Finite Group Databasebeta

← #507659

Subgroups — #507659

Order 512.

399 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 399 conjugacy classes of subgroups — too many to draw. Only its 146 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#33261C211yesmaximal, characteristic
2#21735C211yesmaximal, characteristic
3#13510C212yesmaximal
4#13454C212yesmaximal
5#13187C212yesmaximal
6#13181C212yesmaximal
7#13174C214yesmaximal
8#12660C211yesmaximal, characteristic
9#3162C2211yescharacteristic
10#2960—21no—
11#2647—21no—
12#2159C2214yes—
13C2×#365C2214yes—
14C2×#364C2212yes—
15C2×#347C2212yes—
16#1995—21no—
17#1946—21no—
18#1946C2212yes—
19#1944C2212yes—
20#1943C2212yes—
21#1942—21no—
22#1942—21no—
23#1849C2211yescharacteristic
24#1846C2211yescharacteristic
25#1845C2211yescharacteristic
26#1836—21no—
27#1772—21no—
28#1772C2212yes—
29#1726C2211yescharacteristic
30#1387C2214yes—
31#1386C2214yes—
32#1377C2212yes—
33#533—21no—
34C2×M32C2314yes—
35#501—22no—
36#501—22no—
37#446—41no—
38#446—41no—
39C8×Q8—41no—
40C8×Q8—41no—
41C2×#108—21no—
42C2×#108C2312yes—
43#427—21no—
44#427C2312yes—
45C2×#107—21no—
46C2×#107—21no—
47C2×#107C2312yes—
48C2×#106C2312yes—
49#424—22no—
50#423—21no—
51#405—22no—
52C2×#97D411yescharacteristic
53C2×#97C2311yescharacteristic
54C2×#97C2311yescharacteristic
55C2×C4×C8D411yescharacteristic
56C2×C4×C8D411yescharacteristic
57C2×C4×C8C2311yescharacteristic
58#400—42no—
59#398—42no—
60#389—22no—
61#389—22no—
62#384—21no—
63#382—21no—
64#376D411yescharacteristic
65#367—24no—
66#366—24no—
67#365—44no—
68#364Q818yescomplemented
69#347—42no—
70#340—22no—
71#340—22no—
72#339—22no—
73#339—22no—
74C2×M16—22no—
75C22×C8C2×D411yescharacteristic
76C22×C8C2×D411yescharacteristic
77C22×C8C2×D411yescharacteristic
78C22×C8C2411yescharacteristic
79C4⋊Q8—21no—
80C4⋊Q8—21no—
81#125—22no—
82C22⋊Q8—41no—
83C22⋊Q8—41no—
84C4×Q8—41no—
85C4×Q8—41no—
86#117—22no—
87C2×C4⋊C4—22no—
88C2×C4⋊C4—22no—
89C2×C4⋊C4—42no—
90C2×C4⋊C4—42no—
91C2×C4⋊C4—42no—
92C2×C42C2×D411yescharacteristic
93C2×C42C2×D411yescharacteristic
94C2×C42C2×D411yescharacteristic
95M32—24no—
96M32C2×Q818yes—
97C2×C16—24no—
98#108—22no—
99#108—22no—
100#108—42no—
101#107—21no—
102#107—21no—
103#107—24no—
104#107—41no—
105#107—41no—
106#107—42no—
107#107#4112yes—
108#107#4112yes—
109#106—22no—
110#106—22no—
111#106—22no—
112#106—22no—
113C4⋊C8—41no—
114C4⋊C8—41no—
115C4⋊C8—42no—
116C4⋊C8—42no—
117#97—22no—
118#97—22no—
119#97—21no—
120#97—21no—
121#97—42no—
122C4×C8—21no—
123C4×C8—21no—
124C4×C8—22no—
125C4×C8—42no—
126C4×C8C2×Q814yes—
127#95—21no—
128#95—21no—
129#95—42no—
130#95—42no—
131#95—42no—
132C2×Q8—41no—
133C2×Q8—41no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—42no—
137C22×C4—42no—
138C22×C4AGL(1,ℤ/8)11yescharacteristic
139C22×C4AGL(1,ℤ/8)11yescharacteristic
140C22×C4C22×D411yescharacteristic
141M16—42no—
142M16—42no—
143C2×C8—21no—
144C2×C8—21no—
145C2×C8—21no—
146C2×C8—21no—
147C2×C8—21no—
148C2×C8—42no—
149C2×C8C4⋊Q812yes—
150C2×C8AGL(1,ℤ/8)11yescharacteristic
151C2×C8AGL(1,ℤ/8)11yescharacteristic
152C2×C8#13712yes—
153C2×C8#13712yes—
154C2×C8C22×Q812yes—
155C2×C8C2×#4112yes—
156C2×C8H(5,𝔽2)11yescharacteristic
157C2×C82−1+411yescharacteristic
158C4⋊C4—22no—
159C4⋊C4—22no—
160C4⋊C4—42no—
161C4⋊C4—42no—
162C4⋊C4—41no—
163C4⋊C4—41no—
164C4⋊C4—42no—
165C4⋊C4—42no—
166C4⋊C4—42no—
167C4⋊C4—42no—
168C4⋊C4—162no—
169C4⋊C4—162no—
170C4⋊C4—162no—
171C22⋊C4—42no—
172C42—21no—
173C42—21no—
174C42—21no—
175C42—21no—
176C42—21no—
177C42—42no—
178C42C4⋊Q812yes—
179C16—28no—
180C23C2×AGL(1,ℤ/8)11yescharacteristic
181Q8—82no—
182Q8—82no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—22no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—41no—
195C2×C4—82no—
196C2×C4—82no—
197C2×C4—82no—
198C2×C4—82no—
199C2×C4—82no—
200C2×C4—82no—
201C2×C4#52511yescharacteristic
202C2×C4C2×C4⋊Q811yescharacteristic
203C2×C4#53711yescharacteristic
204C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
205C2×C4C2×#13711yescharacteristic
206C2×C4C2×#13711yescharacteristic
207C8—22no—
208C8—41no—
209C8—82no—
210C8#50112yes—
211C8#50112yes—
212C8#55412yes—
213C8#55712yes—
214C22—21no—
215C22—21no—
216C22#310911yescharacteristic
217C22#316511yescharacteristic
218C22#325411yescharacteristic
219C4—42no—
220C4—41no—
221C4—41no—
222C4—81no—
223C4—82no—
224C4—82no—
225C4—162no—
226C4—162no—
227C4#258411yescharacteristic
228C4#311311yescharacteristic
229C4C2×#50111yescharacteristic
230C4#325411yescharacteristic
231C2—21no—
232C2—21no—
233C2#2931911yescharacteristic
234C2#3369211yescharacteristic
235C2#3369311yescharacteristic
subgroupG/H
center, socleC22#3109
commutator, Frattini, cosocleC22×C8C24
prime radicalC23C2×AGL(1,ℤ/8)
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1726 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
68. #364Q8semidirect

Maximal subgroups

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

subgroupindex
#332612
#217352
#135102
#134542
#131872
#131812
#131742
#126602