Finite Group Databasebeta

← #192737

Subgroups — #192737

Order 512.

1342 conjugacy classes of subgroups.

The subgroup lattice has 495 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2354C211yesmaximal, characteristic
2#21982C211yesmaximal, characteristic, complemented
3#21359C211yesmaximal, characteristic, complemented
4#20646C211yesmaximal, characteristic
5C2×#1478C211yesmaximal, characteristic
6#8893C214yesmaximal, complemented
7#8885C214yesmaximal, complemented
8#8043C211yesmaximal, characteristic, complemented
9#7901C211yesmaximal, characteristic, complemented
10C22×C4×D4C2211yescharacteristic
11C22×#117—21no—
12C2×#431C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C2×#423C2211yescharacteristic
15C2×#418—21no—
16#2354—22no—
17#2354—22no—
18#2354—22no—
19#2354C2214yes—
20C2×#390—21no—
21C2×#388—22no—
22#2285C2211yescharacteristic
23C2×#384C2211yescharacteristic
24C2×#380—21no—
25C2×C4×C4⋊C4—21no—
26C2×C4×C4⋊C4C2211yescharacteristic
27#1948C2212yescomplemented
28#1948C2212yescomplemented
29#1947—22no—
30#1853—21no—
31#1853—21no—
32#1852—22no—
33#1845C2212yescomplemented
34#1834C2212yescomplemented
35#1804—22no—
36#1802C2212yes—
37#1802C2212yes—
38C4×#106—22no—
39#1736—21no—
40#1478C418yescomplemented
41C2×D4⋊C8C2214yes—
42#1296C2214yescomplemented
43#1279C2214yescomplemented
44C23×D4D411yescharacteristic
45C24×C4—21no—
46C22×M16—21no—
47C2×C4×D4—21no—
48C2×C4×D4—21no—
49C2×C4×D4—22no—
50C2×C4×D4—22no—
51C2×C4×D4C2×C414yes—
52C2×C4×D4D412yes—
53C2×C4×D4Q814yes—
54C2×C4×D4C2312yes—
55C2×#117—24no—
56C2×#117—24no—
57C2×#117—21no—
58C2×#117—48no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4D411yescharacteristic
62C22×C4⋊C4D411yescharacteristic
63C22×C4⋊C4D411yescharacteristic
64C22×C4⋊C4D412yes—
65C22×C22⋊C4—21no—
66C22×C22⋊C4—22no—
67C22×C22⋊C4—22no—
68C22×C42—22no—
69C22×C42—21no—
70C22×C42C2311yescharacteristic
71#431—22no—
72#431—22no—
73#431C2×C414yes—
74#431C2×C414yes—
75C2×#107—24no—
76C2×#106—22no—
77C2×#106—22no—
78#423—22no—
79#423—22no—
80C2×C4⋊C8—21no—
81C2×C4⋊C8—21no—
82C2×C4⋊C8C2312yes—
83C2×C4⋊C8C2312yes—
84#418—44no—
85C2×#102—22no—
86C2×#102—22no—
87C2×C22⋊C8—21no—
88C2×C22⋊C8—21no—
89C2×C4×C8—21no—
90C2×C4×C8C2312yes—
91#390—24no—
92#390—24no—
93#388—28no—
94#388—28no—
95#384—21no—
96#384—21no—
97#384D412yes—
98#384C2311yescharacteristic
99#384C2311yescharacteristic
100#384C2312yes—
101#382—21no—
102#380—24no—
103#380—24no—
104#380—21no—
105#380—21no—
106C4×C4⋊C4—24no—
107C4×C4⋊C4—24no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4—21no—
110C4×C4⋊C4—21no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4D412yes—
113C4×C4⋊C4C2312yes—
114C2×#95—22no—
115C2×#95—21no—
116#340—22no—
117#340—22no—
118#340—24no—
119#340—22no—
120#340—22no—
121D4⋊C8C2×C4116yescomplemented
122C25—21no—
123C22×D4—21no—
124C22×D4—22no—
125C22×D4—22no—
126C22×D4—22no—
127C22×D4—22no—
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—22no—
138C23×C4—21no—
139C23×C4—44no—
140C23×C4—44no—
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C2×M16—42no—
145C2×M16—42no—
146C2×M16—42no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C4×D4—24no—
151C4×D4—24no—
152C4×D4—24no—
153C4×D4—24no—
154C4×D4—24no—
155C4×D4—22no—
156C4×D4—22no—
157C4×D4C4⋊C418yes—
158C4×D4C22×C414yes—
159C4×D4C2×Q814yes—
160#117—416no—
161#117—48no—
162#117—48no—
163#117—42no—
164#117—42no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—42no—
182C2×C4⋊C4—44no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—82no—
186C2×C4⋊C4C4⋊C414yes—
187C2×C4⋊C4C2×D412yes—
188C2×C4⋊C4C2×D412yes—
189C2×C4⋊C4C2×D412yes—
190C2×C4⋊C4C2×D414yes—
191C2×C22⋊C4—24no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—24no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—28no—
196C2×C22⋊C4—28no—
197C2×C22⋊C4—48no—
198C2×C22⋊C4—41no—
199C2×C22⋊C4—84no—
200C2×C42—28no—
201C2×C42—22no—
202C2×C42—22no—
203C2×C42—48no—
204C2×C42—42no—
205C2×C42—42no—
206C2×C42—41no—
207C2×C42C22×C412yes—
208C2×C42C2×D411yescharacteristic
209C2×C42C2×D411yescharacteristic
210C2×C42C2×D411yescharacteristic
211C2×C42C2×Q812yes—
212C2×C42#4112yes—
213C2×C42#4112yes—
214C2×C42C2411yescharacteristic
215#107—84no—
216#106—44no—
217#106—44no—
218C4⋊C8—24no—
219C4⋊C8—22no—
220C4⋊C8—22no—
221C4⋊C8—22no—
222C4⋊C8—42no—
223C4⋊C8C22×C414yes—
224#102—44no—
225#102—44no—
226C22⋊C8—24no—
227C22⋊C8—24no—
228C4×C8—24no—
229C4×C8—22no—
230C4×C8C22×C414yes—
231#95—28no—
232#95—22no—
233#95—22no—
234#95—41no—
235#95—41no—
236C24—22no—
237C24—24no—
238C24—42no—
239C24—42no—
240C24—42no—
241C24—42no—
242C24—42no—
243C24—42no—
244C24#12011yescharacteristic
245C2×D4—22no—
246C2×D4—22no—
247C2×D4—22no—
248C2×D4—22no—
249C2×D4—22no—
250C2×D4—24no—
251C2×D4—28no—
252C2×D4—28no—
253C2×D4—42no—
254C2×D4—42no—
255C2×D4—44no—
256C2×D4—44no—
257C2×D4C2×C4⋊C414yes—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—24no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—28no—
267C22×C4—22no—
268C22×C4—24no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—44no—
272C22×C4—44no—
273C22×C4—416no—
274C22×C4—416no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—44no—
280C22×C4—44no—
281C22×C4—44no—
282C22×C4—42no—
283C22×C4—44no—
284C22×C4—42no—
285C22×C4—44no—
286C22×C4—42no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4C2×C4⋊C412yes—
290C22×C4C4×D412yes—
291C22×C4C4×D412yes—
292C22×C4#12011yescharacteristic
293C22×C4#12011yescharacteristic
294C22×C4#12011yescharacteristic
295C22×C4C22⋊Q812yes—
296C22×C4C22⋊Q812yes—
297C22×C4#12312yes—
298C22×C4#13512yes—
299C22×C4#13512yes—
300C22×C4AGL(1,ℤ/8)12yes—
301C22×C4#13712yes—
302C22×C4C22×D411yescharacteristic
303C22×C4C22×D411yescharacteristic
304C22×C4C22×D411yescharacteristic
305M16—84no—
306C2×C8—24no—
307C2×C8—24no—
308C2×C8—42no—
309C2×C8—42no—
310C2×C8—42no—
311C2×C8—42no—
312C2×C8—42no—
313C2×C8—42no—
314C2×C8—42no—
315C4⋊C4—24no—
316C4⋊C4—22no—
317C4⋊C4—22no—
318C4⋊C4—24no—
319C4⋊C4—22no—
320C4⋊C4—22no—
321C4⋊C4—44no—
322C4⋊C4—42no—
323C4⋊C4—44no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C4⋊C4—42no—
328C4⋊C4—44no—
329C4⋊C4—44no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—82no—
337C4⋊C4C2×C4⋊C414yes—
338C4⋊C4AGL(1,ℤ/8)14yes—
339C22⋊C4—24no—
340C22⋊C4—24no—
341C22⋊C4—28no—
342C22⋊C4—416no—
343C22⋊C4—88no—
344C22⋊C4—82no—
345C42—416no—
346C42—42no—
347C42—42no—
348C42—44no—
349C42—82no—
350C42C2×C4⋊C412yes—
351C42C4×D412yes—
352C42C4×D412yes—
353C42C22⋊Q812yes—
354C42C22⋊Q812yes—
355C42#12312yes—
356C42C23×C411yescharacteristic
357C42C22×Q811yescharacteristic
358C42C2×#4111yescharacteristic
359C42C2×#4111yescharacteristic
360C23—21no—
361C23—21no—
362C23—21no—
363C23—21no—
364C23—24no—
365C23—48no—
366C23—48no—
367C23—48no—
368C23—42no—
369C23—42no—
370C23—42no—
371C23—42no—
372C23—42no—
373C23—42no—
374C23—84no—
375C23—84no—
376C23—84no—
377C23—84no—
378C23#38512yes—
379C23#44912yes—
380C23#45212yes—
381C23C2×#12011yescharacteristic
382D4—24no—
383D4—24no—
384D4—42no—
385D4—42no—
386D4—48no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—24no—
394C2×C4—24no—
395C2×C4—24no—
396C2×C4—24no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—416no—
400C2×C4—48no—
401C2×C4—48no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—48no—
407C2×C4—48no—
408C2×C4—44no—
409C2×C4—42no—
410C2×C4—44no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—44no—
414C2×C4—88no—
415C2×C4—84no—
416C2×C4—81no—
417C2×C4—81no—
418C2×C4—81no—
419C2×C4—81no—
420C2×C4—81no—
421C2×C4—81no—
422C2×C4#38512yes—
423C2×C4#38512yes—
424C2×C4#38512yes—
425C2×C4#42714yes—
426C2×C4#42814yes—
427C2×C4#44912yes—
428C2×C4#45212yes—
429C2×C4#48214yes—
430C2×C4#48514yes—
431C2×C4C22×C4⋊C411yescharacteristic
432C2×C4C2×C4×D411yescharacteristic
433C2×C4C2×C4×D411yescharacteristic
434C2×C4C2×C22⋊Q811yescharacteristic
435C2×C4C2×C22⋊Q811yescharacteristic
436C2×C4C2×#12311yescharacteristic
437C2×C4C2×#13511yescharacteristic
438C2×C4C2×#13511yescharacteristic
439C2×C4C2×AGL(1,ℤ/8)12yes—
440C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
441C2×C4C2×#13711yescharacteristic
442C2×C4#57512yes—
443C8—44no—
444C8—44no—
445C8—82no—
446C22—22no—
447C22—22no—
448C22—22no—
449C22—22no—
450C22—22no—
451C22—22no—
452C22—44no—
453C22—44no—
454C22—44no—
455C22—816no—
456C22—82no—
457C22—82no—
458C22—82no—
459C22—82no—
460C22#185414yes—
461C22C2×#38511yescharacteristic
462C22C2×#44911yescharacteristic
463C22C2×#45211yescharacteristic
464C22#300311yescharacteristic
465C22#300511yescharacteristic
466C22#301911yescharacteristic
467C22#302011yescharacteristic
468C4—21no—
469C4—21no—
470C4—22no—
471C4—22no—
472C4—44no—
473C4—44no—
474C4—84no—
475C4—82no—
476C4—161no—
477C4—161no—
478C4#185414yes—
479C4C2×#42711yescharacteristic
480C4C2×#42811yescharacteristic
481C4C4×AGL(1,ℤ/8)12yes—
482C4#307712yes—
483C4C2×#48211yescharacteristic
484C4C2×#48511yescharacteristic
485C2—24no—
486C2—84no—
487C2#1877011yescharacteristic
488C2#1877311yescharacteristic
489C2#1878011yescharacteristic
490C2#1878311yescharacteristic
491C2C2×#185411yescharacteristic
492C2#2086411yescharacteristic
493C2#2087311yescharacteristic
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. #21982C2semidirect
3. #21359C2semidirect
6. #8893C2semidirect
7. #8885C2semidirect
8. #8043C2semidirect
9. #7901C2semidirect
27. #1948C22semidirect
28. #1948C22semidirect
33. #1845C22semidirect
34. #1834C22semidirect
40. #1478C4semidirect
42. #1296C22semidirect
43. #1279C22semidirect
121. D4⋊C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23542
#219822
#213592
#206462
C2×#14782
#88932
#88852
#80432
#79012