Finite Group Databasebeta

← #415044

Subgroups — #415044

Order 512.

1366 conjugacy classes of subgroups.

The subgroup lattice has 649 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2628C211yesmaximal, characteristic
2C2×#1993C211yesmaximal, characteristic, complemented
3C2×#1990C211yesmaximal, characteristic, complemented
4C2×#1846C211yesmaximal, characteristic, complemented
5C2×#1750C211yesmaximal, characteristic
6#14340C211yesmaximal, characteristic
7#11220C212yesmaximal, complemented
8#11219C212yesmaximal, complemented
9#11096C212yesmaximal
10#11095C212yesmaximal
11#10843C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C22×#107C2212yes—
15C22×#106C2212yes—
16C22×C22⋊C8C2211yescharacteristic
17C22×C22⋊C8C2211yescharacteristic
18C22×#97C2211yescharacteristic
19C22×C4×C8C2211yescharacteristic
20#2628—24no—
21#2628—24no—
22C2×#400—21no—
23C2×#399—21no—
24C2×#395—21no—
25C2×#393—21no—
26C2×#389C2212yes—
27C2×#384C2211yescharacteristic
28C2×#382C2211yescharacteristic
29C2×#382C2211yescharacteristic
30C2×#380C2212yes—
31C2×#380C2212yes—
32#1993—24no—
33#1990—24no—
34#1846—24no—
35#1750—24no—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#336C2211yescharacteristic
43C2×#336C2211yescharacteristic
44#1441—21no—
45#1441—21no—
46#1439—21no—
47#1439—21no—
48C4⋊#95C2212yes—
49C23×Q8—21no—
50C24×C4C2311yescharacteristic
51C23×C8C2311yescharacteristic
52C23×C8C2311yescharacteristic
53C23×C8C2311yescharacteristic
54C23×C8C2311yescharacteristic
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8—22no—
60C2×C4⋊Q8D414yes—
61C2×#117—22no—
62C2×#117—22no—
63C2×#117—22no—
64C2×#117—22no—
65C2×#117—22no—
66C2×#117—22no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4D411yescharacteristic
72C22×C4⋊C4D411yescharacteristic
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2312yes—
77C22×C22⋊C4—22no—
78C22×C22⋊C4—22no—
79C22×C22⋊C4D411yescharacteristic
80C22×C22⋊C4D411yescharacteristic
81C22×C42C2311yescharacteristic
82C22×C42C2311yescharacteristic
83C2×#107—24no—
84C2×#107—24no—
85C2×#107—42no—
86C2×#106—24no—
87C2×#106—24no—
88C2×#106—42no—
89C2×C22⋊C8—22no—
90C2×C22⋊C8—22no—
91C2×C22⋊C8—22no—
92C2×C22⋊C8—22no—
93C2×C22⋊C8Q814yescomplemented
94C2×C22⋊C8Q814yescomplemented
95C2×#97—22no—
96C2×#97—22no—
97C2×#97—22no—
98C2×C4×C8—22no—
99C2×C4×C8—22no—
100C2×C4×C8D414yes—
101#400—28no—
102#399—28no—
103#395—28no—
104#393—28no—
105#389—24no—
106#389—24no—
107#384—22no—
108#384—22no—
109#382—22no—
110#382—22no—
111#382—22no—
112#382D414yes—
113#380—24no—
114#380—24no—
115#380—42no—
116#380—42no—
117C2×#95—21no—
118C2×#95—22no—
119C2×#95—22no—
120C2×#95—21no—
121C2×#95—21no—
122C2×#95—21no—
123C2×#95—22no—
124C2×#95—22no—
125C2×#95—22no—
126#340—24no—
127#340—24no—
128#340—24no—
129#340—24no—
130#340—42no—
131#340—42no—
132#336—22no—
133#336—22no—
134C25C2×D411yescharacteristic
135C22×Q8—24no—
136C22×Q8—42no—
137C22×Q8—42no—
138C22×Q8—82no—
139C22×Q8—82no—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2×Q812yes—
162C23×C4C2×Q812yes—
163C23×C4C2411yescharacteristic
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8C2×D412yes—
185C22×C8C2×D412yes—
186C22×C8C2×Q812yes—
187C22×C8C2×Q812yes—
188C22×C8C2×Q812yes—
189C22×C8C2×Q812yes—
190C22×C8#4112yes—
191C22×C8#4112yes—
192C4⋊Q8—28no—
193C4⋊Q8—28no—
194C4⋊Q8—44no—
195C4⋊Q8—44no—
196#117—44no—
197#117—44no—
198#117—44no—
199#117—44no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
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—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—82no—
235C2×C4⋊C4—82no—
236C2×C4⋊C4—82no—
237C2×C4⋊C4—82no—
238C2×C4⋊C4C2×D414yes—
239C2×C4⋊C4C2×D414yes—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—82no—
249C2×C22⋊C4—82no—
250C2×C22⋊C4—82no—
251C2×C22⋊C4—82no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42—21no—
255C2×C42—21no—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42C2×D412yes—
259C2×C42C2×D412yes—
260C2×C42C2×D412yes—
261C2×C42#4112yes—
262C2×C42#4112yes—
263C2×C42#4112yes—
264#107—48no—
265#106—48no—
266C22⋊C8—28no—
267C22⋊C8—28no—
268#97—44no—
269C4×C8—28no—
270#95—28no—
271#95—28no—
272#95—24no—
273#95—24no—
274#95—44no—
275#95—42no—
276#95—42no—
277#95—82no—
278#95—82no—
279C24—21no—
280C24—21no—
281C24—21no—
282C24—21no—
283C24—21no—
284C24—21no—
285C24—22no—
286C24—22no—
287C24—22no—
288C24—22no—
289C24C4⋊Q812yes—
290C24C22×D411yescharacteristic
291C2×Q8—48no—
292C2×Q8—88no—
293C2×Q8—88no—
294C2×Q8—84no—
295C2×Q8—84no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
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—21no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4#12012yes—
373C22×C4#12012yes—
374C22×C4#12112yes—
375C22×C4#12112yes—
376C22×C4#12312yes—
377C22×C4#12312yes—
378C22×C4C4⋊Q812yes—
379C22×C4C22×D411yescharacteristic
380C22×C4C22×D411yescharacteristic
381C22×C4C22×D411yescharacteristic
382C22×C4C22×Q811yescharacteristic
383C22×C4C22×Q811yescharacteristic
384C22×C4C2×#4111yescharacteristic
385C22×C4C2×#4111yescharacteristic
386C22×C4C2×#4111yescharacteristic
387C22×C4C2×#4111yescharacteristic
388C22×C4C2×#4111yescharacteristic
389C22×C4H(5,𝔽2)11yescharacteristic
390C22×C4H(5,𝔽2)11yescharacteristic
391C22×C42−1+411yescharacteristic
392C22×C42−1+411yescharacteristic
393C2×C8—22no—
394C2×C8—22no—
395C2×C8—22no—
396C2×C8—22no—
397C2×C8—22no—
398C2×C8—22no—
399C2×C8—22no—
400C2×C8—22no—
401C2×C8—22no—
402C2×C8—22no—
403C2×C8—22no—
404C2×C8—22no—
405C2×C8—22no—
406C2×C8—22no—
407C2×C8—22no—
408C2×C8—22no—
409C2×C8—22no—
410C2×C8—22no—
411C2×C8—41no—
412C2×C8—41no—
413C2×C8—41no—
414C2×C8—41no—
415C2×C8—41no—
416C2×C8—41no—
417C2×C8—41no—
418C2×C8—41no—
419C2×C8C22⋊Q814yes—
420C2×C8C22⋊Q814yes—
421C4⋊C4—24no—
422C4⋊C4—24no—
423C4⋊C4—24no—
424C4⋊C4—24no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C4⋊C4—44no—
430C4⋊C4—44no—
431C4⋊C4—48no—
432C4⋊C4—42no—
433C4⋊C4—42no—
434C4⋊C4—42no—
435C4⋊C4—42no—
436C4⋊C4—42no—
437C4⋊C4—42no—
438C4⋊C4—42no—
439C4⋊C4—42no—
440C4⋊C4—84no—
441C4⋊C4—84no—
442C4⋊C4—88no—
443C22⋊C4—44no—
444C22⋊C4—44no—
445C22⋊C4—88no—
446C22⋊C4—88no—
447C42—41no—
448C42—41no—
449C42—41no—
450C42—41no—
451C42#12114yes—
452C42#12114yes—
453C42#12314yes—
454C42#12414yes—
455C23—22no—
456C23—24no—
457C23—24no—
458C23—22no—
459C23—22no—
460C23—42no—
461C23—42no—
462C23—41no—
463C23—41no—
464C23—42no—
465C23—42no—
466C23—41no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—41no—
472C23—42no—
473C23—42no—
474C23—41no—
475C23—41no—
476C23—41no—
477C23—41no—
478C23C2×#12011yescharacteristic
479C23C2×#12111yescharacteristic
480C23C2×#12311yescharacteristic
481C23C2×C4⋊Q811yescharacteristic
482C23#53211yescharacteristic
483C23#53511yescharacteristic
484C23#53611yescharacteristic
485C23#57612yes—
486C23#57712yes—
487C23#57712yes—
488C23#57812yes—
489Q8—816no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—42no—
522C2×C4—44no—
523C2×C4—42no—
524C2×C4—44no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—82no—
530C2×C4—82no—
531C2×C4—82no—
532C2×C4—82no—
533C2×C4—82no—
534C2×C4—82no—
535C2×C4—82no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4—82no—
541C2×C4—82no—
542C2×C4—82no—
543C2×C4—82no—
544C2×C4—82no—
545C2×C4—82no—
546C2×C4—82no—
547C2×C4—82no—
548C2×C4—82no—
549C2×C4—82no—
550C2×C4—82no—
551C2×C4#39414yes—
552C2×C4#39414yes—
553C2×C4C2×#12111yescharacteristic
554C2×C4C2×#12111yescharacteristic
555C2×C4C2×C22⋊Q811yescharacteristic
556C2×C4C2×C22⋊Q811yescharacteristic
557C2×C4C2×#12311yescharacteristic
558C2×C4C2×#12411yescharacteristic
559C2×C4#53311yescharacteristic
560C2×C4#53311yescharacteristic
561C2×C4#53711yescharacteristic
562C2×C4#53711yescharacteristic
563C2×C4#54211yescharacteristic
564C2×C4#54211yescharacteristic
565C2×C4#54611yescharacteristic
566C2×C4#54611yescharacteristic
567C2×C4D4×Q811yescharacteristic
568C2×C4D4×Q811yescharacteristic
569C2×C4D4×Q811yescharacteristic
570C2×C4D4×Q811yescharacteristic
571C2×C4#55011yescharacteristic
572C2×C4#55011yescharacteristic
573C2×C4#55211yescharacteristic
574C2×C4#55211yescharacteristic
575C2×C4#55411yescharacteristic
576C2×C4#55411yescharacteristic
577C2×C4#55411yescharacteristic
578C2×C4#55411yescharacteristic
579C2×C4#55611yescharacteristic
580C2×C4#55611yescharacteristic
581C8—24no—
582C8—24no—
583C8—42no—
584C8—42no—
585C22—44no—
586C22—42no—
587C22—42no—
588C22—42no—
589C22—44no—
590C22—82no—
591C22—81no—
592C22—81no—
593C22—81no—
594C22—81no—
595C22—82no—
596C22C2×#39411yescharacteristic
597C22#243111yescharacteristic
598C22#243411yescharacteristic
599C22#249911yescharacteristic
600C22#250211yescharacteristic
601C22#262011yescharacteristic
602C22#285711yescharacteristic
603C22#301212yes—
604C22#301312yes—
605C22#305711yescharacteristic
606C22#305711yescharacteristic
607C22#305811yescharacteristic
608C22#305911yescharacteristic
609C22#309612yes—
610C22#309712yes—
611C22#316312yes—
612C22#316412yes—
613C22#317312yes—
614C22#317512yes—
615C22#322412yes—
616C22#322512yes—
617C22#324112yes—
618C22#324212yes—
619C4—24no—
620C4—24no—
621C4—42no—
622C4—84no—
623C4—81no—
624C4—84no—
625C4—81no—
626C4—162no—
627C4—162no—
628C4#241711yescharacteristic
629C4#241711yescharacteristic
630C4#242311yescharacteristic
631C4#242311yescharacteristic
632C4#249211yescharacteristic
633C4#249211yescharacteristic
634C4#253811yescharacteristic
635C4#253811yescharacteristic
636C2—82no—
637C2#1518211yescharacteristic
638C2#2348611yescharacteristic
639C2#2348711yescharacteristic
640C2#2561012yes—
641C2#2582212yes—
642C2#2788612yes—
643C2#2795912yes—
644C2#2984711yescharacteristic
645C2#2984811yescharacteristic
646C2#3321311yescharacteristic
647C2#3321511yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ 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×#1990C2semidirect
4. C2×#1846C2semidirect
7. #11220C2semidirect
8. #11219C2semidirect
11. #10843C2semidirect
93. C2×C22⋊C8Q8semidirect
94. C2×C22⋊C8Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#26282
C2×#19932
C2×#19902
C2×#18462
C2×#17502
#143402
#112202
#112192
#110962
#110952
#108432