Finite Group Databasebeta

← #29596568

Subgroups — #29596568

Order 15616.

458 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 117 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
1C2×#106×D61C211yesmaximal, characteristic
2C2×#16839586C211yesmaximal, characteristic, complemented
3#21447429C211yesmaximal, characteristic
4#21445854C211yesmaximal, characteristic
5#21443918C211yesmaximal, characteristic, complemented
6#21443917C211yesmaximal, characteristic, complemented
7C2×C61×#365C211yesmaximal, characteristic, complemented
8C22×C8×D61C2211yescharacteristic
9C2×C4⋊C4×D61—21no—
10C2×#14519717C2211yescharacteristic
11C2×#14519713C2211yescharacteristic
12C2×#14519709C2211yescharacteristic
13C2×#14519708—21no—
14#106×D61—22no—
15#106×D61—22no—
16#16839725—21no—
17#16839586—22no—
18#16839541C414yescomplemented
19#16839526—21no—
20C2×M32×C61C2211yescharacteristic
21C2×#108×C61C2211yescharacteristic
22C2×#106×C61C2211yescharacteristic
23C61×#365—22no—
24C22×C4×D61—41no—
25C22×C4×D61D411yescharacteristic
26C2×#11758194—21no—
27C22×#92418D411yescharacteristic
28C2×C8×D61—21no—
29C2×C8×D61—21no—
30C2×C8×D61—21no—
31C2×C8×D61—21no—
32C2×C8×D61C2×C412yes—
33C2×C8×D61Q812yes—
34C2×C4⋊Dic61—21no—
35C2×#11758196—41no—
36C4⋊C4×D61—24no—
37C4⋊C4×D61—42no—
38#14519717—21no—
39#14519717—21no—
40C2×#11758185C2×C412yescomplemented
41#14519713—21no—
42#14519713—21no—
43#14519709—21no—
44#14519709D412yes—
45#14519708—24no—
46#14519689—42no—
47C2×M16×C61—21no—
48C22×C8×C61C2311yescharacteristic
49C2×C4⋊C4×C61—21no—
50M32×C61—21no—
51M32×C61—21no—
52C2×C16×C61C2×C412yescomplemented
53#108×C61—21no—
54#108×C61—21no—
55#106×C61—21no—
56#106×C61D412yes—
57C23×D61SD1611yescharacteristic
58C22×C4×C61—41no—
59C22×C4×C61C2×D411yescharacteristic
60C22×Dic61—41no—
61C22×Dic61SD1611yescharacteristic
62C2×C4×D61—21no—
63C2×C4×D61—21no—
64C2×C4×D61—21no—
65C2×C4×D61—21no—
66C2×C4×D61—44no—
67C2×C4×D61—84no—
68C2×C4×D61C4⋊C412yes—
69C2×C4×D61SD1612yes—
70M16×C61—41no—
71M16×C61—41no—
72C2×C8×C61—41no—
73C2×C8×C61C22×C411yescharacteristic
74C2×C8×C61C2×D411yescharacteristic
75C2×C8×C61C2×D411yescharacteristic
76C2×C8×C61C2×Q811yescharacteristic
77C2×C8×C61#4111yescharacteristic
78C2×C8×C61#4111yescharacteristic
79C4⋊C4×C61—22no—
80C4⋊C4×C61—41no—
81C4⋊Dic61—22no—
82C4⋊Dic61—41no—
83#11758196—44no—
84#11758194—41no—
85#11758194—41no—
86C2×#92418—21no—
87C2×#92418—21no—
88C2×#92418—41no—
89C2×#92418C4⋊C412yes—
90C8×D61—22no—
91C8×D61—22no—
92C8×D61—22no—
93C8×D61—22no—
94C16×C61—22no—
95#11758185—22no—
96C23×C61C2×SD1611yescharacteristic
97C22×D61—21no—
98C22×D61—21no—
99C22×D61—24no—
100C22×D61#10612yes—
101C2×C4×C61—42no—
102C2×C4×C61—82no—
103C2×C4×C61C2×C4⋊C411yescharacteristic
104C2×C4×C61#12011yescharacteristic
105C2×C4×C61#12311yescharacteristic
106C2×C4×C61C2×SD1611yescharacteristic
107C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
108C2×C4×C61#13711yescharacteristic
109C2×Dic61—21no—
110C2×Dic61—21no—
111C2×Dic61—42no—
112C2×Dic61—82no—
113C2×Dic61#10612yes—
114C4×D61—22no—
115C4×D61—22no—
116C4×D61—22no—
117C4×D61—22no—
118C4×D61—88no—
119C8×C61—81no—
120C8×C61C4×D411yescharacteristic
121C8×C61C4×D411yescharacteristic
122C8×C61C22⋊Q811yescharacteristic
123C8×C61C22⋊Q811yescharacteristic
124#92418—22no—
125#92418—81no—
126#12929—611nomaximal, Sylow
127C22×C61—22no—
128C22×C61C2×#10611yescharacteristic
129C22×C61#45011yescharacteristic
130C22×C61#48311yescharacteristic
131C2×D61—24no—
132C2×D61—24no—
133C2×D61—22no—
134C2×D61—22no—
135C2×D61#36514yescomplemented
136C4×C61—82no—
137C4×C61#38511yescharacteristic
138C4×C61#42811yescharacteristic
139C4×C61#44811yescharacteristic
140C4×C61#48111yescharacteristic
141Dic61—22no—
142Dic61—82no—
143C22×#106—611no—
144C2×#365—611no—
145C2×#365D6111yescharacteristic, complemented
146#2112—611no—
147#1810—611no—
148#1383—611no—
149#1383—611no—
150C2×C61—21no—
151C2×C61—21no—
152C2×C61#185311yescharacteristic
153C2×C61C2×#36512yes—
154D61—24no—
155C23×C8—611no—
156C22×C4⋊C4—1221no—
157C2×M32—611no—
158C2×M32C2×D6111yescharacteristic
159C2×#108—611no—
160C2×#108C2×D6111yescharacteristic
161C2×#106—611no—
162C2×#106—1222no—
163C2×#106—1221no—
164C2×#106—1222no—
165C2×#106C2×D6111yescharacteristic
166#407—1221no—
167#365—22no—
168#365—1222no—
169C22⋊C16—614no—
170#341—1221no—
171C61#1292911yescharacteristic, complemented, Sylow
172C23×C4—611no—
173C23×C4—2441no—
174C2×M16—21no—
175C2×M16—1221no—
176C22×C8—611no—
177C22×C8—612no—
178C22×C8—612no—
179C22×C8—1221no—
180C22×C8—1221no—
181C22×C8—1221no—
182C22×C8—1221no—
183C22×C8C22×D6111yescharacteristic
184C2×C4⋊C4—21no—
185C2×C4⋊C4—1224no—
186C2×C4⋊C4—1221no—
187C2×C4⋊C4—2442no—
188C2×C4⋊C4—2441no—
189M32—21no—
190M32—21no—
191M32—1221no—
192M32—1221no—
193C2×C16—612no—
194C2×C16C4×D6112yescomplemented
195#108—21no—
196#108—21no—
197#108—1221no—
198#108—1221no—
199#106—21no—
200#106—612no—
201#106—1224no—
202#106—1221no—
203#106C22⋊D6112yes—
204C22⋊C8—2442no—
205C24—611no—
206C22×C4—41no—
207C22×C4—611no—
208C22×C4—612no—
209C22×C4—612no—
210C22×C4—1221no—
211C22×C4—1221no—
212C22×C4—1221no—
213C22×C4—1221no—
214C22×C4—4884no—
215C22×C4—2444no—
216C22×C4—2441no—
217C22×C4D4×D6111yescharacteristic
218M16—41no—
219M16—41no—
220M16—2441no—
221M16—2441no—
222C2×C8—41no—
223C2×C8—612no—
224C2×C8—1221no—
225C2×C8—1221no—
226C2×C8—1222no—
227C2×C8—1222no—
228C2×C8—1222no—
229C2×C8—1222no—
230C2×C8—2441no—
231C2×C8C2×C4×D6111yescharacteristic
232C2×C8#1175822211yescharacteristic
233C2×C8D4×D6111yescharacteristic
234C2×C8Q8×D6111yescharacteristic
235C2×C8#1175822611yescharacteristic
236C2×C8C2×C22⋊D6111yescharacteristic
237C4⋊C4—22no—
238C4⋊C4—41no—
239C4⋊C4—1222no—
240C4⋊C4—2444no—
241C4⋊C4—2441no—
242C16—22no—
243C16—1222no—
244C23—612no—
245C23—1224no—
246C23—1221no—
247C23—1221no—
248C23SD16×D6111yescharacteristic
249C2×C4—42no—
250C2×C4—82no—
251C2×C4—612no—
252C2×C4—1221no—
253C2×C4—1221no—
254C2×C4—1222no—
255C2×C4—1222no—
256C2×C4—1222no—
257C2×C4—1222no—
258C2×C4—4888no—
259C2×C4—4882no—
260C2×C4—2442no—
261C2×C4C4⋊C4×D6111yescharacteristic
262C2×C4#1451974511yescharacteristic
263C2×C4#1451976211yescharacteristic
264C2×C4SD16×D6111yescharacteristic
265C2×C4#1451980411yescharacteristic
266C2×C4#1451980611yescharacteristic
267C8—81no—
268C8—1222no—
269C8—4881no—
270C8C4×C22⋊D6111yescharacteristic
271C8#1451973911yescharacteristic
272C8#1451975311yescharacteristic
273C8#1451975511yescharacteristic
274C22—22no—
275C22—614no—
276C22—1224no—
277C22—1224no—
278C22—1222no—
279C22—1222no—
280C22#106×D6111yescharacteristic
281C22#1683991111yescharacteristic
282C22#1684009411yescharacteristic
283C4—82no—
284C4—1222no—
285C4—4882no—
286C4#1683963311yescharacteristic
287C4#1683980111yescharacteristic
288C4#1683989911yescharacteristic
289C4#1684008211yescharacteristic
290C2—21no—
291C2—21no—
292C2—1224no—
293C2#2144601011yescharacteristic
294C2#365×D6112yes—
subgroupG/H
centerC22#106×D61
commutatorC2×C8×C61C22×C4
FrattiniC22×C8C22×D61
FittingC2×C61×#365C2
socleC22×C61C2×#106
prime radicalC23×D61SD16
cosocleC22×C8×C61C23
hypercenterC2×#365D61
hypocenterC61#12929
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C2×#365

Complements

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

NH ≅ G/Ntype
2. C2×#16839586C2semidirect
5. #21443918C2semidirect
6. #21443917C2semidirect
7. C2×C61×#365C2semidirect
18. #16839541C4semidirect
40. C2×#11758185C2×C4semidirect
52. C2×C16×C61C2×C4semidirect
135. C2×D61#365semidirect
145. C2×#365D61semidirect
171. C61#12929semidirect
194. C2×C16C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#106×D612
C2×#168395862
#214474292
#214458542
#214439182
#214439172
C2×C61×#3652
#1292961