Finite Group Databasebeta

← #135243

Subgroups — #135243

Order 512.

594 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 594 conjugacy classes of subgroups — too many to draw. Only its 94 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#13926C211yesmaximal, characteristic
2#8303C211yesmaximal, characteristic
3#8303C211yesmaximal, characteristic
4#7189C214yesmaximal
5C22×C4×Q8C2211yescharacteristic
6C2×#391C412yes—
7C2×#384C411yescharacteristic
8C2×C4×C4⋊C4C411yescharacteristic
9#1817—21no—
10#1817—21no—
11C2×#324—21no—
12C2×#324—21no—
13C2×#324C2212yes—
14C2×#324C2212yes—
15#1441—21no—
16#1441—21no—
17C4⋊#95—21no—
18C4⋊#95—21no—
19C4⋊#95C2212yes—
20C23×Q8C2×C411yescharacteristic
21C2×C4×Q8—24no—
22C2×C4×Q8—41no—
23C2×C4×Q8—41no—
24C2×C4×Q8—42no—
25C22×C4⋊C4—21no—
26C22×C4⋊C4—21no—
27C22×C4⋊C4—21no—
28C22×C4⋊C4C2×C412yes—
29C22×C4⋊C4D411yescharacteristic
30C22×C4⋊C4D411yescharacteristic
31C22×C4⋊C4C2311yescharacteristic
32C22×C42C2×C411yescharacteristic
33C22×C42C2×C412yes—
34C22×C42D411yescharacteristic
35C22×C42D411yescharacteristic
36C2×C22⋊C8—22no—
37C2×C22⋊C8—22no—
38#391—44no—
39#384—41no—
40#384—41no—
41C4×C4⋊C4—42no—
42C2×#95—22no—
43C2×#95—22no—
44C2×#95—21no—
45C2×#95—22no—
46C2×#95—22no—
47#336—41no—
48#336—41no—
49#336—41no—
50#336—41no—
51#324—24no—
52#324—24no—
53#324—42no—
54#324—42no—
55C22×Q8—22no—
56C22×Q8—22no—
57C22×Q8—22no—
58C22×Q8—22no—
59C22×Q8—22no—
60C22×Q8—41no—
61C22×Q8—41no—
62C23×C4—22no—
63C23×C4—22no—
64C23×C4C22⋊C411yescharacteristic
65C23×C4C22⋊C411yescharacteristic
66C23×C4C22⋊C411yescharacteristic
67C23×C4C22⋊C411yescharacteristic
68C23×C4C22×C411yescharacteristic
69C23×C4C2×D411yescharacteristic
70C23×C4C2×D411yescharacteristic
71C22×C8—42no—
72C22×C8—42no—
73C4×Q8—48no—
74C4×Q8—48no—
75C2×C4⋊C4—24no—
76C2×C4⋊C4—22no—
77C2×C4⋊C4—22no—
78C2×C4⋊C4—22no—
79C2×C4⋊C4—22no—
80C2×C4⋊C4—24no—
81C2×C4⋊C4—41no—
82C2×C4⋊C4—41no—
83C2×C4⋊C4—41no—
84C2×C4⋊C4—41no—
85C2×C4⋊C4—42no—
86C2×C4⋊C4—42no—
87C2×C4⋊C4—42no—
88C2×C4⋊C4—42no—
89C2×C4⋊C4—42no—
90C2×C4⋊C4—42no—
91C2×C4⋊C4—42no—
92C2×C4⋊C4—41no—
93C2×C4⋊C4—41no—
94C2×C4⋊C4—82no—
95C2×C4⋊C4#4114yes—
96C2×C42—22no—
97C2×C42—21no—
98C2×C42—21no—
99C2×C42—22no—
100C2×C42—21no—
101C2×C42—21no—
102C2×C42—21no—
103C2×C42—21no—
104C2×C42—22no—
105C2×C42—22no—
106C2×C42—24no—
107C2×C42—41no—
108C2×C42—41no—
109C2×C42—42no—
110C2×C42—42no—
111C22⋊C8—42no—
112C22⋊C8—42no—
113C22⋊C8—42no—
114C22⋊C8—42no—
115#95—44no—
116#95—42no—
117#95—44no—
118#95—82no—
119#95—82no—
120C24C2×C22⋊C411yescharacteristic
121C2×Q8—28no—
122C2×Q8—44no—
123C2×Q8—44no—
124C2×Q8—44no—
125C2×Q8—44no—
126C2×Q8—44no—
127C2×Q8—44no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—22no—
135C22×C4—22no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—24no—
155C22×C4—41no—
156C22×C4—41no—
157C22×C4—41no—
158C22×C4—41no—
159C22×C4—41no—
160C22×C4—41no—
161C22×C4—42no—
162C22×C4—42no—
163C22×C4—42no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4—42no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—42no—
173C22×C4—42no—
174C22×C4#9911yescharacteristic
175C22×C4#9911yescharacteristic
176C22×C4#9912yes—
177C22×C4#10012yes—
178C22×C4#10112yes—
179C22×C4#11712yes—
180C22×C4#12312yes—
181C22×C4#12312yes—
182C22×C4#13112yes—
183C22×C4#13112yes—
184C2×C8—82no—
185C2×C8—82no—
186C2×C8—82no—
187C2×C8—82no—
188C2×C8—82no—
189C2×C8—82no—
190C4⋊C4—44no—
191C4⋊C4—44no—
192C4⋊C4—44no—
193C4⋊C4—48no—
194C4⋊C4—84no—
195C4⋊C4—82no—
196C4⋊C4—84no—
197C4⋊C4—82no—
198C42—24no—
199C42—24no—
200C42—42no—
201C42—42no—
202C42—42no—
203C42—42no—
204C42—84no—
205C23—21no—
206C23—21no—
207C23—21no—
208C23—21no—
209C23#38111yescharacteristic
210C23#40811yescharacteristic
211C23#40811yescharacteristic
212C23C2×#9911yescharacteristic
213C23C2×#9911yescharacteristic
214C23C2×#10011yescharacteristic
215C23C2×#10111yescharacteristic
216Q8—416no—
217C2×C4—24no—
218C2×C4—24no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—42no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—44no—
257C2×C4—44no—
258C2×C4—44no—
259C2×C4—81no—
260C2×C4—81no—
261C2×C4—82no—
262C2×C4—82no—
263C2×C4—82no—
264C2×C4—82no—
265C2×C4—82no—
266C2×C4—82no—
267C2×C4—82no—
268C2×C4—82no—
269C2×C4—82no—
270C2×C4—82no—
271C2×C4—82no—
272C2×C4#35512yes—
273C2×C4#35612yes—
274C2×C4#43314yes—
275C8—162no—
276C8—162no—
277C22—21no—
278C22—21no—
279C22—21no—
280C22—21no—
281C22—21no—
282C22—21no—
283C22—21no—
284C22—21no—
285C22—21no—
286C22—21no—
287C22—21no—
288C22—21no—
289C22#147111yescharacteristic
290C22#147111yescharacteristic
291C22#147211yescharacteristic
292C22#147211yescharacteristic
293C22#181811yescharacteristic
294C22#181911yescharacteristic
295C22#182111yescharacteristic
296C22#212811yescharacteristic
297C22#212911yescharacteristic
298C22C2×#35511yescharacteristic
299C22C2×#35611yescharacteristic
300C4—24no—
301C4—22no—
302C4—22no—
303C4—42no—
304C4—42no—
305C4—42no—
306C4—42no—
307C4—42no—
308C4—84no—
309C4—162no—
310C2—21no—
311C2—21no—
312C2—21no—
313C2—21no—
314C2#870411yescharacteristic
315C2#1156711yescharacteristic
316C2#1156811yescharacteristic
317C2#1157311yescharacteristic
318C2#1157411yescharacteristic
319C2#1285911yescharacteristic
320C2#1286011yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC23×C4C22×C4
Frattini, cosocleC22×C4⋊C4C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C23×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#139262
#83032
#71892