Finite Group Databasebeta

← #496272

Subgroups — #496272

Order 512.

1072 conjugacy classes of subgroups.

The subgroup lattice has 633 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3061C211yesmaximal, characteristic
2C2×#2113C211yesmaximal, characteristic, complemented
3#20663C211yesmaximal, characteristic, complemented
4#13062C211yesmaximal, characteristic, complemented
5#13061C211yesmaximal, characteristic, complemented
6#13060C211yesmaximal, characteristic, complemented
7#13059C211yesmaximal, characteristic, complemented
8#12994C211yesmaximal, characteristic, complemented
9#12993C211yesmaximal, characteristic, complemented
10#12992C211yesmaximal, characteristic, complemented
11#12991C211yesmaximal, characteristic, complemented
12#12867C211yesmaximal, characteristic, complemented
13#12866C211yesmaximal, characteristic, complemented
14#12733C211yesmaximal, characteristic, complemented
15#12733C211yesmaximal, characteristic, complemented
16C22×#135C2211yescharacteristic
17C22×#131C2211yescharacteristic
18C2×#537—21no—
19C22×M32C2211yescharacteristic
20C23×C16C2211yescharacteristic
21#3061—24no—
22#3061C418yescomplemented
23C2×#468—21no—
24C2×#468—21no—
25C2×#467C2211yescharacteristic
26C2×#467C2211yescharacteristic
27C2×#466C2211yescharacteristic
28C2×#466C2211yescharacteristic
29C2×#427C2211yescharacteristic
30C22×#107C2211yescharacteristic
31C22×#107C2211yescharacteristic
32C2×#417—21no—
33#2311—21no—
34C2×#358C2211yescharacteristic
35C2×#358C2211yescharacteristic
36C2×#358C2211yescharacteristic
37C2×#358C2211yescharacteristic
38C2×#357C2211yescharacteristic
39C2×#357C2211yescharacteristic
40C2×#357C2211yescharacteristic
41C2×#357C2211yescharacteristic
42#2113—24no—
43C2×C22⋊C16C2211yescharacteristic
44C2×C22⋊C16C2211yescharacteristic
45C2×C22⋊C16C2211yescharacteristic
46C2×C22⋊C16C2211yescharacteristic
47#1809—21no—
48#1809—21no—
49#1808C2211yescharacteristic
50#1808C2211yescharacteristic
51#1808C2211yescharacteristic
52#1808C2211yescharacteristic
53#1736—21no—
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1380C2211yescharacteristic
59#1380C2211yescharacteristic
60#1380C2211yescharacteristic
61#1380C2211yescharacteristic
62C22×#41—21no—
63C22×#41D411yescharacteristic
64C2×#135—22no—
65C2×#135—41no—
66C2×#135—41no—
67C2×#135—41no—
68C2×#135—41no—
69C2×#135C2×C414yescomplemented
70C22×Q16C2311yescharacteristic
71C22×SD16—21no—
72C22×D8C2311yescharacteristic
73C2×#131—41no—
74C2×#131—41no—
75C2×#131—41no—
76C2×#131—41no—
77C2×#131C2×C414yes—
78C2×#131D414yes—
79C22×M16D411yescharacteristic
80C22×M16D411yescharacteristic
81C22×M16D411yescharacteristic
82C23×C8C2311yescharacteristic
83C23×C8C2311yescharacteristic
84C23×C8C2311yescharacteristic
85#537—28no—
86#537—44no—
87C2×C22⋊Q8—21no—
88C2×C22⋊Q8—21no—
89C2×C22⋊Q8—41no—
90C2×#121—21no—
91C2×#121—21no—
92C2×#121—41no—
93C2×#121—41no—
94C2×#121—41no—
95C2×#117—21no—
96C22×C4⋊C4—21no—
97C22×C4⋊C4—21no—
98C2×M32—22no—
99C2×M32—41no—
100C2×M32—41no—
101C22×C16—22no—
102C22×C16—41no—
103C22×C16—41no—
104C22×C16C2311yescharacteristic
105C22×C16C2311yescharacteristic
106C22×C16C2311yescharacteristic
107C22×C16C2311yescharacteristic
108#468—28no—
109#468—28no—
110#467—22no—
111#467—22no—
112#467C2×C414yes—
113#467C2×C414yescomplemented
114#466—22no—
115#466—22no—
116#466C2×C414yes—
117#466C2×C414yescomplemented
118#427—41no—
119#427—41no—
120C2×#107—21no—
121C2×#107—22no—
122C2×#107—21no—
123C2×#107—41no—
124C2×#107—41no—
125C2×#107C2×C414yescomplemented
126C2×#107C2311yescharacteristic
127C2×#107C2311yescharacteristic
128C2×#107C2311yescharacteristic
129C2×#107C2311yescharacteristic
130C2×#107C2311yescharacteristic
131C2×#107C2311yescharacteristic
132C2×#106—21no—
133#417—28no—
134C2×#103—21no—
135C2×#103—21no—
136C2×#102—21no—
137C2×#102—21no—
138#391—41no—
139#390—41no—
140#386—41no—
141#386—41no—
142#380—21no—
143#380—21no—
144#380—21no—
145#380—21no—
146#358—41no—
147#358—41no—
148#358—41no—
149#358—41no—
150#357—41no—
151#357—41no—
152#357—41no—
153#357—41no—
154C22⋊C16—22no—
155C22⋊C16—22no—
156C22⋊C16—22no—
157C22⋊C16—22no—
158#340—21no—
159#340—21no—
160#340—21no—
161#340—21no—
162C2×#41—24no—
163C2×#41—41no—
164C2×#41—41no—
165C2×#41—41no—
166C2×#41—41no—
167C2×#41—42no—
168C2×#41—42no—
169C2×#41—42no—
170C2×#41—42no—
171C2×#41—42no—
172C2×#41C22⋊C414yes—
173C2×#41SD1614yes—
174C22×Q8—21no—
175C22×Q8D811yescharacteristic
176C22×D4—21no—
177C22×D4—41no—
178C22×D4—41no—
179C22×D4—41no—
180C22×D4D811yescharacteristic
181C22×D4D811yescharacteristic
182C22×D4D811yescharacteristic
183C23×C4—41no—
184C23×C4—41no—
185C23×C4—41no—
186C23×C4C2×D411yescharacteristic
187C23×C4C2×D411yescharacteristic
188C23×C4C2×D411yescharacteristic
189#135—44no—
190#135—44no—
191#135—44no—
192#135—44no—
193C2×Q16—21no—
194C2×Q16—21no—
195C2×Q16—21no—
196C2×Q16—21no—
197C2×Q16—21no—
198C2×Q16C22×C412yes—
199C2×SD16—24no—
200C2×SD16—42no—
201C2×SD16—42no—
202C2×D8—21no—
203C2×D8—21no—
204C2×D8—21no—
205C2×D8—21no—
206C2×D8—21no—
207C2×D8C22×C412yes—
208#131—44no—
209#131—44no—
210#131—44no—
211#131—44no—
212C2×M16—41no—
213C2×M16—41no—
214C2×M16—41no—
215C2×M16—41no—
216C2×M16—41no—
217C2×M16—41no—
218C2×M16C22⋊C414yes—
219C2×M16C22⋊C414yes—
220C2×M16C22⋊C414yes—
221C22×C8—21no—
222C22×C8—21no—
223C22×C8—21no—
224C22×C8—21no—
225C22×C8—21no—
226C22×C8—21no—
227C22×C8—21no—
228C22×C8—21no—
229C22×C8—21no—
230C22×C8—21no—
231C22×C8—21no—
232C22×C8—21no—
233C22×C8C22×C412yes—
234C22×C8C22×C412yes—
235C22×C8C22×C412yes—
236C22×C8C2×D411yescharacteristic
237C22×C8C2×D411yescharacteristic
238C22×C8C2×D411yescharacteristic
239C22×C8C2×D412yes—
240C22×C8C2×D412yes—
241C22×C8C2×D412yes—
242C22×C8C2411yescharacteristic
243C22⋊Q8—24no—
244C22⋊Q8—24no—
245C22⋊Q8—48no—
246C22⋊Q8—42no—
247C22⋊Q8—42no—
248#121—24no—
249#121—24no—
250#121—48no—
251#121—48no—
252#121—48no—
253#121—42no—
254#121—42no—
255#117—42no—
256#117—42no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—41no—
265C2×C4⋊C4—41no—
266C2×C4⋊C4—41no—
267C2×C4⋊C4—42no—
268C2×C4⋊C4—42no—
269C2×C4⋊C4—42no—
270C2×C4⋊C4—41no—
271C2×C22⋊C4—41no—
272C2×C22⋊C4—41no—
273C2×C22⋊C4—41no—
274C2×C22⋊C4—41no—
275C2×C22⋊C4—41no—
276C2×C22⋊C4—41no—
277C2×C22⋊C4—41no—
278C2×C22⋊C4—41no—
279C2×C22⋊C4—41no—
280C2×C42—41no—
281M32—44no—
282C2×C16—21no—
283C2×C16—21no—
284C2×C16—21no—
285C2×C16—21no—
286C2×C16—21no—
287C2×C16—21no—
288C2×C16—21no—
289C2×C16—21no—
290C2×C16—21no—
291C2×C16—21no—
292C2×C16—21no—
293C2×C16—21no—
294C2×C16—44no—
295#107—24no—
296#107—21no—
297#107—21no—
298#107—21no—
299#107—21no—
300#107—21no—
301#107—21no—
302#107—21no—
303#107—21no—
304#107—21no—
305#107—21no—
306#107—42no—
307#107C22×C412yes—
308#107C22×C412yes—
309#106—42no—
310#103—24no—
311#103—24no—
312#102—24no—
313#102—24no—
314#95—41no—
315#95—41no—
316#95—41no—
317#95—41no—
318C24—41no—
319C24C2×D811yescharacteristic
320C24C2×D811yescharacteristic
321C24C2×D811yescharacteristic
322#41—44no—
323#41—44no—
324#41—44no—
325#41—44no—
326#41—48no—
327#41—48no—
328#41—84no—
329#41—84no—
330C2×Q8—22no—
331C2×Q8—41no—
332C2×Q8—41no—
333C2×Q8—41no—
334C2×Q8—41no—
335C2×Q8—41no—
336C2×Q8—41no—
337C2×Q8—41no—
338C2×Q8#10214yes—
339C2×D4—22no—
340C2×D4—41no—
341C2×D4—41no—
342C2×D4—41no—
343C2×D4—41no—
344C2×D4—41no—
345C2×D4—41no—
346C2×D4—44no—
347C2×D4—44no—
348C2×D4—44no—
349C2×D4—41no—
350C2×D4—41no—
351C2×D4—41no—
352C2×D4—41no—
353C2×D4—41no—
354C2×D4—82no—
355C2×D4—82no—
356C2×D4—161no—
357C2×D4—161no—
358C2×D4—161no—
359C2×D4—161no—
360C2×D4#10214yes—
361C2×D4#10214yes—
362C2×D4#10214yes—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—44no—
379C22×C4—44no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—82no—
385C22×C4—82no—
386C22×C4—82no—
387C22×C4—161no—
388C22×C4—161no—
389C22×C4—161no—
390C22×C4—161no—
391C22×C4C2×C22⋊C412yes—
392C22×C4C2×C22⋊C412yes—
393C22×C4C2×C22⋊C412yes—
394C22×C4C2×D811yescharacteristic
395C22×C4C2×D811yescharacteristic
396C22×C4C2×D811yescharacteristic
397C22×C4C2×SD1612yes—
398C22×C4C2×SD1612yes—
399C22×C4C2×SD1612yes—
400C22×C4C22×D411yescharacteristic
401Q16—22no—
402Q16—22no—
403Q16—41no—
404Q16—41no—
405SD16—48no—
406D8—22no—
407D8—22no—
408D8—41no—
409D8—41no—
410M16—44no—
411M16—44no—
412M16—44no—
413C2×C8—21no—
414C2×C8—21no—
415C2×C8—21no—
416C2×C8—21no—
417C2×C8—21no—
418C2×C8—21no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8C2×C22⋊C412yes—
432C2×C8C2×C22⋊C412yes—
433C2×C8C2×C22⋊C412yes—
434C2×C8C23×C411yescharacteristic
435C2×C8C22×D411yescharacteristic
436C2×C8H(5,𝔽2)11yescharacteristic
437C2×C8H(5,𝔽2)11yescharacteristic
438C2×C8H(5,𝔽2)11yescharacteristic
439C2×C8H(5,𝔽2)11yescharacteristic
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—41no—
443C4⋊C4—41no—
444C4⋊C4—44no—
445C4⋊C4—44no—
446C4⋊C4—44no—
447C4⋊C4—41no—
448C4⋊C4—41no—
449C4⋊C4—41no—
450C4⋊C4—41no—
451C4⋊C4—41no—
452C4⋊C4—41no—
453C4⋊C4—41no—
454C4⋊C4—41no—
455C4⋊C4—82no—
456C22⋊C4—44no—
457C22⋊C4—44no—
458C22⋊C4—44no—
459C22⋊C4—44no—
460C22⋊C4—82no—
461C22⋊C4—161no—
462C22⋊C4—161no—
463C22⋊C4—161no—
464C22⋊C4—161no—
465C42—82no—
466C16—41no—
467C16—41no—
468C16—41no—
469C16—41no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23—21no—
475C23—21no—
476C23—21no—
477C23—21no—
478C23—21no—
479C23—21no—
480C23—21no—
481C23—21no—
482C23—21no—
483C23—21no—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—42no—
489C23—81no—
490C23—81no—
491C23—81no—
492C23—81no—
493C23—81no—
494C23—81no—
495C23C2×#10212yes—
496C23C2×#10212yes—
497C23C2×#10212yes—
498C23C22×D811yescharacteristic
499Q8—44no—
500Q8—42no—
501Q8—42no—
502Q8—81no—
503Q8—81no—
504D4—44no—
505D4—44no—
506D4—44no—
507D4—42no—
508D4—42no—
509D4—88no—
510D4—81no—
511D4—81no—
512D4—164no—
513D4—164no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—88no—
536C2×C4—81no—
537C2×C4—81no—
538C2×C4—81no—
539C2×C4—81no—
540C2×C4—81no—
541C2×C4—81no—
542C2×C4—81no—
543C2×C4—81no—
544C2×C4—81no—
545C2×C4—81no—
546C2×C4—81no—
547C2×C4—81no—
548C2×C4—81no—
549C2×C4—81no—
550C2×C4—81no—
551C2×C4—81no—
552C2×C4—81no—
553C2×C4—81no—
554C2×C4—164no—
555C2×C4—164no—
556C2×C4C2×#10212yes—
557C2×C4C2×#10212yes—
558C2×C4C2×#10212yes—
559C2×C4C22×C22⋊C411yescharacteristic
560C2×C4#53411yescharacteristic
561C2×C4#53411yescharacteristic
562C2×C4C22×SD1611yescharacteristic
563C2×C4#57511yescharacteristic
564C2×C4#57511yescharacteristic
565C8—22no—
566C8—22no—
567C8—22no—
568C8#51811yescharacteristic
569C8#51811yescharacteristic
570C8#53411yescharacteristic
571C8#53411yescharacteristic
572C22—22no—
573C22—22no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—22no—
580C22—22no—
581C22—41no—
582C22—41no—
583C22—41no—
584C22—41no—
585C22—41no—
586C22—41no—
587C22—41no—
588C22—41no—
589C22—41no—
590C22—41no—
591C22—41no—
592C22—41no—
593C22—82no—
594C22—82no—
595C22—82no—
596C22—161no—
597C22—161no—
598C22—161no—
599C22—161no—
600C22C22×#10211yescharacteristic
601C22#318611yescharacteristic
602C22#318611yescharacteristic
603C22#341511yescharacteristic
604C22#341611yescharacteristic
605C22#341611yescharacteristic
606C22#341711yescharacteristic
607C4—22no—
608C4—22no—
609C4—22no—
610C4—82no—
611C4—82no—
612C4—82no—
613C4—161no—
614C4—161no—
615C4—161no—
616C4—161no—
617C4#230811yescharacteristic
618C4#289611yescharacteristic
619C4#318711yescharacteristic
620C4#318711yescharacteristic
621C2—42no—
622C2—42no—
623C2—42no—
624C2—162no—
625C2#2175811yescharacteristic
626C2#3338111yescharacteristic
627C2#3338211yescharacteristic
628C2#3376411yescharacteristic
629C2#3376711yescharacteristic
630C2#3376911yescharacteristic
631C2#3377011yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radicalC2×#3061C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#41 ◁ C22×#131 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2113C2semidirect
3. #20663C2semidirect
4. #13062C2semidirect
5. #13061C2semidirect
6. #13060C2semidirect
7. #13059C2semidirect
8. #12994C2semidirect
9. #12993C2semidirect
10. #12992C2semidirect
11. #12991C2semidirect
12. #12867C2semidirect
13. #12866C2semidirect
14. #12733C2semidirect
15. #12733C2semidirect
22. #3061C4semidirect
69. C2×#135C2×C4semidirect
113. #467C2×C4semidirect
117. #466C2×C4semidirect
125. C2×#107C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#30612
C2×#21132
#206632
#130622
#130612
#130602
#130592
#129942
#129932
#129922
#129912
#128672
#128662
#127332