Finite Group Databasebeta

← #411144

Subgroups — #411144

Order 512.

768 conjugacy classes of subgroups; only the 210 normal subgroups are shown.

normal subgroups only

This group has 768 conjugacy classes of subgroups — too many to draw. Only its 210 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#23657C212yesmaximal
2C2×#1943C211yesmaximal, characteristic
3#20438C211yesmaximal, characteristic
4#11366C212yesmaximal
5#11331C212yesmaximal
6#11204C212yesmaximal
7#10896C214yesmaximal
8#10823C211yesmaximal, characteristic
9C2×#428C2214yes—
10C22×#107C2211yescharacteristic
11C22×#106C2211yescharacteristic
12C2×#423C2211yescharacteristic
13C2×#405C2212yes—
14C22×#97C2211yescharacteristic
15C2×#389C2212yes—
16#2285C2211yescharacteristic
17#2085—24no—
18#2084—22no—
19#2083—22no—
20#1943—24no—
21#1771—24no—
22C2×#343C2214yes—
23C2×#343C2214yes—
24C2×#341C2212yes—
25C2×#341C2212yes—
26#1736C2214yes—
27C2×#340C2212yes—
28#1729C2212yes—
29#1728C2212yes—
30C22×M16C2314yes—
31C22×M16C2312yes—
32C23×C8C2311yescharacteristic
33C23×C8C2311yescharacteristic
34C2×#117C2312yes—
35C22×C4⋊C4—22no—
36C22×C4⋊C4C2312yes—
37C22×C4⋊C4C2312yes—
38C22×C42C2311yescharacteristic
39#428—28no—
40#428—44no—
41C2×#107—22no—
42C2×#107—24no—
43C2×#107—24no—
44C2×#106—22no—
45C2×#106—24no—
46C2×#106—24no—
47#423—24no—
48C2×C4⋊C8—22no—
49#405—44no—
50C2×#97—24no—
51C2×#97—22no—
52C2×#97D414yes—
53C2×C4×C8—22no—
54#389—22no—
55#389—22no—
56#389—22no—
57#389—22no—
58#384D411yescharacteristic
59#384D411yescharacteristic
60#382D412yes—
61#380—22no—
62C4×C4⋊C4—22no—
63C2×#95—22no—
64#343—28no—
65#343—44no—
66#341—24no—
67#341Q818yes—
68#340—24no—
69#340—24no—
70#340—24no—
71#336—24no—
72#336—22no—
73#336—22no—
74C23×C4—22no—
75C23×C4—22no—
76C23×C4#4112yes—
77C23×C4C2411yescharacteristic
78C2×M16—22no—
79C2×M16—24no—
80C2×M16—24no—
81C2×M16—24no—
82C2×M16—24no—
83C2×M16—24no—
84C2×M16—24no—
85C2×M16—24no—
86C2×M16C2×Q818yes—
87C2×M16#4114yes—
88C22×C8—22no—
89C22×C8—22no—
90C22×C8—22no—
91C22×C8—22no—
92C22×C8—24no—
93C22×C8—22no—
94C22×C8C2×D412yes—
95C22×C8C2×D412yes—
96C22×C8C2×Q814yes—
97C22×C8#4114yes—
98#117—24no—
99#117—24no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—24no—
108C2×C4⋊C4—24no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—42no—
112C2×C4⋊C4—42no—
113C2×C4⋊C4—84no—
114C2×C4⋊C4C2×D412yes—
115C2×C4⋊C4C2×D412yes—
116C2×C4⋊C4#4114yes—
117C2×C22⋊C4—22no—
118C2×C42—22no—
119C2×C42—22no—
120C2×C42—22no—
121C2×C42C2×D411yescharacteristic
122C2×C42C2×D411yescharacteristic
123C2×C42C2×D412yes—
124#107—28no—
125#107—48no—
126#106—48no—
127#106—44no—
128C4⋊C8—28no—
129#97—28no—
130#97—44no—
131C4×C8—44no—
132#95—24no—
133#95—42no—
134#95—42no—
135C24C2×#4111yescharacteristic
136C22×C4—24no—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—24no—
140C22×C4—42no—
141C22×C4—42no—
142C22×C4—42no—
143C22×C4—42no—
144C22×C4—42no—
145C22×C4—44no—
146C22×C4—82no—
147C22×C4—82no—
148C22×C4—82no—
149C22×C4—82no—
150C22×C4#12112yes—
151C22×C4#12112yes—
152C22×C4C22×D411yescharacteristic
153C22×C4C22×D411yescharacteristic
154C22×C4C22×Q812yes—
155C22×C4C2×#4112yes—
156C22×C4C2×#4112yes—
157C22×C4C2×#4112yes—
158C22×C4H(5,𝔽2)12yes—
159C22×C42−1+412yes—
160M16—28no—
161M16—28no—
162M16—28no—
163M16—44no—
164M16—44no—
165M16—44no—
166C2×C8—22no—
167C2×C8—22no—
168C2×C8—22no—
169C2×C8—22no—
170C2×C8—28no—
171C2×C8—28no—
172C2×C8—24no—
173C2×C8—28no—
174C2×C8—44no—
175C2×C8—42no—
176C2×C8—42no—
177C2×C8—44no—
178C2×C8#12314yes—
179C2×C8C4⋊Q814yes—
180C4⋊C4—24no—
181C4⋊C4—24no—
182C4⋊C4—24no—
183C4⋊C4—24no—
184C4⋊C4—42no—
185C4⋊C4—42no—
186C4⋊C4—42no—
187C4⋊C4—42no—
188C4⋊C4—44no—
189C4⋊C4—82no—
190C4⋊C4—82no—
191C4⋊C4—84no—
192C4⋊C4—84no—
193C22⋊C4—44no—
194C42—22no—
195C42—22no—
196C42—22no—
197C42—22no—
198C42—42no—
199C42—42no—
200C23—22no—
201C23—22no—
202C23C2×#12111yescharacteristic
203C23#52912yes—
204C23#55312yes—
205C23#55712yes—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—44no—
215C2×C4—44no—
216C2×C4—44no—
217C2×C4—84no—
218C2×C4—84no—
219C2×C4—82no—
220C2×C4—82no—
221C2×C4—84no—
222C2×C4—82no—
223C2×C4—82no—
224C2×C4—82no—
225C2×C4—164no—
226C2×C4C2×#12311yescharacteristic
227C2×C4#52511yescharacteristic
228C2×C4#52511yescharacteristic
229C2×C4C2×C4⋊Q811yescharacteristic
230C2×C4#53212yes—
231C2×C4#53314yes—
232C2×C4#53511yescharacteristic
233C2×C4#53611yescharacteristic
234C2×C4#53914yes—
235C2×C4#54612yes—
236C2×C4D4×Q812yes—
237C2×C4#55214yes—
238C2×C4#55414yes—
239C8—24no—
240C8—24no—
241C8—44no—
242C8—48no—
243C22—22no—
244C22—22no—
245C22—24no—
246C22—24no—
247C22#253511yescharacteristic
248C22#254611yescharacteristic
249C22#259014yes—
250C22#285411yescharacteristic
251C22#306812yes—
252C22#306912yes—
253C4—42no—
254C4—42no—
255C4—82no—
256C4—84no—
257C4—82no—
258C4—162no—
259C4#239911yescharacteristic
260C4#248411yescharacteristic
261C4#250211yescharacteristic
262C4#255614yes—
263C4#263311yescharacteristic
264C2—24no—
265C2#1510611yescharacteristic
266C2#2817711yescharacteristic
267C2#2817911yescharacteristic
268C2#2870312yes—
269C2#3312712yes—
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC23×C4C24
prime radicalC24C2×#41
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#236572
C2×#19432
#204382
#113662
#113312
#112042
#108962
#108232