Finite Group Databasebeta

← #412489

Subgroups — #412489

Order 512.

1306 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1944C211yesmaximal, characteristic, complemented
2C2×#1942C211yesmaximal, characteristic, complemented
3C2×#1844C211yesmaximal, characteristic, complemented
4C2×#1809C211yesmaximal, characteristic
5C2×#1809C211yesmaximal, characteristic
6C2×#1753C211yesmaximal, characteristic
7#15131C212yesmaximal
8#11220C212yesmaximal, complemented
9#11207C212yesmaximal, complemented
10#11095C212yesmaximal
11#10863C211yesmaximal, characteristic, complemented
12C23×M16C2211yescharacteristic
13C22×#107C2211yescharacteristic
14C22×#107C2211yescharacteristic
15C22×#106C2211yescharacteristic
16C22×#106C2212yes—
17C22×#106C2211yescharacteristic
18C22×C22⋊C8C2211yescharacteristic
19C22×#97C2211yescharacteristic
20C22×C4×C8C2211yescharacteristic
21C2×#396—22no—
22C2×#389C2212yes—
23C2×#389C2212yes—
24C2×#384C2212yes—
25C2×#380C2212yes—
26C2×#380C2212yes—
27C2×C4×C22⋊C4C2211yescharacteristic
28#1944—24no—
29#1942—24no—
30#1844—24no—
31#1809—24no—
32#1809—24no—
33#1753—24no—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#336C2211yescharacteristic
40C2×#336C2211yescharacteristic
41#1441—22no—
42#1441—22no—
43#1439—22no—
44C4⋊#95C2212yes—
45C24×C4C2311yescharacteristic
46C22×M16—22no—
47C22×M16—22no—
48C22×M16—22no—
49C22×M16—22no—
50C22×M16—22no—
51C22×M16—22no—
52C22×M16Q814yes—
53C23×C8C2311yescharacteristic
54C23×C8C2311yescharacteristic
55C23×C8C2311yescharacteristic
56C23×C8C2311yescharacteristic
57C22×C4⋊C4—22no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4C2312yes—
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2312yes—
65C22×C22⋊C4—22no—
66C22×C22⋊C4—22no—
67C22×C22⋊C4D411yescharacteristic
68C22×C22⋊C4D411yescharacteristic
69C22×C42C2311yescharacteristic
70C22×C42C2311yescharacteristic
71C2×#107—22no—
72C2×#107—22no—
73C2×#107—22no—
74C2×#107—22no—
75C2×#107—22no—
76C2×#107D414yes—
77C2×#106—22no—
78C2×#106—22no—
79C2×#106—22no—
80C2×#106—22no—
81C2×#106—24no—
82C2×#106—22no—
83C2×#106—24no—
84C2×#106—42no—
85C2×#106D414yes—
86C2×C22⋊C8—22no—
87C2×C22⋊C8—22no—
88C2×C22⋊C8Q814yes—
89C2×#97—22no—
90C2×#97—22no—
91C2×#97—22no—
92C2×C4×C8—22no—
93C2×C4×C8—22no—
94C2×C4×C8D414yes—
95#396—48no—
96#389—24no—
97#389—24no—
98#389—24no—
99#389—42no—
100#384—24no—
101#384—42no—
102#380—24no—
103#380—24no—
104#380—42no—
105#380—42no—
106C4×C22⋊C4—22no—
107C4×C22⋊C4—22no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95D411yescharacteristic
115C2×#95D411yescharacteristic
116#340—24no—
117#340—24no—
118#340—24no—
119#340—24no—
120#340—42no—
121#336—22no—
122#336—22no—
123C25C2×D411yescharacteristic
124C23×C4—21no—
125C23×C4—21no—
126C23×C4—21no—
127C23×C4—21no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2×Q812yes—
146C23×C4C2×Q812yes—
147C23×C4C2411yescharacteristic
148C2×M16—28no—
149C2×M16—28no—
150C2×M16—28no—
151C2×M16—44no—
152C2×M16—44no—
153C2×M16—44no—
154C2×M16—44no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8C2×D412yes—
172C22×C8C2×D412yes—
173C22×C8C2×D412yes—
174C22×C8C2×D412yes—
175C22×C8C2×Q812yes—
176C22×C8C2×Q812yes—
177C22×C8C2×Q812yes—
178C22×C8C2×Q812yes—
179C22×C8#4112yes—
180C22×C8#4112yes—
181C22×C8#4112yes—
182C22×C8#4112yes—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
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—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—44no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—82no—
215C2×C4⋊C4—82no—
216C2×C4⋊C4—82no—
217C2×C4⋊C4C2×D414yes—
218C2×C4⋊C4C2×D414yes—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—44no—
224C2×C22⋊C4—44no—
225C2×C22⋊C4—44no—
226C2×C22⋊C4—44no—
227C2×C22⋊C4—82no—
228C2×C22⋊C4—82no—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—21no—
233C2×C42—21no—
234C2×C42—21no—
235C2×C42—21no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42C2×D412yes—
240C2×C42#4112yes—
241#107—28no—
242#107—44no—
243#106—28no—
244#106—48no—
245#106—44no—
246C22⋊C8—28no—
247#97—44no—
248C4×C8—28no—
249#95—22no—
250#95—22no—
251#95—44no—
252#95—44no—
253#95—44no—
254#95—44no—
255#95—44no—
256#95—82no—
257C24—21no—
258C24—21no—
259C24—21no—
260C24—21no—
261C24—21no—
262C24—21no—
263C24—22no—
264C24—22no—
265C24—22no—
266C24—22no—
267C24C4⋊Q812yes—
268C24C22×D411yescharacteristic
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
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—42no—
347C22×C4#12112yes—
348C22×C4#12112yes—
349C22×C4C4⋊Q812yes—
350C22×C4AGL(1,ℤ/8)14yes—
351C22×C4#13714yes—
352C22×C4C22×D411yescharacteristic
353C22×C4C22×D411yescharacteristic
354C22×C4C22×D411yescharacteristic
355C22×C4C22×Q811yescharacteristic
356C22×C4C22×Q811yescharacteristic
357C22×C4C2×#4111yescharacteristic
358C22×C4C2×#4111yescharacteristic
359C22×C4C2×#4111yescharacteristic
360C22×C4C2×#4111yescharacteristic
361C22×C4C2×#4111yescharacteristic
362C22×C4H(5,𝔽2)11yescharacteristic
363C22×C4H(5,𝔽2)11yescharacteristic
364C22×C42−1+411yescharacteristic
365C22×C42−1+411yescharacteristic
366M16—416no—
367C2×C8—22no—
368C2×C8—22no—
369C2×C8—22no—
370C2×C8—22no—
371C2×C8—22no—
372C2×C8—22no—
373C2×C8—22no—
374C2×C8—22no—
375C2×C8—22no—
376C2×C8—22no—
377C2×C8—22no—
378C2×C8—22no—
379C2×C8—41no—
380C2×C8—41no—
381C2×C8—41no—
382C2×C8—41no—
383C2×C8—41no—
384C2×C8—41no—
385C2×C8—41no—
386C2×C8—41no—
387C2×C8#12114yes—
388C2×C8#12114yes—
389C2×C8#12114yes—
390C2×C8C22⋊Q814yes—
391C2×C8C22⋊Q814yes—
392C2×C8C22⋊Q814yes—
393C2×C8C22⋊Q814yes—
394C2×C8#12614yes—
395C4⋊C4—24no—
396C4⋊C4—24no—
397C4⋊C4—24no—
398C4⋊C4—24no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—88no—
412C4⋊C4—88no—
413C4⋊C4—88no—
414C4⋊C4—88no—
415C22⋊C4—44no—
416C22⋊C4—44no—
417C22⋊C4—88no—
418C22⋊C4—88no—
419C42—22no—
420C42—22no—
421C42—22no—
422C42—22no—
423C42—41no—
424C42—41no—
425C42—41no—
426C42—41no—
427C23—22no—
428C23—24no—
429C23—24no—
430C23—22no—
431C23—22no—
432C23—42no—
433C23—42no—
434C23—42no—
435C23—42no—
436C23—42no—
437C23—42no—
438C23—41no—
439C23—41no—
440C23—41no—
441C23—42no—
442C23—42no—
443C23—42no—
444C23—42no—
445C23—41no—
446C23—41no—
447C23—41no—
448C23—41no—
449C23—41no—
450C23C2×#12111yescharacteristic
451C23C2×C4⋊Q811yescharacteristic
452C23#53211yescharacteristic
453C23D4211yescharacteristic
454C23D4211yescharacteristic
455C23#54711yescharacteristic
456C23#54711yescharacteristic
457C23C2×AGL(1,ℤ/8)12yes—
458C23C2×#13712yes—
459C23#57712yes—
460C23#57712yes—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—42no—
491C2×C4—44no—
492C2×C4—42no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—44no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—82no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4—82no—
504C2×C4—82no—
505C2×C4—82no—
506C2×C4—82no—
507C2×C4—82no—
508C2×C4—82no—
509C2×C4—82no—
510C2×C4—82no—
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#46914yes—
523C2×C4#47014yes—
524C2×C4C2×#12111yescharacteristic
525C2×C4C2×#12111yescharacteristic
526C2×C4C2×#12111yescharacteristic
527C2×C4C2×C22⋊Q811yescharacteristic
528C2×C4C2×C22⋊Q811yescharacteristic
529C2×C4C2×C22⋊Q811yescharacteristic
530C2×C4C2×C22⋊Q811yescharacteristic
531C2×C4C2×#12611yescharacteristic
532C2×C4#53311yescharacteristic
533C2×C4#53311yescharacteristic
534C2×C4#53711yescharacteristic
535C2×C4#53711yescharacteristic
536C2×C4#54011yescharacteristic
537C2×C4#54111yescharacteristic
538C2×C4#54411yescharacteristic
539C2×C4#54411yescharacteristic
540C2×C4#54611yescharacteristic
541C2×C4#54711yescharacteristic
542C2×C4#54811yescharacteristic
543C2×C4D4×Q811yescharacteristic
544C2×C4D4×Q811yescharacteristic
545C2×C4#55011yescharacteristic
546C2×C4#55211yescharacteristic
547C2×C4#55311yescharacteristic
548C2×C4#55411yescharacteristic
549C2×C4#55411yescharacteristic
550C2×C4#55511yescharacteristic
551C2×C4#55611yescharacteristic
552C8—24no—
553C8—42no—
554C8—42no—
555C8#39618yescomplemented
556C22—42no—
557C22—42no—
558C22—42no—
559C22—44no—
560C22—44no—
561C22—81no—
562C22—81no—
563C22—82no—
564C22—82no—
565C22—81no—
566C22—81no—
567C22#241611yescharacteristic
568C22#242811yescharacteristic
569C22#244011yescharacteristic
570C22#244011yescharacteristic
571C22#249611yescharacteristic
572C22#253711yescharacteristic
573C22#285711yescharacteristic
574C22C2×#46911yescharacteristic
575C22C2×#47011yescharacteristic
576C22#306011yescharacteristic
577C22#306011yescharacteristic
578C22#316312yes—
579C22#316512yes—
580C22#317412yes—
581C22#318212yes—
582C22#326512yes—
583C22#326612yes—
584C22#327212yes—
585C22#327812yes—
586C22#334412yes—
587C22#334512yes—
588C22#335412yes—
589C22#335612yes—
590C4—24no—
591C4—24no—
592C4—42no—
593C4—81no—
594C4—84no—
595C4—81no—
596C4—84no—
597C4—162no—
598C4—162no—
599C4C2×#39611yescharacteristic
600C4#241711yescharacteristic
601C4#242911yescharacteristic
602C4#243911yescharacteristic
603C4#244111yescharacteristic
604C4#249711yescharacteristic
605C4#253811yescharacteristic
606C4#262911yescharacteristic
607C2—82no—
608C2#1513111yescharacteristic
609C2#2478211yescharacteristic
610C2#2478311yescharacteristic
611C2#2542812yes—
612C2#2607212yes—
613C2#2607812yes—
614C2#2785312yes—
615C2#2823111yescharacteristic
616C2#2823211yescharacteristic
617C2#3321211yescharacteristic
618C2#3321411yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#1944C2semidirect
2. C2×#1942C2semidirect
3. C2×#1844C2semidirect
8. #11220C2semidirect
9. #11207C2semidirect
11. #10863C2semidirect
555. C8#396semidirect

Maximal subgroups

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

subgroupindex
C2×#19442
C2×#19422
C2×#18442
C2×#18092
C2×#17532
#151312
#112202
#112072
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