Finite Group Databasebeta

← #26126744

Subgroups — #26126744

Order 12032.

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

normal subgroups only

This group has 586 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
2#20241715C211yesmaximal, characteristic
3#20241412C212yesmaximal, complemented
4#20241388C212yesmaximal, complemented
5C47×#2482C211yesmaximal, characteristic
6C22×C22⋊Dic47—21no—
7#15689087—21no—
8#15688890—21no—
9#15688890C2212yes—
10#15688847—21no—
11#15688847C2212yes—
12C2×#117×C47C411yescharacteristic, complemented
13C22×C22⋊C4×C47C2211yescharacteristic
14C47×#399C411yescharacteristic, complemented
15C47×#399C2211yescharacteristic
16C47×#397C411yescharacteristic, complemented
17C47×#397C2211yescharacteristic
18C47×#388—21no—
19C47×#388—21no—
20C47×#385—21no—
21C47×#381C411yescharacteristic, complemented
22C47×#379—21no—
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
32C23×C4×C47C2×C411yescharacteristic
33C23×C4×C47D411yescharacteristic
34#117×C47—41no—
35#117×C47—41no—
36C2×C4⋊C4×C47—21no—
37C2×C4⋊C4×C47—21no—
38C2×C4⋊C4×C47C2×C411yescharacteristic, complemented
39C2×C4⋊C4×C47C2×C411yescharacteristic, complemented
40C2×C22⋊C4×C47—21no—
41C2×C22⋊C4×C47—21no—
42C2×C22⋊C4×C47—21no—
43C2×C22⋊C4×C47—21no—
44C2×C22⋊C4×C47—21no—
45C2×C22⋊C4×C47—21no—
46C2×C22⋊C4×C47—41no—
47C2×C22⋊C4×C47—41no—
48C2×C22⋊C4×C47C2×C411yescharacteristic
49C2×C22⋊C4×C47D411yescharacteristic
50C2×C22⋊C4×C47D411yescharacteristic
51C2×C22⋊C4×C47C2311yescharacteristic
52C2×C42×C47—21no—
53#95×C47—21no—
54#95×C47—21no—
55#95×C47—21no—
56#95×C47C2×C411yescharacteristic, complemented
57#95×C47C2×C411yescharacteristic, complemented
58C24×C47—21no—
59C24×C47—21no—
60C24×C47—22no—
61C24×C47—21no—
62C24×C47—21no—
63C24×C47—42no—
64C24×C47—81no—
65C24×C47C22⋊C411yescharacteristic
66C24×C47C22⋊C411yescharacteristic
67C24×C47C2×D411yescharacteristic
68C22×C4×C47—21no—
69C22×C4×C47—21no—
70C22×C4×C47—21no—
71C22×C4×C47—21no—
72C22×C4×C47—21no—
73C22×C4×C47—21no—
74C22×C4×C47—21no—
75C22×C4×C47—21no—
76C22×C4×C47—42no—
77C22×C4×C47—81no—
78C22×C4×C47C22⋊C411yescharacteristic
79C22×C4×C47C22⋊C411yescharacteristic
80C22×C4×C47C22×C411yescharacteristic
81C22×C4×C47C2×D411yescharacteristic
82C22×Dic47—44no—
83C22×Dic47—42no—
84C22×Dic47—82no—
85C22×Dic47—82no—
86C4⋊C4×C47—41no—
87C4⋊C4×C47—41no—
88C4⋊C4×C47—81no—
89C4⋊C4×C47—81no—
90C22⋊C4×C47—41no—
91C22⋊C4×C47—41no—
92C22⋊C4×C47—41no—
93C22⋊C4×C47—41no—
94C22⋊C4×C47—42no—
95C22⋊C4×C47—41no—
96C22⋊C4×C47—41no—
97C22⋊C4×C47—42no—
98C22⋊C4×C47—41no—
99C22⋊C4×C47—41no—
100C22⋊C4×C47—81no—
101C22⋊C4×C47—81no—
102C42×C47—42no—
103C22⋊Dic47—24no—
104C22⋊Dic47—44no—
105C22⋊Dic47—42no—
106#10625329—82no—
107C2×C47⋊C8—42no—
108C23×C47—22no—
109C23×C47—22no—
110C23×C47—22no—
111C23×C47—22no—
112C23×C47—22no—
113C23×C47—21no—
114C23×C47—21no—
115C23×C47—21no—
116C23×C47—21no—
117C23×C47—21no—
118C23×C47—21no—
119C23×C47—41no—
120C23×C47—42no—
121C23×C47—41no—
122C23×C47—41no—
123C23×C47—41no—
124C23×C47—41no—
125C23×C47—42no—
126C23×C47—81no—
127C23×C47—81no—
128C23×C47—84no—
129C23×C47—82no—
130C23×C47—82no—
131C23×C47#9911yescharacteristic
132C23×C47#9911yescharacteristic
133C23×C47#10412yes—
134C23×C47C2×C22⋊C411yescharacteristic
135C23×C47AGL(1,ℤ/8)11yescharacteristic
136C23×C47#13711yescharacteristic
137C2×C4×C47—21no—
138C2×C4×C47—21no—
139C2×C4×C47—21no—
140C2×C4×C47—21no—
141C2×C4×C47—42no—
142C2×C4×C47—42no—
143C2×C4×C47—42no—
144C2×C4×C47—41no—
145C2×C4×C47—41no—
146C2×C4×C47—41no—
147C2×C4×C47—41no—
148C2×C4×C47—41no—
149C2×C4×C47—41no—
150C2×C4×C47—41no—
151C2×C4×C47—41no—
152C2×C4×C47—41no—
153C2×C4×C47—41no—
154C2×C4×C47—41no—
155C2×C4×C47—41no—
156C2×C4×C47—41no—
157C2×C4×C47—41no—
158C2×C4×C47—41no—
159C2×C4×C47—41no—
160C2×C4×C47—81no—
161C2×Dic47—44no—
162C2×Dic47—88no—
163C2×Dic47—82no—
164C2×Dic47—82no—
165C47⋊C8—82no—
166#11755—471nomaximal, Sylow
167C22×C47—21no—
168C22×C47—21no—
169C22×C47—21no—
170C22×C47—21no—
171C22×C47—21no—
172C22×C47—21no—
173C22×C47—21no—
174C22×C47—21no—
175C22×C47—22no—
176C22×C47—21no—
177C22×C47—42no—
178C22×C47—42no—
179C22×C47—42no—
180C22×C47—41no—
181C22×C47—41no—
182C22×C47—44no—
183C22×C47—44no—
184C22×C47—41no—
185C22×C47—41no—
186C22×C47—41no—
187C22×C47—41no—
188C22×C47—81no—
189C22×C47—81no—
190C22×C47—82no—
191C22×C47—82no—
192C22×C47#35212yes—
193C22×C47C2×#9911yescharacteristic
194C22×C47#41711yescharacteristic
195C22×C47C2×#10411yescharacteristic
196C4×C47—41no—
197C4×C47—41no—
198C4×C47—41no—
199C4×C47—41no—
200C4×C47—81no—
201C4×C47—81no—
202C4×C47—81no—
203C4×C47—81no—
204Dic47—84no—
205#2482D4711yescharacteristic
206C2×#342—471no—
207#1462—471no—
208#1352—472no—
209#1344—472no—
210C2×C47—21no—
211C2×C47—21no—
212C2×C47—22no—
213C2×C47—41no—
214C2×C47—41no—
215C2×C47—42no—
216C2×C47—41no—
217C2×C47#150411yescharacteristic
218C2×C47C2×#35211yescharacteristic
219C2×C47#212711yescharacteristic
220C2×#117Dic4711yescharacteristic, complemented
221C22×C22⋊C4—941no—
222C22×C22⋊C4C2×D4711yescharacteristic
223#407—941no—
224#399Dic4711yescharacteristic, complemented
225#399C2×D4711yescharacteristic
226#397Dic4711yescharacteristic, complemented
227#397C2×D4711yescharacteristic
228#388—21no—
229#388—21no—
230#385—21no—
231#381Dic4711yescharacteristic, complemented
232#379—21no—
233#342—472no—
234#342—941no—
235#323—472no—
236#323—941no—
237C47#1175511yescharacteristic, complemented, Sylow
238C25C22⋊D4711yescharacteristic
239C23×C4—1881no—
240C23×C4C2×Dic4711yescharacteristic
241C23×C4C22⋊D4711yescharacteristic
242C2×M16—1881no—
243#117—41no—
244#117—41no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4C2×Dic4711yescharacteristic, complemented
248C2×C4⋊C4C2×Dic4711yescharacteristic, complemented
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—41no—
256C2×C22⋊C4—41no—
257C2×C22⋊C4—944no—
258C2×C22⋊C4—942no—
259C2×C22⋊C4—1881no—
260C2×C22⋊C4—1882no—
261C2×C22⋊C4C2×Dic4711yescharacteristic
262C2×C22⋊C4C22⋊D4711yescharacteristic
263C2×C22⋊C4C22⋊D4711yescharacteristic
264C2×C22⋊C4C22×D4711yescharacteristic
265C2×C42—21no—
266C22⋊C8—942no—
267C22⋊C8—1881no—
268#95—21no—
269#95—21no—
270#95—21no—
271#95C2×Dic4711yescharacteristic, complemented
272#95C2×Dic4711yescharacteristic, complemented
273C24—21no—
274C24—21no—
275C24—22no—
276C24—21no—
277C24—21no—
278C24—42no—
279C24—81no—
280C24C22⋊Dic4711yescharacteristic
281C24C22⋊Dic4711yescharacteristic
282C24C2×C22⋊D4711yescharacteristic
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—42no—
292C22×C4—81no—
293C22×C4—3762no—
294C22×C4—3762no—
295C22×C4—1884no—
296C22×C4—1882no—
297C22×C4C22⋊Dic4711yescharacteristic
298C22×C4C22⋊Dic4711yescharacteristic
299C22×C4C22×Dic4711yescharacteristic
300C22×C4C2×C22⋊D4711yescharacteristic
301M16—3762no—
302C2×C8—1882no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—81no—
306C4⋊C4—81no—
307C22⋊C4—41no—
308C22⋊C4—41no—
309C22⋊C4—41no—
310C22⋊C4—41no—
311C22⋊C4—42no—
312C22⋊C4—41no—
313C22⋊C4—41no—
314C22⋊C4—42no—
315C22⋊C4—41no—
316C22⋊C4—41no—
317C22⋊C4—81no—
318C22⋊C4—81no—
319C22⋊C4—944no—
320C22⋊C4—1884no—
321C22⋊C4—1882no—
322C42—42no—
323C23—22no—
324C23—22no—
325C23—22no—
326C23—22no—
327C23—21no—
328C23—21no—
329C23—21no—
330C23—21no—
331C23—22no—
332C23—21no—
333C23—21no—
334C23—41no—
335C23—42no—
336C23—41no—
337C23—41no—
338C23—41no—
339C23—41no—
340C23—42no—
341C23—81no—
342C23—81no—
343C23—84no—
344C23—82no—
345C23—82no—
346C23#1309393911yescharacteristic
347C23#1309393911yescharacteristic
348C23#1309395112yes—
349C23C2×C22⋊Dic4711yescharacteristic
350C23#1309404911yescharacteristic
351C23#1309405411yescharacteristic
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—81no—
376C2×C4—3768no—
377C2×C4—3762no—
378C2×C4—3762no—
379C2×C4—1884no—
380C8—3762no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—22no—
390C22—21no—
391C22—42no—
392C22—42no—
393C22—42no—
394C22—41no—
395C22—41no—
396C22—44no—
397C22—44no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—81no—
403C22—81no—
404C22—82no—
405C22—82no—
406C22#1568891312yes—
407C22C2×#1309393911yescharacteristic
408C22#1568912411yescharacteristic
409C22C2×#1309395111yescharacteristic
410C4—41no—
411C4—41no—
412C4—41no—
413C4—41no—
414C4—81no—
415C4—81no—
416C4—81no—
417C4—81no—
418C4—3764no—
419C2—21no—
420C2—21no—
421C2—22no—
422C2—41no—
423C2—41no—
424C2—42no—
425C2—41no—
426C2#2024189211yescharacteristic
427C2C2×#1568891311yescharacteristic
428C2#2024505711yescharacteristic
subgroupG/H
centerC22C2×#13093939
commutatorC22×C4×C47C22×C4
FrattiniC2×C22⋊C4C22×D47
FittingC47×#2482C2
socleC22×C47C2×#99
prime radicalC25×C47D4
cosocleC2×C22⋊C4×C47C23
hypercenter#2482D47
hypocenterC47#11755
LCSG ▷ C22×C4×C47 ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ #2482

Complements

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

NH ≅ G/Ntype
3. #20241412C2semidirect
4. #20241388C2semidirect
12. C2×#117×C47C4semidirect
14. C47×#399C4semidirect
16. C47×#397C4semidirect
21. C47×#381C4semidirect
38. C2×C4⋊C4×C47C2×C4semidirect
39. C2×C4⋊C4×C47C2×C4semidirect
56. #95×C47C2×C4semidirect
57. #95×C47C2×C4semidirect
220. C2×#117Dic47semidirect
224. #399Dic47semidirect
226. #397Dic47semidirect
231. #381Dic47semidirect
237. C47#11755semidirect
247. C2×C4⋊C4C2×Dic47semidirect
248. C2×C4⋊C4C2×Dic47semidirect
271. #95C2×Dic47semidirect
272. #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
#202413882
C47×#24822
#1175547