Finite Group Databasebeta

← #29689281

Subgroups — #29689281

Order 15616.

1334 conjugacy classes of subgroups; only the 315 normal subgroups are shown.

normal subgroups only

This group has 1334 conjugacy classes of subgroups — too many to draw. Only its 315 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#21448422C212yesmaximal, complemented
2#21446656C218yesmaximal, complemented
3#21445737C214yesmaximal, complemented
4C2×C61×#537C211yesmaximal, characteristic
5C2×#14519692C22116yes—
6#16839648—28no—
7#16839625C2214yes—
8#16839622C2214yes—
9C4×C22⋊Dic61C2214yes—
10C22×#41×C61C2212yes—
11C61×#537—28no—
12C2×C22⋊Q8×C61C414yescomplemented
13C2×#121×C61C414yescomplemented
14C2×#121×C61C2214yes—
15C22×C4⋊C4×C61C2211yescharacteristic
16C2×C4⋊Dic61—24no—
17C2×C22⋊Dic61C2318yes—
18C2×C4×Dic61—24no—
19#14519692—416no—
20#14519684—24no—
21#14519684—24no—
22#14519684—28no—
23C2×#41×C61—28no—
24C2×#41×C61—48no—
25C22×Q8×C61C2×C412yes—
26C22×D4×C61C2×C412yes—
27C22×D4×C61C2×C414yes—
28C22×D4×C61C2314yes—
29C23×C4×C61C2×C414yes—
30C23×C4×C61C2311yescharacteristic
31C23×C4×C61C2312yes—
32C22⋊Q8×C61—216no—
33#121×C61—28no—
34#121×C61—28no—
35#121×C61—216no—
36C2×C4⋊C4×C61C2×C414yes—
37C2×C4⋊C4×C61C2×C414yes—
38C2×C4⋊C4×C61D418yes—
39C2×C22⋊C4×C61C2×C418yes—
40C24×C61C22×C414yes—
41C24×C61C2411yescharacteristic
42#41×C61—432no—
43C2×Q8×C61—24no—
44C2×Q8×C61—44no—
45C2×D4×C61—28no—
46C2×D4×C61—216no—
47C2×D4×C61—24no—
48C2×D4×C61—28no—
49C2×D4×C61—44no—
50C2×D4×C61—84no—
51C22×C4×C61—24no—
52C22×C4×C61—28no—
53C22×C4×C61—48no—
54C22×C4×C61—82no—
55C22×C4×C61C22×C414yes—
56C22×C4×C61C22×C412yes—
57C22×C4×C61C22×C414yes—
58C22×C4×C61C2×D414yes—
59C22×C4×C61C2×D414yes—
60C22×C4×C61C2×D414yes—
61C22×Dic61—28no—
62C4⋊C4×C61—28no—
63C4⋊C4×C61C22⋊C4116yes—
64C22⋊C4×C61—216no—
65C22⋊Dic61—216no—
66C4⋊Dic61—84no—
67C4×Dic61—84no—
68C23×C61—24no—
69C23×C61—28no—
70C23×C61—41no—
71C23×C61—41no—
72C23×C61—44no—
73C23×C61—44no—
74C23×C61C23×C411yescharacteristic
75C23×C61C22×D412yes—
76C23×C61H(5,𝔽2)14yes—
77Q8×C61—48no—
78D4×C61—48no—
79D4×C61—416no—
80D4×C61—88no—
81C2×C4×C61—24no—
82C2×C4×C61—28no—
83C2×C4×C61—24no—
84C2×C4×C61—28no—
85C2×C4×C61—44no—
86C2×C4×C61—44no—
87C2×C4×C61—44no—
88C2×C4×C61—416no—
89C2×C4×C61C2×C22⋊C418yes—
90C2×C4×C61C2×C22⋊C414yes—
91C2×Dic61—48no—
92C2×Dic61—48no—
93C2×Dic61—48no—
94#21796—611nomaximal, Sylow
95C22×C61—22no—
96C22×C61—21no—
97C22×C61—22no—
98C22×C61—21no—
99C22×C61—44no—
100C22×C61—44no—
101C22×C61—44no—
102C22×C61—44no—
103C22×C61—88no—
104C22×C61C22×C22⋊C411yescharacteristic
105C22×C61#51812yes—
106C22×C61#53514yes—
107C4×C61—24no—
108C4×C61—28no—
109C4×C61—44no—
110Dic61—88no—
111C2×#537D6111yescharacteristic
112#2313—612no—
113#1964—618no—
114#1782—614no—
115C2×C61—24no—
116C2×C61—22no—
117C2×C61—84no—
118C2×C61#230911yescharacteristic
119C2×C61#288311yescharacteristic
120C2×C61#288411yescharacteristic
121C22×#41C2×D6112yes—
122#537—28no—
123C2×C22⋊Q8Dic6114yescomplemented
124C2×#121Dic6114yescomplemented
125C2×#121C2×D6114yes—
126C22×C4⋊C4C2×D6111yescharacteristic
127C2×#99—6116no—
128#387—1228no—
129#381—614no—
130#380—614no—
131C4×C22⋊C4—614no—
132C61#2179611yescharacteristic, complemented, Sylow
133C2×#41—28no—
134C2×#41—48no—
135C22×Q8C2×Dic6112yes—
136C22×D4C2×Dic6112yes—
137C22×D4C2×Dic6114yes—
138C22×D4C22×D6114yes—
139C23×C4C2×Dic6114yes—
140C23×C4C22×D6111yescharacteristic
141C23×C4C22×D6112yes—
142C22⋊Q8—216no—
143#121—28no—
144#121—28no—
145#121—216no—
146C2×C4⋊C4—1224no—
147C2×C4⋊C4C2×Dic6114yes—
148C2×C4⋊C4C2×Dic6114yes—
149C2×C4⋊C4C22⋊D6118yes—
150C2×C22⋊C4—618no—
151C2×C22⋊C4C2×Dic6118yes—
152C2×C42—1224no—
153#99—24416no—
154#95—1224no—
155#95—1224no—
156#95—1228no—
157C24C22×Dic6114yes—
158C24C23×D6111yescharacteristic
159#41—432no—
160C2×Q8—24no—
161C2×Q8—44no—
162C2×D4—28no—
163C2×D4—216no—
164C2×D4—24no—
165C2×D4—28no—
166C2×D4—44no—
167C2×D4—84no—
168C22×C4—24no—
169C22×C4—28no—
170C22×C4—48no—
171C22×C4—82no—
172C22×C4—1228no—
173C22×C4C22×Dic6114yes—
174C22×C4C22×Dic6112yes—
175C22×C4C22×Dic6114yes—
176C22×C4C2×C22⋊D6114yes—
177C22×C4C2×C22⋊D6114yes—
178C22×C4C2×C22⋊D6114yes—
179C4⋊C4—28no—
180C4⋊C4—4884no—
181C4⋊C4C22⋊Dic61116yes—
182C22⋊C4—216no—
183C22⋊C4—12216no—
184C42—4884no—
185C23—24no—
186C23—28no—
187C23—41no—
188C23—41no—
189C23—44no—
190C23—44no—
191C23C23×Dic6111yescharacteristic
192C23C22×C22⋊D6112yes—
193C23#1451982214yes—
194Q8—48no—
195D4—48no—
196D4—416no—
197D4—88no—
198C2×C4—24no—
199C2×C4—28no—
200C2×C4—24no—
201C2×C4—28no—
202C2×C4—44no—
203C2×C4—44no—
204C2×C4—44no—
205C2×C4—416no—
206C2×C4—2448no—
207C2×C4—2448no—
208C2×C4—2448no—
209C2×C4C2×C22⋊Dic6118yes—
210C2×C4C2×C22⋊Dic6114yes—
211C22—22no—
212C22—21no—
213C22—22no—
214C22—21no—
215C22—44no—
216C22—44no—
217C22—44no—
218C22—44no—
219C22—88no—
220C22C22×C22⋊Dic6111yescharacteristic
221C22#1684023712yes—
222C22#1684032914yes—
223C4—28no—
224C4—24no—
225C4—44no—
226C4—4888no—
227C2—24no—
228C2—22no—
229C2—84no—
230C2#2144840111yescharacteristic
231C2#2145428011yescharacteristic
232C2#2145428311yescharacteristic
subgroupG/H
centerC22C22×C22⋊Dic61
commutatorC23×C61C23×C4
FrattiniC24C23×D61
Fitting, prime radicalC2×C61×#537C2
socleC22×C61C22×C22⋊C4
cosocleC24×C61C24
hypercenterC2×#537D61
hypocenterC61#21796
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4⋊C4 ◁ C2×#537

Complements

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

NH ≅ G/Ntype
1. #21448422C2semidirect
2. #21446656C2semidirect
3. #21445737C2semidirect
12. C2×C22⋊Q8×C61C4semidirect
13. C2×#121×C61C4semidirect
123. C2×C22⋊Q8Dic61semidirect
124. C2×#121Dic61semidirect
132. C61#21796semidirect

Maximal subgroups

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

subgroupindex
#214484222
#214466562
#214457372
C2×C61×#5372
#2179661