Finite Group Databasebeta

← #393368

Subgroups — #393368

Order 512.

1712 conjugacy classes of subgroups.

The subgroup lattice has 589 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3079C211yesmaximal, characteristic
2C4×#520C211yesmaximal, characteristic
3#21313C211yesmaximal, characteristic, complemented
4C2×#1806C211yesmaximal, characteristic
5#14161C211yesmaximal, characteristic, complemented
6#10899C214yesmaximal, complemented
7#10852C211yesmaximal, characteristic, complemented
8#10847C211yesmaximal, characteristic, complemented
9#10781C214yesmaximal
10C22×C4⋊Q8C2211yescharacteristic
11C2×#520C2212yes—
12C2×C4×#41C2211yescharacteristic
13#3079—24no—
14#3079C418yescomplemented
15C2×#477—21no—
16C2×#475—21no—
17C2×#476—21no—
18C2×#474—21no—
19C2×#428C2212yes—
20C2×#423C2211yescharacteristic
21C2×#423C2211yescharacteristic
22C22×#104C2212yes—
23C2×#417C2212yes—
24#2276—21no—
25#2275—22no—
26#2275—21no—
27#2274—22no—
28#2274C2212yes—
29C42×Q8—21no—
30C42×D4—22no—
31C42×D4—21no—
32C4×#117—21no—
33C4×#117C2211yescharacteristic
34C2×C4×C4⋊C4—21no—
35C2×C4×C4⋊C4C2211yescharacteristic
36C2×C4×C4⋊C4C2211yescharacteristic
37#1988—22no—
38#1987—21no—
39#1987—21no—
40#1806—24no—
41C2×#343C2214yes—
42#1736C2214yes—
43C2×#339C2214yes—
44C2×#339C2212yes—
45#1446—22no—
46C4⋊#95—22no—
47C4⋊#95C2214yes—
48C22×#41C2311yescharacteristic
49C23×Q8—21no—
50C22×M16C2312yes—
51C22×M16C2312yes—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8C2×C414yescomplemented
60#520—24no—
61#520—24no—
62#520—24no—
63#520Q818yescomplemented
64C4×#41—22no—
65C4×#41—22no—
66C4×#41C2×C414yes—
67C4×#41D414yes—
68C2×C4×Q8—22no—
69C2×C4×Q8D411yescharacteristic
70C2×C4×Q8D411yescharacteristic
71C2×C4×D4—22no—
72C2×C4×D4—21no—
73C2×C4×D4—21no—
74C2×C4×D4—24no—
75C2×C4×D4D411yescharacteristic
76C2×C4×D4D411yescharacteristic
77C2×#117—22no—
78C2×#117—21no—
79C2×#117C2311yescharacteristic
80C2×#117C2312yes—
81C22×C4⋊C4—22no—
82C22×C4⋊C4—21no—
83C22×C4⋊C4—21no—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2312yes—
86C22×C42—21no—
87C22×C42—22no—
88C22×C42D412yes—
89C22×C42C2312yes—
90C22×C42C2311yescharacteristic
91#477—28no—
92#476—28no—
93#475—28no—
94#474—28no—
95#428—24no—
96#428C2×C418yescomplemented
97C2×#107—22no—
98C2×#106—22no—
99#423—22no—
100#423—22no—
101#423—22no—
102#423C2×C414yescomplemented
103C2×C4⋊C8—21no—
104C2×C4⋊C8—21no—
105C2×C4⋊C8—21no—
106C2×C4⋊C8—21no—
107C2×#104—24no—
108C2×#104—28no—
109#417—24no—
110#417C2×C418yes—
111C2×#103—22no—
112C2×#102—22no—
113C4×C4⋊C4—22no—
114C4×C4⋊C4—22no—
115C4×C4⋊C4—24no—
116C4×C4⋊C4—24no—
117C4×C4⋊C4—22no—
118C4×C4⋊C4—22no—
119C4×C4⋊C4—22no—
120C4×C4⋊C4—22no—
121C4×C4⋊C4—21no—
122C4×C4⋊C4—21no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4—21no—
125C4×C4⋊C4—22no—
126C4×C4⋊C4—22no—
127C4×C4⋊C4D411yescharacteristic
128C4×C4⋊C4D411yescharacteristic
129C4×C4⋊C4D412yes—
130C4×C22⋊C4—22no—
131C4×C22⋊C4—22no—
132C4×C22⋊C4—24no—
133C4×C22⋊C4—24no—
134C4×C22⋊C4—21no—
135C4×C22⋊C4—21no—
136C4×C22⋊C4—24no—
137#376—22no—
138#376—21no—
139C2×#95—22no—
140C2×#95—21no—
141C2×#95—28no—
142C2×#95D412yes—
143C43—22no—
144C43—21no—
145#343—28no—
146#340—24no—
147#340—24no—
148#339—24no—
149#339—28no—
150C2×#41—21no—
151C2×#41—21no—
152C2×#41—22no—
153C2×#41—22no—
154C2×#41C22×C412yes—
155C2×#41C2×D412yes—
156C2×#41C2×Q814yes—
157C2×#41#4114yes—
158C22×Q8—24no—
159C22×Q8—41no—
160C22×Q8—41no—
161C22×Q8—42no—
162C22×Q8—84no—
163C22×Q8C2×D411yescharacteristic
164C22×D4—21no—
165C22×D4C2×D411yescharacteristic
166C23×C4—21no—
167C23×C4—24no—
168C23×C4—21no—
169C23×C4—21no—
170C23×C4—22no—
171C23×C4—22no—
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D414yes—
174C23×C4C2×D411yescharacteristic
175C23×C4C2411yescharacteristic
176C2×M16—22no—
177C2×M16—24no—
178C2×M16—24no—
179C2×M16—22no—
180C2×M16—24no—
181C2×M16—24no—
182C2×M16—22no—
183C2×M16C22×C414yes—
184C22×C8—22no—
185C22×C8—22no—
186C4⋊Q8—24no—
187C4⋊Q8—24no—
188C4⋊Q8—24no—
189C4⋊Q8—24no—
190C4⋊Q8—42no—
191C4⋊Q8—42no—
192C4⋊Q8—42no—
193C4⋊Q8—42no—
194C4×Q8—28no—
195C4×Q8—28no—
196C4×Q8—22no—
197C4×Q8—22no—
198C4×Q8C22⋊C414yes—
199C4×Q8C22⋊C414yes—
200C4×D4—216no—
201C4×D4—28no—
202C4×D4—28no—
203C4×D4—24no—
204C4×D4—24no—
205C4×D4—24no—
206C4×D4—24no—
207C4×D4—216no—
208C4×D4—22no—
209C4×D4—22no—
210C4×D4C22⋊C414yes—
211C4×D4C22⋊C414yes—
212#117—28no—
213#117—21no—
214#117—21no—
215#117—22no—
216#117—22no—
217#117—28no—
218#117—24no—
219#117—24no—
220#117C22×C412yes—
221#117C2×D412yes—
222#117C2×Q814yes—
223#117#4114yes—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—41no—
242C2×C4⋊C4—41no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4C22×C414yes—
246C2×C4⋊C4C2×D411yescharacteristic
247C2×C4⋊C4C2×D411yescharacteristic
248C2×C4⋊C4C2×D412yes—
249C2×C4⋊C4C2×D412yes—
250C2×C4⋊C4C2×Q814yes—
251C2×C4⋊C4#4114yes—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—21no—
257C2×C42—24no—
258C2×C42—21no—
259C2×C42—21no—
260C2×C42—21no—
261C2×C42—22no—
262C2×C42—24no—
263C2×C42—24no—
264C2×C42—21no—
265C2×C42—22no—
266C2×C42—21no—
267C2×C42—22no—
268C2×C42—24no—
269C2×C42—22no—
270C2×C42—24no—
271C2×C42—24no—
272C2×C42—24no—
273C2×C42—22no—
274C2×C42—22no—
275C2×C42—44no—
276C2×C42—84no—
277C2×C42C22×C412yes—
278C2×C42C2×D412yes—
279C2×C42C2×D411yescharacteristic
280C2×C42C2×D411yescharacteristic
281C2×C42C2×D411yescharacteristic
282C2×C42C2×D411yescharacteristic
283C2×C42#4112yes—
284C2×C42#4112yes—
285#107—28no—
286#106—28no—
287C4⋊C8—24no—
288C4⋊C8—24no—
289C4⋊C8—42no—
290C4⋊C8—42no—
291#104—48no—
292#103—28no—
293#102—28no—
294#95—24no—
295#95—24no—
296#95—21no—
297#95—22no—
298#95—22no—
299#95—21no—
300#95—22no—
301#95—42no—
302#95—88no—
303C24—21no—
304C24C22×D411yescharacteristic
305#41—24no—
306#41—24no—
307#41—24no—
308#41—22no—
309#41—22no—
310#41—24no—
311#41—24no—
312#41C4×Q818yes—
313#41#12414yes—
314#41C4⋊Q814yes—
315C2×Q8—21no—
316C2×Q8—22no—
317C2×Q8—44no—
318C2×Q8—44no—
319C2×Q8—84no—
320C2×Q8—84no—
321C2×Q8—88no—
322C2×Q8—88no—
323C2×Q8C2×C22⋊C412yes—
324C2×Q8C22⋊Q814yes—
325C2×D4—22no—
326C2×D4—22no—
327C2×D4—24no—
328C2×D4—24no—
329C2×D4—21no—
330C2×D4—22no—
331C2×D4C2×C22⋊C412yes—
332C2×D4C22⋊Q814yes—
333C22×C4—21no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—24no—
344C22×C4—22no—
345C22×C4—216no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—44no—
355C22×C4—44no—
356C22×C4—44no—
357C22×C4—48no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—44no—
363C22×C4—82no—
364C22×C4—84no—
365C22×C4—84no—
366C22×C4C4×D412yes—
367C22×C4C4×D412yes—
368C22×C4#12011yescharacteristic
369C22×C4#12011yescharacteristic
370C22×C4#12011yescharacteristic
371C22×C4#12011yescharacteristic
372C22×C4#12112yes—
373C22×C4#12112yes—
374C22×C4C22⋊Q818yes—
375C22×C4C23×C411yescharacteristic
376C22×C4C22×D411yescharacteristic
377C22×C4C22×D411yescharacteristic
378C22×C4C22×D411yescharacteristic
379C22×C4C22×Q812yes—
380C22×C4C2×#4111yescharacteristic
381C22×C4C2×#4112yes—
382C22×C4C2×#4112yes—
383C22×C4C2×#4112yes—
384C22×C4C2×#4111yescharacteristic
385M16—28no—
386M16—42no—
387M16—42no—
388M16—42no—
389M16—44no—
390M16—42no—
391C2×C8—24no—
392C2×C8—44no—
393C2×C8—42no—
394C2×C8—44no—
395C4⋊C4—216no—
396C4⋊C4—24no—
397C4⋊C4—24no—
398C4⋊C4—21no—
399C4⋊C4—21no—
400C4⋊C4—22no—
401C4⋊C4—24no—
402C4⋊C4—24no—
403C4⋊C4—22no—
404C4⋊C4—24no—
405C4⋊C4—24no—
406C4⋊C4—24no—
407C4⋊C4—24no—
408C4⋊C4—48no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—44no—
412C4⋊C4—44no—
413C4⋊C4C2×C22⋊C412yes—
414C4⋊C4C2×C22⋊C412yes—
415C4⋊C4C22⋊Q814yes—
416C4⋊C4C22⋊Q814yes—
417C22⋊C4—216no—
418C22⋊C4—22no—
419C22⋊C4—22no—
420C22⋊C4—24no—
421C22⋊C4—24no—
422C22⋊C4—24no—
423C22⋊C4—24no—
424C42—216no—
425C42—22no—
426C42—22no—
427C42—24no—
428C42—24no—
429C42—24no—
430C42—24no—
431C42—44no—
432C42—42no—
433C42—42no—
434C42—42no—
435C42—44no—
436C42—42no—
437C42—82no—
438C42—82no—
439C42—84no—
440C42—84no—
441C42C2×C22⋊C412yes—
442C42C2×C22⋊C412yes—
443C42C4×D412yes—
444C42C4×D412yes—
445C42#12112yes—
446C42#12112yes—
447C42#12312yes—
448C42#12312yes—
449C23—21no—
450C23—21no—
451C23—22no—
452C23—24no—
453C23—22no—
454C23—41no—
455C23—41no—
456C23—42no—
457C23C2×C4×D411yescharacteristic
458C23C2×#12011yescharacteristic
459C23C2×#12111yescharacteristic
460C23C2×C22⋊Q812yes—
461C23#52512yes—
462Q8—24no—
463Q8—22no—
464Q8—88no—
465Q8—88no—
466Q8#39114yes—
467D4—24no—
468D4—24no—
469D4—28no—
470D4—24no—
471D4—22no—
472D4#39114yes—
473C2×C4—24no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—28no—
484C2×C4—24no—
485C2×C4—28no—
486C2×C4—24no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—44no—
492C2×C4—42no—
493C2×C4—41no—
494C2×C4—42no—
495C2×C4—41no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—44no—
499C2×C4—42no—
500C2×C4—48no—
501C2×C4—48no—
502C2×C4—44no—
503C2×C4—48no—
504C2×C4—48no—
505C2×C4—44no—
506C2×C4—84no—
507C2×C4—84no—
508C2×C4—816no—
509C2×C4—84no—
510C2×C4—82no—
511C2×C4—84no—
512C2×C4—82no—
513C2×C4—84no—
514C2×C4#38612yes—
515C2×C4#38612yes—
516C2×C4#38612yes—
517C2×C4#38612yes—
518C2×C4#45512yes—
519C2×C4#45612yes—
520C2×C4C22×C22⋊C411yescharacteristic
521C2×C4C2×C4×D411yescharacteristic
522C2×C4C2×C4×Q812yes—
523C2×C4C4×#4112yes—
524C2×C4C2×#12111yescharacteristic
525C2×C4C2×C22⋊Q812yes—
526C2×C4C2×C22⋊Q812yes—
527C2×C4C2×#12311yescharacteristic
528C2×C4#52512yes—
529C2×C4#52512yes—
530C2×C4C2×#12411yescharacteristic
531C2×C4#52912yes—
532C2×C4#52912yes—
533C2×C4C2×C4⋊Q811yescharacteristic
534C2×C4#53211yescharacteristic
535C2×C4#53211yescharacteristic
536C2×C4#53312yes—
537C2×C4#53312yes—
538C8—44no—
539C8—82no—
540C22—22no—
541C22—22no—
542C22—21no—
543C22—21no—
544C22—22no—
545C22—22no—
546C22—21no—
547C22—21no—
548C22—24no—
549C22—44no—
550C22—42no—
551C22—44no—
552C22—42no—
553C22C2×#38611yescharacteristic
554C22C4×C22⋊Q812yes—
555C22#284812yes—
556C22#285011yescharacteristic
557C22#285711yescharacteristic
558C22C2×#45511yescharacteristic
559C22C2×#45611yescharacteristic
560C22#302112yes—
561C4—24no—
562C4—22no—
563C4—22no—
564C4—22no—
565C4—48no—
566C4—44no—
567C4—42no—
568C4—44no—
569C4—42no—
570C4—84no—
571C4—84no—
572C4#188214yes—
573C4C2×#39111yescharacteristic
574C4#229711yescharacteristic
575C4C4×C22⋊Q812yes—
576C4#285011yescharacteristic
577C4#285512yes—
578C4#285711yescharacteristic
579C2—21no—
580C2—21no—
581C2—22no—
582C2—44no—
583C2#1470211yescharacteristic
584C2C2×#188211yescharacteristic
585C2#2094511yescharacteristic
586C2#3297112yes—
587C2#3297212yes—
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #21313C2semidirect
5. #14161C2semidirect
6. #10899C2semidirect
7. #10852C2semidirect
8. #10847C2semidirect
14. #3079C4semidirect
59. C2×C4⋊Q8C2×C4semidirect
63. #520Q8semidirect
96. #428C2×C4semidirect
102. #423C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#30792
C4×#5202
#213132
C2×#18062
#141612
#108992
#108522
#108472
#107812