Finite Group Databasebeta

← #141178

Subgroups — #141178

Order 512.

431 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 431 conjugacy classes of subgroups — too many to draw. Only its 88 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#33176C211yesmaximal, characteristic
2#12791C211yesmaximal, characteristic
3C2×#1342C212yesmaximal
4C2×#1340C212yesmaximal
5#10813C211yesmaximal, characteristic
6C2×#494C411yescharacteristic, complemented
7C2×#470—21no—
8C2×#470C2212yes—
9C2×C4×M16C2211yescharacteristic
10#2857—21no—
11C2×#364—21no—
12C2×#349C2212yes—
13#1971—21no—
14#1971C412yescomplemented
15#1956—21no—
16#1844C411yescharacteristic, complemented
17C2×#344—21no—
18C2×#339—21no—
19C2×#338C2211yescharacteristic
20C2×H(3,ℤ/4)C2211yescharacteristic
21#1375—21no—
22#1342—24no—
23#1340—24no—
24C22×Q16—21no—
25C22×Q16C2×C412yes—
26C22×M16—21no—
27C22×M16D411yescharacteristic
28C22×M16C2311yescharacteristic
29C2×C4⋊Q8—21no—
30C2×C22⋊Q8—22no—
31C2×#117—21no—
32C22×C42D411yescharacteristic
33C2×M32—22no—
34#494—42no—
35#470—22no—
36#470—22no—
37#470—22no—
38#470—22no—
39#470—42no—
40#470—42no—
41C2×#106—21no—
42C2×#106C2×C412yes—
43C2×#103—22no—
44C2×#103—22no—
45C4×M16—42no—
46C2×#97C2×C411yescharacteristic
47C2×#97D411yescharacteristic
48C2×C4×C8C2×C411yescharacteristic
49C2×C4×C8D411yescharacteristic
50#391—22no—
51#389—42no—
52#384—21no—
53#380—42no—
54#364—42no—
55#349—22no—
56#349—22no—
57#344—42no—
58#339—42no—
59#338—21no—
60#338D412yes—
61H(3,ℤ/4)—21no—
62H(3,ℤ/4)D412yes—
63C22×Q8—22no—
64C23×C4—21no—
65C23×C4C2×D411yescharacteristic
66C2×Q16—22no—
67C2×Q16—22no—
68C2×Q16—42no—
69C2×Q16—42no—
70C2×Q16—42no—
71C2×Q16—42no—
72C2×Q16—42no—
73C2×M16—22no—
74C2×M16—41no—
75C2×M16—41no—
76C2×M16—41no—
77C2×M16—41no—
78C2×M16—41no—
79C2×M16—41no—
80C2×M16—41no—
81C2×M16—41no—
82C2×M16—41no—
83C2×M16D812yes—
84C2×M16SD1612yes—
85C2×M16C2×D411yescharacteristic
86C2×M16C2×D411yescharacteristic
87C2×M16#4111yescharacteristic
88C2×M16#4111yescharacteristic
89C22×C8—41no—
90C22×C8C22⋊C411yescharacteristic
91C22×C8C22⋊C411yescharacteristic
92C22×C8C22×C411yescharacteristic
93C22×C8C2×D411yescharacteristic
94C4⋊Q8—42no—
95C4⋊Q8—42no—
96C22⋊Q8—42no—
97C22⋊Q8—42no—
98C22⋊Q8—42no—
99C22⋊Q8—42no—
100#117—22no—
101#117—41no—
102#117—41no—
103#117—41no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—41no—
106C2×C4⋊C4—42no—
107C2×C4⋊C4—42no—
108C2×C4⋊C4—42no—
109C2×C22⋊C4—41no—
110C2×C22⋊C4—42no—
111C2×C42—22no—
112C2×C42—21no—
113C2×C42—41no—
114C2×C42—41no—
115C2×C42C22⋊C411yescharacteristic
116C2×C42C22⋊C411yescharacteristic
117M32—42no—
118M32—42no—
119C2×C16—42no—
120#106—22no—
121#106—22no—
122#106—42no—
123#103—44no—
124#103—42no—
125#103—42no—
126#97—41no—
127#97—41no—
128C4×C8—22no—
129C4×C8—22no—
130#95—44no—
131C24AGL(1,ℤ/8)11yescharacteristic
132C2×Q8—42no—
133C2×Q8—42no—
134C2×Q8—42no—
135C2×Q8—42no—
136C2×Q8—42no—
137C2×Q8—42no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—42no—
145C22×C4—44no—
146C22×C4—42no—
147C22×C4—42no—
148C22×C4—81no—
149C22×C4C2×C22⋊C411yescharacteristic
150C22×C4#12111yescharacteristic
151C22×C4#12111yescharacteristic
152C22×C4C2×D811yescharacteristic
153C22×C4C2×SD1611yescharacteristic
154C22×C4AGL(1,ℤ/8)11yescharacteristic
155Q16—82no—
156Q16—82no—
157Q16—84no—
158M16—42no—
159M16—42no—
160M16—41no—
161M16—41no—
162M16—41no—
163M16—41no—
164M16—42no—
165M16—42no—
166M16—81no—
167M16—81no—
168C2×C8—21no—
169C2×C8—21no—
170C2×C8—21no—
171C2×C8—21no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—42no—
177C2×C8—81no—
178C2×C8—81no—
179C2×C8#10212yes—
180C2×C8#10212yes—
181C2×C8C4×D411yescharacteristic
182C2×C8C4×D411yescharacteristic
183C2×C8#12411yescharacteristic
184C2×C8#12711yescharacteristic
185C4⋊C4—42no—
186C4⋊C4—42no—
187C4⋊C4—42no—
188C4⋊C4—84no—
189C4⋊C4—81no—
190C4⋊C4—84no—
191C4⋊C4—82no—
192C4⋊C4—82no—
193C22⋊C4—42no—
194C22⋊C4—81no—
195C22⋊C4—82no—
196C22⋊C4—82no—
197C42—21no—
198C42—21no—
199C42—21no—
200C42—42no—
201C42—42no—
202C42—81no—
203C42#9912yes—
204C16—82no—
205C23—21no—
206C23—21no—
207C23—42no—
208C23#41811yescharacteristic
209C23#45911yescharacteristic
210C23#46011yescharacteristic
211Q8—82no—
212Q8—82no—
213Q8—82no—
214Q8—82no—
215C2×C4—24no—
216C2×C4—22no—
217C2×C4—21no—
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—42no—
228C2×C4—41no—
229C2×C4—84no—
230C2×C4—82no—
231C2×C4—84no—
232C2×C4—82no—
233C2×C4—82no—
234C2×C4—82no—
235C2×C4—82no—
236C2×C4—162no—
237C2×C4#39011yescharacteristic
238C2×C4C2×#9911yescharacteristic
239C2×C4#41011yescharacteristic
240C2×C4C2×#10211yescharacteristic
241C2×C4#42111yescharacteristic
242C2×C4#42111yescharacteristic
243C2×C4#45911yescharacteristic
244C2×C4#46011yescharacteristic
245C8—21no—
246C8—21no—
247C8—21no—
248C8—21no—
249C8—22no—
250C8—22no—
251C8—82no—
252C22—22no—
253C22—21no—
254C22—21no—
255C22—41no—
256C22—41no—
257C22—42no—
258C22—81no—
259C22#151011yescharacteristic
260C22#151411yescharacteristic
261C22#198211yescharacteristic
262C4—24no—
263C4—21no—
264C4—21no—
265C4—81no—
266C4—162no—
267C4—162no—
268C4—162no—
269C4#151711yescharacteristic
270C4#151911yescharacteristic
271C4#196311yescharacteristic
272C4#196411yescharacteristic
273C2—21no—
274C2—21no—
275C2—41no—
276C2—41no—
277C2#971511yescharacteristic
278C2#1183812yes—
subgroupG/H
center, socleC22#1514
commutatorC22×C8C22×C4
Frattini, cosocleC22×M16C23
prime radicalC24AGL(1,ℤ/8)
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×M16 ◁ G
LECSG ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
6. C2×#494C4semidirect
14. #1971C4semidirect
16. #1844C4semidirect

Maximal subgroups

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

subgroupindex
#331762
#127912
C2×#13422
C2×#13402
#108132