Finite Group Databasebeta

← #30602363

Subgroups — #30602363

Order 15616.

1374 conjugacy classes of subgroups; only the 220 normal subgroups are shown.

normal subgroups only

This group has 1374 conjugacy classes of subgroups — too many to draw. Only its 220 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
2C2×#16840630C214yesmaximal, complemented
3C23×#11758189C211yesmaximal, characteristic
4C2×#14519856C2214yes—
5#16840705—22no—
6#16840704—24no—
7#16840703—22no—
8C4×C22⋊#6923C418yescomplemented
9#16840630—28no—
10C23×C4×D61C2211yescharacteristic
11C22×#11758189—28no—
12C22×#11758189—24no—
13C22×#11758189C412yescomplemented
14C22×#11758189C2212yes—
15C23×#92418C411yescharacteristic, complemented
16C23×C8×C61C411yescharacteristic, complemented
17#14519856—28no—
18#14519856—28no—
19C2×C4⋊#6923—24no—
20C2×C22⋊#6923—28no—
21C2×C22⋊#6923—24no—
22C2×C4×#6923C2×C418yes—
23(C2×#6923)⋊C4—28no—
24C24×D61D411yescharacteristic
25C23×Dic61Q811yescharacteristic
26C22×C4×D61—24no—
27C22×C4×D61—24no—
28C22×C4×D61C2×C411yescharacteristic
29C22×C4×D61C2×C414yes—
30C22×C4×D61D411yescharacteristic
31C22×C4×D61D414yes—
32C22×C4×D61Q811yescharacteristic
33C22×C4×D61C2311yescharacteristic
34C2×#11758189—232no—
35C2×#11758189—24no—
36C2×#11758189—24no—
37C2×#11758189—24no—
38C2×#11758189—24no—
39C2×#11758189—24no—
40C2×#11758189D418yes—
41C22×#92418—22no—
42C22×#92418—24no—
43C22×#92418C2×C412yes—
44C23×C4×C61C2×C411yescharacteristic
45C22×C8×C61—22no—
46C22×C8×C61—24no—
47C22×C8×C61C2×C412yes—
48C24×C61C4⋊C411yescharacteristic
49C23×D61—24no—
50C23×D61—44no—
51C23×D61C22⋊C414yes—
52C23×D61C4⋊C411yescharacteristic
53C23×D61C2×D411yescharacteristic
54C22×#6923—28no—
55C22×C4×C61—22no—
56C22×C4×C61—22no—
57C22×C4×C61C4212yes—
58C22×C4×C61C22⋊C412yes—
59C22×C4×C61C4⋊C411yescharacteristic
60C22×C4×C61C22×C411yescharacteristic
61C22×Dic61—24no—
62C22×Dic61C4⋊C411yescharacteristic
63C22×Dic61C4⋊C414yes—
64C22×Dic61C2×Q811yescharacteristic
65C2×C4×D61—24no—
66C2×C4×D61—22no—
67C2×C4×D61—22no—
68C2×C4×D61—24no—
69C2×C4×D61—216no—
70C2×C4×D61—28no—
71C2×C4×D61—28no—
72C2×C4×D61C4212yes—
73C2×C4×D61C22⋊C414yes—
74C2×C4×D61C22⋊C412yes—
75C2×C4×D61C4⋊C414yes—
76C2×C4×D61C22×C412yes—
77C2×C4×D61C2×D414yes—
78C2×C4×D61C2×D412yes—
79C2×C4×D61#4114yes—
80C22⋊#6923—416no—
81C22⋊#6923—48no—
82C4⋊#6923—44no—
83C4⋊#6923—44no—
84C4×#6923—216no—
85C2×C8×C61—28no—
86C2×C8×C61—22no—
87C2×C8×C61—22no—
88C2×C8×C61C22⋊C414yescomplemented
89C2×#92418—28no—
90C2×#92418—22no—
91C2×#92418—22no—
92C2×#92418C22⋊C414yescomplemented
93#11758189—216no—
94#11758189—216no—
95C23×C61—24no—
96C23×C61—22no—
97C23×C61#9512yes—
98C23×C61C2×C4⋊C411yescharacteristic
99C22×D61—28no—
100C22×D61—22no—
101C22×D61—28no—
102C22×D61—44no—
103C22×D61—42no—
104C22×D61—42no—
105C22×D61—416no—
106C22×D61#9512yes—
107C22×D61C2×C22⋊C412yes—
108C22×D61#12114yes—
109C2×#6923—416no—
110C2×#6923—48no—
111C2×C4×C61—22no—
112C2×C4×C61—24no—
113C2×C4×C61—22no—
114C2×C4×C61—22no—
115C2×C4×C61#9512yes—
116C2×C4×C61C2×C4211yescharacteristic
117C2×C4×C61C2×C22⋊C412yes—
118C2×C4×C61C2×C22⋊C411yescharacteristic
119C2×C4×C61#11712yes—
120C2×Dic61—22no—
121C2×Dic61—28no—
122C2×Dic61#9512yes—
123C2×Dic61C2×C4⋊C412yes—
124C2×Dic61C22⋊Q814yes—
125C4×D61—28no—
126C4×D61—24no—
127C4×D61—22no—
128C4×D61—22no—
129C4×D61—28no—
130C4×D61C4×D418yes—
131C4×D61#12014yes—
132C4×D61#12314yes—
133C8×C61—24no—
134#92418—24no—
135#10717—611nomaximal, Sylow
136C22×C61—28no—
137C22×C61—22no—
138C22×C61—22no—
139C22×C61—22no—
140C22×C61H(3,ℤ/4)14yescomplemented
141C22×C61C2×#9511yescharacteristic
142C22×C61#38012yes—
143C2×D61—24no—
144C2×D61—22no—
145C2×D61—24no—
146C2×D61—44no—
147C2×D61—416no—
148C2×D61—44no—
149C2×D61#38614yes—
150#6923—88no—
151C4×C61—22no—
152C4×C61C4×C22⋊C412yes—
153C4×C61#37911yescharacteristic
154C4×C61#38111yescharacteristic
155Dic61—22no—
156Dic61#38514yes—
157C23×M16—611no—
158#2280—612no—
159C2×H(3,ℤ/4)—614no—
160C2×C61—24no—
161C2×C61—22no—
162C2×C61C22⋊#9511yescharacteristic
163C2×C61C2×H(3,ℤ/4)12yes—
164D61—24no—
165D61—42no—
166C24×C4—611no—
167C22×M16—612no—
168C22×M16—612no—
169C22×M16—1228no—
170C22×M16—1224no—
171C23×C8—611no—
172C23×C8#692311yescharacteristic, complemented
173C2×#117—614no—
174#381—1222no—
175#380—1224no—
176#379—1222no—
177C4×C22⋊C4—618no—
178H(3,ℤ/4)—1228no—
179C61#1071711yescharacteristic, complemented, Sylow
180C25—611no—
181C23×C4—611no—
182C23×C4—611no—
183C23×C4—614no—
184C23×C4—611no—
185C23×C4—611no—
186C23×C4—614no—
187C23×C4—611no—
188C23×C4—1224no—
189C23×C4—1224no—
190C23×C4C2×#692311yescharacteristic
191C2×M16—618no—
192C2×M16—1224no—
193C2×M16—1224no—
194C2×M16—1224no—
195C2×M16—1224no—
196C2×M16—12232no—
197C2×M16—1224no—
198C22×C8—22no—
199C22×C8—24no—
200C22×C8—612no—
201C22×C8—1224no—
202C22×C8—1222no—
203C22×C8C2×#692312yes—
204#117—1228no—
205#117—1228no—
206C2×C4⋊C4—1224no—
207C2×C22⋊C4—1224no—
208C2×C22⋊C4—1228no—
209C2×C42—618no—
210#95—1228no—
211C24—611no—
212C24—614no—
213C24—611no—
214C24—1224no—
215C24—2444no—
216C24C4⋊#692311yescharacteristic
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—611no—
220C22×C4—612no—
221C22×C4—612no—
222C22×C4—614no—
223C22×C4—614no—
224C22×C4—614no—
225C22×C4—611no—
226C22×C4—614no—
227C22×C4—614no—
228C22×C4—612no—
229C22×C4—612no—
230C22×C4—1228no—
231C22×C4—1224no—
232C22×C4—1228no—
233C22×C4—12216no—
234C22×C4—1224no—
235C22×C4—1224no—
236C22×C4—1222no—
237C22×C4—1222no—
238C22×C4—1228no—
239C22×C4C4×#692312yes—
240C22×C4C4⋊#692311yescharacteristic
241C22×C4C22⋊#692312yes—
242C22×C4C22×#692311yescharacteristic
243M16—12216no—
244M16—12216no—
245C2×C8—28no—
246C2×C8—22no—
247C2×C8—22no—
248C2×C8—614no—
249C2×C8—1228no—
250C2×C8—1222no—
251C2×C8—1222no—
252C2×C8C22⋊#692314yescomplemented
253C4⋊C4—2444no—
254C4⋊C4—2444no—
255C22⋊C4—2448no—
256C22⋊C4—24416no—
257C42—12216no—
258C23—22no—
259C23—24no—
260C23—612no—
261C23—614no—
262C23—612no—
263C23—1228no—
264C23—1222no—
265C23—1228no—
266C23—2444no—
267C23—2442no—
268C23—2442no—
269C23—24416no—
270C23(C2×#6923)⋊C412yes—
271C23C2×C4⋊#692311yescharacteristic
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—24no—
275C2×C4—22no—
276C2×C4—614no—
277C2×C4—612no—
278C2×C4—614no—
279C2×C4—618no—
280C2×C4—614no—
281C2×C4—612no—
282C2×C4—1228no—
283C2×C4—1228no—
284C2×C4—1228no—
285C2×C4—1222no—
286C2×C4—1222no—
287C2×C4—1224no—
288C2×C4—1222no—
289C2×C4—2448no—
290C2×C4—24416no—
291C2×C4(C2×#6923)⋊C412yes—
292C2×C4C2×C4×#692311yescharacteristic
293C2×C4C2×C22⋊#692311yescharacteristic
294C2×C4C2×C22⋊#692312yes—
295C2×C4#1451985612yes—
296C8—24no—
297C8—1224no—
298C22—28no—
299C22—22no—
300C22—22no—
301C22—22no—
302C22—614no—
303C22—1224no—
304C22—1222no—
305C22—1224no—
306C22—2444no—
307C22—24416no—
308C22—2444no—
309C22#1684063014yescomplemented
310C22C2×(C2×#6923)⋊C411yescharacteristic
311C22#1684070412yes—
312C4—22no—
313C4—614no—
314C4—1222no—
315C4—4888no—
316C4C4×C22⋊#692312yes—
317C4#1684070311yescharacteristic
318C4#1684070511yescharacteristic
319C2—22no—
320C2—24no—
321C2—1224no—
322C2—2442no—
323C2#2146099311yescharacteristic
324C2C2×#1684063012yes—
subgroupG/H
centerC2×C4C2×C4×#6923
commutatorC2×C4×C61C2×C42
FrattiniC22×C4C22×#6923
FittingC23×C8×C61C4
socleC22×C61C2×#95
prime radicalC24×D61D4
cosocleC22×C4×D61C23
hypercenterC23×C8#6923
hypocenterC61#10717
LCSG ▷ C2×C4×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C23×C8

Complements

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

NH ≅ G/Ntype
2. C2×#16840630C2semidirect
8. C4×C22⋊#6923C4semidirect
13. C22×#11758189C4semidirect
15. C23×#92418C4semidirect
16. C23×C8×C61C4semidirect
88. C2×C8×C61C22⋊C4semidirect
92. C2×#92418C22⋊C4semidirect
140. C22×C61H(3,ℤ/4)semidirect
172. C23×C8#6923semidirect
179. C61#10717semidirect
252. C2×C8C22⋊#6923semidirect
309. C22#16840630semidirect

Maximal subgroups

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

subgroupindex
#214622632
C2×#168406302
C23×#117581892
#1071761