Finite Group Databasebeta

← #30596943

Subgroups — #30596943

Order 15616.

954 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 954 conjugacy classes of subgroups — too many to draw. Only its 148 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#21462263C212yesmaximal
2#21460699C214yesmaximal, complemented
3#407×D61C211yesmaximal, characteristic
4C2×#14519856—22no—
5#16840705C414yescomplemented
6#16840704—22no—
7#16840704C414yescomplemented
8#16840703—22no—
9C4×C22⋊#6923—22no—
10C4×C22⋊#6923C2214yes—
11C23×C4×D61C2211yescharacteristic
12C2×M16×D61C412yescomplemented
13#16839726C411yescharacteristic, complemented
14#16839725C2212yes—
15#16839724—22no—
16C22⋊C8×D61—22no—
17C61×#407C411yescharacteristic, complemented
18#14519856—48no—
19C2×C4⋊#6923—24no—
20C2×C22⋊#6923—24no—
21C2×C22⋊#6923—24no—
22C2×C22⋊#6923—24no—
23C2×C4×#6923—24no—
24(C2×#6923)⋊C4—24no—
25(C2×#6923)⋊C4C2×C418yes—
26C24×D61D411yescharacteristic
27C23×Dic61Q811yescharacteristic
28C22×C4×D61—44no—
29C22×C4×D61C2×C411yescharacteristic
30C22×C4×D61C2×C412yes—
31C22×C4×D61C2×C412yes—
32C22×C4×D61D412yes—
33C22×C4×D61D411yescharacteristic
34C22×C4×D61D412yes—
35C22×C4×D61Q811yescharacteristic
36C22×C4×D61C2311yescharacteristic
37C2×#11758194C2×C412yes—
38C2×#11758189—22no—
39M16×D61—44no—
40C2×C8×D61—22no—
41C22⋊#92418—22no—
42#14519689—24no—
43C23×C4×C61C2×C411yescharacteristic
44C2×M16×C61C2×C412yes—
45C22⋊C8×C61—22no—
46C24×C61C4⋊C411yescharacteristic
47C23×D61—42no—
48C23×D61—44no—
49C23×D61C22⋊C412yes—
50C23×D61C22⋊C412yes—
51C23×D61C4⋊C411yescharacteristic
52C23×D61C2×D411yescharacteristic
53C22×#6923—28no—
54C22×C4×C61—42no—
55C22×C4×C61C4212yes—
56C22×C4×C61C22⋊C412yes—
57C22×C4×C61C4⋊C411yescharacteristic
58C22×C4×C61C22×C411yescharacteristic
59C22×Dic61—42no—
60C22×Dic61C4⋊C411yescharacteristic
61C22×Dic61C4⋊C412yes—
62C22×Dic61C4⋊C412yes—
63C22×Dic61C2×Q811yescharacteristic
64C2×C4×D61—24no—
65C2×C4×D61—28no—
66C2×C4×D61—44no—
67C2×C4×D61—42no—
68C2×C4×D61—44no—
69C2×C4×D61—42no—
70C2×C4×D61—44no—
71C2×C4×D61—42no—
72C2×C4×D61C4211yescharacteristic
73C2×C4×D61C4211yescharacteristic
74C2×C4×D61C22⋊C412yes—
75C2×C4×D61C22⋊C411yescharacteristic
76C2×C4×D61C22⋊C412yes—
77C2×C4×D61C22⋊C411yescharacteristic
78C2×C4×D61C4⋊C412yes—
79C2×C4×D61C4⋊C412yes—
80C2×C4×D61C22×C412yes—
81C2×C4×D61C2×D412yes—
82C22⋊#6923—48no—
83C22⋊#6923—48no—
84C22⋊#6923—48no—
85C4⋊#6923—48no—
86C4×#6923—48no—
87M16×C61—42no—
88C2×C8×C61—22no—
89#11758194—42no—
90C2×#92418—22no—
91#11758189—44no—
92C8×D61—44no—
93C23×C61—41no—
94C23×C61—42no—
95C23×C61#9512yes—
96C23×C61C2×C4⋊C411yescharacteristic
97C22×D61—24no—
98C22×D61—22no—
99C22×D61—24no—
100C22×D61—24no—
101C22×D61—24no—
102C22×D61—42no—
103C22×D61—41no—
104C22×D61—42no—
105C22×D61—42no—
106C22×D61—42no—
107C22×D61—416no—
108C22×D61#9511yescharacteristic
109C22×D61#9511yescharacteristic
110C22×D61C2×C22⋊C412yes—
111C2×#6923—48no—
112C2×#6923—48no—
113C2×#6923—48no—
114C2×C4×C61—22no—
115C2×C4×C61—24no—
116C2×C4×C61—42no—
117C2×C4×C61—41no—
118C2×C4×C61#9512yes—
119C2×C4×C61C2×C4211yescharacteristic
120C2×C4×C61C2×C22⋊C411yescharacteristic
121C2×Dic61—22no—
122C2×Dic61—24no—
123C2×Dic61—42no—
124C2×Dic61—41no—
125C2×Dic61#9511yescharacteristic
126C2×Dic61#9511yescharacteristic
127C2×Dic61C2×C4⋊C412yes—
128C4×D61—22no—
129C4×D61—24no—
130C4×D61—22no—
131C4×D61—24no—
132C4×D61—24no—
133C4×D61—22no—
134C4×D61—42no—
135C4×D61—42no—
136C4×D61—42no—
137C4×D61#9914yes—
138C8×C61—42no—
139#92418—42no—
140#8049—611nomaximal, Sylow
141C22×C61—22no—
142C22×C61—22no—
143C22×C61—22no—
144C22×C61—22no—
145C22×C61—21no—
146C22×C61—44no—
147C22×C61C2×#9511yescharacteristic
148C2×D61—22no—
149C2×D61—22no—
150C2×D61—22no—
151C2×D61—22no—
152C2×D61—21no—
153C2×D61—44no—
154C2×D61—48no—
155C2×D61—48no—
156C2×D61—44no—
157C2×D61#34212yes—
158C2×D61#42114yes—
159#6923—88no—
160C4×C61—22no—
161C4×C61—21no—
162C4×C61C2×#9912yes—
163Dic61—21no—
164Dic61—22no—
165Dic61#34212yes—
166C2×#407—611no—
167#2280—612no—
168#1297—614no—
169C2×C61—42no—
170C2×C61—41no—
171C2×C61#173312yes—
172C2×C61C2×#34211yescharacteristic
173D61—24no—
174D61—22no—
175D61—41no—
176C24×C4—611no—
177C22×M16—612no—
178C2×#117—1222no—
179#407—611no—
180#407—612no—
181#407—1222no—
182#407#692311yescharacteristic, complemented
183C2×C22⋊C8—1222no—
184#381—614no—
185#380—614no—
186#380—1222no—
187#379—1222no—
188C4×C22⋊C4—614no—
189C4×C22⋊C4—1222no—
190C61#804911yescharacteristic, complemented, Sylow
191C25—611no—
192C23×C4—611no—
193C23×C4—612no—
194C23×C4—611no—
195C23×C4—612no—
196C23×C4—612no—
197C23×C4—611no—
198C23×C4—611no—
199C23×C4—612no—
200C23×C4—611no—
201C23×C4—2444no—
202C23×C4C2×#692311yescharacteristic
203C2×M16—612no—
204C2×M16—1222no—
205C2×M16—2444no—
206C2×M16C2×#692312yes—
207C22×C8—1222no—
208#117—2448no—
209C2×C4⋊C4—1224no—
210C2×C22⋊C4—1224no—
211C2×C22⋊C4—1224no—
212C2×C22⋊C4—1224no—
213C2×C42—1224no—
214C22⋊C8—22no—
215C22⋊C8—1224no—
216C22⋊C8—1222no—
217#95—618no—
218#95—1224no—
219C24—612no—
220C24—611no—
221C24—612no—
222C24—611no—
223C24—2442no—
224C24—2444no—
225C24C4⋊#692311yescharacteristic
226C22×C4—42no—
227C22×C4—611no—
228C22×C4—612no—
229C22×C4—611no—
230C22×C4—612no—
231C22×C4—611no—
232C22×C4—612no—
233C22×C4—612no—
234C22×C4—611no—
235C22×C4—611no—
236C22×C4—612no—
237C22×C4—611no—
238C22×C4—612no—
239C22×C4—612no—
240C22×C4—612no—
241C22×C4—1228no—
242C22×C4—1224no—
243C22×C4—1228no—
244C22×C4—2442no—
245C22×C4—2444no—
246C22×C4—2444no—
247C22×C4—2442no—
248C22×C4—2444no—
249C22×C4—2442no—
250C22×C4—2442no—
251C22×C4C4×#692312yes—
252C22×C4C4⋊#692311yescharacteristic
253C22×C4C22⋊#692312yes—
254C22×C4C22×#692311yescharacteristic
255M16—42no—
256M16—2444no—
257M16—2442no—
258C2×C8—22no—
259C2×C8—1222no—
260C2×C8—2444no—
261C4⋊C4—2448no—
262C22⋊C4—2448no—
263C22⋊C4—2448no—
264C22⋊C4—2448no—
265C42—2448no—
266C23—41no—
267C23—42no—
268C23—611no—
269C23—612no—
270C23—611no—
271C23—1224no—
272C23—1224no—
273C23—1224no—
274C23—1222no—
275C23—1224no—
276C23—2442no—
277C23—2441no—
278C23—2442no—
279C23—2442no—
280C23—2442no—
281C23—24416no—
282C23(C2×#6923)⋊C412yes—
283C23C2×C4⋊#692311yescharacteristic
284C2×C4—22no—
285C2×C4—24no—
286C2×C4—42no—
287C2×C4—41no—
288C2×C4—611no—
289C2×C4—614no—
290C2×C4—611no—
291C2×C4—612no—
292C2×C4—1224no—
293C2×C4—1222no—
294C2×C4—1224no—
295C2×C4—1222no—
296C2×C4—1224no—
297C2×C4—1222no—
298C2×C4—1224no—
299C2×C4—1222no—
300C2×C4—2442no—
301C2×C4—2442no—
302C2×C4—2442no—
303C2×C4—2442no—
304C2×C4—2441no—
305C2×C4—2448no—
306C2×C4—2448no—
307C2×C4—2448no—
308C2×C4(C2×#6923)⋊C412yes—
309C2×C4C2×C4×#692311yescharacteristic
310C2×C4C2×C22⋊#692311yescharacteristic
311C8—42no—
312C8—2442no—
313C22—22no—
314C22—21no—
315C22—22no—
316C22—22no—
317C22—22no—
318C22—44no—
319C22—612no—
320C22—614no—
321C22—1222no—
322C22—1222no—
323C22—1222no—
324C22—1221no—
325C22—1222no—
326C22—2444no—
327C22—2448no—
328C22—2448no—
329C22—2444no—
330C22C2×(C2×#6923)⋊C411yescharacteristic
331C4—22no—
332C4—21no—
333C4—612no—
334C4—1221no—
335C4—1222no—
336C4—4888no—
337C4D61⋊#9912yes—
338C2—42no—
339C2—41no—
340C2—1224no—
341C2—1222no—
342C2—2441no—
343C2#2146150112yes—
344C2#2146151611yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊#6923
commutatorC2×C4×C61C2×C42
FrattiniC22×C4C22×#6923
FittingC61×#407C4
socleC22×C61C2×#95
prime radicalC24×D61D4
cosocleC22×C4×D61C23
hypercenter#407#6923
hypocenterC61#8049
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
2. #21460699C2semidirect
5. #16840705C4semidirect
7. #16840704C4semidirect
12. C2×M16×D61C4semidirect
13. #16839726C4semidirect
17. C61×#407C4semidirect
182. #407#6923semidirect
190. C61#8049semidirect

Maximal subgroups

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

subgroupindex
#214622632
#214606992
#407×D612
#804961