Finite Group Databasebeta

← #29572389

Subgroups — #29572389

Order 15616.

620 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 620 conjugacy classes of subgroups — too many to draw. Only its 131 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#21446801C211yesmaximal, characteristic
2#21444549C212yesmaximal
3#21444457C211yesmaximal, characteristic
4#21444331C212yesmaximal
5C61×#3556C211yesmaximal, characteristic
6#16839819C2212yes—
7#16839786C2211yescharacteristic
8#16839608C2211yescharacteristic
9#16839521—22no—
10#16839511—22no—
11#16839508—22no—
12#16839504—22no—
13#16839499—24no—
14C61×#564—22no—
15C61×#557—22no—
16C61×#557—22no—
17C61×#557C414yescomplemented
18C61×#541—24no—
19C61×#541—22no—
20C61×#533C2212yes—
21C61×#533C2211yescharacteristic
22C2×#11758194—21no—
23C2×C4×Dic61—21no—
24C4⋊#92418—22no—
25C4⋊#92418—22no—
26C22⋊#92418—22no—
27#14519688—22no—
28#14519684—22no—
29C4⋊Q8×C61—22no—
30C4⋊Q8×C61—24no—
31C4⋊Q8×C61C2×C412yes—
32C4⋊Q8×C61D412yes—
33C4⋊Q8×C61D412yes—
34#126×C61—28no—
35#125×C61—22no—
36#125×C61—28no—
37#125×C61—28no—
38#125×C61—24no—
39#125×C61—28no—
40#125×C61—22no—
41#125×C61D412yes—
42C22⋊Q8×C61—22no—
43C22⋊Q8×C61—22no—
44C22⋊Q8×C61D414yes—
45C4×Q8×C61—24no—
46C4×Q8×C61—24no—
47C4×Q8×C61—24no—
48C4×Q8×C61—22no—
49C4×Q8×C61—22no—
50C4×Q8×C61C2×C414yes—
51#117×C61—22no—
52#117×C61D412yes—
53C2×C42×C61C2311yescharacteristic
54C2×Q8×C61—22no—
55C2×Q8×C61—22no—
56C2×Q8×C61C22⋊C414yes—
57C22×C4×C61C2×D411yescharacteristic
58C22×C4×C61C2×D412yes—
59C22×Dic61—41no—
60C4⋊C4×C61—28no—
61C4⋊C4×C61—28no—
62C4⋊C4×C61—28no—
63C4⋊C4×C61—22no—
64C4⋊C4×C61—24no—
65C4⋊C4×C61—22no—
66C4⋊C4×C61—22no—
67C4⋊C4×C61—22no—
68C4⋊C4×C61—24no—
69C4⋊C4×C61—24no—
70C4⋊C4×C61C22⋊C414yes—
71C4⋊C4×C61C22⋊C414yes—
72C22⋊C4×C61—24no—
73C22⋊C4×C61—22no—
74C42×C61—28no—
75C42×C61—24no—
76C42×C61—22no—
77C42×C61C22×C411yescharacteristic
78C42×C61C2×D412yes—
79C42×C61C2×D411yescharacteristic
80C4×Dic61—22no—
81C4×Dic61—41no—
82#11758194—42no—
83C2×#92418—42no—
84C2×#92418—41no—
85C23×C61#12011yescharacteristic
86Q8×C61—44no—
87Q8×C61—44no—
88Q8×C61—42no—
89Q8×C61—42no—
90C2×C4×C61—21no—
91C2×C4×C61—21no—
92C2×C4×C61—24no—
93C2×C4×C61—24no—
94C2×C4×C61—24no—
95C2×C4×C61—42no—
96C2×C4×C61C2×C22⋊C412yes—
97C2×C4×C61C2×C22⋊C411yescharacteristic
98C2×C4×C61#12012yes—
99C2×C4×C61#12011yescharacteristic
100C2×Dic61—42no—
101C2×Dic61—82no—
102#92418—82no—
103#92418—81no—
104#8495—611nomaximal, Sylow
105C22×C61—21no—
106C22×C61—22no—
107C22×C61#37911yescharacteristic
108C4×C61—21no—
109C4×C61—22no—
110C4×C61—22no—
111C4×C61—44no—
112C4×C61—44no—
113C4×C61—44no—
114C4×C61#45411yescharacteristic
115C4×C61#45611yescharacteristic
116Dic61—82no—
117#3556D6111yescharacteristic
118#1995—611no—
119#1559—612no—
120#1537—611no—
121#1506—612no—
122C2×C61—41no—
123C2×C61#179311yescharacteristic
124C2×C61#179612yes—
125#564—22no—
126#557—22no—
127#557—22no—
128#557Dic6114yescomplemented
129#541—24no—
130#541—22no—
131#533C2×D6112yes—
132#533C2×D6111yescharacteristic
133#432—612no—
134#423—611no—
135#376—611no—
136#339—1222no—
137#333—1222no—
138#332—1222no—
139#330—1222no—
140#328—1224no—
141C61#849511yescharacteristic, complemented, Sylow
142C2×M16—1221no—
143C4⋊Q8—22no—
144C4⋊Q8—24no—
145C4⋊Q8C2×Dic6112yes—
146C4⋊Q8C22⋊D6112yes—
147C4⋊Q8C22⋊D6112yes—
148#126—28no—
149#125—22no—
150#125—28no—
151#125—28no—
152#125—24no—
153#125—28no—
154#125—22no—
155#125C22⋊D6112yes—
156C22⋊Q8—22no—
157C22⋊Q8—22no—
158C22⋊Q8C22⋊D6114yes—
159C4×Q8—24no—
160C4×Q8—24no—
161C4×Q8—24no—
162C4×Q8—22no—
163C4×Q8—22no—
164C4×Q8C2×Dic6114yes—
165#117—22no—
166#117C22⋊D6112yes—
167C2×C42—1221no—
168C2×C42C22×D6111yescharacteristic
169C4⋊C8—1222no—
170C4⋊C8—1222no—
171C22⋊C8—1222no—
172#97—1222no—
173#95—1222no—
174C2×Q8—22no—
175C2×Q8—22no—
176C2×Q8C22⋊Dic6114yes—
177C22×C4—2441no—
178C22×C4C2×C22⋊D6111yescharacteristic
179C22×C4C2×C22⋊D6112yes—
180M16—2442no—
181C2×C8—2442no—
182C2×C8—2441no—
183C4⋊C4—28no—
184C4⋊C4—28no—
185C4⋊C4—28no—
186C4⋊C4—22no—
187C4⋊C4—24no—
188C4⋊C4—22no—
189C4⋊C4—22no—
190C4⋊C4—22no—
191C4⋊C4—24no—
192C4⋊C4—24no—
193C4⋊C4C22⋊Dic6114yes—
194C4⋊C4C22⋊Dic6114yes—
195C22⋊C4—24no—
196C22⋊C4—22no—
197C42—28no—
198C42—24no—
199C42—22no—
200C42—1222no—
201C42—2441no—
202C42C22×Dic6111yescharacteristic
203C42C2×C22⋊D6111yescharacteristic
204C42C2×C22⋊D6112yes—
205C23#1451974611yescharacteristic
206Q8—44no—
207Q8—44no—
208Q8—42no—
209Q8—42no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—24no—
213C2×C4—24no—
214C2×C4—24no—
215C2×C4—42no—
216C2×C4—4882no—
217C2×C4—2442no—
218C2×C4C2×C22⋊Dic6112yes—
219C2×C4C2×C22⋊Dic6111yescharacteristic
220C2×C4#1451974612yes—
221C2×C4#1451974611yescharacteristic
222C8—4882no—
223C8—4881no—
224C22—21no—
225C22—22no—
226C22#1683961811yescharacteristic
227C4—21no—
228C4—22no—
229C4—22no—
230C4—44no—
231C4—44no—
232C4—44no—
233C4—4882no—
234C4#1683993511yescharacteristic
235C4#1683994511yescharacteristic
236C2—41no—
237C2#2144577811yescharacteristic
238C2#2144579412yes—
subgroupG/H
centerC22#16839618
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#3556C2
socleC22×C61#379
prime radicalC23×C61#120
cosocleC2×C42×C61C23
hypercenter#3556D61
hypocenterC61#8495
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3556

Complements

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

NH ≅ G/Ntype
17. C61×#557C4semidirect
128. #557Dic61semidirect
141. C61#8495semidirect

Maximal subgroups

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

subgroupindex
#214468012
#214445492
#214444572
#214443312
C61×#35562
#849561