Finite Group Databasebeta

← #365582

Subgroups — #365582

Order 512.

1636 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34206C211yesmaximal, characteristic
2C2×#1980C211yesmaximal, characteristic
3#21200C211yesmaximal, characteristic
4C2×#1859C214yesmaximal, complemented
5#20800C211yesmaximal, characteristic, complemented
6#13910C211yesmaximal, characteristic, complemented
7#11113C214yesmaximal, complemented
8#10851C211yesmaximal, characteristic, complemented
9C4×#343C211yesmaximal, characteristic, complemented
10C2×#532C2211yescharacteristic
11C2×#520C2212yes—
12C2×#428C2214yes—
13C22×#104C2214yes—
14C2×#417C2214yes—
15C2×C4×M16C2211yescharacteristic
16C2×C4×M16C2211yescharacteristic
17#2367—22no—
18#2366—21no—
19#2365—21no—
20#2354—21no—
21#2352—22no—
22D4×C4⋊C4—21no—
23#2325—22no—
24#2311C2212yes—
25C4×#121—22no—
26C2×#389—22no—
27#2285C2211yescharacteristic
28C2×#384C2212yes—
29C2×#384C2211yescharacteristic
30C2×C4×C4⋊C4C2211yescharacteristic
31C2×C4×C4⋊C4C2211yescharacteristic
32#1980C418yescomplemented
33#1959—22no—
34#1957—21no—
35#1956—21no—
36#1859—216no—
37#1847—22no—
38#1845—21no—
39#1844—21no—
40C2×#343C2212yes—
41C2×#343C2212yes—
42C2×#339C2214yes—
43#1444—22no—
44C4⋊#95C2212yes—
45C22×#41C2312yes—
46C22×M16C2312yes—
47C22×M16C2312yes—
48#532—22no—
49#532—22no—
50#532C2×C414yes—
51#532Q814yes—
52C2×C4⋊Q8D411yescharacteristic
53C2×#127D411yescharacteristic
54C2×#124D412yes—
55C2×#121—21no—
56C2×#121—21no—
57#520—28no—
58#520—44no—
59C2×C4×Q8—22no—
60C2×C4×D4—22no—
61C2×C4×D4—24no—
62C2×C4×D4—22no—
63C2×C4×D4—21no—
64C2×C4×D4—21no—
65C2×#117—22no—
66C2×#117C2312yes—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—24no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2311yescharacteristic
75C22×C42D412yes—
76C22×C42C2312yes—
77C22×C42C2311yescharacteristic
78#428—28no—
79#428—44no—
80C2×#107—24no—
81C2×#106—24no—
82C2×#104—28no—
83C2×#104—28no—
84C2×#104C2×C4116yescomplemented
85#417—28no—
86#417—44no—
87C2×#103—24no—
88C2×#102—24no—
89C4×M16—22no—
90C4×M16—22no—
91C4×M16—22no—
92C4×M16C2×C414yescomplemented
93C2×#97—21no—
94C2×#97—21no—
95C2×C4×C8—21no—
96C2×C4×C8—21no—
97#391—22no—
98#390—22no—
99#390—22no—
100#389—22no—
101#389—48no—
102#388—24no—
103#388—24no—
104#387—24no—
105#386—24no—
106#386—24no—
107#385—24no—
108#384—28no—
109#384—21no—
110#384—21no—
111#384—21no—
112#384—21no—
113#384D411yescharacteristic
114#384D411yescharacteristic
115#382D412yes—
116#380—21no—
117#380—21no—
118#380—22no—
119#380—22no—
120C4×C4⋊C4—22no—
121C4×C4⋊C4—21no—
122C4×C4⋊C4—21no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4—21no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4—22no—
128C4×C4⋊C4C2×C414yescomplemented
129C4×C22⋊C4—24no—
130C2×#95—21no—
131C2×#95—24no—
132C2×#95D412yes—
133#343—24no—
134#343D418yescomplemented
135#339—28no—
136C2×#41—22no—
137C2×#41—22no—
138C2×#41—24no—
139C2×#41—24no—
140C2×#41—24no—
141C2×#41—24no—
142C2×#41C22×C414yes—
143C2×#41C2×Q814yes—
144C22×Q8C2×D412yes—
145C22×D4—22no—
146C22×D4—21no—
147C22×D4—21no—
148C22×D4C2×D412yes—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—24no—
152C23×C4—22no—
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D412yes—
155C23×C4C2×D412yes—
156C23×C4C2×D411yescharacteristic
157C23×C4C2411yescharacteristic
158C2×M16—22no—
159C2×M16—22no—
160C2×M16—22no—
161C2×M16—22no—
162C2×M16—22no—
163C2×M16—22no—
164C2×M16—22no—
165C2×M16—22no—
166C2×M16C22×C414yes—
167C2×M16C2×D414yes—
168C2×M16C2×D414yes—
169C2×M16#4114yes—
170C22×C8—22no—
171C22×C8—22no—
172C4⋊Q8—22no—
173C4⋊Q8C4⋊C414yescomplemented
174#127—22no—
175#127C4⋊C414yescomplemented
176#124—24no—
177#124C4⋊C418yescomplemented
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—48no—
189C4×D4—48no—
190C4×D4—84no—
191#117—24no—
192#117—24no—
193#117—48no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—48no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—48no—
218C2×C4⋊C4—41no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4—41no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4—48no—
223C2×C4⋊C4—82no—
224C2×C4⋊C4—84no—
225C2×C4⋊C4C2×D412yes—
226C2×C4⋊C4C2×D412yes—
227C2×C4⋊C4C2×D414yes—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—24no—
234C2×C42—24no—
235C2×C42—24no—
236C2×C42—24no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—44no—
242C2×C42C22×C414yes—
243C2×C42C22×C412yes—
244C2×C42C2×D411yescharacteristic
245C2×C42C2×D411yescharacteristic
246C2×C42C2×D411yescharacteristic
247C2×C42C2×D411yescharacteristic
248C2×C42C2×Q812yes—
249C2×C42#4112yes—
250C2×C42#4112yes—
251#107—48no—
252#106—48no—
253#104—232no—
254#103—48no—
255#102—48no—
256#97—24no—
257#97—42no—
258C4×C8—24no—
259C4×C8—42no—
260#95—24no—
261#95—24no—
262#95—24no—
263#95—22no—
264#95—22no—
265#95—22no—
266#95—22no—
267#95—48no—
268C24—22no—
269C24C22×D411yescharacteristic
270#41—28no—
271#41—28no—
272#41—28no—
273#41—28no—
274#41—44no—
275#41—44no—
276#41—44no—
277#41—44no—
278C2×Q8—22no—
279C2×Q8—24no—
280C2×Q8—24no—
281C2×Q8C2×C4⋊C414yes—
282C2×D4—24no—
283C2×D4—24no—
284C2×D4—22no—
285C2×D4—24no—
286C2×D4—24no—
287C2×D4—24no—
288C2×D4—24no—
289C2×D4—42no—
290C2×D4—42no—
291C2×D4—42no—
292C2×D4—42no—
293C2×D4—42no—
294C2×D4—42no—
295C2×D4—81no—
296C2×D4—81no—
297C2×D4C2×C4⋊C414yes—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—48no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—44no—
321C22×C4—44no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—44no—
327C22×C4—44no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—84no—
332C22×C4—84no—
333C22×C4C4×D412yes—
334C22×C4C4×D412yes—
335C22×C4#12011yescharacteristic
336C22×C4#12011yescharacteristic
337C22×C4#12011yescharacteristic
338C22×C4#12011yescharacteristic
339C22×C4C22⋊Q812yes—
340C22×C4C22⋊Q812yes—
341C22×C4#12714yes—
342C22×C4C23×C411yescharacteristic
343C22×C4C22×D411yescharacteristic
344C22×C4C22×D411yescharacteristic
345C22×C4C22×D412yes—
346C22×C4C22×Q811yescharacteristic
347C22×C4C2×#4111yescharacteristic
348C22×C4C2×#4112yes—
349C22×C4C2×#4111yescharacteristic
350C22×C4H(5,𝔽2)12yes—
351C22×C42−1+412yes—
352M16—24no—
353M16—24no—
354M16—24no—
355M16—24no—
356M16—24no—
357M16—42no—
358M16—42no—
359M16C4×D418yescomplemented
360C2×C8—24no—
361C2×C8—24no—
362C2×C8—24no—
363C2×C8—24no—
364C2×C8—42no—
365C2×C8—42no—
366C4⋊C4—216no—
367C4⋊C4—24no—
368C4⋊C4—24no—
369C4⋊C4—24no—
370C4⋊C4—24no—
371C4⋊C4—24no—
372C4⋊C4—24no—
373C4⋊C4—48no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—44no—
379C4⋊C4—44no—
380C4⋊C4—81no—
381C4⋊C4—81no—
382C4⋊C4—82no—
383C4⋊C4—82no—
384C4⋊C4—82no—
385C4⋊C4—816no—
386C4⋊C4—82no—
387C22⋊C4—28no—
388C22⋊C4—28no—
389C22⋊C4—48no—
390C22⋊C4—42no—
391C22⋊C4—42no—
392C22⋊C4—84no—
393C22⋊C4—84no—
394C42—28no—
395C42—48no—
396C42—44no—
397C42—44no—
398C42—44no—
399C42—82no—
400C42C2×C4⋊C412yes—
401C42C2×C4⋊C412yes—
402C42C4×D412yes—
403C42C4×D412yes—
404C42C22⋊Q812yes—
405C42C22⋊Q812yes—
406C42#12312yes—
407C42#12312yes—
408C23—21no—
409C23—21no—
410C23—21no—
411C23—21no—
412C23—24no—
413C23—42no—
414C23—42no—
415C23—42no—
416C23—42no—
417C23—42no—
418C23—42no—
419C23C2×C4×D411yescharacteristic
420C23C2×#12011yescharacteristic
421C23C2×C22⋊Q811yescharacteristic
422C23C2×#12711yescharacteristic
423C23#53211yescharacteristic
424C23#53511yescharacteristic
425C23#53611yescharacteristic
426Q8—28no—
427Q8—44no—
428D4—28no—
429D4—44no—
430D4—44no—
431D4—44no—
432D4—44no—
433D4—48no—
434D4—44no—
435D4—84no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—28no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—44no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—48no—
460C2×C4—48no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—84no—
475C2×C4—84no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4—82no—
479C2×C4—168no—
480C2×C4#38512yes—
481C2×C4#38512yes—
482C2×C4#38512yes—
483C2×C4#38512yes—
484C2×C4#45312yes—
485C2×C4#45612yes—
486C2×C4C22×C4⋊C411yescharacteristic
487C2×C4C2×C4×D412yes—
488C2×C4C2×C4×D411yescharacteristic
489C2×C4#52012yes—
490C2×C4C2×C22⋊Q811yescharacteristic
491C2×C4C2×#12311yescharacteristic
492C2×C4#53212yes—
493C2×C4#54212yes—
494C2×C4D4212yes—
495C2×C4D4212yes—
496C2×C4#54712yes—
497C2×C4#54712yes—
498C2×C4#54812yes—
499C2×C4D4×Q812yes—
500C2×C4#55312yes—
501C2×C4#55412yes—
502C8—28no—
503C8—44no—
504C22—21no—
505C22—21no—
506C22—21no—
507C22—21no—
508C22—21no—
509C22—21no—
510C22—21no—
511C22—21no—
512C22—21no—
513C22—21no—
514C22—21no—
515C22—21no—
516C22—44no—
517C22—44no—
518C22—44no—
519C22—82no—
520C22—82no—
521C22—82no—
522C22—82no—
523C22C2×#38511yescharacteristic
524C22C4×#12711yescharacteristic
525C22#232611yescharacteristic
526C22#243211yescharacteristic
527C22#243311yescharacteristic
528C22#254211yescharacteristic
529C22#256611yescharacteristic
530C22C2×#45311yescharacteristic
531C22C2×#45611yescharacteristic
532C22#302111yescharacteristic
533C22#302211yescharacteristic
534C4—22no—
535C4—22no—
536C4—22no—
537C4—48no—
538C4—44no—
539C4—44no—
540C4—82no—
541C4—82no—
542C4—82no—
543C4—162no—
544C4#185914yescomplemented
545C4D4×C4⋊C412yes—
546C4#236112yes—
547C4#242212yes—
548C4#245212yes—
549C2—21no—
550C2—21no—
551C2—21no—
552C2—21no—
553C2—84no—
554C2#1397711yescharacteristic
555C2C2×#185911yescharacteristic
556C2#2086511yescharacteristic
557C2#2569411yescharacteristic
558C2#2569511yescharacteristic
559C2#2573911yescharacteristic
560C2#2574011yescharacteristic
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. C2×#1859C2semidirect
5. #20800C2semidirect
6. #13910C2semidirect
7. #11113C2semidirect
8. #10851C2semidirect
9. C4×#343C2semidirect
32. #1980C4semidirect
84. C2×#104C2×C4semidirect
92. C4×M16C2×C4semidirect
128. C4×C4⋊C4C2×C4semidirect
134. #343D4semidirect
173. C4⋊Q8C4⋊C4semidirect
175. #127C4⋊C4semidirect
177. #124C4⋊C4semidirect
359. M16C4×D4semidirect
544. C4#1859semidirect

Maximal subgroups

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

subgroupindex
#342062
C2×#19802
#212002
C2×#18592
#208002
#139102
#111132
#108512
C4×#3432