Finite Group Databasebeta

← #360678

Subgroups — #360678

Order 512.

1366 conjugacy classes of subgroups.

The subgroup lattice has 565 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34188C211yesmaximal, characteristic
2#20800C211yesmaximal, characteristic, complemented
3#20648C211yesmaximal, characteristic
4C2×C4×#104C211yesmaximal, characteristic
5#14687C211yesmaximal, characteristic, complemented
6#11113C214yesmaximal
7#10899C214yesmaximal, complemented
8#10857C211yesmaximal, characteristic, complemented
9#10839C211yesmaximal, characteristic, complemented
10C2×C4×#41C2211yescharacteristic
11C2×#428C2214yes—
12C2×#423C2211yescharacteristic
13C22×#104C2212yes—
14C2×#417C2212yes—
15C2×C4×M16C2211yescharacteristic
16#2355—21no—
17#2355—21no—
18#2354—21no—
19#2354—21no—
20#2352—22no—
21#2352—22no—
22#2314—22no—
23#2311C2212yes—
24#2285—21no—
25#2285—21no—
26#2285C2211yescharacteristic
27C2×#384C2211yescharacteristic
28C2×#384C2211yescharacteristic
29C2×#382—22no—
30#2274C2212yes—
31C2×C43C2211yescharacteristic
32#1847—22no—
33#1845—21no—
34#1844—21no—
35#1803—21no—
36#1803—21no—
37#1802—21no—
38#1802—21no—
39C4×#104C418yescomplemented
40C2×#343C2214yes—
41#1736C2212yes—
42C2×#339C2214yes—
43C2×#339C2212yes—
44C4⋊#95—22no—
45C4⋊#95C2214yes—
46C22×#41C2311yescharacteristic
47C22×M16C2312yes—
48C22×M16C2312yes—
49C4×#41—22no—
50C4×#41—22no—
51C4×#41C2×C414yes—
52C4×#41Q814yes—
53C2×C4×Q8D411yescharacteristic
54C2×C4×Q8D411yescharacteristic
55C2×C4×D4—21no—
56C2×C4×D4—21no—
57C2×C4×D4D411yescharacteristic
58C2×C4×D4D411yescharacteristic
59C2×#117—22no—
60C2×#117—22no—
61C2×#117—21no—
62C2×#117C2312yes—
63C2×#117C2311yescharacteristic
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4D412yes—
67C22×C4⋊C4D412yes—
68C22×C4⋊C4C2312yes—
69C22×C4⋊C4C2312yes—
70C22×C42—21no—
71C22×C42—21no—
72C22×C42—21no—
73C22×C42C2312yes—
74C22×C42C2311yescharacteristic
75#428—28no—
76#428—44no—
77C2×#107—24no—
78C2×#106—24no—
79#423—22no—
80#423—22no—
81C2×C4⋊C8—21no—
82C2×C4⋊C8—21no—
83C2×#104—24no—
84C2×#104—24no—
85C2×#104C2×C418yes—
86#417—44no—
87C2×#103—22no—
88C2×#102—22no—
89C4×M16—22no—
90C4×M16C2×C414yes—
91C2×#97—21no—
92C2×C4×C8—21no—
93#391—22no—
94#390—22no—
95#388—24no—
96#388—24no—
97#387—24no—
98#387—24no—
99#386—24no—
100#384—21no—
101#384—21no—
102#384—21no—
103#384—21no—
104#384—21no—
105#384—21no—
106#384—21no—
107#384—21no—
108#384D411yescharacteristic
109#384D411yescharacteristic
110#382—22no—
111#382—22no—
112#382—48no—
113#382D412yes—
114#380—22no—
115#380—22no—
116#380—21no—
117#380—21no—
118#380—22no—
119#380—22no—
120C4×C4⋊C4—22no—
121C4×C4⋊C4—22no—
122C4×C4⋊C4—22no—
123C4×C4⋊C4—22no—
124C4×C4⋊C4—22no—
125C4×C4⋊C4—22no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4—21no—
128C4×C22⋊C4—24no—
129C2×#95—28no—
130C43—22no—
131C43C2×C414yes—
132#343—28no—
133#340—22no—
134#340—22no—
135#339—44no—
136#339C2×C418yes—
137C2×#41—21no—
138C2×#41—21no—
139C2×#41—22no—
140C2×#41—22no—
141C2×#41—22no—
142C2×#41—22no—
143C2×#41C22×C412yes—
144C2×#41C2×Q812yes—
145C22×Q8C2×D411yescharacteristic
146C22×D4—21no—
147C22×D4C2×D411yescharacteristic
148C23×C4—22no—
149C23×C4—21no—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—22no—
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D414yes—
156C23×C4C2411yescharacteristic
157C2×M16—22no—
158C2×M16—22no—
159C2×M16—22no—
160C2×M16—24no—
161C2×M16—22no—
162C2×M16—22no—
163C2×M16—24no—
164C2×M16C22×C414yes—
165C2×M16#4114yes—
166C2×M16#4114yes—
167C22×C8—22no—
168C22×C8—22no—
169C4×Q8—22no—
170C4×Q8—22no—
171C4×Q8C4⋊C414yes—
172C4×Q8C4⋊C414yes—
173C4×D4—24no—
174C4×D4—24no—
175C4×D4—24no—
176C4×D4—24no—
177C4×D4—22no—
178C4×D4—22no—
179C4×D4C4⋊C414yes—
180C4×D4C4⋊C414yes—
181#117—21no—
182#117—21no—
183#117—24no—
184#117—24no—
185#117—24no—
186#117—24no—
187#117—84no—
188#117—84no—
189#117C22×C412yes—
190#117C2×Q812yes—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
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—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4—41no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4C2×D412yes—
218C2×C4⋊C4C2×D412yes—
219C2×C4⋊C4C2×D411yescharacteristic
220C2×C4⋊C4C2×D411yescharacteristic
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—24no—
227C2×C42—24no—
228C2×C42—24no—
229C2×C42—24no—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—21no—
233C2×C42—24no—
234C2×C42—24no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—24no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—24no—
243C2×C42—24no—
244C2×C42—42no—
245C2×C42—42no—
246C2×C42C22×C414yes—
247C2×C42C22×C412yes—
248C2×C42C2×D411yescharacteristic
249C2×C42C2×D411yescharacteristic
250C2×C42C2×D411yescharacteristic
251C2×C42C2×D411yescharacteristic
252C2×C42C2×Q812yes—
253C2×C42#4112yes—
254C2×C42#4112yes—
255#107—48no—
256#106—48no—
257C4⋊C8—42no—
258C4⋊C8—42no—
259#104—216no—
260#103—44no—
261#102—44no—
262#97—24no—
263C4×C8—24no—
264#95—24no—
265#95—24no—
266#95—24no—
267#95—24no—
268#95—22no—
269#95—22no—
270#95—416no—
271C24—21no—
272C24C22×D411yescharacteristic
273#41—24no—
274#41—24no—
275#41—24no—
276#41—24no—
277#41—42no—
278#41—42no—
279#41—42no—
280#41—42no—
281C2×Q8—21no—
282C2×Q8—21no—
283C2×Q8—21no—
284C2×Q8—21no—
285C2×Q8—21no—
286C2×Q8C2×C4⋊C412yes—
287C2×D4—22no—
288C2×D4—22no—
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—21no—
292C2×D4—21no—
293C2×D4—21no—
294C2×D4—42no—
295C2×D4—42no—
296C2×D4C2×C4⋊C412yes—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—24no—
314C22×C4—28no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—44no—
326C22×C4—44no—
327C22×C4—44no—
328C22×C4—44no—
329C22×C4—42no—
330C22×C4—44no—
331C22×C4C4×D412yes—
332C22×C4C4×D412yes—
333C22×C4#12011yescharacteristic
334C22×C4#12011yescharacteristic
335C22×C4#12011yescharacteristic
336C22×C4#12011yescharacteristic
337C22×C4C22⋊Q812yes—
338C22×C4C22⋊Q812yes—
339C22×C4C23×C411yescharacteristic
340C22×C4C22×D411yescharacteristic
341C22×C4C22×D411yescharacteristic
342C22×C4C22×Q811yescharacteristic
343C22×C4C2×#4112yes—
344C22×C4C2×#4112yes—
345C22×C4C2×#4111yescharacteristic
346C22×C4C2×#4111yescharacteristic
347C22×C4H(5,𝔽2)12yes—
348C22×C42−1+412yes—
349M16—24no—
350M16—24no—
351M16—24no—
352M16—44no—
353M16—44no—
354M16#11718yescomplemented
355C2×C8—24no—
356C2×C8—24no—
357C2×C8—24no—
358C2×C8—42no—
359C2×C8—42no—
360C2×C8—42no—
361C4⋊C4—21no—
362C4⋊C4—21no—
363C4⋊C4—24no—
364C4⋊C4—24no—
365C4⋊C4—24no—
366C4⋊C4—24no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—44no—
370C4⋊C4—44no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—44no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—44no—
379C4⋊C4—82no—
380C4⋊C4—82no—
381C4⋊C4—82no—
382C4⋊C4—82no—
383C4⋊C4—82no—
384C4⋊C4—82no—
385C4⋊C4C2×C4⋊C412yes—
386C4⋊C4C2×C4⋊C412yes—
387C22⋊C4—22no—
388C22⋊C4—22no—
389C22⋊C4—24no—
390C22⋊C4—24no—
391C22⋊C4—44no—
392C22⋊C4—84no—
393C22⋊C4—84no—
394C42—216no—
395C42—22no—
396C42—22no—
397C42—24no—
398C42—24no—
399C42—24no—
400C42—24no—
401C42—48no—
402C42—42no—
403C42—42no—
404C42—42no—
405C42—42no—
406C42—84no—
407C42—84no—
408C42C2×C4⋊C412yes—
409C42C2×C4⋊C412yes—
410C42C4×D412yes—
411C42C4×D412yes—
412C42C22⋊Q812yes—
413C42C22⋊Q812yes—
414C42#12312yes—
415C42#12312yes—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—22no—
421C23—41no—
422C23—41no—
423C23—41no—
424C23—41no—
425C23—41no—
426C23—41no—
427C23C2×C4×D411yescharacteristic
428C23C2×#12011yescharacteristic
429C23C2×C22⋊Q811yescharacteristic
430C23#54712yes—
431C23#55012yes—
432Q8—22no—
433Q8—22no—
434Q8—41no—
435Q8—41no—
436D4—22no—
437D4—22no—
438D4—44no—
439D4—44no—
440D4—41no—
441D4—41no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—28no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—44no—
467C2×C4—48no—
468C2×C4—48no—
469C2×C4—44no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—44no—
473C2×C4—42no—
474C2×C4—44no—
475C2×C4—42no—
476C2×C4—44no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—88no—
480C2×C4—88no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4#38512yes—
485C2×C4#38512yes—
486C2×C4#38512yes—
487C2×C4#38512yes—
488C2×C4#45312yes—
489C2×C4#45412yes—
490C2×C4#45512yes—
491C2×C4#45612yes—
492C2×C4C22×C4⋊C411yescharacteristic
493C2×C4C2×#11712yes—
494C2×C4C2×C4×D411yescharacteristic
495C2×C4#52012yes—
496C2×C4C2×C22⋊Q811yescharacteristic
497C2×C4C2×#12311yescharacteristic
498C2×C4#53211yescharacteristic
499C2×C4#53211yescharacteristic
500C2×C4#53312yes—
501C2×C4#53511yescharacteristic
502C2×C4#53611yescharacteristic
503C2×C4#54412yes—
504C2×C4#54612yes—
505C2×C4#54812yes—
506C2×C4#55212yes—
507C2×C4#55312yes—
508C2×C4#55512yes—
509C2×C4#55612yes—
510C8—28no—
511C8—82no—
512C22—22no—
513C22—22no—
514C22—22no—
515C22—22no—
516C22—21no—
517C22—21no—
518C22—21no—
519C22—21no—
520C22—42no—
521C22—42no—
522C22—42no—
523C22—81no—
524C22—81no—
525C22—81no—
526C22—81no—
527C22C2×#38511yescharacteristic
528C22#235412yes—
529C22#243111yescharacteristic
530C22#243211yescharacteristic
531C22#244112yes—
532C22C2×#45311yescharacteristic
533C22C2×#45511yescharacteristic
534C22C2×#45411yescharacteristic
535C22C2×#45611yescharacteristic
536C4—22no—
537C4—22no—
538C4—48no—
539C4—42no—
540C4—42no—
541C4—42no—
542C4—82no—
543C4—82no—
544C4—84no—
545C4—162no—
546C4#185914yes—
547C4#186014yes—
548C4#228511yescharacteristic
549C4#231711yescharacteristic
550C4#235212yes—
551C4#243912yes—
552C4#249911yescharacteristic
553C4#250011yescharacteristic
554C2—22no—
555C2—22no—
556C2—82no—
557C2#1387911yescharacteristic
558C2C2×#185911yescharacteristic
559C2C2×#186011yescharacteristic
560C2#2561511yescharacteristic
561C2#2561611yescharacteristic
562C2#2568711yescharacteristic
563C2#2568811yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20800C2semidirect
5. #14687C2semidirect
7. #10899C2semidirect
8. #10857C2semidirect
9. #10839C2semidirect
39. C4×#104C4semidirect
354. M16#117semidirect

Maximal subgroups

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

subgroupindex
#341882
#208002
#206482
C2×C4×#1042
#146872
#111132
#108992
#108572
#108392