Finite Group Databasebeta

← #107713

Subgroups — #107713

Order 512.

942 conjugacy classes of subgroups.

The subgroup lattice has 597 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34214C211yesmaximal, characteristic
2C2×#1550C211yesmaximal, characteristic
3C2×#1503C211yesmaximal, characteristic
4#10852C211yesmaximal, characteristic
5#10849C211yesmaximal, characteristic
6#8043C211yesmaximal, characteristic
7#7891C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9C2×#431C2211yescharacteristic
10C2×#423C2211yescharacteristic
11C22×C4⋊C8C2211yescharacteristic
12#2360—22no—
13#2360—22no—
14#2355—21no—
15#2355C412yescomplemented
16Q8×C4⋊C4—21no—
17Q8×C4⋊C4C412yescomplemented
18#2330—22no—
19#2317—21no—
20C4×C4⋊Q8—21no—
21C4×C4⋊Q8—21no—
22C2×#384C2211yescharacteristic
23#2274—21no—
24C4×#117C2211yescharacteristic
25C2×C4×C4⋊C4C2211yescharacteristic
26C2×#340—22no—
27C2×#339—21no—
28C2×#339—21no—
29#1550—22no—
30#1550C414yescomplemented
31C2×#333—21no—
32C2×#332—21no—
33#1503—22no—
34#1503C414yescomplemented
35C2×#328—22no—
36#1296—22no—
37#1279—21no—
38#1279—21no—
39C22×M16—21no—
40C23×C8—21no—
41#533—21no—
42#533—21no—
43#533—21no—
44#533—21no—
45#533C2×C412yes—
46#533C2×C412yes—
47#533D412yes—
48#533Q812yes—
49C2×C4⋊Q8D411yescharacteristic
50C2×C4⋊Q8D411yescharacteristic
51C2×#125—21no—
52C2×C22⋊Q8—21no—
53C2×C22⋊Q8D412yes—
54C2×C4×Q8—22no—
55C2×C4×Q8—22no—
56C2×C4×Q8—21no—
57C2×C4×Q8C2×C412yes—
58C2×#117—21no—
59C2×#117D411yescharacteristic
60C2×#117D411yescharacteristic
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4D412yes—
65C22×C42—21no—
66C22×C42C2311yescharacteristic
67#431—21no—
68#431—21no—
69#431—21no—
70#431C2×C412yes—
71#423—21no—
72#423—21no—
73#423—21no—
74#423C2×C412yes—
75C2×C4⋊C8—21no—
76C2×C4⋊C8—21no—
77C2×C4⋊C8—21no—
78C2×C4⋊C8—21no—
79C2×C4⋊C8—21no—
80C2×C4⋊C8—21no—
81C2×C4⋊C8—21no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8C2×C412yes—
84C2×C22⋊C8—21no—
85C2×C4×C8—21no—
86#391—22no—
87#391—22no—
88#389—22no—
89#389—22no—
90#389—22no—
91#389—24no—
92#389—22no—
93#389—24no—
94#387—24no—
95#387—24no—
96#385—24no—
97#384—22no—
98#384—21no—
99#384—21no—
100#384C2×C412yes—
101#384D412yes—
102#382—22no—
103C4×C4⋊C4—22no—
104C4×C4⋊C4—22no—
105C4×C4⋊C4—22no—
106C4×C4⋊C4—22no—
107C4×C4⋊C4—22no—
108C4×C4⋊C4—22no—
109C4×C4⋊C4—21no—
110C4×C4⋊C4—21no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4C2×C412yes—
113C4×C4⋊C4C2×C411yescharacteristic
114C4×C4⋊C4C2×C411yescharacteristic
115C4×C4⋊C4D411yescharacteristic
116C4×C4⋊C4D411yescharacteristic
117C4×C22⋊C4—24no—
118C4×C22⋊C4—21no—
119C4×C22⋊C4—21no—
120#376—21no—
121C2×#95—21no—
122C43—21no—
123#340—44no—
124#339—42no—
125#339—42no—
126#333—24no—
127#332—24no—
128#328—28no—
129C22×Q8—21no—
130C22×Q8C22⋊C412yes—
131C23×C4—22no—
132C23×C4—41no—
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C2×M16—22no—
137C2×M16—41no—
138C2×M16—41no—
139C2×M16—41no—
140C2×M16—41no—
141C2×M16—41no—
142C22×C8—22no—
143C22×C8—42no—
144C22×C8—42no—
145C22×C8—41no—
146C22×C8—41no—
147C22×C8—41no—
148C22×C8—41no—
149C4⋊Q8—21no—
150C4⋊Q8—21no—
151C4⋊Q8—21no—
152C4⋊Q8—21no—
153C4⋊Q8C22⋊C412yes—
154C4⋊Q8C22⋊C412yes—
155C4⋊Q8C4⋊C412yes—
156C4⋊Q8C4⋊C412yes—
157#125—22no—
158#125—22no—
159#125—22no—
160#125—22no—
161C22⋊Q8—22no—
162C22⋊Q8—22no—
163C22⋊Q8—22no—
164C22⋊Q8—22no—
165C22⋊Q8—22no—
166C22⋊Q8—22no—
167C22⋊Q8C22⋊C414yes—
168C22⋊Q8C4⋊C414yes—
169C4×Q8—22no—
170C4×Q8—22no—
171C4×Q8—22no—
172C4×Q8—22no—
173C4×Q8—22no—
174C4×Q8—22no—
175C4×Q8—84no—
176C4×Q8—82no—
177C4×Q8—82no—
178C4×Q8C4214yes—
179C4×Q8C4⋊C414yes—
180#117—22no—
181#117—22no—
182#117—22no—
183#117—22no—
184#117—21no—
185#117—21no—
186#117—21no—
187#117—21no—
188#117—21no—
189#117—21no—
190#117SD1612yes—
191#117Q1612yes—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—41no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4—41no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—82no—
217C2×C4⋊C4C22⋊C411yescharacteristic
218C2×C4⋊C4C22⋊C411yescharacteristic
219C2×C4⋊C4C22⋊C412yes—
220C2×C4⋊C4C22⋊C411yescharacteristic
221C2×C4⋊C4C22⋊C411yescharacteristic
222C2×C4⋊C4C22⋊C414yes—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—21no—
226C2×C42—22no—
227C2×C42—21no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—21no—
233C2×C42—22no—
234C2×C42—24no—
235C2×C42—24no—
236C2×C42—41no—
237C2×C42—41no—
238C2×C42—41no—
239C2×C42—41no—
240C2×C42—41no—
241C2×C42C22×C411yescharacteristic
242C2×C42C22×C411yescharacteristic
243C2×C42C22×C411yescharacteristic
244C2×C42C2×D411yescharacteristic
245C2×C42C2×D411yescharacteristic
246C2×C42C2×D411yescharacteristic
247C2×C42C2×D411yescharacteristic
248C2×C42C2×Q811yescharacteristic
249C2×C42#4111yescharacteristic
250C2×C42#4111yescharacteristic
251C2×C42#4111yescharacteristic
252C2×C42#4111yescharacteristic
253C4⋊C8—22no—
254C4⋊C8—22no—
255C4⋊C8—22no—
256C4⋊C8—22no—
257C4⋊C8—22no—
258C4⋊C8—41no—
259C4⋊C8—41no—
260C4⋊C8—41no—
261C4⋊C8—41no—
262C4⋊C8—41no—
263C22⋊C8—22no—
264C22⋊C8—41no—
265C4×C8—22no—
266C4×C8—41no—
267#95—24no—
268#95—24no—
269#95—24no—
270#95—24no—
271#95—21no—
272#95—21no—
273#95—21no—
274#95—21no—
275#95—22no—
276#95—41no—
277C24#12011yescharacteristic
278C2×Q8—22no—
279C2×Q8—22no—
280C2×Q8—22no—
281C2×Q8—42no—
282C2×Q8—42no—
283C2×Q8—41no—
284C2×Q8—41no—
285C2×Q8—41no—
286C2×Q8—41no—
287C2×Q8#9514yes—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—21no—
297C22×C4—24no—
298C22×C4—21no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—82no—
307C22×C4—82no—
308C22×C4—81no—
309C22×C4—81no—
310C22×C4—81no—
311C22×C4—81no—
312C22×C4C2×C22⋊C411yescharacteristic
313C22×C4C2×C22⋊C411yescharacteristic
314C22×C4C2×C22⋊C411yescharacteristic
315C22×C4C2×C22⋊C411yescharacteristic
316C22×C4C2×C4⋊C411yescharacteristic
317C22×C4C4×D411yescharacteristic
318C22×C4C4×D411yescharacteristic
319C22×C4C4×D411yescharacteristic
320C22×C4C4×D411yescharacteristic
321C22×C4#12011yescharacteristic
322C22×C4#12011yescharacteristic
323C22×C4#12011yescharacteristic
324C22×C4#12111yescharacteristic
325C22×C4#12111yescharacteristic
326C22×C4C22⋊Q811yescharacteristic
327C22×C4C22⋊Q811yescharacteristic
328C22×C4#12311yescharacteristic
329C22×C4#12311yescharacteristic
330C22×C4C2×SD1611yescharacteristic
331C22×C4C2×Q1611yescharacteristic
332C22×C4#13511yescharacteristic
333C22×C4#13511yescharacteristic
334C22×C4AGL(1,ℤ/8)11yescharacteristic
335C22×C4#13711yescharacteristic
336C22×C4#13711yescharacteristic
337C22×C4#13711yescharacteristic
338M16—42no—
339M16—42no—
340M16—81no—
341M16—81no—
342C2×C8—44no—
343C2×C8—42no—
344C2×C8—42no—
345C2×C8—42no—
346C2×C8—81no—
347C2×C8—81no—
348C2×C8—82no—
349C2×C8—81no—
350C2×C8—81no—
351C2×C8—81no—
352C2×C8—81no—
353C4⋊C4—24no—
354C4⋊C4—24no—
355C4⋊C4—22no—
356C4⋊C4—22no—
357C4⋊C4—22no—
358C4⋊C4—22no—
359C4⋊C4—24no—
360C4⋊C4—22no—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—24no—
364C4⋊C4—21no—
365C4⋊C4—21no—
366C4⋊C4—24no—
367C4⋊C4—21no—
368C4⋊C4—21no—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4—44no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—44no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—82no—
383C4⋊C4—84no—
384C4⋊C4—84no—
385C4⋊C4#9514yes—
386C4⋊C4#10312yes—
387C4⋊C4#10312yes—
388C22⋊C4—24no—
389C22⋊C4—24no—
390C22⋊C4—22no—
391C22⋊C4—22no—
392C22⋊C4—41no—
393C22⋊C4—41no—
394C42—24no—
395C42—24no—
396C42—22no—
397C42—22no—
398C42—42no—
399C42—42no—
400C42—41no—
401C42—41no—
402C42—41no—
403C42—41no—
404C42—41no—
405C42—41no—
406C42—42no—
407C42—42no—
408C42—84no—
409C42—84no—
410C42—81no—
411C42—81no—
412C42C2×C4211yescharacteristic
413C42C2×C22⋊C411yescharacteristic
414C42C2×C4⋊C411yescharacteristic
415C42C2×C4⋊C411yescharacteristic
416C42#11711yescharacteristic
417C42#11711yescharacteristic
418C42C4×D411yescharacteristic
419C42C4×D411yescharacteristic
420C42C4×D411yescharacteristic
421C42C4×D411yescharacteristic
422C42#12111yescharacteristic
423C42#12111yescharacteristic
424C42C22⋊Q811yescharacteristic
425C42C22⋊Q811yescharacteristic
426C42#12311yescharacteristic
427C42#12311yescharacteristic
428C23—21no—
429C23—21no—
430C23—21no—
431C23—21no—
432C23#37911yescharacteristic
433C23#38511yescharacteristic
434C23#38611yescharacteristic
435C23#44811yescharacteristic
436C23#44911yescharacteristic
437C23#45111yescharacteristic
438C23#45211yescharacteristic
439Q8—44no—
440Q8—44no—
441Q8—42no—
442Q8—42no—
443Q8—42no—
444Q8—42no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—42no—
469C2×C4—41no—
470C2×C4—42no—
471C2×C4—41no—
472C2×C4—42no—
473C2×C4—41no—
474C2×C4—42no—
475C2×C4—44no—
476C2×C4—41no—
477C2×C4—44no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—44no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—44no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4—162no—
492C2×C4—162no—
493C2×C4C2×#9511yescharacteristic
494C2×C4C4×C22⋊C411yescharacteristic
495C2×C4C4×C22⋊C411yescharacteristic
496C2×C4#38011yescharacteristic
497C2×C4#38011yescharacteristic
498C2×C4#38111yescharacteristic
499C2×C4#38511yescharacteristic
500C2×C4#38511yescharacteristic
501C2×C4#38511yescharacteristic
502C2×C4#38611yescharacteristic
503C2×C4#38611yescharacteristic
504C2×C4#38611yescharacteristic
505C2×C4C2×#10311yescharacteristic
506C2×C4#41611yescharacteristic
507C2×C4#41711yescharacteristic
508C2×C4#41911yescharacteristic
509C2×C4C4×SD1611yescharacteristic
510C2×C4C4×SD1611yescharacteristic
511C2×C4C4×Q1611yescharacteristic
512C2×C4C4×Q1611yescharacteristic
513C2×C4#44011yescharacteristic
514C2×C4#44011yescharacteristic
515C2×C4#44111yescharacteristic
516C2×C4#44111yescharacteristic
517C2×C4#44811yescharacteristic
518C2×C4#44911yescharacteristic
519C2×C4#45111yescharacteristic
520C2×C4#45211yescharacteristic
521C2×C4#45311yescharacteristic
522C2×C4#45411yescharacteristic
523C2×C4#45511yescharacteristic
524C2×C4#45611yescharacteristic
525C2×C4#46011yescharacteristic
526C2×C4#46211yescharacteristic
527C2×C4#46411yescharacteristic
528C2×C4#46411yescharacteristic
529C2×C4#47511yescharacteristic
530C2×C4#47711yescharacteristic
531C2×C4#47911yescharacteristic
532C2×C4#47911yescharacteristic
533C8—82no—
534C8—82no—
535C8—82no—
536C22—21no—
537C22—21no—
538C22—21no—
539C22—21no—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—21no—
544C22—21no—
545C22—21no—
546C22—21no—
547C22—21no—
548C22C22⋊#9511yescharacteristic
549C22#178711yescharacteristic
550C22#178811yescharacteristic
551C22#179211yescharacteristic
552C22#179311yescharacteristic
553C22#179511yescharacteristic
554C22#179611yescharacteristic
555C22#186311yescharacteristic
556C22#186511yescharacteristic
557C22#187811yescharacteristic
558C22#188011yescharacteristic
559C4—21no—
560C4—21no—
561C4—21no—
562C4—21no—
563C4—21no—
564C4—21no—
565C4—44no—
566C4—44no—
567C4—42no—
568C4—42no—
569C4—84no—
570C4—81no—
571C4—84no—
572C4—81no—
573C4C4×#10311yescharacteristic
574C4#176311yescharacteristic
575C4#180311yescharacteristic
576C4#180511yescharacteristic
577C4#185911yescharacteristic
578C4#186011yescharacteristic
579C4#186311yescharacteristic
580C4#186511yescharacteristic
581C4#187811yescharacteristic
582C4#188011yescharacteristic
583C4#188211yescharacteristic
584C4#188311yescharacteristic
585C2—21no—
586C2—21no—
587C2—21no—
588C2—21no—
589C2#842411yescharacteristic
590C2#842511yescharacteristic
591C2#842611yescharacteristic
592C2#842711yescharacteristic
593C2#1076511yescharacteristic
594C2#1078111yescharacteristic
595C2#1078911yescharacteristic
subgroupG/H
center, socleC23#379
commutatorC42C2×C42
Frattini, cosocleC22×C42C23
prime radicalC24#120
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
15. #2355C4semidirect
17. Q8×C4⋊C4C4semidirect
30. #1550C4semidirect
34. #1503C4semidirect

Maximal subgroups

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

subgroupindex
#342142
C2×#15502
C2×#15032
#108522
#108492
#80432
#78912