Finite Group Databasebeta

← #149428

Subgroups — #149428

Order 512.

1352 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 1352 conjugacy classes of subgroups — too many to draw. Only its 154 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×#2542C211yesmaximal, characteristic
2C2×#1393C214yesmaximal
3#10761C212yesmaximal
4C22×C4⋊Q8C411yescharacteristic, complemented
5C22×C22⋊Q8—22no—
6C22×C4×D4C2211yescharacteristic
7C22×#99C2214yes—
8#2542—28no—
9C2×#399—22no—
10C2×#390C2212yes—
11C2×#384C412yescomplemented
12C2×C4×C4⋊C4C411yescharacteristic, complemented
13C2×#342—22no—
14C22⋊#95—22no—
15#1393—28no—
16C23×Q8—21no—
17C23×D4C2311yescharacteristic
18C24×C4D412yes—
19C2×C4⋊Q8—24no—
20C2×C4⋊Q8—42no—
21C2×C4⋊Q8—42no—
22C2×C22⋊Q8—216no—
23C2×C22⋊Q8—416no—
24C2×C4×D4—24no—
25C2×C4×D4—42no—
26C2×C4×D4—42no—
27C22×C4⋊C4—21no—
28C22×C4⋊C4—21no—
29C22×C4⋊C4—24no—
30C22×C4⋊C4C2×C411yescharacteristic
31C22×C4⋊C4C2×C412yes—
32C22×C22⋊C4—24no—
33C22×C22⋊C4—24no—
34C22×C22⋊C4D412yes—
35C22×C42C2×C411yescharacteristic
36C22×C42C2×C412yes—
37C2×#99—28no—
38C2×#99—28no—
39C2×#99—24no—
40C2×#99Q818yescomplemented
41#399—216no—
42#390—24no—
43#390D418yes—
44#384—24no—
45#384—24no—
46C4×C4⋊C4—24no—
47C2×#95—21no—
48C2×#95—44no—
49#342—44no—
50C25C2×D412yes—
51C22×Q8—24no—
52C22×Q8—48no—
53C22×Q8—42no—
54C22×Q8—42no—
55C22×D4—41no—
56C22×D4—41no—
57C22×D4—42no—
58C22×D4C2×D414yes—
59C22×D4C2×Q814yes—
60C22×D4#4114yes—
61C23×C4—22no—
62C23×C4—22no—
63C23×C4—24no—
64C23×C4—24no—
65C23×C4—42no—
66C23×C4—48no—
67C23×C4—44no—
68C23×C4C22⋊C412yes—
69C23×C4C22⋊C411yescharacteristic
70C23×C4C22⋊C411yescharacteristic
71C23×C4C22×C411yescharacteristic
72C4⋊Q8—416no—
73C22⋊Q8—464no—
74C4×D4—416no—
75C2×C4⋊C4—24no—
76C2×C4⋊C4—28no—
77C2×C4⋊C4—216no—
78C2×C4⋊C4—24no—
79C2×C4⋊C4—22no—
80C2×C4⋊C4—22no—
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⋊C4C22⋊C414yes—
89C2×C22⋊C4—216no—
90C2×C22⋊C4—24no—
91C2×C22⋊C4—24no—
92C2×C22⋊C4—28no—
93C2×C22⋊C4—42no—
94C2×C22⋊C4—416no—
95C2×C22⋊C4—44no—
96C2×C22⋊C4—44no—
97C2×C22⋊C4—44no—
98C2×C22⋊C4—44no—
99C2×C22⋊C4—44no—
100C2×C42—22no—
101C2×C42—22no—
102C2×C42—22no—
103C2×C42—22no—
104C2×C42—44no—
105C2×C42C22⋊C414yes—
106#99—216no—
107#99—48no—
108#95—24no—
109#95—88no—
110C24—22no—
111C24—22no—
112C24—22no—
113C24—22no—
114C24—44no—
115C24—44no—
116C24C2×C22⋊C411yescharacteristic
117C24#12014yes—
118C24C22⋊Q814yes—
119C24#12412yes—
120C24C4⋊Q812yes—
121C2×Q8—416no—
122C2×Q8—48no—
123C2×Q8—88no—
124C2×Q8—84no—
125C2×Q8—84no—
126C2×D4—44no—
127C2×D4—44no—
128C2×D4—44no—
129C2×D4—44no—
130C2×D4—44no—
131C2×D4—44no—
132C2×D4C22⋊Q818yes—
133C2×D4#12414yes—
134C2×D4C4⋊Q814yes—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—22no—
140C22×C4—24no—
141C22×C4—28no—
142C22×C4—21no—
143C22×C4—24no—
144C22×C4—21no—
145C22×C4—24no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—48no—
150C22×C4—416no—
151C22×C4—44no—
152C22×C4—44no—
153C22×C4—42no—
154C22×C4—44no—
155C22×C4—48no—
156C22×C4—42no—
157C22×C4—42no—
158C22×C4—48no—
159C22×C4—48no—
160C22×C4—84no—
161C22×C4—88no—
162C22×C4—84no—
163C22×C4—84no—
164C22×C4—88no—
165C22×C4—88no—
166C22×C4#9912yes—
167C22×C4#9912yes—
168C22×C4C2×C22⋊C412yes—
169C22×C4C4×Q814yes—
170C4⋊C4—216no—
171C4⋊C4—44no—
172C4⋊C4—44no—
173C4⋊C4—432no—
174C4⋊C4—44no—
175C4⋊C4—48no—
176C22⋊C4—48no—
177C22⋊C4—432no—
178C22⋊C4—48no—
179C22⋊C4—48no—
180C22⋊C4—84no—
181C22⋊C4—84no—
182C42—28no—
183C42—44no—
184C42—44no—
185C23—22no—
186C23—22no—
187C23—24no—
188C23—22no—
189C23—22no—
190C23—24no—
191C23—24no—
192C23—22no—
193C23—22no—
194C23—24no—
195C23—44no—
196C23—44no—
197C23—48no—
198C23—44no—
199C23—44no—
200C23—44no—
201C23—44no—
202C23—82no—
203C23—82no—
204C23—84no—
205C23—84no—
206C23#37911yescharacteristic
207C23#39112yes—
208C23#39314yes—
209C23#39514yes—
210C23C2×#9911yescharacteristic
211C23C2×#9911yescharacteristic
212C23#41012yes—
213Q8—816no—
214D4—48no—
215D4—44no—
216D4—44no—
217C2×C4—24no—
218C2×C4—24no—
219C2×C4—22no—
220C2×C4—24no—
221C2×C4—24no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—28no—
226C2×C4—22no—
227C2×C4—44no—
228C2×C4—416no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—44no—
232C2×C4—42no—
233C2×C4—48no—
234C2×C4—44no—
235C2×C4—44no—
236C2×C4—42no—
237C2×C4—44no—
238C2×C4—816no—
239C2×C4—84no—
240C2×C4—84no—
241C2×C4—82no—
242C2×C4—88no—
243C2×C4—816no—
244C2×C4—164no—
245C2×C4—164no—
246C2×C4—168no—
247C2×C4#35314yes—
248C2×C4#39114yes—
249C22—21no—
250C22—21no—
251C22—21no—
252C22—21no—
253C22—24no—
254C22—24no—
255C22—24no—
256C22—24no—
257C22—44no—
258C22—44no—
259C22—44no—
260C22—44no—
261C22—48no—
262C22—44no—
263C22—44no—
264C22—84no—
265C22—82no—
266C22—84no—
267C22—82no—
268C22#144611yescharacteristic
269C22#178111yescharacteristic
270C22#178211yescharacteristic
271C22#199612yes—
272C22#199612yes—
273C22C2×#35312yes—
274C22#212912yes—
275C4—24no—
276C4—24no—
277C4—48no—
278C4—42no—
279C4—82no—
280C4—84no—
281C4—168no—
282C2—21no—
283C2—21no—
284C2—22no—
285C2—44no—
286C2—44no—
287C2#1138111yescharacteristic
288C2#1269311yescharacteristic
289C2#1269811yescharacteristic
290C2#1329814yes—
subgroupG/H
center, socleC23C2×#99
commutatorC23×C4C22×C4
Frattini, prime radical, cosocleC23×D4C23
LCSG ▷ C23×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×D4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C22×C4⋊Q8C4semidirect
11. C2×#384C4semidirect
12. C2×C4×C4⋊C4C4semidirect
40. C2×#99Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25422
C2×#13932
#107612