Finite Group Databasebeta

← #116877

Subgroups — #116877

Order 512.

770 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 770 conjugacy classes of subgroups — too many to draw. Only its 146 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#21938C211yesmaximal, characteristic
2#20702C212yesmaximal
3C2×#1818C211yesmaximal, characteristic
4C2×#1293C211yesmaximal, characteristic
5#7915C212yesmaximal
6C22×#117C2211yescharacteristic
7C2×#407C2212yes—
8C22×C22⋊C8C2211yescharacteristic
9C22×C22⋊C8C2211yescharacteristic
10C2×#380C2212yes—
11#1853—24no—
12#1825—24no—
13#1824—22no—
14#1823—22no—
15#1818—22no—
16#1818C414yescomplemented
17#1807—22no—
18#1736—24no—
19C2×#336—22no—
20#1293—22no—
21#1280—24no—
22C24×C4C2311yescharacteristic
23C22×M16—22no—
24C23×C8—21no—
25C23×C8D412yes—
26C2×#117—21no—
27C2×#117—21no—
28C2×#117—22no—
29C2×#117—22no—
30C2×#117—22no—
31C2×#117—22no—
32C2×#117C2×C412yes—
33C2×#117Q812yes—
34C22×C4⋊C4D411yescharacteristic
35C22×C4⋊C4D411yescharacteristic
36C22×C4⋊C4D414yes—
37C22×C22⋊C4—21no—
38C22×C22⋊C4—22no—
39C22×C42—21no—
40C2×#106—24no—
41#407—22no—
42#407—22no—
43#407—22no—
44#407—22no—
45C2×C22⋊C8—22no—
46C2×C22⋊C8—22no—
47C2×C22⋊C8—22no—
48C2×C22⋊C8—22no—
49C2×C22⋊C8—21no—
50C2×C22⋊C8—22no—
51C2×C22⋊C8C2×C412yes—
52C2×C22⋊C8C2×C412yes—
53C2×C22⋊C8D414yes—
54C2×C22⋊C8D412yes—
55#380—22no—
56#380—22no—
57#380—22no—
58#380—22no—
59C2×#95—22no—
60#340—24no—
61#340—28no—
62#340—28no—
63#336—28no—
64C25#4111yescharacteristic
65C23×C4—21no—
66C23×C4—21no—
67C23×C4—24no—
68C23×C4—44no—
69C23×C4C22×C411yescharacteristic
70C23×C4C2×D411yescharacteristic
71C23×C4C2×D412yes—
72C23×C4C2×D411yescharacteristic
73C23×C4C2×D411yescharacteristic
74C23×C4C2×D412yes—
75C23×C4C2×Q811yescharacteristic
76C23×C4#4112yes—
77C23×C4#4112yes—
78C23×C4#4111yescharacteristic
79C2×M16—48no—
80C2×M16—42no—
81C2×M16—42no—
82C22×C8—22no—
83C22×C8—22no—
84C22×C8—22no—
85C22×C8—24no—
86C22×C8—44no—
87C22×C8—44no—
88C22×C8—41no—
89C22×C8—41no—
90C22×C8C22⋊C414yes—
91#117—24no—
92#117—24no—
93#117—24no—
94#117—24no—
95#117—42no—
96#117—42no—
97#117—42no—
98#117—42no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—24no—
106C2×C4⋊C4—24no—
107C2×C4⋊C4—48no—
108C2×C4⋊C4C4⋊C412yes—
109C2×C4⋊C4C4⋊C412yes—
110C2×C22⋊C4—22no—
111C2×C22⋊C4—22no—
112C2×C22⋊C4—42no—
113C2×C22⋊C4—42no—
114C2×C22⋊C4—42no—
115C2×C22⋊C4—42no—
116C2×C22⋊C4—84no—
117C2×C42—22no—
118C2×C42—22no—
119C2×C42—41no—
120C2×C42—41no—
121C2×C42—41no—
122C2×C42—41no—
123#106—84no—
124C22⋊C8—24no—
125C22⋊C8—24no—
126C22⋊C8—24no—
127C22⋊C8—24no—
128C22⋊C8—24no—
129C22⋊C8—42no—
130C22⋊C8—42no—
131C22⋊C8—42no—
132#95—42no—
133#95—42no—
134C24—24no—
135C24—42no—
136C24—42no—
137C24#11711yescharacteristic
138C24#12112yes—
139C24#12111yescharacteristic
140C24C22⋊Q811yescharacteristic
141C24#12612yes—
142C22×C4—28no—
143C22×C4—24no—
144C22×C4—22no—
145C22×C4—22no—
146C22×C4—42no—
147C22×C4—42no—
148C22×C4—42no—
149C22×C4—44no—
150C22×C4—44no—
151C22×C4—42no—
152C22×C4—42no—
153C22×C4—44no—
154C22×C4—44no—
155C22×C4—41no—
156C22×C4—41no—
157C22×C4—41no—
158C22×C4—41no—
159C22×C4—44no—
160C22×C4—44no—
161C22×C4—48no—
162C22×C4—84no—
163C22×C4—84no—
164C22×C4C2×C22⋊C411yescharacteristic
165C22×C4C2×C4⋊C411yescharacteristic
166C22×C4C4×D412yes—
167C22×C4C4×D412yes—
168C22×C4#12012yes—
169C22×C4#12011yescharacteristic
170C22×C4#12112yes—
171C22×C4#12112yes—
172C22×C4#12111yescharacteristic
173C22×C4C22⋊Q812yes—
174C22×C4C22⋊Q811yescharacteristic
175C22×C4C22⋊Q812yes—
176C22×C4#12311yescharacteristic
177C22×C4#12312yes—
178C22×C4#12312yes—
179C22×C4#12411yescharacteristic
180C22×C4#12412yes—
181C22×C4#12711yescharacteristic
182M16—84no—
183M16—84no—
184C2×C8—24no—
185C2×C8—44no—
186C2×C8—44no—
187C2×C8—44no—
188C2×C8—48no—
189C2×C8—42no—
190C2×C8—44no—
191C2×C8—82no—
192C2×C8—82no—
193C2×C8—82no—
194C4⋊C4—24no—
195C4⋊C4—24no—
196C4⋊C4—48no—
197C4⋊C4—42no—
198C4⋊C4—42no—
199C4⋊C4—48no—
200C22⋊C4—44no—
201C22⋊C4—44no—
202C22⋊C4—84no—
203C22⋊C4—84no—
204C42—42no—
205C42—42no—
206C42—42no—
207C42—42no—
208C23—22no—
209C23—22no—
210C23—24no—
211C23—22no—
212C23—22no—
213C23—24no—
214C23—22no—
215C23—22no—
216C23—24no—
217C23—41no—
218C23—41no—
219C23—42no—
220C23—41no—
221C23—41no—
222C23—42no—
223C23—88no—
224C23#38011yescharacteristic
225C23#38812yes—
226C23#39211yescharacteristic
227C23#39612yes—
228C23#39711yescharacteristic
229C23#47114yes—
230C2×C4—24no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—24no—
235C2×C4—24no—
236C2×C4—22no—
237C2×C4—24no—
238C2×C4—24no—
239C2×C4—22no—
240C2×C4—41no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—41no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—44no—
248C2×C4—44no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—82no—
252C2×C4—82no—
253C2×C4—82no—
254C2×C4—82no—
255C2×C4—84no—
256C2×C4—84no—
257C2×C4—84no—
258C2×C4—168no—
259C2×C4#38512yes—
260C2×C4#38611yescharacteristic
261C2×C4#39011yescharacteristic
262C2×C4#39311yescharacteristic
263C2×C4#39412yes—
264C2×C4#39911yescharacteristic
265C2×C4#45814yes—
266C8—44no—
267C8—82no—
268C8—84no—
269C22—21no—
270C22—21no—
271C22—21no—
272C22—21no—
273C22—28no—
274C22—22no—
275C22—22no—
276C22—22no—
277C22—22no—
278C22—22no—
279C22—22no—
280C22—44no—
281C22—44no—
282C22—82no—
283C22—82no—
284C22—84no—
285C22#143911yescharacteristic
286C22#181512yes—
287C22#201712yes—
288C22#201912yes—
289C22#202212yes—
290C22#207912yes—
291C4—21no—
292C4—22no—
293C4—22no—
294C4—21no—
295C4—82no—
296C4—82no—
297C4—164no—
298C4#189012yes—
299C4#203012yes—
300C2—22no—
301C2—22no—
302C2—42no—
303C2—42no—
304C2#997814yes—
305C2#1107011yescharacteristic
306C2#1110611yescharacteristic
307C2#1111111yescharacteristic
subgroupG/H
center, socleC23#392
commutatorC23×C4C22×C4
Frattini, cosocleC24×C4C23
prime radicalC25#41
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
16. #1818C4semidirect

Maximal subgroups

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

subgroupindex
#219382
#207022
C2×#18182
C2×#12932
#79152