Finite Group Databasebeta

← #30606546

Subgroups — #30606546

Order 15616.

628 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 628 conjugacy classes of subgroups — too many to draw. Only its 102 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
1C2×#16840667C212yesmaximal
2C4×#14519836C212yesmaximal, complemented
3#21460913C212yesmaximal, complemented
4#21448544C211yesmaximal, characteristic
5C2×#14519836C2214yes—
6C4×C22⋊#6923—22no—
7#16840667C418yescomplemented
8#16840640—22no—
9C2×#14519774C2211yescharacteristic
10C2×C4×D244C411yescharacteristic, complemented
11C2×C4×D244C2211yescharacteristic
12#16839664C411yescharacteristic, complemented
13#16839664C2211yescharacteristic
14#16839640—22no—
15#16839640—22no—
16#16839628C411yescharacteristic, complemented
17C61×#376C411yescharacteristic, complemented
18C2×C22⋊#6923—24no—
19C2×C4×#6923—42no—
20#14519836—24no—
21#14519836C2×C418yescomplemented
22(C2×#6923)⋊C4—42no—
23C22×D244—22no—
24C22×D244C2×C411yescharacteristic
25C22×D244C2311yescharacteristic
26C22×C4×D61—21no—
27C22×C4×D61D412yes—
28#14519774—22no—
29#14519774C2×C414yes—
30C4×D244—42no—
31C4×D244—42no—
32C2×C4⋊Dic61C2×C411yescharacteristic
33C2×C4⋊Dic61C2×C411yescharacteristic
34C2×#11758198—21no—
35C2×#11758198—22no—
36C2×#11758198—22no—
37C2×#11758198D412yes—
38#14519684—22no—
39C2×C42×C61C2×C411yescharacteristic
40C2×C42×C61C2×C411yescharacteristic
41C2×C42×C61C2×C411yescharacteristic
42#95×C61—21no—
43#95×C61—21no—
44C23×D61—21no—
45C23×D61C2×D412yes—
46C22×#6923—44no—
47C22×C4×C61—21no—
48C22×C4×C61—21no—
49C22×C4×C61C22⋊C411yescharacteristic
50C22×C4×C61C22⋊C411yescharacteristic
51C22×C4×C61C22×C411yescharacteristic
52C22×Dic61—21no—
53C22×Dic61C22⋊C412yes—
54C2×D244—21no—
55C2×D244—28no—
56C2×D244—41no—
57C2×D244—41no—
58C2×D244—41no—
59C2×D244—41no—
60C2×D244—82no—
61C2×D244—82no—
62C2×D244C4212yescomplemented
63C2×D244C22×C412yes—
64C2×D244#4112yes—
65C2×C4×D61—22no—
66C2×C4×D61—22no—
67C2×C4×D61—42no—
68C2×C4×D61—82no—
69C2×C4×D61—82no—
70C22⋊#6923—28no—
71C22⋊#6923—44no—
72C4×#6923—84no—
73C42×C61—22no—
74C42×C61—21no—
75C42×C61—41no—
76C42×C61C4212yescomplemented
77#11758198—42no—
78#11758198—42no—
79#11758198—82no—
80#11758198—82no—
81C4⋊Dic61—41no—
82C4⋊Dic61—41no—
83C23×C61C2×C22⋊C411yescharacteristic
84C22×D61—22no—
85C22×D61—22no—
86C22×D61—22no—
87C22×D61—42no—
88C22×D61—42no—
89C22×D61—42no—
90C22×D61—42no—
91C22×D61—41no—
92C22×D61—41no—
93C22×D61C4×D414yes—
94C2×#6923—48no—
95C2×#6923—84no—
96C2×#6923—84no—
97C2×C4×C61—21no—
98C2×C4×C61—21no—
99C2×C4×C61—22no—
100C2×C4×C61—21no—
101C2×C4×C61—41no—
102C2×C4×C61—41no—
103C2×C4×C61—41no—
104C2×C4×C61—41no—
105C2×C4×C61—41no—
106C2×C4×C61—41no—
107C2×C4×C61—41no—
108C2×C4×C61#9912yes—
109C2×C4×C61C2×C4211yescharacteristic
110C2×C4×C61#11711yescharacteristic
111C2×Dic61—22no—
112C2×Dic61—42no—
113C2×Dic61—42no—
114C2×Dic61—41no—
115D244—42no—
116D244—42no—
117D244—42no—
118D244—42no—
119D244—88no—
120C4×D61—44no—
121C4×D61—82no—
122C4×D61—84no—
123#12628—611nomaximal, Sylow
124C22×C61—21no—
125C22×C61—21no—
126C22×C61C4×C22⋊C411yescharacteristic
127C22×C61C2×#9911yescharacteristic
128C22×C61#41011yescharacteristic
129C2×D61—24no—
130C2×D61—42no—
131C2×D61—42no—
132C2×D61—44no—
133C2×D61—44no—
134C2×D61—42no—
135C2×D61—82no—
136C2×D61—82no—
137C2×D61—81no—
138C2×D61—81no—
139#6923—88no—
140C4×C61—22no—
141C4×C61—21no—
142C4×C61—21no—
143C4×C61—42no—
144C4×C61—41no—
145C4×C61—81no—
146Dic61—42no—
147Dic61—81no—
148#2328—611no—
149C2×#353—612no—
150C4×#99—612no—
151#1393—612no—
152C2×C61—21no—
153C2×C61—21no—
154C2×C61C4×#9911yescharacteristic
155C2×C61#212711yescharacteristic
156C2×C61#212811yescharacteristic
157D61—44no—
158D61—82no—
159C2×#127—611no—
160C2×C4×D4—611no—
161C2×C4×D4—611no—
162C2×#99—614no—
163#390—611no—
164#390—611no—
165#386—1222no—
166#386—1222no—
167#382—611no—
168C4×C22⋊C4—1222no—
169#376#692311yescharacteristic, complemented
170#353—618no—
171#342—1222no—
172C61#1262811yescharacteristic, complemented, Sylow
173C22×D4—611no—
174C22×D4—611no—
175C22×D4—1222no—
176C23×C4—612no—
177C23×C4—1221no—
178#127—614no—
179#127—1222no—
180C4×D4—2442no—
181C4×D4—2442no—
182C2×C4⋊C4—611no—
183C2×C4⋊C4—611no—
184C2×C22⋊C4—612no—
185C2×C22⋊C4—1224no—
186C2×C22⋊C4—1222no—
187C2×C22⋊C4—1222no—
188C2×C22⋊C4—1221no—
189C2×C42—2442no—
190C2×C42C2×#692311yescharacteristic
191C2×C42C2×#692311yescharacteristic
192C2×C42C2×#692311yescharacteristic
193#99—618no—
194#99—1224no—
195#95—21no—
196#95—21no—
197#95—1222no—
198#95—2442no—
199C24—612no—
200C24—1221no—
201C2×D4—612no—
202C2×D4—612no—
203C2×D4—612no—
204C2×D4—1228no—
205C2×D4—1221no—
206C2×D4—4882no—
207C2×D4—4882no—
208C2×D4—2441no—
209C2×D4—2441no—
210C2×D4—2441no—
211C2×D4—2441no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—612no—
215C22×C4—1222no—
216C22×C4—1222no—
217C22×C4—1221no—
218C22×C4—4882no—
219C22×C4—4882no—
220C22×C4—2442no—
221C22×C4—2444no—
222C22×C4C22⋊#692311yescharacteristic
223C22×C4C22⋊#692311yescharacteristic
224C22×C4C22×#692311yescharacteristic
225C4⋊C4—2441no—
226C4⋊C4—2441no—
227C22⋊C4—1228no—
228C22⋊C4—4882no—
229C22⋊C4—4882no—
230C22⋊C4—2442no—
231C22⋊C4—2444no—
232C22⋊C4—2442no—
233C42—22no—
234C42—21no—
235C42—41no—
236C42—4884no—
237C42C4×#692312yescomplemented
238C23—614no—
239C23—1222no—
240C23—1222no—
241C23—1222no—
242C23—2442no—
243C23—2442no—
244C23—2442no—
245C23—2442no—
246C23—2441no—
247C23—2441no—
248C23C2×C22⋊#692311yescharacteristic
249D4—4888no—
250D4—2442no—
251D4—2442no—
252D4—2442no—
253D4—2442no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—1222no—
266C2×C4—4882no—
267C2×C4—4884no—
268C2×C4—4884no—
269C2×C4—4884no—
270C2×C4—2444no—
271C2×C4—2442no—
272C2×C4—2448no—
273C2×C4—2441no—
274C2×C4—2442no—
275C2×C4#1451983612yes—
276C2×C4C2×C4×#692311yescharacteristic
277C2×C4#1451985611yescharacteristic
278C22—21no—
279C22—21no—
280C22—1224no—
281C22—4882no—
282C22—4882no—
283C22—4881no—
284C22—4881no—
285C22—2444no—
286C22—2444no—
287C22—2442no—
288C22—2442no—
289C22—2442no—
290C22C4×C22⋊#692311yescharacteristic
291C22C2×#1451983611yescharacteristic
292C22#1684074111yescharacteristic
293C4—21no—
294C4—21no—
295C4—22no—
296C4—42no—
297C4—41no—
298C4—81no—
299C4—4888no—
300C4—4881no—
301C4—2442no—
302C2—21no—
303C2—21no—
304C2—4882no—
305C2—2444no—
306C2C4×#1451983611yescharacteristic
307C2#2146209611yescharacteristic
308C2#2146210011yescharacteristic
subgroupG/H
centerC22C2×#14519836
commutatorC2×C4×C61C2×C42
FrattiniC22×C4C22×#6923
FittingC61×#376C4
socleC22×C61C2×#99
prime radicalC2×#14519774C22
cosocleC22×D244C23
hypercenter#376#6923
hypocenterC61#12628
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ #376

Complements

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

NH ≅ G/Ntype
2. C4×#14519836C2semidirect
3. #21460913C2semidirect
7. #16840667C4semidirect
10. C2×C4×D244C4semidirect
12. #16839664C4semidirect
16. #16839628C4semidirect
17. C61×#376C4semidirect
21. #14519836C2×C4semidirect
62. C2×D244C42semidirect
76. C42×C61C42semidirect
169. #376#6923semidirect
172. C61#12628semidirect
237. C42C4×#6923semidirect

Maximal subgroups

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

subgroupindex
C2×#168406672
C4×#145198362
#214609132
#214485442
#1262861