Finite Group Databasebeta

← #30606544

Subgroups — #30606544

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×#16840666C212yesmaximal
2C4×#14519836C212yesmaximal, complemented
3#21460914C212yesmaximal, complemented
4#21448541C211yesmaximal, characteristic
5C2×#14519836C2214yes—
6C4×C22⋊#6923—22no—
7#16840666C418yescomplemented
8#16840640—22no—
9C2×#14519773C2211yescharacteristic
10C2×C4×D244C2211yescharacteristic
11#16839665C411yescharacteristic, complemented
12#16839663C2211yescharacteristic
13#16839641—22no—
14#16839638—22no—
15#16839627C411yescharacteristic, complemented
16#16839608C411yescharacteristic, complemented
17C2×C4×D4×C61C411yescharacteristic, complemented
18C2×C22⋊#6923—24no—
19C2×C4×#6923—42no—
20#14519836—24no—
21#14519836C2×C418yescomplemented
22(C2×#6923)⋊C4—42no—
23C22×D244C2311yescharacteristic
24C22×C22⋊D61—22no—
25C22×C4×D61D412yes—
26#14519773—22no—
27#14519773C2×C414yes—
28C4×D244—42no—
29C2×C4⋊Dic61C2×C411yescharacteristic
30C2×C22⋊Dic61—22no—
31C2×#11758198—22no—
32C2×#11758198D412yes—
33C2×C4×Dic61C2×C411yescharacteristic
34C2×C4×Dic61C2×C411yescharacteristic
35#14519684—21no—
36#14519684—21no—
37#14519684—22no—
38C22×D4×C61C2×C411yescharacteristic
39C23×C4×C61—21no—
40C4×D4×C61—42no—
41C2×C4⋊C4×C61C2×C411yescharacteristic
42C2×C22⋊C4×C61—21no—
43C2×C42×C61C2×C411yescharacteristic
44C24×C61—21no—
45C23×D61C2×D412yes—
46C22×#6923—44no—
47C2×D4×C61—21no—
48C2×D4×C61—41no—
49C2×D4×C61—41no—
50C2×D4×C61C4212yescomplemented
51C22×C4×C61—21no—
52C22×C4×C61—42no—
53C22×C4×C61—82no—
54C22×C4×C61C22⋊C411yescharacteristic
55C22×C4×C61C22⋊C411yescharacteristic
56C22×C4×C61C22×C411yescharacteristic
57C2×C22⋊D61—28no—
58C2×C22⋊D61—82no—
59C2×C22⋊D61—82no—
60C22×Dic61—21no—
61C22×Dic61—21no—
62C22×Dic61C22⋊C412yes—
63C2×D244—41no—
64C2×D244—41no—
65C2×D244C22×C412yes—
66C2×D244#4112yes—
67C2×C4×D61—22no—
68C2×C4×D61—22no—
69C2×C4×D61—82no—
70C22⋊#6923—28no—
71C22⋊#6923—44no—
72C4×#6923—84no—
73C4⋊C4×C61—41no—
74C22⋊C4×C61—42no—
75C42×C61—22no—
76C22⋊Dic61—82no—
77#11758198—42no—
78#11758198—82no—
79C4⋊Dic61—41no—
80C4×Dic61—21no—
81C4×Dic61—41no—
82C4×Dic61C4212yescomplemented
83C23×C61—22no—
84C23×C61—42no—
85C23×C61—42no—
86C23×C61—41no—
87C23×C61—41no—
88C23×C61C2×C22⋊C411yescharacteristic
89C22×D61—22no—
90C22×D61—22no—
91C22×D61—42no—
92C22×D61—42no—
93C22×D61C4×D414yes—
94C2×#6923—48no—
95C2×#6923—84no—
96C2×#6923—84no—
97D4×C61—42no—
98D4×C61—42no—
99C2×C4×C61—21no—
100C2×C4×C61—21no—
101C2×C4×C61—21no—
102C2×C4×C61—41no—
103C2×C4×C61—41no—
104C2×C4×C61—42no—
105C2×C4×C61—41no—
106C2×C4×C61—84no—
107C2×C4×C61#9912yes—
108C2×C4×C61C2×C4211yescharacteristic
109C2×C4×C61#11711yescharacteristic
110C22⋊D61—88no—
111C2×Dic61—22no—
112C2×Dic61—22no—
113C2×Dic61—41no—
114C2×Dic61—41no—
115C2×Dic61—42no—
116C2×Dic61—41no—
117C2×Dic61—41no—
118C2×Dic61—41no—
119D244—42no—
120D244—42no—
121C4×D61—44no—
122C4×D61—82no—
123#12628—611nomaximal, Sylow
124C22×C61—21no—
125C22×C61—21no—
126C22×C61—44no—
127C22×C61—42no—
128C22×C61—82no—
129C22×C61—81no—
130C22×C61—81no—
131C22×C61C4×C22⋊C411yescharacteristic
132C22×C61C2×#9911yescharacteristic
133C22×C61#41011yescharacteristic
134C2×D61—24no—
135C2×D61—42no—
136C2×D61—42no—
137C2×D61—44no—
138C2×D61—82no—
139#6923—88no—
140C4×C61—22no—
141C4×C61—21no—
142C4×C61—21no—
143C4×C61—41no—
144C4×C61—81no—
145Dic61—42no—
146Dic61—42no—
147Dic61—81no—
148#2328—611no—
149C2×#353—612no—
150C4×#99—612no—
151#1393—612no—
152C2×C61—21no—
153C2×C61—21no—
154C2×C61—82no—
155C2×C61C4×#9911yescharacteristic
156C2×C61#212711yescharacteristic
157C2×C61#212811yescharacteristic
158D61—44no—
159C2×#127—611no—
160C2×C4×D4—611no—
161C2×C4×D4#692311yescharacteristic, complemented
162C2×#99—614no—
163#390—611no—
164#390—611no—
165#386—1222no—
166#386—1222no—
167#382—611no—
168C4×C22⋊C4—1222no—
169#376—611no—
170#353—618no—
171#342—1222no—
172C61#1262811yescharacteristic, complemented, Sylow
173C22×D4—611no—
174C22×D4—1222no—
175C22×D4C2×#692311yescharacteristic
176C23×C4—21no—
177C23×C4—612no—
178#127—614no—
179#127—1222no—
180C4×D4—42no—
181C4×D4—2442no—
182C2×C4⋊C4—611no—
183C2×C4⋊C4C2×#692311yescharacteristic
184C2×C22⋊C4—21no—
185C2×C22⋊C4—612no—
186C2×C22⋊C4—1224no—
187C2×C22⋊C4—1222no—
188C2×C22⋊C4—1222no—
189C2×C42—611no—
190C2×C42—611no—
191C2×C42—2442no—
192C2×C42C2×#692311yescharacteristic
193#99—618no—
194#99—1224no—
195#95—1222no—
196#95—1221no—
197#95—1221no—
198#95—2442no—
199C24—21no—
200C24—612no—
201C2×D4—21no—
202C2×D4—41no—
203C2×D4—41no—
204C2×D4—612no—
205C2×D4—612no—
206C2×D4—1228no—
207C2×D4—4882no—
208C2×D4—4882no—
209C2×D4—2441no—
210C2×D4—2441no—
211C2×D4C4×#692312yescomplemented
212C22×C4—21no—
213C22×C4—42no—
214C22×C4—82no—
215C22×C4—612no—
216C22×C4—1221no—
217C22×C4—1222no—
218C22×C4—1222no—
219C22×C4—1221no—
220C22×C4—4882no—
221C22×C4—2444no—
222C22×C4C22⋊#692311yescharacteristic
223C22×C4C22⋊#692311yescharacteristic
224C22×C4C22×#692311yescharacteristic
225C4⋊C4—41no—
226C4⋊C4—2441no—
227C22⋊C4—42no—
228C22⋊C4—1228no—
229C22⋊C4—4882no—
230C22⋊C4—4882no—
231C22⋊C4—2442no—
232C22⋊C4—2444no—
233C42—22no—
234C42—612no—
235C42—1221no—
236C42—4884no—
237C42—2441no—
238C23—22no—
239C23—42no—
240C23—42no—
241C23—41no—
242C23—41no—
243C23—614no—
244C23—1222no—
245C23—1222no—
246C23—2442no—
247C23—2442no—
248C23C2×C22⋊#692311yescharacteristic
249D4—42no—
250D4—42no—
251D4—4888no—
252D4—2442no—
253D4—2442no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—42no—
260C2×C4—41no—
261C2×C4—84no—
262C2×C4—1222no—
263C2×C4—1222no—
264C2×C4—4882no—
265C2×C4—4884no—
266C2×C4—4884no—
267C2×C4—2444no—
268C2×C4—2441no—
269C2×C4—2441no—
270C2×C4—2448no—
271C2×C4—2441no—
272C2×C4—2441no—
273C2×C4—2441no—
274C2×C4—2442no—
275C2×C4#1451983512yes—
276C2×C4C2×C4×#692311yescharacteristic
277C2×C4#1451985611yescharacteristic
278C22—21no—
279C22—21no—
280C22—44no—
281C22—42no—
282C22—82no—
283C22—81no—
284C22—81no—
285C22—1224no—
286C22—4882no—
287C22—2444no—
288C22—2442no—
289C22—2442no—
290C22C4×C22⋊#692311yescharacteristic
291C22C2×#1451983511yescharacteristic
292C22#1684074111yescharacteristic
293C4—21no—
294C4—21no—
295C4—22no—
296C4—41no—
297C4—81no—
298C4—4881no—
299C4—4888no—
300C4—2442no—
301C4—2442no—
302C2—21no—
303C2—21no—
304C2—82no—
305C2—2444no—
306C2C4×#1451983511yescharacteristic
307C2#2146209411yescharacteristic
308C2#2146209811yescharacteristic
subgroupG/H
centerC22C2×#14519835
commutatorC2×C4×C61C2×C42
FrattiniC22×C4C22×#6923
FittingC2×C4×D4×C61C4
socleC22×C61C2×#99
prime radicalC2×#14519773C22
cosocleC22×D244C23
hypercenterC2×C4×D4#6923
hypocenterC61#12628
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
2. C4×#14519836C2semidirect
3. #21460914C2semidirect
7. #16840666C4semidirect
11. #16839665C4semidirect
15. #16839627C4semidirect
16. #16839608C4semidirect
17. C2×C4×D4×C61C4semidirect
21. #14519836C2×C4semidirect
50. C2×D4×C61C42semidirect
82. C4×Dic61C42semidirect
161. C2×C4×D4#6923semidirect
172. C61#12628semidirect
211. C2×D4C4×#6923semidirect

Maximal subgroups

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

subgroupindex
C2×#168406662
C4×#145198362
#214609142
#214485412
#1262861