Finite Group Databasebeta

← #142163

Subgroups — #142163

Order 512.

484 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 114 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#33208C211yesmaximal, characteristic
2#12783C211yesmaximal, characteristic
3C4×#340C211yesmaximal, characteristic
4#7258C212yesmaximal
5#7258C212yesmaximal
6C2×#499C411yescharacteristic, complemented
7C22×C4×C8C2211yescharacteristic
8#2851—21no—
9C2×C4×C4⋊C4—21no—
10C2×C4⋊C16—21no—
11C2×C22⋊C16C2212yes—
12#1991—21no—
13#1942C411yescharacteristic, complemented
14#1942C411yescharacteristic, complemented
15#1941—21no—
16#1846C411yescharacteristic, complemented
17#1808C2211yescharacteristic
18#1808C2211yescharacteristic
19#1807—21no—
20C2×#340—21no—
21C2×#340C2212yes—
22#1374—21no—
23C23×C8D411yescharacteristic
24C23×C8C2311yescharacteristic
25C2×#125—21no—
26C22×C4⋊C4—22no—
27C22×C42—41no—
28C22×C42D411yescharacteristic
29C22×C16—22no—
30#499—42no—
31C2×#107C2×C411yescharacteristic
32C2×#107C2×C411yescharacteristic
33C2×#107C2×C411yescharacteristic
34C2×#107C2×C411yescharacteristic
35C2×#106—21no—
36C2×#106—21no—
37C2×C4×C8—42no—
38C2×C4×C8C2×C411yescharacteristic
39C2×C4×C8C2×C411yescharacteristic
40C2×C4×C8D411yescharacteristic
41C2×C4×C8D411yescharacteristic
42#389—21no—
43#389—21no—
44#387—41no—
45#387—41no—
46#385—41no—
47#385—41no—
48#382—21no—
49#380—21no—
50#380—21no—
51C4×C4⋊C4—24no—
52C4×C4⋊C4—42no—
53C2×#95—41no—
54C4⋊C16—42no—
55C22⋊C16—24no—
56#340—24no—
57#340—21no—
58#340—21no—
59#340—21no—
60#340—21no—
61#340—22no—
62#340Q814yes—
63C23×C4—42no—
64C23×C4D811yescharacteristic
65C23×C4D811yescharacteristic
66C23×C4C2×D411yescharacteristic
67C22×C8—22no—
68C22×C8—22no—
69C22×C8—41no—
70C22×C8—41no—
71C22×C8—41no—
72C22×C8C22⋊C411yescharacteristic
73C22×C8C22⋊C411yescharacteristic
74C22×C8C22×C411yescharacteristic
75C22×C8C2×D411yescharacteristic
76C22×C8C2×Q812yes—
77C22×C8#4112yes—
78#125—42no—
79#125—42no—
80C2×C4⋊C4—24no—
81C2×C4⋊C4—24no—
82C2×C4⋊C4—24no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—42no—
88C2×C4⋊C4—42no—
89C2×C4⋊C4—42no—
90C2×C4⋊C4—41no—
91C2×C4⋊C4—41no—
92C2×C4⋊C4—41no—
93C2×C4⋊C4—41no—
94C2×C22⋊C4—41no—
95C2×C22⋊C4—41no—
96C2×C42—22no—
97C2×C42—22no—
98C2×C42—44no—
99C2×C42—82no—
100C2×C42—82no—
101C2×C42C22⋊C411yescharacteristic
102C2×C42C22⋊C411yescharacteristic
103C2×C42SD1611yescharacteristic
104C2×C42SD1611yescharacteristic
105C2×C16—42no—
106C2×C16—42no—
107C2×C16—42no—
108#107—41no—
109#107—41no—
110#107—41no—
111#107—41no—
112#106—42no—
113#106—42no—
114C4×C8—22no—
115C4×C8—22no—
116C4×C8—41no—
117C4×C8—41no—
118#95—44no—
119#95—41no—
120#95—41no—
121#95—41no—
122#95—41no—
123#95—41no—
124#95—41no—
125C24C2×D811yescharacteristic
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—22no—
135C22×C4—22no—
136C22×C4—41no—
137C22×C4—44no—
138C22×C4—44no—
139C22×C4—44no—
140C22×C4—41no—
141C22×C4—41no—
142C22×C4—41no—
143C22×C4—41no—
144C22×C4—82no—
145C22×C4—82no—
146C22×C4—82no—
147C22×C4—82no—
148C22×C4#10211yescharacteristic
149C22×C4#10211yescharacteristic
150C22×C4#10211yescharacteristic
151C22×C4#10211yescharacteristic
152C22×C4C2×C22⋊C411yescharacteristic
153C22×C4C22⋊Q812yes—
154C22×C4C2×SD1611yescharacteristic
155C22×C4#13712yes—
156C2×C8—21no—
157C2×C8—21no—
158C2×C8—21no—
159C2×C8—21no—
160C2×C8—21no—
161C2×C8—21no—
162C2×C8—21no—
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—42no—
166C2×C8—42no—
167C2×C8—44no—
168C2×C8#10412yes—
169C2×C8C4×Q812yes—
170C2×C8#12411yescharacteristic
171C2×C8C4⋊Q811yescharacteristic
172C4⋊C4—28no—
173C4⋊C4—44no—
174C4⋊C4—44no—
175C4⋊C4—44no—
176C4⋊C4—41no—
177C4⋊C4—41no—
178C4⋊C4—41no—
179C4⋊C4—41no—
180C4⋊C4—41no—
181C4⋊C4—41no—
182C4⋊C4—41no—
183C4⋊C4—41no—
184C4⋊C4—82no—
185C4⋊C4—82no—
186C4⋊C4—82no—
187C4⋊C4—82no—
188C22⋊C4—82no—
189C22⋊C4—81no—
190C22⋊C4—81no—
191C42—22no—
192C42—22no—
193C42—21no—
194C42—21no—
195C42—88no—
196C42#9912yes—
197C42#10412yes—
198C16—82no—
199C23—21no—
200C23—21no—
201C23—21no—
202C23—21no—
203C23C2×#10211yescharacteristic
204C23#47412yes—
205C23#50812yes—
206C23#50911yescharacteristic
207C23#51011yescharacteristic
208C2×C4—24no—
209C2×C4—24no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—44no—
226C2×C4—48no—
227C2×C4—84no—
228C2×C4—84no—
229C2×C4—84no—
230C2×C4—84no—
231C2×C4—84no—
232C2×C4—84no—
233C2×C4—81no—
234C2×C4—81no—
235C2×C4—81no—
236C2×C4—81no—
237C2×C4—81no—
238C2×C4—81no—
239C2×C4—81no—
240C2×C4—81no—
241C2×C4—81no—
242C2×C4—81no—
243C2×C4—81no—
244C2×C4—81no—
245C2×C4#32712yes—
246C2×C4#32712yes—
247C2×C4#36012yes—
248C2×C4#39111yescharacteristic
249C2×C4C2×#9911yescharacteristic
250C2×C4#41011yescharacteristic
251C2×C4#41911yescharacteristic
252C2×C4C2×#10411yescharacteristic
253C2×C4C2×#10411yescharacteristic
254C2×C4#47512yes—
255C8—21no—
256C8—22no—
257C8—21no—
258C8—22no—
259C8—41no—
260C22—22no—
261C22—22no—
262C22—21no—
263C22—21no—
264C22—22no—
265C22—22no—
266C22—21no—
267C22—21no—
268C22C2×#32711yescharacteristic
269C22#150211yescharacteristic
270C22#198811yescharacteristic
271C22#213811yescharacteristic
272C22#213911yescharacteristic
273C22C2×#36011yescharacteristic
274C22#214511yescharacteristic
275C22#222812yes—
276C22#222912yes—
277C4—22no—
278C4—22no—
279C4—22no—
280C4—22no—
281C4—41no—
282C4—88no—
283C4—161no—
284C4—161no—
285C4—161no—
286C4—161no—
287C4#150611yescharacteristic
288C4#150811yescharacteristic
289C4#199411yescharacteristic
290C4#199611yescharacteristic
291C2—22no—
292C2—22no—
293C2#990911yescharacteristic
294C2#1170311yescharacteristic
295C2#1170411yescharacteristic
296C2#1174111yescharacteristic
297C2#1174611yescharacteristic
298C2#1327711yescharacteristic
299C2#1327811yescharacteristic
subgroupG/H
center, socleC23C2×#102
commutatorC22×C8C22×C4
Frattini, cosocleC23×C8C23
prime radicalC24C2×D8
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
6. C2×#499C4semidirect
13. #1942C4semidirect
14. #1942C4semidirect
16. #1846C4semidirect

Maximal subgroups

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

subgroupindex
#332082
#127832
C4×#3402
#72582