Finite Group Databasebeta

← #26370550

Subgroups — #26370550

Order 12032.

770 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689558C212yesmaximal, complemented
2C2×C4×D376C211yesmaximal, characteristic, complemented
3#20245440C214yesmaximal, complemented
4#20245430C212yesmaximal, complemented
5#20245420C212yesmaximal, complemented
6#20245144C211yesmaximal, characteristic, complemented
7#20245100C212yesmaximal, complemented
8C47×#3147C211yesmaximal, characteristic, complemented
9C22×D376C2211yescharacteristic
10C2×C4×D188—21no—
11#15689558—28no—
12C2×#13093967C2214yes—
13C2×#13093965C2214yes—
14C2×#13093964C2212yes—
15C4×D376—21no—
16C4×D376—21no—
17C4×D376C2212yes—
18C4×D376C2212yes—
19C2×#13093958C2211yescharacteristic
20C2×#13093946—21no—
21C4⋊(C47⋊C16)C2212yescomplemented
22C4⋊(C47⋊C16)C2212yescomplemented
23#15688927C2214yescomplemented
24#15688924C2214yescomplemented
25C2×#135×C47—21no—
26C2×#135×C47C2212yes—
27C47×#532—21no—
28C47×#495—21no—
29C47×#495C2212yescomplemented
30C47×#494C2211yescharacteristic, complemented
31C47×#493C2211yescharacteristic, complemented
32C47×#492—21no—
33C2×C4×C8×C47C2211yescharacteristic
34C22×D188—21no—
35C22×C4×D47—41no—
36C2×D376—21no—
37C2×D376—22no—
38C2×D376D414yescomplemented
39C2×D376C2312yes—
40C4×D188—22no—
41C4×D188—22no—
42C4×D188—41no—
43C4×D188—41no—
44C2×C4⋊Dic47—21no—
45C2×(C2×D47)⋊C4—41no—
46#13093967—28no—
47#13093965—28no—
48#13093964—24no—
49C2×C47⋊C16C2314yes—
50#13093958—21no—
51#13093958C2312yes—
52#13093946—22no—
53#13093946—22no—
54C2×#41×C47—22no—
55#135×C47—24no—
56#135×C47—24no—
57#135×C47—44no—
58C2×Q16×C47—21no—
59C2×Q16×C47C2312yes—
60C2×SD16×C47—22no—
61C2×SD16×C47—22no—
62C2×D8×C47—21no—
63C2×D8×C47C2312yes—
64C22×C8×C47D411yescharacteristic
65C22×C8×C47C2311yescharacteristic
66C4⋊Q8×C47—21no—
67#127×C47—21no—
68#124×C47—22no—
69#121×C47—41no—
70#121×C47—41no—
71C4×D4×C47—41no—
72C4×D4×C47—41no—
73C2×C42×C47D411yescharacteristic
74C4×C8×C47D411yescharacteristic
75C4×C8×C47D411yescharacteristic
76C4×C8×C47C2311yescharacteristic
77C4×C8×C47C2311yescharacteristic
78C23×D47—41no—
79#41×C47—44no—
80#41×C47—44no—
81C2×Q8×C47—22no—
82C2×D4×C47—22no—
83C2×D4×C47—42no—
84C2×D4×C47—41no—
85C2×D4×C47—41no—
86C22×C4×C47—42no—
87C22×C4×C47D811yescharacteristic
88C22×C4×C47D811yescharacteristic
89C22×C4×C47C2×D411yescharacteristic
90C22×Dic47—41no—
91C2×D188—24no—
92C2×D188—22no—
93C2×D188—41no—
94C2×D188—42no—
95C2×C4×D47—42no—
96C2×C4×D47—42no—
97C2×C4×D47—84no—
98Q16×C47—22no—
99Q16×C47—22no—
100Q16×C47—42no—
101SD16×C47—44no—
102SD16×C47—44no—
103D8×C47—22no—
104D8×C47—22no—
105D8×C47—42no—
106C2×C8×C47—22no—
107C2×C8×C47C2×D411yescharacteristic
108C2×C8×C47C2×D411yescharacteristic
109C2×C8×C47C2×D411yescharacteristic
110C2×C8×C47C2×D412yes—
111C2×C8×C47#4112yes—
112C2×C8×C47C2411yescharacteristic
113C4⋊C4×C47—41no—
114C4⋊C4×C47—41no—
115C22⋊C4×C47—42no—
116C22⋊C4×C47—42no—
117C42×C47D811yescharacteristic
118C42×C47D811yescharacteristic
119C42×C47C2×D411yescharacteristic
120C42×C47C2×D411yescharacteristic
121(C2×D47)⋊C4—42no—
122(C2×D47)⋊C4—42no—
123C4⋊Dic47—22no—
124C4⋊Dic47—41no—
125D376—24no—
126D376—22no—
127D376C2×D414yes—
128C47⋊C16—24no—
129C23×C47—42no—
130C23×C47C2×D811yescharacteristic
131C22×D47—42no—
132C22×D47—44no—
133C22×D47—82no—
134C22×D47—82no—
135Q8×C47—42no—
136Q8×C47—42no—
137D4×C47—42no—
138D4×C47—42no—
139D4×C47—82no—
140D4×C47—84no—
141D4×C47—82no—
142C2×C4×C47—22no—
143C2×C4×C47—41no—
144C2×C4×C47—42no—
145C2×C4×C47—84no—
146C2×C4×C47#12112yes—
147C2×C4×C47C2×D811yescharacteristic
148C2×C4×C47C2×D811yescharacteristic
149C2×C4×C47C2×D811yescharacteristic
150C2×C4×C47C2×D811yescharacteristic
151C2×C4×C47C2×D811yescharacteristic
152C2×C4×C47AGL(1,ℤ/8)12yes—
153C2×C4×C47C22×D411yescharacteristic
154C2×Dic47—42no—
155C2×Dic47—82no—
156D188—24no—
157D188—42no—
158D188—44no—
159C4×D47—84no—
160C4×D47—84no—
161C8×C47—21no—
162C8×C47—21no—
163C8×C47#12112yes—
164C8×C47C22×D411yescharacteristic
165C8×C47C2×#4111yescharacteristic
166#33580—471nomaximal, Sylow
167C22×C47—21no—
168C22×C47—21no—
169C22×C47—82no—
170C22×C47—82no—
171C22×C47—82no—
172C22×C47#45912yes—
173C22×C47C22×D811yescharacteristic
174C2×D47—44no—
175C2×D47—88no—
176C2×D47—82no—
177C2×D47—82no—
178C4×C47—21no—
179C4×C47—21no—
180C4×C47—21no—
181C4×C47—82no—
182C4×C47#45912yes—
183C4×C47#50912yescomplemented
184C4×C47C2×#12111yescharacteristic
185C4×C47C2×AGL(1,ℤ/8)11yescharacteristic
186Dic47—82no—
187C2×#509—472no—
188#3147D4711yescharacteristic, complemented
189C2×C4×D8—471no—
190#2210—474no—
191#2208—472no—
192#2206—472no—
193#2150—471no—
194#2139—472no—
195C2×C47—22no—
196C2×C47—162no—
197C2×C47C2×#45911yescharacteristic
198C2×C47C2×#50911yescharacteristic
199C2×C47#341411yescharacteristic
200D47—84no—
201C2×#135—21no—
202C2×#135C2×D4712yes—
203C22×D8—471no—
204#532—21no—
205C2×C4×D4—941no—
206#509—948no—
207C2×SD32—474no—
208C2×D16—474no—
209C2×M32—472no—
210#495—21no—
211#495C2×D4712yescomplemented
212#494C2×D4711yescharacteristic, complemented
213#493C2×D4711yescharacteristic, complemented
214#492—21no—
215C4×D8—472no—
216C4×D8—472no—
217C4×D8—941no—
218C4×D8—941no—
219C2×#107—471no—
220C2×#102—941no—
221C2×C4×C8C2×D4711yescharacteristic
222C4⋊C16—472no—
223C4⋊C16—472no—
224#358—474no—
225#357—474no—
226C47#3358011yescharacteristic, complemented, Sylow
227C2×#41—22no—
228C22×D4—941no—
229C23×C4—1881no—
230#135—24no—
231#135—24no—
232#135—44no—
233C2×Q16—21no—
234C2×Q16C22×D4712yes—
235C2×SD16—22no—
236C2×SD16—22no—
237C2×D8—21no—
238C2×D8—474no—
239C2×D8—472no—
240C2×D8—942no—
241C2×D8—941no—
242C2×D8C22×D4712yes—
243C22×C8C22⋊D4711yescharacteristic
244C22×C8C22×D4711yescharacteristic
245C4⋊Q8—21no—
246#127—21no—
247#124—22no—
248#121—41no—
249#121—41no—
250C4×D4—41no—
251C4×D4—41no—
252C4×D4—942no—
253C4×D4—942no—
254C4×D4—1881no—
255C4×D4—1881no—
256C2×C4⋊C4—941no—
257C2×C22⋊C4—1881no—
258C2×C42C22⋊D4711yescharacteristic
259SD32—948no—
260D16—948no—
261M32—944no—
262C2×C16—474no—
263#107—472no—
264#107—941no—
265#102—942no—
266#102—942no—
267C4×C8C22⋊D4711yescharacteristic
268C4×C8C22⋊D4711yescharacteristic
269C4×C8C22×D4711yescharacteristic
270C4×C8C22×D4711yescharacteristic
271C24—1881no—
272#41—44no—
273#41—44no—
274C2×Q8—22no—
275C2×D4—22no—
276C2×D4—42no—
277C2×D4—41no—
278C2×D4—41no—
279C2×D4—944no—
280C2×D4—942no—
281C2×D4—1881no—
282C2×D4—1882no—
283C22×C4—42no—
284C22×C4—3764no—
285C22×C4—1881no—
286C22×C4—1882no—
287C22×C4—1882no—
288C22×C4#1062533411yescharacteristic
289C22×C4#1062533411yescharacteristic
290C22×C4C2×C22⋊D4711yescharacteristic
291Q16—22no—
292Q16—22no—
293Q16—42no—
294SD16—44no—
295SD16—44no—
296D8—22no—
297D8—22no—
298D8—42no—
299D8—474no—
300D8—944no—
301D8—942no—
302C2×C8—22no—
303C2×C8D4×D4712yes—
304C2×C8#1062535212yes—
305C2×C8C2×C22⋊D4711yescharacteristic
306C2×C8C2×C22⋊D4711yescharacteristic
307C2×C8C2×C22⋊D4711yescharacteristic
308C2×C8C23×D4711yescharacteristic
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—942no—
312C4⋊C4—1881no—
313C22⋊C4—42no—
314C22⋊C4—42no—
315C22⋊C4—1882no—
316C22⋊C4—1882no—
317C42#1062533411yescharacteristic
318C42#1062533411yescharacteristic
319C42C2×C22⋊D4711yescharacteristic
320C42C2×C22⋊D4711yescharacteristic
321C16—944no—
322C23—42no—
323C23—3762no—
324C23—3762no—
325C23—1884no—
326C23—1882no—
327C23C2×#1062533411yescharacteristic
328Q8—42no—
329Q8—42no—
330D4—42no—
331D4—42no—
332D4—82no—
333D4—84no—
334D4—82no—
335D4—944no—
336D4—1884no—
337D4—1882no—
338C2×C4—22no—
339C2×C4—41no—
340C2×C4—42no—
341C2×C4—84no—
342C2×C4—3764no—
343C2×C4—3762no—
344C2×C4—3764no—
345C2×C4—1882no—
346C2×C4#1309399512yes—
347C2×C4C2×#1062533411yescharacteristic
348C2×C4C2×#1062533411yescharacteristic
349C2×C4C2×#1062533411yescharacteristic
350C2×C4C2×#1062533411yescharacteristic
351C2×C4C2×#1062533411yescharacteristic
352C2×C4#1309405012yes—
353C2×C4C22×C22⋊D4711yescharacteristic
354C8—21no—
355C8—21no—
356C8#1309399512yes—
357C8C2×D4×D4711yescharacteristic
358C8C2×#1062535211yescharacteristic
359C22—21no—
360C22—21no—
361C22—82no—
362C22—82no—
363C22—82no—
364C22—3768no—
365C22—3762no—
366C22—3762no—
367C22—1884no—
368C22#1568931512yes—
369C22C22×#1062533411yescharacteristic
370C4—21no—
371C4—21no—
372C4—21no—
373C4—82no—
374C4—3762no—
375C4#1568931512yes—
376C4#1568955812yescomplemented
377C4C2×#1309399511yescharacteristic
378C4C2×#1309405011yescharacteristic
379C2—22no—
380C2—162no—
381C2—3764no—
382C2C2×#1568931511yescharacteristic
383C2C2×#1568955811yescharacteristic
384C2#2025652211yescharacteristic
subgroupG/H
centerC22C22×#10625334
commutator, cosocleC2×C8×C47C24
FrattiniC2×C8C23×D47
FittingC47×#3147C2
socleC22×C47C22×D8
hypercenter#3147D47
hypocenterC47#33580
LCSG ▷ C2×C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ #3147

Complements

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

NH ≅ G/Ntype
1. C2×#15689558C2semidirect
2. C2×C4×D376C2semidirect
3. #20245440C2semidirect
4. #20245430C2semidirect
5. #20245420C2semidirect
6. #20245144C2semidirect
7. #20245100C2semidirect
8. C47×#3147C2semidirect
21. C4⋊(C47⋊C16)C22semidirect
22. C4⋊(C47⋊C16)C22semidirect
23. #15688927C22semidirect
24. #15688924C22semidirect
29. C47×#495C22semidirect
30. C47×#494C22semidirect
31. C47×#493C22semidirect
38. C2×D376D4semidirect
183. C4×C47#509semidirect
188. #3147D47semidirect
211. #495C2×D47semidirect
212. #494C2×D47semidirect
213. #493C2×D47semidirect
226. C47#33580semidirect
376. C4#15689558semidirect

Maximal subgroups

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

subgroupindex
C2×#156895582
C2×C4×D3762
#202454402
#202454302
#202454202
#202451442
#202451002
C47×#31472
#3358047