Finite Group Databasebeta

← #30608338

Subgroups — #30608338

Order 15616.

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

normal subgroups only

This group has 454 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, complemented
2#21462092C211yesmaximal, characteristic, complemented
3#21460946C214yesmaximal, complemented
4#21456269C211yesmaximal, characteristic
5#16840748—21no—
6#16840741—21no—
7#16840673—21no—
8#16840671C2212yescomplemented
9#16840669C2212yescomplemented
10#16840667—21no—
11C2×#14519803—21no—
12C2×#14519803—21no—
13#16840312C2211yescharacteristic
14#16840149C412yescomplemented
15#16840149C2212yes—
16#16840138—21no—
17#16840134—21no—
18C4×#11758194C411yescharacteristic, complemented
19C2×#127×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#14519803—42no—
29#14519803—42no—
30#14519803—42no—
31#14519803—42no—
32C2×#11758200—22no—
33C2×#11758200—22no—
34C2×#11758199—22no—
35C2×#11758199—22no—
36C2×#11758194—21no—
37#14519777D411yescharacteristic
38#14519774D411yescharacteristic
39#14519765C2×C411yescharacteristic
40#14519765C2311yescharacteristic
41#14519751—41no—
42C4×D244—41no—
43#14519688C2×C412yes—
44C4×#92418—21no—
45C22×D4×C61—21no—
46C22×D4×C61—41no—
47#127×C61—21no—
48#127×C61—41no—
49#127×C61C2×C412yes—
50C2×C42×C61C2×C411yescharacteristic
51C24×C61—41no—
52C2×D4×C61—22no—
53C2×D4×C61—41no—
54C2×D4×C61—41no—
55C2×D4×C61—41no—
56C2×D4×C61—41no—
57C2×D4×C61—42no—
58C2×D4×C61—42no—
59C2×D4×C61—41no—
60C2×D4×C61—84no—
61C22×C4×C61—21no—
62C22×C4×C61C22⋊C411yescharacteristic
63C2×C22⋊D61—21no—
64C2×C22⋊D61—21no—
65#11758222—41no—
66#11758222—41no—
67#11758222—41no—
68#11758222—41no—
69C2×D244—41no—
70C2×D244C22⋊C411yescharacteristic
71C2×D244C2×D411yescharacteristic
72C2×C4×D61—21no—
73C2×C4×D61—21no—
74C2×Dic122C22⋊C411yescharacteristic
75C2×Dic122C2×D411yescharacteristic
76C22⋊#6923—42no—
77#11758217—42no—
78C4⋊#6923—81no—
79C4×#6923—81no—
80#11758213—81no—
81D61⋊C8—81no—
82C42×C61—21no—
83C42×C61C22⋊C411yescharacteristic
84C42×C61C22×C411yescharacteristic
85#11758200—44no—
86#11758200—44no—
87#11758199—44no—
88#11758199—44no—
89#11758198—41no—
90#11758198—41no—
91C4⋊Dic61—41no—
92#11758194—22no—
93#11758194—22no—
94C2×#92418—22no—
95C23×C61—42no—
96C23×C61—81no—
97C23×C61—81no—
98C23×C61—84no—
99C23×C61#9911yescharacteristic
100C22×D61—21no—
101C22×D61—21no—
102C2×#6923—82no—
103D4×C61—42no—
104D4×C61—42no—
105D4×C61—84no—
106D4×C61—84no—
107D4×C61—81no—
108D4×C61—81no—
109C2×C4×C61—21no—
110C2×C4×C61—21no—
111C2×C4×C61—21no—
112C2×C4×C61—21no—
113C2×C4×C61—42no—
114C2×C4×C61#9911yescharacteristic
115C2×C4×C61C2×C22⋊C411yescharacteristic
116C22⋊D61—81no—
117C22⋊D61—81no—
118C2×Dic61—21no—
119C2×Dic61—21no—
120D244—41no—
121D244—41no—
122D244—82no—
123C4×D61—41no—
124C4×D61—41no—
125C4×D61—41no—
126C4×D61—41no—
127Dic122—41no—
128Dic122—41no—
129#92420—82no—
130#92418—24no—
131#13579—611nomaximal, Sylow
132C22×C61—21no—
133C22×C61—41no—
134C22×C61—84no—
135C22×C61—82no—
136C22×C61—162no—
137C22×C61—162no—
138C22×C61C2×#9911yescharacteristic
139C2×D61—41no—
140C2×D61—41no—
141C2×D61—81no—
142C2×D61—81no—
143#6923—162no—
144C4×C61—22no—
145C4×C61—21no—
146C4×C61—41no—
147C4×C61#35311yescharacteristic
148C4×C61#35311yescharacteristic
149Dic61—41no—
150Dic61—41no—
151#3151—611no—
152#2132—611no—
153#2126—611no—
154#1406—614no—
155C2×C61—21no—
156C2×C61—41no—
157C2×C61—162no—
158C2×C61C2×#35311yescharacteristic
159D61—81no—
160D61—81no—
161C2×AGL(1,ℤ/8)—1221no—
162C2×AGL(1,ℤ/8)—1221no—
163#532—611no—
164C2×#127#692311yescharacteristic, complemented
165#496—612no—
166#496—612no—
167#493—1221no—
168#492—1221no—
169#413—1221no—
170#410—1221no—
171C4×M16—611no—
172#356—1221no—
173#355—612no—
174#354—612no—
175#353—1221no—
176C61#1357911yescharacteristic, complemented, Sylow
177C2×#41—611no—
178C2×#41—611no—
179C22×D4—21no—
180C22×D4—41no—
181AGL(1,ℤ/8)—2442no—
182AGL(1,ℤ/8)—2442no—
183AGL(1,ℤ/8)—2442no—
184AGL(1,ℤ/8)—2442no—
185C2×SD16—1222no—
186C2×SD16—1222no—
187C2×D8—1222no—
188C2×D8—1222no—
189C2×M16—1221no—
190C2×M16—2441no—
191C4⋊Q8—611no—
192#127—21no—
193#127—41no—
194#127—611no—
195#127C2×#692312yes—
196#124—611no—
197#124—611no—
198#121—2441no—
199C4×D4—2441no—
200#117—2441no—
201C2×C42C2×#692311yescharacteristic
202#101—1222no—
203#100—2441no—
204#99—1222no—
205#99—2441no—
206#97—612no—
207C4×C8—1221no—
208C24—41no—
209#41—2441no—
210#41—2441no—
211#41—2441no—
212#41—2441no—
213C2×Q8—611no—
214C2×Q8—611no—
215C2×D4—22no—
216C2×D4—41no—
217C2×D4—41no—
218C2×D4—41no—
219C2×D4—41no—
220C2×D4—42no—
221C2×D4—42no—
222C2×D4—41no—
223C2×D4—84no—
224C2×D4—611no—
225C2×D4—611no—
226C2×D4—1221no—
227C2×D4—1221no—
228C2×D4—2441no—
229C22×C4—21no—
230C22×C4—1221no—
231C22×C4—1221no—
232C22×C4C22⋊#692311yescharacteristic
233SD16—2444no—
234SD16—2444no—
235D8—2444no—
236D8—2444no—
237M16—1222no—
238M16—1222no—
239M16—4881no—
240M16—2442no—
241C2×C8—1222no—
242C2×C8—4881no—
243C4⋊C4—4881no—
244C4⋊C4—2441no—
245C22⋊C4—2441no—
246C22⋊C4—2441no—
247C22⋊C4—2442no—
248C42—21no—
249C42—4881no—
250C42C22⋊#692311yescharacteristic
251C42C22×#692311yescharacteristic
252C23—42no—
253C23—81no—
254C23—81no—
255C23—84no—
256C23—1221no—
257C23—1221no—
258C23#1451983511yescharacteristic
259Q8—2441no—
260Q8—2441no—
261D4—42no—
262D4—42no—
263D4—84no—
264D4—84no—
265D4—81no—
266D4—81no—
267D4—4882no—
268D4—4881no—
269D4—4881no—
270D4—2441no—
271D4—2441no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—42no—
277C2×C4—1221no—
278C2×C4—1221no—
279C2×C4—4882no—
280C2×C4—2441no—
281C2×C4—2441no—
282C2×C4—2441no—
283C2×C4—2441no—
284C2×C4#1451983511yescharacteristic
285C2×C4C2×C22⋊#692311yescharacteristic
286C8—1224no—
287C8—4882no—
288C22—21no—
289C22—41no—
290C22—84no—
291C22—82no—
292C22—162no—
293C22—162no—
294C22—4881no—
295C22—4881no—
296C22—2441no—
297C22—2441no—
298C22C2×#1451983511yescharacteristic
299C4—22no—
300C4—21no—
301C4—41no—
302C4—9762no—
303C4—2441no—
304C4—2441no—
305C4#1684066611yescharacteristic
306C4#1684066611yescharacteristic
307C2—21no—
308C2—41no—
309C2—162no—
310C2—4881no—
311C2—4881no—
312C2C2×#1684066611yescharacteristic
subgroupG/H
centerC2C2×#16840666
commutatorC42×C61C22×C4
FrattiniC42C22×#6923
FittingC2×#127×C61C4
socleC2×C61C2×#353
prime radical#21456269C2
cosocle#14519765C23
hypercenterC2×#127#6923
hypocenterC61#13579
LCSG ▷ C42×C61 ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ C2×C42 ◁ C2×#127

Complements

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

NH ≅ G/Ntype
1. #21462112C2semidirect
2. #21462092C2semidirect
3. #21460946C2semidirect
8. #16840671C22semidirect
9. #16840669C22semidirect
14. #16840149C4semidirect
18. C4×#11758194C4semidirect
19. C2×#127×C61C4semidirect
164. C2×#127#6923semidirect
176. C61#13579semidirect

Maximal subgroups

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

subgroupindex
#214621122
#214620922
#214609462
#214562692
#1357961