Finite Group Databasebeta

← #29853054

Subgroups — #29853054

Order 15616.

2650 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519756C212yesmaximal
2#21454171C214yesmaximal, complemented
3#21451933C214yesmaximal, complemented
4#21451194C218yesmaximal, complemented
5#21450297C214yesmaximal, complemented
6#21449463C214yesmaximal, complemented
7#21448433C214yesmaximal
8C2×C4×C22⋊C4×C61C211yesmaximal, characteristic
9C23×Dic122C2212yes—
10C2×#14519777C2214yescomplemented
11C2×#14519756—216no—
12C2×#14519756C2214yes—
13C2×#14519756C2214yes—
14C2×#14519756C2214yes—
15C2×#14519756C2214yes—
16C22×C4⋊Dic61C2212yes—
17C22×#11758196C2214yes—
18C22×C22⋊Dic61C2214yes—
19#16839694C22116yescomplemented
20#16839683C22116yescomplemented
21#16839668C2218yescomplemented
22#16839660C22116yescomplemented
23#16839637C22116yes—
24#16839637C22116yes—
25#16839632C2214yescomplemented
26#16839622C22116yes—
27C24×C4×C61C2211yescharacteristic
28C22×C22⋊C4×C61C2212yes—
29C22×C42×C61C2212yes—
30C4×C22⋊C4×C61C2214yes—
31C4×C22⋊C4×C61C2214yes—
32C2×#95×C61C2212yes—
33C22×Dic122—28no—
34C22×Dic122—48no—
35C22×Dic122C2314yes—
36C22×Dic122C2314yes—
37C23×Dic61C2314yes—
38C23×Dic61C2314yes—
39#14519777—48no—
40#14519756—216no—
41#14519756—216no—
42#14519756—216no—
43#14519756—216no—
44C2×C4⋊Dic61—28no—
45C2×C4⋊Dic61C2314yes—
46C2×C4⋊Dic61C2314yes—
47C2×C4⋊Dic61C23116yes—
48C2×C4⋊Dic61C23116yes—
49C2×#11758196—216no—
50C2×#11758196C2318yes—
51C2×#11758196C2318yes—
52C2×C22⋊Dic61—216no—
53C2×C22⋊Dic61C2318yes—
54C2×C22⋊Dic61C2318yes—
55#14519684C23132yes—
56C25×C61C2311yescharacteristic
57C23×C4×C61—44no—
58C23×C4×C61D418yes—
59C23×C4×C61C2312yes—
60C23×C4×C61C2311yescharacteristic
61C23×C4×C61C2312yes—
62C23×C4×C61C2311yescharacteristic
63C23×C4×C61C2314yes—
64C2×C22⋊C4×C61Q8116yescomplemented
65C2×C22⋊C4×C61C2318yes—
66C2×C42×C61—44no—
67C2×C42×C61C2314yes—
68C2×C42×C61C2314yes—
69#95×C61C2318yes—
70C24×C61—42no—
71C24×C61—42no—
72C24×C61C2×D414yes—
73C24×C61C2×Q818yes—
74C24×C61C2412yes—
75C24×C61C2411yescharacteristic
76C22×C4×C61—28no—
77C22×C4×C61—24no—
78C22×C4×C61—416no—
79C22×C4×C61—44no—
80C22×C4×C61—44no—
81C22×C4×C61—42no—
82C22×C4×C61—44no—
83C22×C4×C61C2×D418yes—
84C22×C4×C61C2×D414yes—
85C22×C4×C61C2×D418yes—
86C22×C4×C61C2×D414yes—
87C22×C4×C61C2×Q8116yescomplemented
88C22×C4×C61C2412yes—
89C22×C4×C61C2411yescharacteristic
90C22×C4×C61C2418yes—
91C22×C4×C61C2411yescharacteristic
92C22×Dic61—28no—
93C22×Dic61—28no—
94C22×Dic61—48no—
95C22×Dic61—48no—
96C22×Dic61C2418yes—
97C22×Dic61C2418yes—
98C2×Dic122—28no—
99C2×Dic122—28no—
100C2×Dic122—216no—
101C2×Dic122—48no—
102C2×Dic122—432no—
103C22⋊C4×C61C2×Q8132yescomplemented
104C42×C61—28no—
105C42×C61—44no—
106C22⋊Dic61—216no—
107C22⋊Dic61—216no—
108C4⋊Dic61—28no—
109C4⋊Dic61—28no—
110C4⋊Dic61—416no—
111C4⋊Dic61—416no—
112#11758196—216no—
113#11758196—216no—
114C23×C61—24no—
115C23×C61—22no—
116C23×C61—24no—
117C23×C61—22no—
118C23×C61—44no—
119C23×C61—44no—
120C23×C61—416no—
121C23×C61C4⋊Q8116yescomplemented
122C23×C61C22×D412yes—
123C23×C61C22×D414yes—
124C23×C61C22×Q818yes—
125C23×C61C22×Q814yes—
126C23×C61C2511yescharacteristic
127C2×C4×C61—28no—
128C2×C4×C61—22no—
129C2×C4×C61—24no—
130C2×C4×C61—28no—
131C2×C4×C61—216no—
132C2×C4×C61—28no—
133C2×C4×C61—44no—
134C2×C4×C61—48no—
135C2×C4×C61—416no—
136C2×C4×C61C22×D412yes—
137C2×C4×C61C22×D414yes—
138C2×C4×C61C22×D412yes—
139C2×C4×C61C22×Q8116yes—
140C2×C4×C61H(5,𝔽2)14yes—
141C2×Dic61—28no—
142C2×Dic61—28no—
143C2×Dic61—216no—
144C2×Dic61—216no—
145C2×Dic61—416no—
146C2×Dic61—416no—
147Dic122—416no—
148Dic122—416no—
149#35011—611nomaximal, Sylow
150C22×C61—24no—
151C22×C61—22no—
152C22×C61—24no—
153C22×C61—22no—
154C22×C61—28no—
155C22×C61—216no—
156C22×C61—416no—
157C22×C61C2×C4⋊Q814yes—
158C22×C61C2×C4⋊Q818yes—
159C22×C61C23×D411yescharacteristic
160C22×C61C23×Q812yes—
161C22×C61C2×H(5,𝔽2)12yes—
162C22×C61#58512yes—
163C4×C61—24no—
164C4×C61—22no—
165C4×C61—216no—
166C4×C61#53414yes—
167Dic61—48no—
168Dic61—48no—
169C22×C22⋊Q8—612no—
170#2857—614no—
171#2659—614no—
172#2583—618no—
173#2502—614no—
174#2434—614no—
175#2315—614no—
176C2×C4×C22⋊C4D6111yescharacteristic
177C2×C61—28no—
178C2×C61—24no—
179C2×C61C22×C4⋊Q811yescharacteristic
180C2×C61C2×#53411yescharacteristic
181C2×C61#344911yescharacteristic
182C2×C61#350114yes—
183C23×Q8—612no—
184C24×C4C2×D6111yescharacteristic
185C2×C4⋊Q8—614no—
186C2×C22⋊Q8—614no—
187C2×C22⋊Q8—614no—
188C2×C22⋊Q8—614no—
189C2×C22⋊Q8—614no—
190C2×C22⋊Q8—12216no—
191C22×C4⋊C4—614no—
192C22×C4⋊C4—612no—
193C22×C22⋊C4—614no—
194C22×C22⋊C4C2×D6112yes—
195C22×C42C2×D6112yes—
196#398—6116no—
197#395—6116no—
198#391—618no—
199#389—6116no—
200#385—6116no—
201#385—6116no—
202#384—614no—
203#380—6116no—
204C4×C22⋊C4C2×D6114yes—
205C4×C22⋊C4C2×D6114yes—
206C2×#95C2×D6112yes—
207C61#3501111yescharacteristic, complemented, Sylow
208C25C22×D6111yescharacteristic
209C22×Q8—614no—
210C22×Q8—614no—
211C22×Q8—1228no—
212C22×Q8—2448no—
213C23×C4—44no—
214C23×C4—614no—
215C23×C4—614no—
216C23×C4D24418yes—
217C23×C4C22×D6112yes—
218C23×C4C22×D6111yescharacteristic
219C23×C4C22×D6114yes—
220C23×C4C22×D6112yes—
221C23×C4C22×D6111yescharacteristic
222C4⋊Q8—2448no—
223C22⋊Q8—12216no—
224C22⋊Q8—12216no—
225C22⋊Q8—12216no—
226C22⋊Q8—12216no—
227C2×C4⋊C4—614no—
228C2×C4⋊C4—618no—
229C2×C4⋊C4—618no—
230C2×C4⋊C4—614no—
231C2×C4⋊C4—6116no—
232C2×C4⋊C4—6116no—
233C2×C4⋊C4—12216no—
234C2×C4⋊C4—1228no—
235C2×C22⋊C4—618no—
236C2×C22⋊C4—618no—
237C2×C22⋊C4—12216no—
238C2×C22⋊C4Dic122116yescomplemented
239C2×C22⋊C4C22×D6118yes—
240C2×C42—44no—
241C2×C42C22×D6114yes—
242C2×C42C22×D6114yes—
243#95—6132no—
244#95C22×D6118yes—
245C24—42no—
246C24—42no—
247C24C2×Dic12218yes—
248C24C2×D24414yes—
249C24C23×D6111yescharacteristic
250C24C23×D6112yes—
251C2×Q8—1228no—
252C2×Q8—1228no—
253C2×Q8—12216no—
254C2×Q8—2448no—
255C2×Q8—24432no—
256C22×C4—24no—
257C22×C4—28no—
258C22×C4—416no—
259C22×C4—44no—
260C22×C4—44no—
261C22×C4—42no—
262C22×C4—44no—
263C22×C4—618no—
264C22×C4—618no—
265C22×C4—1228no—
266C22×C4—1228no—
267C22×C4—2448no—
268C22×C4—2448no—
269C22×C4C2×Dic122116yescomplemented
270C22×C4C2×D24418yes—
271C22×C4C2×D24414yes—
272C22×C4C2×D24418yes—
273C22×C4C2×D24414yes—
274C22×C4C23×D6112yes—
275C22×C4C23×D6111yescharacteristic
276C22×C4C23×D6118yes—
277C22×C4C23×D6111yescharacteristic
278C4⋊C4—12216no—
279C4⋊C4—12216no—
280C4⋊C4—1228no—
281C4⋊C4—1228no—
282C4⋊C4—24416no—
283C4⋊C4—24416no—
284C22⋊C4—12216no—
285C22⋊C4—12216no—
286C22⋊C4C2×Dic122132yescomplemented
287C42—28no—
288C42—44no—
289C23—22no—
290C23—24no—
291C23—22no—
292C23—24no—
293C23—44no—
294C23—416no—
295C23—44no—
296C23#14519777116yescomplemented
297C23C22×D24412yes—
298C23C22×D24414yes—
299C23C22×Dic12218yes—
300C23C22×Dic12214yes—
301C23C24×D6111yescharacteristic
302Q8—24416no—
303Q8—24416no—
304C2×C4—28no—
305C2×C4—22no—
306C2×C4—24no—
307C2×C4—28no—
308C2×C4—28no—
309C2×C4—216no—
310C2×C4—48no—
311C2×C4—44no—
312C2×C4—416no—
313C2×C4—1228no—
314C2×C4—1228no—
315C2×C4—12216no—
316C2×C4—12216no—
317C2×C4—24416no—
318C2×C4—24416no—
319C2×C4C22×D24412yes—
320C2×C4C22×D24414yes—
321C2×C4C22×D24412yes—
322C2×C4C22×Dic122116yes—
323C2×C4#1451982214yes—
324C22—28no—
325C22—216no—
326C22—24no—
327C22—22no—
328C22—24no—
329C22—22no—
330C22—416no—
331C22C2×#1451977714yes—
332C22C2×#1451977718yes—
333C22C23×D24411yescharacteristic
334C22C23×Dic12212yes—
335C22C2×#1451982212yes—
336C22#1684059812yes—
337C4—216no—
338C4—22no—
339C4—24no—
340C4—2448no—
341C4—2448no—
342C4#1684032414yes—
343C2—24no—
344C2—28no—
345C2C22×#1451977711yescharacteristic
346C2C2×#1684032411yescharacteristic
347C2#2145915211yescharacteristic
348C2#2145962214yes—
subgroupG/H
center, FrattiniC23C24×D61
commutator, socle, cosocleC23×C61C25
FittingC2×C4×C22⋊C4×C61C2
prime radicalC25×C61C23
hypercenterC2×C4×C22⋊C4D61
hypocenterC61#35011
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. #21454171C2semidirect
3. #21451933C2semidirect
4. #21451194C2semidirect
5. #21450297C2semidirect
6. #21449463C2semidirect
10. C2×#14519777C22semidirect
19. #16839694C22semidirect
20. #16839683C22semidirect
21. #16839668C22semidirect
22. #16839660C22semidirect
25. #16839632C22semidirect
64. C2×C22⋊C4×C61Q8semidirect
87. C22×C4×C61C2×Q8semidirect
103. C22⋊C4×C61C2×Q8semidirect
121. C23×C61C4⋊Q8semidirect
207. C61#35011semidirect
238. C2×C22⋊C4Dic122semidirect
269. C22×C4C2×Dic122semidirect
286. C22⋊C4C2×Dic122semidirect
296. C23#14519777semidirect

Maximal subgroups

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

subgroupindex
C22×#145197562
#214541712
#214519332
#214511942
#214502972
#214494632
#214484332
C2×C4×C22⋊C4×C612
#3501161