Finite Group Databasebeta

← #30604177

Subgroups — #30604177

Order 15616.

596 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 76 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#21461075C212yesmaximal, complemented
2#21460761C212yesmaximal, complemented
3#21460747C212yesmaximal, complemented
4#127×Dic61C211yesmaximal, characteristic
5#16840732—22no—
6#16840607C2214yes—
7#16840605—22no—
8C2×D4×Dic61—22no—
9C2×D4×Dic61C411yescharacteristic
10C2×D4×Dic61C2211yescharacteristic
11#16839665—22no—
12#16839665C411yescharacteristic
13#16839665C2211yescharacteristic
14#16839632C2211yescharacteristic
15C42×Dic61C411yescharacteristic
16C2×#127×C61C411yescharacteristic
17C2×#11758217—42no—
18C22⋊#92420—24no—
19C22⋊#92420—42no—
20C23×Dic61—21no—
21C23×Dic61D412yes—
22D4×Dic61—41no—
23D4×Dic61—41no—
24D4×Dic61—48no—
25D4×Dic61—41no—
26D4×Dic61—41no—
27C2×C4⋊Dic61—22no—
28C2×C4⋊Dic61C2×C411yescharacteristic
29C2×C4⋊Dic61C2311yescharacteristic
30C2×C22⋊Dic61—22no—
31C2×C22⋊Dic61—21no—
32C2×C22⋊Dic61—22no—
33C2×C22⋊Dic61D412yes—
34C2×C4×Dic61—21no—
35C2×C4×Dic61—21no—
36C2×C4×Dic61C2×C411yescharacteristic
37C2×C4×Dic61C2×C411yescharacteristic
38C22×D4×C61—22no—
39C22×D4×C61C2×C411yescharacteristic
40C22×D4×C61C2×C411yescharacteristic
41#127×C61—22no—
42#127×C61—22no—
43C2×C42×C61C2×C411yescharacteristic
44C24×C61—21no—
45C24×C61C22⋊C412yes—
46C2×D4×C61—28no—
47C2×D4×C61—21no—
48C2×D4×C61—21no—
49C2×D4×C61—21no—
50C2×D4×C61—41no—
51C2×D4×C61—41no—
52C2×D4×C61—44no—
53C2×D4×C61—44no—
54C2×D4×C61—41no—
55C2×D4×C61—41no—
56C2×D4×C61M1612yescomplemented
57C22×C4×C61—21no—
58C22×C4×C61C22×C411yescharacteristic
59C22×Dic61—21no—
60C22×Dic61—22no—
61C22×Dic61—22no—
62C22×Dic61—21no—
63C22×Dic61—44no—
64C22×Dic61—44no—
65C22×Dic61—41no—
66C22×Dic61—41no—
67C22×Dic61C22⋊C411yescharacteristic
68C22×Dic61C22⋊C411yescharacteristic
69C22×Dic61C2×D412yes—
70#11758217—84no—
71C2×#92420—44no—
72C42×C61—21no—
73C42×C61M1612yescomplemented
74C22⋊Dic61—42no—
75C22⋊Dic61—44no—
76C22⋊Dic61—44no—
77C22⋊Dic61—42no—
78C4⋊Dic61—41no—
79C4⋊Dic61—44no—
80C4⋊Dic61—41no—
81C4×Dic61—24no—
82C4×Dic61—21no—
83C4×Dic61—21no—
84C4×Dic61—42no—
85C4×Dic61—41no—
86C23×C61—22no—
87C23×C61—22no—
88C23×C61—22no—
89C23×C61—22no—
90C23×C61—42no—
91C23×C61—42no—
92C23×C61—42no—
93C23×C61—42no—
94C23×C61—41no—
95C23×C61—41no—
96C23×C61C2×C22⋊C411yescharacteristic
97D4×C61—42no—
98D4×C61—416no—
99D4×C61—42no—
100D4×C61—42no—
101D4×C61—42no—
102C2×C4×C61—22no—
103C2×C4×C61—22no—
104C2×C4×C61—22no—
105C2×C4×C61—21no—
106C2×C4×C61—21no—
107C2×C4×C61C2×M1611yescharacteristic
108C2×C4×C61C2×M1611yescharacteristic
109C2×Dic61—22no—
110C2×Dic61—22no—
111C2×Dic61—21no—
112C2×Dic61—22no—
113C2×Dic61—41no—
114C2×Dic61—41no—
115C2×Dic61—44no—
116C2×Dic61—41no—
117C2×Dic61—42no—
118C2×Dic61—42no—
119C2×Dic61—41no—
120C2×Dic61—82no—
121C2×Dic61—82no—
122C2×Dic61—82no—
123C2×Dic61#9911yescharacteristic
124C2×Dic61#9911yescharacteristic
125#92420—84no—
126#11529—611nomaximal, Sylow
127C22×C61—21no—
128C22×C61—21no—
129C22×C61—44no—
130C22×C61—44no—
131C22×C61—42no—
132C22×C61—42no—
133C22×C61—42no—
134C22×C61—42no—
135C22×C61—82no—
136C22×C61—82no—
137C22×C61—81no—
138C22×C61—81no—
139C22×C61#40711yescharacteristic
140C22×C61C2×#9911yescharacteristic
141C22×C61#41311yescharacteristic
142C4×C61—24no—
143C4×C61—21no—
144C4×C61—21no—
145Dic61—21no—
146Dic61—42no—
147Dic61—42no—
148Dic61—41no—
149Dic61—81no—
150Dic61#35312yes—
151C4×#127—611no—
152#1465—612no—
153#1325—612no—
154#1318—612no—
155C2×C61—21no—
156C2×C61—21no—
157C2×C61—82no—
158C2×C61—82no—
159C2×C61#146511yescharacteristic
160C2×C61C2×#35311yescharacteristic
161C2×C61#213211yescharacteristic
162C2×#127#692311yescharacteristic
163C2×C4×D4—611no—
164C2×C4×D4—611no—
165C2×C4×D4—1222no—
166#407—1222no—
167#390—611no—
168#390—611no—
169#390—1222no—
170#384—611no—
171C43—611no—
172#324—614no—
173#323—1222no—
174C61#1152911yescharacteristic, complemented, Sylow
175C22×D4—22no—
176C22×D4C2×#692311yescharacteristic
177C22×D4C2×#692311yescharacteristic
178C23×C4—612no—
179C23×C4—1221no—
180C2×M16—2442no—
181#127—22no—
182#127—22no—
183C4×D4—2441no—
184C4×D4—2441no—
185C4×D4—2448no—
186C4×D4—2441no—
187C4×D4—2441no—
188C2×C4⋊C4—611no—
189C2×C4⋊C4—611no—
190C2×C4⋊C4—1222no—
191C2×C22⋊C4—612no—
192C2×C22⋊C4—1221no—
193C2×C22⋊C4—1222no—
194C2×C22⋊C4—1222no—
195C2×C42—611no—
196C2×C42—611no—
197C2×C42—1221no—
198C2×C42—1221no—
199C2×C42C2×#692311yescharacteristic
200C22⋊C8—1224no—
201C22⋊C8—2442no—
202C24—21no—
203C24C22⋊#692312yes—
204C2×D4—28no—
205C2×D4—21no—
206C2×D4—21no—
207C2×D4—21no—
208C2×D4—41no—
209C2×D4—41no—
210C2×D4—41no—
211C2×D4—41no—
212C2×D4—44no—
213C2×D4—44no—
214C2×D4#1175821712yescomplemented
215C22×C4—21no—
216C22×C4—611no—
217C22×C4—612no—
218C22×C4—611no—
219C22×C4—1222no—
220C22×C4—1222no—
221C22×C4—1221no—
222C22×C4—1221no—
223C22×C4—2441no—
224C22×C4—2441no—
225C22×C4—2444no—
226C22×C4—2444no—
227C22×C4C22×#692311yescharacteristic
228M16—4884no—
229C2×C8—2444no—
230C4⋊C4—2441no—
231C4⋊C4—2441no—
232C4⋊C4—2444no—
233C22⋊C4—2442no—
234C22⋊C4—2444no—
235C22⋊C4—2444no—
236C22⋊C4—2442no—
237C42—21no—
238C42—1221no—
239C42—1221no—
240C42—1224no—
241C42—2441no—
242C42—2442no—
243C42#1175821712yescomplemented
244C23—22no—
245C23—22no—
246C23—22no—
247C23—22no—
248C23—42no—
249C23—42no—
250C23—42no—
251C23—42no—
252C23—41no—
253C23—41no—
254C23C2×C22⋊#692311yescharacteristic
255D4—42no—
256D4—416no—
257D4—42no—
258D4—42no—
259D4—42no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—611no—
266C2×C4—611no—
267C2×C4—1222no—
268C2×C4—1222no—
269C2×C4—1221no—
270C2×C4—1222no—
271C2×C4—4882no—
272C2×C4—4882no—
273C2×C4—4882no—
274C2×C4—2444no—
275C2×C4—2442no—
276C2×C4—2441no—
277C2×C4—2441no—
278C2×C4—2441no—
279C2×C4—2441no—
280C2×C4—2442no—
281C2×C4C2×#1175821711yescharacteristic
282C2×C4C2×#1175821711yescharacteristic
283C8—4884no—
284C22—21no—
285C22—21no—
286C22—44no—
287C22—44no—
288C22—42no—
289C22—42no—
290C22—42no—
291C22—42no—
292C22—82no—
293C22—82no—
294C22—81no—
295C22—81no—
296C22#1684073211yescharacteristic
297C22D61⋊#9911yescharacteristic
298C22#1684074911yescharacteristic
299C4—24no—
300C4—21no—
301C4—21no—
302C4—612no—
303C4—1221no—
304C4—4881no—
305C4—2442no—
306C4—2442no—
307C4—2441no—
308C2—21no—
309C2—21no—
310C2—82no—
311C2—82no—
312C2#2146107511yescharacteristic
313C2D61⋊#35311yescharacteristic
314C2#2146211111yescharacteristic
subgroupG/H
centerC22D61⋊#99
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
Fitting, prime radicalC2×#127×C61C4
socleC22×C61C2×#99
cosocleC2×C4⋊Dic61C23
hypercenterC2×#127#6923
hypocenterC61#11529
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ C2×#127

Complements

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

NH ≅ G/Ntype
1. #21461075C2semidirect
2. #21460761C2semidirect
3. #21460747C2semidirect
56. C2×D4×C61M16semidirect
73. C42×C61M16semidirect
174. C61#11529semidirect
214. C2×D4#11758217semidirect
243. C42#11758217semidirect

Maximal subgroups

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

subgroupindex
#214610752
#214607612
#214607472
#127×Dic612
#1152961