Finite Group Databasebeta

← #29589659

Subgroups — #29589659

Order 15616.

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

normal subgroups only

This group has 424 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#21450992C211yesmaximal, characteristic
2#21444157C211yesmaximal, characteristic
3#21443762C214yesmaximal, complemented
4C61×#1459C211yesmaximal, characteristic
5C2×#14519777C411yescharacteristic
6#16839726—21no—
7#16839674—21no—
8#16839674—21no—
9#16839665C412yes—
10#16839627C2212yes—
11#16839622—21no—
12#16839622—21no—
13C4×C4⋊Dic61C411yescharacteristic
14#16839488C2212yes—
15#16839486—21no—
16C2×C4×D4×C61C2211yescharacteristic
17C2×C22⋊C8×C61—21no—
18C61×#324C2212yes—
19C61×#323—21no—
20C22×Dic122—21no—
21C2×#11758194—41no—
22#14519777—41no—
23#14519777—41no—
24C2×C4⋊Dic61—21no—
25C2×C4⋊Dic61—21no—
26C2×C4⋊Dic61C2×C412yes—
27C2×C22⋊Dic61—22no—
28C2×C22⋊Dic61—22no—
29C2×C4×Dic61C2×C412yes—
30C22⋊#92418—22no—
31C22⋊#92418—41no—
32#14519684—21no—
33#14519684—22no—
34#14519684—22no—
35C22×D4×C61C2×C411yescharacteristic
36C23×C4×C61D411yescharacteristic
37C23×C4×C61D411yescharacteristic
38C22×C8×C61—41no—
39C4×D4×C61—41no—
40C4×D4×C61—41no—
41C2×C4⋊C4×C61C2311yescharacteristic
42C2×C22⋊C4×C61D411yescharacteristic
43C2×C22⋊C4×C61D411yescharacteristic
44C2×C42×C61C2×C411yescharacteristic
45C22⋊C8×C61—22no—
46C22⋊C8×C61—41no—
47C24×C61C22⋊C411yescharacteristic
48C24×C61C22⋊C411yescharacteristic
49C2×D4×C61—22no—
50C2×D4×C61—22no—
51C2×D4×C61—41no—
52C22×C4×C61—21no—
53C22×C4×C61—21no—
54C22×C4×C61—21no—
55C22×C4×C61—21no—
56C22×C4×C61—42no—
57C22×C4×C61—42no—
58C22×C4×C61C22⋊C411yescharacteristic
59C22×C4×C61C22⋊C411yescharacteristic
60C22×C4×C61C22×C411yescharacteristic
61C22×C4×C61C2×D411yescharacteristic
62C22×C4×C61C2×D411yescharacteristic
63C22×Dic61—22no—
64C22×Dic61—22no—
65C2×Dic122—42no—
66C2×Dic122—82no—
67C2×C8×C61—42no—
68C2×C8×C61—82no—
69C4⋊C4×C61—41no—
70C22⋊C4×C61—41no—
71C22⋊C4×C61—41no—
72C42×C61—21no—
73C42×C61—21no—
74C22⋊Dic61—82no—
75C22⋊Dic61—82no—
76C4⋊Dic61—42no—
77C4⋊Dic61—41no—
78C4⋊Dic61—41no—
79C4⋊Dic61—42no—
80C4×Dic61—24no—
81#11758194—82no—
82C2×#92418—42no—
83C23×C61—21no—
84C23×C61—21no—
85C23×C61—21no—
86C23×C61—21no—
87C23×C61—21no—
88C23×C61—41no—
89C23×C61—41no—
90C23×C61—41no—
91C23×C61—41no—
92C23×C61#10112yes—
93C23×C61C2×C22⋊C411yescharacteristic
94D4×C61—44no—
95C2×C4×C61—21no—
96C2×C4×C61—21no—
97C2×C4×C61—22no—
98C2×C4×C61—21no—
99C2×C4×C61—21no—
100C2×C4×C61—21no—
101C2×C4×C61—42no—
102C2×C4×C61—42no—
103C2×C4×C61—41no—
104C2×C4×C61—41no—
105C2×C4×C61—41no—
106C2×C4×C61—81no—
107C2×C4×C61—81no—
108C2×C4×C61#9911yescharacteristic
109C2×C4×C61#9911yescharacteristic
110C2×C4×C61#13112yes—
111C2×Dic61—42no—
112C2×Dic61—42no—
113C2×Dic61—42no—
114C2×Dic61—42no—
115C2×Dic61—42no—
116C2×Dic61—42no—
117Dic122—84no—
118C8×C61—82no—
119#92418—82no—
120#11569—611nomaximal, Sylow
121C22×C61—21no—
122C22×C61—21no—
123C22×C61—22no—
124C22×C61—42no—
125C22×C61—42no—
126C22×C61—41no—
127C22×C61—41no—
128C22×C61—41no—
129C22×C61—41no—
130C22×C61—41no—
131C22×C61—81no—
132C22×C61—81no—
133C22×C61#40811yescharacteristic
134C22×C61C2×#9911yescharacteristic
135C22×C61C2×#10111yescharacteristic
136C4×C61—21no—
137C4×C61—22no—
138C4×C61—41no—
139C4×C61—41no—
140C4×C61—41no—
141C4×C61#35412yes—
142Dic61—82no—
143Dic61—82no—
144#2563—611no—
145#1465—611no—
146#1459D6111yescharacteristic
147#1324—614no—
148C2×C61—21no—
149C2×C61—21no—
150C2×C61—42no—
151C2×C61—81no—
152C2×C61#147211yescharacteristic
153C2×C61C2×#35411yescharacteristic
154C2×C61#213411yescharacteristic
155C2×C4⋊Q8—611no—
156C2×C4×D4C2×D6111yescharacteristic
157#407—1221no—
158C2×C22⋊C8—21no—
159#393—1221no—
160#393—1221no—
161#390—612no—
162#382—612no—
163#380—1221no—
164#380—1221no—
165C4×C4⋊C4—611no—
166#324—612no—
167#324C2×D6112yes—
168#323—21no—
169#323—1221no—
170C61#1156911yescharacteristic, complemented, Sylow
171C22×Q8—1221no—
172C22×D4C4×D6111yescharacteristic
173C23×C4D24411yescharacteristic
174C23×C4C22⋊D6111yescharacteristic
175C2×M16—2441no—
176C22×C8—41no—
177C4⋊Q8—2441no—
178C4⋊Q8—2441no—
179C4×D4—41no—
180C4×D4—41no—
181C2×C4⋊C4—612no—
182C2×C4⋊C4—1221no—
183C2×C4⋊C4—1221no—
184C2×C4⋊C4C22×D6111yescharacteristic
185C2×C22⋊C4—1222no—
186C2×C22⋊C4—1222no—
187C2×C22⋊C4D24411yescharacteristic
188C2×C22⋊C4C22⋊D6111yescharacteristic
189C2×C42—612no—
190C2×C42C4×D6111yescharacteristic
191C22⋊C8—22no—
192C22⋊C8—41no—
193C22⋊C8—1222no—
194C22⋊C8—2441no—
195#95—1221no—
196#95—1222no—
197#95—1222no—
198C24#1175819811yescharacteristic
199C24#1175819811yescharacteristic
200C2×Q8—4882no—
201C2×Q8—2442no—
202C2×D4—22no—
203C2×D4—22no—
204C2×D4—41no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—1222no—
212C22×C4—1222no—
213C22×C4#1175819811yescharacteristic
214C22×C4#1175819811yescharacteristic
215C22×C4C2×C4×D6111yescharacteristic
216C22×C4C2×D24411yescharacteristic
217C22×C4C2×C22⋊D6111yescharacteristic
218M16—4882no—
219C2×C8—42no—
220C2×C8—82no—
221C2×C8—2442no—
222C4⋊C4—41no—
223C4⋊C4—2442no—
224C4⋊C4—2442no—
225C4⋊C4—2441no—
226C4⋊C4—2441no—
227C22⋊C4—41no—
228C22⋊C4—41no—
229C22⋊C4—4882no—
230C22⋊C4—4882no—
231C42—21no—
232C42—21no—
233C42—1224no—
234C23—21no—
235C23—21no—
236C23—21no—
237C23—21no—
238C23—21no—
239C23—41no—
240C23—41no—
241C23—41no—
242C23—41no—
243C23#1451969512yes—
244C23C2×#1175819811yescharacteristic
245Q8—4884no—
246D4—44no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—22no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—81no—
259C2×C4—81no—
260C2×C4—2442no—
261C2×C4—2442no—
262C2×C4—2442no—
263C2×C4—2442no—
264C2×C4—2442no—
265C2×C4—2442no—
266C2×C4#1451969111yescharacteristic
267C2×C4#1451969111yescharacteristic
268C2×C4#1451978312yes—
269C8—82no—
270C8—4882no—
271C22—21no—
272C22—21no—
273C22—22no—
274C22—42no—
275C22—42no—
276C22—41no—
277C22—41no—
278C22—41no—
279C22—41no—
280C22—41no—
281C22—81no—
282C22—81no—
283C22#1683972811yescharacteristic
284C22C2×#1451969111yescharacteristic
285C22C2×#1451969511yescharacteristic
286C4—21no—
287C4—22no—
288C4—41no—
289C4—41no—
290C4—41no—
291C4—4882no—
292C4—4882no—
293C4#1683955612yes—
294C2—21no—
295C2—21no—
296C2—42no—
297C2—81no—
298C2#2144418211yescharacteristic
299C2C2×#1683955611yescharacteristic
300C2#2144751011yescharacteristic
subgroupG/H
centerC22C2×#14519691
commutatorC22×C4×C61C22×C4
FrattiniC2×C4⋊C4C22×D61
FittingC61×#1459C2
socleC22×C61C2×#99
prime radicalC22×D4×C61C2×C4
cosocleC2×C4⋊C4×C61C23
hypercenter#1459D61
hypocenterC61#11569
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1459

Complements

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

NH ≅ G/Ntype
3. #21443762C2semidirect
170. C61#11569semidirect

Maximal subgroups

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

subgroupindex
#214509922
#214441572
#214437622
C61×#14592
#1156961