Finite Group Databasebeta

← #72245885

Subgroups — #72245885

Order 2016.

326 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 59 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
1C7×D4×C3⋊D3C211yesmaximal, characteristic, complemented
2#11759628C211yesmaximal, characteristic, complemented
3C3×#66633C211yesmaximal, characteristic, complemented
4C3×#66632C211yesmaximal, characteristic, complemented
5#11759220C211yesmaximal, characteristic, complemented
6#11759217C211yesmaximal, characteristic, complemented
7#11759216C211yesmaximal, characteristic, complemented
8#10621058—31nomaximal
9#10621056—31nomaximal
10C22×C7×C3⋊D3—21no—
11C32×C7×D4C2211yescharacteristic
12C7×#638—21no—
13C7×C4⋊(C3⋊D3)C2211yescharacteristic
14C4×C7×C3⋊D3C2211yescharacteristic
15C3×D84C2211yescharacteristic, complemented
16C3×Dic42C2211yescharacteristic, complemented
17#92694—21no—
18D21⋊Dic3—21no—
19C3×C3⋊(C7⋊C8)C2211yescharacteristic, complemented
20C3×C3⋊(C7⋊C8)C2211yescharacteristic, complemented
21D3×C7×D4—31no—
22D3×C7×D4—31no—
23D3×C7×D4—61no—
24#66673—31no—
25#66669—31no—
26#66633D311yescharacteristic, complemented
27#66632D311yescharacteristic, complemented
28C3×#1142—31no—
29C3×#1141—31no—
30#66567—31no—
31#66564—31no—
32#66562—31no—
33#66560—31no—
34#66558—31no—
35#66558—31no—
36#63937—71nomaximal, Hall
37C22×C32×C7—21no—
38C2×C7×C3⋊D3—21no—
39C2×C7×C3⋊D3—42no—
40C2×C7×C3⋊D3D411yescharacteristic
41C2×C3×D21—21no—
42C32×C4×C7C2311yescharacteristic
43C7×C3⋊Dic3D411yescharacteristic
44C3×Dic21—21no—
45#6495—91noHall
46C22×D3×C7—61no—
47C22×D3×C7—61no—
48C22×D3×C7—61no—
49C22×D3×C7—61no—
50C3×C7×D4—21no—
51C3×C7×D4C2×D311yescharacteristic
52C3×C7×D4C2×D311yescharacteristic
53D84C2×D311yescharacteristic, complemented
54Dic42C2×D311yescharacteristic, complemented
55C7×#67—61no—
56C7×#67—61no—
57C7×#67—61no—
58C7×#67—61no—
59C7×D12—31no—
60C7×D12—31no—
61C7×D12—61no—
62C4×D3×C7—31no—
63C4×D3×C7—31no—
64C4×D3×C7—61no—
65C3×D28—31no—
66C3×Dic14—31no—
67#4247—61no—
68#4245—61no—
69D21⋊C4—61no—
70Dic3×D7—61no—
71C3⋊(C7⋊C8)C2×D311yescharacteristic, complemented
72C3⋊(C7⋊C8)C2×D311yescharacteristic, complemented
73C3×C7⋊C8—31no—
74C3×C7⋊C8—31no—
75D4×C3⋊D3D711yescharacteristic, complemented
76#3796—71no—
77C3×#214—71no—
78C3×#213—71no—
79#3717—71no—
80#3714—71no—
81#3712—71no—
82C2×C32×C7—41no—
83C2×C32×C7C2×D411yescharacteristic
84C7×C3⋊D3—21no—
85C7×C3⋊D3—41no—
86C3×D21—41no—
87C2×C7×D4—91no—
88#1157—91no—
89#1142—31no—
90#1142—91no—
91#1141—31no—
92#1141—91no—
93#1136—91no—
94#914—211no—
95#911—211no—
96C22×C3×C7—21no—
97C22×C3×C7—21no—
98C22×C3×C7—21no—
99C22×C3×C7—21no—
100C2×D21—21no—
101C2×D3×C7—31no—
102C2×D3×C7—31no—
103C2×D3×C7—61no—
104C2×D3×C7—61no—
105C2×D3×C7—61no—
106C2×D3×C7—61no—
107C2×D3×C7—61no—
108C2×D3×C7—122no—
109C2×D3×C7—122no—
110C2×D3×C7—122no—
111C2×D3×C7—122no—
112C2×C3×D7—61no—
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—21no—
122C32×D4C2×D711yescharacteristic
123#638—21no—
124C4⋊(C3⋊D3)C2×D711yescharacteristic
125C4×C3⋊D3C2×D711yescharacteristic
126C3×D12—71no—
127C3×Dic6—71no—
128#626—141no—
129D3×Dic3—141no—
130C3×C3⋊C8—71no—
131C3×C3⋊C8—71no—
132C32×C7AGL(1,ℤ/8)11yescharacteristic, complemented, Hall
133C23×C7—181no—
134C7×D4—91no—
135C7×D4—181no—
136C7×D4D3211yescharacteristic
137C2×C4×C7—91no—
138C22⋊D7—181no—
139D28—31no—
140C4×D7—181no—
141Dic14—31no—
142C7⋊C8—31no—
143C7⋊C8—31no—
144#239—211no—
145#237—211no—
146D3×D4—31no—
147D3×D4—31no—
148D3×D4—61no—
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)—211no—
158ΓL(1,𝔽25)—211no—
159C2×C3×C7—21no—
160C2×C3×C7—41no—
161C2×C3×C7—41no—
162C2×C3×C7—41no—
163C2×C3×C7—41no—
164C2×C3×C7D3×D411yescharacteristic
165C2×C3×C7D3×D411yescharacteristic
166D21—41no—
167C3×D7—121no—
168D3×C7—61no—
169D3×C7—61no—
170D3×C7—61no—
171D3×C7—61no—
172D3×C7—121no—
173D3×C7—121no—
174D3×C7—121no—
175D3×C7—121no—
176C22×C32—21no—
177C2×C3⋊D3—21no—
178C2×C3⋊D3—42no—
179C2×C3⋊D3C22⋊D711yescharacteristic
180C2×C3×D3—141no—
181C32×C4C22×D711yescharacteristic
182C3⋊Dic3C22⋊D711yescharacteristic
183C3×Dic3—141no—
184AGL(1,ℤ/8)—631noSylow
185C22×C7—21no—
186C22×C7—91no—
187C22×C7—181no—
188C22×C7—362no—
189C2×D7—61no—
190C4×C7—91no—
191C4×C7C2×D3211yescharacteristic
192Dic7—61no—
193C22×D3—61no—
194C22×D3—61no—
195C22×D3—61no—
196C22×D3—61no—
197C3×Q8—211no—
198C3×D4—21no—
199C3×D4—211no—
200C3×D4D3×D711yescharacteristic
201C3×D4D3×D711yescharacteristic
202#67—61no—
203#67—61no—
204#67—61no—
205#67—61no—
206#67—421no—
207C2×Dic3—421no—
208D12—31no—
209D12—31no—
210D12—61no—
211D12—71no—
212D12—421no—
213C4×D3—31no—
214C4×D3—31no—
215C4×D3—61no—
216C4×D3—421no—
217Dic6—71no—
218C3×C8—211no—
219C3×C8—211no—
220C3⋊C8—71no—
221C3⋊C8—71no—
222C3×C7—21no—
223C3×C7#91111yescharacteristic, complemented
224C3×C7#91411yescharacteristic, complemented
225C2×C32—41no—
226C2×C32C2×C22⋊D711yescharacteristic
227C3⋊D3—21no—
228C3⋊D3—41no—
229C3×D3—281no—
230#41—631no—
231C2×D4—91no—
232SD16—211no—
233SD16—631no—
234D8—211no—
235D8—631no—
236M16—631no—
237C2×C7—41no—
238C2×C7—181no—
239C2×C7—361no—
240C2×C7#381111yescharacteristic
241D7—121no—
242C22×C3—21no—
243C22×C3—21no—
244C22×C3—21no—
245C22×C3—21no—
246C22×C3—421no—
247C2×D3—31no—
248C2×D3—31no—
249C2×D3—61no—
250C2×D3—61no—
251C2×D3—61no—
252C2×D3—61no—
253C2×D3—61no—
254C2×D3—122no—
255C2×D3—122no—
256C2×D3—122no—
257C2×D3—122no—
258C2×D3—141no—
259C3×C4—21no—
260C3×C4—421no—
261C3×C4C2×D3×D711yescharacteristic
262C3×C4C2×D3×D711yescharacteristic
263Dic3—31no—
264Dic3—31no—
265Dic3—61no—
266Dic3—141no—
267C32#649511yescharacteristic, complemented, Sylow
268C23—181no—
269Q8—211no—
270D4—91no—
271D4—181no—
272D4—211no—
273D4—1261no—
274D4(C3⋊D3)⋊D711yescharacteristic
275C2×C4—91no—
276C2×C4—1261no—
277C8—211no—
278C8—211no—
279C7#6393711yescharacteristic, complemented, Sylow
280C2×C3—21no—
281C2×C3—41no—
282C2×C3—41no—
283C2×C3—41no—
284C2×C3—41no—
285C2×C3—841no—
286C2×C3D3×C22⋊D711yescharacteristic
287C2×C3D3×C22⋊D711yescharacteristic
288D3—61no—
289D3—61no—
290D3—61no—
291D3—61no—
292D3—121no—
293D3—121no—
294D3—121no—
295D3—121no—
296D3—281no—
297C22—21no—
298C22—91no—
299C22—181no—
300C22—362no—
301C22—421no—
302C4—91no—
303C4—421no—
304C4C2×(C3⋊D3)⋊D711yescharacteristic
305C3—21no—
306C3#1062105611yescharacteristic, complemented
307C3#1062105811yescharacteristic, complemented
308C2—41no—
309C2—181no—
310C2—361no—
311C2—841no—
312C2#1175964411yescharacteristic
subgroupG/H
centerC2#11759644
commutator, cosocleC32×C4×C7C23
FrattiniC4C2×(C3⋊D3)⋊D7
FittingC32×C7×D4C22
socleC2×C32×C7C2×D4
hypercenterD4(C3⋊D3)⋊D7
hypocenterC32×C7AGL(1,ℤ/8)
LCSG ▷ C32×C4×C7 ▷ C2×C32×C7 ▷ C32×C7
UCSC1 ◁ C2 ◁ C4 ◁ D4

Complements

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

NH ≅ G/Ntype
1. C7×D4×C3⋊D3C2semidirect
2. #11759628C2semidirect
3. C3×#66633C2semidirect
4. C3×#66632C2semidirect
5. #11759220C2semidirect
6. #11759217C2semidirect
7. #11759216C2semidirect
15. C3×D84C22semidirect
16. C3×Dic42C22semidirect
19. C3×C3⋊(C7⋊C8)C22semidirect
20. C3×C3⋊(C7⋊C8)C22semidirect
26. #66633D3semidirect
27. #66632D3semidirect
53. D84C2×D3semidirect
54. Dic42C2×D3semidirect
71. C3⋊(C7⋊C8)C2×D3semidirect
72. C3⋊(C7⋊C8)C2×D3semidirect
75. D4×C3⋊D3D7semidirect
132. C32×C7AGL(1,ℤ/8)semidirect
223. C3×C7#911semidirect
224. C3×C7#914semidirect
267. C32#6495semidirect
279. C7#63937semidirect
306. C3#10621056semidirect
307. C3#10621058semidirect

Maximal subgroups

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

subgroupindex
C7×D4×C3⋊D32
#117596282
C3×#666332
C3×#666322
#117592202
#117592172
#117592162
#106210583
#106210563
#639377