Finite Group Databasebeta

← #26134451

Subgroups — #26134451

Order 12032.

466 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689199C211yesmaximal, characteristic
2#20245143C211yesmaximal, characteristic, complemented
3C2×#15688905C211yesmaximal, characteristic, complemented
4C2×#15688904C211yesmaximal, characteristic, complemented
5C2×#15688904C211yesmaximal, characteristic, complemented
6M32×Dic47C211yesmaximal, characteristic, complemented
7C2×C47×#349C211yesmaximal, characteristic, complemented
8C22×#10625324C2211yescharacteristic
9C2×C4×C22⋊D47C411yescharacteristic
10C2×#13093964C2211yescharacteristic
11C2×#13093964C2211yescharacteristic
12#15689199—21no—
13#15689199—21no—
14#15689199—21no—
15#15689199—21no—
16C2×Dic47⋊C8C411yescharacteristic
17C2×C22⋊(C47⋊C8)C411yescharacteristic
18C2×(C2×D47)⋊C8C411yescharacteristic
19C2×C8×Dic47C2211yescharacteristic
20Dic47⋊C16—21no—
21Dic47⋊C16—21no—
22#15688905—22no—
23#15688904—22no—
24#15688904—22no—
25#15688898—21no—
26C16×Dic47—21no—
27C22×M16×C47C2211yescharacteristic
28C2×M32×C47C2211yescharacteristic
29C2×M32×C47C2211yescharacteristic
30C47×#349—22no—
31C22×C22⋊D47C811yescharacteristic
32C22×C4×D47C2×C411yescharacteristic
33C2×#10625324—21no—
34C2×#10625324—21no—
35C2×#10625324—21no—
36C2×#10625324—21no—
37C2×#10625324—21no—
38C2×#10625324—21no—
39C22×C47⋊C8C2×C411yescharacteristic
40C22×C47⋊C8C2×C411yescharacteristic
41C4×C22⋊D47—21no—
42C4×C22⋊D47—21no—
43C4×C22⋊D47—21no—
44C4×C22⋊D47—21no—
45C2×Dic47⋊C4C811yescharacteristic
46C2×C22⋊Dic47C811yescharacteristic
47C2×(C2×D47)⋊C4C811yescharacteristic
48C2×C4×Dic47C2×C411yescharacteristic
49#13093964—21no—
50#13093964—21no—
51#13093964—21no—
52#13093964D412yescomplemented
53C2×C47⋊C16—21no—
54C2×C47⋊C16—21no—
55Dic47⋊C8—21no—
56Dic47⋊C8—21no—
57C22⋊(C47⋊C8)—21no—
58C22⋊(C47⋊C8)—21no—
59(C2×D47)⋊C8—21no—
60(C2×D47)⋊C8—21no—
61C8×Dic47D411yescharacteristic
62C8×Dic47D411yescharacteristic
63C8×Dic47D411yescharacteristic
64C8×Dic47D411yescharacteristic
65C23×C4×C47C2×C411yescharacteristic
66C2×M16×C47—21no—
67C2×M16×C47—21no—
68C2×M16×C47—21no—
69C2×M16×C47—21no—
70C2×M16×C47—21no—
71C2×M16×C47—21no—
72C22×C8×C47C2×C411yescharacteristic
73C22×C8×C47C2311yescharacteristic
74M32×C47—21no—
75M32×C47—21no—
76M32×C47—21no—
77M32×C47D412yescomplemented
78C2×C16×C47—21no—
79C2×C16×C47—21no—
80C24×C47C2×C811yescharacteristic
81C23×D47C2×C811yescharacteristic
82C22×C4×C47—21no—
83C22×C4×C47—21no—
84C22×C4×C47—21no—
85C22×C4×C47—21no—
86C22×C4×C47—21no—
87C22×C4×C47—21no—
88C22×C4×C47C2×C811yescharacteristic
89C22×C4×C47C22×C411yescharacteristic
90C2×C22⋊D47—21no—
91C2×C22⋊D47—21no—
92C2×C22⋊D47—42no—
93C22×Dic47C2×C811yescharacteristic
94C22×Dic47C2×C811yescharacteristic
95C22×Dic47C2×C811yescharacteristic
96C2×C4×D47—21no—
97C2×C4×D47—21no—
98C2×C4×D47—21no—
99C2×C4×D47—21no—
100C2×C4×D47—21no—
101C2×C4×D47—21no—
102M16×C47—41no—
103M16×C47—41no—
104M16×C47—41no—
105M16×C47—41no—
106C2×C8×C47—21no—
107C2×C8×C47—21no—
108C2×C8×C47—21no—
109C2×C8×C47C2×D411yescharacteristic
110C2×C8×C47C2×D411yescharacteristic
111C2×C8×C47C2×D411yescharacteristic
112C2×C8×C47C2×D411yescharacteristic
113C2×C8×C47#4111yescharacteristic
114C2×C8×C47#4111yescharacteristic
115C22⋊Dic47—21no—
116C22⋊Dic47—21no—
117(C2×D47)⋊C4—21no—
118(C2×D47)⋊C4—21no—
119Dic47⋊C4—21no—
120Dic47⋊C4—21no—
121C4×Dic47C22⋊C411yescharacteristic
122C4×Dic47C22⋊C411yescharacteristic
123C4×Dic47M1611yescharacteristic
124C4×Dic47M1611yescharacteristic
125C2×C47⋊C8—21no—
126C2×C47⋊C8—21no—
127C2×C47⋊C8—21no—
128C2×C47⋊C8—21no—
129C2×C47⋊C8—21no—
130C2×C47⋊C8C22⋊C411yescharacteristic
131C2×C47⋊C8C22⋊C411yescharacteristic
132#10625324—41no—
133#10625324—41no—
134#10625324—41no—
135#10625324—41no—
136C16×C47—22no—
137C16×C47—41no—
138C47⋊C16—22no—
139C47⋊C16—41no—
140C23×C47—21no—
141C23×C47—21no—
142C23×C47—21no—
143C23×C47—41no—
144C23×C47—41no—
145C23×C47C22×C811yescharacteristic
146C22×D47—21no—
147C22×D47—21no—
148C22×D47—21no—
149C22×D47—41no—
150C22×D47—41no—
151C2×C4×C47—21no—
152C2×C4×C47—21no—
153C2×C4×C47—21no—
154C2×C4×C47—41no—
155C2×C4×C47—41no—
156C2×C4×C47—41no—
157C2×C4×C47—41no—
158C2×C4×C47C2×C22⋊C411yescharacteristic
159C2×C4×C47C4×D411yescharacteristic
160C2×C4×C47C4×D411yescharacteristic
161C2×C4×C47C2×M1611yescharacteristic
162C2×C4×C47#13111yescharacteristic
163C2×C4×C47#13111yescharacteristic
164C22⋊D47—42no—
165C22⋊D47—42no—
166C2×Dic47—22no—
167C2×Dic47—21no—
168C2×Dic47—21no—
169C2×Dic47—21no—
170C2×Dic47—41no—
171C2×Dic47C22⋊C811yescharacteristic
172C2×Dic47C22⋊C811yescharacteristic
173C2×Dic47C22⋊C811yescharacteristic
174C2×Dic47C22⋊C811yescharacteristic
175C4×D47—41no—
176C4×D47—41no—
177C4×D47—41no—
178C4×D47—41no—
179C8×C47—41no—
180C8×C47#12111yescharacteristic
181C8×C47#12111yescharacteristic
182C8×C47#12411yescharacteristic
183C8×C47#12711yescharacteristic
184C47⋊C8—21no—
185C47⋊C8—21no—
186C47⋊C8—41no—
187#13194—471nomaximal, Sylow
188C22×C47—21no—
189C22×C47—21no—
190C22×C47—41no—
191C22×C47—41no—
192C22×C47—41no—
193C22×C47—81no—
194C22×C47—81no—
195C22×C47C2×C22⋊C811yescharacteristic
196C22×C47C8×D411yescharacteristic
197C22×C47C8×D411yescharacteristic
198C2×D47—41no—
199C2×D47—41no—
200C2×D47—41no—
201C2×D47—81no—
202C2×D47—81no—
203C4×C47—41no—
204C4×C47#39011yescharacteristic
205C4×C47#40811yescharacteristic
206C4×C47#43611yescharacteristic
207C4×C47#43611yescharacteristic
208Dic47—21no—
209Dic47—21no—
210Dic47—21no—
211Dic47—41no—
212Dic47#34912yescomplemented
213C2×#436—471no—
214#2150—471no—
215C2×#349—471no—
216C2×#349—471no—
217C2×#349—471no—
218C2×#349D4711yescharacteristic, complemented
219C4×M32—471no—
220C2×C47—21no—
221C2×C47—21no—
222C2×C47—81no—
223C2×C47#195411yescharacteristic
224C2×C47C2×#34911yescharacteristic
225C2×C47#211511yescharacteristic
226D47—81no—
227C22×M16—471no—
228C22×M16C2×D4711yescharacteristic
229C2×C4×D4—471no—
230C2×M32—471no—
231C2×M32—471no—
232C2×M32C2×D4711yescharacteristic
233C2×M32C2×D4711yescharacteristic
234#436—941no—
235#436—941no—
236#436—941no—
237#436—941no—
238C2×C4⋊C8—471no—
239C2×C22⋊C8—471no—
240C2×C22⋊C8—471no—
241C2×C4×C8—471no—
242C4⋊C16—941no—
243C4⋊C16—941no—
244#349—22no—
245#349—942no—
246#349—942no—
247#349—942no—
248#346—941no—
249C4×C16—941no—
250C47#1319411yescharacteristic, complemented, Sylow
251C22×D4—471no—
252C23×C4—471no—
253C23×C4C4×D4711yescharacteristic
254C2×M16—21no—
255C2×M16—21no—
256C2×M16—21no—
257C2×M16—21no—
258C2×M16—21no—
259C2×M16—21no—
260C2×M16—941no—
261C2×M16—941no—
262C2×M16—941no—
263C2×M16—941no—
264C2×M16—941no—
265C2×M16—941no—
266C22×C8—471no—
267C22×C8—471no—
268C22×C8C4×D4711yescharacteristic
269C22×C8C22×D4711yescharacteristic
270C4×D4—941no—
271C4×D4—941no—
272C4×D4—941no—
273C4×D4—941no—
274C2×C4⋊C4—471no—
275C2×C22⋊C4—471no—
276C2×C22⋊C4—471no—
277C2×C42—471no—
278M32—21no—
279M32—21no—
280M32—21no—
281M32—472no—
282M32—941no—
283M32—941no—
284M32—941no—
285M32C22⋊D4712yescomplemented
286C2×C16—21no—
287C2×C16—21no—
288C2×C16—941no—
289C2×C16—941no—
290C4⋊C8—941no—
291C4⋊C8—941no—
292C22⋊C8—941no—
293C22⋊C8—941no—
294C22⋊C8—941no—
295C22⋊C8—941no—
296C4×C8—471no—
297C4×C8—471no—
298C4×C8—471no—
299C4×C8—471no—
300C24—471no—
301C24C8×D4711yescharacteristic
302C2×D4—941no—
303C2×D4—941no—
304C2×D4—1882no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—471no—
312C22×C4—471no—
313C22×C4—471no—
314C22×C4—941no—
315C22×C4—941no—
316C22×C4—941no—
317C22×C4—941no—
318C22×C4—941no—
319C22×C4—941no—
320C22×C4C8×D4711yescharacteristic
321C22×C4C2×C4×D4711yescharacteristic
322M16—41no—
323M16—41no—
324M16—41no—
325M16—41no—
326M16—1881no—
327M16—1881no—
328M16—1881no—
329M16—1881no—
330C2×C8—21no—
331C2×C8—21no—
332C2×C8—21no—
333C2×C8—471no—
334C2×C8—471no—
335C2×C8—941no—
336C2×C8—941no—
337C2×C8—941no—
338C2×C8—941no—
339C2×C8—941no—
340C2×C8#1062535011yescharacteristic
341C2×C8D4×D4711yescharacteristic
342C2×C8D4×D4711yescharacteristic
343C2×C8D4×D4711yescharacteristic
344C2×C8#1062535211yescharacteristic
345C2×C8C2×C22⋊D4711yescharacteristic
346C4⋊C4—941no—
347C4⋊C4—941no—
348C22⋊C4—941no—
349C22⋊C4—941no—
350C22⋊C4—941no—
351C22⋊C4—941no—
352C42—471no—
353C42—471no—
354C42—471no—
355C42—471no—
356C16—22no—
357C16—41no—
358C16—942no—
359C16—1881no—
360C23—21no—
361C23—21no—
362C23—21no—
363C23—41no—
364C23—41no—
365C23—941no—
366C23—941no—
367C23—941no—
368C23—1881no—
369C23—1881no—
370C23C2×C8×D4711yescharacteristic
371D4—1882no—
372D4—1882no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—471no—
381C2×C4—471no—
382C2×C4—471no—
383C2×C4—471no—
384C2×C4—942no—
385C2×C4—941no—
386C2×C4—941no—
387C2×C4—941no—
388C2×C4—1881no—
389C2×C4—1881no—
390C2×C4—1881no—
391C2×C4—1881no—
392C2×C4—1881no—
393C2×C4C22⋊C4×D4711yescharacteristic
394C2×C4(C22⋊D47)⋊C411yescharacteristic
395C2×C4C4×C22⋊D4711yescharacteristic
396C2×C4M16×D4711yescharacteristic
397C2×C4#1309403011yescharacteristic
398C2×C4#1309403211yescharacteristic
399C8—41no—
400C8—941no—
401C8—941no—
402C8—1881no—
403C8#1309399611yescharacteristic
404C8#1309399711yescharacteristic
405C8#1309401011yescharacteristic
406C8#1309402011yescharacteristic
407C22—21no—
408C22—21no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—81no—
413C22—81no—
414C22—3761no—
415C22—3761no—
416C22—1881no—
417C22—1881no—
418C22—1881no—
419C22C22⋊C8×D4711yescharacteristic
420C22(C22⋊D47)⋊C811yescharacteristic
421C22C8×C22⋊D4711yescharacteristic
422C4—41no—
423C4—472no—
424C4—941no—
425C4—941no—
426C4—941no—
427C4—1881no—
428C4#1568902311yescharacteristic
429C4#1568908811yescharacteristic
430C4#1568919811yescharacteristic
431C4#1568919911yescharacteristic
432C2—21no—
433C2—21no—
434C2—81no—
435C2—3761no—
436C2#2024418011yescharacteristic
437C2#349×D4711yescharacteristic
438C2#2024500711yescharacteristic
subgroupG/H
centerC2×C4C22⋊C4×D47
commutatorC23×C47C22×C8
FrattiniC22×C8C22×D47
FittingC2×C47×#349C2
socleC22×C47C2×C22⋊C8
prime radicalC22×C22⋊D47C8
cosocleC22×C8×C47C23
hypercenterC2×#349D47
hypocenterC47#13194
LCSG ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ C2×#349

Complements

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

NH ≅ G/Ntype
2. #20245143C2semidirect
3. C2×#15688905C2semidirect
4. C2×#15688904C2semidirect
5. C2×#15688904C2semidirect
6. M32×Dic47C2semidirect
7. C2×C47×#349C2semidirect
52. #13093964D4semidirect
77. M32×C47D4semidirect
212. Dic47#349semidirect
218. C2×#349D47semidirect
250. C47#13194semidirect
285. M32C22⋊D47semidirect

Maximal subgroups

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

subgroupindex
C2×#156891992
#202451432
C2×#156889052
C2×#156889042
M32×Dic472
C2×C47×#3492
#1319447