Finite Group Databasebeta

← #188677

Subgroups — #188677

Order 512.

1184 conjugacy classes of subgroups.

The subgroup lattice has 699 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34207C211yesmaximal, characteristic
2#21978C211yesmaximal, characteristic, complemented
3#20815C211yesmaximal, characteristic, complemented
4C2×#1557C211yesmaximal, characteristic
5C2×#1553C211yesmaximal, characteristic
6#8593C211yesmaximal, characteristic, complemented
7#8592C211yesmaximal, characteristic, complemented
8#8591C211yesmaximal, characteristic, complemented
9#8589C211yesmaximal, characteristic, complemented
10#8588C211yesmaximal, characteristic, complemented
11#8588C211yesmaximal, characteristic, complemented
12#8586C211yesmaximal, characteristic, complemented
13#8585C211yesmaximal, characteristic, complemented
14#8043C211yesmaximal, characteristic, complemented
15#8040C211yesmaximal, characteristic, complemented
16C2×#532C2211yescharacteristic
17C2×#520—21no—
18C2×#432C2211yescharacteristic
19C2×#432C2211yescharacteristic
20C2×#431C2211yescharacteristic
21C2×#423C2211yescharacteristic
22#2361—21no—
23#2361—21no—
24#2361C412yes—
25#2361C412yes—
26#2354—21no—
27#2354C412yes—
28D4×C4⋊C4—21no—
29D4×C4⋊C4C412yes—
30#2326—21no—
31#2326—21no—
32#2311—21no—
33C4×C4⋊Q8C2211yescharacteristic
34C4×C4⋊Q8C2211yescharacteristic
35C4×#127C2211yescharacteristic
36C4×#127C2211yescharacteristic
37C2×#384C2211yescharacteristic
38C4×#117C2211yescharacteristic
39#1940—21no—
40#1940—21no—
41#1851—21no—
42#1851—21no—
43#1842C2211yescharacteristic
44#1842C2211yescharacteristic
45#1842C2211yescharacteristic
46#1842C2211yescharacteristic
47#1831C2211yescharacteristic
48#1831C2211yescharacteristic
49#1831C2211yescharacteristic
50#1831C2211yescharacteristic
51#1798—21no—
52#1798—21no—
53#1768—21no—
54#1768—21no—
55#1557—24no—
56#1553—24no—
57C2×#332C2211yescharacteristic
58C2×#332C2211yescharacteristic
59C2×#332C2211yescharacteristic
60C2×#332C2211yescharacteristic
61C2×#331C2211yescharacteristic
62C2×#331C2211yescharacteristic
63C2×#331C2211yescharacteristic
64C2×#331C2211yescharacteristic
65#1296C2211yescharacteristic
66#1296C2211yescharacteristic
67#1296C2211yescharacteristic
68#1296C2211yescharacteristic
69#1296C2211yescharacteristic
70#1296C2211yescharacteristic
71#1296C2211yescharacteristic
72#1296C2211yescharacteristic
73C22×#41—21no—
74C22×M16—21no—
75#532—22no—
76#532—22no—
77#532D414yes—
78#532Q814yes—
79C2×C4⋊Q8C2311yescharacteristic
80C2×#127C2311yescharacteristic
81C2×#124—21no—
82C2×#121—21no—
83C2×#121C2×C412yes—
84#520—44no—
85#520—44no—
86C2×C4×Q8—21no—
87C2×C4×Q8—21no—
88C2×C4×D4—22no—
89C2×C4×D4—22no—
90C2×C4×D4—21no—
91C2×C4×D4—21no—
92C2×C4×D4—21no—
93C2×C4×D4—21no—
94C2×C4×D4—21no—
95C2×C4×D4C2×C412yes—
96C2×C4×D4C2×C412yes—
97C2×C4×D4C2×C412yes—
98C2×#117—22no—
99C2×#117D411yescharacteristic
100C2×#117D411yescharacteristic
101C22×C4⋊C4—21no—
102C22×C4⋊C4—21no—
103C22×C4⋊C4—22no—
104C22×C4⋊C4C2×C412yes—
105C22×C42C2311yescharacteristic
106#432—22no—
107#432—22no—
108#432—22no—
109#432—22no—
110#431—22no—
111#431—22no—
112#423—22no—
113#423—22no—
114C2×C4⋊C8—21no—
115C2×C4⋊C8—21no—
116C2×C4⋊C8—21no—
117C2×C4⋊C8—21no—
118C2×C4⋊C8—21no—
119C2×C4⋊C8—21no—
120C2×C4⋊C8—21no—
121C2×C4⋊C8—21no—
122C2×C4⋊C8—21no—
123C2×C4⋊C8C2311yescharacteristic
124C2×C4⋊C8C2311yescharacteristic
125C2×C4⋊C8C2311yescharacteristic
126C2×C4⋊C8C2311yescharacteristic
127C2×C4⋊C8C2311yescharacteristic
128C2×C4⋊C8C2311yescharacteristic
129C2×C4⋊C8C2311yescharacteristic
130C2×C4⋊C8C2311yescharacteristic
131C2×C22⋊C8—21no—
132C2×C22⋊C8—21no—
133C2×C22⋊C8—21no—
134C2×#97—21no—
135C2×#97—21no—
136C2×C4×C8—21no—
137#391—21no—
138#391—21no—
139#390—21no—
140#390—21no—
141#390—21no—
142#390—21no—
143#390C2×C412yes—
144#390C2×C412yes—
145#389—21no—
146#389—21no—
147#389C2×C412yes—
148#389C2×C412yes—
149#388—22no—
150#388—22no—
151#388—22no—
152#388—22no—
153#386—22no—
154#386—22no—
155#385—22no—
156#385—22no—
157#385—22no—
158#385—22no—
159#384—21no—
160#384—21no—
161#384C2×C412yes—
162#384C2311yescharacteristic
163#384C2311yescharacteristic
164#380—22no—
165#380—22no—
166C4×C4⋊C4—21no—
167C4×C4⋊C4—21no—
168C4×C4⋊C4C2×C412yes—
169C4×C4⋊C4C2×C412yes—
170C4×C4⋊C4C2×C411yescharacteristic
171C4×C4⋊C4C2×C411yescharacteristic
172C4×C4⋊C4D411yescharacteristic
173C4×C4⋊C4D411yescharacteristic
174C4×C22⋊C4C2×C411yescharacteristic
175C4×C22⋊C4C2×C411yescharacteristic
176C4×C22⋊C4D411yescharacteristic
177C4×C22⋊C4D411yescharacteristic
178#376D411yescharacteristic
179#376D411yescharacteristic
180C43C2311yescharacteristic
181C43C2311yescharacteristic
182#336—21no—
183#336—21no—
184#336—21no—
185#336—21no—
186#336—21no—
187#336—21no—
188#336—21no—
189#336—21no—
190#332—22no—
191#332—22no—
192#332—22no—
193#332—22no—
194#331—22no—
195#331—22no—
196#331—22no—
197#331—22no—
198C2×#41—24no—
199C2×#41—24no—
200C2×#41—42no—
201C2×#41—42no—
202C2×#41—42no—
203C2×#41—42no—
204C22×Q8—21no—
205C22×D4—22no—
206C22×D4—21no—
207C22×D4—21no—
208C22×D4C22×C412yes—
209C23×C4—22no—
210C23×C4—22no—
211C23×C4—22no—
212C23×C4—22no—
213C23×C4C22×C411yescharacteristic
214C23×C4C2×D411yescharacteristic
215C23×C4C2×D411yescharacteristic
216C2×M16—42no—
217C2×M16—81no—
218C2×M16—81no—
219C22×C8—21no—
220C22×C8—21no—
221C22×C8—21no—
222C22×C8—21no—
223C22×C8—21no—
224C22×C8—21no—
225C22×C8—21no—
226C22×C8—21no—
227C4⋊Q8—21no—
228C4⋊Q8—21no—
229C4⋊Q8C2×D412yes—
230C4⋊Q8C2×Q812yes—
231#127—21no—
232#127—21no—
233#127C2×D412yes—
234#127C2×Q812yes—
235#124—24no—
236#124—24no—
237#121—24no—
238#121—24no—
239#121—24no—
240#121C4⋊C418yes—
241C4×Q8—42no—
242C4×Q8—42no—
243C4×Q8—42no—
244C4×Q8—42no—
245C4×D4—24no—
246C4×D4—24no—
247C4×D4—24no—
248C4×D4—42no—
249C4×D4—42no—
250C4×D4—48no—
251C4×D4—48no—
252C4×D4—42no—
253C4×D4—42no—
254C4×D4—81no—
255C4×D4—81no—
256C4×D4—81no—
257C4×D4—81no—
258C4×D4—82no—
259C4×D4—82no—
260C4×D4C4⋊C418yes—
261#117—22no—
262#117—22no—
263#117—41no—
264#117—41no—
265#117—48no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—48no—
277C2×C4⋊C4—42no—
278C2×C4⋊C4—42no—
279C2×C4⋊C4—81no—
280C2×C4⋊C4—81no—
281C2×C4⋊C4—82no—
282C2×C4⋊C4C22⋊C411yescharacteristic
283C2×C4⋊C4C22⋊C411yescharacteristic
284C2×C4⋊C4C22⋊C411yescharacteristic
285C2×C4⋊C4C22⋊C411yescharacteristic
286C2×C4⋊C4C22×C412yes—
287C2×C4⋊C4C22×C412yes—
288C2×C4⋊C4C22×C412yes—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4C22⋊C411yescharacteristic
298C2×C22⋊C4C22⋊C411yescharacteristic
299C2×C22⋊C4C22⋊C411yescharacteristic
300C2×C22⋊C4C22⋊C411yescharacteristic
301C2×C42—22no—
302C2×C42—22no—
303C2×C42C22×C411yescharacteristic
304C2×C42C22×C411yescharacteristic
305C2×C42C22×C411yescharacteristic
306C2×C42C22×C411yescharacteristic
307C2×C42C22×C411yescharacteristic
308C2×C42C2×D411yescharacteristic
309C2×C42C2×D411yescharacteristic
310C2×C42C2×D411yescharacteristic
311C2×C42C2×D411yescharacteristic
312C2×C42C2×D411yescharacteristic
313C2×C42C2×D411yescharacteristic
314C2×C42C2×D411yescharacteristic
315C2×C42C2×D412yes—
316C2×C42C2×D411yescharacteristic
317C2×C42C2×D411yescharacteristic
318C2×C42C2×D411yescharacteristic
319C2×C42C2×Q812yes—
320C2×C42#4112yes—
321C2×C42#4112yes—
322C2×C42C2411yescharacteristic
323C4⋊C8—21no—
324C4⋊C8—21no—
325C4⋊C8—21no—
326C4⋊C8—21no—
327C4⋊C8—21no—
328C4⋊C8—21no—
329C4⋊C8—21no—
330C4⋊C8—21no—
331C4⋊C8—21no—
332C4⋊C8—21no—
333C4⋊C8—21no—
334C4⋊C8—21no—
335C4⋊C8—21no—
336C4⋊C8—21no—
337C4⋊C8—21no—
338C4⋊C8—21no—
339C4⋊C8—42no—
340C4⋊C8—42no—
341C4⋊C8—42no—
342C4⋊C8—42no—
343C4⋊C8—42no—
344C4⋊C8—81no—
345C4⋊C8—81no—
346C4⋊C8—81no—
347C4⋊C8—81no—
348C22⋊C8—42no—
349C22⋊C8—42no—
350C22⋊C8—42no—
351#97—42no—
352#97—42no—
353C4×C8—42no—
354#95—22no—
355#95—22no—
356#95—22no—
357#95—22no—
358#95C22⋊C411yescharacteristic
359#95C22⋊C411yescharacteristic
360#95C22⋊C411yescharacteristic
361#95C22⋊C411yescharacteristic
362#95C22⋊C411yescharacteristic
363#95C22⋊C411yescharacteristic
364#95C22⋊C411yescharacteristic
365#95C22⋊C411yescharacteristic
366C24—22no—
367C24C2×C22⋊C411yescharacteristic
368#41—48no—
369#41—48no—
370#41—48no—
371#41—48no—
372C2×Q8—22no—
373C2×Q8—22no—
374C2×Q8—41no—
375C2×Q8—41no—
376C2×Q8—41no—
377C2×Q8—41no—
378C2×D4—28no—
379C2×D4—22no—
380C2×D4—22no—
381C2×D4—22no—
382C2×D4—22no—
383C2×D4—22no—
384C2×D4—42no—
385C2×D4—42no—
386C2×D4—44no—
387C2×D4—44no—
388C2×D4—41no—
389C2×D4—41no—
390C2×D4—41no—
391C2×D4—41no—
392C2×D4—41no—
393C2×D4—41no—
394C2×D4—41no—
395C2×D4—41no—
396C2×D4C2×C4⋊C414yes—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—28no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—44no—
412C22×C4—44no—
413C22×C4—44no—
414C22×C4—44no—
415C22×C4—44no—
416C22×C4—82no—
417C22×C4—82no—
418C22×C4—82no—
419C22×C4—82no—
420C22×C4—82no—
421C22×C4—82no—
422C22×C4C2×C22⋊C411yescharacteristic
423C22×C4C2×C22⋊C411yescharacteristic
424C22×C4C2×C22⋊C411yescharacteristic
425C22×C4C2×C22⋊C411yescharacteristic
426C22×C4C2×C22⋊C411yescharacteristic
427C22×C4C2×C22⋊C411yescharacteristic
428C22×C4C2×C22⋊C411yescharacteristic
429C22×C4C2×C22⋊C411yescharacteristic
430C22×C4C2×C22⋊C411yescharacteristic
431C22×C4C2×C22⋊C411yescharacteristic
432C22×C4C2×C22⋊C411yescharacteristic
433C22×C4C2×C4⋊C412yes—
434C22×C4#11712yes—
435C22×C4#12112yes—
436C22×C4#12112yes—
437C22×C4C22⋊Q812yes—
438C22×C4C22⋊Q812yes—
439C22×C4C23×C411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D411yescharacteristic
442M16—84no—
443C2×C8—41no—
444C2×C8—41no—
445C2×C8—41no—
446C2×C8—41no—
447C2×C8—41no—
448C2×C8—41no—
449C2×C8—41no—
450C2×C8—41no—
451C2×C8—41no—
452C2×C8—41no—
453C2×C8—41no—
454C2×C8—41no—
455C2×C8—41no—
456C2×C8—41no—
457C2×C8—41no—
458C2×C8—41no—
459C2×C8—41no—
460C2×C8—41no—
461C2×C8—41no—
462C2×C8—41no—
463C2×C8—41no—
464C2×C8—41no—
465C2×C8—41no—
466C2×C8—41no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—41no—
473C4⋊C4—41no—
474C4⋊C4—42no—
475C4⋊C4—42no—
476C4⋊C4—44no—
477C4⋊C4—44no—
478C4⋊C4—41no—
479C4⋊C4—41no—
480C4⋊C4—41no—
481C4⋊C4—41no—
482C4⋊C4—41no—
483C4⋊C4—41no—
484C4⋊C4—41no—
485C4⋊C4—41no—
486C4⋊C4—44no—
487C4⋊C4—82no—
488C4⋊C4—82no—
489C4⋊C4C2×C4⋊C414yes—
490C22⋊C4—28no—
491C22⋊C4—28no—
492C22⋊C4—22no—
493C22⋊C4—22no—
494C22⋊C4—41no—
495C22⋊C4—41no—
496C22⋊C4—44no—
497C22⋊C4—44no—
498C22⋊C4—82no—
499C22⋊C4—82no—
500C22⋊C4—82no—
501C22⋊C4—82no—
502C42—21no—
503C42—21no—
504C42—21no—
505C42—21no—
506C42—21no—
507C42—21no—
508C42—21no—
509C42—21no—
510C42—21no—
511C42—21no—
512C42—21no—
513C42—21no—
514C42—21no—
515C42—21no—
516C42—41no—
517C42—41no—
518C42—41no—
519C42—41no—
520C42—44no—
521C42—44no—
522C42C2×C4⋊C412yes—
523C42#11712yes—
524C42#12112yes—
525C42#12112yes—
526C42C22⋊Q812yes—
527C42C22⋊Q812yes—
528C42AGL(1,ℤ/8)12yes—
529C42AGL(1,ℤ/8)12yes—
530C42C22×D411yescharacteristic
531C42C22×Q811yescharacteristic
532C42C2×#4111yescharacteristic
533C42C2×#4111yescharacteristic
534C23—21no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—24no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23#38012yes—
548C23C22×C22⋊C411yescharacteristic
549Q8—42no—
550Q8—42no—
551Q8—42no—
552Q8—42no—
553D4—44no—
554D4—44no—
555D4—416no—
556D4—42no—
557D4—42no—
558D4—42no—
559D4—42no—
560D4—42no—
561D4—42no—
562D4—42no—
563D4—42no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—24no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—416no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—164no—
615C2×C4—164no—
616C2×C4—164no—
617C2×C4#38012yes—
618C2×C4#38012yes—
619C2×C4#38012yes—
620C2×C4#41812yes—
621C2×C4#41812yes—
622C2×C4C22×C4⋊C411yescharacteristic
623C2×C4C2×#11711yescharacteristic
624C2×C4C2×#12111yescharacteristic
625C2×C4C2×#12111yescharacteristic
626C2×C4C2×C22⋊Q811yescharacteristic
627C2×C4C2×C22⋊Q811yescharacteristic
628C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
629C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
630C2×C4#57511yescharacteristic
631C2×C4#57511yescharacteristic
632C2×C4#57611yescharacteristic
633C2×C4#57711yescharacteristic
634C2×C4#57711yescharacteristic
635C2×C4#57811yescharacteristic
636C8—81no—
637C8—81no—
638C8—81no—
639C8—81no—
640C8—81no—
641C8—81no—
642C8—81no—
643C8—81no—
644C22—22no—
645C22—22no—
646C22—22no—
647C22—41no—
648C22—41no—
649C22—41no—
650C22—41no—
651C22—44no—
652C22—44no—
653C22—44no—
654C22—82no—
655C22—82no—
656C22—82no—
657C22—82no—
658C22C2×#38011yescharacteristic
659C22#288811yescharacteristic
660C22#288911yescharacteristic
661C22C2×#41811yescharacteristic
662C22#289611yescharacteristic
663C22#289711yescharacteristic
664C22#289811yescharacteristic
665C4—22no—
666C4—22no—
667C4—21no—
668C4—21no—
669C4—22no—
670C4—22no—
671C4—22no—
672C4—41no—
673C4—41no—
674C4—41no—
675C4—41no—
676C4—41no—
677C4—41no—
678C4—44no—
679C4—82no—
680C4—82no—
681C4#303611yescharacteristic
682C4#303611yescharacteristic
683C4#304511yescharacteristic
684C4#304611yescharacteristic
685C4#307711yescharacteristic
686C4#307711yescharacteristic
687C4#308311yescharacteristic
688C4#308411yescharacteristic
689C2—42no—
690C2—84no—
691C2#1775811yescharacteristic
692C2#1775911yescharacteristic
693C2#1776811yescharacteristic
694C2#1776911yescharacteristic
695C2#2064011yescharacteristic
696C2#2064611yescharacteristic
697C2#2065211yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21978C2semidirect
3. #20815C2semidirect
6. #8593C2semidirect
7. #8592C2semidirect
8. #8591C2semidirect
9. #8589C2semidirect
10. #8588C2semidirect
11. #8588C2semidirect
12. #8586C2semidirect
13. #8585C2semidirect
14. #8043C2semidirect
15. #8040C2semidirect

Maximal subgroups

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

subgroupindex
#342072
#219782
#208152
C2×#15572
C2×#15532
#85932
#85922
#85912
#85892
#85882
#85862
#85852
#80432
#80402