Finite Group Databasebeta

← #129253

Subgroups — #129253

Order 512.

1028 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 1028 conjugacy classes of subgroups — too many to draw. Only its 142 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
1C2×#2317C211yesmaximal, characteristic
2C2×#1325C214yesmaximal, complemented
3#8300C212yesmaximal
4C22×#117C411yescharacteristic
5C22×#117C2211yescharacteristic
6C23×C4⋊C4C411yescharacteristic
7#2317—28no—
8C2×#389—22no—
9C2×#385—22no—
10C2×#384C2212yes—
11C2×#382C412yes—
12#1817—22no—
13C2×#324C2214yes—
14C2×#323—22no—
15#1325—28no—
16C24×C4—21no—
17C24×C4C2×C411yescharacteristic
18C2×#117—22no—
19C2×#117—24no—
20C2×#117—24no—
21C2×#117—24no—
22C2×#117—42no—
23C2×#117—42no—
24C2×#117C814yescomplemented
25C22×C4⋊C4—21no—
26C22×C4⋊C4—21no—
27C22×C4⋊C4—24no—
28C22×C4⋊C4—24no—
29C22×C4⋊C4—42no—
30C22×C4⋊C4—42no—
31C22×C4⋊C4C2×C411yescharacteristic
32C22×C4⋊C4C2×C412yes—
33C22×C4⋊C4C2311yescharacteristic
34C22×C22⋊C4—21no—
35C22×C22⋊C4D412yes—
36C22×C42C2×C412yes—
37C22×C42D412yes—
38C2×C22⋊C8—24no—
39#389—216no—
40#385—216no—
41#384—24no—
42#384D418yes—
43#382—24no—
44#382—24no—
45C2×#95—24no—
46#336—44no—
47#324—28no—
48#323—44no—
49C25C22⋊C411yescharacteristic
50C23×C4—22no—
51C23×C4—22no—
52C23×C4—22no—
53C23×C4—22no—
54C23×C4—22no—
55C23×C4—22no—
56C23×C4—22no—
57C23×C4—22no—
58C23×C4—24no—
59C23×C4—42no—
60C23×C4—42no—
61C23×C4—84no—
62C23×C4C22⋊C412yes—
63C23×C4C22⋊C411yescharacteristic
64C23×C4C2×C812yes—
65C23×C4M1612yes—
66C23×C4C22×C411yescharacteristic
67C23×C4C2×D412yes—
68C22×C8—44no—
69#117—216no—
70#117—216no—
71#117—416no—
72C2×C4⋊C4—24no—
73C2×C4⋊C4—24no—
74C2×C4⋊C4—24no—
75C2×C4⋊C4—24no—
76C2×C4⋊C4—22no—
77C2×C4⋊C4—22no—
78C2×C4⋊C4—216no—
79C2×C4⋊C4—41no—
80C2×C4⋊C4—41no—
81C2×C4⋊C4—41no—
82C2×C4⋊C4—41no—
83C2×C4⋊C4—42no—
84C2×C4⋊C4—42no—
85C2×C4⋊C4—416no—
86C2×C4⋊C4—42no—
87C2×C4⋊C4—42no—
88C2×C4⋊C4—88no—
89C2×C4⋊C4C22⋊C414yes—
90C2×C4⋊C4C2×D414yes—
91C2×C22⋊C4—24no—
92C2×C22⋊C4—24no—
93C2×C22⋊C4—24no—
94C2×C22⋊C4—24no—
95C2×C22⋊C4—24no—
96C2×C22⋊C4—42no—
97C2×C42—22no—
98C2×C42—22no—
99C2×C42—22no—
100C2×C42—24no—
101C2×C42—24no—
102C2×C42—42no—
103C2×C42C22⋊C414yes—
104C2×C42C2×C814yes—
105C2×C42M1614yes—
106C22⋊C8—44no—
107C22⋊C8—44no—
108#95—216no—
109C24—21no—
110C24—21no—
111C24—21no—
112C24—21no—
113C24—42no—
114C24—42no—
115C24C22⋊C812yes—
116C24#9912yes—
117C24#10112yes—
118C24C2×C22⋊C411yescharacteristic
119C22×C4—28no—
120C22×C4—28no—
121C22×C4—28no—
122C22×C4—28no—
123C22×C4—24no—
124C22×C4—24no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—24no—
134C22×C4—24no—
135C22×C4—24no—
136C22×C4—24no—
137C22×C4—24no—
138C22×C4—24no—
139C22×C4—24no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—24no—
144C22×C4—22no—
145C22×C4—22no—
146C22×C4—44no—
147C22×C4—44no—
148C22×C4—44no—
149C22×C4—44no—
150C22×C4—48no—
151C22×C4—48no—
152C22×C4—42no—
153C22×C4—42no—
154C22×C4—88no—
155C22×C4—88no—
156C22×C4—88no—
157C22×C4C22⋊C812yes—
158C22×C4#9912yes—
159C22×C4#10112yes—
160C22×C4C2×C22⋊C412yes—
161C22×C4#12014yes—
162C22×C4C22×C811yescharacteristic
163C22×C4C2×M1612yes—
164C22×C4C2×M1611yescharacteristic
165C2×C8—84no—
166C2×C8—88no—
167C4⋊C4—216no—
168C4⋊C4—44no—
169C4⋊C4—44no—
170C4⋊C4—48no—
171C4⋊C4—44no—
172C4⋊C4—44no—
173C4⋊C4—816no—
174C22⋊C4—216no—
175C22⋊C4—48no—
176C42—24no—
177C42—24no—
178C42—24no—
179C42—48no—
180C42C22⋊C818yes—
181C23—22no—
182C23—22no—
183C23—22no—
184C23—22no—
185C23—22no—
186C23—22no—
187C23—22no—
188C23—22no—
189C23—42no—
190C23—44no—
191C23—42no—
192C23—44no—
193C23—42no—
194C23—42no—
195C23—44no—
196C23—44no—
197C23—42no—
198C23—42no—
199C23#32414yes—
200C23#37911yescharacteristic
201C23C2×C22⋊C811yescharacteristic
202C23#40711yescharacteristic
203C23C2×#9911yescharacteristic
204C23C2×#9911yescharacteristic
205C23C2×#10111yescharacteristic
206C23C2×#10111yescharacteristic
207C2×C4—216no—
208C2×C4—216no—
209C2×C4—22no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—28no—
224C2×C4—24no—
225C2×C4—44no—
226C2×C4—44no—
227C2×C4—48no—
228C2×C4—48no—
229C2×C4—44no—
230C2×C4—44no—
231C2×C4—48no—
232C2×C4—44no—
233C2×C4—44no—
234C2×C4—42no—
235C2×C4—44no—
236C2×C4—82no—
237C2×C4—84no—
238C2×C4—816no—
239C2×C4—84no—
240C2×C4—82no—
241C2×C4#32414yes—
242C2×C4C2×C22⋊C812yes—
243C2×C4#40712yes—
244C8—164no—
245C22—21no—
246C22—21no—
247C22—21no—
248C22—21no—
249C22—22no—
250C22—22no—
251C22—22no—
252C22—22no—
253C22—24no—
254C22—44no—
255C22—44no—
256C22—44no—
257C22—44no—
258C22—42no—
259C22—42no—
260C22—44no—
261C22—44no—
262C22C2×#32411yescharacteristic
263C22C2×#32411yescharacteristic
264C22#146311yescharacteristic
265C22#146311yescharacteristic
266C22#177911yescharacteristic
267C22#178111yescharacteristic
268C22#178411yescharacteristic
269C22#212812yes—
270C22#213512yes—
271C4—24no—
272C4—22no—
273C4—22no—
274C4—48no—
275C4—44no—
276C4—84no—
277C4—82no—
278C2—21no—
279C2—21no—
280C2—21no—
281C2—21no—
282C2—44no—
283C2#829911yescharacteristic
284C2#1145012yes—
285C2#1151612yes—
286C2#1269711yescharacteristic
287C2#1270411yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C22×C8
Frattini, cosocleC22×C4⋊C4C23
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#117 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1325C2semidirect
24. C2×#117C8semidirect

Maximal subgroups

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

subgroupindex
C2×#23172
C2×#13252
#83002