Finite Group Databasebeta

← #244164

Subgroups — #244164

Order 512.

659 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 659 conjugacy classes of subgroups — too many to draw. Only its 146 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21718C211yesmaximal, characteristic, complemented
2#21708C211yesmaximal, characteristic
3#17936C212yesmaximal, complemented
4#17908C212yesmaximal, complemented
5#10667C212yesmaximal, complemented
6#10662C211yesmaximal, characteristic, complemented
7#10662C211yesmaximal, characteristic, complemented
8#10034C211yesmaximal, characteristic
9#10034C211yesmaximal, characteristic
10#10014C212yesmaximal, complemented
11#8139C211yesmaximal, characteristic, complemented
12#3150C2211yescharacteristic
13#3103C2212yes—
14#3103C2212yes—
15C2×#446C2211yescharacteristic
16C2×C8×Q8C2211yescharacteristic
17C2×#443—22no—
18#2931C2212yes—
19#2399C2211yescharacteristic
20#1985C2211yescharacteristic
21#1985C2211yescharacteristic
22#1975—22no—
23#1963C2212yes—
24#1939—22no—
25#1849C2211yescharacteristic
26#1843C2211yescharacteristic
27#1771—22no—
28#1726C2211yescharacteristic
29#1592—24no—
30#1588—24no—
31#1585—24no—
32SD16⋊C8—24no—
33#1571C2212yes—
34#1571C2212yes—
35#1487C2212yes—
36#1487C2212yes—
37#1480C2212yes—
38#1480C2212yes—
39#1452C2212yes—
40#1275C2211yescharacteristic
41#1275C2211yescharacteristic
42#1274C2212yes—
43C2×#135—22no—
44C2×#131—22no—
45#532C2312yes—
46C2×C4×Q8C2311yescharacteristic
47C2×C4×Q8C2311yescharacteristic
48#497—22no—
49#496—22no—
50#481—22no—
51#481—22no—
52#475—22no—
53#475—22no—
54#460—22no—
55#460—22no—
56#448—22no—
57#448—22no—
58#446—22no—
59#446—22no—
60C8×Q8—22no—
61C8×Q8—22no—
62#443—48no—
63C4×SD16—24no—
64C4×SD16—24no—
65#436—22no—
66#436—22no—
67C8×D4—22no—
68C8×D4—22no—
69#431C2312yes—
70#431C2312yes—
71C2×#108—22no—
72#423C2312yes—
73C2×C4⋊C8—21no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8—22no—
76C2×C4⋊C8C2311yescharacteristic
77C2×C4⋊C8C2311yescharacteristic
78C2×#104—24no—
79#408—22no—
80C2×#97—21no—
81C2×#97D412yes—
82C2×#97C2311yescharacteristic
83C2×C4×C8—21no—
84C2×C4×C8D412yes—
85C2×C4×C8C2311yescharacteristic
86#398—21no—
87#398—21no—
88#395—21no—
89#395—21no—
90C4×C4⋊C4—22no—
91#376C2311yescharacteristic
92#336—22no—
93#336—22no—
94#335—24no—
95#335—24no—
96Q8⋊C8—24no—
97Q8⋊C8—24no—
98D4⋊C8—24no—
99D4⋊C8—24no—
100#322D414yes—
101#322D414yes—
102C2×#41—22no—
103C22×Q8#4111yescharacteristic
104C22×Q8#4111yescharacteristic
105#135—44no—
106#135—44no—
107C2×Q16—22no—
108C2×SD16—24no—
109C2×SD16—24no—
110C2×SD16—24no—
111C2×D8—22no—
112#131—48no—
113C2×M16—22no—
114C2×M16—42no—
115C22×C8—21no—
116C22×C8—21no—
117C22×C8—42no—
118C22×C8C2×D411yescharacteristic
119C22×C8C2×D411yescharacteristic
120C22×C8C2×D412yes—
121C4⋊Q8#4112yes—
122#127#4112yes—
123#124—22no—
124#121—22no—
125#121—22no—
126C4×Q8—22no—
127C4×Q8—22no—
128C4×Q8—22no—
129C4×Q8—22no—
130C4×D4—22no—
131C4×D4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4#4111yescharacteristic
137C2×C4⋊C4#4111yescharacteristic
138C2×C42—21no—
139C2×C42—21no—
140C2×C42C2×D412yes—
141C2×C42C2411yescharacteristic
142#108—44no—
143#106—24no—
144#106—24no—
145C4⋊C8—22no—
146C4⋊C8—22no—
147C4⋊C8—22no—
148C4⋊C8—22no—
149C4⋊C8—22no—
150C4⋊C8—22no—
151C4⋊C8—41no—
152C4⋊C8—41no—
153C4⋊C8—41no—
154C4⋊C8—41no—
155C4⋊C8—44no—
156#104—48no—
157#103—24no—
158#103—24no—
159#102—24no—
160#102—24no—
161C22⋊C8—22no—
162C22⋊C8—22no—
163C22⋊C8—42no—
164C22⋊C8—42no—
165#97—24no—
166#97—22no—
167#97—42no—
168C4×C8—24no—
169C4×C8—42no—
170C4×C8C2×D414yes—
171C4×C8C2×D414yes—
172C4×C8C2×D412yes—
173C4×C8C2×D412yes—
174#95—22no—
175#95—22no—
176#95—21no—
177#95—21no—
178#41—44no—
179#41—44no—
180C2×Q8—22no—
181C2×Q8—22no—
182C2×Q8—22no—
183C2×Q8—42no—
184C2×Q8—81no—
185C2×D4—22no—
186C2×D4—22no—
187C2×D4—22no—
188C2×D4—42no—
189C22×C4—22no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—42no—
195C22×C4C22×D411yescharacteristic
196C22×C4C2×#4111yescharacteristic
197C22×C4C2×#4111yescharacteristic
198Q16—44no—
199SD16—48no—
200SD16—48no—
201SD16—84no—
202D8—44no—
203M16—44no—
204M16—84no—
205C2×C8—22no—
206C2×C8—22no—
207C2×C8—22no—
208C2×C8—22no—
209C2×C8—24no—
210C2×C8—24no—
211C2×C8—41no—
212C2×C8—41no—
213C2×C8—42no—
214C2×C8—42no—
215C2×C8—42no—
216C2×C8—84no—
217C2×C8#12712yes—
218C2×C8#12711yescharacteristic
219C2×C8#12711yescharacteristic
220C2×C8#13714yes—
221C4⋊C4—22no—
222C4⋊C4—22no—
223C4⋊C4—22no—
224C4⋊C4—22no—
225C4⋊C4—44no—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C4⋊C4—82no—
229C22⋊C4—42no—
230C22⋊C4—42no—
231C42—22no—
232C42—41no—
233C42—41no—
234C42—41no—
235C42—41no—
236C42AGL(1,ℤ/8)14yes—
237C42C22×D412yes—
238C42C2×#4112yes—
239C23—42no—
240C23#52511yescharacteristic
241Q8—24no—
242Q8—42no—
243Q8—42no—
244Q8—42no—
245Q8—44no—
246D4—44no—
247D4—44no—
248D4—44no—
249D4—84no—
250C2×C4—22no—
251C2×C4—24no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—42no—
259C2×C4—41no—
260C2×C4—42no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—42no—
266C2×C4—82no—
267C2×C4—82no—
268C2×C4#52512yes—
269C2×C4#52512yes—
270C2×C4#52511yescharacteristic
271C2×C4C2×#12711yescharacteristic
272C2×C4C2×AGL(1,ℤ/8)12yes—
273C2×C4C2×#13712yes—
274C8—24no—
275C8—21no—
276C8—21no—
277C8—22no—
278C8—44no—
279C8—44no—
280C8—81no—
281C8—81no—
282C8—82no—
283C22—21no—
284C22—21no—
285C22—82no—
286C22—82no—
287C22—82no—
288C22#239711yescharacteristic
289C22#311212yes—
290C4—22no—
291C4—22no—
292C4—44no—
293C4—42no—
294C4—81no—
295C4—81no—
296C4—81no—
297C4—82no—
298C4#311012yes—
299C4#315111yescharacteristic
300C4#315211yescharacteristic
301C2—22no—
302C2—162no—
303C2#2379411yescharacteristic
304C2#2379511yescharacteristic
305C2#2380311yescharacteristic
subgroupG/H
centerC2×C4C2×#127
commutator, Frattini, cosocleC2×C42C24
socleC22#2397
prime radical#3150C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #21718C2semidirect
3. #17936C2semidirect
4. #17908C2semidirect
5. #10667C2semidirect
6. #10662C2semidirect
7. #10662C2semidirect
10. #10014C2semidirect
11. #8139C2semidirect

Maximal subgroups

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

subgroupindex
#217182
#217082
#179362
#179082
#106672
#106622
#100342
#100142
#81392