Finite Group Databasebeta

← #192789

Subgroups — #192789

Order 512.

1170 conjugacy classes of subgroups.

The subgroup lattice has 461 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34182C211yesmaximal, characteristic
2#21981C211yesmaximal, characteristic
3#21981C211yesmaximal, characteristic
4C2×#1803C211yesmaximal, characteristic
5C2×#1479C211yesmaximal, characteristic
6#8896C214yesmaximal, complemented
7#8888C214yesmaximal, complemented
8#8039C211yesmaximal, characteristic
9#7893C211yesmaximal, characteristic
10C2×#519—21no—
11C22×C4×Q8C2211yescharacteristic
12C2×#431C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×#103—21no—
16#2351—22no—
17#2351—22no—
18Q8×C4⋊C4—21no—
19Q8×C4⋊C4—21no—
20Q8×C4⋊C4C2212yes—
21Q8×C4⋊C4C2212yes—
22#2316—22no—
23#2316—21no—
24#2285—22no—
25#2285C2211yescharacteristic
26#2285C2211yescharacteristic
27C2×#384C2211yescharacteristic
28#1948—22no—
29#1948C2214yes—
30#1853—22no—
31#1853—22no—
32#1844—21no—
33#1844C2212yes—
34#1834—21no—
35#1834C2212yes—
36#1803—22no—
37#1803C2214yes—
38C2×#340—21no—
39#1479C418yescomplemented
40C2×Q8⋊C8C2214yes—
41#1296C2212yes—
42#1296C2212yes—
43#1279C2214yes—
44C23×Q8D411yescharacteristic
45C23×C8—21no—
46#519—44no—
47#519—44no—
48C2×C4×Q8—22no—
49C2×C4×Q8—22no—
50C2×C4×Q8—22no—
51C2×C4×Q8—22no—
52C2×C4×Q8—21no—
53C2×C4×Q8—21no—
54C2×C4×Q8—22no—
55C2×C4×Q8—22no—
56C2×C4×Q8C2×C414yes—
57C2×C4×Q8D412yes—
58C2×C4×Q8Q814yes—
59C2×C4×Q8C2312yes—
60C2×#117—24no—
61C2×#117—21no—
62C2×#117—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4D411yescharacteristic
68C22×C42—21no—
69C22×C42C2311yescharacteristic
70#431—22no—
71#431—22no—
72#431C2×C414yes—
73#431C2×C414yes—
74C2×#106—24no—
75C2×#106—24no—
76C2×C4⋊C8—21no—
77C2×C4⋊C8—22no—
78C2×C4⋊C8—22no—
79C2×C4⋊C8—21no—
80C2×C4⋊C8C2312yes—
81C2×C4⋊C8C2312yes—
82C2×#103—24no—
83C2×#103—44no—
84C2×C22⋊C8—21no—
85C2×C22⋊C8—21no—
86C2×C4×C8—21no—
87C2×C4×C8C2312yes—
88#389—22no—
89#389—22no—
90#389—24no—
91#389—24no—
92#389—22no—
93#389—22no—
94#389—24no—
95#389—24no—
96#384—22no—
97#384—22no—
98#384—21no—
99#384—21no—
100#384—21no—
101#384D411yescharacteristic
102#384D411yescharacteristic
103#384C2312yes—
104#384C2311yescharacteristic
105#384C2311yescharacteristic
106#382—24no—
107#382—21no—
108#382—21no—
109#380—24no—
110#380—21no—
111#380—21no—
112#380D412yes—
113#380D412yes—
114C4×C4⋊C4—24no—
115C4×C4⋊C4—22no—
116C4×C4⋊C4—22no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4D412yes—
120C4×C4⋊C4C2312yes—
121#340—24no—
122#340—24no—
123#340—24no—
124Q8⋊C8C2×C4116yescomplemented
125C22×Q8—21no—
126C22×Q8—22no—
127C22×Q8—22no—
128C22×Q8—22no—
129C22×Q8C4⋊C414yes—
130C22×Q8SD1614yes—
131C22×Q8SD1614yes—
132C22×Q8C2×D412yes—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—41no—
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—42no—
143C22×C8—42no—
144C22×C8—42no—
145C4×Q8—24no—
146C4×Q8—24no—
147C4×Q8—24no—
148C4×Q8—24no—
149C4×Q8—24no—
150C4×Q8—22no—
151C4×Q8—22no—
152C4×Q8—44no—
153C4×Q8—44no—
154C4×Q8—48no—
155C4×Q8—48no—
156C4×Q8—44no—
157C4×Q8—44no—
158C4×Q8C4⋊C418yes—
159C4×Q8C22×C414yes—
160C4×Q8C2×Q814yes—
161#117—416no—
162#117—42no—
163#117—42no—
164#117—42no—
165#117—42no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—24no—
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—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—41no—
190C2×C4⋊C4—41no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4—82no—
193C2×C4⋊C4—82no—
194C2×C4⋊C4C4⋊C414yes—
195C2×C4⋊C4C2×D412yes—
196C2×C4⋊C4C2×D412yes—
197C2×C4⋊C4C2×D412yes—
198C2×C4⋊C4C2×D412yes—
199C2×C4⋊C4C2×D412yes—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C42—24no—
203C2×C42—22no—
204C2×C42—24no—
205C2×C42—22no—
206C2×C42—24no—
207C2×C42—24no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42C22×C412yes—
211C2×C42C2×D411yescharacteristic
212C2×C42C2×D411yescharacteristic
213C2×C42C2×D411yescharacteristic
214C2×C42C2×Q812yes—
215C2×C42#4112yes—
216C2×C42#4112yes—
217C2×C42C2411yescharacteristic
218#106—84no—
219#106—84no—
220C4⋊C8—24no—
221C4⋊C8—22no—
222C4⋊C8—22no—
223C4⋊C8—22no—
224C4⋊C8—42no—
225C4⋊C8C22×C414yes—
226#103—48no—
227C22⋊C8—24no—
228C22⋊C8—24no—
229C4×C8—24no—
230C4×C8—22no—
231C4×C8C22×C414yes—
232#95—24no—
233#95—24no—
234#95—22no—
235#95—22no—
236#95—22no—
237#95—22no—
238C24#12011yescharacteristic
239C2×Q8—22no—
240C2×Q8—22no—
241C2×Q8—22no—
242C2×Q8—24no—
243C2×Q8—28no—
244C2×Q8—24no—
245C2×Q8—24no—
246C2×Q8—28no—
247C2×Q8—28no—
248C2×Q8#106116yescomplemented
249C2×Q8C2×C4⋊C414yes—
250C2×Q8C2×SD1614yes—
251C2×Q8C2×SD1614yes—
252C22×C4—22no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—24no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—24no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—42no—
267C22×C4—81no—
268C22×C4—81no—
269C22×C4—82no—
270C22×C4—82no—
271C22×C4—82no—
272C22×C4C2×C4⋊C412yes—
273C22×C4C4×D412yes—
274C22×C4C4×D412yes—
275C22×C4#12011yescharacteristic
276C22×C4#12011yescharacteristic
277C22×C4#12011yescharacteristic
278C22×C4C22⋊Q812yes—
279C22×C4C22⋊Q812yes—
280C22×C4#12312yes—
281C22×C4C2×SD1612yes—
282C22×C4C2×SD1612yes—
283C22×C4AGL(1,ℤ/8)12yes—
284C22×C4#13712yes—
285C22×C4C22×D411yescharacteristic
286C22×C4C22×D411yescharacteristic
287C22×C4C22×D411yescharacteristic
288C2×C8—24no—
289C2×C8—24no—
290C2×C8—42no—
291C2×C8—42no—
292C2×C8—42no—
293C2×C8—42no—
294C2×C8—42no—
295C2×C8—42no—
296C2×C8—42no—
297C2×C8—84no—
298C4⋊C4—24no—
299C4⋊C4—24no—
300C4⋊C4—22no—
301C4⋊C4—28no—
302C4⋊C4—24no—
303C4⋊C4—24no—
304C4⋊C4—24no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—48no—
308C4⋊C4—48no—
309C4⋊C4—44no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—44no—
321C4⋊C4—81no—
322C4⋊C4—81no—
323C4⋊C4—82no—
324C4⋊C4—82no—
325C4⋊C4—84no—
326C4⋊C4—84no—
327C4⋊C4—82no—
328C4⋊C4C2×C4⋊C414yes—
329C22⋊C4—44no—
330C22⋊C4—44no—
331C42—24no—
332C42—24no—
333C42—28no—
334C42—48no—
335C42—48no—
336C42—42no—
337C42—42no—
338C42—44no—
339C42C2×C4⋊C412yes—
340C42C4×D412yes—
341C42C4×D412yes—
342C42C22⋊Q812yes—
343C42C22⋊Q812yes—
344C42#12312yes—
345C42C23×C411yescharacteristic
346C42C22×Q811yescharacteristic
347C42C2×#4111yescharacteristic
348C42C2×#4111yescharacteristic
349C23—21no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23#38512yes—
354C23#44812yes—
355C23#45012yes—
356C23C2×#12011yescharacteristic
357Q8—24no—
358Q8—24no—
359Q8—216no—
360Q8—24no—
361Q8C2×#10618yes—
362C2×C4—216no—
363C2×C4—24no—
364C2×C4—24no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—24no—
372C2×C4—24no—
373C2×C4—24no—
374C2×C4—24no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—84no—
392C2×C4—82no—
393C2×C4—82no—
394C2×C4—82no—
395C2×C4—164no—
396C2×C4#38512yes—
397C2×C4#38512yes—
398C2×C4#38512yes—
399C2×C4C2×#10614yes—
400C2×C4#42814yes—
401C2×C4#44812yes—
402C2×C4#45012yes—
403C2×C4#48114yes—
404C2×C4#48314yes—
405C2×C4C22×C4⋊C411yescharacteristic
406C2×C4C2×C4×D411yescharacteristic
407C2×C4C2×C4×D411yescharacteristic
408C2×C4C2×C22⋊Q811yescharacteristic
409C2×C4C2×C22⋊Q811yescharacteristic
410C2×C4C2×#12311yescharacteristic
411C2×C4C22×SD1611yescharacteristic
412C2×C4C22×SD1611yescharacteristic
413C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
414C2×C4C2×#13711yescharacteristic
415C2×C4#57712yes—
416C2×C4#57812yes—
417C8—44no—
418C8—44no—
419C8—82no—
420C22—22no—
421C22—22no—
422C22—22no—
423C22—22no—
424C22—22no—
425C22—22no—
426C22#185314yes—
427C22C2×#38511yescharacteristic
428C22C2×#45011yescharacteristic
429C22C2×#44811yescharacteristic
430C22#301011yescharacteristic
431C22#301111yescharacteristic
432C22#302111yescharacteristic
433C22#302211yescharacteristic
434C4—28no—
435C4—21no—
436C4—21no—
437C4—22no—
438C4—22no—
439C4—44no—
440C4—82no—
441C4—82no—
442C4—82no—
443C4—82no—
444C4—162no—
445C4#185314yes—
446C4C22×#10611yescharacteristic
447C4C2×#42811yescharacteristic
448C4#294712yes—
449C4#308212yes—
450C4C2×#48311yescharacteristic
451C4C2×#48111yescharacteristic
452C2—24no—
453C2#1891311yescharacteristic
454C2#1891411yescharacteristic
455C2#1892311yescharacteristic
456C2#1892411yescharacteristic
457C2C2×#185311yescharacteristic
458C2#2086511yescharacteristic
459C2#2087711yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#120
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
6. #8896C2semidirect
7. #8888C2semidirect
39. #1479C4semidirect
124. Q8⋊C8C2×C4semidirect
248. C2×Q8#106semidirect

Maximal subgroups

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

subgroupindex
#341822
#219812
C2×#18032
C2×#14792
#88962
#88882
#80392
#78932