Finite Group Databasebeta

← #29572196

Subgroups — #29572196

Order 15616.

1002 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 1002 conjugacy classes of subgroups — too many to draw. Only its 115 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#21459477C211yesmaximal, characteristic
2#21445737C211yesmaximal, characteristic, complemented
3#21444376C214yesmaximal, complemented
4C61×#1780C211yesmaximal, characteristic, complemented
5C2×#14519825C411yescharacteristic, complemented
6#16840431—24no—
7#16840419—24no—
8#16840329—22no—
9#16840329C412yescomplemented
10#16840273—22no—
11#16840273C2212yes—
12#16840240C411yescharacteristic, complemented
13C2×#14519692—22no—
14#16839625C2212yes—
15#16839500—24no—
16#16839497—24no—
17C22×#41×C61C2211yescharacteristic
18C2×C22⋊C8×C61—22no—
19C2×C22⋊C8×C61C2212yes—
20#14519825—44no—
21C2×#11758226—22no—
22C2×#11758222—22no—
23C2×#11758222C2×C412yes—
24C2×Q8×D61—22no—
25#14519773—24no—
26#14519773—22no—
27#14519764—24no—
28#14519764—22no—
29#14519760—22no—
30#14519756—24no—
31#14519756D414yes—
32#14519751—24no—
33#14519751D414yes—
34#14519748—28no—
35#14519748—24no—
36#14519746—24no—
37#14519746—22no—
38Q8×Dic61—24no—
39C4×C22⋊D61—28no—
40C4×C22⋊D61—24no—
41#14519736—22no—
42#14519736C2×C412yes—
43C2×C22⋊Dic61—22no—
44#14519692—44no—
45#14519684—24no—
46C2×#41×C61—24no—
47C2×#41×C61—44no—
48C22×Q8×C61C2×C411yescharacteristic
49C22×D4×C61C2×C411yescharacteristic
50C22×D4×C61D412yes—
51C23×C4×C61D412yes—
52C23×C4×C61C2311yescharacteristic
53C22×C8×C61—22no—
54C22⋊C8×C61—24no—
55C22⋊C8×C61—44no—
56C24×C61C22⋊C412yes—
57C24×C61C2×D411yescharacteristic
58#41×C61—416no—
59C2×Q8×C61—24no—
60C2×Q8×C61—22no—
61C2×D4×C61—24no—
62C2×D4×C61—24no—
63C2×D4×C61—22no—
64C2×D4×C61—42no—
65C2×D4×C61—42no—
66C22×C4×C61—24no—
67C22×C4×C61—24no—
68C22×C4×C61—42no—
69C22×C4×C61—42no—
70C22×C4×C61C22⋊C412yes—
71C22×C4×C61C22×C411yescharacteristic
72C22×C4×C61C2×D411yescharacteristic
73C22×C4×C61C2×D414yes—
74C2×C22⋊D61—22no—
75C2×C22⋊D61—28no—
76C2×C22⋊D61—24no—
77C22×Dic61—42no—
78#11758226—48no—
79Q8×D61—48no—
80#11758222—44no—
81#11758222—44no—
82#11758222—44no—
83C2×D244—22no—
84C2×D244C22⋊C412yes—
85C2×C4×D61—24no—
86C2×C4×D61—22no—
87C2×Dic122—22no—
88C2×Dic122C22⋊C412yes—
89C2×C8×C61—44no—
90C2×C8×C61—42no—
91C22⋊Dic61—28no—
92C22⋊Dic61—22no—
93C22⋊Dic61—24no—
94C22⋊Dic61—44no—
95#11758198—24no—
96C4⋊Dic61—24no—
97C4⋊Dic61C22⋊C414yes—
98#11758196—24no—
99C4×Dic61—24no—
100C4×Dic61—22no—
101C23×C61—22no—
102C23×C61—22no—
103C23×C61—22no—
104C23×C61—22no—
105C23×C61—42no—
106C23×C61—44no—
107C23×C61C2×C22⋊C411yescharacteristic
108C23×C61#12012yes—
109C22×D61—22no—
110Q8×C61—28no—
111D4×C61—48no—
112D4×C61—44no—
113C2×C4×C61—22no—
114C2×C4×C61—22no—
115C2×C4×C61—22no—
116C2×C4×C61—22no—
117C2×C4×C61—44no—
118C2×C4×C61—48no—
119C2×C4×C61#9914yes—
120C2×C4×C61C2×C22⋊C412yes—
121C2×C4×C61#12012yes—
122C22⋊D61—44no—
123C22⋊D61—84no—
124C22⋊D61—88no—
125C2×Dic61—24no—
126C2×Dic61—22no—
127C2×Dic61—82no—
128C2×Dic61—84no—
129D244—42no—
130D244—44no—
131C4×D61—48no—
132C4×D61—44no—
133Dic122—44no—
134Dic122—42no—
135C8×C61—82no—
136#8464—611nomaximal, Sylow
137C22×C61—22no—
138C22×C61—24no—
139C22×C61—22no—
140C22×C61—44no—
141C22×C61—44no—
142C22×C61—42no—
143C22×C61—82no—
144C22×C61—82no—
145C22×C61#37911yescharacteristic
146C22×C61C2×#9912yes—
147C2×D61—44no—
148C2×D61—42no—
149C4×C61—22no—
150C4×C61—24no—
151C4×C61—22no—
152Dic61—44no—
153Dic61—42no—
154Dic61—162no—
155#3487—611no—
156#1782—611no—
157#1780D6111yescharacteristic, complemented
158#1516—614no—
159C2×C61—22no—
160C2×C61—42no—
161C2×C61—82no—
162C2×C61#178111yescharacteristic
163C2×C61#179412yes—
164D61—82no—
165C2×2−1+4—611no—
166C22×#41C2×D6111yescharacteristic
167#550—1224no—
168#548—1224no—
169#535—612no—
170#535—1222no—
171#525—612no—
172#525—1222no—
173#519—611no—
174C2×#99—1222no—
175C2×C22⋊C8—22no—
176C2×C22⋊C8C2×D6112yes—
177#381—612no—
178#328—1224no—
179#327—1224no—
180C61#846411yescharacteristic, complemented, Sylow
1812−1+4—2444no—
182C2×#41—24no—
183C2×#41—44no—
184C2×#41—612no—
185C2×#41—1222no—
186C2×#41—1222no—
187C22×Q8—1222no—
188C22×Q8C4×D6111yescharacteristic
189C22×D4C4×D6111yescharacteristic
190C22×D4D24412yes—
191C23×C4C22⋊D6112yes—
192C23×C4C22×D6111yescharacteristic
193C22×C8—22no—
194#127—1222no—
195#127—1224no—
196#124—1222no—
197#124—1224no—
198#123—1222no—
199C22⋊Q8—614no—
200C22⋊Q8—1224no—
201#121—614no—
202#121—1224no—
203#121—1224no—
204#121—1228no—
205#120—1224no—
206#120—1222no—
207C4×Q8—1224no—
208C4×D4—1228no—
209C4×D4—1224no—
210#117—612no—
211#117—1222no—
212C2×C22⋊C4—1222no—
213#99—2444no—
214C22⋊C8—24no—
215C22⋊C8—44no—
216#95—1224no—
217C24#1175819812yes—
218C24C2×D24411yescharacteristic
219#41—416no—
220#41—2444no—
221#41—2444no—
222#41—2448no—
223#41—2444no—
224C2×Q8—24no—
225C2×Q8—22no—
226C2×Q8—612no—
227C2×Q8—1222no—
228C2×Q8—2448no—
229C2×D4—24no—
230C2×D4—24no—
231C2×D4—22no—
232C2×D4—42no—
233C2×D4—42no—
234C2×D4—612no—
235C2×D4—1222no—
236C2×D4—1228no—
237C2×D4—1224no—
238C2×D4—1222no—
239C22×C4—24no—
240C22×C4—24no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—1224no—
244C22×C4—1222no—
245C22×C4—2442no—
246C22×C4#1175819812yes—
247C22×C4C2×C4×D6111yescharacteristic
248C22×C4D4×D6114yes—
249C22×C4C2×C22⋊D6111yescharacteristic
250C2×C8—44no—
251C2×C8—42no—
252C4⋊C4—614no—
253C4⋊C4—1224no—
254C4⋊C4—1224no—
255C22⋊C4—1228no—
256C22⋊C4—1222no—
257C22⋊C4—1224no—
258C22⋊C4—1224no—
259C22⋊C4—2444no—
260C42—1224no—
261C42—1222no—
262C23—22no—
263C23—22no—
264C23—22no—
265C23—22no—
266C23—42no—
267C23—44no—
268C23—1222no—
269C23C2×#1175819811yescharacteristic
270C23#1451974712yes—
271Q8—28no—
272Q8—2442no—
273Q8—2444no—
274D4—48no—
275D4—44no—
276D4—4884no—
277D4—4888no—
278D4—2442no—
279D4—2444no—
280D4—2444no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—44no—
286C2×C4—48no—
287C2×C4—1224no—
288C2×C4—1222no—
289C2×C4—4882no—
290C2×C4—4884no—
291C2×C4—2448no—
292C2×C4—2444no—
293C2×C4#1451969114yes—
294C2×C4C22⋊C4×D6112yes—
295C2×C4#1451974512yes—
296C8—82no—
297C22—22no—
298C22—22no—
299C22—24no—
300C22—44no—
301C22—42no—
302C22—44no—
303C22—82no—
304C22—82no—
305C22—2442no—
306C22—2444no—
307C22#1683961711yescharacteristic
308C22C2×#1451969112yes—
309C4—22no—
310C4—24no—
311C4—22no—
312C4—9762no—
313C4—2442no—
314C4—2444no—
315C2—22no—
316C2—42no—
317C2—82no—
318C2—4882no—
319C2#2144573311yescharacteristic
320C2#2144578212yes—
subgroupG/H
centerC22#16839617
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1780C2
socleC22×C61#379
prime radical#21459477C2
cosocleC23×C4×C61C23
hypercenter#1780D61
hypocenterC61#8464
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1780

Complements

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

NH ≅ G/Ntype
2. #21445737C2semidirect
3. #21444376C2semidirect
4. C61×#1780C2semidirect
5. C2×#14519825C4semidirect
9. #16840329C4semidirect
12. #16840240C4semidirect
157. #1780D61semidirect
180. C61#8464semidirect

Maximal subgroups

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

subgroupindex
#214594772
#214457372
#214443762
C61×#17802
#846461