Finite Group Databasebeta

← #29572625

Subgroups — #29572625

Order 15616.

1222 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 1222 conjugacy classes of subgroups — too many to draw. Only its 123 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
1C22×#14519764C211yesmaximal, characteristic
2#21445891C214yesmaximal, complemented
3#21445734C211yesmaximal, characteristic
4C61×#1781C211yesmaximal, characteristic
5C23×Dic122C411yescharacteristic, complemented
6C2×#14519764—28no—
7C2×#14519764—22no—
8C2×#14519764—22no—
9C22×C22⋊Dic61C412yescomplemented
10C22×C22⋊Dic61C2212yes—
11C22×C4×Dic61C411yescharacteristic, complemented
12C2×#14519692—22no—
13#16839618C2212yes—
14#16839528—28no—
15C23×D4×C61C2211yescharacteristic
16C2×#99×C61—22no—
17C61×#379C2212yes—
18C22×Dic122—22no—
19C22×Dic122—24no—
20C22×Dic122—44no—
21C23×Dic61C2×C412yes—
22C23×Dic61C2×C412yes—
23#14519764—216no—
24#14519764—216no—
25C2×C22⋊Dic61—24no—
26C2×C22⋊Dic61—22no—
27C2×C22⋊Dic61—24no—
28C2×C22⋊Dic61—22no—
29C2×C22⋊Dic61—24no—
30C2×C22⋊Dic61—24no—
31C2×C22⋊Dic61—22no—
32C2×C22⋊Dic61—24no—
33C2×C22⋊Dic61D418yes—
34C2×C4×Dic61—22no—
35C2×C4×Dic61—24no—
36#14519692—44no—
37C25×C61C2×C411yescharacteristic
38C25×C61C2311yescharacteristic
39C22×D4×C61—24no—
40C22×D4×C61—44no—
41C22×D4×C61D412yes—
42C22×D4×C61D412yes—
43C23×C4×C61C2×C411yescharacteristic
44C2×C22⋊C4×C61—24no—
45C2×C22⋊C4×C61—22no—
46#99×C61—44no—
47C24×C61—22no—
48C24×C61—42no—
49C24×C61—42no—
50C24×C61—44no—
51C24×C61—44no—
52C24×C61C22⋊C412yes—
53C24×C61C22×C411yescharacteristic
54C24×C61C2×D411yescharacteristic
55C24×C61C2×D414yes—
56C24×C61C2×D411yescharacteristic
57C2×D4×C61—24no—
58C2×D4×C61—24no—
59C2×D4×C61—24no—
60C2×D4×C61—24no—
61C2×D4×C61—42no—
62C2×D4×C61—42no—
63C2×D4×C61—416no—
64C22×C4×C61—22no—
65C22×C4×C61—24no—
66C22×C4×C61—42no—
67C22×C4×C61C22⋊C412yes—
68C22×Dic61—22no—
69C22×Dic61—22no—
70C22×Dic61—22no—
71C22×Dic61—22no—
72C22×Dic61—44no—
73C22×Dic61—44no—
74C22×Dic61—42no—
75C22×Dic61C22⋊C414yes—
76C22×Dic61C22⋊C414yes—
77C2×Dic122—28no—
78C2×Dic122—416no—
79C2×Dic122—44no—
80C2×Dic122—44no—
81C22⋊C4×C61—44no—
82C22⋊C4×C61—48no—
83C22⋊Dic61—216no—
84C22⋊Dic61—28no—
85C22⋊Dic61—28no—
86C22⋊Dic61—44no—
87C22⋊Dic61—48no—
88C4×Dic61—28no—
89C23×C61—22no—
90C23×C61—24no—
91C23×C61—22no—
92C23×C61—22no—
93C23×C61—28no—
94C23×C61—22no—
95C23×C61—22no—
96C23×C61—22no—
97C23×C61—44no—
98C23×C61—42no—
99C23×C61—44no—
100C23×C61—42no—
101C23×C61—44no—
102C23×C61—44no—
103C23×C61—44no—
104C23×C61—44no—
105C23×C61—88no—
106C23×C61—88no—
107C23×C61C2×C22⋊C411yescharacteristic
108C23×C61C2×C22⋊C412yes—
109C23×C61#12012yes—
110C23×C61#12012yes—
111D4×C61—48no—
112D4×C61—48no—
113C2×C4×C61—28no—
114C2×C4×C61—22no—
115C2×C4×C61—22no—
116C2×C4×C61—82no—
117C2×C4×C61—84no—
118C2×C4×C61#9914yes—
119C2×Dic61—24no—
120C2×Dic61—24no—
121C2×Dic61—44no—
122C2×Dic61—44no—
123C2×Dic61—44no—
124C2×Dic61—44no—
125C2×Dic61—44no—
126C2×Dic61—44no—
127C2×Dic61—82no—
128C2×Dic61—84no—
129Dic122—416no—
130#8544—611nomaximal, Sylow
131C22×C61—28no—
132C22×C61—22no—
133C22×C61—22no—
134C22×C61—22no—
135C22×C61—22no—
136C22×C61—22no—
137C22×C61—24no—
138C22×C61—24no—
139C22×C61—42no—
140C22×C61—44no—
141C22×C61—82no—
142C22×C61—82no—
143C22×C61—88no—
144C22×C61—82no—
145C22×C61—82no—
146C22×C61—88no—
147C22×C61#37911yescharacteristic
148C22×C61C2×#9912yes—
149C22×C61#41014yes—
150C4×C61—24no—
151C4×C61—162no—
152Dic61—44no—
153Dic61—44no—
154Dic61—162no—
155C22×#124—611no—
156#1819—614no—
157#1781—611no—
158#1781D6111yescharacteristic
159C2×C61—22no—
160C2×C61—24no—
161C2×C61—42no—
162C2×C61—42no—
163C2×C61—82no—
164C2×C61#178111yescharacteristic
165C2×C61#178212yes—
166C23×Q8—611no—
167C23×D4C2×D6111yescharacteristic
168C2×#124—1228no—
169C2×#124—1222no—
170C2×#124—1222no—
171C22×C22⋊C4—612no—
172C22×C22⋊C4—612no—
173C22×C42—611no—
174C2×#99—22no—
175C2×#99—1222no—
176#379—612no—
177#379C2×D6112yes—
178#342—1228no—
179C61#854411yescharacteristic, complemented, Sylow
180C25C4×D6111yescharacteristic
181C25C22×D6111yescharacteristic
182C22×Q8—1224no—
183C22×Q8—1222no—
184C22×Q8—2444no—
185C22×D4—24no—
186C22×D4—44no—
187C22×D4D24412yes—
188C22×D4C22⋊D6112yes—
189C23×C4—612no—
190C23×C4—612no—
191C23×C4C4×D6111yescharacteristic
192#124—12216no—
193#124—12216no—
194C2×C22⋊C4—24no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—618no—
197C2×C22⋊C4—1224no—
198C2×C22⋊C4—1224no—
199C2×C22⋊C4—1224no—
200C2×C22⋊C4—1222no—
201C2×C22⋊C4—1224no—
202C2×C22⋊C4—1222no—
203C2×C22⋊C4—1222no—
204C2×C22⋊C4—1224no—
205C2×C42—1224no—
206C2×C42—1222no—
207#99—44no—
208#99—2444no—
209C24—22no—
210C24—42no—
211C24—42no—
212C24—44no—
213C24—44no—
214C24#1175819812yes—
215C24C2×C4×D6111yescharacteristic
216C24C2×D24411yescharacteristic
217C24D4×D6114yes—
218C24C2×C22⋊D6111yescharacteristic
219C2×Q8—1228no—
220C2×Q8—24416no—
221C2×Q8—2444no—
222C2×Q8—2444no—
223C2×D4—24no—
224C2×D4—24no—
225C2×D4—24no—
226C2×D4—24no—
227C2×D4—42no—
228C2×D4—42no—
229C2×D4—416no—
230C22×C4—22no—
231C22×C4—24no—
232C22×C4—42no—
233C22×C4—614no—
234C22×C4—614no—
235C22×C4—1222no—
236C22×C4—1222no—
237C22×C4—1222no—
238C22×C4—1222no—
239C22×C4—2442no—
240C22×C4—2444no—
241C22×C4—2444no—
242C22×C4#1175819812yes—
243C22⋊C4—44no—
244C22⋊C4—48no—
245C22⋊C4—12216no—
246C22⋊C4—1228no—
247C22⋊C4—1228no—
248C22⋊C4—2444no—
249C22⋊C4—2448no—
250C42—1228no—
251C23—22no—
252C23—24no—
253C23—22no—
254C23—22no—
255C23—28no—
256C23—22no—
257C23—22no—
258C23—22no—
259C23—44no—
260C23—42no—
261C23—44no—
262C23—42no—
263C23—44no—
264C23—44no—
265C23—44no—
266C23—44no—
267C23—88no—
268C23—88no—
269C23C22⋊C4×D6112yes—
270C23C2×#1175819811yescharacteristic
271C23#1451974512yes—
272C23#1451974712yes—
273Q8—24416no—
274D4—48no—
275D4—48no—
276C2×C4—28no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—82no—
280C2×C4—84no—
281C2×C4—1224no—
282C2×C4—1224no—
283C2×C4—4882no—
284C2×C4—4884no—
285C2×C4—2444no—
286C2×C4—2444no—
287C2×C4—2444no—
288C2×C4—2444no—
289C2×C4—2444no—
290C2×C4—2444no—
291C2×C4#1451969114yes—
292C22—28no—
293C22—22no—
294C22—22no—
295C22—22no—
296C22—22no—
297C22—22no—
298C22—24no—
299C22—24no—
300C22—42no—
301C22—44no—
302C22—82no—
303C22—82no—
304C22—88no—
305C22—82no—
306C22—82no—
307C22—88no—
308C22#1683961711yescharacteristic
309C22C2×#1451969112yes—
310C22#1683973414yes—
311C4—24no—
312C4—162no—
313C4—9762no—
314C4—2444no—
315C4—2444no—
316C2—22no—
317C2—24no—
318C2—42no—
319C2—42no—
320C2—82no—
321C2#2144573311yescharacteristic
322C2#2144573512yes—
subgroupG/H
centerC22#16839617
commutatorC24×C61C22×C4
FrattiniC25C22×D61
FittingC61×#1781C2
socleC22×C61#379
prime radicalC23×D4×C61C22
cosocleC25×C61C23
hypercenter#1781D61
hypocenterC61#8544
LCSG ▷ C24×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C25 ◁ #1781

Complements

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

NH ≅ G/Ntype
2. #21445891C2semidirect
5. C23×Dic122C4semidirect
9. C22×C22⋊Dic61C4semidirect
11. C22×C4×Dic61C4semidirect
179. C61#8544semidirect

Maximal subgroups

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

subgroupindex
C22×#145197642
#214458912
#214457342
C61×#17812
#854461