Finite Group Databasebeta

← #364839

Subgroups — #364839

Order 512.

1296 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3076C211yesmaximal, characteristic
2C2×#2961C211yesmaximal, characteristic, complemented
3C2×#1986C211yesmaximal, characteristic, complemented
4C2×#1806C211yesmaximal, characteristic
5#20375C211yesmaximal, characteristic
6#11221C212yesmaximal, complemented
7#11068C212yesmaximal
8#10934C212yesmaximal
9#10909C212yesmaximal, complemented
10#10853C211yesmaximal, characteristic, complemented
11#10835C211yesmaximal, characteristic, complemented
12C2×#533C2211yescharacteristic
13C2×C4×#41C2211yescharacteristic
14#3076—24no—
15#3076—24no—
16C2×#479C412yescomplemented
17C2×#477C411yescharacteristic, complemented
18C2×#475C411yescharacteristic, complemented
19C2×#478C412yescomplemented
20C2×#476C411yescharacteristic, complemented
21C2×#474C411yescharacteristic, complemented
22#2961—24no—
23#2961—24no—
24C2×#446—22no—
25C2×#446—22no—
26C2×#427C2212yes—
27C2×#424C2212yes—
28C2×#423C2211yescharacteristic
29C22×C4⋊C8C2211yescharacteristic
30C22×#104C2212yes—
31C2×#416C2212yes—
32C2×#408C2211yescharacteristic
33C2×#408C2211yescharacteristic
34C2×#405C2212yes—
35C2×C4×M16C2211yescharacteristic
36C22×#97C2211yescharacteristic
37C2×C4×C4⋊C4C2211yescharacteristic
38#1986—24no—
39#1806—24no—
40#1750—22no—
41#1749—22no—
42C2×#341C2212yes—
43C2×#341C2212yes—
44C2×#339C2212yes—
45C2×#339C2212yes—
46C2×#339C2212yes—
47#1729C2211yescharacteristic
48#1729C2211yescharacteristic
49C2×#336C2212yes—
50C2×#336C2212yes—
51C22×#41C2311yescharacteristic
52C22×M16C2312yes—
53C22×M16C2311yescharacteristic
54C22×M16C2311yescharacteristic
55C23×C8C2312yes—
56C23×C8C2311yescharacteristic
57C23×C8C2311yescharacteristic
58#533—22no—
59#533—22no—
60#533D414yes—
61#533D414yes—
62C2×C4⋊Q8C2×C411yescharacteristic
63C2×C4⋊Q8C2×C411yescharacteristic
64C2×#125C2×C412yes—
65C2×C22⋊Q8—21no—
66C2×C22⋊Q8—21no—
67C4×#41—22no—
68C4×#41—22no—
69C4×#41—22no—
70C4×#41—22no—
71C2×C4×Q8—21no—
72C2×C4×Q8—21no—
73C2×C4×Q8C2×C411yescharacteristic
74C2×C4×Q8C2×C411yescharacteristic
75C2×C4×D4—21no—
76C2×C4×D4—21no—
77C2×C4×D4C2×C411yescharacteristic
78C2×C4×D4C2×C411yescharacteristic
79C2×#117—21no—
80C2×#117C2311yescharacteristic
81C2×#117C2312yes—
82C22×C4⋊C4—21no—
83C22×C42—21no—
84C22×C42C2312yes—
85C22×C42C2311yescharacteristic
86#479—28no—
87#478—28no—
88#477—24no—
89#476—24no—
90#475—24no—
91#474—24no—
92#446—216no—
93#446—216no—
94#427—24no—
95#427—24no—
96C2×#107C2×C412yes—
97C2×#107C2×C412yes—
98C2×#106C2×C412yes—
99C2×#106C2×C412yes—
100#424—24no—
101#424—24no—
102#423—22no—
103#423D414yes—
104C2×C4⋊C8—24no—
105C2×C4⋊C8—22no—
106C2×C4⋊C8—22no—
107C2×C4⋊C8—21no—
108C2×C4⋊C8—21no—
109C2×C4⋊C8C2×C412yes—
110C2×C4⋊C8C2×C411yescharacteristic
111C2×C4⋊C8C2×C411yescharacteristic
112C2×#104—24no—
113C2×#104—24no—
114C2×#104—42no—
115#416—24no—
116#416—24no—
117C2×#103C2×C412yes—
118C2×#103C2×C412yes—
119C2×#102C2×C412yes—
120C2×#102C2×C412yes—
121#408—22no—
122#408—22no—
123#408—22no—
124#408Q814yescomplemented
125C2×C22⋊C8—22no—
126C2×C22⋊C8—22no—
127#405—24no—
128#405—24no—
129C4×M16—22no—
130C4×M16—22no—
131C2×#97—22no—
132C2×#97—22no—
133C2×#97—22no—
134C2×#97—21no—
135C2×#97D412yes—
136C2×#97D412yes—
137C2×C4×C8—22no—
138C2×C4×C8—21no—
139C4×C4⋊C4—22no—
140C4×C4⋊C4—22no—
141C2×#95—21no—
142#341—24no—
143#341—42no—
144#339—24no—
145#339—24no—
146#339—24no—
147#336—22no—
148#336—22no—
149#336—24no—
150#336—42no—
151C2×#41—21no—
152C2×#41—21no—
153C2×#41—21no—
154C2×#41—21no—
155C2×#41—21no—
156C2×#41—21no—
157C2×#41—21no—
158C2×#41—21no—
159C2×#41—21no—
160C2×#41—21no—
161C2×#41C2×Q812yes—
162C2×#41#4112yes—
163C22×Q8—21no—
164C22×Q8—21no—
165C22×Q8C22×C411yescharacteristic
166C22×D4—21no—
167C22×D4C22×C411yescharacteristic
168C23×C4—21no—
169C23×C4—22no—
170C23×C4—21no—
171C23×C4—21no—
172C23×C4#4111yescharacteristic
173C23×C4#4111yescharacteristic
174C23×C4C2411yescharacteristic
175C2×M16—22no—
176C2×M16—21no—
177C2×M16—21no—
178C2×M16—21no—
179C2×M16—21no—
180C2×M16—22no—
181C2×M16—21no—
182C2×M16—21no—
183C2×M16—22no—
184C2×M16—21no—
185C2×M16—22no—
186C2×M16—21no—
187C2×M16—21no—
188C2×M16—22no—
189C2×M16—21no—
190C2×M16C2×D414yes—
191C2×M16C2×Q812yes—
192C2×M16#4112yes—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—21no—
198C22×C8—21no—
199C22×C8—22no—
200C22×C8—22no—
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—22no—
204C22×C8—22no—
205C22×C8—22no—
206C22×C8—21no—
207C22×C8—21no—
208C22×C8—21no—
209C22×C8—22no—
210C22×C8—22no—
211C22×C8—21no—
212C22×C8C22×C412yes—
213C22×C8C22×C412yes—
214C22×C8C2×D412yes—
215C22×C8C2×D412yes—
216C22×C8C2×Q812yes—
217C22×C8#4112yes—
218C4⋊Q8—22no—
219C4⋊Q8—22no—
220C4⋊Q8C22⋊C414yes—
221C4⋊Q8C22⋊C414yes—
222#125—24no—
223#125C22⋊C418yes—
224C22⋊Q8—24no—
225C22⋊Q8—24no—
226C22⋊Q8—24no—
227C22⋊Q8—24no—
228C4×Q8—24no—
229C4×Q8—24no—
230C4×Q8—24no—
231C4×Q8—24no—
232C4×Q8—22no—
233C4×Q8—22no—
234C4×Q8—22no—
235C4×Q8—22no—
236C4×D4—22no—
237C4×D4—22no—
238C4×D4—22no—
239C4×D4—22no—
240C4×D4—42no—
241C4×D4—42no—
242C4×D4—42no—
243C4×D4—42no—
244#117—22no—
245#117—22no—
246#117—21no—
247#117—21no—
248#117—21no—
249#117—21no—
250#117—42no—
251#117—42no—
252#117C2×D414yes—
253#117C2×D414yes—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4C22×C411yescharacteristic
263C2×C4⋊C4C22×C411yescharacteristic
264C2×C4⋊C4C22×C412yes—
265C2×C4⋊C4C22×C412yes—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—21no—
270C2×C42—22no—
271C2×C42—21no—
272C2×C42—21no—
273C2×C42—21no—
274C2×C42—22no—
275C2×C42—22no—
276C2×C42—22no—
277C2×C42—22no—
278C2×C42—22no—
279C2×C42—22no—
280C2×C42—42no—
281C2×C42—42no—
282C2×C42C22×C411yescharacteristic
283C2×C42C22×C411yescharacteristic
284C2×C42C2×D412yes—
285C2×C42C2×D412yes—
286C2×C42C2×D412yes—
287C2×C42C2×D411yescharacteristic
288C2×C42C2×D411yescharacteristic
289C2×C42#4112yes—
290#107—24no—
291#107—24no—
292#106—24no—
293#106—24no—
294C4⋊C8—216no—
295C4⋊C8—22no—
296C4⋊C8—22no—
297C4⋊C8—24no—
298C4⋊C8—24no—
299C4⋊C8—24no—
300#104—48no—
301#103—24no—
302#103—24no—
303#102—24no—
304#102—24no—
305C22⋊C8—28no—
306C22⋊C8—28no—
307#97—28no—
308#97—24no—
309#97—24no—
310#97—42no—
311C4×C8—28no—
312C4×C8—42no—
313#95—42no—
314C24—21no—
315C24C2×#4111yescharacteristic
316#41—22no—
317#41—22no—
318#41—22no—
319#41—22no—
320#41—22no—
321#41—22no—
322#41—22no—
323#41—22no—
324#41—41no—
325#41—41no—
326#41—41no—
327#41—41no—
328#41—41no—
329#41—41no—
330#41—41no—
331#41—41no—
332C2×Q8—21no—
333C2×Q8—24no—
334C2×Q8—24no—
335C2×Q8—21no—
336C2×Q8—21no—
337C2×Q8—21no—
338C2×Q8—21no—
339C2×Q8—82no—
340C2×Q8—82no—
341C2×Q8C4×Q812yes—
342C2×D4—22no—
343C2×D4—22no—
344C2×D4—21no—
345C2×D4—21no—
346C2×D4—21no—
347C2×D4—21no—
348C2×D4—21no—
349C2×D4—41no—
350C2×D4—41no—
351C2×D4—41no—
352C2×D4—41no—
353C2×D4C4×Q812yes—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—82no—
377C22×C4—82no—
378C22×C4C4×D411yescharacteristic
379C22×C4C4×D411yescharacteristic
380C22×C4C4×D411yescharacteristic
381C22×C4C4×D411yescharacteristic
382C22×C4#12112yes—
383C22×C4#12112yes—
384C22×C4#12312yes—
385C22×C4#12312yes—
386C22×C4C23×C411yescharacteristic
387C22×C4C22×D411yescharacteristic
388C22×C4C22×D411yescharacteristic
389C22×C4C22×D411yescharacteristic
390C22×C4C22×D411yescharacteristic
391C22×C4C22×Q811yescharacteristic
392C22×C4C2×#4111yescharacteristic
393C22×C4C2×#4111yescharacteristic
394C22×C4C2×#4111yescharacteristic
395C22×C4C2×#4111yescharacteristic
396C22×C4H(5,𝔽2)11yescharacteristic
397C22×C42−1+411yescharacteristic
398C22×C42−1+411yescharacteristic
399C22×C42−1+411yescharacteristic
400M16—24no—
401M16—24no—
402M16—22no—
403M16—22no—
404M16—22no—
405M16—22no—
406M16—42no—
407M16—41no—
408M16—42no—
409M16—41no—
410M16—41no—
411M16—41no—
412C2×C8—22no—
413C2×C8—22no—
414C2×C8—22no—
415C2×C8—22no—
416C2×C8—22no—
417C2×C8—22no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—24no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—41no—
431C2×C8—41no—
432C2×C8—42no—
433C2×C8—41no—
434C2×C8—42no—
435C2×C8—41no—
436C2×C8—41no—
437C2×C8—41no—
438C2×C8—42no—
439C2×C8—42no—
440C2×C8—41no—
441C2×C8—42no—
442C2×C8—42no—
443C2×C8—41no—
444C2×C8C22⋊Q814yes—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—28no—
448C4⋊C4—28no—
449C4⋊C4—24no—
450C4⋊C4—24no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—48no—
456C4⋊C4—42no—
457C4⋊C4—42no—
458C4⋊C4C2×C22⋊C414yes—
459C4⋊C4C2×C22⋊C414yes—
460C22⋊C4—24no—
461C22⋊C4—24no—
462C22⋊C4—41no—
463C22⋊C4—41no—
464C22⋊C4—42no—
465C22⋊C4—42no—
466C42—24no—
467C42—24no—
468C42—41no—
469C42—41no—
470C42—42no—
471C42—42no—
472C42—42no—
473C42—42no—
474C42—42no—
475C42—42no—
476C42C2×C22⋊C412yes—
477C42C2×C22⋊C412yes—
478C42C4×D412yes—
479C42C4×D412yes—
480C42#12012yes—
481C42#12012yes—
482C42#12112yes—
483C42#12112yes—
484C23—22no—
485C23—22no—
486C23—22no—
487C23—41no—
488C23—42no—
489C23—42no—
490C23—41no—
491C23C2×C4×D411yescharacteristic
492C23C2×#12111yescharacteristic
493C23C2×#12311yescharacteristic
494C23#52911yescharacteristic
495C23#53311yescharacteristic
496C23#54111yescharacteristic
497C23#54211yescharacteristic
498Q8—22no—
499Q8—22no—
500Q8—41no—
501Q8—41no—
502Q8—84no—
503Q8—84no—
504D4—22no—
505D4—22no—
506D4—42no—
507D4—42no—
508D4—42no—
509D4—42no—
510D4—41no—
511D4—41no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—22no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—41no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—41no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—84no—
539C2×C4—84no—
540C2×C4—82no—
541C2×C4—82no—
542C2×C4—82no—
543C2×C4—82no—
544C2×C4#38612yes—
545C2×C4#38612yes—
546C2×C4#38612yes—
547C2×C4#38612yes—
548C2×C4C22×C22⋊C411yescharacteristic
549C2×C4C2×C4×D411yescharacteristic
550C2×C4C2×C4×Q811yescharacteristic
551C2×C4C4×#4111yescharacteristic
552C2×C4#51911yescharacteristic
553C2×C4#52011yescharacteristic
554C2×C4C2×#12011yescharacteristic
555C2×C4C2×#12111yescharacteristic
556C2×C4C2×C22⋊Q811yescharacteristic
557C2×C4#52511yescharacteristic
558C2×C4#53611yescharacteristic
559C2×C4#53711yescharacteristic
560C2×C4#54611yescharacteristic
561C2×C4#54711yescharacteristic
562C2×C4#54811yescharacteristic
563C2×C4#54811yescharacteristic
564C2×C4#54811yescharacteristic
565C2×C4#54811yescharacteristic
566C2×C4#54811yescharacteristic
567C2×C4D4×Q811yescharacteristic
568C2×C4D4×Q811yescharacteristic
569C2×C4D4×Q811yescharacteristic
570C2×C4#55011yescharacteristic
571C2×C4#55011yescharacteristic
572C2×C4#55211yescharacteristic
573C2×C4#55211yescharacteristic
574C2×C4#55411yescharacteristic
575C2×C4#55611yescharacteristic
576C8—24no—
577C8—42no—
578C8—42no—
579C8—42no—
580C8—44no—
581C8—42no—
582C8—42no—
583C22—22no—
584C22—22no—
585C22—22no—
586C22—22no—
587C22—21no—
588C22—21no—
589C22—21no—
590C22—21no—
591C22—44no—
592C22—42no—
593C22—82no—
594C22—81no—
595C22—81no—
596C22C2×#38611yescharacteristic
597C22C4×C22⋊Q811yescharacteristic
598C22#232311yescharacteristic
599C22#242111yescharacteristic
600C22#242311yescharacteristic
601C22#242611yescharacteristic
602C22#242911yescharacteristic
603C22#295612yes—
604C22#295712yes—
605C4—22no—
606C4—22no—
607C4—22no—
608C4—22no—
609C4—42no—
610C4—44no—
611C4—44no—
612C4—84no—
613C4—81no—
614C4—81no—
615C4Q8×C22⋊C411yescharacteristic
616C4#234111yescharacteristic
617C4#236011yescharacteristic
618C4#236211yescharacteristic
619C4#240711yescharacteristic
620C4#240911yescharacteristic
621C4#253611yescharacteristic
622C4#253811yescharacteristic
623C2—21no—
624C2—21no—
625C2—21no—
626C2—21no—
627C2—82no—
628C2#1396911yescharacteristic
629C2#2091811yescharacteristic
630C2#2091911yescharacteristic
631C2#2162912yes—
632C2#2213212yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C2×C4×D4
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×#97 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2961C2semidirect
3. C2×#1986C2semidirect
6. #11221C2semidirect
9. #10909C2semidirect
10. #10853C2semidirect
11. #10835C2semidirect
16. C2×#479C4semidirect
17. C2×#477C4semidirect
18. C2×#475C4semidirect
19. C2×#478C4semidirect
20. C2×#476C4semidirect
21. C2×#474C4semidirect
124. #408Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#30762
C2×#29612
C2×#19862
C2×#18062
#203752
#112212
#110682
#109342
#109092
#108532
#108352