Finite Group Databasebeta

← #29577815

Subgroups — #29577815

Order 15616.

818 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 818 conjugacy classes of subgroups — too many to draw. Only its 107 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#21449154C211yesmaximal, characteristic
2#21448176C211yesmaximal, characteristic, complemented
3#21445762C212yesmaximal
4#21443673C212yesmaximal, complemented
5C61×#1463C211yesmaximal, characteristic, complemented
6C22×#11758222C2211yescharacteristic
7C2×#14519760—21no—
8C2×#14519736—21no—
9C2×#14519703—24no—
10#16839682C412yescomplemented
11#16839679C412yescomplemented
12#16839641—21no—
13#16839633—22no—
14#16839625C2211yescharacteristic
15C4⋊C4×Dic61—24no—
16C4×C22⋊Dic61—21no—
17C4×C22⋊Dic61C2212yes—
18C22×C4⋊C4×C61C2211yescharacteristic
19C61×#407C2212yes—
20C61×#324—22no—
21C2×#11758222—22no—
22C2×#11758222—22no—
23C2×#11758222—44no—
24C22×Dic122C2×C411yescharacteristic
25C22×D244C2×C411yescharacteristic
26C22×C22⋊D61—21no—
27C22×C4×D61—21no—
28#14519760—44no—
29#14519736—28no—
30C2×C4⋊Dic61—22no—
31C2×#11758196—22no—
32C2×C22⋊Dic61—22no—
33C2×C22⋊Dic61—21no—
34C2×#11758198—22no—
35C2×#11758198—22no—
36C2×C4×Dic61—22no—
37C2×C4×Dic61—22no—
38C2×C4×Dic61D414yes—
39#14519703—48no—
40#14519684—22no—
41#14519684—22no—
42#14519684—22no—
43#14519684—21no—
44#14519684C2×C412yes—
45C23×C4×C61D411yescharacteristic
46C23×C4×C61D411yescharacteristic
47C23×C4×C61C2311yescharacteristic
48C2×M16×C61—22no—
49C2×C4⋊C4×C61—22no—
50C2×C4⋊C4×C61—22no—
51C2×C4⋊C4×C61C2×C412yes—
52C2×C4⋊C4×C61D412yes—
53C22⋊C8×C61—22no—
54C22⋊C8×C61—22no—
55C24×C61C2×D411yescharacteristic
56C23×D61—21no—
57C22×C4×C61—21no—
58C22×C4×C61—21no—
59C22×C4×C61—24no—
60C22×C4×C61—21no—
61C22×C4×C61—21no—
62C22×C4×C61—42no—
63C22×C4×C61—42no—
64C22×C4×C61C22⋊C412yes—
65C22×C4×C61C22⋊C412yes—
66C22×C4×C61C22×C411yescharacteristic
67C22×C4×C61C2×D412yes—
68C22×C4×C61C2×D411yescharacteristic
69C22×C4×C61#4112yes—
70C2×C22⋊D61—24no—
71C2×C22⋊D61—42no—
72C2×C22⋊D61—41no—
73C2×C22⋊D61—41no—
74C22×Dic61—21no—
75C22×Dic61—24no—
76C22×Dic61—22no—
77#11758222—28no—
78#11758222—44no—
79#11758222—44no—
80#11758222—44no—
81C2×D244—22no—
82C2×D244—42no—
83C2×C4×D61—42no—
84C2×C4×D61—42no—
85C2×C4×D61—42no—
86C2×Dic122—22no—
87C2×Dic122—42no—
88M16×C61—44no—
89C2×C8×C61—42no—
90C4⋊C4×C61—24no—
91C4⋊C4×C61—24no—
92C22⋊Dic61—28no—
93C22⋊Dic61—42no—
94#11758198—44no—
95#11758198—44no—
96C4⋊Dic61—28no—
97#11758196—44no—
98C4×Dic61—28no—
99C4×Dic61—44no—
100C4×Dic61—82no—
101C23×C61—22no—
102C23×C61—22no—
103C23×C61—41no—
104C23×C61C2×C22⋊C411yescharacteristic
105C23×C61#12011yescharacteristic
106C23×C61#12311yescharacteristic
107C22×D61—22no—
108C22×D61—41no—
109C22×D61—42no—
110C22×D61—41no—
111C22×D61—42no—
112C2×C4×C61—22no—
113C2×C4×C61—22no—
114C2×C4×C61—24no—
115C2×C4×C61—22no—
116C2×C4×C61—22no—
117C2×C4×C61—24no—
118C2×C4×C61—22no—
119C2×C4×C61—41no—
120C2×C4×C61—41no—
121C2×C4×C61—42no—
122C2×C4×C61—42no—
123C2×C4×C61—42no—
124C2×C4×C61—81no—
125C2×C4×C61#10414yescomplemented
126C2×C4×C61C4×D412yes—
127C2×C4×C61#12112yes—
128C22⋊D61—44no—
129C22⋊D61—44no—
130C2×Dic61—28no—
131C2×Dic61—22no—
132C2×Dic61—41no—
133C2×Dic61—44no—
134C2×Dic61—41no—
135C2×Dic61—44no—
136C2×Dic61—42no—
137C2×Dic61—42no—
138C2×Dic61—42no—
139D244—24no—
140D244—42no—
141C4×D61—48no—
142Dic122—24no—
143Dic122—42no—
144C8×C61—82no—
145#9476—611nomaximal, Sylow
146C22×C61—21no—
147C22×C61—21no—
148C22×C61—21no—
149C22×C61—21no—
150C22×C61—21no—
151C22×C61—21no—
152C22×C61—21no—
153C22×C61—44no—
154C22×C61#38611yescharacteristic
155C22×C61C2×#10412yes—
156C22×C61B(4,𝔽2)12yes—
157C2×D61—44no—
158C2×D61—42no—
159C2×D61—82no—
160C2×D61—81no—
161C2×D61—81no—
162C4×C61—22no—
163C4×C61—22no—
164C4×C61—24no—
165C4×C61—42no—
166Dic61—48no—
167Dic61—42no—
168Dic61—82no—
169#2406—611no—
170#2274—611no—
171#1789—612no—
172#1463D6111yescharacteristic, complemented
173#1284—612no—
174C2×C61—22no—
175C2×C61—22no—
176C2×C61—41no—
177C2×C61#188911yescharacteristic
178C2×C61#189611yescharacteristic
179C2×C61#191211yescharacteristic
180D61—82no—
181C22×#41—611no—
182C2×#123—1221no—
183C2×#117—1221no—
184C22×C4⋊C4C2×D6111yescharacteristic
185C2×#104—1224no—
186#407C2×D6112yes—
187#395—612no—
188#394—612no—
189#386—1221no—
190#385—1222no—
191#381—611no—
192C4×C4⋊C4—1224no—
193C4×C22⋊C4—612no—
194C4×C22⋊C4—1221no—
195#324—22no—
196C61#947611yescharacteristic, complemented, Sylow
197C2×#41—1222no—
198C2×#41—1222no—
199C2×#41—2444no—
200C22×Q8—611no—
201C22×D4—611no—
202C22×D4—1221no—
203C23×C4—1221no—
204C23×C4D24411yescharacteristic
205C23×C4D24411yescharacteristic
206C23×C4C22×D6111yescharacteristic
207C2×M16—22no—
208#123—2444no—
209#117—1228no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—1222no—
213C2×C4⋊C4—1222no—
214C2×C4⋊C4C4×D6112yes—
215C2×C4⋊C4C22⋊D6112yes—
216C2×C22⋊C4—1222no—
217C2×C22⋊C4—1221no—
218C2×C22⋊C4—1222no—
219C2×C22⋊C4—1222no—
220C2×C42—614no—
221C2×C42—1222no—
222C2×C42—1222no—
223#104—2448no—
224C22⋊C8—22no—
225C22⋊C8—22no—
226#95—612no—
227#95—1222no—
228#95—1222no—
229#95—1222no—
230#95—1221no—
231C24—1221no—
232C24C2×D24411yescharacteristic
233#41—1228no—
234#41—2444no—
235#41—2444no—
236#41—2444no—
237C2×Q8—1222no—
238C2×Q8—2442no—
239C2×D4—1224no—
240C2×D4—1222no—
241C2×D4—2442no—
242C2×D4—2442no—
243C2×D4—2441no—
244C2×D4—2441no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—24no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—1221no—
253C22×C4—1224no—
254C22×C4—1222no—
255C22×C4—2442no—
256C22×C4—2442no—
257C22×C4—2442no—
258C22×C4#1175819812yes—
259C22×C4#1175819812yes—
260C22×C4C2×C4×D6111yescharacteristic
261C22×C4D4×D6112yes—
262C22×C4#1175822412yes—
263C22×C4C2×C22⋊D6111yescharacteristic
264M16—44no—
265C2×C8—42no—
266C4⋊C4—24no—
267C4⋊C4—24no—
268C4⋊C4—1228no—
269C4⋊C4—2444no—
270C22⋊C4—1228no—
271C22⋊C4—2444no—
272C22⋊C4—2442no—
273C22⋊C4—2444no—
274C42—1228no—
275C42—4882no—
276C42—2444no—
277C23—22no—
278C23—22no—
279C23—41no—
280C23—1222no—
281C23—2441no—
282C23—2442no—
283C23—2442no—
284C23—2441no—
285C23C2×#1175819811yescharacteristic
286C23#1451974711yescharacteristic
287C23#1451976111yescharacteristic
288Q8—1224no—
289Q8—2442no—
290D4—1224no—
291D4—2442no—
292D4—2444no—
293D4—2444no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—24no—
297C2×C4—24no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—81no—
307C2×C4—1228no—
308C2×C4—1222no—
309C2×C4—2448no—
310C2×C4—2442no—
311C2×C4—2442no—
312C2×C4—2442no—
313C2×C4—2444no—
314C2×C4—2441no—
315C2×C4—2444no—
316C2×C4—2441no—
317C2×C4#1451970314yescomplemented
318C2×C4#1451973712yes—
319C2×C4#1451974912yes—
320C8—82no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—44no—
329C22—4882no—
330C22—4881no—
331C22—4881no—
332C22—2444no—
333C22—2442no—
334C22#1683963911yescharacteristic
335C22C2×#1451970312yes—
336C22#1683995012yes—
337C4—22no—
338C4—22no—
339C4—24no—
340C4—42no—
341C4—4882no—
342C4—2448no—
343C4—2442no—
344C2—22no—
345C2—22no—
346C2—41no—
347C2—4882no—
348C2#2144622511yescharacteristic
349C2#2144626011yescharacteristic
350C2#2144635411yescharacteristic
subgroupG/H
centerC22#16839639
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1463C2
socleC22×C61#386
prime radicalC22×#11758222C22
cosocleC23×C4×C61C23
hypercenter#1463D61
hypocenterC61#9476
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1463

Complements

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

NH ≅ G/Ntype
2. #21448176C2semidirect
4. #21443673C2semidirect
5. C61×#1463C2semidirect
10. #16839682C4semidirect
11. #16839679C4semidirect
125. C2×C4×C61#104semidirect
172. #1463D61semidirect
196. C61#9476semidirect
317. C2×C4#14519703semidirect

Maximal subgroups

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

subgroupindex
#214491542
#214481762
#214457622
#214436732
C61×#14632
#947661