Finite Group Databasebeta

← #29572612

Subgroups — #29572612

Order 15616.

718 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 718 conjugacy classes of subgroups — too many to draw. Only its 107 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#21459517C211yesmaximal, characteristic
2#21445902C211yesmaximal, characteristic, complemented
3#21444165C214yesmaximal, complemented
4C61×#1818C211yesmaximal, characteristic, complemented
5#16840445—24no—
6#16840436—24no—
7#16840388C411yescharacteristic
8#16840359—22no—
9#16840324C411yescharacteristic
10#16840296—22no—
11C2×#14519760C2212yes—
12#16840235C412yes—
13#16839726C2212yes—
14#16839531—22no—
15#16839487—24no—
16#16839485—24no—
17C2×#117×C61C2211yescharacteristic
18C2×C22⋊C8×C61C2212yes—
19C61×#336—22no—
20C22×C22⋊D61C2×C412yes—
21C2×#11758194—22no—
22#14519772—24no—
23#14519771—24no—
24#14519769—22no—
25#14519767—24no—
26#14519765—22no—
27#14519762—24no—
28#14519761—22no—
29#14519760—22no—
30#14519760—22no—
31#14519760—24no—
32#14519756—22no—
33#14519755—24no—
34#14519754—24no—
35#14519751—22no—
36#14519747—22no—
37C4×Dic122—22no—
38C4×D244—22no—
39C4×C22⋊D61—24no—
40#14519737—24no—
41#14519734—24no—
42#14519733—22no—
43C2×#11758196D414yes—
44C2×C22⋊Dic61C2×C412yes—
45C2×#11758198D414yes—
46C22⋊C4×D61—22no—
47C22⋊#92418—24no—
48C23×C4×C61C2311yescharacteristic
49C22×C8×C61—22no—
50#117×C61—24no—
51C2×C4⋊C4×C61D412yes—
52C2×C22⋊C4×C61C2×C411yescharacteristic
53C2×C22⋊C4×C61C2×C411yescharacteristic
54C2×C42×C61D412yes—
55C22⋊C8×C61—24no—
56C24×C61C22×C411yescharacteristic
57C23×D61C22⋊C412yes—
58C22×C4×C61—42no—
59C22×C4×C61C22⋊C412yes—
60C22×C4×C61C22⋊C412yes—
61C22×C4×C61C2×D411yescharacteristic
62C22×C4×C61C2×D411yescharacteristic
63C22×C4×C61C2×D414yes—
64C2×C22⋊D61—22no—
65C2×C22⋊D61—22no—
66C2×C22⋊D61—44no—
67C22×Dic61C22⋊C412yes—
68C22×Dic61C22⋊C414yes—
69C2×D244—22no—
70C2×C4×D61—24no—
71C2×Dic122—22no—
72C2×C8×C61—44no—
73C2×C8×C61—42no—
74C4⋊C4×C61—24no—
75C22⋊C4×C61—22no—
76C22⋊C4×C61—22no—
77C42×C61—22no—
78C42×C61—22no—
79C22⋊Dic61—22no—
80C22⋊Dic61—22no—
81#11758198—24no—
82#11758198—24no—
83#11758198—24no—
84#11758198—24no—
85C4⋊Dic61—22no—
86C4⋊Dic61—22no—
87#11758196—24no—
88#11758196—24no—
89#11758196—24no—
90#11758196—24no—
91C4×Dic61—24no—
92#11758194—44no—
93C2×#92418—42no—
94C23×C61—22no—
95C23×C61—42no—
96C23×C61C2×C22⋊C412yes—
97C23×C61C2×C22⋊C411yescharacteristic
98C22×D61—22no—
99C22×D61—42no—
100C22×D61—44no—
101C2×C4×C61—22no—
102C2×C4×C61—22no—
103C2×C4×C61—22no—
104C2×C4×C61—22no—
105C2×C4×C61—44no—
106C2×C4×C61—42no—
107C2×C4×C61—42no—
108C2×C4×C61#12012yes—
109C2×C4×C61#12012yes—
110C22⋊D61—44no—
111C22⋊D61—44no—
112C2×Dic61—22no—
113C2×Dic61—24no—
114C2×Dic61—44no—
115C2×Dic61—42no—
116D244—82no—
117C4×D61—84no—
118Dic122—82no—
119C8×C61—82no—
120#92418—82no—
121#8540—611nomaximal, Sylow
122C22×C61—22no—
123C22×C61—22no—
124C22×C61—24no—
125C22×C61—82no—
126C22×C61#37911yescharacteristic
127C22×C61#41012yes—
128C2×D61—42no—
129C2×D61—44no—
130C2×D61—162no—
131C4×C61—22no—
132C4×C61—22no—
133C4×C61—42no—
134C4×C61—42no—
135Dic61—42no—
136Dic61—44no—
137#3491—611no—
138#1822—611no—
139#1818D6111yescharacteristic, complemented
140#1468—614no—
141C2×C61—22no—
142C2×C61—42no—
143C2×C61#178211yescharacteristic
144C2×C61#179112yes—
145D61—82no—
146#552—1224no—
147#551—1224no—
148#543—611no—
149#539—1222no—
150#534—611no—
151#529—1222no—
152C2×#123—612no—
153#518—612no—
154C2×#117C2×D6111yescharacteristic
155#407—612no—
156C2×C22⋊C8C2×D6112yes—
157#343—1222no—
158#336—22no—
159#324—1224no—
160#323—1224no—
161C61#854011yescharacteristic, complemented, Sylow
162C22×D4—612no—
163C23×C4C22×D6111yescharacteristic
164C2×M16—1222no—
165C22×C8—22no—
166#126—1224no—
167#126—1224no—
168#125—1222no—
169#125—1224no—
170#124—1222no—
171#123—1222no—
172#123—1224no—
173#123—1222no—
174#123—1224no—
175#123—1222no—
176C22⋊Q8—1224no—
177C22⋊Q8—1224no—
178C22⋊Q8—1222no—
179#121—1222no—
180#120—1222no—
181C4×Q8—1222no—
182C4×D4—1222no—
183C4×D4—1224no—
184C4×D4—1224no—
185#117—24no—
186#117—1224no—
187#117—1222no—
188C2×C4⋊C4—614no—
189C2×C4⋊C4D24412yes—
190C2×C22⋊C4—612no—
191C2×C22⋊C4—614no—
192C2×C22⋊C4—1222no—
193C2×C22⋊C4C4×D6111yescharacteristic
194C2×C22⋊C4C4×D6111yescharacteristic
195C2×C42C22⋊D6112yes—
196C22⋊C8—24no—
197C22⋊C8—1224no—
198C24—612no—
199C24C2×C4×D6111yescharacteristic
200C2×Q8—1222no—
201C2×D4—1222no—
202C2×D4—1222no—
203C2×D4—1222no—
204C2×D4—2444no—
205C22×C4—42no—
206C22×C4—614no—
207C22×C4—612no—
208C22×C4—1224no—
209C22×C4#1175819812yes—
210C22×C4#1175819812yes—
211C22×C4C2×D24411yescharacteristic
212C22×C4D4×D6114yes—
213C22×C4C2×C22⋊D6111yescharacteristic
214M16—2444no—
215C2×C8—44no—
216C2×C8—42no—
217C2×C8—2442no—
218C4⋊C4—24no—
219C4⋊C4—1224no—
220C4⋊C4—1224no—
221C4⋊C4—1222no—
222C4⋊C4—1222no—
223C4⋊C4—1224no—
224C4⋊C4—1224no—
225C22⋊C4—22no—
226C22⋊C4—22no—
227C22⋊C4—1224no—
228C22⋊C4—1224no—
229C22⋊C4—1222no—
230C22⋊C4—1222no—
231C22⋊C4—1224no—
232C22⋊C4—1224no—
233C42—22no—
234C42—22no—
235C42—1224no—
236C23—22no—
237C23—42no—
238C23—1222no—
239C23—2442no—
240C23—2444no—
241C23C22⋊C4×D6112yes—
242C23C2×#1175819811yescharacteristic
243Q8—4882no—
244D4—4882no—
245D4—2444no—
246D4—2444no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—44no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—1222no—
255C2×C4—1224no—
256C2×C4—4884no—
257C2×C4—2444no—
258C2×C4—2442no—
259C2×C4#1451974512yes—
260C2×C4#1451974712yes—
261C8—82no—
262C8—4882no—
263C22—22no—
264C22—22no—
265C22—24no—
266C22—82no—
267C22—9762no—
268C22—2442no—
269C22—2444no—
270C22#1683961711yescharacteristic
271C22#1683973512yes—
272C4—22no—
273C4—22no—
274C4—42no—
275C4—42no—
276C4—2442no—
277C4—2444no—
278C2—22no—
279C2—42no—
280C2—4882no—
281C2#2144573611yescharacteristic
282C2#2144576912yes—
subgroupG/H
centerC22#16839617
commutatorC24×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1818C2
socleC22×C61#379
prime radical#16840324C4
cosocleC23×C4×C61C23
hypercenter#1818D61
hypocenterC61#8540
LCSG ▷ C24×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1818

Complements

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

NH ≅ G/Ntype
2. #21445902C2semidirect
3. #21444165C2semidirect
4. C61×#1818C2semidirect
139. #1818D61semidirect
161. C61#8540semidirect

Maximal subgroups

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

subgroupindex
#214595172
#214459022
#214441652
C61×#18182
#854061