Finite Group Databasebeta

← #363384

Subgroups — #363384

Order 512.

2026 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C42×D4C211yesmaximal, characteristic
2C2×#1817C211yesmaximal, characteristic, complemented
3C2×#1749C211yesmaximal, characteristic, complemented
4#11147C212yesmaximal, complemented
5#10964C212yesmaximal
6#10922C212yesmaximal, complemented
7#10875C212yesmaximal, complemented
8#10795C212yesmaximal, complemented
9#10712C212yesmaximal, complemented
10C22×C4×D4C411yescharacteristic
11C22×C4×D4C411yescharacteristic
12C22×C4×D4C2211yescharacteristic
13C23×C42C2211yescharacteristic
14C23×C42C2211yescharacteristic
15C22×C22⋊C8C2212yes—
16C22×C22⋊C8C2214yes—
17C22×C22⋊C8C2212yes—
18C22×#97C2212yes—
19C42×D4—24no—
20C42×D4—24no—
21C2×C4×C4⋊C4C411yescharacteristic
22C2×C4×C4⋊C4C411yescharacteristic
23C2×C4×C4⋊C4C2211yescharacteristic
24C2×C4×C22⋊C4C411yescharacteristic
25C2×C4×C22⋊C4C411yescharacteristic
26C2×C4×C22⋊C4C411yescharacteristic
27C2×C4×C22⋊C4C411yescharacteristic
28C2×C4×C22⋊C4C2211yescharacteristic
29C2×C4×C22⋊C4C2211yescharacteristic
30C2×C43C2211yescharacteristic
31#1817—24no—
32#1749—24no—
33C2×#336C2214yes—
34C2×#336C2214yes—
35C2×#336C2212yes—
36C2×#336C2212yes—
37C2×#336C2212yes—
38C2×#336C2214yes—
39C23×D4C2×C411yescharacteristic
40C24×C4C2×C411yescharacteristic
41C24×C4C2×C411yescharacteristic
42C24×C4C2×C411yescharacteristic
43C24×C4C2×C411yescharacteristic
44C24×C4C2311yescharacteristic
45C24×C4C2311yescharacteristic
46C23×C8C2314yes—
47C23×C8C2314yes—
48C2×C4×D4—24no—
49C2×C4×D4—24no—
50C2×C4×D4—24no—
51C2×C4×D4—24no—
52C2×C4×D4—22no—
53C2×C4×D4—22no—
54C2×C4×D4—24no—
55C2×C4×D4—24no—
56C2×C4×D4—24no—
57C2×C4×D4—24no—
58C22×C4⋊C4C2×C411yescharacteristic
59C22×C4⋊C4C2×C411yescharacteristic
60C22×C4⋊C4C2×C411yescharacteristic
61C22×C22⋊C4C2×C411yescharacteristic
62C22×C22⋊C4C2×C411yescharacteristic
63C22×C22⋊C4C2×C411yescharacteristic
64C22×C22⋊C4C2×C411yescharacteristic
65C22×C22⋊C4C2×C411yescharacteristic
66C22×C22⋊C4C2×C411yescharacteristic
67C22×C42—22no—
68C22×C42—22no—
69C22×C42—44no—
70C22×C42—44no—
71C22×C42C2×C411yescharacteristic
72C22×C42C2×C411yescharacteristic
73C22×C42C2×C411yescharacteristic
74C22×C42C2×C411yescharacteristic
75C22×C42C2×C411yescharacteristic
76C22×C42C2×C411yescharacteristic
77C22×C42C2×C411yescharacteristic
78C22×C42C2×C411yescharacteristic
79C22×C42D414yes—
80C22×C42D414yes—
81C22×C42C2311yescharacteristic
82C22×C42C2311yescharacteristic
83C22×C42C2311yescharacteristic
84C22×C42C2311yescharacteristic
85C22×C42C2311yescharacteristic
86C2×C22⋊C8—24no—
87C2×C22⋊C8—28no—
88C2×C22⋊C8—24no—
89C2×C22⋊C8—28no—
90C2×C22⋊C8—28no—
91C2×C22⋊C8—44no—
92C2×C22⋊C8—44no—
93C2×#97—24no—
94C2×#97—44no—
95C4×C4⋊C4—24no—
96C4×C4⋊C4—24no—
97C4×C4⋊C4—22no—
98C4×C4⋊C4—22no—
99C4×C22⋊C4—24no—
100C4×C22⋊C4—24no—
101C4×C22⋊C4—22no—
102C4×C22⋊C4—22no—
103C4×C22⋊C4—24no—
104C4×C22⋊C4—24no—
105C4×C22⋊C4—22no—
106C4×C22⋊C4—22no—
107C2×#95C2×C411yescharacteristic
108C2×#95C2×C411yescharacteristic
109C2×#95C2×C411yescharacteristic
110C2×#95C2×C411yescharacteristic
111C2×#95C2×C411yescharacteristic
112C2×#95C2×C411yescharacteristic
113C43—22no—
114C43—22no—
115#336—28no—
116#336—24no—
117#336—24no—
118#336—24no—
119#336—28no—
120#336—44no—
121C25C22×C411yescharacteristic
122C25C22×C411yescharacteristic
123C22×D4—22no—
124C22×D4—22no—
125C22×D4—22no—
126C22×D4—22no—
127C22×D4—22no—
128C22×D4—22no—
129C22×D4M1614yes—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—24no—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—24no—
146C23×C4—24no—
147C23×C4—24no—
148C23×C4—24no—
149C23×C4—24no—
150C23×C4—42no—
151C23×C4—42no—
152C23×C4—42no—
153C23×C4—42no—
154C23×C4—42no—
155C23×C4—42no—
156C23×C4C22⋊C414yes—
157C23×C4C22⋊C414yes—
158C23×C4M1614yes—
159C23×C4C22×C411yescharacteristic
160C23×C4C22×C411yescharacteristic
161C23×C4C22×C411yescharacteristic
162C23×C4C22×C411yescharacteristic
163C23×C4C22×C411yescharacteristic
164C23×C4C22×C411yescharacteristic
165C23×C4C22×C411yescharacteristic
166C23×C4C22×C411yescharacteristic
167C23×C4C22×C411yescharacteristic
168C23×C4C22×C411yescharacteristic
169C23×C4C22×C411yescharacteristic
170C23×C4C22×C411yescharacteristic
171C23×C4C2×D412yes—
172C23×C4C2×D412yes—
173C23×C4#4114yes—
174C23×C4C2411yescharacteristic
175C22×C8—24no—
176C22×C8—24no—
177C22×C8—24no—
178C22×C8—24no—
179C22×C8—24no—
180C22×C8—24no—
181C22×C8—24no—
182C22×C8—24no—
183C22×C8—24no—
184C22×C8—24no—
185C22×C8—24no—
186C22×C8—24no—
187C22×C8—24no—
188C22×C8#4118yes—
189C4×D4—232no—
190C4×D4—216no—
191C4×D4—216no—
192C4×D4—232no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4M1614yes—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—21no—
226C2×C42—21no—
227C2×C42—22no—
228C2×C42—24no—
229C2×C42—24no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—24no—
235C2×C42—21no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42—24no—
240C2×C42—24no—
241C2×C42—48no—
242C2×C42—48no—
243C2×C42—48no—
244C2×C42—48no—
245C2×C42—42no—
246C2×C42—42no—
247C2×C42—42no—
248C2×C42—42no—
249C2×C42—42no—
250C2×C42C22⋊C414yes—
251C2×C42C22⋊C414yes—
252C2×C42M1614yes—
253C2×C42C2×D412yes—
254C2×C42C2×D412yes—
255C2×C42C2×D412yes—
256C2×C42C2×D412yes—
257C2×C42#4114yes—
258C22⋊C8—216no—
259C22⋊C8—416no—
260C22⋊C8—48no—
261#97—48no—
262#95—22no—
263#95—22no—
264#95—22no—
265#95—22no—
266#95—22no—
267#95—22no—
268C24—21no—
269C24—21no—
270C24—21no—
271C24—21no—
272C24—22no—
273C24—22no—
274C24—22no—
275C24—22no—
276C24—21no—
277C24—21no—
278C24—21no—
279C24—21no—
280C24—22no—
281C24—22no—
282C24—22no—
283C24—22no—
284C24C2×C22⋊C412yes—
285C24#11712yes—
286C24C2×M1612yes—
287C24C2×M1612yes—
288C24C2×M1612yes—
289C24C2×M1612yes—
290C24C23×C411yescharacteristic
291C2×D4—28no—
292C2×D4—28no—
293C2×D4—28no—
294C2×D4—28no—
295C2×D4—28no—
296C2×D4—28no—
297C2×D4—28no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—22no—
307C22×C4—28no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
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—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—28no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—28no—
354C22×C4—28no—
355C22×C4—28no—
356C22×C4—28no—
357C22×C4—28no—
358C22×C4—44no—
359C22×C4—44no—
360C22×C4—44no—
361C22×C4—48no—
362C22×C4—48no—
363C22×C4—48no—
364C22×C4—48no—
365C22×C4—48no—
366C22×C4—44no—
367C22×C4—44no—
368C22×C4—44no—
369C22×C4—48no—
370C22×C4—48no—
371C22×C4—48no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—44no—
377C22×C4—44no—
378C22×C4—44no—
379C22×C4—44no—
380C22×C4—44no—
381C22×C4—44no—
382C22×C4—44no—
383C22×C4C2×C22⋊C412yes—
384C22×C4C2×C22⋊C412yes—
385C22×C4C2×C22⋊C412yes—
386C22×C4C2×C22⋊C412yes—
387C22×C4C2×C22⋊C412yes—
388C22×C4#11712yes—
389C22×C4#11712yes—
390C22×C4#11712yes—
391C22×C4C2×M1612yes—
392C22×C4C2×M1612yes—
393C22×C4C2×M1612yes—
394C22×C4C2×M1612yes—
395C22×C4C2×M1612yes—
396C22×C4C2×M1612yes—
397C22×C4C2×M1612yes—
398C22×C4C2×M1612yes—
399C22×C4#13114yes—
400C22×C4C22×D411yescharacteristic
401C22×C4C22×D411yescharacteristic
402C22×C4C2×#4112yes—
403C22×C4C2×#4112yes—
404C22×C4C2×#4114yes—
405C22×C4H(5,𝔽2)14yes—
406C2×C8—28no—
407C2×C8—28no—
408C2×C8—28no—
409C2×C8—44no—
410C2×C8—44no—
411C2×C8—44no—
412C2×C8—44no—
413C2×C8—44no—
414C2×C8—44no—
415C2×C8—44no—
416C2×C8—44no—
417C2×C8—44no—
418C2×C8—44no—
419C2×C8—44no—
420C4⋊C4—28no—
421C4⋊C4—28no—
422C4⋊C4—28no—
423C22⋊C4—28no—
424C22⋊C4—28no—
425C22⋊C4—28no—
426C22⋊C4—28no—
427C22⋊C4—28no—
428C22⋊C4—28no—
429C42—28no—
430C42—24no—
431C42—24no—
432C42—24no—
433C42—24no—
434C42—24no—
435C42—24no—
436C42—28no—
437C42—44no—
438C42—44no—
439C42—44no—
440C42—44no—
441C42—44no—
442C42—42no—
443C42—42no—
444C42—42no—
445C42—42no—
446C42—44no—
447C23—22no—
448C23—24no—
449C23—22no—
450C23—22no—
451C23—22no—
452C23—22no—
453C23—22no—
454C23—22no—
455C23—22no—
456C23—22no—
457C23—22no—
458C23—22no—
459C23—22no—
460C23—22no—
461C23—24no—
462C23—24no—
463C23—24no—
464C23—24no—
465C23—24no—
466C23—24no—
467C23—24no—
468C23—24no—
469C23—24no—
470C23—22no—
471C23—22no—
472C23—22no—
473C23—41no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—41no—
478C23—41no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—41no—
491C23#40714yes—
492C23#43214yes—
493C23C22×C22⋊C411yescharacteristic
494C23C2×#11711yescharacteristic
495C23C2×#11711yescharacteristic
496C23C4×#4112yes—
497C23#51812yes—
498C23C22×M1611yescharacteristic
499C23C22×M1611yescharacteristic
500C23C22×M1611yescharacteristic
501C23C22×M1611yescharacteristic
502C23C2×#13112yes—
503C23#56812yes—
504D4—232no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—24no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—416no—
541C2×C4—416no—
542C2×C4—416no—
543C2×C4—416no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—48no—
569C2×C4—48no—
570C2×C4—48no—
571C2×C4—48no—
572C2×C4#40714yes—
573C2×C4#40814yes—
574C2×C4#43214yes—
575C2×C4#52512yes—
576C2×C4#52512yes—
577C2×C4#52512yes—
578C2×C4#53212yes—
579C2×C4#53412yes—
580C2×C4#53412yes—
581C2×C4#55114yes—
582C2×C4#55314yes—
583C2×C4#55314yes—
584C2×C4#55914yes—
585C8—48no—
586C8—84no—
587C22—28no—
588C22—24no—
589C22—24no—
590C22—24no—
591C22—24no—
592C22—44no—
593C22—44no—
594C22—44no—
595C22—44no—
596C22—44no—
597C22—44no—
598C22—44no—
599C22—42no—
600C22—42no—
601C22—42no—
602C22—42no—
603C22—42no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—42no—
608C22—42no—
609C22#228011yescharacteristic
610C22#229711yescharacteristic
611C22#230811yescharacteristic
612C22#235312yes—
613C22#235712yes—
614C22C2×#40711yescharacteristic
615C22C2×#40711yescharacteristic
616C22C2×#40811yescharacteristic
617C22#287911yescharacteristic
618C22C2×#43211yescharacteristic
619C22C2×#43211yescharacteristic
620C22#292011yescharacteristic
621C22#292011yescharacteristic
622C22#292512yes—
623C22#292512yes—
624C22#296512yes—
625C22#296512yes—
626C22#297012yes—
627C22#297012yes—
628C22#297312yes—
629C22#297412yes—
630C22#297412yes—
631C22#297512yes—
632C4—24no—
633C4—24no—
634C4—24no—
635C4—42no—
636C4—42no—
637C4—42no—
638C4—42no—
639C4—42no—
640C4—42no—
641C4—42no—
642C4—42no—
643C4—42no—
644C4#260014yes—
645C4#260214yes—
646C2—44no—
647C2—44no—
648C2#1394011yescharacteristic
649C2#2071511yescharacteristic
650C2#2072111yescharacteristic
651C2#2137411yescharacteristic
652C2#2137911yescharacteristic
653C2#2174011yescharacteristic
654C2#2174211yescharacteristic
655C2#2278612yes—
656C2#2278712yes—
657C2#2283712yes—
658C2#2283912yes—
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
2. C2×#1817C2semidirect
3. C2×#1749C2semidirect
4. #11147C2semidirect
6. #10922C2semidirect
7. #10875C2semidirect
8. #10795C2semidirect
9. #10712C2semidirect

Maximal subgroups

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

subgroupindex
C2×C42×D42
C2×#18172
C2×#17492
#111472
#109642
#109222
#108752
#107952
#107122