Finite Group Databasebeta

← #29592149

Subgroups — #29592149

Order 15616.

410 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 63 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#21447747C212yesmaximal, complemented
2#21446928C211yesmaximal, characteristic
3#21443837C212yesmaximal, complemented
4#21443745C211yesmaximal, characteristic, complemented
5C61×#3202C211yesmaximal, characteristic, complemented
6C2×#14519802—21no—
7C22×D488C2211yescharacteristic
8C2×#14519720—22no—
9C2×#14519693—21no—
10#16839562—22no—
11C22⋊#11758185C2212yes—
12#16839527C2211yescharacteristic
13C61×#536—21no—
14C61×#485—22no—
15C61×#470—21no—
16C61×#467C2212yes—
17C61×#451—22no—
18C61×#408C2211yescharacteristic
19C22×D244—21no—
20#14519802—42no—
21#14519802—42no—
22C2×#11758201—41no—
23C2×D488—22no—
24C2×D488—42no—
25C2×#11758199—41no—
26C2×#11758194—21no—
27#14519720—44no—
28C2×#11758185—22no—
29#14519693—42no—
30C2×#41×C61D411yescharacteristic
31C22×Q8×C61—21no—
32C2×Q16×C61—21no—
33C2×Q16×C61D412yes—
34C2×M16×C61D411yescharacteristic
35C22×C8×C61C2311yescharacteristic
36C4⋊Q8×C61—41no—
37#124×C61—41no—
38#123×C61—41no—
39#123×C61—41no—
40C22⋊Q8×C61—22no—
41C22⋊Q8×C61—22no—
42#117×C61—21no—
43#107×C61D412yes—
44#106×C61—21no—
45#103×C61—22no—
46#103×C61—22no—
47C22⋊C8×C61—22no—
48C23×D61—41no—
49#41×C61—41no—
50#41×C61—41no—
51C2×Q8×C61—22no—
52C2×Q8×C61—41no—
53C2×Q8×C61—81no—
54C2×Q8×C61—81no—
55C2×Q8×C61D811yescharacteristic
56C2×D4×C61—21no—
57C2×D4×C61D811yescharacteristic
58C22×C4×C61—21no—
59C22×C4×C61—41no—
60C22×C4×C61C2×D411yescharacteristic
61C2×D244—22no—
62C2×D244—41no—
63C2×D244—41no—
64C2×D244—41no—
65C2×D244—41no—
66C2×D244—41no—
67Q16×C61—42no—
68Q16×C61—81no—
69M16×C61—41no—
70C2×C8×C61—21no—
71C2×C8×C61—41no—
72C2×C8×C61C2×D412yes—
73C4⋊C4×C61—22no—
74C4⋊C4×C61—42no—
75C4⋊C4×C61—42no—
76C22⋊C4×C61—41no—
77C22⋊C4×C61—42no—
78C22⋊C4×C61—42no—
79C42×C61—41no—
80#11758201—44no—
81#11758199—44no—
82#11758194—22no—
83#11758194—41no—
84C2×#92418—41no—
85D488—24no—
86D488—42no—
87#11758185—42no—
88C23×C61—21no—
89C23×C61AGL(1,ℤ/8)11yescharacteristic
90C22×D61—42no—
91C22×D61—81no—
92C22×D61—81no—
93C22×D61—162no—
94Q8×C61—41no—
95Q8×C61—42no—
96Q8×C61—42no—
97Q8×C61—82no—
98D4×C61—41no—
99D4×C61—81no—
100C2×C4×C61—21no—
101C2×C4×C61—21no—
102C2×C4×C61—21no—
103C2×C4×C61—41no—
104C2×C4×C61—41no—
105C2×C4×C61—42no—
106C2×C4×C61—42no—
107C2×C4×C61—82no—
108C2×C4×C61#12011yescharacteristic
109C2×C4×C61C2×D811yescharacteristic
110D244—42no—
111D244—42no—
112D244—81no—
113D244—81no—
114C8×C61—22no—
115C8×C61—41no—
116#92418—42no—
117#12109—611nomaximal, Sylow
118C22×C61—21no—
119C22×C61—22no—
120C22×C61—41no—
121C22×C61—42no—
122C22×C61#44711yescharacteristic
123C2×D61—82no—
124C2×D61—162no—
125C2×D61—162no—
126C2×D61—162no—
127C4×C61—21no—
128C4×C61—41no—
129C4×C61—82no—
130C4×C61—82no—
131C4×C61#45311yescharacteristic
132C4×C61B(4,𝔽2)11yescharacteristic
133#3202D6111yescharacteristic, complemented
134#2187—612no—
135#2018—611no—
136#1350—612no—
137#1315—611no—
138C2×C61—41no—
139C2×C61—81no—
140C2×C61#159511yescharacteristic
141C2×C61#219312yes—
142D61—162no—
143C2×AGL(1,ℤ/8)—1221no—
144C22×D8—611no—
145#536—21no—
146C2×SD32—1222no—
147#485—22no—
148#470—21no—
149#467C2×D6112yes—
150#451—22no—
151C2×#100—1221no—
152#408C2×D6111yescharacteristic
153#357—1222no—
154C22⋊C16—612no—
155#341—611no—
156C61#1210911yescharacteristic, complemented, Sylow
157C2×#41C22⋊D6111yescharacteristic
158C22×Q8—21no—
159C22×D4—1221no—
160AGL(1,ℤ/8)—2442no—
161AGL(1,ℤ/8)—2442no—
162C2×Q16—21no—
163C2×Q16C22⋊D6112yes—
164C2×SD16—2441no—
165C2×D8—1222no—
166C2×D8—2442no—
167C2×D8—2441no—
168C2×M16—1221no—
169C2×M16C22⋊D6111yescharacteristic
170C22×C8C22×D6111yescharacteristic
171C4⋊Q8—41no—
172#124—41no—
173#123—41no—
174#123—41no—
175C22⋊Q8—22no—
176C22⋊Q8—22no—
177#117—21no—
178SD32—2444no—
179C2×C16—1222no—
180#107C22⋊D6112yes—
181#106—21no—
182#103—22no—
183#103—22no—
184#100—2442no—
185C22⋊C8—22no—
186C24—2441no—
187#41—41no—
188#41—41no—
189C2×Q8—22no—
190C2×Q8—41no—
191C2×Q8—81no—
192C2×Q8—81no—
193C2×Q8#1175819911yescharacteristic
194C2×D4—21no—
195C2×D4—1222no—
196C2×D4—2441no—
197C2×D4—2441no—
198C2×D4—2441no—
199C2×D4—2441no—
200C2×D4—2441no—
201C2×D4#1175819911yescharacteristic
202C22×C4—21no—
203C22×C4—41no—
204C22×C4C2×C22⋊D6111yescharacteristic
205Q16—42no—
206Q16—81no—
207SD16—2444no—
208D8—1224no—
209D8—2442no—
210D8—2444no—
211M16—41no—
212M16—1222no—
213M16—2441no—
214C2×C8—21no—
215C2×C8—41no—
216C2×C8—2441no—
217C2×C8C2×C22⋊D6112yes—
218C4⋊C4—22no—
219C4⋊C4—42no—
220C4⋊C4—42no—
221C22⋊C4—41no—
222C22⋊C4—42no—
223C22⋊C4—42no—
224C42—41no—
225C16—2442no—
226C23—21no—
227C23—9762no—
228C23—4881no—
229C23—4881no—
230C23—2442no—
231C23#1451980311yescharacteristic
232Q8—41no—
233Q8—42no—
234Q8—42no—
235Q8—82no—
236D4—41no—
237D4—81no—
238D4—4881no—
239D4—4881no—
240D4—2442no—
241D4—2442no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—82no—
250C2×C4#1451974611yescharacteristic
251C2×C4C2×#1175819911yescharacteristic
252C8—22no—
253C8—41no—
254C8—2442no—
255C22—22no—
256C22—21no—
257C22—42no—
258C22—41no—
259C22—9762no—
260C22—9762no—
261C22—9762no—
262C22—4882no—
263C22#1683989611yescharacteristic
264C4—21no—
265C4—41no—
266C4—82no—
267C4—82no—
268C4#1683993011yescharacteristic
269C4#1683994911yescharacteristic
270C2—41no—
271C2—81no—
272C2—9762no—
273C2#2144473711yescharacteristic
274C2#2144778312yes—
subgroupG/H
centerC22#16839896
commutator, cosocleC22×C8×C61C23
FrattiniC22×C8C22×D61
FittingC61×#3202C2
socleC22×C61#447
prime radical#21446928C2
hypercenter#3202D61
hypocenterC61#12109
LCSG ▷ C22×C8×C61 ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #408 ◁ #3202

Complements

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

NH ≅ G/Ntype
1. #21447747C2semidirect
3. #21443837C2semidirect
4. #21443745C2semidirect
5. C61×#3202C2semidirect
133. #3202D61semidirect
156. C61#12109semidirect

Maximal subgroups

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

subgroupindex
#214477472
#214469282
#214438372
#214437452
C61×#32022
#1210961