Finite Group Databasebeta

← #110772

Subgroups — #110772

Order 512.

943 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 943 conjugacy classes of subgroups — too many to draw. Only its 166 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#21415C211yesmaximal, characteristic
2#17541C212yesmaximal
3C4×#376C211yesmaximal, characteristic, complemented
4#8072C212yesmaximal, complemented
5#7867C211yesmaximal, characteristic, complemented
6#3444C2211yescharacteristic
7#2920C2212yes—
8#2920C2211yescharacteristic
9C2×#376—21no—
10C2×#376C2212yes—
11C2×C43—21no—
12C2×C43C2211yescharacteristic
13#1994C412yescomplemented
14#1962C412yescomplemented
15#1896—21no—
16#1896—21no—
17C4×#104—22no—
18#1506C414yescomplemented
19#1505C414yescomplemented
20#1504—24no—
21#1500—24no—
22C4×#95—24no—
23#1300—24no—
24#1276—22no—
25#533C2×C411yescharacteristic
26#533C2×C412yes—
27#532C2×C411yescharacteristic
28#532C2×C412yes—
29#529—22no—
30#529—22no—
31#529—22no—
32#525—21no—
33#525—22no—
34C4×#41—21no—
35C4×#41—22no—
36C22×C42—22no—
37C22×C42—24no—
38C22×C42—22no—
39C22×C42D411yescharacteristic
40C22×C42D411yescharacteristic
41C22×C42C2311yescharacteristic
42#432—22no—
43#432—22no—
44#432C2×C414yes—
45#431—22no—
46#431—22no—
47#431—22no—
48#423—22no—
49#423C2×C412yes—
50C2×#104—22no—
51C2×#104—22no—
52#407—22no—
53#407—21no—
54C4×M16—22no—
55C4×M16—21no—
56#376—28no—
57#376—22no—
58#376—22no—
59#376C2×C414yes—
60#376D414yes—
61C2×#95—24no—
62C2×#95—22no—
63C2×#95D414yes—
64C43—28no—
65C43—22no—
66C43C2×C412yes—
67C43D412yes—
68#339—22no—
69#339—22no—
70#328—28no—
71#327—28no—
72C2×#41—21no—
73C2×#41—22no—
74C23×C4—21no—
75C23×C4—21no—
76C23×C4—24no—
77C23×C4C2×D411yescharacteristic
78C23×C4C2×D412yes—
79C2×M16—22no—
80C2×M16—42no—
81C4⋊Q8C4211yescharacteristic, complemented
82C4⋊Q8C4212yescomplemented
83#127C4211yescharacteristic, complemented
84#126—42no—
85#125—21no—
86#125—22no—
87#124—21no—
88#124—22no—
89#123—42no—
90#123—41no—
91C22⋊Q8—22no—
92C22⋊Q8—22no—
93C22⋊Q8—22no—
94#121—22no—
95#121—22no—
96#121—22no—
97#120—41no—
98C4×Q8—22no—
99C4×Q8—22no—
100C4×Q8—22no—
101C4×D4—22no—
102C4×D4—22no—
103C4×D4—22no—
104C4×D4—42no—
105C4×D4—42no—
106C4×D4—42no—
107#117—21no—
108#117—22no—
109#117—42no—
110#117—41no—
111C2×C42—28no—
112C2×C42—28no—
113C2×C42—24no—
114C2×C42—24no—
115C2×C42—21no—
116C2×C42—21no—
117C2×C42—24no—
118C2×C42—24no—
119C2×C42—216no—
120C2×C42—24no—
121C2×C42—24no—
122C2×C42—48no—
123C2×C42—416no—
124C2×C42—41no—
125C2×C42C22⋊C412yes—
126C2×C42C22⋊C412yes—
127C2×C42C22×C411yescharacteristic
128C2×C42C22×C412yes—
129C2×C42C2×D411yescharacteristic
130C2×C42C2×D412yes—
131C2×C42#4112yes—
132C2×C42#4112yes—
133C2×C42#4112yes—
134C4⋊C8—24no—
135C4⋊C8—24no—
136C4⋊C8—42no—
137#104—44no—
138#104—44no—
139C22⋊C8—24no—
140C22⋊C8—42no—
141C22⋊C8—42no—
142#97—24no—
143#97—41no—
144C4×C8—41no—
145C4×C8—42no—
146#95—216no—
147#95—24no—
148#95—24no—
149#95—24no—
150#95C22⋊C418yes—
151C24#12011yescharacteristic
152#41—42no—
153#41—42no—
154#41—84no—
155C2×Q8—21no—
156C2×Q8—22no—
157C2×D4—21no—
158C2×D4—22no—
159C2×D4—41no—
160C2×D4—42no—
161C22×C4—24no—
162C22×C4—28no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—28no—
167C22×C4—24no—
168C22×C4—28no—
169C22×C4—28no—
170C22×C4—22no—
171C22×C4—44no—
172C22×C4—44no—
173C22×C4—48no—
174C22×C4—48no—
175C22×C4—42no—
176C22×C4—41no—
177C22×C4C2×C22⋊C412yes—
178C22×C4C2×C22⋊C411yescharacteristic
179C22×C4C4×D412yes—
180C22×C4C4×D412yes—
181C22×C4C4×D412yes—
182C22×C4#12112yes—
183C22×C4#12112yes—
184C22×C4#12112yes—
185C22×C4#12311yescharacteristic
186C22×C4#12312yes—
187M16—42no—
188M16—42no—
189M16—84no—
190C2×C8—44no—
191C2×C8—42no—
192C4⋊C4—22no—
193C4⋊C4—22no—
194C4⋊C4—22no—
195C4⋊C4—42no—
196C4⋊C4—42no—
197C4⋊C4—42no—
198C22⋊C4—41no—
199C22⋊C4—42no—
200C22⋊C4—42no—
201C22⋊C4—42no—
202C22⋊C4—42no—
203C42—24no—
204C42—24no—
205C42—216no—
206C42—216no—
207C42—22no—
208C42—22no—
209C42—48no—
210C42—48no—
211C42—432no—
212C42—42no—
213C42—41no—
214C42#10414yes—
215C42#10414yes—
216C42C2×C4211yescharacteristic
217C42#11712yes—
218C42#11711yescharacteristic
219C42C4×D412yes—
220C42C4×D412yes—
221C42C4×D412yes—
222C42#12412yes—
223C42#12411yescharacteristic
224C42#12612yes—
225C42#12711yescharacteristic
226C23—21no—
227C23—21no—
228C23—24no—
229C23—41no—
230C23#38611yescharacteristic
231C23#38612yes—
232Q8—42no—
233Q8—82no—
234D4—42no—
235D4—82no—
236D4—81no—
237D4—81no—
238D4—84no—
239C2×C4—216no—
240C2×C4—28no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—24no—
244C2×C4—24no—
245C2×C4—24no—
246C2×C4—24no—
247C2×C4—24no—
248C2×C4—216no—
249C2×C4—24no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—416no—
255C2×C4—44no—
256C2×C4—48no—
257C2×C4—416no—
258C2×C4—48no—
259C2×C4—48no—
260C2×C4—41no—
261C2×C4—42no—
262C2×C4—84no—
263C2×C4—82no—
264C2×C4C4×C22⋊C412yes—
265C2×C4C4×C22⋊C411yescharacteristic
266C2×C4#38812yes—
267C2×C4#38812yes—
268C2×C4#38812yes—
269C2×C4#39012yes—
270C2×C4#39011yescharacteristic
271C2×C4C2×#10412yes—
272C2×C4C2×#10412yes—
273C8—84no—
274C8—82no—
275C22—24no—
276C22—24no—
277C22—22no—
278C22—22no—
279C22—22no—
280C22—81no—
281C22—82no—
282C22#143711yescharacteristic
283C22#189611yescharacteristic
284C22#189611yescharacteristic
285C22#190614yes—
286C4—28no—
287C4—22no—
288C4—22no—
289C4—22no—
290C4—22no—
291C4—22no—
292C4—44no—
293C4—416no—
294C4—81no—
295C4—82no—
296C4C4×#10412yes—
297C4#196212yes—
298C2—21no—
299C2—21no—
300C2—24no—
301C2—161no—
302C2#1092311yescharacteristic
303C2#1094512yes—
subgroupG/H
centerC2×C4#390
commutatorC42C2×C42
Frattini, cosocleC22×C42C23
socleC22#1437
prime radical#3444C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C4×#376C2semidirect
4. #8072C2semidirect
5. #7867C2semidirect
13. #1994C4semidirect
14. #1962C4semidirect
18. #1506C4semidirect
19. #1505C4semidirect
81. C4⋊Q8C42semidirect
82. C4⋊Q8C42semidirect
83. #127C42semidirect

Maximal subgroups

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

subgroupindex
#214152
#175412
C4×#3762
#80722
#78672