Finite Group Databasebeta

← #119552

Subgroups — #119552

Order 512.

742 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 742 conjugacy classes of subgroups — too many to draw. Only its 118 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#21753C211yesmaximal, characteristic
2#17397C211yesmaximal, characteristic
3#14292C211yesmaximal, characteristic
4#8114C212yesmaximal
5#7929C212yesmaximal
6C22×C22⋊Q8C2211yescharacteristic
7C2×#407C2212yes—
8C2×#407C2212yes—
9C2×#381C2211yescharacteristic
10C2×C4×C22⋊C4C2211yescharacteristic
11#1901—24no—
12#1821—21no—
13#1784—21no—
14#1472—24no—
15#1463—22no—
16#1446—21no—
17#1445—21no—
18#1440C414yes—
19#1436—22no—
20#1306—24no—
21#1282—24no—
22C23×Q8D411yescharacteristic
23C24×C4C2311yescharacteristic
24C22×M16—22no—
25C22×M16—22no—
26C2×C22⋊Q8—22no—
27C2×C22⋊Q8—22no—
28C2×C22⋊Q8—22no—
29C2×C22⋊Q8—22no—
30C2×C22⋊Q8—22no—
31C2×C22⋊Q8—22no—
32C22×C4⋊C4D411yescharacteristic
33C22×C4⋊C4D412yes—
34C22×C22⋊C4C2×C412yes—
35C22×C22⋊C4C2×C412yes—
36C22×C22⋊C4C2×C412yes—
37C22×C42—21no—
38C2×#101—24no—
39#407—22no—
40#407—22no—
41#407—22no—
42#407—22no—
43#407—22no—
44#407—22no—
45#407—22no—
46#407D414yes—
47C2×C22⋊C8—22no—
48C2×C22⋊C8—22no—
49C2×C22⋊C8—22no—
50C2×C22⋊C8D414yes—
51#381—22no—
52#381—22no—
53C4×C22⋊C4—22no—
54C4×C22⋊C4—22no—
55C2×#95—24no—
56C2×#95—21no—
57C2×#95—21no—
58C2×#95—21no—
59C2×#95—24no—
60#343—24no—
61#324—28no—
62C25C22×C411yescharacteristic
63C22×Q8—21no—
64C22×Q8—22no—
65C22×Q8—22no—
66C22×Q8—21no—
67C22×Q8—82no—
68C23×C4—22no—
69C23×C4—22no—
70C23×C4—82no—
71C23×C4C22⋊C412yes—
72C23×C4C22⋊C412yes—
73C23×C4C2×D412yes—
74C23×C4C2×D411yescharacteristic
75C23×C4C2×D411yescharacteristic
76C23×C4C2×D412yes—
77C23×C4#4112yes—
78C23×C4#4112yes—
79C23×C4#4112yes—
80C23×C4#4112yes—
81C2×M16—24no—
82C2×M16—42no—
83C2×M16—42no—
84C2×M16—42no—
85C2×M16—84no—
86C2×M16—84no—
87C22×C8—24no—
88C22×C8—42no—
89C22⋊Q8—42no—
90C22⋊Q8—42no—
91C22⋊Q8—42no—
92C22⋊Q8—42no—
93C22⋊Q8—42no—
94C22⋊Q8—42no—
95C22⋊Q8—42no—
96C22⋊Q8—42no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—24no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—24no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—21no—
105C2×C22⋊C4—24no—
106C2×C22⋊C4—22no—
107C2×C22⋊C4—22no—
108C2×C22⋊C4—24no—
109C2×C22⋊C4—24no—
110C2×C22⋊C4—24no—
111C2×C22⋊C4—44no—
112C2×C22⋊C4—82no—
113C2×C42—41no—
114C2×C42—41no—
115C2×C42—82no—
116#101—84no—
117C22⋊C8—24no—
118C22⋊C8—24no—
119C22⋊C8—24no—
120C22⋊C8—24no—
121C22⋊C8—42no—
122C22⋊C8—42no—
123C22⋊C8—42no—
124C22⋊C8—42no—
125#95—48no—
126#95—41no—
127#95—42no—
128#95—42no—
129#95—41no—
130#95—84no—
131C24—42no—
132C24—42no—
133C24—42no—
134C24C2×C22⋊C411yescharacteristic
135C24#11711yescharacteristic
136C24#11711yescharacteristic
137C24C4×D412yes—
138C24C4×D412yes—
139C2×Q8—44no—
140C2×Q8—42no—
141C2×Q8—42no—
142C2×Q8—82no—
143C2×Q8—84no—
144C2×Q8—84no—
145C22×C4—24no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—42no—
153C22×C4—44no—
154C22×C4—44no—
155C22×C4—42no—
156C22×C4—44no—
157C22×C4—44no—
158C22×C4—44no—
159C22×C4—42no—
160C22×C4—44no—
161C22×C4—42no—
162C22×C4—44no—
163C22×C4—44no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4—42no—
167C22×C4—82no—
168C22×C4—82no—
169C22×C4—82no—
170C22×C4—82no—
171C22×C4#12011yescharacteristic
172C22×C4#12011yescharacteristic
173C22×C4#12112yes—
174C22×C4#12112yes—
175C22×C4#12112yes—
176C22×C4#12112yes—
177C22×C4#12311yescharacteristic
178C22×C4#12312yes—
179C22×C4#12312yes—
180C22×C4#12312yes—
181C22×C4#12312yes—
182C22×C4#12311yescharacteristic
183C22×C4#12412yes—
184C22×C4#12412yes—
185C22×C4#12612yes—
186C22×C4#12612yes—
187M16—48no—
188M16—84no—
189M16—84no—
190M16—84no—
191C2×C8—44no—
192C2×C8—44no—
193C2×C8—84no—
194C2×C8—84no—
195C4⋊C4—42no—
196C4⋊C4—44no—
197C4⋊C4—42no—
198C4⋊C4—44no—
199C22⋊C4—28no—
200C22⋊C4—44no—
201C22⋊C4—44no—
202C22⋊C4—44no—
203C22⋊C4—84no—
204C42—82no—
205C42—82no—
206C23—22no—
207C23—22no—
208C23—22no—
209C23—22no—
210C23—22no—
211C23—24no—
212C23—24no—
213C23—41no—
214C23—41no—
215C23—42no—
216C23—41no—
217C23—41no—
218C23—42no—
219C23—42no—
220C23—42no—
221C23—84no—
222C23—84no—
223C23#38111yescharacteristic
224C23#38611yescharacteristic
225C23#38611yescharacteristic
226C23#38812yes—
227C23#38812yes—
228Q8—84no—
229Q8—84no—
230C2×C4—22no—
231C2×C4—24no—
232C2×C4—22no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—48no—
247C2×C4—44no—
248C2×C4—44no—
249C2×C4—44no—
250C2×C4—44no—
251C2×C4—44no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—82no—
255C2×C4—84no—
256C2×C4—82no—
257C2×C4—82no—
258C2×C4—82no—
259C2×C4—84no—
260C2×C4—84no—
261C2×C4—82no—
262C2×C4—82no—
263C2×C4—82no—
264C2×C4#39412yes—
265C2×C4#39412yes—
266C2×C4#39712yes—
267C2×C4#39712yes—
268C8—84no—
269C8—84no—
270C22—24no—
271C22—24no—
272C22—21no—
273C22—21no—
274C22—21no—
275C22—21no—
276C22—21no—
277C22—21no—
278C22—22no—
279C22—22no—
280C22—22no—
281C22—22no—
282C22—22no—
283C22—22no—
284C22—84no—
285C22—84no—
286C22—82no—
287C22—82no—
288C22#144011yescharacteristic
289C22#183012yes—
290C22#189112yes—
291C22#191312yes—
292C4—22no—
293C4—22no—
294C4—22no—
295C4—22no—
296C4—84no—
297C4—82no—
298C4—82no—
299C4—162no—
300C2—22no—
301C2—22no—
302C2—22no—
303C2—42no—
304C2—42no—
305C2#1116811yescharacteristic
306C2#1117211yescharacteristic
307C2#1117511yescharacteristic
subgroupG/H
center, socleC22#1440
commutator, prime radicalC25C22×C4
Frattini, cosocleC24×C4C23
LCSG ▷ C25 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
#217532
#173972
#142922
#81142
#79292