Finite Group Databasebeta

← #16855317

Subgroups — #16855317

Order 3968.

500 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 141 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
1C2×#11758527C211yesmaximal, characteristic, complemented
2C22×#92474C211yesmaximal, characteristic, complemented
3#14521560C211yesmaximal, characteristic, complemented
4#14521305C212yesmaximal, complemented
5#14521300C212yesmaximal, complemented
6#14521297C212yesmaximal, complemented
7#14521293C212yesmaximal, complemented
8#14521085C211yesmaximal, characteristic, complemented
9#14521045C211yesmaximal, characteristic, complemented
10#14521043C211yesmaximal, characteristic, complemented
11C31×#525C211yesmaximal, characteristic, complemented
12C2×#41×C31C2211yescharacteristic
13C23×C4×C31C2211yescharacteristic
14C2×#92495C2211yescharacteristic
15C31×#123—21no—
16C31×C22⋊Q8C2212yes—
17C31×#121C2212yes—
18C31×#120—21no—
19C4×D4×C31—22no—
20C31×#117C2211yescharacteristic
21#11758530—21no—
22#11758527—24no—
23C2×#92482C2211yescharacteristic, complemented
24C2×#92481C2211yescharacteristic, complemented
25C2×#92480C2211yescharacteristic, complemented
26C2×#92479C2211yescharacteristic, complemented
27#11758507C2212yes—
28#11758506—21no—
29C4×C22⋊D31—22no—
30#11758503C2211yescharacteristic
31#11758501C2212yes—
32C2×#92474—24no—
33C2×#92474C2212yes—
34C2×#92474C2212yes—
35C22×C31⋊C8C2211yescharacteristic
36C22×C31⋊C8C2211yescharacteristic
37#11758423C2212yescomplemented
38#11758420C2212yescomplemented
39#11758398C2212yescomplemented
40#11758397C2212yescomplemented
41#11758396C2212yescomplemented
42#11758395C2212yescomplemented
43C24×C31D411yescharacteristic
44#41×C31—22no—
45#41×C31—22no—
46C2×Q8×C31C2311yescharacteristic
47C2×D4×C31—22no—
48C2×D4×C31—21no—
49C2×D4×C31C2311yescharacteristic
50C22×C4×C31—22no—
51C22×C4×C31—22no—
52C22×C4×C31—21no—
53C22×C4×C31D412yes—
54C22×C4×C31D411yescharacteristic
55C22×C4×C31C2312yes—
56C22×C4×C31C2311yescharacteristic
57C2×C22⋊D31—22no—
58C2×C22⋊D31—21no—
59#92495—22no—
60#92495—22no—
61C2×D124C2311yescharacteristic
62C2×C4×D31—21no—
63C2×Dic62C2311yescharacteristic
64C4⋊C4×C31—22no—
65C4⋊C4×C31C2312yes—
66C22⋊C4×C31—21no—
67C22⋊C4×C31—22no—
68C22⋊C4×C31—22no—
69C42×C31—21no—
70C22⋊Dic31—22no—
71C22⋊Dic31—21no—
72#92482—22no—
73#92481—22no—
74#92480—22no—
75#92479—22no—
76(C2×D31)⋊C4—22no—
77C4⋊Dic31C2312yes—
78Dic31⋊C4—22no—
79C4×Dic31—21no—
80#92474—24no—
81#92474—24no—
82C2×C31⋊C8—22no—
83C2×C31⋊C8D414yescomplemented
84C2×C31⋊C8C2312yes—
85C2×C31⋊C8C2311yescharacteristic
86C2×C31⋊C8C2311yescharacteristic
87C23×C31—22no—
88C23×C31—21no—
89C23×C31—42no—
90C23×C31C2×D412yes—
91C23×C31C2×D411yescharacteristic
92C22×D31—21no—
93Q8×C31—21no—
94Q8×C31—21no—
95D4×C31—21no—
96D4×C31—21no—
97D4×C31—44no—
98D4×C31—42no—
99C2×C4×C31—22no—
100C2×C4×C31—24no—
101C2×C4×C31—22no—
102C2×C4×C31—22no—
103C2×C4×C31—22no—
104C2×C4×C31—21no—
105C2×C4×C31—42no—
106C2×C4×C31C2×D412yes—
107C2×C4×C31C2×D412yes—
108C2×C4×C31C2×D411yescharacteristic
109C2×C4×C31#4112yes—
110C2×C4×C31C2411yescharacteristic
111C22⋊D31—44no—
112C22⋊D31—42no—
113C2×Dic31—22no—
114C2×Dic31—21no—
115D124—21no—
116D124—21no—
117C4×D31—42no—
118Dic62—21no—
119Dic62—21no—
120C31⋊C8—22no—
121C31⋊C8C2×D412yes—
122C31⋊C8C2×D412yes—
123#3054—311nomaximal, Sylow
124C22×C31—22no—
125C22×C31—22no—
126C22×C31—22no—
127C22×C31—22no—
128C22×C31—44no—
129C22×C31—41no—
130C22×C31—42no—
131C22×C31#12112yes—
132C22×C31C22×D411yescharacteristic
133C2×D31—41no—
134C2×D31—42no—
135C4×C31—22no—
136C4×C31—41no—
137C4×C31—42no—
138C4×C31#12112yes—
139C4×C31C22×D411yescharacteristic
140C4×C31C2×#4111yescharacteristic
141Dic31—41no—
142Dic31—42no—
143C2×#135—311no—
144C22×M16—311no—
145#525—311no—
146#525D3111yescharacteristic, complemented
147#470—312no—
148#469—312no—
149#468—312no—
150#468—312no—
151#427—311no—
152#416—311no—
153#416—311no—
154C2×C31—22no—
155C2×C31—42no—
156C2×C31—81no—
157C2×C31C2×#12111yescharacteristic
158C2×C31#57512yes—
159D31—81no—
160C2×#41—311no—
161C2×#41C2×D3111yescharacteristic
162C23×C4C2×D3111yescharacteristic
163#135—624no—
164C2×Q16—311no—
165C2×SD16—311no—
166C2×SD16—311no—
167C2×D8—311no—
168C2×M16—312no—
169C2×M16—312no—
170C2×M16—624no—
171C22×C8—311no—
172C22×C8—311no—
173#123—21no—
174#123—621no—
175C22⋊Q8—312no—
176C22⋊Q8C2×D3112yes—
177#121—312no—
178#121C2×D3112yes—
179#120—21no—
180#120—621no—
181C4×D4—22no—
182C4×D4—622no—
183#117—311no—
184#117C2×D3111yescharacteristic
185#107—312no—
186#106—312no—
187#103—312no—
188#103—312no—
189#102—312no—
190#102—312no—
191C31#305411yescharacteristic, complemented, Sylow
192C24C22⋊D3111yescharacteristic
193#41—22no—
194#41—22no—
195#41—622no—
196#41—622no—
197C2×Q8—311no—
198C2×Q8C22×D3111yescharacteristic
199C2×D4—22no—
200C2×D4—21no—
201C2×D4—311no—
202C2×D4—622no—
203C2×D4—621no—
204C2×D4C22×D3111yescharacteristic
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—21no—
208C22×C4—621no—
209C22×C4C22⋊D3111yescharacteristic
210C22×C4C22⋊D3112yes—
211C22×C4C22×D3112yes—
212C22×C4C22×D3111yescharacteristic
213Q16—622no—
214SD16—622no—
215SD16—622no—
216D8—622no—
217M16—624no—
218M16—624no—
219C2×C8—314no—
220C2×C8—311no—
221C2×C8—311no—
222C2×C8—312no—
223C2×C8—622no—
224C4⋊C4—22no—
225C4⋊C4—312no—
226C4⋊C4—622no—
227C4⋊C4C22×D3112yes—
228C22⋊C4—21no—
229C22⋊C4—22no—
230C22⋊C4—22no—
231C22⋊C4—621no—
232C22⋊C4—622no—
233C22⋊C4—622no—
234C42—21no—
235C42—621no—
236C23—22no—
237C23—21no—
238C23—42no—
239C23—621no—
240C23C2×C22⋊D3112yes—
241C23C2×C22⋊D3111yescharacteristic
242Q8—21no—
243Q8—21no—
244Q8—621no—
245Q8—621no—
246D4—21no—
247D4—21no—
248D4—44no—
249D4—42no—
250D4—621no—
251D4—621no—
252D4—1244no—
253D4—1242no—
254C2×C4—22no—
255C2×C4—24no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—42no—
261C2×C4—622no—
262C2×C4—621no—
263C2×C4—1242no—
264C2×C4D4×D3112yes—
265C2×C4#9249712yes—
266C2×C4C2×C22⋊D3112yes—
267C2×C4C2×C22⋊D3111yescharacteristic
268C2×C4C23×D3111yescharacteristic
269C8—312no—
270C8—312no—
271C8—622no—
272C22—22no—
273C22—22no—
274C22—22no—
275C22—22no—
276C22—44no—
277C22—41no—
278C22—42no—
279C22—1241no—
280C22—1242no—
281C22#1175851612yes—
282C22C22×C22⋊D3111yescharacteristic
283C4—22no—
284C4—41no—
285C4—42no—
286C4—1241no—
287C4—1242no—
288C4#1175851612yes—
289C4C2×D4×D3111yescharacteristic
290C4C2×#9249711yescharacteristic
291C2—22no—
292C2—42no—
293C2—81no—
294C2—2481no—
295C2C2×#1175851611yescharacteristic
296C2#1452184712yes—
subgroupG/H
centerC2×C4C2×C22⋊D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#525C2
socleC22×C31C22×D4
hypercenter#525D31
hypocenterC31#3054
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. C2×#11758527C2semidirect
2. C22×#92474C2semidirect
3. #14521560C2semidirect
4. #14521305C2semidirect
5. #14521300C2semidirect
6. #14521297C2semidirect
7. #14521293C2semidirect
8. #14521085C2semidirect
9. #14521045C2semidirect
10. #14521043C2semidirect
11. C31×#525C2semidirect
23. C2×#92482C22semidirect
24. C2×#92481C22semidirect
25. C2×#92480C22semidirect
26. C2×#92479C22semidirect
37. #11758423C22semidirect
38. #11758420C22semidirect
39. #11758398C22semidirect
40. #11758397C22semidirect
41. #11758396C22semidirect
42. #11758395C22semidirect
83. C2×C31⋊C8D4semidirect
146. #525D31semidirect
191. C31#3054semidirect

Maximal subgroups

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

subgroupindex
C2×#117585272
C22×#924742
#145215602
#145213052
#145213002
#145212972
#145212932
#145210852
#145210452
#145210432
C31×#5252
#305431