Finite Group Databasebeta

← #29591548

Subgroups — #29591548

Order 15616.

360 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 69 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#21456273C211yesmaximal, characteristic
2#21447630C212yesmaximal, complemented
3#21443860C211yesmaximal, characteristic, complemented
4#21443857C212yesmaximal, complemented
5C61×#1359C211yesmaximal, characteristic, complemented
6C2×#14519806—22no—
7#16840310C411yescharacteristic
8C4⋊Q8×D61C411yescharacteristic
9#16840153—22no—
10#16840142C2211yescharacteristic
11#16840141C411yescharacteristic
12#16840134C411yescharacteristic
13C4×#11758189C2211yescharacteristic
14#16839583C2212yes—
15#16839543—22no—
16#16839539—22no—
17C61×#492C2211yescharacteristic
18C61×#364C2212yes—
19C2×#11758224—21no—
20C2×Q8×D61—21no—
21#14519806—44no—
22#14519806—44no—
23C2×Dic244—22no—
24C2×#11758202—22no—
25C2×#11758200—22no—
26C2×#11758189D412yes—
27#14519777C2×C411yescharacteristic
28#14519777C2×C411yescharacteristic
29#14519776—41no—
30#14519775—21no—
31#14519764—21no—
32#14519748—41no—
33D4×Dic61—41no—
34C4⋊C4×D61—41no—
35C42×D61C2×C411yescharacteristic
36#14519717—22no—
37#14519687D412yes—
38C4×#92418C2×C411yescharacteristic
39C2×SD16×C61—22no—
40C4⋊Q8×C61C2×C411yescharacteristic
41#127×C61C2×C411yescharacteristic
42M32×C61—22no—
43C4×C8×C61C2311yescharacteristic
44C2×Q8×C61—21no—
45C2×D4×C61—21no—
46C2×D4×C61—41no—
47C2×C22⋊D61—41no—
48C22×Dic61—41no—
49Q8×D61—44no—
50#11758224—44no—
51C2×C4×D61—21no—
52C2×C4×D61—41no—
53C2×C4×D61C22⋊C411yescharacteristic
54C2×Dic122—21no—
55C2×Dic122—21no—
56C2×Dic122—41no—
57SD16×C61—42no—
58SD16×C61—42no—
59C2×C8×C61C2×D412yes—
60C4⋊C4×C61—41no—
61C42×C61C22×C411yescharacteristic
62C22⋊Dic61—42no—
63#11758202—44no—
64#11758200—44no—
65C4⋊Dic61—41no—
66C4⋊Dic61—41no—
67#11758196—42no—
68C4×Dic61—21no—
69C4×Dic61C22⋊C411yescharacteristic
70C2×#92418C22⋊C412yes—
71Dic244—42no—
72Dic244—42no—
73#11758189—22no—
74#11758189—22no—
75C16×C61—42no—
76#11758185—42no—
77C23×C61—41no—
78C22×D61#10011yescharacteristic
79Q8×C61—42no—
80D4×C61—42no—
81D4×C61—82no—
82C2×C4×C61—21no—
83C2×C4×C61—41no—
84C2×C4×C61C2×C22⋊C411yescharacteristic
85C22⋊D61—82no—
86C2×Dic61—21no—
87C2×Dic61—41no—
88C2×Dic61—41no—
89C2×Dic61—82no—
90C2×Dic61#10011yescharacteristic
91C4×D61—22no—
92C4×D61—42no—
93C4×D61—82no—
94Dic122—42no—
95Dic122—42no—
96Dic122—82no—
97C8×C61AGL(1,ℤ/8)12yes—
98C8×C61AGL(1,ℤ/8)12yes—
99#92418—22no—
100#11993—611nomaximal, Sylow
101C22×C61—82no—
102C22×C61—81no—
103C22×C61C2×#10011yescharacteristic
104C2×D61—21no—
105C2×D61—41no—
106C4×C61—22no—
107C4×C61—81no—
108C4×C61#41812yes—
109Dic61—21no—
110Dic61—41no—
111Dic61—81no—
112Dic61—81no—
113#3152—611no—
114#2165—612no—
115#1359—611no—
116#1359D6111yescharacteristic, complemented
117#1358—612no—
118C2×C61—21no—
119C2×C61—161no—
120C2×C61#155511yescharacteristic
121D61—41no—
122C2×#137—1222no—
123#532—611no—
124C2×C4⋊Q8—611no—
125#497—1222no—
126#494—611no—
127#494—611no—
128#492—611no—
129#492C2×D6111yescharacteristic
130C4×M16—611no—
131#364—612no—
132#364C2×D6112yes—
133#349—1222no—
134#347—1222no—
135C61#1199311yescharacteristic, complemented, Sylow
136C2×#41—1221no—
137C22×Q8—1221no—
138#137—2444no—
139#137—2444no—
140C2×Q16—1222no—
141C2×Q16—1222no—
142C2×SD16—22no—
143C2×SD16—1222no—
144C2×M16—612no—
145C4⋊Q8—611no—
146C4⋊Q8—611no—
147C4⋊Q8—1221no—
148C4⋊Q8—2441no—
149C4⋊Q8C4×D6111yescharacteristic
150#127C4×D6111yescharacteristic
151#124—1221no—
152#121—2441no—
153C4×D4—2441no—
154C2×C4⋊C4—2441no—
155C2×C42—611no—
156M32—22no—
157M32—1222no—
158#97—612no—
159C4×C8—611no—
160C4×C8C22×D6111yescharacteristic
161#41—2444no—
162C2×Q8—21no—
163C2×Q8—1221no—
164C2×Q8—1221no—
165C2×Q8—2441no—
166C2×Q8—2444no—
167C2×D4—21no—
168C2×D4—41no—
169C2×D4—2441no—
170C22×C4—611no—
171C22×C4—1221no—
172C22×C4—2441no—
173C22×C4—2441no—
174Q16—2444no—
175Q16—2442no—
176Q16—2442no—
177SD16—42no—
178SD16—42no—
179SD16—2444no—
180M16—1222no—
181M16—1222no—
182C2×C8—612no—
183C2×C8D4×D6112yes—
184C4⋊C4—41no—
185C4⋊C4—2441no—
186C4⋊C4—2441no—
187C4⋊C4—2442no—
188C22⋊C4—2442no—
189C42—611no—
190C42—1221no—
191C42C2×C4×D6111yescharacteristic
192C16—42no—
193C16—2442no—
194C23—41no—
195C23—611no—
196Q8—42no—
197Q8—4882no—
198Q8—2442no—
199Q8—2442no—
200D4—42no—
201D4—82no—
202D4—4882no—
203C2×C4—21no—
204C2×C4—41no—
205C2×C4—611no—
206C2×C4—1222no—
207C2×C4—1221no—
208C2×C4—4882no—
209C2×C4—4882no—
210C2×C4—2442no—
211C2×C4—2441no—
212C2×C4—2441no—
213C2×C4C22⋊C4×D6111yescharacteristic
214C8—1222no—
215C8#1451980112yes—
216C8#1451980412yes—
217C22—82no—
218C22—81no—
219C22—1221no—
220C22—2441no—
221C22#100×D6111yescharacteristic
222C4—22no—
223C4—81no—
224C4—1221no—
225C4—4881no—
226C4—4881no—
227C4—2441no—
228C4#1683976512yes—
229C2—21no—
230C2—161no—
231C2—2441no—
232C2#2144453111yescharacteristic
subgroupG/H
centerC2#21444531
commutatorC42×C61C22×C4
FrattiniC4×C8C22×D61
FittingC61×#1359C2
socleC2×C61#1555
prime radical#16840310C4
cosocleC4×C8×C61C23
hypercenter#1359D61
hypocenterC61#11993
LCSG ▷ C42×C61 ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #1359

Complements

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

NH ≅ G/Ntype
2. #21447630C2semidirect
3. #21443860C2semidirect
4. #21443857C2semidirect
5. C61×#1359C2semidirect
116. #1359D61semidirect
135. C61#11993semidirect

Maximal subgroups

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

subgroupindex
#214562732
#214476302
#214438602
#214438572
C61×#13592
#1199361