Finite Group Databasebeta

← #29575736

Subgroups — #29575736

Order 15616.

694 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 694 conjugacy classes of subgroups — too many to draw. Only its 135 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#21454158C211yesmaximal, characteristic
2C2×#16839608C211yesmaximal, characteristic
3C2×#16839500C212yesmaximal
4#21443699C212yesmaximal
5C61×#1817C211yesmaximal, characteristic
6C2×#14519756C2211yescharacteristic
7C22×C4×Dic61—21no—
8#16839668C412yescomplemented
9#16839660C412yescomplemented
10#16839646—22no—
11#16839635—21no—
12#16839625—21no—
13#16839622C2211yescharacteristic
14#16839608—24no—
15C2×#14519684—21no—
16C2×#14519684C2212yes—
17#16839500—24no—
18C22×C42×C61C2211yescharacteristic
19C2×C22⋊C8×C61C2212yes—
20C61×#336—22no—
21C22×Dic122C2×C411yescharacteristic
22C23×Dic61—22no—
23#14519756—22no—
24#14519756—22no—
25C2×C4⋊Dic61C2×C411yescharacteristic
26C2×C4⋊Dic61C2×C412yes—
27C2×#11758196—22no—
28C2×#11758196—21no—
29C2×C22⋊Dic61—21no—
30C2×C22⋊Dic61—21no—
31C2×C4×Dic61—28no—
32C2×C4×Dic61—24no—
33#14519684—24no—
34#14519684—22no—
35#14519684—22no—
36#14519684—22no—
37#14519684—22no—
38#14519684—21no—
39#14519684D414yes—
40C23×C4×C61D411yescharacteristic
41C23×C4×C61D411yescharacteristic
42C23×C4×C61C2311yescharacteristic
43C22×C8×C61—22no—
44C2×C42×C61—22no—
45C2×C42×C61—22no—
46C2×C42×C61C2×C412yes—
47C2×C42×C61D412yes—
48C22⋊C8×C61—22no—
49C22⋊C8×C61—22no—
50C24×C61C2×D411yescharacteristic
51C22×C4×C61—21no—
52C22×C4×C61—21no—
53C22×C4×C61—24no—
54C22×C4×C61—21no—
55C22×C4×C61—21no—
56C22×C4×C61—22no—
57C22×C4×C61—22no—
58C22×C4×C61—42no—
59C22×C4×C61C22⋊C412yes—
60C22×C4×C61C22⋊C412yes—
61C22×C4×C61C22×C411yescharacteristic
62C22×C4×C61C2×D411yescharacteristic
63C22×C4×C61C2×D412yes—
64C22×C4×C61#4112yes—
65C22×Dic61—28no—
66C22×Dic61—24no—
67C22×Dic61—21no—
68C22×Dic61—22no—
69C22×Dic61—21no—
70C22×Dic61—44no—
71C22×Dic61—42no—
72C22×Dic61—42no—
73C2×Dic122—22no—
74C2×Dic122—42no—
75C2×C8×C61—44no—
76C2×C8×C61—42no—
77C42×C61—24no—
78C42×C61—24no—
79C22⋊Dic61—41no—
80C22⋊Dic61—41no—
81C22⋊Dic61—42no—
82C4⋊Dic61—22no—
83C4⋊Dic61—42no—
84#11758196—44no—
85#11758196—42no—
86C4×Dic61—216no—
87C23×C61—22no—
88C23×C61—22no—
89C23×C61C2×C22⋊C411yescharacteristic
90C23×C61#12011yescharacteristic
91C23×C61#12311yescharacteristic
92C23×C61#13512yes—
93C23×C61#13712yes—
94C2×C4×C61—24no—
95C2×C4×C61—24no—
96C2×C4×C61—22no—
97C2×C4×C61—22no—
98C2×C4×C61—22no—
99C2×C4×C61—22no—
100C2×C4×C61—22no—
101C2×C4×C61—22no—
102C2×C4×C61—22no—
103C2×C4×C61—42no—
104C2×C4×C61—42no—
105C2×C4×C61—42no—
106C2×C4×C61—42no—
107C2×C4×C61#10414yes—
108C2×C4×C61C4×D412yes—
109C2×C4×C61#12112yes—
110C2×Dic61—28no—
111C2×Dic61—44no—
112C2×Dic61—48no—
113C2×Dic61—48no—
114C2×Dic61—44no—
115C2×Dic61—41no—
116C2×Dic61—41no—
117C2×Dic61—42no—
118C2×Dic61—42no—
119C2×Dic61—42no—
120C2×Dic61—42no—
121C2×Dic61—42no—
122Dic122—82no—
123C8×C61—82no—
124#9145—611nomaximal, Sylow
125C22×C61—21no—
126C22×C61—21no—
127C22×C61—21no—
128C22×C61—21no—
129C22×C61—24no—
130C22×C61—24no—
131C22×C61#38611yescharacteristic
132C22×C61#41012yes—
133C22×C61#41611yescharacteristic
134C22×C61#41911yescharacteristic
135C22×C61C2×#10412yes—
136C22×C61#45212yes—
137C22×C61#48512yes—
138C4×C61—24no—
139C4×C61—22no—
140C4×C61—22no—
141C4×C61—42no—
142Dic61—48no—
143Dic61—81no—
144Dic61—81no—
145Dic61—82no—
146#2855—611no—
147C2×#376—611no—
148#1817D6111yescharacteristic
149C2×#328—612no—
150#1294—612no—
151C2×C61—22no—
152C2×C61—22no—
153C2×C61#150012yes—
154C2×C61#150612yes—
155C2×C61#187311yescharacteristic
156C2×C61#189611yescharacteristic
157C2×C61#189911yescharacteristic
158C2×C22⋊Q8—611no—
159C22×C42—1221no—
160C22×C42C2×D6111yescharacteristic
161C2×C22⋊C8C2×D6112yes—
162#391—612no—
163#389—612no—
164#387—1222no—
165#385—1221no—
166#381—1221no—
167#380—611no—
168#376—1224no—
169C2×#95—612no—
170C2×#95—1221no—
171#336—22no—
172#328—1224no—
173C61#914511yescharacteristic, complemented, Sylow
174C22×Q8—611no—
175C23×C4—1222no—
176C23×C4D24411yescharacteristic
177C23×C4D24411yescharacteristic
178C23×C4C22×D6111yescharacteristic
179C22×C8—22no—
180C22⋊Q8—1222no—
181C22⋊Q8—1222no—
182C2×C4⋊C4—611no—
183C2×C4⋊C4—612no—
184C2×C4⋊C4—1222no—
185C2×C4⋊C4—1221no—
186C2×C22⋊C4—1221no—
187C2×C22⋊C4—1221no—
188C2×C42—22no—
189C2×C42—22no—
190C2×C42—1224no—
191C2×C42—1228no—
192C2×C42C4×D6112yes—
193C2×C42C22⋊D6112yes—
194C22⋊C8—22no—
195C22⋊C8—22no—
196#95—614no—
197#95—1224no—
198#95—1222no—
199#95—1222no—
200#95—1222no—
201#95—1222no—
202#95—1221no—
203C24C2×D24411yescharacteristic
204C2×Q8—1222no—
205C2×Q8—2442no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—24no—
209C22×C4—22no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—22no—
213C22×C4—42no—
214C22×C4—1228no—
215C22×C4—1224no—
216C22×C4—1222no—
217C22×C4—1221no—
218C22×C4—1221no—
219C22×C4—2444no—
220C22×C4—2442no—
221C22×C4—2442no—
222C22×C4#1175819812yes—
223C22×C4#1175819812yes—
224C22×C4C2×C4×D6111yescharacteristic
225C22×C4D4×D6112yes—
226C22×C4#1175822412yes—
227C22×C4C2×C22⋊D6111yescharacteristic
228C2×C8—44no—
229C2×C8—42no—
230C4⋊C4—1222no—
231C4⋊C4—2442no—
232C4⋊C4—2444no—
233C4⋊C4—2442no—
234C22⋊C4—2441no—
235C22⋊C4—2441no—
236C22⋊C4—2442no—
237C42—24no—
238C42—24no—
239C42—12216no—
240C23—22no—
241C23—22no—
242C23C2×#1175819811yescharacteristic
243C23#1451974711yescharacteristic
244C23#1451976111yescharacteristic
245C23#1451979912yes—
246C23#1451981012yes—
247Q8—4882no—
248C2×C4—24no—
249C2×C4—24no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—1228no—
262C2×C4—2442no—
263C2×C4—2442no—
264C2×C4—2442no—
265C2×C4—2442no—
266C2×C4—2442no—
267C2×C4—2441no—
268C2×C4—2441no—
269C2×C4—2444no—
270C2×C4—2448no—
271C2×C4—2448no—
272C2×C4—2444no—
273C2×C4#1451970314yes—
274C2×C4#1451973712yes—
275C2×C4#1451974912yes—
276C8—82no—
277C22—21no—
278C22—21no—
279C22—21no—
280C22—21no—
281C22—24no—
282C22—24no—
283C22#1683963911yescharacteristic
284C22#1683973512yes—
285C22#1683975911yescharacteristic
286C22#1683977211yescharacteristic
287C22C2×#1451970312yes—
288C22#1683992612yes—
289C22#1684010412yes—
290C4—24no—
291C4—22no—
292C4—22no—
293C4—42no—
294C4—4881no—
295C4—4881no—
296C4—4882no—
297C4—2448no—
298C2—22no—
299C2—22no—
300C2#2144430612yes—
301C2#2144433212yes—
302C2#2144612911yescharacteristic
303C2#2144626011yescharacteristic
304C2#2144627511yescharacteristic
subgroupG/H
centerC23C2×#11758198
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1817C2
socleC23×C61C2×C22⋊C4
prime radicalC24×C61C2×D4
cosocleC23×C4×C61C23
hypercenter#1817D61
hypocenterC61#9145
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1817

Complements

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

NH ≅ G/Ntype
8. #16839668C4semidirect
9. #16839660C4semidirect
173. C61#9145semidirect

Maximal subgroups

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

subgroupindex
#214541582
C2×#168396082
C2×#168395002
#214436992
C61×#18172
#914561