Finite Group Databasebeta

← #241173

Subgroups — #241173

Order 512.

1218 conjugacy classes of subgroups.

The subgroup lattice has 619 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35020C211yesmaximal, characteristic
2C2×#1853C211yesmaximal, characteristic
3C2×#1852C211yesmaximal, characteristic
4#14130C211yesmaximal, characteristic
5#10572C214yesmaximal, complemented
6#10571C214yesmaximal, complemented
7#8023C211yesmaximal, characteristic
8#8023C211yesmaximal, characteristic
9#7907C211yesmaximal, characteristic
10C23×C4⋊C4C2211yescharacteristic
11C22×#107—21no—
12C22×#106—21no—
13C22×C22⋊C8C2211yescharacteristic
14C22×C22⋊C8C2211yescharacteristic
15#2851—21no—
16#2851—21no—
17#2646C2212yes—
18#2646C2212yes—
19#2569—22no—
20#2569—22no—
21#2442—21no—
22#2442—21no—
23#2442—21no—
24#2442—21no—
25#2315—21no—
26#2315—21no—
27C2×#389—21no—
28C2×#385—21no—
29C2×#380C2211yescharacteristic
30C2×#380C2211yescharacteristic
31C2×#380C2211yescharacteristic
32C2×C4×C22⋊C4C2211yescharacteristic
33#1853—22no—
34#1853C2214yes—
35#1852—22no—
36#1852C2214yes—
37C2×#340—21no—
38C2×#340—21no—
39#1444C2212yes—
40#1442C2212yes—
41C4⋊#95—21no—
42C22⋊#95—21no—
43#1294C2214yes—
44#1294C2214yes—
45#1280C2212yes—
46#1280C2212yes—
47C24×C4—21no—
48C24×C4C2311yescharacteristic
49C23×C8—21no—
50C23×C8—21no—
51C2×#125—21no—
52C2×#125—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—44no—
61C22×C4⋊C4D411yescharacteristic
62C22×C4⋊C4D411yescharacteristic
63C22×C4⋊C4D411yescharacteristic
64C22×C4⋊C4D411yescharacteristic
65C22×C4⋊C4C2312yes—
66C22×C22⋊C4—21no—
67C22×C22⋊C4—21no—
68C22×C22⋊C4—21no—
69C22×C22⋊C4D411yescharacteristic
70C22×C22⋊C4D411yescharacteristic
71C22×C42D411yescharacteristic
72C22×C42D411yescharacteristic
73C2×#107—24no—
74C2×#107—44no—
75C2×#106—24no—
76C2×#106—44no—
77C2×C22⋊C8—21no—
78C2×C22⋊C8—21no—
79C2×C22⋊C8—22no—
80C2×C22⋊C8—22no—
81C2×C22⋊C8Q814yes—
82C2×C22⋊C8Q814yes—
83C2×C22⋊C8C2312yes—
84C2×C22⋊C8C2312yes—
85#400—22no—
86#400—22no—
87#400—22no—
88#400—22no—
89#398—22no—
90#398—22no—
91#398—22no—
92#398—22no—
93#389—21no—
94#389—21no—
95#389—21no—
96#389—21no—
97#389—44no—
98#387—22no—
99#387—22no—
100#387—22no—
101#387—22no—
102#385—22no—
103#385—22no—
104#385—22no—
105#385—22no—
106#385—22no—
107#385—22no—
108#385—22no—
109#385—22no—
110#385—44no—
111#382—21no—
112#382—21no—
113#380—22no—
114#380—22no—
115#380—22no—
116#380—22no—
117#380—21no—
118#380—21no—
119#380—21no—
120#380—21no—
121#380—22no—
122#380D412yes—
123#380D412yes—
124#380Q814yes—
125#380C2312yes—
126#380C2312yes—
127C4×C22⋊C4—21no—
128C4×C22⋊C4—21no—
129C4×C22⋊C4—21no—
130C4×C22⋊C4—21no—
131C2×#95—21no—
132C2×#95—21no—
133C2×#95—22no—
134C2×#95—22no—
135C2×#95—22no—
136C2×#95C2311yescharacteristic
137C2×#95C2311yescharacteristic
138C2×#95C2312yes—
139#340—24no—
140#340—24no—
141C25C2×D411yescharacteristic
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—21no—
148C23×C4—42no—
149C23×C4—42no—
150C23×C4—44no—
151C23×C4—41no—
152C23×C4—41no—
153C23×C4—42no—
154C23×C4—42no—
155C23×C4—82no—
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2×Q812yes—
166C23×C4C2×Q812yes—
167C23×C4C2×Q812yes—
168C23×C4#4112yes—
169C23×C4C2411yescharacteristic
170C22×C8—24no—
171C22×C8—24no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—42no—
175C22×C8—42no—
176C22×C8—42no—
177C22×C8—42no—
178#125—82no—
179#125—82no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—41no—
207C2×C4⋊C4—41no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—82no—
215C2×C4⋊C4C2×D412yes—
216C2×C4⋊C4C2×D412yes—
217C2×C4⋊C4C2×D412yes—
218C2×C4⋊C4C2×D412yes—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—42no—
233C2×C22⋊C4—42no—
234C2×C22⋊C4—42no—
235C2×C22⋊C4—42no—
236C2×C22⋊C4—42no—
237C2×C22⋊C4—42no—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42—21no—
241C2×C42—21no—
242C2×C42—42no—
243C2×C42—42no—
244#107—48no—
245#106—48no—
246C22⋊C8—24no—
247C22⋊C8—24no—
248C22⋊C8—22no—
249C22⋊C8—22no—
250C22⋊C8C2×Q814yes—
251C22⋊C8C2×Q814yes—
252#95—22no—
253#95—22no—
254#95—22no—
255#95—22no—
256#95—22no—
257#95—22no—
258#95—22no—
259#95—22no—
260#95—22no—
261#95—22no—
262#95—22no—
263#95—22no—
264#95—21no—
265#95—21no—
266#95—21no—
267#95—21no—
268#95—22no—
269#95—44no—
270#95—44no—
271#95—82no—
272#95C2×Q814yes—
273C24—41no—
274C24—41no—
275C24—41no—
276C24—41no—
277C24#12011yescharacteristic
278C24#12011yescharacteristic
279C24C22⋊Q812yes—
280C24C4⋊Q812yes—
281C24#13512yes—
282C24#13512yes—
283C24AGL(1,ℤ/8)12yes—
284C24#13712yes—
285C24C22×D411yescharacteristic
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—24no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—24no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—44no—
311C22×C4—44no—
312C22×C4—44no—
313C22×C4—44no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—44no—
332C22×C4—44no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—81no—
347C22×C4—81no—
348C22×C4—81no—
349C22×C4—81no—
350C22×C4—81no—
351C22×C4—81no—
352C22×C4—81no—
353C22×C4—81no—
354C22×C4—88no—
355C22×C4—88no—
356C22×C4—82no—
357C22×C4—82no—
358C22×C4—82no—
359C22×C4—82no—
360C22×C4#12011yescharacteristic
361C22×C4#12011yescharacteristic
362C22×C4C22⋊Q812yes—
363C22×C4C22⋊Q812yes—
364C22×C4C22⋊Q812yes—
365C22×C4C22⋊Q812yes—
366C22×C4C22⋊Q812yes—
367C22×C4C4⋊Q812yes—
368C22×C4C4⋊Q812yes—
369C22×C4C4⋊Q812yes—
370C22×C4C4⋊Q812yes—
371C22×C4C4⋊Q812yes—
372C22×C4C22×D411yescharacteristic
373C22×C4C22×D411yescharacteristic
374C22×C4C22×Q811yescharacteristic
375C22×C4C22×Q811yescharacteristic
376C22×C4C22×Q811yescharacteristic
377C22×C4C2×#4111yescharacteristic
378C2×C8—24no—
379C2×C8—24no—
380C2×C8—42no—
381C2×C8—42no—
382C2×C8—42no—
383C2×C8—42no—
384C2×C8—48no—
385C2×C8—48no—
386C4⋊C4—24no—
387C4⋊C4—24no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—48no—
393C4⋊C4—44no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—44no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—82no—
402C4⋊C4—82no—
403C4⋊C4—82no—
404C4⋊C4—82no—
405C4⋊C4—82no—
406C4⋊C4—82no—
407C4⋊C4—82no—
408C4⋊C4—82no—
409C4⋊C4—88no—
410C22⋊C4—42no—
411C22⋊C4—42no—
412C22⋊C4—42no—
413C22⋊C4—42no—
414C22⋊C4—42no—
415C22⋊C4—42no—
416C22⋊C4—44no—
417C22⋊C4—42no—
418C22⋊C4—42no—
419C22⋊C4—44no—
420C22⋊C4—48no—
421C42—42no—
422C42—42no—
423C42—81no—
424C42—81no—
425C23—21no—
426C23—21no—
427C23—21no—
428C23—21no—
429C23—21no—
430C23—21no—
431C23—21no—
432C23—21no—
433C23—21no—
434C23—21no—
435C23—21no—
436C23—21no—
437C23—42no—
438C23—42no—
439C23—42no—
440C23—42no—
441C23—42no—
442C23—42no—
443C23—42no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23—42no—
448C23—42no—
449C23#39512yes—
450C23#39512yes—
451C23#44912yes—
452C23#44912yes—
453C23#45212yes—
454C23#45212yes—
455C23#47814yes—
456C23#47914yes—
457C23#49914yes—
458C23#50114yes—
459C23C2×#12011yescharacteristic
460C23C2×C22⋊Q811yescharacteristic
461C23C2×C4⋊Q811yescharacteristic
462C23C2×#13511yescharacteristic
463C23C2×#13511yescharacteristic
464C23C2×AGL(1,ℤ/8)11yescharacteristic
465C23C2×#13711yescharacteristic
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—44no—
487C2×C4—42no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—44no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4—88no—
527C2×C4—88no—
528C2×C4—82no—
529C2×C4—82no—
530C2×C4—81no—
531C2×C4—81no—
532C2×C4—81no—
533C2×C4—81no—
534C2×C4—162no—
535C2×C4—162no—
536C2×C4—162no—
537C2×C4—162no—
538C2×C4#39512yes—
539C2×C4#39512yes—
540C2×C4B(4,𝔽2)12yes—
541C2×C4#45812yes—
542C2×C4C2×C22⋊Q811yescharacteristic
543C2×C4C2×C22⋊Q811yescharacteristic
544C2×C4C2×C4⋊Q811yescharacteristic
545C2×C4C2×C4⋊Q811yescharacteristic
546C8—44no—
547C8—44no—
548C22—21no—
549C22—21no—
550C22—22no—
551C22—22no—
552C22—21no—
553C22—21no—
554C22—22no—
555C22—22no—
556C22—21no—
557C22—21no—
558C22—22no—
559C22—22no—
560C22—21no—
561C22—21no—
562C22—22no—
563C22—22no—
564C22—21no—
565C22—21no—
566C22—22no—
567C22—22no—
568C22—21no—
569C22—21no—
570C22—22no—
571C22—22no—
572C22—44no—
573C22—44no—
574C22—44no—
575C22—44no—
576C22#205914yes—
577C22#205914yes—
578C22C2×#39511yescharacteristic
579C22C2×#44911yescharacteristic
580C22C2×#45211yescharacteristic
581C22#302111yescharacteristic
582C22#302211yescharacteristic
583C22C2×B(4,𝔽2)11yescharacteristic
584C22C2×#45811yescharacteristic
585C22C2×#47811yescharacteristic
586C22C2×#47911yescharacteristic
587C22C2×#49911yescharacteristic
588C22C2×#50111yescharacteristic
589C4—21no—
590C4—21no—
591C4—21no—
592C4—21no—
593C4—22no—
594C4—84no—
595C4—81no—
596C4—84no—
597C4—82no—
598C4—81no—
599C4—82no—
600C4—81no—
601C4—82no—
602C4—81no—
603C4—82no—
604C4#206214yes—
605C2—21no—
606C2—21no—
607C2—24no—
608C2—24no—
609C2—21no—
610C2—21no—
611C2#1819811yescharacteristic
612C2#1819911yescharacteristic
613C2#1820611yescharacteristic
614C2#1820711yescharacteristic
615C2C2×#205911yescharacteristic
616C2#2356211yescharacteristic
617C2C2×#206211yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #10572C2semidirect
6. #10571C2semidirect

Maximal subgroups

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

subgroupindex
#350202
C2×#18532
C2×#18522
#141302
#105722
#105712
#80232
#79072