Finite Group Databasebeta

← #474674

Subgroups — #474674

Order 512.

728 conjugacy classes of subgroups; only the 198 normal subgroups are shown.

normal subgroups only

This group has 728 conjugacy classes of subgroups — too many to draw. Only its 198 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×#1944C211yesmaximal, characteristic
3#21144C211yesmaximal, characteristic
4#11330C212yesmaximal
5#11270C212yesmaximal
6#11216C214yesmaximal
7#11204C212yesmaximal
8#10823C211yesmaximal, characteristic
9C2×#428C2214yes—
10C22×#107C2211yescharacteristic
11C22×#106C2211yescharacteristic
12C2×#424C2212yes—
13C2×#423C2211yescharacteristic
14C22×#97C2211yescharacteristic
15C2×#389C2212yes—
16#2285C2211yescharacteristic
17#2085—24no—
18#2084—22no—
19#2083—22no—
20#1944—22no—
21#1944—22no—
22#1940—22no—
23#1940—22no—
24C2×#343C2214yes—
25C2×#343C2214yes—
26C2×#341C2212yes—
27C2×#341C2212yes—
28#1736C2214yes—
29C2×#340C2212yes—
30#1729C2212yes—
31#1728C2212yes—
32C22×M16C2312yes—
33C22×M16C2314yes—
34C23×C8C2311yescharacteristic
35C23×C8C2311yescharacteristic
36C2×#117C2312yes—
37C22×C4⋊C4—22no—
38C22×C4⋊C4C2312yes—
39C22×C4⋊C4C2312yes—
40C22×C42C2311yescharacteristic
41#428—28no—
42#428—44no—
43C2×#107—22no—
44C2×#107—24no—
45C2×#107—42no—
46C2×#106—22no—
47C2×#106—24no—
48C2×#106—24no—
49#424—44no—
50#423—42no—
51C2×C4⋊C8—24no—
52C2×C4⋊C8—21no—
53C2×C4⋊C8—21no—
54C2×#97—42no—
55C2×#97D412yes—
56C2×#97D412yes—
57#389—22no—
58#389—22no—
59#389—22no—
60#389—22no—
61#384D412yes—
62#382D412yes—
63#380—21no—
64#380—21no—
65C4×C4⋊C4—21no—
66C4×C4⋊C4—21no—
67C2×#95—22no—
68#343—44no—
69#343—44no—
70#341—42no—
71#341Q818yes—
72#340—24no—
73#340—24no—
74#340—24no—
75#336—24no—
76#336—22no—
77#336—22no—
78C23×C4—22no—
79C23×C4—22no—
80C23×C4#4111yescharacteristic
81C23×C4#4111yescharacteristic
82C23×C4C2411yescharacteristic
83C2×M16—24no—
84C2×M16—24no—
85C2×M16—24no—
86C2×M16—24no—
87C2×M16—24no—
88C2×M16—24no—
89C2×M16—24no—
90C2×M16—24no—
91C2×M16C2×Q818yes—
92C22×C8—22no—
93C22×C8—22no—
94C22×C8—24no—
95C22×C8—22no—
96C22×C8—22no—
97C22×C8—22no—
98C22×C8—22no—
99C22×C8C2×D412yes—
100C22×C8C2×D412yes—
101C22×C8C2×Q814yes—
102#117—24no—
103#117—24no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—24no—
112C2×C4⋊C4—24no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—42no—
116C2×C4⋊C4—42no—
117C2×C4⋊C4—84no—
118C2×C4⋊C4C2×D414yes—
119C2×C4⋊C4#4114yes—
120C2×C22⋊C4—22no—
121C2×C42—22no—
122C2×C42—22no—
123C2×C42—22no—
124C2×C42C2×D411yescharacteristic
125C2×C42C2×D411yescharacteristic
126C2×C42C2×D411yescharacteristic
127C2×C42C2×D411yescharacteristic
128#107—48no—
129#107—44no—
130#106—28no—
131#106—48no—
132C4⋊C8—42no—
133C4⋊C8—42no—
134C4⋊C8—48no—
135#97—42no—
136#97—42no—
137#95—22no—
138#95—22no—
139#95—42no—
140#95—42no—
141C24C2×#4111yescharacteristic
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—24no—
145C22×C4—24no—
146C22×C4—42no—
147C22×C4—42no—
148C22×C4—42no—
149C22×C4—42no—
150C22×C4—42no—
151C22×C4—44no—
152C22×C4—82no—
153C22×C4—82no—
154C22×C4—82no—
155C22×C4—82no—
156C22×C4#12111yescharacteristic
157C22×C4#12111yescharacteristic
158C22×C4#12111yescharacteristic
159C22×C4#12111yescharacteristic
160C22×C4C22×D411yescharacteristic
161C22×C4C22×D411yescharacteristic
162C22×C4C22×Q812yes—
163C22×C4C2×#4112yes—
164C22×C4H(5,𝔽2)12yes—
165C22×C4H(5,𝔽2)12yes—
166C22×C42−1+412yes—
167C22×C42−1+412yes—
168M16—28no—
169M16—28no—
170M16—44no—
171M16—44no—
172M16—48no—
173C2×C8—22no—
174C2×C8—22no—
175C2×C8—28no—
176C2×C8—22no—
177C2×C8—22no—
178C2×C8—22no—
179C2×C8—28no—
180C2×C8—44no—
181C2×C8—44no—
182C2×C8—44no—
183C2×C8—42no—
184C2×C8—42no—
185C2×C8—42no—
186C2×C8C4⋊Q814yes—
187C4⋊C4—24no—
188C4⋊C4—24no—
189C4⋊C4—24no—
190C4⋊C4—24no—
191C4⋊C4—42no—
192C4⋊C4—42no—
193C4⋊C4—42no—
194C4⋊C4—42no—
195C4⋊C4—44no—
196C4⋊C4—82no—
197C4⋊C4—82no—
198C4⋊C4—84no—
199C4⋊C4—84no—
200C22⋊C4—44no—
201C42—21no—
202C42—21no—
203C42—21no—
204C42—21no—
205C42—22no—
206C42—22no—
207C42—42no—
208C42—42no—
209C23—22no—
210C23—22no—
211C23C2×#12111yescharacteristic
212C23#53912yes—
213C23#55312yes—
214C23#55712yes—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—44no—
224C2×C4—44no—
225C2×C4—44no—
226C2×C4—44no—
227C2×C4—82no—
228C2×C4—82no—
229C2×C4—84no—
230C2×C4—82no—
231C2×C4—82no—
232C2×C4—84no—
233C2×C4—82no—
234C2×C4—82no—
235C2×C4—82no—
236C2×C4—164no—
237C2×C4#52511yescharacteristic
238C2×C4C2×C4⋊Q811yescharacteristic
239C2×C4#53411yescharacteristic
240C2×C4#53511yescharacteristic
241C2×C4#53611yescharacteristic
242C2×C4#53611yescharacteristic
243C2×C4#53712yes—
244C2×C4#54114yes—
245C2×C4#54414yes—
246C2×C4#54612yes—
247C2×C4D4×Q812yes—
248C2×C4#55414yes—
249C2×C4#56214yes—
250C8—24no—
251C8—42no—
252C8—48no—
253C8—82no—
254C22—22no—
255C22—22no—
256C22—24no—
257C22—22no—
258C22—22no—
259C22#253511yescharacteristic
260C22#254611yescharacteristic
261C22#261811yescharacteristic
262C22#280214yes—
263C22#306812yes—
264C22#306912yes—
265C4—41no—
266C4—41no—
267C4—41no—
268C4—41no—
269C4—84no—
270C4—84no—
271C4—162no—
272C4#258311yescharacteristic
273C4#263411yescharacteristic
274C4#266011yescharacteristic
275C4#270414yes—
276C4#281611yescharacteristic
277C2—24no—
278C2#1699211yescharacteristic
279C2#2817812yes—
280C2#2870312yes—
281C2#2979111yescharacteristic
282C2#2979211yescharacteristic
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×#19442
#211442
#113302
#112702
#112162
#112042
#108232