Finite Group Databasebeta

← #126319

Subgroups — #126319

Order 512.

278 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 64 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#32653C211yesmaximal, characteristic
2#13530C211yesmaximal, characteristic
3#13506C211yesmaximal, characteristic
4#10618C211yesmaximal, characteristic
5#7128C211yesmaximal, characteristic
6#7120C211yesmaximal, characteristic
7#7113C211yesmaximal, characteristic
8#2581—21no—
9C2×#367—21no—
10C2×#366—21no—
11C2×#358—21no—
12#2097—21no—
13#2085—21no—
14#2066—21no—
15#1984C2211yescharacteristic
16#1969C2211yescharacteristic
17#1942C2211yescharacteristic
18#1878—21no—
19#1863—21no—
20#1845C2211yescharacteristic
21#1803—21no—
22#1380—21no—
23#1374C2211yescharacteristic
24#1374C2211yescharacteristic
25#1296—21no—
26#1277C2211yescharacteristic
27C22×Q16D411yescharacteristic
28C2×C4×Q8—21no—
29C2×C4×Q8D411yescharacteristic
30C22×C16—21no—
31C22×C16—21no—
32C2×#107—21no—
33C2×#107—21no—
34C2×#107D411yescharacteristic
35C2×#107D411yescharacteristic
36C2×#107D411yescharacteristic
37C2×#106—21no—
38C2×C4⋊C8—21no—
39C2×C4⋊C8D411yescharacteristic
40C2×#103—21no—
41C2×#103—21no—
42C2×#103—21no—
43C2×#103—41no—
44C2×C4×C8C2311yescharacteristic
45#400—41no—
46#395—41no—
47#391—21no—
48#391—21no—
49#389—21no—
50#389—21no—
51#387—41no—
52#387—41no—
53#384—21no—
54#382—21no—
55#367—42no—
56#366—42no—
57#358—42no—
58#340—21no—
59#340—21no—
60#340—41no—
61#336—21no—
62#336—21no—
63C22×Q8—21no—
64C22×Q8D811yescharacteristic
65C2×Q16—21no—
66C2×Q16—41no—
67C2×Q16—41no—
68C2×Q16Q1612yes—
69C22×C8—21no—
70C22×C8—41no—
71C22×C8C2×D411yescharacteristic
72C22×C8C2×D411yescharacteristic
73C4×Q8—41no—
74C4×Q8—41no—
75C4×Q8—81no—
76C4×Q8—81no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—41no—
82C2×C4⋊C4—41no—
83C2×C4⋊C4—41no—
84C2×C4⋊C4—41no—
85C2×C4⋊C4—41no—
86C2×C4⋊C4—41no—
87C2×C4⋊C4D811yescharacteristic
88C2×C42—21no—
89C2×C42—41no—
90C2×C42C2×D411yescharacteristic
91C2×C16—22no—
92C2×C16—22no—
93C2×C16—41no—
94C2×C16—41no—
95C2×C16—41no—
96C2×C16—41no—
97#107—21no—
98#107—21no—
99#107—41no—
100#107—41no—
101#107—41no—
102#107—41no—
103#107—41no—
104#107—41no—
105#106—81no—
106C4⋊C8—41no—
107C4⋊C8—41no—
108C4⋊C8—41no—
109#103—41no—
110#103—41no—
111#103—81no—
112#103—81no—
113#103—82no—
114C4×C8—41no—
115#95—41no—
116#95—41no—
117#95—41no—
118#95—41no—
119#95—41no—
120#95—41no—
121#95—41no—
122#95—41no—
123C2×Q8—22no—
124C2×Q8—21no—
125C2×Q8—21no—
126C2×Q8—41no—
127C2×Q8—41no—
128C2×Q8—41no—
129C2×Q8—41no—
130C2×Q8—41no—
131C2×Q8—41no—
132C2×Q8—41no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—41no—
136C22×C4—41no—
137C22×C4—41no—
138C22×C4—41no—
139C22×C4—81no—
140C22×C4#12011yescharacteristic
141C22×C4C2×D811yescharacteristic
142C22×C4AGL(1,ℤ/8)11yescharacteristic
143Q16—42no—
144Q16—81no—
145C2×C8—21no—
146C2×C8—21no—
147C2×C8—21no—
148C2×C8—21no—
149C2×C8—41no—
150C2×C8—41no—
151C2×C8—41no—
152C2×C8—81no—
153C2×C8—81no—
154C2×C8—81no—
155C2×C8C2×Q1611yescharacteristic
156C2×C8#13511yescharacteristic
157C2×C8AGL(1,ℤ/8)11yescharacteristic
158C2×C8#13711yescharacteristic
159C4⋊C4—41no—
160C4⋊C4—41no—
161C4⋊C4—41no—
162C4⋊C4—41no—
163C4⋊C4—41no—
164C4⋊C4—41no—
165C4⋊C4—41no—
166C4⋊C4—81no—
167C4⋊C4—81no—
168C4⋊C4—81no—
169C4⋊C4—82no—
170C4⋊C4—82no—
171C4⋊C4—81no—
172C4⋊C4—81no—
173C4⋊C4—81no—
174C4⋊C4—81no—
175C4⋊C4—161no—
176C42—21no—
177C42—21no—
178C42—41no—
179C42—41no—
180C42—82no—
181C16—42no—
182C16—42no—
183C23#44711yescharacteristic
184Q8—42no—
185Q8—42no—
186Q8—42no—
187Q8—81no—
188Q8—81no—
189Q8—81no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—42no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—81no—
203C2×C4—81no—
204C2×C4—81no—
205C2×C4—81no—
206C2×C4—81no—
207C2×C4—81no—
208C2×C4—81no—
209C2×C4—81no—
210C2×C4—81no—
211C2×C4—81no—
212C2×C4—81no—
213C2×C4—161no—
214C2×C4—161no—
215C2×C4—161no—
216C2×C4#44911yescharacteristic
217C2×C4#45111yescharacteristic
218C2×C4#45311yescharacteristic
219C2×C4#45511yescharacteristic
220C2×C4#45611yescharacteristic
221C2×C4#45811yescharacteristic
222C2×C4#50811yescharacteristic
223C2×C4#51011yescharacteristic
224C8—21no—
225C8—21no—
226C8—21no—
227C8—21no—
228C8—81no—
229C8—161no—
230C22#160711yescharacteristic
231C22#162111yescharacteristic
232C22#164911yescharacteristic
233C22#218911yescharacteristic
234C22#219011yescharacteristic
235C22#219311yescharacteristic
236C22#219511yescharacteristic
237C4—21no—
238C4—21no—
239C4—41no—
240C4—82no—
241C4—81no—
242C4—81no—
243C4—161no—
244C4—161no—
245C4—161no—
246C4—321no—
247C4#160911yescharacteristic
248C4#161411yescharacteristic
249C4#164411yescharacteristic
250C4#166111yescharacteristic
251C2#778911yescharacteristic
252C2#1228311yescharacteristic
253C2#1228611yescharacteristic
254C2#1233511yescharacteristic
255C2#1233711yescharacteristic
256C2#1240911yescharacteristic
257C2#1241511yescharacteristic
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
#326532
#135302
#135062
#106182
#71282
#71202
#71132