Finite Group Databasebeta

← #453935

Subgroups — #453935

Order 512.

621 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 621 conjugacy classes of subgroups — too many to draw. Only its 162 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#11370C211yesmaximal, characteristic
2#11321C212yesmaximal
3#11282C212yesmaximal
4#11263C212yesmaximal
5#11209C212yesmaximal
6#10894C212yesmaximal
7#10894C212yesmaximal
8#10893C211yesmaximal, characteristic
9#10893C211yesmaximal, characteristic
10C2×#423C2212yes—
11#2873C2211yescharacteristic
12#2873C2211yescharacteristic
13#2296C2212yes—
14#2285C2211yescharacteristic
15C2×#344C2212yes—
16C2×#344C2212yes—
17#1740C2211yescharacteristic
18#1740C2211yescharacteristic
19#1738C2212yes—
20#1733C2212yes—
21#1733C2212yes—
22C2×#339C2212yes—
23C2×#339C2212yes—
24C2×#339C2212yes—
25C2×#339C2212yes—
26C2×#339C2212yes—
27C2×#339C2212yes—
28C2×H(3,ℤ/4)C2212yes—
29#1728C2211yescharacteristic
30C4×#95C2211yescharacteristic
31C22×M16C2312yes—
32C22×M16C2312yes—
33C22×M16C2312yes—
34C2×#117C2312yes—
35C2×#117C2312yes—
36C2×#117C2311yescharacteristic
37C2×#117C2311yescharacteristic
38C22×C42C2311yescharacteristic
39C22×C42C2312yes—
40#423—24no—
41#423—24no—
42C2×C4⋊C8—42no—
43#405—42no—
44#405—42no—
45C4×M16—41no—
46C4×M16—41no—
47#389—22no—
48#389—22no—
49#385—42no—
50#384Q811yescharacteristic
51#384Q811yescharacteristic
52#382Q811yescharacteristic
53#382Q811yescharacteristic
54#380—41no—
55C4×C4⋊C4—41no—
56C4×C4⋊C4—42no—
57C2×#95—41no—
58#344—24no—
59#344—24no—
60#342—41no—
61#342—41no—
62#341—42no—
63#339—24no—
64#339—24no—
65#339—24no—
66#339—24no—
67#339—22no—
68#339—22no—
69#339—22no—
70#339—22no—
71#339—42no—
72#339—42no—
73H(3,ℤ/4)—24no—
74#336—41no—
75C23×C4—21no—
76C23×C4—22no—
77C23×C4C2411yescharacteristic
78C2×M16—22no—
79C2×M16—22no—
80C2×M16—22no—
81C2×M16—22no—
82C2×M16—22no—
83C2×M16—22no—
84C2×M16—22no—
85C2×M16—22no—
86C2×M16—22no—
87C2×M16—22no—
88C2×M16—22no—
89C2×M16—22no—
90C2×M16—22no—
91C2×M16—22no—
92C2×M16—22no—
93C2×M16—22no—
94C2×M16#4114yes—
95C2×M16#4114yes—
96C22×C8—22no—
97C22×C8—22no—
98C22×C8—22no—
99#117—22no—
100#117—22no—
101#117—22no—
102#117—22no—
103#117—22no—
104#117—22no—
105#117—22no—
106#117—22no—
107#117—22no—
108#117—22no—
109#117—22no—
110#117#4114yes—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—41no—
114C2×C4⋊C4—41no—
115C2×C4⋊C4—42no—
116C2×C4⋊C4—42no—
117C2×C4⋊C4C2×Q811yescharacteristic
118C2×C4⋊C4C2×Q811yescharacteristic
119C2×C4⋊C4C2×Q811yescharacteristic
120C2×C4⋊C4C2×Q811yescharacteristic
121C2×C22⋊C4—22no—
122C2×C22⋊C4—21no—
123C2×C22⋊C4—21no—
124C2×C22⋊C4—22no—
125C2×C42—22no—
126C2×C42—22no—
127C2×C42—22no—
128C2×C42—22no—
129C2×C42—22no—
130C2×C42—22no—
131C2×C42—22no—
132C2×C42—21no—
133C2×C42—21no—
134C2×C42—21no—
135C2×C42—22no—
136C2×C42—22no—
137C2×C42C2×Q811yescharacteristic
138C2×C42C2×Q811yescharacteristic
139C2×C42#4112yes—
140C2×C42#4112yes—
141C4⋊C8—48no—
142#97—42no—
143#97—42no—
144C4×C8—42no—
145C4×C8—42no—
146#95—41no—
147#95—41no—
148#95—42no—
149#95—42no—
150#95—44no—
151C24H(5,𝔽2)11yescharacteristic
152C22×C4—22no—
153C22×C4—21no—
154C22×C4—22no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—42no—
161C22×C4—44no—
162C22×C4—42no—
163C22×C4—42no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4—42no—
167C22×C4—81no—
168C22×C4—82no—
169C22×C4C22×Q811yescharacteristic
170C22×C4C2×#4112yes—
171C22×C4C2×#4112yes—
172C22×C4C2×#4112yes—
173C22×C4C2×#4112yes—
174C22×C4H(5,𝔽2)12yes—
175C22×C4H(5,𝔽2)11yescharacteristic
176C22×C42−1+412yes—
177M16—24no—
178M16—24no—
179M16—24no—
180M16—24no—
181M16—42no—
182M16—42no—
183M16—42no—
184M16—42no—
185M16—42no—
186M16—42no—
187M16—42no—
188M16—42no—
189C2×C8—24no—
190C2×C8—24no—
191C2×C8—42no—
192C2×C8—42no—
193C2×C8—42no—
194C2×C8—42no—
195C2×C8—42no—
196C2×C8—42no—
197C2×C8—42no—
198C4⋊C4—24no—
199C4⋊C4—22no—
200C4⋊C4—22no—
201C4⋊C4—22no—
202C4⋊C4—22no—
203C4⋊C4—42no—
204C4⋊C4—42no—
205C4⋊C4—42no—
206C4⋊C4—84no—
207C4⋊C4—82no—
208C4⋊C4—82no—
209C4⋊C4—84no—
210C22⋊C4—24no—
211C22⋊C4—42no—
212C22⋊C4—42no—
213C22⋊C4—42no—
214C22⋊C4—42no—
215C22⋊C4—42no—
216C42—22no—
217C42—22no—
218C42—22no—
219C42—22no—
220C42—21no—
221C42—21no—
222C42—21no—
223C42—21no—
224C42—24no—
225C42—42no—
226C42—42no—
227C42—42no—
228C42—42no—
229C42—41no—
230C42—41no—
231C42—41no—
232C42—41no—
233C42—42no—
234C42—42no—
235C23—41no—
236C23—41no—
237C23#53812yes—
238C23#54311yescharacteristic
239C23#55312yes—
240C23#55511yescharacteristic
241C23#56311yescharacteristic
242C2×C4—24no—
243C2×C4—22no—
244C2×C4—41no—
245C2×C4—42no—
246C2×C4—41no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—42no—
254C2×C4—44no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—84no—
260C2×C4—84no—
261C2×C4—84no—
262C2×C4—82no—
263C2×C4—82no—
264C2×C4—81no—
265C2×C4—81no—
266C2×C4—81no—
267C2×C4—81no—
268C2×C4—82no—
269C2×C4—82no—
270C2×C4—82no—
271C2×C4#52912yes—
272C2×C4#52912yes—
273C2×C4#53311yescharacteristic
274C2×C4#53311yescharacteristic
275C2×C4#53912yes—
276C2×C4#53912yes—
277C2×C4#53912yes—
278C2×C4#55112yes—
279C2×C4#55212yes—
280C2×C4#55312yes—
281C2×C4#55312yes—
282C2×C4#55412yes—
283C2×C4#55612yes—
284C2×C4#55712yes—
285C2×C4#55911yescharacteristic
286C2×C4H(3,𝔽4)11yescharacteristic
287C8—44no—
288C8—44no—
289C8—82no—
290C22—22no—
291C22—21no—
292C22—22no—
293C22—21no—
294C22—42no—
295C22—41no—
296C22—41no—
297C22#251812yes—
298C22#260811yescharacteristic
299C22#278712yes—
300C22#280411yescharacteristic
301C22#280911yescharacteristic
302C4—44no—
303C4—42no—
304C4—82no—
305C4—81no—
306C4—84no—
307C4—81no—
308C4—81no—
309C4—81no—
310C4—81no—
311C4—162no—
312C4#258512yes—
313C4#258812yes—
314C4#259612yes—
315C4#278212yes—
316C2—22no—
317C2—21no—
318C2—22no—
319C2—21no—
320C2#1624211yescharacteristic
321C2#3270411yescharacteristic
322C2#3270511yescharacteristic
subgroupG/H
centerC2×C4H(3,𝔽4)
commutator, Frattini, cosocleC23×C4C24
socleC22#2804
prime radicalC24H(5,𝔽2)
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ 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
#113702
#113212
#112822
#112632
#112092
#108942
#108932