Finite Group Databasebeta

← #188655

Subgroups — #188655

Order 512.

1066 conjugacy classes of subgroups.

The subgroup lattice has 675 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34226C211yesmaximal, characteristic
2#21972C211yesmaximal, characteristic, complemented
3#20815C211yesmaximal, characteristic, complemented
4C2×#1551C211yesmaximal, characteristic
5C2×#1543C211yesmaximal, characteristic
6#8594C211yesmaximal, characteristic, complemented
7#8592C211yesmaximal, characteristic, complemented
8#8590C211yesmaximal, characteristic, complemented
9#8590C211yesmaximal, characteristic, complemented
10#8588C211yesmaximal, characteristic, complemented
11#8588C211yesmaximal, characteristic, complemented
12#8587C211yesmaximal, characteristic, complemented
13#8585C211yesmaximal, characteristic, complemented
14#8046C211yesmaximal, characteristic, complemented
15#8039C211yesmaximal, characteristic, complemented
16C2×#532C2211yescharacteristic
17C2×#432C2211yescharacteristic
18C2×#431C2211yescharacteristic
19C2×#431C2211yescharacteristic
20C22×C4⋊C8C2211yescharacteristic
21#2367C412yes—
22#2367C412yes—
23#2363—21no—
24#2363—21no—
25#2357—21no—
26#2349—21no—
27#2341C412yes—
28#2339C412yes—
29#2324—21no—
30#2324—21no—
31#2319—21no—
32C4×C4⋊Q8C2211yescharacteristic
33C4×C4⋊Q8C2211yescharacteristic
34C4×#127C2211yescharacteristic
35C4×#127C2211yescharacteristic
36#2285C2211yescharacteristic
37#2275—21no—
38C4×#117C2211yescharacteristic
39#1939—21no—
40#1939—21no—
41#1850—21no—
42#1850—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#1551—24no—
56#1543—24no—
57C2×#333C2211yescharacteristic
58C2×#333C2211yescharacteristic
59C2×#333C2211yescharacteristic
60C2×#333C2211yescharacteristic
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×#41C2×C412yes—
74C23×C8—21no—
75#532—22no—
76#532—22no—
77#532D414yes—
78#532Q814yes—
79C2×C4⋊Q8C2311yescharacteristic
80C2×#127C2311yescharacteristic
81C2×#124C2×C412yes—
82C2×#121—21no—
83C2×#121—21no—
84C2×C4×Q8C2×C412yes—
85C2×C4×Q8C2×C412yes—
86C2×C4×D4—21no—
87C2×C4×D4—21no—
88C2×C4×D4—21no—
89C2×C4×D4—21no—
90C2×C4×D4—21no—
91C2×C4×D4—21no—
92C2×C4×D4C2×C412yes—
93C2×C4×D4C2×C412yes—
94C2×#117—21no—
95C2×#117D411yescharacteristic
96C2×#117D411yescharacteristic
97C22×C42C2311yescharacteristic
98#432—22no—
99#432—22no—
100#431—22no—
101#431—22no—
102#431—22no—
103#431—22no—
104C2×C4⋊C8—22no—
105C2×C4⋊C8—22no—
106C2×C4⋊C8—21no—
107C2×C4⋊C8—21no—
108C2×C4⋊C8—21no—
109C2×C4⋊C8—21no—
110C2×C4⋊C8—21no—
111C2×C4⋊C8—21no—
112C2×C4⋊C8—21no—
113C2×C4⋊C8—21no—
114C2×C4⋊C8—21no—
115C2×C4⋊C8C2311yescharacteristic
116C2×C4⋊C8C2311yescharacteristic
117C2×C4⋊C8C2311yescharacteristic
118C2×C4⋊C8C2311yescharacteristic
119C2×C4⋊C8C2311yescharacteristic
120C2×C4⋊C8C2311yescharacteristic
121C2×C4⋊C8C2311yescharacteristic
122C2×C4⋊C8C2311yescharacteristic
123C2×C22⋊C8—21no—
124C2×C22⋊C8—21no—
125C2×C22⋊C8—21no—
126C2×#97—21no—
127C2×C4×C8—21no—
128C2×C4×C8—21no—
129#391C2×C412yes—
130#391C2×C412yes—
131#390—22no—
132#390—22no—
133#390—21no—
134#390—21no—
135#390—21no—
136#390—21no—
137#390C2×C412yes—
138#390C2×C412yes—
139#389—21no—
140#389—21no—
141#389—21no—
142#389—21no—
143#388—22no—
144#388—22no—
145#388—22no—
146#388—22no—
147#386—22no—
148#386—22no—
149#385—22no—
150#385—22no—
151#384C2311yescharacteristic
152#384C2311yescharacteristic
153#382—21no—
154#381—22no—
155#380—21no—
156#380—22no—
157#380C2×C412yes—
158C4×C4⋊C4—21no—
159C4×C4⋊C4—21no—
160C4×C4⋊C4—21no—
161C4×C4⋊C4—21no—
162C4×C4⋊C4—21no—
163C4×C4⋊C4C2×C412yes—
164C4×C4⋊C4C2×C411yescharacteristic
165C4×C4⋊C4C2×C411yescharacteristic
166C4×C4⋊C4D411yescharacteristic
167C4×C4⋊C4D411yescharacteristic
168C4×C22⋊C4—22no—
169C4×C22⋊C4—22no—
170C4×C22⋊C4—22no—
171C4×C22⋊C4C2×C411yescharacteristic
172C4×C22⋊C4C2×C411yescharacteristic
173C4×C22⋊C4D411yescharacteristic
174C4×C22⋊C4D411yescharacteristic
175#376D411yescharacteristic
176#376D411yescharacteristic
177C2×#95—22no—
178C43C2311yescharacteristic
179C43C2311yescharacteristic
180#336—21no—
181#336—21no—
182#336—21no—
183#336—21no—
184#336—21no—
185#336—21no—
186#336—21no—
187#336—21no—
188#333—22no—
189#333—22no—
190#333—22no—
191#333—22no—
192#331—22no—
193#331—22no—
194#331—22no—
195#331—22no—
196C2×#41—24no—
197C2×#41—82no—
198C2×#41—82no—
199C2×#41C4⋊C418yes—
200C22×Q8C22×C412yes—
201C22×D4—22no—
202C22×D4—21no—
203C22×D4—21no—
204C22×D4C22×C412yes—
205C23×C4—22no—
206C23×C4—22no—
207C23×C4—22no—
208C23×C4C22×C411yescharacteristic
209C23×C4C2×D411yescharacteristic
210C23×C4C2×D411yescharacteristic
211C22×C8—21no—
212C22×C8—21no—
213C22×C8—21no—
214C22×C8—21no—
215C22×C8—21no—
216C22×C8—21no—
217C22×C8—21no—
218C22×C8—21no—
219C22×C8—42no—
220C22×C8—81no—
221C22×C8—81no—
222C4⋊Q8—21no—
223C4⋊Q8—21no—
224C4⋊Q8C2×D412yes—
225C4⋊Q8C2×Q812yes—
226#127—21no—
227#127—21no—
228#127C2×D412yes—
229#127C2×Q812yes—
230#124—24no—
231#124C4⋊C418yes—
232#121—24no—
233#121—24no—
234#121—24no—
235#121—24no—
236C4×Q8—82no—
237C4×Q8—82no—
238C4×D4—24no—
239C4×D4—24no—
240C4×D4—24no—
241C4×D4—24no—
242C4×D4—42no—
243C4×D4—42no—
244C4×D4—42no—
245C4×D4—42no—
246C4×D4—81no—
247C4×D4—81no—
248C4×D4—81no—
249C4×D4—81no—
250C4×D4—82no—
251C4×D4—82no—
252#117—22no—
253#117—22no—
254#117—41no—
255#117—41no—
256#117—81no—
257#117—81no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4C22⋊C411yescharacteristic
267C2×C4⋊C4C22⋊C411yescharacteristic
268C2×C4⋊C4C22⋊C411yescharacteristic
269C2×C4⋊C4C22⋊C411yescharacteristic
270C2×C4⋊C4C22×C412yes—
271C2×C4⋊C4C22×C412yes—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
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⋊C4C22⋊C411yescharacteristic
282C2×C22⋊C4C22⋊C411yescharacteristic
283C2×C22⋊C4C22⋊C411yescharacteristic
284C2×C22⋊C4C22⋊C411yescharacteristic
285C2×C42—22no—
286C2×C42—22no—
287C2×C42—22no—
288C2×C42C22×C411yescharacteristic
289C2×C42C22×C411yescharacteristic
290C2×C42C22×C411yescharacteristic
291C2×C42C22×C411yescharacteristic
292C2×C42C22×C411yescharacteristic
293C2×C42C2×D411yescharacteristic
294C2×C42C2×D411yescharacteristic
295C2×C42C2×D411yescharacteristic
296C2×C42C2×D411yescharacteristic
297C2×C42C2×D411yescharacteristic
298C2×C42C2×D411yescharacteristic
299C2×C42C2×D411yescharacteristic
300C2×C42C2×D412yes—
301C2×C42C2×D411yescharacteristic
302C2×C42C2×D411yescharacteristic
303C2×C42C2×D411yescharacteristic
304C2×C42C2×Q812yes—
305C2×C42#4112yes—
306C2×C42#4112yes—
307C2×C42C2411yescharacteristic
308C4⋊C8—21no—
309C4⋊C8—21no—
310C4⋊C8—21no—
311C4⋊C8—21no—
312C4⋊C8—21no—
313C4⋊C8—21no—
314C4⋊C8—21no—
315C4⋊C8—21no—
316C4⋊C8—21no—
317C4⋊C8—21no—
318C4⋊C8—21no—
319C4⋊C8—21no—
320C4⋊C8—21no—
321C4⋊C8—21no—
322C4⋊C8—21no—
323C4⋊C8—21no—
324C4⋊C8—42no—
325C4⋊C8—42no—
326C4⋊C8—42no—
327C4⋊C8—42no—
328C4⋊C8—42no—
329C4⋊C8—81no—
330C4⋊C8—81no—
331C4⋊C8—81no—
332C4⋊C8—81no—
333C22⋊C8—42no—
334C22⋊C8—42no—
335C22⋊C8—42no—
336#97—42no—
337C4×C8—42no—
338C4×C8—42no—
339#95—22no—
340#95—22no—
341#95—22no—
342#95—22no—
343#95—22no—
344#95—22no—
345#95C22⋊C411yescharacteristic
346#95C22⋊C411yescharacteristic
347#95C22⋊C411yescharacteristic
348#95C22⋊C411yescharacteristic
349#95C22⋊C411yescharacteristic
350#95C22⋊C411yescharacteristic
351#95C22⋊C411yescharacteristic
352#95C22⋊C411yescharacteristic
353C24—22no—
354C24C2×C22⋊C411yescharacteristic
355#41—88no—
356#41—88no—
357C2×Q8—22no—
358C2×Q8—42no—
359C2×Q8—42no—
360C2×Q8C2×C4⋊C414yes—
361C2×D4—28no—
362C2×D4—22no—
363C2×D4—22no—
364C2×D4—22no—
365C2×D4—22no—
366C2×D4—22no—
367C2×D4—42no—
368C2×D4—42no—
369C2×D4—41no—
370C2×D4—41no—
371C2×D4—41no—
372C2×D4—41no—
373C2×D4—41no—
374C2×D4—41no—
375C2×D4—41no—
376C2×D4—41no—
377C2×D4—82no—
378C2×D4—82no—
379C2×D4C2×C4⋊C414yes—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—28no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—44no—
395C22×C4—44no—
396C22×C4—82no—
397C22×C4—82no—
398C22×C4—82no—
399C22×C4—82no—
400C22×C4—82no—
401C22×C4—82no—
402C22×C4C2×C22⋊C411yescharacteristic
403C22×C4C2×C22⋊C411yescharacteristic
404C22×C4C2×C22⋊C411yescharacteristic
405C22×C4C2×C22⋊C411yescharacteristic
406C22×C4C2×C22⋊C411yescharacteristic
407C22×C4C2×C22⋊C411yescharacteristic
408C22×C4C2×C22⋊C411yescharacteristic
409C22×C4C2×C22⋊C411yescharacteristic
410C22×C4C2×C22⋊C411yescharacteristic
411C22×C4C2×C22⋊C411yescharacteristic
412C22×C4C2×C22⋊C411yescharacteristic
413C22×C4C2×C4⋊C412yes—
414C22×C4#11712yes—
415C22×C4#12112yes—
416C22×C4#12112yes—
417C22×C4C22⋊Q812yes—
418C22×C4C22⋊Q812yes—
419C22×C4C23×C411yescharacteristic
420C22×C4C22×D411yescharacteristic
421C22×C4C22×D411yescharacteristic
422C2×C8—41no—
423C2×C8—41no—
424C2×C8—41no—
425C2×C8—41no—
426C2×C8—41no—
427C2×C8—41no—
428C2×C8—41no—
429C2×C8—41no—
430C2×C8—41no—
431C2×C8—41no—
432C2×C8—41no—
433C2×C8—41no—
434C2×C8—41no—
435C2×C8—41no—
436C2×C8—41no—
437C2×C8—41no—
438C2×C8—41no—
439C2×C8—41no—
440C2×C8—41no—
441C2×C8—41no—
442C2×C8—41no—
443C2×C8—41no—
444C2×C8—41no—
445C2×C8—41no—
446C2×C8—84no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—41no—
454C4⋊C4—41no—
455C4⋊C4—42no—
456C4⋊C4—42no—
457C4⋊C4—41no—
458C4⋊C4—41no—
459C4⋊C4—41no—
460C4⋊C4—41no—
461C4⋊C4—41no—
462C4⋊C4—41no—
463C4⋊C4—41no—
464C4⋊C4—41no—
465C4⋊C4—82no—
466C4⋊C4—82no—
467C22⋊C4—28no—
468C22⋊C4—28no—
469C22⋊C4—22no—
470C22⋊C4—22no—
471C22⋊C4—41no—
472C22⋊C4—41no—
473C22⋊C4—44no—
474C22⋊C4—44no—
475C22⋊C4—82no—
476C22⋊C4—82no—
477C22⋊C4—82no—
478C22⋊C4—82no—
479C22⋊C4—82no—
480C42—21no—
481C42—21no—
482C42—21no—
483C42—21no—
484C42—21no—
485C42—21no—
486C42—21no—
487C42—21no—
488C42—21no—
489C42—21no—
490C42—21no—
491C42—21no—
492C42—21no—
493C42—21no—
494C42—41no—
495C42—41no—
496C42—41no—
497C42—41no—
498C42—82no—
499C42—82no—
500C42—82no—
501C42C2×C4⋊C412yes—
502C42#11712yes—
503C42#12112yes—
504C42#12112yes—
505C42C22⋊Q812yes—
506C42C22⋊Q812yes—
507C42#13512yes—
508C42#13512yes—
509C42C22×D411yescharacteristic
510C42C22×Q811yescharacteristic
511C42C2×#4111yescharacteristic
512C42C2×#4111yescharacteristic
513C23—21no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—21no—
519C23—24no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23#38012yes—
527C23C22×C22⋊C411yescharacteristic
528Q8—44no—
529Q8—44no—
530D4—44no—
531D4—44no—
532D4—42no—
533D4—42no—
534D4—42no—
535D4—42no—
536D4—42no—
537D4—42no—
538D4—42no—
539D4—42no—
540D4—88no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
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—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—24no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—88no—
591C2×C4—164no—
592C2×C4—164no—
593C2×C4#38012yes—
594C2×C4#38012yes—
595C2×C4#38012yes—
596C2×C4#41612yes—
597C2×C4#41612yes—
598C2×C4C22×C4⋊C411yescharacteristic
599C2×C4C2×#11711yescharacteristic
600C2×C4C2×#12111yescharacteristic
601C2×C4C2×#12111yescharacteristic
602C2×C4C2×C22⋊Q811yescharacteristic
603C2×C4C2×C22⋊Q811yescharacteristic
604C2×C4C2×#13511yescharacteristic
605C2×C4C2×#13511yescharacteristic
606C2×C4#57511yescharacteristic
607C2×C4#57511yescharacteristic
608C2×C4#57611yescharacteristic
609C2×C4#57711yescharacteristic
610C2×C4#57711yescharacteristic
611C2×C4#57811yescharacteristic
612C8—81no—
613C8—81no—
614C8—81no—
615C8—81no—
616C8—81no—
617C8—81no—
618C8—81no—
619C8—81no—
620C22—22no—
621C22—22no—
622C22—22no—
623C22—41no—
624C22—41no—
625C22—41no—
626C22—41no—
627C22—44no—
628C22—44no—
629C22—44no—
630C22—82no—
631C22—82no—
632C22—82no—
633C22—82no—
634C22C2×#38011yescharacteristic
635C22#288811yescharacteristic
636C22#288811yescharacteristic
637C22C2×#41611yescharacteristic
638C22#289611yescharacteristic
639C22#289711yescharacteristic
640C22#289811yescharacteristic
641C4—22no—
642C4—22no—
643C4—22no—
644C4—21no—
645C4—21no—
646C4—22no—
647C4—22no—
648C4—41no—
649C4—41no—
650C4—41no—
651C4—41no—
652C4—41no—
653C4—41no—
654C4—44no—
655C4—82no—
656C4—82no—
657C4#303511yescharacteristic
658C4#303511yescharacteristic
659C4#304511yescharacteristic
660C4#304611yescharacteristic
661C4#307611yescharacteristic
662C4#307611yescharacteristic
663C4#308311yescharacteristic
664C4#308411yescharacteristic
665C2—42no—
666C2—84no—
667C2#1772411yescharacteristic
668C2#1772611yescharacteristic
669C2#1776611yescharacteristic
670C2#1776811yescharacteristic
671C2#2063811yescharacteristic
672C2#2064511yescharacteristic
673C2#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. #21972C2semidirect
3. #20815C2semidirect
6. #8594C2semidirect
7. #8592C2semidirect
8. #8590C2semidirect
9. #8590C2semidirect
10. #8588C2semidirect
11. #8588C2semidirect
12. #8587C2semidirect
13. #8585C2semidirect
14. #8046C2semidirect
15. #8039C2semidirect

Maximal subgroups

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

subgroupindex
#342262
#219722
#208152
C2×#15512
C2×#15432
#85942
#85922
#85902
#85882
#85872
#85852
#80462
#80392