Finite Group Databasebeta

← #459309

Subgroups — #459309

Order 512.

1416 conjugacy classes of subgroups.

The subgroup lattice has 593 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2043C212yesmaximal, complemented
2C2×#1892C212yesmaximal
3#15178C211yesmaximal, characteristic
4#15135C211yesmaximal, characteristic
5#11293C212yesmaximal, complemented
6#11190C212yesmaximal, complemented
7#11050C212yesmaximal, complemented
8#10970C212yesmaximal
9#10714C211yesmaximal, characteristic
10C24×C8C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12C22×C22⋊Q8C2212yes—
13C22×#103C2212yes—
14C22×#103C2212yes—
15C22×#103C2212yes—
16C22×#103C2212yes—
17C22×C22⋊C8C2211yescharacteristic
18C2×#399—21no—
19C2×#391C2212yes—
20C2×#389C2212yes—
21C2×#384C2211yescharacteristic
22C2×#380C2212yes—
23C2×C4×C22⋊C4C2211yescharacteristic
24#2043—28no—
25#1892—28no—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#336C2212yes—
31C2×#336C2212yes—
32#1445—21no—
33#1444—21no—
34#1444—21no—
35#1444—21no—
36#1439—21no—
37#1439—21no—
38#1439—21no—
39C4⋊#95C2211yescharacteristic
40C4⋊#95C2211yescharacteristic
41C23×Q8C2312yes—
42C24×C4C2311yescharacteristic
43C23×C8—22no—
44C23×C8—24no—
45C23×C8—42no—
46C23×C8—41no—
47C23×C8—41no—
48C23×C8C2312yes—
49C23×C8C2312yes—
50C2×C4⋊Q8—24no—
51C2×C4⋊Q8—41no—
52C2×C4⋊Q8—41no—
53C2×C4⋊Q8—42no—
54C2×C22⋊Q8—24no—
55C2×C22⋊Q8—24no—
56C2×C22⋊Q8—24no—
57C2×C22⋊Q8—24no—
58C2×C22⋊Q8—42no—
59C2×C22⋊Q8—42no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2312yes—
69C22×C22⋊C4—22no—
70C22×C22⋊C4—22no—
71C22×C22⋊C4D411yescharacteristic
72C22×C22⋊C4D411yescharacteristic
73C22×C42C2311yescharacteristic
74C22×C42C2311yescharacteristic
75C2×#103—24no—
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×#103D418yes—
87C2×C22⋊C8—22no—
88C2×C22⋊C8—41no—
89C2×C22⋊C8—41no—
90#399—44no—
91#391—24no—
92#391—42no—
93#389—24no—
94#389—42no—
95#384—41no—
96#384—41no—
97#380—24no—
98#380—42no—
99C4×C22⋊C4—41no—
100C4×C22⋊C4—41no—
101C2×#95—21no—
102C2×#95—21no—
103C2×#95—21no—
104C2×#95—21no—
105C2×#95—22no—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95D411yescharacteristic
110C2×#95D411yescharacteristic
111#340—24no—
112#340—24no—
113#340—24no—
114#340—42no—
115#336—24no—
116#336—42no—
117C25C2×D411yescharacteristic
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—22no—
131C22×Q8C2×D414yes—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4#4112yes—
150C23×C4#4112yes—
151C23×C4C2411yescharacteristic
152C22×C8—22no—
153C22×C8—22no—
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—44no—
166C22×C8—48no—
167C22×C8—48no—
168C22×C8—44no—
169C22×C8—44no—
170C22×C8C2×D414yes—
171C4⋊Q8—44no—
172C4⋊Q8—48no—
173C4⋊Q8—44no—
174C22⋊Q8—216no—
175C22⋊Q8—48no—
176C22⋊Q8—48no—
177C22⋊Q8—48no—
178C2×C4⋊C4—22no—
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—22no—
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—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—42no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—82no—
208C2×C4⋊C4—82no—
209C2×C4⋊C4C2×D414yes—
210C2×C4⋊C4#4114yes—
211C2×C22⋊C4—41no—
212C2×C22⋊C4—41no—
213C2×C22⋊C4—41no—
214C2×C22⋊C4—41no—
215C2×C22⋊C4—44no—
216C2×C22⋊C4—44no—
217C2×C22⋊C4—44no—
218C2×C22⋊C4—44no—
219C2×C22⋊C4—44no—
220C2×C22⋊C4—82no—
221C2×C22⋊C4SD1614yes—
222C2×C22⋊C4Q1614yes—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—21no—
226C2×C42—21no—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42—21no—
230C2×C42—21no—
231#103—48no—
232#103—48no—
233#103—48no—
234#103—48no—
235C22⋊C8—44no—
236#95—41no—
237#95—41no—
238#95—44no—
239#95—44no—
240#95—44no—
241#95—44no—
242#95—42no—
243#95—42no—
244#95—42no—
245#95—42no—
246C24—21no—
247C24—21no—
248C24—22no—
249C24—22no—
250C24—21no—
251C24—21no—
252C24—22no—
253C24—22no—
254C24#12412yes—
255C24C2×SD1612yes—
256C24C2×Q1612yes—
257C24C22×D411yescharacteristic
258C2×Q8—24no—
259C2×Q8—24no—
260C2×Q8—24no—
261C2×Q8—24no—
262C2×Q8—24no—
263C2×Q8—24no—
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—
284C2×Q8—42no—
285C2×Q8—42no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—22no—
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—21no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—24no—
307C22×C4—22no—
308C22×C4—24no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—41no—
322C22×C4—42no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—42no—
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#12412yes—
353C22×C4C2×SD1612yes—
354C22×C4C2×SD1612yes—
355C22×C4C2×Q1612yes—
356C22×C4C2×Q1612yes—
357C22×C4C22×D412yes—
358C22×C4C2×#4111yescharacteristic
359C22×C4C2×#4112yes—
360C22×C4C2×#4111yescharacteristic
361C22×C4H(5,𝔽2)11yescharacteristic
362C22×C4H(5,𝔽2)12yes—
363C22×C4H(5,𝔽2)11yescharacteristic
364C22×C4H(5,𝔽2)12yes—
365C22×C4H(5,𝔽2)11yescharacteristic
366C22×C42−1+411yescharacteristic
367C2×C8—24no—
368C2×C8—24no—
369C2×C8—24no—
370C2×C8—416no—
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—
382C4⋊C4—24no—
383C4⋊C4—24no—
384C4⋊C4—24no—
385C4⋊C4—24no—
386C4⋊C4—416no—
387C4⋊C4—48no—
388C4⋊C4—48no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—88no—
398C4⋊C4—84no—
399C4⋊C4—84no—
400C22⋊C4—44no—
401C22⋊C4—44no—
402C22⋊C4—416no—
403C22⋊C4—88no—
404C42—42no—
405C42—42no—
406C42—41no—
407C42—41no—
408C42—41no—
409C42—41no—
410C23—22no—
411C23—24no—
412C23—22no—
413C23—22no—
414C23—22no—
415C23—24no—
416C23—22no—
417C23—22no—
418C23—24no—
419C23—24no—
420C23—24no—
421C23—24no—
422C23—41no—
423C23—42no—
424C23—41no—
425C23—41no—
426C23—41no—
427C23—42no—
428C23—42no—
429C23—42no—
430C23—42no—
431C23—42no—
432C23#48714yes—
433C23C2×#12411yescharacteristic
434C23#53411yescharacteristic
435C23#53711yescharacteristic
436C23D4212yes—
437C23#54612yes—
438C23C22×SD1611yescharacteristic
439C23C22×Q1611yescharacteristic
440C23#57511yescharacteristic
441C23#57511yescharacteristic
442C23#57712yes—
443C23#57812yes—
444Q8—44no—
445Q8—44no—
446Q8—44no—
447Q8—44no—
448Q8—82no—
449Q8—82no—
450Q8—82no—
451Q8—82no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—42no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—44no—
479C2×C4—42no—
480C2×C4—44no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—82no—
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#52512yes—
512C2×C4#53412yes—
513C2×C4#53512yes—
514C2×C4#53812yes—
515C2×C4#53811yescharacteristic
516C2×C4#53811yescharacteristic
517C2×C4#53911yescharacteristic
518C2×C4#53911yescharacteristic
519C2×C4#54012yes—
520C2×C4#54012yes—
521C2×C4D4211yescharacteristic
522C2×C4#54612yes—
523C2×C4#54712yes—
524C2×C4#55112yes—
525C2×C4#55911yescharacteristic
526C2×C4#56011yescharacteristic
527C2×C4#56011yescharacteristic
528C2×C4#56212yes—
529C8—44no—
530C8—82no—
531C22—28no—
532C22—24no—
533C22—42no—
534C22—42no—
535C22—42no—
536C22—42no—
537C22—44no—
538C22—44no—
539C22—44no—
540C22—42no—
541C22—42no—
542C22#247212yes—
543C22#263911yescharacteristic
544C22#266911yescharacteristic
545C22#267112yes—
546C22#270911yescharacteristic
547C22C2×#48711yescharacteristic
548C22#313511yescharacteristic
549C22#313712yes—
550C22#318711yescharacteristic
551C22#318911yescharacteristic
552C22#320011yescharacteristic
553C22#320211yescharacteristic
554C22#324611yescharacteristic
555C22#324911yescharacteristic
556C22#325812yes—
557C22#326912yes—
558C22#327112yes—
559C22D4×SD1612yes—
560C22D4×Q1612yes—
561C22#329812yes—
562C22#329912yes—
563C22#330712yes—
564C22#331512yes—
565C4—42no—
566C4—42no—
567C4—42no—
568C4—42no—
569C4—42no—
570C4—84no—
571C4—84no—
572C4—84no—
573C4—162no—
574C4#245511yescharacteristic
575C4#263512yes—
576C4#266811yescharacteristic
577C4#270612yes—
578C4#275211yescharacteristic
579C4#282611yescharacteristic
580C2—44no—
581C2#1633811yescharacteristic
582C2#2725812yes—
583C2#2726712yes—
584C2#3018711yescharacteristic
585C2#3019311yescharacteristic
586C2#3050611yescharacteristic
587C2#3050711yescharacteristic
588C2#3059712yes—
589C2#3060012yes—
590C2#3202711yescharacteristic
591C2#3202911yescharacteristic
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
1. C2×#2043C2semidirect
5. #11293C2semidirect
6. #11190C2semidirect
7. #11050C2semidirect

Maximal subgroups

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

subgroupindex
C2×#20432
C2×#18922
#151782
#151352
#112932
#111902
#110502
#109702
#107142