Finite Group Databasebeta

← #190800

Subgroups — #190800

Order 512.

971 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34226C211yesmaximal, characteristic
2#21924C211yesmaximal, characteristic, complemented
3#21924C211yesmaximal, characteristic, complemented
4#8727C211yesmaximal, characteristic, complemented
5#8726C211yesmaximal, characteristic, complemented
6#8593C211yesmaximal, characteristic, complemented
7#8588C211yesmaximal, characteristic, complemented
8#8588C211yesmaximal, characteristic, complemented
9#8586C211yesmaximal, characteristic, complemented
10#8341C211yesmaximal, characteristic
11#8341C211yesmaximal, characteristic
12#8237C211yesmaximal, characteristic, complemented
13#8236C211yesmaximal, characteristic, complemented
14#8231C211yesmaximal, characteristic, complemented
15C4×#331C211yesmaximal, characteristic, complemented
16C2×#532C2211yescharacteristic
17#2367—21no—
18#2367—21no—
19#2363—21no—
20#2363—21no—
21#2357—21no—
22#2349—21no—
23#2341—21no—
24#2339—21no—
25#2324C412yes—
26#2324C412yes—
27#2319C412yes—
28C4×C4⋊Q8C2211yescharacteristic
29C4×C4⋊Q8C2211yescharacteristic
30C4×#127C2211yescharacteristic
31C4×#127C2211yescharacteristic
32#2285C2211yescharacteristic
33#2275C412yes—
34C4×#117C2211yescharacteristic
35#1924—21no—
36#1924—21no—
37#1923—21no—
38#1923—21no—
39#1917—21no—
40#1917—21no—
41#1916—21no—
42#1916—21no—
43#1842C2211yescharacteristic
44#1842C2211yescharacteristic
45#1831C2211yescharacteristic
46#1831C2211yescharacteristic
47#1797C2211yescharacteristic
48#1797C2211yescharacteristic
49#1797C2211yescharacteristic
50#1797C2211yescharacteristic
51#1767C2211yescharacteristic
52#1767C2211yescharacteristic
53C4×C4⋊C8C2211yescharacteristic
54C4×C4⋊C8C2211yescharacteristic
55C2×#332C2211yescharacteristic
56C2×#332C2211yescharacteristic
57C2×#332C2211yescharacteristic
58C2×#332C2211yescharacteristic
59C2×#331C2211yescharacteristic
60C2×#331C2211yescharacteristic
61C2×#331C2211yescharacteristic
62C2×#331C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1296C2211yescharacteristic
66#1296C2211yescharacteristic
67#1296C2211yescharacteristic
68#1296C2211yescharacteristic
69#1296C2211yescharacteristic
70#1296C2211yescharacteristic
71C22×#41—21no—
72#532—22no—
73#532—22no—
74#532—22no—
75#532—22no—
76C2×C4⋊Q8C2311yescharacteristic
77C2×#127C2311yescharacteristic
78C2×#124—21no—
79C2×#121—21no—
80C2×#121C2×C412yes—
81C2×C4×Q8—21no—
82C2×C4×Q8—21no—
83C2×C4×D4—21no—
84C2×C4×D4—21no—
85C2×C4×D4—21no—
86C2×C4×D4—21no—
87C2×C4×D4—21no—
88C2×C4×D4C2×C412yes—
89C2×C4×D4C2×C412yes—
90C2×C4×D4C2×C412yes—
91C2×#117—21no—
92C2×#117C2×C411yescharacteristic
93C2×#117C2×C411yescharacteristic
94C22×C42C2311yescharacteristic
95C2×C4⋊C8—21no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8—21no—
98C2×C4⋊C8—21no—
99C2×C4⋊C8C2311yescharacteristic
100C2×C4⋊C8C2311yescharacteristic
101C2×C4⋊C8C2311yescharacteristic
102C2×C4⋊C8C2311yescharacteristic
103C2×C4⋊C8C2311yescharacteristic
104C2×C4⋊C8C2311yescharacteristic
105C2×C4⋊C8C2311yescharacteristic
106C2×C4⋊C8C2311yescharacteristic
107C2×C22⋊C8—21no—
108C2×C22⋊C8—21no—
109C2×C22⋊C8—21no—
110C2×C22⋊C8—21no—
111C2×#97—21no—
112C2×#97—21no—
113C2×C4×C8—21no—
114C2×C4×C8—21no—
115#391—21no—
116#391—21no—
117#390—22no—
118#390—22no—
119#390—21no—
120#390—21no—
121#390—21no—
122#390—21no—
123#390C2×C412yes—
124#390C2×C412yes—
125#389—21no—
126#389—21no—
127#389C2×C412yes—
128#389C2×C412yes—
129#388—22no—
130#388—22no—
131#388—22no—
132#388—22no—
133#386—22no—
134#386—22no—
135#385—22no—
136#385—22no—
137#384C2311yescharacteristic
138#384C2311yescharacteristic
139#382—21no—
140#381—22no—
141#380—21no—
142#380—22no—
143#380C2×C412yes—
144C4×C4⋊C4—21no—
145C4×C4⋊C4—21no—
146C4×C4⋊C4—21no—
147C4×C4⋊C4C2×C412yes—
148C4×C4⋊C4C2×C412yes—
149C4×C4⋊C4C2×C412yes—
150C4×C4⋊C4D411yescharacteristic
151C4×C4⋊C4D411yescharacteristic
152C4×C4⋊C4D411yescharacteristic
153C4×C4⋊C4D411yescharacteristic
154C4×C22⋊C4—22no—
155C4×C22⋊C4—22no—
156C4×C22⋊C4—22no—
157C4×C22⋊C4D411yescharacteristic
158C4×C22⋊C4D411yescharacteristic
159C4×C22⋊C4D411yescharacteristic
160C4×C22⋊C4D411yescharacteristic
161#376C2×C411yescharacteristic
162#376C2×C411yescharacteristic
163C2×#95—22no—
164C43C2311yescharacteristic
165C43C2311yescharacteristic
166#336—21no—
167#336—21no—
168#336—21no—
169#336—21no—
170#336—21no—
171#336—21no—
172#336—21no—
173#336—21no—
174#336—21no—
175#336—21no—
176#336—21no—
177#336—21no—
178#332—22no—
179#332—22no—
180#332—22no—
181#332—22no—
182#331—22no—
183#331—22no—
184#331—22no—
185#331—22no—
186C2×#41—24no—
187C2×#41—24no—
188C2×#41—81no—
189C2×#41—81no—
190C2×#41—81no—
191C2×#41—81no—
192C22×Q8—21no—
193C22×D4—22no—
194C22×D4—21no—
195C22×D4—21no—
196C22×D4C22×C412yes—
197C23×C4—22no—
198C23×C4—22no—
199C23×C4—22no—
200C23×C4C22×C411yescharacteristic
201C23×C4C2×D411yescharacteristic
202C23×C4C2×D411yescharacteristic
203C22×C8—21no—
204C22×C8—21no—
205C22×C8—21no—
206C22×C8—21no—
207C22×C8—21no—
208C22×C8—21no—
209C22×C8—21no—
210C22×C8—21no—
211C4⋊Q8—21no—
212C4⋊Q8—21no—
213C4⋊Q8#4112yes—
214C4⋊Q8#4112yes—
215#127—21no—
216#127—21no—
217#127#4112yes—
218#127#4112yes—
219#124—24no—
220#124—24no—
221#121—24no—
222#121—24no—
223#121—24no—
224#121—24no—
225C4×Q8—81no—
226C4×Q8—81no—
227C4×Q8—81no—
228C4×Q8—81no—
229C4×D4—24no—
230C4×D4—24no—
231C4×D4—24no—
232C4×D4—24no—
233C4×D4—44no—
234C4×D4—44no—
235C4×D4—81no—
236C4×D4—81no—
237C4×D4—81no—
238C4×D4—81no—
239C4×D4—81no—
240C4×D4—81no—
241C4×D4—81no—
242C4×D4—81no—
243#117—42no—
244#117—42no—
245#117—81no—
246#117—81no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4C22⋊C411yescharacteristic
255C2×C4⋊C4C22⋊C411yescharacteristic
256C2×C4⋊C4C22⋊C411yescharacteristic
257C2×C4⋊C4C22⋊C411yescharacteristic
258C2×C4⋊C4C22×C412yes—
259C2×C4⋊C4C22×C412yes—
260C2×C4⋊C4C22×C412yes—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4C22⋊C411yescharacteristic
271C2×C22⋊C4C22⋊C411yescharacteristic
272C2×C22⋊C4C22⋊C411yescharacteristic
273C2×C22⋊C4C22⋊C411yescharacteristic
274C2×C42—21no—
275C2×C42—21no—
276C2×C42—21no—
277C2×C42—21no—
278C2×C42—22no—
279C2×C42—22no—
280C2×C42—22no—
281C2×C42C22×C411yescharacteristic
282C2×C42C22×C411yescharacteristic
283C2×C42C22×C411yescharacteristic
284C2×C42C22×C411yescharacteristic
285C2×C42C22×C411yescharacteristic
286C2×C42C2×D411yescharacteristic
287C2×C42C2×D411yescharacteristic
288C2×C42C2×D411yescharacteristic
289C2×C42C2×D411yescharacteristic
290C2×C42C2×D411yescharacteristic
291C2×C42C2×D411yescharacteristic
292C2×C42C2×D411yescharacteristic
293C2×C42C2×D411yescharacteristic
294C2×C42C2×D411yescharacteristic
295C2×C42C2×D411yescharacteristic
296C2×C42C2411yescharacteristic
297C4⋊C8—21no—
298C4⋊C8—21no—
299C4⋊C8—21no—
300C4⋊C8—21no—
301C4⋊C8—21no—
302C4⋊C8—21no—
303C4⋊C8—21no—
304C4⋊C8—21no—
305C4⋊C8—21no—
306C4⋊C8—21no—
307C4⋊C8—21no—
308C4⋊C8—21no—
309C4⋊C8—21no—
310C4⋊C8—21no—
311C4⋊C8—21no—
312C4⋊C8—21no—
313C4⋊C8—81no—
314C4⋊C8—81no—
315C4⋊C8—81no—
316C4⋊C8—81no—
317C22⋊C8—81no—
318C22⋊C8—81no—
319C22⋊C8—81no—
320C22⋊C8—81no—
321#97—81no—
322#97—81no—
323C4×C8—81no—
324C4×C8—81no—
325#95—22no—
326#95—22no—
327#95—22no—
328#95—22no—
329#95—22no—
330#95—22no—
331#95C22⋊C411yescharacteristic
332#95C22⋊C411yescharacteristic
333#95C22⋊C411yescharacteristic
334#95C22⋊C411yescharacteristic
335#95C22⋊C411yescharacteristic
336#95C22⋊C411yescharacteristic
337#95C22⋊C411yescharacteristic
338#95C22⋊C411yescharacteristic
339C24—22no—
340C24C2×C22⋊C411yescharacteristic
341#41—84no—
342#41—84no—
343#41—84no—
344#41—84no—
345C2×Q8—22no—
346C2×Q8—22no—
347C2×Q8—41no—
348C2×Q8—41no—
349C2×Q8—41no—
350C2×Q8—41no—
351C2×D4—28no—
352C2×D4—22no—
353C2×D4—22no—
354C2×D4—22no—
355C2×D4—22no—
356C2×D4—22no—
357C2×D4—42no—
358C2×D4—42no—
359C2×D4—41no—
360C2×D4—41no—
361C2×D4—41no—
362C2×D4—41no—
363C2×D4—41no—
364C2×D4—41no—
365C2×D4—41no—
366C2×D4—41no—
367C2×D4—82no—
368C2×D4—82no—
369C2×D4#11714yes—
370C22×C4—22no—
371C22×C4—28no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—44no—
391C22×C4—44no—
392C22×C4—82no—
393C22×C4—82no—
394C22×C4—82no—
395C22×C4—82no—
396C22×C4—82no—
397C22×C4—82no—
398C22×C4C2×C22⋊C411yescharacteristic
399C22×C4C2×C22⋊C411yescharacteristic
400C22×C4C2×C22⋊C411yescharacteristic
401C22×C4C2×C22⋊C411yescharacteristic
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×C4C23×C411yescharacteristic
410C22×C4C22×D411yescharacteristic
411C22×C4C22×D411yescharacteristic
412C2×C8—41no—
413C2×C8—41no—
414C2×C8—41no—
415C2×C8—41no—
416C2×C8—41no—
417C2×C8—41no—
418C2×C8—41no—
419C2×C8—41no—
420C2×C8—41no—
421C2×C8—41no—
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—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—41no—
442C4⋊C4—41no—
443C4⋊C4—42no—
444C4⋊C4—42no—
445C4⋊C4—41no—
446C4⋊C4—41no—
447C4⋊C4—41no—
448C4⋊C4—41no—
449C4⋊C4—41no—
450C4⋊C4—41no—
451C4⋊C4—41no—
452C4⋊C4—41no—
453C4⋊C4—82no—
454C4⋊C4—82no—
455C4⋊C4#11714yes—
456C22⋊C4—28no—
457C22⋊C4—28no—
458C22⋊C4—22no—
459C22⋊C4—22no—
460C22⋊C4—41no—
461C22⋊C4—41no—
462C22⋊C4—44no—
463C22⋊C4—44no—
464C22⋊C4—82no—
465C22⋊C4—82no—
466C22⋊C4—82no—
467C22⋊C4—82no—
468C22⋊C4—82no—
469C42—21no—
470C42—21no—
471C42—21no—
472C42—21no—
473C42—21no—
474C42—21no—
475C42—21no—
476C42—21no—
477C42—21no—
478C42—21no—
479C42—21no—
480C42—21no—
481C42—21no—
482C42—21no—
483C42—21no—
484C42—21no—
485C42—21no—
486C42—21no—
487C42—21no—
488C42—21no—
489C42—21no—
490C42—21no—
491C42—22no—
492C42—22no—
493C42—41no—
494C42—41no—
495C42—82no—
496C42—82no—
497C42—82no—
498C42C2×#4111yescharacteristic
499C42C2×#4111yescharacteristic
500C42C2×#4111yescharacteristic
501C42C2×#4111yescharacteristic
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—21no—
507C23—21no—
508C23—24no—
509C23—21no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23C22×C22⋊C411yescharacteristic
517Q8—42no—
518Q8—42no—
519Q8—42no—
520Q8—42no—
521D4—44no—
522D4—44no—
523D4—42no—
524D4—42no—
525D4—42no—
526D4—42no—
527D4—42no—
528D4—42no—
529D4—42no—
530D4—42no—
531D4—88no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
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—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—21no—
565C2×C4—24no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—88no—
585C2×C4—164no—
586C2×C4—164no—
587C2×C4#41312yes—
588C2×C4#41312yes—
589C2×C4C2×#11711yescharacteristic
590C2×C4C2×#11711yescharacteristic
591C2×C4#52511yescharacteristic
592C2×C4#52511yescharacteristic
593C2×C4#52511yescharacteristic
594C2×C4#52511yescharacteristic
595C2×C4#57611yescharacteristic
596C2×C4#57611yescharacteristic
597C2×C4#57611yescharacteristic
598C2×C4#57711yescharacteristic
599C2×C4#57711yescharacteristic
600C2×C4#57711yescharacteristic
601C2×C4#57711yescharacteristic
602C2×C4#57811yescharacteristic
603C8—81no—
604C8—81no—
605C8—81no—
606C8—81no—
607C8—81no—
608C8—81no—
609C8—81no—
610C8—81no—
611C22—41no—
612C22—44no—
613C22—41no—
614C22—44no—
615C22—41no—
616C22—41no—
617C22—41no—
618C22—41no—
619C22—41no—
620C22—44no—
621C22—82no—
622C22—82no—
623C22—82no—
624C22—82no—
625C22#228011yescharacteristic
626C22C2×#41311yescharacteristic
627C22C2×#41311yescharacteristic
628C22#289711yescharacteristic
629C22#289711yescharacteristic
630C22#289711yescharacteristic
631C22#289811yescharacteristic
632C4—22no—
633C4—22no—
634C4—21no—
635C4—21no—
636C4—41no—
637C4—44no—
638C4—41no—
639C4—41no—
640C4—41no—
641C4—41no—
642C4—41no—
643C4—41no—
644C4—41no—
645C4—41no—
646C4—82no—
647C4—82no—
648C4#311811yescharacteristic
649C4#311811yescharacteristic
650C4#311911yescharacteristic
651C4#312011yescharacteristic
652C4#312211yescharacteristic
653C4#312311yescharacteristic
654C4#312411yescharacteristic
655C4#312511yescharacteristic
656C2—81no—
657C2—84no—
658C2#1777111yescharacteristic
659C2#1777211yescharacteristic
660C2#1777511yescharacteristic
661C2#1777611yescharacteristic
662C2#2072811yescharacteristic
663C2#2074411yescharacteristic
664C2#2074611yescharacteristic
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. #21924C2semidirect
3. #21924C2semidirect
4. #8727C2semidirect
5. #8726C2semidirect
6. #8593C2semidirect
7. #8588C2semidirect
8. #8588C2semidirect
9. #8586C2semidirect
12. #8237C2semidirect
13. #8236C2semidirect
14. #8231C2semidirect
15. C4×#331C2semidirect

Maximal subgroups

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

subgroupindex
#342262
#219242
#87272
#87262
#85932
#85882
#85862
#83412
#82372
#82362
#82312
C4×#3312