Finite Group Databasebeta

← #496818

Subgroups — #496818

Order 512.

856 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2187C211yesmaximal, characteristic
2C4×#467C211yesmaximal, characteristic, complemented
3C2×#1970C211yesmaximal, characteristic
4#13238C212yesmaximal, complemented
5#13149C212yesmaximal, complemented
6#13130C212yesmaximal, complemented
7#13013C212yesmaximal, complemented
8#12993C211yesmaximal, characteristic, complemented
9#12919C211yesmaximal, characteristic, complemented
10#12731C211yesmaximal, characteristic
11#12599C211yesmaximal, characteristic, complemented
12C23×D8C2211yescharacteristic
13C22×SD32C2212yes—
14C2×#467C2211yescharacteristic
15C2×C4×Q16C2211yescharacteristic, complemented
16C22×#107C2211yescharacteristic
17C22×#102—21no—
18C22×C4×C8C2211yescharacteristic
19C4×C22⋊Q8—21no—
20C2×#390—21no—
21#2187C418yescomplemented
22C2×#366C2212yescomplemented
23C2×#358C2212yescomplemented
24C2×#357C2212yes—
25C2×#357C2212yes—
26C2×#357C2212yes—
27C2×C22⋊C16C2211yescharacteristic
28C2×C22⋊C16C2211yescharacteristic
29C2×#346C2212yescomplemented
30#1970—21no—
31#1970—21no—
32#1970C2212yes—
33#1970C2212yes—
34#1969C2211yescharacteristic, complemented
35#1942C2211yescharacteristic, complemented
36#1933—21no—
37#1892—21no—
38#1808C2211yescharacteristic
39C4×#107C2211yescharacteristic, complemented
40C4×#103—21no—
41#1380C2212yescomplemented
42#1380C2212yescomplemented
43#1374C2212yescomplemented
44C23×D4—21no—
45C22×Q16C2311yescharacteristic
46C22×D8—21no—
47C22×D8—22no—
48C22×D8—42no—
49C22×D8—42no—
50C22×D8C2×C414yes—
51C22×D8C2312yes—
52C23×C8D411yescharacteristic
53C23×C8C2311yescharacteristic
54C2×C22⋊Q8—21no—
55C2×C4×Q8—21no—
56C22×C4⋊C4—21no—
57C22×C22⋊C4—41no—
58C22×C22⋊C4—41no—
59C22×C42D411yescharacteristic
60C2×SD32—42no—
61C2×SD32—42no—
62C2×SD32C2×C418yescomplemented
63C22×C16C2312yes—
64C22×C16C2312yes—
65#467—22no—
66#467C2×C414yes—
67C4×Q16—42no—
68C2×#107—21no—
69C2×#107—42no—
70C2×#107C2311yescharacteristic
71C2×#107C2311yescharacteristic
72C2×#107C2312yes—
73C2×#107C2311yescharacteristic
74C2×#103—21no—
75C2×#103—21no—
76C2×#103—21no—
77C2×#102—22no—
78C2×#102—22no—
79C2×#102—44no—
80C2×C4×C8—42no—
81C2×C4×C8D411yescharacteristic
82C2×C4×C8D411yescharacteristic
83C2×C4×C8C2311yescharacteristic
84C2×C4×C8C2311yescharacteristic
85#391—21no—
86#390—22no—
87#390—22no—
88#390—41no—
89#390—41no—
90#389—21no—
91#387—41no—
92#385—41no—
93#380—21no—
94C4×C4⋊C4—21no—
95C4×C4⋊C4—41no—
96C4×C22⋊C4—41no—
97#366—42no—
98#358—42no—
99#357—42no—
100#357—42no—
101#357C2×C418yescomplemented
102C22⋊C16—22no—
103C22⋊C16C2×C414yes—
104#346—42no—
105#340—21no—
106#340—21no—
107#340—21no—
108C25—41no—
109C22×Q8—21no—
110C22×D4—24no—
111C22×D4—22no—
112C22×D4—42no—
113C22×D4—42no—
114C22×D4—42no—
115C22×D4—42no—
116C22×D4—41no—
117C22×D4—42no—
118C23×C4—41no—
119C23×C4D811yescharacteristic
120C23×C4D811yescharacteristic
121C23×C4C2×D411yescharacteristic
122C2×Q16—21no—
123C2×Q16—22no—
124C2×Q16—22no—
125C2×Q16C22×C412yes—
126C2×D8—24no—
127C2×D8—22no—
128C2×D8—22no—
129C2×D8—22no—
130C2×D8—22no—
131C2×D8—22no—
132C2×D8—48no—
133C2×D8—42no—
134C2×D8—42no—
135C2×D8—48no—
136C2×D8C22×C414yes—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—41no—
142C22×C8C22×C412yes—
143C22×C8C2×D411yescharacteristic
144C22×C8C2×D411yescharacteristic
145C22×C8C2×D411yescharacteristic
146C22×C8#4112yes—
147C22×C8C2411yescharacteristic
148C22⋊Q8—24no—
149C22⋊Q8—42no—
150C4×Q8—42no—
151C4×Q8—42no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—41no—
157C2×C4⋊C4—42no—
158C2×C4⋊C4—42no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4—41no—
162C2×C22⋊C4—41no—
163C2×C22⋊C4—41no—
164C2×C22⋊C4—42no—
165C2×C22⋊C4—42no—
166C2×C22⋊C4—42no—
167C2×C22⋊C4—42no—
168C2×C22⋊C4—84no—
169C2×C22⋊C4—84no—
170C2×C42—41no—
171C2×C42—41no—
172C2×C42—41no—
173C2×C42—41no—
174C2×C42D811yescharacteristic
175C2×C42D811yescharacteristic
176C2×C42C2×D411yescharacteristic
177C2×C42C2×D411yescharacteristic
178SD32—48no—
179C2×C16—22no—
180C2×C16—22no—
181C2×C16—22no—
182C2×C16—22no—
183C2×C16—22no—
184C2×C16C22×C414yes—
185#107—21no—
186#107—22no—
187#107—22no—
188#107—22no—
189#107—22no—
190#107—42no—
191#107C22×C412yes—
192#103—24no—
193#103—42no—
194#103—42no—
195#102—44no—
196#102—44no—
197C4×C8—22no—
198C4×C8—22no—
199C4×C8—41no—
200C4×C8—41no—
201#95—41no—
202#95—41no—
203#95—41no—
204#95—41no—
205#95—41no—
206C24—42no—
207C24—44no—
208C24—82no—
209C24—82no—
210C24—82no—
211C24—82no—
212C24—82no—
213C24—82no—
214C24C2×D811yescharacteristic
215C2×Q8—22no—
216C2×Q8—41no—
217C2×Q8—41no—
218C2×Q8—41no—
219C2×Q8—41no—
220C2×Q8—41no—
221C2×D4—24no—
222C2×D4—48no—
223C2×D4—48no—
224C2×D4—42no—
225C2×D4—42no—
226C2×D4—42no—
227C2×D4—42no—
228C2×D4—42no—
229C2×D4—44no—
230C2×D4—82no—
231C2×D4—82no—
232C2×D4—84no—
233C2×D4—84no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—42no—
249C22×C4—82no—
250C22×C4—82no—
251C22×C4—82no—
252C22×C4C4×D412yes—
253C22×C4C2×D811yescharacteristic
254C22×C4C2×D811yescharacteristic
255C22×C4C2×D811yescharacteristic
256C22×C4C2×D811yescharacteristic
257C22×C4C2×D811yescharacteristic
258C22×C4#13512yes—
259C22×C4C22×D411yescharacteristic
260Q16—24no—
261Q16—42no—
262D8—24no—
263D8—24no—
264D8—48no—
265D8—42no—
266D8—42no—
267C2×C8—21no—
268C2×C8—21no—
269C2×C8—21no—
270C2×C8—21no—
271C2×C8—24no—
272C2×C8—24no—
273C2×C8—21no—
274C2×C8—21no—
275C2×C8—21no—
276C2×C8—44no—
277C2×C8C4×D412yes—
278C2×C8AGL(1,ℤ/8)12yes—
279C2×C8C23×C411yescharacteristic
280C2×C8C2×#4111yescharacteristic
281C2×C8H(5,𝔽2)12yes—
282C2×C8H(5,𝔽2)12yes—
283C4⋊C4—22no—
284C4⋊C4—41no—
285C4⋊C4—44no—
286C4⋊C4—41no—
287C4⋊C4—42no—
288C4⋊C4—41no—
289C4⋊C4—42no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—82no—
293C4⋊C4—82no—
294C4⋊C4—82no—
295C22⋊C4—44no—
296C22⋊C4—84no—
297C22⋊C4—84no—
298C22⋊C4—84no—
299C22⋊C4—82no—
300C22⋊C4—84no—
301C42—21no—
302C42—21no—
303C42—21no—
304C42—21no—
305C42—41no—
306C42—41no—
307C42—82no—
308C42—82no—
309C16—24no—
310C16—42no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—44no—
316C23—88no—
317C23—88no—
318C23—82no—
319C23—82no—
320C23—82no—
321C23—82no—
322C23—82no—
323C23—88no—
324C23—82no—
325C23—164no—
326C23—164no—
327C23—164no—
328C23—164no—
329C23C4×D812yes—
330C23#50912yes—
331C23#51012yes—
332C23C22×D811yescharacteristic
333Q8—42no—
334Q8—42no—
335Q8—81no—
336Q8—81no—
337D4—44no—
338D4—44no—
339D4—88no—
340D4—82no—
341D4—82no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—44no—
360C2×C4—44no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—82no—
369C2×C4—82no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—82no—
375C2×C4—81no—
376C2×C4—164no—
377C2×C4C4×D812yes—
378C2×C4#50912yes—
379C2×C4C2×C4×D411yescharacteristic
380C2×C4#53411yescharacteristic
381C2×C4#53511yescharacteristic
382C2×C4C2×#13511yescharacteristic
383C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
384C2×C4#57511yescharacteristic
385C8—22no—
386C8—22no—
387C8—41no—
388C8#51812yes—
389C8#53812yes—
390C22—22no—
391C22—22no—
392C22—22no—
393C22—22no—
394C22—22no—
395C22—22no—
396C22—84no—
397C22—84no—
398C22—84no—
399C22—1616no—
400C22—162no—
401C22—162no—
402C22—162no—
403C22—162no—
404C22#217514yes—
405C22C2×C4×D811yescharacteristic
406C22#318611yescharacteristic
407C22#320511yescharacteristic
408C22C2×#50911yescharacteristic
409C22C2×#50911yescharacteristic
410C22C2×#51011yescharacteristic
411C22#341411yescharacteristic
412C4—22no—
413C4—22no—
414C4—41no—
415C4—41no—
416C4—41no—
417C4—82no—
418C4—82no—
419C4—161no—
420C4—161no—
421C4—162no—
422C4#232111yescharacteristic
423C4C4×AGL(1,ℤ/8)11yescharacteristic
424C4#318811yescharacteristic
425C4#321111yescharacteristic
426C2—24no—
427C2—164no—
428C2#2204011yescharacteristic
429C2C2×#217511yescharacteristic
430C2C4×#50911yescharacteristic
431C2#3375911yescharacteristic
432C2#3376011yescharacteristic
433C2#3378111yescharacteristic
434C2#3378311yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radicalC23×D8C22
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C4×#467C2semidirect
4. #13238C2semidirect
5. #13149C2semidirect
6. #13130C2semidirect
7. #13013C2semidirect
8. #12993C2semidirect
9. #12919C2semidirect
11. #12599C2semidirect
15. C2×C4×Q16C22semidirect
21. #2187C4semidirect
22. C2×#366C22semidirect
23. C2×#358C22semidirect
29. C2×#346C22semidirect
34. #1969C22semidirect
35. #1942C22semidirect
39. C4×#107C22semidirect
41. #1380C22semidirect
42. #1380C22semidirect
43. #1374C22semidirect
62. C2×SD32C2×C4semidirect
101. #357C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#21872
C4×#4672
C2×#19702
#132382
#131492
#131302
#130132
#129932
#129192
#127312
#125992