Finite Group Databasebeta

← #355472

Subgroups — #355472

Order 512.

1540 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2929C211yesmaximal, characteristic
2#20989C211yesmaximal, characteristic
3#20983C211yesmaximal, characteristic
4#20385C211yesmaximal, characteristic, complemented
5C2×#1729C211yesmaximal, characteristic, complemented
6#10961C212yesmaximal, complemented
7#10914C212yesmaximal, complemented
8#10803C212yesmaximal, complemented
9#10764C212yesmaximal, complemented
10#10738C211yesmaximal, characteristic, complemented
11#10737C211yesmaximal, characteristic, complemented
12C22×#131C2211yescharacteristic
13C23×M16C2211yescharacteristic
14C24×C8C2211yescharacteristic
15C2×#525C2211yescharacteristic
16C22×#117C2211yescharacteristic
17#2929—22no—
18#2929—22no—
19#2929—22no—
20#2929—22no—
21C2×#435C414yescomplemented
22C2×C8×D4C414yescomplemented
23C2×#424C2211yescharacteristic
24C2×#413C2212yes—
25C2×#410C2212yes—
26C2×#408C2211yescharacteristic
27C2×#407C2211yescharacteristic
28C2×#407C2211yescharacteristic
29C2×#405C2212yes—
30C2×#405C2211yescharacteristic
31#2280C2211yescharacteristic
32#1901C414yescomplemented
33#1900C414yescomplemented
34#1899C414yes—
35#1898C414yes—
36#1752—22no—
37C8×C22⋊C4—22no—
38C2×#344C2212yes—
39C2×#344C2212yes—
40C2×#344C2212yes—
41C2×#344C2212yes—
42#1742C2212yes—
43#1740C2212yes—
44#1729—22no—
45#1729C2212yes—
46#1729C2212yes—
47#1729C2212yes—
48C2×#336—22no—
49C22×#41C2311yescharacteristic
50C24×C4C2311yescharacteristic
51C2×#131—22no—
52C2×#131—22no—
53C2×#131—42no—
54C2×#131—42no—
55C22×M16—21no—
56C22×M16—22no—
57C22×M16—42no—
58C22×M16—42no—
59C22×M16C2×C414yes—
60C22×M16C2×C412yescomplemented
61C22×M16C2312yes—
62C22×M16C2312yes—
63C22×M16C2311yescharacteristic
64C22×M16C2311yescharacteristic
65C23×C8—21no—
66C23×C8—42no—
67C23×C8—42no—
68C23×C8C2×C412yes—
69C23×C8C2×C412yes—
70C23×C8C2×C412yescomplemented
71C23×C8D412yes—
72C23×C8D412yes—
73C23×C8D412yes—
74C23×C8D412yes—
75C23×C8C2311yescharacteristic
76C23×C8C2311yescharacteristic
77#525—21no—
78#525—21no—
79#525—22no—
80#525—21no—
81#525—21no—
82#525—22no—
83C2×#123C2×C412yes—
84C2×C22⋊Q8C2×C412yes—
85C2×#121C2×C412yes—
86C2×#120C2×C412yes—
87C2×C4×D4C2×C414yes—
88C2×#117—21no—
89C2×#117—21no—
90C2×#117—42no—
91C2×#117—42no—
92C2×#117C2×C412yes—
93C2×#117C2312yes—
94C2×#117C2312yes—
95C2×#117C2311yescharacteristic
96C22×C4⋊C4—21no—
97C22×C22⋊C4C2×C412yes—
98C22×C42—21no—
99#435—24no—
100#435—24no—
101#435—24no—
102#435—24no—
103C8×D4—24no—
104C8×D4—24no—
105C8×D4—24no—
106C8×D4—24no—
107#424—22no—
108#424—21no—
109#424—21no—
110C2×C4⋊C8C2×C414yescomplemented
111#413—42no—
112#413—42no—
113C2×#101C2×C414yescomplemented
114C2×#100C2×C414yescomplemented
115#410—42no—
116#410—42no—
117C2×#99C2×C418yes—
118#408—22no—
119#408—21no—
120#408—21no—
121#407—21no—
122#407—21no—
123#407—22no—
124#407—21no—
125#407—21no—
126#407C2×C414yes—
127C2×C22⋊C8—22no—
128C2×C22⋊C8C2×C414yes—
129C2×C22⋊C8C2×C414yescomplemented
130#405—22no—
131#405—22no—
132#405—42no—
133#405—42no—
134C2×#97—22no—
135C2×#97C2×C412yescomplemented
136C2×C4×C8—22no—
137C2×C4×C8C2×C412yescomplemented
138#381C2×C412yes—
139#380—22no—
140#379C2×C412yes—
141C4×C22⋊C4—22no—
142#344—42no—
143#344—42no—
144#344—42no—
145#344—42no—
146#343C2×C418yescomplemented
147#342C2×C418yes—
148#336—24no—
149#336—24no—
150#336—24no—
151C25C22×C411yescharacteristic
152C2×#41—21no—
153C2×#41—21no—
154C2×#41—21no—
155C2×#41—21no—
156C2×#41—22no—
157C2×#41—22no—
158C2×#41—22no—
159C2×#41—22no—
160C22×Q8C22×C411yescharacteristic
161C22×D4C4214yes—
162C22×D4C22×C412yes—
163C22×D4C22×C411yescharacteristic
164C23×C4—22no—
165C23×C4—22no—
166C23×C4—22no—
167C23×C4—22no—
168C23×C4—22no—
169C23×C4C22×C412yes—
170C23×C4C22×C411yescharacteristic
171C23×C4C22×C412yes—
172C23×C4C22×C412yes—
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D412yes—
176C23×C4#4111yescharacteristic
177C23×C4#4111yescharacteristic
178C23×C4#4112yes—
179C23×C4C2411yescharacteristic
180#131—44no—
181#131—44no—
182#131—48no—
183C2×M16—24no—
184C2×M16—21no—
185C2×M16—21no—
186C2×M16—21no—
187C2×M16—21no—
188C2×M16—22no—
189C2×M16—22no—
190C2×M16—22no—
191C2×M16—22no—
192C2×M16—22no—
193C2×M16—22no—
194C2×M16—24no—
195C2×M16—22no—
196C2×M16—22no—
197C2×M16—22no—
198C2×M16—22no—
199C2×M16—22no—
200C2×M16—22no—
201C2×M16—22no—
202C2×M16—22no—
203C2×M16—44no—
204C2×M16—48no—
205C2×M16—48no—
206C2×M16—42no—
207C2×M16—42no—
208C2×M16C22×C414yes—
209C2×M16C22×C414yes—
210C22×C8—22no—
211C22×C8—22no—
212C22×C8—22no—
213C22×C8—22no—
214C22×C8—24no—
215C22×C8—22no—
216C22×C8—22no—
217C22×C8—24no—
218C22×C8—44no—
219C22×C8—44no—
220C22×C8—44no—
221C22×C8—44no—
222C22×C8—42no—
223C22×C8—42no—
224C22×C8—42no—
225C22×C8—42no—
226C22×C8—44no—
227C22×C8—42no—
228C22×C8—42no—
229C22×C8C22⋊C412yes—
230C22×C8C22⋊C412yes—
231C22×C8C22⋊C412yes—
232C22×C8C22⋊C412yes—
233C22×C8C22⋊C412yes—
234C22×C8C22⋊C412yes—
235C22×C8C22⋊C412yes—
236C22×C8C22⋊C412yes—
237C22×C8C22×C412yes—
238C22×C8C22×C412yes—
239C22×C8C22×C411yescharacteristic
240C22×C8C22×C411yescharacteristic
241C22×C8C22×C412yes—
242C22×C8C2×D412yes—
243C22×C8C2×D412yes—
244C22×C8C2×D412yes—
245C22×C8C2×D411yescharacteristic
246C22×C8C2×D411yescharacteristic
247#123—24no—
248#123—22no—
249#123—22no—
250C22⋊Q8—22no—
251C22⋊Q8—22no—
252C22⋊Q8—22no—
253C22⋊Q8—22no—
254#121—22no—
255#121—22no—
256#121—22no—
257#121—22no—
258#120—22no—
259#120—24no—
260#120—22no—
261C4×D4—24no—
262C4×D4—24no—
263C4×D4—24no—
264C4×D4—24no—
265#117—21no—
266#117—21no—
267#117—21no—
268#117—21no—
269#117—22no—
270#117—22no—
271#117—22no—
272#117—22no—
273#117—22no—
274#117—22no—
275#117—22no—
276#117—22no—
277#117—44no—
278#117—44no—
279#117—42no—
280#117—42no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4C4214yes—
286C2×C4⋊C4C22×C412yes—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4C4214yes—
291C2×C22⋊C4C4214yes—
292C2×C22⋊C4C22×C414yes—
293C2×C22⋊C4C22×C414yes—
294C2×C22⋊C4C22×C412yes—
295C2×C42—22no—
296C2×C42—22no—
297C2×C42—22no—
298C2×C42—42no—
299C2×C42—42no—
300C2×C42C22×C412yes—
301C4⋊C8—24no—
302C4⋊C8—24no—
303#101—44no—
304#100—44no—
305#99—48no—
306C22⋊C8—24no—
307C22⋊C8—24no—
308C22⋊C8—24no—
309C22⋊C8—24no—
310C22⋊C8—24no—
311C22⋊C8—24no—
312#97—24no—
313#97—44no—
314C4×C8—24no—
315C4×C8—44no—
316#95—24no—
317C24—22no—
318C24—22no—
319C24—44no—
320C24C2×C4212yes—
321C24C2×C4212yes—
322C24C4×D411yescharacteristic
323C24C4×D412yes—
324C24C4×D411yescharacteristic
325C24C23×C411yescharacteristic
326#41—41no—
327#41—41no—
328#41—42no—
329#41—41no—
330#41—41no—
331#41—42no—
332#41—42no—
333#41—42no—
334#41—42no—
335#41—42no—
336C2×Q8—21no—
337C2×Q8—21no—
338C2×Q8—22no—
339C2×Q8—22no—
340C2×D4—22no—
341C2×D4—22no—
342C2×D4—24no—
343C2×D4—24no—
344C2×D4—24no—
345C2×D4—24no—
346C2×D4—21no—
347C2×D4—21no—
348C2×D4—22no—
349C2×D4—22no—
350C2×D4—48no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—24no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—44no—
380C22×C4—48no—
381C22×C4C2×C4212yes—
382C22×C4C2×C4212yes—
383C22×C4C2×C4214yes—
384C22×C4C2×C22⋊C412yes—
385C22×C4C2×C22⋊C412yes—
386C22×C4#11712yes—
387C22×C4#11712yes—
388C22×C4C4×D412yes—
389C22×C4C4×D412yes—
390C22×C4C4×D412yes—
391C22×C4C4×D412yes—
392C22×C4C4×D412yes—
393C22×C4C4×D411yescharacteristic
394C22×C4C4×D411yescharacteristic
395C22×C4C23×C411yescharacteristic
396C22×C4C23×C411yescharacteristic
397C22×C4C22×D411yescharacteristic
398C22×C4C22×D411yescharacteristic
399C22×C4C2×#4111yescharacteristic
400C22×C4C2×#4111yescharacteristic
401C22×C4H(5,𝔽2)12yes—
402C22×C4H(5,𝔽2)12yes—
403C22×C4H(5,𝔽2)12yes—
404C22×C42−1+412yes—
405M16—24no—
406M16—24no—
407M16—24no—
408M16—24no—
409M16—48no—
410M16—48no—
411M16—41no—
412M16—41no—
413M16—42no—
414M16—41no—
415M16—41no—
416M16—42no—
417M16—42no—
418M16—42no—
419M16—42no—
420M16—42no—
421C2×C8—22no—
422C2×C8—24no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—24no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—24no—
434C2×C8—24no—
435C2×C8—24no—
436C2×C8—24no—
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×C8—48no—
446C2×C8—48no—
447C2×C8—42no—
448C2×C8—42no—
449C2×C8—42no—
450C2×C8—42no—
451C2×C8C2×C22⋊C412yes—
452C2×C8C2×C22⋊C412yes—
453C2×C8C2×C22⋊C412yes—
454C2×C8C2×C22⋊C412yes—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—24no—
458C4⋊C4—24no—
459C4⋊C4—44no—
460C4⋊C4—42no—
461C4⋊C4—42no—
462C4⋊C4—42no—
463C4⋊C4—42no—
464C22⋊C4—22no—
465C22⋊C4—22no—
466C22⋊C4—24no—
467C22⋊C4—24no—
468C22⋊C4—24no—
469C22⋊C4—24no—
470C22⋊C4—24no—
471C22⋊C4—24no—
472C22⋊C4—24no—
473C22⋊C4—24no—
474C22⋊C4—48no—
475C22⋊C4—44no—
476C42—22no—
477C42—22no—
478C42—44no—
479C42—42no—
480C42—42no—
481C42—42no—
482C42—42no—
483C23—21no—
484C23—21no—
485C23—24no—
486C23—21no—
487C23—21no—
488C23—24no—
489C23—24no—
490C23—24no—
491C23—24no—
492C23—22no—
493C23—24no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—48no—
499C23—48no—
500C23—44no—
501C23C4×C22⋊C412yes—
502C23C4×C22⋊C412yes—
503C23C22×C4211yescharacteristic
504C23C2×C4×D411yescharacteristic
505C23C2×C4×D411yescharacteristic
506C23#51812yes—
507C23#52012yes—
508Q8—41no—
509Q8—41no—
510Q8—42no—
511D4—48no—
512D4—42no—
513D4—42no—
514D4—44no—
515D4—48no—
516D4—41no—
517D4—41no—
518D4—42no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—24no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—48no—
545C2×C4—48no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—88no—
549C2×C4C4×C22⋊C412yes—
550C2×C4C4×C22⋊C412yes—
551C2×C4C22×C22⋊C411yescharacteristic
552C2×C4C2×#11711yescharacteristic
553C2×C4C2×C4×D411yescharacteristic
554C2×C4C2×C4×D411yescharacteristic
555C2×C4#51812yes—
556C2×C4#51812yes—
557C2×C4#52012yes—
558C2×C4#52012yes—
559C2×C4#53411yescharacteristic
560C2×C4#53411yescharacteristic
561C2×C4#53412yes—
562C2×C4#53511yescharacteristic
563C2×C4#53611yescharacteristic
564C2×C4#53712yes—
565C2×C4#53812yes—
566C2×C4#53812yes—
567C2×C4#53912yes—
568C2×C4#53912yes—
569C8—22no—
570C8—22no—
571C8—22no—
572C8—22no—
573C8—42no—
574C8—42no—
575C8—44no—
576C8—44no—
577C22—24no—
578C22—24no—
579C22—22no—
580C22—22no—
581C22—22no—
582C22—22no—
583C22—22no—
584C22—22no—
585C22—22no—
586C22—22no—
587C22—42no—
588C22—42no—
589C22—48no—
590C22—48no—
591C22—42no—
592C22—42no—
593C22—44no—
594C22—42no—
595C22—84no—
596C22#175714yes—
597C22C2×C4×C22⋊C411yescharacteristic
598C22#227512yes—
599C22#232111yescharacteristic
600C22#232211yescharacteristic
601C22#232512yes—
602C4—22no—
603C4—22no—
604C4—44no—
605C4—42no—
606C4—84no—
607C4—84no—
608C4#230811yescharacteristic
609C4#231111yescharacteristic
610C4#231812yes—
611C4#232111yescharacteristic
612C4#232211yescharacteristic
613C4#232512yes—
614C2—21no—
615C2—22no—
616C2—21no—
617C2—24no—
618C2—22no—
619C2—42no—
620C2—82no—
621C2#1378411yescharacteristic
622C2C2×#175712yes—
subgroupG/H
centerC2×C4C22×C22⋊C4
commutatorC23C22×C42
Frattini, cosocleC23×C4C24
socleC22C2×C4×C22⋊C4
prime radicalC2×#525C22
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×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. #10961C2semidirect
7. #10914C2semidirect
8. #10803C2semidirect
9. #10764C2semidirect
10. #10738C2semidirect
11. #10737C2semidirect
21. C2×#435C4semidirect
22. C2×C8×D4C4semidirect
32. #1901C4semidirect
33. #1900C4semidirect
60. C22×M16C2×C4semidirect
70. C23×C8C2×C4semidirect
110. C2×C4⋊C8C2×C4semidirect
113. C2×#101C2×C4semidirect
114. C2×#100C2×C4semidirect
129. C2×C22⋊C8C2×C4semidirect
135. C2×#97C2×C4semidirect
137. C2×C4×C8C2×C4semidirect
146. #343C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#29292
#209892
#209832
#203852
C2×#17292
#109612
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