Finite Group Databasebeta

← #30595844

Subgroups — #30595844

Order 15616.

500 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 94 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#21461711C211yesmaximal, characteristic
2#21461546C211yesmaximal, characteristic
3#21460637C214yesmaximal, complemented
4#21454172C211yesmaximal, characteristic
5C2×#14519834—21no—
6C2×#14519834—21no—
7C2×#14519832—21no—
8#16840628—21no—
9#16840628C2212yes—
10#16840628C2212yes—
11C2×#14519777C411yescharacteristic
12C2×#14519757—21no—
13C2×C42×D61C2211yescharacteristic
14#16839667—21no—
15#16839661—21no—
16#16839661C2212yes—
17#16839619—21no—
18#16839619C412yes—
19C61×#384C411yescharacteristic
20C22×#92420—22no—
21C22×#92420—22no—
22#14519834—24no—
23#14519834—42no—
24#14519832—42no—
25C22×Dic122—21no—
26C22×C4×D61C2×C411yescharacteristic
27C22×C4×D61D411yescharacteristic
28C22×C4×D61D411yescharacteristic
29#14519777—42no—
30#14519757—44no—
31C2×C4⋊Dic61—21no—
32C2×C4⋊Dic61—22no—
33C2×C4⋊Dic61C2×C412yes—
34C2×#11758198—22no—
35C2×C4×Dic61D411yescharacteristic
36C2×C4×Dic61D411yescharacteristic
37C2×C4×Dic61C2311yescharacteristic
38C42×D61—42no—
39#14519684—22no—
40#14519684—22no—
41C2×C4⋊C4×C61—21no—
42C2×C4⋊C4×C61—21no—
43C2×C4⋊C4×C61C2×C412yes—
44C2×C42×C61C2×C411yescharacteristic
45C23×D61C22⋊C411yescharacteristic
46C22×C4×C61—22no—
47C22×C4×C61C22⋊C411yescharacteristic
48C22×C4×C61C22⋊C411yescharacteristic
49C22×C4×C61C22×C411yescharacteristic
50C22×Dic61—22no—
51C22×Dic61C22⋊C411yescharacteristic
52C22×Dic61C2×D411yescharacteristic
53C22×Dic61C2×D411yescharacteristic
54C2×C4×D61—22no—
55C2×C4×D61—22no—
56C2×C4×D61—22no—
57C2×C4×D61—22no—
58C2×C4×D61—22no—
59C2×C4×D61—22no—
60C2×C4×D61—41no—
61C2×C4×D61—41no—
62C2×C4×D61—41no—
63C2×Dic122—42no—
64C2×Dic122—42no—
65C2×Dic122—42no—
66C2×#92420—24no—
67C2×#92420—42no—
68C2×#92420—42no—
69C2×#92420—42no—
70C2×#92420—42no—
71C2×#92420—42no—
72C4⋊C4×C61—24no—
73C4⋊C4×C61—42no—
74C4⋊C4×C61—42no—
75C42×C61—22no—
76#11758198—44no—
77C4⋊Dic61—42no—
78C4⋊Dic61—44no—
79C4⋊Dic61—42no—
80C4×Dic61—22no—
81C4×Dic61—41no—
82C4×Dic61—41no—
83C23×C61C2×C22⋊C411yescharacteristic
84C22×D61—21no—
85C22×D61—21no—
86C22×D61—21no—
87C22×D61—21no—
88C22×D61—21no—
89C22×D61—21no—
90C22×D61#10112yes—
91C2×C4×C61—21no—
92C2×C4×C61—21no—
93C2×C4×C61—21no—
94C2×C4×C61—21no—
95C2×C4×C61—21no—
96C2×C4×C61—21no—
97C2×C4×C61—42no—
98C2×C4×C61—42no—
99C2×C4×C61—42no—
100C2×C4×C61#9912yes—
101C2×C4×C61#10412yes—
102C2×C4×C61#13112yes—
103C2×Dic61—21no—
104C2×Dic61—21no—
105C2×Dic61—21no—
106C2×Dic61—21no—
107C2×Dic61—21no—
108C2×Dic61—21no—
109C2×Dic61—21no—
110C2×Dic61—42no—
111C2×Dic61—42no—
112C2×Dic61—42no—
113C2×Dic61AGL(1,ℤ/8)12yes—
114C2×Dic61#13712yes—
115C4×D61—44no—
116C4×D61—44no—
117C4×D61—44no—
118Dic122—84no—
119#92420—44no—
120#92420—82no—
121#7705—611nomaximal, Sylow
122C22×C61#40811yescharacteristic
123C22×C61C2×#9911yescharacteristic
124C22×C61C2×#10111yescharacteristic
125C22×C61#41811yescharacteristic
126C22×C61#41911yescharacteristic
127C22×C61C2×#10411yescharacteristic
128C22×C61#42111yescharacteristic
129C2×D61—22no—
130C2×D61—22no—
131C2×D61—22no—
132C2×D61—41no—
133C2×D61—41no—
134C2×D61—41no—
135C2×D61—41no—
136C4×C61—22no—
137C4×C61—22no—
138C4×C61—22no—
139C4×C61—82no—
140Dic61—22no—
141Dic61—22no—
142Dic61—41no—
143Dic61—82no—
144#2857—611no—
145#1817—611no—
146#1749—611no—
147#1271—614no—
148C2×C61#147211yescharacteristic
149C2×C61#148511yescharacteristic
150C2×C61#149611yescharacteristic
151C2×C61#150711yescharacteristic
152C2×C61#151111yescharacteristic
153C2×C61#153111yescharacteristic
154C2×C61#155611yescharacteristic
155D61—42no—
156C2×C4⋊Q8—611no—
157C2×C22⋊Q8—1221no—
158C22×C42—611no—
159C2×C22⋊C8—1221no—
160C2×C22⋊C8—1221no—
161C2×#97—1221no—
162#391—1221no—
163#389—612no—
164#389—1221no—
165#384#692311yescharacteristic
166#380—612no—
167#380—1221no—
168#336—612no—
169#336—612no—
170#336—1221no—
171C61#770511yescharacteristic, complemented, Sylow
172C22×Q8—1221no—
173C23×C4—611no—
174C23×C4—611no—
175C23×C4—611no—
176C22×C8—1222no—
177C22×C8—1222no—
178C4⋊Q8—2442no—
179C22⋊Q8—2444no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—612no—
183C2×C4⋊C4—1222no—
184C2×C4⋊C4—1221no—
185C2×C4⋊C4C2×#692312yes—
186C2×C22⋊C4—1222no—
187C2×C42—611no—
188C2×C42—611no—
189C2×C42—611no—
190C2×C42—2442no—
191C2×C42C2×#692311yescharacteristic
192C22⋊C8—1224no—
193C22⋊C8—2442no—
194#97—2442no—
195#95—1222no—
196#95—1222no—
197C24—611no—
198C2×Q8—2442no—
199C2×Q8—2442no—
200C2×Q8—2442no—
201C22×C4—22no—
202C22×C4—611no—
203C22×C4—611no—
204C22×C4—611no—
205C22×C4—1222no—
206C22×C4—1222no—
207C22×C4—1222no—
208C22×C4—1222no—
209C22×C4—1222no—
210C22×C4—1222no—
211C22×C4—1222no—
212C22×C4—2441no—
213C22×C4—2441no—
214C22×C4—2441no—
215C22×C4C22⋊#692311yescharacteristic
216C22×C4C22⋊#692311yescharacteristic
217C22×C4C22×#692311yescharacteristic
218C2×C8—1224no—
219C2×C8—2442no—
220C2×C8—2442no—
221C2×C8—2442no—
222C2×C8—2442no—
223C2×C8—2442no—
224C4⋊C4—24no—
225C4⋊C4—42no—
226C4⋊C4—42no—
227C4⋊C4—2442no—
228C4⋊C4—2444no—
229C4⋊C4—2442no—
230C22⋊C4—2444no—
231C42—22no—
232C42—1222no—
233C42—2441no—
234C42—2441no—
235C23—612no—
236C23—1221no—
237C23—1221no—
238C23—1221no—
239C23—1221no—
240C23—1221no—
241C23—1221no—
242C23C2×C22⋊#692311yescharacteristic
243Q8—4884no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—612no—
254C2×C4—612no—
255C2×C4—1221no—
256C2×C4—1221no—
257C2×C4—1221no—
258C2×C4—1221no—
259C2×C4—1221no—
260C2×C4—1221no—
261C2×C4—1221no—
262C2×C4—2444no—
263C2×C4—2444no—
264C2×C4—2444no—
265C2×C4—2442no—
266C2×C4—2442no—
267C2×C4—2442no—
268C2×C4#1451983612yes—
269C2×C4#1451984212yes—
270C2×C4#1451986412yes—
271C8—4882no—
272C8—2444no—
273C22—1222no—
274C22—1222no—
275C22—1222no—
276C22—2441no—
277C22—2441no—
278C22—2441no—
279C22—2441no—
280C22#1684073611yescharacteristic
281C22C2×#1451983611yescharacteristic
282C22D61⋊#10111yescharacteristic
283C22#1684075411yescharacteristic
284C22#1684075511yescharacteristic
285C22C2×#1451984211yescharacteristic
286C22#1684076011yescharacteristic
287C4—22no—
288C4—22no—
289C4—22no—
290C4—82no—
291C4—1222no—
292C4—1222no—
293C4—4882no—
294C4—2441no—
295C2—2442no—
296C2#2146109711yescharacteristic
297C2#2146114211yescharacteristic
298C2#2146117911yescharacteristic
299C2#2146121611yescharacteristic
300C2#2146122711yescharacteristic
301C2#2146129311yescharacteristic
302C2#2146134911yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#384C4
socleC23×C61C2×C22⋊C4
prime radicalC23×D61C22⋊C4
cosocleC2×C4×Dic61C23
hypercenter#384#6923
hypocenterC61#7705
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #384

Complements

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

NH ≅ G/Ntype
3. #21460637C2semidirect
171. C61#7705semidirect

Maximal subgroups

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

subgroupindex
#214617112
#214615462
#214606372
#214541722
#770561