Finite Group Databasebeta

← #136298

Subgroups — #136298

Order 512.

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

normal subgroups only

This group has 283 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#23368C211yesmaximal, 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#1917—21no—
14#1917—21no—
15#1917—21no—
16#1916—21no—
17#1840—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⋊C8D411yescharacteristic
41C2×C4⋊C8D411yescharacteristic
42C2×#101—21no—
43C2×#101—21no—
44C4×M16—21no—
45C4×M16—21no—
46C2×#97—21no—
47C2×#97D411yescharacteristic
48C2×C4×C8D411yescharacteristic
49#398—21no—
50#398C2×C411yescharacteristic
51#398Q811yescharacteristic
52#395—21no—
53#395C2×C411yescharacteristic
54#395Q811yescharacteristic
55C4×C4⋊C4—21no—
56C4×C4⋊C4—21no—
57#376D411yescharacteristic
58#343—21no—
59#343C2×C411yescharacteristic
60#343C2×C411yescharacteristic
61#336—21no—
62#336—21no—
63#336—21no—
64#336—21no—
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—21no—
108C4⋊C8—21no—
109C4⋊C8—21no—
110C4⋊C8—21no—
111C4⋊C8—21no—
112C4⋊C8—21no—
113C4⋊C8—41no—
114C4⋊C8—41no—
115C4⋊C8—41no—
116C4⋊C8—81no—
117#101—22no—
118#101—42no—
119#101—41no—
120C22⋊C8—21no—
121C22⋊C8—21no—
122C22⋊C8—41no—
123#97—41no—
124#97—41no—
125#97—41no—
126#97—41no—
127#97—41no—
128#97—81no—
129C4×C8—41no—
130C4×C8—41no—
131C4×C8—41no—
132C4×C8—41no—
133C4×C8—41no—
134#95—21no—
135#95—21no—
136#95—21no—
137#95—21no—
138#95—21no—
139#95C4⋊C411yescharacteristic
140#95C4⋊C411yescharacteristic
141C2×Q8—21no—
142C2×Q8—21no—
143C2×Q8—41no—
144C2×Q8—41no—
145C2×Q8—41no—
146C2×Q8—81no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4C2×C4⋊C411yescharacteristic
152C22×C4C4×D411yescharacteristic
153C22×C4C4×D411yescharacteristic
154C22×C4#12011yescharacteristic
155C22×C4C22⋊Q811yescharacteristic
156C22×C4C22⋊Q811yescharacteristic
157C22×C4#12311yescharacteristic
158M16—22no—
159M16—41no—
160M16—42no—
161M16—41no—
162M16—41no—
163M16—81no—
164C2×C8—41no—
165C2×C8—41no—
166C2×C8—41no—
167C2×C8—41no—
168C2×C8—41no—
169C2×C8—41no—
170C2×C8—41no—
171C2×C8—41no—
172C2×C8—42no—
173C2×C8—42no—
174C2×C8—41no—
175C2×C8—41no—
176C2×C8—41no—
177C2×C8—41no—
178C2×C8—41no—
179C2×C8—41no—
180C2×C8—81no—
181C4⋊C4—41no—
182C4⋊C4—41no—
183C4⋊C4—41no—
184C4⋊C4—41no—
185C4⋊C4—41no—
186C4⋊C4—41no—
187C4⋊C4—81no—
188C4⋊C4—81no—
189C4⋊C4—81no—
190C4⋊C4—81no—
191C42—21no—
192C42—21no—
193C42—21no—
194C42—21no—
195C42—21no—
196C42—21no—
197C42—41no—
198C42—41no—
199C42—41no—
200C42—41no—
201C23#38511yescharacteristic
202Q8—42no—
203Q8—81no—
204Q8—81no—
205Q8—81no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
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—21no—
219C2×C4—41no—
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×C4—41no—
234C2×C4#44311yescharacteristic
235C2×C4#44411yescharacteristic
236C2×C4B(4,𝔽2)11yescharacteristic
237C2×C4#45811yescharacteristic
238C8—42no—
239C8—81no—
240C8—81no—
241C8—81no—
242C8—81no—
243C8—81no—
244C8—82no—
245C8—81no—
246C8—81no—
247C8—81no—
248C8—81no—
249C22—21no—
250C22—21no—
251C22#186111yescharacteristic
252C22#186611yescharacteristic
253C22#186711yescharacteristic
254C4—21no—
255C4—21no—
256C4—21no—
257C4—41no—
258C4—41no—
259C4—41no—
260C4—41no—
261C4—41no—
262C4—81no—
263C4—81no—
264C4—81no—
265C4—81no—
266C4#220311yescharacteristic
267C4#220311yescharacteristic
268C2—21no—
269C2—21no—
270C2#887511yescharacteristic
271C2#1296311yescharacteristic
272C2#1297111yescharacteristic
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
#233682
#117362
#90302
#88652
#88382
#72842
#72062