Finite Group Databasebeta

← #29599696

Subgroups — #29599696

Order 15616.

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

normal subgroups only

This group has 484 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#21453765C211yesmaximal, characteristic
2#21446102C212yesmaximal
3#21446028C211yesmaximal, characteristic
4#21443936C212yesmaximal
5C61×#1393C211yesmaximal, characteristic
6#16840237C2211yescharacteristic
7#16839802—21no—
8C2×#14519694C2211yescharacteristic
9#16839701—21no—
10#16839674—22no—
11#16839668—21no—
12#16839642C2212yes—
13#16839637—22no—
14#16839637—22no—
15#16839528—22no—
16#16839522—21no—
17C2×#99×C61C2212yes—
18C61×#390C2211yescharacteristic
19C22×Dic122—21no—
20C23×Dic61D412yes—
21C2×#11758194—21no—
22D4×Dic61—41no—
23D4×Dic61—41no—
24#14519736—21no—
25C2×C4⋊Dic61—21no—
26C2×C4⋊Dic61—21no—
27C2×C4⋊Dic61—42no—
28C2×C22⋊Dic61—22no—
29C2×C22⋊Dic61—22no—
30C2×C22⋊Dic61D412yes—
31#14519710—41no—
32#14519708—41no—
33#14519694—21no—
34#14519694Q812yescomplemented
35#14519684—22no—
36#14519684—21no—
37#14519684—22no—
38#14519684—22no—
39#14519684—42no—
40C22×D4×C61C2311yescharacteristic
41C2×C4⋊C4×C61D411yescharacteristic
42C2×C22⋊C4×C61—22no—
43C2×C22⋊C4×C61—22no—
44C2×C42×C61D411yescharacteristic
45#99×C61—22no—
46#99×C61Q814yescomplemented
47C24×C61C2×D412yes—
48C2×D4×C61—41no—
49C2×D4×C61—41no—
50C2×D4×C61C2×Q811yescharacteristic
51C2×D4×C61C2×Q812yes—
52C2×D4×C61#4111yescharacteristic
53C22×C4×C61—21no—
54C22×C4×C61—21no—
55C22×C4×C61—42no—
56C22×C4×C61C2×D411yescharacteristic
57C22×Dic61—22no—
58C22×Dic61—22no—
59C22×Dic61—22no—
60C22×Dic61—22no—
61C22×Dic61—42no—
62C22×Dic61—82no—
63C2×Dic122—42no—
64C2×Dic122—81no—
65C2×Dic122—81no—
66C4⋊C4×C61—41no—
67C22⋊C4×C61—24no—
68C22⋊C4×C61—44no—
69C22⋊C4×C61—42no—
70C42×C61—41no—
71C22⋊Dic61—42no—
72C22⋊Dic61—41no—
73C22⋊Dic61—42no—
74C22⋊Dic61—82no—
75C4⋊Dic61—21no—
76C4⋊Dic61—21no—
77C4⋊Dic61—81no—
78C4⋊Dic61—81no—
79C4⋊Dic61—84no—
80C4×Dic61—41no—
81#11758194—22no—
82#11758194—41no—
83C2×#92418—41no—
84C23×C61—22no—
85C23×C61—22no—
86C23×C61—42no—
87C23×C61—42no—
88C23×C61#12011yescharacteristic
89C23×C61C22⋊Q812yes—
90C23×C61C4⋊Q812yes—
91D4×C61—41no—
92D4×C61—42no—
93D4×C61—41no—
94C2×C4×C61—21no—
95C2×C4×C61—21no—
96C2×C4×C61—22no—
97C2×C4×C61—41no—
98C2×C4×C61—41no—
99C2×C4×C61—44no—
100C2×C4×C61—42no—
101C2×C4×C61—41no—
102C2×C4×C61—82no—
103C2×C4×C61—82no—
104C2×C4×C61C22⋊Q811yescharacteristic
105C2×C4×C61C4⋊Q811yescharacteristic
106C2×Dic61—42no—
107C2×Dic61—42no—
108C2×Dic61—42no—
109C2×Dic61—42no—
110C2×Dic61—44no—
111C2×Dic61—41no—
112C2×Dic61—41no—
113C2×Dic61—42no—
114C2×Dic61—42no—
115C2×Dic61—82no—
116C2×Dic61—82no—
117C2×Dic61—82no—
118Dic122—84no—
119#92418—42no—
120#13463—611nomaximal, Sylow
121C22×C61—21no—
122C22×C61—21no—
123C22×C61—22no—
124C22×C61—22no—
125C22×C61—42no—
126C22×C61—42no—
127C22×C61—42no—
128C22×C61—42no—
129C22×C61—82no—
130C22×C61#39511yescharacteristic
131C22×C61B(4,𝔽2)11yescharacteristic
132C22×C61#45811yescharacteristic
133C4×C61—21no—
134C4×C61—21no—
135C4×C61—42no—
136C4×C61—84no—
137C4×C61—81no—
138Dic61—82no—
139Dic61—81no—
140Dic61—82no—
141Dic61—81no—
142#2804—611no—
143#1867—612no—
144#1856—611no—
145#1393D6111yescharacteristic
146#1390—612no—
147C2×C61—21no—
148C2×C61—21no—
149C2×C61—42no—
150C2×C61—42no—
151C2×C61#206211yescharacteristic
152C2×C61#219811yescharacteristic
153C2×C61#219911yescharacteristic
154#518—611no—
155#428—1221no—
156C2×#100—611no—
157C2×#99C2×D6112yes—
158#400—1221no—
159#393—1222no—
160#391—1221no—
161#390C2×D6111yescharacteristic
162#386—612no—
163#385—1222no—
164#385—1222no—
165#342—1222no—
166#339—1221no—
167C61#1346311yescharacteristic, complemented, Sylow
168C22×Q8—1221no—
169C22×D4C22×D6111yescharacteristic
170C23×C4—612no—
171C2×M16—1221no—
172C4×D4—2441no—
173C4×D4—2441no—
174#117—1221no—
175C2×C4⋊C4—1221no—
176C2×C4⋊C4—1221no—
177C2×C4⋊C4—2442no—
178C2×C4⋊C4D24411yescharacteristic
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—612no—
182C2×C22⋊C4—1222no—
183C2×C22⋊C4—1222no—
184C2×C42D24411yescharacteristic
185#107—2441no—
186#106—2441no—
187#100—612no—
188#100—1221no—
189#99—22no—
190#99Dic12214yescomplemented
191#95—1222no—
192#95—1222no—
193#95—1221no—
194#95—1222no—
195#95—2442no—
196C24D4×D6112yes—
197C2×Q8—4881no—
198C2×Q8—4881no—
199C2×Q8—2442no—
200C2×D4—41no—
201C2×D4—41no—
202C2×D4C2×Dic12212yes—
203C2×D4#1175822411yescharacteristic
204C2×D4Q8×D6111yescharacteristic
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—42no—
208C22×C4—1222no—
209C22×C4—1222no—
210C22×C4—1222no—
211C22×C4—1222no—
212C22×C4—4882no—
213C22×C4—2442no—
214C22×C4C2×D24411yescharacteristic
215M16—1222no—
216M16—2441no—
217C2×C8—2441no—
218C4⋊C4—41no—
219C4⋊C4—1221no—
220C4⋊C4—1221no—
221C4⋊C4—4881no—
222C4⋊C4—4881no—
223C4⋊C4—4884no—
224C22⋊C4—24no—
225C22⋊C4—44no—
226C22⋊C4—42no—
227C22⋊C4—4882no—
228C22⋊C4—2442no—
229C22⋊C4—2441no—
230C22⋊C4—2442no—
231C42—41no—
232C42—2441no—
233C23—22no—
234C23—22no—
235C23—42no—
236C23—42no—
237C23#1451974711yescharacteristic
238C23#1451975412yes—
239C23#1451977612yes—
240Q8—4884no—
241D4—42no—
242D4—41no—
243D4—41no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—22no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—44no—
250C2×C4—42no—
251C2×C4—41no—
252C2×C4—82no—
253C2×C4—82no—
254C2×C4—4882no—
255C2×C4—4882no—
256C2×C4—4882no—
257C2×C4—2442no—
258C2×C4—2442no—
259C2×C4—2442no—
260C2×C4—2442no—
261C2×C4—2444no—
262C2×C4—2441no—
263C2×C4—2441no—
264C2×C4—2442no—
265C2×C4—2442no—
266C2×C4#1451975711yescharacteristic
267C2×C4#1451977711yescharacteristic
268C8—2442no—
269C22—21no—
270C22—21no—
271C22—22no—
272C22—22no—
273C22—42no—
274C22—42no—
275C22—42no—
276C22—42no—
277C22—82no—
278C22#1683968311yescharacteristic
279C22#1683995011yescharacteristic
280C22#1683995311yescharacteristic
281C4—21no—
282C4—21no—
283C4—42no—
284C4—84no—
285C4—81no—
286C4—4882no—
287C4—4881no—
288C4—4881no—
289C4—4882no—
290C2—21no—
291C2—21no—
292C2—42no—
293C2—42no—
294C2#2144717311yescharacteristic
295C2#2144781711yescharacteristic
296C2#2144782211yescharacteristic
subgroupG/H
centerC22#16839950
commutator, prime radical, cosocleC22×D4×C61C23
FrattiniC22×D4C22×D61
FittingC61×#1393C2
socleC22×C61B(4,𝔽2)
hypercenter#1393D61
hypocenterC61#13463
LCSG ▷ C22×D4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ #1393

Complements

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

NH ≅ G/Ntype
34. #14519694Q8semidirect
46. #99×C61Q8semidirect
167. C61#13463semidirect
190. #99Dic122semidirect

Maximal subgroups

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

subgroupindex
#214537652
#214461022
#214460282
#214439362
C61×#13932
#1346361