Finite Group Databasebeta

← #195595

Subgroups — #195595

Order 512.

1436 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×#3208C211yesmaximal, characteristic
2#34213C211yesmaximal, characteristic, complemented
3#21803C211yesmaximal, characteristic
4C2×#1982C211yesmaximal, characteristic
5C2×#1542C211yesmaximal, characteristic, complemented
6#9063C214yesmaximal, complemented
7#9062C214yesmaximal, complemented
8#7893C211yesmaximal, characteristic, complemented
9#7892C211yesmaximal, characteristic, complemented
10C2×#533C2211yescharacteristic
11C22×#127C2211yescharacteristic
12#3208—24no—
13#3208C418yescomplemented
14C2×#492C2212yes—
15C2×#486C2212yes—
16C2×#460—22no—
17C2×#450—22no—
18C2×#431C2211yescharacteristic
19C2×#431C2211yescharacteristic
20C22×C4⋊C8C2211yescharacteristic
21C22×#102—21no—
22C22×#102—21no—
23#2370—22no—
24#2369C2212yescomplemented
25#2365C2212yescomplemented
26#2355—21no—
27#2355—22no—
28Q8×C4⋊C4—21no—
29#2329—22no—
30#2317—21no—
31C4×C22⋊Q8—22no—
32#2285C2211yescharacteristic
33#2274—21no—
34C2×C4×C4⋊C4C2211yescharacteristic
35#1982—22no—
36#1982C2214yes—
37#1982C2214yes—
38#1957—21no—
39#1957—21no—
40#1786—22no—
41#1542—24no—
42C2×#333C2212yescomplemented
43C2×#332C2212yescomplemented
44#1279C2214yescomplemented
45#1279C2214yescomplemented
46C23×D4—21no—
47C23×D4—21no—
48C23×D4—21no—
49C22×SD16—24no—
50C23×C8—21no—
51#533—22no—
52#533—22no—
53#533C2×C414yes—
54#533D414yes—
55C2×C4⋊Q8C2311yescharacteristic
56C2×C4⋊Q8C2311yescharacteristic
57C2×#127—22no—
58C2×#127—21no—
59C2×#127—21no—
60C2×#127—21no—
61C2×#127—22no—
62C2×#127—22no—
63C2×#127C2×C414yes—
64C2×#127C2312yes—
65C2×#125—21no—
66C2×C22⋊Q8—21no—
67C2×C22⋊Q8D412yes—
68C2×C4×Q8—22no—
69C2×C4×Q8—21no—
70C2×C4×Q8D412yes—
71C2×#117—21no—
72C2×#117—21no—
73C22×C4⋊C4D411yescharacteristic
74C22×C4⋊C4D411yescharacteristic
75C22×C42—21no—
76C22×C42C2311yescharacteristic
77#492—24no—
78#492C2×C418yescomplemented
79#486—24no—
80#486C2×C418yescomplemented
81#460—216no—
82#450—216no—
83#431—22no—
84#431—22no—
85#431—22no—
86#431C2×C414yes—
87C2×C4⋊C8—21no—
88C2×C4⋊C8—22no—
89C2×C4⋊C8—22no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8C2312yes—
92C2×C4⋊C8C2312yes—
93C2×#102—24no—
94C2×#102—24no—
95C2×#102—24no—
96C2×#102—24no—
97C2×#102—24no—
98C2×#102—44no—
99C2×#102—44no—
100C2×C22⋊C8—21no—
101C2×C22⋊C8—21no—
102C2×C4×C8—21no—
103C2×C4×C8C2312yes—
104#391—24no—
105#391—22no—
106#389—24no—
107#389—22no—
108#389—22no—
109#389—22no—
110#387—24no—
111#387—24no—
112#387—24no—
113#385—24no—
114#384—22no—
115#384D411yescharacteristic
116#384D411yescharacteristic
117#382—22no—
118#382—21no—
119#380—21no—
120#380D412yes—
121C4×C4⋊C4—22no—
122C4×C4⋊C4—22no—
123C4×C4⋊C4—24no—
124C4×C4⋊C4—22no—
125C4×C4⋊C4—22no—
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#333—24no—
136#332—24no—
137C25—41no—
138C22×Q8—21no—
139C22×Q8C2×D412yes—
140C22×D4—24no—
141C22×D4—24no—
142C22×D4—24no—
143C22×D4—22no—
144C22×D4—22no—
145C22×D4—22no—
146C22×D4—41no—
147C22×D4—41no—
148C22×D4—41no—
149C22×D4—42no—
150C22×D4—42no—
151C22×D4—42no—
152C22×D4—42no—
153C22×D4—42no—
154C22×D4—42no—
155C22×D4—42no—
156C22×D4—82no—
157C22×D4—82no—
158C22×D4—82no—
159C22×D4—82no—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C2×SD16—216no—
166C2×SD16—84no—
167C2×SD16—84no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—42no—
172C22×C8—42no—
173C22×C8—42no—
174C4⋊Q8—21no—
175C4⋊Q8—21no—
176C4⋊Q8—21no—
177C4⋊Q8—21no—
178C4⋊Q8C22×C412yes—
179C4⋊Q8C22×C412yes—
180C4⋊Q8C2×D412yes—
181C4⋊Q8C2×D412yes—
182#127—24no—
183#127—24no—
184#127—24no—
185#127—22no—
186#127—22no—
187#127—22no—
188#127—42no—
189#127—42no—
190#127—42no—
191#127C22×C414yes—
192#125—24no—
193#125—24no—
194C22⋊Q8—24no—
195C22⋊Q8—24no—
196C22⋊Q8—24no—
197C22⋊Q8C22⋊C418yes—
198C4×Q8—24no—
199C4×Q8—24no—
200C4×Q8—24no—
201C4×Q8—84no—
202C4×Q8C22⋊C418yes—
203#117—24no—
204#117—24no—
205#117—42no—
206#117—42no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4SD1614yescomplemented
224C2×C4⋊C4C2×D412yes—
225C2×C4⋊C4C2×D412yes—
226C2×C4⋊C4C2×D412yes—
227C2×C4⋊C4C2×D412yes—
228C2×C4⋊C4C2×D412yes—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—24no—
235C2×C42—24no—
236C2×C42—22no—
237C2×C42—82no—
238C2×C42C22×C412yes—
239C2×C42C2×D411yescharacteristic
240C2×C42C2×D412yes—
241C2×C42C2×D411yescharacteristic
242C2×C42C2×D411yescharacteristic
243C2×C42#4112yes—
244C2×C42#4112yes—
245C2×C42C2411yescharacteristic
246C4⋊C8—24no—
247C4⋊C8—22no—
248C4⋊C8—22no—
249C4⋊C8—22no—
250C4⋊C8—22no—
251C4⋊C8—42no—
252#102—216no—
253#102—48no—
254#102—48no—
255#102—84no—
256#102—84no—
257C22⋊C8—24no—
258C22⋊C8—42no—
259C4×C8—22no—
260C4×C8—42no—
261C4×C8C22×C414yes—
262#95—24no—
263#95—24no—
264#95—24no—
265#95—24no—
266#95—22no—
267#95—22no—
268#95—22no—
269#95—22no—
270C24—42no—
271C24—44no—
272C24—82no—
273C24—82no—
274C24—82no—
275C24—82no—
276C24—82no—
277C24—82no—
278C24#12011yescharacteristic
279C2×Q8—22no—
280C2×Q8—22no—
281C2×Q8—22no—
282C2×Q8—42no—
283C2×Q8—42no—
284C2×Q8—42no—
285C2×Q8—42no—
286C2×Q8C2×C22⋊C414yes—
287C2×D4—24no—
288C2×D4—24no—
289C2×D4—24no—
290C2×D4—44no—
291C2×D4—44no—
292C2×D4—42no—
293C2×D4—42no—
294C2×D4—42no—
295C2×D4—42no—
296C2×D4—42no—
297C2×D4—42no—
298C2×D4—42no—
299C2×D4—44no—
300C2×D4—44no—
301C2×D4—44no—
302C2×D4—44no—
303C2×D4—44no—
304C2×D4—48no—
305C2×D4—48no—
306C2×D4—88no—
307C2×D4—82no—
308C2×D4—82no—
309C2×D4—82no—
310C2×D4—82no—
311C2×D4—88no—
312C2×D4—88no—
313C2×D4—88no—
314C2×D4—82no—
315C2×D4—82no—
316C2×D4—84no—
317C2×D4—84no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—24no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—42no—
330C22×C4—81no—
331C22×C4—81no—
332C22×C4—81no—
333C22×C4—81no—
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
350SD16—816no—
351C2×C8—24no—
352C2×C8—42no—
353C2×C8—42no—
354C2×C8—42no—
355C2×C8—42no—
356C2×C8—42no—
357C2×C8—42no—
358C2×C8—42no—
359C2×C8—42no—
360C2×C8—84no—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—28no—
370C4⋊C4—28no—
371C4⋊C4—24no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—24no—
375C4⋊C4—24no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—81no—
380C4⋊C4—81no—
381C4⋊C4—84no—
382C4⋊C4—84no—
383C4⋊C4C2×C22⋊C414yes—
384C4⋊C4C2×SD1614yes—
385C22⋊C4—28no—
386C22⋊C4—44no—
387C42—28no—
388C42—42no—
389C42—42no—
390C42—44no—
391C42—82no—
392C42—84no—
393C42—84no—
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—44no—
409C23—88no—
410C23—88no—
411C23—88no—
412C23—82no—
413C23—82no—
414C23—82no—
415C23—82no—
416C23—82no—
417C23—82no—
418C23—164no—
419C23—164no—
420C23—164no—
421C23—164no—
422C23#38612yes—
423C23#44812yes—
424C23#44912yes—
425C23C2×#12011yescharacteristic
426Q8—44no—
427Q8—44no—
428Q8—44no—
429Q8—82no—
430D4—48no—
431D4—44no—
432D4—44no—
433D4—48no—
434D4—84no—
435D4—88no—
436D4—88no—
437D4—88no—
438D4—82no—
439D4—82no—
440D4—84no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—24no—
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—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—84no—
470C2×C4—84no—
471C2×C4#38612yes—
472C2×C4#38612yes—
473C2×C4#38612yes—
474C2×C4C4×SD1614yes—
475C2×C4#44014yes—
476C2×C4#44812yes—
477C2×C4#44912yes—
478C2×C4#46014yes—
479C2×C4#46414yes—
480C2×C4C22×C22⋊C411yescharacteristic
481C2×C4C2×C4×D411yescharacteristic
482C2×C4C2×C4×D411yescharacteristic
483C2×C4C2×#12111yescharacteristic
484C2×C4C2×#12111yescharacteristic
485C2×C4C2×#12311yescharacteristic
486C2×C4C22×SD1611yescharacteristic
487C2×C4C2×#13511yescharacteristic
488C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
489C2×C4C2×#13711yescharacteristic
490C2×C4#57612yes—
491C2×C4#57712yes—
492C8—44no—
493C8—82no—
494C8—82no—
495C22—22no—
496C22—22no—
497C22—22no—
498C22—22no—
499C22—22no—
500C22—22no—
501C22—84no—
502C22—84no—
503C22—84no—
504C22—1616no—
505C22—162no—
506C22—162no—
507C22—162no—
508C22—162no—
509C22#188014yes—
510C22C2×#38611yescharacteristic
511C22C2×#44811yescharacteristic
512C22C2×#44911yescharacteristic
513C22#300811yescharacteristic
514C22#301011yescharacteristic
515C22#302111yescharacteristic
516C22#302111yescharacteristic
517C4—22no—
518C4—21no—
519C4—21no—
520C4—22no—
521C4—44no—
522C4—44no—
523C4—44no—
524C4—82no—
525C4—82no—
526C4—82no—
527C4#188014yes—
528C4C2×C4×SD1611yescharacteristic
529C4C2×#44011yescharacteristic
530C4#294612yes—
531C4C2×#46011yescharacteristic
532C4C2×#46411yescharacteristic
533C4#304512yes—
534C2—24no—
535C2—164no—
536C2#1889411yescharacteristic
537C2#1889811yescharacteristic
538C2#1891411yescharacteristic
539C2#1891811yescharacteristic
540C2C2×#188011yescharacteristic
541C2#2093211yescharacteristic
542C2#2095611yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#127C22
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. #34213C2semidirect
5. C2×#1542C2semidirect
6. #9063C2semidirect
7. #9062C2semidirect
8. #7893C2semidirect
9. #7892C2semidirect
13. #3208C4semidirect
24. #2369C22semidirect
25. #2365C22semidirect
42. C2×#333C22semidirect
43. C2×#332C22semidirect
44. #1279C22semidirect
45. #1279C22semidirect
78. #492C2×C4semidirect
80. #486C2×C4semidirect
223. C2×C4⋊C4SD16semidirect

Maximal subgroups

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

subgroupindex
C2×#32082
#342132
#218032
C2×#19822
C2×#15422
#90632
#90622
#78932
#78922