Finite Group Databasebeta

← #479885

Subgroups — #479885

Order 512.

611 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11370C211yesmaximal, characteristic
2#11367C211yesmaximal, characteristic
3#11341C212yesmaximal
4#11341C212yesmaximal
5#11286C211yesmaximal, characteristic
6#11282C212yesmaximal
7#11263C212yesmaximal
8#11212C212yesmaximal
9#11209C212yesmaximal
10C2×#430C2211yescharacteristic
11C2×#430C2211yescharacteristic
12#2905C2212yes—
13C2×#423C2212yes—
14#2296C2212yes—
15#2285C2211yescharacteristic
16C2×#344C2212yes—
17C2×#344C2212yes—
18#1740C2212yes—
19#1733C2212yes—
20#1733C2212yes—
21C2×#339C2212yes—
22C2×#339C2212yes—
23C2×#339C2212yes—
24C2×#339C2212yes—
25C2×#339C2212yes—
26C2×#339C2212yes—
27C2×H(3,ℤ/4)C2211yescharacteristic
28C2×H(3,ℤ/4)C2211yescharacteristic
29#1728C2211yescharacteristic
30C4×#95C2211yescharacteristic
31C22×M16C2312yes—
32C22×M16C2312yes—
33C22×M16C2311yescharacteristic
34C22×M16C2311yescharacteristic
35C2×#117C2312yes—
36C2×#117C2312yes—
37C2×#117C2312yes—
38C22×C42C2311yescharacteristic
39C22×C42C2312yes—
40#430—22no—
41#430—41no—
42#430—42no—
43C2×#108—41no—
44C2×#108—41no—
45#424—42no—
46#424—42no—
47#423—42no—
48#423—42no—
49#423—42no—
50C2×C4⋊C8—42no—
51#389—22no—
52#389—22no—
53#385—42no—
54#384Q811yescharacteristic
55#384Q811yescharacteristic
56#382Q812yes—
57#380—41no—
58C4×C4⋊C4—41no—
59C4×C4⋊C4—42no—
60C2×#95—41no—
61#344—24no—
62#344—42no—
63#342—42no—
64#339—22no—
65#339—22no—
66#339—22no—
67#339—22no—
68#339—42no—
69#339—42no—
70#339—42no—
71#339—42no—
72#339—42no—
73#339—42no—
74H(3,ℤ/4)—41no—
75H(3,ℤ/4)Q814yes—
76#336—41no—
77C23×C4—21no—
78C23×C4—22no—
79C23×C4C2411yescharacteristic
80C2×M16—21no—
81C2×M16—21no—
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—22no—
95C2×M16—22no—
96C2×M16—22no—
97C2×M16—22no—
98C2×M16C2×Q812yes—
99C2×M16#4112yes—
100C22×C8—21no—
101C22×C8—21no—
102C22×C8—22no—
103C22×C8—22no—
104#117—22no—
105#117—22no—
106#117—22no—
107#117—22no—
108#117—22no—
109#117—22no—
110#117—22no—
111#117—22no—
112#117—22no—
113#117—22no—
114#117—22no—
115#117C2×Q814yes—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—41no—
119C2×C4⋊C4—41no—
120C2×C4⋊C4—42no—
121C2×C4⋊C4—42no—
122C2×C4⋊C4C2×Q812yes—
123C2×C4⋊C4C2×Q812yes—
124C2×C22⋊C4—22no—
125C2×C22⋊C4—22no—
126C2×C22⋊C4—22no—
127C2×C42—22no—
128C2×C42—22no—
129C2×C42—22no—
130C2×C42—22no—
131C2×C42—22no—
132C2×C42—22no—
133C2×C42—22no—
134C2×C42—22no—
135C2×C42—21no—
136C2×C42—22no—
137C2×C42—22no—
138C2×C42C2×Q811yescharacteristic
139C2×C42C2×Q811yescharacteristic
140C2×C42#4112yes—
141C2×C42#4112yes—
142#108—44no—
143#108—44no—
144C4⋊C8—48no—
145C4⋊C8—48no—
146#95—42no—
147#95—42no—
148#95—42no—
149#95—44no—
150C24H(5,𝔽2)11yescharacteristic
151C22×C4—22no—
152C22×C4—21no—
153C22×C4—22no—
154C22×C4—21no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—42no—
158C22×C4—44no—
159C22×C4—42no—
160C22×C4—42no—
161C22×C4—42no—
162C22×C4—42no—
163C22×C4—42no—
164C22×C4—81no—
165C22×C4—82no—
166C22×C4C22×Q811yescharacteristic
167C22×C4C22×Q811yescharacteristic
168C22×C4C2×#4111yescharacteristic
169C22×C4C2×#4112yes—
170C22×C4H(5,𝔽2)11yescharacteristic
171C22×C4H(5,𝔽2)11yescharacteristic
172C22×C4H(5,𝔽2)12yes—
173C22×C4H(5,𝔽2)12yes—
174C22×C42−1+411yescharacteristic
175C22×C42−1+412yes—
176M16—24no—
177M16—24no—
178M16—42no—
179M16—42no—
180M16—42no—
181M16—42no—
182M16—42no—
183M16—42no—
184M16—42no—
185M16—42no—
186M16—42no—
187M16—42no—
188C2×C8—22no—
189C2×C8—22no—
190C2×C8—41no—
191C2×C8—41no—
192C2×C8—41no—
193C2×C8—41no—
194C2×C8—42no—
195C2×C8—42no—
196C2×C8—42no—
197C2×C8—42no—
198C2×C8—42no—
199C2×C8—42no—
200C4⋊C4—24no—
201C4⋊C4—22no—
202C4⋊C4—22no—
203C4⋊C4—22no—
204C4⋊C4—22no—
205C4⋊C4—42no—
206C4⋊C4—42no—
207C4⋊C4—42no—
208C4⋊C4—84no—
209C4⋊C4—82no—
210C4⋊C4—82no—
211C4⋊C4—84no—
212C22⋊C4—24no—
213C22⋊C4—42no—
214C22⋊C4—42no—
215C22⋊C4—42no—
216C22⋊C4—42no—
217C22⋊C4—42no—
218C42—22no—
219C42—22no—
220C42—22no—
221C42—22no—
222C42—22no—
223C42—22no—
224C42—24no—
225C42—42no—
226C42—42no—
227C42—42no—
228C42—41no—
229C42—42no—
230C42—42no—
231C42—41no—
232C42—42no—
233C42—42no—
234C23—41no—
235C23—41no—
236C23#53812yes—
237C23#54311yescharacteristic
238C23#55411yescharacteristic
239C23#56211yescharacteristic
240C23#56212yes—
241C2×C4—24no—
242C2×C4—22no—
243C2×C4—41no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—41no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—42no—
252C2×C4—41no—
253C2×C4—44no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—84no—
257C2×C4—84no—
258C2×C4—84no—
259C2×C4—82no—
260C2×C4—82no—
261C2×C4—82no—
262C2×C4—82no—
263C2×C4—82no—
264C2×C4—82no—
265C2×C4—82no—
266C2×C4#53811yescharacteristic
267C2×C4#53912yes—
268C2×C4#53912yes—
269C2×C4#54012yes—
270C2×C4#54412yes—
271C2×C4#54411yescharacteristic
272C2×C4#54412yes—
273C2×C4#55212yes—
274C2×C4#55412yes—
275C2×C4#55412yes—
276C2×C4#55711yescharacteristic
277C2×C4Q8211yescharacteristic
278C2×C4#55911yescharacteristic
279C2×C4#56012yes—
280C2×C4#56011yescharacteristic
281C2×C4H(3,𝔽4)11yescharacteristic
282C2×C4#56311yescharacteristic
283C2×C4#56312yes—
284C8—42no—
285C8—42no—
286C8—82no—
287C8—82no—
288C22—22no—
289C22—21no—
290C22—22no—
291C22—21no—
292C22—41no—
293C22—42no—
294C22—41no—
295C22#276812yes—
296C22#280411yescharacteristic
297C22#280611yescharacteristic
298C22#282712yes—
299C22#284111yescharacteristic
300C4—44no—
301C4—42no—
302C4—82no—
303C4—81no—
304C4—84no—
305C4—82no—
306C4—82no—
307C4—162no—
308C4#277111yescharacteristic
309C4#279112yes—
310C4#279212yes—
311C4#280711yescharacteristic
312C4#282811yescharacteristic
313C4#284411yescharacteristic
314C2—22no—
315C2—21no—
316C2—22no—
317C2—21no—
318C2#1730511yescharacteristic
319C2#3270411yescharacteristic
320C2#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
#113672
#113412
#112862
#112822
#112632
#112122
#112092