Finite Group Databasebeta

← #26111328

Subgroups — #26111328

Order 12032.

768 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
1#20246002C211yesmaximal, characteristic
2#20243474C212yesmaximal, complemented
3C2×#15688847C211yesmaximal, characteristic
4#20241267C212yesmaximal, complemented
5C2×C47×#397C211yesmaximal, characteristic
6C22×C22⋊Dic47—21no—
7C2×C22⋊(C47⋊C8)C2212yes—
8#15688998—24no—
9#15688983—21no—
10#15688983C2212yes—
11#15688885—24no—
12#15688847—22no—
13C22×C4⋊C4×C47C2211yescharacteristic
14C22×C22⋊C4×C47—21no—
15C22×C22⋊C4×C47C2211yescharacteristic
16C47×#397—22no—
17C47×#397C414yescomplemented
18C2×#95×C47—21no—
19C2×#95×C47C2211yescharacteristic
20C23×Dic47—22no—
21C22×C47⋊C8—22no—
22C2×C4⋊Dic47—22no—
23C2×C4⋊Dic47D414yes—
24C2×C22⋊Dic47—22no—
25C2×C22⋊Dic47—22no—
26C2×C22⋊Dic47—42no—
27C2×C22⋊Dic47—42no—
28C22⋊(C47⋊C8)—22no—
29C22⋊(C47⋊C8)—22no—
30C47⋊#95—22no—
31C47⋊#95—22no—
32C25×C47D411yescharacteristic
33C23×C4×C47—21no—
34C23×C4×C47—21no—
35C23×C4×C47D411yescharacteristic
36C23×C4×C47C2311yescharacteristic
37C2×C4⋊C4×C47—22no—
38C2×C4⋊C4×C47—22no—
39C2×C4⋊C4×C47—21no—
40C2×C4⋊C4×C47C2×C412yescomplemented
41C2×C22⋊C4×C47—22no—
42C2×C22⋊C4×C47—22no—
43C2×C22⋊C4×C47—24no—
44C2×C22⋊C4×C47—42no—
45C2×C22⋊C4×C47—42no—
46C2×C22⋊C4×C47C2×C412yes—
47C2×C22⋊C4×C47D412yes—
48#95×C47—24no—
49#95×C47—21no—
50#95×C47C2×C412yescomplemented
51C24×C47—21no—
52C24×C47—21no—
53C24×C47—21no—
54C24×C47—21no—
55C24×C47—42no—
56C24×C47—42no—
57C24×C47C22⋊C412yes—
58C24×C47D812yes—
59C24×C47SD1612yes—
60C24×C47C2×D411yescharacteristic
61C22×C4×C47—21no—
62C22×C4×C47—21no—
63C22×C4×C47—22no—
64C22×C4×C47—22no—
65C22×C4×C47—42no—
66C22×C4×C47—42no—
67C22×C4×C47—42no—
68C22×C4×C47—42no—
69C22×C4×C47—41no—
70C22×C4×C47—41no—
71C22×C4×C47—41no—
72C22×C4×C47—41no—
73C22×C4×C47—41no—
74C22×C4×C47—41no—
75C22×C4×C47—41no—
76C22×C4×C47—41no—
77C22×C4×C47C22⋊C412yes—
78C22×C4×C47C22×C411yescharacteristic
79C22×C4×C47C2×D412yes—
80C22×C4×C47C2×D411yescharacteristic
81C22×C4×C47#4112yes—
82C22×Dic47—24no—
83C22×Dic47—44no—
84C22×Dic47—82no—
85C22×Dic47—82no—
86C4⋊C4×C47—24no—
87C4⋊C4×C47—42no—
88C22⋊C4×C47—24no—
89C22⋊C4×C47—24no—
90C22⋊C4×C47—48no—
91C22⋊Dic47—44no—
92C22⋊Dic47—42no—
93C22⋊Dic47—42no—
94C4⋊Dic47—44no—
95C4⋊Dic47—82no—
96C2×C47⋊C8—44no—
97C2×C47⋊C8—42no—
98C23×C47—22no—
99C23×C47—22no—
100C23×C47—22no—
101C23×C47—22no—
102C23×C47—22no—
103C23×C47—22no—
104C23×C47—22no—
105C23×C47—22no—
106C23×C47—42no—
107C23×C47—42no—
108C23×C47—44no—
109C23×C47—44no—
110C23×C47—42no—
111C23×C47—42no—
112C23×C47—82no—
113C23×C47—82no—
114C23×C47—81no—
115C23×C47—81no—
116C23×C47#10214yes—
117C23×C47C2×C22⋊C411yescharacteristic
118C23×C47#12011yescharacteristic
119C23×C47#12311yescharacteristic
120C23×C47C2×D811yescharacteristic
121C23×C47C2×SD1611yescharacteristic
122C23×C47AGL(1,ℤ/8)11yescharacteristic
123C23×C47#13711yescharacteristic
124C2×C4×C47—24no—
125C2×C4×C47—22no—
126C2×C4×C47—41no—
127C2×C4×C47—41no—
128C2×C4×C47—42no—
129C2×C4×C47—42no—
130C2×C4×C47—42no—
131C2×C4×C47—42no—
132C2×C4×C47—42no—
133C2×C4×C47—42no—
134C2×C4×C47—42no—
135C2×C4×C47—42no—
136C2×C4×C47—42no—
137C2×C4×C47—81no—
138C2×C4×C47—82no—
139C2×C4×C47—82no—
140C2×C4×C47—81no—
141C2×C4×C47—81no—
142C2×C4×C47—81no—
143C2×C4×C47—81no—
144C2×C4×C47C4×D412yes—
145C2×C4×C47#12112yes—
146C2×Dic47—44no—
147C2×Dic47—44no—
148C2×Dic47—44no—
149C2×Dic47—88no—
150C47⋊C8—82no—
151#9025—471nomaximal, Sylow
152C22×C47—22no—
153C22×C47—22no—
154C22×C47—22no—
155C22×C47—22no—
156C22×C47—21no—
157C22×C47—21no—
158C22×C47—21no—
159C22×C47—21no—
160C22×C47—24no—
161C22×C47—44no—
162C22×C47—44no—
163C22×C47—44no—
164C22×C47—42no—
165C22×C47—42no—
166C22×C47—82no—
167C22×C47—82no—
168C22×C47—82no—
169C22×C47#38611yescharacteristic
170C22×C47C2×#10211yescharacteristic
171C22×C47#41711yescharacteristic
172C22×C47#44711yescharacteristic
173C22×C47#44811yescharacteristic
174C22×C47#45512yes—
175C22×C47#45812yes—
176C22×C47#48011yescharacteristic
177C22×C47#48111yescharacteristic
178C4×C47—22no—
179C4×C47—22no—
180C4×C47—42no—
181C4×C47—82no—
182C4×C47—82no—
183Dic47—84no—
184C2×#397D4711yescharacteristic
185#2315—471no—
186#1823—472no—
187C2×#323—471no—
188#1294—472no—
189C2×C47—21no—
190C2×C47—21no—
191C2×C47—21no—
192C2×C47—21no—
193C2×C47—44no—
194C2×C47#160712yes—
195C2×C47#160812yes—
196C2×C47#187411yescharacteristic
197C2×C47#188511yescharacteristic
198C2×C47#189011yescharacteristic
199C22×C4⋊C4C2×D4711yescharacteristic
200C22×C22⋊C4—21no—
201C22×C22⋊C4—941no—
202C22×C22⋊C4C2×D4711yescharacteristic
203C2×C22⋊C8—472no—
204#397—22no—
205#397Dic4714yescomplemented
206#385—944no—
207#380—472no—
208#380—941no—
209C2×#95—21no—
210C2×#95C2×D4711yescharacteristic
211#340—944no—
212#323—942no—
213C47#902511yescharacteristic, complemented, Sylow
214C25C22⋊D4711yescharacteristic
215C23×C4—21no—
216C23×C4—21no—
217C23×C4—942no—
218C23×C4C22⋊D4711yescharacteristic
219C23×C4C22×D4711yescharacteristic
220C22×C8—942no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—474no—
225C2×C4⋊C4—942no—
226C2×C4⋊C4C2×Dic4712yescomplemented
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—42no—
231C2×C22⋊C4—42no—
232C2×C22⋊C4—942no—
233C2×C22⋊C4—942no—
234C2×C22⋊C4—1882no—
235C2×C22⋊C4—1882no—
236C2×C22⋊C4C2×Dic4712yes—
237C2×C22⋊C4C22⋊D4712yes—
238C22⋊C8—942no—
239C22⋊C8—942no—
240#95—24no—
241#95—21no—
242#95—942no—
243#95—942no—
244#95C2×Dic4712yescomplemented
245C24—21no—
246C24—21no—
247C24—21no—
248C24—21no—
249C24—42no—
250C24—42no—
251C24#1062533412yes—
252C24#1062533512yes—
253C24C22⋊Dic4712yes—
254C24C2×C22⋊D4711yescharacteristic
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—944no—
272C22×C4—3762no—
273C22×C4—3762no—
274C22×C4—1884no—
275C22×C4C22⋊Dic4712yes—
276C22×C4D4×D4712yes—
277C22×C4#1062535212yes—
278C22×C4C22×Dic4711yescharacteristic
279C22×C4C2×C22⋊D4711yescharacteristic
280C2×C8—1884no—
281C2×C8—1882no—
282C4⋊C4—24no—
283C4⋊C4—42no—
284C4⋊C4—3762no—
285C4⋊C4—1884no—
286C22⋊C4—24no—
287C22⋊C4—24no—
288C22⋊C4—48no—
289C22⋊C4—1884no—
290C22⋊C4—1882no—
291C22⋊C4—1882no—
292C23—22no—
293C23—22no—
294C23—22no—
295C23—22no—
296C23—22no—
297C23—22no—
298C23—22no—
299C23—22no—
300C23—44no—
301C23—44no—
302C23—42no—
303C23—42no—
304C23—42no—
305C23—42no—
306C23—81no—
307C23—81no—
308C23—82no—
309C23—82no—
310C23#1309394514yes—
311C23C2×C22⋊Dic4711yescharacteristic
312C23#1309399211yescharacteristic
313C23#1309400511yescharacteristic
314C23C2×#1062533411yescharacteristic
315C23C2×#1062533511yescharacteristic
316C23#1309404911yescharacteristic
317C23#1309405411yescharacteristic
318C2×C4—24no—
319C2×C4—22no—
320C2×C4—42no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—81no—
332C2×C4—82no—
333C2×C4—82no—
334C2×C4—81no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4—3768no—
339C2×C4—1884no—
340C2×C4—1884no—
341C2×C4—1884no—
342C2×C4D4×Dic4712yes—
343C2×C4#1309399612yes—
344C8—3762no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—22no—
350C22—22no—
351C22—22no—
352C22—22no—
353C22—24no—
354C22—44no—
355C22—42no—
356C22—42no—
357C22—44no—
358C22—44no—
359C22—82no—
360C22—82no—
361C22—82no—
362C22#1568900311yescharacteristic
363C22C2×#1309394511yescharacteristic
364C22#1568912411yescharacteristic
365C22#1568925611yescharacteristic
366C22#1568926111yescharacteristic
367C22#1568930012yes—
368C22#1568931212yes—
369C22#1568943411yescharacteristic
370C22#1568943911yescharacteristic
371C4—22no—
372C4—22no—
373C4—42no—
374C4—82no—
375C4—82no—
376C4—3764no—
377C2—21no—
378C2—21no—
379C2—21no—
380C2—21no—
381C2—44no—
382C2#2024237312yes—
383C2#2024237812yes—
384C2#2024370611yescharacteristic
385C2#2024377711yescharacteristic
386C2#2024380211yescharacteristic
subgroupG/H
centerC23#13093992
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC2×C47×#397C2
socleC23×C47#120
prime radicalC25×C47D4
cosocleC23×C4×C47C23
hypercenterC2×#397D47
hypocenterC47#9025
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#397

Complements

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

NH ≅ G/Ntype
2. #20243474C2semidirect
4. #20241267C2semidirect
17. C47×#397C4semidirect
40. C2×C4⋊C4×C47C2×C4semidirect
50. #95×C47C2×C4semidirect
205. #397Dic47semidirect
213. C47#9025semidirect
226. C2×C4⋊C4C2×Dic47semidirect
244. #95C2×Dic47semidirect

Maximal subgroups

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

subgroupindex
#202460022
#202434742
C2×#156888472
#202412672
C2×C47×#3972
#902547