Finite Group Databasebeta

← #138633

Subgroups — #138633

Order 512.

362 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 98 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×#2209C211yesmaximal, characteristic
2#17856C211yesmaximal, characteristic
3#12819C211yesmaximal, characteristic
4#11932C211yesmaximal, characteristic
5#9772C211yesmaximal, characteristic
6#7277C211yesmaximal, characteristic
7#7153C211yesmaximal, characteristic
8C22×Q32—21no—
9C2×#494C2211yescharacteristic
10#2982—21no—
11C2×C4×Q16C2211yescharacteristic
12#2209—22no—
13#2209C414yes—
14#2150C2211yescharacteristic
15C2×C4⋊C16C2211yescharacteristic
16C2×#362—21no—
17C2×#358—21no—
18#1848C2211yescharacteristic
19#1579—22no—
20C8×Q16—22no—
21#1564C2211yescharacteristic
22#1546—21no—
23#1546—21no—
24#1479—21no—
25#1450C2211yescharacteristic
26#1387—21no—
27#1362—22no—
28C22×Q16—21no—
29C22×Q16—21no—
30C22×Q16D411yescharacteristic
31C2×C4⋊Q8—21no—
32C2×C4×Q8—21no—
33C2×Q32—24no—
34C2×Q32—42no—
35C2×Q32—42no—
36C2×M32—21no—
37C22×C16—21no—
38#494—21no—
39#494—21no—
40#494—21no—
41#494C2×C412yes—
42C8×Q8—24no—
43C8×Q8—42no—
44C4×Q16—21no—
45C4×Q16—21no—
46C4×Q16C2×C412yes—
47C4×Q16D412yes—
48#431—21no—
49#431—42no—
50#431D411yescharacteristic
51#431D411yescharacteristic
52C2×#108—21no—
53C2×#108—21no—
54C2×#107D411yescharacteristic
55#423—21no—
56C2×#103—21no—
57C2×C4×C8C2311yescharacteristic
58C4⋊C16—21no—
59C4⋊C16—21no—
60C4⋊C16—21no—
61C4⋊C16C2×C412yes—
62#362—42no—
63#358—24no—
64#335D412yes—
65#334—21no—
66#332—42no—
67#332—42no—
68Q8⋊C8—24no—
69#322—21no—
70C82—21no—
71C22×Q8—21no—
72C22×Q8—41no—
73C2×Q16—22no—
74C2×Q16—22no—
75C2×Q16—21no—
76C2×Q16—21no—
77C2×Q16—21no—
78C2×Q16—41no—
79C2×Q16—41no—
80C2×Q16—41no—
81C2×Q16—41no—
82C2×Q16—41no—
83C2×Q16—41no—
84C2×Q16—81no—
85C2×Q16—81no—
86C2×Q16C22⋊C412yes—
87C2×Q16D812yes—
88C2×Q16SD1612yes—
89C2×M16—41no—
90C22×C8C2×D411yescharacteristic
91C22×C8#4111yescharacteristic
92C4⋊Q8—22no—
93C4⋊Q8—41no—
94C4⋊Q8—41no—
95C4⋊Q8—41no—
96C4×Q8—22no—
97C4×Q8—22no—
98C4×Q8—41no—
99C4×Q8—41no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—41no—
102C2×C4⋊C4—41no—
103C2×C4⋊C4—41no—
104C2×C42—41no—
105C2×C42C2×D411yescharacteristic
106Q32—48no—
107M32—41no—
108M32—41no—
109C2×C16—22no—
110C2×C16—41no—
111C2×C16—41no—
112C2×C16—41no—
113#108—41no—
114#108—41no—
115#108—81no—
116#107—21no—
117#107C22⋊C412yes—
118C4⋊C8—22no—
119C4⋊C8—21no—
120C4⋊C8—44no—
121C4⋊C8—41no—
122C4⋊C8—41no—
123C4⋊C8D812yes—
124#103—22no—
125#103—22no—
126C22⋊C8—21no—
127C22⋊C8—21no—
128C22⋊C8—41no—
129C4×C8—21no—
130C4×C8—21no—
131C4×C8—44no—
132C4×C8—41no—
133C4×C8C22×C411yescharacteristic
134C4×C8C2×D411yescharacteristic
135C4×C8C2×D411yescharacteristic
136C4×C8#4111yescharacteristic
137C2×Q8—22no—
138C2×Q8—22no—
139C2×Q8—22no—
140C2×Q8—41no—
141C2×Q8—41no—
142C2×Q8—41no—
143C2×Q8—42no—
144C2×Q8—81no—
145C2×Q8—81no—
146C2×Q8—81no—
147C2×Q8—81no—
148C2×Q8—81no—
149C22×C4—41no—
150C22×C4—41no—
151C22×C4—81no—
152C22×C4#12111yescharacteristic
153C22×C4#13511yescharacteristic
154C22×C4AGL(1,ℤ/8)11yescharacteristic
155Q16—22no—
156Q16—22no—
157Q16—22no—
158Q16—42no—
159Q16—42no—
160Q16—82no—
161Q16—82no—
162Q16—81no—
163Q16—81no—
164Q16#10214yes—
165M16—81no—
166M16—81no—
167C2×C8—21no—
168C2×C8—21no—
169C2×C8—22no—
170C2×C8—42no—
171C2×C8—41no—
172C2×C8—81no—
173C2×C8C2×C22⋊C411yescharacteristic
174C2×C8C4×D411yescharacteristic
175C2×C8#12111yescharacteristic
176C2×C8#12311yescharacteristic
177C2×C8C2×D811yescharacteristic
178C2×C8C2×SD1611yescharacteristic
179C2×C8#13511yescharacteristic
180C2×C8#13511yescharacteristic
181C4⋊C4—22no—
182C4⋊C4—42no—
183C4⋊C4—42no—
184C4⋊C4—41no—
185C4⋊C4—42no—
186C4⋊C4—42no—
187C4⋊C4—81no—
188C4⋊C4—81no—
189C42—42no—
190C42—42no—
191C42C4×D411yescharacteristic
192C42#12011yescharacteristic
193C42C2×D811yescharacteristic
194C42AGL(1,ℤ/8)11yescharacteristic
195C16—42no—
196C16—81no—
197C23#46311yescharacteristic
198Q8—24no—
199Q8—42no—
200Q8—42no—
201Q8—42no—
202Q8—82no—
203Q8—82no—
204Q8—161no—
205Q8—161no—
206C2×C4—22no—
207C2×C4—41no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—82no—
213C2×C4—161no—
214C2×C4—161no—
215C2×C4—161no—
216C2×C4#38611yescharacteristic
217C2×C4C4×D811yescharacteristic
218C2×C4#44211yescharacteristic
219C2×C4#44711yescharacteristic
220C2×C4#44911yescharacteristic
221C2×C4#45911yescharacteristic
222C2×C4#46511yescharacteristic
223C2×C4#46611yescharacteristic
224C2×C4#47111yescharacteristic
225C2×C4#47211yescharacteristic
226C8—21no—
227C8—21no—
228C8—42no—
229C8—81no—
230C8—82no—
231C8—161no—
232C8C2×#10211yescharacteristic
233C8#41611yescharacteristic
234C8#46511yescharacteristic
235C8#46611yescharacteristic
236C22—21no—
237C22—21no—
238C22#167711yescharacteristic
239C22#168111yescharacteristic
240C22#187911yescharacteristic
241C4—21no—
242C4—21no—
243C4—21no—
244C4—44no—
245C4—82no—
246C4—162no—
247C4#166511yescharacteristic
248C4#166711yescharacteristic
249C4#189311yescharacteristic
250C4#190711yescharacteristic
251C4#219512yes—
252C2—21no—
253C2—21no—
254C2#922111yescharacteristic
255C2#1302011yescharacteristic
256C2#1303111yescharacteristic
subgroupG/H
center, socleC22#1879
commutatorC4×C8C22×C4
Frattini, cosocleC2×C4×C8C23
prime radicalC23#463
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1450 ◁ G
LECSG ▷ C2×C4×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×#22092
#178562
#128192
#119322
#97722
#72772
#71532