Finite Group Databasebeta

← #29572958

Subgroups — #29572958

Order 15616.

450 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 101 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×#16839827C211yesmaximal, characteristic
2#21444580C212yesmaximal, complemented
3#21444159C212yesmaximal
4#21443689C211yesmaximal, characteristic, complemented
5C61×#1290C211yesmaximal, characteristic, complemented
6C22×C8×D61C412yes—
7C2×C4×D244C2211yescharacteristic
8#16839827—22no—
9#16839827—22no—
10#16839819C2212yes—
11C2×#14519689C412yes—
12C2×C4×#92418C2211yescharacteristic
13#16839514—22no—
14#16839512—22no—
15#16839485—24no—
16C61×#432C2212yes—
17C2×C4⋊C8×C61C2211yescharacteristic
18C22×D244Q811yescharacteristic
19C22×C4×D61C2×C412yes—
20C22×#92418C2×C411yescharacteristic
21C22×#92418C2×C411yescharacteristic
22C2×C8×D61—22no—
23C2×C8×D61—22no—
24C2×C8×D61—44no—
25C4×D244—22no—
26C4×D244—22no—
27C2×C4⋊Dic61Q811yescharacteristic
28C2×#11758198D412yes—
29C4⋊#92418—22no—
30C22⋊#92418—22no—
31#14519689—24no—
32#14519688—22no—
33C4×#92418D412yes—
34C4×#92418D412yes—
35C22×C8×C61C2×C412yes—
36C2×C42×C61C2311yescharacteristic
37C4⋊C8×C61—22no—
38C4⋊C8×C61—22no—
39C22⋊C8×C61—22no—
40#97×C61—22no—
41C23×D61C4⋊C412yes—
42C22×C4×C61C22×C411yescharacteristic
43C22×C4×C61C2×D411yescharacteristic
44C22×C4×C61C2×Q811yescharacteristic
45C22×Dic61C4⋊C412yes—
46C2×D244—24no—
47C2×D244—22no—
48C2×C4×D61—22no—
49C2×C4×D61—22no—
50C2×C4×D61—42no—
51C2×C4×D61—42no—
52C2×C8×C61—22no—
53C2×C8×C61—42no—
54C2×C8×C61—42no—
55C42×C61C2×D412yes—
56C42×C61#4112yes—
57#11758198—24no—
58C4⋊Dic61—22no—
59C2×#92418—22no—
60C2×#92418—41no—
61C2×#92418—42no—
62C2×#92418C22⋊C412yes—
63C2×#92418C22⋊C411yescharacteristic
64C2×#92418C22⋊C411yescharacteristic
65C8×D61—44no—
66C8×D61—44no—
67C23×C61C2×C4⋊C411yescharacteristic
68C22×D61—22no—
69C22×D61—22no—
70C22×D61—42no—
71C22×D61—42no—
72C22×D61#10814yes—
73C2×C4×C61—22no—
74C2×C4×C61—21no—
75C2×C4×C61—21no—
76C2×C4×C61—41no—
77C2×C4×C61C2×C22⋊C411yescharacteristic
78C2×C4×C61#11711yescharacteristic
79C2×C4×C61#12112yes—
80C2×C4×C61C22⋊Q812yes—
81C2×Dic61—22no—
82C2×Dic61—42no—
83D244—28no—
84C4×D61—44no—
85C4×D61—42no—
86C4×D61—42no—
87C8×C61—42no—
88C8×C61—82no—
89#92418—21no—
90#92418—22no—
91#92418—82no—
92#92418#10012yes—
93#8620—611nomaximal, Sylow
94C22×C61—21no—
95C22×C61—21no—
96C22×C61#38011yescharacteristic
97C22×C61C2×#10812yes—
98C2×D61—24no—
99C2×D61—44no—
100C2×D61—42no—
101C2×D61—42no—
102C2×D61—44no—
103C4×C61—21no—
104C4×C61—22no—
105C4×C61—22no—
106C4×C61C2×#10011yescharacteristic
107C4×C61#41311yescharacteristic
108Dic61—42no—
109C2×C8×D4—611no—
110#1568—612no—
111#1466—612no—
112#1290—611no—
113#1290D6111yescharacteristic, complemented
114C2×C61—22no—
115C2×C61#180011yescharacteristic
116C2×C61#181011yescharacteristic
117C2×C61#181211yescharacteristic
118D61—44no—
119C23×C8—612no—
120C2×C4×D4—611no—
121C8×D4—1222no—
122C8×D4—1222no—
123#432—612no—
124#432C2×D6112yes—
125C2×C4⋊C8C2×D6111yescharacteristic
126C2×C22⋊C8—612no—
127C2×C4×C8—611no—
128#335—1222no—
129#334—1222no—
130#323—1224no—
131C61#862011yescharacteristic, complemented, Sylow
132C22×D4—611no—
133C23×C4—612no—
134C22×C8—611no—
135C22×C8—611no—
136C22×C8—1222no—
137C22×C8—1222no—
138C22×C8—2444no—
139C22×C8C4×D6112yes—
140C4×D4—1222no—
141C4×D4—1222no—
142C2×C4⋊C4—611no—
143C2×C22⋊C4—612no—
144C2×C42C22×D6111yescharacteristic
145C4⋊C8—22no—
146C4⋊C8—22no—
147C4⋊C8—1222no—
148C22⋊C8—22no—
149C22⋊C8—1224no—
150C22⋊C8—1222no—
151#97—22no—
152#97—1222no—
153C4×C8—612no—
154C4×C8—612no—
155C24—612no—
156C2×D4—1224no—
157C2×D4—1222no—
158C22×C4—612no—
159C22×C4—1222no—
160C22×C4—1222no—
161C22×C4—2442no—
162C22×C4—2442no—
163C22×C4C2×C4×D6111yescharacteristic
164C22×C4D4×D6111yescharacteristic
165C22×C4Q8×D6111yescharacteristic
166C2×C8—22no—
167C2×C8—42no—
168C2×C8—42no—
169C2×C8—611no—
170C2×C8—612no—
171C2×C8—611no—
172C2×C8—1222no—
173C2×C8—2444no—
174C2×C8—2444no—
175C2×C8—2441no—
176C2×C8—2442no—
177C4⋊C4—1222no—
178C22⋊C4—1224no—
179C42#1175822212yes—
180C42D4×D6112yes—
181C23—614no—
182C23—1222no—
183C23—1222no—
184C23—2442no—
185C23—2442no—
186C23C4⋊C4×D6111yescharacteristic
187D4—1228no—
188C2×C4—22no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—41no—
192C2×C4—1222no—
193C2×C4—2444no—
194C2×C4—2442no—
195C2×C4—2442no—
196C2×C4—2442no—
197C2×C4C22⋊C4×D6111yescharacteristic
198C2×C4#1451973411yescharacteristic
199C2×C4#1451975012yes—
200C2×C4#1451975512yes—
201C8—42no—
202C8—82no—
203C8—612no—
204C8—1221no—
205C8—1222no—
206C8—4882no—
207C22—21no—
208C22—21no—
209C22—1224no—
210C22—2444no—
211C22—2444no—
212C22—2442no—
213C22—2442no—
214C22#1683962111yescharacteristic
215C22#108×D6112yes—
216C4—21no—
217C4—22no—
218C4—22no—
219C4—2442no—
220C4#100×D6111yescharacteristic
221C4#1683974511yescharacteristic
222C2—22no—
223C2—2444no—
224C2#2144581011yescharacteristic
225C2#2144585611yescharacteristic
226C2#2144586411yescharacteristic
subgroupG/H
centerC22#16839621
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1290C2
socleC22×C61#380
prime radicalC22×D244Q8
cosocleC2×C42×C61C23
hypercenter#1290D61
hypocenterC61#8620
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1290

Complements

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

NH ≅ G/Ntype
2. #21444580C2semidirect
4. #21443689C2semidirect
5. C61×#1290C2semidirect
113. #1290D61semidirect
131. C61#8620semidirect

Maximal subgroups

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

subgroupindex
C2×#168398272
#214445802
#214441592
#214436892
C61×#12902
#862061