Finite Group Databasebeta

← #29583025

Subgroups — #29583025

Order 15616.

504 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 141 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#21446767C212yesmaximal
2#21446620C211yesmaximal, characteristic
3#21443702C211yesmaximal, characteristic
4#21443660C212yesmaximal
5C61×#2551C211yesmaximal, characteristic
6C2×#14519777C2211yescharacteristic
7C2×C4⋊#92418C2212yes—
8C2×#14519700—22no—
9C2×C4×#92418C2211yescharacteristic
10#16839524—22no—
11#16839524—22no—
12C2×C4⋊Q8×C61—22no—
13C61×#395—22no—
14C61×#389—22no—
15C61×#384C2211yescharacteristic
16C4×C4⋊C4×C61C2212yes—
17C22×Dic122—21no—
18C22×#92418—21no—
19C22×#92418—22no—
20#14519777—22no—
21#14519777—21no—
22#14519777Q812yescomplemented
23C2×C4⋊Dic61—22no—
24C4⋊#92418—22no—
25C4⋊#92418—22no—
26#14519700—44no—
27C4×#92418—22no—
28C22×Q8×C61—22no—
29C4⋊Q8×C61—44no—
30C4⋊Q8×C61—44no—
31C2×C4⋊C4×C61—24no—
32C2×C4⋊C4×C61—22no—
33C2×C4⋊C4×C61D412yes—
34C2×C4⋊C4×C61D412yes—
35C2×C4⋊C4×C61D412yes—
36C2×C42×C61—22no—
37C2×C42×C61C2311yescharacteristic
38#95×C61—22no—
39#95×C61—22no—
40C2×Q8×C61—48no—
41C2×Q8×C61—82no—
42C22×C4×C61—22no—
43C22×C4×C61—22no—
44C22×C4×C61—22no—
45C22×C4×C61C2×D411yescharacteristic
46C22×C4×C61C2×D412yes—
47C22×Dic61—41no—
48C2×Dic122—22no—
49C2×Dic122—42no—
50C2×Dic122—42no—
51C2×Dic122—41no—
52C4⋊C4×C61—22no—
53C4⋊C4×C61—22no—
54C4⋊C4×C61—48no—
55C4⋊C4×C61—42no—
56C4⋊C4×C61—42no—
57C4⋊C4×C61—42no—
58C4⋊C4×C61SD1614yes—
59C4⋊C4×C61Q1614yes—
60C42×C61—42no—
61C42×C61—42no—
62C42×C61C2×Q811yescharacteristic
63C42×C61#4112yes—
64C42×C61#4111yescharacteristic
65C4⋊Dic61—24no—
66C4⋊Dic61—42no—
67C2×#92418—22no—
68C2×#92418—41no—
69C2×#92418—41no—
70C2×#92418—42no—
71C2×#92418—42no—
72C2×#92418—42no—
73C23×C61#12011yescharacteristic
74Q8×C61—88no—
75C2×C4×C61—21no—
76C2×C4×C61—21no—
77C2×C4×C61—24no—
78C2×C4×C61—24no—
79C2×C4×C61—42no—
80C2×C4×C61—42no—
81C2×C4×C61—42no—
82C2×C4×C61—42no—
83C2×C4×C61—44no—
84C2×C4×C61—42no—
85C2×C4×C61C22⋊Q811yescharacteristic
86C2×C4×C61C22⋊Q812yes—
87C2×C4×C61#12411yescharacteristic
88C2×C4×C61#12412yes—
89C2×C4×C61C2×SD1612yes—
90C2×C4×C61C2×Q1612yes—
91C2×C4×C61AGL(1,ℤ/8)12yes—
92C2×C4×C61#13712yes—
93C2×Dic61—42no—
94C2×Dic61—81no—
95C2×Dic61—81no—
96Dic122—44no—
97Dic122—82no—
98#92418—42no—
99#92418—82no—
100#10349—611nomaximal, Sylow
101C22×C61#39311yescharacteristic
102C22×C61#45012yes—
103C22×C61#45112yes—
104C22×C61#45311yescharacteristic
105C22×C61#45411yescharacteristic
106C4×C61—21no—
107C4×C61—21no—
108C4×C61—44no—
109C4×C61—44no—
110C4×C61—82no—
111C4×C61#47512yes—
112C4×C61#47712yes—
113C4×C61#48712yes—
114C4×C61#48912yes—
115Dic61—82no—
116#2551D6111yescharacteristic
117#1987—612no—
118#1956—611no—
119#1296—611no—
120#1279—612no—
121C2×C61#162911yescharacteristic
122C2×C61#163311yescharacteristic
123C2×C61#163712yes—
124C2×C61#202412yes—
125C2×C61#202711yescharacteristic
126C2×C4⋊Q8—22no—
127C2×C4⋊Q8—611no—
128C2×C4⋊C8—612no—
129C2×#103—1222no—
130C2×C4×C8—611no—
131#395—22no—
132#389—22no—
133#384C2×D6111yescharacteristic
134C4×C4⋊C4C2×D6112yes—
135#340—1222no—
136#340—1222no—
137C61#1034911yescharacteristic, complemented, Sylow
138C22×Q8—22no—
139C22×Q8—1221no—
140C22×C8—1221no—
141C22×C8—1222no—
142C4⋊Q8—44no—
143C4⋊Q8—44no—
144C4⋊Q8—612no—
145C4⋊Q8—1222no—
146C4⋊Q8—1221no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—1222no—
150C2×C4⋊C4C22⋊D6112yes—
151C2×C4⋊C4C22⋊D6112yes—
152C2×C4⋊C4C22⋊D6112yes—
153C2×C42—22no—
154C2×C42C22×D6111yescharacteristic
155C4⋊C8—1222no—
156C4⋊C8—1222no—
157#103—2444no—
158C4×C8—1222no—
159#95—22no—
160#95—22no—
161C2×Q8—48no—
162C2×Q8—82no—
163C2×Q8—1222no—
164C2×Q8—2442no—
165C2×Q8—2442no—
166C2×Q8—2441no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—2441no—
171C22×C4C2×C22⋊D6111yescharacteristic
172C22×C4C2×C22⋊D6112yes—
173C2×C8—1222no—
174C2×C8—2442no—
175C2×C8—2442no—
176C2×C8—2442no—
177C2×C8—2441no—
178C2×C8—2441no—
179C4⋊C4—22no—
180C4⋊C4—22no—
181C4⋊C4—42no—
182C4⋊C4—48no—
183C4⋊C4—42no—
184C4⋊C4—42no—
185C4⋊C4—1224no—
186C4⋊C4—2442no—
187C4⋊C4#1175820014yes—
188C4⋊C4#1175820214yes—
189C42—42no—
190C42—42no—
191C42Q8×D6111yescharacteristic
192C42#1175822612yes—
193C42#1175822611yescharacteristic
194C23#1451974611yescharacteristic
195Q8—88no—
196Q8—4882no—
197Q8—2444no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—24no—
201C2×C4—24no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—44no—
207C2×C4—42no—
208C2×C4—4881no—
209C2×C4—4881no—
210C2×C4—2442no—
211C2×C4#1451975311yescharacteristic
212C2×C4#1451975312yes—
213C2×C4#1451976611yescharacteristic
214C2×C4#1451976612yes—
215C2×C4C2×#1175820012yes—
216C2×C4C2×#1175820212yes—
217C2×C4#1451980312yes—
218C2×C4#1451980812yes—
219C8—4882no—
220C8—2442no—
221C22#1683967311yescharacteristic
222C22#1683991312yes—
223C22#1683991812yes—
224C22#1683993011yescharacteristic
225C22#1683993511yescharacteristic
226C4—21no—
227C4—21no—
228C4—44no—
229C4—44no—
230C4—82no—
231C4—4882no—
232C4#1684004412yes—
233C4#1684005412yes—
234C4#1684011412yes—
235C4#1684012412yes—
236C2#2144493511yescharacteristic
237C2#2144495911yescharacteristic
238C2#2144498412yes—
239C2#2144695712yes—
240C2#2144697211yescharacteristic
subgroupG/H
centerC23#14519746
commutator, cosocleC2×C42×C61C23
FrattiniC2×C42C22×D61
FittingC61×#2551C2
socle, prime radicalC23×C61#120
hypercenter#2551D61
hypocenterC61#10349
LCSG ▷ C2×C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2551

Complements

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

NH ≅ G/Ntype
22. #14519777Q8semidirect
137. C61#10349semidirect

Maximal subgroups

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

subgroupindex
#214467672
#214466202
#214437022
#214436602
C61×#25512
#1034961