Finite Group Databasebeta

← #29574977

Subgroups — #29574977

Order 15616.

398 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 99 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#21456167C211yesmaximal, characteristic
2#21444484C211yesmaximal, characteristic, complemented
3#21444287C212yesmaximal
4#21443717C212yesmaximal, complemented
5C61×#2987C211yesmaximal, characteristic, complemented
6#16840312C2211yescharacteristic
7#16840132C412yes—
8#16840128C412yes—
9#16840124—22no—
10#16839815C2212yes—
11#16839756—22no—
12C2×#14519688C2211yescharacteristic
13#16839508—22no—
14#16839489—24no—
15C61×#533C2211yescharacteristic
16C61×#446—24no—
17C61×#433—22no—
18C61×#432C2212yes—
19C2×#11758222—21no—
20C22×#92418—21no—
21#14519777C2×C411yescharacteristic
22#14519774C2×C411yescharacteristic
23#14519765—21no—
24#14519751—22no—
25C4×D244—22no—
26C4⋊#92418—22no—
27#14519700—24no—
28#14519697—24no—
29C22⋊#92418—22no—
30#14519688—21no—
31#14519688C2×C412yes—
32C4×#92418—22no—
33C4⋊Q8×C61D412yes—
34#125×C61C2×C412yes—
35C22⋊Q8×C61—22no—
36C4×Q8×C61—22no—
37#117×C61D412yes—
38C2×C42×C61C2311yescharacteristic
39C4⋊C8×C61—24no—
40C4⋊C8×C61D414yes—
41C22⋊C8×C61—22no—
42#97×C61—22no—
43#97×C61—22no—
44C4×C8×C61—22no—
45C2×Q8×C61—22no—
46C22×C4×C61C2×D411yescharacteristic
47C22×C4×C61#4112yes—
48C2×C22⋊D61—41no—
49#11758222—44no—
50C2×D244—21no—
51C2×D244—22no—
52C2×C4×D61—41no—
53C2×Dic122—21no—
54C2×C8×C61—24no—
55C4⋊C4×C61—24no—
56C4⋊C4×C61—22no—
57C4⋊C4×C61C22⋊C414yes—
58C22⋊C4×C61—22no—
59C42×C61—22no—
60C42×C61C22×C411yescharacteristic
61C42×C61C2×D411yescharacteristic
62C42×C61C2×D412yes—
63#11758198—42no—
64C4⋊Dic61—22no—
65C2×#92418—22no—
66C2×#92418—42no—
67C2×#92418—42no—
68C23×C61#12311yescharacteristic
69C22×D61—41no—
70Q8×C61—82no—
71C2×C4×C61—22no—
72C2×C4×C61—24no—
73C2×C4×C61—42no—
74C2×C4×C61C2×C22⋊C411yescharacteristic
75C2×C4×C61C4×D412yes—
76C2×C4×C61#12011yescharacteristic
77C2×C4×C61#12112yes—
78C22⋊D61—82no—
79C2×Dic61—41no—
80D244—44no—
81D244—42no—
82C4×D61—82no—
83Dic122—42no—
84C8×C61—44no—
85#92418—42no—
86#92418—82no—
87#9008—611nomaximal, Sylow
88C22×C61—22no—
89C22×C61—21no—
90C22×C61#38611yescharacteristic
91C2×D61—82no—
92C2×D61—81no—
93C4×C61—22no—
94C4×C61—22no—
95C4×C61—21no—
96C4×C61—44no—
97C4×C61#45412yes—
98Dic61—81no—
99#3135—611no—
100#2987D6111yescharacteristic, complemented
101#1544—611no—
102#1496—612no—
103#1302—612no—
104C2×C61—41no—
105C2×C61#188112yes—
106C2×C61#188811yescharacteristic
107D61—161no—
108#533C2×D6111yescharacteristic
109#532—611no—
110#491—612no—
111#490—612no—
112#489—1222no—
113#446—24no—
114#433—22no—
115#432C2×D6112yes—
116#431—612no—
117#416—1222no—
118C2×#97—611no—
119#332—1222no—
120D4⋊C8—1224no—
121C61#900811yescharacteristic, complemented, Sylow
122C2×#41—1221no—
123C22×C8—1221no—
124C4⋊Q8—611no—
125C4⋊Q8D24412yes—
126#127—611no—
127#125C4×D6112yes—
128#124—1221no—
129C22⋊Q8—22no—
130#121—1222no—
131C4×Q8—22no—
132C4×D4—1222no—
133#117C22⋊D6112yes—
134C2×C42C22×D6111yescharacteristic
135C4⋊C8—24no—
136C4⋊C8—1222no—
137C4⋊C8C22⋊D6114yes—
138#103—1224no—
139#102—1224no—
140C22⋊C8—22no—
141C22⋊C8—1222no—
142#97—22no—
143#97—22no—
144#97—612no—
145#97—1221no—
146C4×C8—22no—
147C4×C8—1222no—
148#41—2444no—
149C2×Q8—22no—
150C2×Q8—1221no—
151C2×D4—1221no—
152C2×D4—1222no—
153C2×D4—2441no—
154C22×C4—2441no—
155C22×C4#1175822212yes—
156C22×C4C2×C22⋊D6111yescharacteristic
157C2×C8—24no—
158C2×C8—1222no—
159C2×C8—2442no—
160C2×C8—2442no—
161C4⋊C4—24no—
162C4⋊C4—22no—
163C4⋊C4—1222no—
164C4⋊C4#1175819814yes—
165C22⋊C4—22no—
166C22⋊C4—2442no—
167C42—22no—
168C42C2×C4×D6111yescharacteristic
169C42C2×D24411yescharacteristic
170C42C2×C22⋊D6112yes—
171C23—2441no—
172C23#1451976011yescharacteristic
173Q8—82no—
174Q8—2442no—
175D4—4882no—
176D4—2444no—
177D4—2442no—
178C2×C4—22no—
179C2×C4—24no—
180C2×C4—42no—
181C2×C4—4882no—
182C2×C4—2441no—
183C2×C4C2×#1175819811yescharacteristic
184C2×C4C4×C22⋊D6112yes—
185C2×C4#1451974611yescharacteristic
186C2×C4#1451975112yes—
187C8—44no—
188C8—4882no—
189C8—2442no—
190C22—22no—
191C22—21no—
192C22—4882no—
193C22—4881no—
194C22#1683964111yescharacteristic
195C4—22no—
196C4—22no—
197C4—21no—
198C4—44no—
199C4—4881no—
200C4#1683993512yes—
201C2—41no—
202C2—9761no—
203C2#2144618112yes—
204C2#2144622211yescharacteristic
subgroupG/H
centerC2×C4#14519746
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#2987C2
socleC22×C61#386
prime radical#16840312C22
cosocleC2×C42×C61C23
hypercenter#2987D61
hypocenterC61#9008
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2987

Complements

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

NH ≅ G/Ntype
2. #21444484C2semidirect
4. #21443717C2semidirect
5. C61×#2987C2semidirect
100. #2987D61semidirect
121. C61#9008semidirect

Maximal subgroups

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

subgroupindex
#214561672
#214444842
#214442872
#214437172
C61×#29872
#900861