Finite Group Databasebeta

← #29580580

Subgroups — #29580580

Order 15616.

772 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 772 conjugacy classes of subgroups — too many to draw. Only its 121 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
1C2×#16840081C212yesmaximal, complemented
2#21451921C211yesmaximal, characteristic
3C2×#16839497C212yesmaximal, complemented
4#21444088C211yesmaximal, characteristic, complemented
5C61×#1280C211yesmaximal, characteristic, complemented
6C2×#14519751—22no—
7C2×#14519751C2212yes—
8C22×C4⋊Dic61C2211yescharacteristic
9#16840081—24no—
10#16840081—24no—
11C2×#14519709—22no—
12C2×#14519699—22no—
13#16839699—22no—
14#16839672—22no—
15C2×#14519684—22no—
16C2×#14519684C2211yescharacteristic
17#16839497—24no—
18C2×C22⋊C8×C61C2212yes—
19C2×#95×C61C2211yescharacteristic
20C22×D244—22no—
21C22×D244D412yes—
22C22×C22⋊D61—22no—
23C23×Dic61—21no—
24C23×Dic61—21no—
25#14519751—22no—
26#14519751—22no—
27#14519751—22no—
28#14519751—22no—
29#14519751—44no—
30#14519751—44no—
31C2×C4⋊Dic61—22no—
32C2×C4⋊Dic61—22no—
33C2×C4⋊Dic61—22no—
34C2×C4⋊Dic61—21no—
35C2×C4⋊Dic61D412yes—
36C2×C4⋊Dic61D412yescomplemented
37C2×#11758198—24no—
38C2×#11758198—22no—
39#14519709—28no—
40#14519699—28no—
41#14519684—22no—
42#14519684—44no—
43C23×C4×C61D412yes—
44C23×C4×C61C2311yescharacteristic
45C22×C8×C61—22no—
46C22⋊C8×C61—22no—
47C22⋊C8×C61D414yescomplemented
48#95×C61—22no—
49C24×C61C2×D411yescharacteristic
50C23×D61—22no—
51C22×C4×C61—22no—
52C22×C4×C61—22no—
53C22×C4×C61—42no—
54C22×C4×C61—42no—
55C22×C4×C61—42no—
56C22×C4×C61SD1614yes—
57C22×C4×C61C2×D412yes—
58C22×C4×C61C2×D411yescharacteristic
59C22×C4×C61C2×D412yes—
60C22×C4×C61#4111yescharacteristic
61C2×C22⋊D61—42no—
62C2×C22⋊D61—42no—
63C2×C22⋊D61—44no—
64C2×C22⋊D61—44no—
65C22×Dic61—22no—
66C22×Dic61—22no—
67C22×Dic61—22no—
68C22×Dic61—41no—
69C22×Dic61—42no—
70C22×Dic61—42no—
71C22×Dic61—42no—
72C22×Dic61—41no—
73C22×Dic61—41no—
74C22×Dic61—41no—
75C22×Dic61—42no—
76C2×D244—22no—
77C2×D244—22no—
78C2×D244—44no—
79C2×D244—44no—
80C2×D244—82no—
81C2×D244—82no—
82C2×C8×C61—24no—
83C2×C8×C61—44no—
84#11758198—48no—
85#11758198—42no—
86#11758198—42no—
87C4⋊Dic61—24no—
88C4⋊Dic61—22no—
89C4⋊Dic61—22no—
90C4⋊Dic61—44no—
91C4⋊Dic61—42no—
92C23×C61—21no—
93C23×C61—21no—
94C23×C61—22no—
95C23×C61#12011yescharacteristic
96C23×C61#12111yescharacteristic
97C23×C61#12711yescharacteristic
98C23×C61C2×SD1612yes—
99C23×C61AGL(1,ℤ/8)12yes—
100C22×D61—42no—
101C22×D61—44no—
102C22×D61—42no—
103C22×D61—42no—
104C22×D61—44no—
105C2×C4×C61—24no—
106C2×C4×C61—24no—
107C2×C4×C61—41no—
108C2×C4×C61—42no—
109C2×C4×C61—41no—
110C2×C4×C61—42no—
111C2×C4×C61—42no—
112C2×C4×C61—44no—
113C2×C4×C61—82no—
114C2×C4×C61#12012yes—
115C2×C4×C61#12112yes—
116C22⋊D61—84no—
117C22⋊D61—82no—
118C22⋊D61—82no—
119C2×Dic61—44no—
120C2×Dic61—42no—
121C2×Dic61—44no—
122C2×Dic61—42no—
123C2×Dic61—42no—
124C2×Dic61—44no—
125C2×Dic61—82no—
126C2×Dic61—81no—
127C2×Dic61—82no—
128C2×Dic61—81no—
129C2×Dic61—81no—
130C2×Dic61—81no—
131D244—44no—
132D244—82no—
133D244—88no—
134C8×C61—44no—
135#9968—611nomaximal, Sylow
136C22×C61—22no—
137C22×C61—22no—
138C22×C61—22no—
139C22×C61—22no—
140C22×C61—21no—
141C22×C61—21no—
142C22×C61—21no—
143C22×C61—21no—
144C22×C61#39211yescharacteristic
145C22×C61#45012yes—
146C22×C61#45311yescharacteristic
147C22×C61#45411yescharacteristic
148C22×C61#45812yes—
149C22×C61#46012yes—
150C22×C61#49211yescharacteristic
151C22×C61#49611yescharacteristic
152C2×D61—84no—
153C2×D61—84no—
154C2×D61—82no—
155C2×D61—82no—
156C2×D61—82no—
157C4×C61—22no—
158C4×C61—22no—
159C4×C61—44no—
160Dic61—82no—
161Dic61—82no—
162Dic61—82no—
163C2×#481—612no—
164#2656—611no—
165C2×#327—612no—
166#1445—611no—
167#1280D6111yescharacteristic, complemented
168C2×C61—21no—
169C2×C61—22no—
170C2×C61—21no—
171C2×C61#161312yes—
172C2×C61#161612yes—
173C2×C61#200611yescharacteristic
174C2×C61#201411yescharacteristic
175C2×C61#202211yescharacteristic
176D61—162no—
177C2×#121—612no—
178C2×#121—1222no—
179C22×C4⋊C4—611no—
180#481—1224no—
181#481—1224no—
182C2×#106—1222no—
183C2×#102—1222no—
184C2×C22⋊C8C2×D6112yes—
185#399—1222no—
186#392—1222no—
187C2×#95—611no—
188C2×#95—1222no—
189C2×#95C2×D6111yescharacteristic
190#327—1224no—
191C61#996811yescharacteristic, complemented, Sylow
192C22×D4—612no—
193C22×D4—1222no—
194C22×D4—1222no—
195C23×C4—1221no—
196C23×C4—1221no—
197C23×C4D24412yes—
198C23×C4C22×D6111yescharacteristic
199C22×C8—22no—
200#121—1222no—
201#121—1222no—
202#121—1222no—
203#121—1222no—
204#121—2444no—
205#121—2444no—
206C2×C4⋊C4—612no—
207C2×C4⋊C4—612no—
208C2×C4⋊C4—1222no—
209C2×C4⋊C4—1222no—
210C2×C4⋊C4—1222no—
211C2×C4⋊C4—1221no—
212C2×C22⋊C4—1224no—
213C2×C22⋊C4—1222no—
214#106—1228no—
215#102—1228no—
216C22⋊C8—22no—
217C22⋊C8D24414yescomplemented
218#95—22no—
219#95—1222no—
220#95—2444no—
221C24—1222no—
222C24C2×D24411yescharacteristic
223C2×D4—1222no—
224C2×D4—1222no—
225C2×D4—4882no—
226C2×D4—4882no—
227C2×D4—2444no—
228C2×D4—2444no—
229C2×D4—2442no—
230C2×D4—2442no—
231C2×D4—2444no—
232C2×D4—2444no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—1222no—
239C22×C4—1222no—
240C22×C4—1222no—
241C22×C4—2442no—
242C22×C4—2441no—
243C22×C4—2442no—
244C22×C4—2442no—
245C22×C4—2441no—
246C22×C4—2441no—
247C22×C4—2441no—
248C22×C4—2442no—
249C22×C4#1175819014yes—
250C22×C4C2×D24412yes—
251C22×C4D4×D6112yes—
252C22×C4D4×D6111yescharacteristic
253C22×C4#1175822611yescharacteristic
254C2×C8—24no—
255C2×C8—44no—
256C4⋊C4—1224no—
257C4⋊C4—1222no—
258C4⋊C4—1222no—
259C4⋊C4—2444no—
260C4⋊C4—2442no—
261C22⋊C4—2448no—
262C22⋊C4—2442no—
263C22⋊C4—2442no—
264C23—21no—
265C23—22no—
266C23—21no—
267C23—2442no—
268C23—2444no—
269C23—2444no—
270C23—2442no—
271C23—2442no—
272C23#1451974711yescharacteristic
273C23#1451975211yescharacteristic
274C23#1451977411yescharacteristic
275C23C2×#1175819012yes—
276C23#1451980512yes—
277D4—4882no—
278D4—4888no—
279D4—4884no—
280D4—4882no—
281D4—4882no—
282D4—2444no—
283C2×C4—24no—
284C2×C4—24no—
285C2×C4—42no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—44no—
291C2×C4—82no—
292C2×C4—4881no—
293C2×C4—4882no—
294C2×C4—4882no—
295C2×C4—4881no—
296C2×C4—4881no—
297C2×C4—4881no—
298C2×C4—2442no—
299C2×C4—2444no—
300C2×C4—2444no—
301C2×C4—2444no—
302C2×C4—2442no—
303C2×C4—2442no—
304C2×C4#1451974712yes—
305C2×C4#1451975212yes—
306C8—44no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—22no—
312C22—22no—
313C22—22no—
314C22—22no—
315C22—4882no—
316C22—4882no—
317C22—4884no—
318C22—4884no—
319C22—4882no—
320C22#1683967211yescharacteristic
321C22#1683991512yes—
322C22#1683993211yescharacteristic
323C22#1683993711yescharacteristic
324C22#1683995312yes—
325C22#1683996312yes—
326C22#1684013511yescharacteristic
327C22#1684015111yescharacteristic
328C4—22no—
329C4—22no—
330C4—44no—
331C4—4882no—
332C4—4882no—
333C4—4882no—
334C2—21no—
335C2—22no—
336C2—21no—
337C2—9762no—
338C2#2144484112yes—
339C2#2144485812yes—
340C2#2144686311yescharacteristic
341C2#2144690911yescharacteristic
342C2#2144695111yescharacteristic
subgroupG/H
centerC23#14519747
commutator, cosocleC23×C4×C61C23
FrattiniC23×C4C22×D61
FittingC61×#1280C2
socleC23×C61#120
prime radical#21451921C2
hypercenter#1280D61
hypocenterC61#9968
LCSG ▷ C23×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1280

Complements

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

NH ≅ G/Ntype
1. C2×#16840081C2semidirect
3. C2×#16839497C2semidirect
4. #21444088C2semidirect
5. C61×#1280C2semidirect
36. C2×C4⋊Dic61D4semidirect
47. C22⋊C8×C61D4semidirect
167. #1280D61semidirect
191. C61#9968semidirect
217. C22⋊C8D244semidirect

Maximal subgroups

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

subgroupindex
C2×#168400812
#214519212
C2×#168394972
#214440882
C61×#12802
#996861