Finite Group Databasebeta

← #414879

Subgroups — #414879

Order 512.

1472 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#476C211yesmaximal, characteristic
2C2×#1993C211yesmaximal, characteristic, complemented
3C2×#1940C211yesmaximal, characteristic, complemented
4C2×#1762C211yesmaximal, characteristic
5#14155C211yesmaximal, characteristic, complemented
6#13941C211yesmaximal, characteristic
7#11257C212yesmaximal, complemented
8#11231C212yesmaximal, complemented
9#11189C212yesmaximal
10#11076C212yesmaximal
11#10850C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4×D4C2211yescharacteristic
14C2×#476—24no—
15C2×#476—24no—
16C2×#476—24no—
17C22×#106C2212yes—
18C22×C4⋊C8C2212yes—
19C22×C4⋊C8C2211yescharacteristic
20C22×#102C2212yes—
21C22×#102C2211yescharacteristic
22C22×#102C2211yescharacteristic
23C22×#97C2211yescharacteristic
24C2×#391—22no—
25C2×#390C2212yes—
26C2×#382C2211yescharacteristic
27C2×C4×C4⋊C4C2211yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29#1993—24no—
30#1940—24no—
31#1762—24no—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2211yescharacteristic
36C2×#340C2211yescharacteristic
37C2×#340C2212yes—
38C2×#336C2212yes—
39#1445—21no—
40#1445—21no—
41#1439—22no—
42C4⋊#95C2212yes—
43C4⋊#95C2212yes—
44C4⋊#95C2212yes—
45C22⋊#95—21no—
46C22⋊#95—21no—
47C23×Q8—21no—
48C23×D4C2311yescharacteristic
49C24×C4—21no—
50C23×C8C2312yes—
51C23×C8C2311yescharacteristic
52C23×C8C2311yescharacteristic
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8D414yes—
59C2×C4×D4—22no—
60C2×C4×D4—22no—
61C2×C4×D4—22no—
62C2×C4×D4—22no—
63C2×C4×D4—22no—
64C2×C4×D4D414yes—
65C22×C4⋊C4—21no—
66C22×C4⋊C4D412yes—
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2312yes—
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2311yescharacteristic
72C22×C22⋊C4—21no—
73C22×C22⋊C4—22no—
74C22×C22⋊C4—22no—
75C22×C42—21no—
76C22×C42C2312yes—
77C22×C42C2311yescharacteristic
78#476—232no—
79C2×#106—24no—
80C2×#106—24no—
81C2×#106—24no—
82C2×C4⋊C8—22no—
83C2×C4⋊C8—22no—
84C2×C4⋊C8—24no—
85C2×C4⋊C8—24no—
86C2×C4⋊C8—42no—
87C2×C4⋊C8D414yes—
88C2×#102—22no—
89C2×#102—22no—
90C2×#102—24no—
91C2×#102—22no—
92C2×#102—24no—
93C2×#102—22no—
94C2×#102—24no—
95C2×#102—22no—
96C2×#102Q814yescomplemented
97C2×#97—22no—
98C2×#97—22no—
99C2×#97—22no—
100#391—48no—
101#390—24no—
102#390—42no—
103#382—22no—
104#382—22no—
105C4×C4⋊C4—22no—
106C4×C4⋊C4—22no—
107C4×C4⋊C4—22no—
108C4×C4⋊C4—22no—
109C2×#95—21no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95—21no—
115C2×#95—21no—
116C2×#95—22no—
117C2×#95—22no—
118C2×#95D411yescharacteristic
119C2×#95D411yescharacteristic
120#340—24no—
121#340—24no—
122#340—22no—
123#340—22no—
124#340—24no—
125#340—42no—
126#336—24no—
127C25—21no—
128C22×Q8—24no—
129C22×Q8—44no—
130C22×Q8—44no—
131C22×Q8—42no—
132C22×Q8—42no—
133C22×D4—21no—
134C22×D4—21no—
135C22×D4—21no—
136C22×D4—21no—
137C22×D4—21no—
138C22×D4—21no—
139C22×D4—21no—
140C22×D4—21no—
141C22×D4—21no—
142C22×D4—21no—
143C22×D4—21no—
144C22×D4C2×D412yes—
145C22×D4C2×Q812yes—
146C22×D4#4112yes—
147C23×C4—21no—
148C23×C4—22no—
149C23×C4—21no—
150C23×C4—21no—
151C23×C4—24no—
152C23×C4—22no—
153C23×C4—42no—
154C23×C4—42no—
155C23×C4—82no—
156C23×C4—82no—
157C23×C4C2×D412yes—
158C23×C4C2×D412yes—
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2411yescharacteristic
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—21no—
176C22×C8—22no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8C2×D414yes—
181C22×C8C2×Q812yes—
182C22×C8#4112yes—
183C4⋊Q8—28no—
184C4⋊Q8—28no—
185C4⋊Q8—44no—
186C4⋊Q8—44no—
187C4×D4—28no—
188C4×D4—28no—
189C4×D4—44no—
190C4×D4—44no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4C2×D412yes—
224C2×C4⋊C4C2×D414yes—
225C2×C4⋊C4C2×Q812yes—
226C2×C4⋊C4#4112yes—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—42no—
229C2×C22⋊C4—42no—
230C2×C22⋊C4—44no—
231C2×C22⋊C4—44no—
232C2×C22⋊C4—44no—
233C2×C22⋊C4—44no—
234C2×C22⋊C4—82no—
235C2×C22⋊C4—82no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—44no—
243C2×C42—42no—
244C2×C42C2×D412yes—
245C2×C42C2×D412yes—
246C2×C42C2×D412yes—
247C2×C42#4112yes—
248C2×C42#4112yes—
249C2×C42#4112yes—
250#106—216no—
251C4⋊C8—28no—
252C4⋊C8—48no—
253#102—28no—
254#102—216no—
255#102—28no—
256#97—44no—
257#95—22no—
258#95—22no—
259#95—44no—
260#95—44no—
261#95—42no—
262#95—42no—
263#95—42no—
264#95—44no—
265#95—44no—
266#95—82no—
267#95—82no—
268C24—22no—
269C24—22no—
270C24—22no—
271C24—41no—
272C24—41no—
273C24—41no—
274C24—41no—
275C24—42no—
276C24—42no—
277C24—42no—
278C24—42no—
279C24C22×D411yescharacteristic
280C2×Q8—416no—
281C2×Q8—48no—
282C2×Q8—84no—
283C2×Q8—84no—
284C2×Q8—88no—
285C2×D4—22no—
286C2×D4—22no—
287C2×D4—22no—
288C2×D4—22no—
289C2×D4—22no—
290C2×D4—22no—
291C2×D4—22no—
292C2×D4—22no—
293C2×D4—22no—
294C2×D4—22no—
295C2×D4—22no—
296C2×D4—22no—
297C2×D4—22no—
298C2×D4—22no—
299C2×D4—22no—
300C2×D4—41no—
301C2×D4—41no—
302C2×D4—41no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—41no—
308C2×D4—41no—
309C2×D4—41no—
310C2×D4—41no—
311C2×D4—41no—
312C2×D4C22⋊Q814yes—
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—24no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—41no—
341C22×C4—41no—
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—42no—
351C22×C4—42no—
352C22×C4—48no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—84no—
361C22×C4—84no—
362C22×C4—88no—
363C22×C4—88no—
364C22×C4#12112yes—
365C22×C4#12112yes—
366C22×C4#12112yes—
367C22×C4#12112yes—
368C22×C4#12112yes—
369C22×C4#12112yes—
370C22×C4AGL(1,ℤ/8)14yes—
371C22×C4C22×D411yescharacteristic
372C22×C4C22×D411yescharacteristic
373C22×C4C22×D411yescharacteristic
374C22×C4C22×Q811yescharacteristic
375C22×C4C2×#4111yescharacteristic
376C22×C4C2×#4111yescharacteristic
377C22×C4C2×#4111yescharacteristic
378C22×C4C2×#4111yescharacteristic
379C22×C4C2×#4111yescharacteristic
380C22×C4C2×#4111yescharacteristic
381C22×C4H(5,𝔽2)11yescharacteristic
382C22×C42−1+411yescharacteristic
383C22×C42−1+411yescharacteristic
384C22×C42−1+411yescharacteristic
385C2×C8—24no—
386C2×C8—22no—
387C2×C8—22no—
388C2×C8—22no—
389C2×C8—24no—
390C2×C8—24no—
391C2×C8—22no—
392C2×C8—22no—
393C2×C8—22no—
394C2×C8—41no—
395C2×C8—41no—
396C2×C8—42no—
397C2×C8—41no—
398C2×C8—41no—
399C2×C8—42no—
400C2×C8—41no—
401C2×C8—42no—
402C2×C8—41no—
403C2×C8—41no—
404C2×C8—41no—
405C2×C8—42no—
406C4⋊C4—24no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—24no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—48no—
415C4⋊C4—41no—
416C4⋊C4—42no—
417C4⋊C4—41no—
418C4⋊C4—42no—
419C4⋊C4—41no—
420C4⋊C4—41no—
421C4⋊C4—41no—
422C4⋊C4—41no—
423C4⋊C4—48no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—42no—
428C22⋊C4—48no—
429C22⋊C4—88no—
430C22⋊C4—88no—
431C42—42no—
432C42—42no—
433C42—42no—
434C42—42no—
435C42—84no—
436C42#12014yes—
437C42#12314yes—
438C42#12314yes—
439C42#12414yes—
440C23—24no—
441C23—44no—
442C23—44no—
443C23—42no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23—42no—
448C23—42no—
449C23—42no—
450C23—42no—
451C23—42no—
452C23—42no—
453C23—44no—
454C23—44no—
455C23—82no—
456C23—82no—
457C23—82no—
458C23—82no—
459C23—82no—
460C23—82no—
461C23—82no—
462C23—82no—
463C23C2×#12111yescharacteristic
464C23C2×#12111yescharacteristic
465C23C2×#12111yescharacteristic
466C23#54611yescharacteristic
467C23#54711yescharacteristic
468C23#54811yescharacteristic
469C23D4×Q811yescharacteristic
470C23C2×AGL(1,ℤ/8)12yes—
471C23#57512yes—
472C23#57712yes—
473C23#57712yes—
474Q8—816no—
475D4—24no—
476D4—24no—
477D4—42no—
478D4—42no—
479D4—42no—
480D4—42no—
481D4—42no—
482D4—42no—
483C2×C4—24no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—42no—
503C2×C4—44no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—816no—
516C2×C4—82no—
517C2×C4—81no—
518C2×C4—81no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—81no—
522C2×C4—81no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4—82no—
529C2×C4—82no—
530C2×C4—82no—
531C2×C4—82no—
532C2×C4—82no—
533C2×C4—82no—
534C2×C4—82no—
535C2×C4#39414yes—
536C2×C4#39414yes—
537C2×C4C2×#12011yescharacteristic
538C2×C4C2×C22⋊Q811yescharacteristic
539C2×C4C2×#12311yescharacteristic
540C2×C4C2×#12311yescharacteristic
541C2×C4#52511yescharacteristic
542C2×C4C2×#12411yescharacteristic
543C2×C4#52911yescharacteristic
544C2×C4#52911yescharacteristic
545C2×C4#53311yescharacteristic
546C2×C4#53311yescharacteristic
547C2×C4#53611yescharacteristic
548C2×C4#53711yescharacteristic
549C2×C4#54111yescharacteristic
550C2×C4#54111yescharacteristic
551C2×C4#54211yescharacteristic
552C2×C4#54211yescharacteristic
553C2×C4#54611yescharacteristic
554C2×C4#54711yescharacteristic
555C2×C4#54811yescharacteristic
556C2×C4#54811yescharacteristic
557C2×C4#54811yescharacteristic
558C2×C4D4×Q811yescharacteristic
559C2×C4D4×Q811yescharacteristic
560C2×C4#55011yescharacteristic
561C2×C4#55211yescharacteristic
562C2×C4#55211yescharacteristic
563C2×C4#55411yescharacteristic
564C2×C4#55611yescharacteristic
565C8—44no—
566C8—42no—
567C8—42no—
568C22—44no—
569C22—44no—
570C22—44no—
571C22—84no—
572C22—82no—
573C22—84no—
574C22—82no—
575C22—84no—
576C22—84no—
577C22—82no—
578C22—82no—
579C22C2×#39411yescharacteristic
580C22#241511yescharacteristic
581C22#241711yescharacteristic
582C22#242611yescharacteristic
583C22#242711yescharacteristic
584C22#242911yescharacteristic
585C22#242911yescharacteristic
586C22#303612yes—
587C22#303811yescharacteristic
588C22#304011yescharacteristic
589C22#304211yescharacteristic
590C22#304211yescharacteristic
591C22#304312yes—
592C22#305512yes—
593C22#305712yes—
594C22#330612yes—
595C22#331112yes—
596C22#334512yes—
597C22#335012yes—
598C22#337112yes—
599C22#337312yes—
600C22#338412yes—
601C22#338812yes—
602C4—24no—
603C4—24no—
604C4—44no—
605C4—82no—
606C4—82no—
607C4—82no—
608C4—84no—
609C4—82no—
610C4—162no—
611C4#240711yescharacteristic
612C4#240911yescharacteristic
613C4#242111yescharacteristic
614C4#242311yescharacteristic
615C4#248911yescharacteristic
616C4#249211yescharacteristic
617C4#262511yescharacteristic
618C4#285511yescharacteristic
619C2—84no—
620C2#1517911yescharacteristic
621C2#2354111yescharacteristic
622C2#2354611yescharacteristic
623C2#2469211yescharacteristic
624C2#2469711yescharacteristic
625C2#2482911yescharacteristic
626C2#2483411yescharacteristic
627C2#2528412yes—
628C2#2535412yes—
629C2#2546112yes—
630C2#2548512yes—
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×#1993C2semidirect
3. C2×#1940C2semidirect
5. #14155C2semidirect
7. #11257C2semidirect
8. #11231C2semidirect
11. #10850C2semidirect
96. C2×#102Q8semidirect

Maximal subgroups

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

subgroupindex
C22×#4762
C2×#19932
C2×#19402
C2×#17622
#141552
#139412
#112572
#112312
#111892
#110762
#108502