Finite Group Databasebeta

← #393374

Subgroups — #393374

Order 512.

1511 conjugacy classes of subgroups.

The subgroup lattice has 581 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3078C211yesmaximal, characteristic
2C4×#519C211yesmaximal, characteristic
3#21332C211yesmaximal, characteristic
4#21329C211yesmaximal, characteristic
5#20647C211yesmaximal, characteristic
6#10903C214yesmaximal
7#10856C211yesmaximal, characteristic
8#10836C211yesmaximal, characteristic
9#10786C214yesmaximal
10C2×#533C2211yescharacteristic
11C2×#519C2212yes—
12C22×C4×Q8C2211yescharacteristic
13#3078—22no—
14#3078—22no—
15#3078—22no—
16#3078—22no—
17C2×#479C414yescomplemented
18C2×#477C412yescomplemented
19C2×#475C412yescomplemented
20C2×#428C2212yes—
21C2×#423C2211yescharacteristic
22C2×#423C2211yescharacteristic
23C2×#419C2212yes—
24C2×#419C2212yes—
25#2285C2211yescharacteristic
26#2276—22no—
27#2276—22no—
28#2274—22no—
29#2274C2212yes—
30C42×Q8—22no—
31C42×Q8—22no—
32C4×#117—22no—
33C4×#117C2211yescharacteristic
34C4×#117C2211yescharacteristic
35#1996—22no—
36#1996—22no—
37#1995—22no—
38#1994—22no—
39#1805—22no—
40#1803—22no—
41C2×#344C2214yes—
42#1740C2214yes—
43#1736C2212yes—
44#1733C2214yes—
45C2×H(3,ℤ/4)C2214yes—
46C23×Q8C2311yescharacteristic
47C22×M16C2312yes—
48C22×M16C2312yes—
49#533—21no—
50#533—21no—
51#533—21no—
52#533—21no—
53#533—21no—
54#533—21no—
55#533—21no—
56#533—21no—
57C2×C4⋊Q8C2×C411yescharacteristic
58C2×C4⋊Q8C2×C411yescharacteristic
59C2×#125C2×C412yes—
60C2×C22⋊Q8—21no—
61C2×C22⋊Q8—21no—
62#519—24no—
63#519—24no—
64#519—24no—
65#519Q818yescomplemented
66C2×C4×Q8—24no—
67C2×C4×Q8—21no—
68C2×C4×Q8—21no—
69C2×C4×Q8—24no—
70C2×C4×Q8—21no—
71C2×C4×Q8—21no—
72C2×C4×Q8—21no—
73C2×C4×Q8—21no—
74C2×C4×Q8—21no—
75C2×C4×Q8—21no—
76C2×C4×Q8C2×C412yes—
77C2×C4×Q8C2×C412yes—
78C2×C4×Q8D412yes—
79C2×C4×Q8D412yes—
80C2×C4×Q8D412yes—
81C2×C4×Q8D412yes—
82C2×#117—24no—
83C2×#117C2312yes—
84C2×#117C2312yes—
85C2×#117C2312yes—
86C2×#117C2312yes—
87C22×C4⋊C4—21no—
88C22×C4⋊C4C2311yescharacteristic
89C22×C42—21no—
90C22×C42C2311yescharacteristic
91#479—28no—
92#479—28no—
93#477—24no—
94#477—24no—
95#475—24no—
96#475—24no—
97#428—24no—
98#428—24no—
99C2×#107C2×C414yes—
100C2×#106C2×C414yes—
101#423—21no—
102#423—21no—
103#423—21no—
104#423—21no—
105#423—21no—
106#423—21no—
107#423—21no—
108#423—21no—
109C2×C4⋊C8—21no—
110C2×C4⋊C8—21no—
111C2×C4⋊C8C2×C412yes—
112C2×C4⋊C8C2×C412yes—
113#419—24no—
114#419—24no—
115#419—28no—
116C2×#103—24no—
117C2×#103C2×C414yes—
118C2×#103C2×C414yes—
119#384—21no—
120#382—21no—
121#380—21no—
122#380—21no—
123C4×C4⋊C4—24no—
124C4×C4⋊C4—24no—
125C4×C4⋊C4—28no—
126C4×C4⋊C4—28no—
127C4×C4⋊C4—24no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4—21no—
132C4×C4⋊C4—24no—
133C4×C4⋊C4—21no—
134C4×C4⋊C4—21no—
135C4×C22⋊C4—21no—
136C4×C22⋊C4—21no—
137C4×C22⋊C4—24no—
138C4×C22⋊C4D412yes—
139C4×C22⋊C4D412yes—
140#376—24no—
141#376—21no—
142#376D412yes—
143C43—24no—
144C43—21no—
145C43D412yes—
146#344—28no—
147#342—28no—
148#340—24no—
149#339—28no—
150H(3,ℤ/4)—28no—
151C22×Q8—21no—
152C22×Q8—21no—
153C22×Q8—21no—
154C22×Q8—21no—
155C22×Q8—21no—
156C22×Q8—22no—
157C22×Q8—22no—
158C22×Q8C22×C412yes—
159C22×Q8C2×D412yes—
160C22×Q8C2×D412yes—
161C22×Q8C2×Q814yes—
162C22×Q8#4114yes—
163C23×C4—21no—
164C23×C4—21no—
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2411yescharacteristic
168C2×M16—22no—
169C2×M16—22no—
170C2×M16—24no—
171C2×M16—24no—
172C2×M16—22no—
173C2×M16—24no—
174C2×M16—24no—
175C2×M16—22no—
176C22×C8—22no—
177C22×C8C22×C414yes—
178C4⋊Q8—21no—
179C4⋊Q8—21no—
180C4⋊Q8—21no—
181C4⋊Q8—21no—
182C4⋊Q8—21no—
183C4⋊Q8—21no—
184C4⋊Q8—21no—
185C4⋊Q8—21no—
186#125—22no—
187#125—22no—
188#125—22no—
189#125—22no—
190C22⋊Q8—41no—
191C22⋊Q8—41no—
192C22⋊Q8—41no—
193C22⋊Q8—41no—
194C22⋊Q8—41no—
195C22⋊Q8—41no—
196C22⋊Q8—41no—
197C22⋊Q8—41no—
198C4×Q8—216no—
199C4×Q8—216no—
200C4×Q8—216no—
201C4×Q8—22no—
202C4×Q8—22no—
203C4×Q8—22no—
204C4×Q8—22no—
205C4×Q8—22no—
206C4×Q8—22no—
207C4×Q8—22no—
208C4×Q8—22no—
209C4×Q8—216no—
210C4×Q8—22no—
211C4×Q8—22no—
212C4×Q8—22no—
213C4×Q8—22no—
214C4×Q8—41no—
215C4×Q8—41no—
216C4×Q8—41no—
217C4×Q8—41no—
218C4×Q8—41no—
219C4×Q8—41no—
220C4×Q8—41no—
221C4×Q8—41no—
222C4×Q8C22⋊C414yes—
223C4×Q8C22⋊C414yes—
224C4×Q8C22⋊C414yes—
225C4×Q8C22⋊C414yes—
226#117—216no—
227#117—22no—
228#117—22no—
229#117—22no—
230#117—22no—
231#117—216no—
232#117—22no—
233#117—22no—
234#117—22no—
235#117—22no—
236#117—24no—
237#117—24no—
238#117C2×Q814yes—
239#117C2×Q814yes—
240#117#4114yes—
241#117#4114yes—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
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—22no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—41no—
260C2×C4⋊C4—41no—
261C2×C4⋊C4C22×C412yes—
262C2×C4⋊C4C22×C412yes—
263C2×C4⋊C4C22×C412yes—
264C2×C4⋊C4C2×D412yes—
265C2×C4⋊C4C2×D412yes—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4C2×D414yes—
270C2×C42—28no—
271C2×C42—22no—
272C2×C42—22no—
273C2×C42—22no—
274C2×C42—22no—
275C2×C42—22no—
276C2×C42—22no—
277C2×C42—22no—
278C2×C42—22no—
279C2×C42—22no—
280C2×C42—22no—
281C2×C42—22no—
282C2×C42—28no—
283C2×C42—22no—
284C2×C42—22no—
285C2×C42C22×C411yescharacteristic
286C2×C42C22×C411yescharacteristic
287C2×C42C2×D414yes—
288C2×C42C2×D411yescharacteristic
289C2×C42C2×D411yescharacteristic
290C2×C42C2×D411yescharacteristic
291C2×C42C2×D411yescharacteristic
292C2×C42C2×D411yescharacteristic
293C2×C42C2×D411yescharacteristic
294C2×C42#4111yescharacteristic
295C2×C42#4111yescharacteristic
296C2×C42#4111yescharacteristic
297C2×C42#4111yescharacteristic
298#107—28no—
299#106—28no—
300C4⋊C8—22no—
301C4⋊C8—22no—
302C4⋊C8—22no—
303C4⋊C8—22no—
304C4⋊C8—41no—
305C4⋊C8—41no—
306C4⋊C8—41no—
307C4⋊C8—41no—
308#103—28no—
309#103—28no—
310#103—48no—
311#95—28no—
312#95—22no—
313#95—22no—
314#95—22no—
315#95—22no—
316#95—41no—
317#95—41no—
318C24C22×D411yescharacteristic
319C2×Q8—22no—
320C2×Q8—24no—
321C2×Q8—22no—
322C2×Q8—24no—
323C2×Q8—22no—
324C2×Q8—24no—
325C2×Q8—24no—
326C2×Q8—24no—
327C2×Q8—24no—
328C2×Q8—22no—
329C2×Q8—22no—
330C2×Q8—24no—
331C2×Q8—24no—
332C2×Q8—41no—
333C2×Q8—41no—
334C2×Q8—41no—
335C2×Q8—41no—
336C2×Q8—82no—
337C2×Q8—82no—
338C2×Q8C2×C22⋊C414yes—
339C2×Q8C4×Q818yes—
340C2×Q8C22⋊Q818yes—
341C2×Q8#12414yes—
342C2×Q8C4⋊Q814yes—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—81no—
358C22×C4—82no—
359C22×C4C4×D411yescharacteristic
360C22×C4C4×D411yescharacteristic
361C22×C4C4×D411yescharacteristic
362C22×C4C4×D411yescharacteristic
363C22×C4#12011yescharacteristic
364C22×C4#12011yescharacteristic
365C22×C4#12011yescharacteristic
366C22×C4#12011yescharacteristic
367C22×C4#12111yescharacteristic
368C22×C4#12111yescharacteristic
369C22×C4#12111yescharacteristic
370C22×C4#12111yescharacteristic
371C22×C4C23×C411yescharacteristic
372C22×C4C22×D411yescharacteristic
373C22×C4C22×D411yescharacteristic
374C22×C4C22×D411yescharacteristic
375C22×C4C22×Q812yes—
376C22×C4C2×#4111yescharacteristic
377C22×C4C2×#4112yes—
378C22×C4C2×#4112yes—
379C22×C4C2×#4112yes—
380C22×C4C2×#4111yescharacteristic
381M16—44no—
382M16—42no—
383M16—42no—
384M16—44no—
385M16—44no—
386C2×C8—24no—
387C2×C8—28no—
388C2×C8—42no—
389C2×C8—42no—
390C2×C8—42no—
391C4⋊C4—232no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—24no—
395C4⋊C4—24no—
396C4⋊C4—22no—
397C4⋊C4—24no—
398C4⋊C4—24no—
399C4⋊C4—24no—
400C4⋊C4—24no—
401C4⋊C4—24no—
402C4⋊C4—24no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—41no—
412C4⋊C4—41no—
413C4⋊C4—41no—
414C4⋊C4—41no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C4⋊C4—82no—
418C4⋊C4—82no—
419C4⋊C4C2×C22⋊C414yes—
420C22⋊C4—24no—
421C22⋊C4—24no—
422C22⋊C4—24no—
423C22⋊C4—42no—
424C22⋊C4—42no—
425C22⋊C4—44no—
426C22⋊C4C22⋊Q818yes—
427C42—232no—
428C42—24no—
429C42—24no—
430C42—24no—
431C42—24no—
432C42—24no—
433C42—24no—
434C42—24no—
435C42—44no—
436C42—44no—
437C42—42no—
438C42—42no—
439C42—44no—
440C42—82no—
441C42—82no—
442C42—88no—
443C42C2×C22⋊C411yescharacteristic
444C42C2×C22⋊C411yescharacteristic
445C42C2×C22⋊C411yescharacteristic
446C42C2×C22⋊C411yescharacteristic
447C42C4×D411yescharacteristic
448C42C4×D411yescharacteristic
449C42C4×D411yescharacteristic
450C42C4×D411yescharacteristic
451C42#12111yescharacteristic
452C42#12111yescharacteristic
453C42#12111yescharacteristic
454C42#12111yescharacteristic
455C42C22⋊Q818yes—
456C42#12311yescharacteristic
457C42#12311yescharacteristic
458C42#12311yescharacteristic
459C42#12311yescharacteristic
460C23—41no—
461C23—41no—
462C23C2×C4×D411yescharacteristic
463C23C2×#12011yescharacteristic
464C23C2×#12111yescharacteristic
465C23C2×C22⋊Q812yes—
466C23#52512yes—
467Q8—24no—
468Q8—28no—
469Q8—24no—
470Q8—24no—
471Q8—28no—
472Q8—82no—
473Q8—82no—
474Q8—82no—
475Q8—82no—
476Q8#39118yes—
477C2×C4—28no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—28no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—28no—
486C2×C4—28no—
487C2×C4—24no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—82no—
505C2×C4—82no—
506C2×C4—82no—
507C2×C4—82no—
508C2×C4—82no—
509C2×C4—82no—
510C2×C4—82no—
511C2×C4#38611yescharacteristic
512C2×C4#38611yescharacteristic
513C2×C4#38611yescharacteristic
514C2×C4#38611yescharacteristic
515C2×C4#38611yescharacteristic
516C2×C4#38611yescharacteristic
517C2×C4#38611yescharacteristic
518C2×C4#38611yescharacteristic
519C2×C4C22×C22⋊C411yescharacteristic
520C2×C4C2×C4×D411yescharacteristic
521C2×C4C2×C4×Q812yes—
522C2×C4C4×#4112yes—
523C2×C4C2×#12111yescharacteristic
524C2×C4C2×C22⋊Q812yes—
525C2×C4C2×C22⋊Q812yes—
526C2×C4C2×#12311yescharacteristic
527C2×C4#52512yes—
528C2×C4#52512yes—
529C2×C4C2×#12411yescharacteristic
530C2×C4#52912yes—
531C2×C4#52912yes—
532C2×C4C2×C4⋊Q811yescharacteristic
533C2×C4#53211yescharacteristic
534C2×C4#53211yescharacteristic
535C2×C4#53312yes—
536C2×C4#53312yes—
537C8—44no—
538C8—82no—
539C22—21no—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—21no—
544C22—21no—
545C22—42no—
546C22—42no—
547C22C2×#38611yescharacteristic
548C22C4×C22⋊Q812yes—
549C22#284812yes—
550C22#285011yescharacteristic
551C22#285711yescharacteristic
552C4—21no—
553C4—21no—
554C4—21no—
555C4—21no—
556C4—21no—
557C4—21no—
558C4—21no—
559C4—28no—
560C4—48no—
561C4—48no—
562C4—44no—
563C4—82no—
564C4—82no—
565C4—162no—
566C4#190512yes—
567C4C2×#39111yescharacteristic
568C4#229711yescharacteristic
569C4C4×C22⋊Q812yes—
570C4#285011yescharacteristic
571C4#285512yes—
572C4#285711yescharacteristic
573C2—22no—
574C2—21no—
575C2—21no—
576C2—22no—
577C2#1470211yescharacteristic
578C2C2×#190511yescharacteristic
579C2#2099811yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
17. C2×#479C4semidirect
18. C2×#477C4semidirect
19. C2×#475C4semidirect
65. #519Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#30782
C4×#5192
#213322
#213292
#206472
#109032
#108562
#108362
#107862