Finite Group Databasebeta

← #29572383

Subgroups — #29572383

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#21444539C212yesmaximal, complemented
3#21444457C211yesmaximal, characteristic, complemented
4#21444326C212yesmaximal, complemented
5C61×#1995C211yesmaximal, characteristic, complemented
6#16840510—22no—
7#16840472—22no—
8#16840434—22no—
9#16840415C414yescomplemented
10#16840367—22no—
11#16840342—24no—
12#16840312C2212yes—
13#16839819C2212yes—
14#16839507—22no—
15#16839505—22no—
16#16839504—22no—
17#16839495—24no—
18C61×#533C2211yescharacteristic
19C61×#423C2211yescharacteristic
20C61×#376C2211yescharacteristic
21C61×#339—22no—
22C2×#11758226—24no—
23C2×#11758222—22no—
24#14519777D412yes—
25#14519774—22no—
26#14519774D412yes—
27#14519769—22no—
28#14519765—22no—
29#14519762—28no—
30#14519755—28no—
31#14519753—24no—
32#14519752—28no—
33#14519752—24no—
34#14519751—22no—
35#14519751D414yes—
36#14519750—28no—
37C4×D244—24no—
38C4×D244—22no—
39C4×D244C2×C414yes—
40C4⋊C4×D61—24no—
41C4⋊#92418—22no—
42C22⋊#92418—22no—
43#14519688—22no—
44C2×M16×C61—21no—
45C4⋊Q8×C61C2×C412yes—
46#125×C61D412yes—
47C22⋊Q8×C61—22no—
48C4×Q8×C61—22no—
49#117×C61D412yes—
50C2×C42×C61—21no—
51C2×C42×C61C2311yescharacteristic
52C4⋊C8×C61—22no—
53#95×C61—22no—
54C2×Q8×C61—22no—
55C22×C4×C61—41no—
56C22×C4×C61C2×D411yescharacteristic
57C22×C4×C61C2×D412yes—
58C2×C22⋊D61—24no—
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—
68M16×C61—42no—
69C2×C8×C61—41no—
70C4⋊C4×C61—22no—
71C4⋊C4×C61—24no—
72C4⋊C4×C61C22⋊C414yes—
73C22⋊C4×C61—22no—
74C42×C61—22no—
75C42×C61—22no—
76C42×C61—41no—
77C42×C61C22×C411yescharacteristic
78C42×C61C2×D411yescharacteristic
79C42×C61C2×D412yes—
80#11758198—28no—
81#11758198—24no—
82#11758198—24no—
83C4⋊Dic61—22no—
84C4⋊Dic61C22⋊C414yes—
85#11758196—28no—
86C2×#92418—42no—
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×C61—42no—
95C2×C4×C61—82no—
96C2×C4×C61C2×C22⋊C412yes—
97C2×C4×C61C2×C22⋊C411yescharacteristic
98C2×C4×C61#12012yes—
99C2×C4×C61#12011yescharacteristic
100C22⋊D61—84no—
101C2×Dic61—24no—
102D244—44no—
103D244—42no—
104D244—42no—
105D244—82no—
106D244—88no—
107C4×D61—48no—
108C4×D61—88no—
109Dic122—82no—
110C8×C61—81no—
111#92418—82no—
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—82no—
124C4×C61#45411yescharacteristic
125C4×C61#45611yescharacteristic
126Dic61—44no—
127#3548—611no—
128#1995D6111yescharacteristic, complemented
129#1557—612no—
130#1537—611no—
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#432—612no—
145#423C2×D6111yescharacteristic
146#376C2×D6111yescharacteristic
147#339—22no—
148#332—1222no—
149#331—1222no—
150#330—1222no—
151#327—1224no—
152C61#849411yescharacteristic, complemented, Sylow
153C2×#41—1224no—
154C2×#41—1222no—
155C2×M16—21no—
156C4⋊Q8—612no—
157C4⋊Q8C4×D6112yes—
158#127—612no—
159#127—1222no—
160#125—1222no—
161#125D24412yes—
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#117C22⋊D6112yes—
177C2×C4⋊C4—1224no—
178C2×C42—21no—
179C2×C42C22×D6111yescharacteristic
180C4⋊C8—22no—
181C4⋊C8—1222no—
182C22⋊C8—1222no—
183#97—1222no—
184#95—22no—
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—41no—
195C22×C4—1228no—
196C22×C4—1224no—
197C22×C4D4×D6112yes—
198C22×C4C2×C22⋊D6111yescharacteristic
199M16—42no—
200C2×C8—41no—
201C2×C8—2442no—
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—22no—
214C42—41no—
215C42C2×C4×D6111yescharacteristic
216C42C2×D24411yescharacteristic
217C42D4×D6112yes—
218C23—1224no—
219C23#1451974511yescharacteristic
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—42no—
233C2×C4—82no—
234C2×C4—1224no—
235C2×C4—4888no—
236C2×C4—2448no—
237C2×C4C22⋊C4×D6112yes—
238C2×C4C2×#1175819811yescharacteristic
239C2×C4#1451974511yescharacteristic
240C2×C4#1451974712yes—
241C8—81no—
242C8—4882no—
243C22—21no—
244C22—22no—
245C22—2444no—
246C22—2444no—
247C22—2444no—
248C22#1683961711yescharacteristic
249C4—21no—
250C4—22no—
251C4—22no—
252C4—44no—
253C4—82no—
254C4—2444no—
255C4#1683993411yescharacteristic
256C4#1683994411yescharacteristic
257C2—41no—
258C2—4884no—
259C2#2144577711yescharacteristic
260C2#2144578812yes—
subgroupG/H
centerC22#16839617
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1995C2
socleC22×C61#379
prime radical#21460219C2
cosocleC2×C42×C61C23
hypercenter#1995D61
hypocenterC61#8494
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1995

Complements

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

NH ≅ G/Ntype
2. #21444539C2semidirect
3. #21444457C2semidirect
4. #21444326C2semidirect
5. C61×#1995C2semidirect
9. #16840415C4semidirect
128. #1995D61semidirect
152. C61#8494semidirect

Maximal subgroups

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

subgroupindex
#214602192
#214445392
#214444572
#214443262
C61×#19952
#849461