Finite Group Databasebeta

← #30604662

Subgroups — #30604662

Order 15616.

360 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 70 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
1#21462189C211yesmaximal, characteristic, complemented
2#21461042C211yesmaximal, characteristic, complemented
3#21460784C212yesmaximal, complemented
4#21460782C212yesmaximal, complemented
5#21454728C211yesmaximal, characteristic
6C4×#11758217—21no—
7#16840684C2212yescomplemented
8#16840658—21no—
9#16840655—21no—
10#92420⋊C8—21no—
11#16840603C2211yescharacteristic, complemented
12C8×#92420C2211yescharacteristic, complemented
13#16840562C411yescharacteristic
14#16840528—21no—
15#16839883—21no—
16#16839883C412yes—
17#16839878—21no—
18#16839878C2212yes—
19C2×#14519703—21no—
20M16×Dic61C2211yescharacteristic
21C61×#430C411yescharacteristic
22C2×#11758217—22no—
23#14519851—22no—
24#14519832—22no—
25C4×#92420—21no—
26C4×#92420—21no—
27C2×#11758222—21no—
28#14519810—22no—
29#14519805—22no—
30#14519799—21no—
31#14519799C2×C412yes—
32#14519785—22no—
33#14519785—22no—
34#14519783—42no—
35C2×#11758194D411yescharacteristic
36M16×D61—42no—
37C2×C4×Dic61D411yescharacteristic
38#14519703—22no—
39#14519703—22no—
40#14519687C2×C411yescharacteristic
41#14519687D411yescharacteristic
42C8×Dic61D411yescharacteristic
43C8×Dic61C2311yescharacteristic
44C2×M16×C61—21no—
45C2×M16×C61C2×C411yescharacteristic
46#108×C61—21no—
47#108×C61C2×C412yes—
48C22×C4×C61C22⋊C411yescharacteristic
49C2×C22⋊D61—41no—
50C22×Dic61—21no—
51#11758222—22no—
52#11758222—22no—
53#11758222—41no—
54#11758222—41no—
55C2×D244—21no—
56C2×C4×D61—41no—
57C2×Dic122—21no—
58#11758217—44no—
59C2×#92420—22no—
60C2×#92420—22no—
61M16×C61—22no—
62M16×C61—22no—
63M16×C61—41no—
64C2×C8×C61—41no—
65C2×C8×C61C22⋊C411yescharacteristic
66C2×C8×C61C22×C411yescharacteristic
67C4×Dic61—21no—
68C4×Dic61C22⋊C411yescharacteristic
69C4×Dic61C2×D411yescharacteristic
70#11758194—22no—
71C2×#92418C22⋊C411yescharacteristic
72C2×#92418C2×D411yescharacteristic
73Dic244—21no—
74Dic244M1612yescomplemented
75D488—21no—
76D488M1612yescomplemented
77#11758190—22no—
78#11758190—22no—
79#11758189—42no—
80#11758189—42no—
81C8×D61—42no—
82C8×D61—42no—
83#11758187—42no—
84C23×C61#10411yescharacteristic
85C22×D61—41no—
86C2×C4×C61—21no—
87C2×C4×C61—21no—
88C2×C4×C61#10411yescharacteristic
89C2×C4×C61C2×C22⋊C411yescharacteristic
90C22⋊D61—42no—
91C22⋊D61—42no—
92C2×Dic61—21no—
93C2×Dic61—21no—
94C2×Dic61—42no—
95C2×Dic61—41no—
96D244—22no—
97D244—41no—
98C4×D61—42no—
99C4×D61—42no—
100Dic122—22no—
101Dic122—41no—
102#92420—24no—
103#92420—42no—
104C8×C61—21no—
105C8×C61—42no—
106C8×C61C2×M1612yes—
107#92418—21no—
108#92418AGL(1,ℤ/8)12yes—
109#11669—611nomaximal, Sylow
110C22×C61—21no—
111C22×C61—41no—
112C22×C61C2×#10411yescharacteristic
113C2×D61—42no—
114C2×D61—161no—
115C4×C61—21no—
116C4×C61#40711yescharacteristic
117C4×C61#41811yescharacteristic
118Dic61—22no—
119Dic61—41no—
120Dic61—42no—
121#2956—611no—
122#2165—611no—
123#1455—611no—
124#1336—612no—
125#1335—612no—
126C2×C61—21no—
127C2×C61—41no—
128C2×C61#149011yescharacteristic
129D61—82no—
130#575—611no—
131#568—1221no—
132#444—612no—
133#444—1221no—
134#443—612no—
135#443—1221no—
136#430#692311yescharacteristic
137C2×#104—1221no—
138C4×M16—611no—
139C4×M16—1221no—
140#364—612no—
141#349—1221no—
142#347—1221no—
143#322—611no—
144#322—1221no—
145C82—611no—
146C61#1166911yescharacteristic, complemented, Sylow
147C2×#41—1221no—
148#137—1222no—
149AGL(1,ℤ/8)—1222no—
150#135—612no—
151#135—1221no—
152#131—1222no—
153#131—1222no—
154#131—2442no—
155C2×M16—21no—
156C2×M16—611no—
157C2×M16—1222no—
158C2×M16—2442no—
159C2×M16C2×#692311yescharacteristic
160C2×C42—611no—
161M32—1222no—
162#108—21no—
163#108C2×#692312yes—
164#104—1222no—
165#104—1222no—
166#97—611no—
167#97—611no—
168#97—1222no—
169C4×C8—611no—
170C4×C8—611no—
171C4×C8—1221no—
172C4×C8—1221no—
173#41—1222no—
174#41—1222no—
175#41—2441no—
176#41—2441no—
177C2×Q8—1221no—
178C2×D4—1221no—
179C2×D4—2441no—
180C22×C4—1221no—
181C22×C4—2441no—
182C22×C4C22⋊#692311yescharacteristic
183Q16—612no—
184Q16—1221no—
185SD16—1222no—
186SD16—1222no—
187D8—612no—
188D8—1221no—
189M16—22no—
190M16—22no—
191M16—41no—
192M16—1222no—
193M16—2442no—
194M16—2442no—
195M16—2444no—
196C2×C8—41no—
197C2×C8—611no—
198C2×C8—611no—
199C2×C8—1222no—
200C2×C8—1222no—
201C2×C8—2442no—
202C2×C8—2442no—
203C2×C8C22⋊#692311yescharacteristic
204C2×C8C22×#692311yescharacteristic
205C42—611no—
206C42—611no—
207C42—1221no—
208C16—2442no—
209C23—2441no—
210C23#1451984211yescharacteristic
211Q8—1222no—
212Q8—2441no—
213D4—1222no—
214D4—2441no—
215D4—2442no—
216D4—2442no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—1221no—
220C2×C4—1221no—
221C2×C4—2442no—
222C2×C4—2442no—
223C2×C4—2441no—
224C2×C4—2442no—
225C2×C4#1451984211yescharacteristic
226C2×C4C2×C22⋊#692311yescharacteristic
227C8—21no—
228C8—42no—
229C8—612no—
230C8—1224no—
231C8—1221no—
232C8—2442no—
233C8D61⋊M1612yes—
234C22—21no—
235C22—41no—
236C22—9761no—
237C22—2442no—
238C22C2×#1451984211yescharacteristic
239C4—21no—
240C4—1222no—
241C4—2441no—
242C4—2442no—
243C4#1684073411yescharacteristic
244C4#1684075411yescharacteristic
245C2—21no—
246C2—41no—
247C2—4882no—
248C2#2146116011yescharacteristic
subgroupG/H
centerC4#16840754
commutatorC2×C8×C61C22×C4
FrattiniC2×C8C22×#6923
FittingC61×#430C4
socleC2×C61#1490
prime radical#16840562C4
cosocleC8×Dic61C23
hypercenter#430#6923
hypocenterC61#11669
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×M16 ◁ #430

Complements

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

NH ≅ G/Ntype
1. #21462189C2semidirect
2. #21461042C2semidirect
3. #21460784C2semidirect
4. #21460782C2semidirect
7. #16840684C22semidirect
11. #16840603C22semidirect
12. C8×#92420C22semidirect
74. Dic244M16semidirect
76. D488M16semidirect
146. C61#11669semidirect

Maximal subgroups

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

subgroupindex
#214621892
#214610422
#214607842
#214607822
#214547282
#1166961