Finite Group Databasebeta

← #29572576

Subgroups — #29572576

Order 15616.

800 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 139 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#21446797C211yesmaximal, characteristic, complemented
2#21446648C212yesmaximal, complemented
3#21444557C211yesmaximal, characteristic, complemented
4#21444554C212yesmaximal, complemented
5C61×#3548C211yesmaximal, characteristic
6#16839786C2211yescharacteristic
7#16839786C2212yes—
8C2×#14519704—24no—
9C42×Dic61C2211yescharacteristic
10#16839521—22no—
11#16839511—22no—
12#16839508—22no—
13#16839506—22no—
14C61×#562C414yescomplemented
15C61×#554—22no—
16C61×#550—22no—
17C61×#547—22no—
18C61×#540—22no—
19C61×#537—24no—
20C61×#533C2211yescharacteristic
21C61×#532C2212yes—
22C2×#11758194—22no—
23C2×#11758194—21no—
24C2×C4×Dic61—22no—
25C2×C4×Dic61—21no—
26C4⋊#92418—22no—
27C4⋊#92418—22no—
28C4⋊#92418—22no—
29#14519704—48no—
30C2×#41×C61—24no—
31C2×#41×C61D414yes—
32C4⋊Q8×C61D412yes—
33C4⋊Q8×C61D412yes—
34#127×C61—22no—
35#127×C61D412yes—
36#125×C61—22no—
37#125×C61C2×C412yes—
38#124×C61C2×C414yes—
39#123×C61—28no—
40C22⋊Q8×C61—24no—
41C22⋊Q8×C61—28no—
42C22⋊Q8×C61—22no—
43#121×C61—28no—
44#121×C61—24no—
45#121×C61—28no—
46#121×C61—22no—
47#121×C61—22no—
48C4×Q8×C61—22no—
49C4×D4×C61—24no—
50C4×D4×C61—22no—
51C4×D4×C61—22no—
52#117×C61D412yes—
53C2×C4⋊C4×C61—24no—
54C2×C42×C61C2311yescharacteristic
55#41×C61—44no—
56#41×C61—44no—
57#41×C61—84no—
58#41×C61—84no—
59C2×Q8×C61—22no—
60C2×Q8×C61C22⋊C414yes—
61C2×D4×C61—28no—
62C2×D4×C61—24no—
63C2×D4×C61—22no—
64C2×D4×C61—22no—
65C2×D4×C61C22⋊C414yes—
66C22×C4×C61—28no—
67C22×C4×C61—24no—
68C22×C4×C61C2×D411yescharacteristic
69C22×C4×C61C2×D412yes—
70C22×Dic61—41no—
71C4⋊C4×C61—28no—
72C4⋊C4×C61—24no—
73C4⋊C4×C61—22no—
74C4⋊C4×C61—22no—
75C4⋊C4×C61—22no—
76C4⋊C4×C61C22⋊C414yes—
77C22⋊C4×C61—28no—
78C22⋊C4×C61—22no—
79C22⋊C4×C61—24no—
80C22⋊C4×C61—24no—
81C42×C61—22no—
82C42×C61C22×C411yescharacteristic
83C42×C61C2×D411yescharacteristic
84C42×C61C2×D412yes—
85C4×Dic61—24no—
86C4×Dic61—22no—
87C4×Dic61—41no—
88C4×Dic61—42no—
89#11758194—44no—
90#11758194—42no—
91C2×#92418—42no—
92C2×#92418—41no—
93C23×C61—24no—
94C23×C61#12011yescharacteristic
95Q8×C61—44no—
96Q8×C61—42no—
97Q8×C61—42no—
98D4×C61—44no—
99D4×C61—42no—
100D4×C61—42no—
101D4×C61—42no—
102D4×C61—42no—
103D4×C61—84no—
104D4×C61—88no—
105C2×C4×C61—22no—
106C2×C4×C61—21no—
107C2×C4×C61—22no—
108C2×C4×C61—21no—
109C2×C4×C61—24no—
110C2×C4×C61—24no—
111C2×C4×C61—44no—
112C2×C4×C61—44no—
113C2×C4×C61—88no—
114C2×C4×C61C2×C22⋊C411yescharacteristic
115C2×C4×C61C2×C22⋊C412yes—
116C2×C4×C61#12012yes—
117C2×C4×C61#12011yescharacteristic
118C2×Dic61—44no—
119C2×Dic61—42no—
120#92418—82no—
121#92418—81no—
122#8532—611nomaximal, Sylow
123C22×C61—22no—
124C22×C61—21no—
125C22×C61—44no—
126C22×C61—44no—
127C22×C61—44no—
128C22×C61#37911yescharacteristic
129C4×C61—22no—
130C4×C61—21no—
131C4×C61—44no—
132C4×C61—44no—
133C4×C61#45312yes—
134C4×C61#45411yescharacteristic
135C4×C61#45512yes—
136C4×C61#45611yescharacteristic
137Dic61—44no—
138#3548D6111yescharacteristic
139#1994—611no—
140#1962—612no—
141#1561—611no—
142#1560—612no—
143C2×C61—41no—
144C2×C61—84no—
145C2×C61#179212yes—
146C2×C61#179311yescharacteristic
147#562Dic6114yescomplemented
148#554—22no—
149#550—22no—
150#547—22no—
151#540—22no—
152#537—24no—
153#533C2×D6111yescharacteristic
154#532C2×D6112yes—
155#423—611no—
156#423—612no—
157C2×#104—1224no—
158C43—611no—
159#339—1222no—
160#333—1222no—
161#332—1222no—
162#331—1222no—
163C61#853211yescharacteristic, complemented, Sylow
164C2×#41—24no—
165C2×#41C22⋊D6114yes—
166C2×M16—1221no—
167C2×M16—1222no—
168C4⋊Q8C22⋊D6112yes—
169C4⋊Q8C22⋊D6112yes—
170#127—22no—
171#127C22⋊D6112yes—
172#125—22no—
173#125C2×Dic6112yes—
174#124C2×Dic6114yes—
175#123—28no—
176C22⋊Q8—24no—
177C22⋊Q8—28no—
178C22⋊Q8—22no—
179#121—28no—
180#121—24no—
181#121—28no—
182#121—22no—
183#121—22no—
184C4×Q8—22no—
185C4×D4—24no—
186C4×D4—22no—
187C4×D4—22no—
188#117C22⋊D6112yes—
189C2×C4⋊C4—24no—
190C2×C42—1221no—
191C2×C42—1222no—
192C2×C42C22×D6111yescharacteristic
193C4⋊C8—1222no—
194C4⋊C8—1222no—
195C4⋊C8—1222no—
196#104—2448no—
197#41—44no—
198#41—44no—
199#41—84no—
200#41—84no—
201C2×Q8—22no—
202C2×Q8C22⋊Dic6114yes—
203C2×D4—28no—
204C2×D4—24no—
205C2×D4—22no—
206C2×D4—22no—
207C2×D4C22⋊Dic6114yes—
208C22×C4—28no—
209C22×C4—24no—
210C22×C4—2441no—
211C22×C4C2×C22⋊D6111yescharacteristic
212C22×C4C2×C22⋊D6112yes—
213M16—2444no—
214M16—2442no—
215C2×C8—2442no—
216C2×C8—2441no—
217C4⋊C4—28no—
218C4⋊C4—24no—
219C4⋊C4—22no—
220C4⋊C4—22no—
221C4⋊C4—22no—
222C4⋊C4C22⋊Dic6114yes—
223C22⋊C4—28no—
224C22⋊C4—22no—
225C22⋊C4—24no—
226C22⋊C4—24no—
227C42—22no—
228C42—1224no—
229C42—1222no—
230C42—2442no—
231C42—2441no—
232C42C22×Dic6111yescharacteristic
233C42C2×C22⋊D6111yescharacteristic
234C42C2×C22⋊D6112yes—
235C23—24no—
236C23#1451974611yescharacteristic
237Q8—44no—
238Q8—42no—
239Q8—42no—
240D4—44no—
241D4—42no—
242D4—42no—
243D4—42no—
244D4—42no—
245D4—84no—
246D4—88no—
247C2×C4—22no—
248C2×C4—21no—
249C2×C4—22no—
250C2×C4—21no—
251C2×C4—24no—
252C2×C4—24no—
253C2×C4—44no—
254C2×C4—44no—
255C2×C4—88no—
256C2×C4—2444no—
257C2×C4—2442no—
258C2×C4C2×C22⋊Dic6111yescharacteristic
259C2×C4C2×C22⋊Dic6112yes—
260C2×C4#1451974612yes—
261C2×C4#1451974611yescharacteristic
262C8—4882no—
263C8—4881no—
264C22—22no—
265C22—21no—
266C22—44no—
267C22—44no—
268C22—44no—
269C22#1683961811yescharacteristic
270C4—22no—
271C4—21no—
272C4—44no—
273C4—44no—
274C4—2444no—
275C4#1683993012yes—
276C4#1683993511yescharacteristic
277C4#1683994012yes—
278C4#1683994511yescharacteristic
279C2—41no—
280C2—84no—
281C2#2144577312yes—
282C2#2144577811yescharacteristic
subgroupG/H
centerC22#16839618
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
Fitting, prime radicalC61×#3548C2
socleC22×C61#379
cosocleC2×C42×C61C23
hypercenter#3548D61
hypocenterC61#8532
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. #21446797C2semidirect
2. #21446648C2semidirect
3. #21444557C2semidirect
4. #21444554C2semidirect
14. C61×#562C4semidirect
147. #562Dic61semidirect
163. C61#8532semidirect

Maximal subgroups

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

subgroupindex
#214467972
#214466482
#214445572
#214445542
C61×#35482
#853261