Finite Group Databasebeta

← #72246816

Subgroups — #72246816

Order 2016.

662 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 662 conjugacy classes of subgroups — too many to draw. Only its 106 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
1C23×C3⋊D21C211yesmaximal, characteristic
2C2×#92687C212yesmaximal, complemented
3C3×#66636C211yesmaximal, characteristic, complemented
4C3×C7×#217C211yesmaximal, characteristic, complemented
5#11759209C212yesmaximal, complemented
6C22⋊C4×D21—31nomaximal
7#10621123—31nomaximal
8C23×C32×C7C2211yescharacteristic
9C22×C3⋊D21—24no—
10C22×C3⋊D21C414yescomplemented
11C22×C3⋊D21C2211yescharacteristic
12C22×C3⋊D21C2211yescharacteristic
13C2×C3×Dic21C2212yescomplemented
14C2×C3×C7×Dic3C2212yescomplemented
15#92687—24no—
16C23×D21—31no—
17C23×D21—31no—
18C23×D21—61no—
19C2×C4×D21—32no—
20C2×D21⋊C4—32no—
21C3×C7×C22⋊C4—31no—
22#66636D311yescharacteristic, complemented
23#66631—32no—
24C7×#217D311yescharacteristic, complemented
25C3×C22⋊Dic7—31no—
26#66575—32no—
27#63974—71nomaximal, Hall
28C22×C32×C7—22no—
29C22×C32×C7C2×C412yescomplemented
30C22×C32×C7C2311yescharacteristic
31C2×C3⋊D21—22no—
32C2×C3⋊D21—28no—
33C2×C3⋊D21C2×C412yes—
34C2×C3⋊D21C2×C412yescomplemented
35C2×C3⋊D21D414yes—
36C3×Dic21—22no—
37C3×C7×Dic3—22no—
38D7×C22⋊C4—91noHall
39C23×C3×C7—21no—
40C23×C3×C7C2×D311yescharacteristic
41C23×C3×C7C2×D311yescharacteristic
42C22×D21—31no—
43C22×D21—31no—
44C22×D21—34no—
45C22×D21—34no—
46C22×D21—31no—
47C22×D21—31no—
48C22×D21—61no—
49C22×D21—64no—
50C22×D21—61no—
51C22×D21—64no—
52C22×D21—64no—
53C22×D21—64no—
54C22×D21—64no—
55C2×C3×C4×C7—32no—
56C2×Dic21C2×D312yescomplemented
57C4×D21—64no—
58C2×C7×Dic3C2×D312yescomplemented
59C2×C3×Dic7—32no—
60D21⋊C4—64no—
61C23×C3⋊D3—71no—
62C2×#624—72no—
63C3×#217—71no—
64C3×#217D711yescharacteristic, complemented
65#3722—72no—
66C2×C32×C7—22no—
67C2×C32×C7C22×C411yescharacteristic
68C2×C32×C7C2×D412yes—
69C3⋊D21—22no—
70C3⋊D21C22⋊C414yescomplemented
71C23×D7—91no—
72C2×C4×D7—92no—
73C7×C22⋊C4—31no—
74C22⋊Dic7—31no—
75(C2×D7)⋊C4—92no—
76D3×C22⋊C4—211no—
77D3×C22⋊C4—211no—
78C22×C3×C7—22no—
79C22×C3×C7—22no—
80C22×C3×C7—22no—
81C22×C3×C7—22no—
82C22×C3×C7—22no—
83C22×C3×C7—21no—
84C22×C3×C7C4×D312yescomplemented
85C22×C3×C7C4×D312yescomplemented
86C22×C3×C7C22×D311yescharacteristic
87C22×C3×C7C22×D311yescharacteristic
88C2×D21—32no—
89C2×D21—32no—
90C2×D21—34no—
91C2×D21—34no—
92C2×D21—32no—
93C2×D21—32no—
94C2×D21—62no—
95C2×D21—64no—
96C2×D21—62no—
97C2×D21—68no—
98C2×D21—68no—
99C2×D21—68no—
100C2×D21—62no—
101C2×D21—68no—
102C2×D21—62no—
103C2×D21—62no—
104C2×D21—62no—
105C3×C4×C7—62no—
106Dic21—22no—
107C3×Dic7—62no—
108C7×Dic3—22no—
109C23×C32C2×D711yescharacteristic
110C22×C3⋊D3—71no—
111C22×C3⋊D3—74no—
112C22×C3⋊D3—71no—
113C22×C3⋊D3—144no—
114C2×C3×Dic3—72no—
115C2×C3×Dic3C2×D712yescomplemented
116#624—144no—
117C32×C7C2×C22⋊C411yescharacteristic, complemented, Hall
118C23×C7D3211yescharacteristic
119C22×D7—91no—
120C22×D7—94no—
121C22×D7—91no—
122C22×D7—184no—
123C2×C4×C7—32no—
124C2×Dic7—32no—
125C4×D7—184no—
126C23×D3—211no—
127C23×D3—211no—
128C23×D3—421no—
129C2×C4×D3—212no—
130C2×C4×D3—212no—
131C3×C22⋊C4—31no—
132C3×C22⋊C4—211no—
133#217—71no—
134#217D2111yescharacteristic, complemented
135(C2×D3)⋊C4—212no—
136(C2×D3)⋊C4—212no—
137C2×C3×C7—22no—
138C2×C3×C7—22no—
139C2×C3×C7—22no—
140C2×C3×C7—22no—
141C2×C3×C7—21no—
142C2×C3×C7—22no—
143C2×C3×C7C2×C4×D311yescharacteristic
144C2×C3×C7C2×C4×D311yescharacteristic
145C2×C3×C7D3×D412yes—
146C2×C3×C7D3×D412yes—
147D21—34no—
148D21—34no—
149D21—64no—
150D21—62no—
151D21—62no—
152D21—62no—
153D21—62no—
154C22×C32—22no—
155C22×C32C4×D712yescomplemented
156C22×C32C22×D711yescharacteristic
157C2×C3⋊D3—72no—
158C2×C3⋊D3—74no—
159C2×C3⋊D3—72no—
160C2×C3⋊D3—148no—
161C2×C3⋊D3—142no—
162C3×Dic3—22no—
163C3×Dic3—142no—
164C2×C22⋊C4—631noSylow
165C22×C7—22no—
166C22×C7#62412yescomplemented
167C22×C7C2×D3211yescharacteristic
168C2×D7—92no—
169C2×D7—94no—
170C2×D7—92no—
171C2×D7—188no—
172C2×D7—182no—
173C4×C7—62no—
174Dic7—62no—
175C23×C3—21no—
176C23×C3D3×D711yescharacteristic
177C23×C3C2×D2111yescharacteristic
178C22×D3—211no—
179C22×D3—211no—
180C22×D3—214no—
181C22×D3—214no—
182C22×D3—211no—
183C22×D3—211no—
184C22×D3—421no—
185C22×D3—424no—
186C22×D3—421no—
187C22×D3—424no—
188C22×D3—424no—
189C22×D3—424no—
190C22×D3—424no—
191C2×C3×C4—32no—
192C2×C3×C4—212no—
193C2×Dic3—72no—
194C2×Dic3C2×D2112yescomplemented
195C4×D3—424no—
196C4×D3—424no—
197C3×C7—21no—
198C3×C7D3×C22⋊C411yescharacteristic, complemented
199C3×C7D3×C22⋊C411yescharacteristic, complemented
200C2×C32—22no—
201C2×C32C2×C4×D711yescharacteristic
202C2×C32D4×D712yes—
203C3⋊D3—74no—
204C3⋊D3—142no—
205C24—631no—
206C22×C4—632no—
207C22⋊C4—31no—
208C22⋊C4—211no—
209C22⋊C4—632no—
210C2×C7—22no—
211C2×C7C2×#62411yescharacteristic
212C2×C7#381112yes—
213D7—94no—
214D7—182no—
215C22×C3—22no—
216C22×C3—22no—
217C22×C3—22no—
218C22×C3—22no—
219C22×C3—22no—
220C22×C3—21no—
221C22×C3D21⋊C412yescomplemented
222C22×C3C4×D2112yescomplemented
223C22×C3C2×D3×D711yescharacteristic
224C22×C3C22×D2111yescharacteristic
225C2×D3—212no—
226C2×D3—214no—
227C2×D3—212no—
228C2×D3—214no—
229C2×D3—212no—
230C2×D3—212no—
231C2×D3—422no—
232C2×D3—424no—
233C2×D3—422no—
234C2×D3—428no—
235C2×D3—428no—
236C2×D3—428no—
237C2×D3—422no—
238C2×D3—428no—
239C2×D3—422no—
240C2×D3—422no—
241C2×D3—422no—
242C3×C4—62no—
243C3×C4—422no—
244Dic3—22no—
245Dic3—142no—
246C32D7×C22⋊C411yescharacteristic, complemented, Sylow
247C23—631no—
248C23—634no—
249C23—631no—
250C23—1264no—
251C23D3×D2111yescharacteristic
252C2×C4—32no—
253C2×C4—212no—
254C2×C4—1264no—
255C7#6397411yescharacteristic, complemented, Sylow
256C2×C3—22no—
257C2×C3—22no—
258C2×C3—22no—
259C2×C3—22no—
260C2×C3—21no—
261C2×C3—22no—
262C2×C3C2×D21⋊C411yescharacteristic
263C2×C3#6669412yes—
264C2×C3C2×C4×D2111yescharacteristic
265C2×C3D4×D2112yes—
266D3—214no—
267D3—214no—
268D3—424no—
269D3—422no—
270D3—422no—
271D3—422no—
272D3—422no—
273C22—22no—
274C22—632no—
275C22—634no—
276C22—632no—
277C22—1268no—
278C22—1262no—
279C22#9268712yescomplemented
280C22C2×D3×D2111yescharacteristic
281C4—62no—
282C4—422no—
283C3—21no—
284C3#1062112311yescharacteristic, complemented
285C3C22⋊C4×D2111yescharacteristic, complemented
286C2—22no—
287C2—634no—
288C2—1262no—
289C2C2×#9268711yescharacteristic
290C2#1175962612yes—
subgroupG/H
center, FrattiniC22C2×D3×D21
commutatorC2×C32×C7C22×C4
FittingC23×C32×C7C22
socle, cosocleC22×C32×C7C23
prime radicalC23×C3⋊D21C2
hypercenterC23D3×D21
hypocenterC32×C7C2×C22⋊C4
LCSG ▷ C2×C32×C7 ▷ C32×C7
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
2. C2×#92687C2semidirect
3. C3×#66636C2semidirect
4. C3×C7×#217C2semidirect
5. #11759209C2semidirect
10. C22×C3⋊D21C4semidirect
13. C2×C3×Dic21C22semidirect
14. C2×C3×C7×Dic3C22semidirect
22. #66636D3semidirect
24. C7×#217D3semidirect
29. C22×C32×C7C2×C4semidirect
34. C2×C3⋊D21C2×C4semidirect
56. C2×Dic21C2×D3semidirect
58. C2×C7×Dic3C2×D3semidirect
64. C3×#217D7semidirect
70. C3⋊D21C22⋊C4semidirect
84. C22×C3×C7C4×D3semidirect
85. C22×C3×C7C4×D3semidirect
115. C2×C3×Dic3C2×D7semidirect
117. C32×C7C2×C22⋊C4semidirect
134. #217D21semidirect
155. C22×C32C4×D7semidirect
166. C22×C7#624semidirect
194. C2×Dic3C2×D21semidirect
198. C3×C7D3×C22⋊C4semidirect
199. C3×C7D3×C22⋊C4semidirect
221. C22×C3D21⋊C4semidirect
222. C22×C3C4×D21semidirect
246. C32D7×C22⋊C4semidirect
255. C7#63974semidirect
279. C22#92687semidirect
284. C3#10621123semidirect
285. C3C22⋊C4×D21semidirect

Maximal subgroups

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

subgroupindex
C23×C3⋊D212
C2×#926872
C3×#666362
C3×C7×#2172
#117592092
C22⋊C4×D213
#106211233
#639747