Finite Group Databasebeta

← #128997

Subgroups — #128997

Order 512.

334 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 92 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#21718C211yesmaximal, characteristic
2#11736C211yesmaximal, characteristic
3#11736C211yesmaximal, characteristic
4#8165C211yesmaximal, characteristic
5C8×#101C211yesmaximal, characteristic
6#7286C211yesmaximal, characteristic
7#7284C211yesmaximal, characteristic
8C2×#446C411yescharacteristic
9C2×C8×Q8C2211yescharacteristic
10#2399C2211yescharacteristic
11#1985C411yescharacteristic
12#1985C2211yescharacteristic
13#1939—21no—
14#1939—21no—
15#1843C411yescharacteristic
16#1771—21no—
17#1771—21no—
18C4×#101C2211yescharacteristic
19C4×#101C2211yescharacteristic
20#1726C411yescharacteristic
21#1461C2211yescharacteristic
22#1461C2211yescharacteristic
23#1451—21no—
24C8×M16—21no—
25#1354C412yes—
26#1354C412yes—
27#1346C412yes—
28#1346C412yes—
29#1283—21no—
30#1283—21no—
31C2×C4×Q8C2×C411yescharacteristic
32C2×C4×Q8C2311yescharacteristic
33#446—41no—
34#446—41no—
35C8×Q8—41no—
36C8×Q8—41no—
37#431—21no—
38#431—21no—
39C2×C4⋊C8—21no—
40C2×C4⋊C8—21no—
41C2×C4⋊C8—21no—
42C2×C4⋊C8C2×C411yescharacteristic
43C2×C4⋊C8C2×C411yescharacteristic
44C2×C4⋊C8D411yescharacteristic
45C2×C4⋊C8D411yescharacteristic
46C2×#101—21no—
47C2×#101—21no—
48C4×M16—21no—
49C4×M16—21no—
50C2×#97—21no—
51C2×#97—21no—
52C2×#97C2×C411yescharacteristic
53C2×C4×C8—21no—
54C2×C4×C8C2×C411yescharacteristic
55C2×C4×C8D411yescharacteristic
56C2×C4×C8D411yescharacteristic
57#398—21no—
58#398C2×C411yescharacteristic
59#398C2×C411yescharacteristic
60#395—21no—
61#395C2×C411yescharacteristic
62#395C2×C411yescharacteristic
63C4×C4⋊C4—21no—
64C4×C4⋊C4—21no—
65#376C2×C411yescharacteristic
66#343C2×C412yes—
67#343C2×C412yes—
68#336—21no—
69#336—21no—
70#336—21no—
71#336—21no—
72#324—21no—
73#324—21no—
74#324C2×C412yes—
75#324C2×C412yes—
76#322—41no—
77#322—41no—
78#322—41no—
79C82—41no—
80C22×Q8C4211yescharacteristic
81C22×Q8#4111yescharacteristic
82C2×M16—22no—
83C2×M16—22no—
84C22×C8—21no—
85C22×C8—21no—
86C22×C8C22⋊C411yescharacteristic
87C22×C8C22⋊C411yescharacteristic
88C22×C8C22⋊C411yescharacteristic
89C22×C8C22⋊C411yescharacteristic
90C4×Q8—41no—
91C4×Q8—41no—
92C4×Q8—41no—
93C4×Q8—41no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4C4211yescharacteristic
100C2×C4⋊C4C22×C411yescharacteristic
101C2×C4⋊C4C22×C411yescharacteristic
102C2×C4⋊C4#4111yescharacteristic
103C2×C42—21no—
104C2×C42—21no—
105C2×C42C22×C411yescharacteristic
106C2×C42C2×D411yescharacteristic
107C2×C42C2×D411yescharacteristic
108C4⋊C8—41no—
109C4⋊C8—41no—
110C4⋊C8—41no—
111C4⋊C8—41no—
112C4⋊C8—42no—
113C4⋊C8—42no—
114C4⋊C8—41no—
115C4⋊C8—41no—
116C4⋊C8—42no—
117#101—24no—
118#101—42no—
119C22⋊C8—22no—
120C22⋊C8—22no—
121#97—24no—
122#97—41no—
123#97—41no—
124#97—41no—
125#97—41no—
126#97—41no—
127C4×C8—21no—
128C4×C8—21no—
129C4×C8—21no—
130C4×C8—21no—
131C4×C8—21no—
132C4×C8—21no—
133C4×C8—41no—
134C4×C8—41no—
135C4×C8—41no—
136C4×C8—41no—
137C4×C8—42no—
138#95—21no—
139#95—21no—
140#95—21no—
141#95—21no—
142#95—21no—
143#95C4211yescharacteristic
144#95C4211yescharacteristic
145C2×Q8—22no—
146C2×Q8—22no—
147C2×Q8—41no—
148C2×Q8—41no—
149C2×Q8—81no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4C2×C4211yescharacteristic
155C22×C4C2×C22⋊C411yescharacteristic
156C22×C4#11711yescharacteristic
157C22×C4C4×D411yescharacteristic
158C22×C4C4×D411yescharacteristic
159C22×C4C4×D411yescharacteristic
160C22×C4C4×D411yescharacteristic
161M16—24no—
162M16—42no—
163M16—42no—
164M16—42no—
165C2×C8—22no—
166C2×C8—22no—
167C2×C8—21no—
168C2×C8—21no—
169C2×C8—21no—
170C2×C8—41no—
171C2×C8—42no—
172C2×C8—41no—
173C2×C8—41no—
174C2×C8—41no—
175C2×C8—41no—
176C2×C8—42no—
177C2×C8—42no—
178C2×C8—42no—
179C2×C8—41no—
180C2×C8—41no—
181C2×C8—41no—
182C2×C8#9911yescharacteristic
183C2×C8#9911yescharacteristic
184C4⋊C4—41no—
185C4⋊C4—41no—
186C4⋊C4—41no—
187C4⋊C4—41no—
188C4⋊C4—42no—
189C4⋊C4—42no—
190C4⋊C4—42no—
191C4⋊C4—81no—
192C4⋊C4—81no—
193C42—22no—
194C42—22no—
195C42—21no—
196C42—21no—
197C42—21no—
198C42—21no—
199C42—21no—
200C42—21no—
201C42—41no—
202C42—41no—
203C23C4×C22⋊C411yescharacteristic
204Q8—44no—
205Q8—82no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4C2×#9911yescharacteristic
234C2×C4#41011yescharacteristic
235C2×C4#44312yes—
236C2×C4#44412yes—
237C8—21no—
238C8—42no—
239C8—44no—
240C8—41no—
241C8—41no—
242C8—41no—
243C8—82no—
244C8—82no—
245C8—82no—
246C8—81no—
247C8#35612yes—
248C22—21no—
249C22—21no—
250C22C4×#9911yescharacteristic
251C22#176411yescharacteristic
252C22#176511yescharacteristic
253C4—22no—
254C4—22no—
255C4—21no—
256C4—42no—
257C4—42no—
258C4—41no—
259C4—41no—
260C4—41no—
261C4—41no—
262C4—81no—
263C4—81no—
264C4C2×#35611yescharacteristic
265C4#213211yescharacteristic
266C2—21no—
267C2—21no—
268C2#824011yescharacteristic
269C2#1262411yescharacteristic
270C2C4×#35611yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC22×C4C2×C42
Frattini, cosocleC2×C4×Q8C23
socleC22C4×#99
prime radicalC23C4×C22⋊C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ C2×C8×Q8 ◁ G
LECSG ▷ C2×C4×Q8 ▷ C23 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#217182
#117362
#81652
C8×#1012
#72862
#72842