Finite Group Databasebeta

← #361273

Subgroups — #361273

Order 512.

1672 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2344C211yesmaximal, characteristic
2C2×#1820C211yesmaximal, characteristic
3C2×#1740C211yesmaximal, characteristic
4#13870C212yesmaximal
5#11353C212yesmaximal
6#11197C212yesmaximal
7#10801C211yesmaximal, characteristic
8#10801C211yesmaximal, characteristic
9#10736C212yesmaximal
10#10726C211yesmaximal, characteristic
11#10726C211yesmaximal, characteristic
12C22×#117C2212yes—
13C22×#117C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16#2344—24no—
17#2344—24no—
18C2×#389C412yescomplemented
19C2×#387C412yescomplemented
20C2×#387C412yescomplemented
21C2×#381C2212yes—
22C2×#381C2212yes—
23C2×#381C2212yes—
24C2×#381C2211yescharacteristic
25C2×#381C2211yescharacteristic
26C2×#380C2211yescharacteristic
27C2×C4×C4⋊C4C412yescomplemented
28C2×C4×C22⋊C4C2211yescharacteristic
29#1820—24no—
30#1740—24no—
31C2×#342C2211yescharacteristic
32C2×#342C2212yes—
33C2×#342C2211yescharacteristic
34C2×#342C2211yescharacteristic
35C2×#342C2212yes—
36C2×#342C2212yes—
37C2×#342C2212yes—
38C2×#342C2212yes—
39C2×#342C2212yes—
40C2×#342C2211yescharacteristic
41#1441—24no—
42#1436—24no—
43C22⋊#95C2212yes—
44C22⋊#95C2212yes—
45C24×C4C2311yescharacteristic
46C24×C4C2311yescharacteristic
47C24×C4C2311yescharacteristic
48C2×#117—24no—
49C2×#117—22no—
50C2×#117—22no—
51C2×#117—22no—
52C2×#117—22no—
53C2×#117—22no—
54C2×#117—22no—
55C2×#117—24no—
56C2×#117—24no—
57C2×#117—24no—
58C2×#117—24no—
59C2×#117—42no—
60C2×#117—42no—
61C2×#117—42no—
62C2×#117—42no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—24no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4C2×C412yes—
68C22×C4⋊C4C2×C412yes—
69C22×C4⋊C4C2×C411yescharacteristic
70C22×C4⋊C4C2×C411yescharacteristic
71C22×C4⋊C4C2×C411yescharacteristic
72C22×C4⋊C4C2×C411yescharacteristic
73C22×C4⋊C4C2×C412yes—
74C22×C4⋊C4D414yes—
75C22×C4⋊C4D414yes—
76C22×C22⋊C4C2312yes—
77C22×C22⋊C4C2312yes—
78C22×C22⋊C4C2311yescharacteristic
79C22×C22⋊C4C2311yescharacteristic
80C22×C22⋊C4C2311yescharacteristic
81C22×C22⋊C4C2312yes—
82C22×C22⋊C4C2312yes—
83C22×C22⋊C4C2311yescharacteristic
84C22×C42—22no—
85C22×C42C2×C412yes—
86C22×C42C2×C412yes—
87C22×C42C2×C412yes—
88#389—24no—
89#389—24no—
90#387—24no—
91#387—24no—
92#387—24no—
93#387—24no—
94#381—24no—
95#381—22no—
96#381—22no—
97#381—24no—
98#381—24no—
99#381—24no—
100#381—24no—
101#381—42no—
102#381—42no—
103#380—24no—
104C4×C4⋊C4—22no—
105C4×C4⋊C4—22no—
106C4×C4⋊C4—22no—
107C4×C4⋊C4—22no—
108C4×C22⋊C4—24no—
109C2×#95—24no—
110C2×#95—24no—
111C2×#95—22no—
112C2×#95—24no—
113C2×#95—24no—
114C2×#95—22no—
115C2×#95C2×C412yes—
116C2×#95C2×C412yes—
117C2×#95C2×C412yes—
118C2×#95C2×C412yes—
119C2×#95C2×C412yes—
120C2×#95C2×C412yes—
121#342—24no—
122#342—22no—
123#342—22no—
124#342—24no—
125#342—24no—
126#342—22no—
127#342—42no—
128#342—42no—
129#342—42no—
130#342D414yes—
131C25C2411yescharacteristic
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—22no—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—22no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—44no—
155C23×C4C22×C411yescharacteristic
156C23×C4C22×C411yescharacteristic
157C23×C4C22×C411yescharacteristic
158C23×C4C22×C411yescharacteristic
159C23×C4C22×C411yescharacteristic
160C23×C4C22×C412yes—
161C23×C4C22×C412yes—
162C23×C4C22×C412yes—
163C23×C4C22×C411yescharacteristic
164C23×C4C22×C412yes—
165C23×C4C2×D414yes—
166C23×C4C2×D414yes—
167C23×C4C2×D412yes—
168C23×C4C2×D412yes—
169C23×C4#4112yes—
170C23×C4#4114yes—
171C23×C4#4112yes—
172#117—216no—
173#117—44no—
174#117—44no—
175#117—48no—
176#117—44no—
177#117—44no—
178#117—48no—
179#117—48no—
180#117—48no—
181#117—48no—
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—28no—
191C2×C4⋊C4—28no—
192C2×C4⋊C4—28no—
193C2×C4⋊C4—28no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4C22⋊C412yes—
207C2×C4⋊C4C22⋊C412yes—
208C2×C4⋊C4C22⋊C414yes—
209C2×C4⋊C4C22⋊C412yes—
210C2×C4⋊C4C22⋊C412yes—
211C2×C4⋊C4C22⋊C414yes—
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⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4C2×D414yes—
249C2×C22⋊C4C2×D412yes—
250C2×C22⋊C4#4114yes—
251C2×C22⋊C4#4112yes—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—24no—
259C2×C42—24no—
260C2×C42—42no—
261C2×C42—42no—
262C2×C42—44no—
263C2×C42—44no—
264C2×C42—44no—
265C2×C42—44no—
266#95—24no—
267#95—22no—
268#95—22no—
269#95—22no—
270#95—22no—
271#95—24no—
272#95—24no—
273#95—24no—
274#95—48no—
275#95—48no—
276#95—48no—
277#95—48no—
278#95—44no—
279#95—44no—
280C24—42no—
281C24—42no—
282C24C23×C411yescharacteristic
283C24C22×D412yes—
284C24C22×D411yescharacteristic
285C24C2×#4112yes—
286C24C2×#4111yescharacteristic
287C24C2×#4111yescharacteristic
288C24C2×#4111yescharacteristic
289C24C2×#4112yes—
290C24H(5,𝔽2)12yes—
291C24H(5,𝔽2)11yescharacteristic
292C242−1+411yescharacteristic
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—24no—
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—24no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—22no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—22no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—28no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—42no—
359C22×C4—44no—
360C22×C4—48no—
361C22×C4—44no—
362C22×C4—44no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—42no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—44no—
376C22×C4—44no—
377C22×C4—44no—
378C22×C4—44no—
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—44no—
390C22×C4—44no—
391C22×C4—44no—
392C22×C4—44no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4C2×C22⋊C412yes—
400C22×C4C2×C22⋊C412yes—
401C22×C4C2×C22⋊C412yes—
402C22×C4C2×C22⋊C411yescharacteristic
403C22×C4C2×C22⋊C411yescharacteristic
404C22×C4C2×C22⋊C412yes—
405C22×C4C2×C22⋊C412yes—
406C22×C4#11712yes—
407C22×C4#11712yes—
408C22×C4#11712yes—
409C22×C4#11712yes—
410C22×C4C4×D414yes—
411C22×C4#12314yes—
412C22×C4#12314yes—
413C22×C4#12314yes—
414C22×C4#12314yes—
415C4⋊C4—24no—
416C4⋊C4—24no—
417C4⋊C4—28no—
418C4⋊C4—28no—
419C4⋊C4—24no—
420C4⋊C4—24no—
421C4⋊C4—48no—
422C4⋊C4—48no—
423C4⋊C4—44no—
424C4⋊C4—416no—
425C4⋊C4—44no—
426C22⋊C4—24no—
427C22⋊C4—24no—
428C22⋊C4—24no—
429C22⋊C4—24no—
430C22⋊C4—24no—
431C22⋊C4—24no—
432C22⋊C4—42no—
433C22⋊C4—42no—
434C22⋊C4—42no—
435C22⋊C4—42no—
436C22⋊C4—42no—
437C22⋊C4—42no—
438C22⋊C4—42no—
439C22⋊C4—42no—
440C22⋊C4—42no—
441C22⋊C4—42no—
442C22⋊C4—42no—
443C22⋊C4—42no—
444C22⋊C4—42no—
445C22⋊C4—42no—
446C22⋊C4—42no—
447C22⋊C4—42no—
448C22⋊C4—42no—
449C22⋊C4—42no—
450C42—44no—
451C42—44no—
452C42—44no—
453C42—44no—
454C42—44no—
455C42—416no—
456C42—44no—
457C23—22no—
458C23—24no—
459C23—22no—
460C23—24no—
461C23—42no—
462C23—41no—
463C23—41no—
464C23—42no—
465C23—42no—
466C23—44no—
467C23—42no—
468C23—41no—
469C23—41no—
470C23—44no—
471C23—42no—
472C23—42no—
473C23C22×C22⋊C411yescharacteristic
474C23C2×#11711yescharacteristic
475C23C2×#11711yescharacteristic
476C23C2×C4×D411yescharacteristic
477C23C4×#4111yescharacteristic
478C23#51811yescharacteristic
479C23#52011yescharacteristic
480C23C2×#12312yes—
481C23C2×#12312yes—
482C23#52512yes—
483C23#52912yes—
484C23#53812yes—
485C23#53912yes—
486C23D4212yes—
487C23#54612yes—
488C23#54612yes—
489C23#54611yescharacteristic
490C23#54712yes—
491C23#54811yescharacteristic
492C23#55112yes—
493C23#55211yescharacteristic
494C23#55311yescharacteristic
495C23#55312yes—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—22no—
503C2×C4—21no—
504C2×C4—22no—
505C2×C4—21no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
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—24no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—48no—
532C2×C4—48no—
533C2×C4—44no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—44no—
551C2×C4—48no—
552C2×C4—48no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—84no—
565C2×C4—84no—
566C2×C4—82no—
567C2×C4—82no—
568C2×C4—82no—
569C2×C4—82no—
570C2×C4—82no—
571C2×C4—82no—
572C2×C4—82no—
573C2×C4—82no—
574C2×C4—82no—
575C2×C4—82no—
576C2×C4—84no—
577C2×C4—84no—
578C2×C4—84no—
579C2×C4—84no—
580C2×C4—82no—
581C2×C4—82no—
582C2×C4—82no—
583C2×C4—82no—
584C2×C4—82no—
585C2×C4—82no—
586C2×C4#38112yes—
587C2×C4#38112yes—
588C2×C4#38114yes—
589C2×C4#41014yes—
590C22—22no—
591C22—22no—
592C22—24no—
593C22—24no—
594C22—24no—
595C22—22no—
596C22—22no—
597C22—24no—
598C22—21no—
599C22—21no—
600C22—22no—
601C22—22no—
602C22—22no—
603C22—22no—
604C22—21no—
605C22—21no—
606C22—42no—
607C22—42no—
608C22—42no—
609C22—42no—
610C22—41no—
611C22—41no—
612C22—42no—
613C22—41no—
614C22—42no—
615C22—41no—
616C22C2×#38111yescharacteristic
617C22#229011yescharacteristic
618C22#231811yescharacteristic
619C22D4×C22⋊C411yescharacteristic
620C22#233911yescharacteristic
621C22#235211yescharacteristic
622C22#235311yescharacteristic
623C22#241812yes—
624C22#241912yes—
625C22#244312yes—
626C22#244412yes—
627C22C2×#41012yes—
628C22#288412yes—
629C4—22no—
630C4—22no—
631C4—24no—
632C4—24no—
633C4—42no—
634C4—42no—
635C4—44no—
636C4—42no—
637C4—84no—
638C4—84no—
639C4—82no—
640C4—82no—
641C4—82no—
642C4—162no—
643C4—162no—
644C2—22no—
645C2—22no—
646C2—22no—
647C2—21no—
648C2—21no—
649C2—21no—
650C2—22no—
651C2—21no—
652C2#1389411yescharacteristic
653C2#2068811yescharacteristic
654C2#2069111yescharacteristic
655C2#2240612yes—
656C2#2247812yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, prime radical, cosocleC25C24
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4⋊C4 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
18. C2×#389C4semidirect
19. C2×#387C4semidirect
20. C2×#387C4semidirect
27. C2×C4×C4⋊C4C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23442
C2×#18202
C2×#17402
#138702
#113532
#111972
#108012
#107362
#107262