Finite Group Databasebeta

← #496762

Subgroups — #496762

Order 512.

509 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20490C211yesmaximal, characteristic
2#13128C212yesmaximal, complemented
3#13107C212yesmaximal, complemented
4#12985C212yesmaximal
5#12914C212yesmaximal, complemented
6#12897C211yesmaximal, characteristic, complemented
7#12881C211yesmaximal, characteristic, complemented
8#12849C211yesmaximal, characteristic, complemented
9#12791C211yesmaximal, characteristic, complemented
10#12606C211yesmaximal, characteristic, complemented
11#12547C211yesmaximal, characteristic, complemented
12C22×M32C2212yes—
13C2×#436C411yescharacteristic
14C2×#435C2211yescharacteristic
15C2×#405C2211yescharacteristic
16C2×C4×M16C2211yescharacteristic
17#2290C411yescharacteristic
18C2×#364C2212yes—
19C2×#364C2212yes—
20C2×#349C2212yes—
21C2×#349C2212yes—
22#2112C2212yes—
23C2×#347C2212yes—
24#1771C2211yescharacteristic
25C8×C4⋊C4C411yescharacteristic
26#1768C411yescharacteristic
27#1753C411yescharacteristic
28#1753C411yescharacteristic
29#1753C2211yescharacteristic
30C8×C22⋊C4C2211yescharacteristic
31#1750C411yescharacteristic
32C4×C22⋊C8C411yescharacteristic
33#1726C2211yescharacteristic
34#1377C2212yes—
35#1377C2212yes—
36#1376C2212yes—
37#1376C2212yes—
38#1375C2211yescharacteristic
39#1375C2212yes—
40#1375C2211yescharacteristic
41#1375C2211yescharacteristic
42#1375C2211yescharacteristic
43C22×M16C2×C411yescharacteristic
44C22×M16C2×C411yescharacteristic
45C22×M16C2311yescharacteristic
46C23×C8C2311yescharacteristic
47C2×C4×D4C2×C411yescharacteristic
48C2×#117C2×C411yescharacteristic
49C22×C42C2×C411yescharacteristic
50C2×M32—22no—
51C2×M32—42no—
52C2×M32—42no—
53C2×M32C2312yes—
54C2×M32C2312yes—
55C2×M32C2312yes—
56C2×M32C2312yes—
57C22×C16—22no—
58#436—24no—
59#435—22no—
60#435—22no—
61C2×C4⋊C8C2×C411yescharacteristic
62C2×C4⋊C8C2×C411yescharacteristic
63C2×C22⋊C8C2×C411yescharacteristic
64C2×C22⋊C8C2×C411yescharacteristic
65C2×C22⋊C8C2×C411yescharacteristic
66C2×C22⋊C8C2×C411yescharacteristic
67#405—42no—
68C4×M16—42no—
69C2×#97C2×C411yescharacteristic
70C2×#97C2×C411yescharacteristic
71C2×#97C2311yescharacteristic
72C2×#97C2311yescharacteristic
73C2×#97C2311yescharacteristic
74C2×C4×C8C2×C411yescharacteristic
75C2×C4×C8C2×C411yescharacteristic
76C2×C4×C8C2×C411yescharacteristic
77C2×C4×C8C2×C411yescharacteristic
78C2×C4×C8C2×C411yescharacteristic
79C2×C4×C8C2311yescharacteristic
80C2×C4×C8C2311yescharacteristic
81#388—21no—
82#387—21no—
83#386—21no—
84#385—21no—
85C4×C4⋊C4C2×C411yescharacteristic
86C4×C22⋊C4C2×C411yescharacteristic
87C4×C22⋊C4C2×C411yescharacteristic
88#376C2×C411yescharacteristic
89#364—42no—
90#364—42no—
91#349—42no—
92#349—42no—
93C22⋊C16—22no—
94C22⋊C16—22no—
95#347—42no—
96#336C2×C411yescharacteristic
97#336C2×C411yescharacteristic
98#336C2×C411yescharacteristic
99#336C2×C411yescharacteristic
100#336C2×C411yescharacteristic
101#336C2×C411yescharacteristic
102C22×D4M1611yescharacteristic
103C23×C4—21no—
104C23×C4C22×C411yescharacteristic
105C23×C4C22×C411yescharacteristic
106C2×M16—22no—
107C2×M16—22no—
108C2×M16—22no—
109C2×M16—22no—
110C2×M16—22no—
111C2×M16—21no—
112C2×M16—22no—
113C2×M16—21no—
114C2×M16—41no—
115C2×M16—41no—
116C22×C8—22no—
117C22×C8—21no—
118C22×C8—21no—
119C22×C8C22×C411yescharacteristic
120C22×C8C22×C411yescharacteristic
121C22×C8C22×C411yescharacteristic
122C22×C8C22×C411yescharacteristic
123C22×C8C22×C411yescharacteristic
124C22×C8C22×C411yescharacteristic
125C22×C8C22×C411yescharacteristic
126C22×C8#4112yes—
127C22×C8#4112yes—
128C22×C8C2411yescharacteristic
129C4×D4—22no—
130C4×D4—22no—
131#117—22no—
132#117—41no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4M1611yescharacteristic
136C2×C22⋊C4—21no—
137C2×C22⋊C4—21no—
138C2×C22⋊C4—21no—
139C2×C22⋊C4M1611yescharacteristic
140C2×C22⋊C4M1611yescharacteristic
141C2×C42—22no—
142C2×C42—21no—
143C2×C42—41no—
144C2×C42C22×C411yescharacteristic
145C2×C42C22×C411yescharacteristic
146C2×C42C22×C411yescharacteristic
147C2×C42C22×C411yescharacteristic
148C2×C42C22×C411yescharacteristic
149M32—22no—
150M32—22no—
151M32—22no—
152M32—22no—
153M32—22no—
154M32—22no—
155M32—22no—
156M32—22no—
157M32—44no—
158C2×C16—22no—
159C2×C16—22no—
160C2×C16—22no—
161C2×C16—22no—
162C2×C16—44no—
163C4⋊C8—21no—
164C4⋊C8—21no—
165C4⋊C8—22no—
166C22⋊C8—22no—
167C22⋊C8—21no—
168C22⋊C8—21no—
169C22⋊C8—22no—
170C22⋊C8M1614yes—
171#97—22no—
172#97—22no—
173#97—22no—
174#97—22no—
175#97—41no—
176C4×C8—22no—
177C4×C8—21no—
178C4×C8—22no—
179C4×C8—21no—
180C4×C8—22no—
181C4×C8—41no—
182C4×C8—41no—
183C4×C8—41no—
184#95—21no—
185#95—21no—
186#95—21no—
187#95—21no—
188#95—21no—
189C24C2×M1611yescharacteristic
190C24C2×M1611yescharacteristic
191C2×D4—22no—
192C2×D4—81no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—81no—
210C22×C4#11712yes—
211C22×C4C2×M1611yescharacteristic
212C22×C4C2×M1612yes—
213C22×C4C2×M1611yescharacteristic
214C22×C4C2×M1611yescharacteristic
215C22×C4C2×M1611yescharacteristic
216C22×C4C23×C411yescharacteristic
217M16—44no—
218M16—44no—
219M16—42no—
220M16—42no—
221C2×C8—21no—
222C2×C8—21no—
223C2×C8—21no—
224C2×C8—21no—
225C2×C8—21no—
226C2×C8—21no—
227C2×C8—21no—
228C2×C8—21no—
229C2×C8—22no—
230C2×C8—22no—
231C2×C8—22no—
232C2×C8—22no—
233C2×C8—21no—
234C2×C8—21no—
235C2×C8—21no—
236C2×C8—21no—
237C2×C8—21no—
238C2×C8—21no—
239C2×C8—21no—
240C2×C8—21no—
241C2×C8—21no—
242C2×C8#11711yescharacteristic
243C2×C8#11711yescharacteristic
244C2×C8C2×M1612yes—
245C2×C8C2×M1612yes—
246C2×C8#13112yes—
247C2×C8C2×#4112yes—
248C2×C8H(5,𝔽2)11yescharacteristic
249C2×C8H(5,𝔽2)12yes—
250C2×C82−1+411yescharacteristic
251C4⋊C4—22no—
252C4⋊C4—42no—
253C4⋊C4—42no—
254C22⋊C4—22no—
255C22⋊C4—22no—
256C22⋊C4—42no—
257C22⋊C4—42no—
258C22⋊C4—81no—
259C42—21no—
260C42—21no—
261C42—21no—
262C42—21no—
263C42—21no—
264C42—21no—
265C42—21no—
266C42—21no—
267C42—21no—
268C42—42no—
269C42#13112yes—
270C16—42no—
271C16—42no—
272C16—42no—
273C16—42no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—41no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23#43212yes—
284C23C22×M1611yescharacteristic
285D4—82no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—22no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—84no—
316C2×C4#43212yes—
317C2×C4C2×#11711yescharacteristic
318C2×C4#51811yescharacteristic
319C2×C4#52011yescharacteristic
320C2×C4C22×M1611yescharacteristic
321C2×C4C2×#13111yescharacteristic
322C2×C4C2×#13111yescharacteristic
323C8—21no—
324C8—21no—
325C8—22no—
326C8—22no—
327C8—22no—
328C8—41no—
329C8—41no—
330C8—41no—
331C8#53812yes—
332C8#53912yes—
333C22—21no—
334C22—21no—
335C22—22no—
336C22—22no—
337C22—22no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—81no—
342C22—81no—
343C22—81no—
344C22—81no—
345C22C2×#43211yescharacteristic
346C22#297411yescharacteristic
347C22#299011yescharacteristic
348C4—22no—
349C4—42no—
350C4—42no—
351C4—41no—
352C4—41no—
353C4—41no—
354C4—81no—
355C4—81no—
356C4#231811yescharacteristic
357C4#292211yescharacteristic
358C4#297411yescharacteristic
359C4#299011yescharacteristic
360C2—21no—
361C2—21no—
362C2—42no—
363C2—81no—
364C2#2200011yescharacteristic
365C2#3342711yescharacteristic
366C2#3342811yescharacteristic
subgroupG/H
centerC2×C4C2×#117
commutatorC22×C4C23×C4
Frattini, cosocleC22×C8C24
socleC22C2×#432
prime radicalC22×D4M16
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×#405 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #13128C2semidirect
3. #13107C2semidirect
5. #12914C2semidirect
6. #12897C2semidirect
7. #12881C2semidirect
8. #12849C2semidirect
9. #12791C2semidirect
10. #12606C2semidirect
11. #12547C2semidirect

Maximal subgroups

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

subgroupindex
#204902
#131282
#131072
#129852
#129142
#128972
#128812
#128492
#127912
#126062
#125472