Finite Group Databasebeta

← #29580579

Subgroups — #29580579

Order 15616.

772 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 772 conjugacy classes of subgroups — too many to draw. Only its 119 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×#16840080C212yesmaximal, complemented
2#21451921C211yesmaximal, characteristic
3C2×#16839495C212yesmaximal, complemented
4#21443662C211yesmaximal, characteristic, complemented
5C61×#1445C211yesmaximal, characteristic, complemented
6C2×#14519751—22no—
7C2×#14519751C2212yes—
8#16840080—24no—
9#16840080—24no—
10C2×#14519708—22no—
11C2×#14519697—22no—
12C2×C22⋊#92418C2212yes—
13#16839699—22no—
14#16839672—22no—
15#16839495—24no—
16C22×C4⋊C4×C61C2211yescharacteristic
17C2×#95×C61—22no—
18C2×#95×C61C2211yescharacteristic
19C2×#95×C61C2211yescharacteristic
20C22×D244—22no—
21C22×D244D412yes—
22C22×C22⋊D61—22no—
23C22×#92418—22no—
24#14519751—22no—
25#14519751—22no—
26#14519751—22no—
27#14519751—22no—
28#14519751—44no—
29#14519751—44no—
30C2×C4⋊Dic61—22no—
31C2×C4⋊Dic61D412yes—
32C2×#11758198—24no—
33C2×#11758198—22no—
34#14519708—28no—
35#14519697—28no—
36C22⋊#92418—22no—
37C22⋊#92418D414yescomplemented
38C23×C4×C61—21no—
39C23×C4×C61—21no—
40C23×C4×C61D412yes—
41C23×C4×C61C2311yescharacteristic
42C2×C4⋊C4×C61—22no—
43C2×C4⋊C4×C61—22no—
44C2×C4⋊C4×C61—21no—
45C2×C4⋊C4×C61D412yescomplemented
46#95×C61—22no—
47#95×C61—22no—
48#95×C61—44no—
49C24×C61C2×D411yescharacteristic
50C23×D61—22no—
51C22×C4×C61—22no—
52C22×C4×C61—22no—
53C22×C4×C61—22no—
54C22×C4×C61—22no—
55C22×C4×C61—42no—
56C22×C4×C61—42no—
57C22×C4×C61—42no—
58C22×C4×C61—42no—
59C22×C4×C61—41no—
60C22×C4×C61—42no—
61C22×C4×C61—42no—
62C22×C4×C61—41no—
63C22×C4×C61—41no—
64C22×C4×C61—41no—
65C22×C4×C61—42no—
66C22×C4×C61SD1614yes—
67C22×C4×C61C2×D412yes—
68C22×C4×C61C2×D411yescharacteristic
69C22×C4×C61C2×D412yes—
70C22×C4×C61#4111yescharacteristic
71C2×C22⋊D61—42no—
72C2×C22⋊D61—42no—
73C2×C22⋊D61—44no—
74C2×C22⋊D61—44no—
75C22×Dic61—22no—
76C2×D244—22no—
77C2×D244—22no—
78C2×D244—44no—
79C2×D244—44no—
80C2×D244—82no—
81C2×D244—82no—
82C4⋊C4×C61—24no—
83C4⋊C4×C61—42no—
84#11758198—48no—
85#11758198—42no—
86#11758198—42no—
87C4⋊Dic61—22no—
88C4⋊Dic61—22no—
89C4⋊Dic61—44no—
90C2×#92418—24no—
91C2×#92418—44no—
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—44no—
114C2×C4×C61—42no—
115C2×C4×C61—44no—
116C2×C4×C61—42no—
117C2×C4×C61—82no—
118C2×C4×C61—82no—
119C2×C4×C61—81no—
120C2×C4×C61—82no—
121C2×C4×C61—81no—
122C2×C4×C61—81no—
123C2×C4×C61—81no—
124C2×C4×C61#12012yes—
125C2×C4×C61#12112yes—
126C22⋊D61—84no—
127C22⋊D61—82no—
128C22⋊D61—82no—
129C2×Dic61—42no—
130C2×Dic61—44no—
131D244—44no—
132D244—82no—
133D244—88no—
134#92418—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—
160C4×C61—82no—
161C4×C61—82no—
162Dic61—82no—
163C2×#481—612no—
164#2656—611no—
165C2×#327—612no—
166#1445D6111yescharacteristic, complemented
167#1280—611no—
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⋊C4C2×D6111yescharacteristic
180#481—1224no—
181#481—1224no—
182C2×#106—1222no—
183C2×#102—1222no—
184C2×C22⋊C8—612no—
185#399—1222no—
186#392—1222no—
187C2×#95—22no—
188C2×#95C2×D6111yescharacteristic
189C2×#95C2×D6111yescharacteristic
190#327—1224no—
191C61#996811yescharacteristic, complemented, Sylow
192C22×D4—612no—
193C22×D4—1222no—
194C22×D4—1222no—
195C23×C4—21no—
196C23×C4—21no—
197C23×C4C22⋊D6112yes—
198C23×C4C22×D6111yescharacteristic
199C22×C8—1222no—
200#121—1222no—
201#121—1222no—
202#121—1222no—
203#121—1222no—
204#121—2444no—
205#121—2444no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—612no—
210C2×C4⋊C4—1222no—
211C2×C4⋊C4D24412yescomplemented
212C2×C22⋊C4—1224no—
213C2×C22⋊C4—1222no—
214#106—1228no—
215#102—1228no—
216C22⋊C8—614no—
217C22⋊C8—1222no—
218#95—22no—
219#95—22no—
220#95—44no—
221C24—1222no—
222C24C2×C22⋊D6111yescharacteristic
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—22no—
236C22×C4—22no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—41no—
243C22×C4—42no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—42no—
248C22×C4—1222no—
249C22×C4#1175820114yes—
250C22×C4C2×D24411yescharacteristic
251C22×C4#1175822211yescharacteristic
252C22×C4D4×D6112yes—
253C22×C4D4×D6112yes—
254C2×C8—1224no—
255C2×C8—2444no—
256C4⋊C4—24no—
257C4⋊C4—42no—
258C4⋊C4—1222no—
259C4⋊C4—1222no—
260C4⋊C4—2444no—
261C22⋊C4—2448no—
262C22⋊C4—2442no—
263C22⋊C4—2442no—
264C23—22no—
265C23—21no—
266C23—21no—
267C23—2444no—
268C23—2444no—
269C23—2442no—
270C23—2442no—
271C23—2442no—
272C23#1451974511yescharacteristic
273C23#1451975111yescharacteristic
274C23#1451977311yescharacteristic
275C23C2×#1175820112yes—
276C23#1451980212yes—
277D4—4882no—
278D4—4888no—
279D4—4884no—
280D4—4882no—
281D4—4882no—
282D4—2444no—
283C2×C4—24no—
284C2×C4—24no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—44no—
291C2×C4—42no—
292C2×C4—44no—
293C2×C4—44no—
294C2×C4—42no—
295C2×C4—82no—
296C2×C4—81no—
297C2×C4—82no—
298C2×C4—82no—
299C2×C4—81no—
300C2×C4—81no—
301C2×C4—81no—
302C2×C4—2444no—
303C2×C4—2442no—
304C2×C4#1451974712yes—
305C2×C4#1451975012yes—
306C8—2444no—
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#1683967111yescharacteristic
321C22#1683991212yes—
322C22#1683992911yescharacteristic
323C22#1683993411yescharacteristic
324C22#1683995112yes—
325C22#1683996212yes—
326C22#1684013611yescharacteristic
327C22#1684015211yescharacteristic
328C4—22no—
329C4—22no—
330C4—44no—
331C4—82no—
332C4—82no—
333C4—4882no—
334C2—21no—
335C2—22no—
336C2—21no—
337C2—9762no—
338C2#2144483812yes—
339C2#2144485512yes—
340C2#2144686011yescharacteristic
341C2#2144690611yescharacteristic
342C2#2144695011yescharacteristic
subgroupG/H
centerC23#14519745
commutator, cosocleC23×C4×C61C23
FrattiniC23×C4C22×D61
FittingC61×#1445C2
socleC23×C61#120
prime radical#21451921C2
hypercenter#1445D61
hypocenterC61#9968
LCSG ▷ C23×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1445

Complements

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

NH ≅ G/Ntype
1. C2×#16840080C2semidirect
3. C2×#16839495C2semidirect
4. #21443662C2semidirect
5. C61×#1445C2semidirect
37. C22⋊#92418D4semidirect
45. C2×C4⋊C4×C61D4semidirect
166. #1445D61semidirect
191. C61#9968semidirect
211. C2×C4⋊C4D244semidirect

Maximal subgroups

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

subgroupindex
C2×#168400802
#214519212
C2×#168394952
#214436622
C61×#14452
#996861