Finite Group Databasebeta

← #192747

Subgroups — #192747

Order 512.

1598 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D4×C4⋊C4C211yesmaximal, characteristic
2#21981C211yesmaximal, characteristic, complemented
3#21170C211yesmaximal, characteristic, complemented
4#20732C211yesmaximal, characteristic
5C2×#1486C211yesmaximal, characteristic
6#8890C214yesmaximal, complemented
7#8886C214yesmaximal, complemented
8#8043C211yesmaximal, characteristic, complemented
9#7892C211yesmaximal, characteristic, complemented
10C22×C4×D4—21no—
11C22×C4×D4C2211yescharacteristic
12C23×C4⋊C4—21no—
13C2×#431C2211yescharacteristic
14C2×#431C2211yescharacteristic
15C2×#428—21no—
16C2×#423C2211yescharacteristic
17D4×C4⋊C4—22no—
18D4×C4⋊C4—22no—
19D4×C4⋊C4—22no—
20D4×C4⋊C4C2214yes—
21C2×#389—21no—
22C2×#385—22no—
23#2285C2211yescharacteristic
24C2×#384C2211yescharacteristic
25C2×#380—21no—
26C2×C4×C4⋊C4C2211yescharacteristic
27#1949—22no—
28#1948—22no—
29#1948C2212yescomplemented
30#1947C2212yescomplemented
31#1853—22no—
32#1844C2212yescomplemented
33#1834C2212yescomplemented
34#1823—21no—
35#1802C2212yes—
36#1802C2212yes—
37#1786—21no—
38C4×#102—22no—
39#1736—21no—
40#1486C418yescomplemented
41C2×D4⋊C8C2214yes—
42#1296C2214yescomplemented
43#1279C2214yescomplemented
44C23×D4D411yescharacteristic
45C24×C4—21no—
46C24×C4—22no—
47C22×M16—21no—
48C2×C4×D4—24no—
49C2×C4×D4—24no—
50C2×C4×D4—21no—
51C2×C4×D4—21no—
52C2×C4×D4—22no—
53C2×C4×D4—22no—
54C2×C4×D4—48no—
55C2×C4×D4C2×C414yes—
56C2×C4×D4D412yes—
57C2×C4×D4Q814yes—
58C2×C4×D4C2312yes—
59C2×#117—21no—
60C22×C4⋊C4—24no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—24no—
65C22×C4⋊C4—44no—
66C22×C4⋊C4—44no—
67C22×C4⋊C4D411yescharacteristic
68C22×C4⋊C4D411yescharacteristic
69C22×C4⋊C4D411yescharacteristic
70C22×C4⋊C4D412yes—
71C22×C22⋊C4—21no—
72C22×C22⋊C4—22no—
73C22×C42—21no—
74C22×C42C2311yescharacteristic
75#431—22no—
76#431—22no—
77#431—22no—
78#431C2×C414yescomplemented
79#428—44no—
80C2×#107—22no—
81C2×#106—22no—
82C2×#106—24no—
83#423—22no—
84#423C2×C414yescomplemented
85C2×C4⋊C8—21no—
86C2×C4⋊C8—21no—
87C2×C4⋊C8C2312yes—
88C2×C4⋊C8C2312yes—
89C2×#102—22no—
90C2×#102—22no—
91C2×C22⋊C8—21no—
92C2×C22⋊C8—21no—
93C2×C4×C8—21no—
94C2×C4×C8C2312yes—
95#389—24no—
96#389—24no—
97#385—28no—
98#385—28no—
99#384—21no—
100#384—21no—
101#384D412yes—
102#384C2311yescharacteristic
103#384C2311yescharacteristic
104#384C2312yes—
105#382—21no—
106#380—24no—
107#380—24no—
108#380—21no—
109#380—21no—
110C4×C4⋊C4—21no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4D412yes—
115C4×C4⋊C4C2312yes—
116C2×#95—22no—
117C2×#95—21no—
118#340—22no—
119#340—22no—
120#340—24no—
121#340—22no—
122#340—22no—
123D4⋊C8C2×C4116yescomplemented
124C25—21no—
125C22×D4—24no—
126C22×D4—24no—
127C22×D4—21no—
128C22×D4—22no—
129C22×D4C4⋊C414yes—
130C22×D4C2×D412yes—
131C23×C4—21no—
132C23×C4—21no—
133C23×C4—24no—
134C23×C4—22no—
135C23×C4—24no—
136C23×C4—22no—
137C23×C4—21no—
138C23×C4—22no—
139C23×C4—28no—
140C23×C4—48no—
141C23×C4—42no—
142C23×C4—42no—
143C23×C4—42no—
144C23×C4—42no—
145C23×C4—48no—
146C23×C4—84no—
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C2×M16—24no—
151C2×M16—42no—
152C2×M16—42no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C4×D4—24no—
157C4×D4—24no—
158C4×D4—24no—
159C4×D4—24no—
160C4×D4—24no—
161C4×D4—22no—
162C4×D4—22no—
163C4×D4—416no—
164C4×D4—48no—
165C4×D4—48no—
166C4×D4C4⋊C418yes—
167C4×D4C22×C414yes—
168C4×D4C2×Q814yes—
169#117—42no—
170#117—42no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—28no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—48no—
191C2×C4⋊C4—416no—
192C2×C4⋊C4—48no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—48no—
197C2×C4⋊C4—82no—
198C2×C4⋊C4C4⋊C414yes—
199C2×C4⋊C4SD1614yes—
200C2×C4⋊C4C2×D412yes—
201C2×C4⋊C4C2×D412yes—
202C2×C4⋊C4C2×D412yes—
203C2×C4⋊C4C2×D414yes—
204C2×C22⋊C4—24no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—24no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—28no—
209C2×C22⋊C4—48no—
210C2×C22⋊C4—41no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—44no—
214C2×C42—41no—
215C2×C42C22×C412yes—
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D411yescharacteristic
219C2×C42C2×Q812yes—
220C2×C42#4112yes—
221C2×C42#4112yes—
222C2×C42C2411yescharacteristic
223#107—44no—
224#106—44no—
225#106—84no—
226C4⋊C8—24no—
227C4⋊C8—24no—
228C4⋊C8—22no—
229C4⋊C8—22no—
230C4⋊C8—22no—
231C4⋊C8C22×C414yescomplemented
232#102—44no—
233#102—44no—
234C22⋊C8—24no—
235C22⋊C8—42no—
236C4×C8—22no—
237C4×C8—42no—
238C4×C8C22×C414yescomplemented
239#95—28no—
240#95—22no—
241#95—22no—
242#95—41no—
243#95—41no—
244C24—24no—
245C24—24no—
246C24—22no—
247C24—24no—
248C24—42no—
249C24—42no—
250C24—42no—
251C24—42no—
252C24#12011yescharacteristic
253C2×D4—24no—
254C2×D4—24no—
255C2×D4—216no—
256C2×D4—216no—
257C2×D4—22no—
258C2×D4—22no—
259C2×D4—22no—
260C2×D4—24no—
261C2×D4C2×C4⋊C414yes—
262C2×D4AGL(1,ℤ/8)14yes—
263C2×D4#13714yes—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—28no—
273C22×C4—22no—
274C22×C4—24no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—416no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—48no—
287C22×C4—48no—
288C22×C4—44no—
289C22×C4—44no—
290C22×C4—48no—
291C22×C4—48no—
292C22×C4—42no—
293C22×C4—44no—
294C22×C4—44no—
295C22×C4—416no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—44no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—88no—
302C22×C4—832no—
303C22×C4C2×C4⋊C412yes—
304C22×C4C4×D412yes—
305C22×C4C4×D412yes—
306C22×C4#12011yescharacteristic
307C22×C4#12011yescharacteristic
308C22×C4#12011yescharacteristic
309C22×C4C22⋊Q812yes—
310C22×C4C22⋊Q812yes—
311C22×C4#12312yes—
312C22×C4C2×SD1612yes—
313C22×C4#13512yes—
314C22×C4AGL(1,ℤ/8)12yes—
315C22×C4AGL(1,ℤ/8)12yes—
316C22×C4C22×D411yescharacteristic
317C22×C4C22×D411yescharacteristic
318C22×C4C22×D411yescharacteristic
319M16—48no—
320C2×C8—24no—
321C2×C8—24no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—42no—
325C2×C8—42no—
326C2×C8—42no—
327C2×C8—42no—
328C2×C8—42no—
329C4⋊C4—22no—
330C4⋊C4—22no—
331C4⋊C4—24no—
332C4⋊C4—24no—
333C4⋊C4—22no—
334C4⋊C4—22no—
335C4⋊C4—44no—
336C4⋊C4—416no—
337C4⋊C4—44no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—44no—
343C4⋊C4—42no—
344C4⋊C4—44no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—42no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—42no—
351C4⋊C4—82no—
352C4⋊C4C2×C4⋊C414yes—
353C4⋊C4C2×SD1614yes—
354C22⋊C4—24no—
355C22⋊C4—24no—
356C22⋊C4—28no—
357C22⋊C4—416no—
358C22⋊C4—82no—
359C42—44no—
360C42—44no—
361C42—82no—
362C42C2×C4⋊C412yes—
363C42C4×D412yes—
364C42C4×D412yes—
365C42C22⋊Q812yes—
366C42C22⋊Q812yes—
367C42#12312yes—
368C42C23×C411yescharacteristic
369C42C22×Q811yescharacteristic
370C42C2×#4111yescharacteristic
371C42C2×#4111yescharacteristic
372C23—21no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—216no—
377C23—24no—
378C23—24no—
379C23—24no—
380C23—48no—
381C23—48no—
382C23—48no—
383C23—48no—
384C23—48no—
385C23—48no—
386C23—42no—
387C23—42no—
388C23—42no—
389C23—42no—
390C23#38512yes—
391C23#44912yes—
392C23#45012yes—
393C23C2×#12011yescharacteristic
394D4—216no—
395D4—24no—
396D4—24no—
397D4—24no—
398D4#42818yescomplemented
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—24no—
406C2×C4—24no—
407C2×C4—24no—
408C2×C4—24no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—48no—
412C2×C4—48no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—44no—
418C2×C4—416no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—42no—
428C2×C4—832no—
429C2×C4—84no—
430C2×C4—88no—
431C2×C4—81no—
432C2×C4—81no—
433C2×C4—81no—
434C2×C4—81no—
435C2×C4—81no—
436C2×C4—81no—
437C2×C4#38512yes—
438C2×C4#38512yes—
439C2×C4#38512yes—
440C2×C4C4×SD1614yes—
441C2×C4#44214yes—
442C2×C4#44912yes—
443C2×C4#45012yes—
444C2×C4#47614yes—
445C2×C4#47814yes—
446C2×C4C22×C4⋊C411yescharacteristic
447C2×C4C2×C4×D411yescharacteristic
448C2×C4C2×C4×D411yescharacteristic
449C2×C4C2×C22⋊Q811yescharacteristic
450C2×C4C2×C22⋊Q811yescharacteristic
451C2×C4C2×#12311yescharacteristic
452C2×C4C22×SD1611yescharacteristic
453C2×C4C2×#13511yescharacteristic
454C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
455C2×C4C2×AGL(1,ℤ/8)12yes—
456C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
457C2×C4C2×#13712yes—
458C8—44no—
459C8—44no—
460C8—82no—
461C22—22no—
462C22—22no—
463C22—22no—
464C22—22no—
465C22—22no—
466C22—22no—
467C22—28no—
468C22—432no—
469C22—44no—
470C22—44no—
471C22—44no—
472C22—44no—
473C22—44no—
474C22—44no—
475C22#186414yes—
476C22C2×#38511yescharacteristic
477C22C2×#45011yescharacteristic
478C22C2×#44911yescharacteristic
479C22#300311yescharacteristic
480C22#300411yescharacteristic
481C22#301911yescharacteristic
482C22#302011yescharacteristic
483C4—22no—
484C4—21no—
485C4—21no—
486C4—22no—
487C4—44no—
488C4—44no—
489C4—84no—
490C4—82no—
491C4—161no—
492C4—161no—
493C4#186414yes—
494C4C2×#42812yes—
495C4C2×C4×SD1611yescharacteristic
496C4C2×#44211yescharacteristic
497C4C2×#47611yescharacteristic
498C4C2×#47811yescharacteristic
499C4#309212yes—
500C2—24no—
501C2—48no—
502C2#1874711yescharacteristic
503C2#1875211yescharacteristic
504C2#1877711yescharacteristic
505C2#1878211yescharacteristic
506C2#2084411yescharacteristic
507C2#2086411yescharacteristic
508C2C2×#186411yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC23×D4D4
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. #21170C2semidirect
6. #8890C2semidirect
7. #8886C2semidirect
8. #8043C2semidirect
9. #7892C2semidirect
29. #1948C22semidirect
30. #1947C22semidirect
32. #1844C22semidirect
33. #1834C22semidirect
40. #1486C4semidirect
42. #1296C22semidirect
43. #1279C22semidirect
78. #431C2×C4semidirect
84. #423C2×C4semidirect
123. D4⋊C8C2×C4semidirect
231. C4⋊C8C22×C4semidirect
238. C4×C8C22×C4semidirect
398. D4#428semidirect

Maximal subgroups

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

subgroupindex
C2×D4×C4⋊C42
#219812
#211702
#207322
C2×#14862
#88902
#88862
#80432
#78922