Finite Group Databasebeta

← C2×#21450297 #29849195

Subgroups — C2×#21450297 #29849195

Order 15616.

3094 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519756C211yesmaximal, characteristic
2C23×C4⋊Dic61C211yesmaximal, characteristic
3#21450297C2116yesmaximal, direct product factor
4C2×#16839694C214yesmaximal, complemented
5C2×#16839668C212yesmaximal, complemented
6C2×#16839637C214yesmaximal
7C2×#16839632C212yesmaximal, complemented
8C2×C4×C22⋊C4×C61C211yesmaximal, characteristic
9C23×Dic122C2211yescharacteristic
10C24×Dic61C2212yes—
11C2×#14519756—28no—
12C2×#14519756C2218yes—
13C22×C4⋊Dic61—28no—
14C22×C4⋊Dic61C2211yescharacteristic
15C22×C4⋊Dic61C2212yes—
16C22×C4⋊Dic61C2218yes—
17C22×C4⋊Dic61C2212yes—
18C22×C4⋊Dic61C2218yes—
19C22×#11758196C2212yes—
20C22×C22⋊Dic61C2212yes—
21#16839694C22132yescomplemented
22#16839668C22116yescomplemented
23#16839637C22132yes—
24#16839632C22116yescomplemented
25C2×#14519684C2218yes—
26C24×C4×C61C2211yescharacteristic
27C22×C22⋊C4×C61C2212yes—
28C22×C42×C61C2212yes—
29C4×C22⋊C4×C61C2218yes—
30C2×#95×C61C2212yes—
31C22×Dic122—24no—
32C22×Dic122—28no—
33C22×Dic122C2314yes—
34C23×Dic61—28no—
35C23×Dic61—48no—
36C23×Dic61C2312yes—
37C23×Dic61C2312yes—
38C23×Dic61C2314yes—
39C23×Dic61C2318yes—
40#14519756—232no—
41C2×C4⋊Dic61—28no—
42C2×C4⋊Dic61—232no—
43C2×C4⋊Dic61—28no—
44C2×C4⋊Dic61—24no—
45C2×C4⋊Dic61—416no—
46C2×C4⋊Dic61C2318yes—
47C2×C4⋊Dic61C2318yes—
48C2×C4⋊Dic61C2314yes—
49C2×C4⋊Dic61C23132yes—
50C2×#11758196—28no—
51C2×#11758196C2318yes—
52C2×C22⋊Dic61—28no—
53C2×C22⋊Dic61C2318yes—
54#14519684C23132yes—
55C25×C61C2311yescharacteristic
56C23×C4×C61—28no—
57C23×C4×C61D418yes—
58C23×C4×C61C2311yescharacteristic
59C23×C4×C61C2314yes—
60C23×C4×C61C2314yes—
61C23×C4×C61C2311yescharacteristic
62C2×C22⋊C4×C61Q8116yescomplemented
63C2×C22⋊C4×C61C2318yes—
64C2×C42×C61—28no—
65C2×C42×C61C2318yes—
66#95×C61C2318yes—
67C24×C61—24no—
68C24×C61—24no—
69C24×C61C2×D414yes—
70C24×C61C2×Q818yes—
71C24×C61C2412yes—
72C24×C61C2411yescharacteristic
73C22×C4×C61—28no—
74C22×C4×C61—24no—
75C22×C4×C61—28no—
76C22×C4×C61—232no—
77C22×C4×C61—216no—
78C22×C4×C61C2×D414yes—
79C22×C4×C61C2×D4116yes—
80C22×C4×C61C2×D414yes—
81C22×C4×C61C2×Q8116yes—
82C22×C4×C61#4118yes—
83C22×C4×C61C2412yes—
84C22×C4×C61C2418yes—
85C22×C4×C61C2412yes—
86C22×Dic61—216no—
87C22×Dic61—28no—
88C22×Dic61—24no—
89C22×Dic61—24no—
90C22×Dic61—28no—
91C22×Dic61—216no—
92C22×Dic61—28no—
93C22×Dic61—416no—
94C22×Dic61—432no—
95C22×Dic61C2418yes—
96C22×Dic61C2414yes—
97C22×Dic61C2414yes—
98C2×Dic122—28no—
99C2×Dic122—216no—
100C2×Dic122—416no—
101C22⋊C4×C61C2×Q8132yescomplemented
102C42×C61—216no—
103C22⋊Dic61—216no—
104C4⋊Dic61—216no—
105C4⋊Dic61—216no—
106C4⋊Dic61—28no—
107C4⋊Dic61—432no—
108#11758196—216no—
109C23×C61—28no—
110C23×C61—28no—
111C23×C61—232no—
112C23×C61—24no—
113C23×C61—24no—
114C23×C61C4⋊Q8116yes—
115C23×C61C22×D414yes—
116C23×C61C22×D412yes—
117C23×C61C22×Q818yes—
118C23×C61C22×Q814yes—
119C23×C61C2×#4114yes—
120C23×C61H(5,𝔽2)14yes—
121C23×C61C2511yescharacteristic
122C2×C4×C61—28no—
123C2×C4×C61—232no—
124C2×C4×C61—28no—
125C2×C4×C61—24no—
126C2×C4×C61—28no—
127C2×C4×C61—232no—
128C2×C4×C61#123116yes—
129C2×C4×C61C22×D414yes—
130C2×C4×C61C22×D414yes—
131C2×C4×C61C22×Q8116yes—
132C2×C4×C61C2×#4118yes—
133C2×Dic61—24no—
134C2×Dic61—24no—
135C2×Dic61—28no—
136C2×Dic61—28no—
137C2×Dic61—28no—
138C2×Dic61—216no—
139C2×Dic61—432no—
140C2×Dic61—48no—
141C2×Dic61—48no—
142C2×Dic61—416no—
143Dic122—416no—
144C2×#2502—611nomaximal, Sylow
145C22×C61—24no—
146C22×C61—24no—
147C22×C61—216no—
148C22×C61—232no—
149C22×C61C2×#12314yes—
150C22×C61C2×C4⋊Q818yes—
151C22×C61C2×C4⋊Q814yes—
152C22×C61#53514yes—
153C22×C61#554116yes—
154C22×C61C23×D411yescharacteristic
155C22×C61C23×Q812yes—
156C22×C61C22×#4112yes—
157C22×C61C2×H(5,𝔽2)12yes—
158C4×C61—24no—
159C4×C61—216no—
160C4×C61C2×#12318yes—
161Dic61—44no—
162Dic61—44no—
163Dic61—48no—
164C22×C22⋊Q8—611no—
165C23×C4⋊C4—611no—
166#2502—6116no—
167C2×#398—614no—
168C2×#391—612no—
169C2×#385—614no—
170C2×#384—612no—
171C2×C4×C22⋊C4D6111yescharacteristic
172C2×C61—28no—
173C2×C61#250218yescomplemented
174C2×C61C22×#12311yescharacteristic
175C2×C61C22×C4⋊Q811yescharacteristic
176C2×C61C2×#53511yescharacteristic
177C2×C61C2×#55414yes—
178C23×Q8—611no—
179C24×C4—612no—
180C24×C4C2×D6111yescharacteristic
181C2×C22⋊Q8—618no—
182C2×C22⋊Q8—1228no—
183C22×C4⋊C4—612no—
184C22×C4⋊C4—612no—
185C22×C4⋊C4—618no—
186C22×C4⋊C4—612no—
187C22×C4⋊C4—611no—
188C22×C4⋊C4—618no—
189C22×C4⋊C4—1228no—
190C22×C22⋊C4—612no—
191C22×C22⋊C4C2×D6112yes—
192C22×C42C2×D6112yes—
193#398—6132no—
194#391—6116no—
195#385—6132no—
196#384—6116no—
197C4×C22⋊C4C2×D6118yes—
198C2×#95—618no—
199C2×#95C2×D6112yes—
200C61C2×#250211yescharacteristic, complemented, Sylow
201C25C22×D6111yescharacteristic
202C22×Q8—614no—
203C22×Q8—1224no—
204C22×Q8—1228no—
205C23×C4—28no—
206C23×C4—612no—
207C23×C4—612no—
208C23×C4—614no—
209C23×C4—618no—
210C23×C4—1228no—
211C23×C4—2448no—
212C23×C4D24418yes—
213C23×C4C22×D6114yes—
214C23×C4C22×D6114yes—
215C23×C4C22×D6111yescharacteristic
216C23×C4C22×D6111yescharacteristic
217C22⋊Q8—12232no—
218C2×C4⋊C4—618no—
219C2×C4⋊C4—618no—
220C2×C4⋊C4—614no—
221C2×C4⋊C4—618no—
222C2×C4⋊C4—6132no—
223C2×C4⋊C4—1228no—
224C2×C4⋊C4—12232no—
225C2×C4⋊C4—1228no—
226C2×C4⋊C4—1228no—
227C2×C4⋊C4—1224no—
228C2×C4⋊C4—24416no—
229C2×C22⋊C4—618no—
230C2×C22⋊C4—1228no—
231C2×C22⋊C4Dic122116yescomplemented
232C2×C22⋊C4C22×D6118yes—
233C2×C42—28no—
234C2×C42C22×D6118yes—
235#95—6132no—
236#95C22×D6118yes—
237C24—24no—
238C24—24no—
239C24C2×Dic12218yes—
240C24C2×D24414yes—
241C24C23×D6112yes—
242C24C23×D6111yescharacteristic
243C2×Q8—1228no—
244C2×Q8—12216no—
245C2×Q8—24416no—
246C22×C4—28no—
247C22×C4—24no—
248C22×C4—28no—
249C22×C4—216no—
250C22×C4—232no—
251C22×C4—614no—
252C22×C4—618no—
253C22×C4—614no—
254C22×C4—1224no—
255C22×C4—1224no—
256C22×C4—12216no—
257C22×C4—1228no—
258C22×C4—1228no—
259C22×C4—1228no—
260C22×C4—12216no—
261C22×C4—24416no—
262C22×C4—24432no—
263C22×C4C2×Dic122116yes—
264C22×C4C2×D24414yes—
265C22×C4C2×D244116yes—
266C22×C4C2×D24414yes—
267C22×C4#1175822418yes—
268C22×C4C23×D6112yes—
269C22×C4C23×D6118yes—
270C22×C4C23×D6112yes—
271C4⋊C4—12216no—
272C4⋊C4—12216no—
273C4⋊C4—12216no—
274C4⋊C4—1228no—
275C4⋊C4—24432no—
276C22⋊C4—12216no—
277C22⋊C4C2×Dic122132yescomplemented
278C42—216no—
279C23—232no—
280C23—28no—
281C23—28no—
282C23—24no—
283C23—24no—
284C23#14519777116yes—
285C23C22×D24414yes—
286C23C22×D24412yes—
287C23C22×Dic12218yes—
288C23C22×Dic12214yes—
289C23C2×#1175822414yes—
290C23#1451982314yes—
291C23C24×D6111yescharacteristic
292Q8—24416no—
293C2×C4—232no—
294C2×C4—28no—
295C2×C4—28no—
296C2×C4—24no—
297C2×C4—28no—
298C2×C4—232no—
299C2×C4—1224no—
300C2×C4—1224no—
301C2×C4—1228no—
302C2×C4—1228no—
303C2×C4—1228no—
304C2×C4—12216no—
305C2×C4—24432no—
306C2×C4—2448no—
307C2×C4—2448no—
308C2×C4—24416no—
309C2×C4#14519761116yes—
310C2×C4C22×D24414yes—
311C2×C4C22×D24414yes—
312C2×C4C22×Dic122116yes—
313C2×C4C2×#1175822418yes—
314C22—232no—
315C22—24no—
316C22—216no—
317C22—24no—
318C22C2×#1451976114yes—
319C22C2×#1451977718yes—
320C22C2×#1451977714yes—
321C22#1684033114yes—
322C22#16840469116yes—
323C22C23×D24411yescharacteristic
324C22C23×Dic12212yes—
325C22C22×#1175822412yes—
326C22C2×#1451982312yes—
327C4—24no—
328C4—216no—
329C4—2444no—
330C4—2444no—
331C4—2448no—
332C4C2×#1451976118yes—
333C2—28no—
334C2#2145029718yesdirect product factor
335C2C22×#1451976111yescharacteristic
336C2C22×#1451977711yescharacteristic
337C2C2×#1684033111yescharacteristic
338C2C2×#1684046914yes—
subgroupG/H
centerC24C23×D61
commutator, cosocleC23×C61C25
FrattiniC23C24×D61
FittingC2×C4×C22⋊C4×C61C2
socleC24×C61C24
prime radicalC25×C61C23
hypercenterC2×C4×C22⋊C4D61
hypocenterC61C2×#2502
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
3. #21450297C2direct (N × H)
4. C2×#16839694C2semidirect
5. C2×#16839668C2semidirect
7. C2×#16839632C2semidirect
21. #16839694C22semidirect
22. #16839668C22semidirect
24. #16839632C22semidirect
62. C2×C22⋊C4×C61Q8semidirect
101. C22⋊C4×C61C2×Q8semidirect
173. C2×C61#2502semidirect
200. C61C2×#2502semidirect
231. C2×C22⋊C4Dic122semidirect
277. C22⋊C4C2×Dic122semidirect
334. C2#21450297direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145197562
C23×C4⋊Dic612
#214502972
C2×#168396942
C2×#168396682
C2×#168396372
C2×#168396322
C2×C4×C22⋊C4×C612
C2×#250261