Finite Group Databasebeta

← #407039

Subgroups — #407039

Order 512.

1850 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1874C211yesmaximal, characteristic
2C2×#1874C211yesmaximal, characteristic
3C2×#1853C211yesmaximal, characteristic, complemented
4C2×#1852C211yesmaximal, characteristic, complemented
5#14698C211yesmaximal, characteristic
6C2×C4⋊#95C211yesmaximal, characteristic, complemented
7#11378C212yesmaximal, complemented
8#11377C212yesmaximal, complemented
9#11123C212yesmaximal, complemented
10#11121C212yesmaximal, complemented
11#10712C211yesmaximal, characteristic, complemented
12C22×#121C2211yescharacteristic
13C23×C4⋊C4C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C23×C42C2211yescharacteristic
16C22×#107C2212yes—
17C22×#106C2212yes—
18C22×#102C2212yes—
19C22×#102C2212yes—
20C22×C22⋊C8C2211yescharacteristic
21C22×C22⋊C8C2211yescharacteristic
22C2×#390C2211yescharacteristic
23C2×#390C2211yescharacteristic
24C2×#386—22no—
25C2×#380C2211yescharacteristic
26C22×#95—22no—
27#1874—24no—
28#1874—24no—
29#1853—24no—
30#1852—24no—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#336C2212yes—
36C2×#336C2212yes—
37#1437—22no—
38C4⋊#95—22no—
39C4⋊#95C2211yescharacteristic
40C4⋊#95C2211yescharacteristic
41C4⋊#95C2212yes—
42C4⋊#95C2212yes—
43C23×D4—21no—
44C23×D4C2311yescharacteristic
45C24×C4—21no—
46C24×C4—21no—
47C24×C4D412yes—
48C24×C4C2311yescharacteristic
49C23×C8C2312yes—
50C23×C8C2312yes—
51C2×#121—22no—
52C2×#121—22no—
53C2×#121—22no—
54C2×#121—22no—
55C2×#121—22no—
56C2×#121Q814yes—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—24no—
60C22×C4⋊C4—24no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4—42no—
66C22×C4⋊C4—42no—
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2312yes—
72C22×C22⋊C4—21no—
73C22×C22⋊C4—21no—
74C22×C22⋊C4—22no—
75C22×C22⋊C4—22no—
76C22×C42—22no—
77C22×C42—22no—
78C22×C42—42no—
79C22×C42—42no—
80C22×C42D412yes—
81C22×C42C2311yescharacteristic
82C22×C42C2311yescharacteristic
83C2×#107—24no—
84C2×#107—24no—
85C2×#107—42no—
86C2×#106—24no—
87C2×#106—24no—
88C2×#106—42no—
89C2×#102—24no—
90C2×#102—24no—
91C2×#102—24no—
92C2×#102—24no—
93C2×#102—42no—
94C2×#102—42no—
95C2×C22⋊C8—22no—
96C2×C22⋊C8—22no—
97C2×C22⋊C8—22no—
98C2×C22⋊C8—22no—
99C2×C22⋊C8D414yes—
100C2×C22⋊C8D414yes—
101#390—42no—
102#390—42no—
103#386—48no—
104#380—22no—
105#380—22no—
106C2×#95—24no—
107C2×#95—24no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—21no—
111C2×#95—48no—
112#340—24no—
113#340—24no—
114#340—24no—
115#340—24no—
116#336—42no—
117#336—42no—
118C25—21no—
119C25C2×D411yescharacteristic
120C22×D4—24no—
121C22×D4—21no—
122C22×D4—21no—
123C22×D4—21no—
124C22×D4—21no—
125C22×D4—22no—
126C22×D4—22no—
127C22×D4—21no—
128C22×D4—22no—
129C22×D4—22no—
130C22×D4—42no—
131C22×D4—42no—
132C22×D4—44no—
133C22×D4—44no—
134C22×D4C2×Q812yes—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—24no—
140C23×C4—24no—
141C23×C4—24no—
142C23×C4—24no—
143C23×C4—24no—
144C23×C4—24no—
145C23×C4—24no—
146C23×C4—21no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—21no—
150C23×C4—21no—
151C23×C4—42no—
152C23×C4—42no—
153C23×C4—42no—
154C23×C4—42no—
155C23×C4—44no—
156C23×C4—44no—
157C23×C4—42no—
158C23×C4—42no—
159C23×C4—44no—
160C23×C4—44no—
161C23×C4C2×D412yes—
162C23×C4C2×D412yes—
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D412yes—
165C23×C4C2×D412yes—
166C23×C4C2×Q812yes—
167C23×C4#4112yes—
168C23×C4#4112yes—
169C23×C4#4112yes—
170C23×C4C2411yescharacteristic
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8C2×D414yes—
184C22×C8C2×D414yes—
185#121—28no—
186#121—28no—
187#121—44no—
188#121—44no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—48no—
215C2×C4⋊C4—48no—
216C2×C4⋊C4—48no—
217C2×C4⋊C4—48no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4C2×Q812yes—
223C2×C4⋊C4#4114yes—
224C2×C4⋊C4#4114yes—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—42no—
227C2×C22⋊C4—42no—
228C2×C22⋊C4—44no—
229C2×C22⋊C4—44no—
230C2×C22⋊C4—44no—
231C2×C22⋊C4—44no—
232C2×C22⋊C4—42no—
233C2×C22⋊C4—44no—
234C2×C22⋊C4—44no—
235C2×C22⋊C4—44no—
236C2×C42—24no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42—21no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—42no—
246C2×C42—42no—
247C2×C42—48no—
248C2×C42—48no—
249#107—48no—
250#106—48no—
251#102—48no—
252#102—48no—
253C22⋊C8—28no—
254C22⋊C8—28no—
255#95—48no—
256#95—44no—
257#95—44no—
258#95—42no—
259#95—48no—
260C24—21no—
261C24—21no—
262C24—21no—
263C24—21no—
264C24—22no—
265C24—22no—
266C24—22no—
267C24—41no—
268C24—41no—
269C24—41no—
270C24—41no—
271C24—41no—
272C24—41no—
273C24—42no—
274C24—42no—
275C24—42no—
276C24—42no—
277C24C22⋊Q812yes—
278C24#12312yes—
279C24#12712yes—
280C24#13514yes—
281C24AGL(1,ℤ/8)14yes—
282C24C22×D411yescharacteristic
283C2×D4—22no—
284C2×D4—22no—
285C2×D4—24no—
286C2×D4—24no—
287C2×D4—48no—
288C2×D4—416no—
289C2×D4—41no—
290C2×D4—41no—
291C2×D4—42no—
292C2×D4—42no—
293C2×D4—42no—
294C2×D4—42no—
295C2×D4—42no—
296C2×D4—42no—
297C2×D4—42no—
298C2×D4—42no—
299C2×D4—42no—
300C2×D4—42no—
301C2×D4—88no—
302C2×D4—84no—
303C2×D4—84no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—216no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
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—24no—
335C22×C4—24no—
336C22×C4—48no—
337C22×C4—48no—
338C22×C4—48no—
339C22×C4—48no—
340C22×C4—48no—
341C22×C4—48no—
342C22×C4—48no—
343C22×C4—48no—
344C22×C4—416no—
345C22×C4—416no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—44no—
353C22×C4—42no—
354C22×C4—44no—
355C22×C4—44no—
356C22×C4—42no—
357C22×C4—44no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—84no—
369C22×C4—84no—
370C22×C4—88no—
371C22×C4—88no—
372C22×C4—84no—
373C22×C4—84no—
374C22×C4C22⋊Q814yes—
375C22×C4C22⋊Q814yes—
376C22×C4C22⋊Q812yes—
377C22×C4#12314yes—
378C22×C4#12314yes—
379C22×C4#12312yes—
380C22×C4#12712yes—
381C22×C4AGL(1,ℤ/8)14yes—
382C22×C4#13714yes—
383C22×C4C22×D411yescharacteristic
384C22×C4C22×D411yescharacteristic
385C22×C4C22×Q811yescharacteristic
386C22×C4C2×#4111yescharacteristic
387C22×C4C2×#4111yescharacteristic
388C22×C4C2×#4111yescharacteristic
389C22×C4C2×#4112yes—
390C22×C4C2×#4112yes—
391C22×C4H(5,𝔽2)12yes—
392C22×C4H(5,𝔽2)12yes—
393C2×C8—24no—
394C2×C8—24no—
395C2×C8—24no—
396C2×C8—24no—
397C2×C8—24no—
398C2×C8—24no—
399C2×C8—42no—
400C2×C8—42no—
401C2×C8—42no—
402C2×C8—42no—
403C2×C8—42no—
404C2×C8—42no—
405C2×C8—42no—
406C2×C8—42no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—24no—
410C4⋊C4—24no—
411C4⋊C4—24no—
412C4⋊C4—24no—
413C4⋊C4—48no—
414C4⋊C4—48no—
415C4⋊C4—41no—
416C4⋊C4—41no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C22⋊C4—48no—
426C22⋊C4—88no—
427C22⋊C4—88no—
428C22⋊C4—84no—
429C42—48no—
430C42—42no—
431C42—42no—
432C42—42no—
433C42—42no—
434C23—24no—
435C23—22no—
436C23—22no—
437C23—24no—
438C23—24no—
439C23—24no—
440C23—22no—
441C23—24no—
442C23—24no—
443C23—22no—
444C23—22no—
445C23—22no—
446C23—24no—
447C23—24no—
448C23—24no—
449C23—24no—
450C23—22no—
451C23—22no—
452C23—44no—
453C23—44no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23—82no—
458C23—82no—
459C23—82no—
460C23—82no—
461C23—81no—
462C23—82no—
463C23—82no—
464C23—81no—
465C23—81no—
466C23—81no—
467C23—81no—
468C23—81no—
469C23—82no—
470C23—82no—
471C23—82no—
472C23—82no—
473C23—81no—
474C23—81no—
475C23#39914yes—
476C23#47818yes—
477C23#48218yes—
478C23#49514yes—
479C23#49614yes—
480C23C2×C22⋊Q811yescharacteristic
481C23C2×#12311yescharacteristic
482C23#52511yescharacteristic
483C23#52511yescharacteristic
484C23C2×#12711yescharacteristic
485C23#53411yescharacteristic
486C23#53411yescharacteristic
487C23C2×#13512yes—
488C23C2×AGL(1,ℤ/8)12yes—
489C23C2×AGL(1,ℤ/8)12yes—
490C23C2×#13712yes—
491D4—44no—
492D4—44no—
493D4—816no—
494D4—82no—
495D4—82no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—28no—
503C2×C4—28no—
504C2×C4—28no—
505C2×C4—28no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—432no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—44no—
522C2×C4—48no—
523C2×C4—48no—
524C2×C4—42no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—816no—
528C2×C4—816no—
529C2×C4—82no—
530C2×C4—82no—
531C2×C4—81no—
532C2×C4—82no—
533C2×C4—81no—
534C2×C4—82no—
535C2×C4—82no—
536C2×C4—84no—
537C2×C4—84no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4—81no—
541C2×C4—81no—
542C2×C4—81no—
543C2×C4—81no—
544C2×C4—82no—
545C2×C4—82no—
546C2×C4—81no—
547C2×C4—81no—
548C2×C4#39914yes—
549C2×C4C2×C22⋊Q811yescharacteristic
550C2×C4C2×C22⋊Q811yescharacteristic
551C2×C4C2×#12311yescharacteristic
552C2×C4C2×#12311yescharacteristic
553C2×C4#52912yes—
554C2×C4#52912yes—
555C2×C4#53812yes—
556C2×C4#53812yes—
557C2×C4#54612yes—
558C2×C4#54612yes—
559C2×C4#54612yes—
560C2×C4#54612yes—
561C2×C4#55312yes—
562C2×C4#55312yes—
563C2×C4#55412yes—
564C2×C4#55412yes—
565C8—44no—
566C8—44no—
567C22—28no—
568C22—28no—
569C22—28no—
570C22—24no—
571C22—24no—
572C22—24no—
573C22—24no—
574C22—24no—
575C22—24no—
576C22—28no—
577C22—84no—
578C22—84no—
579C22—82no—
580C22—82no—
581C22—82no—
582C22—162no—
583C22—161no—
584C22—161no—
585C22—161no—
586C22—161no—
587C22—162no—
588C22#209218yes—
589C22C2×#39911yescharacteristic
590C22#253911yescharacteristic
591C22#253911yescharacteristic
592C22#257911yescharacteristic
593C22#257911yescharacteristic
594C22#262211yescharacteristic
595C22#285311yescharacteristic
596C22C2×#47812yes—
597C22#307912yes—
598C22C2×#48212yes—
599C22#309212yes—
600C22#310512yes—
601C22#310812yes—
602C22#310912yes—
603C22#311212yes—
604C22C2×#49511yescharacteristic
605C22C2×#49611yescharacteristic
606C22#315111yescharacteristic
607C22#315211yescharacteristic
608C22#318812yes—
609C22#319012yes—
610C22#319212yes—
611C22#319312yes—
612C4—24no—
613C4—44no—
614C4—44no—
615C4—84no—
616C4—84no—
617C4—82no—
618C4—161no—
619C4—161no—
620C4#244412yes—
621C4#244412yes—
622C4#245012yes—
623C4#245012yes—
624C2—28no—
625C2—162no—
626C2#1494611yescharacteristic
627C2C2×#209211yescharacteristic
628C2#2373411yescharacteristic
629C2#2830311yescharacteristic
630C2#2830611yescharacteristic
631C2#2831011yescharacteristic
632C2#2831411yescharacteristic
633C2#2923712yes—
634C2#2923812yes—
635C2#2989112yes—
636C2#3309612yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1853C2semidirect
4. C2×#1852C2semidirect
6. C2×C4⋊#95C2semidirect
7. #11378C2semidirect
8. #11377C2semidirect
9. #11123C2semidirect
10. #11121C2semidirect
11. #10712C2semidirect

Maximal subgroups

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

subgroupindex
C2×#18742
C2×#18532
C2×#18522
#146982
C2×C4⋊#952
#113782
#113772
#111232
#111212
#107122