Finite Group Databasebeta

← #116869

Subgroups — #116869

Order 512.

621 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 621 conjugacy classes of subgroups — too many to draw. Only its 124 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#20717C211yesmaximal, characteristic
2#17819C211yesmaximal, characteristic, complemented
3#17357C212yesmaximal
4#7998C212yesmaximal, complemented
5#7987C211yesmaximal, characteristic, complemented
6C22×#117C2211yescharacteristic
7#2920C2212yes—
8#2920C2212yes—
9C2×#432C2211yescharacteristic
10C2×#431C2211yescharacteristic
11#1817C412yes—
12#1798—22no—
13#1779C412yes—
14#1750—22no—
15#1574—24no—
16#1568—22no—
17#1464—24no—
18#1463—22no—
19#1462—22no—
20#1458—24no—
21#1291—24no—
22#1290—22no—
23C24×C4C2×C411yescharacteristic
24C2×#117—21no—
25C2×#117—22no—
26C2×#117—22no—
27C2×#117—21no—
28C2×#117—21no—
29C2×#117—21no—
30C2×#117—22no—
31C2×#117—22no—
32C22×C4⋊C4D411yescharacteristic
33C22×C4⋊C4Q811yescharacteristic
34C22×C22⋊C4D411yescharacteristic
35C22×C22⋊C4Q811yescharacteristic
36C22×C42C2×C411yescharacteristic
37C22×C42C2311yescharacteristic
38#432—22no—
39#432—22no—
40#432—22no—
41#432—22no—
42#432—22no—
43#432D414yes—
44#431—22no—
45#431—22no—
46#431—22no—
47#431—22no—
48#431—22no—
49#431D414yes—
50#423—22no—
51#423—22no—
52C2×C4⋊C8—21no—
53C2×C4⋊C8D412yes—
54#407—22no—
55#407—22no—
56C2×C22⋊C8—24no—
57C2×C22⋊C8C2×C412yes—
58C2×C22⋊C8C2×C412yes—
59C4×M16—22no—
60C4×M16—22no—
61C2×#97—21no—
62C2×C4×C8D412yes—
63#336—24no—
64#335—24no—
65#334—24no—
66#324—28no—
67#323—28no—
68C25C4⋊C411yescharacteristic
69C23×C4—21no—
70C23×C4—22no—
71C23×C4—21no—
72C23×C4—22no—
73C23×C4—44no—
74C23×C4C4⋊C411yescharacteristic
75C23×C4C4⋊C411yescharacteristic
76C23×C4C4⋊C411yescharacteristic
77C23×C4C22×C411yescharacteristic
78C23×C4C2×D411yescharacteristic
79C23×C4C2×Q811yescharacteristic
80C2×M16—42no—
81C2×M16—42no—
82C22×C8—22no—
83C22×C8C22⋊C414yes—
84#117—41no—
85#117—42no—
86#117—42no—
87#117—42no—
88#117—41no—
89#117—42no—
90#117—42no—
91#117—41no—
92#117—42no—
93#117—41no—
94C2×C4⋊C4—22no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—22no—
102C2×C22⋊C4—22no—
103C2×C22⋊C4—21no—
104C2×C22⋊C4—21no—
105C2×C22⋊C4—22no—
106C2×C22⋊C4—21no—
107C2×C22⋊C4—21no—
108C2×C22⋊C4—22no—
109C2×C22⋊C4—22no—
110C2×C42—21no—
111C2×C42—21no—
112C2×C42—22no—
113C2×C42—22no—
114C2×C42C2×D412yes—
115C2×C42C2×D411yescharacteristic
116C2×C42C2×D411yescharacteristic
117C2×C42C2×D412yes—
118C2×C42#4111yescharacteristic
119C2×C42#4112yes—
120C2×C42#4112yes—
121C2×C42#4111yescharacteristic
122C4⋊C8—24no—
123C4⋊C8—24no—
124C4⋊C8—24no—
125C4⋊C8—42no—
126C4⋊C8—42no—
127C4⋊C8—42no—
128C22⋊C8—24no—
129C22⋊C8—24no—
130C22⋊C8—24no—
131C22⋊C8—24no—
132C22⋊C8—48no—
133C22⋊C8—42no—
134C22⋊C8—42no—
135#97—24no—
136#97—41no—
137#97—41no—
138#97—42no—
139C4×C8—22no—
140C4×C8—24no—
141C4×C8—22no—
142C4×C8—42no—
143C24—21no—
144C24—22no—
145C24—42no—
146C24—42no—
147C24—81no—
148C24C2×C4⋊C411yescharacteristic
149C22×C4—24no—
150C22×C4—28no—
151C22×C4—21no—
152C22×C4—22no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—42no—
158C22×C4—42no—
159C22×C4—42no—
160C22×C4—42no—
161C22×C4—44no—
162C22×C4—42no—
163C22×C4—44no—
164C22×C4—42no—
165C22×C4—44no—
166C22×C4—41no—
167C22×C4—42no—
168C22×C4—41no—
169C22×C4—42no—
170C22×C4—81no—
171C22×C4—81no—
172C22×C4C2×C22⋊C411yescharacteristic
173C22×C4#11711yescharacteristic
174C22×C4C4×D412yes—
175C22×C4C4×D412yes—
176C22×C4#12012yes—
177C22×C4#12111yescharacteristic
178C22×C4#12111yescharacteristic
179C22×C4C22⋊Q811yescharacteristic
180C22×C4C22⋊Q811yescharacteristic
181C22×C4C22⋊Q812yes—
182C22×C4C22⋊Q812yes—
183C22×C4#12312yes—
184M16—84no—
185M16—84no—
186C2×C8—24no—
187C2×C8—44no—
188C2×C8—44no—
189C2×C8—42no—
190C2×C8—44no—
191C2×C8—44no—
192C4⋊C4—41no—
193C4⋊C4—41no—
194C4⋊C4—42no—
195C4⋊C4—42no—
196C4⋊C4—41no—
197C4⋊C4—41no—
198C22⋊C4—41no—
199C22⋊C4—41no—
200C22⋊C4—42no—
201C22⋊C4—42no—
202C22⋊C4—41no—
203C22⋊C4—41no—
204C42—41no—
205C42—42no—
206C42—41no—
207C42#12011yescharacteristic
208C42#12112yes—
209C42#12112yes—
210C42#12112yes—
211C42#12311yescharacteristic
212C42#12312yes—
213C42#12412yes—
214C42#12411yescharacteristic
215C42#12612yes—
216C42#12711yescharacteristic
217C23—22no—
218C23—22no—
219C23—24no—
220C23—41no—
221C23—41no—
222C23—41no—
223C23—42no—
224C23—42no—
225C23—42no—
226C23—41no—
227C23—82no—
228C23—82no—
229C23—88no—
230C23#38011yescharacteristic
231C23#38512yes—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—24no—
236C2×C4—22no—
237C2×C4—24no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—41no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—44no—
247C2×C4—48no—
248C2×C4—41no—
249C2×C4—42no—
250C2×C4—41no—
251C2×C4—44no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—82no—
257C2×C4—82no—
258C2×C4#38611yescharacteristic
259C2×C4#38812yes—
260C2×C4#39011yescharacteristic
261C2×C4#39211yescharacteristic
262C2×C4#39311yescharacteristic
263C2×C4#39412yes—
264C2×C4#39612yes—
265C2×C4#39711yescharacteristic
266C2×C4#39911yescharacteristic
267C8—44no—
268C8—82no—
269C8—84no—
270C8—84no—
271C22—21no—
272C22—21no—
273C22—22no—
274C22—22no—
275C22—22no—
276C22—44no—
277C22—44no—
278C22—42no—
279C22—82no—
280C22—84no—
281C22—84no—
282C22—81no—
283C22—81no—
284C22#143911yescharacteristic
285C22#181312yes—
286C4—21no—
287C4—22no—
288C4—21no—
289C4—22no—
290C4—21no—
291C4—21no—
292C4—22no—
293C4—21no—
294C4—42no—
295C4—81no—
296C4—81no—
297C4#189112yes—
298C2—22no—
299C2—42no—
300C2—82no—
301C2#1107111yescharacteristic
302C2#1109911yescharacteristic
303C2#1110211yescharacteristic
subgroupG/H
center, socleC22#1439
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC25C4⋊C4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #17819C2semidirect
4. #7998C2semidirect
5. #7987C2semidirect

Maximal subgroups

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

subgroupindex
#207172
#178192
#173572
#79982
#79872