Finite Group Databasebeta

← #126372

Subgroups — #126372

Order 512.

268 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 68 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#32684C211yesmaximal, characteristic
2#13537C211yesmaximal, characteristic
3#13506C211yesmaximal, characteristic
4#10617C211yesmaximal, characteristic
5#7128C211yesmaximal, characteristic
6#7121C211yesmaximal, characteristic
7#7121C211yesmaximal, characteristic
8#2567—21no—
9C2×#367—21no—
10#2087—21no—
11#2057—21no—
12#2057—21no—
13#1984C2211yescharacteristic
14#1942C2211yescharacteristic
15#1942C2211yescharacteristic
16#1921—21no—
17#1921—21no—
18#1845C2211yescharacteristic
19#1803—21no—
20#1380—21no—
21#1380—21no—
22#1380—21no—
23#1374C2211yescharacteristic
24#1374C2211yescharacteristic
25#1296—21no—
26#1277C2211yescharacteristic
27C2×C4⋊Q8—21no—
28C2×C4×Q8D411yescharacteristic
29C22×C16—21no—
30C22×C16—21no—
31C2×#107—21no—
32C2×#107—21no—
33C2×#107—21no—
34C2×#107D411yescharacteristic
35C2×#107D411yescharacteristic
36C2×#107D411yescharacteristic
37C2×#107D411yescharacteristic
38C2×C4⋊C8—21no—
39C2×C4⋊C8D411yescharacteristic
40C2×#103—41no—
41C2×C4×C8C2311yescharacteristic
42#395—41no—
43#395—41no—
44#391—41no—
45#389—21no—
46#389—21no—
47#389—21no—
48#389—21no—
49#384—21no—
50#384—41no—
51C4×C4⋊C4—21no—
52#367—42no—
53#340—21no—
54#340—21no—
55#340—21no—
56#340—21no—
57#340—21no—
58#340—41no—
59#336—21no—
60#336—21no—
61C22×Q8—41no—
62C22×Q8D811yescharacteristic
63C22×C8—21no—
64C22×C8—41no—
65C22×C8C2×D411yescharacteristic
66C22×C8C2×D411yescharacteristic
67C4⋊Q8—81no—
68C4⋊Q8—81no—
69C4×Q8—41no—
70C4×Q8—41no—
71C2×C4⋊C4—21no—
72C2×C4⋊C4—21no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—41no—
77C2×C4⋊C4—41no—
78C2×C4⋊C4—41no—
79C2×C4⋊C4—41no—
80C2×C4⋊C4—41no—
81C2×C4⋊C4—41no—
82C2×C4⋊C4—41no—
83C2×C4⋊C4—41no—
84C2×C4⋊C4D811yescharacteristic
85C2×C42—21no—
86C2×C42—41no—
87C2×C42C2×D411yescharacteristic
88C2×C16—22no—
89C2×C16—41no—
90C2×C16—41no—
91C2×C16—41no—
92C2×C16—41no—
93C2×C16—41no—
94#107—21no—
95#107—21no—
96#107—41no—
97#107—41no—
98#107—42no—
99#107—41no—
100#107—41no—
101#107—81no—
102#107SD1612yes—
103#107Q1612yes—
104C4⋊C8—41no—
105C4⋊C8—41no—
106C4⋊C8—41no—
107#103—82no—
108C4×C8—41no—
109#95—41no—
110#95—41no—
111#95—41no—
112#95—41no—
113#95—41no—
114C2×Q8—21no—
115C2×Q8—41no—
116C2×Q8—41no—
117C2×Q8—82no—
118C2×Q8—161no—
119C2×Q8—161no—
120C2×Q8SD3212yes—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—41no—
124C22×C4—41no—
125C22×C4—41no—
126C22×C4—41no—
127C22×C4—81no—
128C22×C4#12011yescharacteristic
129C22×C4C2×D811yescharacteristic
130C22×C4AGL(1,ℤ/8)11yescharacteristic
131C2×C8—21no—
132C2×C8—21no—
133C2×C8—21no—
134C2×C8—21no—
135C2×C8—41no—
136C2×C8—41no—
137C2×C8—41no—
138C2×C8—81no—
139C2×C8—81no—
140C2×C8—81no—
141C2×C8C2×SD1611yescharacteristic
142C2×C8C2×Q1611yescharacteristic
143C2×C8#13711yescharacteristic
144C2×C8#13711yescharacteristic
145C4⋊C4—22no—
146C4⋊C4—22no—
147C4⋊C4—41no—
148C4⋊C4—41no—
149C4⋊C4—41no—
150C4⋊C4—41no—
151C4⋊C4—41no—
152C4⋊C4—41no—
153C4⋊C4—41no—
154C4⋊C4—81no—
155C4⋊C4—82no—
156C4⋊C4—81no—
157C4⋊C4—81no—
158C4⋊C4—81no—
159C4⋊C4—81no—
160C4⋊C4—161no—
161C4⋊C4—161no—
162C4⋊C4—161no—
163C4⋊C4—161no—
164C4⋊C4—161no—
165C42—21no—
166C42—21no—
167C42—41no—
168C42—41no—
169C42—82no—
170C16—42no—
171C16—81no—
172C23#44711yescharacteristic
173Q8—42no—
174Q8—81no—
175Q8—164no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—22no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—81no—
190C2×C4—81no—
191C2×C4—81no—
192C2×C4—81no—
193C2×C4—81no—
194C2×C4—81no—
195C2×C4—81no—
196C2×C4—81no—
197C2×C4—81no—
198C2×C4—81no—
199C2×C4—161no—
200C2×C4—161no—
201C2×C4—161no—
202C2×C4#44811yescharacteristic
203C2×C4#45111yescharacteristic
204C2×C4#45411yescharacteristic
205C2×C4#45511yescharacteristic
206C2×C4#45611yescharacteristic
207C2×C4#45811yescharacteristic
208C2×C4C2×SD3211yescharacteristic
209C2×C4#51011yescharacteristic
210C8—21no—
211C8—21no—
212C8—21no—
213C8—21no—
214C8—81no—
215C8—161no—
216C22#160711yescharacteristic
217C22#162711yescharacteristic
218C22#164911yescharacteristic
219C22#218511yescharacteristic
220C22#219111yescharacteristic
221C22#219211yescharacteristic
222C22#219411yescharacteristic
223C4—21no—
224C4—21no—
225C4—42no—
226C4—41no—
227C4—82no—
228C4—82no—
229C4—81no—
230C4—161no—
231C4—161no—
232C4—321no—
233C4#161111yescharacteristic
234C4#161711yescharacteristic
235C4#164511yescharacteristic
236C4#166411yescharacteristic
237C2#779411yescharacteristic
238C2#1227911yescharacteristic
239C2#1228411yescharacteristic
240C2#1235411yescharacteristic
241C2#1236211yescharacteristic
242C2#1240611yescharacteristic
243C2#1241211yescharacteristic
subgroupG/H
center, socle, prime radicalC23#447
commutator, Frattini, cosocleC2×C4×C8C23
LCSG ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1984 ◁ G
LECSG ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#326842
#135372
#135062
#106172
#71282
#71212