Finite Group Databasebeta

← #26112028

Subgroups — #26112028

Order 12032.

1088 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 1088 conjugacy classes of subgroups — too many to draw. Only its 147 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×C4×C22⋊C4×C47C211yesmaximal, characteristic, complemented
6C2×#13093998C412yescomplemented
7C2×#13093998C2211yescharacteristic
8C2×#13093993—21no—
9C2×C22⋊(C47⋊C8)—22no—
10C2×C22⋊(C47⋊C8)C2212yes—
11#15689002—21no—
12#15689002—22no—
13#15688983C412yescomplemented
14#15688983C2211yescharacteristic
15#15688856—24no—
16C24×C4×C47C2211yescharacteristic
17C22×C22⋊C4×C47—21no—
18C22×C42×C47—21no—
19C4×C22⋊C4×C47—24no—
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×C47D411yescharacteristic
47C23×C4×C47D412yes—
48C23×C4×C47C2311yescharacteristic
49C2×C22⋊C4×C47—28no—
50C2×C22⋊C4×C47—24no—
51C2×C42×C47—24no—
52C2×C42×C47—44no—
53#95×C47—22no—
54#95×C47D414yescomplemented
55C24×C47—24no—
56C24×C47—21no—
57C24×C47—21no—
58C24×C47—22no—
59C24×C47—22no—
60C24×C47—42no—
61C24×C47C22⋊C412yes—
62C24×C47C2×D411yescharacteristic
63C23×D47—21no—
64C22×C4×C47—21no—
65C22×C4×C47—21no—
66C22×C4×C47—24no—
67C22×C4×C47—28no—
68C22×C4×C47—24no—
69C22×C4×C47—24no—
70C22×C4×C47—24no—
71C22×C4×C47—24no—
72C22×C4×C47—24no—
73C22×C4×C47—42no—
74C22×C4×C47—42no—
75C22×C4×C47—42no—
76C22×C4×C47—48no—
77C22×C4×C47—48no—
78C22×C4×C47—44no—
79C22×C4×C47—42no—
80C22×C4×C47—42no—
81C22×C4×C47C22⋊C412yes—
82C22×C4×C47C22×C411yescharacteristic
83C22×C4×C47C2×D411yescharacteristic
84C22×C4×C47C2×D412yes—
85C22×C4×C47#4112yes—
86C2×C22⋊D47—44no—
87C2×C22⋊D47—44no—
88C2×C22⋊D47—44no—
89C2×C22⋊D47—41no—
90C2×C22⋊D47—41no—
91C2×C22⋊D47—42no—
92C2×C22⋊D47—42no—
93C22×Dic47—21no—
94C22×Dic47—22no—
95C2×D188—22no—
96C2×D188—42no—
97C22⋊C4×C47—216no—
98C42×C47—48no—
99C22⋊Dic47—44no—
100C22⋊Dic47—42no—
101(C2×D47)⋊C4—44no—
102(C2×D47)⋊C4—42no—
103C4⋊Dic47—22no—
104C4⋊Dic47—42no—
105C2×C47⋊C8—44no—
106C2×C47⋊C8—42no—
107C23×C47—22no—
108C23×C47—22no—
109C23×C47—24no—
110C23×C47—24no—
111C23×C47—22no—
112C23×C47—22no—
113C23×C47—22no—
114C23×C47—22no—
115C23×C47—22no—
116C23×C47—22no—
117C23×C47—48no—
118C23×C47—48no—
119C23×C47—42no—
120C23×C47—42no—
121C23×C47—42no—
122C23×C47—42no—
123C23×C47#10414yes—
124C23×C47C2×C22⋊C411yescharacteristic
125C23×C47#12011yescharacteristic
126C23×C47#12311yescharacteristic
127C23×C47#13512yes—
128C23×C47AGL(1,ℤ/8)12yes—
129C22×D47—41no—
130C22×D47—42no—
131C22×D47—41no—
132C22×D47—41no—
133C22×D47—42no—
134C2×C4×C47—24no—
135C2×C4×C47—24no—
136C2×C4×C47—22no—
137C2×C4×C47—22no—
138C2×C4×C47—22no—
139C2×C4×C47—22no—
140C2×C4×C47—28no—
141C2×C4×C47—42no—
142C2×C4×C47—42no—
143C2×C4×C47—48no—
144C2×C4×C47—48no—
145C2×C4×C47—44no—
146C2×C4×C47—48no—
147C2×C4×C47—48no—
148C2×C4×C47—44no—
149C2×C4×C47#9914yes—
150C2×C4×C47C4×D412yes—
151C2×C4×C47#12112yes—
152C22⋊D47—88no—
153C22⋊D47—82no—
154C22⋊D47—82no—
155C2×Dic47—41no—
156C2×Dic47—42no—
157C2×Dic47—41no—
158C2×Dic47—41no—
159C2×Dic47—42no—
160C2×Dic47—42no—
161D188—82no—
162C47⋊C8—82no—
163#9150—471nomaximal, Sylow
164C22×C47—21no—
165C22×C47—21no—
166C22×C47—21no—
167C22×C47—21no—
168C22×C47—24no—
169C22×C47—24no—
170C22×C47—24no—
171C22×C47—24no—
172C22×C47—24no—
173C22×C47—48no—
174C22×C47—48no—
175C22×C47—42no—
176C22×C47—42no—
177C22×C47—42no—
178C22×C47—42no—
179C22×C47#38611yescharacteristic
180C22×C47C2×#9912yes—
181C22×C47#41611yescharacteristic
182C22×C47#41811yescharacteristic
183C22×C47C2×#10412yes—
184C22×C47#44912yes—
185C22×C47#48212yes—
186C2×D47—81no—
187C2×D47—81no—
188C2×D47—81no—
189C2×D47—82no—
190C2×D47—82no—
191C4×C47—22no—
192C4×C47—22no—
193C4×C47—24no—
194C4×C47—48no—
195Dic47—81no—
196Dic47—82no—
197#2847—471no—
198C2×C4×C22⋊C4D4711yescharacteristic, complemented
199#1779—471no—
200C2×#327—472no—
201#1294—472no—
202C2×C47—22no—
203C2×C47—22no—
204C2×C47—44no—
205C2×C47#150112yes—
206C2×C47#150712yes—
207C2×C47#187511yescharacteristic
208C2×C47#189611yescharacteristic
209C2×C47#189811yescharacteristic
210D47—161no—
211C24×C4C2×D4711yescharacteristic
212C2×#121—471no—
213C2×#121—472no—
214C2×#120—941no—
215C22×C22⋊C4—21no—
216C22×C42—21no—
217C2×C22⋊C8—472no—
218C2×C22⋊C8—942no—
219#386—942no—
220#386—941no—
221#380—472no—
222#380—471no—
223C4×C22⋊C4—24no—
224C2×#95C2×D4712yes—
225#327—944no—
226C47#915011yescharacteristic, complemented, Sylow
227C25C22⋊D4711yescharacteristic
228C22×D4—471no—
229C22×D4—942no—
230C22×D4—941no—
231C23×C4—22no—
232C23×C4—22no—
233C23×C4—22no—
234C23×C4—44no—
235C23×C4C4×D4712yes—
236C23×C4D18812yes—
237C23×C4C22⋊D4711yescharacteristic
238C23×C4C22×D4711yescharacteristic
239C22×C8—942no—
240#121—942no—
241#121—942no—
242#121—1884no—
243#120—1884no—
244C2×C4⋊C4—471no—
245C2×C4⋊C4—472no—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—942no—
249C2×C22⋊C4—941no—
250C2×C22⋊C4—942no—
251C2×C22⋊C4—941no—
252C2×C42—24no—
253C2×C42—44no—
254C22⋊C8—942no—
255C22⋊C8—942no—
256C22⋊C8—1884no—
257#95—22no—
258#95—942no—
259#95—941no—
260#95C22⋊D4714yescomplemented
261C24—24no—
262C24—21no—
263C24—21no—
264C24—22no—
265C24—22no—
266C24—42no—
267C24—941no—
268C24(C2×D47)⋊C412yes—
269C24C2×C22⋊D4711yescharacteristic
270C2×D4—942no—
271C2×D4—1882no—
272C2×D4—1884no—
273C2×D4—1884no—
274C2×D4—1884no—
275C2×D4—1881no—
276C2×D4—1881no—
277C2×D4—1882no—
278C2×D4—1882no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—24no—
282C22×C4—28no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—48no—
292C22×C4—48no—
293C22×C4—44no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—941no—
297C22×C4—942no—
298C22×C4(C2×D47)⋊C412yes—
299C22×C4C2×C4×D4711yescharacteristic
300C22×C4C2×D18811yescharacteristic
301C22×C4#1062535012yes—
302C22×C4C2×C22⋊D4712yes—
303C2×C8—1884no—
304C2×C8—1882no—
305C4⋊C4—942no—
306C4⋊C4—1882no—
307C22⋊C4—216no—
308C22⋊C4—1884no—
309C22⋊C4—1882no—
310C22⋊C4—1884no—
311C22⋊C4—1882no—
312C42—48no—
313C23—22no—
314C23—22no—
315C23—24no—
316C23—24no—
317C23—22no—
318C23—22no—
319C23—22no—
320C23—22no—
321C23—22no—
322C23—22no—
323C23—48no—
324C23—48no—
325C23—42no—
326C23—42no—
327C23—42no—
328C23—42no—
329C23—1881no—
330C23—1882no—
331C23—1881no—
332C23—1881no—
333C23—1882no—
334C23#1309395214yes—
335C23C2×(C2×D47)⋊C411yescharacteristic
336C23#1309399311yescharacteristic
337C23#1309400711yescharacteristic
338C23#1309404412yes—
339C23#1309404912yes—
340D4—3762no—
341D4—3768no—
342D4—3762no—
343D4—3762no—
344C2×C4—24no—
345C2×C4—24no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—28no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—48no—
354C2×C4—48no—
355C2×C4—44no—
356C2×C4—48no—
357C2×C4—48no—
358C2×C4—44no—
359C2×C4—1881no—
360C2×C4—1882no—
361C2×C4—1881no—
362C2×C4—1882no—
363C2×C4—1881no—
364C2×C4—1882no—
365C2×C4#1309393814yes—
366C2×C4C4×C22⋊D4712yes—
367C2×C4#1309399812yes—
368C8—3762no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—24no—
374C22—24no—
375C22—24no—
376C22—24no—
377C22—24no—
378C22—48no—
379C22—48no—
380C22—42no—
381C22—42no—
382C22—42no—
383C22—42no—
384C22—3761no—
385C22—3761no—
386C22—3762no—
387C22—3762no—
388C22—3761no—
389C22#1568900211yescharacteristic
390C22C2×#1309393812yes—
391C22#1568911711yescharacteristic
392C22#1568912711yescharacteristic
393C22C2×#1309395212yes—
394C22#1568926812yes—
395C22#1568944712yes—
396C4—22no—
397C4—24no—
398C4—22no—
399C4—48no—
400C4—3762no—
401C4—3761no—
402C2—22no—
403C2—22no—
404C2—44no—
405C2—7521no—
406C2#2024187712yes—
407C2#2024190312yes—
408C2#2024371011yescharacteristic
409C2#2024383111yescharacteristic
410C2#2024384111yescharacteristic
subgroupG/H
centerC23C2×(C2×D47)⋊C4
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC2×C4×C22⋊C4×C47C2
socleC23×C47C2×C22⋊C4
prime radical#20251670C2
cosocleC23×C4×C47C23
hypercenterC2×C4×C22⋊C4D47
hypocenterC47#9150
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C24×C4 ◁ C2×C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. #20243298C2semidirect
3. C2×#15688856C2semidirect
4. #20241267C2semidirect
5. C2×C4×C22⋊C4×C47C2semidirect
6. C2×#13093998C4semidirect
13. #15688983C4semidirect
54. #95×C47D4semidirect
198. C2×C4×C22⋊C4D47semidirect
226. C47#9150semidirect
260. #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×C4×C22⋊C4×C472
#915047