Finite Group Databasebeta

← #29591543

Subgroups — #29591543

Order 15616.

412 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 79 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#21456271C211yesmaximal, characteristic
2#21447631C211yesmaximal, characteristic, complemented
3#21443859C212yesmaximal, complemented
4#21443855C212yesmaximal, complemented
5C61×#2165C211yesmaximal, characteristic, complemented
6C2×#14519805—21no—
7C2×#14519805—21no—
8#16840312C411yescharacteristic
9C2×#14519774C411yescharacteristic
10#16840151—21no—
11#16840151—21no—
12#16840139C411yescharacteristic
13#16840139C2211yescharacteristic
14#16840135C411yescharacteristic
15#16840135C2211yescharacteristic
16#16839583C2212yescomplemented
17#16839544—21no—
18#16839540—21no—
19C4×M16×C61C2211yescharacteristic
20C61×#364C2212yescomplemented
21C61×#349—21no—
22C61×#347—21no—
23C2×#11758222—21no—
24C22×D244—21no—
25C22×D244—41no—
26#14519805—44no—
27#14519805—44no—
28C2×#11758190—21no—
29C2×#11758190—21no—
30C2×#11758190—21no—
31C2×#11758190—21no—
32C2×D488—21no—
33C2×D488—21no—
34C2×D488—21no—
35C2×D488—21no—
36#14519777C2×C411yescharacteristic
37#14519774—21no—
38#14519774—41no—
39#14519774C2×C411yescharacteristic
40#14519774C2×C411yescharacteristic
41#14519774C2×C411yescharacteristic
42#14519765—21no—
43#14519751—41no—
44C4×D244—41no—
45#14519717—22no—
46C2×M16×C61D411yescharacteristic
47C2×M16×C61D411yescharacteristic
48C2×C42×C61C2×C411yescharacteristic
49M32×C61—22no—
50#97×C61D411yescharacteristic
51#97×C61D411yescharacteristic
52C4×C8×C61C2×C411yescharacteristic
53C4×C8×C61C2311yescharacteristic
54C23×D61—41no—
55C22×C4×C61—21no—
56C22×C4×C61C22⋊C411yescharacteristic
57C2×C22⋊D61—41no—
58#11758222—42no—
59#11758222—42no—
60C2×D244—21no—
61C2×D244—21no—
62C2×D244—21no—
63C2×D244—42no—
64C2×D244—42no—
65C2×D244—42no—
66C2×D244—41no—
67C2×D244—41no—
68C2×D244—41no—
69C2×D244—41no—
70C2×D244—84no—
71C2×C4×D61—41no—
72C2×Dic122—21no—
73M16×C61—22no—
74M16×C61—22no—
75C2×C8×C61C22⋊C411yescharacteristic
76C2×C8×C61C22⋊C411yescharacteristic
77C2×C8×C61C2×D411yescharacteristic
78C2×C8×C61C2×D411yescharacteristic
79C42×C61—21no—
80C42×C61C22⋊C411yescharacteristic
81C42×C61C22×C411yescharacteristic
82#11758198—42no—
83C4⋊Dic61—41no—
84D488—42no—
85D488—42no—
86D488—42no—
87D488—42no—
88#11758190—42no—
89#11758190—42no—
90#11758190—42no—
91#11758190—42no—
92C16×C61—42no—
93#11758185—42no—
94C23×C61#10011yescharacteristic
95C22×D61—41no—
96C22×D61—41no—
97C22×D61—41no—
98C22×D61—82no—
99C22×D61—84no—
100C2×C4×C61—21no—
101C2×C4×C61—21no—
102C2×C4×C61—21no—
103C2×C4×C61—21no—
104C2×C4×C61—42no—
105C2×C4×C61#10011yescharacteristic
106C2×C4×C61C2×C22⋊C411yescharacteristic
107C22⋊D61—82no—
108C2×Dic61—41no—
109D244—41no—
110D244—41no—
111D244—41no—
112D244—41no—
113D244—41no—
114D244—41no—
115D244—82no—
116D244—82no—
117D244—82no—
118D244—82no—
119D244—84no—
120C4×D61—82no—
121Dic122—41no—
122Dic122—41no—
123C8×C61—21no—
124C8×C61—21no—
125C8×C61AGL(1,ℤ/8)12yes—
126C8×C61AGL(1,ℤ/8)12yes—
127#11992—611nomaximal, Sylow
128C22×C61—21no—
129C22×C61—41no—
130C22×C61C2×#10011yescharacteristic
131C2×D61—82no—
132C2×D61—82no—
133C2×D61—82no—
134C2×D61—81no—
135C2×D61—81no—
136C2×D61—81no—
137C2×D61—164no—
138C4×C61—22no—
139C4×C61—21no—
140C4×C61—41no—
141C4×C61#41811yescharacteristic
142C4×C61#41811yescharacteristic
143Dic61—81no—
144#3151—611no—
145#2165—611no—
146#2165D6111yescharacteristic, complemented
147#1359—612no—
148#1357—612no—
149C2×C61—21no—
150C2×C61—41no—
151C2×C61#155511yescharacteristic
152D61—161no—
153D61—161no—
154D61—161no—
155C2×AGL(1,ℤ/8)—1221no—
156C2×AGL(1,ℤ/8)—1221no—
157#532—611no—
158C2×#127—611no—
159#496—1221no—
160#496—1221no—
161#493—611no—
162#493—611no—
163#492—611no—
164#492—611no—
165C4×M16C2×D6111yescharacteristic
166#364—612no—
167#364C2×D6112yescomplemented
168#349—21no—
169#349—1221no—
170#347—21no—
171#347—1221no—
172C61#1199211yescharacteristic, complemented, Sylow
173C2×#41—1221no—
174C22×D4—1221no—
175C22×D4—2441no—
176AGL(1,ℤ/8)—2444no—
177AGL(1,ℤ/8)—2444no—
178C2×SD16—1221no—
179C2×SD16—1221no—
180C2×SD16—1221no—
181C2×SD16—1221no—
182C2×D8—1221no—
183C2×D8—1221no—
184C2×D8—1221no—
185C2×D8—1221no—
186C2×M16D24411yescharacteristic
187C2×M16C22⋊D6111yescharacteristic
188C4⋊Q8—611no—
189#127—611no—
190#127—611no—
191#127—611no—
192#127—1221no—
193#127—2441no—
194#124—1221no—
195#121—2441no—
196C4×D4—2441no—
197C2×C42C4×D6111yescharacteristic
198M32—22no—
199M32—1222no—
200#97D24411yescharacteristic
201#97C22⋊D6111yescharacteristic
202C4×C8C4×D6111yescharacteristic
203C4×C8C22×D6111yescharacteristic
204C24—2441no—
205#41—2442no—
206#41—2442no—
207C2×Q8—1221no—
208C2×D4—1221no—
209C2×D4—1221no—
210C2×D4—1221no—
211C2×D4—4884no—
212C2×D4—2441no—
213C2×D4—2441no—
214C2×D4—2441no—
215C2×D4—2442no—
216C2×D4—2442no—
217C2×D4—2442no—
218C2×D4—2441no—
219C2×D4—2441no—
220C22×C4—21no—
221C22×C4—2441no—
222C22×C4#1175819811yescharacteristic
223SD16—2442no—
224SD16—2442no—
225SD16—2442no—
226SD16—2442no—
227D8—2442no—
228D8—2442no—
229D8—2442no—
230D8—2442no—
231M16—22no—
232M16—22no—
233C2×C8#1175819811yescharacteristic
234C2×C8#1175819811yescharacteristic
235C2×C8C2×D24411yescharacteristic
236C2×C8C2×C22⋊D6111yescharacteristic
237C4⋊C4—2441no—
238C22⋊C4—2442no—
239C42—21no—
240C42#1175819811yescharacteristic
241C42C2×C4×D6111yescharacteristic
242C16—42no—
243C16—2442no—
244C23—4882no—
245C23—4884no—
246C23—2441no—
247C23—2441no—
248C23—2441no—
249C23#1451969311yescharacteristic
250Q8—2441no—
251Q8—2441no—
252D4—4882no—
253D4—4882no—
254D4—4882no—
255D4—4882no—
256D4—4884no—
257D4—4882no—
258D4—2441no—
259D4—2441no—
260D4—2441no—
261D4—2441no—
262D4—2441no—
263D4—2441no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—42no—
269C2×C4—4882no—
270C2×C4—2441no—
271C2×C4#1451969311yescharacteristic
272C2×C4C2×#1175819811yescharacteristic
273C8—21no—
274C8—21no—
275C8#1451980212yes—
276C8#1451980512yes—
277C22—21no—
278C22—41no—
279C22—9764no—
280C22—4882no—
281C22—4882no—
282C22—4882no—
283C22—4881no—
284C22—4881no—
285C22—4881no—
286C22C2×#1451969311yescharacteristic
287C4—22no—
288C4—21no—
289C4—41no—
290C4—4881no—
291C4#1683976611yescharacteristic
292C4#1683976811yescharacteristic
293C2—21no—
294C2—41no—
295C2—9761no—
296C2—9761no—
297C2—9761no—
298C2#2144453211yescharacteristic
subgroupG/H
centerC2#21444532
commutatorC42×C61C22×C4
FrattiniC4×C8C22×D61
FittingC61×#2165C2
socleC2×C61#1555
prime radical#21456271C2
cosocleC4×C8×C61C23
hypercenter#2165D61
hypocenterC61#11992
LCSG ▷ C42×C61 ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×M16 ◁ #2165

Complements

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

NH ≅ G/Ntype
2. #21447631C2semidirect
3. #21443859C2semidirect
4. #21443855C2semidirect
5. C61×#2165C2semidirect
16. #16839583C22semidirect
20. C61×#364C22semidirect
146. #2165D61semidirect
167. #364C2×D61semidirect
172. C61#11992semidirect

Maximal subgroups

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

subgroupindex
#214562712
#214476312
#214438592
#214438552
C61×#21652
#1199261