Finite Group Databasebeta

← #135491

Subgroups — #135491

Order 512.

562 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 562 conjugacy classes of subgroups — too many to draw. Only its 66 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#14292C211yesmaximal, characteristic
2#8555C211yesmaximal, characteristic
3#8550C211yesmaximal, characteristic
4#7215C214yesmaximal, complemented
5C22×C22⋊Q8C2211yescharacteristic
6C2×#381C411yescharacteristic
7C2×C4×C22⋊C4C411yescharacteristic
8#1821C2212yes—
9#1784—21no—
10#1463—21no—
11#1463C2212yes—
12#1462—21no—
13#1462—21no—
14#1446—21no—
15#1445—21no—
16#1440—21no—
17#1440—21no—
18#1436C412yes—
19#1436C2212yes—
20C23×Q8D411yescharacteristic
21C24×C4C2×C411yescharacteristic
22C2×C22⋊Q8—24no—
23C2×C22⋊Q8—42no—
24C2×C22⋊Q8—42no—
25C22×C4⋊C4C2×C411yescharacteristic
26C22×C4⋊C4D411yescharacteristic
27C22×C4⋊C4C2311yescharacteristic
28C22×C22⋊C4—21no—
29C22×C22⋊C4—21no—
30C22×C22⋊C4D411yescharacteristic
31C22×C22⋊C4D411yescharacteristic
32C22×C42C2×C412yes—
33C2×#101—42no—
34#407—22no—
35#407—22no—
36#381—42no—
37C4×C22⋊C4—42no—
38C2×#95—22no—
39C2×#95—21no—
40C2×#95—21no—
41C2×#95—22no—
42C2×#95—22no—
43C2×#95—22no—
44C2×#95C2×C412yes—
45#343—24no—
46#324—24no—
47#324—42no—
48#323—42no—
49#323—42no—
50C25C22⋊C411yescharacteristic
51C22×Q8—22no—
52C22×Q8—22no—
53C22×Q8—41no—
54C22×Q8—82no—
55C23×C4—22no—
56C23×C4—22no—
57C23×C4—22no—
58C23×C4—22no—
59C23×C4—22no—
60C23×C4—82no—
61C23×C4C22⋊C411yescharacteristic
62C23×C4C22⋊C411yescharacteristic
63C23×C4C22⋊C411yescharacteristic
64C23×C4C22×C411yescharacteristic
65C23×C4C2×D411yescharacteristic
66C23×C4C2×D411yescharacteristic
67C2×M16—42no—
68C2×M16—42no—
69C22⋊Q8—416no—
70C2×C4⋊C4—22no—
71C2×C4⋊C4—22no—
72C2×C4⋊C4—22no—
73C2×C4⋊C4—41no—
74C2×C4⋊C4—41no—
75C2×C4⋊C4—41no—
76C2×C4⋊C4—41no—
77C2×C4⋊C4—41no—
78C2×C4⋊C4#4114yes—
79C2×C22⋊C4—22no—
80C2×C22⋊C4—22no—
81C2×C22⋊C4—22no—
82C2×C22⋊C4—22no—
83C2×C22⋊C4—41no—
84C2×C22⋊C4—41no—
85C2×C22⋊C4—44no—
86C2×C22⋊C4—42no—
87C2×C22⋊C4—42no—
88C2×C22⋊C4—82no—
89C2×C42—42no—
90C2×C42—82no—
91#101—84no—
92C22⋊C8—42no—
93C22⋊C8—42no—
94C22⋊C8—42no—
95C22⋊C8—42no—
96#95—42no—
97#95—44no—
98#95—44no—
99#95—42no—
100#95—42no—
101#95—82no—
102#95—82no—
103C24—21no—
104C24—21no—
105C24—42no—
106C24—42no—
107C24—42no—
108C24#9912yes—
109C24C2×C22⋊C411yescharacteristic
110C2×Q8—44no—
111C2×Q8—44no—
112C2×Q8—82no—
113C2×Q8—84no—
114C2×Q8—84no—
115C22×C4—21no—
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—28no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—24no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—42no—
132C22×C4—42no—
133C22×C4—42no—
134C22×C4—42no—
135C22×C4—42no—
136C22×C4—42no—
137C22×C4—42no—
138C22×C4—42no—
139C22×C4—42no—
140C22×C4—42no—
141C22×C4—42no—
142C22×C4—42no—
143C22×C4—42no—
144C22×C4—42no—
145C22×C4—42no—
146C22×C4—42no—
147C22×C4—42no—
148C22×C4—42no—
149C22×C4—42no—
150C22×C4—82no—
151C22×C4—82no—
152C22×C4—84no—
153C22×C4—84no—
154C22×C4#9912yes—
155C22×C4#11712yes—
156C22×C4#12312yes—
157C22×C4#12312yes—
158M16—84no—
159M16—84no—
160C2×C8—82no—
161C2×C8—82no—
162C4⋊C4—44no—
163C4⋊C4—44no—
164C4⋊C4—44no—
165C22⋊C4—44no—
166C22⋊C4—44no—
167C22⋊C4—48no—
168C22⋊C4—84no—
169C42—84no—
170C23—22no—
171C23—22no—
172C23—22no—
173C23—22no—
174C23—24no—
175C23—42no—
176C23—42no—
177C23—42no—
178C23—42no—
179C23—42no—
180C23—42no—
181C23—42no—
182C23—42no—
183C23—82no—
184C23—82no—
185C23—82no—
186C23—82no—
187C23#38111yescharacteristic
188C23C2×#9911yescharacteristic
189C23C2×#9911yescharacteristic
190Q8—88no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—24no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—44no—
206C2×C4—44no—
207C2×C4—44no—
208C2×C4—42no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—81no—
214C2×C4—81no—
215C2×C4—81no—
216C2×C4—88no—
217C2×C4—82no—
218C2×C4—82no—
219C2×C4—82no—
220C2×C4—82no—
221C2×C4—81no—
222C2×C4—81no—
223C2×C4—81no—
224C2×C4—82no—
225C2×C4—82no—
226C2×C4—82no—
227C2×C4—82no—
228C2×C4—82no—
229C2×C4—82no—
230C2×C4—82no—
231C8—162no—
232C8—162no—
233C22—22no—
234C22—22no—
235C22—22no—
236C22—22no—
237C22—21no—
238C22—21no—
239C22—21no—
240C22—21no—
241C22—21no—
242C22—21no—
243C22—42no—
244C22—42no—
245C22—42no—
246C22—84no—
247C22—84no—
248C22—84no—
249C22#181911yescharacteristic
250C22#182712yes—
251C22#212112yes—
252C22#213312yes—
253C4—42no—
254C4—42no—
255C4—42no—
256C4—42no—
257C4—84no—
258C4—81no—
259C4—81no—
260C4—162no—
261C2—22no—
262C2—22no—
263C2—22no—
264C2—82no—
265C2#873911yescharacteristic
266C2#1285411yescharacteristic
267C2#1286211yescharacteristic
subgroupG/H
center, socleC22#1819
commutatorC23×C4C22×C4
Frattini, cosocleC22×C4⋊C4C23
prime radicalC25C22⋊C4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C22×C22⋊Q8 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7215C2semidirect

Maximal subgroups

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

subgroupindex
#142922
#85552
#85502
#72152