Finite Group Databasebeta

← #496776

Subgroups — #496776

Order 512.

490 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20490C211yesmaximal, characteristic
2#13194C212yesmaximal
3#13163C212yesmaximal, complemented
4#13128C212yesmaximal, complemented
5#13128C212yesmaximal, complemented
6#12850C211yesmaximal, characteristic, complemented
7#12849C211yesmaximal, characteristic, complemented
8#12791C211yesmaximal, characteristic, complemented
9#12790C211yesmaximal, characteristic, complemented
10#12657C211yesmaximal, characteristic, complemented
11#12593C211yesmaximal, characteristic, complemented
12C2×#436C2211yescharacteristic
13C2×#435C411yescharacteristic
14C2×#405C2211yescharacteristic
15C2×C4×M16C2211yescharacteristic
16#2290C411yescharacteristic
17C2×#364C2212yes—
18C2×#364C2212yes—
19#2150C2212yes—
20C2×#349C2212yes—
21C2×#349C2212yes—
22C2×#349C2212yes—
23C2×#349C2212yes—
24C2×#347C2212yes—
25C4×M32C2212yes—
26#1771C411yescharacteristic
27C8×C4⋊C4C2211yescharacteristic
28#1768C411yescharacteristic
29#1753C411yescharacteristic
30#1753C2211yescharacteristic
31#1753C2211yescharacteristic
32C8×C22⋊C4C411yescharacteristic
33#1750C411yescharacteristic
34C4×C22⋊C8C411yescharacteristic
35#1726C2211yescharacteristic
36#1376C2212yes—
37#1376C2212yes—
38#1375C2211yescharacteristic
39#1375C2212yes—
40#1375C2211yescharacteristic
41#1375C2211yescharacteristic
42#1375C2211yescharacteristic
43C22×M16C2×C411yescharacteristic
44C22×M16C2311yescharacteristic
45C22×M16C2311yescharacteristic
46C23×C8C2×C411yescharacteristic
47C2×C4×D4C2×C411yescharacteristic
48C2×#117C2×C411yescharacteristic
49C22×C42C2×C411yescharacteristic
50C2×M32C2312yes—
51C2×M32C2312yes—
52C2×M32C2312yes—
53C2×M32C2312yes—
54#436—22no—
55#436—22no—
56#435—24no—
57C2×C4⋊C8C2×C411yescharacteristic
58C2×C4⋊C8C2×C411yescharacteristic
59C2×C22⋊C8C2×C411yescharacteristic
60C2×C22⋊C8C2×C411yescharacteristic
61C2×C22⋊C8C2×C411yescharacteristic
62C2×C22⋊C8C2×C411yescharacteristic
63#405—42no—
64C4×M16—42no—
65C2×#97C2×C411yescharacteristic
66C2×#97C2×C411yescharacteristic
67C2×#97C2×C411yescharacteristic
68C2×#97C2311yescharacteristic
69C2×#97C2311yescharacteristic
70C2×C4×C8C2×C411yescharacteristic
71C2×C4×C8C2×C411yescharacteristic
72C2×C4×C8C2×C411yescharacteristic
73C2×C4×C8C2×C411yescharacteristic
74C2×C4×C8C2311yescharacteristic
75C2×C4×C8C2311yescharacteristic
76C2×C4×C8C2311yescharacteristic
77#388—21no—
78#387—21no—
79#386—21no—
80#385—21no—
81C4×C4⋊C4C2×C411yescharacteristic
82C4×C22⋊C4C2×C411yescharacteristic
83C4×C22⋊C4C2×C411yescharacteristic
84#376C2×C411yescharacteristic
85#364—42no—
86#364—42no—
87C4⋊C16—22no—
88C4⋊C16—22no—
89#349—42no—
90#349—42no—
91#349—42no—
92#349—42no—
93#347—42no—
94#346—22no—
95C4×C16—22no—
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—21no—
111C2×M16—21no—
112C2×M16—22no—
113C2×M16—22no—
114C2×M16—21no—
115C2×M16—21no—
116C2×M16—41no—
117C22×C8—22no—
118C22×C8—22no—
119C22×C8—41no—
120C22×C8C22×C411yescharacteristic
121C22×C8C22×C411yescharacteristic
122C22×C8C22×C411yescharacteristic
123C22×C8C22×C411yescharacteristic
124C22×C8C22×C411yescharacteristic
125C22×C8C22×C411yescharacteristic
126C22×C8C22×C411yescharacteristic
127C22×C8C2411yescharacteristic
128C4×D4—22no—
129C4×D4—22no—
130#117—22no—
131#117—41no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4M1611yescharacteristic
135C2×C22⋊C4—21no—
136C2×C22⋊C4—21no—
137C2×C22⋊C4—21no—
138C2×C22⋊C4M1611yescharacteristic
139C2×C22⋊C4M1611yescharacteristic
140C2×C42—22no—
141C2×C42—21no—
142C2×C42—41no—
143C2×C42C22×C411yescharacteristic
144C2×C42C22×C411yescharacteristic
145C2×C42C22×C411yescharacteristic
146C2×C42C22×C411yescharacteristic
147C2×C42C22×C411yescharacteristic
148M32—22no—
149M32—22no—
150M32—22no—
151M32—22no—
152M32—22no—
153M32—22no—
154M32—22no—
155M32—22no—
156C2×C16—22no—
157C2×C16—22no—
158C2×C16—22no—
159C2×C16—22no—
160C4⋊C8—22no—
161C4⋊C8M1614yes—
162C22⋊C8—21no—
163C22⋊C8—21no—
164C22⋊C8—21no—
165C22⋊C8—21no—
166C22⋊C8—22no—
167C22⋊C8—22no—
168#97—21no—
169#97—22no—
170#97—21no—
171#97—22no—
172#97—41no—
173#97—41no—
174C4×C8—22no—
175C4×C8—22no—
176C4×C8—22no—
177C4×C8—22no—
178C4×C8—41no—
179C4×C8—41no—
180C4×C8#4112yes—
181C4×C8#4112yes—
182#95—21no—
183#95—21no—
184#95—21no—
185#95—21no—
186#95—21no—
187C24C2×M1611yescharacteristic
188C24C2×M1611yescharacteristic
189C2×D4—22no—
190C2×D4—81no—
191C22×C4—21no—
192C22×C4—21no—
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×C4C2×M1611yescharacteristic
211C22×C4C2×M1611yescharacteristic
212C22×C4C2×M1611yescharacteristic
213C22×C4C2×M1611yescharacteristic
214C22×C4C23×C411yescharacteristic
215M16—44no—
216M16—42no—
217M16—42no—
218M16—42no—
219M16—42no—
220C2×C8—21no—
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—21no—
230C2×C8—21no—
231C2×C8—21no—
232C2×C8—21no—
233C2×C8—21no—
234C2×C8—21no—
235C2×C8—21no—
236C2×C8—21no—
237C2×C8—44no—
238C2×C8#11711yescharacteristic
239C2×C8#11711yescharacteristic
240C2×C8C2×M1612yes—
241C2×C8C2×M1612yes—
242C2×C8#13112yes—
243C2×C8C2×#4112yes—
244C2×C8H(5,𝔽2)11yescharacteristic
245C2×C8H(5,𝔽2)12yes—
246C2×C82−1+411yescharacteristic
247C4⋊C4—22no—
248C4⋊C4—42no—
249C4⋊C4—42no—
250C22⋊C4—22no—
251C22⋊C4—22no—
252C22⋊C4—42no—
253C22⋊C4—42no—
254C22⋊C4—81no—
255C42—21no—
256C42—21no—
257C42—21no—
258C42—21no—
259C42—21no—
260C42—21no—
261C42—21no—
262C42—42no—
263C42#11712yes—
264C42C2×M1612yes—
265C42#13112yes—
266C16—42no—
267C16—42no—
268C16—42no—
269C16—42no—
270C23—21no—
271C23—21no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23C22×M1611yescharacteristic
281D4—82no—
282C2×C4—22no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—22no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
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—84no—
311C2×C4#43212yes—
312C2×C4#43212yes—
313C2×C4C2×#11711yescharacteristic
314C2×C4#51811yescharacteristic
315C2×C4#52011yescharacteristic
316C2×C4C22×M1611yescharacteristic
317C2×C4C2×#13111yescharacteristic
318C2×C4C2×#13111yescharacteristic
319C8—21no—
320C8—21no—
321C8—22no—
322C8—22no—
323C8—22no—
324C8—41no—
325C8—41no—
326C8—41no—
327C8#54012yes—
328C8#54212yes—
329C22—21no—
330C22—21no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—81no—
338C22—81no—
339C22—81no—
340C22—81no—
341C22C2×#43211yescharacteristic
342C22#297411yescharacteristic
343C22#299011yescharacteristic
344C4—22no—
345C4—22no—
346C4—42no—
347C4—42no—
348C4—41no—
349C4—41no—
350C4—81no—
351C4—81no—
352C4#231911yescharacteristic
353C4#292211yescharacteristic
354C4#297411yescharacteristic
355C4#299111yescharacteristic
356C2—21no—
357C2—21no—
358C2—81no—
359C2—81no—
360C2#2201211yescharacteristic
361C2#3342711yescharacteristic
362C2#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 ◁ #1726 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #13163C2semidirect
4. #13128C2semidirect
5. #13128C2semidirect
6. #12850C2semidirect
7. #12849C2semidirect
8. #12791C2semidirect
9. #12790C2semidirect
10. #12657C2semidirect
11. #12593C2semidirect

Maximal subgroups

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

subgroupindex
#204902
#131942
#131632
#131282
#128502
#128492
#127912
#127902
#126572
#125932