Finite Group Databasebeta

← #29584997

Subgroups — #29584997

Order 15616.

372 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 97 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#21446617C211yesmaximal, characteristic
2#21444591C212yesmaximal, complemented
3#21444243C212yesmaximal
4#21443650C211yesmaximal, characteristic, complemented
5C61×#1849C211yesmaximal, characteristic, complemented
6C2×C4×D244C2211yescharacteristic
7#16839815C2212yes—
8C2×C4⋊#92418C2211yescharacteristic
9C2×#14519689—22no—
10#16839512—24no—
11#16839489—24no—
12C2×#108×C61—22no—
13C61×#423C2212yes—
14C2×#97×C61C2211yescharacteristic
15C22×D244Q811yescharacteristic
16C22×C4×D61—21no—
17C22×#92418—21no—
18C4×D244—22no—
19C4×D244—22no—
20C2×C4⋊Dic61Q811yescharacteristic
21C2×#11758198—21no—
22C4⋊#92418—22no—
23C4⋊#92418—22no—
24C22⋊#92418—22no—
25#14519689—44no—
26C4×#92418D414yes—
27C2×M16×C61—22no—
28C22×C8×C61D412yes—
29C2×C42×C61C2311yescharacteristic
30#108×C61—44no—
31C4⋊C8×C61—22no—
32C4⋊C8×C61—22no—
33#97×C61—22no—
34C23×D61—21no—
35C22×C4×C61C2×D411yescharacteristic
36C22×C4×C61C2×Q811yescharacteristic
37C22×C4×C61#4111yescharacteristic
38C22×Dic61—21no—
39C2×D244—22no—
40C2×D244—42no—
41C2×C4×D61—41no—
42C2×C4×D61—41no—
43C2×C4×D61—42no—
44C2×C4×D61—42no—
45M16×C61—44no—
46C2×C8×C61—22no—
47C2×C8×C61—22no—
48C2×C8×C61—42no—
49C2×C8×C61SD1614yes—
50C42×C61C2×D412yes—
51C42×C61#4112yes—
52#11758198—42no—
53C4⋊Dic61—22no—
54C2×#92418—24no—
55C2×#92418—41no—
56C2×#92418—42no—
57C23×C61C22⋊Q811yescharacteristic
58C22×D61—22no—
59C22×D61—41no—
60C22×D61—42no—
61C22×D61—41no—
62C22×D61—42no—
63C2×C4×C61—41no—
64C2×C4×C61—41no—
65C2×C4×C61C22⋊Q812yes—
66C2×C4×C61#12312yes—
67C2×C4×C61#12311yescharacteristic
68C2×C4×C61#12711yescharacteristic
69C2×C4×C61C2×SD1612yes—
70C2×C4×C61AGL(1,ℤ/8)12yes—
71C2×Dic61—41no—
72C2×Dic61—42no—
73D244—24no—
74D244—42no—
75C4×D61—82no—
76C4×D61—81no—
77C4×D61—81no—
78C8×C61—24no—
79C8×C61—82no—
80#92418—44no—
81#92418—81no—
82#10661—611nomaximal, Sylow
83C22×C61—21no—
84C22×C61—21no—
85C22×C61#39911yescharacteristic
86C22×C61#47612yes—
87C2×D61—44no—
88C2×D61—42no—
89C2×D61—82no—
90C2×D61—81no—
91C2×D61—81no—
92C4×C61—22no—
93C4×C61—22no—
94C4×C61#48112yes—
95C4×C61#49211yescharacteristic
96C4×C61#49611yescharacteristic
97Dic61—81no—
98#1955—611no—
99#1849D6111yescharacteristic, complemented
100#1570—612no—
101#1486—612no—
102#1275—611no—
103C2×C61—22no—
104C2×C61#209111yescharacteristic
105C2×C61#209411yescharacteristic
106C2×C61#209811yescharacteristic
107D61—82no—
108C2×C4×D4—611no—
109#431—612no—
110C2×#108—22no—
111#423C2×D6112yes—
112C2×C4⋊C8—611no—
113C2×C22⋊C8—1222no—
114C2×#97C2×D6111yescharacteristic
115#334—1224no—
116D4⋊C8—1224no—
117C61#1066111yescharacteristic, complemented, Sylow
118C22×D4—611no—
119C23×C4—1221no—
120C2×M16—22no—
121C22×C8—1221no—
122C22×C8C22⋊D6112yes—
123C4×D4—1222no—
124C4×D4—1222no—
125C2×C4⋊C4—611no—
126C2×C22⋊C4—1221no—
127C2×C42C22×D6111yescharacteristic
128#108—44no—
129C4⋊C8—22no—
130C4⋊C8—22no—
131C4⋊C8—1222no—
132C4⋊C8—1222no—
133C22⋊C8—1222no—
134C22⋊C8—2444no—
135#97—22no—
136C4×C8—614no—
137C24—1221no—
138C2×D4—1222no—
139C2×D4—2442no—
140C22×C4—1221no—
141C22×C4—2441no—
142C22×C4—2441no—
143C22×C4—2442no—
144C22×C4—2442no—
145C22×C4Q8×D6111yescharacteristic
146C22×C4#1175822611yescharacteristic
147C22×C4C2×C22⋊D6111yescharacteristic
148M16—44no—
149C2×C8—22no—
150C2×C8—22no—
151C2×C8—42no—
152C2×C8—1224no—
153C2×C8—2441no—
154C2×C8—2442no—
155C2×C8#1175820114yes—
156C4⋊C4—1222no—
157C22⋊C4—2442no—
158C42#1175822212yes—
159C42D4×D6112yes—
160C23—1222no—
161C23—2441no—
162C23—2442no—
163C23—2441no—
164C23—2442no—
165C23#1451975311yescharacteristic
166D4—1224no—
167D4—2442no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—4882no—
171C2×C4—4881no—
172C2×C4—4881no—
173C2×C4—2441no—
174C2×C4—2442no—
175C2×C4#1451975512yes—
176C2×C4#1451976011yescharacteristic
177C2×C4#1451976212yes—
178C2×C4#1451977311yescharacteristic
179C2×C4C2×#1175820112yes—
180C2×C4#1451980312yes—
181C8—24no—
182C8—82no—
183C8—4881no—
184C8—2444no—
185C22—21no—
186C22—21no—
187C22—4882no—
188C22—4881no—
189C22—4881no—
190C22—2444no—
191C22—2442no—
192C22#1683969711yescharacteristic
193C22#1684004912yes—
194C4—22no—
195C4—22no—
196C4—4881no—
197C4#1684008012yes—
198C4#1684013611yescharacteristic
199C4#1684014911yescharacteristic
200C2—22no—
201C2—4882no—
202C2#2144733411yescharacteristic
203C2#2144735311yescharacteristic
204C2#2144737111yescharacteristic
subgroupG/H
centerC2×C4#14519773
commutator, cosocleC2×C42×C61C23
FrattiniC2×C42C22×D61
FittingC61×#1849C2
socleC22×C61#399
prime radicalC22×D244Q8
hypercenter#1849D61
hypocenterC61#10661
LCSG ▷ C2×C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1849

Complements

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

NH ≅ G/Ntype
2. #21444591C2semidirect
4. #21443650C2semidirect
5. C61×#1849C2semidirect
99. #1849D61semidirect
117. C61#10661semidirect

Maximal subgroups

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

subgroupindex
#214466172
#214445912
#214442432
#214436502
C61×#18492
#1066161