Finite Group Databasebeta

← #30608356

Subgroups — #30608356

Order 15616.

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

normal subgroups only

This group has 474 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
1C2×#16840667C212yesmaximal, complemented
2#21460953C212yesmaximal, complemented
3#21460951C212yesmaximal, complemented
4#21456271C211yesmaximal, characteristic
5C2×#14519836—22no—
6#16840667—22no—
7#16840667C2214yes—
8C2×#14519805—22no—
9#16840312C411yescharacteristic, complemented
10C2×#14519774C2211yescharacteristic
11#16840151—22no—
12#16840139C411yescharacteristic, complemented
13#16840139C2211yescharacteristic
14#16840135C411yescharacteristic, complemented
15#16840135C2211yescharacteristic
16C4×M16×C61C411yescharacteristic, complemented
17C2×C22⋊#6923—42no—
18#14519836—24no—
19#14519836—42no—
20C2×#11758222—21no—
21C22×D244—41no—
22C22×D244D412yes—
23#14519805—44no—
24#14519805—44no—
25C2×#11758190—22no—
26C2×#11758190—22no—
27C2×D488—22no—
28C2×D488—22no—
29#14519777C2×C411yescharacteristic, complemented
30#14519774—41no—
31#14519774C2×C411yescharacteristic
32#14519774C2×C411yescharacteristic, complemented
33#14519774D412yes—
34#14519774C2311yescharacteristic
35#14519765—21no—
36#14519751—41no—
37C4×D244—41no—
38C2×M16×C61—21no—
39C2×C42×C61C2×C411yescharacteristic
40#97×C61—21no—
41C4×C8×C61C2×C411yescharacteristic, complemented
42C4×C8×C61C2×C411yescharacteristic, complemented
43C23×D61—22no—
44C22×#6923—82no—
45C22×C4×C61—21no—
46C22×C4×C61C22⋊C411yescharacteristic
47C2×C22⋊D61—41no—
48#11758222—42no—
49#11758222—42no—
50C2×D244—21no—
51C2×D244—22no—
52C2×D244—42no—
53C2×D244—42no—
54C2×D244—41no—
55C2×D244—42no—
56C2×D244—41no—
57C2×D244—41no—
58C2×D244—84no—
59C2×D244C22⋊C412yes—
60C2×D244C2×D412yes—
61C2×C4×D61—41no—
62C2×Dic122—21no—
63C22⋊#6923—44no—
64C22⋊#6923—82no—
65M16×C61—24no—
66C2×C8×C61—21no—
67C2×C8×C61—21no—
68C42×C61—21no—
69C42×C61C22⋊C411yescharacteristic
70C42×C61C22×C411yescharacteristic
71#11758198—42no—
72C4⋊Dic61—41no—
73D488—44no—
74D488—44no—
75#11758190—44no—
76#11758190—44no—
77C23×C61#9911yescharacteristic
78C22×D61—22no—
79C22×D61—22no—
80C22×D61—42no—
81C22×D61—42no—
82C22×D61—41no—
83C22×D61—84no—
84C2×#6923—84no—
85C2×#6923—164no—
86C2×C4×C61—22no—
87C2×C4×C61—21no—
88C2×C4×C61—21no—
89C2×C4×C61—42no—
90C2×C4×C61#9911yescharacteristic
91C2×C4×C61C2×C22⋊C411yescharacteristic
92C22⋊D61—82no—
93C2×Dic61—41no—
94D244—42no—
95D244—42no—
96D244—42no—
97D244—82no—
98D244—82no—
99D244—82no—
100D244—82no—
101D244—84no—
102C4×D61—81no—
103C4×D61—81no—
104Dic122—42no—
105C8×C61—22no—
106C8×C61—22no—
107#13584—611nomaximal, Sylow
108C22×C61—41no—
109C22×C61#35312yes—
110C22×C61C2×#9911yescharacteristic
111C2×D61—42no—
112C2×D61—42no—
113C2×D61—84no—
114C2×D61—82no—
115C2×D61—82no—
116C2×D61—82no—
117C2×D61—81no—
118C2×D61—162no—
119#6923—164no—
120C4×C61—22no—
121C4×C61—21no—
122C4×C61—21no—
123C4×C61—41no—
124Dic61—81no—
125#3151—611no—
126C2×#353—612no—
127#1409—612no—
128#1408—612no—
129C2×C61—21no—
130C2×C61—22no—
131C2×C61C2×#35311yescharacteristic
132D61—82no—
133D61—82no—
134D61—161no—
135C2×AGL(1,ℤ/8)—1222no—
136#532—611no—
137C2×#127—611no—
138#496—1222no—
139#493—611no—
140#493—611no—
141#492—611no—
142#492—611no—
143C2×#99—1222no—
144C4×M16#692311yescharacteristic, complemented
145#353—614no—
146#353—1222no—
147C61#1358411yescharacteristic, complemented, Sylow
148C2×#41—1221no—
149C22×D4—612no—
150C22×D4—2441no—
151AGL(1,ℤ/8)—2444no—
152AGL(1,ℤ/8)—2444no—
153C2×SD16—1222no—
154C2×SD16—1222no—
155C2×D8—1222no—
156C2×D8—1222no—
157C2×M16—21no—
158C4⋊Q8—611no—
159#127—612no—
160#127—611no—
161#127—611no—
162#127—611no—
163#127—2441no—
164#124—1221no—
165#121—2441no—
166C4×D4—2441no—
167C2×C22⋊C4—2442no—
168C2×C42C2×#692311yescharacteristic
169#99—1224no—
170#99—2442no—
171#97—21no—
172C4×C8C2×#692311yescharacteristic, complemented
173C4×C8C2×#692311yescharacteristic, complemented
174C24—1222no—
175#41—2442no—
176#41—2442no—
177C2×Q8—1221no—
178C2×D4—612no—
179C2×D4—612no—
180C2×D4—1222no—
181C2×D4—1221no—
182C2×D4—4884no—
183C2×D4—2441no—
184C2×D4—2441no—
185C2×D4—2442no—
186C2×D4—2441no—
187C2×D4—2442no—
188C2×D4—2442no—
189C2×D4—2441no—
190C22×C4—21no—
191C22×C4—4882no—
192C22×C4—2441no—
193C22×C4C22⋊#692311yescharacteristic
194SD16—2444no—
195SD16—2444no—
196D8—2444no—
197D8—2444no—
198M16—24no—
199C2×C8—21no—
200C2×C8—21no—
201C4⋊C4—2441no—
202C22⋊C4—4882no—
203C22⋊C4—2442no—
204C22⋊C4—2444no—
205C42—21no—
206C42C22⋊#692311yescharacteristic
207C42C22×#692311yescharacteristic
208C23—1222no—
209C23—1222no—
210C23—4884no—
211C23—2442no—
212C23—2441no—
213C23—2442no—
214C23#1451983611yescharacteristic
215Q8—2442no—
216D4—4882no—
217D4—4882no—
218D4—4882no—
219D4—4882no—
220D4—4884no—
221D4—4882no—
222D4—2442no—
223D4—2442no—
224D4—2442no—
225C2×C4—21no—
226C2×C4—22no—
227C2×C4—21no—
228C2×C4—42no—
229C2×C4—9764no—
230C2×C4—4881no—
231C2×C4—4881no—
232C2×C4—4884no—
233C2×C4—2441no—
234C2×C4#1451983611yescharacteristic
235C2×C4C2×C22⋊#692311yescharacteristic
236C8—22no—
237C8—22no—
238C22—41no—
239C22—9762no—
240C22—4882no—
241C22—4882no—
242C22—4882no—
243C22—4884no—
244C22—4881no—
245C22—2442no—
246C22—2442no—
247C22#1684066712yes—
248C22C2×#1451983611yescharacteristic
249C4—22no—
250C4—21no—
251C4—21no—
252C4—41no—
253C4—9764no—
254C4—4881no—
255C2—21no—
256C2—22no—
257C2—9761no—
258C2—4882no—
259C2—4882no—
260C2C2×#1684066711yescharacteristic
subgroupG/H
centerC2C2×#16840667
commutatorC42×C61C22×C4
FrattiniC42C22×#6923
FittingC4×M16×C61C4
socleC2×C61C2×#353
prime radical#21456271C2
cosocle#14519774C23
hypercenterC4×M16#6923
hypocenterC61#13584
LCSG ▷ C42×C61 ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C23 ◁ C22×C4 ◁ C2×C42 ◁ C4×M16

Complements

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

NH ≅ G/Ntype
1. C2×#16840667C2semidirect
2. #21460953C2semidirect
3. #21460951C2semidirect
9. #16840312C4semidirect
12. #16840139C4semidirect
14. #16840135C4semidirect
16. C4×M16×C61C4semidirect
29. #14519777C2×C4semidirect
32. #14519774C2×C4semidirect
41. C4×C8×C61C2×C4semidirect
42. C4×C8×C61C2×C4semidirect
144. C4×M16#6923semidirect
147. C61#13584semidirect
172. C4×C8C2×#6923semidirect
173. C4×C8C2×#6923semidirect

Maximal subgroups

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

subgroupindex
C2×#168406672
#214609532
#214609512
#214562712
#1358461