Finite Group Databasebeta

← #135481

Subgroups — #135481

Order 512.

658 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 658 conjugacy classes of subgroups — too many to draw. Only its 70 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#14159C211yesmaximal, characteristic
2#8555C211yesmaximal, characteristic
3#8550C211yesmaximal, characteristic
4#7215C214yesmaximal, complemented
5C22×C22⋊Q8C2211yescharacteristic
6C2×C4×C22⋊C4C411yescharacteristic
7C22×#95C411yescharacteristic
8#1821C2212yes—
9#1784—21no—
10#1463—21no—
11#1463C2212yes—
12#1462—21no—
13#1462—21no—
14#1446—21no—
15#1445—21no—
16#1436C412yes—
17#1436C2212yes—
18C22⋊#95—21no—
19C22⋊#95—21no—
20C23×Q8D411yescharacteristic
21C24×C4—21no—
22C24×C4C2×C411yescharacteristic
23C2×C22⋊Q8—24no—
24C2×C22⋊Q8—41no—
25C2×C22⋊Q8—41no—
26C2×C22⋊Q8—42no—
27C22×C4⋊C4C2×C411yescharacteristic
28C22×C4⋊C4D411yescharacteristic
29C22×C4⋊C4C2311yescharacteristic
30C22×C22⋊C4—21no—
31C22×C22⋊C4D411yescharacteristic
32C22×C22⋊C4D411yescharacteristic
33C22×C42C2×C412yes—
34C2×#101—42no—
35#407—22no—
36#407—22no—
37C4×C22⋊C4—42no—
38C2×#95—22no—
39C2×#95—21no—
40C2×#95—21no—
41C2×#95—22no—
42C2×#95—22no—
43C2×#95—22no—
44C2×#95—41no—
45C2×#95—41no—
46C2×#95C2×C412yes—
47#343—24no—
48#324—24no—
49#324—42no—
50#323—41no—
51#323—41no—
52#323—41no—
53#323—41no—
54C25C22⋊C411yescharacteristic
55C22×Q8—22no—
56C22×Q8—22no—
57C22×Q8—41no—
58C22×Q8—82no—
59C23×C4—21no—
60C23×C4—21no—
61C23×C4—22no—
62C23×C4—22no—
63C23×C4—22no—
64C23×C4—22no—
65C23×C4—44no—
66C23×C4—41no—
67C23×C4—41no—
68C23×C4—42no—
69C23×C4—42no—
70C23×C4—82no—
71C23×C4—82no—
72C23×C4C22⋊C411yescharacteristic
73C23×C4C22⋊C411yescharacteristic
74C23×C4C22⋊C411yescharacteristic
75C23×C4C22×C411yescharacteristic
76C23×C4C2×D411yescharacteristic
77C23×C4C2×D411yescharacteristic
78C2×M16—42no—
79C2×M16—42no—
80C22⋊Q8—48no—
81C22⋊Q8—48no—
82C2×C4⋊C4—22no—
83C2×C4⋊C4—22no—
84C2×C4⋊C4—22no—
85C2×C4⋊C4—41no—
86C2×C4⋊C4—41no—
87C2×C4⋊C4—41no—
88C2×C4⋊C4—41no—
89C2×C4⋊C4—41no—
90C2×C4⋊C4#4114yes—
91C2×C22⋊C4—24no—
92C2×C22⋊C4—21no—
93C2×C22⋊C4—21no—
94C2×C22⋊C4—21no—
95C2×C22⋊C4—21no—
96C2×C22⋊C4—22no—
97C2×C22⋊C4—22no—
98C2×C22⋊C4—41no—
99C2×C22⋊C4—41no—
100C2×C22⋊C4—42no—
101C2×C22⋊C4—42no—
102C2×C42—42no—
103C2×C42—82no—
104#101—84no—
105C22⋊C8—42no—
106C22⋊C8—42no—
107C22⋊C8—42no—
108C22⋊C8—42no—
109#95—42no—
110#95—44no—
111#95—44no—
112#95—42no—
113#95—42no—
114#95—82no—
115#95—82no—
116C24—22no—
117C24—22no—
118C24—21no—
119C24—21no—
120C24—42no—
121C24—42no—
122C24#9911yescharacteristic
123C24#9911yescharacteristic
124C24C2×C22⋊C411yescharacteristic
125C2×Q8—44no—
126C2×Q8—44no—
127C2×Q8—82no—
128C2×Q8—84no—
129C2×Q8—84no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—24no—
136C22×C4—24no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—24no—
141C22×C4—24no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—42no—
149C22×C4—42no—
150C22×C4—42no—
151C22×C4—42no—
152C22×C4—42no—
153C22×C4—42no—
154C22×C4—42no—
155C22×C4—42no—
156C22×C4—42no—
157C22×C4—42no—
158C22×C4—42no—
159C22×C4—42no—
160C22×C4—42no—
161C22×C4—42no—
162C22×C4—42no—
163C22×C4—44no—
164C22×C4—44no—
165C22×C4—42no—
166C22×C4—42no—
167C22×C4—42no—
168C22×C4—82no—
169C22×C4—82no—
170C22×C4—82no—
171C22×C4—82no—
172C22×C4—84no—
173C22×C4—88no—
174C22×C4—88no—
175C22×C4—82no—
176C22×C4—82no—
177C22×C4—82no—
178C22×C4—82no—
179C22×C4#9912yes—
180C22×C4#11712yes—
181C22×C4#12312yes—
182C22×C4#12312yes—
183M16—84no—
184M16—84no—
185C2×C8—82no—
186C2×C8—82no—
187C4⋊C4—44no—
188C4⋊C4—44no—
189C4⋊C4—44no—
190C22⋊C4—42no—
191C22⋊C4—42no—
192C22⋊C4—42no—
193C22⋊C4—42no—
194C22⋊C4—48no—
195C42—84no—
196C23—22no—
197C23—22no—
198C23—21no—
199C23—21no—
200C23—24no—
201C23—24no—
202C23—21no—
203C23—21no—
204C23—21no—
205C23—21no—
206C23—41no—
207C23—41no—
208C23—41no—
209C23—41no—
210C23—44no—
211C23—44no—
212C23—42no—
213C23—44no—
214C23—44no—
215C23—42no—
216C23—42no—
217C23—42no—
218C23#35512yes—
219C23#35612yes—
220C23#38111yescharacteristic
221C23C2×#9911yescharacteristic
222C23C2×#9911yescharacteristic
223Q8—88no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—24no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—44no—
239C2×C4—44no—
240C2×C4—44no—
241C2×C4—44no—
242C2×C4—44no—
243C2×C4—44no—
244C2×C4—42no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—81no—
250C2×C4—81no—
251C2×C4—81no—
252C2×C4—84no—
253C2×C4—84no—
254C2×C4—82no—
255C2×C4—82no—
256C2×C4—82no—
257C2×C4—82no—
258C2×C4—82no—
259C2×C4—82no—
260C2×C4—88no—
261C2×C4—88no—
262C2×C4—82no—
263C2×C4—82no—
264C2×C4—81no—
265C2×C4—81no—
266C2×C4—81no—
267C8—162no—
268C8—162no—
269C22—22no—
270C22—22no—
271C22—22no—
272C22—22no—
273C22—21no—
274C22—21no—
275C22—21no—
276C22—21no—
277C22—21no—
278C22—21no—
279C22—22no—
280C22—22no—
281C22—42no—
282C22—42no—
283C22—42no—
284C22—48no—
285C22—48no—
286C22—42no—
287C22—41no—
288C22—41no—
289C22—41no—
290C22—41no—
291C22#181911yescharacteristic
292C22#182712yes—
293C22#212112yes—
294C22C2×#35511yescharacteristic
295C22C2×#35611yescharacteristic
296C4—42no—
297C4—42no—
298C4—42no—
299C4—42no—
300C4—84no—
301C4—84no—
302C4—81no—
303C4—81no—
304C2—21no—
305C2—22no—
306C2—22no—
307C2—21no—
308C2—42no—
309C2—42no—
310C2#873811yescharacteristic
311C2#1285411yescharacteristic
312C2#1286011yescharacteristic
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
#141592
#85552
#85502
#72152