Finite Group Databasebeta

← #26126754

Subgroups — #26126754

Order 12032.

700 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 700 conjugacy classes of subgroups — too many to draw. Only its 95 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
1C2×#15688890C211yesmaximal, characteristic, complemented
2#20241715C211yesmaximal, characteristic, complemented
3#20241412C212yesmaximal, complemented
4#20241385C212yesmaximal, complemented
5C47×#2539C211yesmaximal, characteristic
6C22×C22⋊Dic47—21no—
7#15689087—21no—
8#15688890—21no—
9#15688890C2212yes—
10#15688847—21no—
11#15688847C2212yes—
12C2×#123×C47—21no—
13C2×#121×C47C411yescharacteristic, complemented
14C2×#120×C47—21no—
15C22×C22⋊C4×C47C2211yescharacteristic
16C47×#399C411yescharacteristic, complemented
17C47×#399C2211yescharacteristic
18C47×#392C411yescharacteristic, complemented
19C47×#392C2211yescharacteristic
20C47×#390—21no—
21C47×#390—21no—
22C4×C22⋊C4×C47C411yescharacteristic, complemented
23C23×Dic47—41no—
24C2×#10625329—41no—
25C2×C22⋊Dic47—24no—
26C2×C22⋊Dic47—22no—
27C2×C22⋊Dic47—41no—
28C2×C22⋊Dic47—42no—
29C22⋊(C47⋊C8)—22no—
30C22⋊(C47⋊C8)—41no—
31C25×C47D411yescharacteristic
32C22×D4×C47—21no—
33C22×D4×C47—21no—
34C22×D4×C47—21no—
35C22×D4×C47C2×C411yescharacteristic, complemented
36C23×C4×C47C2×C411yescharacteristic
37C23×C4×C47D411yescharacteristic
38#123×C47—44no—
39#121×C47—41no—
40#121×C47—41no—
41#120×C47—44no—
42C2×C4⋊C4×C47—21no—
43C2×C4⋊C4×C47—21no—
44C2×C4⋊C4×C47C2×C411yescharacteristic, complemented
45C2×C22⋊C4×C47—21no—
46C2×C22⋊C4×C47—21no—
47C2×C22⋊C4×C47—21no—
48C2×C22⋊C4×C47—21no—
49C2×C22⋊C4×C47—21no—
50C2×C22⋊C4×C47—21no—
51C2×C22⋊C4×C47—41no—
52C2×C22⋊C4×C47—41no—
53C2×C22⋊C4×C47C2×C411yescharacteristic
54C2×C22⋊C4×C47D411yescharacteristic
55C2×C22⋊C4×C47D411yescharacteristic
56C2×C22⋊C4×C47C2311yescharacteristic
57C2×C42×C47—21no—
58#95×C47C2×C411yescharacteristic, complemented
59#95×C47C2×C411yescharacteristic, complemented
60C24×C47—21no—
61C24×C47—21no—
62C24×C47—22no—
63C24×C47—21no—
64C24×C47—21no—
65C24×C47—21no—
66C24×C47—42no—
67C24×C47—81no—
68C24×C47C22⋊C411yescharacteristic
69C24×C47C22⋊C411yescharacteristic
70C24×C47C2×D411yescharacteristic
71C2×D4×C47—42no—
72C2×D4×C47—42no—
73C2×D4×C47—41no—
74C2×D4×C47—42no—
75C2×D4×C47—42no—
76C2×D4×C47—41no—
77C2×D4×C47—41no—
78C2×D4×C47—42no—
79C2×D4×C47—41no—
80C2×D4×C47—41no—
81C2×D4×C47—81no—
82C2×D4×C47—81no—
83C2×D4×C47—81no—
84C22×C4×C47—21no—
85C22×C4×C47—21no—
86C22×C4×C47—21no—
87C22×C4×C47—21no—
88C22×C4×C47—21no—
89C22×C4×C47—21no—
90C22×C4×C47—21no—
91C22×C4×C47—42no—
92C22×C4×C47—81no—
93C22×C4×C47C22⋊C411yescharacteristic
94C22×C4×C47C22⋊C411yescharacteristic
95C22×C4×C47C22×C411yescharacteristic
96C22×C4×C47C2×D411yescharacteristic
97C22×Dic47—44no—
98C22×Dic47—42no—
99C22×Dic47—82no—
100C22×Dic47—82no—
101C4⋊C4×C47—42no—
102C4⋊C4×C47—42no—
103C4⋊C4×C47—41no—
104C22⋊C4×C47—24no—
105C22⋊C4×C47—41no—
106C22⋊C4×C47—41no—
107C22⋊C4×C47—41no—
108C22⋊C4×C47—41no—
109C22⋊C4×C47—42no—
110C22⋊C4×C47—42no—
111C22⋊C4×C47—42no—
112C22⋊C4×C47—42no—
113C22⋊C4×C47—42no—
114C42×C47—81no—
115C22⋊Dic47—24no—
116C22⋊Dic47—44no—
117C22⋊Dic47—42no—
118#10625329—82no—
119C2×C47⋊C8—42no—
120C23×C47—22no—
121C23×C47—22no—
122C23×C47—22no—
123C23×C47—22no—
124C23×C47—22no—
125C23×C47—21no—
126C23×C47—21no—
127C23×C47—21no—
128C23×C47—21no—
129C23×C47—21no—
130C23×C47—21no—
131C23×C47—41no—
132C23×C47—42no—
133C23×C47—41no—
134C23×C47—41no—
135C23×C47—41no—
136C23×C47—41no—
137C23×C47—41no—
138C23×C47—41no—
139C23×C47—41no—
140C23×C47—41no—
141C23×C47—42no—
142C23×C47—41no—
143C23×C47—41no—
144C23×C47—41no—
145C23×C47—81no—
146C23×C47—81no—
147C23×C47—84no—
148C23×C47—82no—
149C23×C47—82no—
150C23×C47#9911yescharacteristic
151C23×C47#9911yescharacteristic
152C23×C47#10412yes—
153C23×C47C2×C22⋊C411yescharacteristic
154C23×C47AGL(1,ℤ/8)11yescharacteristic
155C23×C47#13711yescharacteristic
156D4×C47—84no—
157D4×C47—82no—
158D4×C47—82no—
159D4×C47—82no—
160D4×C47—82no—
161D4×C47—82no—
162C2×C4×C47—22no—
163C2×C4×C47—22no—
164C2×C4×C47—21no—
165C2×C4×C47—21no—
166C2×C4×C47—21no—
167C2×C4×C47—21no—
168C2×C4×C47—41no—
169C2×C4×C47—41no—
170C2×C4×C47—42no—
171C2×C4×C47—41no—
172C2×C4×C47—41no—
173C2×C4×C47—41no—
174C2×C4×C47—41no—
175C2×C4×C47—41no—
176C2×C4×C47—41no—
177C2×C4×C47—41no—
178C2×C4×C47—41no—
179C2×C4×C47—41no—
180C2×C4×C47—81no—
181C2×C4×C47—81no—
182C2×C4×C47—81no—
183C2×Dic47—44no—
184C2×Dic47—88no—
185C2×Dic47—82no—
186C2×Dic47—82no—
187C47⋊C8—82no—
188#11757—471nomaximal, Sylow
189C22×C47—21no—
190C22×C47—21no—
191C22×C47—21no—
192C22×C47—21no—
193C22×C47—21no—
194C22×C47—21no—
195C22×C47—21no—
196C22×C47—21no—
197C22×C47—22no—
198C22×C47—21no—
199C22×C47—42no—
200C22×C47—42no—
201C22×C47—42no—
202C22×C47—41no—
203C22×C47—41no—
204C22×C47—44no—
205C22×C47—44no—
206C22×C47—41no—
207C22×C47—41no—
208C22×C47—41no—
209C22×C47—41no—
210C22×C47—81no—
211C22×C47—81no—
212C22×C47—81no—
213C22×C47—81no—
214C22×C47—81no—
215C22×C47—81no—
216C22×C47—82no—
217C22×C47—82no—
218C22×C47—81no—
219C22×C47—81no—
220C22×C47—81no—
221C22×C47#35312yes—
222C22×C47C2×#9911yescharacteristic
223C22×C47#41711yescharacteristic
224C22×C47C2×#10411yescharacteristic
225C4×C47—42no—
226C4×C47—42no—
227C4×C47—41no—
228C4×C47—41no—
229C4×C47—41no—
230C4×C47—41no—
231C4×C47—81no—
232Dic47—84no—
233#2539D4711yescharacteristic
234C2×#342—471no—
235#1462—471no—
236#1352—472no—
237#1343—472no—
238C2×C47—21no—
239C2×C47—21no—
240C2×C47—22no—
241C2×C47—41no—
242C2×C47—41no—
243C2×C47—42no—
244C2×C47—41no—
245C2×C47—161no—
246C2×C47#150411yescharacteristic
247C2×C47#212111yescharacteristic
248C2×C47C2×#35311yescharacteristic
249C2×#123—21no—
250C2×#121Dic4711yescharacteristic, complemented
251C2×#120—21no—
252C22×C22⋊C4—941no—
253C22×C22⋊C4C2×D4711yescharacteristic
254#407—941no—
255#399Dic4711yescharacteristic, complemented
256#399C2×D4711yescharacteristic
257#392Dic4711yescharacteristic, complemented
258#392C2×D4711yescharacteristic
259#390—21no—
260#390—21no—
261C4×C22⋊C4Dic4711yescharacteristic, complemented
262#342—472no—
263#342—941no—
264#323—472no—
265#323—941no—
266C47#1175711yescharacteristic, complemented, Sylow
267C25C22⋊D4711yescharacteristic
268C22×D4—21no—
269C22×D4—21no—
270C22×D4—21no—
271C22×D4C2×Dic4711yescharacteristic, complemented
272C23×C4—1881no—
273C23×C4C2×Dic4711yescharacteristic
274C23×C4C22⋊D4711yescharacteristic
275C2×M16—1881no—
276#123—44no—
277#121—41no—
278#121—41no—
279#120—44no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4C2×Dic4711yescharacteristic, complemented
283C2×C22⋊C4—21no—
284C2×C22⋊C4—21no—
285C2×C22⋊C4—21no—
286C2×C22⋊C4—21no—
287C2×C22⋊C4—21no—
288C2×C22⋊C4—21no—
289C2×C22⋊C4—41no—
290C2×C22⋊C4—41no—
291C2×C22⋊C4—944no—
292C2×C22⋊C4—942no—
293C2×C22⋊C4—1881no—
294C2×C22⋊C4—1882no—
295C2×C22⋊C4C2×Dic4711yescharacteristic
296C2×C22⋊C4C22⋊D4711yescharacteristic
297C2×C22⋊C4C22⋊D4711yescharacteristic
298C2×C22⋊C4C22×D4711yescharacteristic
299C2×C42—21no—
300C22⋊C8—942no—
301C22⋊C8—1881no—
302#95C2×Dic4711yescharacteristic, complemented
303#95C2×Dic4711yescharacteristic, complemented
304C24—21no—
305C24—21no—
306C24—22no—
307C24—21no—
308C24—21no—
309C24—21no—
310C24—42no—
311C24—81no—
312C24C22⋊Dic4711yescharacteristic
313C24C22⋊Dic4711yescharacteristic
314C24C2×C22⋊D4711yescharacteristic
315C2×D4—41no—
316C2×D4—42no—
317C2×D4—42no—
318C2×D4—42no—
319C2×D4—42no—
320C2×D4—41no—
321C2×D4—41no—
322C2×D4—42no—
323C2×D4—41no—
324C2×D4—41no—
325C2×D4—81no—
326C2×D4—81no—
327C2×D4—81no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—42no—
336C22×C4—81no—
337C22×C4—3762no—
338C22×C4—3762no—
339C22×C4—1884no—
340C22×C4—1882no—
341C22×C4C22⋊Dic4711yescharacteristic
342C22×C4C22⋊Dic4711yescharacteristic
343C22×C4C22×Dic4711yescharacteristic
344C22×C4C2×C22⋊D4711yescharacteristic
345M16—3762no—
346C2×C8—1882no—
347C4⋊C4—41no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C22⋊C4—24no—
351C22⋊C4—41no—
352C22⋊C4—41no—
353C22⋊C4—41no—
354C22⋊C4—41no—
355C22⋊C4—42no—
356C22⋊C4—42no—
357C22⋊C4—42no—
358C22⋊C4—42no—
359C22⋊C4—42no—
360C22⋊C4—944no—
361C22⋊C4—1884no—
362C22⋊C4—1882no—
363C42—81no—
364C23—22no—
365C23—22no—
366C23—22no—
367C23—22no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—22no—
373C23—21no—
374C23—21no—
375C23—41no—
376C23—42no—
377C23—41no—
378C23—41no—
379C23—41no—
380C23—41no—
381C23—41no—
382C23—41no—
383C23—41no—
384C23—41no—
385C23—42no—
386C23—41no—
387C23—41no—
388C23—41no—
389C23—81no—
390C23—81no—
391C23—84no—
392C23—82no—
393C23—82no—
394C23#1309393911yescharacteristic
395C23#1309393911yescharacteristic
396C23#1309395112yes—
397C23C2×C22⋊Dic4711yescharacteristic
398C23#1309404911yescharacteristic
399C23#1309405411yescharacteristic
400D4—84no—
401D4—82no—
402D4—82no—
403D4—82no—
404D4—82no—
405D4—82no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—42no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—81no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4—3768no—
428C2×C4—3762no—
429C2×C4—3762no—
430C2×C4—1884no—
431C8—3762no—
432C22—21no—
433C22—21no—
434C22—21no—
435C22—21no—
436C22—21no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—22no—
441C22—21no—
442C22—42no—
443C22—42no—
444C22—42no—
445C22—41no—
446C22—41no—
447C22—44no—
448C22—44no—
449C22—41no—
450C22—41no—
451C22—41no—
452C22—41no—
453C22—81no—
454C22—81no—
455C22—81no—
456C22—81no—
457C22—81no—
458C22—81no—
459C22—82no—
460C22—82no—
461C22—81no—
462C22—81no—
463C22—81no—
464C22#1568891612yes—
465C22C2×#1309393911yescharacteristic
466C22#1568912411yescharacteristic
467C22C2×#1309395111yescharacteristic
468C4—42no—
469C4—42no—
470C4—41no—
471C4—41no—
472C4—41no—
473C4—41no—
474C4—81no—
475C4—3764no—
476C2—21no—
477C2—21no—
478C2—22no—
479C2—41no—
480C2—41no—
481C2—42no—
482C2—41no—
483C2—161no—
484C2#2024189211yescharacteristic
485C2#2024503411yescharacteristic
486C2C2×#1568891611yescharacteristic
subgroupG/H
centerC22C2×#13093939
commutatorC22×C4×C47C22×C4
FrattiniC2×C22⋊C4C22×D47
Fitting, prime radicalC47×#2539C2
socleC22×C47C2×#99
cosocleC2×C22⋊C4×C47C23
hypercenter#2539D47
hypocenterC47#11757
LCSG ▷ C22×C4×C47 ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ #2539

Complements

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

NH ≅ G/Ntype
1. C2×#15688890C2semidirect
2. #20241715C2semidirect
3. #20241412C2semidirect
4. #20241385C2semidirect
13. C2×#121×C47C4semidirect
16. C47×#399C4semidirect
18. C47×#392C4semidirect
22. C4×C22⋊C4×C47C4semidirect
35. C22×D4×C47C2×C4semidirect
44. C2×C4⋊C4×C47C2×C4semidirect
58. #95×C47C2×C4semidirect
59. #95×C47C2×C4semidirect
250. C2×#121Dic47semidirect
255. #399Dic47semidirect
257. #392Dic47semidirect
261. C4×C22⋊C4Dic47semidirect
266. C47#11757semidirect
271. C22×D4C2×Dic47semidirect
282. C2×C4⋊C4C2×Dic47semidirect
302. #95C2×Dic47semidirect
303. #95C2×Dic47semidirect

Maximal subgroups

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

subgroupindex
C2×#156888902
#202417152
#202414122
#202413852
C47×#25392
#1175747