Finite Group Databasebeta

← #7198573

Subgroups — #7198573

Order 512.

1918 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3386C212yesmaximal, complemented
2#59828C211yesmaximal, characteristic, complemented
3#59817C211yesmaximal, characteristic
4#30975C214yesmaximal, complemented
5#30973C214yesmaximal, complemented
6#28205C211yesmaximal, characteristic, complemented
7#28201C211yesmaximal, characteristic
8#28195C212yesmaximal, complemented
9#26919C214yesmaximal, complemented
10#26913C214yesmaximal
11#24330C212yesmaximal, complemented
12#24328C212yesmaximal, complemented
13#23799C211yesmaximal, characteristic
14#22247C211yesmaximal, characteristic, complemented
15#22242C211yesmaximal, characteristic, complemented
16C2×#557C2212yes—
17C2×D4×Q8—21no—
18C2×D4×Q8C2212yes—
19#3386—216no—
20C2×#501C2214yes—
21C2×#500C2211yescharacteristic
22C2×#477C2214yes—
23C2×#475C2212yes—
24C2×#478C2214yes—
25C2×#474C2212yes—
26C2×#446C2212yes—
27C2×#440C2214yes—
28C2×C4×SD16C2212yes—
29C2×C4×SD16C2212yes—
30#2726—22no—
31#2683—22no—
32#2582C2212yes—
33#2552C2212yes—
34#2551C2212yes—
35#2546—21no—
36#2538—21no—
37#2538C2212yes—
38#2506—22no—
39#2453C2214yes—
40#2452—22no—
41#2452C2214yes—
42#2446—22no—
43#2409—22no—
44#2366C2212yes—
45#2365C2212yes—
46Q8×C4⋊C4—21no—
47C4×C4⋊Q8C2211yescharacteristic
48C4×C4⋊Q8C2211yescharacteristic
49C4×C4⋊Q8C2211yescharacteristic
50C4×C4⋊Q8C2211yescharacteristic
51C4×#127C2211yescharacteristic
52#2096C2212yes—
53#2094C2212yes—
54#2088C2214yes—
55#2084C2212yes—
56#2083C2212yes—
57#2059C2214yes—
58#2058C2214yes—
59#2055C2214yes—
60#2043C2212yes—
61#2041C2212yes—
62#2025C2214yes—
63#2024C2214yes—
64#2014C2212yes—
65#2013C2212yes—
66#1990C2211yescharacteristic
67#1990C2211yescharacteristic
68#1984C2212yes—
69#1936C2214yes—
70#1935C2214yes—
71#1927C2214yes—
72#1925C2214yes—
73#1918C2214yes—
74#1918C2214yes—
75#1917C2214yes—
76#1917C2214yes—
77#1880C2214yes—
78#1877C2214yes—
79#1864C2214yes—
80#1863C2214yes—
81#1834C2211yescharacteristic
82#1831C2211yescharacteristic
83#1803C2212yes—
84#1802C2212yes—
85#1744C2211yescharacteristic
86C23×Q8—22no—
87C22×SD16C2312yes—
88C22×SD16C2312yes—
89#557—28no—
90#557—44no—
91D4×Q8—28no—
92D4×Q8—44no—
93D4×Q8—48no—
94C2×C4⋊Q8—22no—
95C2×C4⋊Q8—22no—
96C2×C4⋊Q8D411yescharacteristic
97C2×C4⋊Q8C2311yescharacteristic
98C2×C4⋊Q8C2314yes—
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2311yescharacteristic
101C2×#127C2311yescharacteristic
102C2×#125—24no—
103C2×#125C2314yes—
104C2×C22⋊Q8—22no—
105C2×C22⋊Q8—22no—
106C2×C22⋊Q8—24no—
107C2×C22⋊Q8—24no—
108C2×C4×Q8—22no—
109C2×C4×Q8—21no—
110C2×C4×Q8—21no—
111C2×C4×Q8C2312yes—
112C2×C4×Q8C2314yes—
113C2×C4×Q8C2312yes—
114C2×C4×D4—22no—
115C2×C4×D4—21no—
116C2×C4×D4—21no—
117C2×C4×D4C2312yes—
118C2×C4×D4C2314yes—
119#501—216no—
120#500D418yescomplemented
121#478—216no—
122#477—216no—
123#475—28no—
124#474—28no—
125#446—28no—
126#440—216no—
127C4×SD16—28no—
128C4×SD16—44no—
129C2×#107C2314yes—
130C2×#107C2314yes—
131C2×#106C2314yes—
132C2×#106C2312yes—
133C2×#106C2312yes—
134C2×C4⋊C8C2314yes—
135C2×C4⋊C8C2312yes—
136C2×#103C2314yes—
137C2×#103C2312yes—
138C2×#103C2314yes—
139C2×#103C2312yes—
140C2×#102C2312yes—
141C2×#102C2314yes—
142C2×#102C2314yes—
143C2×#102C2312yes—
144C2×#97C2314yes—
145C2×C4×C8C2311yescharacteristic
146C2×C4×C8C2311yescharacteristic
147#400—24no—
148#398—24no—
149#398—24no—
150#398—24no—
151#396—24no—
152#396—24no—
153#395—24no—
154#393—24no—
155#391—22no—
156#391—21no—
157#391—21no—
158#391D412yes—
159#391C2312yes—
160#391C2314yes—
161#390—22no—
162#390—21no—
163#390—21no—
164#390C2312yes—
165#390C2314yes—
166#389—22no—
167#389—24no—
168#389—22no—
169#389—22no—
170#389—22no—
171#389—21no—
172#389—21no—
173#389—21no—
174#389—21no—
175#389D412yes—
176#389D412yes—
177#389C2312yes—
178#389C2312yes—
179#389C2314yes—
180#389C2314yes—
181#387—24no—
182#387—24no—
183#386—24no—
184#385—24no—
185#384—22no—
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—48no—
217C22×Q8—48no—
218C22×Q8—84no—
219C22×Q8C2×D412yes—
220C22×Q8C2412yes—
221C22×D4—21no—
222C22×D4—21no—
223C22×D4C2412yes—
224C23×C4—22no—
225C23×C4—24no—
226C2×SD16—24no—
227C2×SD16—24no—
228C2×SD16—24no—
229C2×SD16—24no—
230C2×SD16—24no—
231C2×SD16—42no—
232C22×C8C2412yes—
233C22×C8C2414yes—
234C22×C8C2412yes—
235C4⋊Q8—24no—
236C4⋊Q8—28no—
237C4⋊Q8—24no—
238C4⋊Q8—22no—
239C4⋊Q8—22no—
240C4⋊Q8—44no—
241C4⋊Q8—48no—
242C4⋊Q8—48no—
243C4⋊Q8C2×D414yes—
244C4⋊Q8C2×D414yes—
245#127—24no—
246#125—28no—
247#125—44no—
248#125—416no—
249C22⋊Q8—416no—
250C22⋊Q8—48no—
251C22⋊Q8—416no—
252C22⋊Q8—48no—
253C4×Q8—24no—
254C4×Q8—24no—
255C4×Q8—28no—
256C4×Q8—24no—
257C4×Q8—24no—
258C4×Q8—44no—
259C4×Q8—48no—
260C4×Q8—44no—
261C4×Q8—44no—
262C4×D4—28no—
263C4×D4—24no—
264C4×D4—24no—
265C4×D4—44no—
266C4×D4—48no—
267C4×D4—44no—
268C4×D4—82no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—21no—
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—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4C2×D412yes—
291C2×C4⋊C4C2×D412yes—
292C2×C4⋊C4C2×D414yes—
293C2×C4⋊C4C2×D412yes—
294C2×C4⋊C4C2×D412yes—
295C2×C4⋊C4C2×D412yes—
296C2×C4⋊C4C2412yes—
297C2×C4⋊C4C2412yes—
298C2×C4⋊C4C2412yes—
299C2×C4⋊C4C2412yes—
300C2×C4⋊C4C2412yes—
301C2×C4⋊C4C2412yes—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—24no—
308C2×C42—22no—
309C2×C42—24no—
310C2×C42—22no—
311C2×C42—22no—
312C2×C42C2×D411yescharacteristic
313C2×C42C2×D411yescharacteristic
314C2×C42C2×D411yescharacteristic
315C2×C42C2×D411yescharacteristic
316C2×C42C2414yes—
317C2×C42C2411yescharacteristic
318C2×C42C2411yescharacteristic
319C2×C42C2411yescharacteristic
320#107—28no—
321#107C2×D4116yes—
322#106—24no—
323#106—28no—
324#106—42no—
325C4⋊C8—28no—
326C4⋊C8—24no—
327#103—28no—
328#103—24no—
329#103—28no—
330#103—42no—
331#102—28no—
332#102—24no—
333#102—28no—
334#102—42no—
335#97—28no—
336C4×C8—22no—
337C4×C8C2×D414yes—
338#95—22no—
339#95—22no—
340#95—22no—
341#95—24no—
342#95—22no—
343#95—22no—
344#95—22no—
345#95—24no—
346#95—24no—
347#95—24no—
348#95C2×D414yes—
349#95C2×D414yes—
350C24—22no—
351C2×Q8—24no—
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—42no—
365C2×Q8—42no—
366C2×Q8—42no—
367C2×Q8—84no—
368C2×Q8—832no—
369C2×Q82−1+414yes—
370C2×D4—24no—
371C2×D4—24no—
372C2×D4—22no—
373C2×D4—22no—
374C2×D4—22no—
375C2×D4—22no—
376C2×D4—22no—
377C2×D4—42no—
378C2×D4—42no—
379C2×D4—81no—
380C2×D4—81no—
381C2×D42−1+414yes—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—48no—
390C22×C4—48no—
391C22×C4—44no—
392C22×C4—44no—
393C22×C4—44no—
394C22×C4—84no—
395C22×C4C22×D411yescharacteristic
396C22×C4C22×D411yescharacteristic
397C22×C4C22×D414yes—
398C22×C4C22×D414yes—
399C22×C4C22×D411yescharacteristic
400C22×C4C22×D411yescharacteristic
401C22×C4C22×D411yescharacteristic
402C22×C4C22×D411yescharacteristic
403C22×C4C2511yescharacteristic
404SD16—216no—
405SD16—48no—
406C2×C8—22no—
407C2×C8—22no—
408C2×C8—24no—
409C2×C8—22no—
410C2×C8—24no—
411C2×C8—24no—
412C2×C8—22no—
413C2×C8—22no—
414C2×C8C22×D414yes—
415C4⋊C4—24no—
416C4⋊C4—24no—
417C4⋊C4—24no—
418C4⋊C4—24no—
419C4⋊C4—24no—
420C4⋊C4—28no—
421C4⋊C4—24no—
422C4⋊C4—24no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—24no—
430C4⋊C4—24no—
431C4⋊C4—42no—
432C4⋊C4—42no—
433C4⋊C4—48no—
434C4⋊C4—48no—
435C4⋊C4—48no—
436C4⋊C4—48no—
437C4⋊C4—42no—
438C4⋊C4—48no—
439C4⋊C4—44no—
440C4⋊C4—42no—
441C4⋊C4—44no—
442C4⋊C4—42no—
443C4⋊C4—44no—
444C4⋊C4—42no—
445C4⋊C4—81no—
446C4⋊C4—81no—
447C4⋊C4—81no—
448C4⋊C4—84no—
449C4⋊C4—81no—
450C4⋊C4C22×D414yes—
451C4⋊C4C22×D414yes—
452C22⋊C4—48no—
453C22⋊C4—48no—
454C22⋊C4—44no—
455C22⋊C4—44no—
456C22⋊C4—84no—
457C22⋊C4—82no—
458C42—22no—
459C42—24no—
460C42—24no—
461C42—22no—
462C42—22no—
463C42—41no—
464C42—41no—
465C42—48no—
466C42—44no—
467C42—44no—
468C42—44no—
469C42AGL(1,ℤ/8)14yes—
470C42C22×D412yes—
471C42H(5,𝔽2)12yes—
472C23—24no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23C23×D411yescharacteristic
480Q8—24no—
481Q8—24no—
482Q8—42no—
483Q8—48no—
484Q8—48no—
485Q8—48no—
486Q8—42no—
487Q8—816no—
488D4—24no—
489D4—24no—
490D4—48no—
491D4—42no—
492D4—42no—
493D4—84no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—24no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
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—816no—
534C2×C4—88no—
535C2×C4D4212yes—
536C2×C4D4212yes—
537C2×C4C2×AGL(1,ℤ/8)12yes—
538C2×C4C2×AGL(1,ℤ/8)12yes—
539C2×C4C2×AGL(1,ℤ/8)12yes—
540C2×C4#57712yes—
541C2×C4C23×D411yescharacteristic
542C2×C4C2×H(5,𝔽2)11yescharacteristic
543C2×C4C2×2−1+412yes—
544C2×C4C2×2−1+412yes—
545C8—24no—
546C8—44no—
547C8—42no—
548C22—44no—
549C22—44no—
550C22—44no—
551C22—82no—
552C22—82no—
553C22—82no—
554C22—82no—
555C22C2×D4211yescharacteristic
556C22#348611yescharacteristic
557C22#348611yescharacteristic
558C22C22×AGL(1,ℤ/8)11yescharacteristic
559C22C2×#57711yescharacteristic
560C22#358611yescharacteristic
561C22#358611yescharacteristic
562C4—22no—
563C4—22no—
564C4—22no—
565C4—44no—
566C4—44no—
567C4—44no—
568C4—44no—
569C4—41no—
570C4—44no—
571C4—41no—
572C4—82no—
573C4—82no—
574C4—82no—
575C4—82no—
576C4#327214yes—
577C4#348612yes—
578C4#353812yes—
579C2—84no—
580C2#4541811yescharacteristic
581C2C2×#327211yescharacteristic
582C2#6108111yescharacteristic
583C2#6182711yescharacteristic
584C2#6182811yescharacteristic
585C2#6184811yescharacteristic
586C2#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×#3386C2semidirect
2. #59828C2semidirect
4. #30975C2semidirect
5. #30973C2semidirect
6. #28205C2semidirect
8. #28195C2semidirect
9. #26919C2semidirect
11. #24330C2semidirect
12. #24328C2semidirect
14. #22247C2semidirect
15. #22242C2semidirect
120. #500D4semidirect

Maximal subgroups

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

subgroupindex
C2×#33862
#598282
#598172
#309752
#309732
#282052
#282012
#281952
#269192
#269132
#243302
#243282
#237992
#222472
#222422