Finite Group Databasebeta

← #192791

Subgroups — #192791

Order 512.

1170 conjugacy classes of subgroups.

The subgroup lattice has 487 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34186C211yesmaximal, characteristic
2#21982C211yesmaximal, characteristic, complemented
3#21981C211yesmaximal, characteristic, complemented
4C2×#1802C211yesmaximal, characteristic
5C2×#1478C211yesmaximal, characteristic
6#8897C214yesmaximal, complemented
7#8889C214yesmaximal, complemented
8#8039C211yesmaximal, characteristic, complemented
9#7893C211yesmaximal, characteristic, complemented
10C22×C4×D4C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C2×#431C2211yescharacteristic
13C22×C4⋊C8C2211yescharacteristic
14C22×#102—21no—
15#2357—22no—
16#2357—22no—
17#2354—21no—
18#2354—21no—
19#2354C2212yes—
20#2354C2212yes—
21#2312—22no—
22#2309—21no—
23#2285C2211yescharacteristic
24#2285C2211yescharacteristic
25C2×#384C2211yescharacteristic
26#2280—22no—
27#2274—21no—
28#1948—22no—
29#1947C2214yescomplemented
30#1853—22no—
31#1852—22no—
32#1845—21no—
33#1844C2212yescomplemented
34#1834—21no—
35#1834C2212yescomplemented
36#1802—22no—
37#1802C2214yes—
38C2×#340—21no—
39#1478C418yescomplemented
40C2×D4⋊C8C2214yes—
41#1296C2212yescomplemented
42#1296C2212yescomplemented
43#1279C2214yescomplemented
44C23×D4D411yescharacteristic
45C24×C4—21no—
46C23×C8—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—21no—
56C2×#117—21no—
57C2×#117—22no—
58C2×#117—22no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4D411yescharacteristic
63C22×C22⋊C4—21no—
64C22×C42C2311yescharacteristic
65#431—22no—
66#431—22no—
67#431C2×C414yes—
68#431C2×C414yes—
69C2×#107—24no—
70C2×#106—24no—
71C2×C4⋊C8—21no—
72C2×C4⋊C8—22no—
73C2×C4⋊C8—22no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8C2312yes—
76C2×C4⋊C8C2312yes—
77C2×#102—24no—
78C2×#102—44no—
79C2×C22⋊C8—21no—
80C2×C22⋊C8—21no—
81C2×C4×C8—21no—
82C2×C4×C8C2312yes—
83#390—22no—
84#390—22no—
85#390—22no—
86#390—22no—
87#388—24no—
88#388—24no—
89#388—24no—
90#388—24no—
91#384—21no—
92#384—21no—
93#384—21no—
94#384D411yescharacteristic
95#384D411yescharacteristic
96#384C2312yes—
97#384C2311yescharacteristic
98#384C2311yescharacteristic
99#382—21no—
100#382—21no—
101#381—24no—
102#380—21no—
103#380—21no—
104#380—22no—
105#380—22no—
106#380D412yes—
107#380D412yes—
108#379—24no—
109C4×C4⋊C4—22no—
110C4×C4⋊C4—22no—
111C4×C4⋊C4—22no—
112C4×C4⋊C4—22no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4D412yes—
116C4×C4⋊C4C2312yes—
117C4×C22⋊C4—24no—
118C4×C22⋊C4—24no—
119#340—24no—
120#340—24no—
121#340—24no—
122D4⋊C8C2×C4116yescomplemented
123C25—21no—
124C22×D4—21no—
125C22×D4—22no—
126C22×D4—22no—
127C22×D4—22no—
128C22×D4—22no—
129C22×D4—22no—
130C22×D4C4⋊C414yes—
131C22×D4C2×D412yes—
132C23×C4—21no—
133C23×C4—22no—
134C23×C4—24no—
135C23×C4—22no—
136C23×C4—24no—
137C23×C4—41no—
138C23×C4—82no—
139C23×C4—82no—
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—42no—
147C22×C8—42no—
148C22×C8—42no—
149C4×D4—24no—
150C4×D4—24no—
151C4×D4—24no—
152C4×D4—24no—
153C4×D4—24no—
154C4×D4—22no—
155C4×D4—22no—
156C4×D4C4⋊C418yes—
157C4×D4C22×C414yes—
158C4×D4C2×Q814yes—
159#117—42no—
160#117—42no—
161#117—42no—
162#117—42no—
163#117—84no—
164#117—84no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—42no—
179C2×C4⋊C4—42no—
180C2×C4⋊C4—41no—
181C2×C4⋊C4—41no—
182C2×C4⋊C4—41no—
183C2×C4⋊C4—82no—
184C2×C4⋊C4—82no—
185C2×C4⋊C4C4⋊C414yes—
186C2×C4⋊C4C2×D412yes—
187C2×C4⋊C4C2×D412yes—
188C2×C4⋊C4C2×D412yes—
189C2×C4⋊C4C2×D412yes—
190C2×C4⋊C4C2×D412yes—
191C2×C22⋊C4—24no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—24no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—24no—
198C2×C22⋊C4—24no—
199C2×C22⋊C4—24no—
200C2×C22⋊C4—24no—
201C2×C42—22no—
202C2×C42—22no—
203C2×C42—24no—
204C2×C42—24no—
205C2×C42C22×C412yes—
206C2×C42C2×D411yescharacteristic
207C2×C42C2×D411yescharacteristic
208C2×C42C2×D411yescharacteristic
209C2×C42C2×Q812yes—
210C2×C42#4112yes—
211C2×C42#4112yes—
212C2×C42C2411yescharacteristic
213#107—84no—
214#106—84no—
215C4⋊C8—24no—
216C4⋊C8—22no—
217C4⋊C8—22no—
218C4⋊C8—22no—
219C4⋊C8—42no—
220C4⋊C8C22×C414yes—
221#102—48no—
222C22⋊C8—24no—
223C22⋊C8—24no—
224C4×C8—24no—
225C4×C8—22no—
226C4×C8C22×C414yes—
227#95—24no—
228#95—24no—
229#95—22no—
230#95—22no—
231#95—22no—
232#95—22no—
233C24—22no—
234C24—24no—
235C24—42no—
236C24—42no—
237C24—81no—
238C24—81no—
239C24—81no—
240C24—81no—
241C24#12011yescharacteristic
242C2×D4—22no—
243C2×D4—22no—
244C2×D4—22no—
245C2×D4—22no—
246C2×D4—22no—
247C2×D4—24no—
248C2×D4—28no—
249C2×D4—28no—
250C2×D4—42no—
251C2×D4—42no—
252C2×D4—44no—
253C2×D4—44no—
254C2×D4C2×C4⋊C414yes—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—28no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—44no—
269C22×C4—44no—
270C22×C4—42no—
271C22×C4—88no—
272C22×C4—88no—
273C22×C4—82no—
274C22×C4—82no—
275C22×C4—81no—
276C22×C4—81no—
277C22×C4—82no—
278C22×C4—82no—
279C22×C4—82no—
280C22×C4C2×C4⋊C412yes—
281C22×C4C4×D412yes—
282C22×C4C4×D412yes—
283C22×C4#12011yescharacteristic
284C22×C4#12011yescharacteristic
285C22×C4#12011yescharacteristic
286C22×C4C22⋊Q812yes—
287C22×C4C22⋊Q812yes—
288C22×C4#12312yes—
289C22×C4#13512yes—
290C22×C4#13512yes—
291C22×C4AGL(1,ℤ/8)12yes—
292C22×C4#13712yes—
293C22×C4C22×D411yescharacteristic
294C22×C4C22×D411yescharacteristic
295C22×C4C22×D411yescharacteristic
296C2×C8—24no—
297C2×C8—24no—
298C2×C8—42no—
299C2×C8—42no—
300C2×C8—42no—
301C2×C8—42no—
302C2×C8—42no—
303C2×C8—42no—
304C2×C8—42no—
305C2×C8—84no—
306C4⋊C4—22no—
307C4⋊C4—24no—
308C4⋊C4—24no—
309C4⋊C4—24no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—44no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—44no—
324C4⋊C4—81no—
325C4⋊C4—81no—
326C4⋊C4—82no—
327C4⋊C4—82no—
328C4⋊C4—82no—
329C4⋊C4C2×C4⋊C414yes—
330C22⋊C4—24no—
331C22⋊C4—24no—
332C22⋊C4—28no—
333C22⋊C4—44no—
334C22⋊C4—44no—
335C22⋊C4—84no—
336C22⋊C4—84no—
337C22⋊C4—84no—
338C22⋊C4—84no—
339C42—42no—
340C42—42no—
341C42—44no—
342C42—84no—
343C42—84no—
344C42C2×C4⋊C412yes—
345C42C4×D412yes—
346C42C4×D412yes—
347C42C22⋊Q812yes—
348C42C22⋊Q812yes—
349C42#12312yes—
350C42C23×C411yescharacteristic
351C42C22×Q811yescharacteristic
352C42C2×#4111yescharacteristic
353C42C2×#4111yescharacteristic
354C23—21no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—24no—
359C23—48no—
360C23—42no—
361C23—42no—
362C23—42no—
363C23—42no—
364C23—42no—
365C23—42no—
366C23—84no—
367C23—84no—
368C23—84no—
369C23—84no—
370C23—84no—
371C23—84no—
372C23#38512yes—
373C23#44912yes—
374C23#45212yes—
375C23C2×#12011yescharacteristic
376D4—24no—
377D4—24no—
378D4—42no—
379D4—42no—
380D4—48no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—24no—
388C2×C4—24no—
389C2×C4—24no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—48no—
395C2×C4—48no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—88no—
409C2×C4—84no—
410C2×C4—82no—
411C2×C4—82no—
412C2×C4—82no—
413C2×C4—164no—
414C2×C4#38512yes—
415C2×C4#38512yes—
416C2×C4#38512yes—
417C2×C4#42714yes—
418C2×C4#42814yes—
419C2×C4#44912yes—
420C2×C4#45212yes—
421C2×C4#48214yes—
422C2×C4#48514yes—
423C2×C4C22×C4⋊C411yescharacteristic
424C2×C4C2×C4×D411yescharacteristic
425C2×C4C2×C4×D411yescharacteristic
426C2×C4C2×C22⋊Q811yescharacteristic
427C2×C4C2×C22⋊Q811yescharacteristic
428C2×C4C2×#12311yescharacteristic
429C2×C4C2×#13511yescharacteristic
430C2×C4C2×#13511yescharacteristic
431C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
432C2×C4C2×#13711yescharacteristic
433C2×C4#57612yes—
434C2×C4#57712yes—
435C8—44no—
436C8—44no—
437C8—82no—
438C22—22no—
439C22—22no—
440C22—22no—
441C22—22no—
442C22—22no—
443C22—22no—
444C22—44no—
445C22—44no—
446C22—44no—
447C22—816no—
448C22—82no—
449C22—82no—
450C22—82no—
451C22—82no—
452C22#185414yes—
453C22C2×#38511yescharacteristic
454C22C2×#44911yescharacteristic
455C22C2×#45211yescharacteristic
456C22#300811yescharacteristic
457C22#300911yescharacteristic
458C22#302111yescharacteristic
459C22#302211yescharacteristic
460C4—21no—
461C4—21no—
462C4—22no—
463C4—22no—
464C4—44no—
465C4—82no—
466C4—82no—
467C4—82no—
468C4—82no—
469C4—162no—
470C4#185414yes—
471C4C2×#42711yescharacteristic
472C4C2×#42811yescharacteristic
473C4#294612yes—
474C4#308112yes—
475C4C2×#48211yescharacteristic
476C4C2×#48511yescharacteristic
477C2—24no—
478C2—84no—
479C2#1891711yescharacteristic
480C2#1891811yescharacteristic
481C2#1892711yescharacteristic
482C2#1892811yescharacteristic
483C2C2×#185411yescharacteristic
484C2#2086511yescharacteristic
485C2#2087611yescharacteristic
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. #21981C2semidirect
6. #8897C2semidirect
7. #8889C2semidirect
8. #8039C2semidirect
9. #7893C2semidirect
29. #1947C22semidirect
33. #1844C22semidirect
35. #1834C22semidirect
39. #1478C4semidirect
41. #1296C22semidirect
42. #1296C22semidirect
43. #1279C22semidirect
122. D4⋊C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
#341862
#219822
#219812
C2×#18022
C2×#14782
#88972
#88892
#80392
#78932