Finite Group Databasebeta

← #198718

Subgroups — #198718

Order 512.

1557 conjugacy classes of subgroups.

The subgroup lattice has 728 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3187C211yesmaximal, characteristic
2#34205C211yesmaximal, characteristic
3C2×#1996C211yesmaximal, characteristic, complemented
4C2×#1520C211yesmaximal, characteristic, complemented
5C2×#1519C211yesmaximal, characteristic
6#9275C218yesmaximal, complemented
7#8023C211yesmaximal, characteristic, complemented
8C2×#1282C211yesmaximal, characteristic, complemented
9C22×C22⋊Q8C2211yescharacteristic
10C22×#121C2211yescharacteristic
11C2×#518—21no—
12#3187—22no—
13#3187—22no—
14#3187C414yescomplemented
15#3187C2214yes—
16C2×#481C412yescomplemented
17C2×#483—21no—
18C2×#465—22no—
19C2×#448C412yescomplemented
20C2×#450—21no—
21C22×C22⋊C8C2211yescharacteristic
22C22×C22⋊C8C2211yescharacteristic
23#2364C2214yes—
24#2361—22no—
25#2344—21no—
26#2342—22no—
27D4×C22⋊C4—21no—
28#2325—22no—
29#2308—21no—
30C4×#121—22no—
31C2×#380C2211yescharacteristic
32C2×#380C2211yescharacteristic
33C2×C4×C22⋊C4C2211yescharacteristic
34#1996—22no—
35#1996C2214yescomplemented
36C2×#342—21no—
37C2×#342—21no—
38#1520—22no—
39#1520—22no—
40#1519—22no—
41#1519C2214yes—
42C2×#328C2214yescomplemented
43C2×#327—22no—
44#1294C2214yescomplemented
45#1282C2214yescomplemented
46C23×Q8C2×C411yescharacteristic, complemented
47C23×D4—21no—
48C23×D4D411yescharacteristic
49C24×C4C2311yescharacteristic
50C22×SD16—22no—
51C23×C8—21no—
52C23×C8—21no—
53C2×C22⋊Q8—21no—
54C2×C22⋊Q8—21no—
55C2×C22⋊Q8—21no—
56C2×C22⋊Q8—21no—
57C2×C22⋊Q8—21no—
58C2×C22⋊Q8—21no—
59C2×C22⋊Q8—21no—
60C2×C22⋊Q8—21no—
61C2×C22⋊Q8—21no—
62C2×C22⋊Q8—21no—
63C2×C22⋊Q8C2×C412yescomplemented
64C2×C22⋊Q8C2312yes—
65C2×#121—21no—
66C2×#121—21no—
67C2×#121—21no—
68C2×#121—21no—
69C2×#121—21no—
70C2×#121—21no—
71C2×#121C2×C412yes—
72C2×#121C2×C412yes—
73C2×#121D412yes—
74C2×#121D412yes—
75C2×#121D412yes—
76C2×#121C2312yes—
77#518—44no—
78#518—44no—
79C2×C4×D4—24no—
80C2×C4×D4—22no—
81C2×C4×D4—24no—
82C2×#117—22no—
83C22×C4⋊C4—21no—
84C22×C4⋊C4—21no—
85C22×C4⋊C4—21no—
86C22×C4⋊C4C2×C411yescharacteristic, complemented
87C22×C4⋊C4D411yescharacteristic
88C22×C22⋊C4—21no—
89C22×C22⋊C4—22no—
90C22×C22⋊C4—21no—
91C22×C22⋊C4—21no—
92C22×C22⋊C4—21no—
93C22×C22⋊C4D411yescharacteristic
94C22×C22⋊C4D411yescharacteristic
95C22×C42—21no—
96#483—24no—
97#483—24no—
98#481—24no—
99#481C2×C418yescomplemented
100#465—28no—
101#465—28no—
102#450—24no—
103#450—24no—
104#448—24no—
105#448C2×C418yescomplemented
106C2×#106—22no—
107C2×#103—22no—
108C2×#102—22no—
109C2×C22⋊C8—21no—
110C2×C22⋊C8—21no—
111C2×C22⋊C8—21no—
112C2×C22⋊C8—21no—
113C2×C22⋊C8—21no—
114C2×C22⋊C8—21no—
115C2×C22⋊C8—21no—
116C2×C22⋊C8—21no—
117C2×C22⋊C8C2×C412yescomplemented
118C2×C22⋊C8C2×C412yescomplemented
119C2×C22⋊C8C2312yes—
120C2×C22⋊C8C2312yes—
121#390—22no—
122#389—22no—
123#388—24no—
124#388—24no—
125#387—24no—
126#386—24no—
127#386—24no—
128#386—24no—
129#385—24no—
130#381—22no—
131#380—21no—
132#380—21no—
133#380—21no—
134#380—21no—
135#380—21no—
136#380D412yes—
137#380D412yes—
138#380C2312yes—
139#379—22no—
140C4×C4⋊C4—22no—
141C4×C22⋊C4—21no—
142C4×C22⋊C4—21no—
143C4×C22⋊C4—24no—
144C4×C22⋊C4D412yes—
145C4×C22⋊C4C2312yes—
146C2×#95—21no—
147C2×#95—21no—
148C2×#95C2312yes—
149#342—24no—
150#342—24no—
151#328—28no—
152#327—28no—
153C25—21no—
154C25C2×D411yescharacteristic
155C22×Q8—21no—
156C22×Q8—21no—
157C22×Q8—21no—
158C22×Q8—21no—
159C22×Q8—21no—
160C22×Q8—82no—
161C22×Q8C22×C412yes—
162C22×D4—22no—
163C22×D4—22no—
164C22×D4—22no—
165C22×D4—22no—
166C22×D4—22no—
167C22×D4—22no—
168C22×D4—21no—
169C22×D4—21no—
170C22×D4—21no—
171C22×D4—42no—
172C22×D4—42no—
173C22×D4—42no—
174C22×D4—42no—
175C22×D4—82no—
176C22×D4C22⋊C412yes—
177C22×D4C22⋊C412yes—
178C22×D4C2×D412yes—
179C23×C4—21no—
180C23×C4—21no—
181C23×C4—24no—
182C23×C4—21no—
183C23×C4—21no—
184C23×C4—21no—
185C23×C4—22no—
186C23×C4—41no—
187C23×C4—82no—
188C23×C4C22×C411yescharacteristic
189C23×C4C22×C411yescharacteristic
190C23×C4C2×D411yescharacteristic
191C23×C4C2×D411yescharacteristic
192C23×C4C2×D411yescharacteristic
193C23×C4C2×D411yescharacteristic
194C23×C4C2×D411yescharacteristic
195C23×C4C2×D411yescharacteristic
196C23×C4C2×D411yescharacteristic
197C23×C4#4111yescharacteristic
198C23×C4#4111yescharacteristic
199C23×C4#4111yescharacteristic
200C23×C4#4111yescharacteristic
201C23×C4C2411yescharacteristic
202C2×SD16—28no—
203C2×SD16—84no—
204C22×C8—22no—
205C22×C8—22no—
206C22×C8—22no—
207C22×C8—22no—
208C22×C8—41no—
209C22×C8—41no—
210C22×C8—82no—
211C22×C8—82no—
212C22⋊Q8—22no—
213C22⋊Q8—22no—
214C22⋊Q8—22no—
215C22⋊Q8—22no—
216C22⋊Q8—22no—
217C22⋊Q8—22no—
218C22⋊Q8—22no—
219C22⋊Q8—22no—
220C22⋊Q8—41no—
221C22⋊Q8—41no—
222C22⋊Q8—41no—
223C22⋊Q8—41no—
224C22⋊Q8—41no—
225C22⋊Q8—41no—
226C22⋊Q8—41no—
227C22⋊Q8—41no—
228#121—22no—
229#121—22no—
230#121—22no—
231#121—22no—
232#121—22no—
233#121—22no—
234#121—22no—
235#121—22no—
236#121—22no—
237#121—22no—
238#121—22no—
239#121—22no—
240#121C22⋊C414yes—
241#121C22⋊C414yes—
242#121C22×C414yes—
243#121C2×D414yes—
244C4×D4—48no—
245C4×D4—416no—
246C4×D4—48no—
247C4×D4—84no—
248#117—48no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—41no—
264C2×C4⋊C4—41no—
265C2×C4⋊C4—41no—
266C2×C4⋊C4—41no—
267C2×C4⋊C4—41no—
268C2×C4⋊C4—82no—
269C2×C4⋊C4—82no—
270C2×C4⋊C4C22⋊C412yes—
271C2×C4⋊C4C22⋊C412yes—
272C2×C4⋊C4C22×C412yes—
273C2×C4⋊C4C2×D412yes—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—21no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—48no—
294C2×C22⋊C4—41no—
295C2×C22⋊C4—41no—
296C2×C22⋊C4—41no—
297C2×C22⋊C4—41no—
298C2×C22⋊C4—42no—
299C2×C22⋊C4—42no—
300C2×C22⋊C4—42no—
301C2×C22⋊C4—42no—
302C2×C22⋊C4—41no—
303C2×C22⋊C4SD1614yescomplemented
304C2×C22⋊C4C2×D412yes—
305C2×C22⋊C4C2×D412yes—
306C2×C42—24no—
307C2×C42—22no—
308C2×C42—22no—
309C2×C42—82no—
310#106—28no—
311#103—28no—
312#102—28no—
313C22⋊C8—22no—
314C22⋊C8—22no—
315C22⋊C8—22no—
316C22⋊C8—22no—
317C22⋊C8—22no—
318C22⋊C8—41no—
319C22⋊C8—41no—
320C22⋊C8C22×C414yes—
321#95—24no—
322#95—24no—
323#95—22no—
324#95—22no—
325#95—22no—
326#95—22no—
327#95—41no—
328#95C2×D414yes—
329C24—22no—
330C24—22no—
331C24—22no—
332C24—42no—
333C24—42no—
334C24—81no—
335C24—81no—
336C24—82no—
337C24—82no—
338C24C4×D411yescharacteristic
339C24C4×D411yescharacteristic
340C24#12011yescharacteristic
341C24#12011yescharacteristic
342C24#12111yescharacteristic
343C24#12111yescharacteristic
344C24C2×SD1612yes—
345C24#13512yes—
346C24AGL(1,ℤ/8)12yes—
347C24#13712yes—
348C24C22×D411yescharacteristic
349C2×Q8—22no—
350C2×Q8—22no—
351C2×Q8—44no—
352C2×Q8—41no—
353C2×Q8—41no—
354C2×Q8—82no—
355C2×Q8—82no—
356C2×Q8—82no—
357C2×Q8—82no—
358C2×Q8—82no—
359C2×D4—24no—
360C2×D4—24no—
361C2×D4—24no—
362C2×D4—24no—
363C2×D4—22no—
364C2×D4—22no—
365C2×D4—22no—
366C2×D4—44no—
367C2×D4—44no—
368C2×D4—42no—
369C2×D4—42no—
370C2×D4—42no—
371C2×D4—42no—
372C2×D4—44no—
373C2×D4—44no—
374C2×D4—44no—
375C2×D4—44no—
376C2×D4—44no—
377C2×D4—44no—
378C2×D4—44no—
379C2×D4—44no—
380C2×D4—44no—
381C2×D4—44no—
382C2×D4—82no—
383C2×D4—82no—
384C2×D4—82no—
385C2×D4—82no—
386C2×D4C2×C22⋊C414yes—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—48no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—81no—
415C22×C4—81no—
416C22×C4—81no—
417C22×C4—81no—
418C22×C4—81no—
419C22×C4—81no—
420C22×C4—81no—
421C22×C4—81no—
422C22×C4—81no—
423C22×C4—81no—
424C22×C4—81no—
425C22×C4—82no—
426C22×C4—84no—
427C22×C4—84no—
428C22×C4—82no—
429C22×C4—82no—
430C22×C4—82no—
431C22×C4—82no—
432C22×C4—82no—
433C22×C4—82no—
434C22×C4C2×C22⋊C411yescharacteristic
435C22×C4C2×C22⋊C411yescharacteristic
436C22×C4C2×C22⋊C411yescharacteristic
437C22×C4C2×C22⋊C411yescharacteristic
438C22×C4C4×D411yescharacteristic
439C22×C4C4×D411yescharacteristic
440C22×C4C4×D411yescharacteristic
441C22×C4C4×D411yescharacteristic
442C22×C4C4×D411yescharacteristic
443C22×C4C4×D411yescharacteristic
444C22×C4#12011yescharacteristic
445C22×C4#12011yescharacteristic
446C22×C4#12111yescharacteristic
447C22×C4#12111yescharacteristic
448C22×C4#12111yescharacteristic
449C22×C4#12111yescharacteristic
450C22×C4#12111yescharacteristic
451C22×C4#12111yescharacteristic
452C22×C4#12311yescharacteristic
453C22×C4#12311yescharacteristic
454C22×C4#12311yescharacteristic
455C22×C4#12311yescharacteristic
456C22×C4C23×C411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D411yescharacteristic
460C22×C4C2×#4111yescharacteristic
461C22×C4C2×#4111yescharacteristic
462SD16—88no—
463C2×C8—24no—
464C2×C8—44no—
465C2×C8—42no—
466C2×C8—82no—
467C2×C8—82no—
468C2×C8—82no—
469C2×C8—82no—
470C2×C8—82no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—44no—
477C4⋊C4—48no—
478C4⋊C4—42no—
479C4⋊C4—42no—
480C4⋊C4—41no—
481C4⋊C4—41no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—82no—
485C4⋊C4—82no—
486C4⋊C4—82no—
487C4⋊C4C2×C22⋊C414yes—
488C22⋊C4—22no—
489C22⋊C4—24no—
490C22⋊C4—24no—
491C22⋊C4—24no—
492C22⋊C4—24no—
493C22⋊C4—24no—
494C22⋊C4—22no—
495C22⋊C4—22no—
496C22⋊C4—48no—
497C22⋊C4—42no—
498C22⋊C4—42no—
499C22⋊C4—42no—
500C22⋊C4—42no—
501C22⋊C4—44no—
502C22⋊C4—44no—
503C22⋊C4—42no—
504C22⋊C4—42no—
505C22⋊C4—42no—
506C22⋊C4—84no—
507C22⋊C4—84no—
508C22⋊C4—82no—
509C22⋊C4C2×SD1614yescomplemented
510C42—44no—
511C42—48no—
512C42—82no—
513C23—24no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23—44no—
529C23—44no—
530C23—42no—
531C23—42no—
532C23—44no—
533C23—44no—
534C23—82no—
535C23—82no—
536C23—82no—
537C23—82no—
538C23—82no—
539C23—82no—
540C23—82no—
541C23—82no—
542C23—164no—
543C23—164no—
544C23#38611yescharacteristic
545C23#38611yescharacteristic
546C23#38611yescharacteristic
547C23#38611yescharacteristic
548C23C4×SD1612yes—
549C23C4×SD1612yes—
550C23#44012yes—
551C23#44012yes—
552C23#45012yes—
553C23#45012yes—
554C23#45212yes—
555C23#45212yes—
556C23#46112yes—
557C23#46112yes—
558C23#46312yes—
559C23#46312yes—
560C23C2×C4×D411yescharacteristic
561C23C2×#12011yescharacteristic
562C23C2×#12111yescharacteristic
563C23C22×SD1611yescharacteristic
564C23C2×#13511yescharacteristic
565C23C2×AGL(1,ℤ/8)11yescharacteristic
566C23C2×#13711yescharacteristic
567Q8—44no—
568Q8—82no—
569Q8—82no—
570Q8—82no—
571D4—44no—
572D4—44no—
573D4—48no—
574D4—48no—
575D4—44no—
576D4—44no—
577D4—44no—
578D4—44no—
579D4—44no—
580D4—82no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—84no—
626C2×C4—82no—
627C2×C4—82no—
628C2×C4—82no—
629C2×C4—82no—
630C2×C4—84no—
631C2×C4—82no—
632C2×C4—82no—
633C2×C4—82no—
634C2×C4—82no—
635C2×C4—82no—
636C2×C4—168no—
637C2×C4—162no—
638C2×C4—162no—
639C2×C4—164no—
640C2×C4#38611yescharacteristic
641C2×C4#38611yescharacteristic
642C2×C4#38611yescharacteristic
643C2×C4#38611yescharacteristic
644C2×C4C22×C22⋊C411yescharacteristic
645C2×C4C2×C4×D411yescharacteristic
646C2×C4C2×#12111yescharacteristic
647C2×C4C2×#12311yescharacteristic
648C8—44no—
649C8—82no—
650C22—21no—
651C22—21no—
652C22—21no—
653C22—21no—
654C22—21no—
655C22—21no—
656C22—21no—
657C22—21no—
658C22—21no—
659C22—21no—
660C22—21no—
661C22—21no—
662C22—21no—
663C22—21no—
664C22—42no—
665C22—42no—
666C22—41no—
667C22—41no—
668C22—42no—
669C22—42no—
670C22—42no—
671C22—41no—
672C22—41no—
673C22—42no—
674C22—42no—
675C22—42no—
676C22—42no—
677C22—42no—
678C22—42no—
679C22—42no—
680C22—44no—
681C22—44no—
682C22—44no—
683C22—84no—
684C22—84no—
685C22—164no—
686C22—164no—
687C22#187712yes—
688C22#187712yes—
689C22#187712yes—
690C22#187712yes—
691C22C2×#38611yescharacteristic
692C22C2×C4×SD1611yescharacteristic
693C22C2×#44011yescharacteristic
694C22C2×#45011yescharacteristic
695C22C2×#45211yescharacteristic
696C22C2×#46111yescharacteristic
697C22C2×#46311yescharacteristic
698C4—21no—
699C4—21no—
700C4—21no—
701C4—21no—
702C4—21no—
703C4—21no—
704C4—21no—
705C4—44no—
706C4—44no—
707C4—82no—
708C4—84no—
709C4—82no—
710C4—82no—
711C4—162no—
712C4#191312yes—
713C2—21no—
714C2—21no—
715C2—21no—
716C2—22no—
717C2—22no—
718C2—22no—
719C2—22no—
720C2—21no—
721C2—21no—
722C2—21no—
723C2—84no—
724C2C2×#187711yescharacteristic
725C2#2100011yescharacteristic
726C2C2×#191311yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1996C2semidirect
4. C2×#1520C2semidirect
6. #9275C2semidirect
7. #8023C2semidirect
8. C2×#1282C2semidirect
14. #3187C4semidirect
16. C2×#481C4semidirect
19. C2×#448C4semidirect
35. #1996C22semidirect
42. C2×#328C22semidirect
44. #1294C22semidirect
45. #1282C22semidirect
46. C23×Q8C2×C4semidirect
63. C2×C22⋊Q8C2×C4semidirect
86. C22×C4⋊C4C2×C4semidirect
99. #481C2×C4semidirect
105. #448C2×C4semidirect
117. C2×C22⋊C8C2×C4semidirect
118. C2×C22⋊C8C2×C4semidirect
303. C2×C22⋊C4SD16semidirect
509. C22⋊C4C2×SD16semidirect

Maximal subgroups

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

subgroupindex
C2×#31872
#342052
C2×#19962
C2×#15202
C2×#15192
#92752
#80232
C2×#12822