Finite Group Databasebeta

← #30608427

Subgroups — #30608427

Order 15616.

322 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 60 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#21462112C211yesmaximal, characteristic
2#21462092C211yesmaximal, characteristic
3#21460967C214yesmaximal, complemented
4#21456220C211yesmaximal, characteristic
5#16840748—21no—
6#16840741—21no—
7#16840673—21no—
8#16840671C2212yes—
9#16840669C2212yes—
10#16840667—21no—
11#16840312C2211yescharacteristic
12C2×#14519769C411yescharacteristic, complemented
13#16840131—21no—
14#16840127—21no—
15#16840117C412yescomplemented
16#16840117C2212yes—
17#16839764—21no—
18#16839764—21no—
19C4×M16×C61C411yescharacteristic, complemented
20D61⋊M16—41no—
21#14519856—41no—
22#14519838—22no—
23#14519837—41no—
24#14519836—22no—
25#14519835—41no—
26C2×#11758222D411yescharacteristic
27C2×#11758222D411yescharacteristic
28#14519777D411yescharacteristic
29#14519774D411yescharacteristic
30#14519769—21no—
31#14519769—41no—
32#14519769C2×C412yes—
33#14519765C2×C411yescharacteristic
34#14519765C2311yescharacteristic
35#14519751—41no—
36C4×D244—41no—
37C2×C4⋊Dic61—21no—
38C2×#11758196—41no—
39#14519702—22no—
40#14519702—22no—
41#14519699—22no—
42#14519699—22no—
43C2×M16×C61—21no—
44C2×C42×C61C2×C411yescharacteristic
45#97×C61—21no—
46C4×C8×C61C2×C412yes—
47C22×C4×C61—21no—
48C22×C4×C61C22⋊C411yescharacteristic
49C2×C22⋊D61—21no—
50C2×C22⋊D61—21no—
51C22×Dic61—41no—
52#11758222—41no—
53#11758222—41no—
54#11758222—41no—
55#11758222—41no—
56C2×D244—41no—
57C2×D244C22⋊C411yescharacteristic
58C2×D244C2×D411yescharacteristic
59C2×C4×D61—21no—
60C2×C4×D61—21no—
61C2×Dic122C22⋊C411yescharacteristic
62C2×Dic122C2×D411yescharacteristic
63C22⋊#6923—42no—
64#11758217—42no—
65C4⋊#6923—81no—
66C4×#6923—81no—
67#11758213—81no—
68D61⋊C8—81no—
69M16×C61—41no—
70M16×C61—41no—
71C2×C8×C61—22no—
72C42×C61—21no—
73C42×C61C22⋊C411yescharacteristic
74C42×C61C22×C411yescharacteristic
75#11758198—41no—
76#11758198—41no—
77C4⋊Dic61—22no—
78C4⋊Dic61—41no—
79C4⋊Dic61—41no—
80#11758196—42no—
81#11758196—42no—
82C23×C61#9911yescharacteristic
83C22×D61—21no—
84C22×D61—21no—
85C2×#6923—82no—
86C2×C4×C61—21no—
87C2×C4×C61—21no—
88C2×C4×C61—21no—
89C2×C4×C61—21no—
90C2×C4×C61—81no—
91C2×C4×C61#9911yescharacteristic
92C2×C4×C61C2×C22⋊C411yescharacteristic
93C22⋊D61—81no—
94C22⋊D61—81no—
95C2×Dic61—21no—
96C2×Dic61—21no—
97C2×Dic61—42no—
98C2×Dic61—82no—
99D244—41no—
100D244—41no—
101D244—82no—
102C4×D61—41no—
103C4×D61—41no—
104C4×D61—41no—
105C4×D61—41no—
106Dic122—41no—
107Dic122—41no—
108#92420—82no—
109C8×C61—42no—
110#13604—611nomaximal, Sylow
111C22×C61—21no—
112C22×C61—41no—
113C22×C61C2×#9911yescharacteristic
114C2×D61—41no—
115C2×D61—41no—
116C2×D61—81no—
117C2×D61—81no—
118#6923—162no—
119C4×C61—21no—
120C4×C61—41no—
121C4×C61—41no—
122C4×C61#35411yescharacteristic
123C4×C61#35411yescharacteristic
124Dic61—41no—
125Dic61—41no—
126Dic61—82no—
127#3142—611no—
128#2132—611no—
129#2126—611no—
130#1414—614no—
131C2×C61—21no—
132C2×C61—41no—
133C2×C61C2×#35411yescharacteristic
134D61—81no—
135D61—81no—
136#532—611no—
137C2×#125—611no—
138#491—1221no—
139#490—1221no—
140#488—612no—
141#488—612no—
142#417—1221no—
143#417—1221no—
144#413—1221no—
145#410—1221no—
146C4×M16#692311yescharacteristic, complemented
147#356—1221no—
148#355—612no—
149#354—612no—
150#353—1221no—
151C61#1360411yescharacteristic, complemented, Sylow
152C2×#41—611no—
153C2×#41—611no—
154C2×M16—21no—
155C2×M16—2441no—
156C4⋊Q8—611no—
157#127—611no—
158#125—612no—
159#125—1221no—
160#125—2441no—
161#124—611no—
162#124—611no—
163#121—2441no—
164C4×D4—2441no—
165#117—2441no—
166C2×C4⋊C4—1221no—
167C2×C4⋊C4—2441no—
168C2×C42C2×#692311yescharacteristic
169#103—1222no—
170#103—1222no—
171#102—1222no—
172#102—1222no—
173#101—1222no—
174#100—2441no—
175#99—1222no—
176#99—2441no—
177#97—21no—
178C4×C8C2×#692312yes—
179#41—2441no—
180#41—2441no—
181#41—2441no—
182#41—2441no—
183C2×Q8—611no—
184C2×Q8—611no—
185C2×D4—611no—
186C2×D4—611no—
187C2×D4—1221no—
188C2×D4—1221no—
189C2×D4—2441no—
190C22×C4—21no—
191C22×C4—1221no—
192C22×C4—1221no—
193C22×C4—2441no—
194C22×C4C22⋊#692311yescharacteristic
195M16—41no—
196M16—41no—
197M16—4881no—
198M16—2442no—
199C2×C8—22no—
200C2×C8—4881no—
201C4⋊C4—1222no—
202C4⋊C4—4881no—
203C4⋊C4—2441no—
204C4⋊C4—2442no—
205C4⋊C4—2442no—
206C4⋊C4—2441no—
207C22⋊C4—2441no—
208C22⋊C4—2441no—
209C22⋊C4—2442no—
210C42—21no—
211C42—4881no—
212C42C22⋊#692311yescharacteristic
213C42C22×#692311yescharacteristic
214C23—1221no—
215C23—1221no—
216C23#1451983611yescharacteristic
217Q8—2441no—
218Q8—2441no—
219D4—4882no—
220D4—4881no—
221D4—4881no—
222D4—2441no—
223D4—2441no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—81no—
229C2×C4—1221no—
230C2×C4—1221no—
231C2×C4—4882no—
232C2×C4—4882no—
233C2×C4—2442no—
234C2×C4—2441no—
235C2×C4—2441no—
236C2×C4—2441no—
237C2×C4—2441no—
238C2×C4#1451983611yescharacteristic
239C2×C4C2×C22⋊#692311yescharacteristic
240C8—42no—
241C8—4882no—
242C22—21no—
243C22—41no—
244C22—4881no—
245C22—4881no—
246C22—2441no—
247C22—2441no—
248C22C2×#1451983611yescharacteristic
249C4—21no—
250C4—41no—
251C4—41no—
252C4—9762no—
253C4—4882no—
254C4—2441no—
255C4—2441no—
256C4#1684066911yescharacteristic
257C4#1684066911yescharacteristic
258C2—21no—
259C2—41no—
260C2—4881no—
261C2—4881no—
262C2C2×#1684066911yescharacteristic
subgroupG/H
centerC2C2×#16840669
commutatorC42×C61C22×C4
FrattiniC42C22×#6923
FittingC4×M16×C61C4
socleC2×C61C2×#354
prime radical#16840312C22
cosocle#14519765C23
hypercenterC4×M16#6923
hypocenterC61#13604
LCSG ▷ C42×C61 ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ C2×C42 ◁ C4×M16

Complements

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

NH ≅ G/Ntype
3. #21460967C2semidirect
12. C2×#14519769C4semidirect
15. #16840117C4semidirect
19. C4×M16×C61C4semidirect
146. C4×M16#6923semidirect
151. C61#13604semidirect

Maximal subgroups

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

subgroupindex
#214621122
#214620922
#214609672
#214562202
#1360461