Finite Group Databasebeta

← #29577945

Subgroups — #29577945

Order 15616.

660 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 660 conjugacy classes of subgroups — too many to draw. Only its 99 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#21450097C211yesmaximal, characteristic
2#21448417C211yesmaximal, characteristic
3#21445891C212yesmaximal
4#21443723C212yesmaximal
5C61×#1819C211yesmaximal, characteristic
6C22×C22⋊Dic61C2211yescharacteristic
7#16839690—22no—
8#16839655—24no—
9#16839635—21no—
10#16839634C412yescomplemented
11#16839625—21no—
12#16839623C412yescomplemented
13#16839622—21no—
14#16839618C2211yescharacteristic
15#16839618C2212yes—
16C4×C22⋊Dic61—21no—
17#16839528—24no—
18C22×C22⋊C4×C61C2211yescharacteristic
19C61×#379C2212yes—
20C61×#342—22no—
21C23×Dic61C2×C411yescharacteristic
22C23×Dic61C2×C411yescharacteristic
23C2×C4⋊Dic61—22no—
24C2×#11758196—22no—
25C2×C22⋊Dic61—21no—
26C2×C22⋊Dic61—22no—
27C2×C22⋊Dic61—21no—
28C2×C22⋊Dic61—22no—
29C2×C22⋊Dic61—21no—
30C2×C22⋊Dic61—22no—
31C2×C22⋊Dic61—21no—
32C2×C22⋊Dic61—22no—
33C2×C22⋊Dic61C2×C412yes—
34C2×C22⋊Dic61D414yes—
35C2×C4×Dic61—22no—
36#14519684—22no—
37#14519684—22no—
38#14519684—22no—
39#14519684—22no—
40#14519684—22no—
41C25×C61C2311yescharacteristic
42C23×C4×C61D411yescharacteristic
43C23×C4×C61D411yescharacteristic
44C2×C22⋊C4×C61—22no—
45C2×C22⋊C4×C61—22no—
46C2×C22⋊C4×C61—22no—
47C2×C22⋊C4×C61—22no—
48C2×C22⋊C4×C61—22no—
49C2×C22⋊C4×C61C2×C412yes—
50C2×C22⋊C4×C61D412yes—
51C24×C61—42no—
52C24×C61—42no—
53C24×C61—81no—
54C24×C61C22×C411yescharacteristic
55C24×C61C2×D411yescharacteristic
56C24×C61C2×D411yescharacteristic
57C24×C61C2×D412yes—
58C24×C61#4112yes—
59C22×C4×C61—21no—
60C22×C4×C61—21no—
61C22×C4×C61—21no—
62C22×C4×C61—21no—
63C22×C4×C61—42no—
64C22×C4×C61—42no—
65C22×C4×C61—81no—
66C22×C4×C61C22⋊C412yes—
67C22×C4×C61C22⋊C412yes—
68C22×Dic61—21no—
69C22×Dic61—21no—
70C22×Dic61—22no—
71C22×Dic61—24no—
72C22×Dic61—22no—
73C22×Dic61—22no—
74C22×Dic61—42no—
75C22×Dic61—81no—
76C22⋊C4×C61—24no—
77C22⋊C4×C61—24no—
78C22⋊C4×C61—44no—
79C22⋊C4×C61—44no—
80C22⋊C4×C61—82no—
81C22⋊Dic61—44no—
82C22⋊Dic61—42no—
83C22⋊Dic61—41no—
84C22⋊Dic61—42no—
85C22⋊Dic61—41no—
86C22⋊Dic61—42no—
87C22⋊Dic61—44no—
88C22⋊Dic61—82no—
89C22⋊Dic61—81no—
90C4⋊Dic61—82no—
91#11758196—82no—
92C4×Dic61—82no—
93C23×C61—22no—
94C23×C61—22no—
95C23×C61—22no—
96C23×C61—22no—
97C23×C61—24no—
98C23×C61—41no—
99C23×C61—41no—
100C23×C61—41no—
101C23×C61—41no—
102C23×C61—42no—
103C23×C61—42no—
104C23×C61—41no—
105C23×C61—42no—
106C23×C61—42no—
107C23×C61—42no—
108C23×C61—88no—
109C23×C61C2×C22⋊C411yescharacteristic
110C23×C61C4×D412yes—
111C23×C61#12011yescharacteristic
112C23×C61#12112yes—
113C23×C61#12311yescharacteristic
114C2×C4×C61—22no—
115C2×C4×C61—22no—
116C2×C4×C61—42no—
117C2×C4×C61—42no—
118C2×C4×C61—42no—
119C2×C4×C61—42no—
120C2×C4×C61—81no—
121C2×C4×C61—81no—
122C2×C4×C61—82no—
123C2×C4×C61—82no—
124C2×C4×C61—82no—
125C2×Dic61—44no—
126C2×Dic61—42no—
127C2×Dic61—42no—
128C2×Dic61—42no—
129C2×Dic61—44no—
130C2×Dic61—41no—
131C2×Dic61—41no—
132C2×Dic61—42no—
133C2×Dic61—42no—
134C2×Dic61—44no—
135C2×Dic61—82no—
136C2×Dic61—82no—
137#9502—611nomaximal, Sylow
138C22×C61—21no—
139C22×C61—21no—
140C22×C61—21no—
141C22×C61—21no—
142C22×C61—22no—
143C22×C61—21no—
144C22×C61—22no—
145C22×C61—22no—
146C22×C61—22no—
147C22×C61—21no—
148C22×C61—22no—
149C22×C61—22no—
150C22×C61—21no—
151C22×C61—44no—
152C22×C61—44no—
153C22×C61—84no—
154C22×C61—82no—
155C22×C61—84no—
156C22×C61#38611yescharacteristic
157C22×C61#41012yes—
158C22×C61#45812yes—
159C4×C61—42no—
160C4×C61—42no—
161C4×C61—162no—
162Dic61—84no—
163Dic61—82no—
164Dic61—81no—
165Dic61—81no—
166#2483—611no—
167#2312—611no—
168#1819—612no—
169#1819D6111yescharacteristic
170#1304—612no—
171C2×C61—22no—
172C2×C61—22no—
173C2×C61—42no—
174C2×C61—41no—
175C2×C61—42no—
176C2×C61#189011yescharacteristic
177C2×C61#189911yescharacteristic
178C2×C61#191311yescharacteristic
179C22×C22⋊C4—611no—
180C22×C22⋊C4C2×D6111yescharacteristic
181#397—1222no—
182#388—1224no—
183#385—612no—
184#385—1221no—
185#381—612no—
186#381—1221no—
187#380—1221no—
188#379—612no—
189#379—611no—
190#379C2×D6112yes—
191C4×C22⋊C4—1221no—
192#342—22no—
193#342—1224no—
194C61#950211yescharacteristic, complemented, Sylow
195C25C22×D6111yescharacteristic
196C23×C4—611no—
197C23×C4—611no—
198C23×C4D24411yescharacteristic
199C23×C4D24411yescharacteristic
200C2×C4⋊C4—1222no—
201C2×C4⋊C4—1222no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—612no—
208C2×C22⋊C4—614no—
209C2×C22⋊C4—1222no—
210C2×C22⋊C4—1221no—
211C2×C22⋊C4—1221no—
212C2×C22⋊C4—1222no—
213C2×C22⋊C4—1221no—
214C2×C22⋊C4—1222no—
215C2×C22⋊C4—1222no—
216C2×C22⋊C4—1221no—
217C2×C22⋊C4C4×D6112yes—
218C2×C22⋊C4C22⋊D6112yes—
219C2×C42—1222no—
220#95—1222no—
221#95—1222no—
222#95—1222no—
223#95—1222no—
224#95—1222no—
225C24—42no—
226C24—42no—
227C24—81no—
228C24C2×C4×D6111yescharacteristic
229C24C2×D24411yescharacteristic
230C24D4×D6112yes—
231C24#1175822412yes—
232C24C2×C22⋊D6111yescharacteristic
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—81no—
240C22×C4—1222no—
241C22×C4—1222no—
242C22×C4—1221no—
243C22×C4—1222no—
244C22×C4—1221no—
245C22×C4—1224no—
246C22×C4—4881no—
247C22×C4—2442no—
248C22×C4#1175819812yes—
249C22×C4#1175819812yes—
250C4⋊C4—4882no—
251C4⋊C4—4882no—
252C22⋊C4—24no—
253C22⋊C4—24no—
254C22⋊C4—44no—
255C22⋊C4—44no—
256C22⋊C4—82no—
257C22⋊C4—4882no—
258C22⋊C4—4881no—
259C22⋊C4—2442no—
260C22⋊C4—2444no—
261C22⋊C4—2441no—
262C22⋊C4—2442no—
263C22⋊C4—2441no—
264C22⋊C4—2444no—
265C22⋊C4—2442no—
266C42—4882no—
267C23—22no—
268C23—22no—
269C23—22no—
270C23—24no—
271C23—22no—
272C23—41no—
273C23—41no—
274C23—41no—
275C23—42no—
276C23—42no—
277C23—41no—
278C23—42no—
279C23—42no—
280C23—42no—
281C23—41no—
282C23—88no—
283C23C2×#1175819811yescharacteristic
284C23#1451973712yes—
285C23#1451974711yescharacteristic
286C23#1451974912yes—
287C23#1451976111yescharacteristic
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—81no—
295C2×C4—81no—
296C2×C4—82no—
297C2×C4—82no—
298C2×C4—82no—
299C2×C4—4882no—
300C2×C4—4882no—
301C2×C4—2444no—
302C2×C4—2442no—
303C2×C4—2442no—
304C2×C4—2444no—
305C2×C4—2442no—
306C2×C4—2444no—
307C2×C4—2442no—
308C2×C4—2442no—
309C2×C4—2441no—
310C2×C4—2441no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—22no—
316C22—21no—
317C22—22no—
318C22—22no—
319C22—22no—
320C22—21no—
321C22—22no—
322C22—21no—
323C22—22no—
324C22—44no—
325C22—44no—
326C22—84no—
327C22—82no—
328C22—84no—
329C22#1683963911yescharacteristic
330C22#1683973512yes—
331C22#1683995312yes—
332C4—42no—
333C4—42no—
334C4—162no—
335C4—4884no—
336C4—4882no—
337C4—4881no—
338C4—4881no—
339C2—22no—
340C2—22no—
341C2—42no—
342C2—41no—
343C2—42no—
344C2#2144623011yescharacteristic
345C2#2144627511yescharacteristic
346C2#2144635911yescharacteristic
subgroupG/H
centerC22#16839639
commutatorC24×C61C22×C4
FrattiniC25C22×D61
FittingC61×#1819C2
socleC22×C61#386
prime radical, cosocleC25×C61C23
hypercenter#1819D61
hypocenterC61#9502
LCSG ▷ C24×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C25 ◁ #1819

Complements

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

NH ≅ G/Ntype
10. #16839634C4semidirect
12. #16839623C4semidirect
194. C61#9502semidirect

Maximal subgroups

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

subgroupindex
#214500972
#214484172
#214458912
#214437232
C61×#18192
#950261