Finite Group Databasebeta

← #120500

Subgroups — #120500

Order 512.

726 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 726 conjugacy classes of subgroups — too many to draw. Only its 132 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#21939C211yesmaximal, characteristic
2#20836C211yesmaximal, characteristic
3#10717C212yesmaximal
4#8102C212yesmaximal
5#7982C211yesmaximal, characteristic
6C23×M16C2211yescharacteristic
7C2×#424—21no—
8C22×C22⋊C8C2211yescharacteristic
9#2280C2211yescharacteristic
10#2280C2212yes—
11#2280C2212yes—
12#1854—24no—
13#1851—22no—
14#1850C414yescomplemented
15#1808—21no—
16#1807—21no—
17C2×H(3,ℤ/4)—24no—
18#1729—21no—
19#1304—24no—
20#1289—22no—
21C24×C4C2311yescharacteristic
22C22×M16—21no—
23C22×M16—22no—
24C22×M16—21no—
25C22×M16—22no—
26C22×M16—44no—
27C23×C8C2×C412yes—
28C23×C8D411yescharacteristic
29C23×C8D411yescharacteristic
30C2×#117—21no—
31C2×#117—22no—
32C2×#117D414yes—
33C2×#107—22no—
34C2×#106—22no—
35#424—44no—
36C2×C4⋊C8—24no—
37C2×C22⋊C8—22no—
38C2×C22⋊C8—22no—
39C2×C22⋊C8C2×C412yes—
40C2×C22⋊C8D412yes—
41#381—21no—
42#381—22no—
43#381—22no—
44#380—21no—
45#380—22no—
46#380—22no—
47#380—22no—
48#380—22no—
49#380Q812yes—
50#379—22no—
51#379—22no—
52#379D412yes—
53C4×C22⋊C4—21no—
54C4×C22⋊C4—22no—
55C4×C22⋊C4—22no—
56C4×C22⋊C4—22no—
57C4×C22⋊C4C2×C412yes—
58C4×C22⋊C4Q814yes—
59#340—22no—
60#340—22no—
61H(3,ℤ/4)—48no—
62#336—24no—
63C25C2×D411yescharacteristic
64C23×C4—42no—
65C23×C4—42no—
66C23×C4C22×C411yescharacteristic
67C23×C4C2×D411yescharacteristic
68C23×C4C2×D412yes—
69C23×C4C2×D411yescharacteristic
70C23×C4C2×Q812yes—
71C23×C4C2×Q811yescharacteristic
72C23×C4#4111yescharacteristic
73C23×C4#4111yescharacteristic
74C23×C4#4112yes—
75C23×C4#4112yes—
76C2×M16—28no—
77C2×M16—24no—
78C2×M16—44no—
79C2×M16—48no—
80C2×M16—44no—
81C2×M16—42no—
82C2×M16—41no—
83C2×M16—42no—
84C2×M16—41no—
85C22×C8—22no—
86C22×C8—24no—
87C22×C8—21no—
88C22×C8—21no—
89C22×C8—21no—
90C22×C8—21no—
91C22×C8—44no—
92C22×C8—42no—
93C22×C8—42no—
94C22×C8C22⋊C412yes—
95C22×C8C22⋊C412yes—
96#117—42no—
97#117—42no—
98#117—44no—
99#117—44no—
100#117—44no—
101#117—44no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—24no—
104C2×C4⋊C4—42no—
105C2×C22⋊C4—22no—
106C2×C22⋊C4—22no—
107C2×C22⋊C4—24no—
108C2×C22⋊C4—24no—
109C2×C22⋊C4—42no—
110C2×C22⋊C4—42no—
111C2×C22⋊C4—42no—
112C2×C22⋊C4—42no—
113C2×C22⋊C4C4⋊C414yes—
114C2×C42—22no—
115C2×C42—24no—
116C2×C42—24no—
117#107—82no—
118#106—82no—
119C4⋊C8—48no—
120C22⋊C8—24no—
121C22⋊C8—24no—
122#95—22no—
123#95—22no—
124#95—24no—
125#95—42no—
126#95—42no—
127#95—42no—
128C24—42no—
129C24—42no—
130C24—81no—
131C24C2×C4⋊C411yescharacteristic
132C24#12111yescharacteristic
133C24#12111yescharacteristic
134C24#12112yes—
135C24C22⋊Q812yes—
136C22×C4—24no—
137C22×C4—22no—
138C22×C4—24no—
139C22×C4—22no—
140C22×C4—24no—
141C22×C4—41no—
142C22×C4—41no—
143C22×C4—42no—
144C22×C4—41no—
145C22×C4—41no—
146C22×C4—44no—
147C22×C4—42no—
148C22×C4—42no—
149C22×C4—42no—
150C22×C4—42no—
151C22×C4—44no—
152C22×C4—44no—
153C22×C4—44no—
154C22×C4C2×C22⋊C411yescharacteristic
155C22×C4#11711yescharacteristic
156C22×C4C4×D412yes—
157C22×C4C4×Q812yes—
158C22×C4#12011yescharacteristic
159C22×C4#12112yes—
160C22×C4C22⋊Q812yes—
161C22×C4C22⋊Q811yescharacteristic
162C22×C4C22⋊Q812yes—
163C22×C4C22⋊Q811yescharacteristic
164C22×C4#12311yescharacteristic
165C22×C4#12312yes—
166C22×C4#12411yescharacteristic
167C22×C4#12512yes—
168C22×C4#12512yes—
169C22×C4#12612yes—
170C22×C4C4⋊Q811yescharacteristic
171C22×C4C4⋊Q812yes—
172M16—48no—
173M16—44no—
174M16—44no—
175C2×C8—24no—
176C2×C8—28no—
177C2×C8—22no—
178C2×C8—22no—
179C2×C8—42no—
180C2×C8—44no—
181C2×C8—42no—
182C2×C8—42no—
183C2×C8—42no—
184C2×C8—42no—
185C2×C8—42no—
186C2×C8—81no—
187C2×C8—81no—
188C4⋊C4—44no—
189C4⋊C4—44no—
190C4⋊C4—44no—
191C4⋊C4—82no—
192C4⋊C4—82no—
193C22⋊C4—44no—
194C22⋊C4—44no—
195C22⋊C4—82no—
196C22⋊C4—84no—
197C22⋊C4—84no—
198C22⋊C4—84no—
199C22⋊C4—84no—
200C22⋊C4—84no—
201C22⋊C4—84no—
202C42—42no—
203C42—42no—
204C42—48no—
205C42—48no—
206C23—22no—
207C23—22no—
208C23—22no—
209C23—22no—
210C23—24no—
211C23—41no—
212C23—41no—
213C23—41no—
214C23—41no—
215C23—42no—
216C23—42no—
217C23—41no—
218C23—42no—
219C23—42no—
220C23—42no—
221C23—88no—
222C23#38011yescharacteristic
223C23#38912yes—
224C23#39411yescharacteristic
225C23#39612yes—
226C23#39811yescharacteristic
227C2×C4—24no—
228C2×C4—22no—
229C2×C4—24no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—24no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—41no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—41no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—44no—
249C2×C4—44no—
250C2×C4—44no—
251C2×C4—84no—
252C2×C4—84no—
253C2×C4—84no—
254C2×C4—84no—
255C2×C4—84no—
256C2×C4—84no—
257C2×C4#38611yescharacteristic
258C2×C4#38712yes—
259C2×C4#39111yescharacteristic
260C2×C4#39511yescharacteristic
261C2×C4#39812yes—
262C2×C4#40011yescharacteristic
263C2×C4B(4,𝔽2)12yes—
264C2×C4#45812yes—
265C8—44no—
266C8—42no—
267C8—42no—
268C22—21no—
269C22—21no—
270C22—21no—
271C22—21no—
272C22—22no—
273C22—22no—
274C22—21no—
275C22—22no—
276C22—22no—
277C22—21no—
278C22—22no—
279C22—22no—
280C22—21no—
281C22—44no—
282C22—44no—
283C22—82no—
284C22—84no—
285C22—84no—
286C22#144111yescharacteristic
287C22#181212yes—
288C22#204412yes—
289C4—22no—
290C4—21no—
291C4—22no—
292C4—21no—
293C4—84no—
294C4—164no—
295C4—164no—
296C4#188911yescharacteristic
297C4#189011yescharacteristic
298C4#206212yes—
299C2—22no—
300C2—22no—
301C2—42no—
302C2—42no—
303C2—41no—
304C2#1122211yescharacteristic
305C2#1122911yescharacteristic
306C2#1123411yescharacteristic
subgroupG/H
center, socleC22#1441
commutatorC23×C4C22×C4
Frattini, cosocleC24×C4C23
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
14. #1850C4semidirect

Maximal subgroups

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

subgroupindex
#219392
#208362
#107172
#81022
#79822