Finite Group Databasebeta

← #363713

Subgroups — #363713

Order 512.

1486 conjugacy classes of subgroups.

The subgroup lattice has 545 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#432C211yesmaximal, characteristic
2C2×#1949C212yesmaximal, complemented
3#14682C211yesmaximal, characteristic
4C4×#380C211yesmaximal, characteristic
5#11096C212yesmaximal
6#11095C212yesmaximal
7#10863C212yesmaximal, complemented
8#10850C212yesmaximal, complemented
9#10821C212yesmaximal, complemented
10C23×C42C2211yescharacteristic
11C2×#432—28no—
12C2×#432C418yescomplemented
13C22×#107C2212yes—
14C22×#106C2212yes—
15C22×C4⋊C8C2211yescharacteristic
16C22×C4⋊C8C2211yescharacteristic
17C22×C22⋊C8C2212yes—
18C22×#97C2212yes—
19C2×#382C2212yes—
20C2×#380C2212yes—
21C2×#380C2211yescharacteristic
22C2×#380C2211yescharacteristic
23C2×C4×C4⋊C4C2212yes—
24C2×C4×C22⋊C4—21no—
25#1949—28no—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C4⋊#95C2211yescharacteristic
33C4⋊#95C2212yes—
34C4⋊#95C2211yescharacteristic
35#1434—21no—
36#1434—21no—
37#1434—21no—
38C22⋊#95—21no—
39C22⋊#95—21no—
40C22⋊#95—21no—
41C4×#95—21no—
42C24×C4D411yescharacteristic
43C24×C4D411yescharacteristic
44C24×C4C2311yescharacteristic
45C23×C8C2312yes—
46C23×C8C2312yes—
47C22×C4⋊C4C2312yes—
48C22×C4⋊C4C2312yes—
49C22×C4⋊C4C2312yes—
50C22×C4⋊C4C2312yes—
51C22×C22⋊C4—22no—
52C22×C22⋊C4—21no—
53C22×C22⋊C4—21no—
54C22×C42—24no—
55C22×C42—24no—
56C22×C42—22no—
57C22×C42C2×C414yes—
58C22×C42D411yescharacteristic
59C22×C42D411yescharacteristic
60C22×C42Q814yes—
61C22×C42C2311yescharacteristic
62C22×C42C2311yescharacteristic
63#432—232no—
64C2×#107—24no—
65C2×#107—24no—
66C2×#107—42no—
67C2×#106—24no—
68C2×#106—24no—
69C2×#106—42no—
70C2×C4⋊C8—24no—
71C2×C4⋊C8C2×C414yes—
72C2×C4⋊C8C2×C414yes—
73C2×C4⋊C8D418yes—
74C2×C22⋊C8—24no—
75C2×C22⋊C8—24no—
76C2×C22⋊C8C2×C418yes—
77C2×#97—24no—
78C2×#97—24no—
79C2×#97C2×C418yes—
80#382—24no—
81#382—24no—
82#380—24no—
83#380—24no—
84#380—41no—
85#380—41no—
86#380—41no—
87#380—41no—
88C4×C4⋊C4—24no—
89C4×C4⋊C4—24no—
90C4×C22⋊C4—44no—
91C2×#95—22no—
92C2×#95—21no—
93C2×#95—21no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—21no—
100C2×#95—21no—
101C2×#95—21no—
102C2×#95—21no—
103#340—24no—
104#340—24no—
105#340—24no—
106#340—24no—
107#340—42no—
108#340—42no—
109C25C2×D411yescharacteristic
110C23×C4—22no—
111C23×C4—22no—
112C23×C4—24no—
113C23×C4—24no—
114C23×C4—22no—
115C23×C4—22no—
116C23×C4—22no—
117C23×C4—22no—
118C23×C4—21no—
119C23×C4—21no—
120C23×C4—21no—
121C23×C4—21no—
122C23×C4—22no—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4—22no—
126C23×C4—42no—
127C23×C4—41no—
128C23×C4—41no—
129C23×C4—42no—
130C23×C4C4⋊C414yes—
131C23×C4C4⋊C414yes—
132C23×C4C22×C412yes—
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C23×C4C2×Q812yes—
139C23×C4C2411yescharacteristic
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8C22×C414yes—
151C22×C8C22×C414yes—
152C22×C8C2×D414yes—
153C22×C8C2×D414yes—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4#4114yes—
177C2×C4⋊C4#4114yes—
178C2×C22⋊C4—28no—
179C2×C22⋊C4—44no—
180C2×C22⋊C4—44no—
181C2×C22⋊C4—42no—
182C2×C22⋊C4—42no—
183C2×C22⋊C4—42no—
184C2×C22⋊C4—42no—
185C2×C22⋊C4—42no—
186C2×C22⋊C4—42no—
187C2×C42—24no—
188C2×C42—24no—
189C2×C42—22no—
190C2×C42—22no—
191C2×C42—28no—
192C2×C42—28no—
193C2×C42—216no—
194C2×C42—44no—
195C2×C42—44no—
196C2×C42—44no—
197C2×C42C4⋊C414yes—
198C2×C42C4⋊C414yes—
199C2×C42C22×C412yes—
200C2×C42C22×C412yes—
201C2×C42C2×D414yes—
202C2×C42C2×Q812yes—
203C2×C42C2×Q812yes—
204C2×C42#4114yes—
205#107—48no—
206#106—48no—
207C4⋊C8—28no—
208C4⋊C8C22⋊C4116yescomplemented
209C22⋊C8—216no—
210#97—216no—
211#95—28no—
212#95—42no—
213#95—42no—
214#95—44no—
215#95—44no—
216#95—42no—
217#95—42no—
218#95—42no—
219#95—42no—
220#95—44no—
221#95—44no—
222#95—44no—
223C24—22no—
224C24—22no—
225C24—21no—
226C24—21no—
227C24—22no—
228C24—21no—
229C24—21no—
230C24—22no—
231C24C2×C4⋊C412yes—
232C24AGL(1,ℤ/8)12yes—
233C24#13712yes—
234C24C22×D411yescharacteristic
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—22no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—28no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—28no—
265C22×C4—28no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—44no—
269C22×C4—44no—
270C22×C4—44no—
271C22×C4—48no—
272C22×C4—48no—
273C22×C4—48no—
274C22×C4—48no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4C2×C4⋊C412yes—
310C22×C4C2×C4⋊C412yes—
311C22×C4C2×C4⋊C412yes—
312C22×C4C2×C4⋊C412yes—
313C22×C4C2×C4⋊C412yes—
314C22×C4#12114yes—
315C22×C4#12114yes—
316C22×C4C23×C411yescharacteristic
317C22×C4C22×D412yes—
318C22×C4C22×Q811yescharacteristic
319C22×C4C2×#4112yes—
320C22×C4C2×#4112yes—
321C22×C4C2×#4112yes—
322C22×C4H(5,𝔽2)11yescharacteristic
323C22×C4H(5,𝔽2)11yescharacteristic
324C22×C42−1+411yescharacteristic
325C22×C42−1+411yescharacteristic
326C2×C8—24no—
327C2×C8—24no—
328C2×C8—24no—
329C2×C8—24no—
330C2×C8—24no—
331C2×C8—24no—
332C2×C8—24no—
333C2×C8—24no—
334C2×C8—24no—
335C2×C8—42no—
336C2×C8—42no—
337C2×C8—42no—
338C2×C8—42no—
339C2×C8C2×C22⋊C418yes—
340C4⋊C4—24no—
341C4⋊C4—24no—
342C4⋊C4—24no—
343C4⋊C4—24no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C22⋊C4—416no—
357C22⋊C4—84no—
358C22⋊C4—84no—
359C42—22no—
360C42—22no—
361C42—24no—
362C42—28no—
363C42—28no—
364C42—88no—
365C42C2×C22⋊C414yes—
366C42#11714yes—
367C42C22⋊Q818yes—
368C23—22no—
369C23—24no—
370C23—24no—
371C23—22no—
372C23—22no—
373C23—22no—
374C23—22no—
375C23—22no—
376C23—22no—
377C23—22no—
378C23—22no—
379C23—24no—
380C23—24no—
381C23—22no—
382C23—42no—
383C23—42no—
384C23—42no—
385C23—42no—
386C23—42no—
387C23—42no—
388C23—42no—
389C23—42no—
390C23#42814yes—
391C23C22×C4⋊C411yescharacteristic
392C23C2×#12112yes—
393C23#52512yes—
394C23#53511yescharacteristic
395C23#53611yescharacteristic
396C23C2×AGL(1,ℤ/8)11yescharacteristic
397C23C2×#13711yescharacteristic
398C23#57511yescharacteristic
399C23#57511yescharacteristic
400C23#57512yes—
401C23#57512yes—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—24no—
407C2×C4—24no—
408C2×C4—24no—
409C2×C4—24no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—416no—
420C2×C4—416no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4—82no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4#38014yes—
462C2×C4#38014yes—
463C2×C4C22×C22⋊C411yescharacteristic
464C2×C4C2×#11711yescharacteristic
465C2×C4C4×#4111yescharacteristic
466C2×C4C4×#4111yescharacteristic
467C2×C4#52011yescharacteristic
468C2×C4#52011yescharacteristic
469C2×C4C2×C22⋊Q812yes—
470C2×C4#53312yes—
471C2×C4#54411yescharacteristic
472C2×C4#54411yescharacteristic
473C2×C4#54612yes—
474C2×C4#54712yes—
475C2×C4#54812yes—
476C2×C4#55012yes—
477C2×C4#55212yes—
478C2×C4#55312yes—
479C2×C4#55512yes—
480C2×C4#55612yes—
481C8—28no—
482C8—44no—
483C22—24no—
484C22—24no—
485C22—24no—
486C22—44no—
487C22—44no—
488C22—42no—
489C22—42no—
490C22—44no—
491C22—44no—
492C22—42no—
493C22—42no—
494C22C2×#38011yescharacteristic
495C22#229611yescharacteristic
496C22#231711yescharacteristic
497C22#241412yes—
498C22#241512yes—
499C22C2×#42811yescharacteristic
500C22#291111yescharacteristic
501C22#291111yescharacteristic
502C22#291111yescharacteristic
503C22#305412yes—
504C22#305412yes—
505C22#305512yes—
506C22#305612yes—
507C22#311112yes—
508C22#311112yes—
509C22#311412yes—
510C22#311712yes—
511C22#321311yescharacteristic
512C22#321511yescharacteristic
513C22#321912yes—
514C22#323411yescharacteristic
515C22#323611yescharacteristic
516C22#324012yes—
517C4—24no—
518C4—24no—
519C4—42no—
520C4—42no—
521C4—42no—
522C4—84no—
523C4—84no—
524C4—162no—
525C4—162no—
526C4#233911yescharacteristic
527C4#234111yescharacteristic
528C4#234611yescharacteristic
529C4#234711yescharacteristic
530C4#243912yes—
531C4#244112yes—
532C2—44no—
533C2#1395111yescharacteristic
534C2#2066011yescharacteristic
535C2#2066111yescharacteristic
536C2#2136311yescharacteristic
537C2#2136611yescharacteristic
538C2#2198611yescharacteristic
539C2#2198911yescharacteristic
540C2#2465612yes—
541C2#2465712yes—
542C2#2470612yes—
543C2#2470712yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1949C2semidirect
7. #10863C2semidirect
8. #10850C2semidirect
9. #10821C2semidirect
12. C2×#432C4semidirect
208. C4⋊C8C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4322
C2×#19492
#146822
C4×#3802
#110962
#110952
#108632
#108502
#108212