Finite Group Databasebeta

← #498709

Subgroups — #498709

Order 512.

550 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21477C211yesmaximal, characteristic
2#13199C212yesmaximal, complemented
3#13191C211yesmaximal, characteristic, complemented
4#13129C212yesmaximal, complemented
5#13126C212yesmaximal, complemented
6#13124C212yesmaximal, complemented
7#12982C211yesmaximal, characteristic, complemented
8#12979C212yesmaximal, complemented
9#12685C212yesmaximal, complemented
10#3573C2211yescharacteristic
11#3407C2211yescharacteristic, complemented
12#3018C411yescharacteristic
13#2955C2212yes—
14#2931C2211yescharacteristic
15#2151C2212yescomplemented
16#2150C2211yescharacteristic, complemented
17#2117C2212yes—
18#2117C2212yes—
19C2×#350C2212yes—
20C2×#350C2212yes—
21C2×#350C2212yescomplemented
22#2115C2212yes—
23#2115C2211yescharacteristic, complemented
24#2113C2212yes—
25#2110C2212yescomplemented
26C2×C4×C16C2211yescharacteristic, complemented
27#1884C411yescharacteristic
28#1881C412yes—
29#1861C412yes—
30#1858C411yescharacteristic
31#1794C411yescharacteristic
32#1764C2212yes—
33#1757C2211yescharacteristic
34#1379C2212yescomplemented
35#1378C2212yescomplemented
36#1376C2212yescomplemented
37#585C2×C411yescharacteristic
38#575C2×C412yes—
39#568—42no—
40#568—41no—
41#568C2×C411yescharacteristic
42#568C2×C412yes—
43C2×#131—42no—
44C2×#131—41no—
45C2×#131C2312yes—
46C2×#131C2311yescharacteristic
47C2×#131C2311yescharacteristic
48#532C2×C411yescharacteristic
49#504—42no—
50C2×M32—41no—
51C2×M32C2312yes—
52C2×M32C2312yes—
53C2×M32C2312yes—
54C22×C16C2312yes—
55#456—21no—
56#455—21no—
57#454—21no—
58#453—21no—
59#444—42no—
60#443—42no—
61#436—41no—
62C8×D4—41no—
63#430C2×C412yes—
64#423C2×C411yescharacteristic
65#421C2×C412yes—
66#421C2×C412yes—
67C2×#104C2×C412yes—
68#416C2×C412yes—
69#413C2×C411yescharacteristic
70#410C2×C411yescharacteristic
71#408C2×C411yescharacteristic
72#408C2×C411yescharacteristic
73#405C2312yes—
74C2×C4×C8C2311yescharacteristic
75C4⋊C16—41no—
76C4⋊C16—42no—
77#350—42no—
78#350—42no—
79#350—42no—
80#350—42no—
81#350—42no—
82#349—42no—
83#349—41no—
84C22⋊C16—42no—
85#347—42no—
86C4×C16—41no—
87#344C2×C412yes—
88#341C2×C412yes—
89#339C2×C412yes—
902−1+4—21no—
912−1+4—22no—
92H(5,𝔽2)—21no—
93H(5,𝔽2)—22no—
94H(5,𝔽2)—41no—
95C2×#41—41no—
96C2×#41—42no—
97C2×#41C22×C411yescharacteristic
98C2×#41C22×C411yescharacteristic
99C2×#41C22×C412yes—
100#137—22no—
101#137—22no—
102AGL(1,ℤ/8)—22no—
103AGL(1,ℤ/8)—22no—
104#135—42no—
105#131—22no—
106#131—22no—
107#131—22no—
108#131—22no—
109#131—22no—
110#131—22no—
111#131—22no—
112#131—22no—
113#131—42no—
114#131—42no—
115#131—42no—
116#131—42no—
117#131—42no—
118#131—44no—
119C2×M16—22no—
120C2×M16—21no—
121C2×M16—21no—
122C2×M16—42no—
123C2×M16—41no—
124C2×M16—42no—
125C2×M16—44no—
126C2×M16C22×C412yes—
127C2×M16C22×C411yescharacteristic
128C2×M16C22×C411yescharacteristic
129C2×M16C22×C412yes—
130C22×C8—22no—
131C22×C8—21no—
132C22×C8—21no—
133C22×C8—42no—
134C22×C8C22×C411yescharacteristic
135C22×C8C2411yescharacteristic
136C4⋊Q8M1611yescharacteristic
137#127M1611yescharacteristic
138#124M1611yescharacteristic
139#124M1611yescharacteristic
140#121—41no—
141C4×D4—41no—
142#117C22×C412yes—
143C2×C42C22×C411yescharacteristic
144M32—22no—
145M32—22no—
146M32—22no—
147M32—22no—
148M32—22no—
149M32—22no—
150M32—42no—
151C2×C16—22no—
152C2×C16—22no—
153C2×C16—22no—
154C2×C16—22no—
155C2×C16—22no—
156C2×C16—22no—
157C2×C16—42no—
158#108—42no—
159C4⋊C8—41no—
160#104—22no—
161#104—22no—
162#104—42no—
163#104—42no—
164#103—22no—
165#102—22no—
166#101—21no—
167#100—21no—
168#99—41no—
169C22⋊C8—41no—
170C22⋊C8—41no—
171#97—22no—
172C4×C8—21no—
173C4×C8—21no—
174C4×C8—22no—
175#41—22no—
176#41—22no—
177#41—22no—
178#41—22no—
179#41—22no—
180#41—22no—
181#41—22no—
182#41—22no—
183#41—42no—
184#41—44no—
185#41—42no—
186#41—42no—
187#41—42no—
188#41—42no—
189C2×Q8—42no—
190C2×Q8—41no—
191C2×Q8C2×M1611yescharacteristic
192C2×Q8C2×M1611yescharacteristic
193C2×Q8#13112yes—
194C2×D4—21no—
195C2×D4—21no—
196C2×D4—22no—
197C2×D4—41no—
198C2×D4—42no—
199C2×D4—41no—
200C2×D4—42no—
201C2×D4—44no—
202C2×D4C2×M1611yescharacteristic
203C2×D4C2×M1611yescharacteristic
204C2×D4#13112yes—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—22no—
209C22×C4—42no—
210C22×C4C23×C411yescharacteristic
211Q16—42no—
212SD16—42no—
213SD16—42no—
214D8—42no—
215M16—22no—
216M16—22no—
217M16—22no—
218M16—22no—
219M16—22no—
220M16—22no—
221M16—44no—
222M16—42no—
223M16—42no—
224M16—42no—
225M16—42no—
226M16—42no—
227C2×C8—21no—
228C2×C8—22no—
229C2×C8—21no—
230C2×C8—21no—
231C2×C8—22no—
232C2×C8—22no—
233C2×C8—44no—
234C2×C8—42no—
235C2×C8—41no—
236C2×C8—42no—
237C2×C8—41no—
238C2×C8—41no—
239C2×C8—42no—
240C2×C8—41no—
241C2×C8H(5,𝔽2)11yescharacteristic
242C2×C8H(5,𝔽2)12yes—
243C2×C8H(5,𝔽2)12yes—
244C2×C8H(5,𝔽2)11yescharacteristic
245C4⋊C4—22no—
246C4⋊C4—41no—
247C22⋊C4—22no—
248C22⋊C4—41no—
249C22⋊C4—41no—
250C42—21no—
251C42—22no—
252C42C2×M1611yescharacteristic
253C42C2×M1611yescharacteristic
254C16—42no—
255C16—42no—
256C16—42no—
257C16—42no—
258C23—21no—
259C23—21no—
260C23—22no—
261C23—42no—
262C23#56811yescharacteristic
263Q8—22no—
264Q8—22no—
265Q8—22no—
266Q8—22no—
267Q8—42no—
268D4—22no—
269D4—22no—
270D4—22no—
271D4—22no—
272D4—42no—
273D4—44no—
274D4—42no—
275D4—42no—
276D4—42no—
277D4—42no—
278D4—82no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—22no—
284C2×C4—42no—
285C2×C4—44no—
286C2×C4—42no—
287C2×C4—41no—
288C2×C4—42no—
289C2×C4—41no—
290C2×C4—42no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—42no—
294C2×C4#51812yes—
295C2×C4#51811yescharacteristic
296C2×C4C22×M1611yescharacteristic
297C2×C4C2×#13112yes—
298C8—41no—
299C8—42no—
300C8—41no—
301C8—42no—
302C8—41no—
303C8#56011yescharacteristic
304C8#56012yes—
305C8H(3,𝔽4)11yescharacteristic
306C22—41no—
307C22—42no—
308C22—41no—
309C22—41no—
310C22—81no—
311C22—81no—
312C22—82no—
313C22—162no—
314C22#297311yescharacteristic
315C22#297612yes—
316C4—22no—
317C4—41no—
318C4—41no—
319C4—42no—
320C4—41no—
321C4—82no—
322C4#237511yescharacteristic
323C4#297412yes—
324C4#297711yescharacteristic
325C2—21no—
326C2—22no—
327C2—81no—
328C2—81no—
329C2—82no—
330C2#2330711yescharacteristic
subgroupG/H
centerC8H(3,𝔽4)
commutatorC22×C4C23×C4
Frattini, cosocleC22×C8C24
socleC2#23307
prime radical#3018C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C8 ◁ C22×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #13199C2semidirect
3. #13191C2semidirect
4. #13129C2semidirect
5. #13126C2semidirect
6. #13124C2semidirect
7. #12982C2semidirect
8. #12979C2semidirect
9. #12685C2semidirect
11. #3407C22semidirect
15. #2151C22semidirect
16. #2150C22semidirect
21. C2×#350C22semidirect
23. #2115C22semidirect
25. #2110C22semidirect
26. C2×C4×C16C22semidirect
34. #1379C22semidirect
35. #1378C22semidirect
36. #1376C22semidirect

Maximal subgroups

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

subgroupindex
#214772
#131992
#131912
#131292
#131262
#131242
#129822
#129792
#126852