Finite Group Databasebeta

← #192834

Subgroups — #192834

Order 512.

1196 conjugacy classes of subgroups.

The subgroup lattice has 545 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34207C211yesmaximal, characteristic
2#21982C211yesmaximal, characteristic, complemented
3#21869C211yesmaximal, characteristic
4#21364C211yesmaximal, characteristic, complemented
5C2×#1496C211yesmaximal, characteristic
6#8893C212yesmaximal, complemented
7#8890C212yesmaximal, complemented
8#8885C212yesmaximal, complemented
9#8882C212yesmaximal, complemented
10#8040C211yesmaximal, characteristic, complemented
11#7902C211yesmaximal, characteristic, complemented
12C2×#532C2211yescharacteristic
13C2×#520—21no—
14C2×#432C2211yescharacteristic
15C2×#432C2211yescharacteristic
16C2×#431C2211yescharacteristic
17#2361—22no—
18#2361—22no—
19#2354—21no—
20#2354C2212yes—
21D4×C4⋊C4—21no—
22D4×C4⋊C4C2212yes—
23#2326—22no—
24#2311—21no—
25C4×C4⋊Q8—21no—
26C4×#127—21no—
27#2285C2211yescharacteristic
28C2×#384C2211yescharacteristic
29C4×#117C2211yescharacteristic
30#1993—21no—
31#1961—21no—
32#1948C2211yescharacteristic
33#1948C2211yescharacteristic
34#1947—22no—
35#1947C2211yescharacteristic
36#1947C2211yescharacteristic
37#1875—22no—
38#1854—21no—
39#1854—21no—
40#1852—22no—
41#1845C2212yes—
42#1834C2212yes—
43#1802C2212yes—
44#1802C2212yes—
45#1776—21no—
46#1776—21no—
47#1496C418yescomplemented
48C2×D4⋊C8C2212yes—
49C2×D4⋊C8C2212yes—
50#1296C2212yes—
51#1296C2212yes—
52#1279C2212yes—
53#1279C2212yes—
54C22×#41—21no—
55#532—22no—
56#532—22no—
57#532C2×C414yes—
58#532Q814yes—
59C2×C4⋊Q8D411yescharacteristic
60C2×#127D411yescharacteristic
61C2×#124—21no—
62C2×#121—21no—
63C2×#121D412yes—
64#520—44no—
65#520—44no—
66C2×C4×Q8—22no—
67C2×C4×D4—22no—
68C2×C4×D4—24no—
69C2×C4×D4—22no—
70C2×C4×D4—22no—
71C2×C4×D4—21no—
72C2×C4×D4C2312yes—
73C2×#117—22no—
74C2×#117—21no—
75C2×#117D411yescharacteristic
76C2×#117D411yescharacteristic
77C22×C4⋊C4—21no—
78C22×C4⋊C4—21no—
79C22×C4⋊C4—22no—
80C22×C4⋊C4D412yes—
81C22×C42C2311yescharacteristic
82#432—22no—
83#432—22no—
84#432—22no—
85#432C2×C414yescomplemented
86#431—22no—
87#431C2×C414yescomplemented
88C2×#107—21no—
89C2×#107—21no—
90C2×#107—24no—
91C2×#106—21no—
92C2×#106—21no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8C2312yes—
95C2×C4⋊C8C2311yescharacteristic
96C2×C4⋊C8C2311yescharacteristic
97C2×#102—22no—
98C2×#102—22no—
99C2×C22⋊C8—21no—
100C2×C22⋊C8—21no—
101C2×C22⋊C8—21no—
102C2×#97—21no—
103C2×#97—21no—
104C2×C4×C8C2312yes—
105#391—22no—
106#390—22no—
107#390—22no—
108#390—22no—
109#389—22no—
110#389—22no—
111#388—24no—
112#388—24no—
113#386—24no—
114#385—24no—
115#385—24no—
116#384—21no—
117#384—21no—
118#384D412yes—
119#384C2311yescharacteristic
120#384C2311yescharacteristic
121#384C2312yes—
122#382—21no—
123#380—22no—
124#380—22no—
125#380—21no—
126#380—21no—
127C4×C4⋊C4—22no—
128C4×C4⋊C4—22no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4D411yescharacteristic
132C4×C4⋊C4D411yescharacteristic
133C4×C4⋊C4C2311yescharacteristic
134C4×C4⋊C4C2311yescharacteristic
135C4×C22⋊C4—21no—
136C4×C22⋊C4—21no—
137#376—21no—
138C43—21no—
139#340—22no—
140#340—22no—
141#340—24no—
142#340—21no—
143#340—21no—
144#340—21no—
145#340—21no—
146D4⋊C8C2×C418yescomplemented
147D4⋊C8C2×C418yescomplemented
148C2×#41—24no—
149C2×#41—24no—
150C2×#41—42no—
151C2×#41—42no—
152C2×#41—42no—
153C2×#41—42no—
154C22×Q8—21no—
155C22×D4—22no—
156C22×D4—21no—
157C22×D4—21no—
158C22×D4C2×D412yes—
159C23×C4—22no—
160C23×C4—24no—
161C23×C4—22no—
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C22×C8—22no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—22no—
169C4⋊Q8—22no—
170C4⋊Q8C4⋊C414yes—
171#127—22no—
172#127C4⋊C414yes—
173#124—24no—
174#124—24no—
175#121—24no—
176#121—24no—
177#121—24no—
178#121C4⋊C418yes—
179C4×Q8—48no—
180C4×D4—24no—
181C4×D4—24no—
182C4×D4—22no—
183C4×D4—22no—
184C4×D4—48no—
185C4×D4—416no—
186C4×D4—82no—
187C4×D4—82no—
188C4×D4—84no—
189C4×D4C22×C414yes—
190C4×D4C2×Q814yes—
191#117—22no—
192#117—22no—
193#117—41no—
194#117—41no—
195#117—44no—
196#117—44no—
197#117—82no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—82no—
216C2×C4⋊C4—82no—
217C2×C4⋊C4C2×D411yescharacteristic
218C2×C4⋊C4C2×D411yescharacteristic
219C2×C4⋊C4C2×D412yes—
220C2×C4⋊C4C2×D411yescharacteristic
221C2×C4⋊C4C2×D411yescharacteristic
222C2×C4⋊C4C2×D414yes—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—41no—
231C2×C42—21no—
232C2×C42—24no—
233C2×C42—21no—
234C2×C42—21no—
235C2×C42—21no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—41no—
239C2×C42C22×C412yes—
240C2×C42C2×D411yescharacteristic
241C2×C42C2×D411yescharacteristic
242C2×C42C2×D411yescharacteristic
243C2×C42C2×Q812yes—
244C2×C42#4112yes—
245C2×C42#4112yes—
246C2×C42C2411yescharacteristic
247#107—42no—
248#107—42no—
249#107—84no—
250#106—42no—
251#106—42no—
252C4⋊C8—24no—
253C4⋊C8—21no—
254C4⋊C8—21no—
255C4⋊C8—22no—
256C4⋊C8—22no—
257C4⋊C8C22×C412yes—
258C4⋊C8C22×C412yes—
259#102—82no—
260#102—82no—
261C22⋊C8—24no—
262C22⋊C8—24no—
263C22⋊C8—42no—
264#97—24no—
265#97—42no—
266C4×C8—22no—
267C4×C8C22×C414yes—
268#95—24no—
269#95—24no—
270#95—21no—
271#95—21no—
272#95—21no—
273#95—21no—
274#95—41no—
275#95—41no—
276C24—22no—
277C24#12011yescharacteristic
278#41—48no—
279#41—48no—
280#41—48no—
281#41—48no—
282C2×Q8—24no—
283C2×Q8—42no—
284C2×Q8—42no—
285C2×D4—28no—
286C2×D4—22no—
287C2×D4—22no—
288C2×D4—22no—
289C2×D4—22no—
290C2×D4—22no—
291C2×D4—24no—
292C2×D4—24no—
293C2×D4—42no—
294C2×D4—42no—
295C2×D4—42no—
296C2×D4—42no—
297C2×D4—44no—
298C2×D4—44no—
299C2×D4C2×C4⋊C414yes—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—24no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—48no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—44no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—82no—
325C22×C4—82no—
326C22×C4—84no—
327C22×C4—84no—
328C22×C4C2×C4⋊C412yes—
329C22×C4C4×D412yes—
330C22×C4C4×D412yes—
331C22×C4#12011yescharacteristic
332C22×C4#12011yescharacteristic
333C22×C4#12011yescharacteristic
334C22×C4C22⋊Q812yes—
335C22×C4C22⋊Q812yes—
336C22×C4#12312yes—
337C22×C4C22×D411yescharacteristic
338C22×C4C22×D411yescharacteristic
339C22×C4C22×D411yescharacteristic
340C2×C8—24no—
341C2×C8—22no—
342C2×C8—22no—
343C2×C8—42no—
344C2×C8—42no—
345C2×C8—41no—
346C2×C8—41no—
347C2×C8—41no—
348C2×C8—41no—
349C2×C8—42no—
350C2×C8—42no—
351C2×C8—42no—
352C4⋊C4—22no—
353C4⋊C4—21no—
354C4⋊C4—21no—
355C4⋊C4—24no—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—21no—
359C4⋊C4—21no—
360C4⋊C4—44no—
361C4⋊C4—44no—
362C4⋊C4—48no—
363C4⋊C4—44no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—82no—
377C4⋊C4C2×C4⋊C414yes—
378C4⋊C4AGL(1,ℤ/8)12yes—
379C4⋊C4AGL(1,ℤ/8)12yes—
380C22⋊C4—24no—
381C22⋊C4—24no—
382C22⋊C4—24no—
383C22⋊C4—28no—
384C22⋊C4—48no—
385C22⋊C4—42no—
386C22⋊C4—84no—
387C22⋊C4—84no—
388C22⋊C4—82no—
389C42—24no—
390C42—42no—
391C42—42no—
392C42—48no—
393C42—42no—
394C42—42no—
395C42—42no—
396C42—82no—
397C42C2×C4⋊C412yes—
398C42C4×D412yes—
399C42C4×D412yes—
400C42C22⋊Q812yes—
401C42C22⋊Q812yes—
402C42#12312yes—
403C42C23×C411yescharacteristic
404C42C22×Q811yescharacteristic
405C42C2×#4111yescharacteristic
406C42C2×#4111yescharacteristic
407C23—21no—
408C23—21no—
409C23—21no—
410C23—21no—
411C23—21no—
412C23—21no—
413C23—24no—
414C23—42no—
415C23—42no—
416C23—42no—
417C23—42no—
418C23—42no—
419C23—42no—
420C23#38512yes—
421C23C2×#12011yescharacteristic
422Q8—48no—
423D4—24no—
424D4—24no—
425D4—48no—
426D4—48no—
427D4—416no—
428D4—42no—
429D4—42no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—24no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—44no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—44no—
467C2×C4—41no—
468C2×C4—81no—
469C2×C4—81no—
470C2×C4—81no—
471C2×C4—81no—
472C2×C4—81no—
473C2×C4—81no—
474C2×C4—168no—
475C2×C4—164no—
476C2×C4#38512yes—
477C2×C4#38512yes—
478C2×C4#38512yes—
479C2×C4C22×C4⋊C411yescharacteristic
480C2×C4C2×C4×D411yescharacteristic
481C2×C4C2×C4×D411yescharacteristic
482C2×C4C2×C22⋊Q811yescharacteristic
483C2×C4C2×C22⋊Q811yescharacteristic
484C2×C4C2×#12311yescharacteristic
485C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
486C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
487C2×C4#57511yescharacteristic
488C2×C4#57511yescharacteristic
489C2×C4#57612yes—
490C2×C4#57812yes—
491C8—44no—
492C8—42no—
493C8—42no—
494C8—82no—
495C22—22no—
496C22—22no—
497C22—22no—
498C22—41no—
499C22—41no—
500C22—41no—
501C22—41no—
502C22—44no—
503C22—44no—
504C22—44no—
505C22—82no—
506C22—82no—
507C22—82no—
508C22—82no—
509C22C2×#38511yescharacteristic
510C22#300311yescharacteristic
511C22#300511yescharacteristic
512C22#300811yescharacteristic
513C22#301111yescharacteristic
514C22#301911yescharacteristic
515C22#302011yescharacteristic
516C4—22no—
517C4—21no—
518C4—21no—
519C4—22no—
520C4—22no—
521C4—42no—
522C4—44no—
523C4—42no—
524C4—84no—
525C4—81no—
526C4—81no—
527C4—161no—
528C4—161no—
529C4#291312yes—
530C4C4×AGL(1,ℤ/8)11yescharacteristic
531C4C4×AGL(1,ℤ/8)11yescharacteristic
532C4#307711yescharacteristic
533C4#307711yescharacteristic
534C4#309812yes—
535C2—42no—
536C2—84no—
537C2#1881411yescharacteristic
538C2#1882111yescharacteristic
539C2#1882711yescharacteristic
540C2#1883211yescharacteristic
541C2#2084711yescharacteristic
542C2#2086411yescharacteristic
543C2#2087311yescharacteristic
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
2. #21982C2semidirect
4. #21364C2semidirect
6. #8893C2semidirect
7. #8890C2semidirect
8. #8885C2semidirect
9. #8882C2semidirect
10. #8040C2semidirect
11. #7902C2semidirect
47. #1496C4semidirect
85. #432C2×C4semidirect
87. #431C2×C4semidirect
146. D4⋊C8C2×C4semidirect
147. D4⋊C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
#342072
#219822
#218692
#213642
C2×#14962
#88932
#88902
#88852
#88822
#80402
#79022