Finite Group Databasebeta

← #26112068

Subgroups — #26112068

Order 12032.

980 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 980 conjugacy classes of subgroups — too many to draw. Only its 139 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
1#20251670C211yesmaximal, characteristic
2#20243298C211yesmaximal, characteristic, complemented
3C2×#15688856C212yesmaximal, complemented
4#20241267C212yesmaximal, complemented
5C2×C47×#381C211yesmaximal, characteristic, complemented
6C2×#13093998C412yescomplemented
7C2×#13093998C2211yescharacteristic
8C2×#13093993—21no—
9C2×C22⋊(C47⋊C8)—22no—
10C2×C22⋊(C47⋊C8)C2212yes—
11#15689002—22no—
12#15689002—21no—
13#15688983C412yescomplemented
14#15688983C2211yescharacteristic
15#15688856—24no—
16C24×C4×C47C2211yescharacteristic
17C22×C22⋊C4×C47—21no—
18C47×#381—24no—
19C2×#95×C47—21no—
20C2×#95×C47C2212yes—
21C22×D188C2×C411yescharacteristic
22C22×C22⋊D47—22no—
23C22×C22⋊D47—21no—
24C22×C47⋊C8—22no—
25#13093998—22no—
26#13093998—22no—
27#13093998—44no—
28#13093993—44no—
29C2×C4⋊Dic47C2×C411yescharacteristic
30C2×C4⋊Dic47C2×C412yes—
31C2×C22⋊Dic47—22no—
32C2×C22⋊Dic47—21no—
33C2×(C2×D47)⋊C4—22no—
34C2×(C2×D47)⋊C4—21no—
35C22⋊(C47⋊C8)—22no—
36C22⋊(C47⋊C8)—22no—
37C22⋊(C47⋊C8)—44no—
38C47⋊#95—22no—
39C47⋊#95—21no—
40C25×C47D411yescharacteristic
41C23×C4×C47—22no—
42C23×C4×C47—22no—
43C23×C4×C47—22no—
44C23×C4×C47—44no—
45C23×C4×C47C2×C412yes—
46C23×C4×C47D412yes—
47C23×C4×C47D411yescharacteristic
48C23×C4×C47C2311yescharacteristic
49C2×C22⋊C4×C47—24no—
50C2×C22⋊C4×C47—44no—
51#95×C47—24no—
52#95×C47—22no—
53#95×C47D414yescomplemented
54C24×C47—24no—
55C24×C47—21no—
56C24×C47—21no—
57C24×C47—22no—
58C24×C47—22no—
59C24×C47—42no—
60C24×C47C22⋊C412yes—
61C24×C47C2×D411yescharacteristic
62C23×D47—21no—
63C22×C4×C47—24no—
64C22×C4×C47—24no—
65C22×C4×C47—21no—
66C22×C4×C47—21no—
67C22×C4×C47—24no—
68C22×C4×C47—24no—
69C22×C4×C47—24no—
70C22×C4×C47—42no—
71C22×C4×C47—42no—
72C22×C4×C47—42no—
73C22×C4×C47—42no—
74C22×C4×C47—48no—
75C22×C4×C47—48no—
76C22×C4×C47—44no—
77C22×C4×C47—44no—
78C22×C4×C47—42no—
79C22×C4×C47—42no—
80C22×C4×C47C22⋊C412yes—
81C22×C4×C47C22×C411yescharacteristic
82C22×C4×C47C2×D411yescharacteristic
83C22×C4×C47C2×D412yes—
84C22×C4×C47#4112yes—
85C2×C22⋊D47—44no—
86C2×C22⋊D47—44no—
87C2×C22⋊D47—44no—
88C2×C22⋊D47—41no—
89C2×C22⋊D47—41no—
90C2×C22⋊D47—42no—
91C2×C22⋊D47—42no—
92C22×Dic47—21no—
93C22×Dic47—22no—
94C2×D188—22no—
95C2×D188—42no—
96C22⋊C4×C47—48no—
97C22⋊Dic47—44no—
98C22⋊Dic47—42no—
99(C2×D47)⋊C4—44no—
100(C2×D47)⋊C4—42no—
101C4⋊Dic47—22no—
102C4⋊Dic47—42no—
103C2×C47⋊C8—44no—
104C2×C47⋊C8—42no—
105C23×C47—22no—
106C23×C47—22no—
107C23×C47—24no—
108C23×C47—24no—
109C23×C47—22no—
110C23×C47—22no—
111C23×C47—22no—
112C23×C47—22no—
113C23×C47—22no—
114C23×C47—22no—
115C23×C47—48no—
116C23×C47—48no—
117C23×C47—42no—
118C23×C47—42no—
119C23×C47—42no—
120C23×C47—42no—
121C23×C47#9914yes—
122C23×C47C2×C22⋊C411yescharacteristic
123C23×C47#12011yescharacteristic
124C23×C47#12311yescharacteristic
125C23×C47#13512yes—
126C23×C47AGL(1,ℤ/8)12yes—
127C22×D47—41no—
128C22×D47—42no—
129C22×D47—41no—
130C22×D47—41no—
131C22×D47—42no—
132C2×C4×C47—22no—
133C2×C4×C47—22no—
134C2×C4×C47—22no—
135C2×C4×C47—22no—
136C2×C4×C47—22no—
137C2×C4×C47—42no—
138C2×C4×C47—42no—
139C2×C4×C47—48no—
140C2×C4×C47—48no—
141C2×C4×C47—42no—
142C2×C4×C47—42no—
143C2×C4×C47—44no—
144C2×C4×C47—44no—
145C2×C4×C47—44no—
146C2×C4×C47—84no—
147C2×C4×C47—84no—
148C2×C4×C47C4×D412yes—
149C2×C4×C47#12112yes—
150C22⋊D47—88no—
151C22⋊D47—82no—
152C22⋊D47—82no—
153C2×Dic47—41no—
154C2×Dic47—42no—
155C2×Dic47—41no—
156C2×Dic47—41no—
157C2×Dic47—42no—
158C2×Dic47—42no—
159D188—82no—
160C47⋊C8—82no—
161#9158—471nomaximal, Sylow
162C22×C47—21no—
163C22×C47—21no—
164C22×C47—21no—
165C22×C47—21no—
166C22×C47—24no—
167C22×C47—24no—
168C22×C47—24no—
169C22×C47—24no—
170C22×C47—24no—
171C22×C47—48no—
172C22×C47—48no—
173C22×C47—42no—
174C22×C47—42no—
175C22×C47—42no—
176C22×C47—42no—
177C22×C47#38611yescharacteristic
178C22×C47C2×#9912yes—
179C22×C47#41611yescharacteristic
180C22×C47#41811yescharacteristic
181C22×C47#42112yes—
182C22×C47#44912yes—
183C22×C47#48212yes—
184C2×D47—81no—
185C2×D47—81no—
186C2×D47—81no—
187C2×D47—82no—
188C2×D47—82no—
189C4×C47—22no—
190C4×C47—22no—
191C4×C47—42no—
192C4×C47—84no—
193Dic47—81no—
194Dic47—82no—
195#2847—471no—
196C2×#381D4711yescharacteristic, complemented
197#1779—471no—
198C2×#327—472no—
199#1294—472no—
200C2×C47—22no—
201C2×C47—22no—
202C2×C47—44no—
203C2×C47#151012yes—
204C2×C47#151612yes—
205C2×C47#187511yescharacteristic
206C2×C47#189711yescharacteristic
207C2×C47#189811yescharacteristic
208D47—161no—
209C24×C4C2×D4711yescharacteristic
210C2×#121—471no—
211C2×#121—472no—
212C2×#120—941no—
213C22×C22⋊C4—21no—
214C2×C22⋊C8—472no—
215C2×C22⋊C8—942no—
216#386—942no—
217#386—941no—
218#381—24no—
219#380—472no—
220#380—471no—
221C2×#95—21no—
222C2×#95C2×D4712yes—
223#327—944no—
224C47#915811yescharacteristic, complemented, Sylow
225C25C22⋊D4711yescharacteristic
226C22×D4—471no—
227C22×D4—942no—
228C22×D4—941no—
229C23×C4—22no—
230C23×C4—22no—
231C23×C4—22no—
232C23×C4—44no—
233C23×C4C4×D4712yes—
234C23×C4D18812yes—
235C23×C4C22⋊D4711yescharacteristic
236C23×C4C22×D4711yescharacteristic
237C22×C8—942no—
238#121—942no—
239#121—942no—
240#121—1884no—
241#120—1884no—
242C2×C4⋊C4—471no—
243C2×C4⋊C4—472no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—942no—
247C2×C22⋊C4—941no—
248C2×C22⋊C4—942no—
249C2×C22⋊C4—941no—
250C22⋊C8—942no—
251C22⋊C8—942no—
252C22⋊C8—1884no—
253#95—24no—
254#95—22no—
255#95—942no—
256#95—941no—
257#95C22⋊D4714yescomplemented
258C24—24no—
259C24—21no—
260C24—21no—
261C24—22no—
262C24—22no—
263C24—42no—
264C24—941no—
265C24(C2×D47)⋊C412yes—
266C24C2×C22⋊D4711yescharacteristic
267C2×D4—942no—
268C2×D4—1882no—
269C2×D4—1884no—
270C2×D4—1884no—
271C2×D4—1884no—
272C2×D4—1881no—
273C2×D4—1881no—
274C2×D4—1882no—
275C2×D4—1882no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—48no—
288C22×C4—48no—
289C22×C4—44no—
290C22×C4—44no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—941no—
294C22×C4—942no—
295C22×C4(C2×D47)⋊C412yes—
296C22×C4C2×C4×D4711yescharacteristic
297C22×C4C2×D18811yescharacteristic
298C22×C4#1062535012yes—
299C22×C4C2×C22⋊D4712yes—
300C2×C8—1884no—
301C2×C8—1882no—
302C4⋊C4—942no—
303C4⋊C4—1882no—
304C22⋊C4—48no—
305C22⋊C4—1884no—
306C22⋊C4—1884no—
307C22⋊C4—1882no—
308C22⋊C4—1882no—
309C23—22no—
310C23—22no—
311C23—24no—
312C23—24no—
313C23—22no—
314C23—22no—
315C23—22no—
316C23—22no—
317C23—22no—
318C23—22no—
319C23—48no—
320C23—48no—
321C23—42no—
322C23—42no—
323C23—42no—
324C23—42no—
325C23—1881no—
326C23—1882no—
327C23—1881no—
328C23—1881no—
329C23—1882no—
330C23#1309393814yes—
331C23C2×(C2×D47)⋊C411yescharacteristic
332C23#1309399311yescharacteristic
333C23#1309400711yescharacteristic
334C23#1309404412yes—
335C23#1309404912yes—
336D4—3762no—
337D4—3768no—
338D4—3762no—
339D4—3762no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—48no—
348C2×C4—48no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—44no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—84no—
355C2×C4—84no—
356C2×C4—1881no—
357C2×C4—1882no—
358C2×C4—1881no—
359C2×C4—1882no—
360C2×C4—1881no—
361C2×C4—1882no—
362C2×C4C4×C22⋊D4712yes—
363C2×C4#1309399812yes—
364C8—3762no—
365C22—24no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—24no—
371C22—24no—
372C22—24no—
373C22—24no—
374C22—48no—
375C22—48no—
376C22—42no—
377C22—42no—
378C22—42no—
379C22—42no—
380C22—3762no—
381C22—3761no—
382C22—3761no—
383C22—3762no—
384C22—3761no—
385C22#1568900211yescharacteristic
386C22C2×#1309393812yes—
387C22#1568911711yescharacteristic
388C22#1568912711yescharacteristic
389C22#1568914112yes—
390C22#1568926812yes—
391C22#1568944712yes—
392C4—22no—
393C4—22no—
394C4—42no—
395C4—84no—
396C4—3762no—
397C4—3761no—
398C2—22no—
399C2—22no—
400C2—44no—
401C2—7521no—
402C2#2024191612yes—
403C2#2024194212yes—
404C2#2024371011yescharacteristic
405C2#2024383611yescharacteristic
406C2#2024384111yescharacteristic
subgroupG/H
centerC23C2×(C2×D47)⋊C4
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC2×C47×#381C2
socleC23×C47C2×C22⋊C4
prime radical#20251670C2
cosocleC23×C4×C47C23
hypercenterC2×#381D47
hypocenterC47#9158
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C24×C4 ◁ C2×#381

Complements

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

NH ≅ G/Ntype
2. #20243298C2semidirect
3. C2×#15688856C2semidirect
4. #20241267C2semidirect
5. C2×C47×#381C2semidirect
6. C2×#13093998C4semidirect
13. #15688983C4semidirect
53. #95×C47D4semidirect
196. C2×#381D47semidirect
224. C47#9158semidirect
257. #95C22⋊D47semidirect

Maximal subgroups

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

subgroupindex
#202516702
#202432982
C2×#156888562
#202412672
C2×C47×#3812
#915847