Finite Group Databasebeta

← #364270

Subgroups — #364270

Order 512.

1348 conjugacy classes of subgroups.

The subgroup lattice has 613 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2924C211yesmaximal, characteristic
2C2×#1855C211yesmaximal, characteristic
3C2×#1853C211yesmaximal, characteristic
4#20702C211yesmaximal, characteristic
5C2×#1736C211yesmaximal, characteristic
6#11242C212yesmaximal, complemented
7#11238C212yesmaximal, complemented
8#10869C212yesmaximal, complemented
9#10827C212yesmaximal, complemented
10#10739C211yesmaximal, characteristic
11#10727C211yesmaximal, characteristic
12C23×M16C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15#2924—24no—
16#2924C418yescomplemented
17C2×#433—22no—
18C2×#432—21no—
19C2×#431—21no—
20C22×#108C2212yes—
21C22×#106C2212yes—
22C2×#424C2212yes—
23C2×#407C2211yescharacteristic
24C2×#407C2211yescharacteristic
25C22×C22⋊C8C2211yescharacteristic
26C2×#405C2212yes—
27C2×#380C2211yescharacteristic
28#1855—24no—
29#1853—21no—
30#1853—21no—
31#1853—22no—
32#1825—21no—
33#1825—21no—
34#1824—21no—
35#1823—21no—
36C2×#343C2212yes—
37C2×#343C2212yes—
38C2×#343C2212yes—
39C2×#343C2212yes—
40C2×#341C2212yes—
41#1736—21no—
42#1736—21no—
43#1736C2211yescharacteristic
44#1736C2212yes—
45#1736C2211yescharacteristic
46#1736C2212yes—
47#1736C2211yescharacteristic
48#1736C2211yescharacteristic
49C2×#340—21no—
50C2×#340—21no—
51C2×#340C2212yes—
52C24×C4—21no—
53C24×C4C2311yescharacteristic
54C22×M16—21no—
55C22×M16—24no—
56C22×M16—22no—
57C22×M16—22no—
58C22×M16—42no—
59C22×M16C2311yescharacteristic
60C22×M16C2311yescharacteristic
61C22×M16C2312yes—
62C22×M16C2312yes—
63C23×C8—21no—
64C23×C8C2312yes—
65C2×#117—21no—
66C2×#117—22no—
67C2×#117—21no—
68C2×#117—22no—
69C2×#117C2×C414yes—
70C2×#117D411yescharacteristic
71C2×#117D411yescharacteristic
72C2×#117Q814yes—
73C2×#117C2311yescharacteristic
74C2×#117C2311yescharacteristic
75C22×C4⋊C4—21no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4—24no—
78C22×C4⋊C4—21no—
79C22×C4⋊C4—21no—
80C22×C4⋊C4—21no—
81C22×C4⋊C4—42no—
82C22×C4⋊C4D411yescharacteristic
83C22×C4⋊C4D411yescharacteristic
84C22×C4⋊C4C2311yescharacteristic
85C22×C4⋊C4C2311yescharacteristic
86C22×C4⋊C4C2312yes—
87C22×C22⋊C4—21no—
88C22×C22⋊C4—21no—
89C22×C42—21no—
90#433—216no—
91#432—28no—
92#431—28no—
93C2×#108—24no—
94C2×#108—24no—
95C2×#108—42no—
96C2×#106—22no—
97C2×#106—22no—
98C2×#106—24no—
99C2×#106—42no—
100#424—24no—
101#424C2×C418yescomplemented
102C2×C4⋊C8—22no—
103C2×C4⋊C8—22no—
104#407—22no—
105#407—22no—
106#407—22no—
107#407C2×C414yes—
108C2×C22⋊C8—21no—
109C2×C22⋊C8—21no—
110C2×C22⋊C8—21no—
111C2×C22⋊C8—21no—
112C2×C22⋊C8C2×C414yes—
113C2×C22⋊C8D411yescharacteristic
114C2×C22⋊C8D411yescharacteristic
115C2×C22⋊C8D412yes—
116C2×C22⋊C8D414yes—
117#405—24no—
118#405C2×C418yescomplemented
119C2×#97—22no—
120C2×C4×C8—22no—
121#380—21no—
122#380—21no—
123#380—21no—
124#380—21no—
125C2×#95—21no—
126#343—24no—
127#343—24no—
128#343—42no—
129#343—42no—
130#341—24no—
131#340—22no—
132#340—22no—
133#340—21no—
134#340—21no—
135#340—21no—
136#340—21no—
137#340—21no—
138#340—21no—
139#340—21no—
140#340—21no—
141#340—22no—
142#340—22no—
143#340—22no—
144#340—22no—
145C25#4111yescharacteristic
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—22no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—44no—
156C23×C4—41no—
157C23×C4—41no—
158C23×C4—82no—
159C23×C4—82no—
160C23×C4C22×C412yes—
161C23×C4C2×D412yes—
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2×Q812yes—
167C23×C4#4111yescharacteristic
168C23×C4#4112yes—
169C23×C4C2411yescharacteristic
170C2×M16—24no—
171C2×M16—22no—
172C2×M16—22no—
173C2×M16—22no—
174C2×M16—22no—
175C2×M16—22no—
176C2×M16—22no—
177C2×M16—22no—
178C2×M16—22no—
179C2×M16—22no—
180C2×M16—22no—
181C2×M16—21no—
182C2×M16—21no—
183C2×M16—22no—
184C2×M16—21no—
185C2×M16—21no—
186C2×M16—22no—
187C2×M16—22no—
188C2×M16—22no—
189C2×M16—48no—
190C2×M16—48no—
191C2×M16—42no—
192C2×M16—42no—
193C2×M16C22×C414yes—
194C2×M16#4114yes—
195C22×C8—22no—
196C22×C8—22no—
197C22×C8—22no—
198C22×C8—22no—
199C22×C8—24no—
200C22×C8—22no—
201C22×C8—22no—
202C22×C8—21no—
203C22×C8—21no—
204C22×C8—42no—
205C22×C8—42no—
206C22×C8C22×C414yes—
207C22×C8C2×D412yes—
208C22×C8C2×D412yes—
209C22×C8C2×D414yes—
210#117—24no—
211#117—24no—
212#117—22no—
213#117—22no—
214#117—24no—
215#117—24no—
216#117—21no—
217#117—21no—
218#117—21no—
219#117—21no—
220#117C4⋊C414yes—
221#117C4⋊C414yes—
222#117C22×C412yes—
223#117C22×C412yes—
224#117C2×Q812yes—
225#117C2×Q812yes—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—41no—
251C2×C4⋊C4—41no—
252C2×C4⋊C4C4⋊C414yes—
253C2×C4⋊C4C4⋊C414yes—
254C2×C4⋊C4C2×D411yescharacteristic
255C2×C4⋊C4C2×D411yescharacteristic
256C2×C4⋊C4C2×D411yescharacteristic
257C2×C4⋊C4C2×D411yescharacteristic
258C2×C4⋊C4#4114yes—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—21no—
261C2×C22⋊C4—21no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—21no—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—41no—
266C2×C22⋊C4—41no—
267C2×C22⋊C4—82no—
268C2×C42—24no—
269C2×C42—21no—
270C2×C42—21no—
271C2×C42—24no—
272C2×C42—21no—
273C2×C42—21no—
274#108—48no—
275#106—44no—
276#106—44no—
277C4⋊C8—28no—
278C4⋊C8—28no—
279C22⋊C8—24no—
280C22⋊C8—24no—
281C22⋊C8—42no—
282C22⋊C8—42no—
283C22⋊C8C22⋊C414yes—
284C22⋊C8C22⋊C414yes—
285C22⋊C8C22⋊C418yes—
286#97—28no—
287C4×C8—28no—
288#95—41no—
289#95—41no—
290C24—22no—
291C24—22no—
292C24—22no—
293C24—22no—
294C24—41no—
295C24—41no—
296C24#11712yes—
297C24#12111yescharacteristic
298C24#12111yescharacteristic
299C24C22⋊Q812yes—
300C24C2×#4111yescharacteristic
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—24no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—24no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—24no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—44no—
333C22×C4—44no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—81no—
341C22×C4—81no—
342C22×C4—82no—
343C22×C4—82no—
344C22×C4—82no—
345C22×C4—816no—
346C22×C4—82no—
347C22×C4—82no—
348C22×C4—82no—
349C22×C4—81no—
350C22×C4—81no—
351C22×C4—81no—
352C22×C4—81no—
353C22×C4C2×C22⋊C412yes—
354C22×C4C2×C22⋊C412yes—
355C22×C4C2×C4⋊C412yes—
356C22×C4C2×C4⋊C412yes—
357C22×C4#11712yes—
358C22×C4#12112yes—
359C22×C4#12112yes—
360C22×C4#12111yescharacteristic
361C22×C4#12111yescharacteristic
362C22×C4C22⋊Q812yes—
363C22×C4C22⋊Q812yes—
364C22×C4C22⋊Q812yes—
365C22×C4C23×C411yescharacteristic
366C22×C4C22×D411yescharacteristic
367C22×C4C22×D411yescharacteristic
368C22×C4C22×D411yescharacteristic
369C22×C4C22×Q811yescharacteristic
370C22×C4C2×#4112yes—
371C22×C4C2×#4112yes—
372C22×C4C2×#4111yescharacteristic
373C22×C4H(5,𝔽2)12yes—
374C22×C42−1+412yes—
375M16—24no—
376M16—24no—
377M16—24no—
378M16—24no—
379M16—48no—
380M16—42no—
381M16—42no—
382M16—42no—
383M16—42no—
384M16—42no—
385M16—42no—
386M16—42no—
387M16—42no—
388C2×C8—24no—
389C2×C8—24no—
390C2×C8—22no—
391C2×C8—22no—
392C2×C8—24no—
393C2×C8—24no—
394C2×C8—22no—
395C2×C8—22no—
396C2×C8—24no—
397C2×C8—24no—
398C2×C8—48no—
399C2×C8—42no—
400C2×C8—42no—
401C2×C8—42no—
402C2×C8—42no—
403C2×C8—41no—
404C2×C8—41no—
405C2×C8—41no—
406C2×C8—41no—
407C2×C8—41no—
408C2×C8—41no—
409C2×C8C2×C22⋊C414yes—
410C2×C8C2×C22⋊C414yes—
411C4⋊C4—21no—
412C4⋊C4—21no—
413C4⋊C4—21no—
414C4⋊C4—21no—
415C4⋊C4—24no—
416C4⋊C4—24no—
417C4⋊C4—24no—
418C4⋊C4—24no—
419C4⋊C4—44no—
420C4⋊C4—44no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—41no—
424C4⋊C4—41no—
425C4⋊C4—41no—
426C4⋊C4—41no—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C4⋊C4C2×C4⋊C412yes—
430C4⋊C4C2×C4⋊C412yes—
431C4⋊C4C2×C4⋊C412yes—
432C4⋊C4C2×C4⋊C412yes—
433C22⋊C4—22no—
434C22⋊C4—22no—
435C22⋊C4—22no—
436C22⋊C4—22no—
437C22⋊C4—24no—
438C22⋊C4—24no—
439C22⋊C4—82no—
440C22⋊C4—82no—
441C42—22no—
442C42—22no—
443C42—22no—
444C42—22no—
445C42—24no—
446C42—24no—
447C23—21no—
448C23—21no—
449C23—21no—
450C23—21no—
451C23—21no—
452C23—21no—
453C23—21no—
454C23—21no—
455C23—24no—
456C23—24no—
457C23—24no—
458C23—21no—
459C23—21no—
460C23—24no—
461C23—21no—
462C23—21no—
463C23—21no—
464C23—21no—
465C23—42no—
466C23—42no—
467C23—44no—
468C23—44no—
469C23—44no—
470C23—44no—
471C23#38012yes—
472C23#38012yes—
473C23#47112yes—
474C23#47112yes—
475C23C2×#11711yescharacteristic
476C23C2×#12111yescharacteristic
477C23C2×C22⋊Q811yescharacteristic
478C23#55112yes—
479C23#55212yes—
480C2×C4—24no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—24no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—48no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—44no—
512C2×C4—84no—
513C2×C4—84no—
514C2×C4—88no—
515C2×C4—88no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—84no—
519C2×C4—81no—
520C2×C4—81no—
521C2×C4—81no—
522C2×C4—81no—
523C2×C4—81no—
524C2×C4—81no—
525C2×C4—162no—
526C2×C4—162no—
527C2×C4#38012yes—
528C2×C4#38012yes—
529C2×C4C22×C22⋊C411yescharacteristic
530C2×C4C22×C4⋊C411yescharacteristic
531C2×C4C4×#4112yes—
532C2×C4#52012yes—
533C2×C4C2×#12111yescharacteristic
534C2×C4C2×C22⋊Q811yescharacteristic
535C2×C4#52511yescharacteristic
536C2×C4#52511yescharacteristic
537C2×C4#53312yes—
538C2×C4#53511yescharacteristic
539C2×C4#53611yescharacteristic
540C2×C4#54412yes—
541C2×C4#54612yes—
542C2×C4#54612yes—
543C2×C4#54712yes—
544C2×C4#54712yes—
545C2×C4#55112yes—
546C2×C4#55212yes—
547C8—24no—
548C8—24no—
549C8—44no—
550C8—44no—
551C8—81no—
552C8—81no—
553C22—22no—
554C22—22no—
555C22—22no—
556C22—22no—
557C22—22no—
558C22—21no—
559C22—21no—
560C22—22no—
561C22—21no—
562C22—21no—
563C22—22no—
564C22—22no—
565C22—22no—
566C22—22no—
567C22—22no—
568C22—22no—
569C22—22no—
570C22—48no—
571C22—48no—
572C22—41no—
573C22—41no—
574C22—41no—
575C22—41no—
576C22#181514yes—
577C22C2×#38011yescharacteristic
578C22#234312yes—
579C22#241111yescharacteristic
580C22#241311yescharacteristic
581C22#243812yes—
582C22C2×#47111yescharacteristic
583C22C2×#47111yescharacteristic
584C22#306811yescharacteristic
585C22#306911yescharacteristic
586C4—22no—
587C4—22no—
588C4—42no—
589C4—42no—
590C4—42no—
591C4—42no—
592C4—84no—
593C4—161no—
594C4—161no—
595C4#229611yescharacteristic
596C4#231711yescharacteristic
597C4#233912yes—
598C4#241111yescharacteristic
599C4#241311yescharacteristic
600C4#243812yes—
601C2—24no—
602C2—22no—
603C2—22no—
604C2—42no—
605C2#1396211yescharacteristic
606C2C2×#181511yescharacteristic
607C2#2067911yescharacteristic
608C2#2452111yescharacteristic
609C2#2452311yescharacteristic
610C2#2462111yescharacteristic
611C2#2462311yescharacteristic
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
6. #11242C2semidirect
7. #11238C2semidirect
8. #10869C2semidirect
9. #10827C2semidirect
16. #2924C4semidirect
101. #424C2×C4semidirect
118. #405C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#29242
C2×#18552
C2×#18532
#207022
C2×#17362
#112422
#112382
#108692
#108272
#107392
#107272