Finite Group Databasebeta

← #29589651

Subgroups — #29589651

Order 15616.

462 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 87 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#21450818C211yesmaximal, characteristic
2#21444163C211yesmaximal, characteristic, complemented
3#21443750C214yesmaximal, complemented
4C61×#1461C211yesmaximal, characteristic, complemented
5C2×#14519774C411yescharacteristic
6C2×#14519766C412yes—
7#16839819—21no—
8#16839679—21no—
9#16839679—21no—
10#16839664—21no—
11#16839664—21no—
12#16839663C2212yes—
13C4×C4⋊Dic61C411yescharacteristic
14#16839488—21no—
15#16839488C2212yes—
16C2×C4×Q8×C61C2211yescharacteristic
17C61×#431—21no—
18C61×#324—21no—
19C61×#324C2212yes—
20C22×D244—21no—
21C22×D244—21no—
22C22×D244C2×C412yes—
23#14519774—41no—
24#14519774—41no—
25#14519766—44no—
26C2×C4⋊Dic61—21no—
27C2×#11758198—22no—
28C2×#11758198—22no—
29C2×#11758198—22no—
30C2×#11758198—22no—
31C2×C4×Dic61C2×C412yes—
32C4⋊#92418—41no—
33C22⋊#92418—22no—
34#14519688—41no—
35#14519684—21no—
36C22×Q8×C61C2×C411yescharacteristic
37C4×Q8×C61—41no—
38C4×Q8×C61—41no—
39C2×C4⋊C4×C61D411yescharacteristic
40C2×C4⋊C4×C61D411yescharacteristic
41C2×C4⋊C4×C61C2311yescharacteristic
42C2×C42×C61C2×C411yescharacteristic
43C2×C42×C61D411yescharacteristic
44C2×C42×C61D411yescharacteristic
45C4⋊C8×C61—41no—
46C22⋊C8×C61—22no—
47C4×C8×C61—41no—
48C23×D61—22no—
49C2×Q8×C61—22no—
50C2×Q8×C61—22no—
51C2×Q8×C61—41no—
52C22×C4×C61C22⋊C411yescharacteristic
53C22×C4×C61C22⋊C411yescharacteristic
54C22×C4×C61C22⋊C411yescharacteristic
55C22×C4×C61C22⋊C411yescharacteristic
56C22×C4×C61C22×C411yescharacteristic
57C22×C4×C61C2×D411yescharacteristic
58C22×C4×C61C2×D411yescharacteristic
59C22×Dic61—22no—
60C2×D244—24no—
61C2×D244—42no—
62C2×D244—42no—
63C2×D244—42no—
64C2×D244—42no—
65C2×D244—42no—
66C2×D244—41no—
67C2×D244—41no—
68C2×D244—82no—
69C2×C8×C61—42no—
70C4⋊C4×C61—41no—
71C4⋊C4×C61—41no—
72C4⋊C4×C61—41no—
73C42×C61—21no—
74C42×C61—21no—
75C42×C61—21no—
76C42×C61—21no—
77C42×C61—21no—
78C42×C61—21no—
79#11758198—44no—
80#11758198—44no—
81#11758198—82no—
82#11758198—82no—
83C4⋊Dic61—42no—
84C4×Dic61—42no—
85C2×#92418—42no—
86C23×C61C2×C22⋊C411yescharacteristic
87C22×D61—42no—
88C22×D61—42no—
89C22×D61—42no—
90C22×D61—42no—
91C22×D61—42no—
92C22×D61—42no—
93C22×D61—42no—
94Q8×C61—44no—
95C2×C4×C61—21no—
96C2×C4×C61—21no—
97C2×C4×C61—21no—
98C2×C4×C61—21no—
99C2×C4×C61—22no—
100C2×C4×C61—21no—
101C2×C4×C61—21no—
102C2×C4×C61—21no—
103C2×C4×C61—21no—
104C2×C4×C61—21no—
105C2×C4×C61—21no—
106C2×C4×C61—41no—
107C2×C4×C61—41no—
108C2×C4×C61—41no—
109C2×C4×C61#9911yescharacteristic
110C2×C4×C61#9911yescharacteristic
111C2×C4×C61#10012yes—
112C2×C4×C61#13112yes—
113C2×Dic61—42no—
114C2×Dic61—42no—
115C2×Dic61—42no—
116D244—84no—
117D244—82no—
118D244—84no—
119D244—82no—
120C8×C61—82no—
121#92418—82no—
122#11567—611nomaximal, Sylow
123C22×C61—21no—
124C22×C61—21no—
125C22×C61#40811yescharacteristic
126C22×C61C2×#9911yescharacteristic
127C22×C61C2×#10011yescharacteristic
128C2×D61—82no—
129C2×D61—82no—
130C2×D61—82no—
131C2×D61—82no—
132C2×D61—82no—
133C2×D61—82no—
134C2×D61—82no—
135C4×C61—21no—
136C4×C61—22no—
137C4×C61—22no—
138C4×C61—41no—
139C4×C61—41no—
140C4×C61—41no—
141C4×C61—41no—
142C4×C61#35512yes—
143Dic61—82no—
144#2548—611no—
145#1467—611no—
146#1461D6111yescharacteristic, complemented
147#1318—614no—
148C2×C61—21no—
149C2×C61—21no—
150C2×C61#147111yescharacteristic
151C2×C61#212811yescharacteristic
152C2×C61C2×#35511yescharacteristic
153D61—162no—
154C2×#127—611no—
155C2×#124—612no—
156C2×C4×Q8C2×D6111yescharacteristic
157#432—1221no—
158#431—21no—
159#394—1221no—
160#394—1221no—
161#390—612no—
162#390—1221no—
163#390—1221no—
164C4×C4⋊C4—611no—
165#324—21no—
166#324—612no—
167#324—1221no—
168#324C2×D6112yes—
169C61#1156711yescharacteristic, complemented, Sylow
170C22×Q8C4×D6111yescharacteristic
171C22×D4—612no—
172C22×D4—1221no—
173C22×D4—1221no—
174#127—2441no—
175#127—2441no—
176#124—2444no—
177C4×Q8—41no—
178C4×Q8—41no—
179C2×C4⋊C4—1221no—
180C2×C4⋊C4D24411yescharacteristic
181C2×C4⋊C4C22⋊D6111yescharacteristic
182C2×C4⋊C4C22×D6111yescharacteristic
183C2×C22⋊C4—1222no—
184C2×C22⋊C4—1222no—
185C2×C22⋊C4—1222no—
186C2×C22⋊C4—1222no—
187C2×C42—612no—
188C2×C42C4×D6111yescharacteristic
189C2×C42D24411yescharacteristic
190C2×C42C22⋊D6111yescharacteristic
191C4⋊C8—41no—
192C4⋊C8—2441no—
193C22⋊C8—22no—
194C22⋊C8—1222no—
195#97—2441no—
196C4×C8—41no—
197#95—1221no—
198C24—1222no—
199C2×Q8—22no—
200C2×Q8—22no—
201C2×Q8—41no—
202C2×D4—1224no—
203C2×D4—4882no—
204C2×D4—2442no—
205C2×D4—2442no—
206C2×D4—2442no—
207C2×D4—2442no—
208C2×D4—2442no—
209C2×D4—2441no—
210C2×D4—2441no—
211C22×C4—1222no—
212C22×C4#1175819811yescharacteristic
213C22×C4#1175819811yescharacteristic
214C22×C4#1175819811yescharacteristic
215C22×C4#1175819811yescharacteristic
216C22×C4C2×C4×D6111yescharacteristic
217C22×C4C2×D24411yescharacteristic
218C22×C4C2×C22⋊D6111yescharacteristic
219C2×C8—42no—
220C2×C8—2442no—
221C4⋊C4—41no—
222C4⋊C4—41no—
223C4⋊C4—41no—
224C4⋊C4—2442no—
225C22⋊C4—4882no—
226C22⋊C4—4882no—
227C22⋊C4—2444no—
228C22⋊C4—2444no—
229C42—21no—
230C42—21no—
231C42—21no—
232C42—21no—
233C42—21no—
234C42—21no—
235C42—2442no—
236C23—2442no—
237C23—2442no—
238C23—2442no—
239C23—2442no—
240C23—2442no—
241C23—2442no—
242C23—2442no—
243C23C2×#1175819811yescharacteristic
244Q8—44no—
245D4—4884no—
246D4—4882no—
247D4—4884no—
248D4—4882no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—2442no—
264C2×C4—2442no—
265C2×C4—2442no—
266C2×C4#1451969111yescharacteristic
267C2×C4#1451969111yescharacteristic
268C2×C4#1451969312yes—
269C2×C4#1451978312yes—
270C8—82no—
271C8—4882no—
272C22—21no—
273C22—21no—
274C22—4882no—
275C22—4882no—
276C22—4882no—
277C22—4882no—
278C22—4882no—
279C22—4882no—
280C22—4882no—
281C22#1683972811yescharacteristic
282C22C2×#1451969111yescharacteristic
283C22C2×#1451969311yescharacteristic
284C4—21no—
285C4—22no—
286C4—22no—
287C4—41no—
288C4—41no—
289C4—41no—
290C4—41no—
291C4—4882no—
292C4#1683955812yes—
293C2—21no—
294C2—21no—
295C2—9762no—
296C2#2144417811yescharacteristic
297C2#2144749011yescharacteristic
298C2C2×#1683955811yescharacteristic
subgroupG/H
centerC22C2×#14519691
commutatorC22×C4×C61C22×C4
FrattiniC2×C4⋊C4C22×D61
FittingC61×#1461C2
socleC22×C61C2×#99
prime radicalC2×#14519774C4
cosocleC2×C4⋊C4×C61C23
hypercenter#1461D61
hypocenterC61#11567
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #1461

Complements

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

NH ≅ G/Ntype
2. #21444163C2semidirect
3. #21443750C2semidirect
4. C61×#1461C2semidirect
146. #1461D61semidirect
169. C61#11567semidirect

Maximal subgroups

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

subgroupindex
#214508182
#214441632
#214437502
C61×#14612
#1156761