Finite Group Databasebeta

← #29580708

Subgroups — #29580708

Order 15616.

698 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 698 conjugacy classes of subgroups — too many to draw. Only its 89 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
1C2×#16839917C212yesmaximal
2#21445742C211yesmaximal, characteristic
3#21444183C212yesmaximal
4#21443706C211yesmaximal, characteristic
5C61×#3195C211yesmaximal, characteristic
6C23×Dic122C2211yescharacteristic
7C22×#11758202—22no—
8#16839917—24no—
9#16839917—24no—
10C2×#14519700—22no—
11C2×#14519695—22no—
12#16839726C2211yescharacteristic
13C2×C22⋊#92418C2212yes—
14#16839488—24no—
15C2×C22⋊Q8×C61C2212yes—
16C61×#483—22no—
17C61×#470—22no—
18C61×#451—22no—
19C61×#407C2211yescharacteristic
20C22×Dic122—22no—
21C22×Dic122—21no—
22C22×Dic122—22no—
23C22×Dic122—44no—
24C22×Dic122D412yes—
25C23×Dic61—21no—
26C2×#11758202—28no—
27C2×#11758202—48no—
28C2×#11758194—21no—
29C22×#92418—22no—
30#14519700—28no—
31#14519695—44no—
32C22⋊#92418—22no—
33C22⋊#92418—22no—
34C22⋊#92418D414yes—
35C22×Q8×C61D412yes—
36C23×C4×C61C2311yescharacteristic
37C2×Q16×C61—22no—
38C2×M16×C61D412yes—
39C22⋊Q8×C61—22no—
40C22⋊Q8×C61—22no—
41C22⋊Q8×C61—22no—
42C22⋊Q8×C61—22no—
43C2×C4⋊C4×C61—22no—
44C2×C4⋊C4×C61D412yes—
45C2×C22⋊C4×C61—22no—
46#106×C61—22no—
47#103×C61—24no—
48C22⋊C8×C61—22no—
49C24×C61#4111yescharacteristic
50C2×Q8×C61—22no—
51C2×Q8×C61—22no—
52C2×Q8×C61—44no—
53C22×C4×C61—22no—
54C22×C4×C61—22no—
55C22×C4×C61—42no—
56C22×C4×C61C2×D412yes—
57C22×C4×C61C2×D411yescharacteristic
58C22×C4×C61C2×D411yescharacteristic
59C22×C4×C61C2×D412yes—
60C22×Dic61—22no—
61C22×Dic61—44no—
62C22×Dic61—42no—
63C2×Dic122—28no—
64C2×Dic122—24no—
65C2×Dic122—24no—
66C2×Dic122—44no—
67C2×Dic122—48no—
68C2×Dic122—44no—
69C2×Dic122—42no—
70C2×Dic122—41no—
71C2×Dic122—41no—
72Q16×C61—82no—
73M16×C61—42no—
74C2×C8×C61—22no—
75C4⋊C4×C61—22no—
76C4⋊C4×C61—22no—
77C4⋊C4×C61—42no—
78C4⋊C4×C61—42no—
79C22⋊C4×C61—42no—
80C22⋊C4×C61—42no—
81#11758202—416no—
82#11758194—42no—
83C2×#92418—24no—
84C2×#92418—44no—
85C2×#92418—41no—
86C23×C61—24no—
87C23×C61—41no—
88C23×C61#12111yescharacteristic
89C23×C61#12112yes—
90Q8×C61—44no—
91Q8×C61—44no—
92C2×C4×C61—24no—
93C2×C4×C61—22no—
94C2×C4×C61—41no—
95C2×C4×C61—42no—
96C2×C4×C61—42no—
97C2×C4×C61—42no—
98C2×C4×C61—44no—
99C2×C4×C61—82no—
100C2×C4×C61#12012yes—
101C2×C4×C61#12011yescharacteristic
102C2×C4×C61#12711yescharacteristic
103C2×Dic61—24no—
104C2×Dic61—48no—
105C2×Dic61—42no—
106C2×Dic61—42no—
107Dic122—28no—
108Dic122—44no—
109Dic122—44no—
110Dic122—44no—
111C8×C61—42no—
112#92418—44no—
113#92418—81no—
114#9991—611nomaximal, Sylow
115C22×C61—22no—
116C22×C61—22no—
117C22×C61—22no—
118C22×C61—21no—
119C22×C61—44no—
120C22×C61#39211yescharacteristic
121C22×C61#47314yes—
122C4×C61—22no—
123C4×C61—21no—
124C4×C61—82no—
125C4×C61—82no—
126C4×C61B(4,𝔽2)12yes—
127Dic61—44no—
128#3195D6111yescharacteristic
129C2×#451—612no—
130#1784—611no—
131#1472—612no—
132#1298—611no—
133C2×C61—24no—
134C2×C61—41no—
135C2×C61#202012yes—
136C2×C61#202111yescharacteristic
137C23×Q8—611no—
138C22×Q16—1222no—
139C2×C22⋊Q8C2×D6112yes—
140#483—22no—
141#470—22no—
142#451—22no—
143#451—1224no—
144#451—1224no—
145C2×#103—1222no—
146C2×#101—1222no—
147#407—611no—
148#407C2×D6111yescharacteristic
149C2×C22⋊C8—612no—
150#324—1224no—
151C61#999111yescharacteristic, complemented, Sylow
152C22×Q8—612no—
153C22×Q8—1221no—
154C22×Q8—1222no—
155C22×Q8—1222no—
156C22×Q8—2444no—
157C22×Q8C22⋊D6112yes—
158C23×C4—1221no—
159C23×C4C22×D6111yescharacteristic
160C2×Q16—22no—
161C2×Q16—1228no—
162C2×Q16—2448no—
163C2×M16—1221no—
164C2×M16C22⋊D6112yes—
165C22×C8—1222no—
166C22⋊Q8—22no—
167C22⋊Q8—22no—
168C22⋊Q8—22no—
169C22⋊Q8—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4C22⋊D6112yes—
172C2×C22⋊C4—22no—
173#106—22no—
174#103—24no—
175#103—1228no—
176#101—2444no—
177C22⋊C8—22no—
178C22⋊C8—614no—
179C22⋊C8—1222no—
180C22⋊C8—1222no—
181C24#1175822411yescharacteristic
182C2×Q8—22no—
183C2×Q8—22no—
184C2×Q8—44no—
185C2×Q8—1224no—
186C2×Q8—1228no—
187C2×Q8—1224no—
188C2×Q8—2444no—
189C2×Q8—2442no—
190C2×Q8—2444no—
191C2×Q8—2448no—
192C2×Q8—2441no—
193C2×Q8—2441no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—42no—
197C22×C4—1222no—
198C22×C4—2444no—
199C22×C4—2442no—
200C22×C4D4×D6111yescharacteristic
201C22×C4D4×D6112yes—
202C22×C4C2×C22⋊D6112yes—
203C22×C4C2×C22⋊D6111yescharacteristic
204Q16—82no—
205Q16—24416no—
206M16—42no—
207M16—2442no—
208C2×C8—22no—
209C2×C8—1224no—
210C2×C8—2441no—
211C2×C8—2444no—
212C4⋊C4—22no—
213C4⋊C4—22no—
214C4⋊C4—42no—
215C4⋊C4—42no—
216C22⋊C4—42no—
217C22⋊C4—42no—
218C23—24no—
219C23—41no—
220C23#1451974811yescharacteristic
221C23#1451974912yes—
222Q8—44no—
223Q8—44no—
224Q8—1228no—
225Q8—2444no—
226Q8—2444no—
227Q8—2444no—
228C2×C4—24no—
229C2×C4—22no—
230C2×C4—42no—
231C2×C4—41no—
232C2×C4—44no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—82no—
236C2×C4—1224no—
237C2×C4—2442no—
238C2×C4—2448no—
239C2×C4—2442no—
240C2×C4#1451974512yes—
241C2×C4#1451974611yescharacteristic
242C2×C4#1451977311yescharacteristic
243C8—42no—
244C8—4881no—
245C8—2444no—
246C22—22no—
247C22—22no—
248C22—22no—
249C22—21no—
250C22—44no—
251C22#1683967011yescharacteristic
252C22#1684003514yes—
253C4—21no—
254C4—22no—
255C4—82no—
256C4—82no—
257C4—2444no—
258C4#1683994912yes—
259C2—24no—
260C2—41no—
261C2#2144693712yes—
262C2#2144694311yescharacteristic
subgroupG/H
centerC22#16839670
commutator, cosocleC23×C4×C61C23
FrattiniC23×C4C22×D61
FittingC61×#3195C2
socleC22×C61#392
prime radicalC24×C61#41
hypercenter#3195D61
hypocenterC61#9991
LCSG ▷ C23×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3195

Complements

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

NH ≅ G/Ntype
151. C61#9991semidirect

Maximal subgroups

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

subgroupindex
C2×#168399172
#214457422
#214441832
#214437062
C61×#31952
#999161