Finite Group Databasebeta

← #29577850

Subgroups — #29577850

Order 15616.

542 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 127 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#21454843C211yesmaximal, characteristic
2#21451173C211yesmaximal, characteristic
3#21444247C212yesmaximal, complemented
4#21443679C212yesmaximal, complemented
5C61×#1800C211yesmaximal, characteristic
6C2×C4×Dic122C2211yescharacteristic
7D4×#92418—24no—
8#16839815C2212yes—
9C2×C22⋊#92418—22no—
10#16839698C412yes—
11#16839674C412yes—
12#16839665—21no—
13#16839658—21no—
14#16839635—22no—
15#16839628C2211yescharacteristic
16#16839492—24no—
17C2×C4×D4×C61C2211yescharacteristic
18C61×#423C2212yes—
19C2×#100×C61—21no—
20C61×#343—21no—
21C22×Dic122C2×C411yescharacteristic
22C22×#92418—24no—
23C4×Dic122—22no—
24C4×Dic122—22no—
25C2×C4⋊Dic61—21no—
26C2×C4⋊Dic61C2×C411yescharacteristic
27C2×#11758196—21no—
28C2×#11758196—22no—
29C2×C22⋊Dic61—22no—
30C2×C22⋊Dic61—22no—
31C2×C4×Dic61—21no—
32C4⋊#92418—22no—
33C22⋊#92418—24no—
34C22⋊#92418—22no—
35C4×#92418D414yes—
36#14519684—22no—
37#14519684—21no—
38#14519684C2×C412yes—
39C22×D4×C61D411yescharacteristic
40C23×C4×C61D412yes—
41C2×M16×C61—22no—
42C4×D4×C61—22no—
43C4×D4×C61—22no—
44C2×C4⋊C4×C61D411yescharacteristic
45C2×C22⋊C4×C61C2×C412yes—
46C2×C42×C61C2311yescharacteristic
47C4⋊C8×C61—22no—
48C4⋊C8×C61—22no—
49#100×C61—42no—
50C24×C61C22⋊C412yes—
51C2×D4×C61—22no—
52C2×D4×C61—22no—
53C2×D4×C61SD1612yes—
54C2×D4×C61Q1612yes—
55C22×C4×C61—22no—
56C22×C4×C61—22no—
57C22×C4×C61—82no—
58C22×C4×C61C22⋊C412yes—
59C22×C4×C61C22×C411yescharacteristic
60C22×C4×C61C2×D411yescharacteristic
61C22×C4×C61C2×D411yescharacteristic
62C22×Dic61—21no—
63C22×Dic61—22no—
64C22×Dic61—21no—
65C2×Dic122—22no—
66C2×Dic122—42no—
67M16×C61—42no—
68M16×C61—42no—
69C2×C8×C61—42no—
70C4⋊C4×C61—22no—
71C22⋊C4×C61—24no—
72C42×C61C2×D412yes—
73C42×C61#4112yes—
74C22⋊Dic61—44no—
75C22⋊Dic61—82no—
76C4⋊Dic61—22no—
77C4⋊Dic61—42no—
78#11758196—41no—
79#11758196—41no—
80#11758196—82no—
81C4×Dic61—42no—
82C2×#92418—24no—
83C2×#92418—84no—
84C23×C61—22no—
85C23×C61—22no—
86C23×C61—42no—
87C23×C61—42no—
88C23×C61#10314yes—
89C23×C61C2×C22⋊C411yescharacteristic
90D4×C61—24no—
91D4×C61—24no—
92C2×C4×C61—22no—
93C2×C4×C61—41no—
94C2×C4×C61—41no—
95C2×C4×C61—44no—
96C2×C4×C61—42no—
97C2×C4×C61—82no—
98C2×C4×C61C4×D412yes—
99C2×C4×C61#12011yescharacteristic
100C2×C4×C61#12112yes—
101C2×C4×C61#12311yescharacteristic
102C2×C4×C61C2×SD1611yescharacteristic
103C2×C4×C61C2×Q1611yescharacteristic
104C2×C4×C61#13512yes—
105C2×Dic61—22no—
106C2×Dic61—41no—
107C2×Dic61—41no—
108C2×Dic61—41no—
109C2×Dic61—42no—
110C2×Dic61—42no—
111C2×Dic61—41no—
112C2×Dic61—41no—
113C2×Dic61—42no—
114Dic122—24no—
115Dic122—42no—
116C8×C61—82no—
117#92418—44no—
118#9483—611nomaximal, Sylow
119C22×C61—21no—
120C22×C61—21no—
121C22×C61—24no—
122C22×C61—44no—
123C22×C61—42no—
124C22×C61—42no—
125C22×C61—82no—
126C22×C61#38611yescharacteristic
127C22×C61C2×#10311yescharacteristic
128C22×C61#41611yescharacteristic
129C4×C61—22no—
130C4×C61—21no—
131C4×C61—21no—
132C4×C61—42no—
133C4×C61#46512yes—
134C4×C61#46712yes—
135Dic61—42no—
136Dic61—82no—
137Dic61—81no—
138#2972—611no—
139#2580—611no—
140#1800D6111yescharacteristic
141#1487—612no—
142#1286—612no—
143C2×C61—21no—
144C2×C61—21no—
145C2×C61—44no—
146C2×C61#189111yescharacteristic
147C2×C61#189211yescharacteristic
148C2×C61#191111yescharacteristic
149C2×C4×Q8—611no—
150C2×C4×D4C2×D6111yescharacteristic
151C8×D4—1224no—
152#431—612no—
153#423C2×D6112yes—
154C2×#100—21no—
155C2×C22⋊C8—1222no—
156#399—612no—
157#393—612no—
158#390—1221no—
159#389—1221no—
160#385—1222no—
161#382—611no—
162#343—21no—
163Q8⋊C8—1224no—
164C61#948311yescharacteristic, complemented, Sylow
165C22×Q8—611no—
166C22×D4C22⋊D6111yescharacteristic
167C23×C4D24412yes—
168C2×M16—22no—
169C22×C8—1224no—
170C4×Q8—1222no—
171C4×Q8—1222no—
172C4×D4—22no—
173C4×D4—22no—
174C2×C4⋊C4—611no—
175C2×C4⋊C4—1222no—
176C2×C4⋊C4—1221no—
177C2×C4⋊C4—1221no—
178C2×C4⋊C4C22⋊D6111yescharacteristic
179C2×C22⋊C4—1222no—
180C2×C22⋊C4—1222no—
181C2×C22⋊C4C4×D6112yes—
182C2×C42—1221no—
183C2×C42C22×D6111yescharacteristic
184C4⋊C8—22no—
185C4⋊C8—22no—
186C4⋊C8—1222no—
187#100—42no—
188C22⋊C8—1224no—
189C22⋊C8—1222no—
190C4×C8—614no—
191#95—612no—
192#95—1222no—
193#95—1221no—
194C24#1175819812yes—
195C2×Q8—1222no—
196C2×Q8—2442no—
197C2×D4—22no—
198C2×D4—22no—
199C2×D4#1175820012yes—
200C2×D4#1175820212yes—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—82no—
204C22×C4—1221no—
205C22×C4—1222no—
206C22×C4—1221no—
207C22×C4#1175819812yes—
208C22×C4C2×C4×D6111yescharacteristic
209C22×C4C2×D24411yescharacteristic
210C22×C4C2×C22⋊D6111yescharacteristic
211M16—42no—
212M16—42no—
213C2×C8—42no—
214C2×C8—1224no—
215C2×C8—4884no—
216C4⋊C4—22no—
217C4⋊C4—1222no—
218C4⋊C4—4882no—
219C4⋊C4—2441no—
220C4⋊C4—2441no—
221C4⋊C4—2442no—
222C22⋊C4—24no—
223C22⋊C4—4882no—
224C22⋊C4—2444no—
225C42—2442no—
226C42D4×D6112yes—
227C42#1175822412yes—
228C23—22no—
229C23—22no—
230C23—42no—
231C23—42no—
232C23#1451970014yes—
233C23C2×#1175819811yescharacteristic
234Q8—1224no—
235Q8—2442no—
236D4—24no—
237D4—24no—
238C2×C4—22no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—44no—
242C2×C4—42no—
243C2×C4—82no—
244C2×C4—1222no—
245C2×C4—2441no—
246C2×C4—2442no—
247C2×C4—2442no—
248C2×C4—2441no—
249C2×C4—2442no—
250C2×C4—2441no—
251C2×C4—2441no—
252C2×C4—2441no—
253C2×C4#1451973712yes—
254C2×C4#1451974711yescharacteristic
255C2×C4#1451974912yes—
256C2×C4#1451976111yescharacteristic
257C2×C4C2×#1175820011yescharacteristic
258C2×C4C2×#1175820211yescharacteristic
259C2×C4#1451979712yes—
260C8—82no—
261C8—2444no—
262C22—21no—
263C22—21no—
264C22—24no—
265C22—44no—
266C22—42no—
267C22—42no—
268C22—82no—
269C22#1683963911yescharacteristic
270C22C2×#1451970011yescharacteristic
271C22#1683975611yescharacteristic
272C4—22no—
273C4—21no—
274C4—21no—
275C4—42no—
276C4—4882no—
277C4—4881no—
278C4—2442no—
279C4#1683998912yes—
280C4#1684000112yes—
281C2—21no—
282C2—21no—
283C2—44no—
284C2#2144623511yescharacteristic
285C2#2144624011yescharacteristic
286C2#2144634911yescharacteristic
subgroupG/H
centerC22#16839639
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1800C2
socleC22×C61#386
prime radicalC22×D4×C61D4
cosocleC2×C42×C61C23
hypercenter#1800D61
hypocenterC61#9483
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1800

Complements

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

NH ≅ G/Ntype
3. #21444247C2semidirect
4. #21443679C2semidirect
164. C61#9483semidirect

Maximal subgroups

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

subgroupindex
#214548432
#214511732
#214442472
#214436792
C61×#18002
#948361