Finite Group Databasebeta

← #138024

Subgroups — #138024

Order 512.

424 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 102 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
1C2×#2162C211yesmaximal, characteristic
2#33130C211yesmaximal, characteristic
3C2×#1350C212yesmaximal
4#10822C211yesmaximal, characteristic
5#7259C212yesmaximal
6C2×#467C2211yescharacteristic
7C22×C4×C8C2211yescharacteristic
8#2855—21no—
9#2162—24no—
10C2×C4⋊C16—21no—
11C2×C22⋊C16C2212yes—
12C2×C4×C16—21no—
13#1969C411yescharacteristic
14#1969C411yescharacteristic
15#1942C411yescharacteristic
16#1942C411yescharacteristic
17#1933—21no—
18#1933—21no—
19#1892—21no—
20#1808C2211yescharacteristic
21C2×#341—21no—
22C2×#341C2212yes—
23#1728—21no—
24#1350—24no—
25C22×Q16C2×C411yescharacteristic
26C22×M16—22no—
27C23×C8D411yescharacteristic
28C23×C8C2311yescharacteristic
29C2×C22⋊Q8—21no—
30C22×C42D411yescharacteristic
31C22×C16—22no—
32#467—22no—
33#467—22no—
34C2×#107C2×C411yescharacteristic
35C2×#107C2×C411yescharacteristic
36C2×#107C2×C411yescharacteristic
37C2×#103—21no—
38C2×#103—21no—
39C2×#103—21no—
40C2×C4×C8—22no—
41C2×C4×C8—22no—
42C2×C4×C8C2×C411yescharacteristic
43C2×C4×C8C2×C411yescharacteristic
44C2×C4×C8D411yescharacteristic
45C2×C4×C8D411yescharacteristic
46#391—21no—
47#391—21no—
48#389—21no—
49#389—21no—
50#387—41no—
51#387—41no—
52#385—41no—
53#381—41no—
54#380—21no—
55C4⋊C16—24no—
56C22⋊C16—22no—
57C22⋊C16D414yes—
58C4×C16—24no—
59#341—24no—
60#341—22no—
61#341—22no—
62#340—21no—
63#340—21no—
64#340—21no—
65#336—41no—
66#336—41no—
67C22×Q8—21no—
68C23×C4SD1611yescharacteristic
69C23×C4SD1611yescharacteristic
70C23×C4C2×D411yescharacteristic
71C2×Q16—22no—
72C2×Q16—41no—
73C2×Q16—41no—
74C2×M16—24no—
75C2×M16—42no—
76C2×M16—42no—
77C2×M16—42no—
78C2×M16—42no—
79C2×M16—42no—
80C22×C8—22no—
81C22×C8—22no—
82C22×C8—41no—
83C22×C8—41no—
84C22×C8—41no—
85C22×C8—42no—
86C22×C8C22⋊C411yescharacteristic
87C22×C8C22⋊C411yescharacteristic
88C22×C8C22×C411yescharacteristic
89C22×C8C2×D411yescharacteristic
90C22×C8C2×D412yes—
91C22×C8#4112yes—
92C22⋊Q8—42no—
93C22⋊Q8—42no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—41no—
98C2×C4⋊C4—41no—
99C2×C4⋊C4—41no—
100C2×C22⋊C4—41no—
101C2×C22⋊C4—41no—
102C2×C42—22no—
103C2×C42—22no—
104C2×C42C22⋊C411yescharacteristic
105C2×C42C22⋊C411yescharacteristic
106C2×C42D811yescharacteristic
107C2×C42D811yescharacteristic
108C2×C16—24no—
109C2×C16—42no—
110C2×C16—42no—
111#107—22no—
112#107—41no—
113#107—41no—
114#103—42no—
115#103—42no—
116#103—41no—
117#103—41no—
118C4×C8—22no—
119C4×C8—22no—
120C4×C8—21no—
121C4×C8—21no—
122C4×C8—21no—
123C4×C8—21no—
124#95—41no—
125#95—41no—
126#95—41no—
127#95—41no—
128#95—41no—
129#95—41no—
130#95—41no—
131C24C2×SD1611yescharacteristic
132C2×Q8—41no—
133C2×Q8—41no—
134C2×Q8—41no—
135C2×Q8—41no—
136C2×Q8—41no—
137C2×Q8—41no—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—22no—
146C22×C4—22no—
147C22×C4—41no—
148C22×C4—41no—
149C22×C4—41no—
150C22×C4—41no—
151C22×C4—41no—
152C22×C4#10211yescharacteristic
153C22×C4#10211yescharacteristic
154C22×C4#10211yescharacteristic
155C22×C4#10211yescharacteristic
156C22×C4C2×C22⋊C411yescharacteristic
157C22×C4#12112yes—
158C22×C4C2×D811yescharacteristic
159C22×C4#13512yes—
160Q16—82no—
161M16—44no—
162M16—44no—
163M16—82no—
164M16—82no—
165C2×C8—21no—
166C2×C8—21no—
167C2×C8—21no—
168C2×C8—21no—
169C2×C8—21no—
170C2×C8—21no—
171C2×C8—21no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—42no—
175C2×C8—42no—
176C2×C8—44no—
177C2×C8—44no—
178C2×C8—82no—
179C2×C8—82no—
180C2×C8#10412yes—
181C2×C8C4×D412yes—
182C2×C8#12011yescharacteristic
183C2×C8#12311yescharacteristic
184C4⋊C4—41no—
185C4⋊C4—41no—
186C4⋊C4—41no—
187C4⋊C4—41no—
188C4⋊C4—41no—
189C4⋊C4—41no—
190C4⋊C4—82no—
191C4⋊C4—82no—
192C4⋊C4—82no—
193C22⋊C4—82no—
194C22⋊C4—82no—
195C42—21no—
196C42—21no—
197C42—21no—
198C42—21no—
199C42—22no—
200C42—22no—
201C16—44no—
202C23—21no—
203C23—21no—
204C23—21no—
205C23—21no—
206C23—21no—
207C23—21no—
208C23C2×#10211yescharacteristic
209C23#46512yes—
210Q8—81no—
211Q8—81no—
212Q8—81no—
213Q8—81no—
214C2×C4—28no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—44no—
231C2×C4—81no—
232C2×C4—81no—
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#35912yes—
244C2×C4#35912yes—
245C2×C4#36212yes—
246C2×C4#38611yescharacteristic
247C2×C4#41011yescharacteristic
248C2×C4#41011yescharacteristic
249C2×C4#41611yescharacteristic
250C2×C4C2×#10411yescharacteristic
251C2×C4#42111yescharacteristic
252C2×C4#46612yes—
253C8—22no—
254C8—21no—
255C8—22no—
256C8—21no—
257C8—41no—
258C8—84no—
259C22—22no—
260C22—22no—
261C22—22no—
262C22—41no—
263C22—41no—
264C22—41no—
265C22—41no—
266C22#150211yescharacteristic
267C22#151711yescharacteristic
268C22#189311yescharacteristic
269C22C2×#35911yescharacteristic
270C22C2×#35911yescharacteristic
271C22C2×#36211yescharacteristic
272C22#214811yescharacteristic
273C4—22no—
274C4—22no—
275C4—22no—
276C4—22no—
277C4—41no—
278C4—161no—
279C4—161no—
280C4—161no—
281C4—161no—
282C4#150011yescharacteristic
283C4#152111yescharacteristic
284C4#187111yescharacteristic
285C4#189911yescharacteristic
286C2—42no—
287C2#911411yescharacteristic
288C2#1174811yescharacteristic
289C2#1174911yescharacteristic
290C2#1186111yescharacteristic
291C2#1186311yescharacteristic
292C2#1306311yescharacteristic
293C2#1306611yescharacteristic
subgroupG/H
center, socleC23C2×#102
commutatorC22×C8C22×C4
Frattini, cosocleC23×C8C23
prime radicalC24C2×SD16
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
C2×#21622
#331302
C2×#13502
#108222
#72592