Finite Group Databasebeta

← #125476

Subgroups — #125476

Order 512.

556 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 80 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#15180C211yesmaximal, characteristic
2#8701C211yesmaximal, characteristic
3#8153C211yesmaximal, characteristic
4#7122C214yesmaximal
5C22×C4⋊Q8C411yescharacteristic
6C2×#391—21no—
7C2×#391—21no—
8C2×C4×C4⋊C4C2211yescharacteristic
9#1817—21no—
10#1749—21no—
11C2×#324—21no—
12#1442—21no—
13#1442C2212yes—
14#1441—21no—
15#1441C412yes—
16C4⋊#95C411yescharacteristic
17#1280—21no—
18#1280C2212yes—
19#1280C2212yes—
20C23×Q8—21no—
21C2×C4⋊Q8—41no—
22C2×C4⋊Q8—41no—
23C2×C4⋊Q8—44no—
24C22×C4⋊C4—21no—
25C22×C4⋊C4—21no—
26C22×C4⋊C4C2×C411yescharacteristic
27C22×C4⋊C4C2×C412yes—
28C22×C4⋊C4D411yescharacteristic
29C22×C42C2×C411yescharacteristic
30C22×C42D411yescharacteristic
31C22×C42D411yescharacteristic
32C2×C22⋊C8—22no—
33C2×C22⋊C8—22no—
34C2×#97—41no—
35#391—44no—
36#391—44no—
37C4×C4⋊C4—41no—
38C4×C4⋊C4—41no—
39C2×#95—21no—
40C2×#95—22no—
41C2×#95—22no—
42C2×#95—22no—
43C2×#95—22no—
44C2×#95C2×C412yes—
45C2×#95D411yescharacteristic
46C2×#95C2311yescharacteristic
47#336—41no—
48#336—41no—
49#336—41no—
50#324—24no—
51C22×Q8—24no—
52C22×Q8—42no—
53C22×Q8—42no—
54C22×Q8—42no—
55C22×Q8—42no—
56C22×Q8—42no—
57C22×Q8—41no—
58C22×Q8—41no—
59C23×C4—22no—
60C23×C4—22no—
61C23×C4C22⋊C411yescharacteristic
62C23×C4C22⋊C411yescharacteristic
63C23×C4C22⋊C411yescharacteristic
64C23×C4C22⋊C411yescharacteristic
65C23×C4C22×C411yescharacteristic
66C23×C4C2×D411yescharacteristic
67C23×C4C2×D411yescharacteristic
68C22×C8—42no—
69C22×C8—42no—
70C4⋊Q8—44no—
71C4⋊Q8—48no—
72C4⋊Q8—44no—
73C2×C4⋊C4—22no—
74C2×C4⋊C4—22no—
75C2×C4⋊C4—22no—
76C2×C4⋊C4—22no—
77C2×C4⋊C4—41no—
78C2×C4⋊C4—41no—
79C2×C4⋊C4—42no—
80C2×C4⋊C4—42no—
81C2×C4⋊C4—44no—
82C2×C4⋊C4—44no—
83C2×C4⋊C4—42no—
84C2×C4⋊C4—42no—
85C2×C4⋊C4—42no—
86C2×C42—21no—
87C2×C42—21no—
88C2×C42—21no—
89C2×C42—21no—
90C2×C42—21no—
91C2×C42—22no—
92C2×C42—21no—
93C2×C42—41no—
94C2×C42—42no—
95C2×C42—42no—
96C22⋊C8—24no—
97C22⋊C8—42no—
98C22⋊C8—42no—
99C22⋊C8—42no—
100#97—82no—
101#95—24no—
102#95—41no—
103#95—41no—
104#95—44no—
105#95—44no—
106#95—44no—
107#95—42no—
108#95—44no—
109C24C2×C22⋊C411yescharacteristic
110C2×Q8—44no—
111C2×Q8—48no—
112C2×Q8—48no—
113C2×Q8—44no—
114C2×Q8—84no—
115C2×Q8—84no—
116C2×Q8—82no—
117C2×Q8—82no—
118C2×Q8—82no—
119C2×Q8—82no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—22no—
123C22×C4—22no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—21no—
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—21no—
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—42no—
165C22×C4#9911yescharacteristic
166C22×C4#9911yescharacteristic
167C22×C4#10112yes—
168C22×C4#10412yes—
169C22×C4#13112yes—
170C22×C4AGL(1,ℤ/8)12yes—
171C22×C4#13712yes—
172C2×C8—44no—
173C2×C8—82no—
174C2×C8—82no—
175C2×C8—82no—
176C2×C8—82no—
177C2×C8—82no—
178C4⋊C4—44no—
179C4⋊C4—44no—
180C4⋊C4—44no—
181C4⋊C4—44no—
182C4⋊C4—48no—
183C4⋊C4—84no—
184C42—42no—
185C42—42no—
186C42—42no—
187C42—42no—
188C42—42no—
189C42—81no—
190C23—21no—
191C23—21no—
192C23—21no—
193C23—21no—
194C23#40811yescharacteristic
195C23C2×#9911yescharacteristic
196C23C2×#10111yescharacteristic
197C23#41811yescharacteristic
198C23#41911yescharacteristic
199C23C2×#10411yescharacteristic
200C23#42111yescharacteristic
201Q8—88no—
202Q8—88no—
203C2×C4—24no—
204C2×C4—24no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—44no—
236C2×C4—44no—
237C2×C4—41no—
238C2×C4—44no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—81no—
242C2×C4—81no—
243C2×C4—82no—
244C2×C4—82no—
245C2×C4—82no—
246C2×C4—82no—
247C2×C4—81no—
248C2×C4—82no—
249C2×C4—82no—
250C2×C4—82no—
251C2×C4—82no—
252C2×C4—82no—
253C2×C4—82no—
254C2×C4—82no—
255C2×C4AΣL(1,𝔽16)12yes—
256C2×C4#35512yes—
257C8—84no—
258C8—162no—
259C22—21no—
260C22—21no—
261C22—21no—
262C22—21no—
263C22—21no—
264C22—21no—
265C22—21no—
266C22—21no—
267C22—21no—
268C22—21no—
269C22—21no—
270C22—21no—
271C22#147211yescharacteristic
272C22#148411yescharacteristic
273C22#149711yescharacteristic
274C22#150811yescharacteristic
275C22#151011yescharacteristic
276C22#153111yescharacteristic
277C22#155611yescharacteristic
278C22C2×AΣL(1,𝔽16)11yescharacteristic
279C22#212311yescharacteristic
280C22#212911yescharacteristic
281C22C2×#35511yescharacteristic
282C4—22no—
283C4—22no—
284C4—42no—
285C4—42no—
286C4—42no—
287C4—42no—
288C4—42no—
289C4—84no—
290C4—81no—
291C4—162no—
292C2—21no—
293C2—21no—
294C2—21no—
295C2—21no—
296C2#770811yescharacteristic
297C2#1157311yescharacteristic
298C2#1158011yescharacteristic
299C2#1178511yescharacteristic
300C2#1179411yescharacteristic
301C2#1180111yescharacteristic
302C2#1180211yescharacteristic
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
#151802
#87012
#81532
#71222