Finite Group Databasebeta

← #29656134

Subgroups — #29656134

Order 15616.

982 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 982 conjugacy classes of subgroups — too many to draw. Only its 163 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×#16840498C211yesmaximal, characteristic
2#21456193C211yesmaximal, characteristic
3#21455950C212yesmaximal, complemented
4#21445006C218yesmaximal, complemented
5#21444265C212yesmaximal, complemented
6C61×#1555C211yesmaximal, characteristic
7#16840498—24no—
8#16840498—22no—
9#16840498C2214yes—
10#16840319C2211yescharacteristic
11C2×#14519764—22no—
12C2×#14519758—22no—
13C2×C4×Dic122C2212yes—
14C2×D4×Dic61—22no—
15#16840126—22no—
16#16840108C2214yes—
17#16840093—22no—
18#16840048C2214yescomplemented
19#16839973C2214yescomplemented
20#16839920—22no—
21#16839870—24no—
22#16839767—22no—
23C4×#11758194C2211yescharacteristic
24#16839492C2218yescomplemented
25C2×#127×C61C2211yescharacteristic
26C61×#432C2212yes—
27C61×#331C2214yes—
28C22×Dic122D412yes—
29C23×Dic61—22no—
30C2×Dic244—24no—
31C2×#11758194—21no—
32#14519777C2312yes—
33#14519769—21no—
34#14519764—28no—
35#14519764—28no—
36#14519758—28no—
37#14519758—28no—
38#14519756—22no—
39C4×Dic122—22no—
40C4×Dic122—22no—
41C4×Dic122—22no—
42C4×Dic122D414yes—
43C4×Dic122C2314yes—
44D4×Dic61—28no—
45D4×Dic61—28no—
46#14519736—21no—
47C2×C4⋊Dic61D412yes—
48C2×#11758196—22no—
49C2×C22⋊Dic61—24no—
50C2×C22⋊Dic61—22no—
51C2×C4×Dic61—22no—
52#14519709—24no—
53#14519702—24no—
54#14519702—24no—
55#14519698—28no—
56#14519688—21no—
57C4×#92418C2312yes—
58C22×D4×C61—22no—
59C22×D4×C61D412yes—
60#127×C61—22no—
61#127×C61D412yes—
62#127×C61C2312yes—
63C2×C42×C61C2311yescharacteristic
64C4⋊C8×C61C2314yes—
65C22⋊C8×C61—22no—
66#97×C61—22no—
67C24×C61—22no—
68C2×D4×C61—24no—
69C2×D4×C61—22no—
70C2×D4×C61—24no—
71C2×D4×C61—48no—
72C2×D4×C61—44no—
73C2×D4×C61C2×D414yes—
74C22×C4×C61C2×D411yescharacteristic
75C22×C4×C61C2×D412yes—
76C22×Dic61—22no—
77C22×Dic61—22no—
78C22×Dic61—28no—
79C22×Dic61—84no—
80C2×Dic122—22no—
81C2×Dic122—22no—
82C2×Dic122—44no—
83C2×Dic122C2×D414yes—
84C2×C8×C61—24no—
85C42×C61C2×D411yescharacteristic
86C42×C61C2×D412yes—
87C42×C61C2411yescharacteristic
88C22⋊Dic61—216no—
89C22⋊Dic61—28no—
90C22⋊Dic61—41no—
91C4⋊Dic61—22no—
92C4⋊Dic61—22no—
93C4⋊Dic61—22no—
94C4⋊Dic61C2×D414yes—
95#11758196—24no—
96#11758196—24no—
97#11758196—42no—
98C4×Dic61—24no—
99C4×Dic61—24no—
100C4×Dic61—41no—
101#11758194—42no—
102C2×#92418—22no—
103Dic244—48no—
104C23×C61—24no—
105C23×C61—48no—
106C23×C61—44no—
107C23×C61#12011yescharacteristic
108D4×C61—44no—
109D4×C61—48no—
110D4×C61—48no—
111D4×C61—82no—
112C2×C4×C61—22no—
113C2×C4×C61—24no—
114C2×C4×C61#12012yes—
115C2×C4×C61#12011yescharacteristic
116C2×C4×C61C22×D411yescharacteristic
117C2×C4×C61C22×D412yes—
118C2×Dic61—24no—
119C2×Dic61—24no—
120C2×Dic61—44no—
121C2×Dic61—44no—
122C2×Dic61—42no—
123C2×Dic61—88no—
124Dic122—24no—
125Dic122—24no—
126Dic122—44no—
127Dic122—44no—
128C8×C61—44no—
129#92418—42no—
130#18531—611nomaximal, Sylow
131C22×C61—22no—
132C22×C61—48no—
133C22×C61—44no—
134C22×C61—88no—
135C22×C61#45512yes—
136C22×C61C2×#12011yescharacteristic
137C4×C61—22no—
138C4×C61—21no—
139C4×C61—21no—
140C4×C61#57514yes—
141Dic61—44no—
142Dic61—44no—
143Dic61—82no—
144C2×#559—611no—
145#3138—611no—
146#3113—612no—
147#1640—618no—
148#1555D6111yescharacteristic
149#1491—612no—
150C2×C61—22no—
151C2×C61—84no—
152C2×C61#300512yes—
153C2×C61C2×#45511yescharacteristic
154#559—614no—
155#559—1224no—
156#559—1222no—
157#533—611no—
158C2×#127C2×D6111yescharacteristic
159C2×#124—1222no—
160C2×#123—1222no—
161C2×C4×Q8—612no—
162C2×C4×D4—1222no—
163#490—1222no—
164#486—614no—
165#483—1222no—
166#475—614no—
167#462—614no—
168#451—1222no—
169#441—1224no—
170#432C2×D6112yes—
171#418—1222no—
172C4×M16—611no—
173#331C2×D6114yes—
174Q8⋊C8—618no—
175C61#1853111yescharacteristic, complemented, Sylow
176C22×Q8—612no—
177C22×D4—22no—
178C22×D4D24412yes—
179C23×C4—1222no—
180C2×Q16—1224no—
181C2×M16—1221no—
182C4⋊Q8—612no—
183#127—22no—
184#127D24412yes—
185#127C22×D6112yes—
186#125—1221no—
187#124—1228no—
188#124—1228no—
189#123—1228no—
190#123—1228no—
191C22⋊Q8—1222no—
192C4×Q8—614no—
193C4×Q8—614no—
194C4×Q8—1222no—
195C4×Q8—1222no—
196C4×Q8—1222no—
197C4×D4—1228no—
198C4×D4—1228no—
199#117—1221no—
200C2×C4⋊C4—612no—
201C2×C4⋊C4—1222no—
202C2×C22⋊C4—1224no—
203C2×C22⋊C4—1222no—
204C2×C42—1222no—
205C2×C42C22×D6111yescharacteristic
206#106—1224no—
207C4⋊C8C22×D6114yes—
208#103—1224no—
209#103—1224no—
210#102—1228no—
211C22⋊C8—22no—
212#97—22no—
213#97—1221no—
214C4×C8—612no—
215C24—22no—
216C2×Q8—614no—
217C2×Q8—1222no—
218C2×Q8—1222no—
219C2×Q8—2444no—
220C2×D4—24no—
221C2×D4—24no—
222C2×D4—22no—
223C2×D4—48no—
224C2×D4—44no—
225C2×D4C2×D24414yes—
226C22×C4—1228no—
227C22×C4—1222no—
228C22×C4—1222no—
229C22×C4—4884no—
230C22×C4C2×D24411yescharacteristic
231C22×C4D4×D6112yes—
232Q16—2448no—
233M16—2442no—
234C2×C8—24no—
235C2×C8—1222no—
236C4⋊C4—614no—
237C4⋊C4—1224no—
238C4⋊C4—1222no—
239C4⋊C4—1224no—
240C4⋊C4—1222no—
241C4⋊C4—1222no—
242C4⋊C4—2442no—
243C22⋊C4—12216no—
244C22⋊C4—1228no—
245C22⋊C4—2441no—
246C42—1224no—
247C42—1224no—
248C42—2441no—
249C42C2×D24411yescharacteristic
250C42D4×D6112yes—
251C42C23×D6111yescharacteristic
252C23—24no—
253C23—48no—
254C23—44no—
255C23#1451974711yescharacteristic
256Q8—1224no—
257Q8—1224no—
258Q8—2444no—
259Q8—2444no—
260D4—44no—
261D4—48no—
262D4—48no—
263D4—82no—
264C2×C4—22no—
265C2×C4—24no—
266C2×C4—1224no—
267C2×C4—1224no—
268C2×C4—4888no—
269C2×C4—2442no—
270C2×C4—2444no—
271C2×C4—2444no—
272C2×C4#1451974712yes—
273C2×C4#1451974711yescharacteristic
274C2×C4C2×D4×D6112yes—
275C2×C4C22×D24411yescharacteristic
276C8—44no—
277C8—2442no—
278C22—22no—
279C22—44no—
280C22—48no—
281C22—88no—
282C22#1683994212yes—
283C22C2×#1451974711yescharacteristic
284C4—21no—
285C4—21no—
286C4—22no—
287C4—4882no—
288C4—2444no—
289C4—2444no—
290C4#1684055814yes—
291C2—22no—
292C2—84no—
293C2#2145506312yes—
294C2C2×#1683994211yescharacteristic
subgroupG/H
centerC22C2×#14519747
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1555C2
socleC22×C61C2×#120
prime radicalC2×#127×C61C22
hypercenter#1555D61
hypocenterC61#18531
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1555

Complements

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

NH ≅ G/Ntype
3. #21455950C2semidirect
4. #21445006C2semidirect
5. #21444265C2semidirect
18. #16840048C22semidirect
19. #16839973C22semidirect
24. #16839492C22semidirect
175. C61#18531semidirect

Maximal subgroups

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

subgroupindex
C2×#168404982
#214561932
#214559502
#214450062
#214442652
C61×#15552
#1853161