Finite Group Databasebeta

← #29589309

Subgroups — #29589309

Order 15616.

392 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 103 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#21448535C211yesmaximal, characteristic
2#21444143C211yesmaximal, characteristic
3#21443762C212yesmaximal
4#21443752C212yesmaximal
5C61×#1461C211yesmaximal, characteristic
6C2×#14519775C411yescharacteristic
7C2×C4×Dic122C411yescharacteristic
8#16839815—21no—
9#16839669C411yescharacteristic
10#16839667C2211yescharacteristic
11#16839646—21no—
12#16839646—21no—
13#16839643—21no—
14#16839643—21no—
15#16839627C2211yescharacteristic
16#16839608C411yescharacteristic
17#16839488—21no—
18#16839488C2212yes—
19C2×C4×Q8×C61C2211yescharacteristic
20C61×#431—21no—
21C61×#324—21no—
22C61×#324C2212yes—
23C22×Dic122C2×C411yescharacteristic
24#14519775—22no—
25#14519775C814yescomplemented
26C4×Dic122—41no—
27C4×Dic122—41no—
28C2×C4⋊Dic61C2×C411yescharacteristic
29C2×#11758196—21no—
30C2×#11758196—21no—
31C2×#11758196—21no—
32C2×C4×Dic61—21no—
33C2×C4×Dic61C2×C411yescharacteristic
34C2×C4×Dic61C2×C411yescharacteristic
35C4⋊#92418—41no—
36C22⋊#92418—22no—
37C4×#92418—41no—
38#14519684—21no—
39#14519684—21no—
40#14519684—21no—
41#14519684—21no—
42#14519684—21no—
43#14519684—21no—
44#14519684—21no—
45#14519684—21no—
46C22×Q8×C61C2×C411yescharacteristic
47C4×Q8×C61—41no—
48C4×Q8×C61—41no—
49C2×C4⋊C4×C61D411yescharacteristic
50C2×C4⋊C4×C61D411yescharacteristic
51C2×C4⋊C4×C61C2311yescharacteristic
52C2×C42×C61C2×C411yescharacteristic
53C2×C42×C61D411yescharacteristic
54C2×C42×C61D411yescharacteristic
55C4⋊C8×C61—41no—
56C22⋊C8×C61—22no—
57C4×C8×C61—41no—
58C2×Q8×C61—41no—
59C2×Q8×C61—41no—
60C2×Q8×C61C2×C812yes—
61C2×Q8×C61M1612yes—
62C22×C4×C61C22⋊C411yescharacteristic
63C22×C4×C61C22⋊C411yescharacteristic
64C22×C4×C61C22⋊C411yescharacteristic
65C22×C4×C61C22⋊C411yescharacteristic
66C22×C4×C61C22×C411yescharacteristic
67C22×C4×C61C2×D411yescharacteristic
68C22×C4×C61C2×D411yescharacteristic
69C22×Dic61—21no—
70C22×Dic61—21no—
71C22×Dic61—21no—
72C22×Dic61—21no—
73C2×Dic122—21no—
74C2×Dic122—41no—
75C2×Dic122—41no—
76C2×Dic122C2×C812yes—
77C2×C8×C61—42no—
78C4⋊C4×C61—41no—
79C4⋊C4×C61—41no—
80C4⋊C4×C61—41no—
81C42×C61—21no—
82C42×C61—21no—
83C42×C61—21no—
84C42×C61—21no—
85C42×C61—21no—
86C42×C61—21no—
87C4⋊Dic61—41no—
88#11758196—24no—
89#11758196—24no—
90#11758196—41no—
91#11758196—41no—
92C4×Dic61—21no—
93C4×Dic61—42no—
94C4×Dic61—41no—
95C4×Dic61C2×C812yes—
96C2×#92418—42no—
97C23×C61C2×C22⋊C411yescharacteristic
98Q8×C61—42no—
99Q8×C61—42no—
100C2×C4×C61—21no—
101C2×C4×C61—21no—
102C2×C4×C61—21no—
103C2×C4×C61—21no—
104C2×C4×C61—21no—
105C2×C4×C61—21no—
106C2×C4×C61—21no—
107C2×C4×C61—21no—
108C2×C4×C61—21no—
109C2×C4×C61—41no—
110C2×C4×C61—41no—
111C2×C4×C61—41no—
112C2×C4×C61—41no—
113C2×C4×C61C22⋊C812yes—
114C2×C4×C61C22⋊C812yes—
115C2×C4×C61#9911yescharacteristic
116C2×C4×C61#9911yescharacteristic
117C2×C4×C61C22×C811yescharacteristic
118C2×C4×C61C2×M1611yescharacteristic
119C2×Dic61—22no—
120C2×Dic61—22no—
121C2×Dic61—41no—
122C2×Dic61—41no—
123C2×Dic61—41no—
124C2×Dic61—41no—
125C2×Dic61—41no—
126C2×Dic61—41no—
127C2×Dic61—42no—
128C2×Dic61—41no—
129C2×Dic61—41no—
130Dic122—42no—
131Dic122—42no—
132C8×C61—82no—
133#92418—82no—
134#11482—611nomaximal, Sylow
135C22×C61—21no—
136C22×C61—21no—
137C22×C61C2×C22⋊C811yescharacteristic
138C22×C61C2×#9911yescharacteristic
139C22×C61#41311yescharacteristic
140C4×C61—21no—
141C4×C61—21no—
142C4×C61—21no—
143C4×C61—21no—
144C4×C61—22no—
145C4×C61—22no—
146C4×C61—41no—
147C4×C61—41no—
148C4×C61—41no—
149Dic61—42no—
150Dic61—42no—
151Dic61—81no—
152Dic61—81no—
153#2327—611no—
154#1461—611no—
155#1461D6111yescharacteristic
156#1324—612no—
157#1319—612no—
158C2×C61—21no—
159C2×C61—21no—
160C2×C61#145911yescharacteristic
161C2×C61#212911yescharacteristic
162C2×C61#213311yescharacteristic
163C2×C4⋊Q8—611no—
164C2×C4×Q8—611no—
165C2×C4×Q8C2×D6111yescharacteristic
166#431—21no—
167#431—1221no—
168#391—611no—
169#391—611no—
170#387—1221no—
171#387—1221no—
172#387—1221no—
173#387—1221no—
174#382—611no—
175#376—611no—
176#324—21no—
177#324—612no—
178#324—1221no—
179#324C2×D6112yes—
180C61#1148211yescharacteristic, complemented, Sylow
181C22×Q8—611no—
182C22×Q8C4×D6111yescharacteristic
183C4⋊Q8—614no—
184C4⋊Q8—1222no—
185C4×Q8—41no—
186C4×Q8—41no—
187C4×Q8—2441no—
188C4×Q8—2441no—
189C2×C4⋊C4—611no—
190C2×C4⋊C4—1221no—
191C2×C4⋊C4—1221no—
192C2×C4⋊C4—1221no—
193C2×C4⋊C4D24411yescharacteristic
194C2×C4⋊C4C22⋊D6111yescharacteristic
195C2×C4⋊C4C22×D6111yescharacteristic
196C2×C42—611no—
197C2×C42—611no—
198C2×C42—1221no—
199C2×C42C4×D6111yescharacteristic
200C2×C42D24411yescharacteristic
201C2×C42C22⋊D6111yescharacteristic
202C4⋊C8—41no—
203C4⋊C8—2441no—
204C22⋊C8—22no—
205C22⋊C8—1222no—
206C4×C8—41no—
207C4×C8—2441no—
208#95—1221no—
209#95—1221no—
210#95—1221no—
211#95—1221no—
212#95—1221no—
213#95—1221no—
214#95—1221no—
215#95—1221no—
216C2×Q8—41no—
217C2×Q8—41no—
218C2×Q8—612no—
219C2×Q8—1221no—
220C2×Q8—2441no—
221C2×Q8—2441no—
222C2×Q8C8×D6112yes—
223C2×Q8#1175818912yes—
224C22×C4—1221no—
225C22×C4—1221no—
226C22×C4—1221no—
227C22×C4—1221no—
228C22×C4#1175819811yescharacteristic
229C22×C4#1175819811yescharacteristic
230C22×C4#1175819811yescharacteristic
231C22×C4#1175819811yescharacteristic
232C22×C4C2×C4×D6111yescharacteristic
233C22×C4C2×D24411yescharacteristic
234C22×C4C2×C22⋊D6111yescharacteristic
235C2×C8—42no—
236C2×C8—2442no—
237C4⋊C4—41no—
238C4⋊C4—41no—
239C4⋊C4—41no—
240C4⋊C4—1224no—
241C4⋊C4—1224no—
242C4⋊C4—2441no—
243C4⋊C4—2441no—
244C4⋊C4—2441no—
245C42—21no—
246C42—21no—
247C42—21no—
248C42—21no—
249C42—21no—
250C42—21no—
251C42—612no—
252C42—1221no—
253C42—2441no—
254C42—2442no—
255C23C2×#1175819811yescharacteristic
256Q8—42no—
257Q8—42no—
258Q8—2442no—
259Q8—2442no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—1222no—
274C2×C4—1222no—
275C2×C4—2441no—
276C2×C4—2441no—
277C2×C4—2442no—
278C2×C4—2441no—
279C2×C4—2441no—
280C2×C4—2441no—
281C2×C4—2441no—
282C2×C4—2441no—
283C2×C4—2441no—
284C2×C4#1451968912yes—
285C2×C4#1451968912yes—
286C2×C4#1451969111yescharacteristic
287C2×C4#1451969111yescharacteristic
288C2×C4C2×C8×D6111yescharacteristic
289C2×C4C2×#1175818911yescharacteristic
290C8—82no—
291C8—4882no—
292C22—21no—
293C22—21no—
294C22C2×#1451968911yescharacteristic
295C22C2×#1451969111yescharacteristic
296C22#1683974411yescharacteristic
297C4—22no—
298C4—22no—
299C4—21no—
300C4—21no—
301C4—21no—
302C4—21no—
303C4—41no—
304C4—41no—
305C4—41no—
306C4—4881no—
307C4—4881no—
308C4—2442no—
309C4—2442no—
310C2—21no—
311C2—21no—
312C2#2144413511yescharacteristic
313C2#2144749311yescharacteristic
314C2#2144750611yescharacteristic
subgroupG/H
centerC22C2×#14519691
commutatorC2×C4×C61C22×C8
FrattiniC2×C4⋊C4C22×D61
FittingC61×#1461C2
socleC22×C61C2×#99
prime radicalC23×C61C2×C22⋊C4
cosocleC2×C4⋊C4×C61C23
hypercenter#1461D61
hypocenterC61#11482
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #1461

Complements

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

NH ≅ G/Ntype
25. #14519775C8semidirect
180. C61#11482semidirect

Maximal subgroups

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

subgroupindex
#214485352
#214441432
#214437622
#214437522
C61×#14612
#1148261