Finite Group Databasebeta

← #30605678

Subgroups — #30605678

Order 15616.

288 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 78 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
2#21461382C211yesmaximal, characteristic
3#21460838C212yesmaximal, complemented
4#21460837C212yesmaximal, complemented
5#21456332C211yesmaximal, characteristic
6#16840784—21no—
7#16840683C2212yes—
8#16840657—21no—
9C4×#11758187—21no—
10#16840626C2212yes—
11#16840624—21no—
12#16840318—21no—
13#16840166C2211yescharacteristic
14#16840166C2211yescharacteristic
15#16840164C411yescharacteristic
16#16840161—21no—
17#16840159—21no—
18#16839799C411yescharacteristic
19#16839799C411yescharacteristic
20#16839797—21no—
21C4×C8×D61C2211yescharacteristic
22C61×#499C411yescharacteristic
23C2×#11758187—22no—
24C4⋊#92420—22no—
25#14519834—41no—
26#14519832—41no—
27C2×C8×D61C2×C411yescharacteristic
28C2×C8×D61D411yescharacteristic
29#14519776—21no—
30#14519776—21no—
31#14519769—21no—
32#14519755—41no—
33#14519755—41no—
34#14519742—41no—
35#14519742—41no—
36#14519735—21no—
37#14519735—21no—
38C42×D61D411yescharacteristic
39#14519711C2×C411yescharacteristic
40#14519711C2×C411yescharacteristic
41#14519710—21no—
42#14519710—21no—
43#14519709—21no—
44#14519708—21no—
45#14519708—21no—
46C4×#92418D411yescharacteristic
47C8×Dic61D411yescharacteristic
48C8×Dic61C2311yescharacteristic
49#125×C61—21no—
50#107×C61C2×C411yescharacteristic
51#107×C61C2×C411yescharacteristic
52#106×C61—21no—
53C4×C8×C61C2×C411yescharacteristic
54C2×C4×D61C22⋊C411yescharacteristic
55C2×C4×D61D811yescharacteristic
56C2×C4×D61SD1611yescharacteristic
57C2×Dic122—41no—
58C2×Dic122—41no—
59C2×#92420—42no—
60C2×C8×C61C22⋊C411yescharacteristic
61C2×C8×C61C22×C411yescharacteristic
62C4⋊C4×C61—21no—
63C4⋊C4×C61—21no—
64C4⋊C4×C61—41no—
65C4⋊C4×C61—41no—
66C42×C61C22⋊C411yescharacteristic
67#11758198—41no—
68#11758198—41no—
69C4⋊Dic61—21no—
70C4⋊Dic61—21no—
71#11758196—41no—
72#11758196—41no—
73#11758196—41no—
74#11758196—41no—
75C4×Dic61—41no—
76C4×Dic61—41no—
77C4×Dic61D811yescharacteristic
78C4×Dic61SD1611yescharacteristic
79C4×Dic61C2×D411yescharacteristic
80C2×#92418C22⋊C411yescharacteristic
81C2×#92418C2×D411yescharacteristic
82C8×D61—22no—
83C8×D61—22no—
84#11758187—42no—
85C22×D61#10211yescharacteristic
86C2×C4×C61—41no—
87C2×C4×C61—41no—
88C2×C4×C61#10211yescharacteristic
89C2×C4×C61#10211yescharacteristic
90C2×C4×C61C2×C22⋊C411yescharacteristic
91C2×Dic61—41no—
92C2×Dic61—41no—
93C2×Dic61#10211yescharacteristic
94C2×Dic61C2×D811yescharacteristic
95C2×Dic61C2×SD1611yescharacteristic
96C4×D61—21no—
97C4×D61—21no—
98C4×D61—22no—
99C4×D61—41no—
100Dic122—82no—
101Dic122—82no—
102#92420—82no—
103C8×C61—21no—
104C8×C61—21no—
105#92418#10112yes—
106#92418#13712yes—
107#11973—611nomaximal, Sylow
108C22×C61C2×#10211yescharacteristic
109C2×D61—21no—
110C2×D61—21no—
111C2×D61—21no—
112C4×C61—21no—
113C4×C61—21no—
114C4×C61—81no—
115C4×C61—81no—
116C4×C61C2×#10111yescharacteristic
117C4×C61#41911yescharacteristic
118Dic61—21no—
119Dic61—21no—
120Dic61—81no—
121Dic61—81no—
122Dic61#50812yes—
123#3160—611no—
124#2162—611no—
125#1568—611no—
126#1365—612no—
127#1364—612no—
128C2×C61#154611yescharacteristic
129C2×C61#213811yescharacteristic
130C2×C61#214811yescharacteristic
131D61—41no—
132#533—1221no—
133#500—611no—
134#500—611no—
135#499—611no—
136#499—1221no—
137#499#692311yescharacteristic
138#498—1221no—
139#432—1221no—
140#427—611no—
141#427—611no—
142#427—1221no—
143C2×C4×C8—611no—
144C4⋊C16—612no—
145C22⋊C16—1221no—
146C4×C16—1221no—
147#335—612no—
148#334—1221no—
149C61#1197311yescharacteristic, complemented, Sylow
150C22×C8—611no—
151C22×C8—611no—
152C4⋊Q8—1221no—
153C4⋊Q8—1221no—
154#125—21no—
155#125—1221no—
156C22⋊Q8—2441no—
157C22⋊Q8—2441no—
158C4×Q8—2441no—
159C4×Q8—2441no—
160#117—1221no—
161#117—1221no—
162C2×C42—611no—
163C2×C16—1222no—
164#107—611no—
165#107—611no—
166#107—1221no—
167#107—1221no—
168#107C2×#692311yescharacteristic
169#107C2×#692311yescharacteristic
170#106—21no—
171#106—1221no—
172#106—1221no—
173#106—1221no—
174C4⋊C8—1222no—
175C22⋊C8—2441no—
176#97—2441no—
177C4×C8—611no—
178C4×C8—611no—
179C4×C8—611no—
180C4×C8C2×#692311yescharacteristic
181C2×Q8—2441no—
182C2×Q8—2441no—
183C22×C4—611no—
184C22×C4—611no—
185C22×C4—611no—
186C2×C8—611no—
187C2×C8—611no—
188C2×C8—1222no—
189C2×C8—1222no—
190C2×C8—2442no—
191C2×C8C22⋊#692311yescharacteristic
192C2×C8C22×#692311yescharacteristic
193C4⋊C4—21no—
194C4⋊C4—21no—
195C4⋊C4—41no—
196C4⋊C4—41no—
197C4⋊C4—1221no—
198C4⋊C4—1221no—
199C4⋊C4—2441no—
200C4⋊C4—2441no—
201C4⋊C4—2441no—
202C4⋊C4—2441no—
203C22⋊C4—2441no—
204C22⋊C4—2441no—
205C42—611no—
206C42—611no—
207C42—611no—
208C42—2441no—
209C42—2441no—
210C42C22⋊#692311yescharacteristic
211C16—2442no—
212C23—611no—
213Q8—4882no—
214Q8—4882no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—611no—
218C2×C4—611no—
219C2×C4—611no—
220C2×C4—1222no—
221C2×C4—1221no—
222C2×C4—1221no—
223C2×C4—2441no—
224C2×C4—2441no—
225C2×C4—2441no—
226C2×C4#1451983911yescharacteristic
227C2×C4#1451983911yescharacteristic
228C2×C4C2×C22⋊#692311yescharacteristic
229C8—21no—
230C8—21no—
231C8—612no—
232C8—612no—
233C8—4882no—
234C22—1221no—
235C22—1221no—
236C22—1221no—
237C22C2×#1451983911yescharacteristic
238C4—21no—
239C4—21no—
240C4—81no—
241C4—81no—
242C4—612no—
243C4—1221no—
244C4—1221no—
245C4—4881no—
246C4—4881no—
247C4D61⋊#10111yescharacteristic
248C4#1684075511yescharacteristic
249C2—2441no—
250C2#2146132911yescharacteristic
251C2#2146212611yescharacteristic
252C2#2146214611yescharacteristic
subgroupG/H
centerC22C2×#14519839
commutatorC2×C8×C61C22×C4
FrattiniC2×C8C22×#6923
FittingC61×#499C4
socleC22×C61C2×#102
prime radicalC22×D61#102
cosocleC8×Dic61C23
hypercenter#499#6923
hypocenterC61#11973
LCSG ▷ C2×C8×C61 ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #499

Complements

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

NH ≅ G/Ntype
3. #21460838C2semidirect
4. #21460837C2semidirect
149. C61#11973semidirect

Maximal subgroups

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

subgroupindex
#214621782
#214613822
#214608382
#214608372
#214563322
#1197361