Finite Group Databasebeta

← #365720

Subgroups — #365720

Order 512.

1558 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34207C211yesmaximal, characteristic
2C2×#1962C211yesmaximal, characteristic
3#21200C211yesmaximal, characteristic
4#21171C211yesmaximal, characteristic, complemented
5#14687C211yesmaximal, characteristic, complemented
6#11113C214yesmaximal, complemented
7#10857C211yesmaximal, characteristic, complemented
8#10781C214yesmaximal, complemented
9C4×#339C211yesmaximal, characteristic, complemented
10C2×#532C2211yescharacteristic
11C2×#520C2212yes—
12C2×#428C2212yes—
13C2×#423C2211yescharacteristic
14C22×#104C2214yes—
15C2×#417C2214yes—
16C2×C4×M16C2211yescharacteristic
17#2361—22no—
18#2361—22no—
19#2354—21no—
20#2354—21no—
21D4×C4⋊C4—21no—
22D4×C4⋊C4—21no—
23#2326—22no—
24#2311C2212yes—
25C4×C4⋊Q8—21no—
26C4×#127—21no—
27C2×#384C2211yescharacteristic
28C2×#384C2211yescharacteristic
29C2×#382—22no—
30C4×#117C2211yescharacteristic
31C2×C43C2211yescharacteristic
32#1962C418yescomplemented
33#1959—22no—
34#1957—21no—
35#1956—21no—
36#1948—21no—
37#1948—21no—
38#1947—21no—
39#1947—21no—
40C2×#343C2214yes—
41#1736C2212yes—
42C2×#339C2212yes—
43C2×#339C2212yes—
44C4⋊#95—22no—
45C4⋊#95C2214yes—
46C22×#41C2312yes—
47C22×M16C2312yes—
48C22×M16C2312yes—
49#532—22no—
50#532—22no—
51#532C2×C414yes—
52#532Q814yes—
53C2×C4⋊Q8D411yescharacteristic
54C2×#127D411yescharacteristic
55C2×#124D412yes—
56C2×#121—21no—
57C2×#121—21no—
58#520—28no—
59#520—44no—
60C2×C4×Q8—22no—
61C2×C4×D4—22no—
62C2×C4×D4—24no—
63C2×C4×D4—22no—
64C2×C4×D4—22no—
65C2×C4×D4—21no—
66C2×C4×D4—21no—
67C2×#117—22no—
68C2×#117C2311yescharacteristic
69C2×#117C2311yescharacteristic
70C22×C4⋊C4—22no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4D412yes—
74C22×C4⋊C4D412yes—
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2312yes—
77C22×C42—21no—
78C22×C42—21no—
79C22×C42C2312yes—
80C22×C42C2311yescharacteristic
81#428—44no—
82C2×#107—22no—
83C2×#106—22no—
84#423—22no—
85#423C2×C414yescomplemented
86C2×C4⋊C8—21no—
87C2×C4⋊C8—21no—
88C2×#104—44no—
89C2×#104—44no—
90C2×#104C2×C4116yescomplemented
91#417—28no—
92#417—44no—
93C2×#103—24no—
94C2×#102—24no—
95C4×M16—22no—
96C4×M16—22no—
97C2×#97—21no—
98C2×C4×C8—21no—
99#391—22no—
100#390—22no—
101#390—22no—
102#390—22no—
103#389—22no—
104#389—22no—
105#388—24no—
106#388—24no—
107#386—24no—
108#385—24no—
109#385—24no—
110#384—21no—
111#384—21no—
112#384—21no—
113#384—21no—
114#384—21no—
115#384—21no—
116#384—21no—
117#384—21no—
118#382—48no—
119#380—22no—
120#380—22no—
121C4×C4⋊C4—22no—
122C4×C4⋊C4—22no—
123C4×C4⋊C4D411yescharacteristic
124C4×C4⋊C4D411yescharacteristic
125C4×C4⋊C4D411yescharacteristic
126C4×C4⋊C4D411yescharacteristic
127C4×C22⋊C4—21no—
128C4×C22⋊C4—21no—
129#376—21no—
130C2×#95—28no—
131C43—21no—
132C43—22no—
133C43C2×C414yescomplemented
134#343—44no—
135#340—22no—
136#340—22no—
137#339—24no—
138#339—24no—
139C2×#41—22no—
140C2×#41—22no—
141C2×#41—24no—
142C2×#41—24no—
143C2×#41—24no—
144C2×#41—24no—
145C2×#41C22×C414yes—
146C2×#41C2×Q814yes—
147C22×Q8C2×D412yes—
148C22×D4—22no—
149C22×D4—21no—
150C22×D4—21no—
151C22×D4C2×D412yes—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—24no—
155C23×C4—22no—
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D414yes—
159C23×C4C2411yescharacteristic
160C2×M16—22no—
161C2×M16—22no—
162C2×M16—22no—
163C2×M16—22no—
164C2×M16—22no—
165C2×M16—22no—
166C2×M16—24no—
167C2×M16—24no—
168C2×M16C22×C414yes—
169C2×M16#4114yes—
170C22×C8—22no—
171C22×C8—22no—
172C4⋊Q8—22no—
173C4⋊Q8C4⋊C414yes—
174#127—22no—
175#127C4⋊C414yes—
176#124—24no—
177#124C4⋊C418yes—
178#121—24no—
179#121—24no—
180#121—24no—
181#121—24no—
182C4×Q8—48no—
183C4×D4—24no—
184C4×D4—24no—
185C4×D4—24no—
186C4×D4—24no—
187C4×D4—48no—
188C4×D4—416no—
189C4×D4—84no—
190C4×D4—84no—
191#117—22no—
192#117—22no—
193#117—22no—
194#117—44no—
195#117—44no—
196#117#4114yes—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—41no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4—41no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4C2×D411yescharacteristic
223C2×C4⋊C4C2×D411yescharacteristic
224C2×C4⋊C4C2×D411yescharacteristic
225C2×C4⋊C4C2×D411yescharacteristic
226C2×C22⋊C4—21no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—24no—
233C2×C42—24no—
234C2×C42—24no—
235C2×C42—24no—
236C2×C42—21no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—24no—
242C2×C42—24no—
243C2×C42—21no—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42—21no—
248C2×C42—22no—
249C2×C42—42no—
250C2×C42—42no—
251C2×C42C22×C414yes—
252C2×C42C22×C412yes—
253C2×C42C2×D411yescharacteristic
254C2×C42C2×D411yescharacteristic
255C2×C42C2×D411yescharacteristic
256C2×C42C2×D411yescharacteristic
257C2×C42C2×Q812yes—
258C2×C42#4112yes—
259C2×C42#4112yes—
260#107—44no—
261#106—44no—
262C4⋊C8—24no—
263C4⋊C8—24no—
264#104—416no—
265#103—48no—
266#102—48no—
267#97—42no—
268C4×C8—42no—
269#95—24no—
270#95—24no—
271#95—21no—
272#95—21no—
273#95—21no—
274#95—21no—
275#95—416no—
276C24—22no—
277C24C22×D411yescharacteristic
278#41—28no—
279#41—28no—
280#41—28no—
281#41—28no—
282#41—44no—
283#41—44no—
284#41—44no—
285#41—44no—
286C2×Q8—22no—
287C2×Q8—24no—
288C2×Q8—24no—
289C2×Q8C2×C4⋊C414yes—
290C2×D4—24no—
291C2×D4—24no—
292C2×D4—22no—
293C2×D4—24no—
294C2×D4—24no—
295C2×D4—24no—
296C2×D4—24no—
297C2×D4—44no—
298C2×D4—44no—
299C2×D4—42no—
300C2×D4—42no—
301C2×D4—42no—
302C2×D4—42no—
303C2×D4C2×C4⋊C414yes—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—22no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—24no—
319C22×C4—28no—
320C22×C4—48no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—44no—
330C22×C4—44no—
331C22×C4—44no—
332C22×C4—44no—
333C22×C4—42no—
334C22×C4—44no—
335C22×C4—84no—
336C22×C4—84no—
337C22×C4C4×D412yes—
338C22×C4C4×D412yes—
339C22×C4#12011yescharacteristic
340C22×C4#12011yescharacteristic
341C22×C4#12011yescharacteristic
342C22×C4#12011yescharacteristic
343C22×C4C22⋊Q812yes—
344C22×C4C22⋊Q812yes—
345C22×C4C23×C411yescharacteristic
346C22×C4C22×D411yescharacteristic
347C22×C4C22×D411yescharacteristic
348C22×C4C22×Q811yescharacteristic
349C22×C4C2×#4111yescharacteristic
350C22×C4C2×#4112yes—
351C22×C4C2×#4112yes—
352C22×C4C2×#4111yescharacteristic
353C22×C4H(5,𝔽2)12yes—
354C22×C4H(5,𝔽2)12yes—
355M16—28no—
356M16—24no—
357M16—24no—
358M16—44no—
359M16—42no—
360M16—42no—
361C2×C8—24no—
362C2×C8—24no—
363C2×C8—42no—
364C2×C8—42no—
365C2×C8—42no—
366C2×C8—42no—
367C4⋊C4—21no—
368C4⋊C4—21no—
369C4⋊C4—24no—
370C4⋊C4—24no—
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—21no—
374C4⋊C4—21no—
375C4⋊C4—48no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—44no—
379C4⋊C4—44no—
380C4⋊C4—44no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4—44no—
384C4⋊C4—82no—
385C4⋊C4—82no—
386C4⋊C4—82no—
387C4⋊C4—82no—
388C4⋊C4—82no—
389C4⋊C4—82no—
390C4⋊C4#12412yes—
391C4⋊C4#12412yes—
392C22⋊C4—24no—
393C22⋊C4—28no—
394C22⋊C4—28no—
395C22⋊C4—48no—
396C22⋊C4—42no—
397C22⋊C4—84no—
398C22⋊C4—84no—
399C42—216no—
400C42—24no—
401C42—24no—
402C42—24no—
403C42—48no—
404C42—48no—
405C42—42no—
406C42—42no—
407C42—42no—
408C42—42no—
409C42—42no—
410C42—42no—
411C42—42no—
412C42C2×C4⋊C412yes—
413C42C2×C4⋊C412yes—
414C42C4×D412yes—
415C42C4×D412yes—
416C42C22⋊Q812yes—
417C42C22⋊Q812yes—
418C42#12312yes—
419C42#12312yes—
420C23—21no—
421C23—21no—
422C23—21no—
423C23—21no—
424C23—24no—
425C23—42no—
426C23—42no—
427C23—42no—
428C23—42no—
429C23—42no—
430C23—42no—
431C23C2×C4×D411yescharacteristic
432C23C2×#12011yescharacteristic
433C23C2×C22⋊Q811yescharacteristic
434C23#54612yes—
435C23#55012yes—
436Q8—28no—
437Q8—44no—
438D4—28no—
439D4—48no—
440D4—48no—
441D4—48no—
442D4—48no—
443D4—44no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—28no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—24no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—44no—
475C2×C4—41no—
476C2×C4—44no—
477C2×C4—41no—
478C2×C4—44no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—88no—
482C2×C4—88no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4—168no—
487C2×C4#38512yes—
488C2×C4#38512yes—
489C2×C4#38512yes—
490C2×C4#38512yes—
491C2×C4#45312yes—
492C2×C4#45412yes—
493C2×C4#45512yes—
494C2×C4#45612yes—
495C2×C4C22×C4⋊C411yescharacteristic
496C2×C4C2×C4×D411yescharacteristic
497C2×C4C4×#4112yes—
498C2×C4#51812yes—
499C2×C4C2×C22⋊Q811yescharacteristic
500C2×C4C2×#12311yescharacteristic
501C2×C4C2×#12411yescharacteristic
502C2×C4#52912yes—
503C2×C4#53211yescharacteristic
504C2×C4#53511yescharacteristic
505C2×C4#53511yescharacteristic
506C2×C4#54012yes—
507C2×C4#54612yes—
508C2×C4#54612yes—
509C2×C4#55312yes—
510C2×C4#55412yes—
511C2×C4#55412yes—
512C2×C4#55512yes—
513C8—44no—
514C8—44no—
515C22—22no—
516C22—22no—
517C22—22no—
518C22—22no—
519C22—21no—
520C22—21no—
521C22—21no—
522C22—21no—
523C22—44no—
524C22—44no—
525C22—44no—
526C22—82no—
527C22—82no—
528C22—82no—
529C22—82no—
530C22C2×#38511yescharacteristic
531C22#236312yes—
532C22#243111yescharacteristic
533C22#243511yescharacteristic
534C22#245412yes—
535C22C2×#45311yescharacteristic
536C22C2×#45511yescharacteristic
537C22C2×#45411yescharacteristic
538C22C2×#45611yescharacteristic
539C4—22no—
540C4—22no—
541C4—48no—
542C4—42no—
543C4—42no—
544C4—44no—
545C4—81no—
546C4—81no—
547C4—84no—
548C4—162no—
549C4#185914yes—
550C4#186014yes—
551C4C4×#12411yescharacteristic
552C4#232811yescharacteristic
553C4#234912yes—
554C4#242012yes—
555C4#256311yescharacteristic
556C4#256611yescharacteristic
557C2—22no—
558C2—22no—
559C2—84no—
560C2#1397911yescharacteristic
561C2C2×#185911yescharacteristic
562C2C2×#186011yescharacteristic
563C2#2561511yescharacteristic
564C2#2561611yescharacteristic
565C2#2587511yescharacteristic
566C2#2587611yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#532C22
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
4. #21171C2semidirect
5. #14687C2semidirect
6. #11113C2semidirect
7. #10857C2semidirect
8. #10781C2semidirect
9. C4×#339C2semidirect
32. #1962C4semidirect
85. #423C2×C4semidirect
90. C2×#104C2×C4semidirect
133. C43C2×C4semidirect

Maximal subgroups

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

subgroupindex
#342072
C2×#19622
#212002
#211712
#146872
#111132
#108572
#107812
C4×#3392