Finite Group Databasebeta

← #369371

Subgroups — #369371

Order 512.

1704 conjugacy classes of subgroups.

The subgroup lattice has 599 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3141C211yesmaximal, characteristic
2#34207C211yesmaximal, characteristic
3#21313C211yesmaximal, characteristic, complemented
4C2×#1962C211yesmaximal, characteristic
5#14714C211yesmaximal, characteristic, complemented
6#11195C214yesmaximal, complemented
7#10857C211yesmaximal, characteristic, complemented
8#10781C214yesmaximal, complemented
9C4×#339C211yesmaximal, characteristic, complemented
10C2×#532C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12C2×#520C2212yes—
13#3141—24no—
14#3141C418yescomplemented
15C2×#489—22no—
16C2×#487—21no—
17C2×#486—21no—
18C2×#423C2211yescharacteristic
19C22×#104C2214yes—
20C2×#417C2214yes—
21C2×C4×M16C2211yescharacteristic
22#2361—22no—
23#2361—22no—
24#2354—21no—
25#2354—21no—
26D4×C4⋊C4—21no—
27D4×C4⋊C4—21no—
28#2326—22no—
29#2311C2212yes—
30C4×C4⋊Q8—21no—
31C4×#127—21no—
32C2×#391—22no—
33C2×#384C2211yescharacteristic
34C4×#117C2211yescharacteristic
35C2×C43C2211yescharacteristic
36#1988—21no—
37#1988—21no—
38#1987—21no—
39#1987—21no—
40#1962—24no—
41C2×#343C2214yes—
42#1736C2212yes—
43#1736C2212yes—
44C2×#339C2212yes—
45C2×#339C2212yes—
46C4⋊#95—22no—
47C4⋊#95C2214yes—
48C22×#41C2312yes—
49C23×Q8—21no—
50C22×M16C2312yes—
51C22×M16C2312yes—
52#532—22no—
53#532—22no—
54#532C2×C414yes—
55#532D414yes—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8—21no—
60C2×C4⋊Q8—21no—
61C2×C4⋊Q8—22no—
62C2×C4⋊Q8—22no—
63C2×C4⋊Q8C2×C414yescomplemented
64C2×C4⋊Q8D411yescharacteristic
65C2×#127D411yescharacteristic
66C2×#124D412yes—
67C2×#121—21no—
68C2×#121—21no—
69#520—28no—
70#520—44no—
71C2×C4×Q8—22no—
72C2×C4×D4—22no—
73C2×C4×D4—24no—
74C2×C4×D4—22no—
75C2×C4×D4—22no—
76C2×C4×D4—21no—
77C2×C4×D4—21no—
78C2×#117—22no—
79C2×#117C2311yescharacteristic
80C2×#117C2311yescharacteristic
81C22×C4⋊C4—21no—
82C22×C4⋊C4—22no—
83C22×C4⋊C4D412yes—
84C22×C4⋊C4D412yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2312yes—
87C22×C42—21no—
88C22×C42—21no—
89C22×C42C2312yes—
90C22×C42C2311yescharacteristic
91#489—216no—
92#487—28no—
93#486—28no—
94#423—22no—
95#423—22no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8—21no—
98C2×#104—28no—
99C2×#104—28no—
100C2×#104—44no—
101#417—28no—
102#417C2×C4116yescomplemented
103C2×#103—24no—
104C2×#102—24no—
105C4×M16—22no—
106C4×M16C2×C414yescomplemented
107C2×#97—21no—
108C2×C4×C8—21no—
109#391—22no—
110#391—48no—
111#390—22no—
112#390—22no—
113#390—22no—
114#389—22no—
115#389—22no—
116#388—24no—
117#388—24no—
118#386—24no—
119#385—24no—
120#385—24no—
121#384—21no—
122#384—21no—
123#384—21no—
124#384—21no—
125#380—22no—
126#380—22no—
127C4×C4⋊C4—22no—
128C4×C4⋊C4—22no—
129C4×C4⋊C4D411yescharacteristic
130C4×C4⋊C4D411yescharacteristic
131C4×C4⋊C4D411yescharacteristic
132C4×C4⋊C4D411yescharacteristic
133C4×C22⋊C4—21no—
134C4×C22⋊C4—21no—
135#376—21no—
136C2×#95—28no—
137C43—21no—
138C43—22no—
139C43—22no—
140#343—44no—
141#340—22no—
142#340—22no—
143#340—22no—
144#340—22no—
145#339—24no—
146#339—24no—
147C2×#41—22no—
148C2×#41—22no—
149C2×#41—24no—
150C2×#41—24no—
151C2×#41—24no—
152C2×#41—24no—
153C2×#41C22×C414yes—
154C2×#41C2×D414yes—
155C22×Q8—24no—
156C22×Q8—44no—
157C22×Q8—44no—
158C22×Q8—41no—
159C22×Q8—41no—
160C22×Q8—42no—
161C22×Q8C2×D412yes—
162C22×D4—22no—
163C22×D4—21no—
164C22×D4—21no—
165C22×D4C2×D412yes—
166C23×C4—22no—
167C23×C4—22no—
168C23×C4—22no—
169C23×C4—24no—
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D414yes—
173C23×C4C2411yescharacteristic
174C2×M16—22no—
175C2×M16—22no—
176C2×M16—22no—
177C2×M16—22no—
178C2×M16—22no—
179C2×M16—22no—
180C2×M16—24no—
181C2×M16—22no—
182C2×M16—24no—
183C2×M16C22×C414yes—
184C22×C8—22no—
185C22×C8—22no—
186C4⋊Q8—24no—
187C4⋊Q8—24no—
188C4⋊Q8—24no—
189C4⋊Q8—24no—
190C4⋊Q8—22no—
191C4⋊Q8—42no—
192C4⋊Q8—42no—
193C4⋊Q8—42no—
194C4⋊Q8—42no—
195C4⋊Q8C22⋊C414yes—
196#127—22no—
197#127C22⋊C414yes—
198#124—24no—
199#124C22⋊C418yes—
200#121—24no—
201#121—24no—
202#121—24no—
203#121—24no—
204C4×Q8—48no—
205C4×D4—24no—
206C4×D4—24no—
207C4×D4—24no—
208C4×D4—24no—
209C4×D4—48no—
210C4×D4—416no—
211C4×D4—84no—
212C4×D4—84no—
213#117—22no—
214#117—22no—
215#117—22no—
216#117—44no—
217#117—44no—
218#117#4114yes—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—41no—
238C2×C4⋊C4—41no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4C22×C414yes—
244C2×C4⋊C4C2×D411yescharacteristic
245C2×C4⋊C4C2×D411yescharacteristic
246C2×C4⋊C4C2×D411yescharacteristic
247C2×C4⋊C4C2×D411yescharacteristic
248C2×C22⋊C4—21no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C42—21no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—24no—
261C2×C42—24no—
262C2×C42—24no—
263C2×C42—21no—
264C2×C42—21no—
265C2×C42—21no—
266C2×C42—21no—
267C2×C42—21no—
268C2×C42—22no—
269C2×C42—42no—
270C2×C42—42no—
271C2×C42—42no—
272C2×C42—42no—
273C2×C42C22×C412yes—
274C2×C42C2×D412yes—
275C2×C42C2×D411yescharacteristic
276C2×C42C2×D411yescharacteristic
277C2×C42C2×D411yescharacteristic
278C2×C42C2×D411yescharacteristic
279C2×C42#4114yes—
280C2×C42#4112yes—
281C2×C42#4112yes—
282C4⋊C8—42no—
283C4⋊C8—42no—
284#104—416no—
285#103—216no—
286#102—216no—
287#97—24no—
288C4×C8—24no—
289#95—24no—
290#95—24no—
291#95—21no—
292#95—21no—
293#95—21no—
294#95—21no—
295#95—416no—
296C24—22no—
297C24C22×D411yescharacteristic
298#41—28no—
299#41—28no—
300#41—28no—
301#41—28no—
302#41—44no—
303#41—44no—
304#41—44no—
305#41—44no—
306C2×Q8—22no—
307C2×Q8—24no—
308C2×Q8—24no—
309C2×Q8—416no—
310C2×Q8—44no—
311C2×Q8—44no—
312C2×Q8—82no—
313C2×Q8—82no—
314C2×Q8—88no—
315C2×Q8—82no—
316C2×Q8—82no—
317C2×Q8C2×C22⋊C414yes—
318C2×D4—24no—
319C2×D4—24no—
320C2×D4—22no—
321C2×D4—24no—
322C2×D4—24no—
323C2×D4—24no—
324C2×D4—24no—
325C2×D4—44no—
326C2×D4—44no—
327C2×D4—42no—
328C2×D4—42no—
329C2×D4—42no—
330C2×D4—42no—
331C2×D4C2×C22⋊C414yes—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—22no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—24no—
348C22×C4—28no—
349C22×C4—48no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—44no—
359C22×C4—44no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—44no—
363C22×C4—44no—
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×C4C23×C411yescharacteristic
375C22×C4C22×D411yescharacteristic
376C22×C4C22×D411yescharacteristic
377C22×C4C22×D411yescharacteristic
378C22×C4C2×#4111yescharacteristic
379C22×C4C2×#4112yes—
380C22×C4C2×#4112yes—
381C22×C4C2×#4111yescharacteristic
382C22×C4H(5,𝔽2)12yes—
383C22×C42−1+412yes—
384M16—24no—
385M16—24no—
386M16—42no—
387M16—42no—
388M16—44no—
389M16—44no—
390C2×C8—24no—
391C2×C8—42no—
392C2×C8—42no—
393C2×C8—42no—
394C2×C8—42no—
395C2×C8—42no—
396C4⋊C4—21no—
397C4⋊C4—21no—
398C4⋊C4—24no—
399C4⋊C4—24no—
400C4⋊C4—24no—
401C4⋊C4—24no—
402C4⋊C4—21no—
403C4⋊C4—21no—
404C4⋊C4—21no—
405C4⋊C4—21no—
406C4⋊C4—48no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—44no—
412C4⋊C4—44no—
413C4⋊C4—44no—
414C4⋊C4—44no—
415C4⋊C4—44no—
416C4⋊C4—82no—
417C4⋊C4—82no—
418C4⋊C4#12412yes—
419C4⋊C4#12412yes—
420C22⋊C4—24no—
421C22⋊C4—28no—
422C22⋊C4—28no—
423C22⋊C4—48no—
424C22⋊C4—42no—
425C22⋊C4—84no—
426C22⋊C4—84no—
427C42—24no—
428C42—24no—
429C42—24no—
430C42—48no—
431C42—48no—
432C42—48no—
433C42—42no—
434C42—42no—
435C42—42no—
436C42—42no—
437C42—42no—
438C42—42no—
439C42—42no—
440C42C2×C22⋊C412yes—
441C42C2×C22⋊C412yes—
442C42C4×D412yes—
443C42C4×D412yes—
444C42#12112yes—
445C42#12112yes—
446C42#12312yes—
447C42#12312yes—
448C23—21no—
449C23—21no—
450C23—21no—
451C23—21no—
452C23—24no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23C2×C4×D411yescharacteristic
460C23C2×#12011yescharacteristic
461C23C2×#12111yescharacteristic
462C23#54612yes—
463C23#54812yes—
464Q8—28no—
465Q8—44no—
466Q8—88no—
467Q8—88no—
468D4—28no—
469D4—48no—
470D4—48no—
471D4—48no—
472D4—48no—
473D4—44no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—28no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—24no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—48no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—44no—
506C2×C4—41no—
507C2×C4—44no—
508C2×C4—41no—
509C2×C4—44no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—44no—
513C2×C4—84no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—88no—
517C2×C4—168no—
518C2×C4#38612yes—
519C2×C4#38612yes—
520C2×C4#38612yes—
521C2×C4#38612yes—
522C2×C4#45312yes—
523C2×C4#45412yes—
524C2×C4#45512yes—
525C2×C4#45612yes—
526C2×C4C22×C22⋊C411yescharacteristic
527C2×C4C2×C4×D411yescharacteristic
528C2×C4C4×#4112yes—
529C2×C4#52012yes—
530C2×C4C2×#12111yescharacteristic
531C2×C4C2×#12311yescharacteristic
532C2×C4C2×#12411yescharacteristic
533C2×C4#52912yes—
534C2×C4#53211yescharacteristic
535C2×C4#53511yescharacteristic
536C2×C4#53611yescharacteristic
537C2×C4#54212yes—
538C2×C4#54612yes—
539C2×C4#54612yes—
540C2×C4#54712yes—
541C2×C4#54712yes—
542C2×C4#55612yes—
543C2×C4#55912yes—
544C8—44no—
545C8—82no—
546C22—22no—
547C22—22no—
548C22—22no—
549C22—22no—
550C22—21no—
551C22—21no—
552C22—21no—
553C22—21no—
554C22—44no—
555C22—44no—
556C22—44no—
557C22—82no—
558C22—82no—
559C22—82no—
560C22—82no—
561C22C2×#38611yescharacteristic
562C22#236712yes—
563C22#242712yes—
564C22#243211yescharacteristic
565C22#243511yescharacteristic
566C22C2×#45311yescharacteristic
567C22C2×#45511yescharacteristic
568C22C2×#45411yescharacteristic
569C22C2×#45611yescharacteristic
570C4—22no—
571C4—22no—
572C4—48no—
573C4—42no—
574C4—42no—
575C4—44no—
576C4—84no—
577C4—81no—
578C4—81no—
579C4—84no—
580C4#188214yes—
581C4#188314yes—
582C4C4×#12411yescharacteristic
583C4#232611yescharacteristic
584C4#233912yes—
585C4#244712yes—
586C4#254411yescharacteristic
587C4#254711yescharacteristic
588C2—22no—
589C2—22no—
590C2—84no—
591C2#1402711yescharacteristic
592C2C2×#188211yescharacteristic
593C2C2×#188311yescharacteristic
594C2#2566611yescharacteristic
595C2#2566711yescharacteristic
596C2#2587511yescharacteristic
597C2#2587611yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#532C22
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. #14714C2semidirect
6. #11195C2semidirect
7. #10857C2semidirect
8. #10781C2semidirect
9. C4×#339C2semidirect
14. #3141C4semidirect
63. C2×C4⋊Q8C2×C4semidirect
102. #417C2×C4semidirect
106. C4×M16C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#31412
#342072
#213132
C2×#19622
#147142
#111952
#108572
#107812
C4×#3392