Finite Group Databasebeta

← #29572372

Subgroups — #29572372

Order 15616.

692 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 692 conjugacy classes of subgroups — too many to draw. Only its 131 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#21446652C211yesmaximal, characteristic
2#21444549C212yesmaximal, complemented
3#21444454C211yesmaximal, characteristic
4#21444331C212yesmaximal, complemented
5C61×#3550C211yesmaximal, characteristic
6#16839819C2212yes—
7#16839786C2211yescharacteristic
8C2×#14519704—22no—
9#16839608C2211yescharacteristic
10#16839511—22no—
11#16839508—22no—
12#16839502—22no—
13#16839499—24no—
14C61×#563—22no—
15C61×#559C412yescomplemented
16Q82×C61C412yescomplemented
17C61×#556—22no—
18D4×Q8×C61—22no—
19C61×#542—24no—
20C61×#536—22no—
21C61×#533C2212yes—
22C61×#532C2211yescharacteristic
23C2×#11758194—21no—
24C2×C4×Dic61—21no—
25C4⋊#92418—22no—
26C4⋊#92418—22no—
27#14519704—44no—
28C22⋊#92418—22no—
29#14519688—22no—
30#14519684—22no—
31C2×#41×C61D412yes—
32C22×Q8×C61—22no—
33C4⋊Q8×C61—24no—
34C4⋊Q8×C61—24no—
35C4⋊Q8×C61C2×C411yescharacteristic
36C4⋊Q8×C61D412yes—
37C4⋊Q8×C61D412yes—
38#127×C61C2×C411yescharacteristic
39#126×C61—28no—
40#125×C61—22no—
41#124×C61—24no—
42#124×C61—24no—
43#124×C61D412yes—
44#123×C61—24no—
45#123×C61—24no—
46C22⋊Q8×C61—24no—
47C22⋊Q8×C61—28no—
48C22⋊Q8×C61—22no—
49C22⋊Q8×C61D414yes—
50#121×C61—22no—
51C4×Q8×C61—24no—
52C4×Q8×C61—22no—
53C4×Q8×C61C2×C414yes—
54C4×D4×C61—24no—
55C4×D4×C61—22no—
56#117×C61—22no—
57#117×C61—22no—
58C2×C42×C61C2311yescharacteristic
59#41×C61—42no—
60#41×C61—42no—
61C2×Q8×C61—24no—
62C2×Q8×C61—22no—
63C2×Q8×C61—82no—
64C2×Q8×C61—82no—
65C2×Q8×C61C22⋊C414yes—
66C2×Q8×C61C22⋊C412yes—
67C2×D4×C61—22no—
68C2×D4×C61—22no—
69C2×D4×C61C22⋊C412yes—
70C22×C4×C61—24no—
71C22×C4×C61—22no—
72C22×C4×C61C2×D411yescharacteristic
73C22×C4×C61C2×D412yes—
74C22×Dic61—41no—
75C4⋊C4×C61—28no—
76C4⋊C4×C61—22no—
77C4⋊C4×C61—24no—
78C4⋊C4×C61—22no—
79C4⋊C4×C61—22no—
80C4⋊C4×C61—22no—
81C4⋊C4×C61—24no—
82C4⋊C4×C61C22⋊C414yes—
83C22⋊C4×C61—24no—
84C22⋊C4×C61—28no—
85C22⋊C4×C61—24no—
86C22⋊C4×C61—24no—
87C42×C61—24no—
88C42×C61—24no—
89C42×C61C22×C411yescharacteristic
90C42×C61C2×D412yes—
91C42×C61C2×D411yescharacteristic
92C4×Dic61—22no—
93C4×Dic61—41no—
94#11758194—42no—
95C2×#92418—42no—
96C2×#92418—41no—
97C23×C61—22no—
98C23×C61#12011yescharacteristic
99Q8×C61—44no—
100Q8×C61—42no—
101Q8×C61—42no—
102Q8×C61—42no—
103Q8×C61—84no—
104D4×C61—42no—
105D4×C61—44no—
106D4×C61—82no—
107C2×C4×C61—21no—
108C2×C4×C61—21no—
109C2×C4×C61—24no—
110C2×C4×C61—22no—
111C2×C4×C61—24no—
112C2×C4×C61—42no—
113C2×C4×C61—42no—
114C2×C4×C61—42no—
115C2×C4×C61—84no—
116C2×C4×C61C2×C22⋊C412yes—
117C2×C4×C61C2×C22⋊C411yescharacteristic
118C2×C4×C61#12012yes—
119C2×C4×C61#12011yescharacteristic
120C2×Dic61—42no—
121C2×Dic61—82no—
122#92418—82no—
123#92418—81no—
124#8492—611nomaximal, Sylow
125C22×C61—21no—
126C22×C61—22no—
127C22×C61—42no—
128C22×C61—44no—
129C22×C61#37911yescharacteristic
130C4×C61—21no—
131C4×C61—22no—
132C4×C61—22no—
133C4×C61—42no—
134C4×C61—44no—
135C4×C61—44no—
136C4×C61#45311yescharacteristic
137C4×C61#45511yescharacteristic
138Dic61—82no—
139#3550D6111yescharacteristic
140#1963—611no—
141#1559—612no—
142#1536—611no—
143#1506—612no—
144C2×C61—41no—
145C2×C61—82no—
146C2×C61#179211yescharacteristic
147C2×C61#179612yes—
148#563—22no—
149#559Dic6112yescomplemented
150Q82Dic6112yescomplemented
151#556—22no—
152D4×Q8—22no—
153#542—24no—
154#536—22no—
155#533C2×D6112yes—
156#532C2×D6111yescharacteristic
157#432—612no—
158#423—611no—
159C2×#104—1222no—
160#376—611no—
161#333—1222no—
162#332—1222no—
163#329—1222no—
164#328—1224no—
165C61#849211yescharacteristic, complemented, Sylow
166C2×#41C22⋊D6112yes—
167C22×Q8—22no—
168C2×M16—1221no—
169C4⋊Q8—24no—
170C4⋊Q8—24no—
171C4⋊Q8C2×Dic6111yescharacteristic
172C4⋊Q8C22⋊D6112yes—
173C4⋊Q8C22⋊D6112yes—
174#127C2×Dic6111yescharacteristic
175#126—28no—
176#125—22no—
177#124—24no—
178#124—24no—
179#124C22⋊D6112yes—
180#123—24no—
181#123—24no—
182C22⋊Q8—24no—
183C22⋊Q8—28no—
184C22⋊Q8—22no—
185C22⋊Q8C22⋊D6114yes—
186#121—22no—
187C4×Q8—24no—
188C4×Q8—22no—
189C4×Q8C2×Dic6114yes—
190C4×D4—24no—
191C4×D4—22no—
192#117—22no—
193#117—22no—
194C2×C42—1221no—
195C2×C42C22×D6111yescharacteristic
196C4⋊C8—1222no—
197C4⋊C8—1222no—
198#104—2444no—
199C22⋊C8—1222no—
200#97—1222no—
201#95—1222no—
202#41—42no—
203#41—42no—
204C2×Q8—24no—
205C2×Q8—22no—
206C2×Q8—82no—
207C2×Q8—82no—
208C2×Q8C22⋊Dic6114yes—
209C2×Q8C22⋊Dic6112yes—
210C2×D4—22no—
211C2×D4—22no—
212C2×D4C22⋊Dic6112yes—
213C22×C4—24no—
214C22×C4—22no—
215C22×C4—2441no—
216C22×C4C2×C22⋊D6111yescharacteristic
217C22×C4C2×C22⋊D6112yes—
218M16—2442no—
219C2×C8—2442no—
220C2×C8—2441no—
221C4⋊C4—28no—
222C4⋊C4—22no—
223C4⋊C4—24no—
224C4⋊C4—22no—
225C4⋊C4—22no—
226C4⋊C4—22no—
227C4⋊C4—24no—
228C4⋊C4C22⋊Dic6114yes—
229C22⋊C4—24no—
230C22⋊C4—28no—
231C22⋊C4—24no—
232C22⋊C4—24no—
233C42—24no—
234C42—24no—
235C42—1222no—
236C42—2441no—
237C42C22×Dic6111yescharacteristic
238C42C2×C22⋊D6111yescharacteristic
239C42C2×C22⋊D6112yes—
240C23—22no—
241C23#1451974611yescharacteristic
242Q8—44no—
243Q8—42no—
244Q8—42no—
245Q8—42no—
246Q8—84no—
247D4—42no—
248D4—44no—
249D4—82no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—24no—
253C2×C4—22no—
254C2×C4—24no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—84no—
259C2×C4—4882no—
260C2×C4—2442no—
261C2×C4C2×C22⋊Dic6112yes—
262C2×C4C2×C22⋊Dic6111yescharacteristic
263C2×C4#1451974612yes—
264C2×C4#1451974611yescharacteristic
265C8—4882no—
266C8—4881no—
267C22—21no—
268C22—22no—
269C22—42no—
270C22—44no—
271C22#1683961811yescharacteristic
272C4—21no—
273C4—22no—
274C4—22no—
275C4—42no—
276C4—44no—
277C4—44no—
278C4—4882no—
279C4#1683993011yescharacteristic
280C4#1683994011yescharacteristic
281C2—41no—
282C2—82no—
283C2#2144577311yescharacteristic
284C2#2144579412yes—
subgroupG/H
centerC22#16839618
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#3550C2
socleC22×C61#379
prime radicalC61×#532C22
cosocleC2×C42×C61C23
hypercenter#3550D61
hypocenterC61#8492
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3550

Complements

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

NH ≅ G/Ntype
2. #21444549C2semidirect
4. #21444331C2semidirect
15. C61×#559C4semidirect
16. Q82×C61C4semidirect
149. #559Dic61semidirect
150. Q82Dic61semidirect
165. C61#8492semidirect

Maximal subgroups

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

subgroupindex
#214466522
#214445492
#214444542
#214443312
C61×#35502
#849261