Finite Group Databasebeta

← #29577835

Subgroups — #29577835

Order 15616.

726 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 726 conjugacy classes of subgroups — too many to draw. Only its 99 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#21448422C211yesmaximal, characteristic, complemented
3#21445762C212yesmaximal
4#21443673C212yesmaximal, complemented
5C61×#1463C211yesmaximal, characteristic, complemented
6C22×#11758222C2211yescharacteristic
7C2×#14519760—21no—
8C2×#14519703—24no—
9#16839682C412yescomplemented
10#16839679C412yescomplemented
11#16839648—24no—
12#16839641—21no—
13#16839633—22no—
14#16839625—21no—
15#16839625C2211yescharacteristic
16#16839622—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—
29C2×C4⋊Dic61—22no—
30C2×#11758196—22no—
31C2×C22⋊Dic61—21no—
32C2×C22⋊Dic61—22no—
33C2×#11758198—22no—
34C2×#11758198—22no—
35C2×C4×Dic61D414yes—
36#14519703—48no—
37#14519684—22no—
38#14519684—24no—
39#14519684—22no—
40#14519684—22no—
41#14519684—21no—
42#14519684C2×C412yes—
43C23×C4×C61D411yescharacteristic
44C23×C4×C61D411yescharacteristic
45C23×C4×C61C2311yescharacteristic
46C2×M16×C61—22no—
47C2×C4⋊C4×C61—22no—
48C2×C4⋊C4×C61—22no—
49C2×C4⋊C4×C61C2×C412yes—
50C2×C4⋊C4×C61D412yes—
51C22⋊C8×C61—22no—
52C22⋊C8×C61—22no—
53C24×C61C2×D411yescharacteristic
54C23×D61—21no—
55C22×C4×C61—21no—
56C22×C4×C61—21no—
57C22×C4×C61—21no—
58C22×C4×C61—21no—
59C22×C4×C61—42no—
60C22×C4×C61—42no—
61C22×C4×C61—42no—
62C22×C4×C61C22⋊C412yes—
63C22×C4×C61C22⋊C412yes—
64C22×C4×C61C22×C411yescharacteristic
65C22×C4×C61C2×D412yes—
66C22×C4×C61C2×D411yescharacteristic
67C22×C4×C61#4112yes—
68C2×C22⋊D61—24no—
69C2×C22⋊D61—42no—
70C2×C22⋊D61—41no—
71C2×C22⋊D61—41no—
72C22×Dic61—21no—
73C22×Dic61—22no—
74C22×Dic61—24no—
75#11758222—28no—
76#11758222—44no—
77#11758222—44no—
78#11758222—44no—
79C2×D244—22no—
80C2×D244—42no—
81C2×C4×D61—42no—
82C2×C4×D61—42no—
83C2×C4×D61—42no—
84C2×Dic122—22no—
85C2×Dic122—42no—
86M16×C61—44no—
87C2×C8×C61—42no—
88C4⋊C4×C61—24no—
89C4⋊C4×C61—24no—
90C22⋊Dic61—44no—
91C22⋊Dic61—42no—
92#11758198—44no—
93#11758198—44no—
94C4⋊Dic61—82no—
95#11758196—44no—
96C4×Dic61—44no—
97C23×C61—22no—
98C23×C61—22no—
99C23×C61—41no—
100C23×C61C2×C22⋊C411yescharacteristic
101C23×C61#12011yescharacteristic
102C23×C61#12311yescharacteristic
103C22×D61—22no—
104C22×D61—41no—
105C22×D61—42no—
106C22×D61—41no—
107C22×D61—42no—
108C2×C4×C61—24no—
109C2×C4×C61—22no—
110C2×C4×C61—22no—
111C2×C4×C61—22no—
112C2×C4×C61—22no—
113C2×C4×C61—41no—
114C2×C4×C61—41no—
115C2×C4×C61—42no—
116C2×C4×C61—42no—
117C2×C4×C61—42no—
118C2×C4×C61—42no—
119C2×C4×C61—42no—
120C2×C4×C61—81no—
121C2×C4×C61C4×D412yes—
122C2×C4×C61#12112yes—
123C22⋊D61—44no—
124C22⋊D61—44no—
125C2×Dic61—22no—
126C2×Dic61—41no—
127C2×Dic61—41no—
128C2×Dic61—42no—
129C2×Dic61—44no—
130C2×Dic61—44no—
131C2×Dic61—42no—
132C2×Dic61—42no—
133C2×Dic61—44no—
134D244—24no—
135D244—42no—
136C4×D61—48no—
137Dic122—24no—
138Dic122—42no—
139C8×C61—82no—
140#9480—611nomaximal, Sylow
141C22×C61—21no—
142C22×C61—21no—
143C22×C61—21no—
144C22×C61—21no—
145C22×C61—21no—
146C22×C61—21no—
147C22×C61—21no—
148C22×C61—44no—
149C22×C61#38611yescharacteristic
150C22×C61#42112yes—
151C22×C61B(4,𝔽2)12yes—
152C2×D61—44no—
153C2×D61—42no—
154C2×D61—82no—
155C2×D61—81no—
156C2×D61—81no—
157C4×C61—22no—
158C4×C61—22no—
159C4×C61—42no—
160C4×C61—42no—
161Dic61—42no—
162Dic61—84no—
163Dic61—82no—
164#2406—611no—
165#2313—611no—
166#1789—612no—
167#1463D6111yescharacteristic, complemented
168#1284—612no—
169C2×C61—22no—
170C2×C61—22no—
171C2×C61—41no—
172C2×C61#188911yescharacteristic
173C2×C61#189711yescharacteristic
174C2×C61#191211yescharacteristic
175D61—82no—
176C22×#41—611no—
177C2×#123—1221no—
178C22×C4⋊C4C2×D6111yescharacteristic
179C2×#104—1224no—
180#407C2×D6112yes—
181#395—612no—
182#394—612no—
183#387—1224no—
184#386—1221no—
185#385—1222no—
186#381—611no—
187#381—1221no—
188#380—1221no—
189C4×C22⋊C4—612no—
190#324—22no—
191C61#948011yescharacteristic, complemented, Sylow
192C2×#41—1222no—
193C2×#41—1222no—
194C2×#41—2444no—
195C22×Q8—611no—
196C22×D4—611no—
197C22×D4—1221no—
198C23×C4—1221no—
199C23×C4D24411yescharacteristic
200C23×C4D24411yescharacteristic
201C23×C4C22×D6111yescharacteristic
202C2×M16—22no—
203#123—2444no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—1222no—
207C2×C4⋊C4—1222no—
208C2×C4⋊C4C4×D6112yes—
209C2×C4⋊C4C22⋊D6112yes—
210C2×C22⋊C4—1221no—
211C2×C22⋊C4—1222no—
212C2×C22⋊C4—1222no—
213C2×C22⋊C4—1222no—
214C2×C42—614no—
215#104—2448no—
216C22⋊C8—22no—
217C22⋊C8—22no—
218#95—612no—
219#95—1222no—
220#95—1224no—
221#95—1222no—
222#95—1222no—
223#95—1221no—
224C24—1221no—
225C24C2×D24411yescharacteristic
226#41—1228no—
227#41—2444no—
228#41—2444no—
229#41—2444no—
230C2×Q8—1222no—
231C2×Q8—2442no—
232C2×D4—1224no—
233C2×D4—1222no—
234C2×D4—2442no—
235C2×D4—2442no—
236C2×D4—2441no—
237C2×D4—2441no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—1221no—
246C22×C4—1222no—
247C22×C4—1224no—
248C22×C4—2442no—
249C22×C4—2442no—
250C22×C4—2442no—
251C22×C4#1175819812yes—
252C22×C4#1175819812yes—
253C22×C4C2×C4×D6111yescharacteristic
254C22×C4D4×D6112yes—
255C22×C4#1175822412yes—
256C22×C4C2×C22⋊D6111yescharacteristic
257M16—44no—
258C2×C8—42no—
259C4⋊C4—24no—
260C4⋊C4—24no—
261C4⋊C4—4882no—
262C4⋊C4—2444no—
263C22⋊C4—2444no—
264C22⋊C4—2444no—
265C22⋊C4—2442no—
266C22⋊C4—2444no—
267C42—2444no—
268C23—22no—
269C23—22no—
270C23—41no—
271C23—1222no—
272C23—2441no—
273C23—2442no—
274C23—2442no—
275C23—2441no—
276C23C2×#1175819811yescharacteristic
277C23#1451974711yescharacteristic
278C23#1451976111yescharacteristic
279Q8—1224no—
280Q8—2442no—
281D4—1224no—
282D4—2442no—
283D4—2444no—
284D4—2444no—
285C2×C4—24no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—81no—
298C2×C4—1222no—
299C2×C4—2448no—
300C2×C4—2442no—
301C2×C4—2444no—
302C2×C4—2444no—
303C2×C4—2442no—
304C2×C4—2444no—
305C2×C4—2442no—
306C2×C4—2441no—
307C2×C4—2441no—
308C2×C4#1451973712yes—
309C2×C4#1451974912yes—
310C8—82no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—44no—
319C22—4882no—
320C22—4881no—
321C22—4881no—
322C22—2444no—
323C22—2442no—
324C22#1683963911yescharacteristic
325C22#1683977812yes—
326C22#1683995012yes—
327C4—22no—
328C4—22no—
329C4—42no—
330C4—42no—
331C4—4884no—
332C4—4882no—
333C4—2442no—
334C2—22no—
335C2—22no—
336C2—41no—
337C2—4882no—
338C2#2144622511yescharacteristic
339C2#2144626511yescharacteristic
340C2#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#9480
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. #21448422C2semidirect
4. #21443673C2semidirect
5. C61×#1463C2semidirect
9. #16839682C4semidirect
10. #16839679C4semidirect
167. #1463D61semidirect
191. C61#9480semidirect

Maximal subgroups

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

subgroupindex
#214491542
#214484222
#214457622
#214436732
C61×#14632
#948061