Finite Group Databasebeta

← #30598928

Subgroups — #30598928

Order 15616.

516 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 140 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#21461752C212yesmaximal
2C2×#16840622C212yesmaximal
3#21460697C212yesmaximal
4#21448388C211yesmaximal, characteristic
5C2×#14519844—22no—
6C2×#14519844C2214yes—
7C2×C4⋊#92420—22no—
8#16840622—24no—
9C2×#14519775C2211yescharacteristic
10C2×Q8×Dic61C411yescharacteristic
11C2×#14519742C411yescharacteristic
12#16839668C411yescharacteristic
13#16839659—22no—
14#16839631C2211yescharacteristic
15#16839613—22no—
16C42×Dic61C2211yescharacteristic
17C61×#391C411yescharacteristic
18C22×#92420—24no—
19#14519844—24no—
20#14519844—24no—
21#14519844—44no—
22C4⋊#92420—44no—
23C22×Dic122C2×C411yescharacteristic
24#14519775—22no—
25#14519775D412yes—
26#14519775Q812yes—
27Q8×Dic61—42no—
28#14519742—42no—
29C2×C4⋊Dic61—21no—
30C2×C4⋊Dic61C2×C411yescharacteristic
31C2×#11758196—22no—
32C2×#11758196—21no—
33C2×#11758196—22no—
34C2×C4×Dic61—21no—
35C2×C4×Dic61—21no—
36C2×C4×Dic61C2×C411yescharacteristic
37C2×C4×Dic61D412yes—
38C2×C4×Dic61D412yes—
39C2×C4×Dic61C2311yescharacteristic
40#14519684—22no—
41C22×Q8×C61C2×C411yescharacteristic
42C2×C4⋊C4×C61C2×C411yescharacteristic
43C2×C42×C61C2×C411yescharacteristic
44#95×C61—22no—
45C2×Q8×C61—21no—
46C2×Q8×C61—41no—
47C2×Q8×C61—41no—
48C2×Q8×C61C4⋊C412yes—
49C22×C4×C61—21no—
50C22×C4×C61—21no—
51C22×C4×C61C22⋊C412yes—
52C22×C4×C61C22×C411yescharacteristic
53C22×Dic61—21no—
54C22×Dic61—21no—
55C22×Dic61C22⋊C411yescharacteristic
56C22×Dic61C22⋊C411yescharacteristic
57C22×Dic61C2×D412yes—
58C2×Dic122—21no—
59C2×Dic122—41no—
60C2×Dic122—41no—
61C2×Dic122C4⋊C412yes—
62C2×#92420—44no—
63C2×#92420—44no—
64C2×#92420—44no—
65C4⋊C4×C61—41no—
66C42×C61—41no—
67C4⋊Dic61—42no—
68C4⋊Dic61—41no—
69#11758196—24no—
70#11758196—24no—
71#11758196—42no—
72#11758196—42no—
73#11758196—42no—
74C4×Dic61—22no—
75C4×Dic61—22no—
76C4×Dic61—22no—
77C4×Dic61—42no—
78C4×Dic61—42no—
79C4×Dic61SD1614yes—
80C4×Dic61Q1614yes—
81C4×Dic61C2×D411yescharacteristic
82C4×Dic61C2×Q811yescharacteristic
83C4×Dic61#4112yes—
84C23×C61C2×C22⋊C411yescharacteristic
85Q8×C61—42no—
86Q8×C61—42no—
87C2×C4×C61—21no—
88C2×C4×C61—21no—
89C2×C4×C61—22no—
90C2×C4×C61—22no—
91C2×C4×C61—22no—
92C2×C4×C61—41no—
93C2×C4×C61—41no—
94C2×C4×C61—41no—
95C2×C4×C61—42no—
96C2×C4×C61#10314yes—
97C2×C4×C61C2×C4⋊C411yescharacteristic
98C2×C4×C61#11711yescharacteristic
99C2×Dic61—22no—
100C2×Dic61—21no—
101C2×Dic61—22no—
102C2×Dic61—41no—
103C2×Dic61—41no—
104C2×Dic61—41no—
105C2×Dic61—42no—
106C2×Dic61#10012yes—
107C2×Dic61#10314yes—
108C2×Dic61#12112yes—
109C2×Dic61C22⋊Q812yes—
110C2×Dic61C2×SD1612yes—
111C2×Dic61C2×Q1612yes—
112C2×Dic61AGL(1,ℤ/8)12yes—
113C2×Dic61#13712yes—
114Dic122—42no—
115Dic122—42no—
116#92420—84no—
117#8587—611nomaximal, Sylow
118C22×C61#38011yescharacteristic
119C22×C61C2×#10011yescharacteristic
120C22×C61#41311yescharacteristic
121C22×C61C2×#10312yes—
122C22×C61#41712yes—
123C4×C61—21no—
124C4×C61—21no—
125C4×C61—24no—
126C4×C61—42no—
127C4×C61—81no—
128Dic61—42no—
129Dic61—41no—
130Dic61—81no—
131Dic61#33314yescomplemented
132Dic61#46012yes—
133Dic61#46212yes—
134Dic61#47512yes—
135Dic61#47712yes—
136C4×C4⋊Q8—611no—
137#1842—612no—
138C2×#333—612no—
139#1296—612no—
140C2×C61C2×#33311yescharacteristic
141C2×C61#155012yes—
142C2×C61#156011yescharacteristic
143C2×C61#180011yescharacteristic
144C2×C61#180312yes—
145C2×C4⋊Q8—611no—
146C2×C4×Q8—611no—
147C2×C4×Q8—611no—
148C2×C4⋊C8—614no—
149C2×C4⋊C8—1222no—
150C2×C4⋊C8—1222no—
151#391—611no—
152#391#692311yescharacteristic
153#389—1222no—
154#384—611no—
155C4×C4⋊C4—1222no—
156C43—611no—
157#333—1224no—
158C61#858711yescharacteristic, complemented, Sylow
159C22×Q8—611no—
160C22×Q8C2×#692311yescharacteristic
161C22×C8—1224no—
162C4⋊Q8—612no—
163C4⋊Q8—612no—
164C4⋊Q8—1222no—
165C4×Q8—2442no—
166C4×Q8—2442no—
167C2×C4⋊C4—611no—
168C2×C4⋊C4—1222no—
169C2×C4⋊C4—1221no—
170C2×C4⋊C4—1221no—
171C2×C4⋊C4—1222no—
172C2×C4⋊C4C2×#692311yescharacteristic
173C2×C42—612no—
174C2×C42—611no—
175C2×C42—612no—
176C2×C42—611no—
177C2×C42—1221no—
178C2×C42—1221no—
179C2×C42C2×#692311yescharacteristic
180C4⋊C8—1224no—
181C4⋊C8—1224no—
182C4⋊C8—2444no—
183C4⋊C8—2444no—
184#95—22no—
185#95—1222no—
186C2×Q8—21no—
187C2×Q8—41no—
188C2×Q8—41no—
189C2×Q8—612no—
190C2×Q8—1221no—
191C2×Q8—2441no—
192C2×Q8—2441no—
193C2×Q8C4⋊#692312yes—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—611no—
197C22×C4—612no—
198C22×C4—611no—
199C22×C4—1221no—
200C22×C4—1221no—
201C22×C4C22⋊#692312yes—
202C22×C4C22×#692311yescharacteristic
203C2×C8—2444no—
204C2×C8—2444no—
205C2×C8—2444no—
206C4⋊C4—41no—
207C4⋊C4—1224no—
208C4⋊C4—1224no—
209C4⋊C4—2441no—
210C4⋊C4—2442no—
211C4⋊C4—2442no—
212C4⋊C4—2442no—
213C4⋊C4—2442no—
214C42—41no—
215C42—614no—
216C42—614no—
217C42—611no—
218C42—611no—
219C42—612no—
220C42—1222no—
221C42—1222no—
222C42—1222no—
223C42—2442no—
224C42—2442no—
225C23C2×C22⋊#692311yescharacteristic
226Q8—42no—
227Q8—42no—
228Q8—2442no—
229Q8—2442no—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—21no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—42no—
239C2×C4—612no—
240C2×C4—612no—
241C2×C4—612no—
242C2×C4—614no—
243C2×C4—612no—
244C2×C4—612no—
245C2×C4—612no—
246C2×C4—612no—
247C2×C4—1222no—
248C2×C4—1221no—
249C2×C4—1222no—
250C2×C4—2441no—
251C2×C4—2442no—
252C2×C4—2441no—
253C2×C4—2441no—
254C2×C4#1451984014yes—
255C2×C4C2×C4⋊#692311yescharacteristic
256C2×C4#1451985611yescharacteristic
257C8—4884no—
258C22#1684070411yescharacteristic
259C22D61⋊#10011yescharacteristic
260C22#1684074911yescharacteristic
261C22C2×#1451984012yes—
262C22#1684075312yes—
263C4—24no—
264C4—21no—
265C4—21no—
266C4—42no—
267C4—81no—
268C4—614no—
269C4—612no—
270C4—612no—
271C4—612no—
272C4—612no—
273C4—4881no—
274C4—2441no—
275C4—2442no—
276C2D61⋊#33311yescharacteristic
277C2#2146133712yes—
278C2#2146135711yescharacteristic
279C2#2146168411yescharacteristic
280C2#2146168812yes—
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#391C4
socle, prime radicalC23×C61C2×C22⋊C4
cosocleC2×C4×Dic61C23
hypercenter#391#6923
hypocenterC61#8587
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #391

Complements

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

NH ≅ G/Ntype
131. Dic61#333semidirect
158. C61#8587semidirect

Maximal subgroups

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

subgroupindex
#214617522
C2×#168406222
#214606972
#214483882
#858761