Finite Group Databasebeta

← #351311

Subgroups — #351311

Order 512.

638 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33209C211yesmaximal, characteristic
2#29923C212yesmaximal
3#20437C211yesmaximal, characteristic
4#9740C212yesmaximal, complemented
5#9620C212yesmaximal, complemented
6#9566C214yesmaximal, complemented
7#9137C212yesmaximal
8#8595C211yesmaximal, characteristic
9C2×#500C2211yescharacteristic
10C2×#423C2211yescharacteristic
11C2×#405—21no—
12C22×C4×C8C2211yescharacteristic
13#2857C2212yes—
14#2285C2211yescharacteristic
15#2095—22no—
16#2072—22no—
17#2036—24no—
18#1990C2212yes—
19#1981C2214yes—
20#1947C2214yes—
21#1942—21no—
22#1942—21no—
23#1845C2211yescharacteristic
24#1808—21no—
25#1808—21no—
26#1797C2212yes—
27#1770—21no—
28#1770C2212yes—
29C8×C4⋊C4—21no—
30C8×C4⋊C4C2212yes—
31#1729—21no—
32C2×#334C2212yes—
33C2×#332C2214yes—
34#1296C2214yes—
35#1277C2212yes—
36C22×M16—21no—
37C23×C8D411yescharacteristic
38C23×C8D411yescharacteristic
39C2×C4⋊Q8C2312yes—
40C2×C22⋊Q8—22no—
41C2×#117—21no—
42C22×C42C2311yescharacteristic
43#500—24no—
44C2×#107—21no—
45C2×#107—21no—
46C2×#107—21no—
47C2×#107—21no—
48C2×#107—24no—
49#423—24no—
50C2×C4⋊C8—21no—
51C2×C4⋊C8C2314yes—
52C2×C4⋊C8C2312yes—
53C2×#103—24no—
54C2×#103—24no—
55C2×C22⋊C8—22no—
56#405—44no—
57C2×#97—22no—
58C2×C4×C8—24no—
59C2×C4×C8—22no—
60C2×C4×C8D411yescharacteristic
61C2×C4×C8D411yescharacteristic
62C2×C4×C8C2311yescharacteristic
63C2×C4×C8C2311yescharacteristic
64#391—22no—
65#389—22no—
66#389—22no—
67#384D411yescharacteristic
68#384D411yescharacteristic
69#384C2312yes—
70#382—21no—
71#380—21no—
72#380—22no—
73#380—22no—
74#380D412yes—
75C4×C4⋊C4—21no—
76C4×C4⋊C4C2312yes—
77#340—22no—
78#340—22no—
79#340—24no—
80#336—22no—
81#336—22no—
82#336—22no—
83#334—24no—
84#332—28no—
85C22×Q8—22no—
86C23×C4C2×D411yescharacteristic
87C23×C4C2×D411yescharacteristic
88C23×C4#4111yescharacteristic
89C2×M16—24no—
90C2×M16—42no—
91C2×M16—42no—
92C22×C8—22no—
93C22×C8—22no—
94C22×C8—22no—
95C22×C8—22no—
96C22×C8—22no—
97C22×C8—24no—
98C22×C8—41no—
99C22×C8Q1614yes—
100C22×C8C2×D411yescharacteristic
101C22×C8C2×D411yescharacteristic
102C22×C8C2×D411yescharacteristic
103C22×C8C2×D411yescharacteristic
104C4⋊Q8—24no—
105C4⋊Q8—24no—
106C22⋊Q8—84no—
107#117—42no—
108#117—42no—
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—22no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—42no—
119C2×C4⋊C4C2×D412yes—
120C2×C4⋊C4C2×D412yes—
121C2×C22⋊C4—22no—
122C2×C22⋊C4—42no—
123C2×C42—22no—
124C2×C42—22no—
125C2×C42—22no—
126C2×C42C2×D411yescharacteristic
127C2×C42C2×D411yescharacteristic
128C2×C42#4111yescharacteristic
129C2×C42C2411yescharacteristic
130#107—42no—
131#107—42no—
132#107—42no—
133#107—42no—
134#107—84no—
135C4⋊C8—24no—
136C4⋊C8—22no—
137C4⋊C8—24no—
138C4⋊C8—24no—
139C4⋊C8#4114yes—
140#103—84no—
141#103—84no—
142C22⋊C8—82no—
143#97—44no—
144C4×C8—22no—
145C4×C8—22no—
146C4×C8—22no—
147C4×C8—44no—
148C4×C8#4114yes—
149#95—22no—
150#95—22no—
151#95—42no—
152#95—42no—
153C24#12111yescharacteristic
154C2×Q8—44no—
155C2×Q8—44no—
156C2×Q8—42no—
157C2×Q8—42no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—41no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4#12111yescharacteristic
173C22×C4#12111yescharacteristic
174C22×C4#12111yescharacteristic
175C22×C4C2×Q1612yes—
176C22×C4#13712yes—
177C22×C4C22×D411yescharacteristic
178C22×C4C22×D411yescharacteristic
179C22×C4C2×#4111yescharacteristic
180M16—48no—
181C2×C8—21no—
182C2×C8—21no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—28no—
186C2×C8—21no—
187C2×C8—24no—
188C2×C8—21no—
189C2×C8—21no—
190C2×C8—44no—
191C2×C8—42no—
192C2×C8—42no—
193C2×C8—44no—
194C2×C8—44no—
195C2×C8—44no—
196C2×C8#12312yes—
197C2×C8#12312yes—
198C2×C8C2×Q1612yes—
199C2×C8C2×Q1612yes—
200C2×C8AGL(1,ℤ/8)12yes—
201C4⋊C4—24no—
202C4⋊C4—24no—
203C4⋊C4—42no—
204C4⋊C4—42no—
205C4⋊C4—42no—
206C4⋊C4—42no—
207C4⋊C4—42no—
208C4⋊C4—42no—
209C4⋊C4—44no—
210C4⋊C4—42no—
211C4⋊C4—42no—
212C4⋊C4—44no—
213C4⋊C4—82no—
214C4⋊C4—84no—
215C22⋊C4—44no—
216C22⋊C4—84no—
217C42—21no—
218C42—21no—
219C42—22no—
220C42—22no—
221C42—42no—
222C42—42no—
223C42C2×#4112yes—
224C42C2×#4112yes—
225C23—21no—
226C23—21no—
227C23—21no—
228C23—21no—
229C23—21no—
230C23—21no—
231C23#46212yes—
232C23C2×#12111yescharacteristic
233Q8—84no—
234Q8—84no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—44no—
249C2×C4—44no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—82no—
257C2×C4—82no—
258C2×C4—82no—
259C2×C4—82no—
260C2×C4—82no—
261C2×C4—82no—
262C2×C4#46212yes—
263C2×C4#47312yes—
264C2×C4C2×#12311yescharacteristic
265C2×C4#52511yescharacteristic
266C2×C4#52912yes—
267C2×C4#53212yes—
268C2×C4C22×Q1611yescharacteristic
269C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
270C2×C4C2×#13711yescharacteristic
271C2×C4#57511yescharacteristic
272C2×C4#57612yes—
273C2×C4#57712yes—
274C8—22no—
275C8—22no—
276C8—41no—
277C8—44no—
278C8—84no—
279C8#48414yes—
280C22—22no—
281C22—22no—
282C22—22no—
283C22—41no—
284C22—41no—
285C22—41no—
286C22—41no—
287C22#285411yescharacteristic
288C22C2×#46211yescharacteristic
289C22#303611yescharacteristic
290C22#305811yescharacteristic
291C22#306011yescharacteristic
292C22C2×#47311yescharacteristic
293C22#306711yescharacteristic
294C4—22no—
295C4—22no—
296C4—22no—
297C4—41no—
298C4—82no—
299C4—82no—
300C4—162no—
301C4—162no—
302C4C2×#48411yescharacteristic
303C4#309511yescharacteristic
304C4#310411yescharacteristic
305C4#311011yescharacteristic
306C4#317914yes—
307C2—42no—
308C2#1920811yescharacteristic
309C2#1921011yescharacteristic
310C2#1933511yescharacteristic
311C2#1933811yescharacteristic
312C2#3310911yescharacteristic
313C2#3312411yescharacteristic
314C2#3313211yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. #9740C2semidirect
5. #9620C2semidirect
6. #9566C2semidirect

Maximal subgroups

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

subgroupindex
#332092
#299232
#204372
#97402
#96202
#95662
#91372
#85952