Finite Group Databasebeta

← #137528

Subgroups — #137528

Order 512.

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

normal subgroups only

This group has 318 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#33731C211yesmaximal, characteristic
2#23170C211yesmaximal, characteristic
3#13279C211yesmaximal, characteristic
4#11942C211yesmaximal, characteristic
5#9730C211yesmaximal, characteristic
6#7176C211yesmaximal, characteristic
7#7169C211yesmaximal, characteristic
8#3160C2211yescharacteristic
9C2×#500C2211yescharacteristic
10#2962—21no—
11#2255C414yescomplemented
12#2254—21no—
13#2252—21no—
14#2159—22no—
15#2150C2211yescharacteristic
16C4×M32C2211yescharacteristic
17#1987—21no—
18#1986—21no—
19#1945—21no—
20#1922—22no—
21C4×#108C2211yescharacteristic
22#1772—21no—
23C8×C4⋊C4C2211yescharacteristic
24C2×#335C2211yescharacteristic
25C2×#333—21no—
26#1387—21no—
27#1387—21no—
28#1365—21no—
29#1364—21no—
30#533—21no—
31C2×C4⋊Q8—21no—
32C2×M32—21no—
33C2×M32—21no—
34#500—21no—
35#500—21no—
36#500—21no—
37#500C2×C412yes—
38#500D411yescharacteristic
39#500D411yescharacteristic
40#499—21no—
41#499C2×C412yes—
42#498—21no—
43#446—42no—
44#446—41no—
45C8×Q8—41no—
46C2×#108D411yescharacteristic
47C2×#108D411yescharacteristic
48#427—21no—
49#427D412yes—
50C2×#107—21no—
51C2×#107—21no—
52#424—22no—
53C2×C4⋊C8—21no—
54C2×C4⋊C8D411yescharacteristic
55#405—22no—
56C4×M16—21no—
57C2×C4×C8—21no—
58C2×C4×C8C2311yescharacteristic
59C4×C4⋊C4D411yescharacteristic
60#367—24no—
61#366—24no—
62C4⋊C16—21no—
63C4⋊C16—21no—
64#346C2×C412yes—
65C4×C16—21no—
66#341—21no—
67#340—42no—
68#339—22no—
69#339—22no—
70#336—21no—
71#335—21no—
72#335—21no—
73#333—41no—
74#333—41no—
75C22×Q8—41no—
76C2×M16—21no—
77C2×M16—21no—
78C22×C8—21no—
79C22×C8—41no—
80C22×C8C2×D411yescharacteristic
81C22×C8C2×D411yescharacteristic
82C4⋊Q8—22no—
83C4⋊Q8—21no—
84C4⋊Q8—21no—
85C4⋊Q8—41no—
86C4⋊Q8—41no—
87C4⋊Q8—41no—
88#125—22no—
89C22⋊Q8—41no—
90C22⋊Q8—41no—
91C4×Q8—41no—
92C4×Q8—41no—
93#117—22no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—41no—
96C2×C4⋊C4—41no—
97C2×C42—21no—
98C2×C42C2×D411yescharacteristic
99M32—22no—
100M32—22no—
101M32—41no—
102M32—41no—
103C2×C16—22no—
104C2×C16—41no—
105#108—21no—
106#108—21no—
107#108—21no—
108#108Q1612yes—
109#107—21no—
110#107—21no—
111#107—22no—
112#107—22no—
113#107—41no—
114#107—41no—
115#107C22⋊C412yes—
116#107C22⋊C412yes—
117#106—22no—
118#106—22no—
119C4⋊C8—21no—
120C4⋊C8—42no—
121C4⋊C8—42no—
122C4⋊C8—41no—
123C4⋊C8—41no—
124C4⋊C8D811yescharacteristic
125C4⋊C8D811yescharacteristic
126#97—42no—
127#97—41no—
128C4×C8—42no—
129C4×C8—41no—
130C4×C8—41no—
131C4×C8—41no—
132C4×C8C22×C411yescharacteristic
133C4×C8C2×D411yescharacteristic
134C4×C8#4111yescharacteristic
135C4×C8#4111yescharacteristic
136#95—21no—
137C2×Q8—41no—
138C2×Q8—41no—
139C2×Q8—42no—
140C2×Q8—81no—
141C2×Q8—162no—
142C22×C4—41no—
143C22×C4—81no—
144C22×C4#12011yescharacteristic
145C22×C4#13511yescharacteristic
146C22×C4AGL(1,ℤ/8)11yescharacteristic
147M16—22no—
148M16—41no—
149M16—41no—
150M16—41no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—21no—
154C2×C8—21no—
155C2×C8—41no—
156C2×C8—41no—
157C2×C8—41no—
158C2×C8—41no—
159C2×C8—82no—
160C2×C8—81no—
161C2×C8C2×C22⋊C411yescharacteristic
162C2×C8C4×D411yescharacteristic
163C2×C8#12111yescharacteristic
164C2×C8#12311yescharacteristic
165C2×C8C2×Q1611yescharacteristic
166C2×C8#13511yescharacteristic
167C2×C8#13711yescharacteristic
168C2×C8#13711yescharacteristic
169C4⋊C4—22no—
170C4⋊C4—22no—
171C4⋊C4—22no—
172C4⋊C4—41no—
173C4⋊C4—42no—
174C4⋊C4—42no—
175C4⋊C4—41no—
176C4⋊C4—41no—
177C4⋊C4—42no—
178C4⋊C4—42no—
179C4⋊C4—81no—
180C4⋊C4—81no—
181C22⋊C4—42no—
182C42—42no—
183C42—41no—
184C42—41no—
185C42C4×D411yescharacteristic
186C42#12111yescharacteristic
187C42C2×D811yescharacteristic
188C42AGL(1,ℤ/8)11yescharacteristic
189C16—24no—
190C16—81no—
191C23#44911yescharacteristic
192Q8—81no—
193Q8—81no—
194Q8—161no—
195Q8—162no—
196Q8—162no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—41no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—81no—
204C2×C4—81no—
205C2×C4—82no—
206C2×C4—162no—
207C2×C4#38611yescharacteristic
208C2×C4C4×D811yescharacteristic
209C2×C4#44211yescharacteristic
210C2×C4#44711yescharacteristic
211C2×C4#45111yescharacteristic
212C2×C4#45211yescharacteristic
213C2×C4#45611yescharacteristic
214C2×C4#45611yescharacteristic
215C2×C4#45911yescharacteristic
216C2×C4#46311yescharacteristic
217C8—21no—
218C8—21no—
219C8—41no—
220C8—41no—
221C8—42no—
222C8—81no—
223C8—161no—
224C8C4×Q1611yescharacteristic
225C8#44111yescharacteristic
226C8#46211yescharacteristic
227C8#46411yescharacteristic
228C22—21no—
229C22—21no—
230C22#164811yescharacteristic
231C22#165011yescharacteristic
232C22#187911yescharacteristic
233C4—21no—
234C4—21no—
235C4—21no—
236C4—82no—
237C4—82no—
238C4—82no—
239C4—162no—
240C4#164711yescharacteristic
241C4#164911yescharacteristic
242C4#187811yescharacteristic
243C4#188611yescharacteristic
244C4#221211yescharacteristic
245C4#221411yescharacteristic
246C2—21no—
247C2—21no—
248C2#905811yescharacteristic
249C2#1302811yescharacteristic
250C2#1303211yescharacteristic
subgroupG/H
center, socleC22#1879
commutatorC4×C8C22×C4
Frattini, cosocleC2×C4×C8C23
prime radicalC23#449
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C8×C4⋊C4 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
11. #2255C4semidirect

Maximal subgroups

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

subgroupindex
#337312
#231702
#132792
#119422
#97302
#71762
#71692