Finite Group Databasebeta

← C2×#20243030 #26204021

Subgroups — C2×#20243030 #26204021

Order 12032.

966 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689500C211yesmaximal, characteristic
2C2×#15689497C211yesmaximal, characteristic, complemented
3C2×#15689471C211yesmaximal, characteristic, complemented
4C2×#15689471C211yesmaximal, characteristic, complemented
5C2×#15689471C211yesmaximal, characteristic, complemented
6#20243030C218yesmaximal, direct product factor
7C2×#15688845C211yesmaximal, characteristic, complemented
8C2×C47×#487C211yesmaximal, characteristic, complemented
9C22×#10625336—22no—
10C22×D376—22no—
11C2×#13094021C2211yescharacteristic
12C2×#13094021C2211yescharacteristic
13#15689500—22no—
14#15689500C2214yes—
15#15689497—22no—
16#15689497C2214yescomplemented
17#15689471—22no—
18#15689471—22no—
19#15689471—22no—
20#15689471C2214yescomplemented
21#15689471C2214yescomplemented
22#15689471C2214yescomplemented
23C2×#13093944—22no—
24C2×#13093944—22no—
25C2×#13093944—22no—
26C2×C4×C47⋊C8C2211yescharacteristic
27C2×C4×C47⋊C8C2211yescharacteristic
28#15688845C2214yescomplemented
29C2×C4⋊Q8×C47C2211yescharacteristic
30C2×C4⋊Q8×C47C2211yescharacteristic
31C47×#487—22no—
32C47×#487C2214yescomplemented
33C2×#103×C47—22no—
34C2×C4×C8×C47C2211yescharacteristic
35C22×D188—21no—
36C22×D188—21no—
37C22×D188—21no—
38C22×D188—21no—
39C22×D188—21no—
40C22×D188—21no—
41C2×#10625336—28no—
42C2×#10625336—48no—
43C2×D376—28no—
44C2×D376—48no—
45C22×C47⋊C8D412yes—
46C22×C47⋊C8D412yes—
47#13094021—21no—
48#13094021—21no—
49#13094021—21no—
50#13094021—21no—
51#13094021—21no—
52#13094021—21no—
53#13094021C2312yes—
54#13094021C2312yes—
55#13093944—28no—
56#13093944—28no—
57#13093944—28no—
58C4×C47⋊C8D412yes—
59C4×C47⋊C8D412yes—
60C4×C47⋊C8C2312yes—
61C4×C47⋊C8C2312yes—
62C22×Q8×C47—21no—
63C22×Q8×C47—21no—
64C22×C8×C47D412yes—
65C4⋊Q8×C47—21no—
66C4⋊Q8×C47—21no—
67C4⋊Q8×C47—21no—
68C4⋊Q8×C47—21no—
69C4⋊Q8×C47—21no—
70C4⋊Q8×C47—21no—
71C4⋊Q8×C47C2312yes—
72C4⋊Q8×C47C2312yes—
73C2×C4⋊C4×C47—21no—
74C2×C4⋊C4×C47—21no—
75C2×C4⋊C4×C47—21no—
76C2×C4⋊C4×C47—21no—
77C2×C42×C47C2311yescharacteristic
78#103×C47—28no—
79C4×C8×C47D412yes—
80C4×C8×C47C2312yes—
81C23×D47—41no—
82C23×D47—41no—
83C2×Q8×C47—22no—
84C2×Q8×C47—22no—
85C2×Q8×C47—41no—
86C2×Q8×C47—41no—
87C2×Q8×C47—44no—
88C2×Q8×C47—44no—
89C22×C4×C47—41no—
90C22×C4×C47—41no—
91C22×C4×C47C2×D411yescharacteristic
92C22×C4×C47C2×D411yescharacteristic
93C22×C4×C47C2×D411yescharacteristic
94C2×D188—22no—
95C2×D188—22no—
96C2×D188—22no—
97C2×D188—22no—
98C2×D188—22no—
99C2×D188—22no—
100C2×D188—44no—
101C2×D188—44no—
102C2×D188—44no—
103C2×D188—44no—
104C2×D188—44no—
105C2×D188—44no—
106C2×D188—41no—
107C2×D188—41no—
108C2×D188—41no—
109C2×D188—41no—
110C2×D188—41no—
111C2×D188—41no—
112C2×C8×C47—24no—
113C2×C8×C47C2×D414yes—
114C4⋊C4×C47—22no—
115C4⋊C4×C47—22no—
116C4⋊C4×C47—22no—
117C4⋊C4×C47—22no—
118C4⋊C4×C47—41no—
119C4⋊C4×C47—41no—
120C4⋊C4×C47—41no—
121C4⋊C4×C47—41no—
122C42×C47C2×D411yescharacteristic
123C42×C47C2×D411yescharacteristic
124C42×C47C2×D411yescharacteristic
125C42×C47C2411yescharacteristic
126#10625336—416no—
127C2×C47⋊C8—24no—
128C2×C47⋊C8—24no—
129C2×C47⋊C8C2×D414yes—
130C2×C47⋊C8C2×D414yes—
131D376—416no—
132C23×C47#12711yescharacteristic
133C22×D47—42no—
134C22×D47—42no—
135C22×D47—82no—
136C22×D47—82no—
137C22×D47—82no—
138C22×D47—82no—
139C22×D47—82no—
140C22×D47—82no—
141Q8×C47—44no—
142Q8×C47—44no—
143Q8×C47—82no—
144Q8×C47—82no—
145C2×C4×C47—42no—
146C2×C4×C47—42no—
147C2×C4×C47—82no—
148C2×C4×C47—82no—
149C2×C4×C47#12711yescharacteristic
150C2×C4×C47#12711yescharacteristic
151C2×C4×C47#12711yescharacteristic
152C2×C4×C47#13514yes—
153C2×C4×C47AGL(1,ℤ/8)14yes—
154C2×C4×C47AGL(1,ℤ/8)14yes—
155C2×C4×C47C22×D411yescharacteristic
156C2×C4×C47C22×D411yescharacteristic
157C2×C4×C47C22×D411yescharacteristic
158D188—44no—
159D188—44no—
160D188—44no—
161D188—44no—
162D188—44no—
163D188—44no—
164D188—82no—
165D188—82no—
166D188—82no—
167D188—82no—
168D188—82no—
169D188—82no—
170C8×C47—24no—
171C47⋊C8—24no—
172C47⋊C8—24no—
173C2×#1717—471nomaximal, Sylow
174C22×C47#49512yes—
175C22×C47#49612yes—
176C22×C47#49612yes—
177C22×C47C2×#12711yescharacteristic
178C2×D47—82no—
179C2×D47—82no—
180C2×D47—82no—
181C2×D47—82no—
182C2×D47—82no—
183C2×D47—82no—
184C2×D47—164no—
185C2×D47—164no—
186C4×C47—82no—
187C4×C47—82no—
188C4×C47#49512yes—
189C4×C47#49612yes—
190C4×C47#49612yes—
191C4×C47C2×#13512yes—
192C4×C47C2×AGL(1,ℤ/8)12yes—
193C4×C47C2×AGL(1,ℤ/8)12yes—
194C2×#493—471no—
195C2×#492—471no—
196C2×#487D4711yescharacteristic, complemented
197C2×#486—471no—
198C2×#486—471no—
199C2×#486—471no—
200#1717—478no—
201C2×#322—471no—
202C2×C47#171714yescomplemented
203C2×C47C2×#49511yescharacteristic
204C2×C47C2×#49611yescharacteristic
205C2×C47C2×#49611yescharacteristic
206D47—162no—
207D47—162no—
208C22×SD16—942no—
209C22×D8—942no—
210C2×C4⋊Q8C2×D4711yescharacteristic
211C2×C4⋊Q8C2×D4711yescharacteristic
212C2×#127—471no—
213C2×#127—471no—
214#493—474no—
215#493—942no—
216#492—474no—
217#492—942no—
218#487—22no—
219#487C2×D4714yescomplemented
220#486—474no—
221#486—474no—
222#486—474no—
223#486—942no—
224#486—942no—
225#486—942no—
226C2×#103—22no—
227C2×#102—942no—
228C2×#102—942no—
229C2×#102—942no—
230C2×C4×C8—471no—
231C2×C4×C8—471no—
232C2×C4×C8C2×D4711yescharacteristic
233#322—474no—
234C47C2×#171711yescharacteristic, complemented, Sylow
235C22×Q8—21no—
236C22×Q8—21no—
237C22×D4—941no—
238C22×D4—941no—
239C22×D4—941no—
240C22×D4—941no—
241C22×D4—941no—
242C22×D4—941no—
243C2×SD16—948no—
244C2×SD16—1888no—
245C2×D8—948no—
246C2×D8—1888no—
247C22×C8—472no—
248C22×C8—472no—
249C22×C8C22⋊D4712yes—
250C4⋊Q8—21no—
251C4⋊Q8—21no—
252C4⋊Q8—21no—
253C4⋊Q8—21no—
254C4⋊Q8—21no—
255C4⋊Q8—21no—
256C4⋊Q8C22×D4712yes—
257C4⋊Q8C22×D4712yes—
258#127—472no—
259#127—472no—
260#127—941no—
261#127—941no—
262#127—941no—
263#127—941no—
264#127—941no—
265#127—941no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C42C22×D4711yescharacteristic
271#103—28no—
272#102—948no—
273#102—948no—
274#102—948no—
275C4×C8—472no—
276C4×C8—472no—
277C4×C8—472no—
278C4×C8—472no—
279C4×C8C22⋊D4712yes—
280C4×C8C22×D4712yes—
281C24—1881no—
282C24—1881no—
283C2×Q8—22no—
284C2×Q8—22no—
285C2×Q8—41no—
286C2×Q8—41no—
287C2×Q8—44no—
288C2×Q8—44no—
289C2×D4—942no—
290C2×D4—942no—
291C2×D4—942no—
292C2×D4—942no—
293C2×D4—942no—
294C2×D4—942no—
295C2×D4—1881no—
296C2×D4—1881no—
297C2×D4—1881no—
298C2×D4—1881no—
299C2×D4—1881no—
300C2×D4—1881no—
301C2×D4—1884no—
302C2×D4—1884no—
303C2×D4—1884no—
304C2×D4—1884no—
305C2×D4—1884no—
306C2×D4—1884no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4D4×D4711yescharacteristic
310C22×C4D4×D4711yescharacteristic
311C22×C4C2×C22⋊D4711yescharacteristic
312SD16—18816no—
313D8—18816no—
314C2×C8—24no—
315C2×C8—474no—
316C2×C8—474no—
317C2×C8—944no—
318C2×C8—944no—
319C2×C8C2×C22⋊D4714yes—
320C4⋊C4—22no—
321C4⋊C4—22no—
322C4⋊C4—22no—
323C4⋊C4—22no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C42D4×D4711yescharacteristic
329C42D4×D4711yescharacteristic
330C42C2×C22⋊D4711yescharacteristic
331C42C23×D4711yescharacteristic
332C23—3762no—
333C23—3762no—
334C23—3762no—
335C23—3762no—
336C23—3762no—
337C23—3762no—
338C23—1882no—
339C23—1882no—
340C23#1309402011yescharacteristic
341Q8—44no—
342Q8—44no—
343Q8—82no—
344Q8—82no—
345D4—3762no—
346D4—3762no—
347D4—3762no—
348D4—3762no—
349D4—3762no—
350D4—3762no—
351D4—1884no—
352D4—1884no—
353D4—1884no—
354D4—1884no—
355D4—1884no—
356D4—1884no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—82no—
360C2×C4—82no—
361C2×C4#1309402011yescharacteristic
362C2×C4#1309402011yescharacteristic
363C2×C4#1309402011yescharacteristic
364C2×C4#1309404714yes—
365C2×C4#1309404914yes—
366C2×C4#1309405114yes—
367C2×C4C2×D4×D4711yescharacteristic
368C2×C4C2×D4×D4711yescharacteristic
369C2×C4C22×C22⋊D4711yescharacteristic
370C8—24no—
371C8—944no—
372C8—944no—
373C22—3762no—
374C22—3762no—
375C22—3762no—
376C22—3762no—
377C22—3762no—
378C22—3762no—
379C22—7524no—
380C22—7524no—
381C22#1568950812yes—
382C22#1568950912yes—
383C22#1568951312yes—
384C22C2×#1309402011yescharacteristic
385C4—82no—
386C4—82no—
387C4#1568950812yes—
388C4#1568950912yes—
389C4#1568951312yes—
390C4C2×#1309404712yes—
391C4C2×#1309404912yes—
392C4C2×#1309405112yes—
393C2—7522no—
394C2—7522no—
395C2#2024303014yesdirect product factor
396C2C2×#1568950811yescharacteristic
397C2C2×#1568950911yescharacteristic
398C2C2×#1568951311yescharacteristic
subgroupG/H
centerC23#13094020
commutator, cosocleC42×C47C24
FrattiniC42C23×D47
FittingC2×C47×#487C2
socleC23×C47#127
prime radicalC2×#15689500C2
hypercenterC2×#487D47
hypocenterC47C2×#1717
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#487

Complements

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

NH ≅ G/Ntype
2. C2×#15689497C2semidirect
3. C2×#15689471C2semidirect
4. C2×#15689471C2semidirect
5. C2×#15689471C2semidirect
6. #20243030C2direct (N × H)
7. C2×#15688845C2semidirect
8. C2×C47×#487C2semidirect
16. #15689497C22semidirect
20. #15689471C22semidirect
21. #15689471C22semidirect
22. #15689471C22semidirect
28. #15688845C22semidirect
32. C47×#487C22semidirect
196. C2×#487D47semidirect
202. C2×C47#1717semidirect
219. #487C2×D47semidirect
234. C47C2×#1717semidirect
395. C2#20243030direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156895002
C2×#156894972
C2×#156894712
#202430302
C2×#156888452
C2×C47×#4872
C2×#171747