Finite Group Databasebeta

← #192659

Subgroups — #192659

Order 512.

1108 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34188C211yesmaximal, characteristic
2#21982C211yesmaximal, characteristic, complemented
3C2×#1834C211yesmaximal, characteristic, complemented
4#20648C211yesmaximal, characteristic
5C2×#1476C211yesmaximal, characteristic
6#8897C214yesmaximal, complemented
7#8895C214yesmaximal, complemented
8#8043C211yesmaximal, characteristic, complemented
9#7897C211yesmaximal, characteristic, complemented
10C2×C4×#41C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C2×#423C2211yescharacteristic
13C2×#417—21no—
14C22×C4×C8C2211yescharacteristic
15#2355—21no—
16#2355C2212yes—
17#2354—21no—
18#2354C2212yes—
19#2352—22no—
20#2352—22no—
21#2314—22no—
22#2311—21no—
23#2285—21no—
24#2285—21no—
25#2285C2211yescharacteristic
26C2×#384C2211yescharacteristic
27C2×#384C2211yescharacteristic
28#2274—21no—
29#1947C2214yes—
30#1852—22no—
31#1845—21no—
32#1834—22no—
33#1834—21no—
34#1834C2214yes—
35#1803—21no—
36#1803C2212yes—
37#1802—21no—
38#1802C2212yes—
39#1736—21no—
40C2×#340—22no—
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—
75#423—22no—
76#423—22no—
77C2×C4⋊C8—21no—
78C2×C4⋊C8C2312yes—
79C2×C4⋊C8C2312yes—
80#417—44no—
81C2×#103—22no—
82C2×#102—22no—
83C2×C22⋊C8—21no—
84C2×C4×C8—21no—
85C2×C4×C8—21no—
86C2×C4×C8—22no—
87C2×C4×C8C2×C414yes—
88C2×C4×C8C2312yes—
89#391—22no—
90#390—22no—
91#388—24no—
92#388—24no—
93#387—24no—
94#387—24no—
95#386—24no—
96#384—21no—
97#384—21no—
98#384—21no—
99#384—21no—
100#384—21no—
101#384D412yes—
102#384D411yescharacteristic
103#384D411yescharacteristic
104#384C2312yes—
105#384C2312yes—
106#382—22no—
107#382—22no—
108#382—21no—
109#380—22no—
110#380—22no—
111#380—22no—
112#380—21no—
113#380—22no—
114#380D412yes—
115C4×C4⋊C4—22no—
116C4×C4⋊C4—22no—
117C4×C4⋊C4—22no—
118C4×C4⋊C4—22no—
119C4×C4⋊C4—22no—
120C4×C4⋊C4—22no—
121C4×C4⋊C4—21no—
122C4×C4⋊C4C2312yes—
123C4×C22⋊C4—24no—
124#340—22no—
125#340—22no—
126#340—28no—
127#340—24no—
128Q8⋊C8C2×C418yescomplemented
129D4⋊C8C2×C418yescomplemented
130C2×#41—22no—
131C2×#41—22no—
132C2×#41—22no—
133C2×#41—22no—
134C2×#41—22no—
135C2×#41C4⋊C414yes—
136C22×Q8C2×D411yescharacteristic
137C22×D4—21no—
138C22×D4C2×D411yescharacteristic
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—22no—
143C23×C4—41no—
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C2×M16—42no—
148C2×M16—42no—
149C2×M16—42no—
150C22×C8—24no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—42no—
155C22×C8—42no—
156C4×Q8—22no—
157C4×Q8—21no—
158C4×Q8—21no—
159C4×Q8C4⋊C414yes—
160C4×Q8C22×C412yes—
161C4×Q8C2×Q812yes—
162C4×D4—24no—
163C4×D4—24no—
164C4×D4—24no—
165C4×D4—24no—
166C4×D4—22no—
167C4×D4—21no—
168C4×D4—21no—
169C4×D4C4⋊C414yes—
170C4×D4C22×C412yes—
171C4×D4C2×Q812yes—
172#117—22no—
173#117—24no—
174#117—24no—
175#117—42no—
176#117—42no—
177#117—84no—
178#117—84no—
179#117C4⋊C414yes—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—41no—
201C2×C4⋊C4—41no—
202C2×C4⋊C4—41no—
203C2×C4⋊C4—82no—
204C2×C4⋊C4—82no—
205C2×C4⋊C4C2×D412yes—
206C2×C4⋊C4C2×D412yes—
207C2×C4⋊C4C2×D411yescharacteristic
208C2×C4⋊C4C2×D411yescharacteristic
209C2×C4⋊C4C2×D414yes—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—21no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—24no—
216C2×C42—21no—
217C2×C42—21no—
218C2×C42—21no—
219C2×C42—24no—
220C2×C42—24no—
221C2×C42—22no—
222C2×C42—24no—
223C2×C42—24no—
224C2×C42C22×C412yes—
225C2×C42C2×D411yescharacteristic
226C2×C42C2×D411yescharacteristic
227C2×C42C2×D411yescharacteristic
228C2×C42C2×Q812yes—
229C2×C42#4112yes—
230C2×C42#4112yes—
231C2×C42C2411yescharacteristic
232#107—84no—
233C4⋊C8—22no—
234C4⋊C8—22no—
235C4⋊C8—22no—
236C4⋊C8—42no—
237C4⋊C8C22×C414yes—
238#103—44no—
239#102—44no—
240C22⋊C8—24no—
241C4×C8—24no—
242C4×C8—24no—
243C4×C8—22no—
244C4×C8C22×C414yes—
245#95—24no—
246#95—24no—
247#95—24no—
248#95—24no—
249#95—22no—
250#95—22no—
251C24—21no—
252C24#12011yescharacteristic
253#41—28no—
254#41—28no—
255#41—44no—
256#41—44no—
257C2×Q8—21no—
258C2×Q8—21no—
259C2×Q8—21no—
260C2×Q8—21no—
261C2×Q8—21no—
262C2×Q8C2×C4⋊C412yes—
263C2×D4—24no—
264C2×D4—21no—
265C2×D4—21no—
266C2×D4—21no—
267C2×D4—21no—
268C2×D4—21no—
269C2×D4—42no—
270C2×D4—42no—
271C2×D4C2×C4⋊C412yes—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—24no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—44no—
293C22×C4—42no—
294C22×C4—82no—
295C22×C4—82no—
296C22×C4—82no—
297C22×C4—82no—
298C22×C4—82no—
299C22×C4C2×C4⋊C412yes—
300C22×C4C4×D412yes—
301C22×C4C4×D412yes—
302C22×C4#12011yescharacteristic
303C22×C4#12011yescharacteristic
304C22×C4#12011yescharacteristic
305C22×C4C22⋊Q812yes—
306C22×C4C22⋊Q812yes—
307C22×C4#12312yes—
308C22×C4#13512yes—
309C22×C4#13512yes—
310C22×C4AGL(1,ℤ/8)12yes—
311C22×C4#13712yes—
312C22×C4C22×D411yescharacteristic
313C22×C4C22×D411yescharacteristic
314C22×C4C22×D411yescharacteristic
315M16—84no—
316C2×C8—24no—
317C2×C8—24no—
318C2×C8—42no—
319C2×C8—42no—
320C2×C8—48no—
321C2×C8—42no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—42no—
325C2×C8—42no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—21no—
329C4⋊C4—21no—
330C4⋊C4—24no—
331C4⋊C4—24no—
332C4⋊C4—24no—
333C4⋊C4—44no—
334C4⋊C4—44no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—44no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—44no—
341C4⋊C4—42no—
342C4⋊C4—44no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—44no—
347C4⋊C4—82no—
348C4⋊C4—82no—
349C4⋊C4—82no—
350C4⋊C4—82no—
351C4⋊C4C2×C4⋊C412yes—
352C4⋊C4C2×C4⋊C412yes—
353C22⋊C4—24no—
354C22⋊C4—22no—
355C22⋊C4—22no—
356C22⋊C4—24no—
357C22⋊C4—44no—
358C22⋊C4—84no—
359C22⋊C4—84no—
360C42—22no—
361C42—22no—
362C42—24no—
363C42—24no—
364C42—42no—
365C42—42no—
366C42—84no—
367C42—84no—
368C42C2×C4⋊C412yes—
369C42C4×D412yes—
370C42C4×D412yes—
371C42C22⋊Q812yes—
372C42C22⋊Q812yes—
373C42#12312yes—
374C42C23×C411yescharacteristic
375C42C22×Q811yescharacteristic
376C42C2×#4111yescharacteristic
377C42C2×#4111yescharacteristic
378C23—21no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23—22no—
383C23—41no—
384C23—41no—
385C23—41no—
386C23—41no—
387C23—41no—
388C23—41no—
389C23#38512yes—
390C23#44912yes—
391C23#45212yes—
392C23C2×#12011yescharacteristic
393Q8—22no—
394Q8—22no—
395Q8—41no—
396Q8—41no—
397D4—22no—
398D4—22no—
399D4—48no—
400D4—41no—
401D4—41no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—24no—
407C2×C4—24no—
408C2×C4—24no—
409C2×C4—24no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—44no—
420C2×C4—48no—
421C2×C4—48no—
422C2×C4—44no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—44no—
426C2×C4—42no—
427C2×C4—44no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—44no—
431C2×C4—42no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4—82no—
435C2×C4—164no—
436C2×C4—164no—
437C2×C4#38512yes—
438C2×C4#38512yes—
439C2×C4#38512yes—
440C2×C4#42714yes—
441C2×C4#42814yes—
442C2×C4#44912yes—
443C2×C4#45212yes—
444C2×C4#45412yes—
445C2×C4#45512yes—
446C2×C4#48214yes—
447C2×C4#48514yes—
448C2×C4C22×C4⋊C411yescharacteristic
449C2×C4C2×C4×D411yescharacteristic
450C2×C4C2×C4×D411yescharacteristic
451C2×C4C2×C22⋊Q811yescharacteristic
452C2×C4C2×C22⋊Q811yescharacteristic
453C2×C4C2×#12311yescharacteristic
454C2×C4C2×#13511yescharacteristic
455C2×C4C2×#13511yescharacteristic
456C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
457C2×C4C2×#13711yescharacteristic
458C2×C4#57612yes—
459C2×C4#57812yes—
460C8—44no—
461C8—44no—
462C8—82no—
463C22—22no—
464C22—22no—
465C22—22no—
466C22—22no—
467C22—22no—
468C22—22no—
469C22—42no—
470C22—42no—
471C22—42no—
472C22—81no—
473C22—81no—
474C22—81no—
475C22—81no—
476C22#185414yes—
477C22C2×#38511yescharacteristic
478C22C2×#44911yescharacteristic
479C22C2×#45211yescharacteristic
480C22#300811yescharacteristic
481C22#301111yescharacteristic
482C22C2×#45511yescharacteristic
483C22C2×#45411yescharacteristic
484C4—21no—
485C4—21no—
486C4—22no—
487C4—42no—
488C4—42no—
489C4—42no—
490C4—84no—
491C4—82no—
492C4—82no—
493C4—162no—
494C4#185414yes—
495C4#186014yes—
496C4C2×#42711yescharacteristic
497C4C2×#42811yescharacteristic
498C4#295112yes—
499C4#308312yes—
500C4C2×#48211yescharacteristic
501C4C2×#48511yescharacteristic
502C2—24no—
503C2—82no—
504C2#1863511yescharacteristic
505C2#1863611yescharacteristic
506C2#1864511yescharacteristic
507C2#1864611yescharacteristic
508C2C2×#185411yescharacteristic
509C2C2×#186011yescharacteristic
510C2#2088111yescharacteristic
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×#1834C2semidirect
6. #8897C2semidirect
7. #8895C2semidirect
8. #8043C2semidirect
9. #7897C2semidirect
41. #1476C4semidirect
128. Q8⋊C8C2×C4semidirect
129. 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×#18342
#206482
C2×#14762
#88972
#88952
#80432
#78972