Finite Group Databasebeta

← #29600669

Subgroups — #29600669

Order 15616.

324 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 71 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×#16839600C212yesmaximal, complemented
2#21447910C211yesmaximal, characteristic, complemented
3#21443997C212yesmaximal, complemented
4C22⋊#14519683C211yesmaximal, characteristic, complemented
5C22×D16×C61C211yesmaximal, characteristic
6C2×Dic488C2211yescharacteristic, complemented
7#16840003—21no—
8#16839600—24no—
9C2×#14519683C2212yescomplemented
10#16839588C2211yescharacteristic, complemented
11#16839567—21no—
12C22×D8×C61—21no—
13C2×D16×C61—24no—
14C2×D16×C61—21no—
15C2×D16×C61C2212yes—
16C22×C16×C61C2211yescharacteristic
17C2×Dic244—21no—
18#14519756—41no—
19Dic488—22no—
20#14519711—21no—
21#14519702—41no—
22#14519683—22no—
23C22×D4×C61—41no—
24C2×D8×C61—24no—
25C2×D8×C61—22no—
26C2×D8×C61—42no—
27C2×D8×C61—41no—
28C22×C8×C61D411yescharacteristic
29D16×C61—24no—
30D16×C61—22no—
31D16×C61D414yes—
32C2×C16×C61—22no—
33C2×C16×C61D411yescharacteristic
34C2×C16×C61C2311yescharacteristic
35C24×C61—81no—
36C2×D4×C61—44no—
37C2×D4×C61—42no—
38C2×D4×C61—82no—
39C2×D4×C61—81no—
40C22×C4×C61D811yescharacteristic
41C2×Dic122—41no—
42D8×C61—24no—
43D8×C61—44no—
44D8×C61—42no—
45C2×C8×C61—21no—
46C2×C8×C61—21no—
47C2×C8×C61D811yescharacteristic
48C2×C8×C61C2×D411yescharacteristic
49C22⋊Dic61—81no—
50C4⋊Dic61—41no—
51#11758196—81no—
52Dic244—41no—
53Dic244—41no—
54C16×C61—21no—
55C16×C61C2×D412yes—
56C23×C61—84no—
57C23×C61—82no—
58C23×C61—162no—
59C23×C61—162no—
60C23×C61D1611yescharacteristic
61D4×C61—44no—
62D4×C61—84no—
63D4×C61—82no—
64C2×C4×C61—21no—
65C2×C4×C61—21no—
66C2×C4×C61D1611yescharacteristic
67C2×C4×C61C2×D811yescharacteristic
68C2×Dic61—81no—
69C2×Dic61—81no—
70Dic122—81no—
71Dic122—81no—
72C8×C61—21no—
73C8×C61#12011yescharacteristic
74C8×C61AGL(1,ℤ/8)11yescharacteristic
75#13664—611nomaximal, Sylow
76C22×C61—21no—
77C22×C61—21no—
78C22×C61—84no—
79C22×C61—168no—
80C22×C61—162no—
81C22×C61—162no—
82C22×C61SD6412yescomplemented
83C22×C61C2×D1611yescharacteristic
84C4×C61—21no—
85C4×C61#44711yescharacteristic
86C4×C61#50911yescharacteristic
87Dic61—161no—
88Dic61—161no—
89C22×D16D6111yescharacteristic
90C2×SD64—612no—
91#2216—611no—
92#1415—612no—
93C22⋊C32—611no—
94C2×C61—22no—
95C2×C61—164no—
96C2×C61#218411yescharacteristic
97C2×C61C2×SD6411yescharacteristic
98C2×C61#226311yescharacteristic
99C22×D8—21no—
100C2×Q32—611no—
101C2×D16—24no—
102C2×D16—21no—
103C2×D16C2×D6112yes—
104C22×C16C2×D6111yescharacteristic
105#467—1221no—
106SD64—1224no—
107C2×C32—612no—
108#366—611no—
109#358—1221no—
110C61#1366411yescharacteristic, complemented, Sylow
111C22×D4—41no—
112C2×Q16—1221no—
113C2×D8—24no—
114C2×D8—22no—
115C2×D8—42no—
116C2×D8—41no—
117C22×C8C22⋊D6111yescharacteristic
118C22⋊Q8—2441no—
119Q32—1222no—
120D16—24no—
121D16—22no—
122D16C22⋊D6114yes—
123C2×C16—22no—
124C2×C16C22⋊D6111yescharacteristic
125C2×C16C22×D6111yescharacteristic
126#107—1221no—
127#103—2441no—
128C32—1222no—
129C24—81no—
130C2×Q8—2441no—
131C2×D4—44no—
132C2×D4—42no—
133C2×D4—82no—
134C2×D4—81no—
135C22×C4#1175819911yescharacteristic
136Q16—2441no—
137Q16—2441no—
138D8—24no—
139D8—44no—
140D8—42no—
141C2×C8—21no—
142C2×C8—21no—
143C2×C8#1175819911yescharacteristic
144C2×C8C2×C22⋊D6111yescharacteristic
145C4⋊C4—4881no—
146C4⋊C4—2441no—
147C22⋊C4—4881no—
148C16—21no—
149C16C2×C22⋊D6112yes—
150C23—84no—
151C23—82no—
152C23—162no—
153C23—162no—
154C23#1451971811yescharacteristic
155Q8—4881no—
156Q8—4881no—
157D4—44no—
158D4—84no—
159D4—82no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—4881no—
163C2×C4—4881no—
164C2×C4#1451971811yescharacteristic
165C2×C4C2×#1175819911yescharacteristic
166C8—21no—
167C8#1451974611yescharacteristic
168C8#1451980311yescharacteristic
169C22—21no—
170C22—21no—
171C22—84no—
172C22—168no—
173C22—162no—
174C22—162no—
175C22#1683960012yescomplemented
176C22C2×#1451971811yescharacteristic
177C4—21no—
178C4—9761no—
179C4—9761no—
180C4#1683989611yescharacteristic
181C4#1684019811yescharacteristic
182C2—22no—
183C2—164no—
184C2#2144773111yescharacteristic
185C2C2×#1683960011yescharacteristic
186C2#2144813211yescharacteristic
subgroupG/H
centerC22C2×#14519718
commutator, cosocleC2×C16×C61C23
FrattiniC2×C16C22×D61
Fitting, prime radicalC22×D16×C61C2
socleC22×C61C2×D16
hypercenterC22×D16D61
hypocenterC61#13664
LCSG ▷ C2×C16×C61 ▷ C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C22×C16 ◁ C22×D16

Complements

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

NH ≅ G/Ntype
1. C2×#16839600C2semidirect
2. #21447910C2semidirect
3. #21443997C2semidirect
4. C22⋊#14519683C2semidirect
6. C2×Dic488C22semidirect
9. C2×#14519683C22semidirect
10. #16839588C22semidirect
82. C22×C61SD64semidirect
110. C61#13664semidirect
175. C22#16839600semidirect

Maximal subgroups

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

subgroupindex
C2×#168396002
#214479102
#214439972
C22⋊#145196832
C22×D16×C612
#1366461