Finite Group Databasebeta

← #30596935

Subgroups — #30596935

Order 15616.

1226 conjugacy classes of subgroups; only the 172 normal subgroups are shown.

normal subgroups only

This group has 1226 conjugacy classes of subgroups — too many to draw. Only its 172 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×C4×C22⋊#6923C212yesmaximal
2#21460699C214yesmaximal, complemented
3#407×D61C211yesmaximal, characteristic
4C22×C22⋊#6923—22no—
5C22×C4×#6923—22no—
6C4×C22⋊#6923—22no—
7C4×C22⋊#6923—22no—
8C4×C22⋊#6923C414yescomplemented
9C4×C22⋊#6923C2214yes—
10C2×(C2×#6923)⋊C4C414yescomplemented
11C23×C4×D61C2211yescharacteristic
12C2×M16×D61C412yescomplemented
13#16839726C411yescharacteristic, complemented
14#16839725C2212yes—
15#16839724—22no—
16C22⋊C8×D61—22no—
17C61×#407C411yescharacteristic, complemented
18C23×#6923—24no—
19C2×C22⋊#6923—216no—
20C2×C22⋊#6923—24no—
21C2×C22⋊#6923—24no—
22C2×C4×#6923—24no—
23C2×C4×#6923—24no—
24C2×C4×#6923—48no—
25(C2×#6923)⋊C4—24no—
26(C2×#6923)⋊C4C2×C418yescomplemented
27C24×D61D411yescharacteristic
28C23×Dic61Q811yescharacteristic
29C22×C4×D61—44no—
30C22×C4×D61C2×C411yescharacteristic
31C22×C4×D61C2×C412yes—
32C22×C4×D61C2×C412yes—
33C22×C4×D61D412yes—
34C22×C4×D61D411yescharacteristic
35C22×C4×D61D412yes—
36C22×C4×D61Q811yescharacteristic
37C22×C4×D61C2311yescharacteristic
38C2×#11758194C2×C412yescomplemented
39C2×#11758189—22no—
40M16×D61—44no—
41C2×C8×D61—22no—
42C22⋊#92418—22no—
43#14519689—24no—
44C23×C4×C61C2×C411yescharacteristic
45C2×M16×C61C2×C412yescomplemented
46C22⋊C8×C61—22no—
47C24×C61C4⋊C411yescharacteristic
48C23×D61—28no—
49C23×D61—42no—
50C23×D61C22⋊C412yes—
51C23×D61C22⋊C412yes—
52C23×D61C4⋊C411yescharacteristic
53C23×D61C2×D411yescharacteristic
54C22×#6923—216no—
55C22×#6923—28no—
56C22×#6923—48no—
57C22×#6923—48no—
58C22×C4×C61—42no—
59C22×C4×C61C4212yes—
60C22×C4×C61C22⋊C412yes—
61C22×C4×C61C4⋊C411yescharacteristic
62C22×C4×C61C22×C411yescharacteristic
63C22×Dic61—42no—
64C22×Dic61C4⋊C411yescharacteristic
65C22×Dic61C4⋊C412yes—
66C22×Dic61C4⋊C412yes—
67C22×Dic61C2×Q811yescharacteristic
68C2×C4×D61—24no—
69C2×C4×D61—28no—
70C2×C4×D61—44no—
71C2×C4×D61—42no—
72C2×C4×D61—44no—
73C2×C4×D61—42no—
74C2×C4×D61—44no—
75C2×C4×D61—42no—
76C2×C4×D61C4211yescharacteristic
77C2×C4×D61C4211yescharacteristic
78C2×C4×D61C22⋊C412yes—
79C2×C4×D61C22⋊C411yescharacteristic
80C2×C4×D61C22⋊C412yes—
81C2×C4×D61C22⋊C411yescharacteristic
82C2×C4×D61C4⋊C412yes—
83C2×C4×D61C4⋊C412yes—
84C2×C4×D61C22×C412yes—
85C2×C4×D61C2×D412yes—
86C22⋊#6923—216no—
87C22⋊#6923—216no—
88C4×#6923—48no—
89C4×#6923—48no—
90M16×C61—42no—
91C2×C8×C61—22no—
92#11758194—42no—
93C2×#92418—22no—
94#11758189—44no—
95C8×D61—44no—
96C23×C61—24no—
97C23×C61—41no—
98C23×C61#9512yes—
99C23×C61C2×C4⋊C411yescharacteristic
100C22×D61—24no—
101C22×D61—24no—
102C22×D61—24no—
103C22×D61—24no—
104C22×D61—22no—
105C22×D61—24no—
106C22×D61—24no—
107C22×D61—42no—
108C22×D61—41no—
109C22×D61—416no—
110C22×D61#9511yescharacteristic
111C22×D61#9511yescharacteristic
112C22×D61#10418yes—
113C22×D61C2×C22⋊C412yes—
114C2×#6923—216no—
115C2×#6923—48no—
116C2×#6923—432no—
117C2×#6923—48no—
118C2×C4×C61—22no—
119C2×C4×C61—24no—
120C2×C4×C61—42no—
121C2×C4×C61—41no—
122C2×C4×C61#9512yes—
123C2×C4×C61C2×C4211yescharacteristic
124C2×C4×C61C2×C22⋊C411yescharacteristic
125C2×Dic61—22no—
126C2×Dic61—24no—
127C2×Dic61—42no—
128C2×Dic61—41no—
129C2×Dic61#9511yescharacteristic
130C2×Dic61#9511yescharacteristic
131C2×Dic61C2×C4⋊C412yes—
132C4×D61—22no—
133C4×D61—24no—
134C4×D61—22no—
135C4×D61—24no—
136C4×D61—24no—
137C4×D61—22no—
138C4×D61—42no—
139C4×D61—42no—
140C4×D61—42no—
141C4×D61#9914yes—
142C8×C61—42no—
143#92418—42no—
144C2×#1297—611nomaximal, Sylow
145C22×C61—22no—
146C22×C61—22no—
147C22×C61—22no—
148C22×C61—21no—
149C22×C61—44no—
150C22×C61#33914yes—
151C22×C61C2×#9511yescharacteristic
152C2×D61—22no—
153C2×D61—22no—
154C2×D61—216no—
155C2×D61—22no—
156C2×D61—21no—
157C2×D61—44no—
158C2×D61—48no—
159C2×D61—44no—
160C2×D61#33914yes—
161C2×D61#34212yes—
162C2×D61C2×#10414yes—
163#6923—416no—
164C4×C61—22no—
165C4×C61—21no—
166C4×C61C2×#9912yes—
167Dic61—21no—
168Dic61—22no—
169Dic61#34212yes—
170C2×#407—611no—
171C2×C4×C22⋊C4—612no—
172#1297—614no—
173C2×C61—24no—
174C2×C61—41no—
175C2×C61C2×#33912yes—
176C2×C61C2×#34211yescharacteristic
177D61—24no—
178D61—41no—
179D61#129714yescomplemented
180C24×C4—611no—
181C22×M16—612no—
182C22×C22⋊C4—1222no—
183C22×C42—1222no—
184#407—611no—
185#407—612no—
186#407—1222no—
187#407#692311yescharacteristic, complemented
188C2×C22⋊C8—1222no—
189C4×C22⋊C4—614no—
190C4×C22⋊C4—614no—
191C4×C22⋊C4—1222no—
192C4×C22⋊C4—1222no—
193C2×#95—614no—
194C61C2×#129711yescharacteristic, complemented, Sylow
195C25—611no—
196C23×C4—611no—
197C23×C4—612no—
198C23×C4—611no—
199C23×C4—612no—
200C23×C4—612no—
201C23×C4—611no—
202C23×C4—611no—
203C23×C4—612no—
204C23×C4—611no—
205C23×C4—1224no—
206C23×C4—2444no—
207C23×C4C2×#692311yescharacteristic
208C2×M16—612no—
209C2×M16—1222no—
210C2×M16—2444no—
211C2×M16C2×#692312yescomplemented
212C22×C8—1222no—
213C2×C22⋊C4—1224no—
214C2×C22⋊C4—1224no—
215C2×C22⋊C4—12216no—
216C2×C42—1224no—
217C2×C42—1224no—
218C2×C42—2448no—
219C22⋊C8—22no—
220C22⋊C8—1224no—
221C22⋊C8—1222no—
222#95—618no—
223#95—1224no—
224C24—612no—
225C24—611no—
226C24—612no—
227C24—611no—
228C24—1228no—
229C24—2442no—
230C24C4⋊#692311yescharacteristic
231C22×C4—42no—
232C22×C4—611no—
233C22×C4—612no—
234C22×C4—611no—
235C22×C4—612no—
236C22×C4—611no—
237C22×C4—612no—
238C22×C4—612no—
239C22×C4—611no—
240C22×C4—611no—
241C22×C4—612no—
242C22×C4—611no—
243C22×C4—612no—
244C22×C4—612no—
245C22×C4—612no—
246C22×C4—1228no—
247C22×C4—1224no—
248C22×C4—1228no—
249C22×C4—12216no—
250C22×C4—2442no—
251C22×C4—2444no—
252C22×C4—2444no—
253C22×C4—2442no—
254C22×C4—2444no—
255C22×C4—2442no—
256C22×C4—2442no—
257C22×C4—2448no—
258C22×C4—2448no—
259C22×C4C4×#692312yes—
260C22×C4C4⋊#692311yescharacteristic
261C22×C4C22⋊#692312yes—
262C22×C4C22×#692311yescharacteristic
263M16—42no—
264M16—2444no—
265M16—2442no—
266C2×C8—22no—
267C2×C8—1222no—
268C2×C8—2444no—
269C22⋊C4—12216no—
270C22⋊C4—12216no—
271C42—2448no—
272C42—2448no—
273C23—24no—
274C23—41no—
275C23—611no—
276C23—618no—
277C23—612no—
278C23—611no—
279C23—1224no—
280C23—1224no—
281C23—1224no—
282C23—1222no—
283C23—1224no—
284C23—1224no—
285C23—1224no—
286C23—2442no—
287C23—2441no—
288C23—24416no—
289C23(C2×#6923)⋊C412yes—
290C23C2×C4⋊#692311yescharacteristic
291C2×C4—22no—
292C2×C4—24no—
293C2×C4—42no—
294C2×C4—41no—
295C2×C4—611no—
296C2×C4—614no—
297C2×C4—611no—
298C2×C4—612no—
299C2×C4—1224no—
300C2×C4—1222no—
301C2×C4—1224no—
302C2×C4—1222no—
303C2×C4—1224no—
304C2×C4—1222no—
305C2×C4—1224no—
306C2×C4—1222no—
307C2×C4—12216no—
308C2×C4—2442no—
309C2×C4—2442no—
310C2×C4—2442no—
311C2×C4—2442no—
312C2×C4—2441no—
313C2×C4—2448no—
314C2×C4—2448no—
315C2×C4—24432no—
316C2×C4(C2×#6923)⋊C412yes—
317C2×C4C2×C4×#692311yescharacteristic
318C2×C4C2×C22⋊#692311yescharacteristic
319C8—42no—
320C8—2442no—
321C22—22no—
322C22—22no—
323C22—21no—
324C22—22no—
325C22—44no—
326C22—612no—
327C22—614no—
328C22—614no—
329C22—12216no—
330C22—1222no—
331C22—1222no—
332C22—1221no—
333C22—1222no—
334C22—2448no—
335C22—2444no—
336C22—2444no—
337C22#1684063414yes—
338C22C2×(C2×#6923)⋊C411yescharacteristic
339C4—22no—
340C4—21no—
341C4—612no—
342C4—1221no—
343C4—1222no—
344C4—24416no—
345C4D61⋊#9912yes—
346C2—24no—
347C2—41no—
348C2—614no—
349C2—1224no—
350C2—2441no—
351C2C2×#1684063412yes—
352C2#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
hypocenterC61C2×#1297
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
8. C4×C22⋊#6923C4semidirect
10. C2×(C2×#6923)⋊C4C4semidirect
12. C2×M16×D61C4semidirect
13. #16839726C4semidirect
17. C61×#407C4semidirect
26. (C2×#6923)⋊C4C2×C4semidirect
38. C2×#11758194C2×C4semidirect
45. C2×M16×C61C2×C4semidirect
179. D61#1297semidirect
187. #407#6923semidirect
194. C61C2×#1297semidirect
211. C2×M16C2×#6923semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C22⋊#69232
#214606992
#407×D612
C2×#129761