Finite Group Databasebeta

← #105877

Subgroups — #105877

Order 512.

888 conjugacy classes of subgroups; only the 190 normal subgroups are shown.

normal subgroups only

This group has 888 conjugacy classes of subgroups — too many to draw. Only its 190 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#60124C211yesmaximal, characteristic
2#20699C212yesmaximal
3C2×#1298C211yesmaximal, characteristic
4#8049C211yesmaximal, characteristic
5#7925C212yesmaximal
6C22×#117C2211yescharacteristic
7#2924—24no—
8#2924—24no—
9#2921—22no—
10#2920—22no—
11#2905C412yescomplemented
12C2×#407C2211yescharacteristic
13C2×#407C2211yescharacteristic
14C22×C22⋊C8C2212yes—
15#2873C412yescomplemented
16#2280C2212yes—
17#1825C418yescomplemented
18#1824—22no—
19#1823—22no—
20#1735—24no—
21#1298—22no—
22#1297—22no—
23#1282—24no—
24C24×C4C2311yescharacteristic
25C22×M16C2×C412yes—
26C22×M16C2×C412yes—
27C23×C8—22no—
28C2×#117—21no—
29C2×#117—22no—
30C2×#117—22no—
31C2×#117—21no—
32C2×#117—22no—
33C2×#117C2×C414yes—
34C2×#117C2×C411yescharacteristic
35C2×#117C2×C411yescharacteristic
36C2×#117D414yes—
37C2×#117Q812yes—
38C22×C4⋊C4D412yes—
39C22×C22⋊C4—21no—
40C22×C42—21no—
41#433—216no—
42#432—28no—
43#431—28no—
44#424—24no—
45#424—24no—
46#423—24no—
47#407—21no—
48#407—21no—
49#407—22no—
50#407—22no—
51#407D412yes—
52#407D412yes—
53C2×C22⋊C8—22no—
54C2×C22⋊C8—22no—
55C2×C22⋊C8—22no—
56C2×C22⋊C8—22no—
57C2×C22⋊C8—22no—
58C2×C22⋊C8—22no—
59C2×C22⋊C8C2×C414yes—
60C2×C22⋊C8D414yes—
61#405—24no—
62#405—24no—
63C4×M16—24no—
64#381—22no—
65#380—22no—
66#380C2×C414yes—
67#379—22no—
68C4×C22⋊C4—22no—
69C4×C22⋊C4—22no—
70#340—28no—
71#340—28no—
72C25#4111yescharacteristic
73C23×C4—22no—
74C23×C4—44no—
75C23×C4C22×C411yescharacteristic
76C23×C4C22×C412yes—
77C23×C4C2×D412yes—
78C23×C4C2×D411yescharacteristic
79C23×C4C2×D411yescharacteristic
80C23×C4C2×D411yescharacteristic
81C23×C4C2×D412yes—
82C23×C4C2×Q811yescharacteristic
83C23×C4#4112yes—
84C23×C4#4111yescharacteristic
85C2×M16—22no—
86C2×M16—22no—
87C2×M16—22no—
88C2×M16—44no—
89C2×M16—44no—
90C2×M16C22⋊C414yes—
91C22×C8—22no—
92C22×C8—24no—
93C22×C8—48no—
94C22×C8—42no—
95C22×C8—42no—
96C22×C8C22⋊C414yes—
97#117—21no—
98#117—21no—
99#117—21no—
100#117—21no—
101#117—24no—
102#117—24no—
103#117—24no—
104#117—24no—
105#117—24no—
106#117—48no—
107#117C4212yes—
108#117C4212yes—
109#117C22⋊C412yes—
110#117C22⋊C412yes—
111#117C4⋊C418yes—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—24no—
116C2×C4⋊C4C22⋊C414yes—
117C2×C4⋊C4C22⋊C414yes—
118C2×C4⋊C4C4⋊C412yes—
119C2×C4⋊C4C4⋊C412yes—
120C2×C22⋊C4—21no—
121C2×C22⋊C4—21no—
122C2×C22⋊C4—24no—
123C2×C22⋊C4—22no—
124C2×C22⋊C4—24no—
125C2×C22⋊C4—24no—
126C2×C22⋊C4—42no—
127C2×C22⋊C4—42no—
128C2×C42—21no—
129C2×C42—21no—
130C2×C42—24no—
131C2×C42—24no—
132C2×C42—22no—
133C2×C42—42no—
134C4⋊C8—216no—
135C22⋊C8—24no—
136C22⋊C8—24no—
137C22⋊C8—24no—
138C22⋊C8—24no—
139C22⋊C8—24no—
140C22⋊C8—24no—
141C22⋊C8—42no—
142C22⋊C8—42no—
143#97—28no—
144C4×C8—28no—
145#95—24no—
146#95—42no—
147C24—42no—
148C24—44no—
149C24#11711yescharacteristic
150C24C4×D412yes—
151C24#12111yescharacteristic
152C24C22⋊Q811yescharacteristic
153C24#12312yes—
154C22×C4—28no—
155C22×C4—24no—
156C22×C4—24no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—24no—
160C22×C4—44no—
161C22×C4—42no—
162C22×C4—42no—
163C22×C4—44no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4—42no—
167C22×C4—44no—
168C22×C4—44no—
169C22×C4—84no—
170C22×C4C2×C4211yescharacteristic
171C22×C4C2×C22⋊C412yes—
172C22×C4C2×C22⋊C411yescharacteristic
173C22×C4C2×C22⋊C411yescharacteristic
174C22×C4C2×C22⋊C411yescharacteristic
175C22×C4C2×C4⋊C411yescharacteristic
176C22×C4C2×C4⋊C412yes—
177C22×C4#11711yescharacteristic
178C22×C4C4×D412yes—
179C22×C4C4×D412yes—
180C22×C4C4×D412yes—
181C22×C4#12012yes—
182C22×C4#12012yes—
183C22×C4#12112yes—
184C22×C4#12111yescharacteristic
185C22×C4C22⋊Q812yes—
186C22×C4C22⋊Q811yescharacteristic
187C22×C4#12312yes—
188M16—44no—
189M16—44no—
190M16—44no—
191M16—44no—
192C2×C8—24no—
193C2×C8—24no—
194C2×C8—44no—
195C2×C8—44no—
196C2×C8—48no—
197C2×C8—44no—
198C2×C8—84no—
199C2×C8—84no—
200C4⋊C4—22no—
201C4⋊C4—22no—
202C4⋊C4—24no—
203C4⋊C4—24no—
204C4⋊C4—48no—
205C4⋊C4#9514yes—
206C4⋊C4#9514yes—
207C22⋊C4—22no—
208C22⋊C4—22no—
209C22⋊C4—22no—
210C22⋊C4—22no—
211C22⋊C4—28no—
212C22⋊C4—48no—
213C22⋊C4—84no—
214C22⋊C4—84no—
215C42—22no—
216C42—22no—
217C42—22no—
218C42—22no—
219C42—28no—
220C42—84no—
221C23—24no—
222C23—24no—
223C23—24no—
224C23—22no—
225C23—24no—
226C23—41no—
227C23—42no—
228C23—42no—
229C23—41no—
230C23—42no—
231C23—42no—
232C23—42no—
233C23—88no—
234C23C4×C22⋊C411yescharacteristic
235C23#38011yescharacteristic
236C23#38111yescharacteristic
237C23#38512yes—
238C23#38612yes—
239C2×C4—24no—
240C2×C4—22no—
241C2×C4—24no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—24no—
245C2×C4—24no—
246C2×C4—22no—
247C2×C4—24no—
248C2×C4—22no—
249C2×C4—24no—
250C2×C4—24no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—41no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—41no—
258C2×C4—48no—
259C2×C4—84no—
260C2×C4—82no—
261C2×C4—82no—
262C2×C4—84no—
263C2×C4—84no—
264C2×C4—84no—
265C2×C4C2×#9511yescharacteristic
266C2×C4C4×C22⋊C411yescharacteristic
267C2×C4#37911yescharacteristic
268C2×C4#38011yescharacteristic
269C2×C4#38512yes—
270C2×C4#38612yes—
271C2×C4#41014yes—
272C8—44no—
273C8—44no—
274C8—84no—
275C22—22no—
276C22—22no—
277C22—22no—
278C22—22no—
279C22—22no—
280C22—22no—
281C22—22no—
282C22—21no—
283C22—21no—
284C22—22no—
285C22—44no—
286C22—48no—
287C22—84no—
288C22—82no—
289C22—82no—
290C22C22⋊#9511yescharacteristic
291C22#182012yes—
292C22#188114yes—
293C4—22no—
294C4—21no—
295C4—21no—
296C4—22no—
297C4—44no—
298C4—44no—
299C4—164no—
300C4#174012yes—
301C4#178212yes—
302C2—24no—
303C2—22no—
304C2—41no—
305C2—42no—
306C2—41no—
307C2#1072611yescharacteristic
308C2#1078212yes—
subgroupG/H
centerC2×C4#379
commutatorC22×C4C2×C42
Frattini, cosocleC24×C4C23
socleC22C22⋊#95
prime radicalC25#41
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
11. #2905C4semidirect
15. #2873C4semidirect
17. #1825C4semidirect

Maximal subgroups

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

subgroupindex
#601242
#206992
C2×#12982
#80492
#79252