Finite Group Databasebeta

← #30606578

Subgroups — #30606578

Order 15616.

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

normal subgroups only

This group has 452 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#21462112C211yesmaximal, characteristic
2#21462092C211yesmaximal, characteristic
3#21461575C212yesmaximal, complemented
4#21460920C212yesmaximal, complemented
5#21454536C211yesmaximal, characteristic
6#16840748C2212yes—
7#16840741C2212yes—
8#16840725—22no—
9#16840673C412yescomplemented
10#16840671C412yescomplemented
11#16840669C412yes—
12#16840667C412yes—
13#16840644—22no—
14C2×#14519785C411yescharacteristic, complemented
15C2×#14519785C2211yescharacteristic
16#16840312C2211yescharacteristic
17#16839832—22no—
18#16839832—22no—
19#16839730C411yescharacteristic, complemented
20#16839730C2211yescharacteristic
21C2×#14519688C411yescharacteristic, complemented
22C61×#423C411yescharacteristic, complemented
23D61⋊M16—22no—
24#14519856—22no—
25#14519838C2×C414yes—
26#14519837—22no—
27#14519836C2×C414yes—
28#14519835—22no—
29#14519831—42no—
30C8×#6923—42no—
31C2×#11758222C2×C411yescharacteristic
32C2×#11758222C2311yescharacteristic
33#14519785—42no—
34#14519785—42no—
35C2×#11758189—21no—
36C2×#11758189D412yes—
37C22×#92418C2×C411yescharacteristic
38C22×#92418C2×C411yescharacteristic
39C2×C8×D61—21no—
40C2×C8×D61D412yes—
41#14519777C2×C411yescharacteristic
42#14519774C2×C411yescharacteristic
43#14519765C2×C411yescharacteristic
44#14519765C2×C411yescharacteristic
45#14519751—22no—
46C4×D244—22no—
47#14519689—22no—
48#14519689—22no—
49#14519688—21no—
50#14519688—21no—
51C2×M16×C61C2×C411yescharacteristic
52C2×M16×C61C2×C411yescharacteristic
53C2×C42×C61C2×C411yescharacteristic
54C4⋊C8×C61—22no—
55C22×C4×C61—21no—
56C22×C4×C61C22×C411yescharacteristic
57C2×C22⋊D61—21no—
58C2×C22⋊D61#4112yes—
59#11758222—41no—
60#11758222—41no—
61#11758222—41no—
62#11758222—41no—
63C2×D244—22no—
64C2×D244C4211yescharacteristic
65C2×D244C22×C411yescharacteristic
66C2×C4×D61—21no—
67C2×C4×D61C2×D412yes—
68C2×Dic122C4211yescharacteristic
69C2×Dic122C22×C411yescharacteristic
70C22⋊#6923—24no—
71#11758217—24no—
72C4⋊#6923—42no—
73C4×#6923—42no—
74#11758213—42no—
75D61⋊C8—42no—
76M16×C61—41no—
77M16×C61—41no—
78C2×C8×C61—21no—
79C2×C8×C61C22⋊C412yes—
80C42×C61—21no—
81C42×C61C4211yescharacteristic
82C42×C61C4211yescharacteristic
83#11758198—22no—
84#11758198—22no—
85C4⋊Dic61—22no—
86C2×#92418—22no—
87C2×#92418—21no—
88C2×#92418—41no—
89C2×#92418C22⋊C411yescharacteristic
90C2×#92418C22⋊C411yescharacteristic
91#11758189—42no—
92#11758189—42no—
93C8×D61—22no—
94C8×D61—22no—
95C8×D61—42no—
96C23×C61#11711yescharacteristic
97C22×D61—21no—
98C22×D61C4×D412yes—
99C2×#6923—44no—
100C2×C4×C61—21no—
101C2×C4×C61—21no—
102C2×C4×C61—21no—
103C2×C4×C61—21no—
104C2×C4×C61—81no—
105C2×C4×C61C2×C4211yescharacteristic
106C2×C4×C61C2×C22⋊C411yescharacteristic
107C22⋊D61—42no—
108C22⋊D61—42no—
109C2×Dic61—21no—
110C2×Dic61C4×D412yes—
111D244—41no—
112D244—41no—
113D244—82no—
114C4×D61—22no—
115C4×D61—22no—
116C4×D61—41no—
117C4×D61—41no—
118Dic122—41no—
119Dic122—41no—
120#92420—44no—
121C8×C61—22no—
122C8×C61—41no—
123#92418—21no—
124#92418—41no—
125#92418#9912yes—
126#12634—611nomaximal, Sylow
127C22×C61—21no—
128C22×C61—41no—
129C22×C61C4×C22⋊C411yescharacteristic
130C2×D61—22no—
131C2×D61—42no—
132C2×D61—42no—
133C2×D61—41no—
134#6923—84no—
135C4×C61—21no—
136C4×C61—41no—
137C4×C61—41no—
138C4×C61C2×#9911yescharacteristic
139C4×C61#41011yescharacteristic
140Dic61—22no—
141Dic61—41no—
142#2932—611no—
143#2132—611no—
144#2126—611no—
145#1757—612no—
146#1395—612no—
147C2×C61—21no—
148C2×C61—41no—
149C2×C61C4×#9911yescharacteristic
150D61—42no—
151D61—81no—
152C2×#131—611no—
153C2×#131—611no—
154#532—611no—
155#435—1222no—
156#435—1222no—
157#423#692311yescharacteristic, complemented
158#413—612no—
159#410—612no—
160#408—611no—
161#408—611no—
162#405—1222no—
163C2×#97—611no—
164#356—612no—
165#355—612no—
166#354—612no—
167#353—612no—
168#344—1222no—
169C61#1263411yescharacteristic, complemented, Sylow
170C2×#41—611no—
171C2×#41—611no—
172#131—2442no—
173#131—2442no—
174C2×M16—612no—
175C2×M16—1221no—
176C2×M16—1222no—
177C2×M16C2×#692311yescharacteristic
178C2×M16C2×#692311yescharacteristic
179C22×C8—611no—
180C22×C8—612no—
181C22×C8—611no—
182C22×C8—1221no—
183C4⋊Q8—611no—
184#127—611no—
185#124—611no—
186#124—611no—
187#121—1222no—
188C4×D4—1222no—
189#117—1222no—
190C2×C42C2×#692311yescharacteristic
191C4⋊C8—22no—
192#101—614no—
193#100—1222no—
194#99—614no—
195#99—1222no—
196C22⋊C8—1222no—
197C22⋊C8—1222no—
198#97—1221no—
199#97—1221no—
200#97—2442no—
201C4×C8—2442no—
202#41—2441no—
203#41—2441no—
204#41—2441no—
205#41—2441no—
206C2×Q8—611no—
207C2×Q8—611no—
208C2×D4—612no—
209C2×D4—611no—
210C2×D4—611no—
211C2×D4—1222no—
212C2×D4—1221no—
213C22×C4—21no—
214C22×C4—612no—
215C22×C4—1221no—
216C22×C4C22×#692311yescharacteristic
217M16—41no—
218M16—41no—
219M16—1224no—
220M16—2442no—
221M16—2442no—
222M16—2442no—
223C2×C8—21no—
224C2×C8—611no—
225C2×C8—611no—
226C2×C8—1222no—
227C2×C8—1222no—
228C2×C8—1222no—
229C2×C8—1221no—
230C2×C8—2442no—
231C2×C8—2442no—
232C2×C8—2441no—
233C2×C8C22⋊#692312yes—
234C4⋊C4—1222no—
235C4⋊C4—2442no—
236C22⋊C4—1222no—
237C22⋊C4—1222no—
238C22⋊C4—1224no—
239C42—21no—
240C42—2442no—
241C42C4×#692311yescharacteristic
242C42C4×#692311yescharacteristic
243C23—612no—
244C23—1221no—
245C23#1451985611yescharacteristic
246Q8—2441no—
247Q8—2441no—
248D4—4882no—
249D4—2441no—
250D4—2441no—
251D4—2442no—
252D4—2442no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—81no—
258C2×C4—612no—
259C2×C4—1222no—
260C2×C4—1222no—
261C2×C4—1221no—
262C2×C4—2444no—
263C2×C4—2441no—
264C2×C4—2441no—
265C2×C4C2×C4×#692311yescharacteristic
266C2×C4C2×C22⋊#692311yescharacteristic
267C8—22no—
268C8—41no—
269C8—612no—
270C8—1221no—
271C8—2444no—
272C8—2441no—
273C22—21no—
274C22—41no—
275C22—1222no—
276C22—2442no—
277C22—2442no—
278C22—2441no—
279C22C4×C22⋊#692311yescharacteristic
280C4—21no—
281C4—41no—
282C4—41no—
283C4—1222no—
284C4—4884no—
285C4—2441no—
286C4D61⋊#9911yescharacteristic
287C4#1684074311yescharacteristic
288C2—21no—
289C2—41no—
290C2—4881no—
291C2—2442no—
292C2#2146156411yescharacteristic
subgroupG/H
centerC4D61⋊#99
commutatorC2×C4×C61C2×C42
FrattiniC22×C4C22×#6923
FittingC61×#423C4
socleC2×C61C4×#99
prime radical#16840312C22
cosocleC2×#11758222C23
hypercenter#423#6923
hypocenterC61#12634
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×M16 ◁ #423

Complements

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

NH ≅ G/Ntype
3. #21461575C2semidirect
4. #21460920C2semidirect
9. #16840673C4semidirect
10. #16840671C4semidirect
14. C2×#14519785C4semidirect
19. #16839730C4semidirect
21. C2×#14519688C4semidirect
22. C61×#423C4semidirect
157. #423#6923semidirect
169. C61#12634semidirect

Maximal subgroups

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

subgroupindex
#214621122
#214620922
#214615752
#214609202
#214545362
#1263461