Finite Group Databasebeta

← #469655

Subgroups — #469655

Order 512.

1410 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1872C212yesmaximal
2#15178C211yesmaximal, characteristic
3#13904C211yesmaximal, characteristic
4#11315C212yesmaximal, complemented
5#11190C212yesmaximal, complemented
6#11077C212yesmaximal, complemented
7#11042C211yesmaximal, characteristic
8#11010C212yesmaximal, complemented
9#10970C212yesmaximal
10C22×C4⋊Q8C2211yescharacteristic
11C22×C22⋊Q8C2212yes—
12C23×C4⋊C4C2211yescharacteristic
13C22×C4⋊C8C2211yescharacteristic
14C22×#103C2212yes—
15C22×#103C2212yes—
16C22×#103C2212yes—
17C22×#103C2212yes—
18C22×C22⋊C8C2212yes—
19C2×#391C2212yes—
20C2×#381—21no—
21C2×#381—21no—
22C2×#380C2211yescharacteristic
23C2×C4×C22⋊C4C2211yescharacteristic
24#1872—28no—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#336C2212yes—
30C2×#336C2211yescharacteristic
31#1445—21no—
32#1445—21no—
33#1444—21no—
34#1444—21no—
35#1439—21no—
36#1439—21no—
37C4⋊#95C2212yes—
38C4⋊#95C2211yescharacteristic
39C4⋊#95C2212yes—
40C23×Q8C2312yes—
41C24×C4—21no—
42C24×C4C2311yescharacteristic
43C23×C8C2312yes—
44C23×C8C2312yes—
45C2×C4⋊Q8—24no—
46C2×C4⋊Q8—42no—
47C2×C4⋊Q8—41no—
48C2×C4⋊Q8—41no—
49C2×C22⋊Q8—24no—
50C2×C22⋊Q8—24no—
51C2×C22⋊Q8—24no—
52C2×C22⋊Q8—24no—
53C2×C22⋊Q8—42no—
54C2×C22⋊Q8—42no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—41no—
61C22×C4⋊C4—41no—
62C22×C4⋊C4—41no—
63C22×C4⋊C4—41no—
64C22×C4⋊C4D414yes—
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2312yes—
68C22×C22⋊C4—22no—
69C22×C22⋊C4—21no—
70C22×C22⋊C4D411yescharacteristic
71C22×C22⋊C4D411yescharacteristic
72C22×C42C2311yescharacteristic
73C22×C42C2311yescharacteristic
74C2×C4⋊C8—41no—
75C2×C4⋊C8—42no—
76C2×#103—24no—
77C2×#103—24no—
78C2×#103—24no—
79C2×#103—42no—
80C2×#103—42no—
81C2×#103—42no—
82C2×#103—42no—
83C2×#103—42no—
84C2×#103—42no—
85C2×#103—42no—
86C2×#103—42no—
87C2×#103—42no—
88C2×C22⋊C8—24no—
89C2×C22⋊C8—42no—
90C2×C22⋊C8—42no—
91#391—24no—
92#391—42no—
93#381—44no—
94#381—44no—
95#380—41no—
96#380—41no—
97C4×C22⋊C4—41no—
98C4×C22⋊C4—41no—
99C2×#95—21no—
100C2×#95—21no—
101C2×#95—22no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95—22no—
107C2×#95—21no—
108C2×#95D411yescharacteristic
109C2×#95D411yescharacteristic
110#340—24no—
111#340—24no—
112#340—24no—
113#340—42no—
114#336—42no—
115#336—41no—
116C25C2×D411yescharacteristic
117C22×Q8—22no—
118C22×Q8—22no—
119C22×Q8—22no—
120C22×Q8—22no—
121C22×Q8—22no—
122C22×Q8—22no—
123C22×Q8—22no—
124C22×Q8—22no—
125C22×Q8—22no—
126C22×Q8—22no—
127C22×Q8—22no—
128C22×Q8—22no—
129C22×Q8—22no—
130C22×Q8#4114yes—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—24no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—42no—
145C23×C4—42no—
146C23×C4—42no—
147C23×C4—42no—
148C23×C4—42no—
149C23×C4—42no—
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D412yes—
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C23×C4#4112yes—
157C23×C4C2411yescharacteristic
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C4⋊Q8—44no—
173C4⋊Q8—48no—
174C4⋊Q8—44no—
175C22⋊Q8—216no—
176C22⋊Q8—48no—
177C22⋊Q8—48no—
178C22⋊Q8—48no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4C2×D414yes—
210C2×C4⋊C4#4114yes—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—41no—
214C2×C22⋊C4—41no—
215C2×C22⋊C4—41no—
216C2×C22⋊C4—44no—
217C2×C22⋊C4—44no—
218C2×C22⋊C4—44no—
219C2×C22⋊C4—42no—
220C2×C22⋊C4—42no—
221C2×C22⋊C4—42no—
222C2×C22⋊C4SD1614yes—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—21no—
227C2×C42—21no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—22no—
231C4⋊C8—44no—
232#103—48no—
233#103—48no—
234#103—48no—
235#103—48no—
236C22⋊C8—48no—
237#95—41no—
238#95—41no—
239#95—44no—
240#95—44no—
241#95—42no—
242#95—42no—
243#95—44no—
244#95—44no—
245#95—44no—
246#95—42no—
247#95—44no—
248C24—21no—
249C24—21no—
250C24—22no—
251C24—22no—
252C24—21no—
253C24—21no—
254C24—22no—
255C24—22no—
256C24#12112yes—
257C24C2×SD1612yes—
258C24#13712yes—
259C24C22×D411yescharacteristic
260C2×Q8—24no—
261C2×Q8—24no—
262C2×Q8—24no—
263C2×Q8—24no—
264C2×Q8—24no—
265C2×Q8—24no—
266C2×Q8—42no—
267C2×Q8—42no—
268C2×Q8—42no—
269C2×Q8—42no—
270C2×Q8—42no—
271C2×Q8—42no—
272C2×Q8—42no—
273C2×Q8—42no—
274C2×Q8—42no—
275C2×Q8—42no—
276C2×Q8—42no—
277C2×Q8—42no—
278C2×Q8—42no—
279C2×Q8—42no—
280C2×Q8—42no—
281C2×Q8—42no—
282C2×Q8—42no—
283C2×Q8—42no—
284C2×Q8—42no—
285C2×Q8—42no—
286C2×Q8—42no—
287C2×Q8—42no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—24no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—24no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—21no—
314C22×C4—22no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—22no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—41no—
325C22×C4—42no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—42no—
329C22×C4—41no—
330C22×C4—48no—
331C22×C4—48no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—48no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—84no—
359C22×C4—84no—
360C22×C4—84no—
361C22×C4—84no—
362C22×C4#12112yes—
363C22×C4C2×SD1612yes—
364C22×C4C2×SD1612yes—
365C22×C4AGL(1,ℤ/8)12yes—
366C22×C4C22×D411yescharacteristic
367C22×C4C2×#4112yes—
368C22×C4C2×#4112yes—
369C22×C4C2×#4111yescharacteristic
370C22×C4H(5,𝔽2)11yescharacteristic
371C22×C4H(5,𝔽2)11yescharacteristic
372C22×C4H(5,𝔽2)11yescharacteristic
373C22×C4H(5,𝔽2)11yescharacteristic
374C22×C4H(5,𝔽2)12yes—
375C22×C4H(5,𝔽2)12yes—
376C2×C8—42no—
377C2×C8—42no—
378C2×C8—42no—
379C2×C8—42no—
380C2×C8—42no—
381C2×C8—42no—
382C2×C8—42no—
383C2×C8—42no—
384C2×C8—42no—
385C2×C8—42no—
386C2×C8—42no—
387C2×C8—42no—
388C2×C8—42no—
389C2×C8—42no—
390C4⋊C4—24no—
391C4⋊C4—24no—
392C4⋊C4—24no—
393C4⋊C4—24no—
394C4⋊C4—48no—
395C4⋊C4—416no—
396C4⋊C4—48no—
397C4⋊C4—48no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C22⋊C4—44no—
407C22⋊C4—44no—
408C22⋊C4—416no—
409C22⋊C4—84no—
410C42—42no—
411C42—42no—
412C42—41no—
413C42—41no—
414C42—41no—
415C42—41no—
416C23—22no—
417C23—24no—
418C23—22no—
419C23—22no—
420C23—22no—
421C23—22no—
422C23—22no—
423C23—24no—
424C23—24no—
425C23—24no—
426C23—24no—
427C23—24no—
428C23—42no—
429C23—41no—
430C23—41no—
431C23—41no—
432C23—42no—
433C23—41no—
434C23—42no—
435C23—42no—
436C23—42no—
437C23—42no—
438C23#46114yes—
439C23C2×#12111yescharacteristic
440C23#53411yescharacteristic
441C23#53511yescharacteristic
442C23#54612yes—
443C23#54612yes—
444C23C22×SD1611yescharacteristic
445C23C2×AGL(1,ℤ/8)11yescharacteristic
446C23C2×#13711yescharacteristic
447C23#57511yescharacteristic
448C23#57812yes—
449C23#57812yes—
450Q8—44no—
451Q8—44no—
452Q8—44no—
453Q8—44no—
454Q8—82no—
455Q8—82no—
456Q8—82no—
457Q8—82no—
458C2×C4—22no—
459C2×C4—24no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—42no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—41no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—42no—
497C2×C4—44no—
498C2×C4—816no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4—82no—
504C2×C4—82no—
505C2×C4—82no—
506C2×C4—82no—
507C2×C4—82no—
508C2×C4—82no—
509C2×C4—82no—
510C2×C4—82no—
511C2×C4—82no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4#52512yes—
519C2×C4#53411yescharacteristic
520C2×C4#53411yescharacteristic
521C2×C4#53512yes—
522C2×C4#53812yes—
523C2×C4#53812yes—
524C2×C4#53811yescharacteristic
525C2×C4#53811yescharacteristic
526C2×C4#54012yes—
527C2×C4#54012yes—
528C2×C4#55111yescharacteristic
529C2×C4#55112yes—
530C2×C4#55112yes—
531C2×C4#55312yes—
532C2×C4#55911yescharacteristic
533C2×C4#56012yes—
534C2×C4#56011yescharacteristic
535C2×C4#56011yescharacteristic
536C8—82no—
537C8—82no—
538C22—24no—
539C22—28no—
540C22—42no—
541C22—42no—
542C22—42no—
543C22—42no—
544C22—44no—
545C22—44no—
546C22—44no—
547C22—42no—
548C22—42no—
549C22#241112yes—
550C22#261711yescharacteristic
551C22#269711yescharacteristic
552C22#270312yes—
553C22#270911yescharacteristic
554C22C2×#46111yescharacteristic
555C22#303611yescharacteristic
556C22#304412yes—
557C22#318711yescharacteristic
558C22#319311yescharacteristic
559C22#320211yescharacteristic
560C22#320211yescharacteristic
561C22#320811yescharacteristic
562C22#321511yescharacteristic
563C22#322312yes—
564C22#329812yes—
565C22#329812yes—
566C22#330812yes—
567C22#331212yes—
568C22#331212yes—
569C22#331512yes—
570C22#332712yes—
571C22#333012yes—
572C4—42no—
573C4—42no—
574C4—42no—
575C4—42no—
576C4—84no—
577C4—84no—
578C4—81no—
579C4—84no—
580C4—162no—
581C4#260011yescharacteristic
582C4#263512yes—
583C4#274912yes—
584C4#275211yescharacteristic
585C4#276011yescharacteristic
586C4#281511yescharacteristic
587C2—44no—
588C2#1666011yescharacteristic
589C2#2448012yes—
590C2#2448912yes—
591C2#2971411yescharacteristic
592C2#2972411yescharacteristic
593C2#3166811yescharacteristic
594C2#3168311yescharacteristic
595C2#3183612yes—
596C2#3185712yes—
597C2#3202711yescharacteristic
598C2#3203911yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11315C2semidirect
5. #11190C2semidirect
6. #11077C2semidirect
8. #11010C2semidirect

Maximal subgroups

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

subgroupindex
C2×#18722
#151782
#139042
#113152
#111902
#110772
#110422
#110102
#109702