Finite Group Databasebeta

← #29577767

Subgroups — #29577767

Order 15616.

726 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 726 conjugacy classes of subgroups — too many to draw. Only its 111 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#21449473C211yesmaximal, characteristic
2#21445743C211yesmaximal, characteristic, complemented
3#21444347C212yesmaximal
4#21443671C212yesmaximal, complemented
5C61×#2310C211yesmaximal, characteristic, complemented
6C2×#14519766C412yescomplemented
7C2×#14519751C2211yescharacteristic
8C2×#14519696—22no—
9#16839726C2212yes—
10#16839673—22no—
11#16839669C412yescomplemented
12#16839641—21no—
13#16839641—21no—
14C4×C22⋊Dic61C2211yescharacteristic
15#16839495—24no—
16C23×Q8×C61C2211yescharacteristic
17C2×#117×C61—21no—
18C61×#391—24no—
19C61×#381—21no—
20C61×#381C2212yes—
21C22×D244C2×C411yescharacteristic
22C22×C22⋊D61—21no—
23C2×#11758194—22no—
24#14519766—44no—
25#14519751—22no—
26#14519751—22no—
27C2×C4⋊Dic61C2×C411yescharacteristic
28C2×C22⋊Dic61—21no—
29C2×#11758198—22no—
30C2×#11758198—22no—
31C2×#11758198—21no—
32C2×C4×Dic61C2×C412yes—
33#14519696—44no—
34C22⋊#92418—22no—
35C22⋊#92418—22no—
36#14519684—22no—
37#14519684—22no—
38#14519684—21no—
39C22×Q8×C61—22no—
40C22×Q8×C61—22no—
41C22×Q8×C61—44no—
42C22×Q8×C61C2×C412yes—
43C22×Q8×C61D412yes—
44C23×C4×C61D411yescharacteristic
45C23×C4×C61D411yescharacteristic
46C23×C4×C61C2311yescharacteristic
47#117×C61—44no—
48C2×C4⋊C4×C61—22no—
49C2×C22⋊C4×C61—22no—
50C2×C42×C61—22no—
51#95×C61—24no—
52#95×C61—22no—
53#95×C61D414yes—
54C24×C61C2×D411yescharacteristic
55C23×D61—21no—
56C2×Q8×C61—24no—
57C2×Q8×C61—24no—
58C2×Q8×C61—24no—
59C2×Q8×C61—24no—
60C2×Q8×C61—42no—
61C2×Q8×C61—42no—
62C2×Q8×C61—44no—
63C2×Q8×C61—44no—
64C2×Q8×C61—48no—
65C22×C4×C61—21no—
66C22×C4×C61—21no—
67C22×C4×C61—24no—
68C22×C4×C61—21no—
69C22×C4×C61—21no—
70C22×C4×C61—24no—
71C22×C4×C61—42no—
72C22×C4×C61—42no—
73C22×C4×C61C22⋊C412yes—
74C22×C4×C61C22⋊C412yes—
75C22×C4×C61C22×C411yescharacteristic
76C22×C4×C61C2×D411yescharacteristic
77C22×C4×C61C2×D412yes—
78C22×C4×C61#4112yes—
79C2×C22⋊D61—44no—
80C2×C22⋊D61—41no—
81C2×C22⋊D61—41no—
82C22×Dic61—21no—
83C22×Dic61—22no—
84C2×D244—22no—
85C2×D244—42no—
86C4⋊C4×C61—44no—
87C22⋊C4×C61—44no—
88C42×C61—44no—
89C22⋊Dic61—42no—
90#11758198—44no—
91#11758198—44no—
92#11758198—42no—
93C4⋊Dic61—22no—
94C4×Dic61—42no—
95#11758194—44no—
96C2×#92418—42no—
97C23×C61—22no—
98C23×C61—22no—
99C23×C61—41no—
100C23×C61C2×C22⋊C411yescharacteristic
101C23×C61#12011yescharacteristic
102C23×C61#12311yescharacteristic
103C22×D61—42no—
104C22×D61—42no—
105C22×D61—41no—
106C22×D61—42no—
107Q8×C61—48no—
108Q8×C61—48no—
109C2×C4×C61—22no—
110C2×C4×C61—22no—
111C2×C4×C61—24no—
112C2×C4×C61—22no—
113C2×C4×C61—22no—
114C2×C4×C61—22no—
115C2×C4×C61—22no—
116C2×C4×C61—22no—
117C2×C4×C61—22no—
118C2×C4×C61—22no—
119C2×C4×C61—42no—
120C2×C4×C61—42no—
121C2×C4×C61—42no—
122C2×C4×C61—42no—
123C2×C4×C61—44no—
124C2×C4×C61—44no—
125C2×C4×C61—44no—
126C2×C4×C61#9914yes—
127C2×C4×C61C4×D412yes—
128C2×C4×C61#12112yes—
129C22⋊D61—82no—
130C22⋊D61—82no—
131C2×Dic61—42no—
132C2×Dic61—42no—
133C2×Dic61—42no—
134C2×Dic61—41no—
135C2×Dic61—42no—
136D244—82no—
137#92418—82no—
138#9466—611nomaximal, Sylow
139C22×C61—21no—
140C22×C61—21no—
141C22×C61—21no—
142C22×C61—21no—
143C22×C61—21no—
144C22×C61—21no—
145C22×C61—21no—
146C22×C61—44no—
147C22×C61#38611yescharacteristic
148C22×C61C2×#9912yes—
149C22×C61#45412yes—
150C2×D61—84no—
151C2×D61—81no—
152C2×D61—81no—
153C2×D61—81no—
154C4×C61—22no—
155C4×C61—22no—
156C4×C61—24no—
157C4×C61—42no—
158C4×C61—84no—
159Dic61—82no—
160Dic61—81no—
161#2436—611no—
162#2310D6111yescharacteristic, complemented
163#1784—611no—
164#1510—612no—
165#1284—612no—
166C2×C61—22no—
167C2×C61—22no—
168C2×C61—41no—
169C2×C61#188811yescharacteristic
170C2×C61#189811yescharacteristic
171C2×C61#191211yescharacteristic
172D61—161no—
173C23×Q8C2×D6111yescharacteristic
174C2×#124—612no—
175C2×#121—611no—
176C2×#117—21no—
177C2×#101—1222no—
178#407—612no—
179#393—1222no—
180#391—24no—
181#391—612no—
182#386—1221no—
183#386—1221no—
184#381—21no—
185#381C2×D6112yes—
186C4×C22⋊C4—611no—
187#327—1224no—
188C61#946611yescharacteristic, complemented, Sylow
189C22×Q8—22no—
190C22×Q8—22no—
191C22×Q8—44no—
192C22×Q8C4×D6112yes—
193C22×Q8D24412yes—
194C22×D4—611no—
195C22×D4—1221no—
196C23×C4C22⋊D6111yescharacteristic
197C23×C4C22⋊D6111yescharacteristic
198C23×C4C22×D6111yescharacteristic
199C2×M16—1222no—
200#124—2444no—
201#121—1222no—
202#121—1222no—
203#117—44no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—611no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—1221no—
208C2×C22⋊C4—1222no—
209C2×C22⋊C4—1222no—
210C2×C22⋊C4—1221no—
211C2×C42—22no—
212C2×C42—612no—
213#101—2444no—
214C22⋊C8—1222no—
215C22⋊C8—1222no—
216#95—24no—
217#95—22no—
218#95—1222no—
219#95—1222no—
220#95—1221no—
221#95C22⋊D6114yes—
222C24—1221no—
223C24C2×C22⋊D6111yescharacteristic
224C2×Q8—24no—
225C2×Q8—24no—
226C2×Q8—24no—
227C2×Q8—24no—
228C2×Q8—42no—
229C2×Q8—42no—
230C2×Q8—44no—
231C2×Q8—44no—
232C2×Q8—48no—
233C2×D4—1222no—
234C2×D4—2442no—
235C2×D4—2444no—
236C2×D4—2441no—
237C2×D4—2441no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—24no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—24no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—1221no—
247C22×C4—1222no—
248C22×C4#1175819812yes—
249C22×C4#1175819812yes—
250C22×C4C2×C4×D6111yescharacteristic
251C22×C4C2×D24411yescharacteristic
252C22×C4#1175822212yes—
253C22×C4C2×C22⋊D6112yes—
254M16—2444no—
255C2×C8—2442no—
256C4⋊C4—44no—
257C4⋊C4—1222no—
258C22⋊C4—44no—
259C22⋊C4—2444no—
260C22⋊C4—2442no—
261C22⋊C4—2444no—
262C22⋊C4—2442no—
263C42—44no—
264C42—2442no—
265C23—22no—
266C23—22no—
267C23—41no—
268C23—2442no—
269C23—2442no—
270C23—2442no—
271C23—2441no—
272C23C2×#1175819811yescharacteristic
273C23#1451974611yescharacteristic
274C23#1451976011yescharacteristic
275Q8—48no—
276Q8—48no—
277D4—4882no—
278D4—4882no—
279D4—4882no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—24no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—44no—
295C2×C4—44no—
296C2×C4—44no—
297C2×C4—2442no—
298C2×C4—2442no—
299C2×C4—2442no—
300C2×C4—2442no—
301C2×C4—2441no—
302C2×C4#1451969114yes—
303C2×C4C4×C22⋊D6112yes—
304C2×C4#1451975112yes—
305C8—4882no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—44no—
314C22—4884no—
315C22—4881no—
316C22—4881no—
317C22—4881no—
318C22#1683964111yescharacteristic
319C22C2×#1451969112yes—
320C22#1683993512yes—
321C4—22no—
322C4—24no—
323C4—22no—
324C4—42no—
325C4—84no—
326C4—4882no—
327C4—4881no—
328C2—22no—
329C2—22no—
330C2—41no—
331C2—9761no—
332C2#2144622211yescharacteristic
333C2#2144627211yescharacteristic
334C2#2144635611yescharacteristic
subgroupG/H
centerC22#16839641
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#2310C2
socleC22×C61#386
prime radicalC2×#14519751C22
cosocleC23×C4×C61C23
hypercenter#2310D61
hypocenterC61#9466
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2310

Complements

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

NH ≅ G/Ntype
2. #21445743C2semidirect
4. #21443671C2semidirect
5. C61×#2310C2semidirect
6. C2×#14519766C4semidirect
11. #16839669C4semidirect
162. #2310D61semidirect
188. C61#9466semidirect

Maximal subgroups

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

subgroupindex
#214494732
#214457432
#214443472
#214436712
C61×#23102
#946661