Finite Group Databasebeta

← #129467

Subgroups — #129467

Order 512.

944 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 944 conjugacy classes of subgroups — too many to draw. Only its 106 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×#2566C211yesmaximal, characteristic
2C2×#1325C212yesmaximal, complemented
3C2×#1324C212yesmaximal, complemented
4#8301C212yesmaximal
5C22×#125C411yescharacteristic
6C22×C4×D4C2211yescharacteristic
7#2566—24no—
8#2566—24no—
9C2×#396—22no—
10C2×#390C411yescharacteristic
11C2×#390C411yescharacteristic
12C2×#385—22no—
13C2×#384C2211yescharacteristic
14C2×#382C2211yescharacteristic
15C2×C4×C4⋊C4C411yescharacteristic
16#1779—22no—
17C2×#324C2214yes—
18C2×#323—22no—
19#1325—24no—
20#1324—24no—
21C23×D4C2×C411yescharacteristic
22C24×C4D412yes—
23C2×#125—22no—
24C2×#125—22no—
25C2×#125—42no—
26C2×#125—42no—
27C2×C4×D4—22no—
28C2×C4×D4—22no—
29C2×C4×D4—42no—
30C2×C4×D4—42no—
31C22×C4⋊C4—21no—
32C22×C4⋊C4—21no—
33C22×C4⋊C4—21no—
34C22×C4⋊C4—22no—
35C22×C4⋊C4C2×C411yescharacteristic
36C22×C4⋊C4C2×C411yescharacteristic
37C22×C4⋊C4C2311yescharacteristic
38C22×C22⋊C4—22no—
39C22×C22⋊C4—22no—
40C22×C22⋊C4D412yes—
41C22×C42C2×C411yescharacteristic
42C22×C42C2×C411yescharacteristic
43C22×C42C2×C411yescharacteristic
44C2×C22⋊C8—24no—
45C2×C22⋊C8—44no—
46#396—216no—
47#390—22no—
48#390—22no—
49#390—22no—
50#390—22no—
51#385—216no—
52#384—22no—
53#384D414yes—
54#382—22no—
55#382D414yes—
56C4×C4⋊C4—22no—
57C4×C4⋊C4—22no—
58C2×#95—21no—
59C2×#95—22no—
60#324—24no—
61#324—24no—
62#323—44no—
63C25C22⋊C412yes—
64C22×D4—24no—
65C22×D4—21no—
66C22×D4—21no—
67C22×D4—24no—
68C22×D4—41no—
69C22×D4—41no—
70C23×C4—21no—
71C23×C4—21no—
72C23×C4—21no—
73C23×C4—21no—
74C23×C4—24no—
75C23×C4—24no—
76C23×C4—42no—
77C23×C4—84no—
78C23×C4C22⋊C411yescharacteristic
79C23×C4C22⋊C411yescharacteristic
80C23×C4C22×C411yescharacteristic
81C23×C4C2×D412yes—
82C22×C8—44no—
83#125—48no—
84#125—48no—
85C4×D4—48no—
86C4×D4—48no—
87C2×C4⋊C4—24no—
88C2×C4⋊C4—24no—
89C2×C4⋊C4—24no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4—28no—
93C2×C4⋊C4—41no—
94C2×C4⋊C4—41no—
95C2×C4⋊C4—41no—
96C2×C4⋊C4—41no—
97C2×C4⋊C4—41no—
98C2×C4⋊C4—41no—
99C2×C4⋊C4—44no—
100C2×C4⋊C4—42no—
101C2×C4⋊C4—42no—
102C2×C4⋊C4—42no—
103C2×C4⋊C4—42no—
104C2×C4⋊C4—84no—
105C2×C4⋊C4C22⋊C412yes—
106C2×C4⋊C4C22⋊C412yes—
107C2×C4⋊C4C2×D412yes—
108C2×C4⋊C4C2×D412yes—
109C2×C22⋊C4—28no—
110C2×C22⋊C4—24no—
111C2×C22⋊C4—24no—
112C2×C22⋊C4—28no—
113C2×C22⋊C4—42no—
114C2×C22⋊C4—48no—
115C2×C22⋊C4—84no—
116C2×C42—22no—
117C2×C42—22no—
118C2×C42—21no—
119C2×C42—21no—
120C2×C42—24no—
121C2×C42—24no—
122C2×C42—21no—
123C2×C42—21no—
124C2×C42C22⋊C412yes—
125C2×C42C22⋊C412yes—
126C22⋊C8—44no—
127C22⋊C8—44no—
128C22⋊C8—88no—
129#95—24no—
130#95—28no—
131C24—22no—
132C24—22no—
133C24—22no—
134C24—22no—
135C24—22no—
136C24—22no—
137C24—44no—
138C24—44no—
139C24#10114yes—
140C24C2×C22⋊C411yescharacteristic
141C2×D4—24no—
142C2×D4—24no—
143C2×D4—48no—
144C2×D4—48no—
145C2×D4—42no—
146C2×D4—42no—
147C2×D4—42no—
148C2×D4—42no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—22no—
152C22×C4—24no—
153C22×C4—21no—
154C22×C4—22no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—42no—
164C22×C4—48no—
165C22×C4—48no—
166C22×C4—42no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4—44no—
170C22×C4—44no—
171C22×C4—42no—
172C22×C4—44no—
173C22×C4—42no—
174C22×C4—42no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4—41no—
179C22×C4—84no—
180C22×C4—88no—
181C22×C4—88no—
182C22×C4#9912yes—
183C22×C4#9912yes—
184C22×C4C2×C22⋊C411yescharacteristic
185C22×C4C2×C22⋊C411yescharacteristic
186C22×C4#12014yes—
187C22×C4#13114yes—
188C2×C8—84no—
189C2×C8—88no—
190C4⋊C4—42no—
191C4⋊C4—42no—
192C4⋊C4—42no—
193C4⋊C4—42no—
194C4⋊C4—48no—
195C4⋊C4—42no—
196C4⋊C4—42no—
197C4⋊C4—48no—
198C4⋊C4—48no—
199C4⋊C4—88no—
200C22⋊C4—48no—
201C22⋊C4—416no—
202C22⋊C4—88no—
203C42—24no—
204C42—24no—
205C42—24no—
206C42—24no—
207C42—42no—
208C42—42no—
209C23—22no—
210C23—22no—
211C23—24no—
212C23—24no—
213C23—24no—
214C23—48no—
215C23—44no—
216C23—44no—
217C23—48no—
218C23—44no—
219C23—44no—
220C23—42no—
221C23—42no—
222C23—42no—
223C23—42no—
224C23—84no—
225C23—82no—
226C23—84no—
227C23—82no—
228C23#37911yescharacteristic
229C23#40812yes—
230C23C2×#9911yescharacteristic
231C23C2×#9911yescharacteristic
232C23C2×#10112yes—
233D4—48no—
234D4—48no—
235C2×C4—24no—
236C2×C4—24no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—24no—
242C2×C4—24no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—24no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—48no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—44no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—44no—
255C2×C4—42no—
256C2×C4—44no—
257C2×C4—42no—
258C2×C4—44no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—82no—
264C2×C4—816no—
265C2×C4—84no—
266C2×C4—82no—
267C2×C4—82no—
268C2×C4—82no—
269C2×C4—82no—
270C2×C4#40812yes—
271C2×C4#40812yes—
272C8—164no—
273C22—24no—
274C22—24no—
275C22—21no—
276C22—21no—
277C22—21no—
278C22—21no—
279C22—48no—
280C22—48no—
281C22—44no—
282C22—84no—
283C22—84no—
284C22—84no—
285C22—84no—
286C22—82no—
287C22—82no—
288C22#147212yes—
289C22#147212yes—
290C22#178011yescharacteristic
291C22#178111yescharacteristic
292C22#178411yescharacteristic
293C22#212712yes—
294C22#213212yes—
295C4—22no—
296C4—22no—
297C4—44no—
298C4—44no—
299C4—42no—
300C4—42no—
301C4—82no—
302C4—82no—
303C4—82no—
304C2—21no—
305C2—21no—
306C2—22no—
307C2—84no—
308C2#830811yescharacteristic
309C2#1157214yes—
310C2#1269611yescharacteristic
311C2#1270111yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC23×C4C22×C4
Frattini, cosocleC22×C4⋊C4C23
prime radicalC23×D4C2×C4
LCSG ▷ C23×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4×D4 ◁ 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
3. C2×#1324C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25662
C2×#13252
C2×#13242
#83012