Finite Group Databasebeta

← #385138

Subgroups — #385138

Order 512.

1664 conjugacy classes of subgroups.

The subgroup lattice has 635 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1954C211yesmaximal, characteristic
2C2×#1870C211yesmaximal, characteristic
3C2×#1851C212yesmaximal, complemented
4C2×#1797C212yesmaximal, complemented
5C4×#386C211yesmaximal, characteristic
6#11322C212yesmaximal, complemented
7#11042C212yesmaximal, complemented
8#11042C212yesmaximal, complemented
9#10876C212yesmaximal, complemented
10C23×M16C2211yescharacteristic
11C22×C4×D4C2211yescharacteristic
12C23×C42C2211yescharacteristic
13C22×C4⋊C8C2212yes—
14C22×C4⋊C8C2212yes—
15C22×C4⋊C8C2212yes—
16C22×C4⋊C8C2211yescharacteristic
17C22×C22⋊C8C2212yes—
18C22×C22⋊C8C2212yes—
19C22×C22⋊C8C2212yes—
20C22×C22⋊C8C2211yescharacteristic
21C22×C4×C8C2211yescharacteristic
22C2×#386C412yes—
23C2×C4×C22⋊C4C2212yes—
24C2×C4×C22⋊C4C2211yescharacteristic
25#1954—24no—
26#1870—24no—
27#1851—28no—
28#1797—28no—
29C2×#336C2212yes—
30C2×#336C2212yes—
31C2×#336C2212yes—
32C2×#336C2212yes—
33C2×#336C2212yes—
34C2×#336C2212yes—
35#1437C411yescharacteristic
36#1437C411yescharacteristic
37#1436C411yescharacteristic
38#1436C411yescharacteristic
39C22⋊#95C412yes—
40C4×#95C2212yes—
41C23×D4C2×C411yescharacteristic
42C24×C4C2×C412yes—
43C24×C4C2311yescharacteristic
44C24×C4C2311yescharacteristic
45C22×M16—22no—
46C22×M16—24no—
47C22×M16—42no—
48C22×M16—41no—
49C22×M16—41no—
50C23×C8C2312yes—
51C23×C8C2312yes—
52C23×C8C2312yes—
53C23×C8C2312yes—
54C2×C4×D4—41no—
55C2×C4×D4—41no—
56C2×C4×D4—41no—
57C2×C4×D4—41no—
58C2×C4×D4D414yes—
59C2×C4×D4Q814yes—
60C22×C4⋊C4C2×C411yescharacteristic
61C22×C22⋊C4C2×C412yes—
62C22×C22⋊C4C2×C412yes—
63C22×C22⋊C4C2×C412yes—
64C22×C22⋊C4C2×C411yescharacteristic
65C22×C22⋊C4C2×C411yescharacteristic
66C22×C42—24no—
67C22×C42—24no—
68C22×C42—41no—
69C22×C42—41no—
70C22×C42C2×C411yescharacteristic
71C22×C42C2×C411yescharacteristic
72C22×C42C2312yes—
73C22×C42C2311yescharacteristic
74C22×C42C2312yes—
75C2×C4⋊C8—24no—
76C2×C4⋊C8—24no—
77C2×C4⋊C8—24no—
78C2×C4⋊C8—24no—
79C2×C4⋊C8—22no—
80C2×C4⋊C8—24no—
81C2×C4⋊C8—42no—
82C2×C4⋊C8—42no—
83C2×C4⋊C8—41no—
84C2×C4⋊C8—41no—
85C2×C4⋊C8—42no—
86C2×C4⋊C8—42no—
87C2×C22⋊C8—24no—
88C2×C22⋊C8—22no—
89C2×C22⋊C8—24no—
90C2×C22⋊C8—24no—
91C2×C22⋊C8—24no—
92C2×C22⋊C8—24no—
93C2×C22⋊C8—24no—
94C2×C22⋊C8—42no—
95C2×C22⋊C8—42no—
96C2×C22⋊C8—42no—
97C2×C22⋊C8—41no—
98C2×C22⋊C8—41no—
99C2×C4×C8—22no—
100C2×C4×C8—41no—
101C2×C4×C8—41no—
102#386—28no—
103C4×C22⋊C4—24no—
104C4×C22⋊C4—24no—
105C4×C22⋊C4—41no—
106C4×C22⋊C4—41no—
107C2×#95C2×C412yes—
108C2×#95C2×C412yes—
109C2×#95C2×C412yes—
110C2×#95C2×C412yes—
111C2×#95C2×C412yes—
112C2×#95C2×C411yescharacteristic
113C2×#95C2×C411yescharacteristic
114C2×#95C2×C412yes—
115#336—24no—
116#336—24no—
117#336—24no—
118#336—42no—
119#336—42no—
120#336D418yescomplemented
121C25C22×C411yescharacteristic
122C25C22×C411yescharacteristic
123C22×D4—24no—
124C22×D4—24no—
125C22×D4—41no—
126C22×D4—41no—
127C22×D4C4⋊C414yes—
128C23×C4—24no—
129C23×C4—24no—
130C23×C4—24no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—24no—
148C23×C4—24no—
149C23×C4—42no—
150C23×C4—42no—
151C23×C4C22×C412yes—
152C23×C4C22×C411yescharacteristic
153C23×C4C22×C411yescharacteristic
154C23×C4C22×C411yescharacteristic
155C23×C4C22×C412yes—
156C23×C4C22×C412yes—
157C23×C4C22×C411yescharacteristic
158C23×C4C22×C412yes—
159C23×C4C2×D412yes—
160C23×C4C2×D412yes—
161C23×C4C2×Q812yes—
162C23×C4C2×Q812yes—
163C23×C4C2411yescharacteristic
164C2×M16—44no—
165C2×M16—48no—
166C2×M16—48no—
167C2×M16—44no—
168C2×M16—44no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8—22no—
193C22×C8—22no—
194C22×C8—22no—
195C22×C8C2×D414yes—
196C22×C8C2×D414yes—
197C4×D4—44no—
198C4×D4—44no—
199C4×D4—44no—
200C4×D4—44no—
201C2×C4⋊C4—41no—
202C2×C4⋊C4—41no—
203C2×C4⋊C4C4⋊C414yes—
204C2×C22⋊C4—24no—
205C2×C22⋊C4—24no—
206C2×C22⋊C4—24no—
207C2×C22⋊C4—24no—
208C2×C22⋊C4—24no—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—42no—
212C2×C22⋊C4—41no—
213C2×C22⋊C4—41no—
214C2×C22⋊C4—41no—
215C2×C22⋊C4—41no—
216C2×C22⋊C4—42no—
217C2×C22⋊C4C4⋊C414yes—
218C2×C22⋊C4C4⋊C414yes—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—21no—
222C2×C42—21no—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—24no—
228C2×C42—24no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—44no—
236C2×C42—48no—
237C2×C42—44no—
238C2×C42—41no—
239C2×C42—41no—
240C2×C42C2×D412yes—
241C2×C42C2×D414yes—
242C2×C42C2×Q812yes—
243C2×C42#4114yes—
244C4⋊C8—48no—
245C4⋊C8—48no—
246C4⋊C8—44no—
247C4⋊C8—48no—
248C22⋊C8—48no—
249C22⋊C8—48no—
250C22⋊C8—48no—
251C22⋊C8—44no—
252C4×C8—44no—
253#95—24no—
254#95—24no—
255#95—24no—
256#95—24no—
257#95—42no—
258#95—41no—
259#95—41no—
260#95M1618yescomplemented
261C24—22no—
262C24—21no—
263C24—21no—
264C24—22no—
265C24—22no—
266C24—22no—
267C24—22no—
268C24—22no—
269C24—21no—
270C24—21no—
271C24—22no—
272C24—22no—
273C24—22no—
274C24—22no—
275C24—22no—
276C24C2×C4⋊C412yes—
277C24C2×C4⋊C412yes—
278C24#13114yes—
279C24C23×C411yescharacteristic
280C2×D4—28no—
281C2×D4—28no—
282C2×D4—44no—
283C2×D4—48no—
284C2×D4—48no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—216no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—21no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
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—24no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—48no—
341C22×C4—48no—
342C22×C4—48no—
343C22×C4—48no—
344C22×C4—48no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—48no—
366C22×C4C2×C4⋊C412yes—
367C22×C4C2×C4⋊C412yes—
368C22×C4C2×C4⋊C412yes—
369C22×C4C2×C4⋊C412yes—
370C22×C4C4×D414yes—
371C22×C4C4×D414yes—
372C22×C4C2×M1614yes—
373C22×C4C2×M1614yes—
374C22×C4C2×M1614yes—
375C22×C4#13114yes—
376C22×C4C22×D411yescharacteristic
377C22×C4C22×D412yes—
378C22×C4C22×Q811yescharacteristic
379C22×C4C2×#4112yes—
380C22×C4H(5,𝔽2)11yescharacteristic
381C22×C4H(5,𝔽2)11yescharacteristic
382C22×C4H(5,𝔽2)11yescharacteristic
383C22×C4H(5,𝔽2)11yescharacteristic
384C22×C4H(5,𝔽2)12yes—
385C22×C42−1+412yes—
386M16—416no—
387C2×C8—24no—
388C2×C8—24no—
389C2×C8—24no—
390C2×C8—24no—
391C2×C8—24no—
392C2×C8—24no—
393C2×C8—42no—
394C2×C8—42no—
395C2×C8—42no—
396C2×C8—42no—
397C2×C8—42no—
398C2×C8—42no—
399C2×C8—42no—
400C2×C8—42no—
401C2×C8—42no—
402C2×C8—42no—
403C2×C8—42no—
404C2×C8—42no—
405C2×C8—42no—
406C2×C8—42no—
407C2×C8—42no—
408C2×C8—42no—
409C2×C8—42no—
410C2×C8—42no—
411C2×C8—42no—
412C2×C8—42no—
413C2×C8—42no—
414C2×C8—42no—
415C4⋊C4—44no—
416C22⋊C4—44no—
417C22⋊C4—44no—
418C22⋊C4—48no—
419C22⋊C4—48no—
420C22⋊C4—48no—
421C42—22no—
422C42—22no—
423C42—22no—
424C42—22no—
425C42—24no—
426C42—24no—
427C42—24no—
428C42—24no—
429C42—44no—
430C42—44no—
431C42—42no—
432C42—42no—
433C42—42no—
434C42—42no—
435C23—22no—
436C23—24no—
437C23—22no—
438C23—24no—
439C23—24no—
440C23—24no—
441C23—24no—
442C23—24no—
443C23—24no—
444C23—24no—
445C23—22no—
446C23—24no—
447C23—22no—
448C23—22no—
449C23—24no—
450C23—42no—
451C23—41no—
452C23—41no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23—41no—
457C23—41no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—41no—
462C23—41no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—41no—
471C23—41no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23#42414yes—
477C23C22×C4⋊C411yescharacteristic
478C23C2×C4×D412yes—
479C23#51811yescharacteristic
480C23#51811yescharacteristic
481C23#52012yes—
482C23C22×M1612yes—
483C23C2×#13112yes—
484C23C2×#13112yes—
485C23#56812yes—
486D4—416no—
487C2×C4—22no—
488C2×C4—24no—
489C2×C4—22no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—432no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—41no—
524C2×C4—42no—
525C2×C4—41no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4#42314yes—
547C2×C4#43618yes—
548C2×C4#53411yescharacteristic
549C2×C4#53411yescharacteristic
550C2×C4#53512yes—
551C2×C4#53712yes—
552C2×C4#54012yes—
553C2×C4#54212yes—
554C2×C4#54311yescharacteristic
555C2×C4#54311yescharacteristic
556C2×C4D4212yes—
557C2×C4D4211yescharacteristic
558C2×C4#54612yes—
559C2×C4#54712yes—
560C2×C4D4×Q812yes—
561C2×C4#55111yescharacteristic
562C2×C4#55412yes—
563C2×C4#56011yescharacteristic
564C2×C4#56212yes—
565C2×C4#56311yescharacteristic
566C8—44no—
567C8—44no—
568C8—82no—
569C8—82no—
570C22—28no—
571C22—24no—
572C22—44no—
573C22—44no—
574C22—42no—
575C22—44no—
576C22—44no—
577C22—44no—
578C22—42no—
579C22—42no—
580C22—44no—
581C22—44no—
582C22—42no—
583C22—42no—
584C22—82no—
585C22—82no—
586C22—82no—
587C22—82no—
588C22#231511yescharacteristic
589C22#232412yes—
590C22D4×C4⋊C412yes—
591C22#235811yescharacteristic
592C22#237611yescharacteristic
593C22C2×#42311yescharacteristic
594C22C2×#42411yescharacteristic
595C22#290612yes—
596C22C2×#43612yes—
597C22#292912yes—
598C22#293312yes—
599C22D4×M1612yes—
600C22#297412yes—
601C22#297412yes—
602C22#297612yes—
603C22#297911yescharacteristic
604C22#298011yescharacteristic
605C22#298912yes—
606C22#299112yes—
607C22#299212yes—
608C22#299412yes—
609C4—28no—
610C4—24no—
611C4—44no—
612C4—42no—
613C4—42no—
614C4—44no—
615C4—42no—
616C4—82no—
617C4#266911yescharacteristic
618C4#267112yes—
619C4#268311yescharacteristic
620C4#269212yes—
621C4#271711yescharacteristic
622C4#274811yescharacteristic
623C2—44no—
624C2—82no—
625C2#1440111yescharacteristic
626C2#2193211yescharacteristic
627C2#2193611yescharacteristic
628C2#2213712yes—
629C2#2214112yes—
630C2#2264712yes—
631C2#2265012yes—
632C2#2284812yes—
633C2#2332012yes—
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
3. C2×#1851C2semidirect
4. C2×#1797C2semidirect
6. #11322C2semidirect
7. #11042C2semidirect
8. #11042C2semidirect
9. #10876C2semidirect
120. #336D4semidirect
260. #95M16semidirect

Maximal subgroups

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

subgroupindex
C2×#19542
C2×#18702
C2×#18512
C2×#17972
C4×#3862
#113222
#110422
#108762