Finite Group Databasebeta

← #29568844

Subgroups — #29568844

Order 15616.

536 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 83 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#21452845C211yesmaximal, characteristic
2#21446952C212yesmaximal, complemented
3#21443645C211yesmaximal, characteristic, complemented
4#21443639C212yesmaximal, complemented
5C61×#2016C211yesmaximal, characteristic, complemented
6C2×#14519766—21no—
7C2×#14519697—22no—
8#16839679—21no—
9#16839672—22no—
10#16839664C2212yes—
11#16839654—22no—
12#16839524—22no—
13#16839517C2212yes—
14C22×Q16×C61—21no—
15C2×C4⋊Q8×C61C2211yescharacteristic
16C2×#103×C61—21no—
17C61×#391C2211yescharacteristic
18C61×#336C2211yescharacteristic
19C22×D244—21no—
20C22×D244—21no—
21C22×D244D412yes—
22C22×#92418—22no—
23#14519766—42no—
24#14519766—42no—
25C2×C4⋊Dic61D412yes—
26C2×#11758198—22no—
27C2×#11758198—22no—
28C2×#11758198—22no—
29C2×#11758198—22no—
30C2×C4×Dic61—21no—
31#14519697—44no—
32#14519684—21no—
33C22×Q8×C61—21no—
34C22×Q8×C61D411yescharacteristic
35C2×Q16×C61—42no—
36C2×Q16×C61—42no—
37C2×Q16×C61—42no—
38C22×C8×C61—21no—
39C4⋊Q8×C61—42no—
40C2×C4⋊C4×C61—22no—
41C2×C4⋊C4×C61D411yescharacteristic
42C2×C42×C61C2311yescharacteristic
43#103×C61—42no—
44#95×C61—22no—
45C23×D61—22no—
46C2×Q8×C61—21no—
47C2×Q8×C61—22no—
48C2×Q8×C61—41no—
49C2×Q8×C61—41no—
50C2×Q8×C61—41no—
51C2×Q8×C61—41no—
52C2×Q8×C61—41no—
53C2×Q8×C61—41no—
54C2×Q8×C61—41no—
55C2×Q8×C61D812yes—
56C22×C4×C61—21no—
57C22×C4×C61—21no—
58C22×C4×C61—41no—
59C22×C4×C61C2×D411yescharacteristic
60C22×C4×C61C2×D412yes—
61C22×Dic61—22no—
62C2×D244—22no—
63C2×D244—22no—
64C2×D244—42no—
65C2×D244—42no—
66C2×D244—42no—
67C2×D244—81no—
68C2×D244—81no—
69C2×D244—81no—
70C2×D244—82no—
71Q16×C61—48no—
72C2×C8×C61—22no—
73C2×C8×C61—41no—
74C2×C8×C61—41no—
75C4⋊C4×C61—24no—
76C4⋊C4×C61—42no—
77C4⋊C4×C61—41no—
78C42×C61—41no—
79#11758198—44no—
80#11758198—42no—
81#11758198—42no—
82#11758198—42no—
83#11758198—42no—
84#11758198—82no—
85C4⋊Dic61—42no—
86C4×Dic61—42no—
87C2×#92418—42no—
88C2×#92418—42no—
89C2×#92418—42no—
90C23×C61#12011yescharacteristic
91C22×D61—42no—
92C22×D61—42no—
93C22×D61—42no—
94C22×D61—42no—
95C22×D61—42no—
96C22×D61—42no—
97C22×D61—42no—
98Q8×C61—42no—
99Q8×C61—42no—
100Q8×C61—42no—
101Q8×C61—81no—
102Q8×C61—81no—
103Q8×C61—81no—
104C2×C4×C61—21no—
105C2×C4×C61—22no—
106C2×C4×C61—22no—
107C2×C4×C61—21no—
108C2×C4×C61—22no—
109C2×C4×C61—41no—
110C2×C4×C61—41no—
111C2×C4×C61—41no—
112C2×C4×C61—42no—
113C2×C4×C61—42no—
114C2×C4×C61—81no—
115C2×C4×C61—81no—
116C2×C4×C61C2×D811yescharacteristic
117C2×C4×C61#13512yes—
118C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
119C2×C4×C61#13712yes—
120C2×Dic61—42no—
121C2×Dic61—42no—
122C2×Dic61—42no—
123D244—44no—
124D244—82no—
125D244—84no—
126D244—84no—
127C8×C61—42no—
128#92418—82no—
129#7758—611nomaximal, Sylow
130C22×C61#44711yescharacteristic
131C22×C61#45212yes—
132C22×C61#45311yescharacteristic
133C22×C61#45512yes—
134C22×C61B(4,𝔽2)11yescharacteristic
135C2×D61—82no—
136C2×D61—82no—
137C2×D61—82no—
138C2×D61—82no—
139C2×D61—82no—
140C2×D61—82no—
141C2×D61—82no—
142C4×C61—21no—
143C4×C61—21no—
144C4×C61—22no—
145C4×C61—22no—
146C4×C61—42no—
147C4×C61—82no—
148C4×C61—81no—
149Dic61—82no—
150#2733—611no—
151#2023—612no—
152#2016D6111yescharacteristic, complemented
153#1272—611no—
154#1270—612no—
155C2×C61#159511yescharacteristic
156C2×C61#160612yes—
157C2×C61#163411yescharacteristic
158C2×C61#164112yes—
159C2×C61#165911yescharacteristic
160D61—162no—
161C22×Q16—21no—
162C2×C4⋊Q8C2×D6111yescharacteristic
163C2×#124—1221no—
164C2×#103—21no—
165C2×#102—1222no—
166#394—1221no—
167#392—1222no—
168#391C2×D6111yescharacteristic
169#390—612no—
170#388—1222no—
171#340—1222no—
172#336—612no—
173#336C2×D6111yescharacteristic
174C61#775811yescharacteristic, complemented, Sylow
175C22×Q8—21no—
176C22×Q8C22⋊D6111yescharacteristic
177C22×D4—612no—
178C22×D4—1221no—
179C22×D4—1221no—
180C2×Q16—42no—
181C2×Q16—42no—
182C2×Q16—42no—
183C22×C8—21no—
184C22×C8—1222no—
185C4⋊Q8—42no—
186#124—2442no—
187#124—2442no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—612no—
190C2×C4⋊C4C22⋊D6111yescharacteristic
191C2×C22⋊C4—1222no—
192C2×C22⋊C4—1222no—
193C2×C22⋊C4—1222no—
194C2×C22⋊C4—1222no—
195C2×C42—1221no—
196C2×C42C22×D6111yescharacteristic
197#103—42no—
198#102—2444no—
199#95—22no—
200#95—1221no—
201C24—1222no—
202C2×Q8—21no—
203C2×Q8—22no—
204C2×Q8—41no—
205C2×Q8—41no—
206C2×Q8—41no—
207C2×Q8—41no—
208C2×Q8—41no—
209C2×Q8—41no—
210C2×Q8—41no—
211C2×Q8#1175819912yes—
212C2×D4—1222no—
213C2×D4—1222no—
214C2×D4—4881no—
215C2×D4—4881no—
216C2×D4—4882no—
217C2×D4—4881no—
218C2×D4—2442no—
219C2×D4—2442no—
220C2×D4—2442no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—41no—
224C22×C4—1222no—
225C22×C4D4×D6112yes—
226C22×C4C2×C22⋊D6111yescharacteristic
227Q16—48no—
228C2×C8—22no—
229C2×C8—41no—
230C2×C8—41no—
231C2×C8—2442no—
232C2×C8—2442no—
233C2×C8—2442no—
234C4⋊C4—24no—
235C4⋊C4—41no—
236C4⋊C4—42no—
237C4⋊C4—2442no—
238C22⋊C4—4882no—
239C22⋊C4—2444no—
240C22⋊C4—2442no—
241C22⋊C4—2442no—
242C22⋊C4—2442no—
243C22⋊C4—2442no—
244C42—41no—
245C42—2442no—
246C23—2442no—
247C23—2442no—
248C23—2442no—
249C23—2442no—
250C23—2442no—
251C23—2442no—
252C23—2442no—
253C23#1451974511yescharacteristic
254Q8—42no—
255Q8—42no—
256Q8—42no—
257Q8—81no—
258Q8—81no—
259Q8—81no—
260D4—4882no—
261D4—4884no—
262D4—4884no—
263D4—2444no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—22no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—41no—
274C2×C4—81no—
275C2×C4—81no—
276C2×C4—2442no—
277C2×C4—2442no—
278C2×C4—2442no—
279C2×C4C2×#1175819911yescharacteristic
280C2×C4#1451980012yes—
281C2×C4#1451980211yescharacteristic
282C2×C4#1451980912yes—
283C8—42no—
284C8—4882no—
285C22—4882no—
286C22—4882no—
287C22—4882no—
288C22—4882no—
289C22—4882no—
290C22—4882no—
291C22—4882no—
292C22#1683989511yescharacteristic
293C22#1683992312yes—
294C22#1683992911yescharacteristic
295C22#1683993812yes—
296C22#1683994811yescharacteristic
297C4—21no—
298C4—21no—
299C4—22no—
300C4—22no—
301C4—42no—
302C4—81no—
303C4—82no—
304C4—4882no—
305C2—9762no—
306C2#2144473611yescharacteristic
307C2#2144479812yes—
308C2#2144496211yescharacteristic
309C2#2144501012yes—
310C2#2144512311yescharacteristic
subgroupG/H
centerC23#14519745
commutator, cosocleC2×C42×C61C23
FrattiniC2×C42C22×D61
FittingC61×#2016C2
socleC23×C61#120
prime radical#21452845C2
hypercenter#2016D61
hypocenterC61#7758
LCSG ▷ C2×C42×C61 ▷ C23×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2016

Complements

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

NH ≅ G/Ntype
2. #21446952C2semidirect
3. #21443645C2semidirect
4. #21443639C2semidirect
5. C61×#2016C2semidirect
152. #2016D61semidirect
174. C61#7758semidirect

Maximal subgroups

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

subgroupindex
#214528452
#214469522
#214436452
#214436392
C61×#20162
#775861