Finite Group Databasebeta

← #30603820

Subgroups — #30603820

Order 15616.

734 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 734 conjugacy classes of subgroups — too many to draw. Only its 112 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×#16840605C212yesmaximal, complemented
2#21460745C212yesmaximal, complemented
3#21460743C212yesmaximal, complemented
4#21448397C211yesmaximal, characteristic
5C2×C22⋊#92420—22no—
6#16840605—22no—
7#16840605C2214yes—
8C22×C22⋊Dic61C411yescharacteristic
9C22×C22⋊Dic61C2211yescharacteristic
10#16839642—22no—
11#16839642—22no—
12#16839625C411yescharacteristic
13#16839625C2211yescharacteristic
14#16839618C411yescharacteristic
15#16839618C2211yescharacteristic
16C23×D4×C61C411yescharacteristic
17C22×#92420—42no—
18C22⋊#92420—24no—
19C22⋊#92420—42no—
20C23×Dic61—21no—
21C23×Dic61D412yes—
22C2×C22⋊Dic61—21no—
23C2×C22⋊Dic61—22no—
24C2×C22⋊Dic61—22no—
25C2×C22⋊Dic61—42no—
26C2×C22⋊Dic61—42no—
27C2×C22⋊Dic61C2×C411yescharacteristic
28C2×C22⋊Dic61C2×C411yescharacteristic
29C2×C22⋊Dic61C2×C411yescharacteristic
30C2×C22⋊Dic61D412yes—
31C2×C22⋊Dic61C2311yescharacteristic
32#14519684—22no—
33#14519684—22no—
34C25×C61C2×C411yescharacteristic
35C25×C61C2×C411yescharacteristic
36C22×D4×C61—21no—
37C22×D4×C61—21no—
38C22×D4×C61—22no—
39C22×D4×C61—42no—
40C22×D4×C61—42no—
41C22×D4×C61C814yescomplemented
42C23×C4×C61C2×C411yescharacteristic
43C24×C61—22no—
44C24×C61—22no—
45C24×C61—21no—
46C24×C61—21no—
47C24×C61—41no—
48C24×C61—41no—
49C24×C61—82no—
50C24×C61—82no—
51C24×C61C22⋊C411yescharacteristic
52C24×C61C22⋊C411yescharacteristic
53C24×C61C2×C812yes—
54C24×C61C2×C812yescomplemented
55C24×C61M1612yes—
56C24×C61C22×C411yescharacteristic
57C2×D4×C61—24no—
58C2×D4×C61—24no—
59C2×D4×C61—48no—
60C2×D4×C61—42no—
61C2×D4×C61—42no—
62C2×D4×C61—42no—
63C2×D4×C61—42no—
64C2×D4×C61—48no—
65C22×C4×C61—21no—
66C22×C4×C61—21no—
67C22×C4×C61—41no—
68C22×C4×C61—41no—
69C22×C4×C61C2×C812yescomplemented
70C22×Dic61—22no—
71C22×Dic61—22no—
72C22×Dic61—21no—
73C22×Dic61—21no—
74C22×Dic61—42no—
75C22×Dic61—82no—
76C22×Dic61—82no—
77C22×Dic61C22⋊C412yes—
78C22×Dic61C2×D412yes—
79C2×#92420—44no—
80C2×#92420—84no—
81C22⋊Dic61—41no—
82C22⋊Dic61—42no—
83C22⋊Dic61—41no—
84C22⋊Dic61—42no—
85C22⋊Dic61—41no—
86C22⋊Dic61—41no—
87C22⋊Dic61—82no—
88C22⋊Dic61—82no—
89C23×C61—21no—
90C23×C61—21no—
91C23×C61—21no—
92C23×C61—21no—
93C23×C61—24no—
94C23×C61—24no—
95C23×C61—22no—
96C23×C61—42no—
97C23×C61—42no—
98C23×C61—42no—
99C23×C61—42no—
100C23×C61—42no—
101C23×C61—42no—
102C23×C61—42no—
103C23×C61—41no—
104C23×C61—41no—
105C23×C61—82no—
106C23×C61—82no—
107C23×C61—84no—
108C23×C61—82no—
109C23×C61—82no—
110C23×C61—82no—
111C23×C61—82no—
112C23×C61—84no—
113C23×C61—84no—
114C23×C61C22⋊C812yes—
115C23×C61C22⋊C812yes—
116C23×C61#9912yes—
117C23×C61#10012yes—
118C23×C61C2×C22⋊C411yescharacteristic
119C23×C61C22×C811yescharacteristic
120C23×C61C2×M1611yescharacteristic
121D4×C61—48no—
122D4×C61—48no—
123C2×C4×C61—22no—
124C2×C4×C61—42no—
125C2×C4×C61—41no—
126C2×C4×C61—41no—
127C2×C4×C61—42no—
128C2×Dic61—22no—
129C2×Dic61—22no—
130C2×Dic61—44no—
131C2×Dic61—42no—
132C2×Dic61—42no—
133C2×Dic61—42no—
134C2×Dic61—42no—
135C2×Dic61—41no—
136C2×Dic61—41no—
137C2×Dic61—82no—
138C2×Dic61—84no—
139#92420—84no—
140#11426—611nomaximal, Sylow
141C22×C61—22no—
142C22×C61—22no—
143C22×C61—22no—
144C22×C61—21no—
145C22×C61—21no—
146C22×C61—22no—
147C22×C61—42no—
148C22×C61—42no—
149C22×C61—41no—
150C22×C61—41no—
151C22×C61—42no—
152C22×C61—42no—
153C22×C61—42no—
154C22×C61—42no—
155C22×C61—42no—
156C22×C61—42no—
157C22×C61—42no—
158C22×C61—42no—
159C22×C61—84no—
160C22×C61—84no—
161C22×C61—82no—
162C22×C61—84no—
163C22×C61—84no—
164C22×C61—82no—
165C22×C61#32314yes—
166C22×C61C2×C22⋊C811yescharacteristic
167C22×C61C2×#9911yescharacteristic
168C22×C61C2×#10011yescharacteristic
169C4×C61—42no—
170Dic61—42no—
171Dic61—42no—
172Dic61—81no—
173Dic61—81no—
174#2308—611no—
175C2×#323—612no—
176#1317—612no—
177#1316—612no—
178C2×C61—21no—
179C2×C61—21no—
180C2×C61—24no—
181C2×C61—42no—
182C2×C61—41no—
183C2×C61—41no—
184C2×C61—82no—
185C2×C61C2×#32311yescharacteristic
186C2×C61#212112yes—
187C23×D4#692311yescharacteristic
188C22×C22⋊C4—611no—
189C22×C22⋊C4—611no—
190C2×C22⋊C8—1222no—
191#386—1222no—
192#386—1222no—
193#381—611no—
194#381—611no—
195#379—611no—
196#379—611no—
197#323—614no—
198#323—1222no—
199C61#1142611yescharacteristic, complemented, Sylow
200C25C2×#692311yescharacteristic
201C25C2×#692311yescharacteristic
202C22×D4—22no—
203C22×D4—21no—
204C22×D4—21no—
205C22×D4—42no—
206C22×D4—42no—
207C22×D4#9242014yescomplemented
208C23×C4—612no—
209C23×C4—1221no—
210C23×C4C2×#692311yescharacteristic
211C22×C8—2442no—
212C2×C22⋊C4—612no—
213C2×C22⋊C4—611no—
214C2×C22⋊C4—611no—
215C2×C22⋊C4—611no—
216C2×C22⋊C4—611no—
217C2×C22⋊C4—1222no—
218C2×C22⋊C4—1222no—
219C2×C22⋊C4—1221no—
220C2×C22⋊C4—2442no—
221C2×C22⋊C4—2442no—
222C22⋊C8—1224no—
223C22⋊C8—2442no—
224#95—1222no—
225#95—1222no—
226C24—22no—
227C24—22no—
228C24—21no—
229C24—21no—
230C24—41no—
231C24—41no—
232C24—82no—
233C24—82no—
234C24C2×#9242012yes—
235C24C2×#9242012yescomplemented
236C24#1175821712yes—
237C24C22⋊#692311yescharacteristic
238C24C22⋊#692311yescharacteristic
239C24C22×#692311yescharacteristic
240C2×D4—24no—
241C2×D4—24no—
242C2×D4—48no—
243C2×D4—42no—
244C2×D4—42no—
245C2×D4—42no—
246C2×D4—42no—
247C2×D4—48no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—612no—
253C22×C4—612no—
254C22×C4—1222no—
255C22×C4—1222no—
256C22×C4—1221no—
257C22×C4—1221no—
258C22×C4—4882no—
259C22×C4—4882no—
260C22×C4—2442no—
261C22×C4C2×#9242012yescomplemented
262C2×C8—4884no—
263C2×C8—2444no—
264C22⋊C4—4882no—
265C22⋊C4—4882no—
266C22⋊C4—2441no—
267C22⋊C4—2442no—
268C22⋊C4—2441no—
269C22⋊C4—2442no—
270C22⋊C4—2441no—
271C22⋊C4—2441no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—24no—
277C23—24no—
278C23—22no—
279C23—42no—
280C23—42no—
281C23—42no—
282C23—42no—
283C23—42no—
284C23—42no—
285C23—42no—
286C23—41no—
287C23—41no—
288C23—82no—
289C23—82no—
290C23—82no—
291C23—82no—
292C23—82no—
293C23—84no—
294C23—82no—
295C23—84no—
296C23—84no—
297C23C22⋊#9242012yes—
298C23C22⋊#9242012yes—
299C23#1451983512yes—
300C23#1451983712yes—
301C23C2×C22⋊#692311yescharacteristic
302C23C22×#9242011yescharacteristic
303C23C2×#1175821711yescharacteristic
304D4—48no—
305D4—48no—
306C2×C4—22no—
307C2×C4—42no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—42no—
311C2×C4—1222no—
312C2×C4—1222no—
313C2×C4—4884no—
314C2×C4—4882no—
315C2×C4—2442no—
316C2×C4—2442no—
317C2×C4—2441no—
318C2×C4—2441no—
319C2×C4—2444no—
320C2×C4—2442no—
321C2×C4—2442no—
322C8—4884no—
323C22—22no—
324C22—22no—
325C22—22no—
326C22—21no—
327C22—21no—
328C22—22no—
329C22—42no—
330C22—42no—
331C22—41no—
332C22—41no—
333C22—42no—
334C22—42no—
335C22—42no—
336C22—42no—
337C22—42no—
338C22—42no—
339C22—42no—
340C22—42no—
341C22—84no—
342C22—84no—
343C22—82no—
344C22—84no—
345C22—84no—
346C22—82no—
347C22#1684060514yes—
348C22C2×C22⋊#9242011yescharacteristic
349C22C2×#1451983511yescharacteristic
350C22C2×#1451983711yescharacteristic
351C4—42no—
352C4—4881no—
353C4—4881no—
354C4—2442no—
355C4—2442no—
356C2—21no—
357C2—21no—
358C2—24no—
359C2—41no—
360C2—41no—
361C2—42no—
362C2—82no—
363C2C2×#1684060511yescharacteristic
364C2#2146207412yes—
subgroupG/H
centerC22C2×#14519835
commutatorC23×C61C22×C8
FrattiniC24C22×#6923
Fitting, prime radicalC23×D4×C61C4
socleC22×C61C2×#99
cosocleC2×C22⋊Dic61C23
hypercenterC23×D4#6923
hypocenterC61#11426
LCSG ▷ C23×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C24 ◁ C25 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. C2×#16840605C2semidirect
2. #21460745C2semidirect
3. #21460743C2semidirect
41. C22×D4×C61C8semidirect
54. C24×C61C2×C8semidirect
69. C22×C4×C61C2×C8semidirect
199. C61#11426semidirect
207. C22×D4#92420semidirect
235. C24C2×#92420semidirect
261. C22×C4C2×#92420semidirect

Maximal subgroups

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

subgroupindex
C2×#168406052
#214607452
#214607432
#214483972
#1142661