Finite Group Databasebeta

← #29677462

Subgroups — #29677462

Order 15616.

2046 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840529C211yesmaximal, characteristic
2C2×#16840245C211yesmaximal, characteristic
3#21445557C216yesmaximal, complemented
4C2×#16839530C216yesmaximal, complemented
5C2×C61×#568C211yesmaximal, characteristic
6#16840529—24no—
7#16840529C418yescomplemented
8C2×#14519784—24no—
9C22×#11758194C2216yes—
10#16840245—24no—
11#16840245—24no—
12C2×Q8×Dic61C412yes—
13C2×D4×Dic61C416yes—
14C2×#14519736C2213yes—
15C22×C4⋊Dic61C2213yes—
16#16839530C22112yes—
17#16839530C2214yes—
18C22×#41×C61C2211yescharacteristic
19C61×#568—24no—
20C61×#568C418yescomplemented
21C2×#131×C61—24no—
22C22×M16×C61C2216yes—
23C23×Dic61C2×C416yes—
24#14519784—232no—
25C2×#11758194—26no—
26C2×#11758194—212no—
27C2×#11758194C2×C4124yescomplemented
28C2×#11758194C2314yes—
29C22×#92418—26no—
30Q8×Dic61—28no—
31D4×Dic61—224no—
32#14519736—26no—
33#14519736—26no—
34C2×C4⋊Dic61—26no—
35C2×C4⋊Dic61—26no—
36C2×C4⋊Dic61C2×C416yes—
37C2×C4⋊Dic61C2313yes—
38C2×C4⋊Dic61C2311yescharacteristic
39C2×C22⋊Dic61C2×C416yes—
40C2×C4×Dic61C2×C416yes—
41C2×#41×C61C2×C414yes—
42C2×#41×C61C2×C414yes—
43C2×#41×C61D414yes—
44C2×#41×C61D414yes—
45C2×#41×C61D414yes—
46C2×#41×C61Q814yes—
47C22×Q8×C61C2×C411yescharacteristic
48C22×D4×C61C2×C413yes—
49C23×C4×C61C2313yes—
50#131×C61—232no—
51C2×M16×C61—26no—
52C2×M16×C61—212no—
53C2×M16×C61C2×C4124yescomplemented
54C2×M16×C61C2314yes—
55C22×C8×C61—26no—
56C24×C61C22×C413yes—
57#41×C61C22⋊C4116yes—
58#41×C61C22⋊C4116yes—
59#41×C61C4⋊C4116yes—
60#41×C61C4⋊C4116yes—
61C2×Q8×C61C4214yes—
62C2×Q8×C61C22⋊C414yes—
63C2×Q8×C61C4⋊C414yes—
64C2×D4×C61C42112yes—
65C2×D4×C61C22⋊C4112yes—
66C2×D4×C61C4⋊C4112yes—
67C22×C4×C61C22×C413yes—
68C22×C4×C61C22×C416yes—
69C22×C4×C61C22×C416yes—
70C22×C4×C61C2×D416yes—
71C22×C4×C61C2×D416yes—
72C22×C4×C61C2×D416yes—
73C22×C4×C61C2×Q816yes—
74C22×C4×C61C2411yescharacteristic
75C22×Dic61—212no—
76C22×Dic61—86no—
77C22×Dic61C22×C416yes—
78C22×Dic61C22×C412yes—
79M16×C61—224no—
80M16×C61—24no—
81M16×C61C22×C418yes—
82C2×C8×C61—224no—
83C2×C8×C61—24no—
84C22⋊Dic61—212no—
85C4⋊Dic61—212no—
86C4⋊Dic61—23no—
87C4⋊Dic61—23no—
88C4⋊Dic61—21no—
89C4⋊Dic61—21no—
90C4×Dic61—212no—
91#11758194—224no—
92#11758194—24no—
93#11758194C22×C418yes—
94C2×#92418—224no—
95C2×#92418—24no—
96C23×C61—43no—
97C23×C61—43no—
98C23×C61C2×C4216yes—
99C23×C61C2×C22⋊C416yes—
100C23×C61C2×C4⋊C416yes—
101C23×C61C23×C411yescharacteristic
102Q8×C61#95116yes—
103D4×C61#95148yes—
104C2×C4×C61C2×C4216yes—
105C2×C4×C61C2×C22⋊C4112yes—
106C2×C4×C61C2×C22⋊C4112yes—
107C2×C4×C61C2×C22⋊C416yes—
108C2×C4×C61C2×C4⋊C4112yes—
109C2×C4×C61C2×C4⋊C4112yes—
110C2×C4×C61C2×C4⋊C416yes—
111C2×C4×C61C23×C411yescharacteristic
112C2×C4×C61C23×C411yescharacteristic
113C2×C4×C61C22×D411yescharacteristic
114C2×C4×C61C22×D411yescharacteristic
115C2×C4×C61C22×D411yescharacteristic
116C2×C4×C61C22×Q811yescharacteristic
117C2×Dic61—22no—
118C2×Dic61—26no—
119C2×Dic61—46no—
120C2×Dic61—42no—
121C2×Dic61—812no—
122C8×C61—28no—
123#92418—28no—
124#20358—611nomaximal, Sylow
125C22×C61—21no—
126C22×C61—26no—
127C22×C61—21no—
128C22×C61—26no—
129C22×C61—46no—
130C22×C61—46no—
131C22×C61C2×#95112yes—
132C22×C61C22×C4211yescharacteristic
133C22×C61C22×C22⋊C411yescharacteristic
134C22×C61C22×C4⋊C411yescharacteristic
135C4×C61C2×#95112yes—
136C4×C61C22×C22⋊C411yescharacteristic
137C4×C61C22×C22⋊C411yescharacteristic
138C4×C61C22×C4⋊C411yescharacteristic
139C4×C61C22×C4⋊C411yescharacteristic
140Dic61—46no—
141Dic61—42no—
142C2×#568—611no—
143C2×#568D6111yescharacteristic
144C2×#520—611no—
145#1742—616no—
146C2×#343—616no—
147C2×C61—21no—
148C2×C61—21no—
149C2×C61—212no—
150C2×C61C22×#9511yescharacteristic
151C2×C61#288811yescharacteristic
152C2×C61#288911yescharacteristic
153C22×#41C2×D6111yescharacteristic
154#568—24no—
155#568—618no—
156#568—1224no—
157#568Dic6118yescomplemented
158C2×#131—24no—
159C2×#131—1224no—
160C22×M16—616no—
161C22×M16C2×D6116yes—
162#520—1224no—
163#520—1224no—
164C2×C4×Q8—612no—
165C2×C4×D4—616no—
166C2×#117—613no—
167C22×C4⋊C4—613no—
168#343—6112no—
169#343—614no—
170C61#2035811yescharacteristic, complemented, Sylow
171C2×#41Dic12214yes—
172C2×#41C4×D6114yes—
173C2×#41D24414yes—
174C2×#41C2×Dic6114yes—
175C2×#41C22⋊D6114yes—
176C2×#41C22⋊D6114yes—
177C22×Q8C4×D6111yescharacteristic
178C22×D4C4×D6113yes—
179C23×C4—616no—
180C23×C4C22×D6113yes—
181#131—232no—
182#131—12232no—
183C2×M16—26no—
184C2×M16—212no—
185C2×M16—6124no—
186C2×M16—614no—
187C2×M16—1226no—
188C2×M16—12212no—
189C2×M16C2×Dic61124yescomplemented
190C2×M16C22×D6114yes—
191C22×C8—26no—
192C22×C8—1226no—
193C4×Q8—1228no—
194C4×D4—12224no—
195#117—1226no—
196#117—1226no—
197C2×C4⋊C4—616no—
198C2×C4⋊C4—613no—
199C2×C4⋊C4—611no—
200C2×C4⋊C4—1226no—
201C2×C4⋊C4—1226no—
202C2×C22⋊C4—616no—
203C2×C42—616no—
204C24C2×C4×D6113yes—
205#41#11758196116yes—
206#41C4⋊Dic61116yes—
207#41#11758198116yes—
208#41C22⋊Dic61116yes—
209C2×Q8C4×Dic6114yes—
210C2×Q8#1175819614yes—
211C2×Q8#1175819814yes—
212C2×D4C4×Dic61112yes—
213C2×D4#11758196112yes—
214C2×D4#11758198112yes—
215C22×C4—612no—
216C22×C4—616no—
217C22×C4—12212no—
218C22×C4—4886no—
219C22×C4C2×Dic12216yes—
220C22×C4C2×C4×D6116yes—
221C22×C4C2×C4×D6113yes—
222C22×C4C2×D24416yes—
223C22×C4C22×Dic6116yes—
224C22×C4C2×C22⋊D6116yes—
225C22×C4C2×C22⋊D6116yes—
226C22×C4C23×D6111yescharacteristic
227M16—24no—
228M16—224no—
229M16—618no—
230M16—1224no—
231M16—12224no—
232M16C22×Dic6118yes—
233C2×C8—24no—
234C2×C8—224no—
235C2×C8—1224no—
236C2×C8—12224no—
237C4⋊C4—1223no—
238C4⋊C4—1221no—
239C4⋊C4—1223no—
240C4⋊C4—1221no—
241C4⋊C4—12212no—
242C22⋊C4—12212no—
243C42—12212no—
244C23—43no—
245C23—43no—
246C23C2×C4×Dic6116yes—
247C23C2×#1175819816yes—
248C23C2×#1175819616yes—
249C23C22×C4×D6111yescharacteristic
250Q8#14519684116yes—
251D4#14519684148yes—
252C2×C4—1222no—
253C2×C4—1226no—
254C2×C4—48812no—
255C2×C4—2442no—
256C2×C4—2446no—
257C2×C4C2×C4×Dic6116yes—
258C2×C4C2×#11758198112yes—
259C2×C4C2×#1175819816yes—
260C2×C4C2×C22⋊Dic61112yes—
261C2×C4C2×#11758196112yes—
262C2×C4C2×#1175819616yes—
263C2×C4C2×C4⋊Dic61112yes—
264C2×C4C22×C4×D6111yescharacteristic
265C2×C4C23×Dic6111yescharacteristic
266C2×C4C22×C22⋊D6111yescharacteristic
267C2×C4C22×C22⋊D6111yescharacteristic
268C2×C4C22×D24411yescharacteristic
269C2×C4C22×Dic12211yescharacteristic
270C8—28no—
271C8—1228no—
272C22—26no—
273C22—21no—
274C22—21no—
275C22—26no—
276C22—46no—
277C22—46no—
278C22C2×#14519684112yes—
279C22C22×C4×Dic6111yescharacteristic
280C22C22×#1175819811yescharacteristic
281C22C22×#1175819611yescharacteristic
282C4—2446no—
283C4—2442no—
284C4C2×#14519684112yes—
285C4C22×#1175819811yescharacteristic
286C4C22×C22⋊Dic6111yescharacteristic
287C4C22×#1175819611yescharacteristic
288C4C22×C4⋊Dic6111yescharacteristic
289C2—21no—
290C2—21no—
291C2—212no—
292C2C22×#1451968411yescharacteristic
293C2#2145429711yescharacteristic
294C2#2145430111yescharacteristic
subgroupG/H
centerC22C22×#11758198
commutatorC22×C61C22×C42
FrattiniC22×C4C23×D61
FittingC2×C61×#568C2
socleC22×C61C22×C22⋊C4
prime radicalC22×#41×C61C22
cosocleC22×C4×C61C24
hypercenterC2×#568D61
hypocenterC61#20358
LCSG ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×#41 ◁ C2×#568

Complements

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

NH ≅ G/Ntype
3. #21445557C2semidirect
4. C2×#16839530C2semidirect
7. #16840529C4semidirect
20. C61×#568C4semidirect
27. C2×#11758194C2×C4semidirect
53. C2×M16×C61C2×C4semidirect
157. #568Dic61semidirect
170. C61#20358semidirect
189. C2×M16C2×Dic61semidirect

Maximal subgroups

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

subgroupindex
C2×#168405292
C2×#168402452
#214455572
C2×#168395302
C2×C61×#5682
#2035861