Finite Group Databasebeta

← #29580705

Subgroups — #29580705

Order 15616.

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

normal subgroups only

This group has 826 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#21456643C211yesmaximal, characteristic, complemented
2C2×#16839913C212yesmaximal
3#21444184C212yesmaximal, complemented
4#21443706C211yesmaximal, characteristic, complemented
5C61×#1783C211yesmaximal, characteristic
6C22×#11758200—22no—
7C2×#14519756C2212yes—
8#16840093—22no—
9#16840020—22no—
10#16839920—22no—
11#16839913—24no—
12#16839913—24no—
13C2×#14519698—22no—
14C2×C22⋊#92418C2212yes—
15#16839487—24no—
16C23×D4×C61C2211yescharacteristic
17C2×#100×C61—22no—
18C61×#407C2211yescharacteristic
19C61×#407C2211yescharacteristic
20C22×Dic122D412yes—
21C2×Dic244—22no—
22C2×#11758200—28no—
23C2×#11758200—48no—
24C22×#92418—22no—
25#14519756—22no—
26#14519756—22no—
27#14519756—22no—
28#14519756—22no—
29C2×C4⋊Dic61D412yes—
30C2×#11758196—22no—
31C2×C22⋊Dic61—22no—
32#14519709—22no—
33#14519702—24no—
34#14519698—28no—
35C22⋊#92418—22no—
36C22⋊#92418D414yes—
37C25×C61—21no—
38C22×D4×C61—22no—
39C22×D4×C61—21no—
40C22×D4×C61—22no—
41C22×D4×C61—44no—
42C22×D4×C61D412yes—
43C23×C4×C61C2311yescharacteristic
44C2×M16×C61—21no—
45C2×M16×C61D412yes—
46#100×C61—44no—
47C22⋊C8×C61—22no—
48C22⋊C8×C61—22no—
49C24×C61—22no—
50C24×C61—44no—
51C24×C61—42no—
52C24×C61—84no—
53C24×C61#4111yescharacteristic
54C2×D4×C61—28no—
55C2×D4×C61—24no—
56C2×D4×C61—24no—
57C2×D4×C61—44no—
58C2×D4×C61—42no—
59C2×D4×C61—44no—
60C2×D4×C61—48no—
61C2×D4×C61—41no—
62C2×D4×C61—41no—
63C22×C4×C61—42no—
64C22×C4×C61C2×D412yes—
65C22×C4×C61C2×D411yescharacteristic
66C22×C4×C61C2×D411yescharacteristic
67C22×C4×C61C2×D412yes—
68C22×Dic61—22no—
69C22×Dic61—22no—
70C2×Dic122—22no—
71C2×Dic122—22no—
72C2×Dic122—44no—
73M16×C61—42no—
74M16×C61—42no—
75C2×C8×C61—22no—
76C2×C8×C61—41no—
77C22⋊Dic61—42no—
78C22⋊Dic61—42no—
79#11758200—416no—
80C4⋊Dic61—22no—
81C4⋊Dic61—22no—
82#11758196—42no—
83#11758196—42no—
84C2×#92418—24no—
85C2×#92418—44no—
86Dic244—82no—
87C23×C61—24no—
88C23×C61—24no—
89C23×C61—41no—
90C23×C61—48no—
91C23×C61—42no—
92C23×C61—42no—
93C23×C61—84no—
94C23×C61—84no—
95C23×C61—816no—
96C23×C61—84no—
97C23×C61#12111yescharacteristic
98C23×C61#12112yes—
99D4×C61—28no—
100D4×C61—44no—
101D4×C61—44no—
102D4×C61—44no—
103C2×C4×C61—24no—
104C2×C4×C61—22no—
105C2×C4×C61—41no—
106C2×C4×C61—42no—
107C2×C4×C61#12012yes—
108C2×C4×C61#12011yescharacteristic
109C2×C4×C61#12711yescharacteristic
110C2×Dic61—42no—
111C2×Dic61—42no—
112C2×Dic61—44no—
113C2×Dic61—82no—
114Dic122—44no—
115Dic122—44no—
116C8×C61—42no—
117C8×C61—81no—
118#92418—44no—
119#9990—611nomaximal, Sylow
120C22×C61—22no—
121C22×C61—22no—
122C22×C61—22no—
123C22×C61—21no—
124C22×C61—44no—
125C22×C61—44no—
126C22×C61—816no—
127C22×C61—84no—
128C22×C61—84no—
129C22×C61—84no—
130C22×C61#39211yescharacteristic
131C22×C61#47114yes—
132C4×C61—22no—
133C4×C61—21no—
134C4×C61B(4,𝔽2)12yes—
135Dic61—82no—
136Dic61—82no—
137#3195—611no—
138C2×#450—612no—
139#1783D6111yescharacteristic
140#1472—612no—
141#1298—611no—
142C2×C61—24no—
143C2×C61—41no—
144C2×C61—84no—
145C2×C61#201912yes—
146C2×C61#202111yescharacteristic
147C23×D4C2×D6111yescharacteristic
148C22×SD16—1222no—
149C2×C22⋊Q8—612no—
150#483—1222no—
151#470—1222no—
152#451—1222no—
153#450—1224no—
154#450—1224no—
155C2×#102—1222no—
156C2×#100—22no—
157#407C2×D6111yescharacteristic
158#407C2×D6111yescharacteristic
159C2×C22⋊C8—612no—
160#324—1224no—
161C61#999011yescharacteristic, complemented, Sylow
162C25—21no—
163C22×Q8—612no—
164C22×D4—22no—
165C22×D4—22no—
166C22×D4—21no—
167C22×D4—44no—
168C22×D4D24412yes—
169C23×C4C22×D6111yescharacteristic
170C2×Q16—1222no—
171C2×SD16—1228no—
172C2×SD16—2448no—
173C2×M16—21no—
174C2×M16C22⋊D6112yes—
175C22×C8—1222no—
176C22⋊Q8—1222no—
177C22⋊Q8—1222no—
178C22⋊Q8—1222no—
179C22⋊Q8—1222no—
180C2×C4⋊C4—612no—
181C2×C4⋊C4—1222no—
182C2×C22⋊C4—1222no—
183#106—1222no—
184#103—1224no—
185#102—1228no—
186#100—44no—
187C22⋊C8—22no—
188C22⋊C8—22no—
189C22⋊C8—614no—
190C22⋊C8—1222no—
191C24—22no—
192C24—44no—
193C24—42no—
194C24—84no—
195C24#1175822211yescharacteristic
196C2×Q8—1222no—
197C2×Q8—1222no—
198C2×Q8—2444no—
199C2×D4—28no—
200C2×D4—24no—
201C2×D4—24no—
202C2×D4—44no—
203C2×D4—42no—
204C2×D4—44no—
205C2×D4—48no—
206C2×D4—41no—
207C2×D4—41no—
208C22×C4—42no—
209C22×C4—1222no—
210C22×C4—1222no—
211C22×C4C2×D24411yescharacteristic
212C22×C4D4×D6112yes—
213C22×C4D4×D6112yes—
214C22×C4C2×C22⋊D6111yescharacteristic
215Q16—4882no—
216SD16—24416no—
217M16—42no—
218M16—42no—
219C2×C8—22no—
220C2×C8—41no—
221C2×C8—1224no—
222C2×C8—2444no—
223C4⋊C4—1222no—
224C4⋊C4—1222no—
225C4⋊C4—2442no—
226C4⋊C4—2442no—
227C22⋊C4—2442no—
228C22⋊C4—2442no—
229C23—24no—
230C23—24no—
231C23—41no—
232C23—48no—
233C23—42no—
234C23—42no—
235C23—84no—
236C23—84no—
237C23—816no—
238C23—84no—
239C23#1451975012yes—
240C23#1451975111yescharacteristic
241Q8—2444no—
242Q8—2444no—
243D4—28no—
244D4—44no—
245D4—44no—
246D4—44no—
247C2×C4—24no—
248C2×C4—22no—
249C2×C4—41no—
250C2×C4—42no—
251C2×C4—4882no—
252C2×C4—2444no—
253C2×C4—2442no—
254C2×C4—2442no—
255C2×C4#1451974511yescharacteristic
256C2×C4#1451974712yes—
257C2×C4#1451977311yescharacteristic
258C8—42no—
259C8—81no—
260C8—2444no—
261C22—22no—
262C22—22no—
263C22—22no—
264C22—21no—
265C22—44no—
266C22—44no—
267C22—816no—
268C22—84no—
269C22—84no—
270C22—84no—
271C22#1683967111yescharacteristic
272C22#1684002414yes—
273C4—21no—
274C4—22no—
275C4—4882no—
276C4—4882no—
277C4#1683994812yes—
278C2—24no—
279C2—41no—
280C2—84no—
281C2#2144693512yes—
282C2#2144694511yescharacteristic
subgroupG/H
centerC22#16839671
commutator, cosocleC23×C4×C61C23
FrattiniC23×C4C22×D61
FittingC61×#1783C2
socleC22×C61#392
prime radicalC23×D4×C61C22
hypercenter#1783D61
hypocenterC61#9990
LCSG ▷ C23×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1783

Complements

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

NH ≅ G/Ntype
1. #21456643C2semidirect
3. #21444184C2semidirect
4. #21443706C2semidirect
161. C61#9990semidirect

Maximal subgroups

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

subgroupindex
#214566432
C2×#168399132
#214441842
#214437062
C61×#17832
#999061