Finite Group Databasebeta

← #188691

Subgroups — #188691

Order 512.

1066 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34226C211yesmaximal, characteristic
2#21976C211yesmaximal, characteristic, complemented
3#21974C211yesmaximal, characteristic, complemented
4C2×#1557C211yesmaximal, characteristic
5C2×#1540C211yesmaximal, 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#8046C211yesmaximal, characteristic, complemented
15#8045C211yesmaximal, characteristic, complemented
16C2×#532C2211yescharacteristic
17C2×#432C2211yescharacteristic
18C2×#431C2211yescharacteristic
19C2×#423C2211yescharacteristic
20C22×C4⋊C8C2211yescharacteristic
21#2367—21no—
22#2367—21no—
23#2363—21no—
24#2363—21no—
25#2357—21no—
26#2349—21no—
27#2341—21no—
28#2339—21no—
29#2324C412yes—
30#2324C412yes—
31#2319C412yes—
32C4×C4⋊Q8C2211yescharacteristic
33C4×C4⋊Q8C2211yescharacteristic
34C4×#127C2211yescharacteristic
35C4×#127C2211yescharacteristic
36#2285C2211yescharacteristic
37#2275C412yes—
38C4×#117C2211yescharacteristic
39#1940—21no—
40#1940—21no—
41#1939—21no—
42#1939—21no—
43#1851—21no—
44#1851—21no—
45#1850—21no—
46#1850—21no—
47#1842C2211yescharacteristic
48#1842C2211yescharacteristic
49#1842C2211yescharacteristic
50#1842C2211yescharacteristic
51#1831C2211yescharacteristic
52#1831C2211yescharacteristic
53#1831C2211yescharacteristic
54#1831C2211yescharacteristic
55#1557—24no—
56#1540—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—
75C23×C8—21no—
76#532—22no—
77#532—22no—
78#532D414yes—
79#532Q814yes—
80C2×C4⋊Q8C2311yescharacteristic
81C2×#127C2311yescharacteristic
82C2×#124—21no—
83C2×#121—21no—
84C2×#121C2×C412yes—
85C2×C4×Q8—21no—
86C2×C4×Q8—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×C4×D4C2×C412yes—
95C2×#117—21no—
96C2×#117C2×C411yescharacteristic
97C2×#117C2×C411yescharacteristic
98C22×C42C2311yescharacteristic
99#432—22no—
100#432—22no—
101#431—22no—
102#431—22no—
103#423—22no—
104#423—22no—
105C2×C4⋊C8—22no—
106C2×C4⋊C8—22no—
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⋊C8—21no—
116C2×C4⋊C8—21no—
117C2×C4⋊C8C2311yescharacteristic
118C2×C4⋊C8C2311yescharacteristic
119C2×C4⋊C8C2311yescharacteristic
120C2×C4⋊C8C2311yescharacteristic
121C2×C4⋊C8C2311yescharacteristic
122C2×C4⋊C8C2311yescharacteristic
123C2×C4⋊C8C2311yescharacteristic
124C2×C4⋊C8C2311yescharacteristic
125C2×C22⋊C8—21no—
126C2×C22⋊C8—21no—
127C2×#97—21no—
128C2×C4×C8—21no—
129#391—21no—
130#391—21no—
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#389C2×C412yes—
142#389C2×C412yes—
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⋊C4C2×C412yes—
162C4×C4⋊C4C2×C412yes—
163C4×C4⋊C4C2×C412yes—
164C4×C4⋊C4D411yescharacteristic
165C4×C4⋊C4D411yescharacteristic
166C4×C4⋊C4D411yescharacteristic
167C4×C4⋊C4D411yescharacteristic
168C4×C22⋊C4—22no—
169C4×C22⋊C4—22no—
170C4×C22⋊C4—22no—
171C4×C22⋊C4D411yescharacteristic
172C4×C22⋊C4D411yescharacteristic
173C4×C22⋊C4D411yescharacteristic
174C4×C22⋊C4D411yescharacteristic
175#376C2×C411yescharacteristic
176#376C2×C411yescharacteristic
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#332—22no—
189#332—22no—
190#332—22no—
191#332—22no—
192#331—22no—
193#331—22no—
194#331—22no—
195#331—22no—
196C2×#41—24no—
197C2×#41—24no—
198C2×#41—81no—
199C2×#41—81no—
200C2×#41—81no—
201C2×#41—81no—
202C22×Q8—21no—
203C22×D4—22no—
204C22×D4—21no—
205C22×D4—21no—
206C22×D4C22×C412yes—
207C23×C4—22no—
208C23×C4—22no—
209C23×C4—22no—
210C23×C4C22×C411yescharacteristic
211C23×C4C2×D411yescharacteristic
212C23×C4C2×D411yescharacteristic
213C2×M16—42no—
214C2×M16—81no—
215C2×M16—81no—
216C22×C8—21no—
217C22×C8—21no—
218C22×C8—21no—
219C22×C8—21no—
220C22×C8—21no—
221C22×C8—21no—
222C22×C8—21no—
223C22×C8—21no—
224C22×C8—42no—
225C22×C8—81no—
226C22×C8—81no—
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—81no—
242C4×Q8—81no—
243C4×Q8—81no—
244C4×Q8—81no—
245C4×D4—24no—
246C4×D4—24no—
247C4×D4—24no—
248C4×D4—44no—
249C4×D4—44no—
250C4×D4—81no—
251C4×D4—81no—
252C4×D4—81no—
253C4×D4—81no—
254C4×D4—81no—
255C4×D4—81no—
256C4×D4—81no—
257C4×D4—81no—
258C4×D4C4⋊C418yes—
259#117—42no—
260#117—42no—
261#117—81no—
262#117—81no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4C22⋊C411yescharacteristic
271C2×C4⋊C4C22⋊C411yescharacteristic
272C2×C4⋊C4C22⋊C411yescharacteristic
273C2×C4⋊C4C22⋊C411yescharacteristic
274C2×C4⋊C4C22×C412yes—
275C2×C4⋊C4C22×C412yes—
276C2×C4⋊C4C22×C412yes—
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—22no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4C22⋊C411yescharacteristic
287C2×C22⋊C4C22⋊C411yescharacteristic
288C2×C22⋊C4C22⋊C411yescharacteristic
289C2×C22⋊C4C22⋊C411yescharacteristic
290C2×C42—22no—
291C2×C42—22no—
292C2×C42—22no—
293C2×C42C22×C411yescharacteristic
294C2×C42C22×C411yescharacteristic
295C2×C42C22×C411yescharacteristic
296C2×C42C22×C411yescharacteristic
297C2×C42C22×C411yescharacteristic
298C2×C42C2×D411yescharacteristic
299C2×C42C2×D411yescharacteristic
300C2×C42C2×D411yescharacteristic
301C2×C42C2×D411yescharacteristic
302C2×C42C2×D411yescharacteristic
303C2×C42C2×D411yescharacteristic
304C2×C42C2×D411yescharacteristic
305C2×C42C2×D411yescharacteristic
306C2×C42C2×D412yes—
307C2×C42C2×D411yescharacteristic
308C2×C42C2×D411yescharacteristic
309C2×C42C2×Q812yes—
310C2×C42#4112yes—
311C2×C42#4112yes—
312C2×C42C2411yescharacteristic
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—21no—
325C4⋊C8—21no—
326C4⋊C8—21no—
327C4⋊C8—21no—
328C4⋊C8—21no—
329C4⋊C8—42no—
330C4⋊C8—42no—
331C4⋊C8—81no—
332C4⋊C8—81no—
333C4⋊C8—81no—
334C4⋊C8—81no—
335C4⋊C8—81no—
336C4⋊C8—81no—
337C4⋊C8—81no—
338C4⋊C8—81no—
339C22⋊C8—42no—
340C22⋊C8—42no—
341#97—42no—
342C4×C8—42no—
343#95—22no—
344#95—22no—
345#95—22no—
346#95—22no—
347#95—22no—
348#95—22no—
349#95C22⋊C411yescharacteristic
350#95C22⋊C411yescharacteristic
351#95C22⋊C411yescharacteristic
352#95C22⋊C411yescharacteristic
353#95C22⋊C411yescharacteristic
354#95C22⋊C411yescharacteristic
355#95C22⋊C411yescharacteristic
356#95C22⋊C411yescharacteristic
357C24—22no—
358C24C2×C22⋊C411yescharacteristic
359#41—84no—
360#41—84no—
361#41—84no—
362#41—84no—
363C2×Q8—22no—
364C2×Q8—22no—
365C2×Q8—41no—
366C2×Q8—41no—
367C2×Q8—41no—
368C2×Q8—41no—
369C2×D4—28no—
370C2×D4—22no—
371C2×D4—22no—
372C2×D4—22no—
373C2×D4—22no—
374C2×D4—22no—
375C2×D4—42no—
376C2×D4—42no—
377C2×D4—41no—
378C2×D4—41no—
379C2×D4—41no—
380C2×D4—41no—
381C2×D4—41no—
382C2×D4—41no—
383C2×D4—41no—
384C2×D4—41no—
385C2×D4—82no—
386C2×D4—82no—
387C2×D4C2×C4⋊C414yes—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—28no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—44no—
403C22×C4—44no—
404C22×C4—82no—
405C22×C4—82no—
406C22×C4—82no—
407C22×C4—82no—
408C22×C4—82no—
409C22×C4—82no—
410C22×C4C2×C22⋊C411yescharacteristic
411C22×C4C2×C22⋊C411yescharacteristic
412C22×C4C2×C22⋊C411yescharacteristic
413C22×C4C2×C22⋊C411yescharacteristic
414C22×C4C2×C22⋊C411yescharacteristic
415C22×C4C2×C22⋊C411yescharacteristic
416C22×C4C2×C22⋊C411yescharacteristic
417C22×C4C2×C22⋊C411yescharacteristic
418C22×C4C2×C22⋊C411yescharacteristic
419C22×C4C2×C22⋊C411yescharacteristic
420C22×C4C2×C22⋊C411yescharacteristic
421C22×C4C2×C4⋊C412yes—
422C22×C4#11712yes—
423C22×C4#12112yes—
424C22×C4#12112yes—
425C22×C4C22⋊Q812yes—
426C22×C4C22⋊Q812yes—
427C22×C4C23×C411yescharacteristic
428C22×C4C22×D411yescharacteristic
429C22×C4C22×D411yescharacteristic
430M16—84no—
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—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—84no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—41no—
462C4⋊C4—41no—
463C4⋊C4—42no—
464C4⋊C4—42no—
465C4⋊C4—41no—
466C4⋊C4—41no—
467C4⋊C4—41no—
468C4⋊C4—41no—
469C4⋊C4—41no—
470C4⋊C4—41no—
471C4⋊C4—41no—
472C4⋊C4—41no—
473C4⋊C4—82no—
474C4⋊C4—82no—
475C4⋊C4C2×C4⋊C414yes—
476C22⋊C4—28no—
477C22⋊C4—28no—
478C22⋊C4—22no—
479C22⋊C4—22no—
480C22⋊C4—41no—
481C22⋊C4—41no—
482C22⋊C4—44no—
483C22⋊C4—44no—
484C22⋊C4—82no—
485C22⋊C4—82no—
486C22⋊C4—82no—
487C22⋊C4—82no—
488C22⋊C4—82no—
489C42—21no—
490C42—21no—
491C42—21no—
492C42—21no—
493C42—21no—
494C42—21no—
495C42—21no—
496C42—21no—
497C42—21no—
498C42—21no—
499C42—21no—
500C42—21no—
501C42—21no—
502C42—21no—
503C42—21no—
504C42—21no—
505C42—22no—
506C42—22no—
507C42—41no—
508C42—41no—
509C42—82no—
510C42—82no—
511C42—82no—
512C42C2×C4⋊C412yes—
513C42#11712yes—
514C42#12112yes—
515C42#12112yes—
516C42C22⋊Q812yes—
517C42C22⋊Q812yes—
518C42C22×D411yescharacteristic
519C42C22×Q811yescharacteristic
520C42C2×#4111yescharacteristic
521C42C2×#4111yescharacteristic
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—24no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23#38012yes—
536C23C22×C22⋊C411yescharacteristic
537Q8—42no—
538Q8—42no—
539Q8—42no—
540Q8—42no—
541D4—44no—
542D4—44no—
543D4—42no—
544D4—42no—
545D4—42no—
546D4—42no—
547D4—42no—
548D4—42no—
549D4—42no—
550D4—42no—
551D4—88no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—24no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—88no—
602C2×C4—164no—
603C2×C4—164no—
604C2×C4#38012yes—
605C2×C4#38012yes—
606C2×C4#38012yes—
607C2×C4#41312yes—
608C2×C4#41312yes—
609C2×C4C22×C4⋊C411yescharacteristic
610C2×C4C2×#11711yescharacteristic
611C2×C4C2×#12111yescharacteristic
612C2×C4C2×#12111yescharacteristic
613C2×C4C2×C22⋊Q811yescharacteristic
614C2×C4C2×C22⋊Q811yescharacteristic
615C2×C4#57611yescharacteristic
616C2×C4#57611yescharacteristic
617C2×C4#57611yescharacteristic
618C2×C4#57711yescharacteristic
619C2×C4#57711yescharacteristic
620C2×C4#57711yescharacteristic
621C2×C4#57711yescharacteristic
622C2×C4#57811yescharacteristic
623C8—81no—
624C8—81no—
625C8—81no—
626C8—81no—
627C8—81no—
628C8—81no—
629C8—81no—
630C8—81no—
631C22—22no—
632C22—22no—
633C22—22no—
634C22—41no—
635C22—41no—
636C22—41no—
637C22—41no—
638C22—44no—
639C22—44no—
640C22—44no—
641C22—82no—
642C22—82no—
643C22—82no—
644C22—82no—
645C22C2×#38011yescharacteristic
646C22C2×#41311yescharacteristic
647C22C2×#41311yescharacteristic
648C22#289711yescharacteristic
649C22#289711yescharacteristic
650C22#289711yescharacteristic
651C22#289811yescharacteristic
652C4—22no—
653C4—22no—
654C4—21no—
655C4—21no—
656C4—22no—
657C4—22no—
658C4—22no—
659C4—41no—
660C4—41no—
661C4—41no—
662C4—41no—
663C4—41no—
664C4—41no—
665C4—44no—
666C4—82no—
667C4—82no—
668C4#304111yescharacteristic
669C4#304211yescharacteristic
670C4#304511yescharacteristic
671C4#304611yescharacteristic
672C4#308111yescharacteristic
673C4#308111yescharacteristic
674C4#308311yescharacteristic
675C4#308411yescharacteristic
676C2—42no—
677C2—84no—
678C2#1777111yescharacteristic
679C2#1777211yescharacteristic
680C2#1777511yescharacteristic
681C2#1777611yescharacteristic
682C2#2063711yescharacteristic
683C2#2065011yescharacteristic
684C2#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. #21976C2semidirect
3. #21974C2semidirect
6. #8593C2semidirect
7. #8592C2semidirect
8. #8591C2semidirect
9. #8589C2semidirect
10. #8588C2semidirect
11. #8588C2semidirect
12. #8586C2semidirect
13. #8585C2semidirect
14. #8046C2semidirect
15. #8045C2semidirect

Maximal subgroups

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

subgroupindex
#342262
#219762
#219742
C2×#15572
C2×#15402
#85932
#85922
#85912
#85892
#85882
#85862
#85852
#80462
#80452