Finite Group Databasebeta

← #501764

Subgroups — #501764

Order 512.

588 conjugacy classes of subgroups.

The subgroup lattice has 417 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2230C211yesmaximal, characteristic, complemented
2C22×#364C211yesmaximal, characteristic, complemented
3#33207C211yesmaximal, characteristic, complemented
4#21544C211yesmaximal, characteristic
5#13521C212yesmaximal, complemented
6#13467C212yesmaximal, complemented
7#13272C211yesmaximal, characteristic, complemented
8#13186C211yesmaximal, characteristic, complemented
9#13065C212yesmaximal
10#12831C212yesmaximal, complemented
11#12823C211yesmaximal, characteristic, complemented
12C22×M32C2212yes—
13C2×#498C2211yescharacteristic
14#3050C2211yescharacteristic
15C2×#443C2211yescharacteristic
16#2929—21no—
17C22×C4×C8C2211yescharacteristic
18#2857—21no—
19#2230—24no—
20C2×#365C2212yes—
21C2×#365C2212yes—
22C2×#364—24no—
23C2×#364C2211yescharacteristic
24C2×#364C2212yes—
25C2×#364C2211yescharacteristic
26C2×#360C2212yes—
27C2×#360C2212yes—
28#1990—21no—
29#1974C2211yescharacteristic
30#1945C2211yescharacteristic
31#1941—21no—
32#1941C2212yes—
33#1937—21no—
34#1896—21no—
35#1844C2211yescharacteristic
36#1810C2211yescharacteristic
37#1807—21no—
38#1807C2212yes—
39#1772C2211yescharacteristic
40#1764—21no—
41#1386C2212yes—
42#1386C2212yes—
43#1383C2212yes—
44#1383C2212yes—
45C2×#135C2311yescharacteristic
46C2×#131—21no—
47C23×C8D411yescharacteristic
48C23×C8C2311yescharacteristic
49C2×C4⋊Q8—21no—
50#525—21no—
51C2×C22⋊Q8—41no—
52C22×C42D411yescharacteristic
53C2×M32—22no—
54C2×M32—24no—
55C2×M32—42no—
56C2×M32C2312yes—
57C2×M32C2312yes—
58C22×C16—22no—
59#498—22no—
60#498—22no—
61#467Q811yescharacteristic, complemented
62#467Q811yescharacteristic, complemented
63#466Q811yescharacteristic, complemented
64#466Q811yescharacteristic, complemented
65#465—21no—
66#465—21no—
67#443—22no—
68#443—22no—
69#435—41no—
70#435—41no—
71C8×D4—41no—
72C8×D4—41no—
73C2×#108C2311yescharacteristic
74C2×#108C2311yescharacteristic
75#427C2311yescharacteristic
76C2×#106—21no—
77C2×#106—21no—
78C2×#106C2312yes—
79C2×#106C2312yes—
80#424—21no—
81C2×#104—21no—
82C2×#104—21no—
83#416—21no—
84#408—21no—
85#407—21no—
86#405—21no—
87C2×C4×C8—22no—
88C2×C4×C8—22no—
89C2×C4×C8D411yescharacteristic
90C2×C4×C8D411yescharacteristic
91C2×C4×C8C2311yescharacteristic
92C2×C4×C8C2311yescharacteristic
93#391—41no—
94#389—22no—
95#389—41no—
96#384—21no—
97#380—22no—
98#380—41no—
99#365—24no—
100#365—42no—
101#364—24no—
102#364—22no—
103#364D414yescomplemented
104#360—24no—
105#360—42no—
106#341—21no—
107#340—22no—
108#340—22no—
109#339—21no—
110#339—21no—
111C2×#41—21no—
112C22×Q8—41no—
113C23×C4—21no—
114C23×C4C2×D411yescharacteristic
115#135—21no—
116#135—21no—
117#135—21no—
118#135—21no—
119C2×Q16C2×Q811yescharacteristic
120C2×SD16—21no—
121C2×D8C2×Q811yescharacteristic
122#131—42no—
123#131—42no—
124C2×M16—21no—
125C2×M16—21no—
126C2×M16—41no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—21no—
130C22×C8—21no—
131C22×C8—21no—
132C22×C8—21no—
133C22×C8—41no—
134C22×C8SD1611yescharacteristic
135C22×C8SD1611yescharacteristic
136C22×C8SD1611yescharacteristic
137C22×C8SD1611yescharacteristic
138C22×C8C2×D411yescharacteristic
139C22×C8C2×D411yescharacteristic
140C22×C8C2×D411yescharacteristic
141C22×C8C2×Q811yescharacteristic
142C22×C8C2×Q811yescharacteristic
143C22×C8#4111yescharacteristic
144C22×C8#4111yescharacteristic
145C22×C8C2411yescharacteristic
146C4⋊Q8—42no—
147C4⋊Q8—42no—
148#123—41no—
149C22⋊Q8—21no—
150C22⋊Q8—21no—
151C22⋊Q8—84no—
152#121—21no—
153#121—21no—
154#120—41no—
155C4×D4—41no—
156C4×D4—41no—
157#117—21no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4—42no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—41no—
164C2×C22⋊C4—42no—
165C2×C42—22no—
166C2×C42—21no—
167C2×C42—41no—
168C2×C42C2×D411yescharacteristic
169C2×C42C2×D411yescharacteristic
170M32—22no—
171M32—22no—
172M32—22no—
173M32—44no—
174M32C2×D414yes—
175C2×C16—22no—
176C2×C16—22no—
177C2×C16—44no—
178#108—21no—
179#108—21no—
180#108—21no—
181#108—21no—
182#107C2×Q811yescharacteristic
183#107C2×Q811yescharacteristic
184#106—21no—
185#106—22no—
186#106—22no—
187#106—22no—
188#106—22no—
189#106—42no—
190#106—42no—
191C4⋊C8—41no—
192C4⋊C8—41no—
193#104—42no—
194#104—42no—
195#103—21no—
196#103—21no—
197#102—21no—
198#102—21no—
199C22⋊C8—41no—
200C22⋊C8—41no—
201C22⋊C8—41no—
202C22⋊C8—41no—
203#97—41no—
204C4×C8—22no—
205C4×C8—21no—
206C4×C8—21no—
207C4×C8—22no—
208C4×C8—41no—
209C4×C8C2×D412yes—
210C4×C8#4112yes—
211#95—42no—
212#95—42no—
213C24AGL(1,ℤ/8)11yescharacteristic
214#41—41no—
215#41—41no—
216#41—41no—
217#41—41no—
218C2×Q8—21no—
219C2×Q8—82no—
220C2×Q8—82no—
221C2×Q8—82no—
222C2×D4—21no—
223C2×D4—41no—
224C2×D4—41no—
225C2×D4—41no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—24no—
232C22×C4—21no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—41no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4C22⋊Q811yescharacteristic
239C22×C4C22⋊Q811yescharacteristic
240C22×C4C2×SD1611yescharacteristic
241C22×C4C2×SD1611yescharacteristic
242C22×C4AGL(1,ℤ/8)11yescharacteristic
243C22×C4C22×D411yescharacteristic
244Q16—21no—
245Q16—21no—
246SD16—41no—
247SD16—41no—
248D8—21no—
249D8—21no—
250M16—41no—
251M16—41no—
252M16—41no—
253M16—41no—
254M16—82no—
255C2×C8—21no—
256C2×C8—21no—
257C2×C8—21no—
258C2×C8—21no—
259C2×C8—21no—
260C2×C8—21no—
261C2×C8—21no—
262C2×C8—21no—
263C2×C8—21no—
264C2×C8—21no—
265C2×C8—21no—
266C2×C8—21no—
267C2×C8—22no—
268C2×C8—22no—
269C2×C8—22no—
270C2×C8—22no—
271C2×C8—41no—
272C2×C8—41no—
273C2×C8—82no—
274C2×C8#12112yes—
275C2×C8C22⋊Q811yescharacteristic
276C2×C8C22⋊Q811yescharacteristic
277C2×C8C2×SD1611yescharacteristic
278C2×C8C2×SD1611yescharacteristic
279C2×C8C2×SD1611yescharacteristic
280C2×C8C2×SD1611yescharacteristic
281C2×C8#13512yes—
282C2×C8C22×D411yescharacteristic
283C2×C8C22×Q811yescharacteristic
284C2×C8C2×#4111yescharacteristic
285C2×C8C2×#4111yescharacteristic
286C2×C8H(5,𝔽2)11yescharacteristic
287C2×C82−1+411yescharacteristic
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—82no—
297C4⋊C4—82no—
298C4⋊C4—84no—
299C4⋊C4—82no—
300C22⋊C4—41no—
301C22⋊C4—41no—
302C22⋊C4—41no—
303C22⋊C4—41no—
304C22⋊C4—41no—
305C22⋊C4—84no—
306C42—21no—
307C42—21no—
308C42—21no—
309C42—42no—
310C42—41no—
311C42#12112yes—
312C16—24no—
313C16—42no—
314C23—21no—
315C23—22no—
316C23—21no—
317C23—41no—
318C23—41no—
319C23#47611yescharacteristic
320C23#47611yescharacteristic
321C23C2×AGL(1,ℤ/8)11yescharacteristic
322Q8—41no—
323Q8—41no—
324Q8—164no—
325D4—41no—
326D4—41no—
327D4—82no—
328D4—81no—
329D4—82no—
330D4—81no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—21no—
334C2×C4—22no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—22no—
345C2×C4—48no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4—82no—
354C2×C4—82no—
355C2×C4—82no—
356C2×C4—82no—
357C2×C4—82no—
358C2×C4—82no—
359C2×C4#47611yescharacteristic
360C2×C4#47611yescharacteristic
361C2×C4C2×#12111yescharacteristic
362C2×C4C2×C22⋊Q811yescharacteristic
363C2×C4#53711yescharacteristic
364C2×C4C22×SD1611yescharacteristic
365C2×C4C2×#13511yescharacteristic
366C2×C4#57511yescharacteristic
367C8—21no—
368C8—21no—
369C8—21no—
370C8—21no—
371C8—81no—
372C8—81no—
373C8#46512yes—
374C8#46512yes—
375C8#54611yescharacteristic
376C8D4×Q811yescharacteristic
377C8#55211yescharacteristic
378C8#55411yescharacteristic
379C22—21no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—42no—
387C22—42no—
388C22—81no—
389C22—81no—
390C22—81no—
391C22#223012yescomplemented
392C22#305511yescharacteristic
393C22C2×#47611yescharacteristic
394C22#324711yescharacteristic
395C4—24no—
396C4—21no—
397C4—21no—
398C4—42no—
399C4—81no—
400C4—81no—
401C4—81no—
402C4—162no—
403C4—162no—
404C4#241711yescharacteristic
405C4C2×#46511yescharacteristic
406C4#307611yescharacteristic
407C4#324611yescharacteristic
408C2—21no—
409C2—21no—
410C2—22no—
411C2—41no—
412C2—161no—
413C2#2483611yescharacteristic
414C2C2×#223011yescharacteristic
415C2#3363711yescharacteristic
subgroupG/H
center, socleC22C2×#476
commutator, Frattini, cosocleC22×C8C24
prime radical#3050C22
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C4×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2230C2semidirect
2. C22×#364C2semidirect
3. #33207C2semidirect
5. #13521C2semidirect
6. #13467C2semidirect
7. #13272C2semidirect
8. #13186C2semidirect
10. #12831C2semidirect
11. #12823C2semidirect
61. #467Q8semidirect
62. #467Q8semidirect
63. #466Q8semidirect
64. #466Q8semidirect
103. #364D4semidirect
391. C22#2230semidirect

Maximal subgroups

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

subgroupindex
C2×#22302
C22×#3642
#332072
#215442
#135212
#134672
#132722
#131862
#130652
#128312
#128232