Finite Group Databasebeta

← #29575050

Subgroups — #29575050

Order 15616.

616 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 616 conjugacy classes of subgroups — too many to draw. Only its 143 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×#16839701C211yesmaximal, characteristic
2#21448437C211yesmaximal, characteristic
3#21445910C212yesmaximal
4#21443699C212yesmaximal
5C2×C61×#324C211yesmaximal, characteristic
6C2×#14519736—21no—
7C22×C4⋊Dic61C2211yescharacteristic
8#16839701—22no—
9#16839701C414yescomplemented
10#16839661—24no—
11#16839622—21no—
12#16839622C2212yes—
13C2×#14519684—21no—
14C2×#14519684—21no—
15C2×#14519684C2211yescharacteristic
16#16839525—24no—
17C22×C4⋊C4×C61C2211yescharacteristic
18C2×C22⋊C8×C61C2212yes—
19C61×#324—22no—
20C23×Dic61—21no—
21C23×Dic61—21no—
22#14519736—42no—
23#14519736—42no—
24C2×C4⋊Dic61—22no—
25C2×C4⋊Dic61—22no—
26C2×C4⋊Dic61—22no—
27C2×C4⋊Dic61—21no—
28C2×C4⋊Dic61—22no—
29C2×C4⋊Dic61C2×C412yes—
30C2×C4⋊Dic61D414yes—
31C2×C22⋊Dic61—22no—
32C2×C4×Dic61—22no—
33#14519684—22no—
34#14519684—24no—
35#14519684—24no—
36#14519684—21no—
37#14519684—22no—
38#14519684C2×C412yes—
39C23×C4×C61D411yescharacteristic
40C23×C4×C61D411yescharacteristic
41C23×C4×C61C2311yescharacteristic
42C22×C8×C61—22no—
43C2×C4⋊C4×C61—22no—
44C2×C4⋊C4×C61—22no—
45C2×C4⋊C4×C61C2×C412yes—
46C2×C4⋊C4×C61D412yes—
47C22⋊C8×C61—22no—
48C22⋊C8×C61—22no—
49C24×C61C2×D411yescharacteristic
50C22×C4×C61—21no—
51C22×C4×C61—21no—
52C22×C4×C61—21no—
53C22×C4×C61—21no—
54C22×C4×C61—21no—
55C22×C4×C61—21no—
56C22×C4×C61—42no—
57C22×C4×C61—42no—
58C22×C4×C61C22⋊C412yes—
59C22×C4×C61C22⋊C412yes—
60C22×C4×C61SD1612yes—
61C22×C4×C61Q1612yes—
62C22×C4×C61C22×C411yescharacteristic
63C22×C4×C61C2×D412yes—
64C22×C4×C61C2×D411yescharacteristic
65C22×C4×C61#4112yes—
66C22×Dic61—22no—
67C22×Dic61—22no—
68C22×Dic61—24no—
69C22×Dic61—42no—
70C22×Dic61—42no—
71C22×Dic61—44no—
72C22×Dic61—41no—
73C22×Dic61—41no—
74C22×Dic61—41no—
75C22×Dic61—41no—
76C2×C8×C61—44no—
77C2×C8×C61—42no—
78C4⋊C4×C61—24no—
79C4⋊C4×C61—24no—
80C22⋊Dic61—42no—
81C22⋊Dic61—42no—
82C4⋊Dic61—24no—
83C4⋊Dic61—44no—
84C4⋊Dic61—42no—
85C4⋊Dic61—44no—
86C4⋊Dic61—82no—
87C4×Dic61—44no—
88C23×C61—22no—
89C23×C61—22no—
90C23×C61C2×C22⋊C411yescharacteristic
91C23×C61#12011yescharacteristic
92C23×C61#12311yescharacteristic
93C23×C61C2×SD1611yescharacteristic
94C23×C61C2×Q1611yescharacteristic
95C23×C61AGL(1,ℤ/8)11yescharacteristic
96C23×C61#13711yescharacteristic
97C2×C4×C61—22no—
98C2×C4×C61—22no—
99C2×C4×C61—22no—
100C2×C4×C61—22no—
101C2×C4×C61—24no—
102C2×C4×C61—22no—
103C2×C4×C61—22no—
104C2×C4×C61—22no—
105C2×C4×C61—41no—
106C2×C4×C61—41no—
107C2×C4×C61—42no—
108C2×C4×C61—42no—
109C2×C4×C61—42no—
110C2×C4×C61—81no—
111C2×C4×C61#10314yes—
112C2×C4×C61C4×D412yes—
113C2×C4×C61#12112yes—
114C2×Dic61—42no—
115C2×Dic61—42no—
116C2×Dic61—44no—
117C2×Dic61—42no—
118C2×Dic61—42no—
119C2×Dic61—44no—
120C2×Dic61—44no—
121C2×Dic61—44no—
122C2×Dic61—82no—
123C2×Dic61—82no—
124C2×Dic61—82no—
125C2×Dic61—82no—
126C8×C61—82no—
127#9023—611nomaximal, Sylow
128C22×C61—22no—
129C22×C61—22no—
130C22×C61—22no—
131C22×C61—22no—
132C22×C61—21no—
133C22×C61—21no—
134C22×C61—21no—
135C22×C61—21no—
136C22×C61#38611yescharacteristic
137C22×C61C2×#10311yescharacteristic
138C22×C61#41711yescharacteristic
139C22×C61#45011yescharacteristic
140C22×C61#45111yescharacteristic
141C22×C61#45412yes—
142C22×C61#45812yes—
143C22×C61#48311yescharacteristic
144C22×C61#48411yescharacteristic
145C4×C61—24no—
146C4×C61—22no—
147C4×C61—22no—
148C4×C61—42no—
149Dic61—82no—
150Dic61—84no—
151Dic61—82no—
152C2×#400—611no—
153#2316—611no—
154#1824—612no—
155C2×#324D6111yescharacteristic
156#1294—612no—
157C2×C61—21no—
158C2×C61—21no—
159C2×C61—21no—
160C2×C61—21no—
161C2×C61#161612yes—
162C2×C61#161712yes—
163C2×C61#187211yescharacteristic
164C2×C61#188811yescharacteristic
165C2×C61#189011yescharacteristic
166C2×#117—1221no—
167C22×C4⋊C4—611no—
168C22×C4⋊C4C2×D6111yescharacteristic
169C2×C22⋊C8C2×D6112yes—
170#400—614no—
171#400—1222no—
172#389—1224no—
173#380—612no—
174#380—1221no—
175C2×#95—611no—
176C2×#95—1221no—
177C2×#95—1221no—
178#340—1224no—
179#324—22no—
180C61#902311yescharacteristic, complemented, Sylow
181C23×C4—1221no—
182C23×C4—1221no—
183C23×C4D24411yescharacteristic
184C23×C4D24411yescharacteristic
185C23×C4C22×D6111yescharacteristic
186C22×C8—22no—
187#117—2442no—
188#117—2442no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—612no—
192C2×C4⋊C4—614no—
193C2×C4⋊C4—1222no—
194C2×C4⋊C4—1222no—
195C2×C4⋊C4—1222no—
196C2×C4⋊C4—1222no—
197C2×C4⋊C4—1221no—
198C2×C4⋊C4C4×D6112yes—
199C2×C4⋊C4C22⋊D6112yes—
200C2×C22⋊C4—1222no—
201C2×C42—1222no—
202C22⋊C8—22no—
203C22⋊C8—22no—
204#95—612no—
205#95—1222no—
206#95—1224no—
207#95—1224no—
208#95—1222no—
209#95—1221no—
210C24C2×D24411yescharacteristic
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—1222no—
220C22×C4—1222no—
221C22×C4—1224no—
222C22×C4—2442no—
223C22×C4—2442no—
224C22×C4—2444no—
225C22×C4—2441no—
226C22×C4—2441no—
227C22×C4—2441no—
228C22×C4—2441no—
229C22×C4#1175819012yes—
230C22×C4Dic24412yes—
231C22×C4#1175819812yes—
232C22×C4#1175819812yes—
233C22×C4C2×C4×D6111yescharacteristic
234C22×C4D4×D6112yes—
235C22×C4#1175822412yes—
236C22×C4C2×C22⋊D6111yescharacteristic
237C2×C8—44no—
238C2×C8—42no—
239C4⋊C4—24no—
240C4⋊C4—24no—
241C4⋊C4—1224no—
242C4⋊C4—4882no—
243C4⋊C4—2444no—
244C4⋊C4—2444no—
245C4⋊C4—2442no—
246C22⋊C4—2442no—
247C22⋊C4—2442no—
248C42—2444no—
249C23—22no—
250C23—22no—
251C23C2×#1175819811yescharacteristic
252C23#1451974711yescharacteristic
253C23#1451976111yescharacteristic
254C23C2×#1175819011yescharacteristic
255C23C2×Dic24411yescharacteristic
256C23#1451980511yescharacteristic
257C23#1451981011yescharacteristic
258C2×C4—24no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—41no—
267C2×C4—42no—
268C2×C4—41no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—81no—
272C2×C4—4882no—
273C2×C4—4882no—
274C2×C4—4882no—
275C2×C4—4882no—
276C2×C4—2442no—
277C2×C4—2442no—
278C2×C4—2444no—
279C2×C4—2444no—
280C2×C4—2444no—
281C2×C4—2442no—
282C2×C4—2442no—
283C2×C4—2444no—
284C2×C4#1451970214yes—
285C2×C4#1451973712yes—
286C2×C4#1451974912yes—
287C8—82no—
288C22—21no—
289C22—21no—
290C22—21no—
291C22—21no—
292C22—22no—
293C22—22no—
294C22—22no—
295C22—22no—
296C22#1683963911yescharacteristic
297C22C2×#1451970211yescharacteristic
298C22#1683976411yescharacteristic
299C22#1683991511yescharacteristic
300C22#1683992011yescharacteristic
301C22#1683993712yes—
302C22#1683995312yes—
303C22#1684009311yescharacteristic
304C22#1684009811yescharacteristic
305C4—22no—
306C4—22no—
307C4—24no—
308C4—42no—
309C4—4882no—
310C4—4882no—
311C4—4884no—
312C2—21no—
313C2—21no—
314C2—21no—
315C2—21no—
316C2#2144485812yes—
317C2#2144486312yes—
318C2#2144612411yescharacteristic
319C2#2144622011yescharacteristic
320C2#2144623011yescharacteristic
subgroupG/H
centerC23#14519747
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC2×C61×#324C2
socleC23×C61#120
prime radicalC24×C61C2×D4
cosocleC23×C4×C61C23
hypercenterC2×#324D61
hypocenterC61#9023
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#324

Complements

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

NH ≅ G/Ntype
9. #16839701C4semidirect
180. C61#9023semidirect

Maximal subgroups

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

subgroupindex
C2×#168397012
#214484372
#214459102
#214436992
C2×C61×#3242
#902361