Finite Group Databasebeta

← #192653

Subgroups — #192653

Order 512.

1152 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34188C211yesmaximal, characteristic
2#21982C211yesmaximal, characteristic, complemented
3C2×#1844C211yesmaximal, characteristic, complemented
4#20648C211yesmaximal, characteristic
5C2×#1476C211yesmaximal, characteristic
6#8889C214yesmaximal, complemented
7#8887C214yesmaximal, complemented
8#8043C211yesmaximal, characteristic, complemented
9#7897C211yesmaximal, characteristic, complemented
10C2×C4×#41C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C22×#106—22no—
13C2×#423C2211yescharacteristic
14C2×#417—21no—
15C22×C4×C8C2211yescharacteristic
16#2355—21no—
17#2355C2212yes—
18#2354—21no—
19#2354C2212yes—
20#2352—22no—
21#2352—22no—
22#2314—22no—
23#2311—21no—
24#2285—21no—
25#2285—21no—
26#2285C2211yescharacteristic
27C2×#384C2211yescharacteristic
28C2×#384C2211yescharacteristic
29#2274—21no—
30#1947C2214yes—
31#1852—22no—
32#1845—21no—
33#1844—22no—
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—
49C4×#41—22no—
50C4×#41—22no—
51C4×#41C2×C414yes—
52C4×#41Q814yes—
53C2×C4×Q8D411yescharacteristic
54C2×C4×Q8C2311yescharacteristic
55C2×C4×D4—21no—
56C2×C4×D4—21no—
57C2×C4×D4D411yescharacteristic
58C2×C4×D4C2311yescharacteristic
59C2×#117—21no—
60C2×#117—22no—
61C2×#117—22no—
62C2×#117—21no—
63C2×#117D411yescharacteristic
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4D412yes—
70C22×C42—21no—
71C22×C42C2311yescharacteristic
72#431—22no—
73#431C2×C414yes—
74C2×#107—24no—
75C2×#106—28no—
76C2×#106—48no—
77#423—22no—
78#423—22no—
79C2×C4⋊C8—21no—
80C2×C4⋊C8C2312yes—
81C2×C4⋊C8C2312yes—
82#417—44no—
83C2×#103—22no—
84C2×#102—22no—
85C2×C22⋊C8—21no—
86C2×C4×C8—21no—
87C2×C4×C8—21no—
88C2×C4×C8—22no—
89C2×C4×C8C2×C414yes—
90C2×C4×C8C2312yes—
91#391—22no—
92#390—22no—
93#388—24no—
94#388—24no—
95#387—24no—
96#387—24no—
97#386—24no—
98#384—21no—
99#384—21no—
100#384—21no—
101#384—21no—
102#384—21no—
103#384D412yes—
104#384D411yescharacteristic
105#384D411yescharacteristic
106#384C2312yes—
107#384C2312yes—
108#382—22no—
109#382—22no—
110#382—21no—
111#380—22no—
112#380—22no—
113#380—22no—
114#380—21no—
115#380—22no—
116#380D412yes—
117C4×C4⋊C4—22no—
118C4×C4⋊C4—22no—
119C4×C4⋊C4—22no—
120C4×C4⋊C4—22no—
121C4×C4⋊C4—22no—
122C4×C4⋊C4—22no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4C2312yes—
125C4×C22⋊C4—24no—
126#340—22no—
127#340—22no—
128#340—24no—
129Q8⋊C8C2×C418yescomplemented
130D4⋊C8C2×C418yescomplemented
131C2×#41—22no—
132C2×#41—22no—
133C2×#41—22no—
134C2×#41—22no—
135C2×#41—22no—
136C2×#41C4⋊C414yes—
137C22×Q8C2×D411yescharacteristic
138C22×D4—21no—
139C22×D4C2×D411yescharacteristic
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—22no—
144C23×C4—41no—
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C2×M16—42no—
149C2×M16—42no—
150C2×M16—42no—
151C22×C8—24no—
152C22×C8—24no—
153C22×C8—24no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C4×Q8—22no—
158C4×Q8—21no—
159C4×Q8—21no—
160C4×Q8C4⋊C414yes—
161C4×Q8C22×C412yes—
162C4×Q8C2×Q812yes—
163C4×D4—24no—
164C4×D4—24no—
165C4×D4—24no—
166C4×D4—24no—
167C4×D4—22no—
168C4×D4—21no—
169C4×D4—21no—
170C4×D4C4⋊C414yes—
171C4×D4C22×C412yes—
172C4×D4C2×Q812yes—
173#117—22no—
174#117—24no—
175#117—24no—
176#117—42no—
177#117—42no—
178#117—84no—
179#117—84no—
180#117C4⋊C414yes—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—41no—
202C2×C4⋊C4—41no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4—82no—
205C2×C4⋊C4—82no—
206C2×C4⋊C4C2×D412yes—
207C2×C4⋊C4C2×D412yes—
208C2×C4⋊C4C2×D411yescharacteristic
209C2×C4⋊C4C2×D411yescharacteristic
210C2×C4⋊C4C2×D414yes—
211C2×C22⋊C4—21no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—24no—
217C2×C42—21no—
218C2×C42—21no—
219C2×C42—21no—
220C2×C42—24no—
221C2×C42—24no—
222C2×C42—22no—
223C2×C42—24no—
224C2×C42—24no—
225C2×C42C22×C412yes—
226C2×C42C2×D411yescharacteristic
227C2×C42C2×D411yescharacteristic
228C2×C42C2×D411yescharacteristic
229C2×C42C2×Q812yes—
230C2×C42#4112yes—
231C2×C42#4112yes—
232C2×C42C2411yescharacteristic
233#107—84no—
234#106—416no—
235C4⋊C8—22no—
236C4⋊C8—22no—
237C4⋊C8—22no—
238C4⋊C8—42no—
239C4⋊C8C22×C414yes—
240#103—44no—
241#102—44no—
242C22⋊C8—24no—
243C4×C8—24no—
244C4×C8—24no—
245C4×C8—22no—
246C4×C8C22×C414yes—
247#95—24no—
248#95—24no—
249#95—24no—
250#95—24no—
251#95—22no—
252#95—22no—
253C24—21no—
254C24#12011yescharacteristic
255#41—28no—
256#41—28no—
257#41—44no—
258#41—44no—
259C2×Q8—21no—
260C2×Q8—21no—
261C2×Q8—21no—
262C2×Q8—21no—
263C2×Q8—21no—
264C2×Q8C2×C4⋊C412yes—
265C2×D4—24no—
266C2×D4—21no—
267C2×D4—21no—
268C2×D4—21no—
269C2×D4—21no—
270C2×D4—21no—
271C2×D4—42no—
272C2×D4—42no—
273C2×D4C2×C4⋊C412yes—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—24no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—44no—
295C22×C4—42no—
296C22×C4—82no—
297C22×C4—82no—
298C22×C4—82no—
299C22×C4—82no—
300C22×C4—82no—
301C22×C4C2×C4⋊C412yes—
302C22×C4C4×D412yes—
303C22×C4C4×D412yes—
304C22×C4#12011yescharacteristic
305C22×C4#12011yescharacteristic
306C22×C4#12011yescharacteristic
307C22×C4C22⋊Q812yes—
308C22×C4C22⋊Q812yes—
309C22×C4#12312yes—
310C22×C4#13512yes—
311C22×C4#13512yes—
312C22×C4AGL(1,ℤ/8)12yes—
313C22×C4#13712yes—
314C22×C4C22×D411yescharacteristic
315C22×C4C22×D411yescharacteristic
316C22×C4C22×D411yescharacteristic
317M16—84no—
318C2×C8—24no—
319C2×C8—24no—
320C2×C8—216no—
321C2×C8—24no—
322C2×C8—24no—
323C2×C8—42no—
324C2×C8—42no—
325C2×C8—42no—
326C2×C8—42no—
327C2×C8—42no—
328C4⋊C4—22no—
329C4⋊C4—22no—
330C4⋊C4—21no—
331C4⋊C4—21no—
332C4⋊C4—24no—
333C4⋊C4—24no—
334C4⋊C4—24no—
335C4⋊C4—44no—
336C4⋊C4—44no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—44no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—44no—
343C4⋊C4—44no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—44no—
349C4⋊C4—82no—
350C4⋊C4—82no—
351C4⋊C4—82no—
352C4⋊C4—82no—
353C4⋊C4C2×C4⋊C412yes—
354C4⋊C4C2×C4⋊C412yes—
355C22⋊C4—24no—
356C22⋊C4—22no—
357C22⋊C4—22no—
358C22⋊C4—24no—
359C22⋊C4—44no—
360C22⋊C4—84no—
361C22⋊C4—84no—
362C42—22no—
363C42—22no—
364C42—24no—
365C42—24no—
366C42—42no—
367C42—42no—
368C42—84no—
369C42—84no—
370C42C2×C4⋊C412yes—
371C42C4×D412yes—
372C42C4×D412yes—
373C42C22⋊Q812yes—
374C42C22⋊Q812yes—
375C42#12312yes—
376C42C23×C411yescharacteristic
377C42C22×Q811yescharacteristic
378C42C2×#4111yescharacteristic
379C42C2×#4111yescharacteristic
380C23—21no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—22no—
385C23—41no—
386C23—41no—
387C23—41no—
388C23—41no—
389C23—41no—
390C23—41no—
391C23#38512yes—
392C23#44912yes—
393C23#45212yes—
394C23C2×#12011yescharacteristic
395Q8—22no—
396Q8—22no—
397Q8—41no—
398Q8—41no—
399D4—22no—
400D4—22no—
401D4—48no—
402D4—41no—
403D4—41no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—24no—
409C2×C4—24no—
410C2×C4—24no—
411C2×C4—24no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—44no—
422C2×C4—48no—
423C2×C4—48no—
424C2×C4—44no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—44no—
428C2×C4—42no—
429C2×C4—44no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—44no—
433C2×C4—42no—
434C2×C4—82no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4—164no—
438C2×C4—164no—
439C2×C4#38512yes—
440C2×C4#38512yes—
441C2×C4#38512yes—
442C2×C4#42714yes—
443C2×C4#42814yes—
444C2×C4#44912yes—
445C2×C4#45212yes—
446C2×C4#45312yes—
447C2×C4#45612yes—
448C2×C4#48214yes—
449C2×C4#48514yes—
450C2×C4C22×C4⋊C411yescharacteristic
451C2×C4C2×C4×D411yescharacteristic
452C2×C4C2×C4×D411yescharacteristic
453C2×C4C2×C22⋊Q811yescharacteristic
454C2×C4C2×C22⋊Q811yescharacteristic
455C2×C4C2×#12311yescharacteristic
456C2×C4C2×#13511yescharacteristic
457C2×C4C2×#13511yescharacteristic
458C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
459C2×C4C2×#13711yescharacteristic
460C2×C4#57712yes—
461C2×C4#57712yes—
462C8—28no—
463C8—44no—
464C8—82no—
465C22—22no—
466C22—22no—
467C22—22no—
468C22—22no—
469C22—22no—
470C22—22no—
471C22—42no—
472C22—42no—
473C22—42no—
474C22—81no—
475C22—81no—
476C22—81no—
477C22—81no—
478C22#185414yes—
479C22C2×#38511yescharacteristic
480C22C2×#44911yescharacteristic
481C22C2×#45211yescharacteristic
482C22#300911yescharacteristic
483C22#301011yescharacteristic
484C22C2×#45311yescharacteristic
485C22C2×#45611yescharacteristic
486C4—21no—
487C4—21no—
488C4—22no—
489C4—42no—
490C4—42no—
491C4—42no—
492C4—84no—
493C4—82no—
494C4—82no—
495C4—162no—
496C4#185414yes—
497C4#185914yes—
498C4C2×#42711yescharacteristic
499C4C2×#42811yescharacteristic
500C4#295212yes—
501C4#308412yes—
502C4C2×#48211yescharacteristic
503C4C2×#48511yescharacteristic
504C2—24no—
505C2—82no—
506C2#1862511yescharacteristic
507C2#1862611yescharacteristic
508C2#1865511yescharacteristic
509C2#1865611yescharacteristic
510C2C2×#185411yescharacteristic
511C2C2×#185911yescharacteristic
512C2#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. #21982C2semidirect
3. C2×#1844C2semidirect
6. #8889C2semidirect
7. #8887C2semidirect
8. #8043C2semidirect
9. #7897C2semidirect
41. #1476C4semidirect
129. Q8⋊C8C2×C4semidirect
130. D4⋊C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
#341882
#219822
C2×#18442
#206482
C2×#14762
#88892
#88872
#80432
#78972