Finite Group Databasebeta

← #141241

Subgroups — #141241

Order 512.

338 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 114 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×#1710C211yesmaximal, characteristic
2#13276C211yesmaximal, characteristic
3#13275C211yesmaximal, characteristic
4C2×#1370C211yesmaximal, characteristic
5C2×#1365C211yesmaximal, characteristic
6C2×#1364C211yesmaximal, characteristic
7C8×#107C211yesmaximal, characteristic
8C2×#500C2211yescharacteristic
9C2×#500C2211yescharacteristic
10C2×#487—21no—
11C2×#487—21no—
12C2×C4⋊C16C2211yescharacteristic
13C2×C4⋊C16C2211yescharacteristic
14C4×#107C2211yescharacteristic
15C8×C4⋊C4—21no—
16#1710—22no—
17#1710C414yescomplemented
18C2×#335C2211yescharacteristic
19C2×C82C2211yescharacteristic
20#1380—22no—
21#1380—22no—
22#1370—22no—
23#1365—22no—
24#1364—22no—
25#1279—21no—
26C2×C4⋊Q8—21no—
27C2×C4⋊Q8—21no—
28C22×C16—21no—
29C22×C16—21no—
30#500—21no—
31#500—21no—
32#500—21no—
33#500—21no—
34#500—21no—
35#500—21no—
36#500C2×C412yes—
37#500C2×C412yes—
38#487—24no—
39#487—24no—
40#487—42no—
41#487—42no—
42C2×#107—21no—
43C2×#107—21no—
44C2×#107—21no—
45C2×#107—21no—
46C2×#107D412yes—
47C2×C4⋊C8—21no—
48C2×#103—42no—
49C2×#103—41no—
50C2×#103—41no—
51C2×C4×C8—41no—
52C2×C4×C8D411yescharacteristic
53C2×C4×C8D411yescharacteristic
54C2×C4×C8C2311yescharacteristic
55C4×C4⋊C4—21no—
56C4⋊C16—21no—
57C4⋊C16—21no—
58C4⋊C16—21no—
59C4⋊C16—21no—
60#340—21no—
61#336—41no—
62#335—21no—
63#335D412yes—
64C82C2×C412yes—
65C82D412yes—
66C22×Q8—41no—
67C22×Q8—41no—
68C22×C8—21no—
69C22×C8—41no—
70C22×C8D811yescharacteristic
71C22×C8D811yescharacteristic
72C22×C8C2×D411yescharacteristic
73C22×C8#4111yescharacteristic
74C4⋊Q8—22no—
75C4⋊Q8—22no—
76C4⋊Q8—41no—
77C4⋊Q8—41no—
78C4⋊Q8—41no—
79C4⋊Q8—41no—
80C4⋊Q8—41no—
81C4⋊Q8—41no—
82C2×C4⋊C4—22no—
83C2×C4⋊C4—41no—
84C2×C4⋊C4—41no—
85C2×C4⋊C4—41no—
86C2×C4⋊C4—41no—
87C2×C42—41no—
88C2×C42C2×D411yescharacteristic
89C2×C16—41no—
90C2×C16—41no—
91C2×C16—42no—
92C2×C16—42no—
93#107—22no—
94#107—22no—
95#107—22no—
96#107—22no—
97#107—22no—
98#107—41no—
99#107—41no—
100#107—41no—
101#107—41no—
102#107Q1614yes—
103C4⋊C8—22no—
104C4⋊C8—41no—
105#103—48no—
106#103—44no—
107#103—44no—
108C4×C8—21no—
109C4×C8—82no—
110C4×C8C22⋊C412yes—
111C4×C8C22⋊C412yes—
112C4×C8SD1612yes—
113C4×C8C22×C411yescharacteristic
114C4×C8C2×D411yescharacteristic
115C4×C8C2×D411yescharacteristic
116C4×C8#4111yescharacteristic
117#95—41no—
118C2×Q8—42no—
119C2×Q8—42no—
120C2×Q8—84no—
121C2×Q8—81no—
122C2×Q8—81no—
123C2×Q8—81no—
124C2×Q8—81no—
125C2×Q8—81no—
126C2×Q8—81no—
127C22×C4—42no—
128C22×C4—81no—
129C22×C4—81no—
130C22×C4#12111yescharacteristic
131C22×C4C2×D811yescharacteristic
132C22×C4AGL(1,ℤ/8)11yescharacteristic
133C2×C8—22no—
134C2×C8—21no—
135C2×C8—21no—
136C2×C8—21no—
137C2×C8—21no—
138C2×C8—22no—
139C2×C8—42no—
140C2×C8—42no—
141C2×C8—81no—
142C2×C8—81no—
143C2×C8#10212yes—
144C2×C8#10212yes—
145C2×C8C4×D411yescharacteristic
146C2×C8C4×D411yescharacteristic
147C2×C8#12411yescharacteristic
148C2×C8#12711yescharacteristic
149C2×C8C2×Q1612yes—
150C2×C8#13512yes—
151C4⋊C4—24no—
152C4⋊C4—42no—
153C4⋊C4—42no—
154C4⋊C4—42no—
155C4⋊C4—42no—
156C4⋊C4—42no—
157C4⋊C4—81no—
158C4⋊C4—81no—
159C4⋊C4—81no—
160C4⋊C4—81no—
161C42—81no—
162C42—81no—
163C42C2×C22⋊C411yescharacteristic
164C42#12111yescharacteristic
165C42C2×SD1611yescharacteristic
166C42AGL(1,ℤ/8)11yescharacteristic
167C16—81no—
168C16—81no—
169C23#45911yescharacteristic
170Q8—84no—
171Q8—82no—
172Q8—82no—
173Q8—162no—
174Q8—161no—
175Q8—161no—
176C2×C4—22no—
177C2×C4—44no—
178C2×C4—82no—
179C2×C4—82no—
180C2×C4—82no—
181C2×C4—82no—
182C2×C4—161no—
183C2×C4—161no—
184C2×C4—162no—
185C2×C4#39011yescharacteristic
186C2×C4C2×#10211yescharacteristic
187C2×C4#41811yescharacteristic
188C2×C4#45911yescharacteristic
189C2×C4#46011yescharacteristic
190C2×C4#46011yescharacteristic
191C2×C4#46712yes—
192C2×C4#47312yes—
193C2×C4#50811yescharacteristic
194C2×C4#50811yescharacteristic
195C2×C4#50911yescharacteristic
196C2×C4#51011yescharacteristic
197C8—22no—
198C8—22no—
199C8—21no—
200C8—21no—
201C8—42no—
202C8—82no—
203C8C4×Q1612yes—
204C8#49411yescharacteristic
205C8#49511yescharacteristic
206C22#166912yes—
207C22#198211yescharacteristic
208C22#221011yescharacteristic
209C22#221111yescharacteristic
210C22#221212yes—
211C4—21no—
212C4—21no—
213C4—84no—
214C4—162no—
215C4—162no—
216C4#167012yes—
217C4#196911yescharacteristic
218C4#197911yescharacteristic
219C4#213811yescharacteristic
220C4#213911yescharacteristic
221C4#221011yescharacteristic
222C4#221111yescharacteristic
223C2#973311yescharacteristic
224C2#1244912yes—
225C2#1245012yes—
226C2#1325411yescharacteristic
227C2#1326011yescharacteristic
subgroupG/H
center, socle, prime radicalC23#459
commutatorC4×C8C22×C4
Frattini, cosocleC2×C4×C8C23
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C82 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
17. #1710C4semidirect

Maximal subgroups

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

subgroupindex
C2×#17102
#132762
#132752
C2×#13702
C2×#13652
C2×#13642
C8×#1072