Finite Group Databasebeta

← #194157

Subgroups — #194157

Order 512.

1362 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3103C211yesmaximal, characteristic
2C2×#2355C211yesmaximal, characteristic, complemented
3#21859C211yesmaximal, characteristic
4#21258C211yesmaximal, characteristic
5C2×#1487C211yesmaximal, characteristic, complemented
6#8982C214yesmaximal, complemented
7#8978C214yesmaximal, complemented
8#8043C211yesmaximal, characteristic, complemented
9#7892C211yesmaximal, characteristic, complemented
10C2×#532C2211yescharacteristic
11C22×C4×Q8C2211yescharacteristic
12#3103—24no—
13#3103C418yescomplemented
14C2×#481—21no—
15C2×#475C2212yes—
16C2×#460C2212yes—
17C2×#448—21no—
18C2×C4×SD16—22no—
19C2×#431C2211yescharacteristic
20C2×#431C2211yescharacteristic
21C2×#423C2211yescharacteristic
22C2×#417—21no—
23C2×#417—21no—
24#2355—22no—
25#2355—22no—
26#2355—22no—
27#2355C2214yes—
28C2×#391—21no—
29C2×#387—22no—
30C2×#384C2211yescharacteristic
31C2×#382—21no—
32C2×C4×C4⋊C4—21no—
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—21no—
60C2×C4×Q8—21no—
61C2×C4×Q8—22no—
62C2×C4×Q8—22no—
63C2×C4×Q8C2×C414yes—
64C2×C4×Q8D414yes—
65C2×C4×Q8D412yes—
66C2×C4×Q8C2312yes—
67C2×C4×D4—21no—
68C2×C4×D4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4D411yescharacteristic
73C22×C4⋊C4D411yescharacteristic
74C22×C4⋊C4D411yescharacteristic
75C22×C4⋊C4D412yes—
76C22×C42—21no—
77C22×C42—21no—
78C22×C42—22no—
79C22×C42C2311yescharacteristic
80#481—28no—
81#475—24no—
82#475C2×C418yescomplemented
83#460—24no—
84#460C2×C418yescomplemented
85#448—28no—
86C4×SD16—216no—
87#431—22no—
88#431—22no—
89#431—22no—
90#431C2×C414yes—
91C2×#106—22no—
92#423—22no—
93#423—22no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8C2312yes—
97C2×C4⋊C8C2312yes—
98#417—44no—
99#417—44no—
100C2×#103—22no—
101C2×#103—22no—
102C2×#103—22no—
103C2×#102—22no—
104C2×#102—22no—
105C2×#102—22no—
106C2×#102—24no—
107C2×C22⋊C8—21no—
108C2×C22⋊C8—21no—
109C2×C4×C8—21no—
110C2×C4×C8C2312yes—
111#391—24no—
112#391—24no—
113#387—28no—
114#387—28no—
115#384—21no—
116#384—21no—
117#384D412yes—
118#384C2312yes—
119#382—24no—
120#382—24no—
121C4×C4⋊C4—24no—
122C4×C4⋊C4—24no—
123C4×C4⋊C4—24no—
124C4×C4⋊C4—24no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4D412yes—
128C4×C4⋊C4C2312yes—
129C2×#95—22no—
130C2×#95—22no—
131C2×#95—22no—
132C2×#95—21no—
133#340—24no—
134Q8⋊C8—28no—
135C2×#41—24no—
136C2×#41—42no—
137C2×#41—42no—
138C2×#41—42no—
139C2×#41—42no—
140C2×#41—42no—
141C22×Q8—21no—
142C22×Q8—22no—
143C22×Q8—21no—
144C22×Q8—42no—
145C22×Q8—42no—
146C22×Q8C22⋊C414yes—
147C22×Q8C2×D412yes—
148C22×D4—21no—
149C22×D4—21no—
150C22×D4—21no—
151C22×D4—41no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—22no—
156C23×C4—21no—
157C23×C4—41no—
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C2×SD16—28no—
162C2×SD16—44no—
163C2×SD16—44no—
164C2×M16—42no—
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⋊Q8C22×C412yes—
174#127—21no—
175#127—21no—
176#127—21no—
177#127C22×C412yes—
178#124—24no—
179#124—42no—
180#121—24no—
181#121—24no—
182#121—42no—
183#121—42no—
184C4×Q8—24no—
185C4×Q8—24no—
186C4×Q8—24no—
187C4×Q8—24no—
188C4×Q8—24no—
189C4×Q8—22no—
190C4×Q8—22no—
191C4×Q8C22⋊C418yes—
192C4×Q8C22×C414yes—
193C4×Q8C2×D414yes—
194C4×D4—24no—
195C4×D4—24no—
196C4×D4—42no—
197C4×D4—42no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—82no—
220C2×C4⋊C4C22⋊C414yes—
221C2×C4⋊C4SD1614yescomplemented
222C2×C4⋊C4C2×D412yes—
223C2×C4⋊C4C2×D412yes—
224C2×C4⋊C4C2×D412yes—
225C2×C4⋊C4C2×D414yes—
226C2×C22⋊C4—41no—
227C2×C22⋊C4—41no—
228C2×C42—24no—
229C2×C42—22no—
230C2×C42—24no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—28no—
235C2×C42—44no—
236C2×C42—84no—
237C2×C42C22×C412yes—
238C2×C42C2×D411yescharacteristic
239C2×C42C2×D412yes—
240C2×C42C2×D411yescharacteristic
241C2×C42C2×D411yescharacteristic
242C2×C42#4112yes—
243C2×C42#4112yes—
244C2×C42C2411yescharacteristic
245#106—28no—
246C4⋊C8—22no—
247C4⋊C8—22no—
248C4⋊C8—22no—
249C4⋊C8—42no—
250C4⋊C8—42no—
251C4⋊C8C22×C414yes—
252#103—28no—
253#103—44no—
254#103—44no—
255#102—28no—
256#102—44no—
257#102—44no—
258#102—84no—
259C22⋊C8—24no—
260C22⋊C8—42no—
261C4×C8—24no—
262C4×C8—22no—
263C4×C8—22no—
264#95—28no—
265#95—28no—
266#95—28no—
267#95—22no—
268#95—22no—
269C24—41no—
270C24#12011yescharacteristic
271#41—48no—
272#41—48no—
273#41—84no—
274#41—84no—
275C2×Q8—22no—
276C2×Q8—24no—
277C2×Q8—22no—
278C2×Q8—22no—
279C2×Q8—22no—
280C2×Q8—22no—
281C2×Q8—22no—
282C2×Q8—48no—
283C2×Q8—42no—
284C2×Q8—42no—
285C2×Q8—48no—
286C2×Q8—41no—
287C2×Q8—41no—
288C2×Q8—41no—
289C2×Q8—41no—
290C2×Q8—41no—
291C2×Q8C2×C22⋊C414yes—
292C2×D4—22no—
293C2×D4—22no—
294C2×D4—22no—
295C2×D4—42no—
296C2×D4—42no—
297C2×D4—44no—
298C2×D4—41no—
299C2×D4—41no—
300C2×D4—41no—
301C2×D4—41no—
302C2×D4—41no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—42no—
306C2×D4—42no—
307C2×D4—82no—
308C2×D4—82no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—22no—
318C22×C4—24no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—44no—
329C22×C4—44no—
330C22×C4—42no—
331C22×C4—41no—
332C22×C4—42no—
333C22×C4—44no—
334C22×C4—44no—
335C22×C4—82no—
336C22×C4—82no—
337C22×C4C2×C22⋊C412yes—
338C22×C4C4×D412yes—
339C22×C4C4×D412yes—
340C22×C4#12011yescharacteristic
341C22×C4#12011yescharacteristic
342C22×C4#12011yescharacteristic
343C22×C4#12112yes—
344C22×C4#12112yes—
345C22×C4#12312yes—
346C22×C4C2×SD1612yes—
347C22×C4#13512yes—
348C22×C4AGL(1,ℤ/8)12yes—
349C22×C4#13712yes—
350C22×C4C22×D411yescharacteristic
351C22×C4C22×D411yescharacteristic
352C22×C4C22×D411yescharacteristic
353SD16—416no—
354M16—84no—
355C2×C8—24no—
356C2×C8—42no—
357C2×C8—42no—
358C2×C8—42no—
359C2×C8—42no—
360C2×C8—42no—
361C2×C8—42no—
362C2×C8—42no—
363C2×C8—42no—
364C4⋊C4—24no—
365C4⋊C4—24no—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—28no—
369C4⋊C4—24no—
370C4⋊C4—24no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—22no—
375C4⋊C4—44no—
376C4⋊C4—44no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—44no—
382C4⋊C4—42no—
383C4⋊C4—44no—
384C4⋊C4—82no—
385C4⋊C4C2×C22⋊C414yes—
386C4⋊C4C2×SD1614yes—
387C22⋊C4—44no—
388C22⋊C4—44no—
389C42—24no—
390C42—24no—
391C42—28no—
392C42—44no—
393C42—44no—
394C42—88no—
395C42C2×C22⋊C412yes—
396C42C4×D412yes—
397C42C4×D412yes—
398C42#12112yes—
399C42#12112yes—
400C42#12312yes—
401C42C23×C411yescharacteristic
402C42C22×D411yescharacteristic
403C42C2×#4111yescharacteristic
404C42C2×#4111yescharacteristic
405C23—21no—
406C23—21no—
407C23—21no—
408C23—21no—
409C23—42no—
410C23—81no—
411C23—81no—
412C23—81no—
413C23—81no—
414C23—81no—
415C23—81no—
416C23#38612yes—
417C23#44812yes—
418C23#44912yes—
419C23C2×#12011yescharacteristic
420Q8—24no—
421Q8—24no—
422Q8—48no—
423Q8—42no—
424Q8—42no—
425Q8—42no—
426Q8—42no—
427Q8—81no—
428Q8—81no—
429D4—44no—
430D4—44no—
431D4—42no—
432D4—42no—
433D4—82no—
434D4—88no—
435D4—82no—
436D4—81no—
437D4—81no—
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—42no—
450C2×C4—42no—
451C2×C4—48no—
452C2×C4—48no—
453C2×C4—48no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—44no—
462C2×C4—42no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—88no—
468C2×C4—84no—
469C2×C4—81no—
470C2×C4—81no—
471C2×C4—88no—
472C2×C4#38612yes—
473C2×C4#38612yes—
474C2×C4#38612yes—
475C2×C4C4×SD1614yes—
476C2×C4#44014yes—
477C2×C4#44812yes—
478C2×C4#44912yes—
479C2×C4#46014yes—
480C2×C4#46414yes—
481C2×C4C22×C22⋊C411yescharacteristic
482C2×C4C2×C4×D411yescharacteristic
483C2×C4C2×C4×D411yescharacteristic
484C2×C4C2×#12111yescharacteristic
485C2×C4C2×#12111yescharacteristic
486C2×C4C2×#12311yescharacteristic
487C2×C4C22×SD1611yescharacteristic
488C2×C4C2×#13511yescharacteristic
489C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
490C2×C4C2×#13711yescharacteristic
491C2×C4#57512yes—
492C2×C4#57512yes—
493C8—44no—
494C8—82no—
495C8—82no—
496C22—22no—
497C22—22no—
498C22—22no—
499C22—22no—
500C22—22no—
501C22—22no—
502C22—82no—
503C22—82no—
504C22—82no—
505C22—161no—
506C22—161no—
507C22—161no—
508C22—161no—
509C22#188014yes—
510C22C2×#38611yescharacteristic
511C22C2×#44811yescharacteristic
512C22C2×#44911yescharacteristic
513C22#300511yescharacteristic
514C22#300511yescharacteristic
515C22#301911yescharacteristic
516C22#302011yescharacteristic
517C4—22no—
518C4—21no—
519C4—21no—
520C4—22no—
521C4—44no—
522C4—44no—
523C4—44no—
524C4—84no—
525C4—82no—
526C4—82no—
527C4#188014yes—
528C4#289612yes—
529C4C2×C4×SD1611yescharacteristic
530C4C2×#44011yescharacteristic
531C4C2×#46011yescharacteristic
532C4C2×#46411yescharacteristic
533C4#309312yes—
534C2—24no—
535C2—162no—
536C2#1874811yescharacteristic
537C2#1875011yescharacteristic
538C2#1877811yescharacteristic
539C2#1878011yescharacteristic
540C2#2091011yescharacteristic
541C2C2×#188011yescharacteristic
542C2#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×#2355C2semidirect
5. C2×#1487C2semidirect
6. #8982C2semidirect
7. #8978C2semidirect
8. #8043C2semidirect
9. #7892C2semidirect
13. #3103C4semidirect
82. #475C2×C4semidirect
84. #460C2×C4semidirect
221. C2×C4⋊C4SD16semidirect

Maximal subgroups

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

subgroupindex
C2×#31032
C2×#23552
#218592
#212582
C2×#14872
#89822
#89782
#80432
#78922