Finite Group Databasebeta

← #509025

Subgroups — #509025

Order 512.

541 conjugacy classes of subgroups.

The subgroup lattice has 373 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33209C211yesmaximal, characteristic, complemented
2#33129C211yesmaximal, characteristic
3#13537C211yesmaximal, characteristic, complemented
4#13526C211yesmaximal, characteristic, complemented
5#13492C212yesmaximal, complemented
6#13451C211yesmaximal, characteristic, complemented
7#13380C212yesmaximal
8#13363C211yesmaximal, characteristic, complemented
9#13165C212yesmaximal, complemented
10#13147C212yesmaximal, complemented
11C4×C22⋊C16C211yesmaximal, characteristic, complemented
12C2×#500C2211yescharacteristic
13C2×#466C2211yescharacteristic
14C22×C4×C8C2211yescharacteristic
15#2857—21no—
16#2854—21no—
17C2×#366C2212yes—
18C2×#366C2212yes—
19C2×#357C2212yes—
20C2×#357C2212yes—
21C2×C22⋊C16C2212yes—
22C2×C4×C16C2212yes—
23#1990—21no—
24#1970C2211yescharacteristic
25#1970C2211yescharacteristic
26#1942—21no—
27#1942C2212yes—
28#1942C2211yescharacteristic
29#1942C2211yescharacteristic
30#1934—21no—
31#1934—21no—
32#1893—21no—
33#1845C2211yescharacteristic
34#1808—21no—
35#1808C2212yes—
36#1808C2211yescharacteristic
37#1380C2212yes—
38#1380C2212yes—
39#1380C2212yes—
40#1380C2212yes—
41#1374C2211yescharacteristic
42#1374C2211yescharacteristic
43C22×D8C2311yescharacteristic
44C23×C8D411yescharacteristic
45C23×C8C2311yescharacteristic
46C2×C4⋊Q8—21no—
47C2×#123—41no—
48C2×C22⋊Q8—41no—
49C2×#121—21no—
50C22×C42D411yescharacteristic
51C22×C16C2312yes—
52C22×C16C2312yes—
53#500—42no—
54#466—42no—
55C2×#107—21no—
56C2×#107—21no—
57C2×#107C2312yes—
58C2×#107C2312yes—
59C2×#107C2311yescharacteristic
60C2×#107C2311yescharacteristic
61C2×#107C2311yescharacteristic
62C2×#102—21no—
63C2×#102—21no—
64C2×#102—21no—
65C2×C4×C8—42no—
66C2×C4×C8D411yescharacteristic
67C2×C4×C8D411yescharacteristic
68C2×C4×C8C2311yescharacteristic
69C2×C4×C8C2311yescharacteristic
70#391—41no—
71#390—21no—
72#390—21no—
73#389—21no—
74#389—21no—
75#389—22no—
76#389—41no—
77#388—41no—
78#388—41no—
79#386—41no—
80#384—21no—
81#380—22no—
82#380—21no—
83#380—41no—
84#366—24no—
85#366—42no—
86#357—24no—
87#357—42no—
88C22⋊C16—24no—
89C4×C16—42no—
90#340—22no—
91#340—21no—
92#340—22no—
93#340—21no—
94#340—21no—
95C22×Q8—41no—
96C22×D4—21no—
97C22×D4—41no—
98C23×C4D811yescharacteristic
99C23×C4D811yescharacteristic
100C23×C4C2×D411yescharacteristic
101C2×D8—22no—
102C2×D8—22no—
103C2×D8—21no—
104C2×D8—21no—
105C22×C8—22no—
106C22×C8—22no—
107C22×C8—22no—
108C22×C8—22no—
109C22×C8—21no—
110C22×C8—21no—
111C22×C8—41no—
112C22×C8C2×D411yescharacteristic
113C22×C8C2×D411yescharacteristic
114C22×C8C2×D411yescharacteristic
115C22×C8C2411yescharacteristic
116C4⋊Q8—42no—
117C4⋊Q8—42no—
118#123—84no—
119C22⋊Q8—84no—
120#121—42no—
121#121—42no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—41no—
128C2×C4⋊C4—42no—
129C2×C4⋊C4—41no—
130C2×C4⋊C4—41no—
131C2×C4⋊C4—41no—
132C2×C4⋊C4—41no—
133C2×C22⋊C4—42no—
134C2×C22⋊C4—41no—
135C2×C22⋊C4—41no—
136C2×C22⋊C4—41no—
137C2×C22⋊C4—41no—
138C2×C22⋊C4—41no—
139C2×C22⋊C4—41no—
140C2×C42—41no—
141C2×C42—41no—
142C2×C42D811yescharacteristic
143C2×C42D811yescharacteristic
144C2×C42C2×D411yescharacteristic
145C2×C42C2×D411yescharacteristic
146C2×C16—22no—
147C2×C16—22no—
148C2×C16—22no—
149C2×C16—22no—
150C2×C16—22no—
151C2×C16#4114yes—
152#107—22no—
153#107—22no—
154#107—22no—
155#107—22no—
156#107—22no—
157#107—22no—
158#107—21no—
159#107—21no—
160#107—42no—
161#107—42no—
162#102—42no—
163#102—42no—
164#102—42no—
165C4×C8—22no—
166C4×C8—22no—
167C4×C8—41no—
168C4×C8—41no—
169#95—41no—
170#95—42no—
171#95—42no—
172#95—41no—
173#95—41no—
174C24—41no—
175C24C2×D811yescharacteristic
176C2×Q8—82no—
177C2×Q8—82no—
178C2×Q8—82no—
179C2×D4—41no—
180C2×D4—41no—
181C2×D4—41no—
182C2×D4—41no—
183C2×D4—41no—
184C2×D4—41no—
185C2×D4—82no—
186C2×D4—82no—
187C2×D4—82no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—41no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4—41no—
201C22×C4—42no—
202C22×C4—41no—
203C22×C4—42no—
204C22×C4D1614yes—
205C22×C4C2×D811yescharacteristic
206C22×C4C2×D811yescharacteristic
207C22×C4C2×D811yescharacteristic
208C22×C4C2×D811yescharacteristic
209C22×C4C2×D811yescharacteristic
210C22×C4C22×D411yescharacteristic
211D8—42no—
212D8—42no—
213C2×C8—21no—
214C2×C8—21no—
215C2×C8—21no—
216C2×C8—21no—
217C2×C8—21no—
218C2×C8—21no—
219C2×C8—21no—
220C2×C8—44no—
221C2×C8—42no—
222C2×C8—42no—
223C2×C8#13512yes—
224C2×C8#13712yes—
225C2×C8C2×#4112yes—
226C2×C8H(5,𝔽2)12yes—
227C2×C82−1+411yescharacteristic
228C2×C82−1+411yescharacteristic
229C4⋊C4—42no—
230C4⋊C4—41no—
231C4⋊C4—41no—
232C4⋊C4—42no—
233C4⋊C4—41no—
234C4⋊C4—41no—
235C4⋊C4—42no—
236C4⋊C4—41no—
237C4⋊C4—41no—
238C4⋊C4—42no—
239C4⋊C4—82no—
240C4⋊C4—82no—
241C4⋊C4—84no—
242C4⋊C4—82no—
243C4⋊C4—82no—
244C4⋊C4—82no—
245C22⋊C4—82no—
246C22⋊C4—84no—
247C22⋊C4—82no—
248C22⋊C4—82no—
249C22⋊C4—82no—
250C22⋊C4—82no—
251C22⋊C4—82no—
252C42—21no—
253C42—21no—
254C42—21no—
255C42—21no—
256C42—41no—
257C42—41no—
258C16—24no—
259C16—42no—
260C23—21no—
261C23—21no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23—81no—
267C23—81no—
268C23—81no—
269C23—81no—
270C23—81no—
271C23—81no—
272C23—81no—
273C23C2×D1612yes—
274C23C22×D811yescharacteristic
275Q8—164no—
276D4—81no—
277D4—81no—
278D4—81no—
279D4—81no—
280D4—164no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—44no—
293C2×C4—44no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—81no—
299C2×C4—81no—
300C2×C4—82no—
301C2×C4—82no—
302C2×C4—81no—
303C2×C4—81no—
304C2×C4—82no—
305C2×C4—81no—
306C2×C4—81no—
307C2×C4—82no—
308C2×C4—81no—
309C2×C4—82no—
310C2×C4—81no—
311C2×C4—82no—
312C2×C4—81no—
313C2×C4C2×D1612yes—
314C2×C4C2×D1612yes—
315C2×C4#50812yes—
316C2×C4#51012yes—
317C2×C4#53211yescharacteristic
318C2×C4#53411yescharacteristic
319C2×C4#53611yescharacteristic
320C2×C4C2×#13511yescharacteristic
321C2×C4C2×#13711yescharacteristic
322C2×C4#57511yescharacteristic
323C8—22no—
324C8—22no—
325C8—41no—
326C8#53912yes—
327C8#54212yes—
328C22—22no—
329C22—22no—
330C22—22no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22—161no—
336C22—161no—
337C22—161no—
338C22—161no—
339C22—161no—
340C22—161no—
341C22—161no—
342C22#316911yescharacteristic
343C22#318611yescharacteristic
344C22#322611yescharacteristic
345C22C22×D1611yescharacteristic
346C22C2×#50811yescharacteristic
347C22C2×#51011yescharacteristic
348C22#341411yescharacteristic
349C4—22no—
350C4—22no—
351C4—22no—
352C4—22no—
353C4—41no—
354C4—161no—
355C4—161no—
356C4—162no—
357C4—161no—
358C4—162no—
359C4#266311yescharacteristic
360C4#317211yescharacteristic
361C4#319011yescharacteristic
362C4#323311yescharacteristic
363C2—42no—
364C2—321no—
365C2#3035611yescharacteristic
366C2#3373311yescharacteristic
367C2#3373811yescharacteristic
368C2#3375211yescharacteristic
369C2#3375611yescharacteristic
370C2#3379211yescharacteristic
371C2#3379911yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC2×#466C22
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
1. #33209C2semidirect
3. #13537C2semidirect
4. #13526C2semidirect
5. #13492C2semidirect
6. #13451C2semidirect
8. #13363C2semidirect
9. #13165C2semidirect
10. #13147C2semidirect
11. C4×C22⋊C16C2semidirect

Maximal subgroups

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

subgroupindex
#332092
#331292
#135372
#135262
#134922
#134512
#133802
#133632
#131652
#131472
C4×C22⋊C162