Finite Group Databasebeta

← #506286

Subgroups — #506286

Order 512.

647 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 647 conjugacy classes of subgroups — too many to draw. Only its 152 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×#2209C212yesmaximal
2#21698C211yesmaximal, characteristic
3#21698C211yesmaximal, characteristic
4#13483C212yesmaximal
5#13330C212yesmaximal
6#13015C214yesmaximal
7#12880C211yesmaximal, characteristic
8#12814C212yesmaximal
9C22×Q32C2214yes—
10C2×#494C2211yescharacteristic
11C2×#494C2211yescharacteristic
12C2×#441—21no—
13C2×#441—21no—
14C2×C4×Q16C2212yes—
15C2×C4×Q16C2212yes—
16#2327—21no—
17#2327—21no—
18#2209—28no—
19C2×C4⋊C16C2212yes—
20C2×#358C2214yes—
21C2×#358C2214yes—
22#1971—21no—
23#1971—21no—
24#1969C2212yes—
25#1969C2212yes—
26#1960—21no—
27#1960—21no—
28#1846C2211yescharacteristic
29#1846C2211yescharacteristic
30#1726C2211yescharacteristic
31#1380C2214yes—
32#1374C2212yes—
33#1374C2212yes—
34C22×Q16—21no—
35C22×Q16—21no—
36C22×Q16C2312yes—
37C22×Q16C2312yes—
38C2×C4⋊Q8—21no—
39C2×C4⋊Q8—21no—
40C2×C4×Q8—22no—
41C2×C4×Q8—22no—
42C2×Q32—28no—
43C2×Q32—28no—
44C2×Q32—44no—
45C22×C16C2314yes—
46#494—24no—
47#494—22no—
48#494D414yes—
49#441—44no—
50#441—44no—
51C4×Q16—24no—
52C4×Q16—24no—
53C4×Q16—44no—
54C2×#107C2312yes—
55C2×#107C2312yes—
56C2×#106—21no—
57C2×#106—21no—
58C2×#103—22no—
59C2×#103—22no—
60C2×#103—22no—
61C2×#103—22no—
62C2×#97D411yescharacteristic
63C2×#97D411yescharacteristic
64C2×#97D411yescharacteristic
65C2×C4×C8C2311yescharacteristic
66C2×C4×C8C2311yescharacteristic
67C2×C4×C8C2311yescharacteristic
68#391—22no—
69#391—22no—
70#387—44no—
71#387—44no—
72#382—21no—
73#382—21no—
74#376D411yescharacteristic
75C4⋊C16—24no—
76#358—28no—
77#358—44no—
78C22×Q8—22no—
79C22×Q8—22no—
80C2×Q16—24no—
81C2×Q16—24no—
82C2×Q16—22no—
83C2×Q16—22no—
84C2×Q16—24no—
85C2×Q16—22no—
86C2×Q16—22no—
87C2×Q16—22no—
88C2×Q16—24no—
89C2×Q16—22no—
90C2×Q16—42no—
91C2×Q16—42no—
92C2×Q16—42no—
93C2×Q16—42no—
94C2×Q16C2×D414yes—
95C2×Q16#4114yes—
96C22×C8C2×D411yescharacteristic
97C22×C8C2×D411yescharacteristic
98C22×C8C2×D411yescharacteristic
99C22×C8C2411yescharacteristic
100C4⋊Q8—24no—
101C4⋊Q8—24no—
102C4⋊Q8—42no—
103C4⋊Q8—42no—
104C4×Q8—44no—
105C4×Q8—44no—
106C4×Q8—44no—
107C4×Q8—44no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—42no—
111C2×C4⋊C4—42no—
112C2×C4⋊C4—42no—
113C2×C4⋊C4—42no—
114C2×C42—42no—
115C2×C42—42no—
116C2×C42C2×D411yescharacteristic
117C2×C42C2×D411yescharacteristic
118C2×C42C2×D411yescharacteristic
119Q32—416no—
120C2×C16—24no—
121C2×C16—24no—
122C2×C16—24no—
123#107—22no—
124#107—22no—
125#107—24no—
126#106—42no—
127#106—42no—
128#103—44no—
129#103—44no—
130#103—44no—
131#103—44no—
132#97—22no—
133#97—22no—
134#97—41no—
135C4×C8—22no—
136C4×C8—22no—
137C4×C8C2×D412yes—
138C4×C8#4112yes—
139#95—42no—
140#95—42no—
141#95—44no—
142#95—44no—
143#95D814yes—
144C2×Q8—24no—
145C2×Q8—24no—
146C2×Q8—42no—
147C2×Q8—42no—
148C2×Q8—42no—
149C2×Q8—42no—
150C2×Q8—42no—
151C2×Q8—42no—
152C2×Q8—42no—
153C2×Q8—42no—
154C2×Q8—42no—
155C2×Q8—42no—
156C22×C4—42no—
157C22×C4—42no—
158C22×C4—42no—
159C22×C4—42no—
160C22×C4C2×D812yes—
161C22×C4C2×D812yes—
162C22×C4C2×D812yes—
163C22×C4C22×D411yescharacteristic
164Q16—24no—
165Q16—28no—
166Q16—24no—
167Q16—48no—
168Q16—48no—
169Q16—42no—
170Q16—42no—
171Q16—44no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8#12112yes—
180C2×C8AGL(1,ℤ/8)12yes—
181C2×C8C22×D411yescharacteristic
182C2×C8C2×#4112yes—
183C2×C8C2×#4111yescharacteristic
184C2×C8H(5,𝔽2)12yes—
185C4⋊C4—48no—
186C4⋊C4—48no—
187C4⋊C4—42no—
188C4⋊C4—42no—
189C4⋊C4—42no—
190C4⋊C4—42no—
191C4⋊C4—84no—
192C4⋊C4—84no—
193C42—21no—
194C42—21no—
195C42—21no—
196C42—21no—
197C42—84no—
198C42—84no—
199C42#12112yes—
200C42AGL(1,ℤ/8)12yes—
201C16—44no—
202C23C22×D811yescharacteristic
203Q8—44no—
204Q8—44no—
205Q8—44no—
206Q8—44no—
207Q8—82no—
208Q8—82no—
209Q8—82no—
210Q8—82no—
211C2×C4—24no—
212C2×C4—22no—
213C2×C4—24no—
214C2×C4—24no—
215C2×C4—22no—
216C2×C4—44no—
217C2×C4—44no—
218C2×C4—82no—
219C2×C4—82no—
220C2×C4—82no—
221C2×C4—82no—
222C2×C4—84no—
223C2×C4—84no—
224C2×C4—82no—
225C2×C4—82no—
226C2×C4#45914yes—
227C2×C4C2×#12111yescharacteristic
228C2×C4#52511yescharacteristic
229C2×C4#53511yescharacteristic
230C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
231C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
232C2×C4#57511yescharacteristic
233C8—22no—
234C8—22no—
235C8—41no—
236C8#54612yes—
237C8#55312yes—
238C22C2×#45911yescharacteristic
239C22#310011yescharacteristic
240C22#320511yescharacteristic
241C22#341714yes—
242C4—24no—
243C4—42no—
244C4—42no—
245C4—84no—
246C4—84no—
247C4—162no—
248C4—162no—
249C4#253511yescharacteristic
250C4#303611yescharacteristic
251C4#310911yescharacteristic
252C4#321411yescharacteristic
253C2#2815611yescharacteristic
254C2#3358812yes—
255C2#3368712yes—
256C2#3378912yes—
subgroupG/H
center, socle, prime radicalC23C22×D8
commutator, Frattini, cosocleC22×C8C24
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #376 ◁ #1726 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
C2×#22092
#216982
#134832
#133302
#130152
#128802
#128142