Finite Group Databasebeta

← #412505

Subgroups — #412505

Order 512.

1396 conjugacy classes of subgroups.

The subgroup lattice has 623 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2067C212yesmaximal
2C2×#1947C211yesmaximal, characteristic, complemented
3C2×#1940C211yesmaximal, characteristic, complemented
4C2×#1804C211yesmaximal, characteristic
5C2×#1762C211yesmaximal, characteristic
6#13954C211yesmaximal, characteristic, complemented
7#13906C211yesmaximal, characteristic
8#11228C212yesmaximal, complemented
9#11219C212yesmaximal, complemented
10#11066C212yesmaximal
11#10843C211yesmaximal, characteristic, complemented
12C22×C4×D4C2211yescharacteristic
13C22×#107C2212yes—
14C22×C4⋊C8C2212yes—
15C22×C4⋊C8C2211yescharacteristic
16C22×#102C2212yes—
17C22×#102C2211yescharacteristic
18C22×#102C2211yescharacteristic
19C22×#97C2211yescharacteristic
20C2×#390C2212yes—
21C2×#389C2212yes—
22C2×#389C2212yes—
23C2×#384C2211yescharacteristic
24C2×#382C2211yescharacteristic
25C2×#380—22no—
26C2×C4×C4⋊C4C2211yescharacteristic
27C2×C4×C4⋊C4C2211yescharacteristic
28#2067—28no—
29#1947—24no—
30#1940—24no—
31#1804—24no—
32#1762—24no—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2211yescharacteristic
37C2×#340C2211yescharacteristic
38C2×#340C2212yes—
39C2×#336C2212yes—
40#1444—22no—
41#1439—21no—
42#1439—21no—
43C4⋊#95—21no—
44C4⋊#95—21no—
45C4⋊#95C2212yes—
46C23×D4C2311yescharacteristic
47C24×C4—21no—
48C23×C8C2311yescharacteristic
49C23×C8C2312yes—
50C23×C8C2311yescharacteristic
51C2×C4×D4—22no—
52C2×C4×D4—22no—
53C2×C4×D4—22no—
54C2×C4×D4—22no—
55C2×C4×D4—22no—
56C2×C4×D4Q814yes—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4D412yes—
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2311yescharacteristic
67C22×C22⋊C4—21no—
68C22×C22⋊C4—22no—
69C22×C22⋊C4—22no—
70C22×C42—21no—
71C22×C42C2312yes—
72C22×C42C2311yescharacteristic
73C2×#107—24no—
74C2×#107—24no—
75C2×#107—24no—
76C2×C4⋊C8—22no—
77C2×C4⋊C8—22no—
78C2×C4⋊C8—24no—
79C2×C4⋊C8—24no—
80C2×C4⋊C8—42no—
81C2×C4⋊C8D414yes—
82C2×#102—22no—
83C2×#102—22no—
84C2×#102—22no—
85C2×#102—22no—
86C2×#102—24no—
87C2×#102—24no—
88C2×#102—42no—
89C2×#102D414yes—
90C2×#102Q814yes—
91C2×#97—22no—
92C2×#97—22no—
93C2×#97—22no—
94#390—24no—
95#390—42no—
96#389—24no—
97#389—24no—
98#389—42no—
99#389—42no—
100#384—22no—
101#384—22no—
102#382—22no—
103#382—22no—
104#380—48no—
105C4×C4⋊C4—22no—
106C4×C4⋊C4—22no—
107C4×C4⋊C4—22no—
108C4×C4⋊C4D414yes—
109C2×#95—21no—
110C2×#95—21no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95—21no—
116C2×#95D411yescharacteristic
117C2×#95D411yescharacteristic
118#340—24no—
119#340—24no—
120#340—24no—
121#340—22no—
122#340—22no—
123#340—42no—
124#336—24no—
125C25—21no—
126C22×D4—21no—
127C22×D4—21no—
128C22×D4—21no—
129C22×D4—21no—
130C22×D4—21no—
131C22×D4—21no—
132C22×D4—21no—
133C22×D4—21no—
134C22×D4—21no—
135C22×D4—21no—
136C22×D4—21no—
137C22×D4C2×D412yes—
138C22×D4C2×Q812yes—
139C22×D4C2×Q812yes—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—24no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—44no—
147C23×C4—44no—
148C23×C4—42no—
149C23×C4—42no—
150C23×C4C2×D412yes—
151C23×C4C2×D412yes—
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4C2411yescharacteristic
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—22no—
159C22×C8—21no—
160C22×C8—22no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—21no—
168C22×C8—22no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—22no—
172C22×C8—21no—
173C22×C8C2×D414yes—
174C22×C8C2×D412yes—
175C22×C8C2×Q812yes—
176C4×D4—28no—
177C4×D4—28no—
178C4×D4—44no—
179C4×D4—44no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—82no—
214C2×C4⋊C4—82no—
215C2×C4⋊C4—82no—
216C2×C4⋊C4C2×D412yes—
217C2×C4⋊C4C2×D412yes—
218C2×C4⋊C4C2×D412yes—
219C2×C4⋊C4C2×Q812yes—
220C2×C4⋊C4C2×Q812yes—
221C2×C4⋊C4#4112yes—
222C2×C4⋊C4#4112yes—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—42no—
225C2×C22⋊C4—42no—
226C2×C22⋊C4—44no—
227C2×C22⋊C4—44no—
228C2×C22⋊C4—44no—
229C2×C22⋊C4—44no—
230C2×C22⋊C4—82no—
231C2×C22⋊C4—82no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—24no—
239C2×C42—82no—
240C2×C42C2×D412yes—
241C2×C42C2×D412yes—
242C2×C42C2×Q812yes—
243C2×C42#4112yes—
244C2×C42#4112yes—
245C2×C42#4112yes—
246#107—48no—
247C4⋊C8—28no—
248C4⋊C8—48no—
249#102—28no—
250#102—28no—
251#102—48no—
252#97—44no—
253#95—24no—
254#95—22no—
255#95—22no—
256#95—44no—
257#95—44no—
258#95—44no—
259#95—42no—
260#95—42no—
261#95—44no—
262C24—22no—
263C24—22no—
264C24—22no—
265C24—41no—
266C24—41no—
267C24—41no—
268C24—41no—
269C24—42no—
270C24—42no—
271C24—42no—
272C24—42no—
273C24C22×D411yescharacteristic
274C2×D4—22no—
275C2×D4—22no—
276C2×D4—22no—
277C2×D4—22no—
278C2×D4—22no—
279C2×D4—22no—
280C2×D4—22no—
281C2×D4—22no—
282C2×D4—22no—
283C2×D4—22no—
284C2×D4—22no—
285C2×D4—22no—
286C2×D4—22no—
287C2×D4—22no—
288C2×D4—22no—
289C2×D4—41no—
290C2×D4—41no—
291C2×D4—41no—
292C2×D4—41no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4—41no—
296C2×D4—41no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—41no—
300C2×D4—41no—
301C2×D4C4⋊Q814yes—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—416no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—48no—
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—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—84no—
351C22×C4—84no—
352C22×C4—88no—
353C22×C4#12112yes—
354C22×C4#12112yes—
355C22×C4#12112yes—
356C22×C4#12112yes—
357C22×C4C22⋊Q812yes—
358C22×C4C22⋊Q812yes—
359C22×C4AGL(1,ℤ/8)14yes—
360C22×C4C22×D411yescharacteristic
361C22×C4C22×D411yescharacteristic
362C22×C4C22×D411yescharacteristic
363C22×C4C22×Q811yescharacteristic
364C22×C4C22×Q811yescharacteristic
365C22×C4C2×#4111yescharacteristic
366C22×C4C2×#4111yescharacteristic
367C22×C4C2×#4111yescharacteristic
368C22×C4C2×#4111yescharacteristic
369C22×C4C2×#4111yescharacteristic
370C22×C4H(5,𝔽2)11yescharacteristic
371C22×C4H(5,𝔽2)11yescharacteristic
372C22×C42−1+411yescharacteristic
373C22×C42−1+411yescharacteristic
374C2×C8—22no—
375C2×C8—24no—
376C2×C8—22no—
377C2×C8—24no—
378C2×C8—22no—
379C2×C8—24no—
380C2×C8—22no—
381C2×C8—22no—
382C2×C8—22no—
383C2×C8—41no—
384C2×C8—41no—
385C2×C8—42no—
386C2×C8—41no—
387C2×C8—42no—
388C2×C8—41no—
389C2×C8—41no—
390C2×C8—42no—
391C2×C8—41no—
392C2×C8—41no—
393C2×C8—42no—
394C2×C8—41no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—48no—
404C4⋊C4—41no—
405C4⋊C4—41no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—88no—
415C4⋊C4—84no—
416C4⋊C4—84no—
417C4⋊C4—88no—
418C4⋊C4#12114yes—
419C4⋊C4C22⋊Q814yes—
420C22⋊C4—48no—
421C22⋊C4—88no—
422C22⋊C4—88no—
423C42—42no—
424C42—42no—
425C42—42no—
426C42—42no—
427C42—84no—
428C42#12114yes—
429C42C22⋊Q814yes—
430C42C22⋊Q814yes—
431C42#12614yes—
432C23—24no—
433C23—44no—
434C23—44no—
435C23—44no—
436C23—44no—
437C23—42no—
438C23—42no—
439C23—42no—
440C23—42no—
441C23—42no—
442C23—42no—
443C23—44no—
444C23—44no—
445C23—82no—
446C23—82no—
447C23—81no—
448C23—81no—
449C23—81no—
450C23—81no—
451C23—82no—
452C23—82no—
453C23—82no—
454C23—82no—
455C23C2×#12111yescharacteristic
456C23C2×#12111yescharacteristic
457C23C2×C22⋊Q811yescharacteristic
458C23D4211yescharacteristic
459C23#54611yescharacteristic
460C23#54811yescharacteristic
461C23D4×Q811yescharacteristic
462C23C2×AGL(1,ℤ/8)12yes—
463C23#57512yes—
464C23#57612yes—
465C23#57812yes—
466D4—24no—
467D4—24no—
468D4—42no—
469D4—42no—
470D4—42no—
471D4—42no—
472D4—42no—
473D4—42no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—42no—
490C2×C4—44no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—44no—
498C2×C4—42no—
499C2×C4—44no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—816no—
509C2×C4—81no—
510C2×C4—81no—
511C2×C4—82no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4#39614yes—
527C2×C4#39614yes—
528C2×C4C2×#12111yescharacteristic
529C2×C4C2×#12111yescharacteristic
530C2×C4C2×C22⋊Q811yescharacteristic
531C2×C4C2×C22⋊Q811yescharacteristic
532C2×C4C2×C22⋊Q811yescharacteristic
533C2×C4C2×#12611yescharacteristic
534C2×C4C2×C4⋊Q811yescharacteristic
535C2×C4#53211yescharacteristic
536C2×C4#53311yescharacteristic
537C2×C4#53311yescharacteristic
538C2×C4#53711yescharacteristic
539C2×C4#53711yescharacteristic
540C2×C4#54011yescharacteristic
541C2×C4#54111yescharacteristic
542C2×C4#54411yescharacteristic
543C2×C4#54411yescharacteristic
544C2×C4D4211yescharacteristic
545C2×C4#54711yescharacteristic
546C2×C4#54711yescharacteristic
547C2×C4#54711yescharacteristic
548C2×C4D4×Q811yescharacteristic
549C2×C4#55011yescharacteristic
550C2×C4#55211yescharacteristic
551C2×C4#55311yescharacteristic
552C2×C4#55411yescharacteristic
553C2×C4#55411yescharacteristic
554C2×C4#55511yescharacteristic
555C2×C4#55611yescharacteristic
556C8—42no—
557C8—44no—
558C8—42no—
559C22—44no—
560C22—48no—
561C22—84no—
562C22—82no—
563C22—82no—
564C22—84no—
565C22—82no—
566C22—82no—
567C22—84no—
568C22—82no—
569C22—82no—
570C22C2×#39611yescharacteristic
571C22#241611yescharacteristic
572C22#241711yescharacteristic
573C22#242811yescharacteristic
574C22#242911yescharacteristic
575C22#243911yescharacteristic
576C22#244011yescharacteristic
577C22#303811yescharacteristic
578C22#304011yescharacteristic
579C22#304111yescharacteristic
580C22#304411yescharacteristic
581C22#305512yes—
582C22#306112yes—
583C22#307712yes—
584C22#308312yes—
585C22#326412yes—
586C22#326912yes—
587C22#330812yes—
588C22#331912yes—
589C22#336612yes—
590C22#337712yes—
591C22#338612yes—
592C22#338912yes—
593C4—24no—
594C4—24no—
595C4—44no—
596C4—44no—
597C4—44no—
598C4—82no—
599C4—82no—
600C4—162no—
601C4—162no—
602C4#244011yescharacteristic
603C4#244111yescharacteristic
604C4#249611yescharacteristic
605C4#249711yescharacteristic
606C4#253711yescharacteristic
607C4#253811yescharacteristic
608C4#262911yescharacteristic
609C4#285711yescharacteristic
610C2—84no—
611C2#1513111yescharacteristic
612C2#2361111yescharacteristic
613C2#2361811yescharacteristic
614C2#2474311yescharacteristic
615C2#2474411yescharacteristic
616C2#2482711yescharacteristic
617C2#2483011yescharacteristic
618C2#2539712yes—
619C2#2548212yes—
620C2#2603312yes—
621C2#2605212yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1947C2semidirect
3. C2×#1940C2semidirect
6. #13954C2semidirect
8. #11228C2semidirect
9. #11219C2semidirect
11. #10843C2semidirect

Maximal subgroups

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

subgroupindex
C2×#20672
C2×#19472
C2×#19402
C2×#18042
C2×#17622
#139542
#139062
#112282
#112192
#110662
#108432