Finite Group Databasebeta

← #29577620

Subgroups — #29577620

Order 15616.

556 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 103 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#21454886C211yesmaximal, characteristic, complemented
3#21444488C212yesmaximal, complemented
4#21443708C212yesmaximal, complemented
5C61×#1560C211yesmaximal, characteristic, complemented
6C2×#14519802—22no—
7C2×#14519774C2211yescharacteristic
8#16840152C414yes—
9#16840138—21no—
10#16840136—21no—
11#16839840—22no—
12D4×#92418—22no—
13#16839815C2212yes—
14#16839729—22no—
15C4×#11758194C2211yescharacteristic
16#16839505—22no—
17#16839505—22no—
18C61×#532C2211yescharacteristic
19C61×#423C2212yes—
20C61×#333—21no—
21C61×#331—21no—
22C22×D244—21no—
23C22×D244—22no—
24#14519802—44no—
25#14519802—44no—
26C2×#11758201—24no—
27C2×#11758199—24no—
28C2×#11758194—21no—
29C2×#11758194—22no—
30C22×#92418—22no—
31#14519774—22no—
32#14519774—21no—
33#14519774C2×C412yes—
34C4⋊#92418D412yes—
35C4⋊#92418D412yes—
36C22⋊#92418—24no—
37C22⋊#92418—22no—
38#14519688C2×C412yes—
39C4×#92418—21no—
40C4×#92418—22no—
41C2×#41×C61D412yes—
42C2×M16×C61—22no—
43C4⋊Q8×C61D411yescharacteristic
44#127×C61D411yescharacteristic
45#124×C61C2×C412yes—
46#121×C61—22no—
47C4×D4×C61—22no—
48C2×C42×C61C2311yescharacteristic
49C4⋊C8×C61—22no—
50C4⋊C8×C61—22no—
51C23×D61—41no—
52#41×C61—42no—
53#41×C61—42no—
54C2×Q8×C61C22⋊C412yes—
55C2×D4×C61—22no—
56C2×D4×C61—22no—
57C2×D4×C61C22⋊C412yes—
58C22×C4×C61—22no—
59C22×C4×C61C2×D411yescharacteristic
60C22×C4×C61#4112yes—
61C2×D244—22no—
62C2×D244—24no—
63C2×D244—42no—
64C2×D244—42no—
65C2×D244—41no—
66C2×D244—42no—
67C2×D244—82no—
68C2×D244—82no—
69M16×C61—82no—
70C2×C8×C61—42no—
71C4⋊C4×C61—22no—
72C22⋊C4×C61—24no—
73C42×C61C22×C411yescharacteristic
74C42×C61C2×D411yescharacteristic
75C42×C61C2×D412yes—
76#11758201—48no—
77#11758199—48no—
78#11758194—22no—
79#11758194—22no—
80#11758194—82no—
81C2×#92418—22no—
82C2×#92418—22no—
83C2×#92418—22no—
84C2×#92418—82no—
85C23×C61—22no—
86C23×C61#12311yescharacteristic
87C22×D61—42no—
88C22×D61—82no—
89C22×D61—84no—
90Q8×C61—42no—
91D4×C61—42no—
92D4×C61—42no—
93D4×C61—42no—
94D4×C61—82no—
95C2×C4×C61—21no—
96C2×C4×C61—21no—
97C2×C4×C61—22no—
98C2×C4×C61—22no—
99C2×C4×C61—22no—
100C2×C4×C61—42no—
101C2×C4×C61—42no—
102C2×C4×C61C2×C22⋊C411yescharacteristic
103C2×C4×C61C4×D412yes—
104C2×C4×C61#12011yescharacteristic
105C2×C4×C61#12112yes—
106D244—44no—
107D244—84no—
108D244—81no—
109D244—81no—
110D244—82no—
111D244—82no—
112C8×C61—82no—
113#92418—24no—
114#92418—42no—
115#92418—42no—
116#9437—611nomaximal, Sylow
117C22×C61—21no—
118C22×C61—22no—
119C22×C61—42no—
120C22×C61—44no—
121C22×C61#38611yescharacteristic
122C2×D61—82no—
123C2×D61—84no—
124C2×D61—164no—
125C4×C61—21no—
126C4×C61—22no—
127C4×C61—42no—
128C4×C61#45311yescharacteristic
129C4×C61#45311yescharacteristic
130C4×C61#47112yes—
131C4×C61#47212yes—
132#3151—611no—
133#2979—611no—
134#1560D6111yescharacteristic, complemented
135#1545—612no—
136#1299—612no—
137C2×C61—41no—
138C2×C61—82no—
139C2×C61#188711yescharacteristic
140C2×C61#190712yes—
141D61—162no—
142C2×AGL(1,ℤ/8)—1222no—
143#532C2×D6111yescharacteristic
144C2×#127—611no—
145#496—614no—
146#493—1221no—
147#492—1221no—
148#436—1222no—
149C8×D4—1222no—
150#431—612no—
151#423C2×D6112yes—
152#408—1222no—
153C4×M16—611no—
154#333—21no—
155#331—21no—
156#331—1222no—
157#331—1222no—
158C61#943711yescharacteristic, complemented, Sylow
159C2×#41D24412yes—
160C22×D4—1221no—
161C22×D4—1222no—
162AGL(1,ℤ/8)—2444no—
163AGL(1,ℤ/8)—2444no—
164C2×SD16—1224no—
165C2×D8—1224no—
166C2×M16—22no—
167C2×M16—1221no—
168C2×M16—1222no—
169C22×C8—1222no—
170C4⋊Q8C22⋊D6111yescharacteristic
171#127—612no—
172#127—1222no—
173#127—1221no—
174#127C22⋊D6111yescharacteristic
175#124C4×D6112yes—
176#121—22no—
177C4×D4—22no—
178C2×C42C22×D6111yescharacteristic
179C4⋊C8—22no—
180C4⋊C8—22no—
181C4⋊C8—612no—
182C4⋊C8—612no—
183C22⋊C8—1224no—
184C22⋊C8—1222no—
185#97—612no—
186C4×C8—1222no—
187C4×C8—1221no—
188C24—2441no—
189#41—42no—
190#41—42no—
191C2×Q8#1175819812yes—
192C2×D4—22no—
193C2×D4—22no—
194C2×D4—1222no—
195C2×D4—1224no—
196C2×D4—4882no—
197C2×D4—4882no—
198C2×D4—2442no—
199C2×D4—2442no—
200C2×D4—2441no—
201C2×D4—2442no—
202C2×D4#1175819812yes—
203C22×C4—22no—
204C22×C4C2×D24411yescharacteristic
205C22×C4#1175822412yes—
206SD16—2448no—
207D8—2448no—
208M16—82no—
209M16—1222no—
210M16—1222no—
211M16—4882no—
212C2×C8—42no—
213C2×C8—1222no—
214C2×C8—1222no—
215C2×C8—1222no—
216C2×C8—4882no—
217C4⋊C4—22no—
218C22⋊C4—24no—
219C42C2×C4×D6111yescharacteristic
220C42D4×D6112yes—
221C42C2×C22⋊D6111yescharacteristic
222C23—22no—
223C23—4884no—
224C23—4882no—
225C23—2442no—
226C23#1451976111yescharacteristic
227Q8—42no—
228D4—42no—
229D4—42no—
230D4—42no—
231D4—82no—
232D4—4884no—
233D4—4882no—
234D4—4882no—
235D4—4881no—
236D4—4881no—
237D4—2444no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4C2×#1175819811yescharacteristic
246C2×C4#1451973712yes—
247C2×C4#1451974711yescharacteristic
248C2×C4#1451974912yes—
249C8—82no—
250C8—1224no—
251C8—2442no—
252C8—2442no—
253C22—21no—
254C22—22no—
255C22—42no—
256C22—44no—
257C22—9764no—
258C22—4882no—
259C22—4884no—
260C22#1683963911yescharacteristic
261C4—21no—
262C4—22no—
263C4—42no—
264C4#1683993211yescharacteristic
265C4#1683993211yescharacteristic
266C4#1684002712yes—
267C4#1684003012yes—
268C2—41no—
269C2—82no—
270C2—9762no—
271C2#2144621511yescharacteristic
272C2#2144632512yes—
subgroupG/H
centerC22#16839639
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1560C2
socleC22×C61#386
prime radical#21456272C2
cosocleC2×C42×C61C23
hypercenter#1560D61
hypocenterC61#9437
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1560

Complements

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

NH ≅ G/Ntype
2. #21454886C2semidirect
3. #21444488C2semidirect
4. #21443708C2semidirect
5. C61×#1560C2semidirect
134. #1560D61semidirect
158. C61#9437semidirect

Maximal subgroups

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

subgroupindex
#214562722
#214548862
#214444882
#214437082
C61×#15602
#943761