Finite Group Databasebeta

← #26246089

Subgroups — #26246089

Order 12032.

1314 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689166C212yesmaximal, complemented
2C2×#15689127C212yesmaximal
3C2×#15689025C211yesmaximal, characteristic
4#20244922C212yesmaximal, complemented
5#20244917C212yesmaximal, complemented
6#20244912C214yesmaximal, complemented
7#20243082C211yesmaximal, characteristic, complemented
8C47×#2285C211yesmaximal, characteristic, complemented
9C23×D188C2211yescharacteristic
10C22×#10625329C2212yes—
11C22×C4⋊Dic47C2211yescharacteristic
12C22×(C2×D47)⋊C4—22no—
13#15689166—28no—
14C2×#13093957C2214yescomplemented
15C2×#13093955C2214yescomplemented
16#15689127—28no—
17C2×#13093944C2214yes—
18C2×#13093944C2214yes—
19#15689025—22no—
20#15689025C2212yes—
21#15689025C2212yes—
22#15688878C2212yescomplemented
23#15688878C2212yescomplemented
24C2×#117×C47C2212yes—
25C22×C42×C47C2211yescharacteristic
26C47×#384C2211yescharacteristic, complemented
27C47×#384C2211yescharacteristic, complemented
28C47×#382C2212yescomplemented
29C47×#380—22no—
30C4×C4⋊C4×C47—22no—
31C24×D47—21no—
32C22×D188—24no—
33C22×D188—24no—
34C22×D188—21no—
35C22×D188—22no—
36C22×D188Q814yes—
37C22×D188C2312yes—
38C23×Dic47—21no—
39C2×#10625329—24no—
40C2×#10625329—24no—
41C2×#10625329D418yes—
42C22×C47⋊C8C2314yes—
43C2×C4⋊Dic47—24no—
44C2×C4⋊Dic47—21no—
45C2×C4⋊Dic47C2312yes—
46C2×(C2×D47)⋊C4—24no—
47C2×(C2×D47)⋊C4—24no—
48C2×(C2×D47)⋊C4—48no—
49#13093957—28no—
50#13093955—28no—
51#13093944—28no—
52#13093944—28no—
53C23×C4×C47D412yes—
54C23×C4×C47C2311yescharacteristic
55#117×C47—24no—
56#117×C47—24no—
57C2×C4⋊C4×C47—22no—
58C2×C4⋊C4×C47—22no—
59C2×C4⋊C4×C47C2312yes—
60C2×C4⋊C4×C47C2312yes—
61C2×C22⋊C4×C47—22no—
62C2×C42×C47—24no—
63C2×C42×C47—22no—
64C2×C42×C47D412yes—
65C2×C42×C47C2311yescharacteristic
66C2×C42×C47C2311yescharacteristic
67#95×C47—24no—
68C24×C47C2×D411yescharacteristic
69C23×D47—24no—
70C23×D47—22no—
71C23×D47—44no—
72C23×D47—44no—
73C23×D47—42no—
74C23×D47—42no—
75C22×C4×C47—24no—
76C22×C4×C47—24no—
77C22×C4×C47—22no—
78C22×C4×C47—22no—
79C22×C4×C47—42no—
80C22×C4×C47C2×D412yes—
81C22×C4×C47C2×D412yes—
82C22×C4×C47C2×D411yescharacteristic
83C22×C4×C47C2×D414yes—
84C22×C4×C47C2×Q812yes—
85C22×C4×C47#4114yes—
86C22×C4×C47#4112yes—
87C22×C4×C47C2411yescharacteristic
88C22×Dic47—22no—
89C22×Dic47—44no—
90C22×Dic47—42no—
91C2×D188—216no—
92C2×D188—24no—
93C2×D188—24no—
94C2×D188—24no—
95C2×D188—22no—
96C2×D188—48no—
97C2×D188—42no—
98C2×D188—42no—
99C2×D188C2×Q814yes—
100C4⋊C4×C47—22no—
101C4⋊C4×C47—22no—
102C4⋊C4×C47—22no—
103C4⋊C4×C47—22no—
104C4⋊C4×C47—44no—
105C4⋊C4×C47—44no—
106C22⋊C4×C47—44no—
107C42×C47—24no—
108C42×C47—22no—
109C42×C47—22no—
110C42×C47—44no—
111(C2×D47)⋊C4—48no—
112(C2×D47)⋊C4—48no—
113C4⋊Dic47—24no—
114C4⋊Dic47—42no—
115#10625329—216no—
116C2×C47⋊C8—24no—
117C2×C47⋊C8—24no—
118C2×C47⋊C8C2×D418yes—
119C23×C47—22no—
120C23×C47—22no—
121C23×C47C22⋊Q812yes—
122C23×C47#12312yes—
123C23×C47#12712yes—
124C23×C47C22×D411yescharacteristic
125C22×D47—24no—
126C22×D47—416no—
127C22×D47—42no—
128C22×D47—42no—
129C22×D47—44no—
130C22×D47—44no—
131C22×D47—48no—
132C22×D47—88no—
133C22×D47—84no—
134C22×D47—84no—
135C2×C4×C47—22no—
136C2×C4×C47—22no—
137C2×C4×C47—22no—
138C2×C4×C47—22no—
139C2×C4×C47—22no—
140C2×C4×C47—48no—
141C2×C4×C47—42no—
142C2×C4×C47—42no—
143C2×C4×C47—44no—
144C2×C4×C47—44no—
145C2×C4×C47C22⋊Q812yes—
146C2×C4×C47C22⋊Q818yes—
147C2×C4×C47#12312yes—
148C2×C4×C47#12318yes—
149C2×C4×C47#12712yes—
150C2×C4×C47AGL(1,ℤ/8)14yes—
151C2×C4×C47C22×D412yes—
152C2×C4×C47C22×Q811yescharacteristic
153C2×C4×C47C2×#4112yes—
154C2×C4×C47C2×#4111yescharacteristic
155C2×Dic47—42no—
156C2×Dic47—44no—
157C2×Dic47—84no—
158D188—28no—
159D188—44no—
160D188—48no—
161C47⋊C8—28no—
162#23731—471nomaximal, Sylow
163C22×C47—24no—
164C22×C47—24no—
165C22×C47—22no—
166C22×C47—22no—
167C22×C47#39914yes—
168C22×C47C2×C22⋊Q811yescharacteristic
169C22×C47C2×#12311yescharacteristic
170C22×C47C2×#12711yescharacteristic
171C22×C47C2×AGL(1,ℤ/8)12yes—
172C22×C47#57512yes—
173C2×D47—48no—
174C2×D47—44no—
175C2×D47—816no—
176C2×D47—84no—
177C2×D47—82no—
178C2×D47—82no—
179C4×C47—24no—
180C4×C47—42no—
181C4×C47—82no—
182C4×C47—82no—
183C4×C47#39914yes—
184C4×C47C2×C22⋊Q812yes—
185C4×C47C2×#12312yes—
186Dic47—82no—
187C2×#428—472no—
188C2×#418—472no—
189C2×#390—471no—
190#2285D4711yescharacteristic, complemented
191#2094—472no—
192#2093—472no—
193#2092—474no—
194#1728—471no—
195C2×C47—24no—
196C2×C47C2×#39911yescharacteristic
197C2×C47#307712yes—
198C2×C47#309412yes—
199C2×C47#315111yescharacteristic
200C2×C47#315311yescharacteristic
201D47—84no—
202C23×D4—471no—
203C22×M16—472no—
204C2×#117C2×D4712yes—
205C22×C4⋊C4—471no—
206C22×C22⋊C4—942no—
207C22×C42C2×D4711yescharacteristic
208#428—948no—
209C2×#107—474no—
210C2×#106—474no—
211#418—948no—
212C2×#102—474no—
213C2×#102—474no—
214#390—472no—
215#390—472no—
216#390—942no—
217#384C2×D4711yescharacteristic, complemented
218#384C2×D4711yescharacteristic, complemented
219#382C2×D4712yescomplemented
220#380—22no—
221C4×C4⋊C4—22no—
222#336—472no—
223#336—472no—
224C47#2373111yescharacteristic, complemented, Sylow
225C25—941no—
226C22×D4—474no—
227C22×D4—472no—
228C22×D4—944no—
229C22×D4—944no—
230C22×D4—942no—
231C22×D4—941no—
232C23×C4—941no—
233C23×C4C22⋊D4712yes—
234C23×C4C22×D4711yescharacteristic
235C2×M16—478no—
236C2×M16—944no—
237C2×M16—944no—
238C22×C8—474no—
239#117—24no—
240#117—24no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—472no—
244C2×C4⋊C4—944no—
245C2×C4⋊C4—941no—
246C2×C4⋊C4C22×D4712yes—
247C2×C4⋊C4C22×D4712yes—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—944no—
250C2×C22⋊C4—944no—
251C2×C22⋊C4—1888no—
252C2×C42—24no—
253C2×C42—22no—
254C2×C42C22⋊D4712yes—
255C2×C42C22×D4711yescharacteristic
256C2×C42C22×D4711yescharacteristic
257#107—948no—
258#106—948no—
259#102—948no—
260#102—948no—
261#95—24no—
262C24—944no—
263C24—942no—
264C24—1884no—
265C24—1884no—
266C24—1882no—
267C24—1882no—
268C24C2×C22⋊D4711yescharacteristic
269C2×D4—474no—
270C2×D4—944no—
271C2×D4—9416no—
272C2×D4—944no—
273C2×D4—944no—
274C2×D4—942no—
275C2×D4—1882no—
276C2×D4—1888no—
277C2×D4—1882no—
278C22×C4—24no—
279C22×C4—24no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—42no—
283C22×C4—942no—
284C22×C4—1884no—
285C22×C4—1882no—
286C22×C4#1062535014yes—
287C22×C4D4×D4714yes—
288C22×C4Q8×D4712yes—
289C22×C4#1062535412yes—
290C22×C4C2×C22⋊D4712yes—
291C22×C4C2×C22⋊D4712yes—
292C22×C4C2×C22⋊D4711yescharacteristic
293C22×C4C23×D4711yescharacteristic
294M16—9416no—
295C2×C8—478no—
296C2×C8—944no—
297C2×C8—944no—
298C4⋊C4—22no—
299C4⋊C4—22no—
300C4⋊C4—22no—
301C4⋊C4—22no—
302C4⋊C4—44no—
303C4⋊C4—44no—
304C4⋊C4—944no—
305C4⋊C4—1882no—
306C22⋊C4—44no—
307C22⋊C4—1888no—
308C22⋊C4—1888no—
309C42—24no—
310C42—22no—
311C42—22no—
312C42—44no—
313C23—22no—
314C23—22no—
315C23—944no—
316C23—3768no—
317C23—3764no—
318C23—3764no—
319C23—18816no—
320C23—1888no—
321C23—1882no—
322C23—1884no—
323C23—1882no—
324C23—1884no—
325C23#1309400012yes—
326C23#1309400712yes—
327C23#1309402012yes—
328C23C22×C22⋊D4711yescharacteristic
329D4—948no—
330D4—1888no—
331D4—1884no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—48no—
338C2×C4—44no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—44no—
342C2×C4—3764no—
343C2×C4—1884no—
344C2×C4—1882no—
345C2×C4#1309400012yes—
346C2×C4#1309400218yes—
347C2×C4#1309400712yes—
348C2×C4#1309400918yes—
349C2×C4#1309402012yes—
350C2×C4#1309404914yes—
351C2×C4C2×D4×D4712yes—
352C2×C4C2×Q8×D4711yescharacteristic
353C2×C4C2×#1062535012yes—
354C2×C4C2×#1062535411yescharacteristic
355C8—948no—
356C22—24no—
357C22—22no—
358C22—22no—
359C22—24no—
360C22—37616no—
361C22—3764no—
362C22—3762no—
363C22—3762no—
364C22—1884no—
365C22—1888no—
366C22#1568905814yes—
367C22C2×#1309400011yescharacteristic
368C22C2×#1309400711yescharacteristic
369C22C2×#1309402011yescharacteristic
370C22C2×#1309404912yes—
371C22#1568992112yes—
372C4—24no—
373C4—42no—
374C4—82no—
375C4—82no—
376C4—3762no—
377C4#1568905814yes—
378C4C2×#1309400212yes—
379C4C2×#1309400912yes—
380C2—24no—
381C2—3764no—
382C2C2×#1568905811yescharacteristic
383C2#2025318112yes—
384C2#2025332812yes—
385C2#2025384111yescharacteristic
386C2#2025385611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC47×#2285C2
socleC23×C47C22×D4
prime radicalC23×D188C22
hypercenter#2285D47
hypocenterC47#23731
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2285

Complements

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

NH ≅ G/Ntype
1. C2×#15689166C2semidirect
4. #20244922C2semidirect
5. #20244917C2semidirect
6. #20244912C2semidirect
7. #20243082C2semidirect
8. C47×#2285C2semidirect
14. C2×#13093957C22semidirect
15. C2×#13093955C22semidirect
22. #15688878C22semidirect
23. #15688878C22semidirect
26. C47×#384C22semidirect
27. C47×#384C22semidirect
28. C47×#382C22semidirect
190. #2285D47semidirect
217. #384C2×D47semidirect
218. #384C2×D47semidirect
219. #382C2×D47semidirect
224. C47#23731semidirect

Maximal subgroups

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

subgroupindex
C2×#156891662
C2×#156891272
C2×#156890252
#202449222
#202449172
#202449122
#202430822
C47×#22852
#2373147