Finite Group Databasebeta

← C2×#20244716 #26244299

Subgroups — C2×#20244716 #26244299

Order 12032.

1454 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13094024C211yesmaximal, characteristic
2C22×#13093957C212yesmaximal
3#20244716C218yesmaximal, direct product factor
4C2×#15688885C212yesmaximal
5C2×#15688878C211yesmaximal, characteristic
6C2×C47×#389C211yesmaximal, characteristic
7C23×Dic94—21no—
8C23×C47⋊C8C2212yes—
9C2×#13094024—28no—
10C2×#13094024—22no—
11C2×#13094024C2214yes—
12C22×C4⋊Dic47—21no—
13C22×C4⋊Dic47C2212yes—
14C2×#13093957—28no—
15C2×#13093957C2218yes—
16#15688885C2218yes—
17#15688878C2214yes—
18C22×C4⋊C4×C47—21no—
19C22×C4⋊C4×C47C2211yescharacteristic
20C22×C4⋊C4×C47C2211yescharacteristic
21C22×C42×C47C2211yescharacteristic
22C47×#389—22no—
23C47×#389C2214yes—
24C2×#95×C47—21no—
25C22×Dic94—28no—
26C22×Dic94—24no—
27C22×Dic94—44no—
28C22×Dic94—44no—
29C23×Dic47—22no—
30C22×C47⋊C8—24no—
31C22×C47⋊C8—24no—
32C22×C47⋊C8Q818yescomplemented
33C22×C47⋊C8C2314yes—
34#13094024—216no—
35#13094024—216no—
36C2×C4⋊Dic47—28no—
37C2×C4⋊Dic47—24no—
38C2×C4⋊Dic47—22no—
39C2×C4⋊Dic47—24no—
40C2×C4⋊Dic47D418yes—
41C2×C4⋊Dic47C2314yes—
42#13093957—216no—
43C23×C4×C47—21no—
44C23×C4×C47—21no—
45C23×C4×C47D411yescharacteristic
46C23×C4×C47D411yescharacteristic
47C23×C4×C47C2311yescharacteristic
48C2×C4⋊C4×C47—21no—
49C2×C4⋊C4×C47—22no—
50C2×C4⋊C4×C47—24no—
51C2×C4⋊C4×C47—24no—
52C2×C4⋊C4×C47—21no—
53C2×C4⋊C4×C47—44no—
54C2×C4⋊C4×C47Q814yescomplemented
55C2×C4⋊C4×C47C2312yes—
56C2×C4⋊C4×C47C2312yes—
57C2×C42×C47—24no—
58C2×C42×C47D412yes—
59C2×C42×C47C2312yes—
60#95×C47—24no—
61C24×C47C2×D411yescharacteristic
62C22×C4×C47—22no—
63C22×C4×C47—22no—
64C22×C4×C47—22no—
65C22×C4×C47—22no—
66C22×C4×C47—24no—
67C22×C4×C47—22no—
68C22×C4×C47—22no—
69C22×C4×C47—42no—
70C22×C4×C47—42no—
71C22×C4×C47—42no—
72C22×C4×C47—42no—
73C22×C4×C47—42no—
74C22×C4×C47D814yes—
75C22×C4×C47Q1614yes—
76C22×C4×C47C2×D412yes—
77C22×C4×C47C2×D412yes—
78C22×C4×C47C2×D411yescharacteristic
79C22×C4×C47C2×D414yes—
80C22×C4×C47C2×Q814yes—
81C22×C4×C47C2×Q812yes—
82C22×C4×C47#4112yes—
83C22×C4×C47C2411yescharacteristic
84C22×Dic47—28no—
85C22×Dic47—24no—
86C22×Dic47—44no—
87C22×Dic47—44no—
88C2×Dic94—216no—
89C2×Dic94—48no—
90C2×Dic94—432no—
91C2×Dic94—48no—
92C4⋊C4×C47—24no—
93C4⋊C4×C47—22no—
94C4⋊C4×C47—24no—
95C4⋊C4×C47—42no—
96C4⋊C4×C47—48no—
97C4⋊C4×C47C2×Q814yescomplemented
98C42×C47—24no—
99C42×C47—24no—
100C4⋊Dic47—216no—
101C4⋊Dic47—28no—
102C4⋊Dic47—24no—
103C4⋊Dic47C2×D418yes—
104C2×C47⋊C8—216no—
105C2×C47⋊C8—24no—
106C2×C47⋊C8—24no—
107C2×C47⋊C8C2×Q818yescomplemented
108C23×C47#12012yes—
109C23×C47C22⋊Q812yes—
110C23×C47C4⋊Q812yes—
111C23×C47C2×D812yes—
112C23×C47C2×Q1612yes—
113C23×C47AGL(1,ℤ/8)12yes—
114C23×C47#13712yes—
115C23×C47C22×D411yescharacteristic
116C2×C4×C47—22no—
117C2×C4×C47—22no—
118C2×C4×C47—22no—
119C2×C4×C47—22no—
120C2×C4×C47—28no—
121C2×C4×C47—28no—
122C2×C4×C47—24no—
123C2×C4×C47—42no—
124C2×C4×C47—42no—
125C2×C4×C47—48no—
126C2×C4×C47—42no—
127C2×C4×C47—42no—
128C2×C4×C47—42no—
129C2×C4×C47—84no—
130C2×C4×C47#12012yes—
131C2×C4×C47C22⋊Q818yes—
132C2×C4×C47C22⋊Q812yes—
133C2×C4×C47C4⋊Q812yes—
134C2×C4×C47C4⋊Q818yes—
135C2×C4×C47C2×D814yes—
136C2×C4×C47C2×Q1614yes—
137C2×C4×C47C22×D412yes—
138C2×C4×C47C22×Q812yes—
139C2×C4×C47C22×Q811yescharacteristic
140C2×C4×C47C2×#4111yescharacteristic
141C2×Dic47—28no—
142C2×Dic47—44no—
143C2×Dic47—44no—
144C2×Dic47—416no—
145Dic94—432no—
146C47⋊C8—28no—
147C2×#2057—471nomaximal, Sylow
148C22×C47#39514yes—
149C22×C47#44714yes—
150C22×C47#45114yes—
151C22×C47#47414yes—
152C22×C47#47714yes—
153C22×C47#50014yes—
154C22×C47#50114yes—
155C22×C47C2×#12011yescharacteristic
156C22×C47C2×C22⋊Q811yescharacteristic
157C22×C47C2×C4⋊Q811yescharacteristic
158C22×C47C22×D811yescharacteristic
159C22×C47C22×Q1611yescharacteristic
160C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
161C22×C47C2×#13711yescharacteristic
162C4×C47—24no—
163C4×C47—24no—
164C4×C47—44no—
165C4×C47—82no—
166C4×C47#39514yes—
167C4×C47C2×C22⋊Q812yes—
168C4×C47C2×C4⋊Q812yes—
169Dic47—48no—
170C22×C4⋊Q8—471no—
171C22×#107—472no—
172C2×#389D4711yescharacteristic
173#2057—478no—
174C2×#340—472no—
175C2×#336—471no—
176C2×C47#205718yescomplemented
177C2×C47C2×#39511yescharacteristic
178C2×C47C2×#44711yescharacteristic
179C2×C47C2×#45111yescharacteristic
180C2×C47C2×#47411yescharacteristic
181C2×C47C2×#47711yescharacteristic
182C2×C47C2×#50011yescharacteristic
183C2×C47C2×#50111yescharacteristic
184C23×Q8—941no—
185C23×C8—472no—
186C2×C4⋊Q8—474no—
187C2×C4⋊Q8—948no—
188C2×C4⋊Q8—942no—
189C22×C4⋊C4—21no—
190C22×C4⋊C4—472no—
191C22×C4⋊C4—941no—
192C22×C4⋊C4C2×D4711yescharacteristic
193C22×C4⋊C4C2×D4711yescharacteristic
194C22×C42C2×D4711yescharacteristic
195C2×#107—478no—
196C2×#107—948no—
197#389—22no—
198#389C2×D4714yes—
199C2×#95—21no—
200#340—478no—
201#336—474no—
202C47C2×#205711yescharacteristic, complemented, Sylow
203C22×Q8—948no—
204C22×Q8—944no—
205C22×Q8—1884no—
206C22×Q8—1884no—
207C23×C4—21no—
208C23×C4—21no—
209C23×C4—942no—
210C23×C4C22⋊D4711yescharacteristic
211C23×C4C22⋊D4711yescharacteristic
212C23×C4C22×D4711yescharacteristic
213C22×C8—478no—
214C22×C8—474no—
215C22×C8—944no—
216C22×C8—944no—
217C4⋊Q8—9416no—
218C4⋊Q8—9416no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—478no—
226C2×C4⋊C4—474no—
227C2×C4⋊C4—948no—
228C2×C4⋊C4—944no—
229C2×C4⋊C4—942no—
230C2×C4⋊C4—944no—
231C2×C4⋊C4Dic9414yescomplemented
232C2×C4⋊C4C22×D4712yes—
233C2×C4⋊C4C22×D4712yes—
234C2×C42—24no—
235C2×C42C22⋊D4712yes—
236C2×C42C22×D4712yes—
237#107—9416no—
238#95—24no—
239C24C2×C22⋊D4711yescharacteristic
240C2×Q8—9416no—
241C2×Q8—18832no—
242C2×Q8—1888no—
243C2×Q8—1888no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—24no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—948no—
257C22×C4—944no—
258C22×C4—1884no—
259C22×C4—1884no—
260C22×C4#1062533414yes—
261C22×C4#1062533714yes—
262C22×C4C2×Dic9412yes—
263C22×C4#1062535012yes—
264C22×C4D4×D4714yes—
265C22×C4Q8×D4714yes—
266C22×C4C2×C22⋊D4712yes—
267C22×C4C2×C22⋊D4712yes—
268C22×C4C2×C22⋊D4711yescharacteristic
269C22×C4C23×D4711yescharacteristic
270C2×C8—478no—
271C2×C8—9416no—
272C2×C8—944no—
273C2×C8—944no—
274C4⋊C4—24no—
275C4⋊C4—22no—
276C4⋊C4—24no—
277C4⋊C4—42no—
278C4⋊C4—48no—
279C4⋊C4—478no—
280C4⋊C4—9416no—
281C4⋊C4—948no—
282C4⋊C4—944no—
283C4⋊C4C2×Dic9414yescomplemented
284C42—24no—
285C42—24no—
286C23#1309399212yes—
287C23#1309400312yes—
288C23#1309402212yes—
289C23C2×#1062533412yes—
290C23C2×#1062533712yes—
291C23#1309404912yes—
292C23#1309405412yes—
293C23C22×C22⋊D4711yescharacteristic
294Q8—18832no—
295C2×C4—28no—
296C2×C4—28no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—24no—
302C2×C4—42no—
303C2×C4—48no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—84no—
309C2×C4—948no—
310C2×C4—18816no—
311C2×C4—1884no—
312C2×C4—1884no—
313C2×C4#1309399212yes—
314C2×C4#1309400218yes—
315C2×C4#1309400312yes—
316C2×C4#1309402212yes—
317C2×C4#1309402318yes—
318C2×C4C2×#1062533414yes—
319C2×C4C2×#1062533714yes—
320C2×C4C2×D4×D4712yes—
321C2×C4C2×Q8×D4712yes—
322C2×C4C22×Dic9411yescharacteristic
323C2×C4C2×#1062535011yescharacteristic
324C8—948no—
325C22#1568904314yes—
326C22#1568925614yes—
327C22#1568927814yes—
328C22#1568940314yes—
329C22#1568941814yes—
330C22#1568952614yes—
331C22#1568953314yes—
332C22C2×#1309399211yescharacteristic
333C22C2×#1309400311yescharacteristic
334C22C2×#1309402211yescharacteristic
335C22C22×#1062533411yescharacteristic
336C22C22×#1062533711yescharacteristic
337C22C2×#1309404911yescharacteristic
338C22C2×#1309405411yescharacteristic
339C4—24no—
340C4—24no—
341C4—44no—
342C4—82no—
343C4—1888no—
344C4#1568904314yes—
345C4C2×#1309400212yes—
346C4C2×#1309402312yes—
347C2#2024471618yesdirect product factor
348C2C2×#1568904311yescharacteristic
349C2C2×#1568925611yescharacteristic
350C2C2×#1568927811yescharacteristic
351C2C2×#1568940311yescharacteristic
352C2C2×#1568941811yescharacteristic
353C2C2×#1568952611yescharacteristic
354C2C2×#1568953311yescharacteristic
subgroupG/H
centerC24C2×C22⋊D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC2×C47×#389C2
socle, prime radicalC24×C47C2×D4
hypercenterC2×#389D47
hypocenterC47C2×#2057
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C24 ◁ C22×C42 ◁ C2×#389

Complements

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

NH ≅ G/Ntype
3. #20244716C2direct (N × H)
32. C22×C47⋊C8Q8semidirect
54. C2×C4⋊C4×C47Q8semidirect
97. C4⋊C4×C47C2×Q8semidirect
107. C2×C47⋊C8C2×Q8semidirect
176. C2×C47#2057semidirect
202. C47C2×#2057semidirect
231. C2×C4⋊C4Dic94semidirect
283. C4⋊C4C2×Dic94semidirect
347. C2#20244716direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#130940242
C22×#130939572
#202447162
C2×#156888852
C2×#156888782
C2×C47×#3892
C2×#205747