Finite Group Databasebeta

← #30606527

Subgroups — #30606527

Order 15616.

456 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 96 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#21462070C212yesmaximal
2#21461575C212yesmaximal, complemented
3#21460911C211yesmaximal, characteristic, complemented
4#21460908C211yesmaximal, characteristic, complemented
5#21454518C211yesmaximal, characteristic
6#16840748C2212yes—
7#16840741C2212yes—
8#16840725—22no—
9#16840664C414yescomplemented
10#16840662C414yescomplemented
11#16840644—22no—
12C2×#14519785C2211yescharacteristic
13C22×#11758194C411yescharacteristic, complemented
14#16840273C2211yescharacteristic
15#16839837—22no—
16#16839836—22no—
17#16839790C411yescharacteristic, complemented
18#16839730C2211yescharacteristic
19#16839720C411yescharacteristic, complemented
20C61×#407C411yescharacteristic, complemented
21D61⋊M16—22no—
22#14519856—22no—
23#14519838—22no—
24#14519837C2×C414yes—
25#14519836—22no—
26#14519835C2×C414yes—
27#14519831—42no—
28C8×#6923—42no—
29C2×#11758222C2311yescharacteristic
30#14519785—42no—
31C2×#11758194—21no—
32C2×#11758194—21no—
33C2×#11758194—42no—
34C2×#11758189D412yes—
35C22×#92418C2×C411yescharacteristic
36C22×#92418C2×C411yescharacteristic
37C2×C8×D61D412yes—
38#14519760C2×C412yes—
39#14519756—21no—
40#14519751—21no—
41#14519746C2×C412yes—
42C4×C22⋊D61—22no—
43#14519736C2×C411yescharacteristic
44#14519706—22no—
45#14519689—22no—
46#14519687—21no—
47C8×Dic61—21no—
48C23×C4×C61C2×C411yescharacteristic
49C2×M16×C61C2×C411yescharacteristic
50C2×M16×C61C2×C411yescharacteristic
51C22⋊C8×C61—22no—
52C24×C61C4211yescharacteristic
53C22×C4×C61—21no—
54C22×C4×C61—21no—
55C22×C4×C61—41no—
56C22×C4×C61—41no—
57C22×C4×C61C4211yescharacteristic
58C22×C4×C61C22×C411yescharacteristic
59C2×C22⋊D61—22no—
60C2×C22⋊D61C22×C412yes—
61#11758222—41no—
62#11758222—41no—
63C2×D244#4111yescharacteristic
64C2×C4×D61C2×D412yes—
65C2×Dic122#4111yescharacteristic
66C22⋊#6923—24no—
67#11758217—24no—
68C4⋊#6923—42no—
69C4×#6923—42no—
70#11758213—42no—
71D61⋊C8—42no—
72M16×C61—41no—
73M16×C61—41no—
74C2×C8×C61—21no—
75C2×C8×C61C22⋊C412yes—
76C22⋊Dic61—22no—
77C22⋊Dic61C4212yes—
78#11758198—22no—
79C4⋊Dic61—21no—
80#11758196—22no—
81C4×Dic61—21no—
82#11758194—42no—
83#11758194—42no—
84C2×#92418—22no—
85C2×#92418—21no—
86C2×#92418—41no—
87C2×#92418C22⋊C411yescharacteristic
88C2×#92418C22⋊C411yescharacteristic
89#11758189—42no—
90C8×D61—22no—
91C8×D61—22no—
92C23×C61—21no—
93C23×C61—41no—
94C23×C61—81no—
95C23×C61C2×C4211yescharacteristic
96C22×D61C4×D412yes—
97C2×#6923—44no—
98C2×C4×C61—21no—
99C2×C4×C61—21no—
100C2×C4×C61—21no—
101C2×C4×C61—42no—
102C2×C4×C61—41no—
103C2×C4×C61—41no—
104C2×C4×C61—41no—
105C2×C4×C61C2×C22⋊C411yescharacteristic
106C2×C4×C61#11711yescharacteristic
107C22⋊D61—42no—
108C22⋊D61—82no—
109C2×Dic61—22no—
110C2×Dic61C4×D412yes—
111D244—41no—
112C4×D61—22no—
113C4×D61—22no—
114Dic122—41no—
115#92420—44no—
116C8×C61—22no—
117C8×C61—41no—
118#92418—21no—
119#92418—41no—
120#92418#9912yes—
121#12624—611nomaximal, Sylow
122C22×C61—21no—
123C22×C61—41no—
124C22×C61—41no—
125C22×C61—41no—
126C22×C61—41no—
127C22×C61—82no—
128C22×C61C4×C22⋊C411yescharacteristic
129C2×D61—22no—
130C2×D61—42no—
131#6923—84no—
132C4×C61—21no—
133C4×C61—41no—
134C4×C61C2×#9911yescharacteristic
135C4×C61#41011yescharacteristic
136Dic61—22no—
137Dic61—42no—
138#2930—611no—
139#2120—612no—
140#1757—612no—
141#1392—611no—
142#1391—611no—
143C2×C61—21no—
144C2×C61—41no—
145C2×C61—81no—
146C2×C61C4×#9911yescharacteristic
147D61—42no—
148C2×#131—611no—
149C22×M16—611no—
150#525—611no—
151#436—1222no—
152#435—1222no—
153#424—611no—
154#413—612no—
155#410—612no—
156#408—611no—
157#407#692311yescharacteristic, complemented
158#405—611no—
159#405—1222no—
160#352—614no—
161AΣL(1,𝔽16)—614no—
162#344—1222no—
163C61#1262411yescharacteristic, complemented, Sylow
164C2×#41—611no—
165C23×C4C2×#692311yescharacteristic
166#131—2442no—
167C2×M16—612no—
168C2×M16—1221no—
169C2×M16—1221no—
170C2×M16—1222no—
171C2×M16—2442no—
172C2×M16C2×#692311yescharacteristic
173C2×M16C2×#692311yescharacteristic
174C22×C8—611no—
175C22×C8—612no—
176C22×C8—611no—
177#123—612no—
178C22⋊Q8—1221no—
179#121—1221no—
180#120—612no—
181C4×D4—1222no—
182#117—611no—
183#117—1222no—
184C4⋊C8—1222no—
185#101—1222no—
186#100—614no—
187#99—614no—
188#99—1222no—
189C22⋊C8—22no—
190C22⋊C8—1222no—
191#97—1221no—
192#97—2442no—
193C4×C8—1221no—
194C4×C8—2442no—
195C24C4×#692311yescharacteristic
196#41—2441no—
197#41—2441no—
198C2×Q8—611no—
199C2×D4—612no—
200C2×D4—611no—
201C2×D4—1222no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—41no—
205C22×C4—41no—
206C22×C4—612no—
207C22×C4C4×#692311yescharacteristic
208C22×C4C22×#692311yescharacteristic
209M16—41no—
210M16—41no—
211M16—1224no—
212M16—2442no—
213M16—2442no—
214M16—2442no—
215M16—2442no—
216C2×C8—21no—
217C2×C8—611no—
218C2×C8—611no—
219C2×C8—1222no—
220C2×C8—1222no—
221C2×C8—1222no—
222C2×C8—1221no—
223C2×C8—2442no—
224C2×C8—2441no—
225C2×C8C22⋊#692312yes—
226C4⋊C4—1222no—
227C4⋊C4—1221no—
228C4⋊C4—2442no—
229C22⋊C4—612no—
230C22⋊C4—1222no—
231C22⋊C4—1222no—
232C22⋊C4—1224no—
233C42—1221no—
234C42—2442no—
235C23—21no—
236C23—41no—
237C23—81no—
238C23—612no—
239C23C2×C4×#692311yescharacteristic
240Q8—2441no—
241D4—4882no—
242D4—2441no—
243D4—2442no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—42no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—612no—
252C2×C4—1222no—
253C2×C4—1222no—
254C2×C4—1222no—
255C2×C4—2444no—
256C2×C4C2×C22⋊#692311yescharacteristic
257C2×C4#1451985611yescharacteristic
258C8—22no—
259C8—41no—
260C8—612no—
261C8—1221no—
262C8—2444no—
263C8—2441no—
264C22—21no—
265C22—41no—
266C22—41no—
267C22—41no—
268C22—41no—
269C22—82no—
270C22—1222no—
271C22—2442no—
272C22C4×C22⋊#692311yescharacteristic
273C4—21no—
274C4—41no—
275C4—1222no—
276C4—4884no—
277C4—2442no—
278C4D61⋊#9911yescharacteristic
279C4#1684074211yescharacteristic
280C2—21no—
281C2—41no—
282C2—81no—
283C2—2442no—
284C2#2146156611yescharacteristic
subgroupG/H
centerC4D61⋊#99
commutatorC23×C61C2×C42
FrattiniC22×C4C22×#6923
FittingC61×#407C4
socleC2×C61C4×#99
prime radical#16840273C22
cosocleC2×#11758222C23
hypercenter#407#6923
hypocenterC61#12624
LCSG ▷ C23×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×M16 ◁ #407

Complements

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

NH ≅ G/Ntype
2. #21461575C2semidirect
3. #21460911C2semidirect
4. #21460908C2semidirect
9. #16840664C4semidirect
10. #16840662C4semidirect
13. C22×#11758194C4semidirect
17. #16839790C4semidirect
19. #16839720C4semidirect
20. C61×#407C4semidirect
157. #407#6923semidirect
163. C61#12624semidirect

Maximal subgroups

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

subgroupindex
#214620702
#214615752
#214609112
#214609082
#214545182
#1262461