Finite Group Databasebeta

← #192651

Subgroups — #192651

Order 512.

1318 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34183C211yesmaximal, characteristic
2#21981C211yesmaximal, characteristic, complemented
3C2×#1844C211yesmaximal, characteristic, complemented
4#20648C211yesmaximal, characteristic
5C2×#1476C211yesmaximal, characteristic
6#8888C214yesmaximal, complemented
7#8886C214yesmaximal, complemented
8#8043C211yesmaximal, characteristic, complemented
9#7897C211yesmaximal, characteristic, complemented
10C2×#520—21no—
11C2×C4×#41C2211yescharacteristic
12C2×#431C2211yescharacteristic
13C22×#106—22no—
14C2×#423C2211yescharacteristic
15C2×#417—21no—
16C22×C4×C8C2211yescharacteristic
17Q8×C4⋊C4—21no—
18Q8×C4⋊C4C2212yes—
19D4×C4⋊C4—22no—
20D4×C4⋊C4—21no—
21D4×C4⋊C4—22no—
22D4×C4⋊C4C2212yes—
23#2317—22no—
24#2317—21no—
25#2285—21no—
26#2285C2211yescharacteristic
27C2×#384—21no—
28C2×#384C2211yescharacteristic
29C2×#384C2211yescharacteristic
30#1948C2214yes—
31#1853—22no—
32#1844—22no—
33#1844—21no—
34#1844C2214yes—
35#1834—21no—
36#1803—21no—
37#1803C2212yes—
38#1802—21no—
39#1802C2212yes—
40#1736—21no—
41#1476C418yescomplemented
42C2×Q8⋊C8C2212yes—
43C2×D4⋊C8C2212yes—
44#1296C2214yes—
45#1279C2214yes—
46C22×#41D411yescharacteristic
47C22×M16—21no—
48C23×C8—21no—
49#520—44no—
50#520—44no—
51C4×#41—22no—
52C4×#41—22no—
53C4×#41C2×C414yes—
54C4×#41Q814yes—
55C2×C4×Q8—22no—
56C2×C4×Q8D411yescharacteristic
57C2×C4×Q8C2311yescharacteristic
58C2×C4×D4—21no—
59C2×C4×D4—21no—
60C2×C4×D4—24no—
61C2×C4×D4—22no—
62C2×C4×D4D411yescharacteristic
63C2×C4×D4C2311yescharacteristic
64C2×#117—22no—
65C2×#117—21no—
66C2×#117—21no—
67C2×#117D411yescharacteristic
68C22×C4⋊C4—22no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—22no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—22no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4D412yes—
77C22×C42—21no—
78C22×C42C2311yescharacteristic
79#431—22no—
80#431C2×C414yescomplemented
81C2×#106—28no—
82C2×#106—24no—
83C2×#106—48no—
84#423—22no—
85#423—22no—
86C2×C4⋊C8—21no—
87C2×C4⋊C8C2312yes—
88C2×C4⋊C8C2312yes—
89#417—44no—
90C2×#103—22no—
91C2×#102—22no—
92C2×C22⋊C8—21no—
93C2×C4×C8—21no—
94C2×C4×C8—21no—
95C2×C4×C8—22no—
96C2×C4×C8C2×C414yescomplemented
97C2×C4×C8C2312yes—
98#389—24no—
99#389—22no—
100#389—24no—
101#389—22no—
102#385—24no—
103#385—24no—
104#385—24no—
105#384—22no—
106#384—22no—
107#384—22no—
108#384—21no—
109#384—21no—
110#384—21no—
111#384—21no—
112#384—21no—
113#384D412yes—
114#384D411yescharacteristic
115#384D411yescharacteristic
116#384C2312yes—
117#384C2312yes—
118#382—22no—
119#382—21no—
120#380—22no—
121#380—21no—
122#380—22no—
123#380D412yes—
124C4×C4⋊C4—22no—
125C4×C4⋊C4—22no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4C2312yes—
128#340—22no—
129#340—22no—
130#340—24no—
131Q8⋊C8C2×C418yescomplemented
132D4⋊C8C2×C418yescomplemented
133C2×#41—22no—
134C2×#41—22no—
135C2×#41—22no—
136C2×#41C4⋊C414yes—
137C2×#41SD1614yes—
138C2×#41SD1614yes—
139C22×Q8C2×D411yescharacteristic
140C22×D4—21no—
141C22×D4C2×D411yescharacteristic
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—24no—
146C23×C4—22no—
147C23×C4—41no—
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C2×M16—42no—
152C2×M16—42no—
153C2×M16—42no—
154C22×C8—24no—
155C22×C8—24no—
156C22×C8—24no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C4×Q8—22no—
161C4×Q8—21no—
162C4×Q8—21no—
163C4×Q8—44no—
164C4×Q8—44no—
165C4×Q8C4⋊C414yes—
166C4×Q8C22×C412yes—
167C4×Q8C2×Q812yes—
168C4×D4—24no—
169C4×D4—24no—
170C4×D4—24no—
171C4×D4—24no—
172C4×D4—22no—
173C4×D4—21no—
174C4×D4—21no—
175C4×D4—416no—
176C4×D4—44no—
177C4×D4—44no—
178C4×D4C4⋊C414yes—
179C4×D4C22×C412yes—
180C4×D4C2×Q812yes—
181#117—22no—
182#117—24no—
183#117—24no—
184#117—48no—
185#117—42no—
186#117—42no—
187#117C4⋊C414yes—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—48no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4—84no—
216C2×C4⋊C4—82no—
217C2×C4⋊C4—82no—
218C2×C4⋊C4—84no—
219C2×C4⋊C4C2×D412yes—
220C2×C4⋊C4C2×D412yes—
221C2×C4⋊C4C2×D411yescharacteristic
222C2×C4⋊C4C2×D411yescharacteristic
223C2×C4⋊C4C2×D414yes—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—22no—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—24no—
233C2×C42—24no—
234C2×C42—24no—
235C2×C42—22no—
236C2×C42C22×C412yes—
237C2×C42C2×D411yescharacteristic
238C2×C42C2×D411yescharacteristic
239C2×C42C2×D411yescharacteristic
240C2×C42C2×Q812yes—
241C2×C42#4112yes—
242C2×C42#4112yes—
243C2×C42C2411yescharacteristic
244#106—416no—
245#106—84no—
246C4⋊C8—22no—
247C4⋊C8—22no—
248C4⋊C8—22no—
249C4⋊C8—42no—
250C4⋊C8C22×C414yes—
251#103—44no—
252#102—44no—
253C22⋊C8—24no—
254C4×C8—24no—
255C4×C8—24no—
256C4×C8—22no—
257C4×C8C22×C414yes—
258#95—24no—
259#95—24no—
260#95—22no—
261#95—22no—
262C24—21no—
263C24#12011yescharacteristic
264#41—28no—
265#41—28no—
266#41—28no—
267#41#106116yescomplemented
268C2×Q8—21no—
269C2×Q8—21no—
270C2×Q8—21no—
271C2×Q8C2×C4⋊C412yes—
272C2×Q8C2×SD1612yes—
273C2×Q8C2×SD1612yes—
274C2×D4—24no—
275C2×D4—24no—
276C2×D4—24no—
277C2×D4—21no—
278C2×D4—21no—
279C2×D4—21no—
280C2×D4C2×C4⋊C412yes—
281C2×D4C2×SD1612yes—
282C2×D4C2×SD1612yes—
283C22×C4—21no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—24no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—24no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—48no—
304C22×C4—44no—
305C22×C4—42no—
306C22×C4—82no—
307C22×C4—82no—
308C22×C4—84no—
309C22×C4—84no—
310C22×C4—82no—
311C22×C4—82no—
312C22×C4—82no—
313C22×C4C2×C4⋊C412yes—
314C22×C4C4×D412yes—
315C22×C4C4×D412yes—
316C22×C4#12011yescharacteristic
317C22×C4#12011yescharacteristic
318C22×C4#12011yescharacteristic
319C22×C4C22⋊Q812yes—
320C22×C4C22⋊Q812yes—
321C22×C4#12312yes—
322C22×C4C2×SD1612yes—
323C22×C4C2×SD1612yes—
324C22×C4AGL(1,ℤ/8)12yes—
325C22×C4#13712yes—
326C22×C4C22×D411yescharacteristic
327C22×C4C22×D411yescharacteristic
328C22×C4C22×D411yescharacteristic
329M16—84no—
330C2×C8—24no—
331C2×C8—24no—
332C2×C8—216no—
333C2×C8—24no—
334C2×C8—24no—
335C2×C8—42no—
336C2×C8—42no—
337C2×C8—42no—
338C2×C8—42no—
339C2×C8—42no—
340C4⋊C4—22no—
341C4⋊C4—22no—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—24no—
345C4⋊C4—24no—
346C4⋊C4—24no—
347C4⋊C4—44no—
348C4⋊C4—48no—
349C4⋊C4—44no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—44no—
353C4⋊C4—42no—
354C4⋊C4—44no—
355C4⋊C4—42no—
356C4⋊C4—44no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—44no—
362C4⋊C4—82no—
363C4⋊C4—82no—
364C4⋊C4—84no—
365C4⋊C4—84no—
366C4⋊C4—82no—
367C4⋊C4—82no—
368C4⋊C4C2×C4⋊C412yes—
369C4⋊C4C2×C4⋊C412yes—
370C22⋊C4—24no—
371C22⋊C4—22no—
372C22⋊C4—22no—
373C22⋊C4—24no—
374C22⋊C4—48no—
375C22⋊C4—44no—
376C42—22no—
377C42—22no—
378C42—24no—
379C42—24no—
380C42—48no—
381C42—42no—
382C42—42no—
383C42C2×C4⋊C412yes—
384C42C4×D412yes—
385C42C4×D412yes—
386C42C22⋊Q812yes—
387C42C22⋊Q812yes—
388C42#12312yes—
389C42C23×C411yescharacteristic
390C42C22×Q811yescharacteristic
391C42C2×#4111yescharacteristic
392C42C2×#4111yescharacteristic
393C23—21no—
394C23—21no—
395C23—21no—
396C23—21no—
397C23—22no—
398C23—22no—
399C23—22no—
400C23—41no—
401C23—41no—
402C23—41no—
403C23—41no—
404C23#38512yes—
405C23#44812yes—
406C23#45012yes—
407C23C2×#12011yescharacteristic
408Q8—22no—
409Q8—22no—
410Q8—22no—
411Q8C2×#10614yescomplemented
412D4—216no—
413D4—22no—
414D4—22no—
415D4—22no—
416D4C2×#10614yescomplemented
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—22no—
424C2×C4—24no—
425C2×C4—216no—
426C2×C4—24no—
427C2×C4—24no—
428C2×C4—24no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—44no—
436C2×C4—48no—
437C2×C4—44no—
438C2×C4—42no—
439C2×C4—44no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—44no—
443C2×C4—42no—
444C2×C4—44no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4—164no—
451C2×C4—168no—
452C2×C4—164no—
453C2×C4#38512yes—
454C2×C4#38512yes—
455C2×C4#38512yes—
456C2×C4C2×#10614yes—
457C2×C4#42814yes—
458C2×C4#44812yes—
459C2×C4#45012yes—
460C2×C4#45312yes—
461C2×C4#45612yes—
462C2×C4#48114yes—
463C2×C4#48314yes—
464C2×C4C22×C4⋊C411yescharacteristic
465C2×C4C2×C4×D411yescharacteristic
466C2×C4C2×C4×D411yescharacteristic
467C2×C4C2×C22⋊Q811yescharacteristic
468C2×C4C2×C22⋊Q811yescharacteristic
469C2×C4C2×#12311yescharacteristic
470C2×C4C22×SD1611yescharacteristic
471C2×C4C22×SD1611yescharacteristic
472C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
473C2×C4C2×#13711yescharacteristic
474C2×C4#57712yes—
475C2×C4#57712yes—
476C8—28no—
477C8—44no—
478C8—82no—
479C22—22no—
480C22—22no—
481C22—22no—
482C22—22no—
483C22—22no—
484C22—22no—
485C22—24no—
486C22—42no—
487C22—42no—
488C22—42no—
489C22—42no—
490C22—42no—
491C22—42no—
492C22#185314yes—
493C22C2×#38511yescharacteristic
494C22C2×#45011yescharacteristic
495C22C2×#44811yescharacteristic
496C22#300911yescharacteristic
497C22#301011yescharacteristic
498C22C2×#45311yescharacteristic
499C22C2×#45611yescharacteristic
500C4—24no—
501C4—21no—
502C4—21no—
503C4—22no—
504C4—42no—
505C4—42no—
506C4—82no—
507C4—84no—
508C4—82no—
509C4—162no—
510C4#185314yes—
511C4#185914yes—
512C4C22×#10611yescharacteristic
513C4C2×#42811yescharacteristic
514C4#295212yes—
515C4#308412yes—
516C4C2×#48311yescharacteristic
517C4C2×#48111yescharacteristic
518C2—24no—
519C2—44no—
520C2#1862111yescharacteristic
521C2#1862211yescharacteristic
522C2#1865111yescharacteristic
523C2#1865211yescharacteristic
524C2C2×#185311yescharacteristic
525C2C2×#185911yescharacteristic
526C2#2088211yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#41D4
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. #21981C2semidirect
3. C2×#1844C2semidirect
6. #8888C2semidirect
7. #8886C2semidirect
8. #8043C2semidirect
9. #7897C2semidirect
41. #1476C4semidirect
80. #431C2×C4semidirect
96. C2×C4×C8C2×C4semidirect
131. Q8⋊C8C2×C4semidirect
132. D4⋊C8C2×C4semidirect
267. #41#106semidirect
411. Q8C2×#106semidirect
416. D4C2×#106semidirect

Maximal subgroups

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

subgroupindex
#341832
#219812
C2×#18442
#206482
C2×#14762
#88882
#88862
#80432
#78972