Finite Group Databasebeta

← #133146

Subgroups — #133146

Order 512.

636 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 636 conjugacy classes of subgroups — too many to draw. Only its 82 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
1C2×#2822C211yesmaximal, characteristic
2#11814C212yesmaximal
3#11803C212yesmaximal
4#8554C211yesmaximal, characteristic
5#8554C211yesmaximal, characteristic
6C22×#125C411yescharacteristic
7C22×#125C2211yescharacteristic
8#2822—24no—
9#2822—24no—
10C2×#400—21no—
11C2×#400C411yescharacteristic
12C2×#400C2211yescharacteristic
13C2×#398—21no—
14C2×#398—21no—
15C2×#398—21no—
16C2×#398C411yescharacteristic
17C2×#398C2211yescharacteristic
18C2×#382C411yescharacteristic
19#1821—21no—
20#1821C2212yes—
21#1463—21no—
22#1463—21no—
23#1463—21no—
24#1463C2212yes—
25#1353—24no—
26#1346—24no—
27C2×#125—22no—
28C2×#125—22no—
29C2×#125—22no—
30C2×#125—22no—
31C2×#125—42no—
32C2×#125—42no—
33C2×#125—42no—
34C2×#125—42no—
35C22×C4⋊C4—21no—
36C22×C4⋊C4—21no—
37C22×C4⋊C4—21no—
38C22×C4⋊C4—21no—
39C22×C4⋊C4—21no—
40C22×C4⋊C4C2×C411yescharacteristic
41C22×C4⋊C4C2×C411yescharacteristic
42C22×C4⋊C4C2×C411yescharacteristic
43C22×C4⋊C4D411yescharacteristic
44C22×C4⋊C4D411yescharacteristic
45C22×C4⋊C4D411yescharacteristic
46C22×C4⋊C4D411yescharacteristic
47C22×C4⋊C4C2311yescharacteristic
48C22×C42C2×C411yescharacteristic
49#407—22no—
50#407—22no—
51#400—28no—
52#400—22no—
53#400—22no—
54#400—22no—
55#400D414yes—
56#398—28no—
57#398—28no—
58#398—28no—
59#398—22no—
60#398—22no—
61#398—22no—
62#398D414yes—
63#382—22no—
64#382—22no—
65C2×#95—21no—
66C2×#95—21no—
67C2×#95—21no—
68C2×#95—21no—
69C2×#95—21no—
70C2×#95—21no—
71C2×#95—21no—
72C2×#95C2×C411yescharacteristic
73C2×#95C2×C411yescharacteristic
74#343—22no—
75#343—22no—
76#343—42no—
77#324—22no—
78#324—22no—
79#324—42no—
80#324—42no—
81#324—42no—
82C23×C4—21no—
83C23×C4—21no—
84C23×C4—21no—
85C23×C4—21no—
86C23×C4C22⋊C411yescharacteristic
87C23×C4C22⋊C411yescharacteristic
88C23×C4C22⋊C411yescharacteristic
89C23×C4C22⋊C411yescharacteristic
90C23×C4C22×C411yescharacteristic
91C23×C4C2×D411yescharacteristic
92C23×C4C2×D411yescharacteristic
93C2×M16—42no—
94C2×M16—42no—
95#125—48no—
96#125—48no—
97#125—48no—
98#125—48no—
99C2×C4⋊C4—24no—
100C2×C4⋊C4—24no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—24no—
114C2×C4⋊C4—24no—
115C2×C4⋊C4—24no—
116C2×C4⋊C4—41no—
117C2×C4⋊C4—41no—
118C2×C4⋊C4—41no—
119C2×C4⋊C4—41no—
120C2×C4⋊C4—41no—
121C2×C4⋊C4—41no—
122C2×C4⋊C4—41no—
123C2×C4⋊C4—41no—
124C2×C4⋊C4—41no—
125C2×C4⋊C4—41no—
126C2×C4⋊C4—41no—
127C2×C4⋊C4—41no—
128C2×C4⋊C4—42no—
129C2×C4⋊C4—42no—
130C2×C4⋊C4—42no—
131C2×C4⋊C4—42no—
132C2×C4⋊C4—82no—
133C2×C4⋊C4—82no—
134C2×C4⋊C4—82no—
135C2×C4⋊C4C22⋊C412yes—
136C2×C4⋊C4C22⋊C412yes—
137C2×C4⋊C4C2×D412yes—
138C2×C4⋊C4C2×D412yes—
139C2×C42—22no—
140C2×C42—22no—
141C2×C42—21no—
142C2×C42—21no—
143C22⋊C8—42no—
144C22⋊C8—42no—
145C22⋊C8—42no—
146C22⋊C8—42no—
147#95—24no—
148#95—24no—
149#95—24no—
150#95—24no—
151#95—21no—
152#95—21no—
153#95—24no—
154#95—24no—
155#95—24no—
156#95C22⋊C412yes—
157#95C22⋊C412yes—
158C24C2×C22⋊C411yescharacteristic
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—21no—
172C22×C4—22no—
173C22×C4—21no—
174C22×C4—22no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—42no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4—44no—
195C22×C4—42no—
196C22×C4—41no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4—41no—
201C22×C4—41no—
202C22×C4#9912yes—
203C22×C4#9912yes—
204C22×C4C2×C22⋊C411yescharacteristic
205C22×C4C2×C22⋊C411yescharacteristic
206C22×C4#12012yes—
207C22×C4#12012yes—
208M16—82no—
209M16—82no—
210M16—84no—
211C2×C8—82no—
212C2×C8—82no—
213C4⋊C4—42no—
214C4⋊C4—42no—
215C4⋊C4—44no—
216C4⋊C4—44no—
217C4⋊C4—44no—
218C4⋊C4—44no—
219C4⋊C4—42no—
220C4⋊C4—42no—
221C4⋊C4—48no—
222C4⋊C4—48no—
223C4⋊C4—42no—
224C4⋊C4—42no—
225C4⋊C4—42no—
226C4⋊C4—42no—
227C4⋊C4—84no—
228C4⋊C4—84no—
229C4⋊C4—84no—
230C42—24no—
231C42—24no—
232C23—21no—
233C23—21no—
234C23—22no—
235C23—22no—
236C23#37911yescharacteristic
237C23C2×#9911yescharacteristic
238C23C2×#9911yescharacteristic
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—24no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—44no—
266C2×C4—41no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—41no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—81no—
278C2×C4—81no—
279C2×C4—81no—
280C2×C4—84no—
281C2×C4—84no—
282C2×C4—82no—
283C2×C4—82no—
284C2×C4—82no—
285C2×C4—82no—
286C2×C4—81no—
287C2×C4—81no—
288C2×C4—81no—
289C8—162no—
290C8—162no—
291C22—24no—
292C22—24no—
293C22—21no—
294C22—21no—
295C22—21no—
296C22—21no—
297C22—21no—
298C22—21no—
299C22#178111yescharacteristic
300C22#179511yescharacteristic
301C22#179611yescharacteristic
302C22#212312yes—
303C22#213312yes—
304C4—42no—
305C4—42no—
306C4—42no—
307C4—41no—
308C4—41no—
309C4—42no—
310C4—82no—
311C4—82no—
312C4—81no—
313C4—82no—
314C4—81no—
315C4—82no—
316C2—21no—
317C2—21no—
318C2—22no—
319C2—22no—
320C2#851711yescharacteristic
321C2#1269211yescharacteristic
322C2#1270211yescharacteristic
subgroupG/H
center, socleC22#1781
commutatorC23×C4C22×C4
Frattini, cosocleC22×C4⋊C4C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C22×#125 ◁ 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
C2×#28222
#118142
#118032
#85542