Finite Group Databasebeta

← #136283

Subgroups — #136283

Order 512.

303 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 303 conjugacy classes of subgroups — too many to draw. Only its 66 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#23252C211yesmaximal, characteristic
2#11736C211yesmaximal, characteristic
3#9030C211yesmaximal, characteristic
4#8865C211yesmaximal, characteristic
5#8838C211yesmaximal, characteristic
6#7284C211yesmaximal, characteristic
7#7206C211yesmaximal, characteristic
8#2983C2211yescharacteristic
9#2399C2211yescharacteristic
10#1985—21no—
11#1985C2211yescharacteristic
12#1984C2211yescharacteristic
13#1940—21no—
14#1939—21no—
15#1771—21no—
16#1770—21no—
17#1768—21no—
18C4×#101C2211yescharacteristic
19C4×#101C2211yescharacteristic
20#1573—21no—
21#1565—21no—
22#1491—21no—
23#1461C2211yescharacteristic
24#1354C411yescharacteristic
25#1354C411yescharacteristic
26#1346C411yescharacteristic
27#1346C411yescharacteristic
28#1302—21no—
29#1300—21no—
30#1278—21no—
31C2×C4×Q8D411yescharacteristic
32C2×C4×Q8C2311yescharacteristic
33#446—41no—
34#446—41no—
35#432—21no—
36#432—21no—
37#431—21no—
38#431—21no—
39C2×C4⋊C8—21no—
40C2×C4⋊C8—21no—
41C2×C4⋊C8—21no—
42C2×C4⋊C8D411yescharacteristic
43C2×C4⋊C8D411yescharacteristic
44C2×#101—21no—
45C2×#101—21no—
46C4×M16—21no—
47C4×M16—21no—
48C2×#97—21no—
49C2×#97—21no—
50C2×#97D411yescharacteristic
51C2×C4×C8—21no—
52C2×C4×C8D411yescharacteristic
53#398—21no—
54#398C2×C411yescharacteristic
55#398Q811yescharacteristic
56#395—21no—
57#395C2×C411yescharacteristic
58#395Q811yescharacteristic
59C4×C4⋊C4—21no—
60C4×C4⋊C4—21no—
61#376D411yescharacteristic
62#343—21no—
63#343C2×C411yescharacteristic
64#343C2×C411yescharacteristic
65#336—21no—
66#336—21no—
67#336—21no—
68#336—21no—
69#336—21no—
70#336—21no—
71#335—41no—
72#335—41no—
73#334—41no—
74#334—41no—
75Q8⋊C8—41no—
76Q8⋊C8—41no—
77#324—21no—
78#324C2×C411yescharacteristic
79#324C2×C411yescharacteristic
80C22×Q8C4⋊C411yescharacteristic
81C22×Q8#4111yescharacteristic
82C2×M16—21no—
83C2×M16—21no—
84C2×M16—41no—
85C22×C8—21no—
86C22×C8—21no—
87C22×C8—21no—
88C22×C8—21no—
89C4×Q8—41no—
90C4×Q8—41no—
91C4×Q8—41no—
92C4×Q8—41no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4C4⋊C411yescharacteristic
99C2×C4⋊C4C22×C411yescharacteristic
100C2×C4⋊C4C2×Q811yescharacteristic
101C2×C4⋊C4#4111yescharacteristic
102C2×C42—21no—
103C2×C42—21no—
104C2×C42C2×D411yescharacteristic
105C2×C42C2×D411yescharacteristic
106C2×C42C2×D411yescharacteristic
107C4⋊C8—24no—
108C4⋊C8—21no—
109C4⋊C8—21no—
110C4⋊C8—21no—
111C4⋊C8—21no—
112C4⋊C8—21no—
113C4⋊C8—21no—
114C4⋊C8—41no—
115C4⋊C8—41no—
116C4⋊C8—42no—
117C4⋊C8—41no—
118C4⋊C8—81no—
119#101—22no—
120#101—42no—
121#101—41no—
122C22⋊C8—21no—
123C22⋊C8—21no—
124C22⋊C8—41no—
125#97—41no—
126#97—42no—
127#97—41no—
128#97—41no—
129#97—41no—
130#97—41no—
131#97—81no—
132C4×C8—41no—
133C4×C8—41no—
134C4×C8—41no—
135C4×C8—42no—
136C4×C8—41no—
137C4×C8—41no—
138#95—21no—
139#95—21no—
140#95—21no—
141#95—21no—
142#95—21no—
143#95C4⋊C411yescharacteristic
144#95C4⋊C411yescharacteristic
145C2×Q8—21no—
146C2×Q8—21no—
147C2×Q8—41no—
148C2×Q8—41no—
149C2×Q8—41no—
150C2×Q8—81no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4C2×C4⋊C411yescharacteristic
156C22×C4C4×D411yescharacteristic
157C22×C4C4×D411yescharacteristic
158C22×C4#12011yescharacteristic
159C22×C4C22⋊Q811yescharacteristic
160C22×C4C22⋊Q811yescharacteristic
161C22×C4#12311yescharacteristic
162M16—22no—
163M16—41no—
164M16—42no—
165M16—41no—
166M16—41no—
167M16—81no—
168C2×C8—22no—
169C2×C8—22no—
170C2×C8—41no—
171C2×C8—41no—
172C2×C8—41no—
173C2×C8—41no—
174C2×C8—41no—
175C2×C8—41no—
176C2×C8—41no—
177C2×C8—41no—
178C2×C8—41no—
179C2×C8—41no—
180C2×C8—41no—
181C2×C8—41no—
182C2×C8—41no—
183C2×C8—41no—
184C2×C8—41no—
185C2×C8—41no—
186C2×C8—81no—
187C4⋊C4—41no—
188C4⋊C4—41no—
189C4⋊C4—41no—
190C4⋊C4—41no—
191C4⋊C4—41no—
192C4⋊C4—41no—
193C4⋊C4—81no—
194C4⋊C4—81no—
195C4⋊C4—81no—
196C4⋊C4—81no—
197C42—22no—
198C42—22no—
199C42—21no—
200C42—21no—
201C42—21no—
202C42—21no—
203C42—21no—
204C42—21no—
205C42—41no—
206C42—41no—
207C23#38511yescharacteristic
208Q8—42no—
209Q8—81no—
210Q8—81no—
211Q8—81no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4#44311yescharacteristic
241C2×C4#44411yescharacteristic
242C2×C4B(4,𝔽2)11yescharacteristic
243C2×C4#45811yescharacteristic
244C8—42no—
245C8—42no—
246C8—42no—
247C8—81no—
248C8—81no—
249C8—81no—
250C8—82no—
251C8—81no—
252C8—81no—
253C8—81no—
254C8—81no—
255C22—21no—
256C22—21no—
257C22#186111yescharacteristic
258C22#186611yescharacteristic
259C22#186711yescharacteristic
260C4—21no—
261C4—21no—
262C4—21no—
263C4—42no—
264C4—42no—
265C4—41no—
266C4—41no—
267C4—41no—
268C4—41no—
269C4—41no—
270C4—81no—
271C4—81no—
272C4#220411yescharacteristic
273C4#220511yescharacteristic
274C2—21no—
275C2—21no—
276C2#887411yescharacteristic
277C2#1296311yescharacteristic
278C2#1297311yescharacteristic
subgroupG/H
centerC2×C4B(4,𝔽2)
commutatorC2×C4⋊C4C22×C4
Frattini, cosocleC2×C4×Q8C23
socleC22#1867
prime radicalC23#385
LCSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C4×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
#232522
#117362
#90302
#88652
#88382
#72842
#72062