Finite Group Databasebeta

← #30608361

Subgroups — #30608361

Order 15616.

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

normal subgroups only

This group has 406 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#21462092C212yesmaximal
2#21460952C214yesmaximal, complemented
3#21456276C211yesmaximal, characteristic
4#16840741—22no—
5#16840669—22no—
6#16840667C2214yes—
7C2×#14519808—22no—
8#16840312C2211yescharacteristic
9#16840154—22no—
10#16840141C412yescomplemented
11#16840136C2212yes—
12C4×#11758194C411yescharacteristic, complemented
13C2×C4⋊Q8×C61C411yescharacteristic, complemented
14#14519856—42no—
15#14519836—24no—
16#14519835—42no—
17C2×#11758222D412yes—
18#14519808—44no—
19#14519808—44no—
20C2×#11758202—24no—
21C2×#11758201—24no—
22C2×#11758194—21no—
23#14519777C2×C411yescharacteristic
24#14519774C2311yescharacteristic
25#14519765D412yes—
26#14519751—41no—
27C4×D244—41no—
28#14519688—21no—
29C4×#92418C2×C412yes—
30C22×Q8×C61—21no—
31C4⋊Q8×C61—21no—
32C4⋊Q8×C61—41no—
33C4⋊Q8×C61C2×C412yes—
34C2×C4⋊C4×C61—41no—
35C2×C42×C61C2×C411yescharacteristic
36C2×Q8×C61—22no—
37C2×Q8×C61—42no—
38C2×Q8×C61—42no—
39C2×Q8×C61—41no—
40C22×C4×C61—21no—
41C22×C4×C61—41no—
42C22×C4×C61C22⋊C411yescharacteristic
43C2×C22⋊D61—22no—
44#11758222—42no—
45#11758222—42no—
46C2×D244—41no—
47C2×D244C2×D412yes—
48C2×C4×D61—22no—
49C2×Dic122C22⋊C412yes—
50C22⋊#6923—44no—
51C4⋊#6923—82no—
52C4×#6923—82no—
53C4⋊C4×C61—42no—
54C4⋊C4×C61—42no—
55C42×C61—21no—
56C42×C61C22⋊C411yescharacteristic
57C42×C61C22×C411yescharacteristic
58#11758202—48no—
59#11758201—48no—
60#11758198—42no—
61C4⋊Dic61—41no—
62#11758194—24no—
63C2×#92418—22no—
64C23×C61#9911yescharacteristic
65C22×D61—22no—
66C2×#6923—84no—
67Q8×C61—44no—
68Q8×C61—82no—
69C2×C4×C61—21no—
70C2×C4×C61—22no—
71C2×C4×C61—21no—
72C2×C4×C61—42no—
73C2×C4×C61—42no—
74C2×C4×C61—81no—
75C2×C4×C61—81no—
76C2×C4×C61#9911yescharacteristic
77C2×C4×C61C2×C22⋊C411yescharacteristic
78C22⋊D61—82no—
79C2×Dic61—22no—
80D244—42no—
81D244—82no—
82C4×D61—42no—
83C4×D61—42no—
84Dic122—42no—
85#92418—24no—
86#13586—611nomaximal, Sylow
87C22×C61—21no—
88C22×C61—41no—
89C22×C61C2×#9911yescharacteristic
90C2×D61—42no—
91C2×D61—82no—
92#6923—164no—
93C4×C61—22no—
94C4×C61—21no—
95C4×C61—41no—
96C4×C61—82no—
97C4×C61#35312yes—
98Dic61—42no—
99#3152—611no—
100#2126—612no—
101#1409—614no—
102C2×C61—21no—
103C2×C61—41no—
104C2×C61C2×#35311yescharacteristic
105D61—82no—
106C2×#137—1222no—
107#532—611no—
108C2×C4⋊Q8#692311yescharacteristic, complemented
109#497—1222no—
110#494—612no—
111#492—612no—
112#410—1222no—
113C4×M16—611no—
114#354—1222no—
115#353—614no—
116C61#1358611yescharacteristic, complemented, Sylow
117C2×#41—612no—
118C22×Q8—21no—
119#137—2444no—
120#137—2444no—
121C2×Q16—1224no—
122C2×SD16—1224no—
123C2×M16—1221no—
124C4⋊Q8—21no—
125C4⋊Q8—41no—
126C4⋊Q8—611no—
127C4⋊Q8C2×#692312yes—
128#127—611no—
129#124—612no—
130#121—2441no—
131C4×D4—2441no—
132#117—2442no—
133C2×C4⋊C4—41no—
134C2×C42C2×#692311yescharacteristic
135#99—1224no—
136#99—2442no—
137#97—1221no—
138C4×C8—612no—
139#41—2442no—
140#41—2442no—
141C2×Q8—22no—
142C2×Q8—42no—
143C2×Q8—42no—
144C2×Q8—41no—
145C2×Q8—612no—
146C2×D4—612no—
147C2×D4—1222no—
148C2×D4—2441no—
149C22×C4—21no—
150C22×C4—41no—
151C22×C4—1222no—
152C22×C4C22⋊#692311yescharacteristic
153Q16—2448no—
154SD16—2448no—
155M16—1224no—
156C2×C8—1222no—
157C4⋊C4—42no—
158C4⋊C4—42no—
159C4⋊C4—4882no—
160C4⋊C4—2441no—
161C22⋊C4—2442no—
162C22⋊C4—2444no—
163C42—21no—
164C42—4882no—
165C42C22⋊#692311yescharacteristic
166C42C22×#692311yescharacteristic
167C23—1222no—
168C23#1451983511yescharacteristic
169Q8—44no—
170Q8—82no—
171Q8—2442no—
172D4—4882no—
173D4—4882no—
174D4—2442no—
175C2×C4—21no—
176C2×C4—22no—
177C2×C4—21no—
178C2×C4—42no—
179C2×C4—42no—
180C2×C4—81no—
181C2×C4—81no—
182C2×C4—1222no—
183C2×C4—4884no—
184C2×C4—2442no—
185C2×C4—2442no—
186C2×C4#1451983511yescharacteristic
187C2×C4C2×C22⋊#692311yescharacteristic
188C8—1224no—
189C22—21no—
190C22—41no—
191C22—4882no—
192C22—2442no—
193C22C2×#1451983511yescharacteristic
194C4—22no—
195C4—21no—
196C4—41no—
197C4—82no—
198C4—9764no—
199C4—2442no—
200C4#1684066612yes—
201C2—21no—
202C2—41no—
203C2—4882no—
204C2C2×#1684066611yescharacteristic
subgroupG/H
centerC2C2×#16840666
commutatorC42×C61C22×C4
FrattiniC42C22×#6923
FittingC2×C4⋊Q8×C61C4
socleC2×C61C2×#353
prime radical#16840312C22
cosocle#14519774C23
hypercenterC2×C4⋊Q8#6923
hypocenterC61#13586
LCSG ▷ C42×C61 ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ C2×C42 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
2. #21460952C2semidirect
10. #16840141C4semidirect
12. C4×#11758194C4semidirect
13. C2×C4⋊Q8×C61C4semidirect
108. C2×C4⋊Q8#6923semidirect
116. C61#13586semidirect

Maximal subgroups

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

subgroupindex
#214620922
#214609522
#214562762
#1358661