Finite Group Databasebeta

← #358473

Subgroups — #358473

Order 512.

1998 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2911C211yesmaximal, characteristic
2C22×#428C211yesmaximal, characteristic
3C22×#428C211yesmaximal, characteristic
4C2×#1742C211yesmaximal, characteristic
5C22×#343C211yesmaximal, characteristic
6C2×#1738C211yesmaximal, characteristic
7#20335C211yesmaximal, characteristic
8#11331C214yesmaximal, complemented
9#11330C214yesmaximal, complemented
10C23×M16C2211yescharacteristic
11C23×M16C2211yescharacteristic
12C23×M16C2211yescharacteristic
13C23×M16C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C22×#117C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17#2911—24no—
18#2911—24no—
19C2×#428—24no—
20C2×#428—22no—
21C2×#428—22no—
22C2×#428—22no—
23C2×#428—24no—
24C2×#428—24no—
25C2×#428—24no—
26C2×#428—21no—
27C2×#428—21no—
28C2×#428C2212yes—
29C2×#428C2214yes—
30C2×#428C2212yes—
31C2×#427—22no—
32C22×#107—21no—
33C22×#107C2212yes—
34C22×#106—21no—
35C22×#106C2212yes—
36#1742—24no—
37C2×#343—24no—
38C2×#343C2212yes—
39C2×#343C2214yes—
40C2×#343C2212yes—
41#1738C418yescomplemented
42C2×#341C2214yes—
43#1729—22no—
44#1729C2214yes—
45#1728—22no—
46C24×C4—21no—
47C24×C4C2311yescharacteristic
48C22×M16—22no—
49C22×M16—21no—
50C22×M16—21no—
51C22×M16—22no—
52C22×M16—22no—
53C22×M16—22no—
54C22×M16—22no—
55C22×M16—22no—
56C22×M16—22no—
57C22×M16—22no—
58C22×M16—22no—
59C22×M16—22no—
60C22×M16—21no—
61C22×M16—22no—
62C22×M16—22no—
63C22×M16—22no—
64C22×M16—22no—
65C22×M16—22no—
66C22×M16C2×C414yes—
67C22×M16C2×C414yes—
68C22×M16C2×C414yes—
69C22×M16D414yes—
70C22×M16D414yes—
71C22×M16D412yes—
72C22×M16D412yes—
73C22×M16Q814yes—
74C22×M16Q814yes—
75C22×M16Q814yes—
76C22×M16Q814yes—
77C22×M16C2312yes—
78C22×M16C2312yes—
79C22×M16C2312yes—
80C23×C8—21no—
81C23×C8—21no—
82C23×C8—21no—
83C23×C8C2312yes—
84C2×#117—21no—
85C2×#117—22no—
86C2×#117—21no—
87C2×#117—21no—
88C2×#117—22no—
89C2×#117—22no—
90C2×#117—21no—
91C2×#117—21no—
92C2×#117—21no—
93C2×#117—22no—
94C2×#117—22no—
95C2×#117—22no—
96C2×#117—22no—
97C2×#117D412yes—
98C2×#117D414yes—
99C2×#117C2312yes—
100C22×C4⋊C4—21no—
101C22×C4⋊C4—22no—
102C22×C4⋊C4—21no—
103C22×C4⋊C4—22no—
104C22×C4⋊C4—21no—
105C22×C4⋊C4—21no—
106C22×C4⋊C4—22no—
107C22×C4⋊C4—21no—
108C22×C4⋊C4D411yescharacteristic
109C22×C4⋊C4D411yescharacteristic
110C22×C4⋊C4D414yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2311yescharacteristic
114C22×C22⋊C4—21no—
115C22×C22⋊C4—21no—
116C22×C42—21no—
117C22×C42—21no—
118#428—216no—
119#428—216no—
120#428—216no—
121#428—28no—
122#428—28no—
123#428—28no—
124#428—24no—
125#428—24no—
126#428—28no—
127#428—24no—
128#428—24no—
129#427—216no—
130C2×#107—28no—
131C2×#107—24no—
132C2×#107—24no—
133C2×#107—24no—
134C2×#107—24no—
135C2×#107—22no—
136C2×#107—22no—
137C2×#106—28no—
138C2×#106—24no—
139C2×#106—24no—
140C2×#106—24no—
141C2×#106—24no—
142C2×#106—22no—
143C2×#106—22no—
144#343—28no—
145#343—24no—
146#343—24no—
147#341C2×C4116yescomplemented
148#336—28no—
149C25#4111yescharacteristic
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—24no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—22no—
156C23×C4—42no—
157C23×C4—41no—
158C23×C4—41no—
159C23×C4—84no—
160C23×C4C22×C412yes—
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D412yes—
163C23×C4C2×D412yes—
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D412yes—
168C23×C4C2×D412yes—
169C23×C4C2×Q812yes—
170C23×C4C2×Q812yes—
171C23×C4C2×Q812yes—
172C23×C4C2×Q812yes—
173C23×C4#4112yes—
174C23×C4#4112yes—
175C23×C4#4112yes—
176C23×C4#4111yescharacteristic
177C23×C4C2411yescharacteristic
178C2×M16—24no—
179C2×M16—24no—
180C2×M16—28no—
181C2×M16—28no—
182C2×M16—28no—
183C2×M16—28no—
184C2×M16—24no—
185C2×M16—24no—
186C2×M16—24no—
187C2×M16—24no—
188C2×M16—24no—
189C2×M16—24no—
190C2×M16—28no—
191C2×M16—28no—
192C2×M16—28no—
193C2×M16—28no—
194C2×M16—24no—
195C2×M16—24no—
196C2×M16—28no—
197C2×M16—28no—
198C2×M16—22no—
199C2×M16—22no—
200C2×M16—22no—
201C2×M16—22no—
202C2×M16—22no—
203C2×M16—22no—
204C2×M16—24no—
205C2×M16—22no—
206C2×M16—28no—
207C2×M16—28no—
208C2×M16—22no—
209C2×M16—22no—
210C2×M16—22no—
211C2×M16—22no—
212C2×M16—42no—
213C2×M16—44no—
214C2×M16—44no—
215C2×M16—42no—
216C2×M16C4⋊C4116yes—
217C2×M16C4⋊C4116yes—
218C2×M16C4⋊C418yes—
219C2×M16C4⋊C418yes—
220C2×M16C22×C414yes—
221C2×M16C22×C414yes—
222C2×M16C2×D414yes—
223C2×M16C2×D414yes—
224C2×M16C2×Q814yes—
225C2×M16C2×Q814yes—
226C2×M16C2×Q814yes—
227C22×C8—24no—
228C22×C8—24no—
229C22×C8—24no—
230C22×C8—24no—
231C22×C8—24no—
232C22×C8—24no—
233C22×C8—22no—
234C22×C8—22no—
235C22×C8—22no—
236C22×C8—22no—
237C22×C8—22no—
238C22×C8—22no—
239C22×C8—24no—
240C22×C8—22no—
241C22×C8—42no—
242C22×C8—42no—
243C22×C8C22×C414yes—
244C22×C8C2×D414yes—
245C22×C8C2×Q814yes—
246#117—24no—
247#117—24no—
248#117—24no—
249#117—24no—
250#117—24no—
251#117—24no—
252#117—22no—
253#117—22no—
254#117—22no—
255#117—22no—
256#117—42no—
257#117—42no—
258#117—42no—
259#117—42no—
260#117—42no—
261#117—42no—
262#117—42no—
263#117—42no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—42no—
291C2×C4⋊C4—42no—
292C2×C4⋊C4—42no—
293C2×C4⋊C4—41no—
294C2×C4⋊C4—41no—
295C2×C4⋊C4—42no—
296C2×C4⋊C4—42no—
297C2×C4⋊C4C2×D414yes—
298C2×C4⋊C4C2×D412yes—
299C2×C4⋊C4C2×D412yes—
300C2×C4⋊C4C2×D412yes—
301C2×C4⋊C4C2×D412yes—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—42no—
306C2×C22⋊C4—42no—
307C2×C22⋊C4—42no—
308C2×C22⋊C4—41no—
309C2×C22⋊C4—41no—
310C2×C42—22no—
311C2×C42—24no—
312C2×C42—22no—
313C2×C42—42no—
314C2×C42—42no—
315C2×C42—41no—
316C2×C42—41no—
317C2×C42—42no—
318#107—216no—
319#107—216no—
320#107—28no—
321#107—28no—
322#106—216no—
323#106—216no—
324#106—28no—
325#106—28no—
326C24—42no—
327C24—42no—
328C24C4×Q812yes—
329C24#12111yescharacteristic
330C24#12111yescharacteristic
331C24#12112yes—
332C24#12112yes—
333C24#12512yes—
334C24#12512yes—
335C24#12512yes—
336C24C2×#4111yescharacteristic
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—24no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—42no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—44no—
351C22×C4—44no—
352C22×C4—82no—
353C22×C4—82no—
354C22×C4—84no—
355C22×C4—84no—
356C22×C4—88no—
357C22×C4—82no—
358C22×C4—82no—
359C22×C4—84no—
360C22×C4—88no—
361C22×C4C2×C4⋊C412yes—
362C22×C4C2×C4⋊C412yes—
363C22×C4C2×C4⋊C414yes—
364C22×C4C2×C4⋊C414yes—
365C22×C4C4×D412yes—
366C22×C4C4×D412yes—
367C22×C4C4×D414yes—
368C22×C4C4×D414yes—
369C22×C4C4×Q812yes—
370C22×C4#12012yes—
371C22×C4#12012yes—
372C22×C4#12012yes—
373C22×C4#12012yes—
374C22×C4#12012yes—
375C22×C4#12012yes—
376C22×C4#12112yes—
377C22×C4#12112yes—
378C22×C4#12111yescharacteristic
379C22×C4#12111yescharacteristic
380C22×C4C22⋊Q812yes—
381C22×C4C22⋊Q812yes—
382C22×C4C22⋊Q812yes—
383C22×C4C22⋊Q812yes—
384C22×C4C22⋊Q814yes—
385C22×C4C22⋊Q814yes—
386C22×C4C22⋊Q814yes—
387C22×C4C22⋊Q814yes—
388C22×C4C22⋊Q812yes—
389C22×C4C22⋊Q812yes—
390C22×C4C22⋊Q814yes—
391C22×C4C22⋊Q814yes—
392C22×C4#12312yes—
393C22×C4#12312yes—
394C22×C4#12314yes—
395C22×C4#12314yes—
396C22×C4#12512yes—
397C22×C4#12512yes—
398C22×C4#12512yes—
399C22×C4#12712yes—
400C22×C4#12712yes—
401C22×C4C4⋊Q812yes—
402C22×C4C4⋊Q812yes—
403C22×C4C4⋊Q814yes—
404C22×C4C4⋊Q814yes—
405C22×C4C4⋊Q812yes—
406C22×C4C4⋊Q812yes—
407C22×C4C4⋊Q814yes—
408C22×C4C4⋊Q814yes—
409C22×C4C23×C411yescharacteristic
410C22×C4C22×D411yescharacteristic
411C22×C4C22×D411yescharacteristic
412C22×C4C22×D411yescharacteristic
413C22×C4C22×D411yescharacteristic
414C22×C4C22×D411yescharacteristic
415C22×C4C22×D411yescharacteristic
416C22×C4C22×Q811yescharacteristic
417C22×C4C22×Q811yescharacteristic
418C22×C4C22×Q811yescharacteristic
419C22×C4C22×Q811yescharacteristic
420C22×C4C2×#4111yescharacteristic
421C22×C4C2×#4111yescharacteristic
422C22×C4C2×#4111yescharacteristic
423M16—24no—
424M16—24no—
425M16—24no—
426M16—24no—
427M16—24no—
428M16—24no—
429M16—216no—
430M16—216no—
431M16—216no—
432M16—216no—
433M16—24no—
434M16—24no—
435M16—42no—
436M16—42no—
437M16—48no—
438M16C2×C4⋊C418yes—
439M16C2×C4⋊C418yes—
440C2×C8—24no—
441C2×C8—24no—
442C2×C8—24no—
443C2×C8—24no—
444C2×C8—24no—
445C2×C8—24no—
446C2×C8—24no—
447C2×C8—24no—
448C2×C8—24no—
449C2×C8—24no—
450C2×C8—24no—
451C2×C8—24no—
452C2×C8—216no—
453C2×C8—216no—
454C2×C8—24no—
455C2×C8—42no—
456C2×C8—42no—
457C2×C8—48no—
458C2×C8C2×C4⋊C418yes—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—24no—
462C4⋊C4—24no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—42no—
470C4⋊C4—41no—
471C4⋊C4—41no—
472C4⋊C4—42no—
473C4⋊C4—41no—
474C4⋊C4—41no—
475C4⋊C4—44no—
476C4⋊C4—44no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C22⋊C4—42no—
480C22⋊C4—42no—
481C22⋊C4—44no—
482C22⋊C4—44no—
483C22⋊C4—44no—
484C42—42no—
485C42—42no—
486C42—44no—
487C42—44no—
488C42—44no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23—42no—
496C23—42no—
497C23—44no—
498C23—42no—
499C23—42no—
500C23—44no—
501C23—44no—
502C23—44no—
503C23#38912yes—
504C23#38912yes—
505C23#38914yes—
506C23#38914yes—
507C23#39212yes—
508C23#39212yes—
509C23#39812yes—
510C23#39812yes—
511C23#39814yes—
512C23#39814yes—
513C23C2×C4×Q811yescharacteristic
514C23C2×#12111yescharacteristic
515C23C2×#12111yescharacteristic
516C23C2×#12111yescharacteristic
517C23C2×#12511yescharacteristic
518C23C2×#12511yescharacteristic
519C23C2×#12511yescharacteristic
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—84no—
528C2×C4—82no—
529C2×C4—84no—
530C2×C4—82no—
531C2×C4—88no—
532C2×C4—84no—
533C2×C4—88no—
534C2×C4—82no—
535C2×C4—84no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4—84no—
540C2×C4#38414yes—
541C2×C4#38414yes—
542C2×C4#38514yes—
543C2×C4#38514yes—
544C2×C4#38514yes—
545C2×C4#38514yes—
546C2×C4#38514yes—
547C2×C4#38514yes—
548C2×C4#38914yes—
549C2×C4#38914yes—
550C2×C4#38912yes—
551C2×C4#38912yes—
552C2×C4#39212yes—
553C2×C4#39212yes—
554C2×C4#39514yes—
555C2×C4#39514yes—
556C2×C4#39514yes—
557C2×C4#39514yes—
558C2×C4#39514yes—
559C2×C4#39514yes—
560C2×C4#39814yes—
561C2×C4#39814yes—
562C2×C4#39812yes—
563C2×C4#39812yes—
564C2×C4#39914yes—
565C2×C4#39914yes—
566C2×C4C22×C4⋊C411yescharacteristic
567C2×C4C22×C4⋊C411yescharacteristic
568C2×C4C22×C4⋊C411yescharacteristic
569C2×C4C2×C4×D411yescharacteristic
570C2×C4C2×C4×D411yescharacteristic
571C2×C4C2×C4×D411yescharacteristic
572C2×C4C2×#12011yescharacteristic
573C2×C4C2×#12011yescharacteristic
574C2×C4C2×#12011yescharacteristic
575C2×C4C2×C22⋊Q811yescharacteristic
576C2×C4C2×C22⋊Q811yescharacteristic
577C2×C4C2×C22⋊Q811yescharacteristic
578C2×C4C2×C22⋊Q811yescharacteristic
579C2×C4C2×C22⋊Q811yescharacteristic
580C2×C4C2×C22⋊Q811yescharacteristic
581C2×C4C2×C22⋊Q811yescharacteristic
582C2×C4C2×C22⋊Q811yescharacteristic
583C2×C4C2×C22⋊Q811yescharacteristic
584C2×C4C2×#12311yescharacteristic
585C2×C4C2×#12311yescharacteristic
586C2×C4C2×#12311yescharacteristic
587C2×C4C2×#12711yescharacteristic
588C2×C4C2×C4⋊Q811yescharacteristic
589C2×C4C2×C4⋊Q811yescharacteristic
590C2×C4C2×C4⋊Q811yescharacteristic
591C2×C4C2×C4⋊Q811yescharacteristic
592C2×C4C2×C4⋊Q811yescharacteristic
593C2×C4C2×C4⋊Q811yescharacteristic
594C8—28no—
595C8—28no—
596C8—28no—
597C8—44no—
598C22—22no—
599C22—22no—
600C22—24no—
601C22—24no—
602C22—21no—
603C22—21no—
604C22—22no—
605C22—22no—
606C22—21no—
607C22—21no—
608C22—22no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—24no—
613C22—24no—
614C22—44no—
615C22—44no—
616C22—44no—
617C22—44no—
618C22#144414yes—
619C22#144414yes—
620C22C2×#38911yescharacteristic
621C22C2×#38911yescharacteristic
622C22C2×#38911yescharacteristic
623C22C2×#39211yescharacteristic
624C22C2×#39811yescharacteristic
625C22C2×#39811yescharacteristic
626C22C2×#39811yescharacteristic
627C22#306812yes—
628C22#306912yes—
629C4—82no—
630C4—82no—
631C4—84no—
632C4—82no—
633C4—82no—
634C4#144414yes—
635C4#144414yes—
636C4C2×#38411yescharacteristic
637C4C2×#38511yescharacteristic
638C4C2×#38511yescharacteristic
639C4C2×#38511yescharacteristic
640C4C2×#39511yescharacteristic
641C4C2×#39511yescharacteristic
642C4C2×#39511yescharacteristic
643C4C2×#39911yescharacteristic
644C2—21no—
645C2—21no—
646C2—24no—
647C2—24no—
648C2—21no—
649C2—21no—
650C2C2×#144411yescharacteristic
651C2#2118212yes—
652C2#2343911yescharacteristic
653C2#2344011yescharacteristic
654C2#2367512yes—
subgroupG/H
center, socleC23C2×#121
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25#41
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
8. #11331C2semidirect
9. #11330C2semidirect
41. #1738C4semidirect
147. #341C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#29112
C22×#4282
C2×#17422
C22×#3432
C2×#17382
#203352
#113312
#113302