Finite Group Databasebeta

← #30595212

Subgroups — #30595212

Order 15616.

270 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 42 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#21462093C211yesmaximal, characteristic
2#21461955C211yesmaximal, characteristic
3#21447503C211yesmaximal, characteristic
4#16840767C411yescharacteristic, complemented
5C2×#14519842—21no—
6C2×#14519842—21no—
7#16840743—21no—
8#16840698C411yescharacteristic, complemented
9#16840668—21no—
10#16840667C412yescomplemented
11#16840313C2211yescharacteristic
12#16839745—21no—
13#16839561C411yescharacteristic, complemented
14#16839558—21no—
15C61×#356C411yescharacteristic, complemented
16C2×#11758213—21no—
17#14519856—41no—
18C2×C4×#6923—21no—
19#14519844—41no—
20#14519842—42no—
21#14519842—42no—
22#14519836—22no—
23#14519836—41no—
24(C2×#6923)⋊C4—41no—
25C2×#11758226D411yescharacteristic
26C2×#11758226D411yescharacteristic
27M16×D61—41no—
28#14519774C2×C411yescharacteristic
29#14519766D411yescharacteristic
30#14519766Q811yescharacteristic
31#14519752—41no—
32#14519739—41no—
33C42×D61C2×C411yescharacteristic
34#14519696—21no—
35#14519693—41no—
36C4⋊Q8×C61C2×C411yescharacteristic
37#101×C61—21no—
38C22×#6923—41no—
39C2×Q8×C61C22⋊C411yescharacteristic
40C2×Q8×C61C4⋊C411yescharacteristic
41#11758226—41no—
42#11758226—41no—
43#11758226—41no—
44#11758226—41no—
45C2×D244—21no—
46C2×D244—21no—
47C2×D244—41no—
48C2×D244C22⋊C411yescharacteristic
49C2×D244C4⋊C411yescharacteristic
50C2×C4×D61—21no—
51C2×C4×D61—21no—
52C2×C4×D61—21no—
53C2×C4×D61C22⋊C411yescharacteristic
54C22⋊#6923—42no—
55C2×#92420—41no—
56C4⋊#6923—81no—
57C4×#6923—22no—
58C4×#6923—41no—
59C4×#6923—81no—
60#11758213—41no—
61#11758213—41no—
62M16×C61—41no—
63C4⋊C4×C61—41no—
64C42×C61C4211yescharacteristic
65#11758198—41no—
66#11758198—41no—
67C4×Dic61—21no—
68C4×Dic61C4⋊C411yescharacteristic
69#11758194—41no—
70#11758189—81no—
71C8×D61—81no—
72C22×D61—21no—
73C22×D61—21no—
74C22×D61#9911yescharacteristic
75C2×#6923—42no—
76C2×#6923—82no—
77C2×#6923—82no—
78Q8×C61—41no—
79Q8×C61—41no—
80C2×C4×C61—21no—
81C2×C4×C61—21no—
82C2×C4×C61—21no—
83C2×C4×C61#9511yescharacteristic
84C2×Dic61—21no—
85C2×Dic61#9911yescharacteristic
86D244—41no—
87D244—41no—
88D244—81no—
89D244—81no—
90D244—82no—
91C4×D61—21no—
92C4×D61—21no—
93C4×D61—41no—
94C4×D61—41no—
95C4×D61—41no—
96C4×D61—41no—
97C4×D61—81no—
98#92420—81no—
99C8×C61—81no—
100#92418—81no—
101#7503—611nomaximal, Sylow
102C22×C61#34211yescharacteristic
103C2×D61—21no—
104C2×D61—41no—
105C2×D61—41no—
106C2×D61—41no—
107C2×D61—81no—
108C2×D61—81no—
109#6923—82no—
110#6923—162no—
111C4×C61—21no—
112C4×C61—41no—
113C4×C61—41no—
114C4×C61—41no—
115Dic61—41no—
116Dic61#35311yescharacteristic
117Dic61#35411yescharacteristic
118#2132—611no—
119#2126—611no—
120#1963—611no—
121C2×C61—21no—
122C2×C61#139311yescharacteristic
123D61—41no—
124D61—81no—
125D61—81no—
126#532—611no—
127#423—611no—
128C2×#104—1221no—
129C2×#104—1221no—
130#413—1221no—
131#410—1221no—
132#376—611no—
133#356—611no—
134#356#692311yescharacteristic, complemented
135#355—1221no—
136#354—1221no—
137#353—612no—
138C61#750311yescharacteristic, complemented, Sylow
139C2×#41—611no—
140C2×#41—611no—
141C2×M16—1221no—
142C2×M16—2441no—
143C4⋊Q8C2×#692311yescharacteristic
144#127—611no—
145#124—611no—
146#124—611no—
147#121—2441no—
148C4×D4—2441no—
149#117—2441no—
150C2×C42—611no—
151C2×C42—1221no—
152C4⋊C8—2441no—
153#104—2442no—
154#104—2442no—
155#101—21no—
156#101—1221no—
157#100—2441no—
158#99—1222no—
159#99—2441no—
160#95—2441no—
161#41—2441no—
162#41—2441no—
163#41—2441no—
164#41—2441no—
165C2×Q8C4⋊#692311yescharacteristic
166C2×Q8C22⋊#692311yescharacteristic
167C2×D4—611no—
168C2×D4—611no—
169C2×D4—1221no—
170C2×D4—1221no—
171C2×D4—2441no—
172C22×C4—611no—
173C22×C4—1221no—
174C22×C4—1221no—
175C22×C4—1221no—
176C22×C4—2441no—
177M16—41no—
178M16—4881no—
179M16—2441no—
180M16—2441no—
181M16—2441no—
182C2×C8—4881no—
183C2×C8—2441no—
184C4⋊C4—41no—
185C4⋊C4—4881no—
186C22⋊C4—2441no—
187C22⋊C4—2441no—
188C22⋊C4—2442no—
189C42—611no—
190C42—1221no—
191C42—1222no—
192C42—4881no—
193C42—2441no—
194C42C4×#692311yescharacteristic
195C23—611no—
196C23—1221no—
197C23—1221no—
198Q8—41no—
199Q8—41no—
200D4—4881no—
201D4—4881no—
202D4—4882no—
203D4—2441no—
204D4—2441no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—611no—
209C2×C4—1221no—
210C2×C4—1221no—
211C2×C4—1221no—
212C2×C4—4881no—
213C2×C4—4882no—
214C2×C4—4882no—
215C2×C4—2441no—
216C2×C4—2441no—
217C2×C4—2441no—
218C2×C4—2441no—
219C2×C4—2442no—
220C2×C4(C2×#6923)⋊C411yescharacteristic
221C8—81no—
222C8—4881no—
223C8—4881no—
224C22—1221no—
225C22—4881no—
226C22—4881no—
227C22—2441no—
228C22—2441no—
229C22—2441no—
230C22#1684063911yescharacteristic
231C4—21no—
232C4—41no—
233C4—41no—
234C4—41no—
235C4—611no—
236C4—611no—
237C4—9762no—
238C4—4882no—
239C4—2441no—
240C2—21no—
241C2—4881no—
242C2—4881no—
243C2—2441no—
244C2#2146091211yescharacteristic
subgroupG/H
centerC2#21460912
commutatorC42×C61C42
FrattiniC4⋊Q8C2×#6923
FittingC61×#356C4
socleC2×C61#1393
prime radical, cosocle#16840313C22
hypercenter#356#6923
hypocenterC61#7503
LCSG ▷ C42×C61 ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C4⋊Q8 ◁ #356

Complements

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

NH ≅ G/Ntype
4. #16840767C4semidirect
8. #16840698C4semidirect
10. #16840667C4semidirect
13. #16839561C4semidirect
15. C61×#356C4semidirect
134. #356#6923semidirect
138. C61#7503semidirect

Maximal subgroups

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

subgroupindex
#214620932
#214619552
#214475032
#750361