Finite Group Databasebeta

← #501765

Subgroups — #501765

Order 512.

566 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2231C211yesmaximal, characteristic, complemented
2C22×#364C211yesmaximal, characteristic, complemented
3#33208C211yesmaximal, characteristic, complemented
4#21544C211yesmaximal, characteristic
5#13521C212yesmaximal, complemented
6#13467C212yesmaximal, complemented
7#13273C211yesmaximal, characteristic, complemented
8#13187C211yesmaximal, characteristic, complemented
9#13065C212yesmaximal
10#12831C212yesmaximal, complemented
11#12824C211yesmaximal, characteristic, complemented
12C22×M32C2212yes—
13C2×#499C2211yescharacteristic
14#3050C2211yescharacteristic
15C2×#443C2211yescharacteristic
16#2929—21no—
17C22×C4×C8C2211yescharacteristic
18#2851—21no—
19#2231—24no—
20C2×#365C2212yes—
21C2×#365C2212yes—
22C2×#364—24no—
23C2×#364C2211yescharacteristic
24C2×#364C2212yes—
25C2×#364C2211yescharacteristic
26C2×#360C2212yes—
27C2×#360C2212yes—
28#1991—21no—
29#1974C2211yescharacteristic
30#1945C2211yescharacteristic
31#1942—21no—
32#1941C2212yes—
33#1937—21no—
34#1896—21no—
35#1846C2211yescharacteristic
36#1810C2211yescharacteristic
37#1808—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×#125—21no—
50#525—21no—
51C22×C42D411yescharacteristic
52C2×M32—22no—
53C2×M32—24no—
54C2×M32—42no—
55C2×M32C2312yes—
56C2×M32C2312yes—
57C22×C16—22no—
58#499—22no—
59#499—22no—
60#467Q811yescharacteristic
61#467Q811yescharacteristic
62#466Q811yescharacteristic
63#466Q811yescharacteristic
64#465—21no—
65#465—21no—
66#443—22no—
67#443—22no—
68#435—41no—
69#435—41no—
70C8×D4—41no—
71C8×D4—41no—
72C2×#108C2311yescharacteristic
73C2×#108C2311yescharacteristic
74#427C2311yescharacteristic
75C2×#107—21no—
76C2×#107—21no—
77C2×#106C2312yes—
78C2×#106C2312yes—
79#424—21no—
80C2×#104—21no—
81C2×#104—21no—
82#416—21no—
83#408—21no—
84#407—21no—
85#405—21no—
86C2×C4×C8—22no—
87C2×C4×C8—22no—
88C2×C4×C8D411yescharacteristic
89C2×C4×C8D411yescharacteristic
90C2×C4×C8C2311yescharacteristic
91C2×C4×C8C2311yescharacteristic
92#389—22no—
93#387—42no—
94#385—42no—
95#382—21no—
96#380—22no—
97#365—24no—
98#365—42no—
99#364—24no—
100#364—22no—
101#364D414yescomplemented
102#360—24no—
103#360—42no—
104#341—21no—
105#340—22no—
106#340—22no—
107#339—21no—
108#339—21no—
109C2×#41—21no—
110C23×C4—21no—
111C23×C4C2×D411yescharacteristic
112#135—21no—
113#135—21no—
114#135—21no—
115#135—21no—
116C2×Q16C2×Q811yescharacteristic
117C2×SD16—21no—
118C2×D8C2×Q811yescharacteristic
119#131—42no—
120#131—42no—
121C2×M16—21no—
122C2×M16—21no—
123C2×M16—41no—
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—21no—
127C22×C8—21no—
128C22×C8—21no—
129C22×C8—21no—
130C22×C8—41no—
131C22×C8SD1611yescharacteristic
132C22×C8SD1611yescharacteristic
133C22×C8SD1611yescharacteristic
134C22×C8SD1611yescharacteristic
135C22×C8C2×D411yescharacteristic
136C22×C8C2×D411yescharacteristic
137C22×C8C2×D411yescharacteristic
138C22×C8C2×Q811yescharacteristic
139C22×C8C2×Q811yescharacteristic
140C22×C8#4111yescharacteristic
141C22×C8#4111yescharacteristic
142C22×C8C2411yescharacteristic
143#125—42no—
144#125—42no—
145#123—41no—
146C22⋊Q8—21no—
147C22⋊Q8—21no—
148#121—21no—
149#121—21no—
150#120—41no—
151C4×D4—41no—
152C4×D4—41no—
153#117—21no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—42no—
157C2×C4⋊C4—42no—
158C2×C22⋊C4—42no—
159C2×C42—22no—
160C2×C42—21no—
161C2×C42—41no—
162C2×C42C2×D411yescharacteristic
163C2×C42C2×D411yescharacteristic
164M32—22no—
165M32—22no—
166M32—22no—
167M32—44no—
168M32C2×D414yes—
169C2×C16—22no—
170C2×C16—22no—
171C2×C16—44no—
172#108—21no—
173#108—21no—
174#108—21no—
175#108—21no—
176#107—42no—
177#107—42no—
178#107C2×Q811yescharacteristic
179#107C2×Q811yescharacteristic
180#106—21no—
181#106—22no—
182#106—22no—
183#106—22no—
184#106—22no—
185C4⋊C8—41no—
186C4⋊C8—41no—
187#104—42no—
188#104—42no—
189#103—21no—
190#103—21no—
191#102—21no—
192#102—21no—
193C22⋊C8—41no—
194C22⋊C8—41no—
195C22⋊C8—41no—
196C22⋊C8—41no—
197#97—41no—
198C4×C8—22no—
199C4×C8—21no—
200C4×C8—21no—
201C4×C8—22no—
202C4×C8—41no—
203C4×C8C2×D412yes—
204C4×C8#4112yes—
205#95—42no—
206#95—42no—
207#95—42no—
208C24AGL(1,ℤ/8)11yescharacteristic
209#41—41no—
210#41—41no—
211#41—41no—
212#41—41no—
213C2×Q8—21no—
214C2×D4—21no—
215C2×D4—41no—
216C2×D4—41no—
217C2×D4—41no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—41no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4C22⋊Q811yescharacteristic
231C22×C4C22⋊Q811yescharacteristic
232C22×C4C2×SD1611yescharacteristic
233C22×C4C2×SD1611yescharacteristic
234C22×C4AGL(1,ℤ/8)11yescharacteristic
235C22×C4C22×D411yescharacteristic
236Q16—21no—
237Q16—21no—
238SD16—41no—
239SD16—41no—
240D8—21no—
241D8—21no—
242M16—41no—
243M16—41no—
244M16—41no—
245M16—41no—
246M16—82no—
247C2×C8—21no—
248C2×C8—21no—
249C2×C8—21no—
250C2×C8—21no—
251C2×C8—21no—
252C2×C8—21no—
253C2×C8—21no—
254C2×C8—21no—
255C2×C8—21no—
256C2×C8—21no—
257C2×C8—21no—
258C2×C8—21no—
259C2×C8—22no—
260C2×C8—22no—
261C2×C8—22no—
262C2×C8—22no—
263C2×C8—41no—
264C2×C8—41no—
265C2×C8—82no—
266C2×C8#12112yes—
267C2×C8C22⋊Q811yescharacteristic
268C2×C8C22⋊Q811yescharacteristic
269C2×C8C2×SD1611yescharacteristic
270C2×C8C2×SD1611yescharacteristic
271C2×C8C2×SD1611yescharacteristic
272C2×C8C2×SD1611yescharacteristic
273C2×C8#13512yes—
274C2×C8C22×D411yescharacteristic
275C2×C8C22×Q811yescharacteristic
276C2×C8C2×#4111yescharacteristic
277C2×C8C2×#4111yescharacteristic
278C2×C8H(5,𝔽2)11yescharacteristic
279C2×C82−1+411yescharacteristic
280C4⋊C4—21no—
281C4⋊C4—21no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—42no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4—84no—
289C4⋊C4—84no—
290C22⋊C4—41no—
291C22⋊C4—41no—
292C22⋊C4—41no—
293C22⋊C4—41no—
294C22⋊C4—41no—
295C22⋊C4—84no—
296C42—21no—
297C42—21no—
298C42—21no—
299C42—42no—
300C42—41no—
301C42#12112yes—
302C16—24no—
303C16—42no—
304C23—21no—
305C23—22no—
306C23—21no—
307C23—41no—
308C23—41no—
309C23#47611yescharacteristic
310C23#47611yescharacteristic
311C23C2×AGL(1,ℤ/8)11yescharacteristic
312Q8—41no—
313Q8—41no—
314D4—41no—
315D4—41no—
316D4—82no—
317D4—81no—
318D4—82no—
319D4—81no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—48no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—81no—
341C2×C4—81no—
342C2×C4—82no—
343C2×C4—82no—
344C2×C4—82no—
345C2×C4—82no—
346C2×C4—82no—
347C2×C4—82no—
348C2×C4#47611yescharacteristic
349C2×C4#47611yescharacteristic
350C2×C4C2×#12111yescharacteristic
351C2×C4C2×C22⋊Q811yescharacteristic
352C2×C4#53711yescharacteristic
353C2×C4C22×SD1611yescharacteristic
354C2×C4C2×#13511yescharacteristic
355C2×C4#57511yescharacteristic
356C8—21no—
357C8—21no—
358C8—21no—
359C8—21no—
360C8—81no—
361C8—81no—
362C8#46512yes—
363C8#46512yes—
364C8#54611yescharacteristic
365C8D4×Q811yescharacteristic
366C8#55211yescharacteristic
367C8#55411yescharacteristic
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—42no—
376C22—42no—
377C22—81no—
378C22—81no—
379C22—81no—
380C22#223112yescomplemented
381C22#305511yescharacteristic
382C22C2×#47611yescharacteristic
383C22#324711yescharacteristic
384C4—21no—
385C4—21no—
386C4—42no—
387C4—42no—
388C4—81no—
389C4—81no—
390C4—81no—
391C4—162no—
392C4—162no—
393C4#241711yescharacteristic
394C4C2×#46511yescharacteristic
395C4#307611yescharacteristic
396C4#324611yescharacteristic
397C2—21no—
398C2—21no—
399C2—22no—
400C2—41no—
401C2—161no—
402C2#2483611yescharacteristic
403C2C2×#223111yescharacteristic
404C2#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×#2231C2semidirect
2. C22×#364C2semidirect
3. #33208C2semidirect
5. #13521C2semidirect
6. #13467C2semidirect
7. #13273C2semidirect
8. #13187C2semidirect
10. #12831C2semidirect
11. #12824C2semidirect
101. #364D4semidirect
380. C22#2231semidirect

Maximal subgroups

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

subgroupindex
C2×#22312
C22×#3642
#332082
#215442
#135212
#134672
#132732
#131872
#130652
#128312
#128242