Finite Group Databasebeta

← #6508438

Subgroups — #6508438

Order 512.

1632 conjugacy classes of subgroups.

The subgroup lattice has 615 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35015C211yesmaximal, characteristic
2C2×#2451C211yesmaximal, characteristic
3#29105C211yesmaximal, characteristic
4#29073C214yesmaximal, complemented
5#28945C214yesmaximal, complemented
6#28304C211yesmaximal, characteristic
7#26822C214yesmaximal, complemented
8#26485C214yesmaximal, complemented
9#25902C211yesmaximal, characteristic
10#25813C211yesmaximal, characteristic
11#24602C212yesmaximal
12#24502C212yesmaximal
13#22079C212yesmaximal
14C4×#451C212yesmaximal
15#20533C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×C4×Q8C2211yescharacteristic
18C2×#500C2212yes—
19C2×#487C2212yes—
20C2×#484C2212yes—
21C2×#451C2212yes—
22C2×C4×Q16C2214yes—
23C2×#431C2212yes—
24#2857C2211yescharacteristic
25#2857C2211yescharacteristic
26#2855C2212yes—
27#2646—21no—
28#2627—21no—
29#2590—22no—
30#2588—22no—
31#2511C2212yes—
32#2507C2212yes—
33#2451—22no—
34#2451—22no—
35#2451C2214yes—
36#2451C2212yes—
37#2451C2214yes—
38#2446C2212yes—
39C2×#398—21no—
40C2×#395—21no—
41#2329—21no—
42#2313—21no—
43C4×C22⋊Q8C2212yes—
44C2×#391—21no—
45C2×#391C2211yescharacteristic
46C2×#391C2211yescharacteristic
47C2×#389C2211yescharacteristic
48C2×#389C2211yescharacteristic
49C2×#387—21no—
50C2×C4×C4⋊C4C2211yescharacteristic
51#2095C2212yes—
52#2087C2214yes—
53#2072C2212yes—
54#2068C2214yes—
55#2068C2214yes—
56#2057C2214yes—
57#2056C2214yes—
58#2024C2214yes—
59#2024C2212yes—
60#2016C2212yes—
61#1990C2212yes—
62#1990C2212yes—
63#1969C2214yes—
64#1969C2214yes—
65#1942C2214yes—
66#1942C2214yes—
67#1933C2214yes—
68#1933C2214yes—
69#1933C2214yes—
70#1933C2214yes—
71#1921C2214yes—
72#1878C2214yes—
73#1872C2212yes—
74#1872C2212yes—
75#1863C2214yes—
76#1863C2214yes—
77#1852C2212yes—
78#1852C2212yes—
79#1824C2212yes—
80#1787C2212yes—
81C8×C4⋊C4C2214yes—
82C8×C4⋊C4C2214yes—
83C4×#103C2214yes—
84C4×C22⋊C8C2211yescharacteristic
85C4×C22⋊C8C2212yes—
86C4×C22⋊C8C2211yescharacteristic
87C23×Q8—21no—
88C23×Q8C2311yescharacteristic
89C22×Q16C2314yes—
90C2×C4⋊Q8—21no—
91C2×C4⋊Q8—21no—
92C2×C4⋊Q8—44no—
93C2×C4⋊Q8C2311yescharacteristic
94C2×C4⋊Q8C2312yes—
95C2×C4⋊Q8C2312yes—
96C2×C4⋊Q8C2311yescharacteristic
97C2×C22⋊Q8—21no—
98C2×C22⋊Q8C2312yes—
99C2×C4×Q8—22no—
100C2×C4×Q8—44no—
101C2×C4×Q8C2314yes—
102C2×C4×Q8C2312yes—
103C22×C4⋊C4—21no—
104C22×C4⋊C4—21no—
105C22×C4⋊C4—21no—
106C22×C4⋊C4—21no—
107C22×C4⋊C4D411yescharacteristic
108C22×C4⋊C4D411yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C42—21no—
113C22×C42C2311yescharacteristic
114C22×C42C2311yescharacteristic
115C22×C42C2311yescharacteristic
116#500—44no—
117#487—44no—
118#484—28no—
119#451—28no—
120C4×Q16—48no—
121#431—28no—
122C2×#107C2314yes—
123C2×#107C2314yes—
124C2×#107C2314yes—
125C2×C4⋊C8C2314yes—
126C2×#103C2314yes—
127C2×#103C2314yes—
128C2×#103C2314yes—
129C2×#103C2314yes—
130C2×#103C2314yes—
131C2×#103C2314yes—
132C2×C22⋊C8C2312yes—
133C2×C22⋊C8C2312yes—
134C2×C4×C8C2314yes—
135C2×C4×C8C2312yes—
136C2×C4×C8C2312yes—
137C2×C4×C8C2314yes—
138#400—22no—
139#400—22no—
140#400—22no—
141#398—22no—
142#398—22no—
143#398—22no—
144#398—22no—
145#398—24no—
146#398—24no—
147#395—22no—
148#395—24no—
149#395—24no—
150#391—21no—
151#391—22no—
152#391—21no—
153#391—21no—
154#391—22no—
155#391—24no—
156#391—24no—
157#391C2312yes—
158#391C2314yes—
159#391C2312yes—
160#391C2312yes—
161#391C2312yes—
162#391C2312yes—
163#391C2312yes—
164#389—21no—
165#389—21no—
166#389—21no—
167#389—22no—
168#389—22no—
169#389—22no—
170#389C2312yes—
171#389C2312yes—
172#389C2314yes—
173#389C2312yes—
174#389C2312yes—
175#389C2314yes—
176#387—22no—
177#387—22no—
178#387—22no—
179#387—22no—
180#387—24no—
181#387—24no—
182#387—22no—
183#387—22no—
184#387—22no—
185#387—22no—
186#387—22no—
187#385—22no—
188#385—22no—
189#384C2311yescharacteristic
190#384C2311yescharacteristic
191#381—22no—
192#380—21no—
193#380—21no—
194#380—21no—
195#380C2312yes—
196C4×C4⋊C4—22no—
197C4×C4⋊C4D412yes—
198C4×C4⋊C4D412yes—
199C4×C4⋊C4C2312yes—
200C4×C4⋊C4C2312yes—
201C4×C4⋊C4C2312yes—
202C4×C22⋊C4—22no—
203C2×#95—21no—
204C2×#95—21no—
205C2×#95—21no—
206C2×#95—21no—
207C2×#95—21no—
208C2×#95—21no—
209C2×#95D411yescharacteristic
210C2×#95D411yescharacteristic
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#336C2312yes—
218#336C2312yes—
219#336C2314yes—
220C22×Q8—24no—
221C22×Q8—21no—
222C22×Q8—22no—
223C22×Q8—22no—
224C22×Q8—21no—
225C22×Q8—24no—
226C22×Q8—42no—
227C22×Q8—44no—
228C22×Q8—44no—
229C22×Q8—82no—
230C22×Q8C2412yes—
231C22×Q8C2412yes—
232C23×C4—21no—
233C23×C4—21no—
234C23×C4—21no—
235C23×C4—21no—
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2411yescharacteristic
243C2×Q16—28no—
244C2×Q16—28no—
245C2×Q16—44no—
246C22×C8C2414yes—
247C22×C8C2414yes—
248C4⋊Q8—24no—
249C4⋊Q8—24no—
250C4⋊Q8—42no—
251C4⋊Q8—42no—
252C4⋊Q8—42no—
253C4⋊Q8—42no—
254C4⋊Q8—44no—
255C4⋊Q8—44no—
256C22⋊Q8—24no—
257C22⋊Q8—24no—
258C22⋊Q8—44no—
259C4×Q8—28no—
260C4×Q8—44no—
261C4×Q8—44no—
262C4×Q8—48no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—42no—
288C2×C4⋊C4—44no—
289C2×C4⋊C4—44no—
290C2×C4⋊C4—82no—
291C2×C4⋊C4—82no—
292C2×C4⋊C4Q1618yescomplemented
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—22no—
304C2×C22⋊C4—22no—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—22no—
308C2×C42—22no—
309C2×C42—42no—
310C2×C42—42no—
311C2×C42—44no—
312C2×C42C2×D412yes—
313C2×C42C2×D412yes—
314C2×C42C2×D411yescharacteristic
315C2×C42C2×D411yescharacteristic
316C2×C42C2412yes—
317C2×C42C2411yescharacteristic
318C2×C42C2411yescharacteristic
319C2×C42C2412yes—
320#107—28no—
321#107—28no—
322#107—44no—
323C4⋊C8—28no—
324#103—28no—
325#103—28no—
326#103—28no—
327#103—28no—
328#103—44no—
329#103—44no—
330C22⋊C8—24no—
331C22⋊C8#4118yes—
332C4×C8—28no—
333C4×C8—44no—
334C4×C8—42no—
335C4×C8—42no—
336#95—24no—
337#95—24no—
338#95—24no—
339#95—22no—
340#95—22no—
341#95—22no—
342#95—22no—
343#95—24no—
344#95—22no—
345#95—22no—
346#95—22no—
347#95—22no—
348#95—22no—
349#95—22no—
350#95—22no—
351#95—22no—
352#95—22no—
353#95—22no—
354#95—22no—
355#95—22no—
356#95—22no—
357#95—22no—
358#95C2×D412yes—
359#95C2×D412yes—
360#95C2×D412yes—
361#95C2×D412yes—
362C24C22×D411yescharacteristic
363C2×Q8—24no—
364C2×Q8—22no—
365C2×Q8—24no—
366C2×Q8—22no—
367C2×Q8—22no—
368C2×Q8—24no—
369C2×Q8—24no—
370C2×Q8—24no—
371C2×Q8—44no—
372C2×Q8—44no—
373C2×Q8—416no—
374C2×Q8—42no—
375C2×Q8—42no—
376C2×Q8—44no—
377C2×Q8—44no—
378C2×Q8—42no—
379C2×Q8—82no—
380C2×Q8—82no—
381C2×Q8—81no—
382C2×Q8—81no—
383C2×Q8—816no—
384C2×Q82−1+414yes—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4C2×Q1614yes—
422C22×C4C2×Q1614yes—
423C22×C4C2×Q1614yes—
424C22×C4#13514yes—
425C22×C4C22×D412yes—
426C22×C4C22×D412yes—
427C22×C4C22×D411yescharacteristic
428C22×C4C22×D412yes—
429C22×C4C22×D412yes—
430C22×C4C22×D411yescharacteristic
431C22×C4C22×D411yescharacteristic
432C22×C4C22×D411yescharacteristic
433C22×C4C22×D411yescharacteristic
434C22×C4C2×#4114yes—
435C22×C4C2511yescharacteristic
436Q16—416no—
437C2×C8—24no—
438C2×C8—24no—
439C2×C8—24no—
440C2×C8—24no—
441C2×C8—24no—
442C2×C8C2×#4118yes—
443C4⋊C4—24no—
444C4⋊C4—24no—
445C4⋊C4—24no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—24no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—24no—
452C4⋊C4—22no—
453C4⋊C4—24no—
454C4⋊C4—48no—
455C4⋊C4—44no—
456C4⋊C4—44no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C4⋊C4—44no—
460C4⋊C4—42no—
461C4⋊C4—44no—
462C4⋊C4—81no—
463C4⋊C4—81no—
464C4⋊C4—81no—
465C4⋊C4—82no—
466C4⋊C4—82no—
467C4⋊C4—82no—
468C4⋊C4—84no—
469C4⋊C4—84no—
470C4⋊C4—82no—
471C4⋊C4—81no—
472C4⋊C4—81no—
473C4⋊C4—81no—
474C4⋊C4C2×Q1618yes—
475C4⋊C4C2×Q1618yes—
476C4⋊C4H(5,𝔽2)14yes—
477C22⋊C4—44no—
478C22⋊C4—44no—
479C42—24no—
480C42—22no—
481C42—22no—
482C42—24no—
483C42—22no—
484C42—22no—
485C42—42no—
486C42—41no—
487C42—41no—
488C42—48no—
489C42—82no—
490C42—82no—
491C23—21no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23#53212yes—
496C23C22×Q1612yes—
497C23C2×#13512yes—
498C23C23×D411yescharacteristic
499Q8—28no—
500Q8—48no—
501Q8—44no—
502Q8—44no—
503Q8—44no—
504Q8—88no—
505Q8—88no—
506Q8—162no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—22no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—28no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—22no—
522C2×C4—28no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—22no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—48no—
535C2×C4—48no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—82no—
559C2×C4—82no—
560C2×C4—84no—
561C2×C4—84no—
562C2×C4#53212yes—
563C2×C4C22×Q1614yes—
564C2×C4C22×Q1614yes—
565C2×C4C22×Q1612yes—
566C2×C4C22×Q1612yes—
567C2×C4C2×#13512yes—
568C2×C4C2×#13512yes—
569C2×C4C22×#4112yes—
570C2×C4C2×H(5,𝔽2)12yes—
571C2×C4C2×2−1+412yes—
572C8—28no—
573C8—44no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—22no—
580C22#317014yes—
581C22C2×#53211yescharacteristic
582C22#348411yescharacteristic
583C22#348611yescharacteristic
584C22C23×Q1611yescharacteristic
585C22C22×#13511yescharacteristic
586C22#358511yescharacteristic
587C22#358611yescharacteristic
588C4—24no—
589C4—22no—
590C4—22no—
591C4—22no—
592C4—24no—
593C4—22no—
594C4—22no—
595C4—42no—
596C4—44no—
597C4—42no—
598C4—44no—
599C4—82no—
600C4—82no—
601C4—82no—
602C4—82no—
603C4—82no—
604C4—82no—
605C4#352014yes—
606C2—24no—
607C2#3539711yescharacteristic
608C2C2×#317011yescharacteristic
609C2#6070311yescharacteristic
610C2#6156111yescharacteristic
611C2#6156311yescharacteristic
612C2#6182111yescharacteristic
613C2#6182211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #29073C2semidirect
5. #28945C2semidirect
7. #26822C2semidirect
8. #26485C2semidirect
292. C2×C4⋊C4Q16semidirect

Maximal subgroups

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

subgroupindex
#350152
C2×#24512
#291052
#290732
#289452
#283042
#268222
#264852
#259022
#258132
#246022
#245022
#220792
C4×#4512
#205332