Finite Group Databasebeta

← #6436199

Subgroups — #6436199

Order 512.

1872 conjugacy classes of subgroups.

The subgroup lattice has 603 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3165C211yesmaximal, characteristic, complemented
2C2×#2922C211yesmaximal, characteristic
3#35280C211yesmaximal, characteristic
4#35279C211yesmaximal, characteristic
5#29885C218yesmaximal, complemented
6#29878C212yesmaximal, complemented
7#29877C212yesmaximal, complemented
8#29877C212yesmaximal, complemented
9#29876C212yesmaximal, complemented
10#23671C214yesmaximal, complemented
11#23643C214yesmaximal
12#21318C211yesmaximal, characteristic, complemented
13C2×#1990C211yesmaximal, characteristic, complemented
14#20764C211yesmaximal, characteristic, complemented
15C2×#533C2211yescharacteristic
16C22×C4⋊Q8C2211yescharacteristic
17C2×C4×#41C2211yescharacteristic
18#3165—28no—
19C2×#501C2214yes—
20C2×#500C2212yes—
21C2×#498C2212yes—
22#2922—28no—
23C2×#433C2214yes—
24C2×#431C2212yes—
25C2×#431C2212yes—
26C2×#428C2214yes—
27C2×#424C2212yes—
28C2×#408C2212yes—
29C2×C4×M16C2211yescharacteristic
30C22×C4×C8C2211yescharacteristic
31#2857C2212yes—
32#2857C2212yes—
33#2857C2212yes—
34#2857C2212yes—
35#2629C2214yes—
36#2628C2214yes—
37#2582—21no—
38#2582—21no—
39#2568—21no—
40#2567—21no—
41#2563—21no—
42#2542—21no—
43#2538—21no—
44#2538—21no—
45#2487—22no—
46#2417—22no—
47#2417—22no—
48#2409—22no—
49#2317C2212yes—
50#2316C2212yes—
51#2311C2212yes—
52#2311C2212yes—
53#2285C2211yescharacteristic
54C2×#384C2211yescharacteristic
55#2088C2218yes—
56#2088C2218yes—
57#2087C2218yes—
58#2086C2218yes—
59#2077C2218yes—
60#2077C2218yes—
61#2076C2218yes—
62#2075C2218yes—
63#1991C2214yes—
64#1990C2212yes—
65#1990C2212yes—
66#1990C2214yes—
67#1990C2211yescharacteristic
68#1990C2211yescharacteristic
69#1990C2211yescharacteristic
70#1990C2211yescharacteristic
71#1836C2214yes—
72#1834C2212yes—
73#1834C2212yes—
74#1736C2214yes—
75#1735C2214yes—
76C2×#340C2214yes—
77C22×#41C2311yescharacteristic
78C23×Q8—21no—
79C22×M16C2312yes—
80C23×C8C2312yes—
81#533—24no—
82#533—24no—
83C2×C4⋊Q8—21no—
84C2×C4⋊Q8—21no—
85C2×C4⋊Q8—24no—
86C2×C4⋊Q8—24no—
87C2×C4⋊Q8C2311yescharacteristic
88C2×C4⋊Q8C2312yes—
89C2×C4⋊Q8C2311yescharacteristic
90C2×C4⋊Q8C2312yes—
91C2×#125C2312yes—
92C2×C22⋊Q8—21no—
93C2×C22⋊Q8—21no—
94C2×C22⋊Q8—22no—
95C2×C22⋊Q8—22no—
96C2×C22⋊Q8—22no—
97C2×C22⋊Q8—22no—
98C4×#41—24no—
99C4×#41—24no—
100C2×C4×Q8—21no—
101C2×C4×Q8—21no—
102C2×C4×Q8D411yescharacteristic
103C2×C4×Q8D411yescharacteristic
104C2×C4×D4D411yescharacteristic
105C2×C4×D4D412yes—
106C2×C4×D4D412yes—
107C2×C4×D4D411yescharacteristic
108C2×#117C2312yes—
109C2×#117C2312yes—
110C2×#117C2311yescharacteristic
111C2×#117C2312yes—
112C22×C4⋊C4—21no—
113C22×C4⋊C4—21no—
114C22×C4⋊C4—21no—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2312yes—
117C22×C42C2312yes—
118C22×C42C2311yescharacteristic
119#501—216no—
120#500—28no—
121#498—28no—
122#433—216no—
123#431—28no—
124#431—28no—
125#428—216no—
126C2×#107C2318yes—
127C2×#106C2318yes—
128#424—28no—
129C2×C4⋊C8C2314yes—
130C2×C4⋊C8C2314yes—
131#408—28no—
132C2×C22⋊C8C2314yes—
133C2×C22⋊C8C2314yes—
134C4×M16D418yes—
135C2×#97C2312yes—
136C2×C4×C8—24no—
137C2×C4×C8C2311yescharacteristic
138C2×C4×C8C2311yescharacteristic
139C2×C4×C8C2312yes—
140C2×C4×C8C2312yes—
141#400—24no—
142#399—24no—
143#398—24no—
144#398—24no—
145#396—24no—
146#396—24no—
147#395—24no—
148#393—24no—
149#391—22no—
150#391—22no—
151#391—22no—
152#391—22no—
153#391—22no—
154#391—22no—
155#390—22no—
156#390—22no—
157#389—22no—
158#389—22no—
159#389—22no—
160#389—22no—
161#389—22no—
162#389—22no—
163#389C2314yes—
164#389C2314yes—
165#389C2314yes—
166#389C2314yes—
167#386—24no—
168#386—24no—
169#385—24no—
170#384—22no—
171#384—21no—
172#384—21no—
173#384C2312yes—
174#384C2311yescharacteristic
175#384C2311yescharacteristic
176#384C2312yes—
177#382—22no—
178#382C2312yes—
179#380—22no—
180#380—22no—
181#380—22no—
182#380—22no—
183#380—21no—
184#380—21no—
185#380—24no—
186#380C2314yes—
187#380C2314yes—
188#380C2314yes—
189#380C2314yes—
190C4×C4⋊C4—21no—
191C4×C4⋊C4—21no—
192#340C2314yes—
193#340C2318yes—
194#340C2318yes—
195#340C2314yes—
196#340C2318yes—
197#340C2318yes—
198#340C2314yes—
199#340C2314yes—
200C2×#41—22no—
201C2×#41—22no—
202C2×#41—24no—
203C2×#41—24no—
204C22×Q8—21no—
205C22×Q8—21no—
206C22×Q8—22no—
207C22×Q8—22no—
208C22×Q8—44no—
209C22×Q8—82no—
210C22×Q8—82no—
211C22×Q8C2×D411yescharacteristic
212C22×D4C2×D412yes—
213C22×D4C2×D411yescharacteristic
214C23×C4—22no—
215C23×C4—22no—
216C23×C4C2×D412yes—
217C23×C4C2×D412yes—
218C23×C4C2×D411yescharacteristic
219C23×C4C2×D411yescharacteristic
220C23×C4C2412yes—
221C23×C4C2411yescharacteristic
222C2×M16—24no—
223C2×M16—24no—
224C2×M16C2×D418yes—
225C22×C8—24no—
226C22×C8—24no—
227C22×C8—24no—
228C22×C8C2412yes—
229C22×C8C2412yes—
230C22×C8C2414yes—
231C4⋊Q8—24no—
232C4⋊Q8—24no—
233C4⋊Q8—22no—
234C4⋊Q8—22no—
235C4⋊Q8—24no—
236C4⋊Q8—24no—
237C4⋊Q8—22no—
238C4⋊Q8—22no—
239C4⋊Q8—44no—
240C4⋊Q8—44no—
241#125—24no—
242#125—24no—
243C22⋊Q8—44no—
244C22⋊Q8—44no—
245C22⋊Q8—84no—
246C22⋊Q8—84no—
247C22⋊Q8—84no—
248C22⋊Q8—84no—
249C4×Q8—24no—
250C4×Q8—24no—
251C4×Q8—44no—
252C4×Q8—44no—
253C4×D4—28no—
254C4×D4—28no—
255C4×D4—24no—
256C4×D4—24no—
257#117—22no—
258#117—22no—
259#117—24no—
260#117—24no—
261#117—24no—
262#117—24no—
263#117—28no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—28no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—44no—
288C2×C4⋊C4—82no—
289C2×C4⋊C4—82no—
290C2×C4⋊C4C2×D411yescharacteristic
291C2×C4⋊C4C2×D411yescharacteristic
292C2×C4⋊C4C2×D412yes—
293C2×C4⋊C4C2×D412yes—
294C2×C4⋊C4C2414yes—
295C2×C4⋊C4C2414yes—
296C2×C4⋊C4C2412yes—
297C2×C4⋊C4C2412yes—
298C2×C4⋊C4C2412yes—
299C2×C4⋊C4C2412yes—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4C2×D412yes—
307C2×C22⋊C4C2×D412yes—
308C2×C22⋊C4C2×D412yes—
309C2×C42—24no—
310C2×C42—24no—
311C2×C42—22no—
312C2×C42—22no—
313C2×C42C2×D412yes—
314C2×C42C2×D412yes—
315C2×C42C2×D411yescharacteristic
316C2×C42C2×D411yescharacteristic
317C2×C42C2×D414yes—
318C2×C42#4114yes—
319C2×C42C2412yes—
320C2×C42C2411yescharacteristic
321C2×C42C2411yescharacteristic
322#107—216no—
323#106—216no—
324C4⋊C8—28no—
325C4⋊C8—28no—
326C22⋊C8—28no—
327C22⋊C8—28no—
328#97C2×D418yes—
329C4×C8—22no—
330C4×C8—22no—
331C4×C8—24no—
332C4×C8C2×D418yes—
333#95—22no—
334#95—22no—
335#95—24no—
336#95—24no—
337#95—24no—
338#95—24no—
339#95—24no—
340#95—24no—
341#95—24no—
342#95—24no—
343C24C22×D412yes—
344C24C22×D411yescharacteristic
345#41—48no—
346#41—48no—
347C2×Q8—22no—
348C2×Q8—22no—
349C2×Q8—22no—
350C2×Q8—42no—
351C2×Q8—42no—
352C2×Q8—44no—
353C2×Q8—44no—
354C2×Q8—42no—
355C2×Q8—44no—
356C2×Q8—42no—
357C2×Q8—44no—
358C2×Q8—42no—
359C2×Q8—42no—
360C2×Q8—82no—
361C2×Q8—82no—
362C2×Q8—816no—
363C2×D4—24no—
364C2×D4—24no—
365C2×D4—24no—
366C2×D4—22no—
367C2×D4—22no—
368C2×D4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—44no—
390C22×C4—44no—
391C22×C4—82no—
392C22×C4—82no—
393C22×C4—82no—
394C22×C4—82no—
395C22×C4#12114yes—
396C22×C4#12114yes—
397C22×C4C22×D412yes—
398C22×C4C22×D412yes—
399C22×C4C22×D412yes—
400C22×C4C22×D411yescharacteristic
401C22×C4C22×D411yescharacteristic
402C22×C4C22×D411yescharacteristic
403C22×C4C22×D411yescharacteristic
404C22×C4C22×D411yescharacteristic
405C22×C4C22×D412yes—
406C22×C4C2×#4112yes—
407C22×C42−1+414yes—
408C22×C42−1+414yes—
409C22×C4C2511yescharacteristic
410M16—216no—
411C2×C8—24no—
412C2×C8—24no—
413C2×C8—22no—
414C2×C8—22no—
415C2×C8—22no—
416C2×C8—22no—
417C2×C8—22no—
418C2×C8—22no—
419C2×C8—48no—
420C2×C8C22×D418yes—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—24no—
424C4⋊C4—24no—
425C4⋊C4—22no—
426C4⋊C4—24no—
427C4⋊C4—22no—
428C4⋊C4—24no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—28no—
432C4⋊C4—24no—
433C4⋊C4—24no—
434C4⋊C4—44no—
435C4⋊C4—44no—
436C4⋊C4—44no—
437C4⋊C4—44no—
438C4⋊C4—44no—
439C4⋊C4—42no—
440C4⋊C4—42no—
441C4⋊C4—44no—
442C4⋊C4—44no—
443C4⋊C4—42no—
444C4⋊C4—42no—
445C4⋊C4—44no—
446C4⋊C4—42no—
447C4⋊C4—42no—
448C4⋊C4—82no—
449C4⋊C4—82no—
450C4⋊C4—84no—
451C4⋊C4—84no—
452C4⋊C4—84no—
453C4⋊C4—84no—
454C22⋊C4—24no—
455C22⋊C4—24no—
456C22⋊C4—24no—
457C22⋊C4—44no—
458C22⋊C4—44no—
459C22⋊C4—84no—
460C22⋊C4—84no—
461C22⋊C4—84no—
462C22⋊C4—84no—
463C42—24no—
464C42—24no—
465C42—22no—
466C42—22no—
467C42—44no—
468C42—44no—
469C42#12114yes—
470C42#12114yes—
471C42C22×D412yes—
472C42C22×D412yes—
473C42C2×#4112yes—
474C42C2×#4112yes—
475C23—21no—
476C23—21no—
477C23—21no—
478C23—21no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—22no—
483C23—22no—
484C23—22no—
485C23—22no—
486C23C2×#12112yes—
487C23#53612yes—
488C23#53612yes—
489C23C23×D411yescharacteristic
490Q8—44no—
491Q8—88no—
492Q8—84no—
493Q8—88no—
494Q8—84no—
495D4—48no—
496D4—44no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—28no—
502C2×C4—28no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—48no—
515C2×C4—48no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—44no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—44no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—88no—
535C2×C4—164no—
536C2×C4C2×#12112yes—
537C2×C4C2×#12112yes—
538C2×C4C2×#12112yes—
539C2×C4C2×#12112yes—
540C2×C4C2×#12112yes—
541C2×C4#53612yes—
542C2×C4#53612yes—
543C2×C4#53618yes—
544C2×C4#54118yes—
545C2×C4#57612yes—
546C2×C4#57712yes—
547C2×C4#57712yes—
548C2×C4#57812yes—
549C2×C4C23×D411yescharacteristic
550C2×C4C22×#4111yescharacteristic
551C2×C4C2×2−1+412yes—
552C2×C4C2×2−1+412yes—
553C8—28no—
554C8—42no—
555C8—42no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—42no—
563C22—42no—
564C22—42no—
565C22—42no—
566C22—42no—
567C22—42no—
568C22—42no—
569C22#262114yes—
570C22C22×#12111yescharacteristic
571C22C2×#53611yescharacteristic
572C22C2×#53611yescharacteristic
573C22C2×#57611yescharacteristic
574C22C2×#57711yescharacteristic
575C22C2×#57711yescharacteristic
576C22C2×#57811yescharacteristic
577C4—22no—
578C4—22no—
579C4—42no—
580C4—42no—
581C4—42no—
582C4—84no—
583C4—82no—
584C4—82no—
585C4—84no—
586C4—82no—
587C4—82no—
588C4#262114yes—
589C4#306214yes—
590C4#306314yes—
591C4C2×#53612yes—
592C4C2×#54112yes—
593C2—24no—
594C2—82no—
595C2C2×#262111yescharacteristic
596C2C2×#306211yescharacteristic
597C2C2×#306311yescharacteristic
598C2#6087711yescharacteristic
599C2#6087811yescharacteristic
600C2#6088011yescharacteristic
601C2#6088111yescharacteristic
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
1. C2×#3165C2semidirect
5. #29885C2semidirect
6. #29878C2semidirect
7. #29877C2semidirect
8. #29877C2semidirect
9. #29876C2semidirect
10. #23671C2semidirect
12. #21318C2semidirect
13. C2×#1990C2semidirect
14. #20764C2semidirect

Maximal subgroups

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

subgroupindex
C2×#31652
C2×#29222
#352802
#352792
#298852
#298782
#298772
#298762
#236712
#236432
#213182
C2×#19902
#207642