Finite Group Databasebeta

← #72245849

Subgroups — #72245849

Order 2016.

326 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 58 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11759628C211yesmaximal, characteristic, complemented
2C3⋊D3×D28C211yesmaximal, characteristic, complemented
3#11759200C211yesmaximal, characteristic, complemented
4#11759195C211yesmaximal, characteristic, complemented
5C7×#3712C211yesmaximal, characteristic, complemented
6C3×#66565C211yesmaximal, characteristic, complemented
7C3×#66561C211yesmaximal, characteristic, complemented
8#10620816—31nomaximal
9#10620815—31nomaximal
10C2×D7×C3⋊D3—21no—
11C4⋊(C3⋊D21)C2211yescharacteristic
12C4×C7×C3⋊D3C2211yescharacteristic
13C3×D84C2211yescharacteristic, complemented
14C3×Dic42C2211yescharacteristic, complemented
15C32×D28C2211yescharacteristic
16#92694—21no—
17D21⋊Dic3—21no—
18#92677—21no—
19C3×C7×C3⋊C8C2211yescharacteristic, complemented
20C3×C7×C3⋊C8C2211yescharacteristic, complemented
21D3×D28—31no—
22D3×D28—31no—
23D3×D28—61no—
24#66673—31no—
25#66669—31no—
26D168—31no—
27#66623—31no—
28C7×ΓL(1,𝔽25)—31no—
29C7×ΓL(1,𝔽25)—31no—
30C3×D56—31no—
31C3×#1132—31no—
32#66566—31no—
33#66565D311yescharacteristic, complemented
34#66561—31no—
35#66561D311yescharacteristic, complemented
36#63937—71nomaximal, Hall
37C2×C3⋊D21—21no—
38C2×C7×C3⋊D3D411yescharacteristic
39C2×C3×D21—21no—
40C2×C32×D7—21no—
41D7×C3⋊D3—42no—
42C32×C4×C7C2311yescharacteristic
43C7×C3⋊Dic3D411yescharacteristic
44C3×Dic21—21no—
45#6471—91noHall
46C2×D3×D7—61no—
47C2×D3×D7—61no—
48C2×D3×D7—61no—
49C2×D3×D7—61no—
50D84—31no—
51D84—31no—
52D84—61no—
53D84C2×D311yescharacteristic, complemented
54Dic42C2×D311yescharacteristic, complemented
55C4×D3×C7—31no—
56C4×D3×C7—31no—
57C4×D3×C7—61no—
58C3×D28—21no—
59C3×D28—31no—
60C3×D28C2×D311yescharacteristic
61C3×D28C2×D311yescharacteristic
62C3×Dic14—31no—
63#4247—61no—
64#4246—61no—
65#4246—61no—
66#4246—61no—
67#4246—61no—
68#4245—61no—
69D21⋊C4—61no—
70Dic3×D7—61no—
71C3×C7×C8—31no—
72C3×C7×C8—31no—
73C7×C3⋊C8C2×D311yescharacteristic, complemented
74C7×C3⋊C8C2×D311yescharacteristic, complemented
75D4×C3⋊D3—71no—
76#3796—71no—
77C3×#214—71no—
78C3×#213—71no—
79#3717—71no—
80#3714—71no—
81#3712D711yescharacteristic, complemented
82C2×C32×C7C2×D411yescharacteristic
83C3⋊D21—41no—
84C7×C3⋊D3—21no—
85C3×D21—41no—
86C32×D7—41no—
87#1157—91no—
88C2×D28—91no—
89C7×M16—91no—
90D56—31no—
91D56—91no—
92#1132—31no—
93#1132—91no—
94#914—211no—
95#911—211no—
96C2×D21—21no—
97C2×D21—61no—
98C2×D21—61no—
99C2×D21—61no—
100C2×D21—61no—
101C2×D3×C7—31no—
102C2×D3×C7—31no—
103C2×D3×C7—61no—
104C2×C3×D7—21no—
105C2×C3×D7—21no—
106C2×C3×D7—21no—
107C2×C3×D7—21no—
108C2×C3×D7—61no—
109D3×D7—122no—
110D3×D7—122no—
111D3×D7—122no—
112D3×D7—122no—
113C3×C4×C7—21no—
114C3×C4×C7C22×D311yescharacteristic
115C3×C4×C7C22×D311yescharacteristic
116Dic21—21no—
117C3×Dic7—61no—
118C7×Dic3—31no—
119C7×Dic3—31no—
120C7×Dic3—61no—
121C22×C3⋊D3—141no—
122C32×D4—71no—
123#638—141no—
124C4⋊(C3⋊D3)—71no—
125C4×C3⋊D3C2×D711yescharacteristic
126C3×D12—71no—
127C3×Dic6—71no—
128#626—141no—
129D3×Dic3—141no—
130C3×C3⋊C8C2×D711yescharacteristic, complemented
131C3×C3⋊C8C2×D711yescharacteristic, complemented
132C32×C7AGL(1,ℤ/8)11yescharacteristic, complemented, Hall
133C22×D7—181no—
134C2×C4×C7—91no—
135C22⋊D7—181no—
136D28—31no—
137D28—91no—
138D28—181no—
139D28D3211yescharacteristic
140C4×D7—181no—
141Dic14—31no—
142C7×C8—31no—
143C7×C8—31no—
144#239—211no—
145#237—211no—
146D3×D4—211no—
147D3×D4—211no—
148D3×D4—421no—
149C3×SD16—211no—
150C3×D8—211no—
151#215—211no—
152#214—71no—
153#213—71no—
154#213—211no—
155D24—211no—
156#204—211no—
157ΓL(1,𝔽25)—31no—
158ΓL(1,𝔽25)—31no—
159C2×C3×C7—21no—
160C2×C3×C7D3×D411yescharacteristic
161C2×C3×C7D3×D411yescharacteristic
162D21—41no—
163D21—121no—
164D21—121no—
165D21—121no—
166D21—121no—
167C3×D7—41no—
168C3×D7—41no—
169C3×D7—41no—
170C3×D7—41no—
171C3×D7—121no—
172D3×C7—61no—
173D3×C7—61no—
174D3×C7—61no—
175D3×C7—61no—
176C22×C32—141no—
177C2×C3⋊D3—141no—
178C2×C3⋊D3—282no—
179C2×C3⋊D3D2811yescharacteristic
180C2×C3×D3—141no—
181C32×C4C22×D711yescharacteristic
182C3⋊Dic3D2811yescharacteristic
183C3×Dic3—141no—
184AGL(1,ℤ/8)—631noSylow
185C22×C7—91no—
186C2×D7—21no—
187C2×D7—61no—
188C2×D7—181no—
189C2×D7—362no—
190C4×C7—91no—
191C4×C7C2×D3211yescharacteristic
192Dic7—61no—
193C22×D3—421no—
194C22×D3—421no—
195C22×D3—421no—
196C22×D3—421no—
197C3×Q8—211no—
198C3×D4—71no—
199C3×D4—71no—
200C3×D4—141no—
201C3×D4—211no—
202#67—421no—
203#67—421no—
204#67—421no—
205#67—421no—
206#67—421no—
207C2×Dic3—421no—
208D12—71no—
209D12—211no—
210D12—211no—
211D12—421no—
212D12—421no—
213C4×D3—31no—
214C4×D3—31no—
215C4×D3—61no—
216C4×D3—421no—
217Dic6—71no—
218C3×C8—31no—
219C3×C8—31no—
220C3⋊C8D3×D711yescharacteristic, complemented
221C3⋊C8D3×D711yescharacteristic, complemented
222C3×C7—21no—
223C3×C7#91111yescharacteristic, complemented
224C3×C7#91411yescharacteristic, complemented
225C2×C32—281no—
226C2×C32C2×D2811yescharacteristic
227C3⋊D3—21no—
228C3⋊D3—281no—
229C3×D3—281no—
230#41—631no—
231C2×D4—631no—
232SD16—211no—
233SD16—631no—
234D8—211no—
235D8—631no—
236M16—91no—
237C2×C7—181no—
238C2×C7#381111yescharacteristic
239D7—41no—
240D7—121no—
241D7—361no—
242C22×C3—141no—
243C22×C3—141no—
244C22×C3—141no—
245C22×C3—141no—
246C22×C3—421no—
247C2×D3—31no—
248C2×D3—31no—
249C2×D3—61no—
250C2×D3—141no—
251C2×D3—421no—
252C2×D3—421no—
253C2×D3—421no—
254C2×D3—421no—
255C2×D3—842no—
256C2×D3—842no—
257C2×D3—842no—
258C2×D3—842no—
259C3×C4—21no—
260C3×C4—421no—
261C3×C4C2×D3×D711yescharacteristic
262C3×C4C2×D3×D711yescharacteristic
263Dic3—31no—
264Dic3—31no—
265Dic3—61no—
266Dic3—141no—
267C32#647111yescharacteristic, complemented, Sylow
268C23—1261no—
269Q8—211no—
270D4—71no—
271D4—211no—
272D4—631no—
273D4—1261no—
274D4—1261no—
275C2×C4—91no—
276C2×C4—1261no—
277C8—31no—
278C8—31no—
279C7#6393711yescharacteristic, complemented, Sylow
280C2×C3—21no—
281C2×C3—281no—
282C2×C3—281no—
283C2×C3—281no—
284C2×C3—281no—
285C2×C3—841no—
286C2×C3D3×D2811yescharacteristic
287C2×C3D3×D2811yescharacteristic
288D3—61no—
289D3—61no—
290D3—61no—
291D3—61no—
292D3—281no—
293D3—841no—
294D3—841no—
295D3—841no—
296D3—841no—
297C22—91no—
298C22—141no—
299C22—421no—
300C22—1261no—
301C22—2522no—
302C4—91no—
303C4—421no—
304C4D32×D711yescharacteristic
305C3—21no—
306C3#1062081511yescharacteristic, complemented
307C3#1062081611yescharacteristic, complemented
308C2—181no—
309C2—281no—
310C2—841no—
311C2—2521no—
312C2#1175942811yescharacteristic
subgroupG/H
centerC2#11759428
commutator, Fitting, cosocleC32×C4×C7C23
Frattini, hypercenterC4D32×D7
socleC2×C32×C7C2×D4
hypocenterC32×C7AGL(1,ℤ/8)
LCSG ▷ C32×C4×C7 ▷ C2×C32×C7 ▷ C32×C7
UCSC1 ◁ C2 ◁ C4

Complements

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

NH ≅ G/Ntype
1. #11759628C2semidirect
2. C3⋊D3×D28C2semidirect
3. #11759200C2semidirect
4. #11759195C2semidirect
5. C7×#3712C2semidirect
6. C3×#66565C2semidirect
7. C3×#66561C2semidirect
13. C3×D84C22semidirect
14. C3×Dic42C22semidirect
19. C3×C7×C3⋊C8C22semidirect
20. C3×C7×C3⋊C8C22semidirect
33. #66565D3semidirect
35. #66561D3semidirect
53. D84C2×D3semidirect
54. Dic42C2×D3semidirect
73. C7×C3⋊C8C2×D3semidirect
74. C7×C3⋊C8C2×D3semidirect
81. #3712D7semidirect
130. C3×C3⋊C8C2×D7semidirect
131. C3×C3⋊C8C2×D7semidirect
132. C32×C7AGL(1,ℤ/8)semidirect
220. C3⋊C8D3×D7semidirect
221. C3⋊C8D3×D7semidirect
223. C3×C7#911semidirect
224. C3×C7#914semidirect
267. C32#6471semidirect
279. C7#63937semidirect
306. C3#10620815semidirect
307. C3#10620816semidirect

Maximal subgroups

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

subgroupindex
#117596282
C3⋊D3×D282
#117592002
#117591952
C7×#37122
C3×#665652
C3×#665612
#106208163
#106208153
#639377