Finite Group Databasebeta

← #30604912

Subgroups — #30604912

Order 15616.

356 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 80 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#21462178C211yesmaximal, characteristic, complemented
2C4×#14519845C211yesmaximal, characteristic, complemented
3#21460807C212yesmaximal, complemented
4#21460806C212yesmaximal
5#21456392C211yesmaximal, characteristic
6C2×#14519845—21no—
7C4×C4⋊#6923—21no—
8#16840683—21no—
9#16840657C2212yes—
10C4×#11758187—21no—
11#16840636C2212yes—
12#16840299—21no—
13#16840163C411yescharacteristic, complemented
14#16840147C411yescharacteristic, complemented
15#16840118—21no—
16#16840000C2211yescharacteristic
17#16839999C2211yescharacteristic
18#16839994—21no—
19C4×#11758201—21no—
20C4×D488C411yescharacteristic, complemented
21C4×C8×D61C2211yescharacteristic
22C4×Q16×C61C411yescharacteristic, complemented
23C2×C4⋊#6923—22no—
24C2×C4×#6923—41no—
25C2×#11758187—22no—
26#14519845—24no—
27(C2×#6923)⋊C4—41no—
28C2×#11758201—21no—
29C2×D488C2×C411yescharacteristic
30C2×C8×D61D411yescharacteristic
31C2×C8×D61C2311yescharacteristic
32#14519772—41no—
33#14519768—21no—
34#14519766—21no—
35#14519762—41no—
36#14519753—21no—
37#14519752—21no—
38C4×D244—21no—
39C4×D244—41no—
40#14519734—41no—
41C42×D61D411yescharacteristic
42#14519711C2×C411yescharacteristic
43#14519708—21no—
44#14519701—21no—
45#14519699—21no—
46#14519697—21no—
47C4×#92418D411yescharacteristic
48C8×Dic61C2×C411yescharacteristic
49C8×Dic61D411yescharacteristic
50C2×Q16×C61C2×C411yescharacteristic
51C4×Q8×C61—21no—
52#107×C61C2×C411yescharacteristic
53#103×C61—21no—
54C4×C8×C61C2×C411yescharacteristic
55C22×#6923—42no—
56C2×Q8×C61—21no—
57C2×D244—21no—
58C2×D244—41no—
59C2×C4×D61—41no—
60C2×C4×D61D811yescharacteristic
61C2×C4×D61D811yescharacteristic
62C2×C4×D61C2×D411yescharacteristic
63C4⋊#6923—24no—
64C4⋊#6923—42no—
65C4×#6923—82no—
66Q16×C61—41no—
67C2×C8×C61C22⋊C411yescharacteristic
68C2×C8×C61C22×C411yescharacteristic
69C4⋊C4×C61—21no—
70C4⋊C4×C61—41no—
71C42×C61—41no—
72C42×C61C22⋊C411yescharacteristic
73#11758201—42no—
74#11758198—41no—
75#11758198—41no—
76#11758198—41no—
77#11758198—41no—
78#11758198—41no—
79C4⋊Dic61—21no—
80C4⋊Dic61—41no—
81#11758196—41no—
82#11758196—41no—
83C4×Dic61C22⋊C411yescharacteristic
84C4×Dic61SD1611yescharacteristic
85C4×Dic61SD1611yescharacteristic
86C2×#92418C22⋊C411yescharacteristic
87C2×#92418C2×D411yescharacteristic
88D488—41no—
89C8×D61—22no—
90C8×D61—41no—
91#11758187—42no—
92C22×D61—41no—
93C22×D61C2×D811yescharacteristic
94C2×#6923—44no—
95C2×#6923—82no—
96C2×#6923—82no—
97Q8×C61—41no—
98Q8×C61—41no—
99C2×C4×C61—41no—
100C2×C4×C61—41no—
101C2×C4×C61#10211yescharacteristic
102C2×C4×C61#10211yescharacteristic
103C2×C4×C61C2×C22⋊C411yescharacteristic
104C2×Dic61—41no—
105C2×Dic61#10211yescharacteristic
106C2×Dic61#10211yescharacteristic
107C2×Dic61C2×SD1611yescharacteristic
108D244—41no—
109D244—41no—
110D244—82no—
111C4×D61—22no—
112C4×D61—22no—
113C4×D61—21no—
114C4×D61—21no—
115C4×D61—82no—
116C8×C61—21no—
117C8×C61—21no—
118#92418—21no—
119#92418#10412yes—
120#11741—611nomaximal, Sylow
121C22×C61C2×#10211yescharacteristic
122C2×D61—21no—
123C2×D61—21no—
124C2×D61—81no—
125C2×D61—81no—
126C2×D61—81no—
127C2×D61#50812yes—
128#6923—84no—
129C4×C61—21no—
130C4×C61—81no—
131C4×C61—81no—
132C4×C61#36012yes—
133C4×C61#41011yescharacteristic
134C4×C61C2×#10411yescharacteristic
135Dic61—21no—
136Dic61—21no—
137Dic61—21no—
138Dic61—81no—
139#3168—611no—
140#2162—611no—
141C4×#106—611no—
142#1348—612no—
143#1347—612no—
144C2×C61#150211yescharacteristic
145C2×C61#213811yescharacteristic
146C2×C61C2×#36011yescharacteristic
147D61—22no—
148D61—161no—
149#529—1221no—
150#499—611no—
151#495—611no—
152#488—1221no—
153#467—611no—
154#466—611no—
155#465—1221no—
156C4×Q16#692311yescharacteristic, complemented
157C4×SD16—1221no—
158C4×D8—611no—
159C2×#106—1221no—
160C2×C4×C8—611no—
161C4×C4⋊C4—1221no—
162C4⋊C16—1221no—
163C22⋊C16—612no—
164C4×C16—1221no—
165#340—612no—
166C61#1174111yescharacteristic, complemented, Sylow
167C2×Q16C2×#692311yescharacteristic
168C2×SD16—1221no—
169C2×D8—611no—
170C22×C8—611no—
171C22×C8—611no—
172#126—2441no—
173#125—1221no—
174#124—1221no—
175#123—2441no—
176C22⋊Q8—1221no—
177#121—1221no—
178C4×Q8—21no—
179C4×D4—1221no—
180C4×D4—2441no—
181#117—2441no—
182C2×C4⋊C4—1222no—
183C2×C42—611no—
184C2×C42—2441no—
185C2×C16—1222no—
186#107—611no—
187#107C2×#692311yescharacteristic
188#106—1221no—
189#106—1224no—
190#103—21no—
191#103—1221no—
192#102—1221no—
193#102—1221no—
194C4×C8—611no—
195C4×C8—611no—
196C4×C8—611no—
197C4×C8C2×#692311yescharacteristic
198#95—2441no—
199C2×Q8—21no—
200C2×D4—1221no—
201C2×D4—2441no—
202C22×C4—611no—
203C22×C4—611no—
204C22×C4—611no—
205C22×C4—2442no—
206C22×C4—2441no—
207Q16—41no—
208SD16—2442no—
209D8—2441no—
210C2×C8—611no—
211C2×C8—611no—
212C2×C8—1222no—
213C2×C8—2441no—
214C2×C8C22⋊#692311yescharacteristic
215C2×C8C22×#692311yescharacteristic
216C4⋊C4—21no—
217C4⋊C4—41no—
218C4⋊C4—1224no—
219C4⋊C4—1221no—
220C4⋊C4—2441no—
221C4⋊C4—2441no—
222C4⋊C4—2441no—
223C4⋊C4—2442no—
224C22⋊C4—2441no—
225C22⋊C4—2441no—
226C22⋊C4—2441no—
227C22⋊C4—2441no—
228C22⋊C4—2441no—
229C42—41no—
230C42—611no—
231C42—611no—
232C42—611no—
233C42—4882no—
234C42C22⋊#692311yescharacteristic
235C16—2442no—
236C23—611no—
237C23—2441no—
238Q8—41no—
239Q8—41no—
240D4—4882no—
241D4—2441no—
242D4—2441no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—611no—
246C2×C4—611no—
247C2×C4—611no—
248C2×C4—1222no—
249C2×C4—1221no—
250C2×C4—1221no—
251C2×C4—1222no—
252C2×C4—4882no—
253C2×C4—4882no—
254C2×C4—4882no—
255C2×C4—2444no—
256C2×C4—2441no—
257C2×C4#1451983911yescharacteristic
258C2×C4#1451983911yescharacteristic
259C2×C4C2×C22⋊#692311yescharacteristic
260C8—21no—
261C8—21no—
262C8—612no—
263C8—1221no—
264C22—612no—
265C22—1221no—
266C22—1221no—
267C22—4881no—
268C22—4881no—
269C22—4881no—
270C22C2×#1451983911yescharacteristic
271C4—21no—
272C4—81no—
273C4—81no—
274C4—1221no—
275C4—1221no—
276C4—1221no—
277C4—4884no—
278C4—4881no—
279C4#1684067912yes—
280C4#1684074111yescharacteristic
281C4D61⋊#10411yescharacteristic
282C2—1222no—
283C2—9761no—
284C2#2146119811yescharacteristic
285C2#2146212611yescharacteristic
286C2C2×#1684067911yescharacteristic
subgroupG/H
centerC2×C4#14519839
commutatorC2×C8×C61C22×C4
FrattiniC2×C8C22×#6923
FittingC4×Q16×C61C4
socleC22×C61C2×#102
prime radical#16839999C22
cosocleC2×C8×D61C23
hypercenterC4×Q16#6923
hypocenterC61#11741
LCSG ▷ C2×C8×C61 ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C42 ◁ C4×C8 ◁ C4×Q16

Complements

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

NH ≅ G/Ntype
1. #21462178C2semidirect
2. C4×#14519845C2semidirect
3. #21460807C2semidirect
13. #16840163C4semidirect
14. #16840147C4semidirect
20. C4×D488C4semidirect
22. C4×Q16×C61C4semidirect
156. C4×Q16#6923semidirect
166. C61#11741semidirect

Maximal subgroups

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

subgroupindex
#214621782
C4×#145198452
#214608072
#214608062
#214563922
#1174161