Finite Group Databasebeta

← #195591

Subgroups — #195591

Order 512.

1424 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3227C211yesmaximal, characteristic
2#34206C211yesmaximal, characteristic
3#21804C211yesmaximal, characteristic, complemented
4C2×#1988C211yesmaximal, characteristic, complemented
5C2×#1540C211yesmaximal, characteristic
6#9059C214yesmaximal, complemented
7#9057C214yesmaximal, complemented
8#7893C211yesmaximal, characteristic, complemented
9#7892C211yesmaximal, characteristic, complemented
10C2×#532C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12C2×#520—21no—
13#3227—24no—
14#3227C418yescomplemented
15C2×#498C2212yes—
16C2×#486C2212yes—
17C2×#481—22no—
18C2×#476—22no—
19C2×#431C2211yescharacteristic
20C2×#431C2211yescharacteristic
21C22×C4⋊C8C2211yescharacteristic
22#2367—22no—
23#2366C2212yes—
24#2365C2212yes—
25#2354—21no—
26#2352—22no—
27D4×C4⋊C4—21no—
28#2325—22no—
29#2311—21no—
30C4×#121—22no—
31#2285C2211yescharacteristic
32C2×C4×C4⋊C4C2211yescharacteristic
33#1988—22no—
34#1988C2214yes—
35#1981C2214yes—
36#1956—21no—
37#1956—21no—
38#1787—22no—
39C2×#340—21no—
40C2×#340—21no—
41#1540—24no—
42C2×#332C2212yes—
43C2×#331C2212yes—
44#1279C2214yes—
45#1279C2214yes—
46C22×#41—21no—
47C23×Q8—21no—
48C23×C8—21no—
49#532—22no—
50#532—22no—
51#532C2×C414yes—
52#532D414yes—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8C2×C414yescomplemented
60C2×C4⋊Q8C2312yes—
61C2×C4⋊Q8C2311yescharacteristic
62C2×#127C2311yescharacteristic
63C2×#124—21no—
64C2×#121—21no—
65C2×#121D412yes—
66#520—44no—
67#520—44no—
68C2×C4×Q8—22no—
69C2×C4×D4—24no—
70C2×C4×D4—22no—
71C2×C4×D4—22no—
72C2×C4×D4—21no—
73C2×C4×D4D412yes—
74C2×#117—22no—
75C2×#117—21no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4—21no—
78C22×C4⋊C4—22no—
79C22×C4⋊C4D411yescharacteristic
80C22×C4⋊C4D411yescharacteristic
81C22×C42—21no—
82C22×C42C2311yescharacteristic
83#498—24no—
84#498C2×C418yescomplemented
85#486—24no—
86#486C2×C418yescomplemented
87#481—216no—
88#476—216no—
89#431—22no—
90#431—22no—
91#431—22no—
92#431C2×C414yescomplemented
93C2×#106—24no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8—22no—
96C2×C4⋊C8—22no—
97C2×C4⋊C8—21no—
98C2×C4⋊C8C2312yes—
99C2×C4⋊C8C2312yes—
100C2×#103—24no—
101C2×#103—24no—
102C2×#102—24no—
103C2×C22⋊C8—21no—
104C2×C22⋊C8—21no—
105C2×C4×C8—21no—
106C2×C4×C8C2312yes—
107#391—22no—
108#390—22no—
109#390—22no—
110#389—22no—
111#388—24no—
112#388—24no—
113#387—24no—
114#386—24no—
115#386—24no—
116#385—24no—
117#384D411yescharacteristic
118#384D411yescharacteristic
119#382—21no—
120#380—22no—
121#380—22no—
122#380—21no—
123#380D412yes—
124C4×C4⋊C4—22no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4—21no—
128C4×C4⋊C4D412yes—
129C4×C4⋊C4C2312yes—
130C4×C4⋊C4C2312yes—
131C4×C22⋊C4—24no—
132C2×#95—21no—
133#340—24no—
134#340—24no—
135#332—24no—
136#331—24no—
137C2×#41—24no—
138C2×#41—24no—
139C2×#41—42no—
140C2×#41—42no—
141C2×#41—42no—
142C2×#41—42no—
143C22×Q8—21no—
144C22×Q8—24no—
145C22×Q8—22no—
146C22×Q8—41no—
147C22×Q8—42no—
148C22×Q8—82no—
149C22×Q8—82no—
150C22×D4—22no—
151C22×D4—21no—
152C22×D4—21no—
153C22×D4C2×D412yes—
154C23×C4—21no—
155C23×C4—22no—
156C23×C4—24no—
157C23×C4—21no—
158C23×C4—41no—
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—42no—
166C22×C8—42no—
167C22×C8—42no—
168C4⋊Q8—24no—
169C4⋊Q8—24no—
170C4⋊Q8—24no—
171C4⋊Q8—22no—
172C4⋊Q8—21no—
173C4⋊Q8—21no—
174C4⋊Q8—22no—
175C4⋊Q8—22no—
176C4⋊Q8—42no—
177C4⋊Q8—42no—
178C4⋊Q8—42no—
179C4⋊Q8C22×C414yes—
180C4⋊Q8C22×C412yes—
181C4⋊Q8C2×D412yes—
182#127—21no—
183#127—21no—
184#127C22×C412yes—
185#127C2×D412yes—
186#124—24no—
187#124—24no—
188#121—24no—
189#121—24no—
190#121—24no—
191#121C22⋊C418yes—
192C4×Q8—44no—
193C4×Q8—44no—
194C4×D4—24no—
195C4×D4—24no—
196C4×D4—24no—
197C4×D4—44no—
198C4×D4—416no—
199C4×D4—44no—
200C4×D4—84no—
201C4×D4C22⋊C418yes—
202#117—48no—
203#117—42no—
204#117—42no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—48no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4SD1614yes—
226C2×C4⋊C4C2×D412yes—
227C2×C4⋊C4C2×D412yes—
228C2×C4⋊C4C2×D412yes—
229C2×C4⋊C4C2×D412yes—
230C2×C4⋊C4C2×D412yes—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—24no—
237C2×C42—24no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—82no—
242C2×C42C22×C412yes—
243C2×C42C2×D411yescharacteristic
244C2×C42C2×D412yes—
245C2×C42C2×D411yescharacteristic
246C2×C42C2×D411yescharacteristic
247C2×C42#4112yes—
248C2×C42#4112yes—
249C2×C42C2411yescharacteristic
250#106—216no—
251C4⋊C8—24no—
252C4⋊C8—22no—
253C4⋊C8—22no—
254C4⋊C8—22no—
255C4⋊C8—22no—
256C4⋊C8—42no—
257#103—84no—
258#103—84no—
259#102—216no—
260C22⋊C8—24no—
261C22⋊C8—42no—
262C4×C8—22no—
263C4×C8—42no—
264C4×C8C22×C414yes—
265#95—24no—
266#95—24no—
267#95—22no—
268#95—22no—
269#95—22no—
270#95—22no—
271C24—22no—
272C24#12011yescharacteristic
273#41—48no—
274#41—48no—
275#41—48no—
276#41—48no—
277C2×Q8—22no—
278C2×Q8—22no—
279C2×Q8—24no—
280C2×Q8—42no—
281C2×Q8—41no—
282C2×Q8—41no—
283C2×Q8—41no—
284C2×Q8—41no—
285C2×Q8—44no—
286C2×Q8—44no—
287C2×Q8—44no—
288C2×Q8—88no—
289C2×Q8—82no—
290C2×Q8—82no—
291C2×Q8—88no—
292C2×D4—28no—
293C2×D4—22no—
294C2×D4—22no—
295C2×D4—22no—
296C2×D4—22no—
297C2×D4—22no—
298C2×D4—42no—
299C2×D4—42no—
300C2×D4—44no—
301C2×D4—44no—
302C2×D4—41no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—42no—
307C2×D4—42no—
308C2×D4C2×C22⋊C414yes—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—24no—
314C22×C4—28no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—48no—
321C22×C4—44no—
322C22×C4—44no—
323C22×C4—42no—
324C22×C4—44no—
325C22×C4—42no—
326C22×C4—81no—
327C22×C4—81no—
328C22×C4—81no—
329C22×C4—81no—
330C22×C4—84no—
331C22×C4—84no—
332C22×C4—82no—
333C22×C4—82no—
334C22×C4C2×C22⋊C412yes—
335C22×C4C4×D412yes—
336C22×C4C4×D412yes—
337C22×C4#12011yescharacteristic
338C22×C4#12011yescharacteristic
339C22×C4#12011yescharacteristic
340C22×C4#12112yes—
341C22×C4#12112yes—
342C22×C4#12312yes—
343C22×C4C2×SD1612yes—
344C22×C4#13512yes—
345C22×C4AGL(1,ℤ/8)12yes—
346C22×C4#13712yes—
347C22×C4C22×D411yescharacteristic
348C22×C4C22×D411yescharacteristic
349C22×C4C22×D411yescharacteristic
350C2×C8—24no—
351C2×C8—42no—
352C2×C8—42no—
353C2×C8—42no—
354C2×C8—42no—
355C2×C8—42no—
356C2×C8—42no—
357C2×C8—42no—
358C2×C8—42no—
359C2×C8—84no—
360C4⋊C4—22no—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—22no—
364C4⋊C4—24no—
365C4⋊C4—24no—
366C4⋊C4—24no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—24no—
370C4⋊C4—24no—
371C4⋊C4—48no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4—44no—
375C4⋊C4—44no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—81no—
380C4⋊C4—81no—
381C4⋊C4C2×C22⋊C414yes—
382C4⋊C4C2×SD1614yes—
383C22⋊C4—28no—
384C22⋊C4—28no—
385C22⋊C4—48no—
386C22⋊C4—44no—
387C22⋊C4—84no—
388C22⋊C4—84no—
389C42—48no—
390C42—42no—
391C42—42no—
392C42—44no—
393C42—82no—
394C42C2×C22⋊C412yes—
395C42C4×D412yes—
396C42C4×D412yes—
397C42#12112yes—
398C42#12112yes—
399C42#12312yes—
400C42C23×C411yescharacteristic
401C42C22×D411yescharacteristic
402C42C2×#4111yescharacteristic
403C42C2×#4111yescharacteristic
404C23—21no—
405C23—21no—
406C23—21no—
407C23—21no—
408C23—24no—
409C23—42no—
410C23—42no—
411C23—42no—
412C23—42no—
413C23—42no—
414C23—42no—
415C23#38612yes—
416C23#45012yes—
417C23#45212yes—
418C23C2×#12011yescharacteristic
419Q8—42no—
420Q8—42no—
421Q8—42no—
422Q8—42no—
423Q8—48no—
424Q8—88no—
425Q8—84no—
426D4—44no—
427D4—416no—
428D4—42no—
429D4—42no—
430D4—44no—
431D4—42no—
432D4—42no—
433D4—44no—
434D4—82no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—416no—
452C2×C4—42no—
453C2×C4—42no—
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—84no—
463C2×C4—84no—
464C2×C4—88no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—168no—
468C2×C4#38612yes—
469C2×C4#38612yes—
470C2×C4#38612yes—
471C2×C4C4×SD1614yes—
472C2×C4#44014yes—
473C2×C4#45012yes—
474C2×C4#45212yes—
475C2×C4#46114yes—
476C2×C4#46314yes—
477C2×C4C22×C22⋊C411yescharacteristic
478C2×C4C2×C4×D411yescharacteristic
479C2×C4C2×C4×D411yescharacteristic
480C2×C4C2×#12111yescharacteristic
481C2×C4C2×#12111yescharacteristic
482C2×C4C2×#12311yescharacteristic
483C2×C4C22×SD1611yescharacteristic
484C2×C4C2×#13511yescharacteristic
485C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
486C2×C4C2×#13711yescharacteristic
487C2×C4#57612yes—
488C2×C4#57712yes—
489C8—44no—
490C8—82no—
491C8—82no—
492C22—22no—
493C22—22no—
494C22—22no—
495C22—22no—
496C22—22no—
497C22—22no—
498C22—44no—
499C22—44no—
500C22—44no—
501C22—82no—
502C22—82no—
503C22—82no—
504C22—82no—
505C22#187714yes—
506C22C2×#38611yescharacteristic
507C22C2×#45011yescharacteristic
508C22C2×#45211yescharacteristic
509C22#300811yescharacteristic
510C22#301011yescharacteristic
511C22#302211yescharacteristic
512C22#302211yescharacteristic
513C4—22no—
514C4—21no—
515C4—21no—
516C4—22no—
517C4—44no—
518C4—44no—
519C4—84no—
520C4—82no—
521C4—82no—
522C4—82no—
523C4#187714yes—
524C4C2×C4×SD1611yescharacteristic
525C4C2×#44011yescharacteristic
526C4#294612yes—
527C4C2×#46111yescharacteristic
528C4C2×#46311yescharacteristic
529C4#304512yes—
530C2—24no—
531C2—84no—
532C2#1890311yescharacteristic
533C2#1890711yescharacteristic
534C2#1892311yescharacteristic
535C2#1892711yescharacteristic
536C2C2×#187711yescharacteristic
537C2#2093211yescharacteristic
538C2#2095511yescharacteristic
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
3. #21804C2semidirect
4. C2×#1988C2semidirect
6. #9059C2semidirect
7. #9057C2semidirect
8. #7893C2semidirect
9. #7892C2semidirect
14. #3227C4semidirect
59. C2×C4⋊Q8C2×C4semidirect
84. #498C2×C4semidirect
86. #486C2×C4semidirect
92. #431C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#32272
#342062
#218042
C2×#19882
C2×#15402
#90592
#90572
#78932
#78922