Finite Group Databasebeta

← #194161

Subgroups — #194161

Order 512.

1514 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3103C211yesmaximal, characteristic
2C2×Q8×C4⋊C4C211yesmaximal, characteristic, complemented
3#21859C211yesmaximal, characteristic
4#21258C211yesmaximal, characteristic
5C2×#1487C211yesmaximal, characteristic, complemented
6#8983C214yesmaximal, complemented
7#8979C214yesmaximal, complemented
8#8043C211yesmaximal, characteristic, complemented
9#7892C211yesmaximal, characteristic, complemented
10C2×#532C2211yescharacteristic
11C22×C4×Q8—21no—
12C22×C4×Q8C2211yescharacteristic
13#3103—24no—
14#3103C418yescomplemented
15C2×#481—21no—
16C2×#475C2212yes—
17C2×#460C2212yes—
18C2×#448—21no—
19C2×C4×SD16—22no—
20C2×#431C2211yescharacteristic
21C2×#431C2211yescharacteristic
22C2×#423C2211yescharacteristic
23C2×#417—21no—
24C2×#417—21no—
25Q8×C4⋊C4—22no—
26Q8×C4⋊C4—22no—
27Q8×C4⋊C4—22no—
28Q8×C4⋊C4C2214yes—
29C2×#389—22no—
30C2×#389—21no—
31C2×#384—21no—
32C2×#384C2211yescharacteristic
33C2×C4×C4⋊C4—21no—
34C2×C4×C4⋊C4C2211yescharacteristic
35#1990C2212yes—
36#1983—22no—
37#1982C2212yes—
38#1981C2212yes—
39#1957C2212yes—
40#1874—22no—
41#1802—22no—
42#1487—24no—
43C2×Q8⋊C8C2214yes—
44#1296C2214yes—
45#1279C2214yes—
46C22×#41—21no—
47C23×Q8D411yescharacteristic
48C22×SD16—22no—
49C22×M16—21no—
50#532—22no—
51#532—22no—
52#532—22no—
53#532C2×C414yes—
54C2×C4⋊Q8C2311yescharacteristic
55C2×#127C2311yescharacteristic
56C2×#124—21no—
57C2×#121—21no—
58C2×#121—21no—
59C2×C4×Q8—24no—
60C2×C4×Q8—24no—
61C2×C4×Q8—21no—
62C2×C4×Q8—21no—
63C2×C4×Q8—22no—
64C2×C4×Q8—22no—
65C2×C4×Q8—48no—
66C2×C4×Q8C2×C414yes—
67C2×C4×Q8D414yes—
68C2×C4×Q8D412yes—
69C2×C4×Q8C2312yes—
70C2×C4×D4—21no—
71C2×C4×D4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—22no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4—22no—
77C22×C4⋊C4D411yescharacteristic
78C22×C4⋊C4D411yescharacteristic
79C22×C4⋊C4D411yescharacteristic
80C22×C4⋊C4D412yes—
81C22×C42—21no—
82C22×C42—22no—
83C22×C42—21no—
84C22×C42C2311yescharacteristic
85#481—28no—
86#475—24no—
87#475C2×C418yescomplemented
88#460—24no—
89#460C2×C418yescomplemented
90#448—28no—
91C4×SD16—216no—
92#431—22no—
93#431—22no—
94#431—22no—
95#431C2×C414yes—
96C2×#106—22no—
97#423—22no—
98#423—22no—
99C2×C4⋊C8—21no—
100C2×C4⋊C8—21no—
101C2×C4⋊C8C2312yes—
102C2×C4⋊C8C2312yes—
103#417—44no—
104#417—44no—
105C2×#103—22no—
106C2×#103—22no—
107C2×#103—22no—
108C2×#102—22no—
109C2×#102—22no—
110C2×#102—22no—
111C2×#102—24no—
112C2×C22⋊C8—21no—
113C2×C22⋊C8—21no—
114C2×C4×C8—21no—
115C2×C4×C8C2312yes—
116#389—24no—
117#389—28no—
118#389—24no—
119#389—28no—
120#384—24no—
121#384—24no—
122#384—21no—
123#384—21no—
124#384D412yes—
125#384C2312yes—
126C4×C4⋊C4—24no—
127C4×C4⋊C4—24no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4D412yes—
131C4×C4⋊C4C2312yes—
132C2×#95—22no—
133C2×#95—21no—
134#340—24no—
135Q8⋊C8—28no—
136C2×#41—24no—
137C2×#41—42no—
138C2×#41—42no—
139C2×#41—42no—
140C2×#41—42no—
141C2×#41—42no—
142C22×Q8—24no—
143C22×Q8—24no—
144C22×Q8—21no—
145C22×Q8—22no—
146C22×Q8—21no—
147C22×Q8C22⋊C414yes—
148C22×Q8C2×D412yes—
149C22×D4—21no—
150C22×D4—21no—
151C22×D4—21no—
152C22×D4—41no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—22no—
158C23×C4—41no—
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C2×SD16—28no—
163C2×SD16—44no—
164C2×SD16—44no—
165C2×M16—42no—
166C2×M16—42no—
167C2×M16—42no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C4⋊Q8—21no—
172C4⋊Q8—21no—
173C4⋊Q8—21no—
174C4⋊Q8C22×C412yes—
175#127—21no—
176#127—21no—
177#127—21no—
178#127C22×C412yes—
179#124—24no—
180#124—42no—
181#121—24no—
182#121—24no—
183#121—42no—
184#121—42no—
185C4×Q8—24no—
186C4×Q8—24no—
187C4×Q8—24no—
188C4×Q8—24no—
189C4×Q8—24no—
190C4×Q8—22no—
191C4×Q8—22no—
192C4×Q8—416no—
193C4×Q8—48no—
194C4×Q8—48no—
195C4×Q8C22⋊C418yes—
196C4×Q8C22×C414yes—
197C4×Q8C2×D414yes—
198C4×D4—24no—
199C4×D4—24no—
200C4×D4—42no—
201C4×D4—42no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—48no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—82no—
227C2×C4⋊C4—84no—
228C2×C4⋊C4C22⋊C414yes—
229C2×C4⋊C4SD1614yescomplemented
230C2×C4⋊C4C2×D412yes—
231C2×C4⋊C4C2×D412yes—
232C2×C4⋊C4C2×D412yes—
233C2×C4⋊C4C2×D414yes—
234C2×C22⋊C4—41no—
235C2×C22⋊C4—41no—
236C2×C42—24no—
237C2×C42—22no—
238C2×C42—24no—
239C2×C42—22no—
240C2×C42—28no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—48no—
244C2×C42—44no—
245C2×C42C22×C412yes—
246C2×C42C2×D411yescharacteristic
247C2×C42C2×D412yes—
248C2×C42C2×D411yescharacteristic
249C2×C42C2×D411yescharacteristic
250C2×C42#4112yes—
251C2×C42#4112yes—
252C2×C42C2411yescharacteristic
253#106—28no—
254C4⋊C8—22no—
255C4⋊C8—22no—
256C4⋊C8—22no—
257C4⋊C8—42no—
258C4⋊C8—42no—
259C4⋊C8C22×C414yes—
260#103—28no—
261#103—44no—
262#103—44no—
263#102—28no—
264#102—44no—
265#102—44no—
266#102—84no—
267C22⋊C8—24no—
268C22⋊C8—42no—
269C4×C8—24no—
270C4×C8—22no—
271C4×C8—22no—
272#95—28no—
273#95—22no—
274#95—22no—
275C24—41no—
276C24#12011yescharacteristic
277#41—48no—
278#41—48no—
279#41—84no—
280#41—84no—
281C2×Q8—24no—
282C2×Q8—216no—
283C2×Q8—24no—
284C2×Q8—216no—
285C2×Q8—22no—
286C2×Q8—22no—
287C2×Q8—22no—
288C2×Q8—24no—
289C2×Q8—22no—
290C2×Q8—41no—
291C2×Q8—41no—
292C2×Q8—41no—
293C2×Q8—41no—
294C2×Q8—41no—
295C2×Q8C2×C22⋊C414yes—
296C2×Q8AGL(1,ℤ/8)14yes—
297C2×Q8#13714yes—
298C2×D4—22no—
299C2×D4—22no—
300C2×D4—22no—
301C2×D4—42no—
302C2×D4—42no—
303C2×D4—44no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—41no—
308C2×D4—41no—
309C2×D4—41no—
310C2×D4—42no—
311C2×D4—42no—
312C2×D4—41no—
313C2×D4—82no—
314C2×D4—82no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—22no—
318C22×C4—24no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—22no—
326C22×C4—24no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—44no—
334C22×C4—42no—
335C22×C4—44no—
336C22×C4—42no—
337C22×C4—41no—
338C22×C4—42no—
339C22×C4—44no—
340C22×C4—44no—
341C22×C4—82no—
342C22×C4—82no—
343C22×C4C2×C22⋊C412yes—
344C22×C4C4×D412yes—
345C22×C4C4×D412yes—
346C22×C4#12011yescharacteristic
347C22×C4#12011yescharacteristic
348C22×C4#12011yescharacteristic
349C22×C4#12112yes—
350C22×C4#12112yes—
351C22×C4#12312yes—
352C22×C4C2×SD1612yes—
353C22×C4#13512yes—
354C22×C4AGL(1,ℤ/8)12yes—
355C22×C4#13712yes—
356C22×C4C22×D411yescharacteristic
357C22×C4C22×D411yescharacteristic
358C22×C4C22×D411yescharacteristic
359SD16—416no—
360M16—84no—
361C2×C8—24no—
362C2×C8—42no—
363C2×C8—42no—
364C2×C8—42no—
365C2×C8—42no—
366C2×C8—42no—
367C2×C8—42no—
368C2×C8—42no—
369C2×C8—42no—
370C4⋊C4—24no—
371C4⋊C4—24no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—28no—
375C4⋊C4—24no—
376C4⋊C4—24no—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—44no—
382C4⋊C4—416no—
383C4⋊C4—44no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—44no—
389C4⋊C4—42no—
390C4⋊C4—44no—
391C4⋊C4—82no—
392C4⋊C4—88no—
393C4⋊C4C2×C22⋊C414yes—
394C4⋊C4C2×SD1614yes—
395C22⋊C4—44no—
396C22⋊C4—44no—
397C42—24no—
398C42—24no—
399C42—28no—
400C42—416no—
401C42—44no—
402C42—44no—
403C42C2×C22⋊C412yes—
404C42C4×D412yes—
405C42C4×D412yes—
406C42#12112yes—
407C42#12112yes—
408C42#12312yes—
409C42C23×C411yescharacteristic
410C42C22×D411yescharacteristic
411C42C2×#4111yescharacteristic
412C42C2×#4111yescharacteristic
413C23—21no—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—42no—
418C23—81no—
419C23—81no—
420C23—81no—
421C23—81no—
422C23—81no—
423C23—81no—
424C23#38612yes—
425C23#44812yes—
426C23#44912yes—
427C23C2×#12011yescharacteristic
428Q8—216no—
429Q8—24no—
430Q8—24no—
431Q8—24no—
432Q8—42no—
433Q8—42no—
434Q8—81no—
435Q8—81no—
436Q8#41718yescomplemented
437D4—44no—
438D4—44no—
439D4—42no—
440D4—42no—
441D4—82no—
442D4—88no—
443D4—82no—
444D4—81no—
445D4—81no—
446C2×C4—216no—
447C2×C4—24no—
448C2×C4—21no—
449C2×C4—24no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—48no—
463C2×C4—48no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—42no—
472C2×C4—44no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—84no—
476C2×C4—88no—
477C2×C4—81no—
478C2×C4—81no—
479C2×C4—88no—
480C2×C4#38612yes—
481C2×C4#38612yes—
482C2×C4#38612yes—
483C2×C4C4×SD1614yes—
484C2×C4#44014yes—
485C2×C4#44812yes—
486C2×C4#44912yes—
487C2×C4#46014yes—
488C2×C4#46414yes—
489C2×C4C22×C22⋊C411yescharacteristic
490C2×C4C2×C4×D411yescharacteristic
491C2×C4C2×C4×D411yescharacteristic
492C2×C4C2×#12111yescharacteristic
493C2×C4C2×#12111yescharacteristic
494C2×C4C2×#12311yescharacteristic
495C2×C4C22×SD1611yescharacteristic
496C2×C4C2×#13511yescharacteristic
497C2×C4C2×AGL(1,ℤ/8)12yes—
498C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
499C2×C4C2×#13712yes—
500C2×C4C2×#13711yescharacteristic
501C8—44no—
502C8—82no—
503C8—82no—
504C22—22no—
505C22—22no—
506C22—22no—
507C22—22no—
508C22—22no—
509C22—22no—
510C22—82no—
511C22—82no—
512C22—82no—
513C22—161no—
514C22—161no—
515C22—161no—
516C22—161no—
517C22#188014yes—
518C22C2×#38611yescharacteristic
519C22C2×#44811yescharacteristic
520C22C2×#44911yescharacteristic
521C22#300311yescharacteristic
522C22#300411yescharacteristic
523C22#301911yescharacteristic
524C22#302011yescharacteristic
525C4—28no—
526C4—22no—
527C4—21no—
528C4—21no—
529C4—22no—
530C4—44no—
531C4—44no—
532C4—82no—
533C4—84no—
534C4—82no—
535C4#188014yes—
536C4C2×#41712yes—
537C4C2×C4×SD1611yescharacteristic
538C4C2×#44011yescharacteristic
539C4C2×#46011yescharacteristic
540C4C2×#46411yescharacteristic
541C4#309212yes—
542C2—24no—
543C2—162no—
544C2#1875811yescharacteristic
545C2#1876011yescharacteristic
546C2#1876811yescharacteristic
547C2#1877011yescharacteristic
548C2#2090911yescharacteristic
549C2C2×#188011yescharacteristic
550C2#2094311yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#532C22
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
2. C2×Q8×C4⋊C4C2semidirect
5. C2×#1487C2semidirect
6. #8983C2semidirect
7. #8979C2semidirect
8. #8043C2semidirect
9. #7892C2semidirect
14. #3103C4semidirect
87. #475C2×C4semidirect
89. #460C2×C4semidirect
229. C2×C4⋊C4SD16semidirect
436. Q8#417semidirect

Maximal subgroups

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

subgroupindex
C2×#31032
C2×Q8×C4⋊C42
#218592
#212582
C2×#14872
#89832
#89792
#80432
#78922