Finite Group Databasebeta

← #503958

Subgroups — #503958

Order 512.

654 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2235C212yesmaximal, complemented
2#28486C211yesmaximal, characteristic
3D4×#107C211yesmaximal, characteristic
4#13382C212yesmaximal, complemented
5#13382C212yesmaximal, complemented
6#13253C211yesmaximal, characteristic, complemented
7#13203C211yesmaximal, characteristic, complemented
8#13189C211yesmaximal, characteristic
9#13177C211yesmaximal, characteristic, complemented
10#12993C212yesmaximal, complemented
11C4×#367C211yesmaximal, characteristic, complemented
12C2×#494C2211yescharacteristic
13C2×#467C2212yes—
14C2×#467C2212yes—
15C2×C8×D4C2211yescharacteristic
16C22×#107C2212yes—
17#2542—21no—
18D4×C4⋊C4—21no—
19#2235—28no—
20C2×#367C2211yescharacteristic
21C2×#367C2212yes—
22C2×#367C2211yescharacteristic
23C2×C4⋊C16C2211yescharacteristic
24C2×#358C2212yes—
25C2×#358C2212yes—
26C2×#358C2212yes—
27C2×#358C2212yes—
28C2×C22⋊C16C2212yes—
29C2×C22⋊C16C2212yes—
30C2×C4×C16C2211yescharacteristic
31#2001—22no—
32#1981—21no—
33#1947—21no—
34#1852—22no—
35#1845C2211yescharacteristic
36#1808C2212yes—
37C4×#107C2211yescharacteristic
38C4×#107C2211yescharacteristic
39#1380C2211yescharacteristic
40#1380C2212yes—
41#1380C2211yescharacteristic
42C22×Q16—21no—
43C22×Q16C2312yes—
44C23×C8C2312yes—
45C2×C4⋊Q8—21no—
46C2×C22⋊Q8—22no—
47C2×C22⋊Q8—22no—
48C2×C4×D4—41no—
49C2×C4×D4D411yescharacteristic
50C22×C4⋊C4—22no—
51C22×C16C2311yescharacteristic
52C22×C16C2312yes—
53C22×C16C2311yescharacteristic
54#494—22no—
55#494—22no—
56#467—24no—
57#467—24no—
58#467—42no—
59#467—42no—
60C8×D4—22no—
61C8×D4—22no—
62C2×#107—24no—
63C2×#107—22no—
64C2×#107—21no—
65C2×#107—42no—
66C2×#107C2311yescharacteristic
67C2×#107C2312yes—
68C2×#107C2311yescharacteristic
69C2×#107C2311yescharacteristic
70C2×#107C2311yescharacteristic
71C2×C4⋊C8D411yescharacteristic
72C2×#103—22no—
73C2×#103—22no—
74C2×C22⋊C8D412yes—
75C2×C4×C8C2311yescharacteristic
76#399—42no—
77#390—41no—
78#389—41no—
79#385—42no—
80#384—21no—
81#384—41no—
82#380—22no—
83C4×C4⋊C4—21no—
84C4×C4⋊C4—21no—
85#367—24no—
86#367—22no—
87#367D414yescomplemented
88C4⋊C16—22no—
89#358—24no—
90#358—24no—
91#358—42no—
92#358—42no—
93C22⋊C16—24no—
94C22⋊C16—42no—
95C4×C16—22no—
96#340—22no—
97#340—21no—
98#340—21no—
99C22×Q8—22no—
100C22×D4D811yescharacteristic
101C23×C4—42no—
102C23×C4C2×D412yes—
103C2×Q16—22no—
104C2×Q16—22no—
105C2×Q16—22no—
106C2×Q16—22no—
107C2×Q16—22no—
108C2×Q16—22no—
109C2×Q16—42no—
110C2×Q16—81no—
111C2×Q16—81no—
112C22×C8—22no—
113C22×C8—22no—
114C22×C8—22no—
115C22×C8—22no—
116C22×C8—22no—
117C22×C8—22no—
118C22×C8C2×D412yes—
119C22×C8C2×D411yescharacteristic
120C22×C8C2411yescharacteristic
121C4⋊Q8—42no—
122C4⋊Q8—42no—
123C22⋊Q8—44no—
124C22⋊Q8—44no—
125C22⋊Q8—82no—
126C22⋊Q8—82no—
127C4×D4—22no—
128C4×D4—22no—
129C4×D4—84no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—44no—
136C2×C4⋊C4—42no—
137C2×C4⋊C4—41no—
138C2×C4⋊C4—42no—
139C2×C4⋊C4—42no—
140C2×C4⋊C4—41no—
141C2×C4⋊C4—41no—
142C2×C4⋊C4—41no—
143C2×C4⋊C4—82no—
144C2×C4⋊C4D811yescharacteristic
145C2×C22⋊C4—42no—
146C2×C22⋊C4—42no—
147C2×C22⋊C4—42no—
148C2×C22⋊C4D812yes—
149C2×C42—41no—
150C2×C42—41no—
151C2×C42C2×D411yescharacteristic
152C2×C16—21no—
153C2×C16—21no—
154C2×C16—22no—
155C2×C16—21no—
156C2×C16—22no—
157C2×C16—21no—
158C2×C16—22no—
159C2×C16—21no—
160C2×C16C2×D412yes—
161#107—21no—
162#107—21no—
163#107—21no—
164#107—22no—
165#107—21no—
166#107—21no—
167#107—22no—
168#107—44no—
169#107—81no—
170#107C2×D412yes—
171#107C2×D412yes—
172#107#4112yes—
173C4⋊C8—22no—
174#103—44no—
175#103—44no—
176C22⋊C8—24no—
177C4×C8#4112yes—
178C4×C8#4112yes—
179#95—41no—
180#95—41no—
181#95—42no—
182C24C2×D812yes—
183C2×Q8—42no—
184C2×Q8—42no—
185C2×Q8—42no—
186C2×Q8—42no—
187C2×Q8—42no—
188C2×Q8—42no—
189C2×D4—22no—
190C2×D4—22no—
191C2×D4SD3214yes—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4—41no—
201C22×C4—41no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—84no—
205C22×C4—84no—
206C22×C4—162no—
207C22×C4C2×D812yes—
208C22×C4C2×D811yescharacteristic
209C22×C4C2×D811yescharacteristic
210C22×C4C22×D411yescharacteristic
211Q16—42no—
212Q16—42no—
213Q16—42no—
214Q16—42no—
215Q16—84no—
216C2×C8—21no—
217C2×C8—21no—
218C2×C8—22no—
219C2×C8—22no—
220C2×C8—22no—
221C2×C8—42no—
222C2×C8—42no—
223C2×C8—42no—
224C2×C8—42no—
225C2×C8#12312yes—
226C2×C8C22×D411yescharacteristic
227C2×C8C2×#4111yescharacteristic
228C2×C8C2×#4111yescharacteristic
229C2×C8C2×#4111yescharacteristic
230C2×C8H(5,𝔽2)11yescharacteristic
231C2×C8H(5,𝔽2)11yescharacteristic
232C4⋊C4—22no—
233C4⋊C4—22no—
234C4⋊C4—22no—
235C4⋊C4—22no—
236C4⋊C4—41no—
237C4⋊C4—41no—
238C4⋊C4—41no—
239C4⋊C4—42no—
240C4⋊C4—41no—
241C4⋊C4—42no—
242C4⋊C4—41no—
243C4⋊C4—84no—
244C4⋊C4—84no—
245C4⋊C4—84no—
246C4⋊C4—82no—
247C4⋊C4—82no—
248C4⋊C4—82no—
249C4⋊C4—161no—
250C22⋊C4—24no—
251C22⋊C4—84no—
252C22⋊C4—84no—
253C22⋊C4—84no—
254C42—82no—
255C42—82no—
256C42#12312yes—
257C42#13712yes—
258C16—22no—
259C16—41no—
260C16—42no—
261C23—22no—
262C23—22no—
263C23—22no—
264C23—22no—
265C23—22no—
266C23—22no—
267C23C2×SD3214yes—
268C23C22×D811yescharacteristic
269Q8—82no—
270Q8—82no—
271Q8—82no—
272Q8—82no—
273D4—28no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—81no—
287C2×C4—81no—
288C2×C4—81no—
289C2×C4—81no—
290C2×C4—81no—
291C2×C4—81no—
292C2×C4—82no—
293C2×C4—81no—
294C2×C4—82no—
295C2×C4—82no—
296C2×C4—81no—
297C2×C4—82no—
298C2×C4—82no—
299C2×C4—82no—
300C2×C4—81no—
301C2×C4—168no—
302C2×C4#48012yes—
303C2×C4#48012yes—
304C2×C4C2×SD3212yes—
305C2×C4#51012yes—
306C2×C4C2×#12311yescharacteristic
307C2×C4D4211yescharacteristic
308C2×C4#54611yescharacteristic
309C2×C4C2×#13711yescharacteristic
310C2×C4#57611yescharacteristic
311C2×C4#57711yescharacteristic
312C8—22no—
313C8—42no—
314C8#53211yescharacteristic
315C8#54011yescharacteristic
316C8#55011yescharacteristic
317C8#55911yescharacteristic
318C22—24no—
319C22—24no—
320C22—24no—
321C22—42no—
322C22—42no—
323C22—42no—
324C22—42no—
325C22C2×#48011yescharacteristic
326C22D4×D811yescharacteristic
327C22#329411yescharacteristic
328C22C22×SD3211yescharacteristic
329C22C2×#51011yescharacteristic
330C22#341611yescharacteristic
331C22#341711yescharacteristic
332C4—22no—
333C4—42no—
334C4—82no—
335C4—82no—
336C4—82no—
337C4—162no—
338C4—161no—
339C4—162no—
340C4#223514yescomplemented
341C4#244511yescharacteristic
342C4#309611yescharacteristic
343C4#327811yescharacteristic
344C4#333311yescharacteristic
345C2—44no—
346C2#2623011yescharacteristic
347C2C2×#223511yescharacteristic
348C2#3365111yescharacteristic
349C2#3384211yescharacteristic
350C2D4×SD3211yescharacteristic
351C2#3386511yescharacteristic
352C2#3389811yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC22×D4D8
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×C8×D4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2235C2semidirect
4. #13382C2semidirect
5. #13382C2semidirect
6. #13253C2semidirect
7. #13203C2semidirect
9. #13177C2semidirect
10. #12993C2semidirect
11. C4×#367C2semidirect
87. #367D4semidirect
340. C4#2235semidirect

Maximal subgroups

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

subgroupindex
C2×#22352
#284862
D4×#1072
#133822
#132532
#132032
#131892
#131772
#129932
C4×#3672