Finite Group Databasebeta

← #474673

Subgroups — #474673

Order 512.

742 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23657C212yesmaximal
2C2×#1942C211yesmaximal, characteristic
3#21143C211yesmaximal, characteristic
4#11331C212yesmaximal
5#11270C212yesmaximal
6#11216C214yesmaximal
7#11204C212yesmaximal
8#10822C211yesmaximal, characteristic
9C2×#428C2214yes—
10C22×#107C2211yescharacteristic
11C22×#107C2211yescharacteristic
12C2×#424C2212yes—
13C2×#423C2211yescharacteristic
14C22×C4×C8C2211yescharacteristic
15C2×#389C2212yes—
16#2285C2211yescharacteristic
17#2085—24no—
18#2084—22no—
19#2083—22no—
20#1942—22no—
21#1942—22no—
22#1939—22no—
23#1939—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—22no—
46C2×#107—24no—
47C2×#107—42no—
48C2×#106—24no—
49#424—44no—
50#423—42no—
51C2×C4⋊C8—24no—
52C2×C4⋊C8—21no—
53C2×C4⋊C8—21no—
54C2×C4×C8—42no—
55C2×C4×C8D412yes—
56C2×C4×C8D412yes—
57#389—22no—
58#389—22no—
59#389—22no—
60#389—22no—
61#384D411yescharacteristic
62#384D411yescharacteristic
63#382D412yes—
64#380—21no—
65#380—21no—
66C4×C4⋊C4—21no—
67C4×C4⋊C4—21no—
68C2×#95—22no—
69#343—44no—
70#343—44no—
71#341—42no—
72#341Q818yes—
73#340—24no—
74#340—24no—
75#340—24no—
76#336—24no—
77#336—22no—
78#336—22no—
79C23×C4—22no—
80C23×C4—22no—
81C23×C4#4111yescharacteristic
82C23×C4#4111yescharacteristic
83C23×C4C2411yescharacteristic
84C2×M16—24no—
85C2×M16—24no—
86C2×M16—24no—
87C2×M16—24no—
88C2×M16—24no—
89C2×M16—24no—
90C2×M16—24no—
91C2×M16—24no—
92C2×M16C2×Q818yes—
93C22×C8—22no—
94C22×C8—22no—
95C22×C8—24no—
96C22×C8—22no—
97C22×C8—22no—
98C22×C8—22no—
99C22×C8—22no—
100C22×C8C2×D412yes—
101C22×C8C2×D412yes—
102C22×C8C2×Q814yes—
103#117—24no—
104#117—24no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—24no—
117C2×C4⋊C4—24no—
118C2×C4⋊C4—42no—
119C2×C4⋊C4—42no—
120C2×C4⋊C4—84no—
121C2×C4⋊C4C2×D412yes—
122C2×C4⋊C4C2×D412yes—
123C2×C4⋊C4#4114yes—
124C2×C22⋊C4—22no—
125C2×C42—22no—
126C2×C42—22no—
127C2×C42—22no—
128C2×C42C2×D411yescharacteristic
129C2×C42C2×D411yescharacteristic
130C2×C42C2×D411yescharacteristic
131C2×C42C2×D411yescharacteristic
132#107—28no—
133#107—48no—
134#107—44no—
135#106—48no—
136C4⋊C8—42no—
137C4⋊C8—42no—
138C4⋊C8—48no—
139C4×C8—42no—
140C4×C8—42no—
141#95—22no—
142#95—22no—
143#95—42no—
144#95—42no—
145C24C2×#4111yescharacteristic
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—24no—
153C22×C4—41no—
154C22×C4—41no—
155C22×C4—42no—
156C22×C4—42no—
157C22×C4—42no—
158C22×C4—44no—
159C22×C4—82no—
160C22×C4—82no—
161C22×C4—82no—
162C22×C4—82no—
163C22×C4#12111yescharacteristic
164C22×C4#12111yescharacteristic
165C22×C4#12111yescharacteristic
166C22×C4#12111yescharacteristic
167C22×C4C22×D411yescharacteristic
168C22×C4C22×D411yescharacteristic
169C22×C4C22×Q812yes—
170C22×C4C2×#4112yes—
171C22×C4H(5,𝔽2)12yes—
172C22×C4H(5,𝔽2)12yes—
173C22×C42−1+412yes—
174C22×C42−1+412yes—
175M16—28no—
176M16—28no—
177M16—44no—
178M16—44no—
179M16—48no—
180C2×C8—22no—
181C2×C8—22no—
182C2×C8—28no—
183C2×C8—22no—
184C2×C8—22no—
185C2×C8—22no—
186C2×C8—28no—
187C2×C8—44no—
188C2×C8—44no—
189C2×C8—44no—
190C2×C8—42no—
191C2×C8—42no—
192C2×C8—42no—
193C2×C8C4⋊Q814yes—
194C4⋊C4—24no—
195C4⋊C4—24no—
196C4⋊C4—24no—
197C4⋊C4—24no—
198C4⋊C4—42no—
199C4⋊C4—42no—
200C4⋊C4—42no—
201C4⋊C4—42no—
202C4⋊C4—42no—
203C4⋊C4—42no—
204C4⋊C4—42no—
205C4⋊C4—42no—
206C4⋊C4—44no—
207C4⋊C4—84no—
208C4⋊C4—84no—
209C22⋊C4—44no—
210C42—21no—
211C42—21no—
212C42—21no—
213C42—21no—
214C42—22no—
215C42—22no—
216C42—42no—
217C42—42no—
218C23—21no—
219C23—21no—
220C23—21no—
221C23—21no—
222C23C2×#12111yescharacteristic
223C23#53912yes—
224C23#55312yes—
225C23#55712yes—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—44no—
237C2×C4—44no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—44no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—84no—
246C2×C4—82no—
247C2×C4—82no—
248C2×C4—84no—
249C2×C4—82no—
250C2×C4—82no—
251C2×C4—82no—
252C2×C4—164no—
253C2×C4#47212yes—
254C2×C4#47312yes—
255C2×C4#52511yescharacteristic
256C2×C4C2×C4⋊Q811yescharacteristic
257C2×C4#53411yescharacteristic
258C2×C4#53511yescharacteristic
259C2×C4#53611yescharacteristic
260C2×C4#53611yescharacteristic
261C2×C4#53712yes—
262C2×C4#54114yes—
263C2×C4#54414yes—
264C2×C4#54612yes—
265C2×C4D4×Q812yes—
266C2×C4#55414yes—
267C2×C4#56214yes—
268C8—24no—
269C8—42no—
270C8—48no—
271C8—82no—
272C22—22no—
273C22—22no—
274C22—22no—
275C22—22no—
276C22—22no—
277C22—22no—
278C22#253511yescharacteristic
279C22#254611yescharacteristic
280C22#261811yescharacteristic
281C22#280214yes—
282C22C2×#47211yescharacteristic
283C22C2×#47311yescharacteristic
284C22#306711yescharacteristic
285C22#306711yescharacteristic
286C4—22no—
287C4—22no—
288C4—41no—
289C4—41no—
290C4—82no—
291C4—84no—
292C4—82no—
293C4—162no—
294C4#258311yescharacteristic
295C4#263411yescharacteristic
296C4#266011yescharacteristic
297C4#270414yes—
298C4#281611yescharacteristic
299C2—24no—
300C2#1699211yescharacteristic
301C2#2817411yescharacteristic
302C2#2817611yescharacteristic
303C2#2869911yescharacteristic
304C2#2870211yescharacteristic
305C2#2978811yescharacteristic
306C2#2978911yescharacteristic
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×#19422
#211432
#113312
#112702
#112162
#112042
#108222