Finite Group Databasebeta

← #507647

Subgroups — #507647

Order 512.

464 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33257C211yesmaximal, characteristic
2C4×#500C211yesmaximal, characteristic
3#13506C211yesmaximal, characteristic
4#13505C211yesmaximal, characteristic
5#13452C211yesmaximal, characteristic
6#13451C211yesmaximal, characteristic
7#13178C212yesmaximal
8#13178C212yesmaximal
9#13168C212yesmaximal
10#13167C212yesmaximal
11#12653C211yesmaximal, characteristic
12C2×#500C2211yescharacteristic
13#2864—21no—
14C4×C4⋊Q8—21no—
15C2×#367C2212yes—
16C2×#367C2212yes—
17C2×#366C2212yes—
18C2×#366C2212yes—
19C2×C4⋊C16C2212yes—
20C2×#346C2212yes—
21#1990—21no—
22#1942—21no—
23#1942C2211yescharacteristic
24#1942C2212yes—
25#1942C2211yescharacteristic
26#1942C2211yescharacteristic
27#1942C2211yescharacteristic
28#1941—21no—
29#1941—21no—
30#1846C2211yescharacteristic
31#1846C2211yescharacteristic
32#1845C2211yescharacteristic
33#1835—21no—
34C4×#107—21no—
35C4×#107C2212yes—
36C42×C8C2211yescharacteristic
37#1380C2212yes—
38#1380C2212yes—
39#1380C2212yes—
40#1380C2212yes—
41#1374C2211yescharacteristic
42#1374C2211yescharacteristic
43C2×C4⋊Q8—21no—
44C2×C4×Q8—41no—
45C22×C16C2312yes—
46C22×C16C2312yes—
47#500—24no—
48C2×#107—21no—
49C2×#107—21no—
50C2×#107C2312yes—
51C2×#107C2311yescharacteristic
52C2×#107C2311yescharacteristic
53C2×#107C2311yescharacteristic
54C2×#107C2311yescharacteristic
55C2×#107C2312yes—
56C2×#106—21no—
57C2×#106—21no—
58C2×C4×C8D411yescharacteristic
59C2×C4×C8D411yescharacteristic
60C2×C4×C8D411yescharacteristic
61C2×C4×C8C2311yescharacteristic
62C2×C4×C8C2311yescharacteristic
63C2×C4×C8C2311yescharacteristic
64#391—41no—
65#389—22no—
66#389—21no—
67#389—21no—
68#389—21no—
69#389—21no—
70#389—41no—
71#387—41no—
72#387—41no—
73#387—41no—
74#387—41no—
75#384—21no—
76#382—21no—
77#382—21no—
78C4×C4⋊C4—22no—
79C4×C4⋊C4—41no—
80C43D411yescharacteristic
81#367—24no—
82#367—42no—
83#366—24no—
84#366—42no—
85C4⋊C16Q818yescomplemented
86#346—42no—
87#340—21no—
88#340—21no—
89#340—21no—
90#340—21no—
91#340—22no—
92#340—22no—
93C22×Q8—41no—
94C22×C8C2×D411yescharacteristic
95C22×C8C2×D411yescharacteristic
96C22×C8C2×D411yescharacteristic
97C22×C8C2411yescharacteristic
98C4⋊Q8—24no—
99C4⋊Q8—42no—
100C4×Q8—84no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—41no—
108C2×C4⋊C4—41no—
109C2×C4⋊C4—42no—
110C2×C4⋊C4—41no—
111C2×C4⋊C4—41no—
112C2×C4⋊C4—41no—
113C2×C4⋊C4—41no—
114C2×C4⋊C4—41no—
115C2×C42—42no—
116C2×C42D811yescharacteristic
117C2×C42D811yescharacteristic
118C2×C42D811yescharacteristic
119C2×C42D811yescharacteristic
120C2×C42C2×D411yescharacteristic
121C2×C42C2×D411yescharacteristic
122C2×C42C2×D411yescharacteristic
123C2×C16—22no—
124C2×C16—22no—
125C2×C16—22no—
126C2×C16—22no—
127C2×C16C2×Q814yes—
128C2×C16C2×Q814yes—
129#107—24no—
130#107—22no—
131#107—22no—
132#107—22no—
133#107—21no—
134#107—22no—
135#107—21no—
136#107—22no—
137#107—21no—
138#107—21no—
139#107—42no—
140#107#4114yes—
141#106—42no—
142#106—42no—
143C4×C8—22no—
144C4×C8—21no—
145C4×C8—21no—
146C4×C8—41no—
147C4×C8—41no—
148C4×C8Q1614yes—
149C4×C8C2×Q814yes—
150#95—42no—
151#95—42no—
152#95—41no—
153#95—41no—
154#95—41no—
155#95—41no—
156#95—41no—
157#95—41no—
158#95—41no—
159#95—41no—
160C2×Q8—44no—
161C2×Q8—82no—
162C2×Q8—82no—
163C22×C4—42no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4—41no—
167C22×C4—41no—
168C22×C4—42no—
169C22×C4C2×D811yescharacteristic
170C22×C4C2×D811yescharacteristic
171C22×C4C2×D811yescharacteristic
172C22×C4C2×D811yescharacteristic
173C22×C4C2×D811yescharacteristic
174C22×C4C2×D811yescharacteristic
175C22×C4C22×D411yescharacteristic
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8C4⋊Q812yes—
183C2×C8C2×Q1612yes—
184C2×C8C2×Q1612yes—
185C2×C8C22×Q812yes—
186C2×C8C2×#4112yes—
187C2×C8H(5,𝔽2)11yescharacteristic
188C2×C82−1+411yescharacteristic
189C4⋊C4—24no—
190C4⋊C4—44no—
191C4⋊C4—42no—
192C4⋊C4—41no—
193C4⋊C4—41no—
194C4⋊C4—41no—
195C4⋊C4—42no—
196C4⋊C4—41no—
197C4⋊C4—41no—
198C4⋊C4—41no—
199C4⋊C4—41no—
200C4⋊C4—41no—
201C4⋊C4—42no—
202C4⋊C4—82no—
203C4⋊C4—82no—
204C4⋊C4—82no—
205C4⋊C4—84no—
206C4⋊C4—82no—
207C4⋊C4—82no—
208C4⋊C4—82no—
209C42—21no—
210C42—21no—
211C42—21no—
212C42—21no—
213C42—22no—
214C42—22no—
215C42—41no—
216C42—41no—
217C42—84no—
218C42C4⋊Q812yes—
219C42C2×Q1612yes—
220C16—24no—
221C16—24no—
222C23C22×D811yescharacteristic
223Q8—88no—
224C2×C4—21no—
225C2×C4—21no—
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—21no—
237C2×C4—21no—
238C2×C4—44no—
239C2×C4—82no—
240C2×C4—82no—
241C2×C4—81no—
242C2×C4—81no—
243C2×C4—82no—
244C2×C4—81no—
245C2×C4—81no—
246C2×C4—81no—
247C2×C4—81no—
248C2×C4—81no—
249C2×C4—82no—
250C2×C4—81no—
251C2×C4—81no—
252C2×C4—81no—
253C2×C4—81no—
254C2×C4—82no—
255C2×C4—81no—
256C2×C4#50012yes—
257C2×C4#50012yes—
258C2×C4#50912yes—
259C2×C4#51012yes—
260C2×C4#52511yescharacteristic
261C2×C4C2×C4⋊Q811yescharacteristic
262C2×C4#53711yescharacteristic
263C2×C4C22×Q1611yescharacteristic
264C2×C4#57511yescharacteristic
265C2×C4#57511yescharacteristic
266C8—22no—
267C8—22no—
268C8—22no—
269C8#55412yes—
270C8#55712yes—
271C22#310011yescharacteristic
272C22C2×#50011yescharacteristic
273C22#324511yescharacteristic
274C22C2×#50911yescharacteristic
275C22C2×#51011yescharacteristic
276C22#341411yescharacteristic
277C22#341411yescharacteristic
278C4—22no—
279C4—22no—
280C4—22no—
281C4—22no—
282C4—41no—
283C4—41no—
284C4—84no—
285C4—162no—
286C4—161no—
287C4—161no—
288C4—161no—
289C4—161no—
290C4#258411yescharacteristic
291C4#310411yescharacteristic
292C4#316611yescharacteristic
293C4#324911yescharacteristic
294C2#2931211yescharacteristic
295C2#3367411yescharacteristic
296C2#3367711yescharacteristic
297C2#3372811yescharacteristic
298C2#3373111yescharacteristic
299C2#3381711yescharacteristic
300C2#3381811yescharacteristic
subgroupG/H
center, socle, prime radicalC23C22×D8
commutator, Frattini, cosocleC22×C8C24
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ C42×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
85. C4⋊C16Q8semidirect

Maximal subgroups

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

subgroupindex
#332572
C4×#5002
#135062
#135052
#134522
#134512
#131782
#131682
#131672
#126532