Finite Group Databasebeta

← #72245869

Subgroups — #72245869

Order 2016.

326 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 60 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
1D4×C3⋊D21C211yesmaximal, characteristic, complemented
2#11759604C211yesmaximal, characteristic, complemented
3C3×#66633C211yesmaximal, characteristic, complemented
4C3×C7×#213C211yesmaximal, characteristic, complemented
5#11759200C211yesmaximal, characteristic, complemented
6#11759196C211yesmaximal, characteristic, complemented
7#11759193C211yesmaximal, characteristic, complemented
8#10621394—31nomaximal
9#10621062—31nomaximal
10C22×C3⋊D21—21no—
11C32×C7×D4C2211yescharacteristic
12#92726—21no—
13C4⋊(C3⋊D21)C2211yescharacteristic
14C4×C3⋊D21C2211yescharacteristic
15C3×Dic42C2211yescharacteristic, complemented
16C3×C7×D12C2211yescharacteristic, complemented
17#92690—21no—
18D3×Dic21—21no—
19C3×C3⋊(C7⋊C8)C2211yescharacteristic, complemented
20C3×C7×C3⋊C8C2211yescharacteristic, complemented
21#66730—31no—
22D4×D21—31no—
23D4×D21—31no—
24D4×D21—61no—
25#66672—31no—
26C3×C7×D8—31no—
27#66633D311yescharacteristic, complemented
28#66632—31no—
29#66623—31no—
30#66622—31no—
31C7×#213D311yescharacteristic, complemented
32C3×#1142—31no—
33#66566—31no—
34#66562—31no—
35#66559—31no—
36#63937—71nomaximal, Hall
37C22×C32×C7—21no—
38C2×C3⋊D21—21no—
39C2×C3⋊D21—42no—
40C2×C3⋊D21D411yescharacteristic
41C2×C3×D3×C7—21no—
42C32×C4×C7C2311yescharacteristic
43C3⋊Dic21D411yescharacteristic
44C3×Dic21—21no—
45#6474—91noHall
46C22×D21—61no—
47C22×D21—61no—
48C22×D21—61no—
49C22×D21—61no—
50C3×C7×D4—21no—
51C3×C7×D4—31no—
52C3×C7×D4C2×D311yescharacteristic
53C3×C7×D4C2×D311yescharacteristic
54#4268—61no—
55#4268—61no—
56#4268—61no—
57#4268—61no—
58#4268—61no—
59C2×Dic21—61no—
60D84—31no—
61D84—31no—
62D84—61no—
63C4×D21—31no—
64C4×D21—31no—
65C4×D21—61no—
66Dic42C2×D311yescharacteristic, complemented
67C7×D12C2×D311yescharacteristic, complemented
68C3×Dic14—31no—
69#4247—61no—
70D3×Dic7—61no—
71C3×C7×C8—31no—
72C3⋊(C7⋊C8)C2×D311yescharacteristic, complemented
73C3×C7⋊C8—31no—
74C7×C3⋊C8C2×D311yescharacteristic, complemented
75D4×C3⋊D3—71no—
76#3796—71no—
77C3×#214—71no—
78C3×#213D711yescharacteristic, complemented
79#3717—71no—
80#3714—71no—
81#3712—71no—
82C2×C32×C7—41no—
83C2×C32×C7C2×D411yescharacteristic
84C3⋊D21—21no—
85C3⋊D21—41no—
86C3×D3×C7—41no—
87#1159—91no—
88D4×D7—91no—
89C7×D8—31no—
90#1142—31no—
91#1141—91no—
92#1132—91no—
93#1131—91no—
94#914—211no—
95#911—211no—
96C22×C3×C7—21no—
97C22×C3×C7—21no—
98C22×C3×C7—21no—
99C22×C3×C7—21no—
100C22×C3×C7—61no—
101C2×D21—31no—
102C2×D21—31no—
103C2×D21—61no—
104C2×D21—61no—
105C2×D21—61no—
106C2×D21—61no—
107C2×D21—61no—
108C2×D21—122no—
109C2×D21—122no—
110C2×D21—122no—
111C2×D21—122no—
112C2×D3×C7—21no—
113C3×C4×C7—21no—
114C3×C4×C7C22×D311yescharacteristic
115C3×C4×C7C22×D311yescharacteristic
116Dic21—21no—
117Dic21—31no—
118Dic21—31no—
119Dic21—61no—
120C3×Dic7—61no—
121C22×C3⋊D3—141no—
122C32×D4C2×D711yescharacteristic
123#638—141no—
124C4⋊(C3⋊D3)—71no—
125C4×C3⋊D3—71no—
126C3×D12C2×D711yescharacteristic, complemented
127C3×Dic6—71no—
128#626—141no—
129D3×Dic3—141no—
130C3×C3⋊C8—71no—
131C3×C3⋊C8C2×D711yescharacteristic, complemented
132C32×C7AGL(1,ℤ/8)11yescharacteristic, complemented, Hall
133C22×D7—181no—
134C7×D4—31no—
135C7×D4D3211yescharacteristic
136C22⋊D7—181no—
137C22⋊D7—181no—
138C2×Dic7—181no—
139D28—91no—
140C4×D7—91no—
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—31no—
151#215—211no—
152#214—71no—
153#213—211no—
154#213D2111yescharacteristic, complemented
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×C7—121no—
165C2×C3×C7D3×D411yescharacteristic
166C2×C3×C7D3×D411yescharacteristic
167D21—61no—
168D21—61no—
169D21—61no—
170D21—61no—
171D21—121no—
172D21—121no—
173D21—121no—
174D21—121no—
175D3×C7—41no—
176C22×C32—21no—
177C2×C3⋊D3—71no—
178C2×C3⋊D3—141no—
179C2×C3⋊D3—282no—
180C2×C3×D3—21no—
181C32×C4C22×D711yescharacteristic
182C3⋊Dic3—71no—
183C3×Dic3—141no—
184AGL(1,ℤ/8)—631noSylow
185C22×C7—21no—
186C22×C7—61no—
187C2×D7—91no—
188C2×D7—181no—
189C2×D7—362no—
190C4×C7C2×D3211yescharacteristic
191Dic7—61no—
192Dic7—91no—
193C22×D3—421no—
194C22×D3—421no—
195C22×D3—421no—
196C22×D3—421no—
197C3×Q8—211no—
198C3×D4—21no—
199C3×D4—31no—
200C3×D4D3×D711yescharacteristic
201C3×D4C2×D2111yescharacteristic
202#67—421no—
203#67—421no—
204#67—421no—
205#67—421no—
206#67—421no—
207C2×Dic3—421no—
208D12—211no—
209D12—211no—
210D12—421no—
211D12—421no—
212D12C2×D2111yescharacteristic, complemented
213C4×D3—211no—
214C4×D3—211no—
215C4×D3—421no—
216C4×D3—421no—
217Dic6—71no—
218C3×C8—31no—
219C3×C8—211no—
220C3⋊C8—71no—
221C3⋊C8C2×D2111yescharacteristic, complemented
222C3×C7—21no—
223C3×C7#91111yescharacteristic, complemented
224C3×C7#91411yescharacteristic, complemented
225C2×C32—41no—
226C2×C32D4×D711yescharacteristic
227C3⋊D3—141no—
228C3⋊D3—281no—
229C3×D3—41no—
230#41—631no—
231C2×D4—631no—
232SD16—211no—
233SD16—631no—
234D8—31no—
235D8—631no—
236M16—631no—
237C2×C7—41no—
238C2×C7—121no—
239C2×C7#381111yescharacteristic
240D7—181no—
241D7—361no—
242C22×C3—21no—
243C22×C3—21no—
244C22×C3—21no—
245C22×C3—21no—
246C22×C3—61no—
247C2×D3—21no—
248C2×D3—211no—
249C2×D3—211no—
250C2×D3—421no—
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×C4C22×D2111yescharacteristic
263Dic3—141no—
264Dic3—211no—
265Dic3—211no—
266Dic3—421no—
267C32#647411yescharacteristic, complemented, Sylow
268C23—1261no—
269Q8—211no—
270D4—31no—
271D4—631no—
272D4—1261no—
273D4—1261no—
274D4D3×D2111yescharacteristic
275C2×C4—631no—
276C2×C4—1261no—
277C8—31no—
278C8—211no—
279C7#6393711yescharacteristic, complemented, Sylow
280C2×C3—21no—
281C2×C3—41no—
282C2×C3—41no—
283C2×C3—41no—
284C2×C3—41no—
285C2×C3—121no—
286C2×C3#6669411yescharacteristic
287C2×C3D4×D2111yescharacteristic
288D3—41no—
289D3—421no—
290D3—421no—
291D3—421no—
292D3—421no—
293D3—841no—
294D3—841no—
295D3—841no—
296D3—841no—
297C22—21no—
298C22—61no—
299C22—631no—
300C22—1261no—
301C22—2522no—
302C4—421no—
303C4—631no—
304C4C2×D3×D2111yescharacteristic
305C3—21no—
306C3#1062106211yescharacteristic, complemented
307C3#1062139411yescharacteristic, complemented
308C2—41no—
309C2—121no—
310C2—1261no—
311C2—2521no—
312C2#1175962611yescharacteristic
subgroupG/H
centerC2#11759626
commutator, cosocleC32×C4×C7C23
FrattiniC4C2×D3×D21
FittingC32×C7×D4C22
socleC2×C32×C7C2×D4
hypercenterD4D3×D21
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. D4×C3⋊D21C2semidirect
2. #11759604C2semidirect
3. C3×#66633C2semidirect
4. C3×C7×#213C2semidirect
5. #11759200C2semidirect
6. #11759196C2semidirect
7. #11759193C2semidirect
15. C3×Dic42C22semidirect
16. C3×C7×D12C22semidirect
19. C3×C3⋊(C7⋊C8)C22semidirect
20. C3×C7×C3⋊C8C22semidirect
27. #66633D3semidirect
31. C7×#213D3semidirect
66. Dic42C2×D3semidirect
67. C7×D12C2×D3semidirect
72. C3⋊(C7⋊C8)C2×D3semidirect
74. C7×C3⋊C8C2×D3semidirect
78. C3×#213D7semidirect
126. C3×D12C2×D7semidirect
131. C3×C3⋊C8C2×D7semidirect
132. C32×C7AGL(1,ℤ/8)semidirect
154. #213D21semidirect
212. D12C2×D21semidirect
221. C3⋊C8C2×D21semidirect
223. C3×C7#911semidirect
224. C3×C7#914semidirect
267. C32#6474semidirect
279. C7#63937semidirect
306. C3#10621062semidirect
307. C3#10621394semidirect

Maximal subgroups

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

subgroupindex
D4×C3⋊D212
#117596042
C3×#666332
C3×C7×#2132
#117592002
#117591962
#117591932
#106213943
#106210623
#639377