Finite Group Databasebeta

← #365581

Subgroups — #365581

Order 512.

1564 conjugacy classes of subgroups.

The subgroup lattice has 558 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×#1860C214yesmaximal, complemented
5#20800C211yesmaximal, characteristic, complemented
6#13911C211yesmaximal, 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—
26#2285C2211yescharacteristic
27C2×#384C2211yescharacteristic
28C2×#382C2212yes—
29C2×C4×C4⋊C4C2211yescharacteristic
30C2×C4×C4⋊C4C2211yescharacteristic
31#1980C418yescomplemented
32#1959—22no—
33#1957—21no—
34#1956—21no—
35#1860—216no—
36#1847—22no—
37#1845—21no—
38#1844—21no—
39C2×#343C2212yes—
40C2×#343C2212yes—
41C2×#339C2214yes—
42#1442—22no—
43C4⋊#95—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—21no—
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2311yescharacteristic
74C22×C42D412yes—
75C22×C42C2312yes—
76C22×C42C2311yescharacteristic
77#428—28no—
78#428—44no—
79C2×#107—24no—
80C2×#106—24no—
81C2×#104—28no—
82C2×#104—28no—
83C2×#104C2×C4116yescomplemented
84#417—28no—
85#417—44no—
86C2×#103—24no—
87C2×#102—24no—
88C4×M16—22no—
89C4×M16—22no—
90C4×M16—22no—
91C4×M16C2×C414yescomplemented
92C2×#97—21no—
93C2×#97—21no—
94C2×C4×C8—21no—
95C2×C4×C8—21no—
96#391—22no—
97#390—22no—
98#390—22no—
99#389—22no—
100#388—24no—
101#388—24no—
102#387—24no—
103#386—24no—
104#386—24no—
105#385—24no—
106#384—21no—
107#384—21no—
108#384—21no—
109#384—21no—
110#384D411yescharacteristic
111#384D411yescharacteristic
112#382—28no—
113#382D412yes—
114#380—21no—
115#380—21no—
116#380—22no—
117#380—22no—
118C4×C4⋊C4—22no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4—21no—
121C4×C4⋊C4—21no—
122C4×C4⋊C4—21no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4—21no—
125C4×C4⋊C4—22no—
126C4×C4⋊C4C2×C414yescomplemented
127C4×C22⋊C4—24no—
128C2×#95—21no—
129C2×#95—24no—
130C2×#95—24no—
131C2×#95D412yes—
132#343—24no—
133#343D418yescomplemented
134#339—28no—
135C2×#41—22no—
136C2×#41—22no—
137C2×#41—24no—
138C2×#41—24no—
139C2×#41—24no—
140C2×#41—24no—
141C2×#41C22×C414yes—
142C2×#41C2×Q814yes—
143C22×Q8C2×D412yes—
144C22×D4—22no—
145C22×D4—21no—
146C22×D4—21no—
147C22×D4C2×D412yes—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—24no—
151C23×C4—22no—
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D412yes—
154C23×C4C2×D412yes—
155C23×C4C2×D411yescharacteristic
156C23×C4C2411yescharacteristic
157C2×M16—22no—
158C2×M16—22no—
159C2×M16—22no—
160C2×M16—22no—
161C2×M16—22no—
162C2×M16—22no—
163C2×M16—22no—
164C2×M16—22no—
165C2×M16C22×C414yes—
166C2×M16C2×D414yes—
167C2×M16C2×D414yes—
168C2×M16#4114yes—
169C22×C8—22no—
170C22×C8—22no—
171C4⋊Q8—22no—
172C4⋊Q8C4⋊C414yes—
173#127—22no—
174#127C4⋊C414yes—
175#124—24no—
176#124C4⋊C418yes—
177#121—24no—
178#121—24no—
179#121—24no—
180#121—24no—
181C4×Q8—48no—
182C4×D4—24no—
183C4×D4—24no—
184C4×D4—24no—
185C4×D4—24no—
186C4×D4—48no—
187C4×D4—48no—
188C4×D4—48no—
189C4×D4—84no—
190#117—24no—
191#117—24no—
192#117—48no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—48no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—41no—
218C2×C4⋊C4—41no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4—82no—
221C2×C4⋊C4C2×D412yes—
222C2×C4⋊C4C2×D412yes—
223C2×C4⋊C4C2×D414yes—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—24no—
230C2×C42—24no—
231C2×C42—24no—
232C2×C42—24no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—44no—
238C2×C42C22×C414yes—
239C2×C42C22×C412yes—
240C2×C42C2×D411yescharacteristic
241C2×C42C2×D411yescharacteristic
242C2×C42C2×D411yescharacteristic
243C2×C42C2×D411yescharacteristic
244C2×C42C2×Q812yes—
245C2×C42#4112yes—
246C2×C42#4112yes—
247#107—48no—
248#106—48no—
249#104—232no—
250#103—48no—
251#102—48no—
252#97—24no—
253#97—42no—
254C4×C8—24no—
255C4×C8—42no—
256#95—24no—
257#95—24no—
258#95—24no—
259#95—22no—
260#95—22no—
261#95—22no—
262#95—22no—
263#95—48no—
264#95—84no—
265C24—22no—
266C24C22×D411yescharacteristic
267#41—28no—
268#41—28no—
269#41—28no—
270#41—28no—
271#41—44no—
272#41—44no—
273#41—44no—
274#41—44no—
275C2×Q8—22no—
276C2×Q8—24no—
277C2×Q8—24no—
278C2×Q8C2×C4⋊C414yes—
279C2×D4—24no—
280C2×D4—24no—
281C2×D4—22no—
282C2×D4—24no—
283C2×D4—24no—
284C2×D4—24no—
285C2×D4—24no—
286C2×D4—42no—
287C2×D4—42no—
288C2×D4—42no—
289C2×D4—42no—
290C2×D4—42no—
291C2×D4—42no—
292C2×D4—81no—
293C2×D4—81no—
294C2×D4C2×C4⋊C414yes—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
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—22no—
307C22×C4—22no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—48no—
311C22×C4—44no—
312C22×C4—44no—
313C22×C4—42no—
314C22×C4—42no—
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—44no—
324C22×C4—44no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—84no—
329C22×C4—84no—
330C22×C4C4×D412yes—
331C22×C4C4×D412yes—
332C22×C4#12011yescharacteristic
333C22×C4#12011yescharacteristic
334C22×C4#12011yescharacteristic
335C22×C4#12011yescharacteristic
336C22×C4C22⋊Q812yes—
337C22×C4C22⋊Q812yes—
338C22×C4#12714yes—
339C22×C4C23×C411yescharacteristic
340C22×C4C22×D411yescharacteristic
341C22×C4C22×D411yescharacteristic
342C22×C4C22×D412yes—
343C22×C4C22×Q811yescharacteristic
344C22×C4C2×#4111yescharacteristic
345C22×C4C2×#4112yes—
346C22×C4C2×#4111yescharacteristic
347C22×C4H(5,𝔽2)12yes—
348C22×C42−1+412yes—
349M16—24no—
350M16—24no—
351M16—24no—
352M16—24no—
353M16—24no—
354M16—42no—
355M16—42no—
356M16C4×D418yescomplemented
357C2×C8—24no—
358C2×C8—24no—
359C2×C8—24no—
360C2×C8—24no—
361C2×C8—42no—
362C2×C8—42no—
363C4⋊C4—24no—
364C4⋊C4—24no—
365C4⋊C4—24no—
366C4⋊C4—24no—
367C4⋊C4—24no—
368C4⋊C4—24no—
369C4⋊C4—48no—
370C4⋊C4—48no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—44no—
376C4⋊C4—44no—
377C4⋊C4—81no—
378C4⋊C4—81no—
379C4⋊C4—82no—
380C4⋊C4—82no—
381C4⋊C4—82no—
382C4⋊C4—82no—
383C22⋊C4—28no—
384C22⋊C4—28no—
385C22⋊C4—48no—
386C22⋊C4—42no—
387C22⋊C4—42no—
388C22⋊C4—84no—
389C22⋊C4—84no—
390C42—28no—
391C42—48no—
392C42—44no—
393C42—44no—
394C42—44no—
395C42—82no—
396C42C2×C4⋊C412yes—
397C42C2×C4⋊C412yes—
398C42C4×D412yes—
399C42C4×D412yes—
400C42C22⋊Q812yes—
401C42C22⋊Q812yes—
402C42#12312yes—
403C42#12312yes—
404C23—21no—
405C23—21no—
406C23—21no—
407C23—21no—
408C23—24no—
409C23—42no—
410C23—42no—
411C23—42no—
412C23—42no—
413C23—42no—
414C23—42no—
415C23C2×C4×D411yescharacteristic
416C23C2×#12011yescharacteristic
417C23C2×C22⋊Q811yescharacteristic
418C23C2×#12711yescharacteristic
419C23#53211yescharacteristic
420C23#53511yescharacteristic
421C23#53611yescharacteristic
422Q8—28no—
423Q8—44no—
424D4—28no—
425D4—44no—
426D4—44no—
427D4—44no—
428D4—44no—
429D4—48no—
430D4—44no—
431D4—84no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—24no—
439C2×C4—24no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—48no—
449C2×C4—48no—
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—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—82no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4—84no—
471C2×C4—84no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4—168no—
476C2×C4#38512yes—
477C2×C4#38512yes—
478C2×C4#38512yes—
479C2×C4#38512yes—
480C2×C4#45412yes—
481C2×C4#45512yes—
482C2×C4C22×C4⋊C411yescharacteristic
483C2×C4C2×C4×D412yes—
484C2×C4C2×C4×D411yescharacteristic
485C2×C4#52012yes—
486C2×C4C2×C22⋊Q811yescharacteristic
487C2×C4C2×#12311yescharacteristic
488C2×C4#53212yes—
489C2×C4#54212yes—
490C2×C4D4212yes—
491C2×C4D4212yes—
492C2×C4#54712yes—
493C2×C4#54712yes—
494C2×C4#54812yes—
495C2×C4D4×Q812yes—
496C2×C4#55312yes—
497C2×C4#55412yes—
498C8—28no—
499C8—44no—
500C22—21no—
501C22—21no—
502C22—21no—
503C22—21no—
504C22—21no—
505C22—21no—
506C22—21no—
507C22—21no—
508C22—21no—
509C22—21no—
510C22—21no—
511C22—21no—
512C22—44no—
513C22—44no—
514C22—44no—
515C22—82no—
516C22—82no—
517C22—82no—
518C22—82no—
519C22C2×#38511yescharacteristic
520C22C4×#12711yescharacteristic
521C22#232611yescharacteristic
522C22#243211yescharacteristic
523C22#243311yescharacteristic
524C22#254211yescharacteristic
525C22#256611yescharacteristic
526C22C2×#45511yescharacteristic
527C22C2×#45411yescharacteristic
528C22#302111yescharacteristic
529C22#302211yescharacteristic
530C4—22no—
531C4—22no—
532C4—22no—
533C4—44no—
534C4—44no—
535C4—84no—
536C4—82no—
537C4—82no—
538C4—82no—
539C4—162no—
540C4#186014yescomplemented
541C4D4×C4⋊C412yes—
542C4#236112yes—
543C4#242212yes—
544C4#245212yes—
545C2—21no—
546C2—21no—
547C2—21no—
548C2—21no—
549C2—84no—
550C2#1397711yescharacteristic
551C2C2×#186011yescharacteristic
552C2#2086511yescharacteristic
553C2#2569311yescharacteristic
554C2#2569611yescharacteristic
555C2#2573811yescharacteristic
556C2#2574111yescharacteristic
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×#1860C2semidirect
5. #20800C2semidirect
6. #13911C2semidirect
7. #11113C2semidirect
8. #10851C2semidirect
9. C4×#343C2semidirect
31. #1980C4semidirect
83. C2×#104C2×C4semidirect
91. C4×M16C2×C4semidirect
126. C4×C4⋊C4C2×C4semidirect
133. #343D4semidirect
356. M16C4×D4semidirect
540. C4#1860semidirect

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×#18602
#208002
#139112
#111132
#108512
C4×#3432