Finite Group Databasebeta

← #30608378

Subgroups — #30608378

Order 15616.

450 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 60 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#21462112C212yesmaximal, complemented
2#21460954C214yesmaximal, complemented
3#21456269C211yesmaximal, characteristic
4#16840748—22no—
5#16840673C2214yescomplemented
6#16840671—22no—
7C2×#14519803—22no—
8#16840312C2211yescharacteristic
9#16840149—22no—
10#16840138C412yes—
11#16840134C2212yes—
12C4×#11758194C411yescharacteristic
13C2×#127×C61C411yescharacteristic
14D61⋊M16—42no—
15#14519838—24no—
16#14519837—42no—
17C2×#11758222D412yes—
18#14519803—44no—
19#14519803—44no—
20C2×#11758200—24no—
21C2×#11758199—24no—
22C2×#11758194—21no—
23#14519777C2311yescharacteristic
24#14519774C2×C411yescharacteristic
25#14519765D412yes—
26#14519751—41no—
27C4×D244—41no—
28#14519688—21no—
29C4×#92418C2×C412yes—
30C22×D4×C61—21no—
31C22×D4×C61—41no—
32#127×C61—21no—
33#127×C61—41no—
34#127×C61C2×C412yes—
35C2×C42×C61C2×C411yescharacteristic
36C24×C61—41no—
37C2×D4×C61—22no—
38C2×D4×C61—42no—
39C2×D4×C61—42no—
40C2×D4×C61—42no—
41C2×D4×C61—42no—
42C2×D4×C61—41no—
43C2×D4×C61—84no—
44C22×C4×C61—21no—
45C22×C4×C61C22⋊C411yescharacteristic
46C2×C22⋊D61—22no—
47#11758222—42no—
48#11758222—42no—
49C2×D244—41no—
50C2×D244C22⋊C412yes—
51C2×C4×D61—22no—
52C2×Dic122C2×D412yes—
53#11758217—44no—
54#11758213—82no—
55D61⋊C8—82no—
56C42×C61—21no—
57C42×C61C22⋊C411yescharacteristic
58C42×C61C22×C411yescharacteristic
59#11758200—48no—
60#11758199—48no—
61#11758198—42no—
62C4⋊Dic61—41no—
63#11758194—24no—
64C2×#92418—22no—
65C23×C61—42no—
66C23×C61—81no—
67C23×C61—81no—
68C23×C61—84no—
69C23×C61#9911yescharacteristic
70C22×D61—22no—
71D4×C61—44no—
72D4×C61—84no—
73D4×C61—84no—
74D4×C61—82no—
75C2×C4×C61—21no—
76C2×C4×C61—22no—
77C2×C4×C61—21no—
78C2×C4×C61—42no—
79C2×C4×C61#9911yescharacteristic
80C2×C4×C61C2×C22⋊C411yescharacteristic
81C22⋊D61—82no—
82C2×Dic61—22no—
83D244—42no—
84D244—82no—
85C4×D61—42no—
86C4×D61—42no—
87Dic122—42no—
88#92420—84no—
89#92418—24no—
90#13591—611nomaximal, Sylow
91C22×C61—21no—
92C22×C61—41no—
93C22×C61—84no—
94C22×C61—82no—
95C22×C61—162no—
96C22×C61—162no—
97C22×C61C2×#9911yescharacteristic
98C2×D61—42no—
99C2×D61—82no—
100C4×C61—22no—
101C4×C61—21no—
102C4×C61—41no—
103C4×C61#35312yes—
104Dic61—42no—
105#3151—611no—
106#2132—612no—
107#1410—614no—
108C2×C61—21no—
109C2×C61—41no—
110C2×C61—162no—
111C2×C61C2×#35311yescharacteristic
112D61—82no—
113C2×AGL(1,ℤ/8)—1222no—
114#532—611no—
115C2×#127#692311yescharacteristic
116#496—1222no—
117#493—612no—
118#492—612no—
119#413—1222no—
120C4×M16—611no—
121#356—614no—
122#355—1222no—
123C61#1359111yescharacteristic, complemented, Sylow
124C2×#41—612no—
125C22×D4—21no—
126C22×D4—41no—
127AGL(1,ℤ/8)—2444no—
128AGL(1,ℤ/8)—2444no—
129C2×SD16—1224no—
130C2×D8—1224no—
131C2×M16—1221no—
132C2×M16—2442no—
133C4⋊Q8—611no—
134#127—21no—
135#127—41no—
136#127—611no—
137#127C2×#692312yes—
138#124—612no—
139#121—2441no—
140C4×D4—2441no—
141C2×C42C2×#692311yescharacteristic
142#101—1224no—
143#100—2442no—
144#97—1221no—
145C4×C8—612no—
146C24—41no—
147#41—2442no—
148#41—2442no—
149C2×Q8—612no—
150C2×D4—22no—
151C2×D4—42no—
152C2×D4—42no—
153C2×D4—42no—
154C2×D4—42no—
155C2×D4—41no—
156C2×D4—84no—
157C2×D4—612no—
158C2×D4—1222no—
159C2×D4—2441no—
160C22×C4—21no—
161C22×C4—1222no—
162C22×C4C22⋊#692311yescharacteristic
163SD16—2448no—
164D8—2448no—
165M16—1224no—
166M16—4882no—
167M16—2444no—
168C2×C8—1222no—
169C2×C8—4882no—
170C4⋊C4—2441no—
171C22⋊C4—2442no—
172C42—21no—
173C42C22⋊#692311yescharacteristic
174C42C22×#692311yescharacteristic
175C23—42no—
176C23—81no—
177C23—81no—
178C23—84no—
179C23—1222no—
180C23#1451983511yescharacteristic
181Q8—2442no—
182D4—44no—
183D4—84no—
184D4—84no—
185D4—82no—
186D4—4882no—
187D4—4882no—
188D4—2442no—
189C2×C4—21no—
190C2×C4—22no—
191C2×C4—21no—
192C2×C4—42no—
193C2×C4—1222no—
194C2×C4—2442no—
195C2×C4—2442no—
196C2×C4#1451983511yescharacteristic
197C2×C4C2×C22⋊#692311yescharacteristic
198C8—1224no—
199C8—4884no—
200C22—21no—
201C22—41no—
202C22—84no—
203C22—82no—
204C22—162no—
205C22—162no—
206C22—4882no—
207C22—2442no—
208C22C2×#1451983511yescharacteristic
209C4—22no—
210C4—21no—
211C4—41no—
212C4—2442no—
213C4#1684066612yes—
214C2—21no—
215C2—41no—
216C2—162no—
217C2—4882no—
218C2C2×#1684066611yescharacteristic
subgroupG/H
centerC2C2×#16840666
commutatorC42×C61C22×C4
FrattiniC42C22×#6923
FittingC2×#127×C61C4
socleC2×C61C2×#353
prime radical#21456269C2
cosocle#14519777C23
hypercenterC2×#127#6923
hypocenterC61#13591
LCSG ▷ C42×C61 ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ C2×C42 ◁ C2×#127

Complements

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

NH ≅ G/Ntype
1. #21462112C2semidirect
2. #21460954C2semidirect
5. #16840673C22semidirect
123. C61#13591semidirect

Maximal subgroups

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

subgroupindex
#214621122
#214609542
#214562692
#1359161