Finite Group Databasebeta

← #316838

Subgroups — #316838

Order 512.

632 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28281C212yesmaximal
2#22270C211yesmaximal, characteristic, complemented
3#22208C211yesmaximal, characteristic
4#10624C211yesmaximal, characteristic, complemented
5#10622C211yesmaximal, characteristic
6#10276C211yesmaximal, characteristic, complemented
7#10274C211yesmaximal, characteristic
8#9220C212yesmaximal, complemented
9#9219C212yesmaximal
10#8787C211yesmaximal, characteristic, complemented
11#8586C212yesmaximal, complemented
12C2×#494C2211yescharacteristic
13C2×#475C2212yes—
14C2×#474C2212yes—
15C2×C8×Q8C2211yescharacteristic
16C2×C8×D4C2211yescharacteristic
17C4×C4⋊Q8C2212yes—
18C4×#124C2211yescharacteristic
19#2068—22no—
20#2065—22no—
21#1984C2211yescharacteristic
22#1983—22no—
23#1956C2212yes—
24#1954C2211yescharacteristic
25#1949—22no—
26#1923—22no—
27#1878—22no—
28#1878—22no—
29#1850—22no—
30#1842C2212yes—
31#1842C2211yescharacteristic
32#1834C2211yescharacteristic
33#1803C2212yes—
34#1802C2212yes—
35#1744C2211yescharacteristic
36C2×#332C2212yes—
37C2×#332C2212yes—
38#1296C2212yes—
39#1296C2212yes—
40#1277C2212yes—
41#1277C2211yescharacteristic
42#1277C2211yescharacteristic
43C22×Q16—22no—
44C23×C8—21no—
45C2×C4⋊Q8C2312yes—
46C2×#124D411yescharacteristic
47C2×C4×Q8—22no—
48C2×C4×Q8C2311yescharacteristic
49C2×C4×D4C2311yescharacteristic
50#494—24no—
51#475—24no—
52#475—42no—
53#474—24no—
54#474—42no—
55C8×Q8—42no—
56C8×D4—42no—
57C2×#107—22no—
58C2×#106—22no—
59C2×C4⋊C8—21no—
60C2×C4⋊C8—22no—
61C2×C4⋊C8—22no—
62C2×C4⋊C8—22no—
63C2×C4⋊C8C2312yes—
64C2×C4⋊C8C2311yescharacteristic
65C2×C4⋊C8C2311yescharacteristic
66C2×C4⋊C8C2312yes—
67C2×#103—22no—
68C2×#103—22no—
69C2×#103—22no—
70C2×#103—22no—
71C2×#102—22no—
72C2×#102—22no—
73C2×C22⋊C8—21no—
74C2×C22⋊C8—22no—
75C2×C4×C8—21no—
76C2×C4×C8C2311yescharacteristic
77C2×C4×C8C2311yescharacteristic
78#391—22no—
79#391D411yescharacteristic
80#390D411yescharacteristic
81#389—22no—
82#389—22no—
83#388—22no—
84#384C2312yes—
85#382D411yescharacteristic
86C4×C4⋊C4—22no—
87C4×C4⋊C4—22no—
88C4×C22⋊C4—22no—
89C43C2311yescharacteristic
90#340—22no—
91#340—22no—
92#340—22no—
93#336—22no—
94#336D412yes—
95#336D412yes—
96#332—42no—
97#332—42no—
98C22×Q8—22no—
99C22×Q8C2×D411yescharacteristic
100C22×D4C2×D411yescharacteristic
101C23×C4—21no—
102C2×Q16—28no—
103C2×Q16—82no—
104C2×Q16—82no—
105C22×C8—21no—
106C22×C8—22no—
107C22×C8—22no—
108C22×C8—21no—
109C22×C8—42no—
110C22×C8—82no—
111C22×C8C2×D411yescharacteristic
112C22×C8C2×D412yes—
113C22×C8C2×D411yescharacteristic
114C4⋊Q8—22no—
115C4⋊Q8—22no—
116C4⋊Q8—22no—
117C4⋊Q8#4114yes—
118#124—22no—
119#124Q1614yescomplemented
120C4×Q8—22no—
121C4×Q8—22no—
122C4×Q8—82no—
123C4×Q8—82no—
124C4×D4—22no—
125C4×D4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—42no—
133C2×C4⋊C4C2×D411yescharacteristic
134C2×C4⋊C4C2×D411yescharacteristic
135C2×C22⋊C4—22no—
136C2×C22⋊C4—21no—
137C2×C22⋊C4—22no—
138C2×C42—21no—
139C2×C42—42no—
140C2×C42C2×D412yes—
141C2×C42C2×D411yescharacteristic
142C2×C42C2×D411yescharacteristic
143C2×C42#4112yes—
144C2×C42C2411yescharacteristic
145#107—44no—
146#106—44no—
147C4⋊C8—22no—
148C4⋊C8—22no—
149C4⋊C8—22no—
150C4⋊C8—22no—
151C4⋊C8—22no—
152C4⋊C8—22no—
153C4⋊C8—42no—
154C4⋊C8—44no—
155C4⋊C8—82no—
156C4⋊C8—82no—
157#103—44no—
158#103—82no—
159#103—82no—
160#103—82no—
161#102—44no—
162#102—82no—
163C22⋊C8—42no—
164C22⋊C8—82no—
165C4×C8—22no—
166C4×C8—22no—
167C4×C8—42no—
168#95—22no—
169#95—22no—
170#95—42no—
171#95—42no—
172C24—21no—
173C2×Q8—21no—
174C2×Q8—24no—
175C2×Q8—22no—
176C2×Q8—22no—
177C2×Q8—42no—
178C2×Q8—42no—
179C2×Q8—42no—
180C2×Q8—42no—
181C2×Q8—42no—
182C2×Q8C2×Q1612yes—
183C2×D4—22no—
184C2×D4—21no—
185C2×D4—22no—
186C2×D4C2×Q1612yes—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—42no—
191C22×C4—41no—
192C22×C4—41no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4#12112yes—
197C22×C4#12112yes—
198C22×C4C22×D411yescharacteristic
199C22×C4C22×D411yescharacteristic
200C22×C4C2×#4111yescharacteristic
201Q16—88no—
202C2×C8—22no—
203C2×C8—21no—
204C2×C8—21no—
205C2×C8—22no—
206C2×C8—22no—
207C2×C8—22no—
208C2×C8—22no—
209C2×C8—41no—
210C2×C8—41no—
211C2×C8—42no—
212C2×C8—42no—
213C2×C8—42no—
214C2×C8—42no—
215C2×C8—42no—
216C2×C8—42no—
217C2×C8—42no—
218C2×C8—42no—
219C2×C8—84no—
220C4⋊C4—24no—
221C4⋊C4—22no—
222C4⋊C4—22no—
223C4⋊C4—24no—
224C4⋊C4—42no—
225C4⋊C4—42no—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C4⋊C4—42no—
229C4⋊C4—42no—
230C4⋊C4—42no—
231C4⋊C4—82no—
232C4⋊C4—84no—
233C22⋊C4—28no—
234C22⋊C4—42no—
235C22⋊C4—82no—
236C42—22no—
237C42—22no—
238C42—22no—
239C42—42no—
240C42—41no—
241C42—41no—
242C42—84no—
243C42C2×Q1612yes—
244C42#13512yes—
245C42C2×#4112yes—
246C42H(5,𝔽2)12yes—
247C23—22no—
248C23—42no—
249C23—41no—
250C23—41no—
251C23—42no—
252C23C2×#12111yescharacteristic
253Q8—24no—
254Q8—42no—
255Q8—44no—
256Q8—44no—
257Q8—82no—
258Q8—82no—
259D4—24no—
260D4—42no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—42no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—44no—
274C2×C4—42no—
275C2×C4—82no—
276C2×C4—82no—
277C2×C4—82no—
278C2×C4—81no—
279C2×C4—81no—
280C2×C4—82no—
281C2×C4—82no—
282C2×C4—82no—
283C2×C4#46714yes—
284C2×C4#52511yescharacteristic
285C2×C4#53511yescharacteristic
286C2×C4#54612yes—
287C2×C4#55312yes—
288C2×C4C22×Q1611yescharacteristic
289C2×C4C2×#13511yescharacteristic
290C2×C4#57612yes—
291C2×C4#57712yes—
292C2×C4#57712yes—
293C8—42no—
294C8—41no—
295C8—41no—
296C8—81no—
297C8—81no—
298C8—82no—
299C8—82no—
300C22—44no—
301C22—42no—
302C22—81no—
303C22—81no—
304C22—82no—
305C22#253511yescharacteristic
306C22#304512yes—
307C22C2×#46711yescharacteristic
308C22#306011yescharacteristic
309C22#306912yes—
310C4—22no—
311C4—24no—
312C4—42no—
313C4—42no—
314C4—81no—
315C4—84no—
316C4—81no—
317C4—162no—
318C4#311812yes—
319C4#322011yescharacteristic
320C4#322111yescharacteristic
321C4#329912yes—
322C4#330412yes—
323C2—82no—
324C2#1941112yes—
325C2#1954112yes—
326C2#2814211yescharacteristic
327C2#2817111yescharacteristic
328C2#2817911yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC22×D4C2×D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22270C2semidirect
4. #10624C2semidirect
6. #10276C2semidirect
8. #9220C2semidirect
10. #8787C2semidirect
11. #8586C2semidirect
119. #124Q16semidirect

Maximal subgroups

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

subgroupindex
#282812
#222702
#222082
#106242
#106222
#102762
#102742
#92202
#92192
#87872
#85862