Finite Group Databasebeta

← #29577543

Subgroups — #29577543

Order 15616.

436 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 95 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#21456195C211yesmaximal, characteristic
2#21454925C211yesmaximal, characteristic, complemented
3#21444532C212yesmaximal
4#21443692C212yesmaximal, complemented
5C61×#1537C211yesmaximal, characteristic, complemented
6C2×#14519774C2211yescharacteristic
7#16840128—22no—
8#16840110C414yes—
9#16839893—24no—
10#16839823—22no—
11#16839819C2212yes—
12#16839766—22no—
13C4×#11758194C2211yescharacteristic
14#16839505—24no—
15C61×#533C2211yescharacteristic
16C61×#432C2212yes—
17C61×#330—22no—
18C22×D244—21no—
19C22×D244—22no—
20C2×#11758194—21no—
21#14519774—22no—
22#14519774—21no—
23#14519774C2×C412yes—
24C4⋊#92418—24no—
25C4⋊#92418D414yes—
26#14519697—28no—
27C22⋊#92418—22no—
28#14519688—22no—
29#14519688—21no—
30#14519688—22no—
31C4×#92418—22no—
32C4×#92418C2×C412yes—
33C4⋊Q8×C61C2×C412yes—
34#125×C61D412yes—
35C22⋊Q8×C61—22no—
36C4×Q8×C61—22no—
37#117×C61D412yes—
38C2×C42×C61C2311yescharacteristic
39C4⋊C8×C61—22no—
40C22⋊C8×C61—22no—
41#97×C61—22no—
42C23×D61—41no—
43C2×Q8×C61—22no—
44C22×C4×C61C2×D411yescharacteristic
45C22×C4×C61#4112yes—
46C2×D244—22no—
47C2×D244—24no—
48C2×D244—44no—
49C2×D244—41no—
50C2×D244—42no—
51C2×D244—84no—
52C2×C8×C61—42no—
53C4⋊C4×C61—22no—
54C4⋊C4×C61—24no—
55C4⋊C4×C61C22⋊C414yes—
56C22⋊C4×C61—22no—
57C42×C61—22no—
58C42×C61C22×C411yescharacteristic
59C42×C61C2×D411yescharacteristic
60C42×C61C2×D412yes—
61#11758194—42no—
62C2×#92418—22no—
63C2×#92418—24no—
64C23×C61#12311yescharacteristic
65C22×D61—42no—
66C22×D61—82no—
67C22×D61—84no—
68Q8×C61—82no—
69C2×C4×C61—22no—
70C2×C4×C61—24no—
71C2×C4×C61—42no—
72C2×C4×C61C2×C22⋊C411yescharacteristic
73C2×C4×C61C4×D412yes—
74C2×C4×C61#12011yescharacteristic
75C2×C4×C61#12112yes—
76D244—44no—
77D244—84no—
78D244—84no—
79D244—82no—
80C8×C61—82no—
81#92418—42no—
82#92418—44no—
83#9424—611nomaximal, Sylow
84C22×C61—21no—
85C22×C61—22no—
86C22×C61#38611yescharacteristic
87C2×D61—82no—
88C2×D61—84no—
89C2×D61—164no—
90C4×C61—21no—
91C4×C61—22no—
92C4×C61—22no—
93C4×C61—44no—
94C4×C61#45412yes—
95#3138—611no—
96#2987—611no—
97#1555—612no—
98#1537D6111yescharacteristic, complemented
99#1291—612no—
100C2×C61—41no—
101C2×C61#188811yescharacteristic
102C2×C61#190312yes—
103D61—162no—
104#533C2×D6111yescharacteristic
105C2×#127—611no—
106#490—1222no—
107#486—614no—
108#446—1224no—
109#433—1222no—
110#432—612no—
111#432C2×D6112yes—
112#418—1222no—
113C4×M16—611no—
114#331—1224no—
115#330—22no—
116C61#942411yescharacteristic, complemented, Sylow
117C22×D4—1221no—
118C22×D4—1222no—
119C2×M16—1221no—
120C4⋊Q8C4×D6112yes—
121#127—612no—
122#127—1222no—
123#127—1221no—
124#125C22⋊D6112yes—
125C22⋊Q8—22no—
126C4×Q8—22no—
127#117D24412yes—
128C2×C42C22×D6111yescharacteristic
129C4⋊C8—22no—
130C4⋊C8—614no—
131C4⋊C8—1224no—
132#102—1228no—
133C22⋊C8—22no—
134C22⋊C8—1222no—
135#97—22no—
136#97—1222no—
137#97—1222no—
138#97—1221no—
139C4×C8—612no—
140C4×C8—1222no—
141C24—2441no—
142C2×Q8—22no—
143C2×D4—1222no—
144C2×D4—1224no—
145C2×D4—4884no—
146C2×D4—2444no—
147C2×D4—2441no—
148C2×D4—2442no—
149C22×C4C2×D24411yescharacteristic
150C22×C4#1175822412yes—
151M16—2442no—
152C2×C8—42no—
153C2×C8—1222no—
154C2×C8—1224no—
155C4⋊C4—22no—
156C4⋊C4—24no—
157C4⋊C4#1175819814yes—
158C22⋊C4—22no—
159C42—22no—
160C42C2×C4×D6111yescharacteristic
161C42D4×D6112yes—
162C42C2×C22⋊D6111yescharacteristic
163C23—4884no—
164C23—4882no—
165C23—2442no—
166C23#1451976111yescharacteristic
167Q8—82no—
168D4—4884no—
169D4—4884no—
170D4—4882no—
171D4—2444no—
172C2×C4—22no—
173C2×C4—24no—
174C2×C4—42no—
175C2×C4C2×#1175819811yescharacteristic
176C2×C4#1451973712yes—
177C2×C4#1451974711yescharacteristic
178C2×C4#1451974912yes—
179C8—82no—
180C8—2442no—
181C8—2444no—
182C22—21no—
183C22—22no—
184C22—9764no—
185C22—4882no—
186C22—4884no—
187C22#1683963911yescharacteristic
188C4—21no—
189C4—22no—
190C4—22no—
191C4—44no—
192C4#1683993712yes—
193C2—41no—
194C2—9762no—
195C2#2144622011yescharacteristic
196C2#2144629712yes—
subgroupG/H
centerC22#16839639
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1537C2
socleC22×C61#386
prime radicalC2×#14519774C22
cosocleC2×C42×C61C23
hypercenter#1537D61
hypocenterC61#9424
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1537

Complements

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

NH ≅ G/Ntype
2. #21454925C2semidirect
4. #21443692C2semidirect
5. C61×#1537C2semidirect
98. #1537D61semidirect
116. C61#9424semidirect

Maximal subgroups

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

subgroupindex
#214561952
#214549252
#214445322
#214436922
C61×#15372
#942461