48no—
394C22×C4—44no—
395C22×C4—48no—
396C22×C4—44no—
397C22×C4—44no—
398C22×C4—84no—
399C22×C4C22×D411yescharacteristic
400C22×C4C22×D411yescharacteristic
401C22×C4C22×D414yes—
402C22×C4C22×D411yescharacteristic
403C22×C4C22×D411yescharacteristic
404C22×C4C22×D414yes—
405C22×C4C22×D411yescharacteristic
406C22×C4C22×D411yescharacteristic
407C22×C4C2511yescharacteristic
408SD16—216no—
409SD16—48no—
410C2×C8—22no—
411C2×C8—22no—
412C2×C8—24no—
413C2×C8—22no—
414C2×C8—24no—
415C2×C8—24no—
416C2×C8—22no—
417C2×C8—22no—
418C2×C8C22×D414yes—
419C4⋊C4—24no—
420C4⋊C4—24no—
421C4⋊C4—24no—
422C4⋊C4—24no—
423C4⋊C4—28no—
424C4⋊C4—24no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—24no—
432C4⋊C4—24no—
433C4⋊C4—42no—
434C4⋊C4—42no—
435C4⋊C4—48no—
436C4⋊C4—48no—
437C4⋊C4—48no—
438C4⋊C4—42no—
439C4⋊C4—44no—
440C4⋊C4—42no—
441C4⋊C4—44no—
442C4⋊C4—42no—
443C4⋊C4—48no—
444C4⋊C4—42no—
445C4⋊C4—81no—
446C4⋊C4—81no—
447C4⋊C4—81no—
448C4⋊C4—81no—
449C4⋊C4—82no—
450C4⋊C4—84no—
451C4⋊C4C2×SD1618yes—
452C4⋊C4C2×SD1618yes—
453C4⋊C4C22×D414yes—
454C4⋊C4C22×D414yes—
455C22⋊C4—48no—
456C22⋊C4—44no—
457C22⋊C4—48no—
458C22⋊C4—44no—
459C22⋊C4—84no—
460C22⋊C4—82no—
461C42—22no—
462C42—24no—
463C42—24no—
464C42—22no—
465C42—22no—
466C42—41no—
467C42—41no—
468C42—48no—
469C42—44no—
470C42—44no—
471C42—44no—
472C42C2×SD1614yes—
473C42C22×D412yes—
474C42H(5,𝔽2)12yes—
475C23—24no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23C23×D411yescharacteristic
483Q8—216no—
484Q8—24no—
485Q8—24no—
486Q8—42no—
487Q8—48no—
488Q8—42no—
489Q8—84no—
490Q8—816no—
491Q8—168no—
492D4—24no—
493D4—24no—
494D4—48no—
495D4—42no—
496D4—42no—
497D4—84no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—24no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—24no—
506C2×C4—21no—
507C2×C4—24no—
508C2×C4—21no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—24no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—816no—
537C2×C4—88no—
538C2×C4D4212yes—
539C2×C4D4212yes—
540C2×C4C22×SD1612yes—
541C2×C4C22×SD1612yes—
542C2×C4C22×SD1618yes—
543C2×C4C22×SD1612yes—
544C2×C4#57712yes—
545C2×C4C23×D411yescharacteristic
546C2×C4C2×H(5,𝔽2)11yescharacteristic
547C2×C4C2×2−1+412yes—
548C2×C4C2×2−1+412yes—
549C8—24no—
550C8—44no—
551C8—42no—
552C22—44no—
553C22—44no—
554C22—44no—
555C22—82no—
556C22—82no—
557C22—82no—
558C22—82no—
559C22C2×D4211yescharacteristic
560C22#348611yescharacteristic
561C22#348611yescharacteristic
562C22C23×SD1611yescharacteristic
563C22C2×#57711yescharacteristic
564C22#358511yescharacteristic
565C22#358611yescharacteristic
566C4—28no—
567C4—22no—
568C4—22no—
569C4—22no—
570C4—44no—
571C4—44no—
572C4—44no—
573C4—44no—
574C4—41no—
575C4—41no—
576C4—82no—
577C4—82no—
578C4—82no—
579C4—82no—
580C4D4×SD1614yes—
581C4#348612yes—
582C4#353812yes—
583C2—84no—
584C2#4541811yescharacteristic
585C2C2×D4×SD1611yescharacteristic
586C2#6108011yescharacteristic
587C2#6181811yescharacteristic
588C2#6182011yescharacteristic
589C2#6185011yescharacteristic
590C2#6185311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#127C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×Q8×SD16C2semidirect
2. #59832C2semidirect
4. #30954C2semidirect
5. #30949C2semidirect
6. #28205C2semidirect
8. #28186C2semidirect
9. #26902C2semidirect
11. #24330C2semidirect
12. #24329C2semidirect
14. #22246C2semidirect
15. #22238C2semidirect
121. #498D4semidirect
247. C4⋊Q8SD16semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×SD162
#598322
#598172
#309542
#309492
#282052
#282022
#281862
#269022
#268972
#243302
#243292
#237992
#222462
#222382