Finite Group Databasebeta

← #101020

Subgroups — #101020

Order 512.

644 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 644 conjugacy classes of subgroups — too many to draw. Only its 104 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#21916C211yesmaximal, characteristic
2#15128C211yesmaximal, characteristic
3#8558C211yesmaximal, characteristic
4#8342C212yesmaximal
5#7039C212yesmaximal
6C22×C4⋊Q8C2211yescharacteristic
7C2×#395C412yes—
8C2×#389—21no—
9C2×C4×C4⋊C4C2211yescharacteristic
10#1917—24no—
11#1817—21no—
12#1797C2212yes—
13#1797C2212yes—
14#1749—21no—
15C2×#333—24no—
16C2×#324—21no—
17#1446C412yes—
18#1444—21no—
19#1441—21no—
20C4⋊#95—21no—
21C4⋊#95C2211yescharacteristic
22#1280—21no—
23C23×Q8C2×C412yes—
24C2×C4⋊Q8—22no—
25C2×C4⋊Q8—22no—
26C2×C4⋊Q8—44no—
27C22×C4⋊C4—21no—
28C22×C4⋊C4—21no—
29C22×C4⋊C4—21no—
30C22×C4⋊C4—22no—
31C22×C4⋊C4C2×C411yescharacteristic
32C22×C4⋊C4D411yescharacteristic
33C22×C42D411yescharacteristic
34C22×C42D411yescharacteristic
35C22×C42C2311yescharacteristic
36C2×C4⋊C8—22no—
37C2×C4⋊C8D414yes—
38C2×C22⋊C8—22no—
39C2×C22⋊C8—22no—
40C2×#97—22no—
41#395—44no—
42#389—44no—
43C4×C4⋊C4—22no—
44C4×C4⋊C4—22no—
45C2×#95—22no—
46C2×#95—22no—
47C2×#95—21no—
48C2×#95—22no—
49C2×#95C2×C411yescharacteristic
50C2×#95C2×C412yes—
51C2×#95D411yescharacteristic
52#336—24no—
53#336—22no—
54#336—22no—
55#336—22no—
56#333—48no—
57#324—42no—
58C22×Q8—24no—
59C22×Q8—44no—
60C22×Q8—42no—
61C22×Q8—42no—
62C22×Q8—82no—
63C23×C4—22no—
64C23×C4—22no—
65C23×C4C22⋊C411yescharacteristic
66C23×C4C22⋊C411yescharacteristic
67C23×C4C22⋊C411yescharacteristic
68C23×C4C22⋊C411yescharacteristic
69C23×C4C22×C411yescharacteristic
70C23×C4C2×D411yescharacteristic
71C23×C4C2×D411yescharacteristic
72C22×C8—24no—
73C22×C8—42no—
74C4⋊Q8—28no—
75C4⋊Q8—44no—
76C4⋊Q8—48no—
77C2×C4⋊C4—24no—
78C2×C4⋊C4—24no—
79C2×C4⋊C4—22no—
80C2×C4⋊C4—22no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—22no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—41no—
86C2×C4⋊C4—42no—
87C2×C4⋊C4—44no—
88C2×C4⋊C4—42no—
89C2×C4⋊C4—41no—
90C2×C4⋊C4—42no—
91C2×C4⋊C4—41no—
92C2×C4⋊C4—42no—
93C2×C4⋊C4—44no—
94C2×C4⋊C4—44no—
95C2×C4⋊C4—82no—
96C2×C42—21no—
97C2×C42—21no—
98C2×C42—21no—
99C2×C42—22no—
100C2×C42—22no—
101C2×C42—21no—
102C2×C42—42no—
103C2×C42—42no—
104C2×C42C2×D412yes—
105C2×C42#4112yes—
106C4⋊C8—44no—
107C4⋊C8—44no—
108C22⋊C8—42no—
109C22⋊C8—42no—
110C22⋊C8—42no—
111C22⋊C8—42no—
112#97—82no—
113#95—22no—
114#95—22no—
115#95—42no—
116#95—44no—
117#95—44no—
118#95—41no—
119#95—41no—
120#95—82no—
121C24C2×C22⋊C411yescharacteristic
122C2×Q8—28no—
123C2×Q8—44no—
124C2×Q8—84no—
125C2×Q8—84no—
126C2×Q8—88no—
127C2×Q8—88no—
128C22×C4—22no—
129C22×C4—22no—
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—24no—
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—41no—
150C22×C4—41no—
151C22×C4—41no—
152C22×C4—41no—
153C22×C4—41no—
154C22×C4—41no—
155C22×C4—41no—
156C22×C4—41no—
157C22×C4—41no—
158C22×C4—41no—
159C22×C4—42no—
160C22×C4—42no—
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#10012yes—
172C22×C4C4×D412yes—
173C22×C4#12011yescharacteristic
174C22×C4#12112yes—
175C22×C4#12311yescharacteristic
176C22×C4#13512yes—
177C22×C4AGL(1,ℤ/8)11yescharacteristic
178C22×C4#13712yes—
179C22×C4#13712yes—
180C22×C4#13711yescharacteristic
181C2×C8—44no—
182C2×C8—44no—
183C2×C8—44no—
184C2×C8—82no—
185C2×C8—82no—
186C2×C8—82no—
187C4⋊C4—44no—
188C4⋊C4—48no—
189C4⋊C4—44no—
190C4⋊C4—48no—
191C4⋊C4—44no—
192C4⋊C4—42no—
193C4⋊C4—42no—
194C4⋊C4—88no—
195C42—24no—
196C42—42no—
197C42—42no—
198C42—42no—
199C42—42no—
200C42—81no—
201C23—21no—
202C23—21no—
203C23—21no—
204C23—21no—
205C23#38611yescharacteristic
206C23C2×#10011yescharacteristic
207C23#41311yescharacteristic
208C23#41611yescharacteristic
209C23#41711yescharacteristic
210C23#41911yescharacteristic
211C23#41911yescharacteristic
212Q8—88no—
213Q8—88no—
214C2×C4—21no—
215C2×C4—22no—
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—24no—
219C2×C4—24no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—41no—
227C2×C4—42no—
228C2×C4—41no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—44no—
236C2×C4—42no—
237C2×C4—44no—
238C2×C4—44no—
239C2×C4—44no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—44no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—44no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—82no—
254C2×C4—82no—
255C2×C4—82no—
256C2×C4—82no—
257C2×C4—82no—
258C2×C4—82no—
259C2×C4—82no—
260C2×C4—82no—
261C2×C4—81no—
262C2×C4—81no—
263C2×C4—81no—
264C2×C4#45212yes—
265C2×C4#45612yes—
266C2×C4B(4,𝔽2)12yes—
267C2×C4#46812yes—
268C2×C4#47012yes—
269C2×C4#48512yes—
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#155111yescharacteristic
285C22#155411yescharacteristic
286C22#155811yescharacteristic
287C22#155911yescharacteristic
288C22#187311yescharacteristic
289C22#188611yescharacteristic
290C22#188911yescharacteristic
291C22#189411yescharacteristic
292C22#190011yescharacteristic
293C22#190911yescharacteristic
294C22#191411yescharacteristic
295C4—22no—
296C4—22no—
297C4—22no—
298C4—42no—
299C4—42no—
300C4—84no—
301C4—81no—
302C4—84no—
303C4—81no—
304C4—81no—
305C4#160314yes—
306C2—21no—
307C2—21no—
308C2—21no—
309C2—21no—
310C2#901411yescharacteristic
311C2#913611yescharacteristic
312C2#919211yescharacteristic
313C2#940011yescharacteristic
314C2#945211yescharacteristic
315C2#947111yescharacteristic
316C2#950411yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C23×C4 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Maximal subgroups

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

subgroupindex
#219162
#151282
#85582
#83422
#70392