Finite Group Databasebeta

← #125210

Subgroups — #125210

Order 512.

610 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 610 conjugacy classes of subgroups — too many to draw. Only its 108 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#15128C211yesmaximal, characteristic
2#8558C211yesmaximal, characteristic
3#7908C211yesmaximal, characteristic
4#7131C214yesmaximal
5C22×C4⋊Q8C411yescharacteristic
6C2×#431—21no—
7C2×#395C412yes—
8C2×#389—21no—
9C2×C4×C4⋊C4C2211yescharacteristic
10#1797—21no—
11C2×#324—21no—
12#1446C2212yes—
13#1444—21no—
14#1441—21no—
15C4⋊#95—21no—
16C4⋊#95C411yescharacteristic
17#1280—21no—
18#1280C2212yes—
19#1280C2212yes—
20C23×Q8C2×C412yes—
21C2×C4⋊Q8—41no—
22C2×C4⋊Q8—41no—
23C2×C4⋊Q8—44no—
24C22×C4⋊C4—21no—
25C22×C4⋊C4—21no—
26C22×C4⋊C4—21no—
27C22×C4⋊C4—22no—
28C22×C4⋊C4C2×C411yescharacteristic
29C22×C4⋊C4D411yescharacteristic
30C22×C42C2×C411yescharacteristic
31C22×C42D411yescharacteristic
32C22×C42D411yescharacteristic
33#431—24no—
34#431—42no—
35C2×C4⋊C8—41no—
36C2×C4⋊C8—41no—
37C2×C22⋊C8—22no—
38C2×C22⋊C8—22no—
39C2×C4×C8—41no—
40#395—44no—
41#389—44no—
42C4×C4⋊C4—41no—
43C4×C4⋊C4—41no—
44C2×#95—22no—
45C2×#95—21no—
46C2×#95—22no—
47C2×#95—22no—
48C2×#95C2×C412yes—
49C2×#95D411yescharacteristic
50C2×#95C2311yescharacteristic
51#336—41no—
52#324—42no—
53C22×Q8—24no—
54C22×Q8—44no—
55C22×Q8—42no—
56C22×Q8—42no—
57C22×Q8—82no—
58C23×C4—22no—
59C23×C4—22no—
60C23×C4C22⋊C411yescharacteristic
61C23×C4C22⋊C411yescharacteristic
62C23×C4C22⋊C411yescharacteristic
63C23×C4C22⋊C411yescharacteristic
64C23×C4C22×C411yescharacteristic
65C23×C4C2×D411yescharacteristic
66C23×C4C2×D411yescharacteristic
67C22×C8—42no—
68C22×C8—42no—
69C4⋊Q8—48no—
70C4⋊Q8—44no—
71C4⋊Q8—44no—
72C2×C4⋊C4—24no—
73C2×C4⋊C4—24no—
74C2×C4⋊C4—22no—
75C2×C4⋊C4—22no—
76C2×C4⋊C4—41no—
77C2×C4⋊C4—41no—
78C2×C4⋊C4—41no—
79C2×C4⋊C4—42no—
80C2×C4⋊C4—42no—
81C2×C4⋊C4—44no—
82C2×C4⋊C4—42no—
83C2×C4⋊C4—41no—
84C2×C4⋊C4—41no—
85C2×C4⋊C4—42no—
86C2×C4⋊C4—44no—
87C2×C4⋊C4—44no—
88C2×C4⋊C4—82no—
89C2×C4⋊C4SD1612yes—
90C2×C4⋊C4Q1612yes—
91C2×C42—21no—
92C2×C42—21no—
93C2×C42—22no—
94C2×C42—21no—
95C2×C42—21no—
96C2×C42—21no—
97C2×C42—21no—
98C2×C42—42no—
99C2×C42—42no—
100C2×C42SD1612yes—
101C2×C42Q1612yes—
102C4⋊C8—44no—
103C4⋊C8—82no—
104C22⋊C8—24no—
105C22⋊C8—42no—
106C22⋊C8—42no—
107C22⋊C8—42no—
108C4×C8—44no—
109#95—41no—
110#95—41no—
111#95—42no—
112#95—42no—
113#95—44no—
114#95—44no—
115#95—82no—
116C24C2×C22⋊C411yescharacteristic
117C2×Q8—44no—
118C2×Q8—44no—
119C2×Q8—84no—
120C2×Q8—88no—
121C2×Q8—88no—
122C2×Q8—84no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—24no—
138C22×C4—24no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—41no—
143C22×C4—41no—
144C22×C4—41no—
145C22×C4—41no—
146C22×C4—41no—
147C22×C4—41no—
148C22×C4—41no—
149C22×C4—41no—
150C22×C4—41no—
151C22×C4—41no—
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—42no—
164C22×C4#9911yescharacteristic
165C22×C4#9911yescharacteristic
166C22×C4#10012yes—
167C22×C4#10312yes—
168C22×C4#10312yes—
169C22×C4#10312yes—
170C22×C4#10312yes—
171C22×C4#10412yes—
172C22×C4#13112yes—
173C22×C4C2×SD1611yescharacteristic
174C22×C4C2×SD1611yescharacteristic
175C22×C4C2×Q1611yescharacteristic
176C22×C4C2×Q1611yescharacteristic
177C2×C8—44no—
178C2×C8—82no—
179C2×C8—82no—
180C2×C8—82no—
181C2×C8—82no—
182C2×C8—82no—
183C4⋊C4—42no—
184C4⋊C4—42no—
185C4⋊C4—44no—
186C4⋊C4—48no—
187C4⋊C4—44no—
188C4⋊C4—48no—
189C4⋊C4—44no—
190C4⋊C4—88no—
191C42—22no—
192C42—22no—
193C42—22no—
194C42—22no—
195C42—42no—
196C42—42no—
197C42—42no—
198C42—81no—
199C23—21no—
200C23—21no—
201C23—21no—
202C23—21no—
203C23#40811yescharacteristic
204C23C2×#9911yescharacteristic
205C23C2×#10011yescharacteristic
206C23C2×#10311yescharacteristic
207C23C2×#10311yescharacteristic
208C23C2×#10411yescharacteristic
209C23#42111yescharacteristic
210Q8—88no—
211Q8—88no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—24no—
221C2×C4—24no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—22no—
227C2×C4—21no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—44no—
245C2×C4—42no—
246C2×C4—44no—
247C2×C4—44no—
248C2×C4—44no—
249C2×C4—44no—
250C2×C4—41no—
251C2×C4—44no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—81no—
255C2×C4—81no—
256C2×C4—82no—
257C2×C4—82no—
258C2×C4—82no—
259C2×C4—82no—
260C2×C4—82no—
261C2×C4—82no—
262C2×C4—82no—
263C2×C4—82no—
264C2×C4—81no—
265C2×C4#32812yes—
266C2×C4#32812yes—
267C2×C4#33314yes—
268C2×C4AΣL(1,𝔽16)12yes—
269C2×C4#35612yes—
270C8—84no—
271C8—162no—
272C22—21no—
273C22—21no—
274C22—21no—
275C22—21no—
276C22—21no—
277C22—21no—
278C22—21no—
279C22—21no—
280C22—21no—
281C22—21no—
282C22—21no—
283C22—21no—
284C22#147111yescharacteristic
285C22#148111yescharacteristic
286C22#149311yescharacteristic
287C22C2×#32811yescharacteristic
288C22#152011yescharacteristic
289C22#153011yescharacteristic
290C22C2×#33311yescharacteristic
291C22C2×AΣL(1,𝔽16)11yescharacteristic
292C22#212211yescharacteristic
293C22#212911yescharacteristic
294C22C2×#35611yescharacteristic
295C4—22no—
296C4—24no—
297C4—42no—
298C4—41no—
299C4—42no—
300C4—41no—
301C4—41no—
302C4—41no—
303C4—84no—
304C4—81no—
305C4—84no—
306C4#135414yes—
307C2—21no—
308C2—21no—
309C2—21no—
310C2—21no—
311C2#766711yescharacteristic
312C2#1156111yescharacteristic
313C2#1156811yescharacteristic
314C2#1171711yescharacteristic
315C2C2×#135411yescharacteristic
316C2#1189811yescharacteristic
317C2#1190811yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC23×C4C22×C4
Frattini, cosocleC2×#95C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C2×#95 ▷ C24 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#151282
#85582
#79082
#71312