Finite Group Databasebeta

← #194133

Subgroups — #194133

Order 512.

1422 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3037C211yesmaximal, characteristic
2C2×#2355C211yesmaximal, characteristic, complemented
3#21859C211yesmaximal, characteristic
4#21386C211yesmaximal, characteristic
5C2×#1479C211yesmaximal, characteristic, complemented
6#8986C214yesmaximal, complemented
7#8978C214yesmaximal, complemented
8#8043C211yesmaximal, characteristic, complemented
9#7901C211yesmaximal, characteristic, complemented
10C22×#137—21no—
11C2×#532C2211yescharacteristic
12C22×C4×Q8C2211yescharacteristic
13#3037—24no—
14#3037C418yescomplemented
15C2×#464—22no—
16C2×#462C2212yes—
17C2×#460C2212yes—
18C2×#431C2211yescharacteristic
19C2×#431C2211yescharacteristic
20C2×#423C2211yescharacteristic
21C2×#417—21no—
22#2355—22no—
23#2355—22no—
24#2355—22no—
25#2355C2214yes—
26C2×#391—21no—
27C2×#387—22no—
28C2×#384C2211yescharacteristic
29C2×#382—21no—
30C2×C4×C4⋊C4—21no—
31C2×C4×C4⋊C4—21no—
32C2×C4×C4⋊C4C2211yescharacteristic
33#1990C2212yes—
34#1988C2212yes—
35#1982C2212yes—
36#1957C2212yes—
37#1874—22no—
38#1874—21no—
39#1874—21no—
40#1802—22no—
41C4×#102—22no—
42#1479—24no—
43C2×Q8⋊C8C2214yes—
44#1296C2214yes—
45#1279C2214yes—
46C22×#41—21no—
47C23×Q8D411yescharacteristic
48C2×#137—28no—
49C2×#137—44no—
50C2×#137—44no—
51C22×Q16—22no—
52C22×SD16—22no—
53C22×M16—21no—
54#532—22no—
55#532—22no—
56#532—22no—
57#532C2×C414yes—
58C2×C4⋊Q8C2311yescharacteristic
59C2×#127C2311yescharacteristic
60C2×#124—21no—
61C2×#121—21no—
62C2×#121—21no—
63C2×C4×Q8—21no—
64C2×C4×Q8—21no—
65C2×C4×Q8—22no—
66C2×C4×Q8—22no—
67C2×C4×Q8C2×C414yes—
68C2×C4×Q8D414yes—
69C2×C4×Q8D412yes—
70C2×C4×Q8C2312yes—
71C2×C4×D4—21no—
72C2×C4×D4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4D411yescharacteristic
77C22×C4⋊C4D411yescharacteristic
78C22×C4⋊C4D411yescharacteristic
79C22×C4⋊C4D412yes—
80C22×C42—21no—
81C22×C42—21no—
82C22×C42—22no—
83C22×C42C2311yescharacteristic
84#464—216no—
85#462—24no—
86#462C2×C418yescomplemented
87#460—24no—
88#460C2×C418yescomplemented
89#431—22no—
90#431—22no—
91#431—22no—
92#431—22no—
93#423—22no—
94#423C2×C414yes—
95C2×C4⋊C8—21no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8C2312yes—
98C2×C4⋊C8C2312yes—
99#417—28no—
100C2×#103—22no—
101C2×#102—22no—
102C2×#102—22no—
103C2×#102—22no—
104C2×#102—24no—
105C2×C22⋊C8—21no—
106C2×C22⋊C8—21no—
107C2×C4×C8—21no—
108C2×C4×C8C2312yes—
109#391—24no—
110#391—24no—
111#387—28no—
112#387—28no—
113#384—21no—
114#384—21no—
115#384D412yes—
116#384C2312yes—
117#382—24no—
118#382—24no—
119C4×C4⋊C4—24no—
120C4×C4⋊C4—24no—
121C4×C4⋊C4—24no—
122C4×C4⋊C4—24no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4—21no—
125C4×C4⋊C4D412yes—
126C4×C4⋊C4C2312yes—
127C2×#95—22no—
128C2×#95—22no—
129C2×#95—22no—
130C2×#95—21no—
131#340—24no—
132#340—22no—
133#340—22no—
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—22no—
144C22×Q8—22no—
145C22×Q8—22no—
146C22×Q8—22no—
147C22×Q8—21no—
148C22×Q8C22⋊C414yes—
149C22×Q8C2×D412yes—
150C22×D4—21no—
151C22×D4—21no—
152C22×D4—21no—
153C22×D4—41no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—22no—
158C23×C4—21no—
159C23×C4—41no—
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163#137—432no—
164C2×Q16—28no—
165C2×Q16—44no—
166C2×Q16—44no—
167C2×SD16—28no—
168C2×SD16—44no—
169C2×SD16—44no—
170C2×M16—24no—
171C2×M16—42no—
172C2×M16—42no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C4⋊Q8—21no—
177C4⋊Q8—21no—
178C4⋊Q8—21no—
179C4⋊Q8C22×C412yes—
180#127—21no—
181#127—21no—
182#127—21no—
183#127C22×C412yes—
184#124—24no—
185#124—42no—
186#121—24no—
187#121—24no—
188#121—42no—
189#121—42no—
190C4×Q8—24no—
191C4×Q8—24no—
192C4×Q8—24no—
193C4×Q8—24no—
194C4×Q8—24no—
195C4×Q8—22no—
196C4×Q8—22no—
197C4×Q8C22⋊C418yes—
198C4×Q8C22×C414yes—
199C4×Q8C2×D414yes—
200C4×D4—24no—
201C4×D4—24no—
202C4×D4—42no—
203C4×D4—42no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—82no—
225C2×C4⋊C4C22⋊C414yes—
226C2×C4⋊C4C2×D412yes—
227C2×C4⋊C4C2×D412yes—
228C2×C4⋊C4C2×D412yes—
229C2×C4⋊C4C2×D414yes—
230C2×C22⋊C4—41no—
231C2×C22⋊C4—41no—
232C2×C42—24no—
233C2×C42—22no—
234C2×C42—24no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—28no—
239C2×C42—42no—
240C2×C42—42no—
241C2×C42—84no—
242C2×C42C22×C412yes—
243C2×C42C2×D411yescharacteristic
244C2×C42C2×D412yes—
245C2×C42C2×D411yescharacteristic
246C2×C42C2×D411yescharacteristic
247C2×C42#4112yes—
248C2×C42#4112yes—
249C2×C42C2411yescharacteristic
250C4⋊C8—24no—
251C4⋊C8—22no—
252C4⋊C8—22no—
253C4⋊C8—22no—
254C4⋊C8—42no—
255C4⋊C8C22×C414yes—
256#103—28no—
257#102—28no—
258#102—44no—
259#102—44no—
260#102—84no—
261C22⋊C8—42no—
262C22⋊C8—42no—
263C4×C8—22no—
264C4×C8—22no—
265C4×C8—42no—
266#95—28no—
267#95—28no—
268#95—28no—
269#95—22no—
270#95—22no—
271C24—41no—
272C24#12011yescharacteristic
273#41—48no—
274#41—48no—
275#41—84no—
276#41—84no—
277C2×Q8—22no—
278C2×Q8—24no—
279C2×Q8—22no—
280C2×Q8—22no—
281C2×Q8—22no—
282C2×Q8—22no—
283C2×Q8—22no—
284C2×Q8—28no—
285C2×Q8—28no—
286C2×Q8—41no—
287C2×Q8—42no—
288C2×Q8—42no—
289C2×Q8—44no—
290C2×Q8—44no—
291C2×Q8—41no—
292C2×Q8—41no—
293C2×Q8—41no—
294C2×Q8—41no—
295C2×Q8C2×C22⋊C414yes—
296C2×D4—22no—
297C2×D4—22no—
298C2×D4—22no—
299C2×D4—42no—
300C2×D4—42no—
301C2×D4—44no—
302C2×D4—41no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—41no—
308C2×D4—41no—
309C2×D4—42no—
310C2×D4—42no—
311C2×D4—82no—
312C2×D4—82no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—24no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—22no—
322C22×C4—24no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—44no—
332C22×C4—42no—
333C22×C4—44no—
334C22×C4—42no—
335C22×C4—41no—
336C22×C4—42no—
337C22×C4—44no—
338C22×C4—44no—
339C22×C4—82no—
340C22×C4—82no—
341C22×C4C2×C22⋊C412yes—
342C22×C4C4×D412yes—
343C22×C4C4×D412yes—
344C22×C4#12011yescharacteristic
345C22×C4#12011yescharacteristic
346C22×C4#12011yescharacteristic
347C22×C4#12112yes—
348C22×C4#12112yes—
349C22×C4#12312yes—
350C22×C4#13512yes—
351C22×C4#13512yes—
352C22×C4AGL(1,ℤ/8)12yes—
353C22×C4AGL(1,ℤ/8)12yes—
354C22×C4C22×D411yescharacteristic
355C22×C4C22×D411yescharacteristic
356C22×C4C22×D411yescharacteristic
357Q16—416no—
358SD16—416no—
359M16—48no—
360C2×C8—24no—
361C2×C8—42no—
362C2×C8—42no—
363C2×C8—42no—
364C2×C8—42no—
365C2×C8—42no—
366C2×C8—42no—
367C2×C8—42no—
368C2×C8—42no—
369C4⋊C4—24no—
370C4⋊C4—24no—
371C4⋊C4—24no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—28no—
375C4⋊C4—24no—
376C4⋊C4—22no—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—44no—
381C4⋊C4—42no—
382C4⋊C4—44no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—44no—
388C4⋊C4—44no—
389C4⋊C4—82no—
390C4⋊C4C2×C22⋊C414yes—
391C4⋊C4#13714yescomplemented
392C22⋊C4—44no—
393C22⋊C4—44no—
394C42—24no—
395C42—24no—
396C42—28no—
397C42—42no—
398C42—42no—
399C42—44no—
400C42—88no—
401C42C2×C22⋊C412yes—
402C42C4×D412yes—
403C42C4×D412yes—
404C42#12112yes—
405C42#12112yes—
406C42#12312yes—
407C42C23×C411yescharacteristic
408C42C22×D411yescharacteristic
409C42C2×#4111yescharacteristic
410C42C2×#4111yescharacteristic
411C23—21no—
412C23—21no—
413C23—21no—
414C23—21no—
415C23—42no—
416C23—81no—
417C23—81no—
418C23—81no—
419C23—81no—
420C23—81no—
421C23—81no—
422C23#38612yes—
423C23#44912yes—
424C23#44912yes—
425C23C2×#12011yescharacteristic
426Q8—24no—
427Q8—24no—
428Q8—42no—
429Q8—42no—
430Q8—48no—
431Q8—42no—
432Q8—42no—
433Q8—81no—
434Q8—81no—
435D4—44no—
436D4—42no—
437D4—42no—
438D4—44no—
439D4—82no—
440D4—88no—
441D4—81no—
442D4—81no—
443D4—82no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—48no—
458C2×C4—48no—
459C2×C4—48no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—44no—
468C2×C4—42no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—88no—
474C2×C4—84no—
475C2×C4—81no—
476C2×C4—81no—
477C2×C4—88no—
478C2×C4#38612yes—
479C2×C4#38612yes—
480C2×C4#38612yes—
481C2×C4#41614yes—
482C2×C4#41814yes—
483C2×C4#44912yes—
484C2×C4#44912yes—
485C2×C4#48214yes—
486C2×C4#48214yes—
487C2×C4C22×C22⋊C411yescharacteristic
488C2×C4C2×C4×D411yescharacteristic
489C2×C4C2×C4×D411yescharacteristic
490C2×C4C2×#12111yescharacteristic
491C2×C4C2×#12111yescharacteristic
492C2×C4C2×#12311yescharacteristic
493C2×C4C2×#13511yescharacteristic
494C2×C4C2×#13511yescharacteristic
495C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
496C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
497C2×C4C2×#13712yes—
498C2×C4#57512yes—
499C8—44no—
500C8—82no—
501C8—82no—
502C22—22no—
503C22—22no—
504C22—22no—
505C22—22no—
506C22—22no—
507C22—22no—
508C22—82no—
509C22—82no—
510C22—82no—
511C22—161no—
512C22—161no—
513C22—161no—
514C22—161no—
515C22#187514yes—
516C22C2×#38611yescharacteristic
517C22C2×#44911yescharacteristic
518C22C2×#44911yescharacteristic
519C22#300411yescharacteristic
520C22#300511yescharacteristic
521C22#301911yescharacteristic
522C22#302011yescharacteristic
523C4—22no—
524C4—21no—
525C4—21no—
526C4—22no—
527C4—44no—
528C4—44no—
529C4—44no—
530C4—84no—
531C4—82no—
532C4—82no—
533C4#187514yes—
534C4C2×#41611yescharacteristic
535C4C2×#41811yescharacteristic
536C4C4×#13712yes—
537C4#303712yes—
538C4C2×#48211yescharacteristic
539C4C2×#48211yescharacteristic
540C2—24no—
541C2—162no—
542C2#1875011yescharacteristic
543C2#1875211yescharacteristic
544C2#1876011yescharacteristic
545C2#1876211yescharacteristic
546C2C2×#187511yescharacteristic
547C2#2092611yescharacteristic
548C2#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×#1479C2semidirect
6. #8986C2semidirect
7. #8978C2semidirect
8. #8043C2semidirect
9. #7901C2semidirect
14. #3037C4semidirect
86. #462C2×C4semidirect
88. #460C2×C4semidirect
391. C4⋊C4#137semidirect

Maximal subgroups

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

subgroupindex
C2×#30372
C2×#23552
#218592
#213862
C2×#14792
#89862
#89782
#80432
#79012