Finite Group Databasebeta

← #198713

Subgroups — #198713

Order 512.

1141 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60943C211yesmaximal, characteristic
2#34145C211yesmaximal, characteristic
3#21332C211yesmaximal, characteristic
4#17575C211yesmaximal, characteristic
5C2×#1503C211yesmaximal, characteristic
6#9276C218yesmaximal
7#8014C211yesmaximal, characteristic
8#7924C211yesmaximal, characteristic
9C2×#533C2211yescharacteristic
10C2×#533C2211yescharacteristic
11#3249—22no—
12#3230—21no—
13#3230C412yescomplemented
14#3209—21no—
15#3209C412yescomplemented
16#3189C2214yes—
17#3170—22no—
18#3104—22no—
19#3104C414yescomplemented
20C2×#431C2211yescharacteristic
21C2×#431C2211yescharacteristic
22#2370—22no—
23#2369—21no—
24#2368C2214yes—
25#2365—21no—
26#2360—22no—
27#2296—21no—
28#2295—21no—
29#2295—22no—
30#2285C2211yescharacteristic
31C42×Q8—21no—
32C4×#117—21no—
33C4×#117C2211yescharacteristic
34C2×C4×C4⋊C4C2211yescharacteristic
35#1996—22no—
36#1996C2214yes—
37#1740—21no—
38#1740—21no—
39#1520C2214yes—
40#1518—22no—
41#1503—22no—
42#1503—21no—
43#1503—21no—
44#1503—22no—
45C2×#328C2214yes—
46#1294C2214yes—
47#1282C2214yes—
48#533—21no—
49#533—21no—
50#533—21no—
51#533—21no—
52#533—21no—
53#533—21no—
54#533—21no—
55#533—21no—
56#533—21no—
57#533—21no—
58#533—21no—
59#533—21no—
60#533C2×C412yes—
61#533C2×C412yes—
62#533D412yes—
63#533D412yes—
64C2×C4⋊Q8C2×C411yescharacteristic
65C2×C4⋊Q8C2×C411yescharacteristic
66C2×C4⋊Q8D411yescharacteristic
67C2×C4⋊Q8D411yescharacteristic
68C2×#125—21no—
69C2×#125—21no—
70C2×C22⋊Q8—21no—
71C2×C22⋊Q8—21no—
72C2×C22⋊Q8C2312yes—
73C2×C22⋊Q8C2312yes—
74C2×C4×Q8—22no—
75C2×C4×Q8—21no—
76C2×C4×Q8—21no—
77C2×C4×Q8C2×C412yes—
78C2×C4×Q8D412yes—
79C2×#117—21no—
80C2×#117—21no—
81C2×#117—21no—
82C2×#117D411yescharacteristic
83C2×#117D411yescharacteristic
84C22×C4⋊C4—21no—
85C22×C42—21no—
86C22×C42C2311yescharacteristic
87#500—22no—
88#494—22no—
89#487—22no—
90#487—22no—
91#484—24no—
92#484C2×C418yes—
93#477—24no—
94#477—24no—
95#467—28no—
96#462—24no—
97#462—24no—
98#451—24no—
99#451C2×C418yes—
100C4×Q16—28no—
101#431—21no—
102#431—21no—
103#431—21no—
104#431—21no—
105#431—21no—
106#431—21no—
107#431C2×C412yes—
108#431C2×C412yes—
109C2×C4⋊C8—21no—
110C2×C4⋊C8—21no—
111C2×C22⋊C8C2312yes—
112C2×C22⋊C8C2312yes—
113C2×C4×C8—21no—
114C2×C4×C8—21no—
115#391—22no—
116#391—24no—
117#389—22no—
118#389—24no—
119#389—22no—
120#389—22no—
121#387—24no—
122#387—24no—
123#387—24no—
124#385—24no—
125#384—21no—
126#382—21no—
127#380—21no—
128#380C2312yes—
129C4×C4⋊C4—22no—
130C4×C4⋊C4—22no—
131C4×C4⋊C4—22no—
132C4×C4⋊C4—22no—
133C4×C4⋊C4—22no—
134C4×C4⋊C4—21no—
135C4×C4⋊C4—21no—
136C4×C4⋊C4—21no—
137C4×C4⋊C4—21no—
138C4×C4⋊C4—21no—
139C4×C4⋊C4—24no—
140C4×C4⋊C4—21no—
141C4×C4⋊C4D412yes—
142C4×C4⋊C4D411yescharacteristic
143C4×C4⋊C4D411yescharacteristic
144C4×C22⋊C4—24no—
145C4×C22⋊C4D412yes—
146C4×C22⋊C4C2312yes—
147#376—22no—
148#376—21no—
149C2×#95C2312yes—
150C43—22no—
151C43—21no—
152#342—24no—
153#342—24no—
154#328—28no—
155#328—24no—
156#328—24no—
157C22×Q8—21no—
158C22×Q8—21no—
159C22×Q8C22×C412yes—
160C22×Q8C2×D412yes—
161C23×C4—22no—
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2411yescharacteristic
165C2×Q16—28no—
166C22×C8—22no—
167C22×C8—22no—
168C4⋊Q8—21no—
169C4⋊Q8—21no—
170C4⋊Q8—21no—
171C4⋊Q8—21no—
172C4⋊Q8—21no—
173C4⋊Q8—21no—
174C4⋊Q8—21no—
175C4⋊Q8—21no—
176C4⋊Q8—21no—
177C4⋊Q8—21no—
178C4⋊Q8—21no—
179C4⋊Q8—21no—
180C4⋊Q8C22⋊C412yes—
181C4⋊Q8C22⋊C412yes—
182C4⋊Q8C22⋊C412yes—
183C4⋊Q8C22⋊C412yes—
184#125—22no—
185#125—22no—
186#125—22no—
187#125—22no—
188#125—41no—
189#125—41no—
190#125—41no—
191#125—41no—
192C22⋊Q8—22no—
193C22⋊Q8—22no—
194C22⋊Q8—22no—
195C22⋊Q8—22no—
196C22⋊Q8—22no—
197C22⋊Q8—22no—
198C22⋊Q8—22no—
199C22⋊Q8—22no—
200C22⋊Q8—22no—
201C22⋊Q8—22no—
202C22⋊Q8—41no—
203C22⋊Q8—41no—
204C22⋊Q8—41no—
205C22⋊Q8—41no—
206C22⋊Q8C22×C414yes—
207C22⋊Q8C2×D414yes—
208C4×Q8—22no—
209C4×Q8—22no—
210C4×Q8—22no—
211C4×Q8—22no—
212C4×Q8—22no—
213C4×Q8—22no—
214C4×Q8—22no—
215C4×Q8—22no—
216C4×Q8—22no—
217C4×Q8—22no—
218C4×Q8—48no—
219C4×Q8—41no—
220C4×Q8—41no—
221C4×Q8—41no—
222C4×Q8—41no—
223C4×Q8C22⋊C414yes—
224C4×Q8C22⋊C414yes—
225#117—22no—
226#117—22no—
227#117—22no—
228#117—22no—
229#117—22no—
230#117—22no—
231#117—22no—
232#117—41no—
233#117—41no—
234#117—41no—
235#117—41no—
236#117—42no—
237#117—42no—
238#117Q1614yescomplemented
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—41no—
259C2×C4⋊C4—41no—
260C2×C4⋊C4C22×C412yes—
261C2×C4⋊C4C2×D412yes—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4C2×D412yes—
266C2×C22⋊C4C2×D412yes—
267C2×C42—22no—
268C2×C42—21no—
269C2×C42—24no—
270C2×C42—22no—
271C2×C42—21no—
272C2×C42—22no—
273C2×C42—21no—
274C2×C42—21no—
275C2×C42—22no—
276C2×C42—24no—
277C2×C42—22no—
278C2×C42—42no—
279C2×C42—42no—
280C2×C42—41no—
281C2×C42C22×C411yescharacteristic
282C2×C42C22×C411yescharacteristic
283C2×C42C2×D411yescharacteristic
284C2×C42C2×D411yescharacteristic
285C2×C42C2×D411yescharacteristic
286C2×C42C2×D411yescharacteristic
287C2×C42C2×D411yescharacteristic
288C2×C42C2×D411yescharacteristic
289C2×C42#4111yescharacteristic
290C2×C42#4111yescharacteristic
291C2×C42#4111yescharacteristic
292C2×C42#4111yescharacteristic
293#107—28no—
294C4⋊C8—22no—
295C4⋊C8—22no—
296C4⋊C8—41no—
297C4⋊C8—41no—
298#103—28no—
299#103—28no—
300C22⋊C8—22no—
301C22⋊C8—22no—
302C22⋊C8—22no—
303C22⋊C8C22×C414yes—
304C4×C8—22no—
305C4×C8—22no—
306C4×C8—41no—
307C4×C8—41no—
308#95—24no—
309#95—24no—
310#95—24no—
311#95—24no—
312#95—22no—
313#95—22no—
314#95—22no—
315#95—22no—
316#95—41no—
317#95C2×D412yes—
318#95C2×D412yes—
319C24C22×D411yescharacteristic
320C2×Q8—22no—
321C2×Q8—22no—
322C2×Q8—22no—
323C2×Q8—22no—
324C2×Q8—22no—
325C2×Q8—44no—
326C2×Q8—41no—
327C2×Q8—41no—
328C2×Q8—82no—
329C2×Q8—82no—
330C2×Q8—82no—
331C2×Q8C2×C22⋊C414yes—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—42no—
341C22×C4—44no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—81no—
346C22×C4—82no—
347C22×C4C4×D411yescharacteristic
348C22×C4C4×D411yescharacteristic
349C22×C4C4×D411yescharacteristic
350C22×C4C4×D411yescharacteristic
351C22×C4#12011yescharacteristic
352C22×C4#12011yescharacteristic
353C22×C4#12011yescharacteristic
354C22×C4#12011yescharacteristic
355C22×C4#12111yescharacteristic
356C22×C4#12111yescharacteristic
357C22×C4#12111yescharacteristic
358C22×C4#12111yescharacteristic
359C22×C4C2×Q1612yes—
360C22×C4#13512yes—
361C22×C4#13712yes—
362C22×C4#13712yes—
363C22×C4C23×C411yescharacteristic
364C22×C4C22×D411yescharacteristic
365C22×C4C22×D411yescharacteristic
366C22×C4C22×D411yescharacteristic
367C22×C4C2×#4111yescharacteristic
368C22×C4C2×#4111yescharacteristic
369Q16—88no—
370C2×C8—24no—
371C2×C8—42no—
372C2×C8—82no—
373C2×C8—82no—
374C4⋊C4—24no—
375C4⋊C4—24no—
376C4⋊C4—24no—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—24no—
385C4⋊C4—24no—
386C4⋊C4—24no—
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—42no—
390C4⋊C4—44no—
391C4⋊C4—48no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—41no—
395C4⋊C4—41no—
396C4⋊C4—41no—
397C4⋊C4—41no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—41no—
401C4⋊C4—41no—
402C4⋊C4—42no—
403C4⋊C4—82no—
404C4⋊C4—84no—
405C4⋊C4—82no—
406C4⋊C4—82no—
407C4⋊C4C2×C22⋊C414yes—
408C22⋊C4—24no—
409C22⋊C4—24no—
410C22⋊C4—22no—
411C22⋊C4—22no—
412C22⋊C4—22no—
413C22⋊C4—42no—
414C22⋊C4—42no—
415C22⋊C4—42no—
416C22⋊C4—42no—
417C22⋊C4C2×Q1614yes—
418C42—24no—
419C42—24no—
420C42—24no—
421C42—44no—
422C42—48no—
423C42—42no—
424C42—42no—
425C42—42no—
426C42—44no—
427C42—42no—
428C42—42no—
429C42—82no—
430C42—84no—
431C42—82no—
432C42C2×C22⋊C411yescharacteristic
433C42C2×C22⋊C411yescharacteristic
434C42C2×C22⋊C411yescharacteristic
435C42C2×C22⋊C411yescharacteristic
436C42C4×D411yescharacteristic
437C42C4×D411yescharacteristic
438C42C4×D411yescharacteristic
439C42C4×D411yescharacteristic
440C42#12111yescharacteristic
441C42#12111yescharacteristic
442C42#12111yescharacteristic
443C42#12111yescharacteristic
444C42#12311yescharacteristic
445C42#12311yescharacteristic
446C42#12311yescharacteristic
447C42#12311yescharacteristic
448C23—41no—
449C23—41no—
450C23C2×C4×D411yescharacteristic
451C23C2×#12011yescharacteristic
452C23C2×#12111yescharacteristic
453C23C22×Q1611yescharacteristic
454C23C2×#13511yescharacteristic
455C23C2×#13711yescharacteristic
456C23C2×#13711yescharacteristic
457Q8—44no—
458Q8—44no—
459Q8—44no—
460Q8—44no—
461Q8—82no—
462Q8—82no—
463Q8—82no—
464Q8—82no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—42no—
482C2×C4—44no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—44no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—84no—
496C2×C4—84no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—82no—
502C2×C4#38611yescharacteristic
503C2×C4#38611yescharacteristic
504C2×C4#38611yescharacteristic
505C2×C4#38611yescharacteristic
506C2×C4#38611yescharacteristic
507C2×C4#38611yescharacteristic
508C2×C4#38611yescharacteristic
509C2×C4#38611yescharacteristic
510C2×C4C4×Q1612yes—
511C2×C4C4×Q1612yes—
512C2×C4#44112yes—
513C2×C4#44112yes—
514C2×C4#45112yes—
515C2×C4#45112yes—
516C2×C4#45212yes—
517C2×C4#45212yes—
518C2×C4#46212yes—
519C2×C4#46212yes—
520C2×C4#46412yes—
521C2×C4#46412yes—
522C2×C4C22×C22⋊C411yescharacteristic
523C2×C4C2×C4×D411yescharacteristic
524C2×C4C2×#12111yescharacteristic
525C2×C4C2×#12311yescharacteristic
526C8—44no—
527C8—82no—
528C22—21no—
529C22—21no—
530C22—21no—
531C22—21no—
532C22—21no—
533C22—21no—
534C22—41no—
535C22—41no—
536C22—41no—
537C22—41no—
538C22C2×#38611yescharacteristic
539C22C2×C4×Q1611yescharacteristic
540C22C2×#44111yescharacteristic
541C22C2×#45111yescharacteristic
542C22C2×#45211yescharacteristic
543C22C2×#46211yescharacteristic
544C22C2×#46411yescharacteristic
545C4—21no—
546C4—21no—
547C4—21no—
548C4—21no—
549C4—21no—
550C4—21no—
551C4—21no—
552C4—44no—
553C4—44no—
554C4—44no—
555C4—44no—
556C4—82no—
557C4—82no—
558C4—82no—
559C4—82no—
560C4—162no—
561C4#187812yes—
562C4#187812yes—
563C4#187812yes—
564C4#187812yes—
565C4#190612yes—
566C2—21no—
567C2—21no—
568C2—21no—
569C2—21no—
570C2—21no—
571C2—21no—
572C2C2×#187811yescharacteristic
573C2C2×#190611yescharacteristic
574C2#2101411yescharacteristic
subgroupG/H
centerC2×C4#386
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC22C2×#386
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
13. #3230C4semidirect
15. #3209C4semidirect
19. #3104C4semidirect
238. #117Q16semidirect

Maximal subgroups

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

subgroupindex
#609432
#341452
#213322
#175752
C2×#15032
#92762
#80142
#79242