Finite Group Databasebeta

← #390427

Subgroups — #390427

Order 512.

1568 conjugacy classes of subgroups.

The subgroup lattice has 635 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23386C211yesmaximal, characteristic
2C2×#1896C212yesmaximal, complemented
3C2×#1876C212yesmaximal, complemented
4#20899C212yesmaximal, complemented
5#11317C212yesmaximal, complemented
6#11125C212yesmaximal, complemented
7#11097C211yesmaximal, characteristic, complemented
8#10791C212yesmaximal, complemented
9#10712C211yesmaximal, characteristic, complemented
10C22×#135C2211yescharacteristic
11C22×#131C2212yes—
12C2×#525C2212yes—
13C23×C42C2211yescharacteristic
14C22×#108C2211yescharacteristic
15C22×#104C2212yes—
16C22×#104C2212yes—
17C2×#416C2212yes—
18C2×#408C2212yes—
19C2×#407C2212yes—
20C22×C22⋊C8C2211yescharacteristic
21C22×C22⋊C8C2211yescharacteristic
22#2001C411yescharacteristic, complemented
23#2001C411yescharacteristic, complemented
24#2000C412yescomplemented
25#2000C412yescomplemented
26#1999C411yescharacteristic, complemented
27#1999C411yescharacteristic, complemented
28#1896—28no—
29#1876—28no—
30#1870—22no—
31#1870—22no—
32#1869—22no—
33#1869—22no—
34C2×#341C2212yes—
35C2×#341C2211yescharacteristic
36#1735C2211yescharacteristic
37C2×#339C2212yes—
38C2×#339C2212yes—
39#1729C2212yes—
40#1729C2212yes—
41C2×#336C2212yes—
42C2×#336C2212yes—
43C22×#41C2312yes—
44C24×C4—21no—
45C24×C4C2311yescharacteristic
46C2×#135—24no—
47C2×#135—24no—
48C2×#135—41no—
49C2×#135—41no—
50C22×Q16C2×C411yescharacteristic, complemented
51C22×SD16C2×C412yescomplemented
52C22×D8C2×C411yescharacteristic, complemented
53C2×#131—24no—
54C2×#131—24no—
55C2×#131—42no—
56C2×#131—42no—
57C2×#131—42no—
58C2×#131—42no—
59C22×M16—22no—
60C22×M16C2312yes—
61C22×M16C2312yes—
62C23×C8—22no—
63C23×C8C2311yescharacteristic
64C23×C8C2311yescharacteristic
65C23×C8C2312yes—
66#525—24no—
67#525—24no—
68#525—42no—
69#525—42no—
70C2×#123—22no—
71C2×C22⋊Q8C2×C412yes—
72C2×C22⋊Q8C2×C412yes—
73C2×#121C2×C412yes—
74C2×#121C2×C412yes—
75C2×#120—22no—
76C2×C4×D4—22no—
77C2×C4×D4—22no—
78C2×#117C2312yes—
79C22×C42—24no—
80C22×C42—21no—
81C22×C42—41no—
82C22×C42—41no—
83C22×C42—42no—
84C22×C42C2312yes—
85C2×#108—41no—
86C2×#108—42no—
87C2×#104—24no—
88C2×#104—24no—
89C2×#104—42no—
90C2×#104—42no—
91C2×#104—42no—
92C2×#104D418yescomplemented
93#416—42no—
94#416—42no—
95C2×#103C2×C412yescomplemented
96C2×#103C2×C412yescomplemented
97C2×#102C2×C412yescomplemented
98C2×#102C2×C412yescomplemented
99#408—24no—
100#408—42no—
101#407—24no—
102#407—42no—
103C2×C22⋊C8—22no—
104C2×C22⋊C8—22no—
105C2×C22⋊C8—22no—
106C2×C22⋊C8—22no—
107C2×C22⋊C8—41no—
108C2×C22⋊C8—41no—
109C2×C22⋊C8—41no—
110C2×C22⋊C8—41no—
111C2×C22⋊C8C2×C411yescharacteristic
112C2×C22⋊C8C2×C411yescharacteristic
113C2×C22⋊C8C2×C411yescharacteristic
114C2×C22⋊C8C2×C411yescharacteristic
115C2×C22⋊C8D411yescharacteristic
116C2×C22⋊C8D412yes—
117C2×C22⋊C8D411yescharacteristic
118#341—24no—
119#341—41no—
120#340C2×C411yescharacteristic, complemented
121#340C2×C411yescharacteristic, complemented
122#340C2×C412yescomplemented
123#339—24no—
124#339—24no—
125#336—22no—
126#336—22no—
127#336—22no—
128#336—22no—
129#336—24no—
130#336—42no—
131C25#4111yescharacteristic
132C2×#41—22no—
133C2×#41—22no—
134C2×#41—22no—
135C2×#41—22no—
136C2×#41—22no—
137C2×#41—22no—
138C2×#41—22no—
139C2×#41—22no—
140C2×#41—22no—
141C2×#41—22no—
142C2×#41C2×D414yes—
143C2×#41#4114yes—
144C22×Q8C22×C412yes—
145C22×D4—22no—
146C22×D4—22no—
147C22×D4—22no—
148C22×D4C22×C412yes—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—24no—
160C23×C4—44no—
161C23×C4—44no—
162C23×C4—42no—
163C23×C4—42no—
164C23×C4C22×C411yescharacteristic
165C23×C4C22×C411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4#4111yescharacteristic
169C23×C4C2411yescharacteristic
170#135—44no—
171#135—48no—
172#135—44no—
173C2×Q16—24no—
174C2×Q16—41no—
175C2×SD16—24no—
176C2×SD16—24no—
177C2×SD16—42no—
178C2×D8—24no—
179C2×D8—41no—
180#131—48no—
181#131—48no—
182#131—48no—
183#131—48no—
184C2×M16—22no—
185C2×M16—22no—
186C2×M16—22no—
187C2×M16—22no—
188C2×M16—22no—
189C2×M16—22no—
190C2×M16—22no—
191C2×M16—22no—
192C2×M16—22no—
193C2×M16—22no—
194C2×M16—22no—
195C2×M16—44no—
196C2×M16—44no—
197C2×M16—44no—
198C2×M16C2×D414yes—
199C22×C8—22no—
200C22×C8—22no—
201C22×C8—22no—
202C22×C8—22no—
203C22×C8—22no—
204C22×C8—22no—
205C22×C8—22no—
206C22×C8—22no—
207C22×C8—22no—
208C22×C8—21no—
209C22×C8—21no—
210C22×C8—22no—
211C22×C8—21no—
212C22×C8—21no—
213C22×C8—22no—
214C22×C8—22no—
215C22×C8—44no—
216C22×C8—44no—
217C22×C8—44no—
218C22×C8C22×C411yescharacteristic
219C22×C8C22×C411yescharacteristic
220C22×C8C22×C411yescharacteristic
221C22×C8C22×C411yescharacteristic
222C22×C8C2×D412yes—
223C22×C8C2×D412yes—
224#123—44no—
225#123—44no—
226C22⋊Q8—24no—
227C22⋊Q8—24no—
228C22⋊Q8—42no—
229C22⋊Q8—42no—
230#121—24no—
231#121—24no—
232#121—42no—
233#121—42no—
234#120—44no—
235#120—44no—
236C4×D4—44no—
237C4×D4—44no—
238C4×D4—44no—
239C4×D4—44no—
240#117—22no—
241#117—22no—
242#117—22no—
243#117—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4C22×C412yes—
247C2×C4⋊C4C22×C412yes—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—22no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—44no—
260C2×C42—48no—
261C2×C42—44no—
262C2×C42—42no—
263C2×C42—44no—
264C2×C42C2×D414yes—
265#108—44no—
266#104—48no—
267#104—48no—
268#103—42no—
269#103—42no—
270#102—42no—
271#102—42no—
272C22⋊C8—41no—
273C22⋊C8—41no—
274C22⋊C8—41no—
275C22⋊C8—41no—
276C22⋊C8—41no—
277C22⋊C8—42no—
278C22⋊C8—41no—
279C22⋊C8—44no—
280C22⋊C8—44no—
281C22⋊C8—44no—
282C22⋊C8—44no—
283C24—21no—
284C24—21no—
285C24—22no—
286C24—22no—
287C24—22no—
288C24—22no—
289C24—22no—
290C24—42no—
291C24C4×D411yescharacteristic
292C24C4×D411yescharacteristic
293C24C2×#4111yescharacteristic
294#41—24no—
295#41—24no—
296#41—24no—
297#41—24no—
298#41—24no—
299#41—24no—
300#41—24no—
301#41—24no—
302#41—42no—
303#41—42no—
304#41—42no—
305#41—42no—
306#41—42no—
307#41—42no—
308#41—42no—
309#41—42no—
310C2×Q8—22no—
311C2×Q8—22no—
312C2×Q8—22no—
313C2×Q8—22no—
314C2×Q8—22no—
315C2×Q8C4×D414yes—
316C2×D4—24no—
317C2×D4—24no—
318C2×D4—22no—
319C2×D4—22no—
320C2×D4—22no—
321C2×D4—22no—
322C2×D4—22no—
323C2×D4—42no—
324C2×D4—42no—
325C2×D4—42no—
326C2×D4—42no—
327C2×D4—44no—
328C2×D4—44no—
329C2×D4—44no—
330C2×D4—44no—
331C2×D4—82no—
332C2×D4—82no—
333C2×D4C4×D414yes—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—22no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—24no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—416no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—48no—
366C22×C4—48no—
367C22×C4—416no—
368C22×C4—48no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—41no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—42no—
384C22×C4—41no—
385C22×C4C4×D411yescharacteristic
386C22×C4C4×D411yescharacteristic
387C22×C4C23×C411yescharacteristic
388C22×C4C22×D412yes—
389C22×C4C22×D411yescharacteristic
390C22×C4C2×#4112yes—
391C22×C4H(5,𝔽2)12yes—
392C22×C4H(5,𝔽2)11yescharacteristic
393C22×C4H(5,𝔽2)12yes—
394C22×C4H(5,𝔽2)11yescharacteristic
395C22×C4H(5,𝔽2)11yescharacteristic
396C22×C42−1+411yescharacteristic
397Q16—44no—
398SD16—48no—
399D8—44no—
400M16—24no—
401M16—24no—
402M16—416no—
403M16—42no—
404M16—42no—
405M16—42no—
406M16—42no—
407M16—42no—
408M16—42no—
409C2×C8—22no—
410C2×C8—22no—
411C2×C8—21no—
412C2×C8—21no—
413C2×C8—21no—
414C2×C8—21no—
415C2×C8—22no—
416C2×C8—22no—
417C2×C8—22no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—24no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—416no—
425C2×C8—42no—
426C2×C8—42no—
427C2×C8—42no—
428C2×C8—41no—
429C2×C8—42no—
430C2×C8—42no—
431C2×C8—42no—
432C2×C8—41no—
433C2×C8—41no—
434C2×C8—42no—
435C2×C8—41no—
436C2×C8—42no—
437C2×C8—42no—
438C2×C8—42no—
439C2×C8—42no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—44no—
445C4⋊C4—44no—
446C22⋊C4—44no—
447C22⋊C4—44no—
448C22⋊C4—44no—
449C22⋊C4—44no—
450C22⋊C4—42no—
451C22⋊C4—42no—
452C42—24no—
453C42—24no—
454C42—48no—
455C42—42no—
456C42—42no—
457C42—42no—
458C42—42no—
459C23—21no—
460C23—21no—
461C23—22no—
462C23—24no—
463C23—24no—
464C23—21no—
465C23—21no—
466C23—22no—
467C23—22no—
468C23—21no—
469C23—21no—
470C23—22no—
471C23—22no—
472C23—24no—
473C23—44no—
474C23—44no—
475C23—44no—
476C23—44no—
477C23—42no—
478C23—42no—
479C23—41no—
480C23—42no—
481C23—41no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23#44312yes—
489C23#44412yes—
490C23C2×C4×D411yescharacteristic
491C23#53811yescharacteristic
492C23#53911yescharacteristic
493C23#54612yes—
494C23#55112yes—
495Q8—24no—
496Q8—24no—
497Q8—42no—
498Q8—42no—
499D4—24no—
500D4—24no—
501D4—44no—
502D4—44no—
503D4—44no—
504D4—44no—
505D4—42no—
506D4—42no—
507D4—88no—
508D4—88no—
509C2×C4—28no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—24no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—432no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—44no—
524C2×C4—41no—
525C2×C4—42no—
526C2×C4—41no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—44no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—44no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—44no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—44no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4C2×C4×D412yes—
550C2×C4#51812yes—
551C2×C4#51811yescharacteristic
552C2×C4#52011yescharacteristic
553C2×C4#53412yes—
554C2×C4#53411yescharacteristic
555C2×C4#53712yes—
556C2×C4#53711yescharacteristic
557C2×C4#53812yes—
558C2×C4#53912yes—
559C2×C4D4211yescharacteristic
560C2×C4D4212yes—
561C2×C4#54612yes—
562C2×C4#54612yes—
563C2×C4#55111yescharacteristic
564C2×C4#56012yes—
565C2×C4#56011yescharacteristic
566C2×C4#56011yescharacteristic
567C8—41no—
568C8—41no—
569C8—41no—
570C8—41no—
571C8—42no—
572C8—42no—
573C8—44no—
574C8—82no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—21no—
580C22—21no—
581C22—21no—
582C22—21no—
583C22—24no—
584C22—24no—
585C22—22no—
586C22—22no—
587C22—44no—
588C22—48no—
589C22—44no—
590C22—42no—
591C22—42no—
592C22—41no—
593C22—41no—
594C22—41no—
595C22—41no—
596C22—44no—
597C22—44no—
598C22—44no—
599C22—82no—
600C22—82no—
601C22—82no—
602C22—82no—
603C22#232511yescharacteristic
604C22#235812yes—
605C22#266811yescharacteristic
606C22#269312yes—
607C22#276011yescharacteristic
608C22C2×#44311yescharacteristic
609C22C2×#44411yescharacteristic
610C22#295612yes—
611C4—48no—
612C4—41no—
613C4—41no—
614C4—44no—
615C4—82no—
616C4—82no—
617C4#232512yes—
618C4#235811yescharacteristic
619C4#238011yescharacteristic
620C4#266812yes—
621C4#267511yescharacteristic
622C4#269311yescharacteristic
623C2—21no—
624C2—22no—
625C2—21no—
626C2—42no—
627C2—42no—
628C2—84no—
629C2#1457211yescharacteristic
630C2#2220411yescharacteristic
631C2#2220711yescharacteristic
632C2#2288612yes—
633C2#2288812yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C2×C4×D4
prime radical#23386C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C22⋊C8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1896C2semidirect
3. C2×#1876C2semidirect
4. #20899C2semidirect
5. #11317C2semidirect
6. #11125C2semidirect
7. #11097C2semidirect
8. #10791C2semidirect
9. #10712C2semidirect
22. #2001C4semidirect
23. #2001C4semidirect
24. #2000C4semidirect
25. #2000C4semidirect
26. #1999C4semidirect
27. #1999C4semidirect
50. C22×Q16C2×C4semidirect
51. C22×SD16C2×C4semidirect
52. C22×D8C2×C4semidirect
92. C2×#104D4semidirect
95. C2×#103C2×C4semidirect
96. C2×#103C2×C4semidirect
97. C2×#102C2×C4semidirect
98. C2×#102C2×C4semidirect
120. #340C2×C4semidirect
121. #340C2×C4semidirect
122. #340C2×C4semidirect

Maximal subgroups

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

subgroupindex
#233862
C2×#18962
C2×#18762
#208992
#113172
#111252
#110972
#107912
#107122