Finite Group Databasebeta

← #7196738

Subgroups — #7196738

Order 512.

1780 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×#3398C212yesmaximal, complemented
2#59828C211yesmaximal, characteristic, complemented
3#59821C211yesmaximal, characteristic
4#30762C214yesmaximal
5#30760C214yesmaximal, complemented
6#28195C212yesmaximal, complemented
7#28161C211yesmaximal, characteristic, complemented
8#28157C211yesmaximal, characteristic
9#26187C214yesmaximal, complemented
10#26185C214yesmaximal, complemented
11#24189C212yesmaximal, complemented
12#24187C212yesmaximal, complemented
13#23799C211yesmaximal, characteristic
14#22270C211yesmaximal, characteristic, complemented
15#22242C211yesmaximal, characteristic, complemented
16C2×#557C2212yes—
17C2×#556C2212yes—
18#3398—216no—
19C2×#497C2214yes—
20C2×#494C2211yescharacteristic
21C2×#464C2214yes—
22C2×#462C2212yes—
23C2×#462C2214yes—
24C2×#461C2212yes—
25C2×#446C2212yes—
26C2×#441C2214yes—
27C2×C4×SD16C2212yes—
28C2×C4×SD16C2212yes—
29#2727—22no—
30#2726—22no—
31#2582C2212yes—
32#2552—21no—
33#2548C2212yes—
34#2546C2212yes—
35#2542—21no—
36#2538C2212yes—
37#2512—22no—
38#2506—22no—
39#2453—22no—
40#2448C2214yes—
41#2446C2214yes—
42#2415—22no—
43#2365—21no—
44#2354—21no—
45Q8×C4⋊C4C2212yes—
46Q8×C4⋊C4C2212yes—
47C4×C4⋊Q8C2211yescharacteristic
48C4×C4⋊Q8C2211yescharacteristic
49C4×C4⋊Q8C2211yescharacteristic
50C4×C4⋊Q8C2211yescharacteristic
51C4×#127C2211yescharacteristic
52#2096C2212yes—
53#2094C2212yes—
54#2088C2214yes—
55#2072C2212yes—
56#2072C2212yes—
57#2055C2214yes—
58#2043C2214yes—
59#2042C2214yes—
60#2016C2212yes—
61#2015C2214yes—
62#2014C2212yes—
63#2014C2212yes—
64#2013C2212yes—
65#2013C2214yes—
66#1984C2212yes—
67#1956C2211yescharacteristic
68#1956C2211yescharacteristic
69#1936C2214yes—
70#1935C2214yes—
71#1917C2214yes—
72#1917C2214yes—
73#1880C2214yes—
74#1880C2214yes—
75#1878C2214yes—
76#1878C2214yes—
77#1878C2214yes—
78#1878C2214yes—
79#1865C2214yes—
80#1865C2214yes—
81#1834C2211yescharacteristic
82#1831C2211yescharacteristic
83#1803C2212yes—
84#1802C2212yes—
85#1744C2211yescharacteristic
86C22×Q16C2314yes—
87C22×Q16C2314yes—
88C22×SD16C2312yes—
89C22×SD16C2314yes—
90C22×SD16C2312yes—
91#557—28no—
92#557—44no—
93#556—28no—
94#556—44no—
95C2×C4⋊Q8D411yescharacteristic
96C2×C4⋊Q8C2314yes—
97C2×C4⋊Q8C2311yescharacteristic
98C2×C4⋊Q8C2311yescharacteristic
99C2×C4⋊Q8C2311yescharacteristic
100C2×#127C2311yescharacteristic
101C2×#126—24no—
102C2×#125—22no—
103C2×#125—24no—
104C2×#124C2314yes—
105C2×C22⋊Q8—22no—
106C2×C22⋊Q8—22no—
107C2×C4×Q8—22no—
108C2×C4×Q8C2312yes—
109C2×C4×Q8C2314yes—
110C2×C4×Q8C2314yes—
111C2×C4×Q8C2312yes—
112C2×C4×Q8C2312yes—
113C2×C4×Q8C2312yes—
114C2×C4×D4—21no—
115C2×C4×D4—21no—
116C2×C4×D4C2312yes—
117C2×#117—22no—
118#497—216no—
119#494D418yescomplemented
120#464—216no—
121#462—216no—
122#462—28no—
123#461—28no—
124#446—28no—
125#441—216no—
126C4×SD16—28no—
127C4×SD16—44no—
128C2×#106C2312yes—
129C2×#106C2312yes—
130C2×C4⋊C8C2314yes—
131C2×C4⋊C8C2312yes—
132C2×#103C2314yes—
133C2×#103C2314yes—
134C2×#103C2312yes—
135C2×#103C2314yes—
136C2×#103C2312yes—
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×#102C2312yes—
140C2×#102C2314yes—
141C2×#102C2312yes—
142C2×#97C2314yes—
143C2×C4×C8C2311yescharacteristic
144C2×C4×C8C2311yescharacteristic
145#400—24no—
146#399—24no—
147#398—24no—
148#398—24no—
149#398—24no—
150#397—24no—
151#396—24no—
152#394—24no—
153#391—22no—
154#391D412yes—
155#391C2312yes—
156#391C2312yes—
157#391C2312yes—
158#391C2314yes—
159#390—22no—
160#390—22no—
161#390—21no—
162#390—21no—
163#390C2312yes—
164#390C2314yes—
165#389—22no—
166#389—22no—
167#389—22no—
168#389—21no—
169#389—21no—
170#389—21no—
171#389—21no—
172#389—21no—
173#389—21no—
174#389—24no—
175#389—24no—
176#389D412yes—
177#389D412yes—
178#389C2314yes—
179#389C2314yes—
180#388—24no—
181#388—24no—
182#387—24no—
183#387—24no—
184#384—22no—
185#384—22no—
186#384D411yescharacteristic
187#384C2311yescharacteristic
188#384C2311yescharacteristic
189#384C2311yescharacteristic
190#384C2311yescharacteristic
191#380—22no—
192C4×C4⋊C4—22no—
193C4×C4⋊C4—22no—
194C4×C4⋊C4—21no—
195C4×C4⋊C4—21no—
196C4×C4⋊C4—21no—
197C4×C4⋊C4—21no—
198C4×C4⋊C4—21no—
199C4×C4⋊C4—21no—
200C4×C4⋊C4C2312yes—
201C4×C4⋊C4C2312yes—
202C43C2311yescharacteristic
203#340C2314yes—
204#340C2314yes—
205#340C2314yes—
206#336C2312yes—
207#336C2312yes—
208#336C2312yes—
209#336C2312yes—
210#336C2314yes—
211#336C2314yes—
212C22×Q8C2×D412yes—
213C22×Q8C2412yes—
214C22×Q8C2412yes—
215C22×Q8C2412yes—
216C22×D4—21no—
217C22×D4—21no—
218C22×D4C2412yes—
219C23×C4—22no—
220C2×Q16—28no—
221C2×Q16—44no—
222C2×Q16—44no—
223C2×Q16—44no—
224C2×Q16—44no—
225C2×Q16C2×D4116yes—
226C2×SD16—28no—
227C2×SD16—24no—
228C2×SD16—42no—
229C2×SD16—42no—
230C2×SD16—42no—
231C2×SD16—42no—
232C2×SD16—42no—
233C2×SD16—44no—
234C2×SD16—44no—
235C22×C8C2414yes—
236C22×C8C2412yes—
237C22×C8C2412yes—
238C4⋊Q8—28no—
239C4⋊Q8—24no—
240C4⋊Q8—22no—
241C4⋊Q8—22no—
242C4⋊Q8—42no—
243C4⋊Q8—44no—
244C4⋊Q8C2×D414yes—
245C4⋊Q8C2×D414yes—
246#127—42no—
247#126—416no—
248#125—416no—
249#125—48no—
250#124—28no—
251#124—44no—
252C22⋊Q8—48no—
253C22⋊Q8—84no—
254C4×Q8—24no—
255C4×Q8—24no—
256C4×Q8—28no—
257C4×Q8—28no—
258C4×Q8—24no—
259C4×Q8—24no—
260C4×Q8—44no—
261C4×Q8—44no—
262C4×Q8—44no—
263C4×Q8—44no—
264C4×Q8—48no—
265C4×D4—24no—
266C4×D4—24no—
267C4×D4—44no—
268C4×D4—82no—
269#117—48no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4C2×D412yes—
294C2×C4⋊C4C2×D414yes—
295C2×C4⋊C4C2×D412yes—
296C2×C4⋊C4C2×D412yes—
297C2×C4⋊C4C2×D412yes—
298C2×C4⋊C4C2×D412yes—
299C2×C4⋊C4C2412yes—
300C2×C4⋊C4C2412yes—
301C2×C4⋊C4C2412yes—
302C2×C4⋊C4C2412yes—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—24no—
310C2×C42—24no—
311C2×C42—22no—
312C2×C42—24no—
313C2×C42—22no—
314C2×C42—22no—
315C2×C42C2×D411yescharacteristic
316C2×C42C2×D411yescharacteristic
317C2×C42C2×D411yescharacteristic
318C2×C42C2×D411yescharacteristic
319C2×C42C2414yes—
320C2×C42C2411yescharacteristic
321C2×C42C2411yescharacteristic
322C2×C42C2411yescharacteristic
323#106—24no—
324#106—42no—
325C4⋊C8—28no—
326C4⋊C8—24no—
327#103—28no—
328#103—28no—
329#103—24no—
330#103—44no—
331#103—44no—
332#103—44no—
333#103—42no—
334#102—24no—
335#102—44no—
336#102—42no—
337#97—28no—
338C4×C8—22no—
339C4×C8C2×D414yes—
340#95—24no—
341#95—22no—
342#95—22no—
343#95—22no—
344#95—22no—
345#95—24no—
346#95—22no—
347#95—22no—
348#95—24no—
349#95—24no—
350#95C2×D414yes—
351#95C2×D414yes—
352C24—22no—
353C2×Q8—24no—
354C2×Q8—24no—
355C2×Q8—24no—
356C2×Q8—22no—
357C2×Q8—22no—
358C2×Q8—22no—
359C2×Q8—22no—
360C2×Q8—24no—
361C2×Q8—24no—
362C2×Q8—24no—
363C2×Q8—24no—
364C2×Q8—22no—
365C2×Q8—22no—
366C2×Q8—22no—
367C2×Q8—22no—
368C2×Q8C22×D414yes—
369C2×Q8C22×D414yes—
370C2×D4—22no—
371C2×D4—22no—
372C2×D4—22no—
373C2×D4—22no—
374C2×D4—22no—
375C2×D4—22no—
376C2×D4—42no—
377C2×D4—42no—
378C2×D4—42no—
379C2×D4—42no—
380C2×D4—81no—
381C2×D4—81no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—44no—
390C22×C4—44no—
391C22×C4—44no—
392C22×C4C22×D411yescharacteristic
393C22×C4C22×D411yescharacteristic
394C22×C4C22×D414yes—
395C22×C4C22×D414yes—
396C22×C4C22×D411yescharacteristic
397C22×C4C22×D411yescharacteristic
398C22×C4C22×D411yescharacteristic
399C22×C4C22×D411yescharacteristic
400C22×C4C2511yescharacteristic
401Q16—416no—
402Q16—88no—
403SD16—48no—
404SD16—48no—
405SD16—88no—
406C2×C8—22no—
407C2×C8—22no—
408C2×C8—24no—
409C2×C8—22no—
410C2×C8—24no—
411C2×C8—22no—
412C2×C8—22no—
413C2×C8—24no—
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—24no—
428C4⋊C4—24no—
429C4⋊C4—42no—
430C4⋊C4—42no—
431C4⋊C4—48no—
432C4⋊C4—48no—
433C4⋊C4—48no—
434C4⋊C4—48no—
435C4⋊C4—42no—
436C4⋊C4—44no—
437C4⋊C4—44no—
438C4⋊C4—42no—
439C4⋊C4—42no—
440C4⋊C4—44no—
441C4⋊C4—42no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C4⋊C4—81no—
445C4⋊C4—81no—
446C4⋊C4—81no—
447C4⋊C4—84no—
448C4⋊C4—81no—
449C4⋊C4—84no—
450C22⋊C4—48no—
451C22⋊C4—48no—
452C22⋊C4—44no—
453C22⋊C4—44no—
454C22⋊C4—84no—
455C22⋊C4—82no—
456C22⋊C4—84no—
457C42—22no—
458C42—24no—
459C42—24no—
460C42—22no—
461C42—22no—
462C42—41no—
463C42—41no—
464C42—48no—
465C42—48no—
466C42—44no—
467C42—44no—
468C42—44no—
469C42#13714yes—
470C42C22×D412yes—
471C42H(5,𝔽2)12yes—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23C23×D411yescharacteristic
480Q8—28no—
481Q8—28no—
482Q8—48no—
483Q8—44no—
484Q8—44no—
485Q8—42no—
486Q8—42no—
487Q8—42no—
488Q8—42no—
489D4—42no—
490D4—42no—
491D4—42no—
492D4—42no—
493D4—84no—
494D4—84no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—21no—
510C2×C4—21no—
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—42no—
534C2×C4—88no—
535C2×C4D4212yes—
536C2×C4D4212yes—
537C2×C4C2×#13712yes—
538C2×C4C2×#13712yes—
539C2×C4C2×#13712yes—
540C2×C4#57712yes—
541C2×C4C23×D411yescharacteristic
542C2×C4C2×H(5,𝔽2)11yescharacteristic
543C2×C4#58512yes—
544C2×C4#58512yes—
545C8—24no—
546C8—44no—
547C8—42no—
548C22—82no—
549C22—82no—
550C22—82no—
551C22—82no—
552C22—82no—
553C22—82no—
554C22—82no—
555C22C2×D4211yescharacteristic
556C22#348711yescharacteristic
557C22#348711yescharacteristic
558C22C22×#13711yescharacteristic
559C22C2×#57711yescharacteristic
560C22#358611yescharacteristic
561C22#358611yescharacteristic
562C4—22no—
563C4—22no—
564C4—22no—
565C4—44no—
566C4—44no—
567C4—44no—
568C4—44no—
569C4—41no—
570C4—41no—
571C4—82no—
572C4—82no—
573C4—82no—
574C4—82no—
575C4—82no—
576C4#327814yes—
577C4#348512yes—
578C4#353912yes—
579C2—162no—
580C2#4538711yescharacteristic
581C2C2×#327811yescharacteristic
582C2#6108111yescharacteristic
583C2#6188911yescharacteristic
584C2#6189311yescharacteristic
585C2#6191111yescharacteristic
586C2#6191611yescharacteristic
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×#3398C2semidirect
2. #59828C2semidirect
5. #30760C2semidirect
6. #28195C2semidirect
7. #28161C2semidirect
9. #26187C2semidirect
10. #26185C2semidirect
11. #24189C2semidirect
12. #24187C2semidirect
14. #22270C2semidirect
15. #22242C2semidirect
119. #494D4semidirect

Maximal subgroups

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

subgroupindex
C2×#33982
#598282
#598212
#307622
#307602
#281952
#281612
#281572
#261872
#261852
#241892
#241872
#237992
#222702
#222422