Finite Group Databasebeta

← #30606832

Subgroups — #30606832

Order 15616.

990 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 990 conjugacy classes of subgroups — too many to draw. Only its 100 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×#16840669C214yesmaximal, complemented
2#21461648C212yesmaximal
3C2×#16840338C211yesmaximal, characteristic
4C2×#14519836C2214yes—
5C2×#14519835—22no—
6#16840705—22no—
7#16840669—28no—
8C22×#11758222C2211yescharacteristic
9C23×Dic122C411yescharacteristic, complemented
10#16840338—28no—
11C2×#14519777C412yescomplemented
12C2×#14519765C2212yes—
13C2×#14519761—22no—
14C2×#14519757—22no—
15C2×#117×C61C411yescharacteristic, complemented
16C2×C22⋊#6923—24no—
17#14519836—28no—
18#14519835—44no—
19(C2×#6923)⋊C4—44no—
20C2×#11758222—24no—
21C2×#11758222—42no—
22C2×#11758222—42no—
23C22×Dic122—21no—
24C22×Dic122—24no—
25C22×Dic122—42no—
26C22×Dic122—42no—
27C22×Dic122C2×C411yescharacteristic
28C22×Dic122C2×C412yes—
29C22×D244C2311yescharacteristic
30C22×C22⋊D61D412yes—
31C23×Dic61—21no—
32C22×C4×D61D412yes—
33#14519777—24no—
34#14519777—24no—
35#14519765—24no—
36#14519765D418yes—
37#14519761—216no—
38#14519757—216no—
39C2×C4⋊Dic61—24no—
40C2×#11758198—24no—
41C23×C4×C61C2×C411yescharacteristic
42#117×C61—24no—
43C2×C4⋊C4×C61—21no—
44C2×C22⋊C4×C61—21no—
45C2×C42×C61C2×C412yes—
46C24×C61C22⋊C411yescharacteristic
47C23×D61C2×D412yes—
48C22×#6923—44no—
49C22×C4×C61—22no—
50C22×C4×C61—22no—
51C22×C4×C61—22no—
52C22×C4×C61—22no—
53C22×C4×C61C22⋊C411yescharacteristic
54C22×C4×C61C22×C411yescharacteristic
55C2×C22⋊D61—24no—
56C2×C22⋊D61—24no—
57C2×C22⋊D61—42no—
58C22×Dic61—24no—
59C22×Dic61—22no—
60C22×Dic61—42no—
61C22×Dic61—42no—
62C22×Dic61C22⋊C412yes—
63#11758222—416no—
64C2×D244—41no—
65C2×D244—41no—
66C2×D244C2×D414yes—
67C2×C4×D61—24no—
68C2×C4×D61—24no—
69C2×C4×D61—42no—
70C2×Dic122—24no—
71C2×Dic122—22no—
72C2×Dic122—22no—
73C2×Dic122—41no—
74C2×Dic122—41no—
75C2×Dic122—42no—
76C2×Dic122—416no—
77C2×Dic122—42no—
78C2×Dic122—42no—
79C2×Dic122—42no—
80C2×Dic122C22⋊C414yes—
81C22⋊#6923—44no—
82C22⋊#6923—44no—
83C4⋊C4×C61—24no—
84C22⋊C4×C61—24no—
85C42×C61—22no—
86C42×C61C22⋊C414yes—
87#11758198—216no—
88C4⋊Dic61—216no—
89C23×C61—21no—
90C23×C61—21no—
91C23×C61—42no—
92C23×C61#9912yes—
93C23×C61C2×C22⋊C411yescharacteristic
94C22×D61—22no—
95C22×D61—22no—
96C22×D61—44no—
97C22×D61#12014yes—
98C2×#6923—84no—
99C2×#6923—88no—
100C2×C4×C61—28no—
101C2×C4×C61—21no—
102C2×C4×C61—21no—
103C2×C4×C61—21no—
104C2×C4×C61—21no—
105C2×C4×C61—24no—
106C2×C4×C61—44no—
107C2×C4×C61#9912yes—
108C2×C4×C61C2×C22⋊C412yes—
109C22⋊D61—48no—
110C2×Dic61—22no—
111C2×Dic61—24no—
112C2×Dic61—22no—
113C2×Dic61—22no—
114C2×Dic61—48no—
115C2×Dic61—42no—
116C2×Dic61—42no—
117D244—44no—
118C4×D61—48no—
119Dic122—44no—
120Dic122—48no—
121Dic122—44no—
122Dic122—44no—
123#12698—611nomaximal, Sylow
124C22×C61—22no—
125C22×C61—22no—
126C22×C61—42no—
127C22×C61—42no—
128C22×C61—42no—
129C22×C61#37911yescharacteristic
130C22×C61C2×#9911yescharacteristic
131C22×C61C2×#9911yescharacteristic
132C2×D61—24no—
133C2×D61—44no—
134C2×D61—44no—
135C2×D61—82no—
136#6923—164no—
137C4×C61—22no—
138C4×C61—22no—
139C4×C61—44no—
140Dic61—44no—
141Dic61—42no—
142C2×#536—611no—
143C2×#354—614no—
144#1782—612no—
145C2×C61—21no—
146C2×C61—21no—
147C2×C61—42no—
148C2×C61#178111yescharacteristic
149C2×C61#212912yes—
150D61—44no—
151C22×#41—611no—
152C23×Q8—611no—
153#536—1228no—
154C2×C4⋊Q8—612no—
155C2×#124—612no—
156C2×#123—1222no—
157C2×C22⋊Q8—1222no—
158C2×#117#692311yescharacteristic, complemented
159C2×#99—614no—
160C2×#99—1222no—
161#381—1222no—
162#354—1228no—
163C61#1269811yescharacteristic, complemented, Sylow
164C2×#41—1224no—
165C2×#41—2442no—
166C2×#41—2442no—
167C22×Q8—611no—
168C22×Q8—612no—
169C22×Q8—1224no—
170C22×Q8—1221no—
171C22×Q8—2442no—
172C22×Q8—2442no—
173C22×D4—612no—
174C22×D4—611no—
175C23×C4—612no—
176C23×C4—1221no—
177C23×C4C2×#692311yescharacteristic
178C4⋊Q8—1224no—
179C4⋊Q8—1224no—
180#124—618no—
181#124—1224no—
182#123—12216no—
183C22⋊Q8—12216no—
184#117—24no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—1224no—
187C2×C22⋊C4—21no—
188C2×C22⋊C4—1224no—
189C2×C22⋊C4—1224no—
190C2×C42C2×#692312yes—
191#99—1228no—
192#99—2444no—
193#95—2444no—
194C24—612no—
195C24C22⋊#692311yescharacteristic
196#41—24416no—
197C2×Q8—614no—
198C2×Q8—1224no—
199C2×Q8—1222no—
200C2×Q8—1222no—
201C2×Q8—2441no—
202C2×Q8—2441no—
203C2×Q8—24416no—
204C2×Q8—2442no—
205C2×Q8—2442no—
206C2×Q8—2442no—
207C2×Q8—2442no—
208C2×D4—614no—
209C2×D4—1224no—
210C2×D4—1224no—
211C2×D4—2441no—
212C2×D4—2441no—
213C2×D4—2442no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—612no—
219C22×C4—1224no—
220C22×C4—1224no—
221C22×C4—1222no—
222C22×C4—1224no—
223C22×C4—2442no—
224C22×C4—2444no—
225C22×C4—2442no—
226C22×C4—2442no—
227C22×C4C22⋊#692311yescharacteristic
228C22×C4C22×#692311yescharacteristic
229C4⋊C4—24no—
230C4⋊C4—12216no—
231C22⋊C4—24no—
232C22⋊C4—12216no—
233C22⋊C4—2444no—
234C22⋊C4—2444no—
235C42—22no—
236C42C22⋊#692314yes—
237C23—21no—
238C23—21no—
239C23—42no—
240C23—614no—
241C23—1222no—
242C23—1222no—
243C23—2444no—
244C23#1451983612yes—
245C23C2×C22⋊#692311yescharacteristic
246Q8—2444no—
247Q8—2448no—
248Q8—2444no—
249Q8—2444no—
250D4—2444no—
251D4—2448no—
252C2×C4—28no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—24no—
258C2×C4—44no—
259C2×C4—1222no—
260C2×C4—1222no—
261C2×C4—1224no—
262C2×C4—1222no—
263C2×C4—4884no—
264C2×C4—4888no—
265C2×C4—2448no—
266C2×C4—2448no—
267C2×C4—2442no—
268C2×C4—2442no—
269C2×C4#1451983612yes—
270C2×C4C2×C22⋊#692312yes—
271C22—22no—
272C22—22no—
273C22—42no—
274C22—42no—
275C22—42no—
276C22—1224no—
277C22—4882no—
278C22—2444no—
279C22—2444no—
280C22#1684070311yescharacteristic
281C22C2×#1451983611yescharacteristic
282C22C2×#1451983611yescharacteristic
283C4—22no—
284C4—22no—
285C4—44no—
286C4—9764no—
287C4—2442no—
288C4—2444no—
289C2—21no—
290C2—21no—
291C2—42no—
292C2—2444no—
293C2#2146164711yescharacteristic
294C2#2146210312yes—
subgroupG/H
centerC22C2×#14519836
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC2×#117×C61C4
socleC22×C61C2×#99
prime radicalC22×#11758222C22
cosocleC22×D244C23
hypercenterC2×#117#6923
hypocenterC61#12698
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
1. C2×#16840669C2semidirect
9. C23×Dic122C4semidirect
11. C2×#14519777C4semidirect
15. C2×#117×C61C4semidirect
158. C2×#117#6923semidirect
163. C61#12698semidirect

Maximal subgroups

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

subgroupindex
C2×#168406692
#214616482
C2×#168403382
#1269861