Finite Group Databasebeta

← #370102

Subgroups — #370102

Order 512.

1414 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
1C22×#428C211yesmaximal, characteristic
2C2×#1952C212yesmaximal, complemented
3C2×#1855C211yesmaximal, characteristic
4#20846C211yesmaximal, characteristic
5C2×#1852C211yesmaximal, characteristic
6#20379C211yesmaximal, characteristic
7#11331C212yesmaximal, complemented
8#11249C212yesmaximal, complemented
9#10896C212yesmaximal, complemented
10#10739C211yesmaximal, characteristic
11#10727C211yesmaximal, characteristic
12C23×M16C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C22×#108C2212yes—
16C2×#428—28no—
17C2×#428—21no—
18C2×#428—21no—
19C2×#428C2212yes—
20C2×#428C2211yescharacteristic
21C2×#428C2212yes—
22C2×#428C2211yescharacteristic
23C22×#107—21no—
24C22×#107C2212yes—
25C22×#106—21no—
26C2×#407C2211yescharacteristic
27C2×#407C2211yescharacteristic
28C22×C22⋊C8C2211yescharacteristic
29C2×#405C2212yes—
30C2×#380C2211yescharacteristic
31#1952—28no—
32#1855C418yescomplemented
33#1854—21no—
34#1854—21no—
35#1853—21no—
36#1852—21no—
37#1852—21no—
38#1852—22no—
39#1852—21no—
40#1750—22no—
41#1749—21no—
42C4×C22⋊C8—21no—
43C2×#343C2212yes—
44C2×#343C2212yes—
45C2×#343C2212yes—
46C2×#343C2212yes—
47C2×#341C2212yes—
48#1736C2211yescharacteristic
49#1736C2211yescharacteristic
50C2×#340C2212yes—
51#1729C2212yes—
52C24×C4—21no—
53C24×C4C2311yescharacteristic
54C22×M16—21no—
55C22×M16—22no—
56C22×M16—24no—
57C22×M16—24no—
58C22×M16C2×C414yes—
59C22×M16C2311yescharacteristic
60C22×M16C2311yescharacteristic
61C22×M16C2312yes—
62C22×M16C2312yes—
63C23×C8—21no—
64C23×C8C2312yes—
65C2×#117—21no—
66C2×#117—21no—
67C2×#117—24no—
68C2×#117—42no—
69C2×#117D411yescharacteristic
70C2×#117D411yescharacteristic
71C2×#117C2311yescharacteristic
72C2×#117C2311yescharacteristic
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—24no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4—21no—
78C22×C4⋊C4—21no—
79C22×C4⋊C4—42no—
80C22×C4⋊C4D411yescharacteristic
81C22×C4⋊C4D411yescharacteristic
82C22×C4⋊C4C2311yescharacteristic
83C22×C4⋊C4C2311yescharacteristic
84C22×C4⋊C4C2312yes—
85C22×C22⋊C4—21no—
86C22×C22⋊C4—21no—
87C22×C42—21no—
88C2×#108—24no—
89C2×#108—24no—
90C2×#108C2×C418yescomplemented
91#428—28no—
92#428—28no—
93#428—24no—
94#428—24no—
95#428—24no—
96#428—24no—
97#428—42no—
98#428D418yes—
99C2×#107—28no—
100C2×#107—22no—
101C2×#107—22no—
102C2×#107—22no—
103C2×#107—22no—
104C2×#107—21no—
105C2×#107—21no—
106C2×#107—24no—
107C2×#107—42no—
108C2×#106—28no—
109C2×#106—22no—
110C2×#106—22no—
111C2×#106—21no—
112C2×#106—21no—
113#407—22no—
114#407—22no—
115#407C2×C414yes—
116#407Q814yes—
117C2×C22⋊C8—21no—
118C2×C22⋊C8—21no—
119C2×C22⋊C8—21no—
120C2×C22⋊C8—21no—
121C2×C22⋊C8C2×C414yes—
122C2×C22⋊C8D411yescharacteristic
123C2×C22⋊C8D411yescharacteristic
124C2×C22⋊C8D412yes—
125C2×C22⋊C8Q814yes—
126#405—24no—
127#405—42no—
128C2×#97—22no—
129C2×C4×C8—22no—
130#380—21no—
131#380—21no—
132#380—21no—
133#380—21no—
134C2×#95—21no—
135#343—24no—
136#343—24no—
137#343—24no—
138#343—42no—
139#341C2×C418yescomplemented
140#340—22no—
141#340—22no—
142#340—21no—
143#340—21no—
144#340—21no—
145#340—21no—
146#336—22no—
147#336—22no—
148C25#4111yescharacteristic
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—22no—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—44no—
159C23×C4—41no—
160C23×C4—41no—
161C23×C4—82no—
162C23×C4—82no—
163C23×C4C22×C412yes—
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2×Q812yes—
169C23×C4C2×Q812yes—
170C23×C4#4112yes—
171C23×C4#4111yescharacteristic
172C23×C4C2411yescharacteristic
173C2×M16—216no—
174C2×M16—24no—
175C2×M16—24no—
176C2×M16—24no—
177C2×M16—22no—
178C2×M16—22no—
179C2×M16—22no—
180C2×M16—216no—
181C2×M16—21no—
182C2×M16—21no—
183C2×M16—22no—
184C2×M16—22no—
185C2×M16—22no—
186C2×M16—22no—
187C2×M16—22no—
188C2×M16—22no—
189C2×M16—22no—
190C2×M16—22no—
191C2×M16—22no—
192C2×M16C22×C412yes—
193C2×M16C22×C412yes—
194C2×M16C22×C414yes—
195C2×M16C2×D414yes—
196C2×M16C2×Q814yes—
197C2×M16#4114yes—
198C22×C8—24no—
199C22×C8—24no—
200C22×C8—24no—
201C22×C8—22no—
202C22×C8—22no—
203C22×C8—22no—
204C22×C8—22no—
205C22×C8—22no—
206C22×C8—22no—
207C22×C8—21no—
208C22×C8—21no—
209C22×C8C22×C414yes—
210C22×C8C2×D412yes—
211C22×C8C2×D412yes—
212C22×C8C2×Q814yes—
213#117—22no—
214#117—22no—
215#117—22no—
216#117—22no—
217#117—22no—
218#117—44no—
219#117—44no—
220#117—41no—
221#117—41no—
222#117C2×D414yes—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—41no—
243C2×C4⋊C4—41no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—41no—
250C2×C4⋊C4—41no—
251C2×C4⋊C4C2×D414yes—
252C2×C4⋊C4C2×D411yescharacteristic
253C2×C4⋊C4C2×D411yescharacteristic
254C2×C4⋊C4C2×D411yescharacteristic
255C2×C4⋊C4C2×D411yescharacteristic
256C2×C22⋊C4—21no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—21no—
259C2×C22⋊C4—21no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—41no—
262C2×C22⋊C4—41no—
263C2×C22⋊C4—82no—
264C2×C42—21no—
265C2×C42—21no—
266C2×C42—21no—
267C2×C42—21no—
268C2×C42—44no—
269#108—216no—
270#107—28no—
271#107—28no—
272#107—44no—
273#107—44no—
274#107—42no—
275#107—42no—
276#106—28no—
277#106—28no—
278#106—42no—
279#106—42no—
280C22⋊C8—24no—
281C22⋊C8—24no—
282C22⋊C8—24no—
283C22⋊C8—24no—
284C22⋊C8C4⋊C414yes—
285C22⋊C8C4⋊C414yes—
286C22⋊C8C4⋊C418yes—
287#97—44no—
288C4×C8—44no—
289#95—41no—
290#95—41no—
291C24—22no—
292C24—22no—
293C24—22no—
294C24—22no—
295C24—41no—
296C24—41no—
297C24C4×Q812yes—
298C24#12111yescharacteristic
299C24#12111yescharacteristic
300C24#12512yes—
301C24C2×#4111yescharacteristic
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—24no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—24no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—24no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—44no—
332C22×C4—44no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—81no—
342C22×C4—81no—
343C22×C4—82no—
344C22×C4—82no—
345C22×C4—82no—
346C22×C4—816no—
347C22×C4—82no—
348C22×C4—82no—
349C22×C4—82no—
350C22×C4—81no—
351C22×C4—81no—
352C22×C4—81no—
353C22×C4—81no—
354C22×C4C2×C4⋊C412yes—
355C22×C4C2×C4⋊C412yes—
356C22×C4C4×D412yes—
357C22×C4C4×D412yes—
358C22×C4C4×Q812yes—
359C22×C4#12111yescharacteristic
360C22×C4#12111yescharacteristic
361C22×C4C22⋊Q812yes—
362C22×C4C22⋊Q812yes—
363C22×C4#12512yes—
364C22×C4C4⋊Q812yes—
365C22×C4C4⋊Q812yes—
366C22×C4C23×C411yescharacteristic
367C22×C4C22×D411yescharacteristic
368C22×C4C22×D411yescharacteristic
369C22×C4C22×D412yes—
370C22×C4C22×Q811yescharacteristic
371C22×C4C22×Q811yescharacteristic
372C22×C4C2×#4111yescharacteristic
373C22×C4C2×#4112yes—
374C22×C4H(5,𝔽2)12yes—
375C22×C42−1+412yes—
376M16—216no—
377M16—24no—
378M16—24no—
379M16—24no—
380M16—24no—
381M16—24no—
382M16—24no—
383M16—24no—
384M16—42no—
385M16—42no—
386M16—42no—
387M16—42no—
388M16C4×D418yes—
389C2×C8—216no—
390C2×C8—24no—
391C2×C8—24no—
392C2×C8—24no—
393C2×C8—24no—
394C2×C8—24no—
395C2×C8—22no—
396C2×C8—22no—
397C2×C8—22no—
398C2×C8—22no—
399C2×C8—24no—
400C2×C8—24no—
401C2×C8—22no—
402C2×C8—22no—
403C2×C8—24no—
404C2×C8—41no—
405C2×C8—41no—
406C2×C8—41no—
407C2×C8—41no—
408C2×C8—42no—
409C2×C8—42no—
410C2×C8C2×C4⋊C414yes—
411C2×C8C2×C4⋊C414yes—
412C4⋊C4—21no—
413C4⋊C4—21no—
414C4⋊C4—21no—
415C4⋊C4—24no—
416C4⋊C4—21no—
417C4⋊C4—21no—
418C4⋊C4—21no—
419C4⋊C4—24no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4—44no—
423C4⋊C4—44no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—41no—
427C4⋊C4—41no—
428C4⋊C4—41no—
429C4⋊C4—41no—
430C4⋊C4—42no—
431C4⋊C4—42no—
432C4⋊C4#12012yes—
433C4⋊C4#12012yes—
434C22⋊C4—24no—
435C22⋊C4—42no—
436C22⋊C4—42no—
437C22⋊C4—42no—
438C22⋊C4—82no—
439C22⋊C4—82no—
440C42—24no—
441C42—42no—
442C42—42no—
443C42—42no—
444C23—21no—
445C23—21no—
446C23—21no—
447C23—21no—
448C23—21no—
449C23—21no—
450C23—21no—
451C23—21no—
452C23—24no—
453C23—21no—
454C23—21no—
455C23—24no—
456C23—24no—
457C23—24no—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—42no—
463C23—42no—
464C23—44no—
465C23—44no—
466C23—44no—
467C23—44no—
468C23#38912yes—
469C23#38912yes—
470C23#47212yes—
471C23#47312yes—
472C23C2×C4×Q811yescharacteristic
473C23C2×#12111yescharacteristic
474C23C2×#12511yescharacteristic
475C23#54612yes—
476C23#55212yes—
477C2×C4—24no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—24no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—44no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—84no—
509C2×C4—84no—
510C2×C4—88no—
511C2×C4—84no—
512C2×C4—88no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4—84no—
516C2×C4—81no—
517C2×C4—81no—
518C2×C4—81no—
519C2×C4—81no—
520C2×C4—81no—
521C2×C4—81no—
522C2×C4—162no—
523C2×C4—162no—
524C2×C4#38912yes—
525C2×C4#38912yes—
526C2×C4C22×C4⋊C411yescharacteristic
527C2×C4C2×C4×D411yescharacteristic
528C2×C4C2×C4×D412yes—
529C2×C4#52012yes—
530C2×C4C2×#12011yescharacteristic
531C2×C4C2×C22⋊Q811yescharacteristic
532C2×C4#52511yescharacteristic
533C2×C4#52512yes—
534C2×C4C2×C4⋊Q811yescharacteristic
535C2×C4#53411yescharacteristic
536C2×C4#53611yescharacteristic
537C2×C4#53912yes—
538C2×C4D4212yes—
539C2×C4#54712yes—
540C2×C4D4×Q812yes—
541C2×C4D4×Q812yes—
542C2×C4#55412yes—
543C2×C4#55412yes—
544C8—24no—
545C8—24no—
546C8—28no—
547C8—42no—
548C8—44no—
549C8—42no—
550C22—22no—
551C22—22no—
552C22—22no—
553C22—22no—
554C22—22no—
555C22—22no—
556C22—22no—
557C22—22no—
558C22—21no—
559C22—21no—
560C22—21no—
561C22—21no—
562C22—22no—
563C22—22no—
564C22—22no—
565C22—22no—
566C22—22no—
567C22—48no—
568C22—48no—
569C22—41no—
570C22—41no—
571C22—41no—
572C22—41no—
573C22#195214yescomplemented
574C22C2×#38911yescharacteristic
575C22#236012yes—
576C22#241011yescharacteristic
577C22#241311yescharacteristic
578C22#247312yes—
579C22C2×#47211yescharacteristic
580C22C2×#47311yescharacteristic
581C22#306811yescharacteristic
582C22#306911yescharacteristic
583C4—22no—
584C4—22no—
585C4—42no—
586C4—42no—
587C4—81no—
588C4—84no—
589C4—81no—
590C4—161no—
591C4—161no—
592C4C4×#12011yescharacteristic
593C4#232211yescharacteristic
594C4D4×C4⋊C412yes—
595C4#243411yescharacteristic
596C4#244912yes—
597C4#258311yescharacteristic
598C2—22no—
599C2—24no—
600C2—22no—
601C2—42no—
602C2#1404311yescharacteristic
603C2C2×#195211yescharacteristic
604C2#2118211yescharacteristic
605C2#2439811yescharacteristic
606C2#2440011yescharacteristic
607C2#2462011yescharacteristic
608C2#2462211yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25#41
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1952C2semidirect
7. #11331C2semidirect
8. #11249C2semidirect
9. #10896C2semidirect
32. #1855C4semidirect
90. C2×#108C2×C4semidirect
139. #341C2×C4semidirect
573. C22#1952semidirect

Maximal subgroups

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

subgroupindex
C22×#4282
C2×#19522
C2×#18552
#208462
C2×#18522
#203792
#113312
#112492
#108962
#107392
#107272