Finite Group Databasebeta

← #30595565

Subgroups — #30595565

Order 15616.

874 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 874 conjugacy classes of subgroups — too many to draw. Only its 116 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#21461545C212yesmaximal, complemented
2#21460635C214yesmaximal, complemented
3C2×#16839665C211yesmaximal, characteristic
4C2×C22⋊#92420—22no—
5C2×#14519832—22no—
6#16840628C2214yescomplemented
7C22×C4⋊Dic61C411yescharacteristic
8C22×C22⋊Dic61—22no—
9C22×C4×Dic61C2211yescharacteristic
10#16839665—22no—
11#16839665C412yes—
12#16839665C2212yes—
13C23×D4×C61C411yescharacteristic
14C22×#92420—24no—
15C22⋊#92420—28no—
16#14519832—44no—
17C23×Dic61—21no—
18C23×Dic61D412yes—
19C2×C4⋊Dic61—21no—
20C2×C4⋊Dic61—42no—
21C2×C4⋊Dic61C2×C412yes—
22C2×C22⋊Dic61—24no—
23C2×C22⋊Dic61—24no—
24C2×C22⋊Dic61—48no—
25C2×C4×Dic61—42no—
26C2×C4×Dic61C2×C411yescharacteristic
27C2×C4×Dic61D412yes—
28C2×C4×Dic61C2311yescharacteristic
29C25×C61—21no—
30C22×D4×C61—24no—
31C22×D4×C61—21no—
32C22×D4×C61—42no—
33C22×D4×C61—42no—
34C22×D4×C61C2×C412yes—
35C23×C4×C61C2×C411yescharacteristic
36C24×C61—24no—
37C24×C61—22no—
38C24×C61—42no—
39C24×C61—44no—
40C24×C61—42no—
41C24×C61—44no—
42C24×C61C22⋊C411yescharacteristic
43C2×D4×C61—24no—
44C2×D4×C61—22no—
45C2×D4×C61—416no—
46C2×D4×C61—42no—
47C2×D4×C61—42no—
48C2×D4×C61—42no—
49C2×D4×C61—42no—
50C2×D4×C61M1614yes—
51C22×C4×C61—22no—
52C22×C4×C61—22no—
53C22×C4×C61—22no—
54C22×C4×C61C22⋊C411yescharacteristic
55C22×C4×C61C22×C411yescharacteristic
56C22×Dic61—24no—
57C22×Dic61—24no—
58C22×Dic61—22no—
59C22×Dic61—42no—
60C22×Dic61—44no—
61C22×Dic61—42no—
62C22×Dic61C22⋊C412yes—
63C22×Dic61C2×D412yes—
64C2×#92420—28no—
65C2×#92420—44no—
66C2×#92420—44no—
67C22⋊Dic61—48no—
68C22⋊Dic61—48no—
69C4⋊Dic61—42no—
70C4⋊Dic61—42no—
71C4×Dic61—24no—
72C4×Dic61—41no—
73C4×Dic61—41no—
74C23×C61—21no—
75C23×C61—21no—
76C23×C61—22no—
77C23×C61—24no—
78C23×C61—48no—
79C23×C61—416no—
80C23×C61—42no—
81C23×C61—42no—
82C23×C61—44no—
83C23×C61—44no—
84C23×C61—84no—
85C23×C61—88no—
86C23×C61—84no—
87C23×C61#10012yes—
88C23×C61#10414yes—
89C23×C61C2×C22⋊C411yescharacteristic
90D4×C61—44no—
91D4×C61—48no—
92D4×C61—44no—
93C2×C4×C61—28no—
94C2×C4×C61—21no—
95C2×C4×C61—21no—
96C2×C4×C61—21no—
97C2×C4×C61—21no—
98C2×C4×C61#9912yes—
99C2×C4×C61C2×M1612yes—
100C2×Dic61—22no—
101C2×Dic61—22no—
102C2×Dic61—22no—
103C2×Dic61—22no—
104C2×Dic61—22no—
105C2×Dic61—48no—
106C2×Dic61—42no—
107C2×Dic61—44no—
108C2×Dic61—84no—
109C2×Dic61AGL(1,ℤ/8)14yes—
110#92420—48no—
111#7626—611nomaximal, Sylow
112C22×C61—22no—
113C22×C61—22no—
114C22×C61—24no—
115C22×C61—24no—
116C22×C61—44no—
117C22×C61—48no—
118C22×C61—816no—
119C22×C61—82no—
120C22×C61—84no—
121C22×C61—82no—
122C22×C61#32914yes—
123C22×C61#40711yescharacteristic
124C22×C61C2×#9911yescharacteristic
125C22×C61C2×#10011yescharacteristic
126C22×C61#41812yes—
127C22×C61C2×#10412yes—
128C4×C61—22no—
129C4×C61—22no—
130Dic61—24no—
131Dic61—42no—
132Dic61—82no—
133C2×#390—611no—
134#1749—612no—
135#1270—614no—
136C2×C61—24no—
137C2×C61—84no—
138C2×C61#146211yescharacteristic
139C2×C61#149012yes—
140C2×C61#150712yes—
141C2×C61C2×#32911yescharacteristic
142C2×C61#155511yescharacteristic
143C23×D4#692311yescharacteristic
144C22×C4⋊C4—611no—
145C22×C22⋊C4—1222no—
146C22×C42—611no—
147C2×C22⋊C8—1222no—
148C2×#97—1222no—
149#390—612no—
150#390—612no—
151#390—1222no—
152#336—614no—
153C61#762611yescharacteristic, complemented, Sylow
154C25—21no—
155C22×D4—24no—
156C22×D4—21no—
157C22×D4—42no—
158C22×D4—42no—
159C22×D4C2×#692312yes—
160C23×C4—612no—
161C23×C4—1221no—
162C23×C4C2×#692311yescharacteristic
163C22×C8—1224no—
164C2×C4⋊C4—612no—
165C2×C4⋊C4—1221no—
166C2×C4⋊C4—2442no—
167C2×C22⋊C4—1224no—
168C2×C22⋊C4—1224no—
169C2×C22⋊C4—2448no—
170C2×C42—612no—
171C2×C42—611no—
172C2×C42—611no—
173C2×C42—2442no—
174C22⋊C8—1228no—
175#97—2444no—
176C24—24no—
177C24—22no—
178C24—44no—
179C24—42no—
180C24—42no—
181C24—44no—
182C24C22⋊#692311yescharacteristic
183C2×D4—24no—
184C2×D4—22no—
185C2×D4—42no—
186C2×D4—42no—
187C2×D4—416no—
188C2×D4—42no—
189C2×D4—42no—
190C2×D4#1175821714yes—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—612no—
195C22×C4—612no—
196C22×C4—1224no—
197C22×C4—1224no—
198C22×C4—1222no—
199C22×C4—2442no—
200C22×C4—2444no—
201C22×C4—2442no—
202C22×C4C22⋊#692311yescharacteristic
203C22×C4C22×#692311yescharacteristic
204C2×C8—1228no—
205C2×C8—2444no—
206C2×C8—2444no—
207C4⋊C4—2442no—
208C4⋊C4—2442no—
209C22⋊C4—2448no—
210C22⋊C4—2448no—
211C42—1224no—
212C42—2441no—
213C42—2441no—
214C23—21no—
215C23—21no—
216C23—22no—
217C23—24no—
218C23—48no—
219C23—416no—
220C23—42no—
221C23—42no—
222C23—44no—
223C23—44no—
224C23—84no—
225C23—88no—
226C23—84no—
227C23#1451983712yes—
228C23#1451984114yes—
229C23C2×C22⋊#692311yescharacteristic
230D4—44no—
231D4—48no—
232D4—44no—
233C2×C4—28no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—614no—
239C2×C4—1222no—
240C2×C4—1222no—
241C2×C4—1222no—
242C2×C4—1222no—
243C2×C4—1222no—
244C2×C4—4884no—
245C2×C4—2448no—
246C2×C4—2442no—
247C2×C4—2444no—
248C2×C4#1451983512yes—
249C2×C4C2×#1175821712yes—
250C8—2448no—
251C22—22no—
252C22—22no—
253C22—24no—
254C22—24no—
255C22—44no—
256C22—48no—
257C22—816no—
258C22—82no—
259C22—82no—
260C22—84no—
261C22#1684061714yes—
262C22#1684073211yescharacteristic
263C22C2×#1451983511yescharacteristic
264C22C2×#1451983711yescharacteristic
265C22#1684075412yes—
266C22C2×#1451984112yes—
267C4—22no—
268C4—22no—
269C4—1224no—
270C4—4882no—
271C4—2442no—
272C2—24no—
273C2—84no—
274C2#2146106411yescharacteristic
275C2#2146115812yes—
276C2#2146121512yes—
277C2C2×#1684061711yescharacteristic
278C2#2146134611yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
Fitting, prime radicalC23×D4×C61C4
socleC23×C61C2×C22⋊C4
cosocleC2×C4×Dic61C23
hypercenterC23×D4#6923
hypocenterC61#7626
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. #21461545C2semidirect
2. #21460635C2semidirect
6. #16840628C22semidirect
153. C61#7626semidirect

Maximal subgroups

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

subgroupindex
#214615452
#214606352
C2×#168396652
#762661