Finite Group Databasebeta

← #369421

Subgroups — #369421

Order 512.

1646 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3141C211yesmaximal, characteristic
2#34206C211yesmaximal, characteristic
3#21324C211yesmaximal, characteristic, complemented
4C2×#1980C211yesmaximal, characteristic
5#14155C211yesmaximal, characteristic, complemented
6#11195C214yesmaximal, complemented
7#10851C211yesmaximal, characteristic, complemented
8#10781C214yesmaximal, complemented
9C4×#343C211yesmaximal, characteristic, complemented
10C2×#532C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12C2×#520C2212yes—
13#3141—24no—
14#3141C418yescomplemented
15C2×#489—22no—
16C2×#487—21no—
17C2×#486—21no—
18C22×#104C2214yes—
19C2×#417C2214yes—
20C2×C4×M16C2211yescharacteristic
21C2×C4×M16C2211yescharacteristic
22#2367—22no—
23#2366—21no—
24#2365—21no—
25#2354—21no—
26#2352—22no—
27D4×C4⋊C4—21no—
28#2325—22no—
29#2311C2212yes—
30C4×#121—22no—
31C2×#391—22no—
32#2285C2211yescharacteristic
33C2×C4×C4⋊C4C2211yescharacteristic
34C2×C4×C4⋊C4C2211yescharacteristic
35#1993—22no—
36#1990—21no—
37#1990—21no—
38#1980—24no—
39C2×#343C2212yes—
40C2×#343C2212yes—
41#1736C2214yes—
42C2×#339C2214yes—
43#1445—22no—
44C4⋊#95C2212yes—
45C4⋊#95C2212yes—
46C22×#41C2312yes—
47C23×Q8—21no—
48C22×M16C2312yes—
49C22×M16C2312yes—
50#532—22no—
51#532—22no—
52#532C2×C414yes—
53#532D414yes—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8—21no—
59C2×C4⋊Q8—22no—
60C2×C4⋊Q8—22no—
61C2×C4⋊Q8C2×C414yescomplemented
62C2×C4⋊Q8D411yescharacteristic
63C2×#127D411yescharacteristic
64C2×#124D412yes—
65C2×#121—21no—
66C2×#121—21no—
67#520—28no—
68#520—44no—
69C2×C4×Q8—22no—
70C2×C4×D4—22no—
71C2×C4×D4—24no—
72C2×C4×D4—22no—
73C2×C4×D4—21no—
74C2×C4×D4—21no—
75C2×#117—22no—
76C2×#117C2312yes—
77C22×C4⋊C4—21no—
78C22×C4⋊C4—21no—
79C22×C4⋊C4—22no—
80C22×C4⋊C4C2311yescharacteristic
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2311yescharacteristic
83C22×C42D412yes—
84C22×C42C2312yes—
85C22×C42C2311yescharacteristic
86#489—216no—
87#487—28no—
88#486—28no—
89C2×#104—28no—
90C2×#104—28no—
91C2×#104—44no—
92#417—28no—
93#417C2×C4116yescomplemented
94C2×#103—24no—
95C2×#102—24no—
96C4×M16—22no—
97C4×M16—22no—
98C4×M16—22no—
99C4×M16C2×C414yescomplemented
100C2×#97—21no—
101C2×#97—21no—
102C2×C4×C8—21no—
103C2×C4×C8—21no—
104#391—22no—
105#391—48no—
106#390—22no—
107#390—22no—
108#389—22no—
109#388—24no—
110#388—24no—
111#387—24no—
112#386—24no—
113#386—24no—
114#385—24no—
115#384D411yescharacteristic
116#384D411yescharacteristic
117#382D412yes—
118#380—21no—
119#380—21no—
120#380—22no—
121#380—22no—
122C4×C4⋊C4—22no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4—21no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4—21no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4—22no—
130C4×C4⋊C4—22no—
131C4×C22⋊C4—24no—
132C2×#95—24no—
133C2×#95—24no—
134C2×#95—21no—
135C2×#95D412yes—
136#343—24no—
137#343—24no—
138#340—24no—
139#340—24no—
140#339—44no—
141C2×#41—22no—
142C2×#41—22no—
143C2×#41—24no—
144C2×#41—24no—
145C2×#41—24no—
146C2×#41—24no—
147C2×#41C22×C414yes—
148C2×#41C2×D414yes—
149C22×Q8—24no—
150C22×Q8—44no—
151C22×Q8—44no—
152C22×Q8—41no—
153C22×Q8—41no—
154C22×Q8—42no—
155C22×Q8C2×D412yes—
156C22×D4—22no—
157C22×D4—21no—
158C22×D4—21no—
159C22×D4C2×D412yes—
160C23×C4—22no—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4—24no—
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D412yes—
166C23×C4C2×D412yes—
167C23×C4C2×D411yescharacteristic
168C23×C4C2411yescharacteristic
169C2×M16—22no—
170C2×M16—22no—
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×M16C22×C414yes—
180C2×M16#4114yes—
181C22×C8—22no—
182C22×C8—22no—
183C4⋊Q8—24no—
184C4⋊Q8—24no—
185C4⋊Q8—24no—
186C4⋊Q8—24no—
187C4⋊Q8—22no—
188C4⋊Q8—42no—
189C4⋊Q8—42no—
190C4⋊Q8—42no—
191C4⋊Q8—42no—
192C4⋊Q8C22⋊C414yes—
193#127—22no—
194#127C22⋊C414yes—
195#124—24no—
196#124C22⋊C418yes—
197#121—24no—
198#121—24no—
199#121—24no—
200#121—24no—
201C4×Q8—48no—
202C4×D4—24no—
203C4×D4—24no—
204C4×D4—24no—
205C4×D4—24no—
206C4×D4—48no—
207C4×D4—48no—
208C4×D4—48no—
209C4×D4—84no—
210#117—24no—
211#117—24no—
212#117—48no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—48no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—41no—
237C2×C4⋊C4—41no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4C22×C414yes—
241C2×C4⋊C4C2×D412yes—
242C2×C4⋊C4C2×D412yes—
243C2×C4⋊C4#4114yes—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C42—24no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—24no—
256C2×C42—24no—
257C2×C42—22no—
258C2×C42—44no—
259C2×C42C22×C412yes—
260C2×C42C2×D412yes—
261C2×C42C2×D411yescharacteristic
262C2×C42C2×D411yescharacteristic
263C2×C42C2×D411yescharacteristic
264C2×C42C2×D411yescharacteristic
265C2×C42#4112yes—
266C2×C42#4112yes—
267#104—416no—
268#103—216no—
269#102—216no—
270#97—24no—
271#97—42no—
272C4×C8—24no—
273C4×C8—42no—
274#95—24no—
275#95—24no—
276#95—22no—
277#95—22no—
278#95—22no—
279#95—22no—
280#95—48no—
281#95—48no—
282#95—42no—
283C24—22no—
284C24C22×D411yescharacteristic
285#41—28no—
286#41—28no—
287#41—28no—
288#41—28no—
289#41—44no—
290#41—44no—
291#41—44no—
292#41—44no—
293C2×Q8—22no—
294C2×Q8—24no—
295C2×Q8—24no—
296C2×Q8—416no—
297C2×Q8—44no—
298C2×Q8—44no—
299C2×Q8—82no—
300C2×Q8—82no—
301C2×Q8—88no—
302C2×Q8—82no—
303C2×Q8—82no—
304C2×Q8C2×C22⋊C414yes—
305C2×D4—24no—
306C2×D4—24no—
307C2×D4—22no—
308C2×D4—24no—
309C2×D4—24no—
310C2×D4—24no—
311C2×D4—24no—
312C2×D4—42no—
313C2×D4—42no—
314C2×D4—42no—
315C2×D4—42no—
316C2×D4—42no—
317C2×D4—42no—
318C2×D4—81no—
319C2×D4—81no—
320C2×D4C2×C22⋊C414yes—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—48no—
338C22×C4—44no—
339C22×C4—44no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—44no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—44no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—84no—
355C22×C4—84no—
356C22×C4C4×D412yes—
357C22×C4C4×D412yes—
358C22×C4#12011yescharacteristic
359C22×C4#12011yescharacteristic
360C22×C4#12011yescharacteristic
361C22×C4#12011yescharacteristic
362C22×C4#12112yes—
363C22×C4#12112yes—
364C22×C4#12414yes—
365C22×C4C23×C411yescharacteristic
366C22×C4C22×D411yescharacteristic
367C22×C4C22×D411yescharacteristic
368C22×C4C22×D411yescharacteristic
369C22×C4C2×#4111yescharacteristic
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—42no—
380M16—42no—
381M16—42no—
382M16—42no—
383C2×C8—24no—
384C2×C8—24no—
385C2×C8—42no—
386C2×C8—42no—
387C2×C8—42no—
388C2×C8—42no—
389C4⋊C4—24no—
390C4⋊C4—24no—
391C4⋊C4—24no—
392C4⋊C4—24no—
393C4⋊C4—24no—
394C4⋊C4—24no—
395C4⋊C4—24no—
396C4⋊C4—24no—
397C4⋊C4—48no—
398C4⋊C4—48no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—44no—
404C4⋊C4—44no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—81no—
408C4⋊C4—81no—
409C22⋊C4—28no—
410C22⋊C4—28no—
411C22⋊C4—48no—
412C22⋊C4—42no—
413C22⋊C4—42no—
414C22⋊C4—84no—
415C22⋊C4—84no—
416C42—48no—
417C42—44no—
418C42—44no—
419C42—44no—
420C42—44no—
421C42—82no—
422C42C2×C22⋊C412yes—
423C42C2×C22⋊C412yes—
424C42C4×D412yes—
425C42C4×D412yes—
426C42#12112yes—
427C42#12112yes—
428C42#12312yes—
429C42#12312yes—
430C23—21no—
431C23—21no—
432C23—21no—
433C23—21no—
434C23—24no—
435C23—42no—
436C23—42no—
437C23—42no—
438C23—42no—
439C23—42no—
440C23—42no—
441C23C2×C4×D411yescharacteristic
442C23C2×#12011yescharacteristic
443C23C2×#12111yescharacteristic
444C23C2×#12411yescharacteristic
445C23#53211yescharacteristic
446C23#53511yescharacteristic
447C23#53611yescharacteristic
448Q8—28no—
449Q8—44no—
450Q8—88no—
451Q8—88no—
452D4—28no—
453D4—44no—
454D4—44no—
455D4—44no—
456D4—44no—
457D4—48no—
458D4—44no—
459D4—84no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—48no—
477C2×C4—48no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—48no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—44no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—84no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4—168no—
504C2×C4#38612yes—
505C2×C4#38612yes—
506C2×C4#38612yes—
507C2×C4#38612yes—
508C2×C4#45312yes—
509C2×C4#45412yes—
510C2×C4C22×C22⋊C411yescharacteristic
511C2×C4C2×C4×D411yescharacteristic
512C2×C4C4×#4112yes—
513C2×C4#52012yes—
514C2×C4C2×#12111yescharacteristic
515C2×C4C2×#12311yescharacteristic
516C2×C4#52912yes—
517C2×C4#54212yes—
518C2×C4#54612yes—
519C2×C4#54612yes—
520C2×C4#54612yes—
521C2×C4#54712yes—
522C2×C4#54712yes—
523C2×C4#54812yes—
524C2×C4#55612yes—
525C2×C4#55912yes—
526C8—44no—
527C8—44no—
528C22—21no—
529C22—21no—
530C22—21no—
531C22—21no—
532C22—21no—
533C22—21no—
534C22—21no—
535C22—21no—
536C22—21no—
537C22—21no—
538C22—21no—
539C22—21no—
540C22—44no—
541C22—44no—
542C22—44no—
543C22—82no—
544C22—82no—
545C22—82no—
546C22—82no—
547C22C2×#38611yescharacteristic
548C22C4×#12411yescharacteristic
549C22#232611yescharacteristic
550C22#243211yescharacteristic
551C22#243511yescharacteristic
552C22#254411yescharacteristic
553C22#254711yescharacteristic
554C22C2×#45311yescharacteristic
555C22C2×#45411yescharacteristic
556C22#302211yescharacteristic
557C22#302211yescharacteristic
558C4—22no—
559C4—22no—
560C4—22no—
561C4—44no—
562C4—44no—
563C4—84no—
564C4—82no—
565C4—82no—
566C4—82no—
567C4—84no—
568C4#188314yes—
569C4#233912yes—
570C4#236712yes—
571C4#242712yes—
572C4#244712yes—
573C2—21no—
574C2—21no—
575C2—21no—
576C2—21no—
577C2—84no—
578C2#1402711yescharacteristic
579C2C2×#188311yescharacteristic
580C2#2094411yescharacteristic
581C2#2569411yescharacteristic
582C2#2569611yescharacteristic
583C2#2587011yescharacteristic
584C2#2587211yescharacteristic
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. #21324C2semidirect
5. #14155C2semidirect
6. #11195C2semidirect
7. #10851C2semidirect
8. #10781C2semidirect
9. C4×#343C2semidirect
14. #3141C4semidirect
61. C2×C4⋊Q8C2×C4semidirect
93. #417C2×C4semidirect
99. 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
#342062
#213242
C2×#19802
#141552
#111952
#108512
#107812
C4×#3432