Finite Group Databasebeta

← #29572386

Subgroups — #29572386

Order 15616.

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

normal subgroups only

This group has 732 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#21460219C211yesmaximal, characteristic
2#21446801C211yesmaximal, characteristic, complemented
3#21444542C212yesmaximal, complemented
4#21444328C212yesmaximal, complemented
5C61×#1537C211yesmaximal, characteristic, complemented
6#16840510—22no—
7#16840472—22no—
8#16840434—22no—
9#16840415C414yescomplemented
10#16840367—22no—
11#16840342—24no—
12#16840312C2212yes—
13#16839786C2211yescharacteristic
14#16839608C2211yescharacteristic
15#16839521—22no—
16#16839509—22no—
17#16839505—22no—
18#16839497—24no—
19C61×#533C2211yescharacteristic
20C61×#432C2212yes—
21C61×#330—22no—
22C2×#11758226—24no—
23C2×#11758222—22no—
24C2×#11758194—21no—
25#14519777D412yes—
26#14519774—22no—
27#14519774D412yes—
28#14519769—22no—
29#14519765—22no—
30#14519762—28no—
31#14519755—28no—
32#14519753—24no—
33#14519752—28no—
34#14519752—24no—
35#14519751—22no—
36#14519751D414yes—
37#14519750—28no—
38C4×D244—24no—
39C4×D244—22no—
40C4×D244C2×C414yes—
41C4⋊C4×D61—24no—
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—
57C2×C22⋊D61—24no—
58C22×Dic61—41no—
59#11758226—88no—
60#11758222—84no—
61C2×D244—22no—
62C2×D244—28no—
63C2×D244—22no—
64C2×D244C22⋊C414yes—
65C2×C4×D61—24no—
66C2×C4×D61—28no—
67C2×Dic122—22no—
68C2×C8×C61—42no—
69C4⋊C4×C61—22no—
70C4⋊C4×C61—24no—
71C4⋊C4×C61C22⋊C414yes—
72C22⋊C4×C61—22no—
73C42×C61—22no—
74C42×C61C22×C411yescharacteristic
75C42×C61C2×D412yes—
76C42×C61C2×D411yescharacteristic
77#11758198—28no—
78#11758198—24no—
79#11758198—24no—
80C4⋊Dic61—22no—
81C4⋊Dic61C22⋊C414yes—
82#11758196—28no—
83C4×Dic61—22no—
84C4×Dic61—41no—
85#11758194—42no—
86C2×#92418—41no—
87C23×C61#12011yescharacteristic
88C22×D61—24no—
89Q8×C61—44no—
90C2×C4×C61—21no—
91C2×C4×C61—21no—
92C2×C4×C61—24no—
93C2×C4×C61—42no—
94C2×C4×C61C2×C22⋊C412yes—
95C2×C4×C61C2×C22⋊C411yescharacteristic
96C2×C4×C61#12012yes—
97C2×C4×C61#12011yescharacteristic
98C22⋊D61—84no—
99C2×Dic61—24no—
100C2×Dic61—42no—
101C2×Dic61—82no—
102D244—44no—
103D244—42no—
104D244—42no—
105D244—82no—
106D244—88no—
107C4×D61—48no—
108C4×D61—88no—
109Dic122—82no—
110C8×C61—82no—
111#92418—81no—
112#8494—611nomaximal, Sylow
113C22×C61—21no—
114C22×C61—22no—
115C22×C61#37911yescharacteristic
116C2×D61—44no—
117C2×D61—44no—
118C2×D61—44no—
119C4×C61—21no—
120C4×C61—22no—
121C4×C61—22no—
122C4×C61—44no—
123C4×C61#45411yescharacteristic
124C4×C61#45611yescharacteristic
125Dic61—44no—
126Dic61—82no—
127#3548—611no—
128#1995—611no—
129#1557—612no—
130#1537D6111yescharacteristic, complemented
131#1505—612no—
132C2×C61—41no—
133C2×C61#179311yescharacteristic
134C2×C61#179512yes—
135D61—84no—
136#562—1222no—
137#554—1222no—
138#550—1222no—
139#547—614no—
140#540—1222no—
141#537—1224no—
142#533C2×D6111yescharacteristic
143#532—612no—
144#432C2×D6112yes—
145#423—611no—
146#376—611no—
147#339—1222no—
148#332—1222no—
149#331—1222no—
150#330—22no—
151#327—1224no—
152C61#849411yescharacteristic, complemented, Sylow
153C2×#41—1224no—
154C2×#41—1222no—
155C2×M16—1221no—
156C4⋊Q8—612no—
157C4⋊Q8C4×D6112yes—
158#127—612no—
159#127—1222no—
160#125—1222no—
161#125C22⋊D6112yes—
162#124—1222no—
163#123—1228no—
164C22⋊Q8—22no—
165C22⋊Q8—1228no—
166C22⋊Q8—1224no—
167#121—614no—
168#121—1228no—
169#121—1224no—
170#121—1228no—
171#121—1222no—
172C4×Q8—22no—
173C4×D4—614no—
174C4×D4—1224no—
175C4×D4—1222no—
176#117D24412yes—
177C2×C4⋊C4—1224no—
178C2×C42—1221no—
179C2×C42C22×D6111yescharacteristic
180C4⋊C8—22no—
181C4⋊C8—1222no—
182C22⋊C8—22no—
183#97—22no—
184#95—1222no—
185#41—4884no—
186#41—4888no—
187C2×Q8—22no—
188C2×Q8—1222no—
189C2×D4—614no—
190C2×D4—1228no—
191C2×D4—1224no—
192C2×D4—1222no—
193C2×D4—1222no—
194C22×C4—1228no—
195C22×C4—1224no—
196C22×C4—2441no—
197C22×C4C2×D24411yescharacteristic
198C22×C4D4×D6112yes—
199M16—2442no—
200C2×C8—42no—
201C2×C8—2441no—
202C4⋊C4—22no—
203C4⋊C4—24no—
204C4⋊C4—614no—
205C4⋊C4—1228no—
206C4⋊C4—1222no—
207C4⋊C4#1175819814yes—
208C22⋊C4—22no—
209C22⋊C4—1228no—
210C22⋊C4—1224no—
211C22⋊C4—1224no—
212C42—22no—
213C42—1222no—
214C42—2441no—
215C42C2×C4×D6111yescharacteristic
216C42D4×D6112yes—
217C42C2×C22⋊D6111yescharacteristic
218C23—1224no—
219C23#1451974711yescharacteristic
220Q8—44no—
221Q8—4882no—
222D4—4882no—
223D4—4888no—
224D4—4884no—
225D4—2444no—
226D4—2442no—
227D4—2442no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—24no—
231C2×C4—42no—
232C2×C4—1224no—
233C2×C4—4888no—
234C2×C4—4882no—
235C2×C4—2448no—
236C2×C4—2442no—
237C2×C4C22⋊C4×D6112yes—
238C2×C4C2×#1175819811yescharacteristic
239C2×C4#1451974512yes—
240C2×C4#1451974711yescharacteristic
241C8—82no—
242C8—4881no—
243C22—21no—
244C22—22no—
245C22—2444no—
246C22—2444no—
247C22—2444no—
248C22#1683961711yescharacteristic
249C4—21no—
250C4—22no—
251C4—22no—
252C4—44no—
253C4—4882no—
254C4—2444no—
255C4#1683993711yescharacteristic
256C4#1683994711yescharacteristic
257C2—41no—
258C2—4884no—
259C2#2144578011yescharacteristic
260C2#2144578812yes—
subgroupG/H
centerC22#16839617
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1537C2
socleC22×C61#379
prime radical#21460219C2
cosocleC2×C42×C61C23
hypercenter#1537D61
hypocenterC61#8494
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
2. #21446801C2semidirect
3. #21444542C2semidirect
4. #21444328C2semidirect
5. C61×#1537C2semidirect
9. #16840415C4semidirect
130. #1537D61semidirect
152. C61#8494semidirect

Maximal subgroups

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

subgroupindex
#214602192
#214468012
#214445422
#214443282
C61×#15372
#849461