Finite Group Databasebeta

← #356071

Subgroups — #356071

Order 512.

2271 conjugacy classes of subgroups.

The subgroup lattice has 569 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2898C212yesmaximal
2C2×#1793C211yesmaximal, characteristic
3C2×#1765C212yesmaximal, complemented
4#20415C212yesmaximal, complemented
5#10957C211yesmaximal, characteristic, complemented
6#10942C212yesmaximal, complemented
7#10928C211yesmaximal, characteristic, complemented
8#10781C212yesmaximal, complemented
9#10774C212yesmaximal, complemented
10C22×2−1+4C2211yescharacteristic
11C22×#131C2212yes—
12C2×#520C2212yes—
13C22×#104C2212yes—
14#2898—28no—
15#2898C4116yescomplemented
16C2×#417—22no—
17C2×#417C2212yes—
18C2×#417C2212yes—
19C2×#419—22no—
20C2×#419C2212yes—
21C2×#416—22no—
22C2×#416C2212yes—
23C22×#103—22no—
24C22×#103C2212yes—
25C22×#101C2211yescharacteristic
26C2×#405C2212yes—
27C22×#97C2211yescharacteristic
28C2×#391C2211yescharacteristic
29#2274C2211yescharacteristic
30#1793C418yescomplemented
31#1765—28no—
32#1763—22no—
33#1762—22no—
34C4×#103—22no—
35C4×#102—22no—
36C2×#343C2211yescharacteristic
37C2×#343C2212yes—
38C2×#341C2212yes—
39#1736C2212yes—
40C2×#340C2212yes—
41C2×#339C2212yes—
42C4⋊#95C2211yescharacteristic
43C2×2−1+4—24no—
44C2×2−1+4—24no—
45C2×2−1+4—24no—
46C2×2−1+4—41no—
47C2×2−1+4—42no—
48C2×2−1+4C2×C414yes—
49C2×2−1+4C2×C418yes—
50C22×#41—22no—
51C22×#41—41no—
52C22×#41C2312yes—
53C23×Q8—22no—
54C23×Q8C2311yescharacteristic
55C2×#131—24no—
56C2×#131—24no—
57C2×#131—24no—
58C2×#131—42no—
59C2×#131—42no—
60C2×#131C2×C418yescomplemented
61C22×M16—22no—
62C22×M16C2312yes—
63C22×M16C2312yes—
64C23×C8—22no—
65C23×C8C2312yes—
66#520—24no—
67#520—24no—
68#520—42no—
69#520C2×C418yescomplemented
70C2×C4×Q8—22no—
71C2×C4×D4—22no—
72C2×C4×D4—42no—
73C2×#117—22no—
74C2×#117C2312yes—
75C22×C4⋊C4—22no—
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2311yescharacteristic
78C22×C42C2311yescharacteristic
79C2×#104—24no—
80C2×#104—24no—
81C2×#104C2×C418yescomplemented
82#419—24no—
83#419—216no—
84#419C2×C418yescomplemented
85#417—24no—
86#417—24no—
87#417—216no—
88#417C2×C418yes—
89#417C2×C418yescomplemented
90#416—216no—
91#416C2×C418yes—
92#416D418yes—
93C2×#103—22no—
94C2×#103—22no—
95C2×#103—22no—
96C2×#103—22no—
97C2×#103—24no—
98C2×#103—22no—
99C2×#103—22no—
100C2×#103—216no—
101C2×#103C2×C418yescomplemented
102C2×#103D412yes—
103C2×#103D412yes—
104C2×#102—22no—
105C2×#102—22no—
106C2×#102D412yes—
107C2×#102D412yes—
108C2×#101—42no—
109C2×#101C2×C414yescomplemented
110#405—24no—
111#405—42no—
112C2×#97—24no—
113C2×#97—22no—
114C2×#97—41no—
115C2×C4×C8—22no—
116#391—41no—
117#391C2×C414yescomplemented
118C4×C4⋊C4—22no—
119C4×C4⋊C4—22no—
120C4×C22⋊C4—41no—
121C2×#95—41no—
122C2×#95—41no—
123#343—42no—
124#343—41no—
125#341—24no—
126#340—22no—
127#340—22no—
128#340—22no—
129#340—22no—
130#339—24no—
1312−1+4—216no—
1322−1+4—216no—
1332−1+4—216no—
1342−1+4—44no—
1352−1+4—48no—
1362−1+4—44no—
1372−1+4—48no—
1382−1+4—44no—
1392−1+4—48no—
1402−1+4C42116yescomplemented
141C2×#41—22no—
142C2×#41—22no—
143C2×#41—22no—
144C2×#41—22no—
145C2×#41—22no—
146C2×#41—22no—
147C2×#41—28no—
148C2×#41—28no—
149C2×#41—28no—
150C2×#41—44no—
151C2×#41—48no—
152C2×#41—44no—
153C2×#41—44no—
154C2×#41—44no—
155C2×#41—48no—
156C2×#41—44no—
157C2×#41C22×C414yes—
158C2×#41C22×C414yes—
159C2×#41C22×C414yes—
160C2×#41C2×D414yes—
161C2×#41#4114yes—
162C2×#41#4114yes—
163C22×Q8—21no—
164C22×Q8—22no—
165C22×Q8—28no—
166C22×Q8—28no—
167C22×Q8—28no—
168C22×Q8—22no—
169C22×Q8—22no—
170C22×Q8—22no—
171C22×Q8—22no—
172C22×Q8—41no—
173C22×Q8—44no—
174C22×Q8—44no—
175C22×Q8—44no—
176C22×Q8C22×C412yes—
177C22×Q8C22×C414yes—
178C22×Q8C2×D412yes—
179C22×D4—22no—
180C22×D4—41no—
181C22×D4C2×D412yes—
182C23×C4—22no—
183C23×C4—21no—
184C23×C4—21no—
185C23×C4—22no—
186C23×C4—21no—
187C23×C4—42no—
188C23×C4—42no—
189C23×C4C2411yescharacteristic
190#131—216no—
191#131—48no—
192#131—48no—
193#131—48no—
194C2×M16—22no—
195C2×M16—22no—
196C2×M16—28no—
197C2×M16—22no—
198C2×M16—22no—
199C2×M16—22no—
200C2×M16—22no—
201C2×M16—22no—
202C2×M16—22no—
203C2×M16—22no—
204C2×M16—22no—
205C2×M16—44no—
206C2×M16—44no—
207C2×M16C22×C414yes—
208C2×M16C22×C414yes—
209C22×C8—22no—
210C22×C8—22no—
211C22×C8—22no—
212C22×C8—22no—
213C22×C8—22no—
214C22×C8—22no—
215C22×C8—28no—
216C22×C8—44no—
217C22×C8—44no—
218C22×C8C22×C414yes—
219C22×C8C2×D412yes—
220C22×C8C2×D412yes—
221C22×C8C2×D414yes—
222C4×Q8—28no—
223C4×Q8—44no—
224C4×D4—28no—
225C4×D4—44no—
226C4×D4—48no—
227C4×D4—48no—
228#117—22no—
229#117—22no—
230#117—28no—
231#117—44no—
232#117C22×C414yes—
233#117C2×D414yes—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4C22×C414yes—
246C2×C4⋊C4C22×C412yes—
247C2×C4⋊C4C2×D412yes—
248C2×C4⋊C4C2×D412yes—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—42no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—21no—
255C2×C42—42no—
256C2×C42—41no—
257C2×C42—41no—
258C2×C42C22×C412yes—
259#104—216no—
260#103—28no—
261#103—28no—
262#103—28no—
263#103—28no—
264#103—28no—
265#103—28no—
266#103C22⋊C418yes—
267#103C22⋊C418yes—
268#102—28no—
269#102—28no—
270#102C22⋊C418yes—
271#102C22⋊C418yes—
272#101—44no—
273#97—44no—
274#97—44no—
275C4×C8—44no—
276#95—44no—
277#95—42no—
278#95—82no—
279C24—22no—
280C24H(5,𝔽2)11yescharacteristic
281#41—24no—
282#41—24no—
283#41—24no—
284#41—24no—
285#41—24no—
286#41—24no—
287#41—24no—
288#41—24no—
289#41—24no—
290#41—24no—
291#41—24no—
292#41—24no—
293#41—232no—
294#41—416no—
295#41—416no—
296#41—416no—
297#41—416no—
298#41—432no—
299#41—416no—
300#41C2×C4218yes—
301#41#11718yes—
302#41C4×D418yes—
303#41C4×D418yes—
304C2×Q8—22no—
305C2×Q8—24no—
306C2×Q8—22no—
307C2×Q8—22no—
308C2×Q8—22no—
309C2×Q8—232no—
310C2×Q8—24no—
311C2×Q8—24no—
312C2×Q8—24no—
313C2×Q8—24no—
314C2×Q8—24no—
315C2×Q8—24no—
316C2×Q8—48no—
317C2×Q8—44no—
318C2×Q8—416no—
319C2×Q8—416no—
320C2×Q8—416no—
321C2×Q8—42no—
322C2×Q8—41no—
323C2×Q8—42no—
324C2×Q8—42no—
325C2×Q8—42no—
326C2×Q8—42no—
327C2×Q8—41no—
328C2×Q8C2×C4214yes—
329C2×Q8C2×C22⋊C414yes—
330C2×Q8C4×D414yes—
331C2×Q8C4×D414yes—
332C2×D4—24no—
333C2×D4—24no—
334C2×D4—24no—
335C2×D4—24no—
336C2×D4—24no—
337C2×D4—24no—
338C2×D4—22no—
339C2×D4—22no—
340C2×D4—22no—
341C2×D4—48no—
342C2×D4—44no—
343C2×D4C2×C22⋊C414yes—
344C2×D4C4×D414yes—
345C2×D4C4×D414yes—
346C22×C4—22no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—48no—
361C22×C4—48no—
362C22×C4—48no—
363C22×C4—48no—
364C22×C4—41no—
365C22×C4—42no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—42no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—41no—
375C22×C4—82no—
376C22×C4—81no—
377C22×C4—81no—
378C22×C4—82no—
379C22×C4—162no—
380C22×C4—162no—
381C22×C4C23×C412yes—
382C22×C4C23×C411yescharacteristic
383C22×C4C22×D412yes—
384C22×C4C22×D412yes—
385C22×C4C2×#4112yes—
386C22×C4C2×#4112yes—
387C22×C4H(5,𝔽2)11yescharacteristic
388C22×C4H(5,𝔽2)12yes—
389M16—24no—
390M16—24no—
391M16—24no—
392M16—24no—
393M16—416no—
394M16—42no—
395M16—42no—
396M16—42no—
397M16—42no—
398C2×C8—24no—
399C2×C8—24no—
400C2×C8—22no—
401C2×C8—22no—
402C2×C8—24no—
403C2×C8—22no—
404C2×C8—22no—
405C2×C8—24no—
406C2×C8—24no—
407C2×C8—24no—
408C2×C8—416no—
409C2×C8—42no—
410C2×C8—42no—
411C2×C8—42no—
412C2×C8—42no—
413C2×C8C2×C22⋊C414yes—
414C2×C8C2×C22⋊C414yes—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—24no—
418C4⋊C4—24no—
419C4⋊C4—24no—
420C4⋊C4—48no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4C2×C22⋊C414yes—
425C4⋊C4C2×C22⋊C414yes—
426C22⋊C4—24no—
427C22⋊C4—24no—
428C22⋊C4—48no—
429C42—24no—
430C42—24no—
431C42—24no—
432C42—48no—
433C42—41no—
434C42—41no—
435C42—161no—
436C42—161no—
437C23—21no—
438C23—22no—
439C23—21no—
440C23—24no—
441C23—24no—
442C23—24no—
443C23—42no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23#51812yes—
448C23#51811yescharacteristic
449C23D4211yescharacteristic
450C23#54612yes—
451C23#55111yescharacteristic
452Q8—24no—
453Q8—24no—
454Q8—24no—
455Q8—28no—
456Q8—416no—
457Q8—44no—
458Q8—44no—
459Q8—44no—
460Q8C4×C22⋊C418yes—
461D4—28no—
462D4—28no—
463D4—28no—
464D4—28no—
465D4—24no—
466D4—24no—
467D4—24no—
468D4—416no—
469D4C4×C22⋊C418yes—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—28no—
474C2×C4—28no—
475C2×C4—28no—
476C2×C4—28no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—432no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—44no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—81no—
497C2×C4—82no—
498C2×C4—81no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—84no—
502C2×C4—168no—
503C2×C4—161no—
504C2×C4—162no—
505C2×C4C22×C4211yescharacteristic
506C2×C4C22×C22⋊C412yes—
507C2×C4C2×#11712yes—
508C2×C4C2×C4×D412yes—
509C2×C4C2×C4×D412yes—
510C2×C4C2×C4×D412yes—
511C2×C4C2×C4×D412yes—
512C2×C4#51812yes—
513C2×C4#51811yescharacteristic
514C2×C4D4211yescharacteristic
515C2×C4D4212yes—
516C2×C4#54612yes—
517C2×C4#54612yes—
518C2×C4#54612yes—
519C2×C4#55111yescharacteristic
520C2×C4#55112yes—
521C8—24no—
522C8—24no—
523C8—44no—
524C8—44no—
525C22—21no—
526C22—21no—
527C22—21no—
528C22—21no—
529C22—22no—
530C22—22no—
531C22—22no—
532C22—22no—
533C22—28no—
534C22—44no—
535C22—44no—
536C22—44no—
537C22—44no—
538C22—44no—
539C22—44no—
540C22#227311yescharacteristic
541C22D4×C22⋊C412yes—
542C22#234212yes—
543C22#235811yescharacteristic
544C22#235811yescharacteristic
545C22#328811yescharacteristic
546C22#328912yes—
547C22#328911yescharacteristic
548C4—28no—
549C4—44no—
550C4—44no—
551C4—44no—
552C4—82no—
553C4—82no—
554C4—84no—
555C4C2×C4×C22⋊C412yes—
556C4D4×C22⋊C412yes—
557C4#234212yes—
558C4#235812yes—
559C2—21no—
560C2—22no—
561C2—21no—
562C2—48no—
563C2C22⋊C4211yescharacteristic
564C2#2246412yes—
565C2#2246512yes—
566C2#2289211yescharacteristic
567C2#2289311yescharacteristic
subgroupG/H
center, socleC23D42
commutatorC2×C4C22×C42
Frattini, cosocleC23×C4C24
prime radicalC22×2−1+4C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1765C2semidirect
4. #20415C2semidirect
5. #10957C2semidirect
6. #10942C2semidirect
7. #10928C2semidirect
8. #10781C2semidirect
9. #10774C2semidirect
15. #2898C4semidirect
30. #1793C4semidirect
60. C2×#131C2×C4semidirect
69. #520C2×C4semidirect
81. C2×#104C2×C4semidirect
84. #419C2×C4semidirect
89. #417C2×C4semidirect
101. C2×#103C2×C4semidirect
109. C2×#101C2×C4semidirect
117. #391C2×C4semidirect
140. 2−1+4C42semidirect

Maximal subgroups

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

subgroupindex
C2×#28982
C2×#17932
C2×#17652
#204152
#109572
#109422
#109282
#107812
#107742