Finite Group Databasebeta

← #388053

Subgroups — #388053

Order 512.

1416 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1954C211yesmaximal, characteristic
2C2×#1869C211yesmaximal, characteristic
3#13839C211yesmaximal, characteristic
4#11255C212yesmaximal, complemented
5#11042C212yesmaximal, complemented
6#11042C212yesmaximal, complemented
7#10876C212yesmaximal, complemented
8#10714C212yesmaximal, complemented
9#10713C212yesmaximal, complemented
10C24×C8C2211yescharacteristic
11C22×C4×D4C2211yescharacteristic
12C22×#117C2211yescharacteristic
13C22×C4⋊C8C2212yes—
14C22×C4⋊C8C2211yescharacteristic
15C22×C22⋊C8C2212yes—
16C22×C22⋊C8C2212yes—
17C22×C22⋊C8C2212yes—
18C22×C22⋊C8C2211yescharacteristic
19C22×C4×C8C2211yescharacteristic
20C2×C4×C22⋊C4C2211yescharacteristic
21C2×C4×C22⋊C4C2212yes—
22#1954—24no—
23#1869—24no—
24C2×#336C2212yes—
25C2×#336C2212yes—
26C2×#336C2212yes—
27C2×#336C2212yes—
28C2×#336C2212yes—
29C2×#336C2212yes—
30C2×#336C2212yes—
31C2×#336C2212yes—
32#1441C412yes—
33#1440C411yescharacteristic
34#1440C411yescharacteristic
35#1439C412yes—
36#1438C411yescharacteristic
37#1438C411yescharacteristic
38C4×#95C2212yes—
39C23×D4C2×C411yescharacteristic
40C24×C4C2311yescharacteristic
41C24×C4C2311yescharacteristic
42C23×C8—22no—
43C23×C8—24no—
44C23×C8—41no—
45C23×C8—41no—
46C23×C8—42no—
47C23×C8C2312yes—
48C23×C8C2312yes—
49C23×C8C2312yes—
50C23×C8C2312yes—
51C2×C4×D4—41no—
52C2×C4×D4—41no—
53C2×C4×D4—41no—
54C2×C4×D4—41no—
55C2×C4×D4D414yes—
56C2×C4×D4Q814yes—
57C2×#117—41no—
58C2×#117—41no—
59C2×#117—42no—
60C2×#117—42no—
61C22×C4⋊C4C2×C412yes—
62C22×C4⋊C4C2×C411yescharacteristic
63C22×C22⋊C4C2×C412yes—
64C22×C22⋊C4C2×C411yescharacteristic
65C22×C22⋊C4C2×C412yes—
66C22×C22⋊C4C2×C411yescharacteristic
67C22×C22⋊C4C2×C411yescharacteristic
68C22×C22⋊C4C2×C411yescharacteristic
69C22×C22⋊C4C2×C411yescharacteristic
70C22×C22⋊C4C2×C411yescharacteristic
71C22×C42C2312yes—
72C22×C42C2312yes—
73C22×C42C2311yescharacteristic
74C2×C4⋊C8—24no—
75C2×C4⋊C8—22no—
76C2×C4⋊C8—41no—
77C2×C4⋊C8—41no—
78C2×C4⋊C8—42no—
79C2×C4⋊C8—42no—
80C2×C22⋊C8—24no—
81C2×C22⋊C8—22no—
82C2×C22⋊C8—24no—
83C2×C22⋊C8—24no—
84C2×C22⋊C8—24no—
85C2×C22⋊C8—24no—
86C2×C22⋊C8—24no—
87C2×C22⋊C8—42no—
88C2×C22⋊C8—42no—
89C2×C22⋊C8—42no—
90C2×C22⋊C8—41no—
91C2×C22⋊C8—41no—
92C2×C4×C8—22no—
93C2×C4×C8—41no—
94C2×C4×C8—41no—
95C4×C22⋊C4—42no—
96C4×C22⋊C4—42no—
97C4×C22⋊C4—41no—
98C4×C22⋊C4—41no—
99C2×#95C2×C412yes—
100C2×#95C2×C412yes—
101C2×#95C2×C412yes—
102C2×#95C2×C412yes—
103C2×#95C2×C412yes—
104C2×#95C2×C412yes—
105C2×#95C2×C412yes—
106#336—24no—
107#336—24no—
108#336—42no—
109#336—42no—
110#336—42no—
111#336—42no—
112#336—42no—
113#336—42no—
114C25C22×C411yescharacteristic
115C25C22×C411yescharacteristic
116C22×D4—24no—
117C22×D4—24no—
118C22×D4—41no—
119C22×D4—41no—
120C22×D4C4⋊C414yes—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4—21no—
126C23×C4—21no—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4C22×C412yes—
138C23×C4C22×C412yes—
139C23×C4C22×C412yes—
140C23×C4C22×C411yescharacteristic
141C23×C4C22×C411yescharacteristic
142C23×C4C22×C411yescharacteristic
143C23×C4C22×C412yes—
144C23×C4C22×C411yescharacteristic
145C23×C4C2×D412yes—
146C23×C4C2×D412yes—
147C23×C4C2×Q812yes—
148C23×C4C2×Q812yes—
149C23×C4C2411yescharacteristic
150C22×C8—22no—
151C22×C8—22no—
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—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—44no—
177C22×C8—48no—
178C22×C8—48no—
179C22×C8—44no—
180C22×C8—44no—
181C22×C8#4114yes—
182C22×C8#4114yes—
183C4×D4—44no—
184C4×D4—44no—
185C4×D4—44no—
186C4×D4—44no—
187#117—48no—
188#117—44no—
189#117—44no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—41no—
194C2×C4⋊C4—41no—
195C2×C4⋊C4C4⋊C414yes—
196C2×C22⋊C4—24no—
197C2×C22⋊C4—24no—
198C2×C22⋊C4—24no—
199C2×C22⋊C4—24no—
200C2×C22⋊C4—42no—
201C2×C22⋊C4—42no—
202C2×C22⋊C4—42no—
203C2×C22⋊C4—42no—
204C2×C22⋊C4—41no—
205C2×C22⋊C4—41no—
206C2×C22⋊C4—41no—
207C2×C22⋊C4—41no—
208C2×C22⋊C4—41no—
209C2×C22⋊C4—41no—
210C2×C22⋊C4—41no—
211C2×C22⋊C4—42no—
212C2×C22⋊C4—42no—
213C2×C22⋊C4—41no—
214C2×C22⋊C4C4⋊C414yes—
215C2×C22⋊C4C4⋊C414yes—
216C2×C42—21no—
217C2×C42—21no—
218C2×C42—21no—
219C2×C42—21no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42C2×D412yes—
233C2×C42C2×Q812yes—
234C4⋊C8—44no—
235C4⋊C8—48no—
236C22⋊C8—48no—
237C22⋊C8—48no—
238C22⋊C8—48no—
239C22⋊C8—44no—
240C4×C8—44no—
241#95—24no—
242#95—24no—
243#95—24no—
244#95—42no—
245#95—42no—
246#95—42no—
247#95—42no—
248C24—22no—
249C24—22no—
250C24—21no—
251C24—21no—
252C24—22no—
253C24—22no—
254C24—22no—
255C24—22no—
256C24—21no—
257C24—21no—
258C24—22no—
259C24—22no—
260C24—22no—
261C24—22no—
262C24—22no—
263C24C2×C4⋊C412yes—
264C24C2×C4⋊C412yes—
265C24#13114yes—
266C24C23×C411yescharacteristic
267C2×D4—28no—
268C2×D4—28no—
269C2×D4—44no—
270C2×D4—48no—
271C2×D4—48no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—21no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
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—41no—
339C22×C4—41no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—42no—
345C22×C4—41no—
346C22×C4—42no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4C2×C4⋊C412yes—
351C22×C4C2×C4⋊C412yes—
352C22×C4C2×C4⋊C412yes—
353C22×C4C2×C4⋊C412yes—
354C22×C4#13114yes—
355C22×C4#13114yes—
356C22×C4C22×D411yescharacteristic
357C22×C4C22×Q811yescharacteristic
358C22×C4C2×#4112yes—
359C22×C4C2×#4112yes—
360C22×C4H(5,𝔽2)11yescharacteristic
361C22×C4H(5,𝔽2)11yescharacteristic
362C22×C4H(5,𝔽2)11yescharacteristic
363C22×C4H(5,𝔽2)11yescharacteristic
364C22×C4H(5,𝔽2)12yes—
365C22×C4H(5,𝔽2)12yes—
366C2×C8—24no—
367C2×C8—24no—
368C2×C8—24no—
369C2×C8—24no—
370C2×C8—24no—
371C2×C8—24no—
372C2×C8—416no—
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—
384C2×C8—42no—
385C2×C8—42no—
386C2×C8—42no—
387C2×C8—42no—
388C2×C8—42no—
389C2×C8—42no—
390C2×C8—42no—
391C2×C8—42no—
392C2×C8—42no—
393C2×C8—42no—
394C2×C8—42no—
395C4⋊C4—44no—
396C4⋊C4—48no—
397C22⋊C4—44no—
398C22⋊C4—44no—
399C22⋊C4—44no—
400C22⋊C4—48no—
401C22⋊C4—48no—
402C22⋊C4—44no—
403C22⋊C4—44no—
404C22⋊C4—44no—
405C42—22no—
406C42—22no—
407C42—22no—
408C42—22no—
409C42—42no—
410C42—42no—
411C42—42no—
412C42—42no—
413C42—42no—
414C42—42no—
415C42—42no—
416C42—42no—
417C23—22no—
418C23—24no—
419C23—22no—
420C23—24no—
421C23—24no—
422C23—24no—
423C23—24no—
424C23—24no—
425C23—24no—
426C23—24no—
427C23—22no—
428C23—24no—
429C23—22no—
430C23—22no—
431C23—24no—
432C23—42no—
433C23—41no—
434C23—41no—
435C23—42no—
436C23—42no—
437C23—41no—
438C23—41no—
439C23—42no—
440C23—42no—
441C23—42no—
442C23—42no—
443C23—41no—
444C23—41no—
445C23—42no—
446C23—42no—
447C23—42no—
448C23—42no—
449C23—42no—
450C23—41no—
451C23—42no—
452C23—42no—
453C23—42no—
454C23—41no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23#42414yes—
459C23C22×C4⋊C411yescharacteristic
460C23C4×#4112yes—
461C23#51811yescharacteristic
462C23#51811yescharacteristic
463C23#51812yes—
464C23C2×#13112yes—
465C23C2×#13112yes—
466C23C2×#13112yes—
467C23#56812yes—
468D4—416no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4#42414yes—
529C2×C4#53411yescharacteristic
530C2×C4#53411yescharacteristic
531C2×C4#53812yes—
532C2×C4#53812yes—
533C2×C4#54012yes—
534C2×C4#54012yes—
535C2×C4#54311yescharacteristic
536C2×C4#54311yescharacteristic
537C2×C4#54612yes—
538C2×C4#54612yes—
539C2×C4#55111yescharacteristic
540C2×C4#55312yes—
541C2×C4#55412yes—
542C2×C4#55412yes—
543C2×C4#55911yescharacteristic
544C2×C4#56012yes—
545C2×C4#56011yescharacteristic
546C2×C4H(3,𝔽4)11yescharacteristic
547C8—44no—
548C8—44no—
549C8—82no—
550C8—82no—
551C22—28no—
552C22—24no—
553C22—44no—
554C22—44no—
555C22—44no—
556C22—44no—
557C22—44no—
558C22—42no—
559C22—42no—
560C22—44no—
561C22—42no—
562C22—44no—
563C22—42no—
564C22—42no—
565C22—82no—
566C22—82no—
567C22—82no—
568C22—82no—
569C22#231511yescharacteristic
570C22#233312yes—
571C22#234912yes—
572C22#237411yescharacteristic
573C22#237511yescharacteristic
574C22C2×#42411yescharacteristic
575C22C2×#42411yescharacteristic
576C22#290612yes—
577C22#296612yes—
578C22#296612yes—
579C22#296812yes—
580C22#296912yes—
581C22#297612yes—
582C22#297712yes—
583C22#297712yes—
584C22#297711yescharacteristic
585C22#297712yes—
586C22#297811yescharacteristic
587C22#297912yes—
588C22#298112yes—
589C22#298112yes—
590C4—24no—
591C4—44no—
592C4—42no—
593C4—42no—
594C4—42no—
595C4—82no—
596C4—82no—
597C4—82no—
598C4#268411yescharacteristic
599C4#268812yes—
600C4#270911yescharacteristic
601C4#274811yescharacteristic
602C4#278012yes—
603C4#280411yescharacteristic
604C2—44no—
605C2—82no—
606C2#1451311yescharacteristic
607C2#2193111yescharacteristic
608C2#2193711yescharacteristic
609C2#2233012yes—
610C2#2233612yes—
611C2#2269712yes—
612C2#2270112yes—
613C2#2329912yes—
614C2#2331012yes—
subgroupG/H
center, Frattini, cosocleC23×C4C24
commutator, socleC24C23×C4
prime radicalC23×D4C2×C4
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11255C2semidirect
5. #11042C2semidirect
6. #11042C2semidirect
7. #10876C2semidirect
8. #10714C2semidirect
9. #10713C2semidirect

Maximal subgroups

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

subgroupindex
C2×#19542
C2×#18692
#138392
#112552
#110422
#108762
#107142
#107132