Finite Group Databasebeta

← #29568127

Subgroups — #29568127

Order 15616.

492 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 111 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#21448362C211yesmaximal, characteristic
2#21445934C212yesmaximal, complemented
3#21443645C211yesmaximal, characteristic, complemented
4#21443639C212yesmaximal
5C61×#1272C211yesmaximal, characteristic, complemented
6C2×#14519766C411yescharacteristic
7C2×Q8×Dic61C411yescharacteristic
8C2×#14519739C411yescharacteristic
9C2×#14519706—22no—
10#16839664C2211yescharacteristic
11#16839654—22no—
12#16839627C411yescharacteristic
13#16839609—22no—
14C42×Dic61C2211yescharacteristic
15#16839517C2212yes—
16#16839516—22no—
17C61×#391C2211yescharacteristic
18C61×#336C2212yes—
19C22×D244C2×C411yescharacteristic
20C22×C4×D61—21no—
21C22×#92418—22no—
22#14519766—22no—
23#14519766C814yescomplemented
24Q8×Dic61—42no—
25#14519739—42no—
26C2×C4⋊Dic61—21no—
27C2×C4⋊Dic61C2×C411yescharacteristic
28C2×#11758198—22no—
29C2×#11758198—22no—
30C2×#11758198—21no—
31C2×C4×Dic61—21no—
32C2×C4×Dic61C2×C411yescharacteristic
33C2×C4×Dic61C2×C411yescharacteristic
34C2×C4×Dic61D412yes—
35C2×C4×Dic61D412yes—
36#14519706—44no—
37#14519684—22no—
38C22×Q8×C61C2×C411yescharacteristic
39C22×C8×C61—22no—
40C2×C4⋊C4×C61C2×C411yescharacteristic
41C2×C42×C61C2311yescharacteristic
42#95×C61—22no—
43C23×D61—21no—
44C2×Q8×C61—21no—
45C2×Q8×C61—41no—
46C2×Q8×C61—41no—
47C2×Q8×C61C2×C812yes—
48C22×C4×C61—21no—
49C22×C4×C61—21no—
50C22×C4×C61C22×C411yescharacteristic
51C22×C4×C61C2×D412yes—
52C22×Dic61—21no—
53C22×Dic61C22⋊C411yescharacteristic
54C22×Dic61C22⋊C412yes—
55C22×Dic61C22⋊C411yescharacteristic
56C2×D244—21no—
57C2×D244—41no—
58C2×D244—41no—
59C2×D244C2×C812yes—
60C2×C4×D61—44no—
61C2×C4×D61—42no—
62C2×C8×C61—42no—
63C2×C8×C61—42no—
64C2×C8×C61—42no—
65C4⋊C4×C61—41no—
66C42×C61—41no—
67#11758198—28no—
68#11758198—42no—
69#11758198—42no—
70#11758198—42no—
71C4⋊Dic61—42no—
72C4⋊Dic61—41no—
73C4×Dic61—22no—
74C4×Dic61—22no—
75C4×Dic61—22no—
76C4×Dic61—42no—
77C4×Dic61—42no—
78C4×Dic61C2×C812yes—
79C4×Dic61M1612yes—
80C2×#92418—42no—
81C2×#92418—42no—
82C2×#92418—42no—
83C23×C61C2×C22⋊C411yescharacteristic
84C22×D61—22no—
85C22×D61—41no—
86C22×D61—41no—
87C22×D61—42no—
88C22×D61—42no—
89Q8×C61—42no—
90Q8×C61—42no—
91C2×C4×C61—22no—
92C2×C4×C61—21no—
93C2×C4×C61—21no—
94C2×C4×C61—22no—
95C2×C4×C61—22no—
96C2×C4×C61—41no—
97C2×C4×C61—41no—
98C2×C4×C61—41no—
99C2×C4×C61—42no—
100C2×C4×C61C22×C811yescharacteristic
101C2×C4×C61C2×M1611yescharacteristic
102C2×C4×C61AGL(1,ℤ/8)12yes—
103C2×C4×C61#13712yes—
104C2×Dic61—22no—
105C2×Dic61—22no—
106C2×Dic61—22no—
107C2×Dic61—21no—
108C2×Dic61—22no—
109C2×Dic61—41no—
110C2×Dic61—42no—
111C2×Dic61C22⋊C814yes—
112C2×Dic61#10012yes—
113D244—42no—
114D244—42no—
115C4×D61—84no—
116C8×C61—82no—
117#92418—82no—
118#7601—611nomaximal, Sylow
119C22×C61C2×C22⋊C811yescharacteristic
120C22×C61#41011yescharacteristic
121C22×C61C2×#10011yescharacteristic
122C22×C61#41712yes—
123C22×C61#42112yes—
124C2×D61—42no—
125C2×D61—44no—
126C2×D61—81no—
127C2×D61—82no—
128C2×D61—81no—
129C4×C61—21no—
130C4×C61—21no—
131C4×C61—22no—
132C4×C61—22no—
133C4×C61—42no—
134C4×C61—81no—
135Dic61—24no—
136Dic61—22no—
137Dic61—41no—
138Dic61—81no—
139C4×#124—611no—
140#1831—612no—
141#1272—611no—
142#1272D6111yescharacteristic, complemented
143#1270—612no—
144C2×C61#146011yescharacteristic
145C2×C61#147712yes—
146C2×C61#151212yes—
147C2×C61#153311yescharacteristic
148C2×C61#156011yescharacteristic
149D61—82no—
150C2×#124—611no—
151C2×C4×Q8—611no—
152C2×C4×D4—611no—
153C2×C4⋊C8—1222no—
154#391C2×D6111yescharacteristic
155#390—611no—
156#388—1222no—
157#382—611no—
158C4×C22⋊C4—1222no—
159C43—611no—
160#336—612no—
161#336—1222no—
162#336C2×D6112yes—
163C61#760111yescharacteristic, complemented, Sylow
164C22×Q8C4×D6111yescharacteristic
165C22×D4—611no—
166C23×C4—1221no—
167C22×C8—22no—
168C22×C8—1222no—
169#124—614no—
170#124—1222no—
171C4×Q8—2442no—
172C4×D4—2442no—
173C2×C4⋊C4—611no—
174C2×C4⋊C4—1221no—
175C2×C4⋊C4C4×D6111yescharacteristic
176C2×C22⋊C4—1222no—
177C2×C22⋊C4—1221no—
178C2×C22⋊C4—1222no—
179C2×C42—611no—
180C2×C42—611no—
181C2×C42—612no—
182C2×C42—612no—
183C2×C42—1221no—
184C2×C42C22×D6111yescharacteristic
185C4⋊C8—2444no—
186#95—22no—
187#95—1222no—
188C24—1221no—
189C2×Q8—21no—
190C2×Q8—41no—
191C2×Q8—41no—
192C2×Q8C8×D6112yes—
193C2×D4—612no—
194C2×D4—1221no—
195C2×D4—2441no—
196C2×D4—2441no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—612no—
200C22×C4—611no—
201C22×C4—611no—
202C22×C4—1221no—
203C22×C4—2442no—
204C22×C4—2444no—
205C22×C4C2×C4×D6111yescharacteristic
206C22×C4D4×D6112yes—
207C2×C8—42no—
208C2×C8—42no—
209C2×C8—42no—
210C2×C8—2442no—
211C2×C8—2442no—
212C2×C8—2442no—
213C4⋊C4—41no—
214C4⋊C4—2441no—
215C4⋊C4—2442no—
216C22⋊C4—1228no—
217C22⋊C4—2442no—
218C22⋊C4—2442no—
219C22⋊C4—2442no—
220C42—41no—
221C42—612no—
222C42—612no—
223C42—1222no—
224C42—1222no—
225C42—1222no—
226C42—2442no—
227C42—2442no—
228C23—1222no—
229C23—2441no—
230C23—2441no—
231C23—2442no—
232C23—2442no—
233C23C22⋊C4×D6111yescharacteristic
234Q8—42no—
235Q8—42no—
236D4—2442no—
237D4—2442no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—42no—
247C2×C4—614no—
248C2×C4—612no—
249C2×C4—1222no—
250C2×C4—1222no—
251C2×C4—1222no—
252C2×C4—1221no—
253C2×C4—1222no—
254C2×C4—4884no—
255C2×C4—2441no—
256C2×C4—2442no—
257C2×C4C2×C8×D6111yescharacteristic
258C2×C4M16×D6111yescharacteristic
259C2×C4#1451980412yes—
260C2×C4#1451980912yes—
261C8—82no—
262C8—4882no—
263C22—4882no—
264C22—4881no—
265C22—4881no—
266C22—2442no—
267C22—2444no—
268C22C22⋊C8×D6111yescharacteristic
269C22#1683973411yescharacteristic
270C22#100×D6111yescharacteristic
271C22#1683976212yes—
272C22#1683977712yes—
273C4—22no—
274C4—22no—
275C4—21no—
276C4—21no—
277C4—42no—
278C4—81no—
279C4—1222no—
280C4—1224no—
281C4—4881no—
282C4—2441no—
283C2—4882no—
284C2#2144413811yescharacteristic
285C2#2144420512yes—
286C2#2144435612yes—
287C2#2144444111yescharacteristic
288C2#2144455311yescharacteristic
subgroupG/H
centerC23C22⋊C4×D61
commutatorC2×C4×C61C22×C8
FrattiniC2×C42C22×D61
FittingC61×#1272C2
socleC23×C61C2×C22⋊C4
prime radicalC22×D244C2×C4
cosocleC2×C42×C61C23
hypercenter#1272D61
hypocenterC61#7601
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1272

Complements

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

NH ≅ G/Ntype
2. #21445934C2semidirect
3. #21443645C2semidirect
5. C61×#1272C2semidirect
23. #14519766C8semidirect
142. #1272D61semidirect
163. C61#7601semidirect

Maximal subgroups

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

subgroupindex
#214483622
#214459342
#214436452
#214436392
C61×#12722
#760161