Finite Group Databasebeta

← #497124

Subgroups — #497124

Order 512.

558 conjugacy classes of subgroups.

The subgroup lattice has 327 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33442C211yesmaximal, characteristic, complemented
2#33432C211yesmaximal, characteristic, complemented
3#23794C211yesmaximal, characteristic
4#13200C211yesmaximal, characteristic, complemented
5#13199C211yesmaximal, characteristic, complemented
6#12987C212yesmaximal, complemented
7#12983C212yesmaximal, complemented
8#12981C212yesmaximal, complemented
9#12979C212yesmaximal, complemented
10#12885C211yesmaximal, characteristic, complemented
11#12597C211yesmaximal, characteristic, complemented
12#3407C2212yes—
13C2×#504C2211yescharacteristic
14C23×C16C2211yescharacteristic
15#2955C2211yescharacteristic
16#2955C2212yes—
17#2929C2211yescharacteristic
18#2929C2212yes—
19#2171—22no—
20#2170—22no—
21#2168—22no—
22D4×C16—22no—
23#2153C2211yescharacteristic
24#2151C2211yescharacteristic
25#2117C2212yes—
26C2×#350C2212yes—
27C2×#350C2212yes—
28#2115C2212yes—
29#2113C2211yescharacteristic
30#2112C2211yescharacteristic
31#2112C2212yes—
32#2110C2211yescharacteristic
33#2108C2211yescharacteristic
34#2098C411yescharacteristic
35#2098C412yes—
36#2091C411yescharacteristic
37#2017C411yescharacteristic
38#2010C412yes—
39#2010C411yescharacteristic
40#1747C2211yescharacteristic
41#1379C2212yes—
42#1379C2212yes—
43#1378C2212yes—
44#1378C2212yes—
45#575C2×C411yescharacteristic
46#575C2×C412yes—
47C2×#131C2311yescharacteristic
48C2×#131C2312yes—
49C23×C8C2311yescharacteristic
50#525C2×C411yescharacteristic
51#525C2×C412yes—
52#504—24no—
53#504—42no—
54#504—42no—
55C2×M32—21no—
56C2×M32—42no—
57C2×M32C2312yes—
58C2×M32C2311yescharacteristic
59C2×M32C2312yes—
60C2×M32C2311yescharacteristic
61C22×C16—21no—
62C22×C16—21no—
63C22×C16—21no—
64C22×C16—42no—
65C22×C16C2311yescharacteristic
66C22×C16C2311yescharacteristic
67#444—22no—
68#444—42no—
69#443—22no—
70#443—42no—
71#435—22no—
72#435—42no—
73C8×D4—22no—
74C8×D4—42no—
75#430C2×C411yescharacteristic
76#430C2×C412yes—
77#424C2×C411yescharacteristic
78#424C2×C412yes—
79#421C2×C412yes—
80#421C2×C412yes—
81#421C2×C412yes—
82#408C2×C411yescharacteristic
83#408C2×C412yes—
84#407C2×C411yescharacteristic
85#407C2×C412yes—
86#405C2312yes—
87#405C2311yescharacteristic
88#364—22no—
89C4⋊C16—22no—
90#350—22no—
91#350—22no—
92#350—42no—
93#350—42no—
94#349—42no—
95C22⋊C16—22no—
96C22⋊C16—22no—
97C22⋊C16—42no—
98#347—22no—
99#346D412yescomplemented
100C4×C16D412yescomplemented
101#338C2×C411yescharacteristic
102H(3,ℤ/4)C2×C411yescharacteristic
103H(3,ℤ/4)C2×C412yes—
104C2×#41C22×C411yescharacteristic
105C2×#41C22×C412yes—
106C23×C4C22×C411yescharacteristic
107#137—22no—
108#137—22no—
109#137—22no—
110AGL(1,ℤ/8)—22no—
111AGL(1,ℤ/8)—22no—
112AGL(1,ℤ/8)—22no—
113#135—22no—
114#135—42no—
115#131—22no—
116#131—22no—
117#131—22no—
118#131—22no—
119#131—22no—
120#131—22no—
121C2×M16—21no—
122C2×M16—22no—
123C2×M16C22×C411yescharacteristic
124C2×M16C22×C412yes—
125C2×M16C22×C412yes—
126C2×M16C22×C411yescharacteristic
127C22×C8—21no—
128C22×C8—21no—
129C22×C8—21no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8C22×C411yescharacteristic
134C22×C8C2411yescharacteristic
135#123—21no—
136#123—22no—
137C22⋊Q8—21no—
138C22⋊Q8—22no—
139#121—21no—
140#121—22no—
141#120—21no—
142#120—22no—
143C4×D4—22no—
144C4×D4—42no—
145#117C22×C412yes—
146#117C22×C411yescharacteristic
147M32—22no—
148M32—22no—
149M32—22no—
150M32—22no—
151M32—22no—
152M32—22no—
153M32—42no—
154M32—44no—
155C2×C16—22no—
156C2×C16—21no—
157C2×C16—21no—
158C2×C16—22no—
159C2×C16—22no—
160C2×C16—22no—
161C2×C16—42no—
162C2×C16—42no—
163C2×C16—42no—
164C2×C16—44no—
165C2×C16C2×D411yescharacteristic
166C2×C16C2×D412yes—
167C2×C16C2×D411yescharacteristic
168#108—22no—
169#108—42no—
170C4⋊C8—22no—
171C4⋊C8—42no—
172#104—24no—
173#104—42no—
174#104—42no—
175C22⋊C8—22no—
176C22⋊C8—22no—
177C22⋊C8—42no—
178C22⋊C8—42no—
179#97—22no—
180#97#4112yes—
181C4×C8—22no—
182C4×C8C2×D412yes—
183C24#13111yescharacteristic
184#41—22no—
185#41—22no—
186#41—22no—
187#41—22no—
188#41—22no—
189#41—22no—
190C2×Q8#13111yescharacteristic
191C2×Q8#13112yes—
192C2×D4—22no—
193C2×D4—21no—
194C2×D4—22no—
195C2×D4—42no—
196C2×D4#13112yes—
197C2×D4#13111yescharacteristic
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—21no—
203C22×C4—22no—
204C22×C4#13111yescharacteristic
205C22×C4C23×C411yescharacteristic
206Q16—22no—
207Q16—42no—
208SD16—24no—
209SD16—42no—
210SD16—42no—
211D8—22no—
212D8—42no—
213M16—22no—
214M16—22no—
215M16—22no—
216M16—22no—
217M16—22no—
218M16—22no—
219M16—42no—
220M16—42no—
221M16—42no—
222C2×C8—21no—
223C2×C8—21no—
224C2×C8—22no—
225C2×C8—22no—
226C2×C8—22no—
227C2×C8—41no—
228C2×C8—41no—
229C2×C8—42no—
230C2×C8—41no—
231C2×C8—41no—
232C2×C8—42no—
233C2×C8—42no—
234C2×C8C4×D412yes—
235C2×C8C22×D411yescharacteristic
236C2×C8C2×#4111yescharacteristic
237C2×C8H(5,𝔽2)12yes—
238C2×C8H(5,𝔽2)12yes—
239C4⋊C4—22no—
240C4⋊C4—21no—
241C4⋊C4—22no—
242C4⋊C4—42no—
243C22⋊C4—22no—
244C22⋊C4—22no—
245C22⋊C4—21no—
246C22⋊C4—22no—
247C22⋊C4—42no—
248C22⋊C4—42no—
249C42—22no—
250C42C4×D412yes—
251C16—21no—
252C16—21no—
253C16—22no—
254C16—41no—
255C16—41no—
256C16—42no—
257C16—42no—
258C23—21no—
259C23—22no—
260C23—21no—
261C23—22no—
262C23—42no—
263C23C2×#13111yescharacteristic
264Q8—22no—
265Q8—22no—
266Q8—22no—
267D4—22no—
268D4—22no—
269D4—22no—
270D4—44no—
271D4—42no—
272D4—42no—
273D4—42no—
274D4—84no—
275C2×C4—22no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—42no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4C2×C4×D411yescharacteristic
289C2×C4#51812yes—
290C2×C4C2×#13112yes—
291C2×C4C2×#13111yescharacteristic
292C8—22no—
293C8—41no—
294C8—41no—
295C8—42no—
296C8—42no—
297C8#53511yescharacteristic
298C8#53511yescharacteristic
299C8#54012yes—
300C22—22no—
301C22—24no—
302C22—41no—
303C22—41no—
304C22—42no—
305C22—42no—
306C22—41no—
307C22—41no—
308C22—42no—
309C22—82no—
310C22#292911yescharacteristic
311C22#297712yes—
312C4—42no—
313C4—41no—
314C4—41no—
315C4—42no—
316C4—82no—
317C4#232811yescharacteristic
318C4#292911yescharacteristic
319C4#297712yes—
320C2—21no—
321C2—22no—
322C2—42no—
323C2—81no—
324C2—82no—
325C2#2226111yescharacteristic
subgroupG/H
centerC8#535
commutatorC22×C4C23×C4
Frattini, cosocleC22×C8C24
socleC2#22261
prime radical#23794C2
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
1. #33442C2semidirect
2. #33432C2semidirect
4. #13200C2semidirect
5. #13199C2semidirect
6. #12987C2semidirect
7. #12983C2semidirect
8. #12981C2semidirect
9. #12979C2semidirect
10. #12885C2semidirect
11. #12597C2semidirect
99. #346D4semidirect
100. C4×C16D4semidirect

Maximal subgroups

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

subgroupindex
#334422
#334322
#237942
#132002
#131992
#129872
#129832
#129812
#129792
#128852
#125972