Finite Group Databasebeta

← #29828670

Subgroups — #29828670

Order 15616.

3798 conjugacy classes of subgroups; only the 323 normal subgroups are shown.

normal subgroups only

This group has 3798 conjugacy classes of subgroups — too many to draw. Only its 323 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
1C24×Dic122C211yesmaximal, characteristic
2C2×#16839920C216yesmaximal, complemented
3#21454106C211yesmaximal, characteristic
4#21445755C216yesmaximal, complemented
5C61×#1779C211yesmaximal, characteristic
6C23×Dic122—23no—
7C23×Dic122—224no—
8C23×Dic122C2216yes—
9C24×Dic61—21no—
10C22×Dic244C2214yescomplemented
11C2×#14519756C2213yes—
12#16839920—224no—
13C2×#14519702C22112yescomplemented
14#16839635—23no—
15#16839622C2213yes—
16#16839618—21no—
17C24×C4×C61C2211yescharacteristic
18C2×C22⋊C8×C61C2216yes—
19C22×Dic122—224no—
20C22×Dic122—248no—
21C22×Dic122—2128no—
22C22×Dic122—26no—
23C22×Dic122—212no—
24C22×Dic122—248no—
25C22×Dic122D4124yes—
26C22×Dic122C2311yescharacteristic
27C22×Dic122C2314yes—
28C23×Dic61—224no—
29C23×Dic61—26no—
30C2×Dic244—224no—
31#14519756—212no—
32C2×C4⋊Dic61C2313yes—
33C2×#11758196—23no—
34C2×C22⋊Dic61—23no—
35C2×C22⋊Dic61—23no—
36#14519702—224no—
37#14519684—23no—
38C25×C61D411yescharacteristic
39C23×C4×C61—212no—
40C23×C4×C61D413yes—
41C23×C4×C61C2313yes—
42C22×C8×C61C2314yes—
43C22⋊C8×C61—212no—
44C24×C61—26no—
45C24×C61—26no—
46C24×C61Q1614yescomplemented
47C24×C61C2×D413yes—
48C22×C4×C61—232no—
49C22×C4×C61—26no—
50C22×C4×C61—212no—
51C22×C4×C61—212no—
52C22×C4×C61C2×D413yes—
53C22×C4×C61C2×D4112yes—
54C22×C4×C61C2411yescharacteristic
55C22×Dic61—264no—
56C22×Dic61—248no—
57C22×Dic61—224no—
58C22×Dic61—24no—
59C22×Dic61—21no—
60C22×Dic61—23no—
61C2×Dic122—2192no—
62C2×Dic122—248no—
63C2×Dic122—296no—
64C2×Dic122—224no—
65C2×Dic122—212no—
66C2×Dic122—296no—
67C2×Dic122—26no—
68C2×Dic122C2×D4124yes—
69C2×C8×C61—212no—
70C22⋊Dic61—46no—
71C22⋊Dic61—46no—
72C4⋊Dic61—26no—
73#11758196—46no—
74Dic244—416no—
75C23×C61—28no—
76C23×C61—224no—
77C23×C61—26no—
78C23×C61—26no—
79C23×C61—224no—
80C23×C61#12016yes—
81C23×C61C2×Q1616yes—
82C23×C61#13716yes—
83C23×C61C22×D411yescharacteristic
84C2×C4×C61—224no—
85C2×C4×C61—26no—
86C2×C4×C61—224no—
87C2×C4×C61#12016yes—
88C2×C4×C61C22×D416yes—
89C2×Dic61—296no—
90C2×Dic61—224no—
91C2×Dic61—43no—
92C2×Dic61—43no—
93C2×Dic61—46no—
94Dic122—296no—
95Dic122—224no—
96Dic122—44no—
97Dic122#120116yes—
98C8×C61—44no—
99#32882—611nomaximal, Sylow
100C22×C61—232no—
101C22×C61—26no—
102C22×C61—212no—
103C22×C61—26no—
104C22×C61—212no—
105C22×C61#451112yescomplemented
106C22×C61C2×#12013yes—
107C22×C61C22×Q1611yescharacteristic
108C22×C61C2×#13713yes—
109C4×C61—26no—
110C4×C61C2×#12014yes—
111Dic61—216no—
112Dic61—81no—
113Dic61—83no—
114C24×Q8—611no—
115C2×#451—616no—
116#2848—611no—
117#1787—616no—
118#1779D6111yescharacteristic
119C2×C61—212no—
120C2×C61#284611yescharacteristic
121C2×C61C2×#45113yes—
122C2×C61#300413yes—
123C23×Q8—616no—
124C23×Q8—12224no—
125C23×Q8—1223no—
126C24×C4—1221no—
127C24×C4C2×D6111yescharacteristic
128C22×Q16—614no—
129C2×C22⋊Q8—613no—
130#451—12224no—
131C2×#103—6112no—
132C2×C22⋊C8C2×D6116yes—
133#385—1223no—
134#380—613no—
135#379—1221no—
136C61#3288211yescharacteristic, complemented, Sylow
137C25D24411yescharacteristic
138C22×Q8—6124no—
139C22×Q8—614no—
140C22×Q8—611no—
141C22×Q8—12248no—
142C22×Q8—122128no—
143C22×Q8—12224no—
144C22×Q8—12248no—
145C22×Q8—1226no—
146C22×Q8—12212no—
147C23×C4—212no—
148C23×C4—12224no—
149C23×C4—1226no—
150C23×C4D24413yes—
151C23×C4C22×D6113yes—
152C2×Q16—12224no—
153C22×C8C22×D6114yes—
154C22⋊Q8—12212no—
155C2×C4⋊C4—613no—
156C2×C4⋊C4—1223no—
157C2×C22⋊C4—1223no—
158C2×C22⋊C4—1223no—
159#103—12224no—
160C22⋊C8—212no—
161#95—1223no—
162C24—26no—
163C24—26no—
164C24Dic24414yescomplemented
165C24C2×D24413yes—
166C2×Q8—6124no—
167C2×Q8—12296no—
168C2×Q8—122192no—
169C2×Q8—12248no—
170C2×Q8—12296no—
171C2×Q8—12212no—
172C2×Q8—12224no—
173C2×Q8—1226no—
174C22×C4—232no—
175C22×C4—26no—
176C22×C4—212no—
177C22×C4—212no—
178C22×C4—12264no—
179C22×C4—12248no—
180C22×C4—1224no—
181C22×C4—12224no—
182C22×C4—1223no—
183C22×C4—1221no—
184C22×C4C2×D24413yes—
185C22×C4D4×D61112yes—
186C22×C4C23×D6111yescharacteristic
187Q16—24416no—
188C2×C8—212no—
189C4⋊C4—1226no—
190C4⋊C4—2446no—
191C22⋊C4—2446no—
192C22⋊C4—2446no—
193C23—28no—
194C23—224no—
195C23—26no—
196C23—26no—
197C23—224no—
198C23#1451974716yes—
199C23C2×Dic24416yes—
200C23#1451981016yes—
201C23C22×D24411yescharacteristic
202Q8—6116no—
203Q8—12296no—
204Q8—12224no—
205Q8—2444no—
206C2×C4—224no—
207C2×C4—26no—
208C2×C4—224no—
209C2×C4—12296no—
210C2×C4—12224no—
211C2×C4—2443no—
212C2×C4—2443no—
213C2×C4—2446no—
214C2×C4#1451974716yes—
215C2×C4C2×D4×D6116yes—
216C8—44no—
217C22—232no—
218C22—26no—
219C22—212no—
220C22—212no—
221C22—26no—
222C22#16839920112yescomplemented
223C22C2×#1451974713yes—
224C22C22×Dic24411yescharacteristic
225C22C2×#1451981013yes—
226C4—26no—
227C4—12216no—
228C4—4881no—
229C4—4883no—
230C4#120×D6114yes—
231C2—212no—
232C2#2145409711yescharacteristic
233C2C2×#1683992013yes—
234C2#2145505613yes—
subgroupG/H
centerC23C22×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1779C2
socleC23×C61C22×D4
prime radicalC25×C61D4
hypercenter#1779D61
hypocenterC61#32882
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C24×C4 ◁ #1779

Complements

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

NH ≅ G/Ntype
2. C2×#16839920C2semidirect
4. #21445755C2semidirect
10. C22×Dic244C22semidirect
13. C2×#14519702C22semidirect
46. C24×C61Q16semidirect
105. C22×C61#451semidirect
136. C61#32882semidirect
164. C24Dic244semidirect
222. C22#16839920semidirect

Maximal subgroups

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

subgroupindex
C24×Dic1222
C2×#168399202
#214541062
#214457552
C61×#17792
#3288261