Finite Group Databasebeta

← #365604

Subgroups — #365604

Order 512.

1536 conjugacy classes of subgroups.

The subgroup lattice has 757 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×Q16C211yesmaximal, characteristic
2C2×#1971C211yesmaximal, characteristic
3C2×#1971C211yesmaximal, characteristic
4C2×#1941C211yesmaximal, characteristic
5C2×#1865C211yesmaximal, characteristic
6C2×#1863C211yesmaximal, characteristic
7C2×#1847C211yesmaximal, characteristic
8#13978C211yesmaximal, characteristic
9#13976C211yesmaximal, characteristic
10#11124C211yesmaximal, characteristic
11#11122C211yesmaximal, characteristic
12#11052C211yesmaximal, characteristic
13#11050C211yesmaximal, characteristic
14#10863C211yesmaximal, characteristic
15#10696C211yesmaximal, characteristic
16C23×Q16C2211yescharacteristic
17C22×C4×Q8C2211yescharacteristic
18C22×C4×Q8C2211yescharacteristic
19C2×C4×Q16—24no—
20C2×C4×Q16—24no—
21C2×C4×Q16—24no—
22C22×#107C2211yescharacteristic
23C22×#107C2211yescharacteristic
24C22×#106C2211yescharacteristic
25C22×#106C2211yescharacteristic
26C22×#106C2211yescharacteristic
27C22×#103C2211yescharacteristic
28C22×#103C2211yescharacteristic
29C22×#103C2211yescharacteristic
30C22×#103C2211yescharacteristic
31C22×#103C2211yescharacteristic
32C22×#103C2211yescharacteristic
33C22×#97C2211yescharacteristic
34C22×#97C2211yescharacteristic
35C22×C4×C8C2211yescharacteristic
36C2×#391C2211yescharacteristic
37C2×#391C2211yescharacteristic
38C2×#391C2211yescharacteristic
39C2×#391C2211yescharacteristic
40C2×#389C2211yescharacteristic
41C2×#389C2211yescharacteristic
42C2×#384C2211yescharacteristic
43C2×#382C2211yescharacteristic
44#1971—24no—
45#1971C418yescomplemented
46#1941—24no—
47#1865—24no—
48#1863—24no—
49#1847—24no—
50C2×#340C2211yescharacteristic
51C2×#340C2211yescharacteristic
52C2×#340C2211yescharacteristic
53C2×#340C2211yescharacteristic
54C2×#336C2211yescharacteristic
55C2×#336C2211yescharacteristic
56C2×#336C2211yescharacteristic
57#1444—22no—
58#1442—22no—
59#1436—22no—
60#1436—22no—
61C4⋊#95C2211yescharacteristic
62C4⋊#95C2211yescharacteristic
63C4×#95C2211yescharacteristic
64C23×Q8C2311yescharacteristic
65C23×Q8C2311yescharacteristic
66C22×Q16—22no—
67C22×Q16—22no—
68C22×Q16—22no—
69C22×Q16—22no—
70C22×Q16C2×C414yes—
71C22×Q16D414yes—
72C22×Q16Q814yes—
73C23×C8C2311yescharacteristic
74C23×C8C2311yescharacteristic
75C23×C8C2311yescharacteristic
76C23×C8C2311yescharacteristic
77C2×C4×Q8—22no—
78C2×C4×Q8—22no—
79C2×C4×Q8—22no—
80C2×C4×Q8—22no—
81C2×C4×Q8—22no—
82C2×C4×Q8—22no—
83C2×C4×Q8—22no—
84C2×C4×Q8—22no—
85C2×C4×Q8—41no—
86C2×C4×Q8—41no—
87C2×C4×Q8—41no—
88C2×C4×Q8—41no—
89C22×C4⋊C4—21no—
90C22×C4⋊C4—21no—
91C22×C4⋊C4—21no—
92C22×C4⋊C4—22no—
93C22×C4⋊C4—21no—
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2311yescharacteristic
100C22×C42—21no—
101C22×C42—21no—
102C22×C42C2311yescharacteristic
103C22×C42C2311yescharacteristic
104C22×C42C2311yescharacteristic
105C4×Q16—232no—
106C2×#107—22no—
107C2×#107—22no—
108C2×#107—22no—
109C2×#107—22no—
110C2×#107—41no—
111C2×#107D414yes—
112C2×#106—22no—
113C2×#106—22no—
114C2×#106—22no—
115C2×#106—22no—
116C2×#106—22no—
117C2×#106—22no—
118C2×#106—22no—
119C2×#106—41no—
120C2×#106C2×C414yes—
121C2×#103—22no—
122C2×#103—22no—
123C2×#103—22no—
124C2×#103—22no—
125C2×#103—22no—
126C2×#103—22no—
127C2×#103—22no—
128C2×#103—22no—
129C2×#103—22no—
130C2×#103—22no—
131C2×#103—22no—
132C2×#103—22no—
133C2×#103—22no—
134C2×#103—22no—
135C2×#103—41no—
136C2×#103—41no—
137C2×#103C2×C414yes—
138C2×#103C2×C414yes—
139C2×#97—22no—
140C2×#97—22no—
141C2×#97—22no—
142C2×#97—22no—
143C2×#97—22no—
144C2×#97C2×C414yes—
145C2×C4×C8—22no—
146C2×C4×C8—22no—
147C2×C4×C8D414yes—
148#391—22no—
149#391—22no—
150#391—22no—
151#391—22no—
152#391—41no—
153#391—41no—
154#391C2×C414yes—
155#391C2×C414yes—
156#389—22no—
157#389—22no—
158#389—22no—
159#389—22no—
160#384—22no—
161#384—41no—
162#382—22no—
163#382—41no—
164C2×#95—22no—
165C2×#95—22no—
166C2×#95—22no—
167C2×#95—21no—
168C2×#95—22no—
169C2×#95—22no—
170C2×#95—22no—
171C2×#95—21no—
172C2×#95—22no—
173C2×#95D412yes—
174C2×#95D412yes—
175#340—22no—
176#340—22no—
177#340—22no—
178#340—22no—
179#336—22no—
180#336—22no—
181#336D414yes—
182C22×Q8—21no—
183C22×Q8—21no—
184C22×Q8—21no—
185C22×Q8—21no—
186C22×Q8—21no—
187C22×Q8—21no—
188C22×Q8—21no—
189C22×Q8—21no—
190C22×Q8—21no—
191C22×Q8—21no—
192C22×Q8—21no—
193C22×Q8—21no—
194C22×Q8—21no—
195C22×Q8—21no—
196C22×Q8—21no—
197C22×Q8—21no—
198C22×Q8—21no—
199C22×Q8—21no—
200C22×Q8—21no—
201C22×Q8—21no—
202C22×Q8—21no—
203C22×Q8—21no—
204C22×Q8C22×C412yes—
205C22×Q8C22×C412yes—
206C22×Q8C2×D412yes—
207C22×Q8C2×D412yes—
208C22×Q8C2×Q812yes—
209C22×Q8C2×Q812yes—
210C23×C4—21no—
211C23×C4—21no—
212C23×C4—21no—
213C23×C4—21no—
214C23×C4—21no—
215C23×C4—21no—
216C23×C4—21no—
217C23×C4—21no—
218C23×C4C2×D412yes—
219C23×C4C2×D412yes—
220C23×C4C2×D411yescharacteristic
221C23×C4C2×D411yescharacteristic
222C23×C4C2411yescharacteristic
223C2×Q16—28no—
224C2×Q16—28no—
225C2×Q16—28no—
226C2×Q16—28no—
227C2×Q16—28no—
228C2×Q16—28no—
229C2×Q16C4⋊C4116yescomplemented
230C22×C8—21no—
231C22×C8—21no—
232C22×C8—21no—
233C22×C8—21no—
234C22×C8—21no—
235C22×C8—21no—
236C22×C8—21no—
237C22×C8—21no—
238C22×C8—21no—
239C22×C8—21no—
240C22×C8—21no—
241C22×C8—21no—
242C22×C8—21no—
243C22×C8—21no—
244C22×C8—21no—
245C22×C8—21no—
246C22×C8C22×C412yes—
247C22×C8C22×C412yes—
248C22×C8C2×D412yes—
249C22×C8C2×D412yes—
250C22×C8C2×D412yes—
251C22×C8C2×D412yes—
252C22×C8C2×D412yes—
253C22×C8C2×D412yes—
254C22×C8C2×Q812yes—
255C22×C8#4112yes—
256C22×C8#4112yes—
257C22×C8#4112yes—
258C4×Q8—28no—
259C4×Q8—28no—
260C4×Q8—44no—
261C4×Q8—44no—
262C4×Q8—44no—
263C4×Q8—44no—
264C4×Q8—44no—
265C4×Q8—44no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
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—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—42no—
299C2×C4⋊C4—42no—
300C2×C4⋊C4—42no—
301C2×C4⋊C4—42no—
302C2×C4⋊C4—42no—
303C2×C4⋊C4—42no—
304C2×C4⋊C4—44no—
305C2×C4⋊C4—44no—
306C2×C4⋊C4—42no—
307C2×C4⋊C4—42no—
308C2×C4⋊C4—44no—
309C2×C4⋊C4—82no—
310C2×C4⋊C4—82no—
311C2×C4⋊C4C22×C412yes—
312C2×C4⋊C4C22×C412yes—
313C2×C4⋊C4C2×D412yes—
314C2×C4⋊C4C2×D412yes—
315C2×C42—21no—
316C2×C42—21no—
317C2×C42—21no—
318C2×C42—21no—
319C2×C42—21no—
320C2×C42—21no—
321C2×C42—21no—
322C2×C42—21no—
323C2×C42—21no—
324C2×C42—21no—
325C2×C42—21no—
326C2×C42—21no—
327C2×C42—21no—
328C2×C42—21no—
329C2×C42—42no—
330C2×C42—42no—
331C2×C42—42no—
332C2×C42—42no—
333C2×C42—42no—
334C2×C42—42no—
335C2×C42C22×C412yes—
336C2×C42C2×D412yes—
337C2×C42C2×D412yes—
338C2×C42#4112yes—
339#107—28no—
340#107—44no—
341#106—28no—
342#106—44no—
343#106—44no—
344#103—28no—
345#103—28no—
346#103—28no—
347#103—28no—
348#103—44no—
349#103—44no—
350#97—28no—
351#97—44no—
352C4×C8—28no—
353#95—28no—
354#95—28no—
355#95—24no—
356#95—42no—
357#95—44no—
358#95—44no—
359#95—44no—
360#95—44no—
361#95—42no—
362#95—42no—
363#95—82no—
364C24C22×D411yescharacteristic
365C2×Q8—22no—
366C2×Q8—22no—
367C2×Q8—22no—
368C2×Q8—22no—
369C2×Q8—22no—
370C2×Q8—22no—
371C2×Q8—22no—
372C2×Q8—22no—
373C2×Q8—22no—
374C2×Q8—22no—
375C2×Q8—22no—
376C2×Q8—22no—
377C2×Q8—22no—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8—22no—
381C2×Q8—22no—
382C2×Q8—22no—
383C2×Q8—22no—
384C2×Q8—22no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—41no—
396C2×Q8—41no—
397C2×Q8—41no—
398C2×Q8—41no—
399C2×Q8—41no—
400C2×Q8—41no—
401C2×Q8—41no—
402C2×Q8—41no—
403C2×Q8—41no—
404C2×Q8—41no—
405C2×Q8—41no—
406C2×Q8—41no—
407C2×Q8—41no—
408C2×Q8—41no—
409C2×Q8—41no—
410C2×Q8—41no—
411C2×Q8—41no—
412C2×Q8—41no—
413C2×Q8—41no—
414C2×Q8—41no—
415C2×Q8—41no—
416C2×Q8—41no—
417C2×Q8—41no—
418C2×Q8—41no—
419C2×Q8C2×C4⋊C414yes—
420C2×Q8C2×C4⋊C414yes—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—41no—
452C22×C4—41no—
453C22×C4—41no—
454C22×C4—41no—
455C22×C4—41no—
456C22×C4—41no—
457C22×C4—41no—
458C22×C4—41no—
459C22×C4—41no—
460C22×C4—41no—
461C22×C4—41no—
462C22×C4—41no—
463C22×C4—41no—
464C22×C4—41no—
465C22×C4—41no—
466C22×C4—41no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—41no—
476C22×C4—41no—
477C22×C4—41no—
478C22×C4—41no—
479C22×C4—41no—
480C22×C4—41no—
481C22×C4—42no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4—42no—
485C22×C4C4×D414yes—
486C22×C4#12712yes—
487C22×C4#12712yes—
488C22×C4#13714yes—
489C22×C4C23×C411yescharacteristic
490C22×C4C22×D411yescharacteristic
491C22×C4C22×D411yescharacteristic
492C22×C4C22×D411yescharacteristic
493C22×C4C22×D411yescharacteristic
494C22×C4C22×Q811yescharacteristic
495C22×C4C2×#4111yescharacteristic
496C22×C4C2×#4111yescharacteristic
497C22×C4C2×#4111yescharacteristic
498C22×C4C2×#4111yescharacteristic
499C22×C4H(5,𝔽2)11yescharacteristic
500C22×C4H(5,𝔽2)11yescharacteristic
501C22×C42−1+411yescharacteristic
502C22×C42−1+411yescharacteristic
503Q16—232no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—22no—
507C2×C8—22no—
508C2×C8—22no—
509C2×C8—22no—
510C2×C8—22no—
511C2×C8—22no—
512C2×C8—22no—
513C2×C8—22no—
514C2×C8—22no—
515C2×C8—22no—
516C2×C8—41no—
517C2×C8—41no—
518C2×C8—41no—
519C2×C8—41no—
520C2×C8—41no—
521C2×C8—41no—
522C2×C8—41no—
523C2×C8—41no—
524C2×C8C2×C4⋊C414yes—
525C2×C8C4×D414yes—
526C2×C8C4×D414yes—
527C2×C8C4×D414yes—
528C2×C8#12014yes—
529C2×C8C22⋊Q814yes—
530C2×C8C22⋊Q814yes—
531C2×C8#12314yes—
532C4⋊C4—22no—
533C4⋊C4—22no—
534C4⋊C4—22no—
535C4⋊C4—22no—
536C4⋊C4—22no—
537C4⋊C4—22no—
538C4⋊C4—22no—
539C4⋊C4—22no—
540C4⋊C4—48no—
541C4⋊C4—48no—
542C4⋊C4—41no—
543C4⋊C4—41no—
544C4⋊C4—41no—
545C4⋊C4—41no—
546C4⋊C4—41no—
547C4⋊C4—41no—
548C4⋊C4—41no—
549C4⋊C4—41no—
550C4⋊C4—41no—
551C4⋊C4—41no—
552C4⋊C4—41no—
553C4⋊C4—41no—
554C4⋊C4—41no—
555C4⋊C4—41no—
556C4⋊C4—41no—
557C4⋊C4—41no—
558C4⋊C4—84no—
559C4⋊C4—88no—
560C4⋊C4—84no—
561C42—22no—
562C42—22no—
563C42—22no—
564C42—22no—
565C42—22no—
566C42—22no—
567C42—22no—
568C42—22no—
569C42—48no—
570C42—48no—
571C42—41no—
572C42—41no—
573C42—41no—
574C42—41no—
575C23C2×C4×D411yescharacteristic
576C23C2×#12711yescharacteristic
577C23#53211yescharacteristic
578C23D4211yescharacteristic
579C23D4211yescharacteristic
580C23#54711yescharacteristic
581C23D4×Q811yescharacteristic
582C23C2×#13712yes—
583C23#57512yes—
584C23#57712yes—
585C23#57812yes—
586Q8—24no—
587Q8—24no—
588Q8—24no—
589Q8—24no—
590Q8—42no—
591Q8—42no—
592Q8—42no—
593Q8—42no—
594Q8—42no—
595Q8—42no—
596Q8—42no—
597Q8—42no—
598Q8—42no—
599Q8—42no—
600Q8—42no—
601Q8—42no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—24no—
608C2×C4—24no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—24no—
613C2×C4—24no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—81no—
648C2×C4—81no—
649C2×C4—81no—
650C2×C4—81no—
651C2×C4—81no—
652C2×C4—81no—
653C2×C4—81no—
654C2×C4—81no—
655C2×C4—82no—
656C2×C4—82no—
657C2×C4—82no—
658C2×C4—82no—
659C2×C4—81no—
660C2×C4—81no—
661C2×C4—81no—
662C2×C4—81no—
663C2×C4—81no—
664C2×C4—81no—
665C2×C4—82no—
666C2×C4—82no—
667C2×C4—82no—
668C2×C4—82no—
669C2×C4#49714yes—
670C2×C4C22×C4⋊C411yescharacteristic
671C2×C4C2×C4×D411yescharacteristic
672C2×C4C2×C4×D411yescharacteristic
673C2×C4C2×C4×D411yescharacteristic
674C2×C4#52011yescharacteristic
675C2×C4#52011yescharacteristic
676C2×C4C2×#12011yescharacteristic
677C2×C4C2×C22⋊Q811yescharacteristic
678C2×C4C2×C22⋊Q811yescharacteristic
679C2×C4C2×#12311yescharacteristic
680C2×C4#53211yescharacteristic
681C2×C4#53211yescharacteristic
682C2×C4#53511yescharacteristic
683C2×C4#53611yescharacteristic
684C2×C4#54211yescharacteristic
685C2×C4#54211yescharacteristic
686C2×C4D4211yescharacteristic
687C2×C4D4211yescharacteristic
688C2×C4#54711yescharacteristic
689C2×C4#54711yescharacteristic
690C2×C4#54711yescharacteristic
691C2×C4#54811yescharacteristic
692C2×C4#54811yescharacteristic
693C2×C4D4×Q811yescharacteristic
694C2×C4#55311yescharacteristic
695C2×C4#55311yescharacteristic
696C2×C4#55411yescharacteristic
697C2×C4#55411yescharacteristic
698C8—24no—
699C8—42no—
700C8—42no—
701C8#38518yes—
702C22C4×#12711yescharacteristic
703C22D4×C4⋊C411yescharacteristic
704C22#236111yescharacteristic
705C22#242211yescharacteristic
706C22#243311yescharacteristic
707C22#245211yescharacteristic
708C22#254211yescharacteristic
709C22C4×#13712yes—
710C22#294712yes—
711C22C2×#49711yescharacteristic
712C22#315011yescharacteristic
713C22#315411yescharacteristic
714C22#315611yescharacteristic
715C22#317212yes—
716C22#317812yes—
717C22#326712yes—
718C22#327012yes—
719C22#327012yes—
720C22#327112yes—
721C22#335212yes—
722C22#335612yes—
723C22#338712yes—
724C22#339012yes—
725C4—24no—
726C4—44no—
727C4—44no—
728C4—42no—
729C4—44no—
730C4—82no—
731C4—82no—
732C4—82no—
733C4—82no—
734C4—82no—
735C4—161no—
736C4—161no—
737C4C2×#38511yescharacteristic
738C4#232611yescharacteristic
739C4D4×C4⋊C411yescharacteristic
740C4#236111yescharacteristic
741C4#242211yescharacteristic
742C4#243211yescharacteristic
743C4#245211yescharacteristic
744C4#256611yescharacteristic
745C2#1397711yescharacteristic
746C2C4×#49711yescharacteristic
747C2#2170311yescharacteristic
748C2#2267912yes—
749C2#2303612yes—
750C2#2503912yes—
751C2#2578711yescharacteristic
752C2#2578911yescharacteristic
753C2#2648912yes—
754C2#2853811yescharacteristic
755C2#2854211yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C4×#95 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
45. #1971C4semidirect
229. C2×Q16C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×C4×Q162
C2×#19712
C2×#19412
C2×#18652
C2×#18632
C2×#18472
#139782
#139762
#111242
#111222
#110522
#110502
#108632
#106962