Finite Group Databasebeta

← #501311

Subgroups — #501311

Order 512.

836 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
2C2×#2163C211yesmaximal, characteristic, complemented
3#33130C211yesmaximal, characteristic, complemented
4C2×#1970C211yesmaximal, characteristic
5#13474C212yesmaximal, complemented
6#13449C212yesmaximal, complemented
7#13238C212yesmaximal, complemented
8#13149C212yesmaximal, complemented
9#12993C211yesmaximal, characteristic, complemented
10#12919C211yesmaximal, characteristic, complemented
11#12731C211yesmaximal, characteristic
12C23×D8C2211yescharacteristic
13C22×SD32C2212yes—
14C2×#467C2211yescharacteristic
15C22×#107C2211yescharacteristic
16C22×#102—21no—
17C22×C4×C8C2211yescharacteristic
18#2855—21no—
19C2×#390—21no—
20#2187—24no—
21#2163—24no—
22C2×#366C2212yescomplemented
23C2×C4⋊C16C2212yescomplemented
24C2×#358C2212yescomplemented
25C2×#357C2212yes—
26C2×#357C2212yes—
27C2×#357C2212yes—
28C2×C22⋊C16C2211yescharacteristic
29C2×C22⋊C16C2211yescharacteristic
30C2×#346C2212yescomplemented
31#1970—21no—
32#1970—21no—
33#1970C2212yes—
34#1970C2212yes—
35#1969C2211yescharacteristic, complemented
36#1969C2211yescharacteristic, complemented
37#1942C2211yescharacteristic, complemented
38#1942C2211yescharacteristic, complemented
39#1933—21no—
40#1933—21no—
41#1892—21no—
42#1808C2211yescharacteristic
43#1380C2212yescomplemented
44#1380C2212yescomplemented
45C23×D4—21no—
46C22×Q16C2311yescharacteristic
47C22×D8—24no—
48C22×D8—21no—
49C22×D8—22no—
50C22×D8—22no—
51C22×D8—42no—
52C22×D8C2312yes—
53C23×C8D411yescharacteristic
54C23×C8C2311yescharacteristic
55C2×C22⋊Q8—21no—
56C22×C4⋊C4—21no—
57C22×C22⋊C4—41no—
58C22×C22⋊C4—41no—
59C22×C42D411yescharacteristic
60C2×SD32—24no—
61C2×SD32—24no—
62C2×SD32—42no—
63C22×C16C2312yes—
64C22×C16C2312yes—
65#467—22no—
66#467—22no—
67C2×#107—21no—
68C2×#107—42no—
69C2×#107C2311yescharacteristic
70C2×#107C2311yescharacteristic
71C2×#107C2311yescharacteristic
72C2×#107C2312yes—
73C2×#103—21no—
74C2×#103—21no—
75C2×#103—21no—
76C2×#102—22no—
77C2×#102—22no—
78C2×#102—44no—
79C2×C4×C8—22no—
80C2×C4×C8—22no—
81C2×C4×C8D411yescharacteristic
82C2×C4×C8D411yescharacteristic
83C2×C4×C8C2311yescharacteristic
84C2×C4×C8C2311yescharacteristic
85#391—21no—
86#391—21no—
87#390—22no—
88#390—22no—
89#390—41no—
90#390—41no—
91#389—21no—
92#389—21no—
93#387—41no—
94#387—41no—
95#385—41no—
96#381—41no—
97#380—21no—
98#366—42no—
99C4⋊C16—24no—
100#358—42no—
101#357—24no—
102#357—24no—
103#357—42no—
104C22⋊C16—22no—
105C22⋊C16D414yes—
106#346—24no—
107#340—21no—
108#340—21no—
109#340—21no—
110C25—41no—
111C22×Q8—21no—
112C22×D4—24no—
113C22×D4—22no—
114C22×D4—42no—
115C22×D4—42no—
116C22×D4—42no—
117C22×D4—42no—
118C22×D4—41no—
119C22×D4—42no—
120C23×C4—41no—
121C23×C4D811yescharacteristic
122C23×C4D811yescharacteristic
123C23×C4C2×D411yescharacteristic
124C2×Q16—21no—
125C2×Q16—21no—
126C2×Q16—22no—
127C2×Q16—22no—
128C2×D8—24no—
129C2×D8—22no—
130C2×D8—22no—
131C2×D8—22no—
132C2×D8—22no—
133C2×D8—22no—
134C2×D8—48no—
135C2×D8—48no—
136C2×D8—42no—
137C2×D8—42no—
138C2×D8#4114yes—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—41no—
144C22×C8C2×D411yescharacteristic
145C22×C8C2×D411yescharacteristic
146C22×C8C2×D412yes—
147C22×C8C2×D411yescharacteristic
148C22×C8#4112yes—
149C22×C8C2411yescharacteristic
150C22⋊Q8—42no—
151C22⋊Q8—42no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—42no—
157C2×C4⋊C4—42no—
158C2×C4⋊C4—41no—
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—22no—
171C2×C42—22no—
172C2×C42D811yescharacteristic
173C2×C42D811yescharacteristic
174C2×C42C2×D411yescharacteristic
175C2×C42C2×D411yescharacteristic
176SD32—48no—
177C2×C16—22no—
178C2×C16—22no—
179C2×C16—22no—
180C2×C16—22no—
181C2×C16—22no—
182C2×C16C2×D414yes—
183#107—21no—
184#107—22no—
185#107—21no—
186#107—22no—
187#107—22no—
188#107—22no—
189#107—42no—
190#103—42no—
191#103—42no—
192#103—42no—
193#102—44no—
194#102—44no—
195C4×C8—22no—
196C4×C8—22no—
197C4×C8—21no—
198C4×C8—21no—
199C4×C8—21no—
200C4×C8—21no—
201#95—41no—
202#95—41no—
203#95—41no—
204#95—41no—
205#95—41no—
206#95—41no—
207#95—41no—
208C24—44no—
209C24—42no—
210C24—82no—
211C24—82no—
212C24—82no—
213C24—82no—
214C24—82no—
215C24—82no—
216C24C2×D811yescharacteristic
217C2×Q8—41no—
218C2×Q8—41no—
219C2×Q8—41no—
220C2×Q8—41no—
221C2×Q8—41no—
222C2×Q8—41no—
223C2×D4—24no—
224C2×D4—48no—
225C2×D4—48no—
226C2×D4—44no—
227C2×D4—42no—
228C2×D4—42no—
229C2×D4—42no—
230C2×D4—42no—
231C2×D4—42no—
232C2×D4—82no—
233C2×D4—84no—
234C2×D4—84no—
235C2×D4—82no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—42no—
251C22×C4—82no—
252C22×C4—82no—
253C22×C4—82no—
254C22×C4#12112yes—
255C22×C4C2×D811yescharacteristic
256C22×C4C2×D811yescharacteristic
257C22×C4C2×D811yescharacteristic
258C22×C4C2×D811yescharacteristic
259C22×C4C2×D811yescharacteristic
260C22×C4#13512yes—
261C22×C4C22×D411yescharacteristic
262Q16—42no—
263Q16—42no—
264D8—24no—
265D8—24no—
266D8—48no—
267D8—42no—
268D8—42no—
269C2×C8—21no—
270C2×C8—21no—
271C2×C8—21no—
272C2×C8—21no—
273C2×C8—24no—
274C2×C8—24no—
275C2×C8—21no—
276C2×C8—21no—
277C2×C8—21no—
278C2×C8—44no—
279C2×C8#12112yes—
280C2×C8AGL(1,ℤ/8)12yes—
281C2×C8C22×D411yescharacteristic
282C2×C8C2×#4112yes—
283C2×C8C2×#4111yescharacteristic
284C2×C8H(5,𝔽2)12yes—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—42no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4—42no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—82no—
294C4⋊C4—82no—
295C4⋊C4—82no—
296C4⋊C4—82no—
297C22⋊C4—84no—
298C22⋊C4—84no—
299C22⋊C4—84no—
300C22⋊C4—82no—
301C22⋊C4—82no—
302C22⋊C4—84no—
303C42—21no—
304C42—21no—
305C42—21no—
306C42—21no—
307C42—22no—
308C42—22no—
309C16—24no—
310C16—42no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—44no—
316C23—88no—
317C23—88no—
318C23—88no—
319C23—82no—
320C23—82no—
321C23—82no—
322C23—82no—
323C23—82no—
324C23—82no—
325C23—164no—
326C23—164no—
327C23—164no—
328C23—164no—
329C23#46612yes—
330C23#50912yes—
331C23#51012yes—
332C23C22×D811yescharacteristic
333Q8—81no—
334Q8—81no—
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—81no—
362C2×C4—81no—
363C2×C4—82no—
364C2×C4—82no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—82no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—164no—
377C2×C4#46612yes—
378C2×C4#50912yes—
379C2×C4C2×#12111yescharacteristic
380C2×C4#52511yescharacteristic
381C2×C4#53511yescharacteristic
382C2×C4C2×#13511yescharacteristic
383C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
384C2×C4#57511yescharacteristic
385C8—22no—
386C8—22no—
387C8—41no—
388C8#54612yes—
389C8#55112yes—
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#222014yes—
405C22C2×#46611yescharacteristic
406C22#310011yescharacteristic
407C22#320511yescharacteristic
408C22C2×#50911yescharacteristic
409C22C2×#50911yescharacteristic
410C22C2×#51011yescharacteristic
411C22#341411yescharacteristic
412C4—22no—
413C4—22no—
414C4—41no—
415C4—41no—
416C4—41no—
417C4—161no—
418C4—161no—
419C4—161no—
420C4—161no—
421C4—162no—
422C4#241111yescharacteristic
423C4#305511yescharacteristic
424C4#310511yescharacteristic
425C4#321111yescharacteristic
426C2—24no—
427C2—164no—
428C2#2450011yescharacteristic
429C2C2×#222011yescharacteristic
430C2#3360511yescharacteristic
431C2#3367011yescharacteristic
432C2#3367411yescharacteristic
433C2#3378111yescharacteristic
434C2#3378311yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC23×D8C22
LCSG ▷ C22×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. C2×#2163C2semidirect
3. #33130C2semidirect
5. #13474C2semidirect
6. #13449C2semidirect
7. #13238C2semidirect
8. #13149C2semidirect
9. #12993C2semidirect
10. #12919C2semidirect
22. C2×#366C22semidirect
23. C2×C4⋊C16C22semidirect
24. C2×#358C22semidirect
30. C2×#346C22semidirect
35. #1969C22semidirect
36. #1969C22semidirect
37. #1942C22semidirect
38. #1942C22semidirect
43. #1380C22semidirect
44. #1380C22semidirect

Maximal subgroups

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

subgroupindex
C2×#21872
C2×#21632
#331302
C2×#19702
#134742
#134492
#132382
#131492
#129932
#129192
#127312