Finite Group Databasebeta

← #29584186

Subgroups — #29584186

Order 15616.

402 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 81 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#21457200C211yesmaximal, characteristic
2#21445990C211yesmaximal, characteristic, complemented
3#21444552C212yesmaximal, complemented
4#21444269C212yesmaximal, complemented
5C61×#1309C211yesmaximal, characteristic, complemented
6#16840312C2212yes—
7#16840053—22no—
8#16840015—22no—
9#16839958—22no—
10C2×#14519712—22no—
11#16839786C2211yescharacteristic
12C2×C4×#92418C2211yescharacteristic
13#16839510—22no—
14#16839505—22no—
15#16839489—24no—
16C61×#423C2212yes—
17C61×#423C2211yescharacteristic
18C2×#11758222—22no—
19C2×D488—22no—
20C2×#11758194—21no—
21C22×#92418—21no—
22#14519777D412yes—
23#14519774D412yes—
24#14519765—22no—
25#14519751—22no—
26#14519751—22no—
27C4×D244—22no—
28C4×D244—22no—
29#14519712—44no—
30#14519709—22no—
31C4⋊#92418—22no—
32#14519699—24no—
33C4×#92418—21no—
34C4×#92418D412yes—
35C2×M16×C61—22no—
36C2×M16×C61D412yes—
37C2×C42×C61C2311yescharacteristic
38C4⋊C8×C61—22no—
39C4⋊C8×C61—22no—
40C4⋊C8×C61—22no—
41C22×C4×C61C2×D411yescharacteristic
42C22×C4×C61#4112yes—
43C2×C22⋊D61—42no—
44#11758222—82no—
45#11758222—82no—
46C2×D244—22no—
47C2×D244—22no—
48C2×D244—22no—
49C2×C4×D61—42no—
50C2×Dic122—22no—
51M16×C61—42no—
52M16×C61—82no—
53C2×C8×C61—22no—
54C2×C8×C61—42no—
55C42×C61C2×D412yes—
56C42×C61C2×D411yescharacteristic
57C42×C61#4111yescharacteristic
58#11758198—42no—
59#11758198—42no—
60C4⋊Dic61—22no—
61C4⋊Dic61—22no—
62#11758194—42no—
63C2×#92418—22no—
64C2×#92418—22no—
65C2×#92418—22no—
66C2×#92418—41no—
67D488—82no—
68C23×C61#12411yescharacteristic
69C22×D61—42no—
70C2×C4×C61—21no—
71C2×C4×C61—21no—
72C2×C4×C61—22no—
73C2×C4×C61—22no—
74C2×C4×C61#12011yescharacteristic
75C2×C4×C61#12111yescharacteristic
76C2×C4×C61#12112yes—
77C2×C4×C61#12312yes—
78C22⋊D61—84no—
79C2×Dic61—42no—
80D244—42no—
81D244—42no—
82D244—42no—
83D244—42no—
84D244—82no—
85C4×D61—82no—
86C4×D61—82no—
87Dic122—82no—
88C8×C61—42no—
89C8×C61—82no—
90#92418—24no—
91#92418—81no—
92#10530—611nomaximal, Sylow
93C22×C61—22no—
94C22×C61—21no—
95C22×C61#39411yescharacteristic
96C2×D61—82no—
97C2×D61—82no—
98C2×D61—82no—
99C4×C61—22no—
100C4×C61—21no—
101C4×C61#45311yescharacteristic
102C4×C61#45611yescharacteristic
103C4×C61#47112yes—
104C4×C61#47212yes—
105Dic61—82no—
106#3253—611no—
107#1848—611no—
108#1560—612no—
109#1492—612no—
110#1309D6111yescharacteristic, complemented
111C2×C61—41no—
112C2×C61#203911yescharacteristic
113C2×C61#205112yes—
114D61—162no—
115#532—612no—
116#476—1222no—
117#469—1222no—
118#459—1222no—
119C2×#108—1222no—
120#423—611no—
121#423C2×D6112yes—
122#423C2×D6111yescharacteristic
123C2×C4×C8—611no—
124#333—1222no—
125#331—1222no—
126D4⋊C8—1224no—
127C61#1053011yescharacteristic, complemented, Sylow
128C2×#41—1222no—
129C2×D8—1222no—
130C2×M16—22no—
131C2×M16—1221no—
132C2×M16D24412yes—
133C22×C8—1221no—
134C4⋊Q8—612no—
135#127—612no—
136#124—1222no—
137#121—1222no—
138#121—1222no—
139C4×D4—1222no—
140C4×D4—1222no—
141C2×C42C22×D6111yescharacteristic
142#108—2444no—
143#106—1222no—
144C4⋊C8—22no—
145C4⋊C8—22no—
146C4⋊C8—22no—
147C4⋊C8—1222no—
148#102—1224no—
149C4×C8—612no—
150C4×C8—1221no—
151#41—4882no—
152#41—4882no—
153C2×Q8—1222no—
154C2×D4—1222no—
155C2×D4—1222no—
156C2×D4—1222no—
157C2×D4—2442no—
158C22×C4—2442no—
159C22×C4C2×D24411yescharacteristic
160C22×C4#1175822212yes—
161D8—4882no—
162M16—42no—
163M16—82no—
164M16—2442no—
165C2×C8—22no—
166C2×C8—42no—
167C2×C8—1222no—
168C2×C8—1222no—
169C2×C8—1222no—
170C2×C8—2441no—
171C4⋊C4—1222no—
172C4⋊C4—1222no—
173C22⋊C4—2442no—
174C22⋊C4—2442no—
175C42D4×D6112yes—
176C42D4×D6111yescharacteristic
177C42#1175822611yescharacteristic
178C23—2442no—
179C23#1451976511yescharacteristic
180Q8—4882no—
181D4—4882no—
182D4—4884no—
183D4—2442no—
184D4—2442no—
185D4—2442no—
186D4—2442no—
187C2×C4—21no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—21no—
191C2×C4—4882no—
192C2×C4—4882no—
193C2×C4—2442no—
194C2×C4#1451974711yescharacteristic
195C2×C4#1451975012yes—
196C2×C4#1451975211yescharacteristic
197C2×C4#1451976212yes—
198C8—42no—
199C8—82no—
200C8—1224no—
201C8—4881no—
202C22—21no—
203C22—22no—
204C22—4882no—
205C22—4882no—
206C22—4882no—
207C22#1683968011yescharacteristic
208C4—21no—
209C4—22no—
210C4—4882no—
211C4#1683993211yescharacteristic
212C4#1683994711yescharacteristic
213C4#1684002812yes—
214C4#1684003112yes—
215C2—41no—
216C2—9762no—
217C2#2144704411yescharacteristic
218C2#2144711412yes—
subgroupG/H
centerC22#16839680
commutator, cosocleC2×C42×C61C23
FrattiniC2×C42C22×D61
FittingC61×#1309C2
socleC22×C61#394
prime radical#21457200C2
hypercenter#1309D61
hypocenterC61#10530
LCSG ▷ C2×C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1309

Complements

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

NH ≅ G/Ntype
2. #21445990C2semidirect
3. #21444552C2semidirect
4. #21444269C2semidirect
5. C61×#1309C2semidirect
110. #1309D61semidirect
127. C61#10530semidirect

Maximal subgroups

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

subgroupindex
#214572002
#214459902
#214445522
#214442692
C61×#13092
#1053061