Finite Group Databasebeta

← #124759

Subgroups — #124759

Order 512.

420 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 102 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
1C4×#467C211yesmaximal, characteristic
2#12783C211yesmaximal, characteristic
3#7874C211yesmaximal, characteristic
4#7120C212yesmaximal, complemented
5#7113C212yesmaximal, complemented
6C2×#467C411yescharacteristic
7C2×C4×Q16C411yescharacteristic
8C2×#423—21no—
9C22×C4×C8C2211yescharacteristic
10C4×C22⋊Q8—21no—
11C2×C4⋊C16—21no—
12C2×C22⋊C16—21no—
13#1969C2211yescharacteristic
14#1942C2211yescharacteristic
15#1933—21no—
16#1892—21no—
17#1808C411yescharacteristic
18C4×#107C411yescharacteristic
19C4×#103—21no—
20#1374C2212yes—
21#1277—21no—
22#1277C2212yes—
23C22×Q16C2×C411yescharacteristic
24C22×M16—41no—
25C23×C8C2×C411yescharacteristic
26C23×C8D411yescharacteristic
27C2×C22⋊Q8—21no—
28C2×C4×Q8—21no—
29C22×C42D411yescharacteristic
30C22×C16—22no—
31#467—22no—
32#467—22no—
33C4×Q16—42no—
34C2×#107C2×C411yescharacteristic
35C2×#107C2×C411yescharacteristic
36C2×#107C2×C411yescharacteristic
37#423—24no—
38#423—42no—
39C2×C4⋊C8—22no—
40C2×C4⋊C8—41no—
41C2×#103—21no—
42C2×#103—21no—
43C2×#103—21no—
44C2×C4×C8—42no—
45C2×C4×C8C2×C411yescharacteristic
46C2×C4×C8D411yescharacteristic
47C2×C4×C8D411yescharacteristic
48C2×C4×C8C2311yescharacteristic
49#391—21no—
50#389—21no—
51#387—41no—
52#385—41no—
53#380—21no—
54C4×C4⋊C4—21no—
55C4×C4⋊C4—41no—
56C4×C22⋊C4—41no—
57C4⋊C16—42no—
58C22⋊C16—42no—
59#340—21no—
60#340—21no—
61#340—21no—
62C22×Q8—21no—
63C23×C4C22⋊C411yescharacteristic
64C23×C4D811yescharacteristic
65C23×C4SD1611yescharacteristic
66C2×Q16—21no—
67C2×Q16—41no—
68C2×Q16—41no—
69C2×Q16M1612yescomplemented
70C2×M16—44no—
71C2×M16—82no—
72C2×M16—82no—
73C22×C8—21no—
74C22×C8—21no—
75C22×C8—21no—
76C22×C8—21no—
77C22×C8—41no—
78C22×C8—41no—
79C22×C8—41no—
80C22×C8—41no—
81C22×C8—42no—
82C22×C8C22⋊C411yescharacteristic
83C22×C8C22⋊C411yescharacteristic
84C22×C8C22×C411yescharacteristic
85C22×C8C2×D411yescharacteristic
86C22⋊Q8—24no—
87C22⋊Q8—42no—
88C4×Q8—42no—
89C4×Q8—42no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—41no—
94C2×C4⋊C4—41no—
95C2×C4⋊C4—41no—
96C2×C22⋊C4—41no—
97C2×C22⋊C4—41no—
98C2×C42—22no—
99C2×C42—22no—
100C2×C42—41no—
101C2×C42—41no—
102C2×C42C22⋊C411yescharacteristic
103C2×C42D811yescharacteristic
104C2×C42SD1611yescharacteristic
105C2×C42C2×D411yescharacteristic
106C2×C16—42no—
107C2×C16—42no—
108C2×C16—42no—
109#107—21no—
110#107—41no—
111#107—41no—
112#107M1612yescomplemented
113C4⋊C8—24no—
114C4⋊C8—42no—
115C4⋊C8—44no—
116#103—24no—
117#103—42no—
118#103—41no—
119#103—41no—
120C4×C8—22no—
121C4×C8—22no—
122C4×C8—41no—
123C4×C8—41no—
124#95—41no—
125#95—41no—
126#95—41no—
127#95—41no—
128#95—41no—
129C24#10211yescharacteristic
130C2×Q8—22no—
131C2×Q8—41no—
132C2×Q8—41no—
133C2×Q8—41no—
134C2×Q8—41no—
135C2×Q8—41no—
136C22×C4—22no—
137C22×C4—22no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—41no—
150C22×C4—41no—
151C22×C4—41no—
152C22×C4—41no—
153C22×C4#10211yescharacteristic
154C22×C4#10211yescharacteristic
155C22×C4#10211yescharacteristic
156C22×C4C2×C22⋊C411yescharacteristic
157C22×C4C2×D811yescharacteristic
158C22×C4C2×SD1611yescharacteristic
159Q16—42no—
160Q16—42no—
161M16—88no—
162C2×C8—21no—
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—21no—
166C2×C8—21no—
167C2×C8—21no—
168C2×C8—21no—
169C2×C8—21no—
170C2×C8—21no—
171C2×C8—21no—
172C2×C8—42no—
173C2×C8—42no—
174C2×C8—42no—
175C2×C8—42no—
176C2×C8—44no—
177C2×C8—82no—
178C2×C8—82no—
179C2×C8C2×M1611yescharacteristic
180C2×C8C2×M1611yescharacteristic
181C2×C8AGL(1,ℤ/8)11yescharacteristic
182C2×C8#13711yescharacteristic
183C4⋊C4—22no—
184C4⋊C4—44no—
185C4⋊C4—41no—
186C4⋊C4—41no—
187C4⋊C4—41no—
188C4⋊C4—41no—
189C4⋊C4—41no—
190C4⋊C4—82no—
191C4⋊C4—81no—
192C4⋊C4—81no—
193C22⋊C4—44no—
194C22⋊C4—82no—
195C42—22no—
196C42—22no—
197C42—21no—
198C42—21no—
199C42—82no—
200C42—82no—
201C42#9912yes—
202C42#10412yes—
203C16—82no—
204C23—21no—
205C23—21no—
206C23—21no—
207C23—21no—
208C23—21no—
209C23—21no—
210C23#36012yes—
211C23C2×#10211yescharacteristic
212Q8—42no—
213Q8—42no—
214Q8—81no—
215Q8—81no—
216C2×C4—24no—
217C2×C4—24no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—44no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—81no—
236C2×C4—81no—
237C2×C4—81no—
238C2×C4—81no—
239C2×C4—81no—
240C2×C4—81no—
241C2×C4—81no—
242C2×C4—81no—
243C2×C4—81no—
244C2×C4—81no—
245C2×C4#32712yes—
246C2×C4#32712yes—
247C2×C4#35912yes—
248C2×C4#36212yes—
249C2×C4#40711yescharacteristic
250C2×C4C2×#9911yescharacteristic
251C2×C4#41311yescharacteristic
252C2×C4#41711yescharacteristic
253C2×C4C2×#10411yescharacteristic
254C2×C4#42111yescharacteristic
255C2×C4#50911yescharacteristic
256C2×C4#51011yescharacteristic
257C8—21no—
258C8—21no—
259C8—21no—
260C8—21no—
261C8—41no—
262C8—41no—
263C8—84no—
264C22—22no—
265C22—22no—
266C22—22no—
267C22—41no—
268C22—41no—
269C22—41no—
270C22—41no—
271C22#148611yescharacteristic
272C22C2×#32711yescharacteristic
273C22#154711yescharacteristic
274C22#213911yescharacteristic
275C22C2×#35911yescharacteristic
276C22C2×#36011yescharacteristic
277C22C2×#36211yescharacteristic
278C4—21no—
279C4—22no—
280C4—21no—
281C4—41no—
282C4—41no—
283C4—82no—
284C4—82no—
285C4—161no—
286C4—161no—
287C4#135014yes—
288C4#146511yescharacteristic
289C4#148911yescharacteristic
290C4#151311yescharacteristic
291C4#152911yescharacteristic
292C2—42no—
293C2#761611yescharacteristic
294C2#1166011yescharacteristic
295C2#1166411yescharacteristic
296C2#1170411yescharacteristic
297C2C2×#135011yescharacteristic
298C2#1197711yescharacteristic
299C2#1198811yescharacteristic
subgroupG/H
center, socleC23C2×#102
commutatorC22×C8C22×C4
Frattini, cosocleC2×C4×C8C23
prime radicalC24#102
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G
LECSG ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7120C2semidirect
5. #7113C2semidirect
69. C2×Q16M16semidirect
112. #107M16semidirect

Maximal subgroups

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

subgroupindex
C4×#4672
#127832
#78742
#71202
#71132