Finite Group Databasebeta

← #368193

Subgroups — #368193

Order 512.

1654 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2323C211yesmaximal, characteristic
2C2×#1873C211yesmaximal, characteristic
3C2×#1872C211yesmaximal, characteristic
4C2×#1853C211yesmaximal, characteristic
5#13795C211yesmaximal, characteristic
6#11293C212yesmaximal, complemented
7#11122C212yesmaximal, complemented
8#10908C212yesmaximal, complemented
9#10789C212yesmaximal, complemented
10#10731C211yesmaximal, characteristic
11#10713C211yesmaximal, characteristic
12C22×C22⋊Q8C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C22×#117C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17C22×#106C2212yes—
18C22×#103C2212yes—
19C22×#103C2212yes—
20C22×C22⋊C8C2211yescharacteristic
21C22×C22⋊C8C2211yescharacteristic
22#2323—24no—
23#2323—24no—
24C2×#391C2211yescharacteristic
25C2×#389—21no—
26C2×#389C2211yescharacteristic
27C2×#388—21no—
28C2×#388—21no—
29C2×#387—21no—
30C2×#380C2211yescharacteristic
31C2×C4×C4⋊C4C2211yescharacteristic
32#1873C418yescomplemented
33#1872—24no—
34#1853—24no—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#336C2212yes—
41C2×#336C2212yes—
42C4⋊#95—21no—
43C4⋊#95—21no—
44C4⋊#95C2212yes—
45C4⋊#95C2212yes—
46C22⋊#95—21no—
47C22⋊#95—21no—
48C23×Q8C2311yescharacteristic
49C24×C4—21no—
50C24×C4C2311yescharacteristic
51C23×C8C2312yes—
52C23×C8C2312yes—
53C2×C22⋊Q8—22no—
54C2×C22⋊Q8—22no—
55C2×C22⋊Q8—22no—
56C2×C22⋊Q8C2×C414yes—
57C2×C22⋊Q8D414yes—
58C2×C22⋊Q8Q814yes—
59C2×C4×Q8—22no—
60C2×C4×Q8—22no—
61C2×C4×Q8—24no—
62C2×C4×Q8—42no—
63C2×#117—22no—
64C2×#117—22no—
65C2×#117—22no—
66C2×#117—24no—
67C2×#117—42no—
68C2×#117—42no—
69C2×#117—42no—
70C2×#117C2×C414yes—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—24no—
76C22×C4⋊C4—22no—
77C22×C4⋊C4—22no—
78C22×C4⋊C4—42no—
79C22×C4⋊C4D411yescharacteristic
80C22×C4⋊C4D411yescharacteristic
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2311yescharacteristic
83C22×C4⋊C4C2311yescharacteristic
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2311yescharacteristic
86C22×C22⋊C4—21no—
87C22×C22⋊C4—21no—
88C22×C22⋊C4—21no—
89C22×C22⋊C4D411yescharacteristic
90C22×C22⋊C4D411yescharacteristic
91C22×C42—21no—
92C22×C42—21no—
93C22×C42—21no—
94C22×C42C2311yescharacteristic
95C22×C42C2311yescharacteristic
96C2×#106—24no—
97C2×#106—24no—
98C2×#106—42no—
99C2×#103—24no—
100C2×#103—24no—
101C2×#103—42no—
102C2×#103—42no—
103C2×#103—42no—
104C2×#103C2×C418yescomplemented
105C2×C22⋊C8—22no—
106C2×C22⋊C8—22no—
107C2×C22⋊C8—22no—
108C2×C22⋊C8—22no—
109C2×C22⋊C8—22no—
110C2×C22⋊C8C2×C414yes—
111#391—22no—
112#391—22no—
113#389—28no—
114#389—22no—
115#389—22no—
116#388—28no—
117#388—28no—
118#387—28no—
119#380—22no—
120#380D414yes—
121C4×C4⋊C4—22no—
122C4×C4⋊C4—22no—
123C2×#95—21no—
124C2×#95—21no—
125C2×#95—21no—
126C2×#95—21no—
127C2×#95—21no—
128C2×#95—22no—
129C2×#95—22no—
130C2×#95—22no—
131C2×#95—22no—
132#340—24no—
133#340—24no—
134#340—42no—
135#340—42no—
136#340C2×C418yescomplemented
137#336—24no—
138#336—42no—
139C25C2×D411yescharacteristic
140C22×Q8—21no—
141C22×Q8—21no—
142C22×Q8—21no—
143C22×Q8—22no—
144C22×Q8—22no—
145C22×Q8—22no—
146C22×Q8—22no—
147C22×Q8C22×C412yes—
148C22×Q8C2×D412yes—
149C22×Q8C2×Q812yes—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—22no—
158C23×C4—21no—
159C23×C4—22no—
160C23×C4—24no—
161C23×C4—42no—
162C23×C4—42no—
163C23×C4—82no—
164C23×C4—82no—
165C23×C4C22×C412yes—
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D412yes—
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D412yes—
173C23×C4C2×Q812yes—
174C23×C4#4112yes—
175C23×C4#4112yes—
176C23×C4C2411yescharacteristic
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×C8C22×C414yes—
191C22⋊Q8—28no—
192C22⋊Q8—28no—
193C22⋊Q8—28no—
194C22⋊Q8C4⋊C4116yes—
195C4×Q8—48no—
196C4×Q8—48no—
197#117—48no—
198#117—48no—
199#117—48no—
200#117—48no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—48no—
231C2×C4⋊C4—48no—
232C2×C4⋊C4—41no—
233C2×C4⋊C4—41no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—82no—
239C2×C4⋊C4C22×C414yes—
240C2×C4⋊C4C22×C412yes—
241C2×C4⋊C4C2×D412yes—
242C2×C4⋊C4C2×D412yes—
243C2×C4⋊C4C2×D412yes—
244C2×C4⋊C4C2×Q812yes—
245C2×C4⋊C4#4114yes—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—41no—
256C2×C22⋊C4—41no—
257C2×C22⋊C4—42no—
258C2×C22⋊C4—42no—
259C2×C22⋊C4—82no—
260C2×C42—21no—
261C2×C42—21no—
262C2×C42—22no—
263C2×C42—24no—
264C2×C42—24no—
265C2×C42—21no—
266C2×C42—21no—
267C2×C42—22no—
268C2×C42—22no—
269C2×C42—24no—
270C2×C42—42no—
271C2×C42—42no—
272C2×C42—42no—
273C2×C42—42no—
274C2×C42—82no—
275C2×C42#4114yes—
276#106—48no—
277#103—48no—
278#103—48no—
279C22⋊C8—28no—
280C22⋊C8—44no—
281#95—24no—
282#95—24no—
283#95—24no—
284#95—24no—
285#95—24no—
286#95—82no—
287#95—82no—
288#95—82no—
289#95—82no—
290C24—21no—
291C24—21no—
292C24—21no—
293C24—21no—
294C24—22no—
295C24—22no—
296C24—22no—
297C24—22no—
298C24C4×D412yes—
299C24#12012yes—
300C24C22⋊Q812yes—
301C24C22×D411yescharacteristic
302C2×Q8—22no—
303C2×Q8—22no—
304C2×Q8—22no—
305C2×Q8—24no—
306C2×Q8—24no—
307C2×Q8—24no—
308C2×Q8—24no—
309C2×Q8—24no—
310C2×Q8—24no—
311C2×Q8—42no—
312C2×Q8—42no—
313C2×Q8—42no—
314C2×Q8—42no—
315C2×Q8—42no—
316C2×Q8—42no—
317C2×Q8C2×C4⋊C414yes—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—24no—
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—24no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—21no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—21no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—21no—
350C22×C4—22no—
351C22×C4—21no—
352C22×C4—22no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—24no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—48no—
369C22×C4—42no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—42no—
373C22×C4—41no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—41no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—88no—
390C22×C4—84no—
391C22×C4—84no—
392C22×C4—88no—
393C22×C4C2×C4⋊C414yes—
394C22×C4C4×D414yes—
395C22×C4C4×D412yes—
396C22×C4#12012yes—
397C22×C4C22⋊Q814yes—
398C22×C4C22⋊Q812yes—
399C22×C4#12312yes—
400C22×C4#12312yes—
401C22×C4#12314yes—
402C22×C4C23×C411yescharacteristic
403C22×C4C22×D411yescharacteristic
404C22×C4C22×D411yescharacteristic
405C22×C4C22×Q811yescharacteristic
406C22×C4C2×#4111yescharacteristic
407C22×C4C2×#4112yes—
408C22×C4C2×#4112yes—
409C22×C4C2×#4111yescharacteristic
410C22×C4H(5,𝔽2)12yes—
411C22×C4H(5,𝔽2)12yes—
412C2×C8—24no—
413C2×C8—24no—
414C2×C8—24no—
415C2×C8—42no—
416C2×C8—42no—
417C2×C8—42no—
418C2×C8—42no—
419C2×C8—42no—
420C2×C8—42no—
421C2×C8—42no—
422C2×C8—42no—
423C2×C8—42no—
424C2×C8—42no—
425C2×C8—42no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—216no—
430C4⋊C4—24no—
431C4⋊C4—24no—
432C4⋊C4—24no—
433C4⋊C4—24no—
434C4⋊C4—24no—
435C4⋊C4—24no—
436C4⋊C4—44no—
437C4⋊C4—44no—
438C4⋊C4—48no—
439C4⋊C4—48no—
440C4⋊C4—42no—
441C4⋊C4—42no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C4⋊C4—42no—
445C4⋊C4—42no—
446C4⋊C4—84no—
447C4⋊C4C2×C4⋊C414yes—
448C22⋊C4—216no—
449C22⋊C4—44no—
450C22⋊C4—44no—
451C22⋊C4—48no—
452C22⋊C4—84no—
453C42—24no—
454C42—24no—
455C42—48no—
456C42—48no—
457C42—42no—
458C42—42no—
459C42—84no—
460C23—24no—
461C23—24no—
462C23—22no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—24no—
467C23—24no—
468C23—22no—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23#38514yes—
483C23C2×C4×D411yescharacteristic
484C23C2×#12011yescharacteristic
485C23C2×C22⋊Q811yescharacteristic
486C23C2×#12311yescharacteristic
487C23#52511yescharacteristic
488C23#53411yescharacteristic
489C23#53511yescharacteristic
490C23#57712yes—
491C23#57712yes—
492C23#57712yes—
493C23#57812yes—
494Q8—28no—
495Q8—44no—
496Q8—44no—
497Q8—44no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—28no—
505C2×C4—28no—
506C2×C4—28no—
507C2×C4—28no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—44no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—44no—
522C2×C4—42no—
523C2×C4—41no—
524C2×C4—42no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—42no—
528C2×C4—41no—
529C2×C4—42no—
530C2×C4—44no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—42no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—44no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—42no—
550C2×C4—41no—
551C2×C4—42no—
552C2×C4—816no—
553C2×C4—82no—
554C2×C4—82no—
555C2×C4—82no—
556C2×C4—82no—
557C2×C4—82no—
558C2×C4—82no—
559C2×C4—82no—
560C2×C4—82no—
561C2×C4—82no—
562C2×C4—82no—
563C2×C4—82no—
564C2×C4—82no—
565C2×C4#38514yes—
566C2×C4C22×C4⋊C411yescharacteristic
567C2×C4C2×C4×D411yescharacteristic
568C2×C4C4×#4112yes—
569C2×C4#51812yes—
570C2×C4C2×C22⋊Q811yescharacteristic
571C2×C4C2×#12311yescharacteristic
572C2×C4#52912yes—
573C2×C4#53812yes—
574C2×C4#54612yes—
575C2×C4#54612yes—
576C2×C4#54612yes—
577C2×C4#54612yes—
578C2×C4#55312yes—
579C2×C4#55312yes—
580C2×C4#55412yes—
581C2×C4#55412yes—
582C8—44no—
583C8—82no—
584C22—24no—
585C22—24no—
586C22—24no—
587C22—44no—
588C22—44no—
589C22—44no—
590C22—44no—
591C22—42no—
592C22—42no—
593C22—42no—
594C22—42no—
595C22C2×#38511yescharacteristic
596C22C4×#12311yescharacteristic
597C22#232111yescharacteristic
598C22#240411yescharacteristic
599C22#240511yescharacteristic
600C22#248711yescharacteristic
601C22#257911yescharacteristic
602C22#294712yes—
603C22#295212yes—
604C22#300911yescharacteristic
605C22#301011yescharacteristic
606C22#301011yescharacteristic
607C22#301111yescharacteristic
608C22#308212yes—
609C22#308412yes—
610C22#309712yes—
611C22#309912yes—
612C22#312112yes—
613C22#312312yes—
614C22#320112yes—
615C22#320412yes—
616C22#322212yes—
617C22#322512yes—
618C4—24no—
619C4—44no—
620C4—44no—
621C4—42no—
622C4—42no—
623C4—84no—
624C4—84no—
625C4—82no—
626C4—81no—
627C4—82no—
628C4—81no—
629C4#234912yes—
630C4#236412yes—
631C4#244412yes—
632C4#245012yes—
633C2—44no—
634C2#1401611yescharacteristic
635C2#2087711yescharacteristic
636C2#2088211yescharacteristic
637C2#2159112yes—
638C2#2205912yes—
639C2#2394211yescharacteristic
640C2#2394611yescharacteristic
641C2#2401211yescharacteristic
642C2#2401411yescharacteristic
643C2#2768912yes—
644C2#2925012yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #11293C2semidirect
7. #11122C2semidirect
8. #10908C2semidirect
9. #10789C2semidirect
32. #1873C4semidirect
104. C2×#103C2×C4semidirect
136. #340C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23232
C2×#18732
C2×#18722
C2×#18532
#137952
#112932
#111222
#109082
#107892
#107312
#107132