Finite Group Databasebeta

← #29599684

Subgroups — #29599684

Order 15616.

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

normal subgroups only

This group has 540 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#21452191C211yesmaximal, characteristic
2#21446102C212yesmaximal
3#21446028C211yesmaximal, characteristic
4#21443936C212yesmaximal
5C61×#1393C211yesmaximal, characteristic
6C2×#14519754—22no—
7#16840237C2211yescharacteristic
8#16839802—21no—
9C2×#14519694C2211yescharacteristic
10#16839685—22no—
11#16839682—21no—
12#16839660—21no—
13#16839642C2212yes—
14#16839637—22no—
15#16839528—22no—
16#16839522—21no—
17C2×#99×C61C2212yes—
18C61×#390C2211yescharacteristic
19C22×Dic122—21no—
20C23×Dic61D412yes—
21C2×#11758194—21no—
22#14519754—48no—
23D4×Dic61—41no—
24D4×Dic61—41no—
25#14519736—21no—
26C2×C4⋊Dic61—21no—
27C2×C4⋊Dic61—21no—
28C2×C4⋊Dic61—42no—
29C2×#11758196—22no—
30C2×#11758196—22no—
31C2×C22⋊Dic61—22no—
32C2×C22⋊Dic61—22no—
33C2×C22⋊Dic61D412yes—
34#14519710—41no—
35#14519708—41no—
36#14519694—21no—
37#14519694Q812yescomplemented
38#14519684—21no—
39#14519684—22no—
40#14519684—42no—
41C22×D4×C61C2311yescharacteristic
42C2×C4⋊C4×C61D411yescharacteristic
43C2×C22⋊C4×C61—22no—
44C2×C22⋊C4×C61—22no—
45C2×C42×C61D411yescharacteristic
46#99×C61—22no—
47#99×C61Q814yescomplemented
48C24×C61C2×D412yes—
49C2×D4×C61—41no—
50C2×D4×C61—41no—
51C2×D4×C61C2×Q811yescharacteristic
52C2×D4×C61C2×Q812yes—
53C2×D4×C61#4111yescharacteristic
54C22×C4×C61—21no—
55C22×C4×C61—21no—
56C22×C4×C61—42no—
57C22×C4×C61C2×D411yescharacteristic
58C22×Dic61—22no—
59C22×Dic61—22no—
60C22×Dic61—22no—
61C22×Dic61—22no—
62C22×Dic61—44no—
63C22×Dic61—42no—
64C2×Dic122—44no—
65C2×Dic122—81no—
66C4⋊C4×C61—41no—
67C22⋊C4×C61—24no—
68C22⋊C4×C61—44no—
69C22⋊C4×C61—42no—
70C42×C61—41no—
71C22⋊Dic61—44no—
72C22⋊Dic61—42no—
73C22⋊Dic61—41no—
74C22⋊Dic61—42no—
75C4⋊Dic61—24no—
76C4⋊Dic61—21no—
77C4⋊Dic61—21no—
78C4⋊Dic61—81no—
79C4⋊Dic61—84no—
80#11758196—44no—
81#11758196—44no—
82C4×Dic61—41no—
83#11758194—22no—
84#11758194—41no—
85C2×#92418—41no—
86C23×C61—22no—
87C23×C61—22no—
88C23×C61—42no—
89C23×C61—42no—
90C23×C61#12011yescharacteristic
91C23×C61C22⋊Q812yes—
92C23×C61C4⋊Q812yes—
93D4×C61—41no—
94D4×C61—42no—
95D4×C61—41no—
96C2×C4×C61—21no—
97C2×C4×C61—21no—
98C2×C4×C61—22no—
99C2×C4×C61—41no—
100C2×C4×C61—41no—
101C2×C4×C61—42no—
102C2×C4×C61—44no—
103C2×C4×C61—41no—
104C2×C4×C61—82no—
105C2×C4×C61—82no—
106C2×C4×C61C22⋊Q811yescharacteristic
107C2×C4×C61C4⋊Q811yescharacteristic
108C2×Dic61—24no—
109C2×Dic61—42no—
110C2×Dic61—42no—
111C2×Dic61—44no—
112C2×Dic61—42no—
113C2×Dic61—42no—
114C2×Dic61—41no—
115C2×Dic61—41no—
116C2×Dic61—42no—
117C2×Dic61—82no—
118C2×Dic61—82no—
119C2×Dic61—82no—
120Dic122—84no—
121#92418—42no—
122#13461—611nomaximal, Sylow
123C22×C61—21no—
124C22×C61—21no—
125C22×C61—22no—
126C22×C61—22no—
127C22×C61—42no—
128C22×C61—42no—
129C22×C61—42no—
130C22×C61—42no—
131C22×C61—82no—
132C22×C61#39511yescharacteristic
133C22×C61B(4,𝔽2)11yescharacteristic
134C22×C61#45811yescharacteristic
135C4×C61—21no—
136C4×C61—21no—
137C4×C61—42no—
138C4×C61—84no—
139C4×C61—81no—
140Dic61—44no—
141Dic61—82no—
142Dic61—81no—
143Dic61—81no—
144#2683—611no—
145#1867—612no—
146#1856—611no—
147#1393D6111yescharacteristic
148#1390—612no—
149C2×C61—21no—
150C2×C61—21no—
151C2×C61—42no—
152C2×C61—42no—
153C2×C61#206211yescharacteristic
154C2×C61#219611yescharacteristic
155C2×C61#220111yescharacteristic
156C2×C22⋊Q8—1222no—
157#518—611no—
158#428—1221no—
159C2×#100—611no—
160C2×#99C2×D6112yes—
161#396—1222no—
162#395—1221no—
163#390C2×D6111yescharacteristic
164#389—1221no—
165#386—612no—
166#385—1222no—
167#342—1222no—
168#339—1221no—
169C61#1346111yescharacteristic, complemented, Sylow
170C22×Q8—1221no—
171C22×D4C22×D6111yescharacteristic
172C23×C4—612no—
173C2×M16—1221no—
174C22⋊Q8—2448no—
175C4×D4—2441no—
176C4×D4—2441no—
177#117—1221no—
178C2×C4⋊C4—1221no—
179C2×C4⋊C4—1221no—
180C2×C4⋊C4—1222no—
181C2×C4⋊C4—1222no—
182C2×C4⋊C4—2442no—
183C2×C4⋊C4D24411yescharacteristic
184C2×C22⋊C4—22no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—612no—
187C2×C22⋊C4—1222no—
188C2×C22⋊C4—1222no—
189C2×C42D24411yescharacteristic
190#107—2441no—
191#106—2441no—
192#100—612no—
193#100—1221no—
194#99—22no—
195#99Dic12214yescomplemented
196#95—1221no—
197#95—1222no—
198#95—2442no—
199C24D4×D6112yes—
200C2×Q8—4881no—
201C2×Q8—2444no—
202C2×D4—41no—
203C2×D4—41no—
204C2×D4C2×Dic12212yes—
205C2×D4#1175822411yescharacteristic
206C2×D4Q8×D6111yescharacteristic
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—42no—
210C22×C4—1222no—
211C22×C4—1222no—
212C22×C4—1222no—
213C22×C4—1222no—
214C22×C4—2444no—
215C22×C4—2442no—
216C22×C4C2×D24411yescharacteristic
217M16—1222no—
218M16—2441no—
219C2×C8—2441no—
220C4⋊C4—41no—
221C4⋊C4—1224no—
222C4⋊C4—1221no—
223C4⋊C4—1221no—
224C4⋊C4—4881no—
225C4⋊C4—4884no—
226C4⋊C4—2444no—
227C4⋊C4—2444no—
228C22⋊C4—24no—
229C22⋊C4—44no—
230C22⋊C4—42no—
231C22⋊C4—2444no—
232C22⋊C4—2442no—
233C22⋊C4—2441no—
234C22⋊C4—2442no—
235C42—41no—
236C42—2441no—
237C23—22no—
238C23—22no—
239C23—42no—
240C23—42no—
241C23#1451974711yescharacteristic
242C23#1451975412yes—
243C23#1451977612yes—
244Q8—4884no—
245D4—42no—
246D4—41no—
247D4—41no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—22no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—44no—
254C2×C4—42no—
255C2×C4—41no—
256C2×C4—82no—
257C2×C4—82no—
258C2×C4—1224no—
259C2×C4—4882no—
260C2×C4—4882no—
261C2×C4—4882no—
262C2×C4—2442no—
263C2×C4—2442no—
264C2×C4—2441no—
265C2×C4—2441no—
266C2×C4—2442no—
267C2×C4—2442no—
268C2×C4—2442no—
269C2×C4—2444no—
270C2×C4#1451975711yescharacteristic
271C2×C4#1451977711yescharacteristic
272C8—2442no—
273C22—21no—
274C22—21no—
275C22—22no—
276C22—22no—
277C22—42no—
278C22—42no—
279C22—42no—
280C22—42no—
281C22—82no—
282C22#1683968311yescharacteristic
283C22#1683995011yescharacteristic
284C22#1683995311yescharacteristic
285C4—21no—
286C4—21no—
287C4—42no—
288C4—84no—
289C4—81no—
290C4—4882no—
291C4—4881no—
292C4—4881no—
293C4—2444no—
294C2—21no—
295C2—21no—
296C2—42no—
297C2—42no—
298C2#2144717311yescharacteristic
299C2#2144780511yescharacteristic
300C2#2144783411yescharacteristic
subgroupG/H
centerC22#16839950
commutator, prime radical, cosocleC22×D4×C61C23
FrattiniC22×D4C22×D61
FittingC61×#1393C2
socleC22×C61B(4,𝔽2)
hypercenter#1393D61
hypocenterC61#13461
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
37. #14519694Q8semidirect
47. #99×C61Q8semidirect
169. C61#13461semidirect
195. #99Dic122semidirect

Maximal subgroups

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

subgroupindex
#214521912
#214461022
#214460282
#214439362
C61×#13932
#1346161