Finite Group Databasebeta

← C2×#20242692 #26197396

Subgroups — C2×#20242692 #26197396

Order 12032.

1150 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 1150 conjugacy classes of subgroups — too many to draw. Only its 171 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
1C2×#15689865C211yesmaximal, characteristic
2C2×#15689485C212yesmaximal, complemented
3#20242692C218yesmaximal, direct product factor
4C2×#15688863C211yesmaximal, characteristic, complemented
5C2×#15688862C212yesmaximal, complemented
6C2×C47×#497C211yesmaximal, characteristic, complemented
7#15689865—24no—
8#15689865—22no—
9#15689865C2214yes—
10C2×#13094013—21no—
11C2×#13094012C2212yes—
12C2×#13094010—22no—
13C2×#13093994—22no—
14C2×#13093992—21no—
15#15689485—24no—
16#15689485C2218yescomplemented
17C2×#13093947—22no—
18C2×#13093944—22no—
19C2×#13093935C2212yescomplemented
20#15688863C2214yescomplemented
21#15688862C2218yescomplemented
22C22×Q16×C47—21no—
23C22×SD16×C47—21no—
24C2×C4⋊Q8×C47C2211yescharacteristic, complemented
25C2×#124×C47C2211yescharacteristic
26C47×#497—22no—
27C47×#497C2214yescomplemented
28C2×#97×C47C2211yescharacteristic, complemented
29C24×D47—21no—
30C22×Dic94D412yes—
31C22×D188—21no—
32C22×D188D412yes—
33C22×C22⋊D47—22no—
34C22×C47⋊C8—22no—
35#13094013—24no—
36#13094013—24no—
37#13094012—22no—
38#13094012—22no—
39#13094012D414yes—
40#13094012C2314yes—
41#13094010—28no—
42#13094010—28no—
43#13093994—28no—
44#13093994—28no—
45#13093992—24no—
46#13093992—24no—
47C2×C22⋊Dic47—21no—
48C2×(C2×D47)⋊C4—22no—
49C2×(C2×D47)⋊C4—22no—
50C2×(C2×D47)⋊C4—22no—
51C2×C4×Dic47—21no—
52#13093947—28no—
53#13093944—28no—
54#13093935—22no—
55#13093935C2314yescomplemented
56C22×Q8×C47—21no—
57C22×Q8×C47D411yescharacteristic
58C22×D4×C47D411yescharacteristic
59C2×Q16×C47—24no—
60C2×Q16×C47—44no—
61C2×SD16×C47—24no—
62C2×SD16×C47—44no—
63C22×C8×C47—21no—
64C4⋊Q8×C47—22no—
65C4⋊Q8×C47—21no—
66C4⋊Q8×C47C2312yescomplemented
67#124×C47—22no—
68#124×C47D412yes—
69#124×C47C2312yes—
70C2×C4⋊C4×C47—22no—
71C2×C22⋊C4×C47—22no—
72C2×C42×C47C2311yescharacteristic
73#97×C47—21no—
74#97×C47C2312yescomplemented
75C24×C47—21no—
76C23×D47—22no—
77C23×D47—24no—
78C23×D47—82no—
79C23×D47—84no—
80C2×Q8×C47—21no—
81C2×Q8×C47—22no—
82C2×Q8×C47—42no—
83C2×Q8×C47—42no—
84C2×Q8×C47—42no—
85C2×Q8×C47—41no—
86C2×Q8×C47C2×D412yes—
87C2×D4×C47—21no—
88C2×D4×C47—42no—
89C2×D4×C47C2×D412yes—
90C22×C4×C47—21no—
91C22×C4×C47—41no—
92C22×C4×C47C2×D412yes—
93C22×C4×C47C2×D411yescharacteristic
94C2×C22⋊D47—28no—
95C2×C22⋊D47—84no—
96C22×Dic47—22no—
97C2×D188—24no—
98C2×D188—22no—
99C2×D188—44no—
100C2×D188—82no—
101C2×D188C2×D414yes—
102C2×Dic94—22no—
103C2×Dic94—44no—
104C2×Dic94C2×D414yes—
105Q16×C47—48no—
106SD16×C47—48no—
107C2×C8×C47—24no—
108C2×C8×C47—22no—
109C4⋊C4×C47—24no—
110C4⋊C4×C47—42no—
111C22⋊C4×C47—24no—
112C22⋊C4×C47—24no—
113C42×C47C2×D412yes—
114C42×C47C2×D411yescharacteristic
115C42×C47C2411yescharacteristic
116C22⋊Dic47—24no—
117(C2×D47)⋊C4—28no—
118(C2×D47)⋊C4—24no—
119(C2×D47)⋊C4—24no—
120(C2×D47)⋊C4—24no—
121(C2×D47)⋊C4—24no—
122C4×Dic47—24no—
123C2×C47⋊C8—24no—
124C2×C47⋊C8—44no—
125C23×C47—22no—
126C23×C47—42no—
127C23×C47—44no—
128C23×C47#12011yescharacteristic
129C22×D47—24no—
130C22×D47—44no—
131C22×D47—44no—
132C22×D47—44no—
133C22×D47—84no—
134C22×D47—88no—
135C22×D47—816no—
136Q8×C47—42no—
137Q8×C47—42no—
138Q8×C47—42no—
139Q8×C47—81no—
140Q8×C47—82no—
141D4×C47—42no—
142D4×C47—81no—
143C2×C4×C47—22no—
144C2×C4×C47—24no—
145C2×C4×C47—22no—
146C2×C4×C47—42no—
147C2×C4×C47—42no—
148C2×C4×C47—82no—
149C2×C4×C47#12011yescharacteristic
150C2×C4×C47#12012yes—
151C2×C4×C47C22×D412yes—
152C2×C4×C47C22×D411yescharacteristic
153C22⋊D47—88no—
154C2×Dic47—24no—
155C2×Dic47—44no—
156D188—44no—
157D188—82no—
158D188—84no—
159Dic94—44no—
160Dic94—82no—
161C8×C47—24no—
162C47⋊C8—44no—
163C2×#1659—471nomaximal, Sylow
164C22×C47—42no—
165C22×C47—44no—
166C22×C47—84no—
167C22×C47#45312yes—
168C22×C47#45514yes—
169C22×C47C2×#12011yescharacteristic
170C2×D47—44no—
171C2×D47—44no—
172C2×D47—44no—
173C2×D47—816no—
174C2×D47—88no—
175C4×C47—21no—
176C4×C47—21no—
177C4×C47—22no—
178C4×C47—22no—
179C4×C47—42no—
180C4×C47—82no—
181Dic47—44no—
182C2×H(3,𝔽4)—471no—
183C2×#497D4711yescharacteristic, complemented
184C2×#489—472no—
185#1659—478no—
186C2×#329—472no—
187C2×#329—471no—
188C2×C47—82no—
189C2×C47#165914yescomplemented
190C2×C47C2×#45311yescharacteristic
191C2×C47C2×#45512yes—
192D47—84no—
193C22×Q16—21no—
194C22×SD16—21no—
195H(3,𝔽4)—474no—
196H(3,𝔽4)—944no—
197H(3,𝔽4)—942no—
198C2×C4⋊Q8C2×D4711yescharacteristic, complemented
199C2×#124—472no—
200C2×#124—942no—
201C2×#124—941no—
202C2×#124C2×D4711yescharacteristic
203C2×#120—942no—
204C2×#120—941no—
205#497—22no—
206#497C2×D4714yescomplemented
207#489—478no—
208#489—944no—
209C2×#103—942no—
210C2×#102—942no—
211C2×#97—472no—
212C2×#97C2×D4711yescharacteristic, complemented
213#329—474no—
214#329—478no—
215C47C2×#165911yescharacteristic, complemented, Sylow
216C25—941no—
217C22×Q8—21no—
218C22×Q8—472no—
219C22×Q8C22⋊D4711yescharacteristic
220C22×D4—472no—
221C22×D4—941no—
222C22×D4—942no—
223C22×D4C22⋊D4711yescharacteristic
224C2×Q16—24no—
225C2×Q16—44no—
226C2×SD16—24no—
227C2×SD16—44no—
228C22×C8—21no—
229C22×C8—942no—
230C4⋊Q8—22no—
231C4⋊Q8—21no—
232C4⋊Q8C22×D4712yescomplemented
233#124—22no—
234#124—474no—
235#124—474no—
236#124—948no—
237#124—948no—
238#124—944no—
239#124—944no—
240#124—942no—
241#124—942no—
242#124C22⋊D4712yes—
243#124C22×D4712yes—
244#120—948no—
245#120—948no—
246#120—944no—
247#120—944no—
248C2×C4⋊C4—22no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—942no—
251C2×C22⋊C4—941no—
252C2×C22⋊C4—942no—
253C2×C22⋊C4—942no—
254C2×C42—941no—
255C2×C42C22×D4711yescharacteristic
256#103—948no—
257#102—948no—
258#97—21no—
259#97—474no—
260#97—942no—
261#97C22×D4712yescomplemented
262C24—21no—
263C24—944no—
264C24—942no—
265C24—3762no—
266C24—3764no—
267C2×Q8—21no—
268C2×Q8—22no—
269C2×Q8—42no—
270C2×Q8—42no—
271C2×Q8—42no—
272C2×Q8—41no—
273C2×Q8—474no—
274C2×Q8—942no—
275C2×Q8—1884no—
276C2×Q8C2×C22⋊D4712yes—
277C2×D4—21no—
278C2×D4—42no—
279C2×D4—474no—
280C2×D4—944no—
281C2×D4—948no—
282C2×D4—942no—
283C2×D4—3762no—
284C2×D4—3764no—
285C2×D4—1884no—
286C2×D4C2×C22⋊D4712yes—
287C22×C4—21no—
288C22×C4—41no—
289C22×C4—942no—
290C22×C4D4×D4712yes—
291C22×C4C2×C22⋊D4711yescharacteristic
292Q16—48no—
293SD16—48no—
294C2×C8—24no—
295C2×C8—22no—
296C2×C8—944no—
297C2×C8—1884no—
298C4⋊C4—24no—
299C4⋊C4—42no—
300C22⋊C4—24no—
301C22⋊C4—24no—
302C22⋊C4—948no—
303C22⋊C4—944no—
304C22⋊C4—944no—
305C22⋊C4—944no—
306C22⋊C4—944no—
307C22⋊C4—944no—
308C42—944no—
309C42D4×D4712yes—
310C42C2×C22⋊D4711yescharacteristic
311C42C23×D4711yescharacteristic
312C23—22no—
313C23—42no—
314C23—44no—
315C23—944no—
316C23—3764no—
317C23—3768no—
318C23—37616no—
319C23—1884no—
320C23—1884no—
321C23—1884no—
322C23#1309399211yescharacteristic
323Q8—42no—
324Q8—42no—
325Q8—42no—
326Q8—81no—
327Q8—82no—
328Q8—3762no—
329Q8—1884no—
330D4—42no—
331D4—81no—
332D4—3762no—
333D4—3764no—
334D4—3768no—
335D4—1884no—
336C2×C4—22no—
337C2×C4—24no—
338C2×C4—22no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—82no—
342C2×C4—944no—
343C2×C4—1884no—
344C2×C4#1309399211yescharacteristic
345C2×C4#1309399212yes—
346C2×C4C2×D4×D4712yes—
347C2×C4C22×C22⋊D4711yescharacteristic
348C8—24no—
349C8—1884no—
350C22—42no—
351C22—44no—
352C22—84no—
353C22—37616no—
354C22—3768no—
355C22—1884no—
356C22—1884no—
357C22—1884no—
358C22#1568929012yes—
359C22#1568929914yes—
360C22C2×#1309399211yescharacteristic
361C4—21no—
362C4—21no—
363C4—22no—
364C4—22no—
365C4—42no—
366C4—82no—
367C4—1884no—
368C2—82no—
369C2—3764no—
370C2#2024269214yesdirect product factor
371C2C2×#1568929011yescharacteristic
372C2C2×#1568929912yes—
subgroupG/H
centerC23#13093992
commutator, cosocleC42×C47C24
FrattiniC42C23×D47
FittingC2×C47×#497C2
socleC23×C47#120
prime radicalC2×#15689865C2
hypercenterC2×#497D47
hypocenterC47C2×#1659
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#497

Complements

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

NH ≅ G/Ntype
2. C2×#15689485C2semidirect
3. #20242692C2direct (N × H)
4. C2×#15688863C2semidirect
5. C2×#15688862C2semidirect
6. C2×C47×#497C2semidirect
16. #15689485C22semidirect
19. C2×#13093935C22semidirect
20. #15688863C22semidirect
21. #15688862C22semidirect
24. C2×C4⋊Q8×C47C22semidirect
27. C47×#497C22semidirect
28. C2×#97×C47C22semidirect
55. #13093935C23semidirect
66. C4⋊Q8×C47C23semidirect
74. #97×C47C23semidirect
183. C2×#497D47semidirect
189. C2×C47#1659semidirect
198. C2×C4⋊Q8C2×D47semidirect
206. #497C2×D47semidirect
212. C2×#97C2×D47semidirect
215. C47C2×#1659semidirect
232. C4⋊Q8C22×D47semidirect
261. #97C22×D47semidirect
370. C2#20242692direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156898652
C2×#156894852
#202426922
C2×#156888632
C2×#156888622
C2×C47×#4972
C2×#165947