Finite Group Databasebeta

← #135343

Subgroups — #135343

Order 512.

620 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 620 conjugacy classes of subgroups — too many to draw. Only its 100 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#13942C211yesmaximal, characteristic
2#10763C211yesmaximal, characteristic
3#8303C211yesmaximal, characteristic
4#7251C214yesmaximal
5C22×C4×Q8C2211yescharacteristic
6C22×#101—21no—
7C2×#391—21no—
8C2×#382C411yescharacteristic
9C2×C4×C4⋊C4C411yescharacteristic
10#1817—21no—
11C2×#343C2212yes—
12#1728—21no—
13C2×#324—21no—
14C2×#324C2212yes—
15#1441C412yes—
16#1441C2212yes—
17C4⋊#95—21no—
18#1434—21no—
19#1434—21no—
20C23×Q8D411yescharacteristic
21C22×M16—22no—
22C2×C4×Q8—24no—
23C2×C4×Q8—42no—
24C2×C4×Q8—42no—
25C22×C4⋊C4—21no—
26C22×C4⋊C4C2×C411yescharacteristic
27C22×C4⋊C4C2×C412yes—
28C22×C4⋊C4D411yescharacteristic
29C22×C4⋊C4C2311yescharacteristic
30C22×C42—21no—
31C22×C42C2×C411yescharacteristic
32C22×C42D411yescharacteristic
33C22×C42D411yescharacteristic
34C2×#101—24no—
35C2×#101—42no—
36C2×#101—42no—
37C2×C22⋊C8—22no—
38#391—44no—
39#382—42no—
40C4×C4⋊C4—42no—
41C2×#95—22no—
42C2×#95—22no—
43C2×#95—21no—
44C2×#95—21no—
45C2×#95—22no—
46C2×#95—22no—
47C2×#95C2×C412yes—
48#343—24no—
49#336—42no—
50#336—42no—
51#324—24no—
52#324—42no—
53C22×Q8—22no—
54C22×Q8—22no—
55C22×Q8—22no—
56C22×Q8—22no—
57C22×Q8—41no—
58C22×Q8—41no—
59C22×Q8SD1612yes—
60C22×Q8Q1612yes—
61C23×C4—22no—
62C23×C4—22no—
63C23×C4C22⋊C411yescharacteristic
64C23×C4C22⋊C411yescharacteristic
65C23×C4C22⋊C411yescharacteristic
66C23×C4C22⋊C411yescharacteristic
67C23×C4C22×C411yescharacteristic
68C23×C4C2×D411yescharacteristic
69C23×C4C2×D411yescharacteristic
70C2×M16—24no—
71C2×M16—42no—
72C2×M16—42no—
73C2×M16—42no—
74C2×M16—42no—
75C2×M16—42no—
76C22×C8—42no—
77C22×C8—42no—
78C4×Q8—416no—
79C2×C4⋊C4—22no—
80C2×C4⋊C4—22no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4—41no—
83C2×C4⋊C4—41no—
84C2×C4⋊C4—41no—
85C2×C4⋊C4—41no—
86C2×C4⋊C4—41no—
87C2×C4⋊C4—42no—
88C2×C4⋊C4—44no—
89C2×C4⋊C4—42no—
90C2×C4⋊C4—42no—
91C2×C4⋊C4—42no—
92C2×C4⋊C4#4114yes—
93C2×C42—22no—
94C2×C42—22no—
95C2×C42—22no—
96C2×C42—22no—
97C2×C42—22no—
98C2×C42—22no—
99C2×C42—22no—
100C2×C42—41no—
101C2×C42—41no—
102C2×C42—42no—
103C2×C42—42no—
104C2×C42—42no—
105#101—48no—
106C22⋊C8—42no—
107C22⋊C8—42no—
108#95—42no—
109#95—44no—
110#95—44no—
111#95—42no—
112#95—44no—
113#95—42no—
114#95—44no—
115C24C2×C22⋊C411yescharacteristic
116C2×Q8—24no—
117C2×Q8—24no—
118C2×Q8—24no—
119C2×Q8—24no—
120C2×Q8—42no—
121C2×Q8—42no—
122C2×Q8—44no—
123C2×Q8—44no—
124C2×Q8—44no—
125C2×Q8—44no—
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—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—24no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—42no—
151C22×C4—42no—
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—42no—
166C22×C4—42no—
167C22×C4#9912yes—
168C22×C4#9912yes—
169C22×C4#10312yes—
170C22×C4#10312yes—
171C22×C4#10412yes—
172C22×C4#11712yes—
173C22×C4#12312yes—
174C22×C4#12312yes—
175C22×C4C2×SD1611yescharacteristic
176C22×C4C2×Q1611yescharacteristic
177C22×C4AGL(1,ℤ/8)12yes—
178M16—44no—
179M16—44no—
180M16—82no—
181M16—82no—
182C2×C8—44no—
183C2×C8—82no—
184C2×C8—82no—
185C2×C8—82no—
186C2×C8—84no—
187C4⋊C4—44no—
188C4⋊C4—44no—
189C4⋊C4—44no—
190C4⋊C4—44no—
191C4⋊C4—44no—
192C4⋊C4—84no—
193C42—28no—
194C42—44no—
195C42—44no—
196C42—84no—
197C23—22no—
198C23—22no—
199C23#38111yescharacteristic
200C23C2×#9911yescharacteristic
201C23C2×#9911yescharacteristic
202C23C2×#10311yescharacteristic
203C23#41811yescharacteristic
204C23C2×#10411yescharacteristic
205C23#42111yescharacteristic
206Q8—48no—
207Q8—48no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—21no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—24no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—44no—
241C2×C4—44no—
242C2×C4—42no—
243C2×C4—41no—
244C2×C4—44no—
245C2×C4—42no—
246C2×C4—81no—
247C2×C4—82no—
248C2×C4—82no—
249C2×C4—82no—
250C2×C4—82no—
251C2×C4—81no—
252C2×C4—82no—
253C2×C4—82no—
254C2×C4—82no—
255C2×C4—82no—
256C2×C4—82no—
257C2×C4—82no—
258C2×C4—82no—
259C2×C4#32814yes—
260C2×C4#48112yes—
261C2×C4#48412yes—
262C8—84no—
263C8—162no—
264C22—22no—
265C22—22no—
266C22—21no—
267C22—21no—
268C22—21no—
269C22—21no—
270C22—22no—
271C22—22no—
272C22C2×#32811yescharacteristic
273C22#150711yescharacteristic
274C22#151011yescharacteristic
275C22#152011yescharacteristic
276C22#181911yescharacteristic
277C22#182411yescharacteristic
278C22#182611yescharacteristic
279C22#212012yes—
280C22#213312yes—
281C4—22no—
282C4—24no—
283C4—22no—
284C4—42no—
285C4—42no—
286C4—42no—
287C4—42no—
288C4—84no—
289C4—81no—
290C4—162no—
291C2—21no—
292C2—21no—
293C2—21no—
294C2—21no—
295C2#871311yescharacteristic
296C2#1177812yes—
297C2#1190512yes—
298C2#1285311yescharacteristic
299C2#1286211yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC23×C4C22×C4
Frattini, cosocleC22×C4⋊C4C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C23×C4 ▷ C23 ▷ 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
#139422
#107632
#83032
#72512