Finite Group Databasebeta

← #188422

Subgroups — #188422

Order 512.

1234 conjugacy classes of subgroups.

The subgroup lattice has 561 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34141C211yesmaximal, characteristic
2#21350C211yesmaximal, characteristic, complemented
3#20815C211yesmaximal, characteristic, complemented
4C2×#1496C211yesmaximal, characteristic
5C2×#1484C211yesmaximal, characteristic
6#8569C212yesmaximal, complemented
7#8567C212yesmaximal, complemented
8#8565C212yesmaximal, complemented
9#8563C212yesmaximal, complemented
10#7904C211yesmaximal, characteristic, complemented
11#7892C211yesmaximal, characteristic, complemented
12C2×#532C2211yescharacteristic
13C2×C4×#41—21no—
14C2×#432C2211yescharacteristic
15C2×#431C2211yescharacteristic
16C2×#431C2211yescharacteristic
17C2×C4×M16C2211yescharacteristic
18#2367—22no—
19#2366C412yes—
20#2365C412yes—
21#2364—22no—
22#2361C412yes—
23#2361C412yes—
24#2297—21no—
25#2293C2211yescharacteristic
26#2293C2211yescharacteristic
27#2290—21no—
28#2290—21no—
29C42×D4C2211yescharacteristic
30C42×D4C2211yescharacteristic
31C4×#117C2211yescharacteristic
32C2×C4×C4⋊C4C2211yescharacteristic
33#1940—22no—
34#1851—22no—
35#1842C2212yes—
36#1831C2212yes—
37#1798—21no—
38#1798—21no—
39#1768—21no—
40#1768—21no—
41C4×C4⋊C8C2212yes—
42C4×C4⋊C8C2212yes—
43#1496—24no—
44#1484—24no—
45C2×D4⋊C8C2212yes—
46C2×D4⋊C8C2212yes—
47C2×D4⋊C8C2212yes—
48C2×D4⋊C8C2212yes—
49#1279C2212yes—
50#1279C2212yes—
51#1279C2212yes—
52#1279C2212yes—
53C22×#41—21no—
54C22×M16—21no—
55#532—22no—
56#532—22no—
57#532D414yes—
58#532Q814yes—
59C2×C4⋊Q8C2×C411yescharacteristic
60C2×#127C2×C411yescharacteristic
61C2×#124—21no—
62C2×#121—21no—
63C2×#121C2×C411yescharacteristic
64C2×#121C2×C411yescharacteristic
65C4×#41—48no—
66C2×C4×Q8—22no—
67C2×C4×D4—22no—
68C2×C4×D4—22no—
69C2×C4×D4—22no—
70C2×C4×D4—21no—
71C2×C4×D4C2×C412yes—
72C2×C4×D4C2×C412yes—
73C2×C4×D4C2311yescharacteristic
74C2×C4×D4C2311yescharacteristic
75C2×#117—21no—
76C2×#117—21no—
77C2×#117D411yescharacteristic
78C2×#117D411yescharacteristic
79C22×C4⋊C4C2×C412yes—
80C22×C42—21no—
81C22×C42—21no—
82C22×C42—21no—
83C22×C42C2311yescharacteristic
84#432—22no—
85#432—22no—
86#431—22no—
87#431—22no—
88#431—22no—
89#431—22no—
90C2×C4⋊C8—22no—
91C2×C4⋊C8—22no—
92C2×C4⋊C8—22no—
93C2×C4⋊C8—22no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8C2312yes—
96C2×C4⋊C8C2312yes—
97C2×C22⋊C8—21no—
98C2×C22⋊C8—21no—
99C2×C22⋊C8—21no—
100C4×M16—22no—
101C4×M16—22no—
102C2×#97—21no—
103C2×#97—21no—
104C2×C4×C8—21no—
105C2×C4×C8C2312yes—
106C2×C4×C8C2312yes—
107#391—22no—
108#390—22no—
109#390C2×C412yes—
110#390C2×C412yes—
111#389C2×C412yes—
112#389C2×C412yes—
113#388—22no—
114#388—22no—
115#388—22no—
116#388—22no—
117#387—22no—
118#387—22no—
119#386—22no—
120#386—22no—
121#386—22no—
122#386—22no—
123#385—22no—
124#385—22no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4C2×C412yes—
128C4×C4⋊C4C2×C412yes—
129C4×C4⋊C4C2×C412yes—
130C4×C4⋊C4C2×C412yes—
131C4×C4⋊C4D412yes—
132C4×C4⋊C4C2311yescharacteristic
133C4×C4⋊C4C2311yescharacteristic
134C4×C22⋊C4—22no—
135C4×C22⋊C4—22no—
136C4×C22⋊C4—21no—
137C4×C22⋊C4—21no—
138C4×C22⋊C4—21no—
139C4×C22⋊C4—21no—
140C4×C22⋊C4C2×C412yes—
141C4×C22⋊C4D412yes—
142#376D411yescharacteristic
143#376D411yescharacteristic
144C2×#95—21no—
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—
155C2×#41—24no—
156C2×#41—24no—
157C2×#41—44no—
158C2×#41—44no—
159C22×Q8—21no—
160C22×D4—21no—
161C22×D4—21no—
162C22×D4—21no—
163C22×D4—21no—
164C22×D4C22×C411yescharacteristic
165C22×D4C22×C411yescharacteristic
166C23×C4—21no—
167C23×C4—21no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—22no—
171C23×C4C22×C411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C2×M16—44no—
175C2×M16—42no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C4⋊Q8—22no—
181C4⋊Q8C4⋊C414yes—
182#127—22no—
183#127C4⋊C414yes—
184#124—24no—
185#124—24no—
186#121—22no—
187#121—22no—
188#121—24no—
189#121—24no—
190#121C4⋊C414yes—
191#121C4⋊C414yes—
192C4×Q8—48no—
193C4×D4—24no—
194C4×D4—24no—
195C4×D4—24no—
196C4×D4—24no—
197C4×D4—21no—
198C4×D4—21no—
199C4×D4—21no—
200C4×D4—21no—
201C4×D4—42no—
202C4×D4—42no—
203C4×D4—48no—
204C4×D4—48no—
205C4×D4—48no—
206C4×D4C2×D412yes—
207C4×D4C2×D412yes—
208C4×D4C2×Q812yes—
209C4×D4C2×Q812yes—
210#117—24no—
211#117—42no—
212#117—44no—
213#117—44no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4C22⋊C412yes—
221C2×C4⋊C4C22⋊C412yes—
222C2×C4⋊C4C22×C412yes—
223C2×C4⋊C4C22×C411yescharacteristic
224C2×C4⋊C4C22×C411yescharacteristic
225C2×C4⋊C4C22×C412yes—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—21no—
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⋊C4C22⋊C412yes—
237C2×C22⋊C4C22⋊C412yes—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—21no—
242C2×C42—21no—
243C2×C42—44no—
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—
294C24C2×C22⋊C411yescharacteristic
295#41—416no—
296#41—416no—
297C2×Q8—24no—
298C2×Q8—42no—
299C2×Q8—42no—
300C2×D4—24no—
301C2×D4—24no—
302C2×D4—21no—
303C2×D4—21no—
304C2×D4—22no—
305C2×D4—22no—
306C2×D4—22no—
307C2×D4—22no—
308C2×D4—24no—
309C2×D4—24no—
310C2×D4—42no—
311C2×D4—42no—
312C2×D4—42no—
313C2×D4—42no—
314C2×D4—42no—
315C2×D4—42no—
316C2×D4—82no—
317C2×D4C2×C4⋊C412yes—
318C2×D4C2×C4⋊C412yes—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—42no—
332C22×C4—44no—
333C22×C4—44no—
334C22×C4—44no—
335C22×C4—44no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—44no—
341C22×C4—44no—
342C22×C4—82no—
343C22×C4—82no—
344C22×C4—82no—
345C22×C4C2×C22⋊C412yes—
346C22×C4C2×C22⋊C412yes—
347C22×C4C2×C22⋊C412yes—
348C22×C4C2×C22⋊C411yescharacteristic
349C22×C4C2×C22⋊C411yescharacteristic
350C22×C4C2×C22⋊C411yescharacteristic
351C22×C4C2×C22⋊C412yes—
352C22×C4C2×C4⋊C412yes—
353C22×C4#11712yes—
354C22×C4#12112yes—
355C22×C4#12112yes—
356C22×C4C22⋊Q812yes—
357C22×C4C22⋊Q812yes—
358C22×C4C23×C411yescharacteristic
359C22×C4C22×D411yescharacteristic
360C22×C4C22×D411yescharacteristic
361M16—48no—
362C2×C8—24no—
363C2×C8—42no—
364C2×C8—42no—
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—
374C4⋊C4—21no—
375C4⋊C4—21no—
376C4⋊C4—22no—
377C4⋊C4—24no—
378C4⋊C4—24no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—42no—
382C4⋊C4—44no—
383C4⋊C4—44no—
384C4⋊C4—44no—
385C4⋊C4C2×C4⋊C412yes—
386C4⋊C4C2×C4⋊C412yes—
387C4⋊C4#13114yes—
388C22⋊C4—22no—
389C22⋊C4—22no—
390C22⋊C4—22no—
391C22⋊C4—22no—
392C22⋊C4—24no—
393C22⋊C4—24no—
394C22⋊C4—24no—
395C22⋊C4—42no—
396C22⋊C4—44no—
397C22⋊C4—44no—
398C22⋊C4—44no—
399C22⋊C4—44no—
400C22⋊C4—82no—
401C22⋊C4—82no—
402C42—22no—
403C42—22no—
404C42—22no—
405C42—22no—
406C42—22no—
407C42—22no—
408C42—22no—
409C42—42no—
410C42—42no—
411C42—44no—
412C42—44no—
413C42—42no—
414C42—42no—
415C42—44no—
416C42C2×C4⋊C412yes—
417C42#11712yes—
418C42#12112yes—
419C42#12112yes—
420C42C22⋊Q812yes—
421C42C22⋊Q812yes—
422C42AGL(1,ℤ/8)14yes—
423C42C22×D411yescharacteristic
424C42C22×Q811yescharacteristic
425C42C2×#4111yescharacteristic
426C42C2×#4111yescharacteristic
427C23—22no—
428C23—21no—
429C23—21no—
430C23—22no—
431C23—22no—
432C23—22no—
433C23—42no—
434C23—42no—
435C23—41no—
436C23—41no—
437C23—41no—
438C23—41no—
439C23—42no—
440C23—42no—
441C23#38012yes—
442C23C22×C22⋊C411yescharacteristic
443Q8—48no—
444D4—24no—
445D4—24no—
446D4—48no—
447D4—48no—
448D4—48no—
449D4—41no—
450D4—41no—
451D4—41no—
452D4—41no—
453D4—84no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—24no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
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—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—42no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—88no—
494C2×C4—88no—
495C2×C4—88no—
496C2×C4#38012yes—
497C2×C4#38012yes—
498C2×C4#38012yes—
499C2×C4#41814yes—
500C2×C4C22×C4⋊C411yescharacteristic
501C2×C4C2×#11711yescharacteristic
502C2×C4C2×#12111yescharacteristic
503C2×C4C2×#12111yescharacteristic
504C2×C4C2×C22⋊Q811yescharacteristic
505C2×C4C2×C22⋊Q811yescharacteristic
506C2×C4C2×#13112yes—
507C2×C4#56812yes—
508C2×C4C2×AGL(1,ℤ/8)12yes—
509C2×C4#57512yes—
510C8—44no—
511C8—82no—
512C8—82no—
513C8—82no—
514C22—22no—
515C22—22no—
516C22—22no—
517C22—42no—
518C22—42no—
519C22—42no—
520C22—42no—
521C22—42no—
522C22—42no—
523C22—42no—
524C22—42no—
525C22—82no—
526C22—82no—
527C22—82no—
528C22—82no—
529C22C2×#38011yescharacteristic
530C22#288012yes—
531C22C2×#41811yescharacteristic
532C22#289611yescharacteristic
533C22#290112yes—
534C4—24no—
535C4—22no—
536C4—22no—
537C4—21no—
538C4—21no—
539C4—22no—
540C4—42no—
541C4—44no—
542C4—42no—
543C4—42no—
544C4—42no—
545C4—42no—
546C4—82no—
547C4#290612yes—
548C4#292211yescharacteristic
549C4#292311yescharacteristic
550C4#303612yes—
551C4#307712yes—
552C2—42no—
553C2—82no—
554C2—82no—
555C2#1751112yes—
556C2#1751612yes—
557C2#2063011yescharacteristic
558C2#2064611yescharacteristic
559C2#2065511yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC2×#532C22
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. #7904C2semidirect
11. #7892C2semidirect

Maximal subgroups

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

subgroupindex
#341412
#213502
#208152
C2×#14962
C2×#14842
#85692
#85672
#85652
#85632
#79042
#78922