Finite Group Databasebeta

← #500278

Subgroups — #500278

Order 512.

852 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3170C211yesmaximal, characteristic
2C2×#2236C212yesmaximal
3C2×#1942C211yesmaximal, characteristic
4#13451C212yesmaximal, complemented
5#13409C212yesmaximal, complemented
6#12993C212yesmaximal
7#12936C214yesmaximal, complemented
8#12844C211yesmaximal, characteristic
9C2×#533—21no—
10#3170—28no—
11C2×#500C2211yescharacteristic
12C2×#494C2211yescharacteristic
13C2×#487—21no—
14C2×#467—21no—
15C2×#467C2212yes—
16C2×C4×Q16—21no—
17C2×C4×Q16C2212yes—
18C22×#107C2211yescharacteristic
19C22×#107C2211yescharacteristic
20C22×C4×C8C2211yescharacteristic
21C2×#389—21no—
22#2236—28no—
23C2×#366C2214yes—
24C2×#358C2214yes—
25C2×#358C2214yes—
26C2×C22⋊C16C2212yes—
27#1942—21no—
28#1942—21no—
29#1942C2212yes—
30#1942C2212yes—
31C2×#340—21no—
32#1380C2214yes—
33#1374C2212yes—
34#1374C2212yes—
35C22×Q16—21no—
36C22×Q16C2312yes—
37C23×C8D411yescharacteristic
38C23×C8C2311yescharacteristic
39#533—28no—
40#533—44no—
41C2×C4⋊Q8—21no—
42C2×C4⋊Q8—21no—
43C2×#125—41no—
44C2×C22⋊Q8—22no—
45C2×C22⋊Q8—41no—
46C2×C4×Q8—22no—
47C2×C4×Q8—41no—
48C2×#117—41no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—41no—
52C22×C42D411yescharacteristic
53C22×C16C2314yes—
54#500—24no—
55#494—24no—
56#487—28no—
57#467—28no—
58#467—24no—
59#467D418yes—
60C4×Q16—28no—
61C4×Q16—24no—
62C4×Q16—24no—
63C2×#107—21no—
64C2×#107—21no—
65C2×#107—24no—
66C2×#107—24no—
67C2×#107—21no—
68C2×#107C2312yes—
69C2×#107C2312yes—
70C2×#107C2312yes—
71C2×#103—22no—
72C2×#103—22no—
73C2×C4×C8—24no—
74C2×C4×C8D411yescharacteristic
75C2×C4×C8D411yescharacteristic
76C2×C4×C8C2311yescharacteristic
77C2×C4×C8C2311yescharacteristic
78#389—22no—
79#389—22no—
80#389—41no—
81#389—41no—
82C2×#95—41no—
83#366—28no—
84#358—28no—
85#358—28no—
86C22⋊C16—24no—
87#340—22no—
88#340—22no—
89C22×Q8—22no—
90C22×Q8—41no—
91C23×C4—41no—
92C23×C4—41no—
93C23×C4D811yescharacteristic
94C23×C4D811yescharacteristic
95C23×C4C2×D411yescharacteristic
96C2×Q16—24no—
97C2×Q16—22no—
98C2×Q16—24no—
99C2×Q16—24no—
100C2×Q16—42no—
101C2×Q16—42no—
102C2×Q16C2×D414yes—
103C22×C8—22no—
104C22×C8—22no—
105C22×C8—22no—
106C22×C8—22no—
107C22×C8—41no—
108C22×C8C2×D411yescharacteristic
109C22×C8C2×D411yescharacteristic
110C22×C8C2×D414yes—
111C22×C8C2×D411yescharacteristic
112C22×C8C2411yescharacteristic
113C4⋊Q8—24no—
114C4⋊Q8—24no—
115C4⋊Q8—42no—
116C4⋊Q8—42no—
117#125—48no—
118C22⋊Q8—28no—
119C22⋊Q8—48no—
120C22⋊Q8—44no—
121C4×Q8—28no—
122C4×Q8—44no—
123C4×Q8—48no—
124#117—48no—
125C2×C4⋊C4—24no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—42no—
130C2×C4⋊C4—41no—
131C2×C4⋊C4—41no—
132C2×C4⋊C4—41no—
133C2×C4⋊C4—42no—
134C2×C4⋊C4—42no—
135C2×C4⋊C4—41no—
136C2×C4⋊C4—42no—
137C2×C4⋊C4—42no—
138C2×C4⋊C4—41no—
139C2×C4⋊C4—42no—
140C2×C4⋊C4—42no—
141C2×C4⋊C4—84no—
142C2×C22⋊C4—42no—
143C2×C42—22no—
144C2×C42—22no—
145C2×C42—42no—
146C2×C42D811yescharacteristic
147C2×C42D811yescharacteristic
148C2×C42C2×D411yescharacteristic
149C2×C42C2×D411yescharacteristic
150C2×C16—24no—
151C2×C16—24no—
152C2×C16—24no—
153#107—24no—
154#107—24no—
155#107—22no—
156#107—22no—
157#107—24no—
158#107—42no—
159#107C2×D414yes—
160#107#4114yes—
161#103—28no—
162#103—28no—
163C4×C8—22no—
164C4×C8—22no—
165C4×C8—22no—
166C4×C8—22no—
167#95—42no—
168#95—42no—
169C24C2×D811yescharacteristic
170C2×Q8—24no—
171C2×Q8—44no—
172C2×Q8—42no—
173C2×Q8—42no—
174C2×Q8—42no—
175C2×Q8—42no—
176C2×Q8—42no—
177C2×Q8—82no—
178C2×Q8—82no—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—41no—
188C22×C4—44no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4—82no—
194C22×C4—82no—
195C22×C4—82no—
196C22×C4—82no—
197C22×C4—82no—
198C22×C4Q3214yes—
199C22×C4#12012yes—
200C22×C4C2×D811yescharacteristic
201C22×C4C2×D811yescharacteristic
202C22×C4C2×D811yescharacteristic
203C22×C4C2×D811yescharacteristic
204C22×C4C2×D811yescharacteristic
205C22×C4AGL(1,ℤ/8)12yes—
206C22×C4C22×D411yescharacteristic
207Q16—28no—
208Q16—48no—
209Q16—44no—
210C2×C8—21no—
211C2×C8—21no—
212C2×C8—21no—
213C2×C8—21no—
214C2×C8—28no—
215C2×C8—21no—
216C2×C8—21no—
217C2×C8—21no—
218C2×C8—44no—
219C2×C8#12012yes—
220C2×C8#13712yes—
221C2×C8C22×D412yes—
222C2×C8C2×#4112yes—
223C2×C82−1+412yes—
224C4⋊C4—24no—
225C4⋊C4—24no—
226C4⋊C4—44no—
227C4⋊C4—44no—
228C4⋊C4—44no—
229C4⋊C4—48no—
230C4⋊C4—44no—
231C4⋊C4—42no—
232C4⋊C4—42no—
233C4⋊C4—48no—
234C4⋊C4—42no—
235C4⋊C4—42no—
236C4⋊C4—84no—
237C4⋊C4—82no—
238C4⋊C4—84no—
239C22⋊C4—48no—
240C42—21no—
241C42—21no—
242C42—21no—
243C42—21no—
244C42—22no—
245C42—22no—
246C42—48no—
247C16—44no—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—21no—
252C23#44712yes—
253C23C2×Q3212yes—
254C23#51012yes—
255C23C22×D811yescharacteristic
256Q8—44no—
257Q8—44no—
258Q8—88no—
259Q8—82no—
260Q8—82no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—28no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—44no—
278C2×C4—44no—
279C2×C4—44no—
280C2×C4—44no—
281C2×C4—88no—
282C2×C4—82no—
283C2×C4—82no—
284C2×C4—82no—
285C2×C4—82no—
286C2×C4—82no—
287C2×C4—82no—
288C2×C4—82no—
289C2×C4—82no—
290C2×C4—82no—
291C2×C4—164no—
292C2×C4#44712yes—
293C2×C4C2×Q3212yes—
294C2×C4C2×Q3212yes—
295C2×C4#50912yes—
296C2×C4C2×#12011yescharacteristic
297C2×C4#52511yescharacteristic
298C2×C4#53611yescharacteristic
299C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
300C2×C4C2×#13711yescharacteristic
301C2×C4#57511yescharacteristic
302C8—22no—
303C8—22no—
304C8—41no—
305C8#54714yes—
306C22—22no—
307C22—22no—
308C22—22no—
309C22—22no—
310C22—22no—
311C22—22no—
312C22#218814yes—
313C22C2×#44711yescharacteristic
314C22#310011yescharacteristic
315C22#322611yescharacteristic
316C22C22×Q3211yescharacteristic
317C22C2×#50911yescharacteristic
318C22C2×#51011yescharacteristic
319C22#341411yescharacteristic
320C4—22no—
321C4—22no—
322C4—22no—
323C4—22no—
324C4—41no—
325C4—84no—
326C4—84no—
327C4—84no—
328C4—162no—
329C4#240611yescharacteristic
330C4#300711yescharacteristic
331C4#311211yescharacteristic
332C4#323411yescharacteristic
333C2—24no—
334C2#2406011yescharacteristic
335C2C2×#218811yescharacteristic
336C2#3352011yescharacteristic
337C2#3366411yescharacteristic
338C2#3367011yescharacteristic
339C2#3379611yescharacteristic
340C2#3380211yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC24C2×D8
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
4. #13451C2semidirect
5. #13409C2semidirect
7. #12936C2semidirect

Maximal subgroups

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

subgroupindex
C2×#31702
C2×#22362
C2×#19422
#134512
#134092
#129932
#129362
#128442