Finite Group Databasebeta

← #29577540

Subgroups — #29577540

Order 15616.

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

normal subgroups only

This group has 540 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#21456272C211yesmaximal, characteristic
2#21444532C212yesmaximal, complemented
3#21444454C211yesmaximal, characteristic, complemented
4#21443690C212yesmaximal, complemented
5C61×#2980C211yesmaximal, characteristic, complemented
6C2×#14519802—22no—
7C2×#14519774C2211yescharacteristic
8#16840152—22no—
9#16840138C412yes—
10#16840136C412yes—
11#16839819C2212yes—
12C4×#11758194C2211yescharacteristic
13#16839505—24no—
14#16839502—22no—
15C61×#532C2211yescharacteristic
16C61×#435—24no—
17C61×#432C2212yes—
18C61×#408—22no—
19C22×D244—21no—
20C22×D244—22no—
21#14519802—44no—
22#14519802—44no—
23C2×#11758201—24no—
24C2×#11758199—24no—
25C2×#11758194—21no—
26#14519774—22no—
27#14519774—21no—
28#14519774C2×C412yes—
29C4⋊#92418—22no—
30C22⋊#92418—22no—
31#14519688—21no—
32#14519688—22no—
33C4×#92418C2×C412yes—
34C2×#41×C61D412yes—
35C2×M16×C61—22no—
36C22×C8×C61—22no—
37C4⋊Q8×C61C2×C411yescharacteristic
38#127×C61C2×C411yescharacteristic
39#124×C61D412yes—
40#121×C61—22no—
41C4×D4×C61—22no—
42C2×C42×C61C2311yescharacteristic
43C4⋊C8×C61D414yes—
44C22⋊C8×C61—24no—
45C22⋊C8×C61—22no—
46#97×C61—22no—
47C23×D61—41no—
48#41×C61—42no—
49#41×C61—42no—
50C2×Q8×C61C22⋊C412yes—
51C2×D4×C61—22no—
52C2×D4×C61—22no—
53C2×D4×C61C22⋊C412yes—
54C22×C4×C61—22no—
55C22×C4×C61C2×D411yescharacteristic
56C22×C4×C61#4112yes—
57C2×D244—22no—
58C2×D244—24no—
59C2×D244—42no—
60C2×D244—42no—
61C2×D244—41no—
62C2×D244—42no—
63C2×D244—84no—
64M16×C61—82no—
65C2×C8×C61—24no—
66C2×C8×C61—82no—
67C4⋊C4×C61—22no—
68C22⋊C4×C61—24no—
69C42×C61C22×C411yescharacteristic
70C42×C61C2×D411yescharacteristic
71C42×C61C2×D412yes—
72#11758201—48no—
73#11758199—48no—
74#11758194—24no—
75C2×#92418—22no—
76C2×#92418—42no—
77C23×C61—22no—
78C23×C61#12311yescharacteristic
79C22×D61—42no—
80C22×D61—81no—
81C22×D61—81no—
82C22×D61—84no—
83Q8×C61—42no—
84D4×C61—42no—
85D4×C61—82no—
86D4×C61—82no—
87C2×C4×C61—22no—
88C2×C4×C61—22no—
89C2×C4×C61—42no—
90C2×C4×C61—42no—
91C2×C4×C61—42no—
92C2×C4×C61C2×C22⋊C411yescharacteristic
93C2×C4×C61C4×D412yes—
94C2×C4×C61#12011yescharacteristic
95C2×C4×C61#12112yes—
96D244—44no—
97D244—84no—
98D244—84no—
99D244—82no—
100C8×C61—44no—
101#92418—24no—
102#92418—82no—
103#9423—611nomaximal, Sylow
104C22×C61—21no—
105C22×C61—22no—
106C22×C61—42no—
107C22×C61—44no—
108C22×C61#38611yescharacteristic
109C2×D61—82no—
110C2×D61—84no—
111C2×D61—162no—
112C2×D61—162no—
113C4×C61—21no—
114C4×C61—22no—
115C4×C61—22no—
116C4×C61—42no—
117C4×C61#45312yes—
118#3151—611no—
119#2980D6111yescharacteristic, complemented
120#1555—612no—
121#1536—611no—
122#1291—612no—
123C2×C61—41no—
124C2×C61—82no—
125C2×C61#188711yescharacteristic
126C2×C61#190312yes—
127D61—162no—
128C2×AGL(1,ℤ/8)—1222no—
129#532C2×D6111yescharacteristic
130C2×#127—611no—
131#496—1222no—
132#493—612no—
133#492—612no—
134#435—24no—
135#432—612no—
136#432C2×D6112yes—
137#408—22no—
138C4×M16—611no—
139#331—1224no—
140#329—1222no—
141C61#942311yescharacteristic, complemented, Sylow
142C2×#41C22⋊D6112yes—
143C22×D4—1222no—
144C22×D4—1221no—
145AGL(1,ℤ/8)—2444no—
146AGL(1,ℤ/8)—2444no—
147C2×SD16—1224no—
148C2×D8—1224no—
149C2×M16—22no—
150C2×M16—1221no—
151C22×C8—22no—
152C4⋊Q8C4×D6111yescharacteristic
153#127—612no—
154#127—1221no—
155#127—1222no—
156#127C4×D6111yescharacteristic
157#124D24412yes—
158#121—22no—
159C4×D4—22no—
160C2×C42C22×D6111yescharacteristic
161C4⋊C8—1222no—
162C4⋊C8C22⋊D6114yes—
163C22⋊C8—24no—
164C22⋊C8—22no—
165C22⋊C8—1222no—
166#97—22no—
167#97—1221no—
168#97—1222no—
169C4×C8—612no—
170C24—2441no—
171#41—42no—
172#41—42no—
173C2×Q8#1175819812yes—
174C2×D4—22no—
175C2×D4—22no—
176C2×D4—1222no—
177C2×D4—1224no—
178C2×D4—4884no—
179C2×D4—2442no—
180C2×D4—2442no—
181C2×D4—2441no—
182C2×D4—2442no—
183C2×D4#1175819812yes—
184C22×C4—22no—
185C22×C4#1175822212yes—
186C22×C4C2×C22⋊D6111yescharacteristic
187SD16—2448no—
188D8—2448no—
189M16—82no—
190M16—1224no—
191C2×C8—24no—
192C2×C8—82no—
193C2×C8—1222no—
194C2×C8—2442no—
195C4⋊C4—22no—
196C22⋊C4—24no—
197C42C2×C4×D6111yescharacteristic
198C42C2×D24411yescharacteristic
199C42C2×C22⋊D6112yes—
200C23—22no—
201C23—4884no—
202C23—4881no—
203C23—4881no—
204C23—2442no—
205C23#1451976011yescharacteristic
206Q8—42no—
207D4—42no—
208D4—82no—
209D4—82no—
210D4—4884no—
211D4—4884no—
212D4—4882no—
213D4—2444no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4C2×#1175819811yescharacteristic
220C2×C4C4×C22⋊D6112yes—
221C2×C4#1451974611yescharacteristic
222C2×C4#1451975112yes—
223C8—44no—
224C8—1224no—
225C8—4882no—
226C22—21no—
227C22—22no—
228C22—42no—
229C22—44no—
230C22—9762no—
231C22—9762no—
232C22—4882no—
233C22—4884no—
234C22#1683964111yescharacteristic
235C4—21no—
236C4—22no—
237C4—22no—
238C4—42no—
239C4#1683993012yes—
240C2—41no—
241C2—82no—
242C2—9762no—
243C2#2144621711yescharacteristic
244C2#2144629912yes—
subgroupG/H
centerC22#16839641
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#2980C2
socleC22×C61#386
prime radical#21456272C2
cosocleC2×C42×C61C23
hypercenter#2980D61
hypocenterC61#9423
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2980

Complements

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

NH ≅ G/Ntype
2. #21444532C2semidirect
3. #21444454C2semidirect
4. #21443690C2semidirect
5. C61×#2980C2semidirect
119. #2980D61semidirect
141. C61#9423semidirect

Maximal subgroups

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

subgroupindex
#214562722
#214445322
#214444542
#214436902
C61×#29802
#942361