Finite Group Databasebeta

← #29572384

Subgroups — #29572384

Order 15616.

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

normal subgroups only

This group has 732 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#21446801C211yesmaximal, characteristic, complemented
2#21444541C212yesmaximal, complemented
3#21444457C211yesmaximal, characteristic, complemented
4#21444327C212yesmaximal, complemented
5C61×#3548C211yesmaximal, characteristic
6#16839819C2212yes—
7#16839786C2211yescharacteristic
8#16839608C2211yescharacteristic
9#16839521—22no—
10#16839508—22no—
11#16839506—22no—
12#16839504—22no—
13#16839496—24no—
14C61×#562—22no—
15C61×#554—22no—
16C61×#550—22no—
17C61×#547C414yescomplemented
18C61×#540—22no—
19C61×#537—24no—
20C61×#533C2211yescharacteristic
21C61×#532C2212yes—
22C2×#11758194—21no—
23C2×C4×Dic61—21no—
24C4⋊#92418—22no—
25C4⋊#92418—22no—
26C22⋊#92418—22no—
27#14519688—22no—
28#14519684—22no—
29C2×#41×C61—24no—
30C2×#41×C61—22no—
31C4⋊Q8×C61C2×C412yes—
32C4⋊Q8×C61D412yes—
33#127×C61—22no—
34#127×C61D412yes—
35#125×C61—22no—
36#125×C61D412yes—
37#124×C61—22no—
38#123×C61—28no—
39C22⋊Q8×C61—28no—
40C22⋊Q8×C61—22no—
41C22⋊Q8×C61—24no—
42#121×C61—28no—
43#121×C61—28no—
44#121×C61—24no—
45#121×C61—22no—
46#121×C61D414yes—
47C4×Q8×C61—22no—
48C4×D4×C61—24no—
49C4×D4×C61—22no—
50C4×D4×C61C2×C414yes—
51#117×C61D412yes—
52C2×C4⋊C4×C61—24no—
53C2×C42×C61C2311yescharacteristic
54#41×C61—84no—
55#41×C61—88no—
56C2×Q8×C61—22no—
57C2×Q8×C61—22no—
58C2×D4×C61—28no—
59C2×D4×C61—24no—
60C2×D4×C61—22no—
61C2×D4×C61—22no—
62C2×D4×C61C22⋊C414yes—
63C22×C4×C61—28no—
64C22×C4×C61—24no—
65C22×C4×C61C2×D411yescharacteristic
66C22×C4×C61C2×D412yes—
67C22×Dic61—41no—
68C4⋊C4×C61—28no—
69C4⋊C4×C61—22no—
70C4⋊C4×C61—22no—
71C4⋊C4×C61—24no—
72C4⋊C4×C61C22⋊C414yes—
73C4⋊C4×C61C22⋊C414yes—
74C22⋊C4×C61—28no—
75C22⋊C4×C61—22no—
76C22⋊C4×C61—24no—
77C22⋊C4×C61—24no—
78C42×C61—22no—
79C42×C61C22×C411yescharacteristic
80C42×C61C2×D412yes—
81C42×C61C2×D411yescharacteristic
82C4×Dic61—22no—
83C4×Dic61—41no—
84#11758194—42no—
85C2×#92418—42no—
86C2×#92418—41no—
87C23×C61—24no—
88C23×C61#12011yescharacteristic
89Q8×C61—44no—
90Q8×C61—82no—
91D4×C61—44no—
92D4×C61—42no—
93D4×C61—42no—
94D4×C61—82no—
95D4×C61—84no—
96D4×C61—88no—
97C2×C4×C61—21no—
98C2×C4×C61—21no—
99C2×C4×C61—24no—
100C2×C4×C61—24no—
101C2×C4×C61—42no—
102C2×C4×C61—48no—
103C2×C4×C61—88no—
104C2×C4×C61C2×C22⋊C411yescharacteristic
105C2×C4×C61C2×C22⋊C412yes—
106C2×C4×C61#12012yes—
107C2×C4×C61#12011yescharacteristic
108C2×Dic61—42no—
109C2×Dic61—82no—
110#92418—82no—
111#92418—81no—
112#8494—611nomaximal, Sylow
113C22×C61—21no—
114C22×C61—22no—
115C22×C61—44no—
116C22×C61—44no—
117C22×C61—44no—
118C22×C61#37911yescharacteristic
119C4×C61—21no—
120C4×C61—22no—
121C4×C61—22no—
122C4×C61—44no—
123C4×C61—44no—
124C4×C61#45411yescharacteristic
125C4×C61#45611yescharacteristic
126Dic61—82no—
127#3548D6111yescharacteristic
128#1995—611no—
129#1557—612no—
130#1537—611no—
131#1505—612no—
132C2×C61—41no—
133C2×C61—84no—
134C2×C61#179311yescharacteristic
135C2×C61#179512yes—
136#562—22no—
137#554—22no—
138#550—22no—
139#547Dic6114yescomplemented
140#540—22no—
141#537—24no—
142#533C2×D6111yescharacteristic
143#532C2×D6112yes—
144#432—612no—
145#423—611no—
146#376—611no—
147#339—1222no—
148#332—1222no—
149#331—1222no—
150#330—1222no—
151#327—1224no—
152C61#849411yescharacteristic, complemented, Sylow
153C2×#41—24no—
154C2×#41—22no—
155C2×M16—1221no—
156C4⋊Q8C2×Dic6112yes—
157C4⋊Q8C22⋊D6112yes—
158#127—22no—
159#127C22⋊D6112yes—
160#125—22no—
161#125C22⋊D6112yes—
162#124—22no—
163#123—28no—
164C22⋊Q8—28no—
165C22⋊Q8—22no—
166C22⋊Q8—24no—
167#121—28no—
168#121—28no—
169#121—24no—
170#121—22no—
171#121C22⋊D6114yes—
172C4×Q8—22no—
173C4×D4—24no—
174C4×D4—22no—
175C4×D4C2×Dic6114yes—
176#117C22⋊D6112yes—
177C2×C4⋊C4—24no—
178C2×C42—1221no—
179C2×C42C22×D6111yescharacteristic
180C4⋊C8—1222no—
181C4⋊C8—1222no—
182C22⋊C8—1222no—
183#97—1222no—
184#95—1222no—
185#41—84no—
186#41—88no—
187C2×Q8—22no—
188C2×Q8—22no—
189C2×D4—28no—
190C2×D4—24no—
191C2×D4—22no—
192C2×D4—22no—
193C2×D4C22⋊Dic6114yes—
194C22×C4—28no—
195C22×C4—24no—
196C22×C4—2441no—
197C22×C4C2×C22⋊D6111yescharacteristic
198C22×C4C2×C22⋊D6112yes—
199M16—2442no—
200C2×C8—2442no—
201C2×C8—2441no—
202C4⋊C4—28no—
203C4⋊C4—22no—
204C4⋊C4—22no—
205C4⋊C4—24no—
206C4⋊C4C22⋊Dic6114yes—
207C4⋊C4C22⋊Dic6114yes—
208C22⋊C4—28no—
209C22⋊C4—22no—
210C22⋊C4—24no—
211C22⋊C4—24no—
212C42—22no—
213C42—1222no—
214C42—2441no—
215C42C22×Dic6111yescharacteristic
216C42C2×C22⋊D6111yescharacteristic
217C42C2×C22⋊D6112yes—
218C23—24no—
219C23#1451974611yescharacteristic
220Q8—44no—
221Q8—82no—
222D4—44no—
223D4—42no—
224D4—42no—
225D4—82no—
226D4—84no—
227D4—88no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—24no—
231C2×C4—24no—
232C2×C4—42no—
233C2×C4—48no—
234C2×C4—88no—
235C2×C4—4882no—
236C2×C4—2442no—
237C2×C4C2×C22⋊Dic6111yescharacteristic
238C2×C4C2×C22⋊Dic6112yes—
239C2×C4#1451974612yes—
240C2×C4#1451974611yescharacteristic
241C8—4882no—
242C8—4881no—
243C22—21no—
244C22—22no—
245C22—44no—
246C22—44no—
247C22—44no—
248C22#1683961811yescharacteristic
249C4—21no—
250C4—22no—
251C4—22no—
252C4—44no—
253C4—44no—
254C4—4882no—
255C4#1683993511yescharacteristic
256C4#1683994511yescharacteristic
257C2—41no—
258C2—84no—
259C2#2144577811yescharacteristic
260C2#2144578912yes—
subgroupG/H
centerC22#16839618
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
Fitting, prime radicalC61×#3548C2
socleC22×C61#379
cosocleC2×C42×C61C23
hypercenter#3548D61
hypocenterC61#8494
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3548

Complements

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

NH ≅ G/Ntype
1. #21446801C2semidirect
2. #21444541C2semidirect
3. #21444457C2semidirect
4. #21444327C2semidirect
17. C61×#547C4semidirect
139. #547Dic61semidirect
152. C61#8494semidirect

Maximal subgroups

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

subgroupindex
#214468012
#214445412
#214444572
#214443272
C61×#35482
#849461