Finite Group Databasebeta

← #195586

Subgroups — #195586

Order 512.

1452 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×#3226C211yesmaximal, characteristic
2#34206C211yesmaximal, characteristic
3#21833C211yesmaximal, characteristic, complemented
4C2×#1981C211yesmaximal, characteristic, complemented
5C2×#1540C211yesmaximal, characteristic
6#9058C214yesmaximal, complemented
7#9057C214yesmaximal, complemented
8#7893C211yesmaximal, characteristic, complemented
9#7892C211yesmaximal, characteristic, complemented
10C2×#532C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12C2×#520—21no—
13#3226—24no—
14#3226C418yescomplemented
15C2×#500C2212yes—
16C2×#486C2212yes—
17C2×#480—22no—
18C2×#474—22no—
19C2×#431C2211yescharacteristic
20C2×#431C2211yescharacteristic
21C22×C4⋊C8C2211yescharacteristic
22C22×#103—21no—
23C22×#103—21no—
24#2367—22no—
25#2366C2212yes—
26#2365C2212yes—
27#2354—21no—
28#2352—22no—
29D4×C4⋊C4—21no—
30#2325—22no—
31#2311—21no—
32C4×#121—22no—
33#2285C2211yescharacteristic
34C2×C4×C4⋊C4C2211yescharacteristic
35#1990—21no—
36#1990—21no—
37#1988C2214yes—
38#1981—22no—
39#1981C2214yes—
40#1823—22no—
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#500—24no—
84#500C2×C418yescomplemented
85#486—24no—
86#486C2×C418yescomplemented
87#480—216no—
88#474—216no—
89#431—22no—
90#431—22no—
91#431—22no—
92#431C2×C414yescomplemented
93C2×#107—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×#103—44no—
103C2×#103—44no—
104C2×#102—24no—
105C2×C22⋊C8—21no—
106C2×C22⋊C8—21no—
107C2×C4×C8—21no—
108C2×C4×C8C2312yes—
109#391—22no—
110#390—22no—
111#390—22no—
112#389—22no—
113#388—24no—
114#388—24no—
115#387—24no—
116#386—24no—
117#386—24no—
118#385—24no—
119#384D411yescharacteristic
120#384D411yescharacteristic
121#382—21no—
122#380—22no—
123#380—22no—
124#380—21no—
125#380D412yes—
126C4×C4⋊C4—22no—
127C4×C4⋊C4—21no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4D412yes—
131C4×C4⋊C4C2312yes—
132C4×C4⋊C4C2312yes—
133C4×C22⋊C4—24no—
134C2×#95—21no—
135#340—24no—
136#340—24no—
137#332—24no—
138#331—24no—
139C2×#41—24no—
140C2×#41—24no—
141C2×#41—42no—
142C2×#41—42no—
143C2×#41—42no—
144C2×#41—42no—
145C22×Q8—21no—
146C22×Q8—24no—
147C22×Q8—22no—
148C22×Q8—41no—
149C22×Q8—42no—
150C22×Q8—42no—
151C22×Q8—42no—
152C22×Q8—42no—
153C22×Q8—42no—
154C22×D4—22no—
155C22×D4—21no—
156C22×D4—21no—
157C22×D4C2×D412yes—
158C23×C4—21no—
159C23×C4—22no—
160C23×C4—24no—
161C23×C4—21no—
162C23×C4—41no—
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—42no—
170C22×C8—42no—
171C22×C8—42no—
172C4⋊Q8—24no—
173C4⋊Q8—24no—
174C4⋊Q8—24no—
175C4⋊Q8—22no—
176C4⋊Q8—21no—
177C4⋊Q8—21no—
178C4⋊Q8—22no—
179C4⋊Q8—22no—
180C4⋊Q8—42no—
181C4⋊Q8—42no—
182C4⋊Q8—42no—
183C4⋊Q8C22×C414yes—
184C4⋊Q8C22×C412yes—
185C4⋊Q8C2×D412yes—
186#127—21no—
187#127—21no—
188#127C22×C412yes—
189#127C2×D412yes—
190#124—24no—
191#124—24no—
192#121—24no—
193#121—24no—
194#121—24no—
195#121C22⋊C418yes—
196C4×Q8—44no—
197C4×Q8—44no—
198C4×D4—24no—
199C4×D4—24no—
200C4×D4—24no—
201C4×D4—44no—
202C4×D4—416no—
203C4×D4—44no—
204C4×D4—84no—
205C4×D4C22⋊C418yes—
206#117—48no—
207#117—42no—
208#117—42no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—48no—
225C2×C4⋊C4—41no—
226C2×C4⋊C4—41no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4D814yes—
230C2×C4⋊C4C2×D412yes—
231C2×C4⋊C4C2×D412yes—
232C2×C4⋊C4C2×D412yes—
233C2×C4⋊C4C2×D412yes—
234C2×C4⋊C4C2×D412yes—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—24no—
241C2×C42—24no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—82no—
246C2×C42C22×C412yes—
247C2×C42C2×D411yescharacteristic
248C2×C42C2×D412yes—
249C2×C42C2×D411yescharacteristic
250C2×C42C2×D411yescharacteristic
251C2×C42#4112yes—
252C2×C42#4112yes—
253C2×C42C2411yescharacteristic
254#107—216no—
255C4⋊C8—24no—
256C4⋊C8—22no—
257C4⋊C8—22no—
258C4⋊C8—22no—
259C4⋊C8—22no—
260C4⋊C8—42no—
261#103—48no—
262#103—48no—
263#102—216no—
264C22⋊C8—24no—
265C22⋊C8—42no—
266C4×C8—22no—
267C4×C8—42no—
268C4×C8C22×C414yes—
269#95—24no—
270#95—24no—
271#95—22no—
272#95—22no—
273#95—22no—
274#95—22no—
275C24—22no—
276C24#12011yescharacteristic
277#41—48no—
278#41—48no—
279#41—48no—
280#41—48no—
281C2×Q8—22no—
282C2×Q8—22no—
283C2×Q8—24no—
284C2×Q8—42no—
285C2×Q8—41no—
286C2×Q8—41no—
287C2×Q8—41no—
288C2×Q8—41no—
289C2×Q8—44no—
290C2×Q8—42no—
291C2×Q8—42no—
292C2×Q8—42no—
293C2×Q8—42no—
294C2×Q8—48no—
295C2×Q8—48no—
296C2×Q8—82no—
297C2×Q8—82no—
298C2×Q8—84no—
299C2×Q8—84no—
300C2×D4—28no—
301C2×D4—22no—
302C2×D4—22no—
303C2×D4—22no—
304C2×D4—22no—
305C2×D4—22no—
306C2×D4—42no—
307C2×D4—42no—
308C2×D4—44no—
309C2×D4—44no—
310C2×D4—41no—
311C2×D4—41no—
312C2×D4—41no—
313C2×D4—41no—
314C2×D4—42no—
315C2×D4—42no—
316C2×D4C2×C22⋊C414yes—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—24no—
322C22×C4—28no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—48no—
329C22×C4—44no—
330C22×C4—44no—
331C22×C4—42no—
332C22×C4—44no—
333C22×C4—42no—
334C22×C4—81no—
335C22×C4—81no—
336C22×C4—81no—
337C22×C4—81no—
338C22×C4—84no—
339C22×C4—84no—
340C22×C4—82no—
341C22×C4—82no—
342C22×C4C2×C22⋊C412yes—
343C22×C4C4×D412yes—
344C22×C4C4×D412yes—
345C22×C4#12011yescharacteristic
346C22×C4#12011yescharacteristic
347C22×C4#12011yescharacteristic
348C22×C4#12112yes—
349C22×C4#12112yes—
350C22×C4#12312yes—
351C22×C4C2×D812yes—
352C22×C4#13512yes—
353C22×C4AGL(1,ℤ/8)12yes—
354C22×C4AGL(1,ℤ/8)12yes—
355C22×C4C22×D411yescharacteristic
356C22×C4C22×D411yescharacteristic
357C22×C4C22×D411yescharacteristic
358C2×C8—24no—
359C2×C8—42no—
360C2×C8—42no—
361C2×C8—42no—
362C2×C8—42no—
363C2×C8—42no—
364C2×C8—42no—
365C2×C8—42no—
366C2×C8—42no—
367C2×C8—84no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—24no—
373C4⋊C4—24no—
374C4⋊C4—24no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—24no—
378C4⋊C4—24no—
379C4⋊C4—48no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—44no—
383C4⋊C4—44no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—81no—
388C4⋊C4—81no—
389C4⋊C4C2×C22⋊C414yes—
390C4⋊C4C2×D814yes—
391C22⋊C4—28no—
392C22⋊C4—28no—
393C22⋊C4—48no—
394C22⋊C4—44no—
395C22⋊C4—84no—
396C22⋊C4—84no—
397C42—48no—
398C42—42no—
399C42—42no—
400C42—44no—
401C42—82no—
402C42C2×C22⋊C412yes—
403C42C4×D412yes—
404C42C4×D412yes—
405C42#12112yes—
406C42#12112yes—
407C42#12312yes—
408C42C23×C411yescharacteristic
409C42C22×D411yescharacteristic
410C42C2×#4111yescharacteristic
411C42C2×#4111yescharacteristic
412C23—21no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—24no—
417C23—42no—
418C23—42no—
419C23—42no—
420C23—42no—
421C23—42no—
422C23—42no—
423C23#38612yes—
424C23#44712yes—
425C23#44912yes—
426C23C2×#12011yescharacteristic
427Q8—42no—
428Q8—42no—
429Q8—42no—
430Q8—42no—
431Q8—44no—
432Q8—44no—
433Q8—88no—
434Q8—82no—
435Q8—82no—
436D4—44no—
437D4—416no—
438D4—42no—
439D4—42no—
440D4—44no—
441D4—42no—
442D4—42no—
443D4—44no—
444D4—82no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—416no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—44no—
472C2×C4—84no—
473C2×C4—84no—
474C2×C4—88no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—168no—
478C2×C4#38612yes—
479C2×C4#38612yes—
480C2×C4#38612yes—
481C2×C4C4×D814yes—
482C2×C4#44214yes—
483C2×C4#44712yes—
484C2×C4#44912yes—
485C2×C4#45914yes—
486C2×C4#46314yes—
487C2×C4C22×C22⋊C411yescharacteristic
488C2×C4C2×C4×D411yescharacteristic
489C2×C4C2×C4×D411yescharacteristic
490C2×C4C2×#12111yescharacteristic
491C2×C4C2×#12111yescharacteristic
492C2×C4C2×#12311yescharacteristic
493C2×C4C22×D811yescharacteristic
494C2×C4C2×#13511yescharacteristic
495C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
496C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
497C2×C4#57712yes—
498C2×C4#57812yes—
499C8—44no—
500C8—82no—
501C8—82no—
502C22—22no—
503C22—22no—
504C22—22no—
505C22—22no—
506C22—22no—
507C22—22no—
508C22—44no—
509C22—44no—
510C22—44no—
511C22—82no—
512C22—82no—
513C22—82no—
514C22—82no—
515C22#187914yes—
516C22C2×#38611yescharacteristic
517C22C2×#44711yescharacteristic
518C22C2×#44911yescharacteristic
519C22#300911yescharacteristic
520C22#301111yescharacteristic
521C22#302211yescharacteristic
522C22#302211yescharacteristic
523C4—22no—
524C4—21no—
525C4—21no—
526C4—22no—
527C4—44no—
528C4—44no—
529C4—84no—
530C4—82no—
531C4—82no—
532C4—82no—
533C4#187914yes—
534C4C2×C4×D811yescharacteristic
535C4C2×#44211yescharacteristic
536C4#294712yes—
537C4C2×#45911yescharacteristic
538C4C2×#46311yescharacteristic
539C4#304612yes—
540C2—24no—
541C2—84no—
542C2#1890211yescharacteristic
543C2#1890611yescharacteristic
544C2#1892211yescharacteristic
545C2#1892611yescharacteristic
546C2C2×#187911yescharacteristic
547C2#2093111yescharacteristic
548C2#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. #21833C2semidirect
4. C2×#1981C2semidirect
6. #9058C2semidirect
7. #9057C2semidirect
8. #7893C2semidirect
9. #7892C2semidirect
14. #3226C4semidirect
59. C2×C4⋊Q8C2×C4semidirect
84. #500C2×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×#32262
#342062
#218332
C2×#19812
C2×#15402
#90582
#90572
#78932
#78922