Finite Group Databasebeta

← #469656

Subgroups — #469656

Order 512.

1382 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1872C212yesmaximal
2#15178C211yesmaximal, characteristic
3#13924C211yesmaximal, characteristic
4#11316C212yesmaximal, 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×#380C2211yescharacteristic
21C2×C4×C22⋊C4C2211yescharacteristic
22#1872—28no—
23C2×#340C2212yes—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#336C2212yes—
28C2×#336C2211yescharacteristic
29#1445—21no—
30#1445—21no—
31#1444—21no—
32#1444—21no—
33#1439—21no—
34#1439—21no—
35C4⋊#95C2212yes—
36C4⋊#95C2211yescharacteristic
37C4⋊#95C2212yes—
38C22⋊#95—21no—
39C22⋊#95—21no—
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#380—41no—
94#380—41no—
95C4×C22⋊C4—41no—
96C4×C22⋊C4—41no—
97C2×#95—21no—
98C2×#95—21no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95—21no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95D411yescharacteristic
107C2×#95D411yescharacteristic
108#340—24no—
109#340—24no—
110#340—24no—
111#340—42no—
112#336—42no—
113#336—41no—
114C25C2×D411yescharacteristic
115C22×Q8—22no—
116C22×Q8—22no—
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#4114yes—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—21no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—24no—
141C23×C4—22no—
142C23×C4—42no—
143C23×C4—42no—
144C23×C4—82no—
145C23×C4—82no—
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D412yes—
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4#4112yes—
153C23×C4C2411yescharacteristic
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
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—
168C4⋊Q8—44no—
169C4⋊Q8—48no—
170C4⋊Q8—44no—
171C22⋊Q8—216no—
172C22⋊Q8—48no—
173C22⋊Q8—48no—
174C22⋊Q8—48no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4C2×D414yes—
206C2×C4⋊C4#4114yes—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—41no—
210C2×C22⋊C4—41no—
211C2×C22⋊C4—41no—
212C2×C22⋊C4—44no—
213C2×C22⋊C4—44no—
214C2×C22⋊C4—44no—
215C2×C22⋊C4—42no—
216C2×C22⋊C4—42no—
217C2×C22⋊C4—42no—
218C2×C22⋊C4Q1614yes—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—21no—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—21no—
226C2×C42—22no—
227C4⋊C8—44no—
228#103—48no—
229#103—48no—
230#103—48no—
231#103—48no—
232C22⋊C8—48no—
233#95—41no—
234#95—41no—
235#95—44no—
236#95—44no—
237#95—42no—
238#95—42no—
239#95—42no—
240#95—82no—
241#95—82no—
242#95—82no—
243#95—82no—
244C24—21no—
245C24—21no—
246C24—22no—
247C24—22no—
248C24—21no—
249C24—21no—
250C24—22no—
251C24—22no—
252C24#12112yes—
253C24C2×Q1612yes—
254C24#13712yes—
255C24C22×D411yescharacteristic
256C2×Q8—24no—
257C2×Q8—24no—
258C2×Q8—24no—
259C2×Q8—24no—
260C2×Q8—24no—
261C2×Q8—24no—
262C2×Q8—42no—
263C2×Q8—42no—
264C2×Q8—42no—
265C2×Q8—42no—
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—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—24no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—24no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—21no—
310C22×C4—22no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—22no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—41no—
321C22×C4—42no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—42no—
325C22×C4—41no—
326C22×C4—48no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
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—42no—
351C22×C4—42no—
352C22×C4—88no—
353C22×C4—84no—
354C22×C4—84no—
355C22×C4—88no—
356C22×C4#12112yes—
357C22×C4C2×Q1612yes—
358C22×C4C2×Q1612yes—
359C22×C4#13712yes—
360C22×C4C22×D411yescharacteristic
361C22×C4C2×#4112yes—
362C22×C4C2×#4112yes—
363C22×C4C2×#4111yescharacteristic
364C22×C4H(5,𝔽2)11yescharacteristic
365C22×C4H(5,𝔽2)11yescharacteristic
366C22×C4H(5,𝔽2)11yescharacteristic
367C22×C4H(5,𝔽2)11yescharacteristic
368C22×C4H(5,𝔽2)12yes—
369C22×C4H(5,𝔽2)12yes—
370C2×C8—42no—
371C2×C8—42no—
372C2×C8—42no—
373C2×C8—42no—
374C2×C8—42no—
375C2×C8—42no—
376C2×C8—42no—
377C2×C8—42no—
378C2×C8—42no—
379C2×C8—42no—
380C2×C8—42no—
381C2×C8—42no—
382C2×C8—42no—
383C2×C8—42no—
384C4⋊C4—24no—
385C4⋊C4—24no—
386C4⋊C4—24no—
387C4⋊C4—24no—
388C4⋊C4—48no—
389C4⋊C4—416no—
390C4⋊C4—48no—
391C4⋊C4—48no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C22⋊C4—44no—
401C22⋊C4—44no—
402C22⋊C4—416no—
403C22⋊C4—84no—
404C42—42no—
405C42—42no—
406C42—41no—
407C42—41no—
408C42—41no—
409C42—41no—
410C23—22no—
411C23—24no—
412C23—22no—
413C23—22no—
414C23—22no—
415C23—22no—
416C23—22no—
417C23—24no—
418C23—24no—
419C23—24no—
420C23—24no—
421C23—24no—
422C23—42no—
423C23—41no—
424C23—41no—
425C23—41no—
426C23—42no—
427C23—41no—
428C23—42no—
429C23—42no—
430C23—42no—
431C23—42no—
432C23#46214yes—
433C23C2×#12111yescharacteristic
434C23#53411yescharacteristic
435C23#53511yescharacteristic
436C23#54612yes—
437C23#54612yes—
438C23C22×Q1611yescharacteristic
439C23C2×#13711yescharacteristic
440C23C2×#13711yescharacteristic
441C23#57511yescharacteristic
442C23#57712yes—
443C23#57712yes—
444Q8—44no—
445Q8—44no—
446Q8—44no—
447Q8—44no—
448Q8—82no—
449Q8—82no—
450Q8—82no—
451Q8—82no—
452C2×C4—22no—
453C2×C4—24no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—42no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—41no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—42no—
492C2×C4—816no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—82no—
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#52512yes—
513C2×C4#53411yescharacteristic
514C2×C4#53411yescharacteristic
515C2×C4#53512yes—
516C2×C4#53812yes—
517C2×C4#53812yes—
518C2×C4#53811yescharacteristic
519C2×C4#53811yescharacteristic
520C2×C4#54012yes—
521C2×C4#54012yes—
522C2×C4#55111yescharacteristic
523C2×C4#55112yes—
524C2×C4#55112yes—
525C2×C4#55312yes—
526C2×C4#55911yescharacteristic
527C2×C4#56012yes—
528C2×C4#56011yescharacteristic
529C2×C4#56011yescharacteristic
530C8—82no—
531C8—82no—
532C22—24no—
533C22—28no—
534C22—42no—
535C22—42no—
536C22—42no—
537C22—42no—
538C22—44no—
539C22—44no—
540C22—44no—
541C22—42no—
542C22—42no—
543C22#241112yes—
544C22#261711yescharacteristic
545C22#269711yescharacteristic
546C22#270312yes—
547C22#270911yescharacteristic
548C22C2×#46211yescharacteristic
549C22#303711yescharacteristic
550C22#304212yes—
551C22#318911yescharacteristic
552C22#319511yescharacteristic
553C22#320011yescharacteristic
554C22#320011yescharacteristic
555C22#320911yescharacteristic
556C22#321511yescharacteristic
557C22#322212yes—
558C22#329912yes—
559C22#329912yes—
560C22#330712yes—
561C22#331612yes—
562C22#332012yes—
563C22#332012yes—
564C22#332612yes—
565C22#333112yes—
566C4—42no—
567C4—42no—
568C4—42no—
569C4—42no—
570C4—84no—
571C4—84no—
572C4—81no—
573C4—84no—
574C4—162no—
575C4#260011yescharacteristic
576C4#263512yes—
577C4#274912yes—
578C4#275211yescharacteristic
579C4#276011yescharacteristic
580C4#281511yescharacteristic
581C2—44no—
582C2#1666011yescharacteristic
583C2#2448212yes—
584C2#2449112yes—
585C2#2971511yescharacteristic
586C2#2972311yescharacteristic
587C2#3166911yescharacteristic
588C2#3168211yescharacteristic
589C2#3184012yes—
590C2#3185312yes—
591C2#3202911yescharacteristic
592C2#3203711yescharacteristic
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. #11316C2semidirect
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
#139242
#113162
#111902
#110772
#110422
#110102
#109702