Finite Group Databasebeta

← #6436953

Subgroups — #6436953

Order 512.

1500 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35297C211yesmaximal, characteristic
2#34060C211yesmaximal, characteristic
3#29941C212yesmaximal, complemented
4#29936C212yesmaximal, complemented
5#29926C212yesmaximal, complemented
6#29923C212yesmaximal, complemented
7#29918C214yesmaximal, complemented
8#29901C214yesmaximal
9#23773C214yesmaximal, complemented
10#23768C214yesmaximal, complemented
11#21394C211yesmaximal, characteristic, complemented
12#21389C211yesmaximal, characteristic, complemented
13#21364C211yesmaximal, characteristic, complemented
14#21358C211yesmaximal, characteristic, complemented
15#20721C211yesmaximal, characteristic
16C2×#533C2211yescharacteristic
17C2×C4×#41C2211yescharacteristic
18C2×#432C2211yescharacteristic
19C2×#431C2211yescharacteristic
20#2857C2212yes—
21#2857C2212yes—
22#2625C2212yes—
23#2616C2212yes—
24#2584—21no—
25#2581—21no—
26#2562—21no—
27#2536—21no—
28#2485—22no—
29#2415—22no—
30#2323C2214yes—
31#2320—21no—
32#2319—21no—
33#2318—22no—
34#2317C2212yes—
35#2314C2212yes—
36#2314C2212yes—
37#2311C2212yes—
38#2295—21no—
39#2293—21no—
40#2290—22no—
41#2285C2212yes—
42#2285C2211yescharacteristic
43#2283C2212yes—
44C4×#117C2211yescharacteristic
45C2×C4×C4⋊C4C2211yescharacteristic
46#2101C2214yes—
47#2099C2214yes—
48#2097C2214yes—
49#2095C2214yes—
50#2085C2214yes—
51#2083C2214yes—
52#2077C2214yes—
53#2076C2214yes—
54#2075C2214yes—
55#2075C2214yes—
56#2074C2214yes—
57#2072C2214yes—
58#2036C2214yes—
59#2036C2214yes—
60#2036C2214yes—
61#2035C2214yes—
62#1988C2211yescharacteristic
63#1988C2211yescharacteristic
64#1988C2212yes—
65#1981C2211yescharacteristic
66#1981C2211yescharacteristic
67#1981C2212yes—
68#1948C2212yes—
69#1947C2212yes—
70#1947C2212yes—
71#1947C2212yes—
72#1873C2214yes—
73#1872C2214yes—
74#1854C2214yes—
75#1852C2214yes—
76#1818C2212yes—
77#1797C2212yes—
78#1797C2212yes—
79#1780C2212yes—
80#1776C2214yes—
81C4×#107C2212yes—
82C4×#107C2212yes—
83#1763C2212yes—
84#1763C2212yes—
85C4×#103C2214yes—
86#1750C2214yes—
87C22×#41C2311yescharacteristic
88#533—24no—
89#533—24no—
90C2×C4⋊Q8C2311yescharacteristic
91C2×C4⋊Q8C2311yescharacteristic
92C2×#125—21no—
93C2×C22⋊Q8—21no—
94C2×C22⋊Q8—22no—
95C2×C22⋊Q8C2312yes—
96C2×C22⋊Q8C2314yes—
97C4×#41—24no—
98C4×#41—24no—
99C2×C4×Q8—21no—
100C2×C4×Q8D411yescharacteristic
101C2×C4×Q8D411yescharacteristic
102C2×C4×Q8C2312yes—
103C2×C4×D4D411yescharacteristic
104C2×C4×D4D412yes—
105C2×C4×D4D412yes—
106C2×C4×D4D411yescharacteristic
107C2×#117—21no—
108C2×#117—21no—
109C2×#117—22no—
110C2×#117C2312yes—
111C2×#117C2312yes—
112C2×#117C2311yescharacteristic
113C22×C4⋊C4C2312yes—
114C22×C42—21no—
115C22×C42C2312yes—
116C22×C42C2311yescharacteristic
117#432—24no—
118#431—24no—
119C2×#107C2314yes—
120C2×#107C2314yes—
121C2×#107C2314yes—
122C2×#106C2314yes—
123C2×C4⋊C8C2311yescharacteristic
124C2×C4⋊C8C2311yescharacteristic
125C2×C4⋊C8C2312yes—
126C2×#103C2314yes—
127C2×#103C2314yes—
128C2×#103C2314yes—
129C2×#103C2314yes—
130C2×C22⋊C8C2312yes—
131C2×C22⋊C8C2312yes—
132C2×C22⋊C8C2314yes—
133C2×C22⋊C8C2314yes—
134C2×#97C2312yes—
135C2×C4×C8C2312yes—
136#400—22no—
137#398—22no—
138#397—24no—
139#395—22no—
140#394—22no—
141#394—24no—
142#391—22no—
143#391—22no—
144#391C2314yes—
145#390—22no—
146#389—22no—
147#389—22no—
148#389—24no—
149#389—22no—
150#389—22no—
151#389—22no—
152#389C2314yes—
153#388—24no—
154#388—22no—
155#388—22no—
156#388—22no—
157#388—22no—
158#388—24no—
159#387—22no—
160#387—22no—
161#387—22no—
162#387—22no—
163#387—24no—
164#386—22no—
165#386—22no—
166#385—22no—
167#385—22no—
168#384C2312yes—
169#384C2312yes—
170#384C2312yes—
171#383—22no—
172#382—22no—
173#382—21no—
174#382C2312yes—
175#382C2312yes—
176#381—22no—
177#380—22no—
178#380—22no—
179#380—22no—
180#380—22no—
181#380—21no—
182#380C2312yes—
183#380C2312yes—
184#380C2312yes—
185#380C2312yes—
186#380C2314yes—
187#380C2312yes—
188C4×C4⋊C4—21no—
189C4×C4⋊C4—21no—
190C4×C4⋊C4—21no—
191C4×C4⋊C4—22no—
192C4×C4⋊C4C2311yescharacteristic
193C4×C4⋊C4C2311yescharacteristic
194C4×C4⋊C4C2312yes—
195C4×C4⋊C4C2312yes—
196C4×C4⋊C4C2312yes—
197C4×C4⋊C4C2311yescharacteristic
198C4×C4⋊C4C2311yescharacteristic
199C4×C22⋊C4—21no—
200C4×C22⋊C4—21no—
201C4×C22⋊C4—22no—
202#376—21no—
203#376—22no—
204C2×#95—21no—
205C43—21no—
206#340C2314yes—
207#340C2314yes—
208#340C2314yes—
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#336C2314yes—
215#336C2314yes—
216C2×#41—22no—
217C2×#41—22no—
218C2×#41—42no—
219C2×#41—42no—
220C22×Q8—21no—
221C22×Q8C2×D411yescharacteristic
222C22×Q8C2412yes—
223C22×D4C2×D412yes—
224C22×D4C2×D411yescharacteristic
225C23×C4—22no—
226C23×C4C2×D412yes—
227C23×C4C2×D411yescharacteristic
228C23×C4C2×D412yes—
229C23×C4C2×D411yescharacteristic
230C23×C4C2412yes—
231C23×C4C2411yescharacteristic
232C22×C8C2412yes—
233C22×C8C2412yes—
234C22×C8C2414yes—
235C4⋊Q8—22no—
236C4⋊Q8—22no—
237C4⋊Q8—22no—
238C4⋊Q8—22no—
239#125—44no—
240C22⋊Q8—24no—
241C22⋊Q8—24no—
242C22⋊Q8—44no—
243C22⋊Q8—44no—
244C22⋊Q8—44no—
245C22⋊Q8—84no—
246C4×Q8—24no—
247C4×Q8—24no—
248C4×Q8—24no—
249C4×Q8—24no—
250C4×Q8—44no—
251C4×D4—28no—
252C4×D4—28no—
253C4×D4—24no—
254C4×D4—24no—
255#117—22no—
256#117—22no—
257#117—24no—
258#117—24no—
259#117—42no—
260#117—42no—
261#117—44no—
262#117—84no—
263#117—82no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—42no—
279C2×C4⋊C4—42no—
280C2×C4⋊C4C2×D412yes—
281C2×C4⋊C4C2×D412yes—
282C2×C4⋊C4C2×D411yescharacteristic
283C2×C4⋊C4C2×D411yescharacteristic
284C2×C4⋊C4C2412yes—
285C2×C4⋊C4C2412yes—
286C2×C4⋊C4C2412yes—
287C2×C4⋊C4C2412yes—
288C2×C4⋊C4C2412yes—
289C2×C4⋊C4C2414yes—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4C2×D412yes—
300C2×C22⋊C4C2×D412yes—
301C2×C22⋊C4C2×D412yes—
302C2×C42—24no—
303C2×C42—24no—
304C2×C42—21no—
305C2×C42—22no—
306C2×C42—22no—
307C2×C42—22no—
308C2×C42—21no—
309C2×C42—22no—
310C2×C42—22no—
311C2×C42—21no—
312C2×C42—21no—
313C2×C42—82no—
314C2×C42C2×D412yes—
315C2×C42C2×D412yes—
316C2×C42C2×D411yescharacteristic
317C2×C42C2×D411yescharacteristic
318C2×C42#4114yes—
319C2×C42#4114yes—
320C2×C42C2412yes—
321C2×C42C2411yescharacteristic
322C2×C42C2411yescharacteristic
323#107—44no—
324#107—44no—
325#107—44no—
326#106—44no—
327C4⋊C8—22no—
328C4⋊C8—22no—
329C4⋊C8—24no—
330#103—44no—
331#103—44no—
332#103—44no—
333#103—44no—
334C22⋊C8—24no—
335C22⋊C8—24no—
336C22⋊C8—44no—
337C22⋊C8—44no—
338#97—24no—
339C4×C8—24no—
340#95—22no—
341#95—21no—
342#95—21no—
343#95—21no—
344#95—21no—
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—24no—
357#95—24no—
358#95—24no—
359#95—24no—
360C24C22×D412yes—
361C24C22×D411yescharacteristic
362#41—44no—
363#41—44no—
364#41—48no—
365C2×Q8—22no—
366C2×Q8—22no—
367C2×Q8—22no—
368C2×Q8—24no—
369C2×Q8—24no—
370C2×Q8—42no—
371C2×Q8—42no—
372C2×Q8—42no—
373C2×Q8—42no—
374C2×D4—24no—
375C2×D4—22no—
376C2×D4—42no—
377C2×D4—42no—
378C2×D4—42no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—44no—
404C22×C4—82no—
405C22×C4—82no—
406C22×C4#12314yes—
407C22×C4#12314yes—
408C22×C4C22×D412yes—
409C22×C4C22×D412yes—
410C22×C4C22×D412yes—
411C22×C4C22×D411yescharacteristic
412C22×C4C22×D411yescharacteristic
413C22×C4C22×D411yescharacteristic
414C22×C4C22×D411yescharacteristic
415C22×C4C22×D411yescharacteristic
416C22×C4C2×#4112yes—
417C22×C4C2×#4112yes—
418C22×C4C2511yescharacteristic
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—24no—
422C2×C8—24no—
423C2×C8—24no—
424C2×C8—24no—
425C2×C8—24no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—24no—
429C4⋊C4—24no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—24no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—24no—
442C4⋊C4—44no—
443C4⋊C4—44no—
444C4⋊C4—48no—
445C4⋊C4—42no—
446C4⋊C4—42no—
447C4⋊C4—44no—
448C4⋊C4—44no—
449C4⋊C4—42no—
450C4⋊C4—44no—
451C4⋊C4—82no—
452C4⋊C4—82no—
453C4⋊C4—82no—
454C4⋊C4—82no—
455C4⋊C4—84no—
456C22⋊C4—24no—
457C22⋊C4—24no—
458C22⋊C4—24no—
459C22⋊C4—44no—
460C22⋊C4—48no—
461C22⋊C4—44no—
462C22⋊C4—82no—
463C22⋊C4—82no—
464C22⋊C4—82no—
465C22⋊C4—82no—
466C22⋊C4—82no—
467C22⋊C4—84no—
468C42—24no—
469C42—24no—
470C42—22no—
471C42—22no—
472C42—44no—
473C42—44no—
474C42—44no—
475C42—42no—
476C42—42no—
477C42—82no—
478C42—81no—
479C42—81no—
480C42—82no—
481C42#12314yes—
482C42#12314yes—
483C42C2×#4112yes—
484C42C2×#4112yes—
485C42C2×#4112yes—
486C42C2×#4112yes—
487C23—22no—
488C23—22no—
489C23—21no—
490C23—21no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—22no—
498C23C2×#12312yes—
499C23C23×D411yescharacteristic
500Q8—44no—
501Q8—42no—
502Q8—42no—
503Q8—44no—
504Q8—84no—
505D4—44no—
506D4—48no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—28no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—24no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—44no—
526C2×C4—48no—
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—44no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—84no—
552C2×C4—88no—
553C2×C4C2×#12312yes—
554C2×C4C2×#12312yes—
555C2×C4C2×#12312yes—
556C2×C4C2×#12312yes—
557C2×C4C2×#12312yes—
558C2×C4#57612yes—
559C2×C4#57812yes—
560C2×C4C22×#4111yescharacteristic
561C2×C4C22×#4111yescharacteristic
562C2×C4#58512yes—
563C2×C4#58512yes—
564C8—42no—
565C8—42no—
566C8—44no—
567C22—22no—
568C22—22no—
569C22—22no—
570C22—42no—
571C22—42no—
572C22—42no—
573C22—42no—
574C22—42no—
575C22—42no—
576C22—42no—
577C22—42no—
578C22—42no—
579C22C22×#12311yescharacteristic
580C22#344912yes—
581C22C2×#57611yescharacteristic
582C22C2×#57811yescharacteristic
583C22#358612yes—
584C4—22no—
585C4—22no—
586C4—22no—
587C4—22no—
588C4—42no—
589C4—42no—
590C4—42no—
591C4—82no—
592C4—84no—
593C4—82no—
594C4—82no—
595C4—82no—
596C4—82no—
597C4—82no—
598C4#309814yes—
599C4#345511yescharacteristic
600C4#345511yescharacteristic
601C4#345512yes—
602C2—42no—
603C2—82no—
604C2#3463511yescharacteristic
605C2C2×#309811yescharacteristic
606C2#6051911yescharacteristic
607C2#6098812yes—
608C2#6099512yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #29941C2semidirect
4. #29936C2semidirect
5. #29926C2semidirect
6. #29923C2semidirect
7. #29918C2semidirect
9. #23773C2semidirect
10. #23768C2semidirect
11. #21394C2semidirect
12. #21389C2semidirect
13. #21364C2semidirect
14. #21358C2semidirect

Maximal subgroups

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

subgroupindex
#352972
#340602
#299412
#299362
#299262
#299232
#299182
#299012
#237732
#237682
#213942
#213892
#213642
#213582
#207212