Finite Group Databasebeta

← #194002

Subgroups — #194002

Order 512.

1770 conjugacy classes of subgroups.

The subgroup lattice has 537 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3110C211yesmaximal, characteristic
2C2×D4×C4⋊C4C211yesmaximal, characteristic
3#21394C211yesmaximal, characteristic, complemented
4#21322C211yesmaximal, characteristic, complemented
5C2×#1490C211yesmaximal, characteristic
6#8985C214yesmaximal, complemented
7#8981C214yesmaximal, complemented
8#8040C211yesmaximal, characteristic, complemented
9#7904C211yesmaximal, characteristic, complemented
10C2×#533C2211yescharacteristic
11C22×C4×D4—21no—
12C22×C4×D4C2211yescharacteristic
13C23×C4⋊C4—21no—
14#3110—24no—
15#3110C418yescomplemented
16C2×#484—21no—
17C2×#474C2212yes—
18C2×#461C2212yes—
19C2×#450—21no—
20C2×#440—22no—
21C2×#432C2211yescharacteristic
22C2×#432C2211yescharacteristic
23C2×C4×M16C2211yescharacteristic
24D4×C4⋊C4—22no—
25D4×C4⋊C4—22no—
26D4×C4⋊C4—22no—
27D4×C4⋊C4C2214yes—
28C2×#389—21no—
29C2×#385—22no—
30C2×#384C2211yescharacteristic
31C2×#380—21no—
32C2×C4×C4⋊C4C2211yescharacteristic
33#1992—22no—
34#1990C2212yescomplemented
35#1990C2212yescomplemented
36#1988C2212yescomplemented
37#1981C2212yescomplemented
38#1872—21no—
39#1872—21no—
40#1763—22no—
41#1736—22no—
42#1490—24no—
43C2×D4⋊C8C2214yes—
44#1296C2214yescomplemented
45#1279C2214yescomplemented
46C23×D4D411yescharacteristic
47C24×C4—21no—
48C24×C4—22no—
49C22×SD16—22no—
50C22×M16—21no—
51#533—22no—
52#533—22no—
53#533—22no—
54#533C2×C414yescomplemented
55C2×C4⋊Q8C2311yescharacteristic
56C2×C4⋊Q8C2311yescharacteristic
57C2×#125—21no—
58C2×C22⋊Q8—21no—
59C2×C22⋊Q8—21no—
60C2×C4×Q8—21no—
61C2×C4×Q8—21no—
62C2×C4×D4—24no—
63C2×C4×D4—24no—
64C2×C4×D4—21no—
65C2×C4×D4—21no—
66C2×C4×D4—22no—
67C2×C4×D4—22no—
68C2×C4×D4—48no—
69C2×C4×D4C2×C414yes—
70C2×C4×D4D414yes—
71C2×C4×D4D412yes—
72C2×C4×D4C2312yes—
73C2×#117—21no—
74C22×C4⋊C4—24no—
75C22×C4⋊C4—22no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4—21no—
78C22×C4⋊C4—24no—
79C22×C4⋊C4—48no—
80C22×C4⋊C4D411yescharacteristic
81C22×C4⋊C4D411yescharacteristic
82C22×C4⋊C4D411yescharacteristic
83C22×C4⋊C4D412yes—
84C22×C22⋊C4—21no—
85C22×C22⋊C4—22no—
86C22×C42—21no—
87C22×C42C2311yescharacteristic
88#484—28no—
89#474—24no—
90#474C2×C418yescomplemented
91#461—24no—
92#461C2×C418yescomplemented
93#450—28no—
94#440—216no—
95#432—22no—
96#432—22no—
97#432—22no—
98#432C2×C414yescomplemented
99C2×#107—22no—
100C2×C4⋊C8C2312yes—
101C2×C4⋊C8C2312yes—
102C2×#103—22no—
103C2×#103—22no—
104C2×#103—22no—
105C2×#102—22no—
106C2×C22⋊C8—21no—
107C2×C22⋊C8—21no—
108C4×M16—22no—
109C4×M16—22no—
110C2×#97—21no—
111C2×#97—21no—
112C2×#97—21no—
113C2×C4×C8C2312yes—
114#389—24no—
115#389—24no—
116#385—28no—
117#385—28no—
118#384—21no—
119#384—21no—
120#384D412yes—
121#384C2312yes—
122#380—24no—
123#380—24no—
124C4×C4⋊C4—21no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4D412yes—
127C4×C4⋊C4C2312yes—
128C2×#95—22no—
129C2×#95—21no—
130#340—22no—
131#340—22no—
132#340—24no—
133#340—24no—
134D4⋊C8—28no—
135C25—21no—
136C22×Q8—21no—
137C22×Q8—21no—
138C22×D4—24no—
139C22×D4—24no—
140C22×D4—21no—
141C22×D4—22no—
142C22×D4C22⋊C414yes—
143C22×D4C2×D412yes—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—24no—
147C23×C4—22no—
148C23×C4—24no—
149C23×C4—22no—
150C23×C4—21no—
151C23×C4—22no—
152C23×C4—28no—
153C23×C4—44no—
154C23×C4—44no—
155C23×C4—48no—
156C23×C4—48no—
157C23×C4—84no—
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C2×SD16—28no—
162C2×SD16—44no—
163C2×SD16—44no—
164C2×M16—24no—
165C2×M16—42no—
166C2×M16—42no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C4⋊Q8—21no—
171C4⋊Q8—21no—
172C4⋊Q8—21no—
173C4⋊Q8—21no—
174C4⋊Q8—21no—
175C4⋊Q8—21no—
176C4⋊Q8C22×C412yescomplemented
177C4⋊Q8C22×C412yescomplemented
178#125—24no—
179#125—42no—
180C22⋊Q8—24no—
181C22⋊Q8—24no—
182C22⋊Q8—42no—
183C22⋊Q8—42no—
184C4×Q8—24no—
185C4×Q8—24no—
186C4×Q8—42no—
187C4×Q8—42no—
188C4×D4—24no—
189C4×D4—24no—
190C4×D4—24no—
191C4×D4—24no—
192C4×D4—24no—
193C4×D4—22no—
194C4×D4—22no—
195C4×D4—416no—
196C4×D4—48no—
197C4×D4—48no—
198C4×D4C22⋊C418yes—
199C4×D4C22×C414yes—
200C4×D4C2×D414yes—
201#117—24no—
202#117—42no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—28no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—416no—
220C2×C4⋊C4—48no—
221C2×C4⋊C4—48no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—41no—
225C2×C4⋊C4—41no—
226C2×C4⋊C4—48no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4C22⋊C414yes—
229C2×C4⋊C4C2×D412yes—
230C2×C4⋊C4C2×D412yes—
231C2×C4⋊C4C2×D412yes—
232C2×C4⋊C4C2×D414yes—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—28no—
238C2×C22⋊C4—48no—
239C2×C22⋊C4—41no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—44no—
243C2×C42—41no—
244C2×C42C22×C412yes—
245C2×C42C2×D411yescharacteristic
246C2×C42C2×D412yes—
247C2×C42C2×D411yescharacteristic
248C2×C42C2×D411yescharacteristic
249C2×C42#4112yes—
250C2×C42#4112yes—
251C2×C42C2411yescharacteristic
252#107—28no—
253C4⋊C8—22no—
254C4⋊C8—22no—
255C4⋊C8—22no—
256C4⋊C8C22×C414yescomplemented
257#103—28no—
258#103—44no—
259#103—44no—
260#102—28no—
261C22⋊C8—24no—
262C22⋊C8—42no—
263#97—24no—
264#97—42no—
265#97—42no—
266C4×C8—22no—
267C4×C8—22no—
268#95—28no—
269#95—22no—
270#95—22no—
271C24—24no—
272C24—24no—
273C24—22no—
274C24—24no—
275C24—42no—
276C24—42no—
277C24—42no—
278C24—42no—
279C24#12011yescharacteristic
280C2×Q8—22no—
281C2×Q8—22no—
282C2×Q8—41no—
283C2×Q8—41no—
284C2×Q8—42no—
285C2×Q8—42no—
286C2×Q8—42no—
287C2×Q8—42no—
288C2×D4—24no—
289C2×D4—24no—
290C2×D4—216no—
291C2×D4—216no—
292C2×D4—22no—
293C2×D4—22no—
294C2×D4—22no—
295C2×D4—24no—
296C2×D4C2×C22⋊C414yes—
297C2×D4AGL(1,ℤ/8)14yes—
298C2×D4#13714yes—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—24no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—28no—
312C22×C4—22no—
313C22×C4—24no—
314C22×C4—44no—
315C22×C4—44no—
316C22×C4—416no—
317C22×C4—416no—
318C22×C4—416no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—44no—
324C22×C4—44no—
325C22×C4—42no—
326C22×C4—44no—
327C22×C4—44no—
328C22×C4—42no—
329C22×C4—416no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—44no—
333C22×C4—42no—
334C22×C4—88no—
335C22×C4—832no—
336C22×C4C2×C22⋊C412yes—
337C22×C4C4×D412yes—
338C22×C4C4×D412yes—
339C22×C4#12011yescharacteristic
340C22×C4#12011yescharacteristic
341C22×C4#12011yescharacteristic
342C22×C4#12112yes—
343C22×C4#12112yes—
344C22×C4#12312yes—
345C22×C4C22×D411yescharacteristic
346C22×C4C22×D411yescharacteristic
347C22×C4C22×D411yescharacteristic
348SD16—416no—
349M16—48no—
350C2×C8—24no—
351C2×C8—24no—
352C2×C8—42no—
353C2×C8—42no—
354C2×C8—42no—
355C2×C8—42no—
356C2×C8—42no—
357C2×C8—42no—
358C2×C8—42no—
359C4⋊C4—24no—
360C4⋊C4—22no—
361C4⋊C4—22no—
362C4⋊C4—24no—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—44no—
370C4⋊C4—42no—
371C4⋊C4—44no—
372C4⋊C4—416no—
373C4⋊C4—44no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—44no—
379C4⋊C4—44no—
380C4⋊C4—44no—
381C4⋊C4—44no—
382C4⋊C4—82no—
383C4⋊C4C2×C22⋊C414yes—
384C4⋊C4AGL(1,ℤ/8)14yes—
385C22⋊C4—24no—
386C22⋊C4—24no—
387C22⋊C4—28no—
388C22⋊C4—416no—
389C22⋊C4—44no—
390C42—44no—
391C42—44no—
392C42—44no—
393C42C2×C22⋊C412yes—
394C42C4×D412yes—
395C42C4×D412yes—
396C42#12112yes—
397C42#12112yes—
398C42#12312yes—
399C42C23×C411yescharacteristic
400C42C22×D411yescharacteristic
401C42C2×#4111yescharacteristic
402C42C2×#4111yescharacteristic
403C23—21no—
404C23—21no—
405C23—21no—
406C23—21no—
407C23—216no—
408C23—24no—
409C23—24no—
410C23—24no—
411C23—48no—
412C23—48no—
413C23—48no—
414C23—48no—
415C23—48no—
416C23—48no—
417C23—42no—
418C23—42no—
419C23—42no—
420C23—42no—
421C23#38612yes—
422C23#45512yes—
423C23#45612yes—
424C23C2×#12011yescharacteristic
425Q8—44no—
426Q8—44no—
427Q8—82no—
428Q8—82no—
429D4—216no—
430D4—24no—
431D4—24no—
432D4—24no—
433D4#41718yes—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—416no—
451C2×C4—48no—
452C2×C4—48no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—48no—
458C2×C4—48no—
459C2×C4—42no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—42no—
463C2×C4—44no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—832no—
468C2×C4—84no—
469C2×C4—88no—
470C2×C4—81no—
471C2×C4—81no—
472C2×C4—81no—
473C2×C4—81no—
474C2×C4#38612yes—
475C2×C4#38612yes—
476C2×C4#38612yes—
477C2×C4C22×C22⋊C411yescharacteristic
478C2×C4C2×C4×D411yescharacteristic
479C2×C4C2×C4×D411yescharacteristic
480C2×C4C2×#12111yescharacteristic
481C2×C4C2×#12111yescharacteristic
482C2×C4C2×#12311yescharacteristic
483C2×C4C2×AGL(1,ℤ/8)12yes—
484C2×C4C2×AGL(1,ℤ/8)12yes—
485C2×C4C2×#13712yes—
486C2×C4#57512yes—
487C8—44no—
488C8—44no—
489C8—82no—
490C22—22no—
491C22—22no—
492C22—22no—
493C22—22no—
494C22—22no—
495C22—22no—
496C22—28no—
497C22—432no—
498C22—44no—
499C22—44no—
500C22—44no—
501C22—44no—
502C22—44no—
503C22—44no—
504C22#188214yes—
505C22C2×#38611yescharacteristic
506C22#300311yescharacteristic
507C22#300311yescharacteristic
508C22#300411yescharacteristic
509C22#300511yescharacteristic
510C22C2×#45511yescharacteristic
511C22C2×#45611yescharacteristic
512C4—22no—
513C4—22no—
514C4—21no—
515C4—21no—
516C4—22no—
517C4—44no—
518C4—44no—
519C4—84no—
520C4—82no—
521C4—82no—
522C4—82no—
523C4C2×#41712yes—
524C4C4×AGL(1,ℤ/8)12yes—
525C4#303612yes—
526C4#309212yes—
527C2—24no—
528C2—48no—
529C2#1852911yescharacteristic
530C2#1853011yescharacteristic
531C2#1853911yescharacteristic
532C2#1854011yescharacteristic
533C2#2090911yescharacteristic
534C2#2092511yescharacteristic
535C2C2×#188211yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC23×D4D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #21394C2semidirect
4. #21322C2semidirect
6. #8985C2semidirect
7. #8981C2semidirect
8. #8040C2semidirect
9. #7904C2semidirect
15. #3110C4semidirect
34. #1990C22semidirect
35. #1990C22semidirect
36. #1988C22semidirect
37. #1981C22semidirect
44. #1296C22semidirect
45. #1279C22semidirect
54. #533C2×C4semidirect
90. #474C2×C4semidirect
92. #461C2×C4semidirect
98. #432C2×C4semidirect
176. C4⋊Q8C22×C4semidirect
177. C4⋊Q8C22×C4semidirect
256. C4⋊C8C22×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#31102
C2×D4×C4⋊C42
#213942
#213222
C2×#14902
#89852
#89812
#80402
#79042