Finite Group Databasebeta

← #119558

Subgroups — #119558

Order 512.

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

normal subgroups only

This group has 718 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#14660C211yesmaximal, characteristic
4#8114C212yesmaximal
5#8107C212yesmaximal
6C22×C22⋊Q8C2211yescharacteristic
7C2×#407C2212yes—
8C2×#407C2212yes—
9C2×#381C2211yescharacteristic
10C2×#380C2211yescharacteristic
11#1901—24no—
12#1821—21no—
13#1784—21no—
14#1472—24no—
15#1463—22no—
16#1446—21no—
17#1445—21no—
18#1441—22no—
19#1440C414yes—
20#1306—24no—
21#1305—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⋊C4—21no—
33C22×C4⋊C4D411yescharacteristic
34C22×C4⋊C4D412yes—
35C22×C22⋊C4C2×C412yes—
36C22×C22⋊C4C2×C412yes—
37C22×C22⋊C4C2×C412yes—
38C2×#101—24no—
39#407—22no—
40#407—22no—
41#407—22no—
42#407—22no—
43#407—22no—
44#407—22no—
45#407D414yes—
46#407D414yes—
47C2×C22⋊C8—22no—
48C2×C22⋊C8—22no—
49C2×C22⋊C8—22no—
50C2×C22⋊C8—22no—
51#381—22no—
52#381—22no—
53#380—22no—
54#380—22no—
55C2×#95—21no—
56C2×#95—21no—
57C2×#95—21no—
58C2×#95—24no—
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—24no—
83C2×M16—42no—
84C2×M16—42no—
85C2×M16—84no—
86C2×M16—84no—
87C22×C8—42no—
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×C4⋊C4—41no—
106C2×C4⋊C4—41no—
107C2×C4⋊C4—82no—
108C2×C22⋊C4—22no—
109C2×C22⋊C4—22no—
110C2×C22⋊C4—24no—
111C2×C22⋊C4—24no—
112C2×C22⋊C4—24no—
113C2×C22⋊C4—24no—
114C2×C22⋊C4—82no—
115C2×C22⋊C4—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—41no—
126#95—42no—
127#95—42no—
128#95—41no—
129#95—84no—
130#95—84no—
131C24—41no—
132C24—42no—
133C24—42no—
134C24—41no—
135C24C2×C22⋊C411yescharacteristic
136C24#11711yescharacteristic
137C24#11711yescharacteristic
138C24C4×D412yes—
139C24C4×D412yes—
140C2×Q8—44no—
141C2×Q8—42no—
142C2×Q8—42no—
143C2×Q8—82no—
144C2×Q8—84no—
145C2×Q8—84no—
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—42no—
157C22×C4—44no—
158C22×C4—42no—
159C22×C4—42no—
160C22×C4—44no—
161C22×C4—44no—
162C22×C4—44no—
163C22×C4—44no—
164C22×C4—44no—
165C22×C4—42no—
166C22×C4—42no—
167C22×C4—42no—
168C22×C4—82no—
169C22×C4—82no—
170C22×C4—82no—
171C22×C4—82no—
172C22×C4#12011yescharacteristic
173C22×C4#12011yescharacteristic
174C22×C4#12112yes—
175C22×C4#12112yes—
176C22×C4#12112yes—
177C22×C4#12112yes—
178C22×C4#12311yescharacteristic
179C22×C4#12312yes—
180C22×C4#12312yes—
181C22×C4#12312yes—
182C22×C4#12312yes—
183C22×C4#12311yescharacteristic
184C22×C4#12412yes—
185C22×C4#12412yes—
186C22×C4#12612yes—
187C22×C4#12612yes—
188M16—48no—
189M16—84no—
190M16—84no—
191M16—84no—
192C2×C8—44no—
193C2×C8—44no—
194C2×C8—84no—
195C2×C8—84no—
196C4⋊C4—42no—
197C4⋊C4—44no—
198C4⋊C4—42no—
199C4⋊C4—44no—
200C4⋊C4—82no—
201C4⋊C4—82no—
202C22⋊C4—44no—
203C22⋊C4—44no—
204C22⋊C4—84no—
205C22⋊C4—84no—
206C23—22no—
207C23—22no—
208C23—21no—
209C23—21no—
210C23—22no—
211C23—22no—
212C23—24no—
213C23—22no—
214C23—22no—
215C23—41no—
216C23—41no—
217C23—42no—
218C23—41no—
219C23—41no—
220C23—42no—
221C23—42no—
222C23—42no—
223C23—84no—
224C23—84no—
225C23#38111yescharacteristic
226C23#38611yescharacteristic
227C23#38611yescharacteristic
228C23#38812yes—
229C23#38812yes—
230Q8—84no—
231Q8—84no—
232C2×C4—22no—
233C2×C4—24no—
234C2×C4—22no—
235C2×C4—41no—
236C2×C4—41no—
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—42no—
247C2×C4—42no—
248C2×C4—44no—
249C2×C4—44no—
250C2×C4—44no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—82no—
256C2×C4—82no—
257C2×C4—82no—
258C2×C4—84no—
259C2×C4—88no—
260C2×C4—84no—
261C2×C4—82no—
262C2×C4—82no—
263C2×C4—82no—
264C2×C4—82no—
265C2×C4—82no—
266C2×C4#39412yes—
267C2×C4#39412yes—
268C2×C4#39712yes—
269C2×C4#39712yes—
270C8—84no—
271C8—84no—
272C22—24no—
273C22—24no—
274C22—21no—
275C22—21no—
276C22—21no—
277C22—21no—
278C22—21no—
279C22—21no—
280C22—22no—
281C22—22no—
282C22—22no—
283C22—22no—
284C22—22no—
285C22—22no—
286C22—82no—
287C22—84no—
288C22—82no—
289C22—82no—
290C22—82no—
291C22#144011yescharacteristic
292C22#183012yes—
293C22#191312yes—
294C22#191411yescharacteristic
295C22#191511yescharacteristic
296C4—22no—
297C4—22no—
298C4—22no—
299C4—22no—
300C4—82no—
301C4—82no—
302C4—162no—
303C4—162no—
304C2—21no—
305C2—22no—
306C2—22no—
307C2—21no—
308C2—42no—
309C2—42no—
310C2#1117111yescharacteristic
311C2#1117211yescharacteristic
312C2#1117311yescharacteristic
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
#146602
#81142
#81072