Finite Group Databasebeta

← #29577725

Subgroups — #29577725

Order 15616.

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

normal subgroups only

This group has 536 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#21454904C211yesmaximal, characteristic, complemented
2#21450818C211yesmaximal, characteristic
3#21444532C212yesmaximal
4#21443674C212yesmaximal, complemented
5C61×#1467C211yesmaximal, characteristic, complemented
6C2×#14519774C2211yescharacteristic
7C2×#14519766C412yes—
8#16839893—24no—
9#16839819—21no—
10#16839819C2212yes—
11#16839815—21no—
12#16839679—21no—
13#16839679—21no—
14#16839664—21no—
15#16839664—21no—
16#16839663C412yes—
17C4×C4⋊Dic61C2211yescharacteristic
18#16839505—24no—
19C2×C4×Q8×C61C2211yescharacteristic
20C61×#432C2212yes—
21C61×#324—21no—
22C61×#324—21no—
23C22×D244—21no—
24C22×D244—21no—
25C22×D244C2×C412yes—
26#14519774—22no—
27#14519774—22no—
28#14519766—44no—
29C2×C4⋊Dic61—21no—
30C2×#11758198—22no—
31C2×#11758198—22no—
32C2×#11758198—22no—
33C2×#11758198—22no—
34C2×C4×Dic61C2×C412yes—
35C4⋊#92418—22no—
36C4⋊#92418—22no—
37C4⋊#92418D414yes—
38C22⋊#92418—22no—
39#14519688—22no—
40#14519688—22no—
41C4×#92418—22no—
42#14519684—21no—
43C22×Q8×C61C2×C411yescharacteristic
44C4×Q8×C61—22no—
45C4×Q8×C61—22no—
46C2×C4⋊C4×C61C2×C411yescharacteristic
47C2×C4⋊C4×C61D411yescharacteristic
48C2×C4⋊C4×C61D411yescharacteristic
49C2×C42×C61D411yescharacteristic
50C2×C42×C61D411yescharacteristic
51C2×C42×C61C2311yescharacteristic
52C4⋊C8×C61—22no—
53C22⋊C8×C61—22no—
54#97×C61—22no—
55C23×D61—22no—
56C2×Q8×C61—22no—
57C2×Q8×C61—22no—
58C2×Q8×C61—41no—
59C22×C4×C61C22⋊C411yescharacteristic
60C22×C4×C61C22⋊C411yescharacteristic
61C22×C4×C61C22⋊C411yescharacteristic
62C22×C4×C61C22⋊C411yescharacteristic
63C22×C4×C61C22×C411yescharacteristic
64C22×C4×C61C2×D411yescharacteristic
65C22×C4×C61C2×D411yescharacteristic
66C22×Dic61—22no—
67C2×D244—24no—
68C2×D244—42no—
69C2×D244—42no—
70C2×D244—42no—
71C2×D244—42no—
72C2×D244—41no—
73C2×D244—42no—
74C2×D244—41no—
75C2×D244—82no—
76C2×C8×C61—42no—
77C4⋊C4×C61—22no—
78C4⋊C4×C61—22no—
79C4⋊C4×C61—22no—
80C42×C61—21no—
81C42×C61—21no—
82C42×C61—21no—
83C42×C61—21no—
84C42×C61C2×D412yes—
85C42×C61#4112yes—
86#11758198—44no—
87#11758198—44no—
88#11758198—82no—
89#11758198—82no—
90C4⋊Dic61—42no—
91C4×Dic61—42no—
92C2×#92418—24no—
93C23×C61C2×C22⋊C411yescharacteristic
94C22×D61—42no—
95C22×D61—42no—
96C22×D61—42no—
97C22×D61—42no—
98C22×D61—42no—
99C22×D61—42no—
100C22×D61—42no—
101Q8×C61—44no—
102C2×C4×C61—21no—
103C2×C4×C61—21no—
104C2×C4×C61—21no—
105C2×C4×C61—21no—
106C2×C4×C61—22no—
107C2×C4×C61—21no—
108C2×C4×C61—21no—
109C2×C4×C61—21no—
110C2×C4×C61—21no—
111C2×C4×C61—21no—
112C2×C4×C61—41no—
113C2×C4×C61—41no—
114C2×C4×C61—41no—
115C2×C4×C61#10012yes—
116C2×C4×C61C4×D412yes—
117C2×C4×C61#12011yescharacteristic
118C2×C4×C61#12112yes—
119C2×C4×C61#12311yescharacteristic
120C2×Dic61—42no—
121C2×Dic61—42no—
122C2×Dic61—42no—
123D244—84no—
124D244—82no—
125D244—84no—
126D244—82no—
127C8×C61—82no—
128#92418—44no—
129#9458—611nomaximal, Sylow
130C22×C61—21no—
131C22×C61—21no—
132C22×C61#38611yescharacteristic
133C22×C61C2×#10011yescharacteristic
134C22×C61#41311yescharacteristic
135C2×D61—82no—
136C2×D61—82no—
137C2×D61—82no—
138C2×D61—82no—
139C2×D61—82no—
140C2×D61—82no—
141C2×D61—82no—
142C4×C61—21no—
143C4×C61—22no—
144C4×C61—22no—
145C4×C61—22no—
146C4×C61—41no—
147C4×C61—41no—
148C4×C61—41no—
149C4×C61#45412yes—
150Dic61—82no—
151#2983—611no—
152#2548—611no—
153#1555—612no—
154#1467D6111yescharacteristic, complemented
155#1285—612no—
156C2×C61—21no—
157C2×C61—21no—
158C2×C61#188811yescharacteristic
159C2×C61#190011yescharacteristic
160C2×C61#191011yescharacteristic
161D61—162no—
162C2×#127—611no—
163C2×#124—612no—
164C2×C4×Q8C2×D6111yescharacteristic
165#446—1224no—
166#432—612no—
167#432—1221no—
168#432C2×D6112yes—
169#431—1221no—
170#394—1221no—
171#394—1221no—
172#390—612no—
173#390—1221no—
174#390—1221no—
175C4×C4⋊C4—611no—
176#331—1224no—
177#324—21no—
178#324—21no—
179C61#945811yescharacteristic, complemented, Sylow
180C22×Q8C4×D6111yescharacteristic
181C22×D4—612no—
182C22×D4—1221no—
183C22×D4—1221no—
184#127—1222no—
185#127—1222no—
186#124—2444no—
187C4×Q8—22no—
188C4×Q8—22no—
189C2×C4⋊C4—1221no—
190C2×C4⋊C4C4×D6111yescharacteristic
191C2×C4⋊C4C22⋊D6111yescharacteristic
192C2×C4⋊C4C22⋊D6111yescharacteristic
193C2×C22⋊C4—1222no—
194C2×C22⋊C4—1222no—
195C2×C22⋊C4—1222no—
196C2×C22⋊C4—1222no—
197C2×C42—612no—
198C2×C42D24411yescharacteristic
199C2×C42D24411yescharacteristic
200C2×C42C22×D6111yescharacteristic
201C4⋊C8—22no—
202C4⋊C8—614no—
203C4⋊C8—1222no—
204C4⋊C8—1222no—
205C22⋊C8—22no—
206C22⋊C8—1222no—
207#97—22no—
208#97—1222no—
209#97—1222no—
210C4×C8—1222no—
211#95—1221no—
212C24—1222no—
213C2×Q8—22no—
214C2×Q8—22no—
215C2×Q8—41no—
216C2×D4—1224no—
217C2×D4—4882no—
218C2×D4—2442no—
219C2×D4—2442no—
220C2×D4—2442no—
221C2×D4—2442no—
222C2×D4—2441no—
223C2×D4—2442no—
224C2×D4—2441no—
225C22×C4—1222no—
226C22×C4#1175819811yescharacteristic
227C22×C4#1175819811yescharacteristic
228C22×C4#1175819811yescharacteristic
229C22×C4#1175819811yescharacteristic
230C22×C4C2×C4×D6111yescharacteristic
231C22×C4C2×D24411yescharacteristic
232C22×C4C2×C22⋊D6111yescharacteristic
233C2×C8—42no—
234C2×C8—1224no—
235C4⋊C4—22no—
236C4⋊C4—22no—
237C4⋊C4—22no—
238C4⋊C4—2442no—
239C22⋊C4—4882no—
240C22⋊C4—4882no—
241C22⋊C4—2444no—
242C22⋊C4—2444no—
243C42—21no—
244C42—21no—
245C42—21no—
246C42—21no—
247C42—2442no—
248C42D4×D6112yes—
249C42#1175822412yes—
250C23—2442no—
251C23—2442no—
252C23—2442no—
253C23—2442no—
254C23—2442no—
255C23—2442no—
256C23—2442no—
257C23C2×#1175819811yescharacteristic
258Q8—44no—
259D4—4884no—
260D4—4884no—
261D4—4882no—
262D4—4882no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—22no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—2442no—
277C2×C4—2442no—
278C2×C4—2442no—
279C2×C4#1451969312yes—
280C2×C4#1451973712yes—
281C2×C4#1451974711yescharacteristic
282C2×C4#1451974912yes—
283C2×C4#1451976111yescharacteristic
284C8—82no—
285C8—2444no—
286C22—21no—
287C22—21no—
288C22—4882no—
289C22—4882no—
290C22—4882no—
291C22—4882no—
292C22—4882no—
293C22—4882no—
294C22—4882no—
295C22#1683963911yescharacteristic
296C22C2×#1451969311yescharacteristic
297C22#1683974411yescharacteristic
298C4—21no—
299C4—22no—
300C4—22no—
301C4—22no—
302C4—41no—
303C4—41no—
304C4—41no—
305C4—4882no—
306C4#1683993712yes—
307C2—21no—
308C2—21no—
309C2—9762no—
310C2#2144622011yescharacteristic
311C2#2144628011yescharacteristic
312C2#2144634411yescharacteristic
subgroupG/H
centerC22#16839639
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1467C2
socleC22×C61#386
prime radicalC2×#14519774C22
cosocleC2×C42×C61C23
hypercenter#1467D61
hypocenterC61#9458
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1467

Complements

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

NH ≅ G/Ntype
1. #21454904C2semidirect
4. #21443674C2semidirect
5. C61×#1467C2semidirect
154. #1467D61semidirect
179. C61#9458semidirect

Maximal subgroups

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

subgroupindex
#214549042
#214508182
#214445322
#214436742
C61×#14672
#945861