Finite Group Databasebeta

← #355461

Subgroups — #355461

Order 512.

1540 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2929C211yesmaximal, characteristic
2C2×#1897C211yesmaximal, characteristic
3#20970C211yesmaximal, characteristic
4#20385C211yesmaximal, characteristic, complemented
5C2×#1729C211yesmaximal, characteristic, complemented
6#10934C212yesmaximal, complemented
7#10887C212yesmaximal, complemented
8#10809C212yesmaximal, complemented
9#10791C212yesmaximal, complemented
10#10724C211yesmaximal, characteristic, complemented
11#10720C211yesmaximal, characteristic, complemented
12C22×#131C2211yescharacteristic
13C23×M16C2211yescharacteristic
14C24×C8C2211yescharacteristic
15C2×#525C2211yescharacteristic
16C22×#117C2211yescharacteristic
17#2929—24no—
18#2929C418yescomplemented
19C2×#435—21no—
20C2×#435—21no—
21C2×C8×D4—21no—
22C2×C8×D4—21no—
23C2×#424C2211yescharacteristic
24C22×#104C2212yes—
25C2×#416C2212yes—
26C2×#408C2211yescharacteristic
27C2×#407C2211yescharacteristic
28C2×#407C2211yescharacteristic
29C2×#405C2211yescharacteristic
30C2×#405C2212yes—
31#2280C2211yescharacteristic
32#1897C418yescomplemented
33#1895C412yescomplemented
34#1895C412yescomplemented
35#1894C412yescomplemented
36#1894C412yescomplemented
37#1752—21no—
38#1752—21no—
39C8×C22⋊C4—21no—
40C8×C22⋊C4—21no—
41C2×#341C2212yes—
42C2×#341C2212yes—
43#1735C2212yes—
44C2×#339C2212yes—
45C2×#339C2212yes—
46C2×#339C2212yes—
47#1729—24no—
48#1729C2212yes—
49C2×#336C2212yes—
50C2×#336C2212yes—
51C22×#41C2311yescharacteristic
52C24×C4C2311yescharacteristic
53C2×#131—22no—
54C2×#131—42no—
55C2×#131—42no—
56C2×#131C2×C414yes—
57C22×M16—22no—
58C22×M16—22no—
59C22×M16—21no—
60C22×M16—42no—
61C22×M16—42no—
62C22×M16C2×C412yes—
63C22×M16C2×C412yes—
64C22×M16C2311yescharacteristic
65C22×M16C2312yes—
66C22×M16C2311yescharacteristic
67C23×C8—21no—
68C23×C8—42no—
69C23×C8—42no—
70C23×C8C2×C414yes—
71C23×C8D411yescharacteristic
72C23×C8D411yescharacteristic
73C23×C8D411yescharacteristic
74C23×C8D411yescharacteristic
75C23×C8D414yes—
76C23×C8C2312yes—
77C23×C8C2311yescharacteristic
78C23×C8C2311yescharacteristic
79#525—22no—
80#525—22no—
81#525C2×C414yes—
82#525C2×C414yes—
83C2×#123—21no—
84C2×C22⋊Q8C2×C411yescharacteristic
85C2×C22⋊Q8C2×C411yescharacteristic
86C2×#121C2×C411yescharacteristic
87C2×#121C2×C411yescharacteristic
88C2×#120—21no—
89C2×C4×D4—21no—
90C2×C4×D4—21no—
91C2×#117—22no—
92C2×#117—42no—
93C2×#117—42no—
94C2×#117C2×C414yes—
95C2×#117C2312yes—
96C2×#117C2311yescharacteristic
97C22×C4⋊C4—21no—
98C22×C22⋊C4—21no—
99C22×C42C2312yes—
100#435—28no—
101#435—28no—
102C8×D4—28no—
103C8×D4—28no—
104#424—22no—
105#424C2×C414yes—
106C2×C4⋊C8—21no—
107C2×C4⋊C8—21no—
108C2×#104—42no—
109C2×#104—42no—
110C2×#104C2×C418yes—
111#416—42no—
112#416—42no—
113C2×#103C2×C414yes—
114C2×#102C2×C414yes—
115#408—22no—
116#408C2×C414yes—
117#407—22no—
118#407—22no—
119#407C2×C414yes—
120#407C2×C414yes—
121C2×C22⋊C8—21no—
122C2×C22⋊C8—21no—
123C2×C22⋊C8—21no—
124C2×C22⋊C8—21no—
125C2×C22⋊C8—21no—
126C2×C22⋊C8—21no—
127#405—22no—
128#405—42no—
129#405—42no—
130#405C2×C414yes—
131C2×#97—21no—
132C2×#97—22no—
133C2×C4×C8—21no—
134C2×C4×C8—22no—
135#381—21no—
136#380C2×C412yes—
137#380C2×C412yes—
138#379—21no—
139C4×C22⋊C4—21no—
140C4×C22⋊C4—21no—
141#341—42no—
142#341—42no—
143#340C2×C414yes—
144#340C2×C414yes—
145#339—42no—
146#339—42no—
147#339C2×C418yes—
148#336—24no—
149#336—24no—
150#336—22no—
151#336—22no—
152C25#4111yescharacteristic
153C2×#41—21no—
154C2×#41—21no—
155C2×#41—22no—
156C2×#41—22no—
157C2×#41—22no—
158C2×#41—22no—
159C2×#41C22×C412yes—
160C2×#41C22×C412yes—
161C22×Q8C22×C411yescharacteristic
162C22×D4—21no—
163C22×D4—21no—
164C22×D4—21no—
165C22×D4C22×C411yescharacteristic
166C23×C4—22no—
167C23×C4—22no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—21no—
171C23×C4—22no—
172C23×C4C22×C412yes—
173C23×C4C22×C412yes—
174C23×C4C22×C411yescharacteristic
175C23×C4C22×C411yescharacteristic
176C23×C4C2×D412yes—
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C23×C4#4111yescharacteristic
180C23×C4#4112yes—
181C23×C4C2411yescharacteristic
182#131—48no—
183#131—44no—
184#131—44no—
185C2×M16—24no—
186C2×M16—24no—
187C2×M16—21no—
188C2×M16—21no—
189C2×M16—22no—
190C2×M16—24no—
191C2×M16—22no—
192C2×M16—22no—
193C2×M16—22no—
194C2×M16—22no—
195C2×M16—22no—
196C2×M16—22no—
197C2×M16—22no—
198C2×M16—22no—
199C2×M16—44no—
200C2×M16—44no—
201C2×M16—44no—
202C2×M16—44no—
203C2×M16—44no—
204C2×M16—48no—
205C2×M16C22×C412yes—
206C2×M16C22×C414yes—
207C2×M16C22×C412yes—
208C22×C8—22no—
209C22×C8—22no—
210C22×C8—22no—
211C22×C8—22no—
212C22×C8—22no—
213C22×C8—21no—
214C22×C8—22no—
215C22×C8—21no—
216C22×C8—22no—
217C22×C8—22no—
218C22×C8—22no—
219C22×C8—22no—
220C22×C8—22no—
221C22×C8—24no—
222C22×C8—48no—
223C22×C8—42no—
224C22×C8—42no—
225C22×C8—42no—
226C22×C8—42no—
227C22×C8—44no—
228C22×C8—44no—
229C22×C8—44no—
230C22×C8C22⋊C414yes—
231C22×C8C22⋊C414yes—
232C22×C8C22⋊C414yes—
233C22×C8C22⋊C414yes—
234C22×C8C22×C414yes—
235C22×C8C22×C412yes—
236C22×C8C22×C412yes—
237C22×C8C2×D412yes—
238C22×C8C2×D412yes—
239C22×C8C2×D411yescharacteristic
240C22×C8C2×D411yescharacteristic
241C22×C8C2×D411yescharacteristic
242C22×C8C2×D411yescharacteristic
243#123—24no—
244#123—24no—
245C22⋊Q8—22no—
246C22⋊Q8—22no—
247C22⋊Q8C4214yes—
248C22⋊Q8C4214yes—
249#121—22no—
250#121—22no—
251#121C4214yes—
252#121C4214yes—
253#120—24no—
254#120—24no—
255C4×D4—24no—
256C4×D4—24no—
257C4×D4—24no—
258C4×D4—24no—
259#117—21no—
260#117—21no—
261#117—22no—
262#117—22no—
263#117—22no—
264#117—22no—
265#117—48no—
266#117—44no—
267#117—44no—
268#117C22×C412yes—
269#117C22×C412yes—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4C22×C414yes—
275C2×C4⋊C4C22×C411yescharacteristic
276C2×C4⋊C4C22×C411yescharacteristic
277C2×C22⋊C4—21no—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—21no—
280C2×C22⋊C4—21no—
281C2×C22⋊C4—21no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—44no—
287C2×C42—21no—
288C2×C42—22no—
289C2×C42—22no—
290C2×C42—22no—
291C2×C42—22no—
292C2×C42—22no—
293C2×C42—22no—
294C2×C42C22×C414yes—
295C4⋊C8—24no—
296C4⋊C8—24no—
297#104—48no—
298#103—44no—
299#102—44no—
300C22⋊C8—24no—
301C22⋊C8—24no—
302C22⋊C8—24no—
303C22⋊C8—24no—
304C22⋊C8—24no—
305C22⋊C8—24no—
306#97—24no—
307#97—44no—
308C4×C8—24no—
309C4×C8—44no—
310#95—22no—
311#95—22no—
312C24—22no—
313C24—22no—
314C24—21no—
315C24—42no—
316C24—42no—
317C24#11712yes—
318C24C4×D411yescharacteristic
319C24C4×D411yescharacteristic
320C24C4×D412yes—
321C24C2×#4111yescharacteristic
322#41—24no—
323#41—24no—
324#41—24no—
325#41—24no—
326#41—41no—
327#41—41no—
328#41—41no—
329#41—41no—
330#41—42no—
331#41—42no—
332C2×Q8—21no—
333C2×Q8—22no—
334C2×Q8—22no—
335C2×Q8C2×C4212yes—
336C2×D4—22no—
337C2×D4—22no—
338C2×D4—24no—
339C2×D4—24no—
340C2×D4—21no—
341C2×D4—22no—
342C2×D4—22no—
343C2×D4—44no—
344C2×D4—44no—
345C2×D4—42no—
346C2×D4—42no—
347C2×D4C2×C4212yes—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—44no—
378C22×C4—44no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4C2×C4212yes—
382C22×C4C2×C4212yes—
383C22×C4C2×C22⋊C412yes—
384C22×C4C2×C22⋊C412yes—
385C22×C4#11712yes—
386C22×C4C4×D412yes—
387C22×C4C4×D412yes—
388C22×C4C4×D412yes—
389C22×C4C4×D412yes—
390C22×C4C4×D411yescharacteristic
391C22×C4C4×D411yescharacteristic
392C22×C4C4×D412yes—
393C22×C4C23×C411yescharacteristic
394C22×C4C23×C411yescharacteristic
395C22×C4C23×C411yescharacteristic
396C22×C4C22×D411yescharacteristic
397C22×C4C22×D411yescharacteristic
398C22×C4C2×#4111yescharacteristic
399C22×C4H(5,𝔽2)12yes—
400C22×C4H(5,𝔽2)12yes—
401C22×C4H(5,𝔽2)12yes—
402C22×C42−1+412yes—
403M16—24no—
404M16—24no—
405M16—24no—
406M16—24no—
407M16—48no—
408M16—48no—
409M16—48no—
410M16—41no—
411M16—41no—
412M16—41no—
413M16—41no—
414M16—42no—
415M16—42no—
416C2×C8—24no—
417C2×C8—24no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—24no—
429C2×C8—24no—
430C2×C8—24no—
431C2×C8—24no—
432C2×C8—44no—
433C2×C8—44no—
434C2×C8—44no—
435C2×C8—44no—
436C2×C8—48no—
437C2×C8—42no—
438C2×C8—42no—
439C2×C8—42no—
440C2×C8—42no—
441C2×C8—42no—
442C2×C8—42no—
443C2×C8—42no—
444C2×C8—42no—
445C2×C8C2×C22⋊C412yes—
446C2×C8C2×C22⋊C412yes—
447C2×C8C2×C22⋊C412yes—
448C2×C8C2×C22⋊C412yes—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—24no—
452C4⋊C4—24no—
453C4⋊C4—24no—
454C4⋊C4—24no—
455C4⋊C4—48no—
456C4⋊C4C2×C4212yes—
457C4⋊C4C2×C4212yes—
458C22⋊C4—22no—
459C22⋊C4—22no—
460C22⋊C4—24no—
461C22⋊C4—24no—
462C22⋊C4—24no—
463C22⋊C4—24no—
464C22⋊C4—44no—
465C22⋊C4—44no—
466C22⋊C4—48no—
467C22⋊C4—42no—
468C22⋊C4—42no—
469C42—22no—
470C42—22no—
471C42—24no—
472C42—24no—
473C42—42no—
474C42—42no—
475C42—42no—
476C42—42no—
477C23—22no—
478C23—22no—
479C23—24no—
480C23—22no—
481C23—22no—
482C23—22no—
483C23—22no—
484C23—22no—
485C23—22no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—48no—
494C23—48no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23C4×C22⋊C412yes—
499C23C4×C22⋊C412yes—
500C23C2×#11711yescharacteristic
501C23C2×C4×D411yescharacteristic
502C23C2×C4×D411yescharacteristic
503C23#53812yes—
504C23#53912yes—
505Q8—24no—
506Q8—41no—
507Q8—41no—
508D4—24no—
509D4—48no—
510D4—48no—
511D4—44no—
512D4—42no—
513D4—42no—
514D4—41no—
515D4—41no—
516C2×C4—24no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—84no—
549C2×C4—84no—
550C2×C4C4×C22⋊C412yes—
551C2×C4C4×C22⋊C412yes—
552C2×C4C22×C4211yescharacteristic
553C2×C4C22×C22⋊C411yescharacteristic
554C2×C4C2×C4×D411yescharacteristic
555C2×C4C2×C4×D411yescharacteristic
556C2×C4#51812yes—
557C2×C4#51812yes—
558C2×C4#51812yes—
559C2×C4#52012yes—
560C2×C4#52012yes—
561C2×C4#52012yes—
562C2×C4#53411yescharacteristic
563C2×C4#53411yescharacteristic
564C2×C4#53412yes—
565C2×C4#53511yescharacteristic
566C2×C4#53611yescharacteristic
567C2×C4#53712yes—
568C2×C4#53812yes—
569C2×C4#53912yes—
570C8—22no—
571C8—22no—
572C8—22no—
573C8—22no—
574C8—42no—
575C8—42no—
576C8—44no—
577C8—44no—
578C22—24no—
579C22—24no—
580C22—21no—
581C22—21no—
582C22—21no—
583C22—21no—
584C22—22no—
585C22—22no—
586C22—22no—
587C22—22no—
588C22—22no—
589C22—22no—
590C22—44no—
591C22—48no—
592C22—44no—
593C22—44no—
594C22—44no—
595C22—42no—
596C22—42no—
597C22—42no—
598C22—84no—
599C22C2×C4×C22⋊C411yescharacteristic
600C22#231812yes—
601C22#232111yescharacteristic
602C22#232211yescharacteristic
603C22#232512yes—
604C22#295512yes—
605C22#295512yes—
606C4—22no—
607C4—22no—
608C4—42no—
609C4—42no—
610C4—42no—
611C4—84no—
612C4—84no—
613C4#227512yes—
614C4#230811yescharacteristic
615C4#231111yescharacteristic
616C4#232111yescharacteristic
617C4#232211yescharacteristic
618C4#232512yes—
619C2—21no—
620C2—22no—
621C2—21no—
622C2—42no—
623C2—42no—
624C2—82no—
625C2#1378411yescharacteristic
626C2#2042011yescharacteristic
627C2#2042011yescharacteristic
628C2#2220514yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC2×C4C22×C42
Frattini, cosocleC23×C4C24
socleC23C2×C4×D4
prime radicalC2×#525C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C24×C8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #20385C2semidirect
5. C2×#1729C2semidirect
6. #10934C2semidirect
7. #10887C2semidirect
8. #10809C2semidirect
9. #10791C2semidirect
10. #10724C2semidirect
11. #10720C2semidirect
18. #2929C4semidirect
32. #1897C4semidirect
33. #1895C4semidirect
34. #1895C4semidirect
35. #1894C4semidirect
36. #1894C4semidirect

Maximal subgroups

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

subgroupindex
C2×#29292
C2×#18972
#209702
#203852
C2×#17292
#109342
#108872
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