Finite Group Databasebeta

← #197887

Subgroups — #197887

Order 512.

1274 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3037C211yesmaximal, characteristic
2C2×#2984C211yesmaximal, characteristic, complemented
3#21863C211yesmaximal, characteristic
4#20734C211yesmaximal, characteristic, complemented
5C2×#1553C211yesmaximal, characteristic
6#9230C212yesmaximal, complemented
7#9226C212yesmaximal, complemented
8#9222C212yesmaximal, complemented
9#9218C212yesmaximal, complemented
10#8042C211yesmaximal, characteristic, complemented
11#7880C211yesmaximal, characteristic, complemented
12C22×#137—21no—
13C2×#532C2211yescharacteristic
14C22×C4×Q8C2211yescharacteristic
15#3037—24no—
16#3037C418yescomplemented
17C2×#464—21no—
18C2×#464—21no—
19C2×#462C2211yescharacteristic
20C2×#462C2211yescharacteristic
21C2×#460C2211yescharacteristic
22C2×#460C2211yescharacteristic
23#2984—24no—
24#2984—24no—
25C2×#446—22no—
26C2×#446C2212yes—
27C2×C8×Q8C2212yes—
28C2×#432—21no—
29C2×#432C2211yescharacteristic
30C2×#431—21no—
31C2×#431C2211yescharacteristic
32C2×#431C2211yescharacteristic
33C2×#423C2211yescharacteristic
34C2×#417—21no—
35#2285C2211yescharacteristic
36#1990C2212yes—
37#1957C2212yes—
38#1875—21no—
39#1875—21no—
40#1824—21no—
41#1804—22no—
42#1803C2212yes—
43#1803C2212yes—
44#1787—21no—
45C4×#103—21no—
46C4×#103—21no—
47#1553—24no—
48C2×#332C2212yes—
49C2×#331C2212yes—
50#1296C2212yes—
51#1296C2212yes—
52#1277C2212yes—
53#1277C2212yes—
54C22×#41—21no—
55C23×Q8D411yescharacteristic
56C2×#137—28no—
57C2×#137—48no—
58C22×Q16—21no—
59C22×Q16—21no—
60C22×SD16—21no—
61C22×SD16—21no—
62C22×M16—21no—
63#532—22no—
64#532—22no—
65#532—22no—
66#532C2×C414yes—
67C2×C4⋊Q8C2311yescharacteristic
68C2×#127C2311yescharacteristic
69C2×#124—21no—
70C2×#121—21no—
71C2×#121—21no—
72C2×C4×Q8—21no—
73C2×C4×Q8—22no—
74C2×C4×Q8—22no—
75C2×C4×Q8—21no—
76C2×C4×Q8C2×C414yes—
77C2×C4×Q8D414yes—
78C2×C4×Q8D411yescharacteristic
79C2×C4×Q8D411yescharacteristic
80C2×C4×Q8C2311yescharacteristic
81C2×C4×Q8C2311yescharacteristic
82C2×C4×D4—21no—
83C2×C4×D4—21no—
84C2×#117—21no—
85C22×C4⋊C4—21no—
86C22×C4⋊C4D411yescharacteristic
87C22×C42—21no—
88C22×C42C2311yescharacteristic
89#464—28no—
90#464—28no—
91#462—22no—
92#462—22no—
93#462C2×C414yes—
94#462C2×C414yes—
95#460—22no—
96#460—22no—
97#460C2×C414yes—
98#460C2×C414yes—
99#446—216no—
100#446—24no—
101#446—24no—
102C8×Q8—24no—
103C8×Q8—24no—
104#432—28no—
105#432—22no—
106#432D414yes—
107#431—28no—
108#431—22no—
109#431—22no—
110#431—22no—
111#431—22no—
112#423—22no—
113#423C2×C414yes—
114C2×C4⋊C8—22no—
115C2×C4⋊C8—22no—
116C2×C4⋊C8—21no—
117C2×C4⋊C8D412yes—
118C2×C4⋊C8C2312yes—
119C2×C4⋊C8C2311yescharacteristic
120C2×C4⋊C8C2311yescharacteristic
121C2×C4⋊C8C2312yes—
122#417—28no—
123C2×#103—21no—
124C2×#103—21no—
125C2×#103—22no—
126C2×#103—22no—
127C2×#102—21no—
128C2×#102—21no—
129C2×#102—22no—
130C2×#102—22no—
131C2×C22⋊C8—21no—
132C2×C22⋊C8—21no—
133C2×C22⋊C8D411yescharacteristic
134C2×C22⋊C8D411yescharacteristic
135C2×#97—22no—
136C2×#97—21no—
137C2×C4×C8—22no—
138C2×C4×C8—21no—
139C2×C4×C8C2312yes—
140#384C2312yes—
141#382—21no—
142#380—21no—
143#380—21no—
144C4×C4⋊C4—21no—
145C4×C4⋊C4—21no—
146#340—22no—
147#340—22no—
148#340—22no—
149#340—22no—
150#332—24no—
151#331—24no—
152C2×#41—24no—
153C2×#41—44no—
154C2×#41—44no—
155C2×#41—42no—
156C22×Q8—21no—
157C22×Q8—22no—
158C22×Q8—24no—
159C22×Q8—21no—
160C22×Q8—42no—
161C22×Q8C22⋊C414yes—
162C22×Q8C2×D411yescharacteristic
163C22×Q8C2×D411yescharacteristic
164C22×D4—21no—
165C22×D4—21no—
166C22×D4—21no—
167C22×D4—41no—
168C23×C4—21no—
169C23×C4—21no—
170C23×C4—41no—
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4#4111yescharacteristic
174#137—432no—
175C2×Q16—24no—
176C2×Q16—24no—
177C2×Q16—44no—
178C2×Q16—44no—
179C2×SD16—24no—
180C2×SD16—24no—
181C2×SD16—44no—
182C2×SD16—44no—
183C2×M16—24no—
184C2×M16—44no—
185C22×C8—21no—
186C22×C8—21no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8C2×D412yes—
190C22×C8C2×D412yes—
191C4⋊Q8—21no—
192C4⋊Q8—21no—
193C4⋊Q8—21no—
194C4⋊Q8C22×C412yes—
195#127—21no—
196#127—21no—
197#127—21no—
198#127C22×C412yes—
199#124—24no—
200#124—42no—
201#121—24no—
202#121—24no—
203#121—42no—
204#121—42no—
205C4×Q8—24no—
206C4×Q8—24no—
207C4×Q8—24no—
208C4×Q8—22no—
209C4×Q8—22no—
210C4×Q8—24no—
211C4×Q8—21no—
212C4×Q8—21no—
213C4×Q8—21no—
214C4×Q8—21no—
215C4×Q8C22⋊C414yes—
216C4×Q8C22⋊C414yes—
217C4×Q8C22×C412yes—
218C4×Q8C22×C412yes—
219C4×Q8C2×D412yes—
220C4×Q8C2×D412yes—
221C4×D4—24no—
222C4×D4—24no—
223C4×D4—42no—
224C4×D4—42no—
225#117—82no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4C22⋊C414yes—
240C2×C4⋊C4C2×D411yescharacteristic
241C2×C4⋊C4C2×D411yescharacteristic
242C2×C22⋊C4—41no—
243C2×C22⋊C4—41no—
244C2×C22⋊C4—41no—
245C2×C42—24no—
246C2×C42—24no—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42—41no—
252C2×C42C22×C412yes—
253C2×C42C2×D411yescharacteristic
254C2×C42C2×D412yes—
255C2×C42C2×D411yescharacteristic
256C2×C42C2×D412yes—
257C2×C42#4111yescharacteristic
258C2×C42#4112yes—
259C2×C42C2411yescharacteristic
260C4⋊C8—28no—
261C4⋊C8—28no—
262C4⋊C8—24no—
263C4⋊C8—24no—
264C4⋊C8—21no—
265C4⋊C8—21no—
266C4⋊C8—22no—
267C4⋊C8—22no—
268C4⋊C8—22no—
269C4⋊C8C22×C412yes—
270C4⋊C8C22×C412yes—
271C4⋊C8C2×D414yes—
272#103—24no—
273#103—24no—
274#103—44no—
275#103—44no—
276#102—24no—
277#102—24no—
278#102—82no—
279#102—82no—
280C22⋊C8—24no—
281C22⋊C8—22no—
282C22⋊C8—22no—
283C22⋊C8—42no—
284#97—28no—
285#97—24no—
286C4×C8—28no—
287C4×C8—22no—
288C4×C8—22no—
289C4×C8—42no—
290#95—41no—
291#95—41no—
292C24—41no—
293C24#12111yescharacteristic
294#41—48no—
295#41—48no—
296#41—88no—
297C2×Q8—21no—
298C2×Q8—21no—
299C2×Q8—22no—
300C2×Q8—24no—
301C2×Q8—28no—
302C2×Q8—22no—
303C2×Q8—22no—
304C2×Q8—22no—
305C2×Q8—44no—
306C2×Q8—44no—
307C2×Q8—48no—
308C2×Q8—41no—
309C2×Q8—42no—
310C2×Q8—42no—
311C2×Q8C2×C22⋊C412yes—
312C2×Q8C2×C22⋊C412yes—
313C2×Q8AGL(1,ℤ/8)14yes—
314C2×D4—22no—
315C2×D4—22no—
316C2×D4—22no—
317C2×D4—44no—
318C2×D4—41no—
319C2×D4—41no—
320C2×D4—41no—
321C2×D4—42no—
322C2×D4—42no—
323C2×D4—42no—
324C2×D4—42no—
325C2×D4—82no—
326C2×D4—84no—
327C22×C4—21no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—42no—
338C22×C4—44no—
339C22×C4—44no—
340C22×C4—41no—
341C22×C4—44no—
342C22×C4—42no—
343C22×C4—84no—
344C22×C4C2×C22⋊C412yes—
345C22×C4C4×D412yes—
346C22×C4C4×D412yes—
347C22×C4#12012yes—
348C22×C4#12112yes—
349C22×C4#12111yescharacteristic
350C22×C4#12111yescharacteristic
351C22×C4#12111yescharacteristic
352C22×C4#12312yes—
353C22×C4C22×D411yescharacteristic
354C22×C4C22×D411yescharacteristic
355C22×C4C2×#4111yescharacteristic
356Q16—48no—
357Q16—48no—
358SD16—48no—
359SD16—48no—
360M16—48no—
361C2×C8—22no—
362C2×C8—22no—
363C2×C8—22no—
364C2×C8—22no—
365C2×C8—24no—
366C2×C8—22no—
367C2×C8—22no—
368C2×C8—22no—
369C2×C8—22no—
370C2×C8—42no—
371C2×C8—42no—
372C2×C8—42no—
373C2×C8—42no—
374C2×C8—42no—
375C2×C8—42no—
376C2×C8—42no—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—21no—
380C4⋊C4—21no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—24no—
384C4⋊C4—24no—
385C4⋊C4—24no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—41no—
389C4⋊C4—41no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—82no—
395C4⋊C4C2×C22⋊C412yes—
396C4⋊C4C2×C22⋊C412yes—
397C22⋊C4—44no—
398C22⋊C4—44no—
399C22⋊C4—82no—
400C42—22no—
401C42—22no—
402C42—22no—
403C42—22no—
404C42—24no—
405C42—24no—
406C42—82no—
407C42C2×C22⋊C412yes—
408C42C4×D412yes—
409C42C4×D412yes—
410C42#12012yes—
411C42#12112yes—
412C42#12312yes—
413C42C23×C411yescharacteristic
414C42C22×D411yescharacteristic
415C42C22×D411yescharacteristic
416C42C2×#4111yescharacteristic
417C23—21no—
418C23—21no—
419C23—22no—
420C23—22no—
421C23—42no—
422C23—81no—
423C23—81no—
424C23—82no—
425C23—82no—
426C23#38612yes—
427C23C2×#12111yescharacteristic
428Q8—22no—
429Q8—24no—
430Q8—22no—
431Q8—22no—
432Q8—48no—
433Q8—42no—
434Q8—42no—
435Q8—42no—
436Q8—82no—
437Q8#41814yes—
438D4—42no—
439D4—42no—
440D4—44no—
441D4—82no—
442D4—82no—
443D4—88no—
444D4—82no—
445D4—82no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—24no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—48no—
464C2×C4—48no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—88no—
470C2×C4—82no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4#38612yes—
475C2×C4#38612yes—
476C2×C4#38612yes—
477C2×C4C22×C22⋊C411yescharacteristic
478C2×C4C2×C4×D411yescharacteristic
479C2×C4C2×C4×D411yescharacteristic
480C2×C4C2×#12011yescharacteristic
481C2×C4C2×#12111yescharacteristic
482C2×C4C2×#12311yescharacteristic
483C2×C4C2×AGL(1,ℤ/8)12yes—
484C2×C4#57512yes—
485C2×C4#57712yes—
486C2×C4#57812yes—
487C8—42no—
488C8—42no—
489C8—42no—
490C8—42no—
491C8—44no—
492C8—82no—
493C22—22no—
494C22—22no—
495C22—22no—
496C22—42no—
497C22—42no—
498C22—82no—
499C22—82no—
500C22—82no—
501C22—162no—
502C22—162no—
503C22C2×#38611yescharacteristic
504C22#304311yescharacteristic
505C22#304411yescharacteristic
506C22#305512yes—
507C22#306712yes—
508C4—21no—
509C4—21no—
510C4—22no—
511C4—22no—
512C4—22no—
513C4—24no—
514C4—42no—
515C4—42no—
516C4—42no—
517C4—82no—
518C4—162no—
519C4C2×#41811yescharacteristic
520C4#289611yescharacteristic
521C4#294712yes—
522C4#301312yes—
523C4#305512yes—
524C2—42no—
525C2—162no—
526C2#1947312yes—
527C2#1947912yes—
528C2#2093011yescharacteristic
529C2#2097311yescharacteristic
530C2#2100311yescharacteristic
subgroupG/H
center, socleC23C2×#121
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×#2984C2semidirect
4. #20734C2semidirect
6. #9230C2semidirect
7. #9226C2semidirect
8. #9222C2semidirect
9. #9218C2semidirect
10. #8042C2semidirect
11. #7880C2semidirect
16. #3037C4semidirect

Maximal subgroups

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

subgroupindex
C2×#30372
C2×#29842
#218632
#207342
C2×#15532
#92302
#92262
#92222
#92182
#80422
#78802