Finite Group Databasebeta

← #30607499

Subgroups — #30607499

Order 15616.

488 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 112 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#21462786C211yesmaximal, characteristic, complemented
2#21462055C211yesmaximal, characteristic, complemented
3#21462048C212yesmaximal, complemented
4#21460870C212yesmaximal, complemented
5#21454516C211yesmaximal, characteristic
6#16840820—24no—
7C2×#14519851—22no—
8C2×#14519851C2212yes—
9C2×#14519851C2212yes—
10#16840658—22no—
11#16840657—24no—
12C2×#14519784C2211yescharacteristic
13C22×C8×D61C2211yescharacteristic
14#16840269C411yescharacteristic
15#16839836—22no—
16#16839825—22no—
17#16839788C411yescharacteristic
18#16839725C411yescharacteristic
19#16839720C2211yescharacteristic
20C61×#408C411yescharacteristic
21#14519851—24no—
22#14519851—22no—
23#14519851—22no—
24#14519851—22no—
25#14519851—22no—
26C2×#11758187—24no—
27C2×#11758187—22no—
28C2×#11758187D414yescomplemented
29C22×C4×D61C2×C411yescharacteristic
30#14519784—22no—
31#14519784—22no—
32C2×#11758194C2×C411yescharacteristic
33C2×#11758194D412yes—
34C22×#92418D412yes—
35C22×#92418C2311yescharacteristic
36C2×C8×D61—21no—
37C2×C8×D61—21no—
38C2×C8×D61—22no—
39C2×C8×D61—22no—
40#14519758C812yes—
41#14519753—21no—
42#14519748—21no—
43#14519745C812yes—
44C4×C22⋊D61—22no—
45#14519736C2×C411yescharacteristic
46#14519706—22no—
47#14519689—22no—
48#14519687—21no—
49C8×Dic61—21no—
50C2×#41×C61C2×C411yescharacteristic
51C2×M16×C61C2×C411yescharacteristic
52C22×C8×C61C2×C411yescharacteristic
53C22⋊C8×C61—22no—
54C23×D61C2×C811yescharacteristic
55#41×C61—22no—
56#41×C61—22no—
57C2×Q8×C61M1611yescharacteristic
58C2×D4×C61C2×C812yes—
59C2×D4×C61M1611yescharacteristic
60C22×C4×C61C22⋊C412yes—
61C22×C4×C61C22×C411yescharacteristic
62C2×C22⋊D61—22no—
63C22×Dic61C2×C811yescharacteristic
64C2×C4×D61—21no—
65C2×C4×D61—22no—
66C2×C4×D61—21no—
67C2×C4×D61—22no—
68M16×C61—22no—
69C2×C8×C61—21no—
70C2×C8×C61—21no—
71C2×C8×C61—22no—
72C22⋊Dic61—22no—
73C22⋊Dic61C2×C812yes—
74#11758198—22no—
75C4⋊Dic61—21no—
76#11758196—22no—
77C4×Dic61—21no—
78#11758194—22no—
79#11758194—24no—
80C2×#92418—22no—
81C2×#92418—22no—
82C2×#92418C22⋊C412yes—
83C2×#92418C2×D412yes—
84C2×#92418C2×D411yescharacteristic
85C2×#92418C2×D411yescharacteristic
86C2×#92418#4112yes—
87C8×D61—41no—
88C8×D61—42no—
89C8×D61—41no—
90#11758187—24no—
91#11758187—42no—
92C23×C61C22⋊C812yes—
93C23×C61C22×C811yescharacteristic
94C22×D61—21no—
95C22×D61—22no—
96C22×D61—42no—
97Q8×C61—22no—
98D4×C61—24no—
99D4×C61—22no—
100C2×C4×C61—22no—
101C2×C4×C61—22no—
102C2×C4×C61C22⋊C812yes—
103C2×C4×C61C2×C22⋊C411yescharacteristic
104C2×C4×C61C4×D412yes—
105C2×C4×C61C2×M1611yescharacteristic
106C2×C4×C61#13112yes—
107C22⋊D61—44no—
108C2×Dic61—22no—
109C2×Dic61—21no—
110C2×Dic61—22no—
111C4×D61—41no—
112C4×D61—41no—
113C4×D61—42no—
114C8×C61—41no—
115C8×C61—41no—
116#92418—22no—
117#92418#12011yescharacteristic
118#92418#12112yes—
119#92418#12311yescharacteristic
120#12981—611nomaximal, Sylow
121C22×C61—22no—
122C22×C61—41no—
123C22×C61—42no—
124C22×C61C2×C22⋊C811yescharacteristic
125C22×C61C8×D412yes—
126C2×D61—22no—
127C2×D61—24no—
128C2×D61—41no—
129C2×D61—42no—
130C2×D61—41no—
131C4×C61—22no—
132C4×C61#38611yescharacteristic
133C4×C61#40811yescharacteristic
134C4×C61#43512yes—
135Dic61—42no—
136Dic61—41no—
137#3407—611no—
138#2929—611no—
139#2115—611no—
140#2112—612no—
141#1378—612no—
142C2×C61—21no—
143C2×C61—22no—
144C2×C61—42no—
145C2×C61#186911yescharacteristic
146D61—42no—
147C2×#131—611no—
148C23×C8—611no—
149#525—611no—
150#504—1224no—
151C2×M32—612no—
152C2×M32—612no—
153C2×M32—1222no—
154#435—1222no—
155C8×D4—1222no—
156#424—611no—
157#408#692311yescharacteristic
158#407—611no—
159#405—611no—
160#349—1222no—
161C22⋊C16—1224no—
162C61#1298111yescharacteristic, complemented, Sylow
163C2×#41C2×#692311yescharacteristic
164C23×C4—611no—
165#131—1222no—
166#131—1222no—
167C2×M16—612no—
168C2×M16—611no—
169C2×M16C2×#692311yescharacteristic
170C22×C8—611no—
171C22×C8—612no—
172C22×C8—1222no—
173C22×C8—1221no—
174C22×C8—1221no—
175C22×C8—1222no—
176C22×C8C2×#692311yescharacteristic
177#123—612no—
178C22⋊Q8—1221no—
179#121—1221no—
180#120—612no—
181C4×D4—1222no—
182#117—611no—
183M32—1222no—
184M32—1222no—
185M32—1224no—
186M32—1222no—
187M32—1222no—
188C2×C16—614no—
189C2×C16—1224no—
190C2×C16—1222no—
191C4⋊C8—1222no—
192C22⋊C8—22no—
193C22⋊C8—1222no—
194#97—1221no—
195C4×C8—1221no—
196C24—611no—
197#41—22no—
198#41—22no—
199C2×Q8#1175821311yescharacteristic
200C2×D4—1222no—
201C2×D4D61⋊C812yes—
202C2×D4#1175821311yescharacteristic
203C22×C4—611no—
204C22×C4—1221no—
205C22×C4—1222no—
206C22×C4—1221no—
207C22×C4—1222no—
208C22×C4C22⋊#692312yes—
209C22×C4C22×#692311yescharacteristic
210M16—22no—
211M16—1224no—
212M16—1222no—
213C2×C8—21no—
214C2×C8—21no—
215C2×C8—22no—
216C2×C8—611no—
217C2×C8—611no—
218C2×C8—612no—
219C2×C8—612no—
220C2×C8—612no—
221C2×C8—1222no—
222C2×C8—1222no—
223C2×C8—2441no—
224C2×C8—2442no—
225C2×C8—2441no—
226C4⋊C4—1222no—
227C4⋊C4—1221no—
228C22⋊C4—612no—
229C22⋊C4—1222no—
230C22⋊C4—1222no—
231C42—1221no—
232C16—1224no—
233C16—2442no—
234C23—1221no—
235C23—1222no—
236C23—2442no—
237C23#1451983412yes—
238C23C2×D61⋊C811yescharacteristic
239Q8—22no—
240D4—24no—
241D4—22no—
242D4—2444no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—1222no—
246C2×C4—1221no—
247C2×C4—1222no—
248C2×C4—2441no—
249C2×C4—2441no—
250C2×C4—2442no—
251C2×C4#1451983412yes—
252C2×C4C2×C22⋊#692311yescharacteristic
253C2×C4D4×#692312yes—
254C2×C4C2×#1175821311yescharacteristic
255C2×C4#1451986412yes—
256C8—41no—
257C8—41no—
258C8—611no—
259C8—612no—
260C8—611no—
261C8—1222no—
262C22—22no—
263C22—41no—
264C22—42no—
265C22—1222no—
266C22—1224no—
267C22—2441no—
268C22—2442no—
269C22—2441no—
270C22C2×#1451983411yescharacteristic
271C22#1684078912yes—
272C4—22no—
273C4—2442no—
274C4—2441no—
275C4#1684071011yescharacteristic
276C4#1684073511yescharacteristic
277C4#1684079112yes—
278C2—21no—
279C2—22no—
280C2—42no—
281C2—2442no—
282C2#2146179411yescharacteristic
subgroupG/H
centerC4#16840710
commutatorC23×C61C22×C8
FrattiniC22×C4C22×#6923
FittingC61×#408C4
socleC2×C61#1869
prime radical#16840269C4
cosocleC22×#92418C23
hypercenter#408#6923
hypocenterC61#12981
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C4 ◁ C22×C4 ◁ #408

Complements

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

NH ≅ G/Ntype
1. #21462786C2semidirect
2. #21462055C2semidirect
3. #21462048C2semidirect
4. #21460870C2semidirect
28. C2×#11758187D4semidirect
162. C61#12981semidirect

Maximal subgroups

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

subgroupindex
#214627862
#214620552
#214620482
#214608702
#214545162
#1298161