Finite Group Databasebeta

← #188402

Subgroups — #188402

Order 512.

1304 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D4×#117C211yesmaximal, characteristic
2#21350C211yesmaximal, characteristic, complemented
3#20815C211yesmaximal, characteristic, complemented
4C2×#1490C211yesmaximal, characteristic
5C2×#1478C211yesmaximal, characteristic
6#8569C212yesmaximal, complemented
7#8567C212yesmaximal, complemented
8#8565C212yesmaximal, complemented
9#8563C212yesmaximal, complemented
10#7900C211yesmaximal, characteristic, complemented
11#7896C211yesmaximal, characteristic, complemented
12C22×C4×D4C2211yescharacteristic
13C22×#117—21no—
14C2×#432C2211yescharacteristic
15C2×#431C2211yescharacteristic
16C2×#431C2211yescharacteristic
17C2×C4×M16C2211yescharacteristic
18#2354C412yes—
19#2352—22no—
20D4×C4⋊C4C412yes—
21#2344C412yes—
22#2342—22no—
23D4×C22⋊C4C412yes—
24#2293C2211yescharacteristic
25#2293C2211yescharacteristic
26#2290—21no—
27#2290—21no—
28#2285C2211yescharacteristic
29#2280—21no—
30C42×D4C2211yescharacteristic
31C42×D4C2211yescharacteristic
32C4×#117C2211yescharacteristic
33#1940—22no—
34#1851—22no—
35#1842C2212yescomplemented
36#1831C2212yescomplemented
37#1798—21no—
38#1798—21no—
39#1768—21no—
40#1768—21no—
41C4×C4⋊C8C2212yescomplemented
42C4×C4⋊C8C2212yescomplemented
43#1490—24no—
44#1478—24no—
45C2×D4⋊C8C2212yes—
46C2×D4⋊C8C2212yes—
47C2×D4⋊C8C2212yes—
48C2×D4⋊C8C2212yes—
49#1279C2212yescomplemented
50#1279C2212yescomplemented
51#1279C2212yescomplemented
52#1279C2212yescomplemented
53C23×D4C2×C411yescharacteristic
54C24×C4—21no—
55C22×M16—21no—
56C2×C4×D4—21no—
57C2×C4×D4—21no—
58C2×C4×D4—22no—
59C2×C4×D4—22no—
60C2×C4×D4C2×C412yes—
61C2×C4×D4C2×C412yes—
62C2×C4×D4C2×C411yescharacteristic
63C2×C4×D4C2×C411yescharacteristic
64C2×C4×D4D414yes—
65C2×C4×D4Q814yes—
66C2×C4×D4C2311yescharacteristic
67C2×C4×D4C2311yescharacteristic
68C2×#117—21no—
69C2×#117—22no—
70C2×#117—22no—
71C2×#117—21no—
72C2×#117—21no—
73C2×#117—48no—
74C2×#117D411yescharacteristic
75C2×#117D411yescharacteristic
76C22×C4⋊C4—22no—
77C22×C4⋊C4C2×C411yescharacteristic
78C22×C22⋊C4—21no—
79C22×C22⋊C4—22no—
80C22×C42—21no—
81C22×C42—21no—
82C22×C42C2311yescharacteristic
83#432—22no—
84#432—22no—
85#431—22no—
86#431—22no—
87#431—22no—
88#431—22no—
89C2×C4⋊C8—22no—
90C2×C4⋊C8—22no—
91C2×C4⋊C8—22no—
92C2×C4⋊C8—22no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8C2312yes—
95C2×C4⋊C8C2312yes—
96C2×C22⋊C8—21no—
97C2×C22⋊C8—21no—
98C2×C22⋊C8—21no—
99C4×M16—22no—
100C4×M16—22no—
101C2×#97—21no—
102C2×#97—21no—
103C2×C4×C8—21no—
104C2×C4×C8C2312yes—
105C2×C4×C8C2312yes—
106#390C2×C412yes—
107#390C2×C412yes—
108#389C2×C412yes—
109#389C2×C412yes—
110#388—22no—
111#388—22no—
112#388—22no—
113#388—22no—
114#387—22no—
115#387—22no—
116#386—22no—
117#386—22no—
118#385—22no—
119#385—22no—
120#384C2×C412yes—
121#382—21no—
122#381—22no—
123#380—21no—
124#380—22no—
125#380—22no—
126#380C2×C412yes—
127#379—22no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4C2×C412yes—
130C4×C4⋊C4C2×C412yes—
131C4×C4⋊C4C2×C412yes—
132C4×C4⋊C4D412yes—
133C4×C4⋊C4C2311yescharacteristic
134C4×C4⋊C4C2311yescharacteristic
135C4×C22⋊C4—22no—
136C4×C22⋊C4—22no—
137C4×C22⋊C4—21no—
138C4×C22⋊C4—21no—
139C4×C22⋊C4—21no—
140C4×C22⋊C4—21no—
141C4×C22⋊C4C2×C412yes—
142C4×C22⋊C4D412yes—
143#376D411yescharacteristic
144#376D411yescharacteristic
145C43C2311yescharacteristic
146C43C2311yescharacteristic
147#336—22no—
148#336—22no—
149#336—22no—
150#336—22no—
151D4⋊C8—24no—
152D4⋊C8—24no—
153D4⋊C8—24no—
154D4⋊C8—24no—
155C25—21no—
156C22×D4—24no—
157C22×D4—21no—
158C22×D4—22no—
159C22×D4—42no—
160C22×D4C4⋊C414yes—
161C22×D4C22×C411yescharacteristic
162C22×D4C22×C411yescharacteristic
163C23×C4—21no—
164C23×C4—21no—
165C23×C4—21no—
166C23×C4—24no—
167C23×C4—24no—
168C23×C4—21no—
169C23×C4—21no—
170C23×C4—22no—
171C23×C4—22no—
172C23×C4—44no—
173C23×C4—44no—
174C23×C4C22×C411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D411yescharacteristic
177C2×M16—44no—
178C2×M16—42no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C4×D4—24no—
184C4×D4—24no—
185C4×D4—22no—
186C4×D4—22no—
187C4×D4—24no—
188C4×D4—24no—
189C4×D4—24no—
190C4×D4—24no—
191C4×D4—21no—
192C4×D4—21no—
193C4×D4—21no—
194C4×D4—21no—
195C4×D4—42no—
196C4×D4—42no—
197C4×D4C4⋊C414yes—
198C4×D4C4⋊C414yes—
199C4×D4C2×D412yes—
200C4×D4C2×D412yes—
201C4×D4C2×Q812yes—
202C4×D4C2×Q812yes—
203#117—24no—
204#117—42no—
205#117—44no—
206#117—48no—
207#117—48no—
208#117—44no—
209#117—44no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—48no—
217C2×C4⋊C4C22⋊C412yes—
218C2×C4⋊C4C22⋊C412yes—
219C2×C4⋊C4C4⋊C414yes—
220C2×C4⋊C4C22×C412yes—
221C2×C4⋊C4C22×C412yes—
222C2×C4⋊C4C22×C411yescharacteristic
223C2×C4⋊C4C22×C411yescharacteristic
224C2×C22⋊C4—21no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—48no—
238C2×C22⋊C4C22⋊C412yes—
239C2×C22⋊C4C22⋊C412yes—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—21no—
243C2×C42—21no—
244C2×C42—44no—
245C2×C42—44no—
246C2×C42C22×C412yes—
247C2×C42C22×C412yes—
248C2×C42C22×C411yescharacteristic
249C2×C42C2×D411yescharacteristic
250C2×C42C2×D411yescharacteristic
251C2×C42C2×D411yescharacteristic
252C2×C42C2×D412yes—
253C2×C42C2×D412yes—
254C2×C42C2×D412yes—
255C2×C42C2×D411yescharacteristic
256C2×C42C2×D411yescharacteristic
257C2×C42C2×D411yescharacteristic
258C2×C42C2×Q812yes—
259C2×C42#4112yes—
260C2×C42#4112yes—
261C2×C42C2411yescharacteristic
262C4⋊C8—22no—
263C4⋊C8—22no—
264C4⋊C8—22no—
265C4⋊C8—22no—
266C4⋊C8—44no—
267C4⋊C8—44no—
268C4⋊C8—44no—
269C4⋊C8—42no—
270C4⋊C8—44no—
271C22⋊C8—42no—
272C22⋊C8—42no—
273C22⋊C8—42no—
274#97—42no—
275#97—42no—
276C4×C8—22no—
277C4×C8—22no—
278C4×C8—22no—
279C4×C8—22no—
280C4×C8—42no—
281#95—22no—
282#95—22no—
283#95—22no—
284#95—22no—
285#95—22no—
286#95—21no—
287#95—21no—
288#95C22⋊C412yes—
289#95C22⋊C412yes—
290#95C22⋊C412yes—
291#95C22⋊C412yes—
292C24—21no—
293C24—21no—
294C24—24no—
295C24—44no—
296C24—42no—
297C24—42no—
298C24—82no—
299C24C2×C22⋊C411yescharacteristic
300C2×D4—28no—
301C2×D4—21no—
302C2×D4—21no—
303C2×D4—22no—
304C2×D4—22no—
305C2×D4—22no—
306C2×D4—24no—
307C2×D4—44no—
308C2×D4—48no—
309C2×D4—44no—
310C2×D4C2×C4⋊C412yes—
311C2×D4C2×C4⋊C412yes—
312C2×D4#13114yes—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—24no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—416no—
334C22×C4—416no—
335C22×C4—44no—
336C22×C4—44no—
337C22×C4—44no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—44no—
345C22×C4—82no—
346C22×C4—82no—
347C22×C4C2×C22⋊C412yes—
348C22×C4C2×C22⋊C412yes—
349C22×C4C2×C22⋊C412yes—
350C22×C4C2×C22⋊C411yescharacteristic
351C22×C4C2×C22⋊C411yescharacteristic
352C22×C4C2×C22⋊C411yescharacteristic
353C22×C4C2×C22⋊C412yes—
354C22×C4C2×C4⋊C412yes—
355C22×C4#11712yes—
356C22×C4#12112yes—
357C22×C4#12112yes—
358C22×C4C22⋊Q812yes—
359C22×C4C22⋊Q812yes—
360C22×C4C23×C411yescharacteristic
361C22×C4C22×D411yescharacteristic
362C22×C4C22×D411yescharacteristic
363M16—48no—
364C2×C8—24no—
365C2×C8—42no—
366C2×C8—42no—
367C2×C8—42no—
368C2×C8—42no—
369C2×C8—42no—
370C2×C8—42no—
371C2×C8—42no—
372C2×C8—42no—
373C2×C8—42no—
374C2×C8—42no—
375C2×C8—42no—
376C4⋊C4—21no—
377C4⋊C4—21no—
378C4⋊C4—24no—
379C4⋊C4—24no—
380C4⋊C4—22no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—42no—
385C4⋊C4—44no—
386C4⋊C4—44no—
387C4⋊C4—44no—
388C4⋊C4—82no—
389C4⋊C4C2×C4⋊C412yes—
390C4⋊C4C2×C4⋊C412yes—
391C22⋊C4—22no—
392C22⋊C4—22no—
393C22⋊C4—22no—
394C22⋊C4—22no—
395C22⋊C4—24no—
396C22⋊C4—24no—
397C22⋊C4—24no—
398C22⋊C4—42no—
399C22⋊C4—44no—
400C22⋊C4—44no—
401C22⋊C4—44no—
402C22⋊C4—44no—
403C22⋊C4—44no—
404C22⋊C4—82no—
405C22⋊C4—82no—
406C42—22no—
407C42—22no—
408C42—22no—
409C42—22no—
410C42—22no—
411C42—22no—
412C42—22no—
413C42—42no—
414C42—42no—
415C42—44no—
416C42—42no—
417C42—42no—
418C42—44no—
419C42C2×C4⋊C412yes—
420C42#11712yes—
421C42#12112yes—
422C42#12112yes—
423C42C22⋊Q812yes—
424C42C22⋊Q812yes—
425C42AGL(1,ℤ/8)14yes—
426C42C22×D411yescharacteristic
427C42C22×Q811yescharacteristic
428C42C2×#4111yescharacteristic
429C42C2×#4111yescharacteristic
430C23—24no—
431C23—22no—
432C23—21no—
433C23—21no—
434C23—22no—
435C23—22no—
436C23—22no—
437C23—48no—
438C23—48no—
439C23—41no—
440C23—41no—
441C23—42no—
442C23—42no—
443C23—41no—
444C23—41no—
445C23—42no—
446C23—84no—
447C23—88no—
448C23—88no—
449C23#38012yes—
450C23C22×C22⋊C411yescharacteristic
451D4—22no—
452D4—22no—
453D4—22no—
454D4—24no—
455D4—48no—
456D4—42no—
457D4#42414yes—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—24no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—48no—
484C2×C4—48no—
485C2×C4—48no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—88no—
501C2×C4—88no—
502C2×C4#38012yes—
503C2×C4#38012yes—
504C2×C4#38012yes—
505C2×C4#41814yes—
506C2×C4C22×C4⋊C411yescharacteristic
507C2×C4C2×#11711yescharacteristic
508C2×C4C2×#12111yescharacteristic
509C2×C4C2×#12111yescharacteristic
510C2×C4C2×C22⋊Q811yescharacteristic
511C2×C4C2×C22⋊Q811yescharacteristic
512C2×C4C2×#13112yes—
513C2×C4#56812yes—
514C2×C4C2×AGL(1,ℤ/8)12yes—
515C2×C4#57512yes—
516C8—44no—
517C8—82no—
518C8—82no—
519C8—82no—
520C22—22no—
521C22—24no—
522C22—22no—
523C22—22no—
524C22—44no—
525C22—42no—
526C22—42no—
527C22—44no—
528C22—42no—
529C22—42no—
530C22—42no—
531C22—42no—
532C22—42no—
533C22—816no—
534C22—82no—
535C22—82no—
536C22C2×#38011yescharacteristic
537C22#288012yes—
538C22C2×#41811yescharacteristic
539C22#289611yescharacteristic
540C22#290112yes—
541C4—24no—
542C4—22no—
543C4—22no—
544C4—21no—
545C4—21no—
546C4—22no—
547C4—42no—
548C4—42no—
549C4—42no—
550C4—42no—
551C4—42no—
552C4—82no—
553C4—82no—
554C4C2×#42411yescharacteristic
555C4#290511yescharacteristic
556C4#292412yes—
557C4#303612yes—
558C4#307712yes—
559C2—42no—
560C2—44no—
561C2—82no—
562C2#1751112yes—
563C2#1751312yes—
564C2#2062811yescharacteristic
565C2#2064611yescharacteristic
566C2#2065511yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC23×D4C2×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #21350C2semidirect
3. #20815C2semidirect
6. #8569C2semidirect
7. #8567C2semidirect
8. #8565C2semidirect
9. #8563C2semidirect
10. #7900C2semidirect
11. #7896C2semidirect
35. #1842C22semidirect
36. #1831C22semidirect
41. C4×C4⋊C8C22semidirect
42. C4×C4⋊C8C22semidirect
49. #1279C22semidirect
50. #1279C22semidirect
51. #1279C22semidirect
52. #1279C22semidirect

Maximal subgroups

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

subgroupindex
D4×#1172
#213502
#208152
C2×#14902
C2×#14782
#85692
#85672
#85652
#85632
#79002
#78962