Finite Group Databasebeta

← #29572391

Subgroups — #29572391

Order 15616.

620 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 620 conjugacy classes of subgroups — too many to draw. Only its 107 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#21460314C211yesmaximal, characteristic
2#21446801C211yesmaximal, characteristic
3#21444550C212yesmaximal
4#21444332C212yesmaximal
5C61×#1537C211yesmaximal, characteristic
6C61⋊#564—22no—
7#16840491—22no—
8#16840490—22no—
9#16840487C414yescomplemented
10#16840374—22no—
11#16840372—24no—
12#16840319C2212yes—
13#16839786C2211yescharacteristic
14#16839608C2211yescharacteristic
15#16839521—22no—
16#16839510—22no—
17#16839507—22no—
18#16839500—24no—
19C61×#533C2211yescharacteristic
20C61×#432C2212yes—
21C61×#330—22no—
22C2×#11758194—21no—
23#14519777—22no—
24#14519777D412yes—
25#14519777D412yes—
26#14519775—24no—
27#14519770—28no—
28#14519769—22no—
29#14519769—22no—
30#14519768—28no—
31#14519768—24no—
32#14519767—28no—
33#14519767—28no—
34#14519756—22no—
35#14519756D414yes—
36Q8×Dic61—24no—
37C4×Dic122—24no—
38C4×Dic122—22no—
39C4×Dic122C2×C414yes—
40#14519742—24no—
41#14519736—22no—
42C2×C4×Dic61—21no—
43C4⋊#92418—22no—
44#14519684—22no—
45C4⋊Q8×C61C2×C412yes—
46#125×C61D412yes—
47C22⋊Q8×C61—22no—
48C4×Q8×C61—22no—
49#117×C61D412yes—
50C2×C42×C61C2311yescharacteristic
51C4⋊C8×C61—22no—
52C22⋊C8×C61—22no—
53#97×C61—22no—
54C2×Q8×C61—22no—
55C22×C4×C61C2×D411yescharacteristic
56C22×C4×C61C2×D412yes—
57C22×Dic61—41no—
58C2×Dic122—22no—
59C2×Dic122C22⋊C414yes—
60C2×C8×C61—42no—
61C4⋊C4×C61—22no—
62C4⋊C4×C61—24no—
63C4⋊C4×C61C22⋊C414yes—
64C22⋊C4×C61—22no—
65C42×C61—22no—
66C42×C61C22×C411yescharacteristic
67C42×C61C2×D412yes—
68C42×C61C2×D411yescharacteristic
69C22⋊Dic61—24no—
70C4⋊Dic61—28no—
71C4⋊Dic61—22no—
72C4⋊Dic61—22no—
73C4⋊Dic61—22no—
74C4⋊Dic61C22⋊C414yes—
75#11758196—28no—
76#11758196—28no—
77#11758196—24no—
78#11758196—24no—
79C4×Dic61—28no—
80C4×Dic61—24no—
81C4×Dic61—22no—
82C4×Dic61—41no—
83#11758194—42no—
84C2×#92418—41no—
85C23×C61#12011yescharacteristic
86Q8×C61—44no—
87C2×C4×C61—21no—
88C2×C4×C61—21no—
89C2×C4×C61—24no—
90C2×C4×C61—42no—
91C2×C4×C61C2×C22⋊C412yes—
92C2×C4×C61C2×C22⋊C411yescharacteristic
93C2×C4×C61#12012yes—
94C2×C4×C61#12011yescharacteristic
95C2×Dic61—24no—
96C2×Dic61—24no—
97C2×Dic61—42no—
98C2×Dic61—82no—
99Dic122—42no—
100Dic122—42no—
101Dic122—44no—
102C8×C61—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#45411yescharacteristic
113C4×C61#45611yescharacteristic
114Dic61—44no—
115Dic61—44no—
116Dic61—82no—
117#3556—611no—
118#1995—611no—
119#1559—612no—
120#1537D6111yescharacteristic
121#1506—612no—
122C2×C61—41no—
123C2×C61#179311yescharacteristic
124C2×C61#179612yes—
125#564—1222no—
126#557—614no—
127#557—1222no—
128#557—1222no—
129#541—1224no—
130#541—1222no—
131#533—612no—
132#533C2×D6111yescharacteristic
133#432C2×D6112yes—
134#423—611no—
135#376—611no—
136#339—1222no—
137#333—1222no—
138#332—1222no—
139#330—22no—
140#328—1224no—
141C61#849511yescharacteristic, complemented, Sylow
142C2×M16—1221no—
143C4⋊Q8—612no—
144C4⋊Q8—612no—
145C4⋊Q8—1222no—
146C4⋊Q8—1224no—
147C4⋊Q8C4×D6112yes—
148#126—1228no—
149#125—1222no—
150#125—1224no—
151#125—1228no—
152#125—1228no—
153#125—1228no—
154#125—1222no—
155#125C22⋊D6112yes—
156C22⋊Q8—22no—
157C22⋊Q8—614no—
158C22⋊Q8—1222no—
159C4×Q8—22no—
160C4×Q8—614no—
161C4×Q8—1224no—
162C4×Q8—1224no—
163C4×Q8—1224no—
164C4×Q8—1222no—
165#117—1222no—
166#117D24412yes—
167C2×C42—1221no—
168C2×C42C22×D6111yescharacteristic
169C4⋊C8—22no—
170C4⋊C8—1222no—
171C22⋊C8—22no—
172#97—22no—
173#95—1222no—
174C2×Q8—22no—
175C2×Q8—614no—
176C2×Q8—1222no—
177C22×C4—2441no—
178C22×C4C2×D24411yescharacteristic
179C22×C4D4×D6112yes—
180M16—2442no—
181C2×C8—42no—
182C2×C8—2441no—
183C4⋊C4—22no—
184C4⋊C4—24no—
185C4⋊C4—614no—
186C4⋊C4—1228no—
187C4⋊C4—1228no—
188C4⋊C4—1228no—
189C4⋊C4—1224no—
190C4⋊C4—1224no—
191C4⋊C4—1222no—
192C4⋊C4—1222no—
193C4⋊C4—1222no—
194C4⋊C4#1175819814yes—
195C22⋊C4—22no—
196C22⋊C4—1224no—
197C42—22no—
198C42—1228no—
199C42—1222no—
200C42—1224no—
201C42—2441no—
202C42C2×C4×D6111yescharacteristic
203C42D4×D6112yes—
204C42C2×C22⋊D6111yescharacteristic
205C23#1451974711yescharacteristic
206Q8—44no—
207Q8—2444no—
208Q8—2442no—
209Q8—2442no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—24no—
213C2×C4—42no—
214C2×C4—1224no—
215C2×C4—1224no—
216C2×C4—4882no—
217C2×C4—2442no—
218C2×C4C22⋊C4×D6112yes—
219C2×C4C2×#1175819811yescharacteristic
220C2×C4#1451974512yes—
221C2×C4#1451974711yescharacteristic
222C8—82no—
223C8—4881no—
224C22—21no—
225C22—22no—
226C22#1683961711yescharacteristic
227C4—21no—
228C4—22no—
229C4—22no—
230C4—44no—
231C4—4882no—
232C4—2444no—
233C4—2444no—
234C4#1683993711yescharacteristic
235C4#1683994711yescharacteristic
236C2—41no—
237C2#2144578011yescharacteristic
238C2#2144579312yes—
subgroupG/H
centerC22#16839617
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1537C2
socleC22×C61#379
prime radicalC23×C61#120
cosocleC2×C42×C61C23
hypercenter#1537D61
hypocenterC61#8495
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1537

Complements

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

NH ≅ G/Ntype
9. #16840487C4semidirect
141. C61#8495semidirect

Maximal subgroups

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

subgroupindex
#214603142
#214468012
#214445502
#214443322
C61×#15372
#849561