Finite Group Databasebeta

← #26114278

Subgroups — #26114278

Order 12032.

628 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 628 conjugacy classes of subgroups — too many to draw. Only its 107 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#20252417C211yesmaximal, characteristic, complemented
2#20241726C211yesmaximal, characteristic, complemented
3#20241715C212yesmaximal, complemented
4#20241294C212yesmaximal, complemented
5C47×#2534C211yesmaximal, characteristic
6#15689195—24no—
7#15689180—21no—
8#15689087C2212yes—
9#15689087C2212yes—
10C2×C22⋊(C47⋊C8)—21no—
11#15688849—21no—
12#15688847—24no—
13#15688847—21no—
14C2×#123×C47C411yescharacteristic
15C2×#120×C47C411yescharacteristic
16C2×C4×D4×C47C2211yescharacteristic
17C22×C22⋊C4×C47C2211yescharacteristic
18C47×#396—21no—
19C47×#394—21no—
20C47×#388—21no—
21C47×#386—21no—
22C47×#381C411yescharacteristic
23C47×#380C2211yescharacteristic
24C47×#379C411yescharacteristic
25C2×#10625329—22no—
26C2×#10625329—22no—
27C22×C47⋊C8—22no—
28C4⋊(C47⋊C8)—22no—
29C22⋊(C47⋊C8)—22no—
30C22⋊(C47⋊C8)—22no—
31C22⋊(C47⋊C8)—22no—
32C22⋊(C47⋊C8)—22no—
33C22⋊(C47⋊C8)D414yescomplemented
34#13093935—22no—
35C25×C47C2×C411yescharacteristic
36C22×D4×C47—21no—
37C22×D4×C47C2×C411yescharacteristic
38C23×C4×C47C2×C411yescharacteristic
39C23×C4×C47D411yescharacteristic
40C23×C4×C47C2311yescharacteristic
41#123×C47—42no—
42#120×C47—42no—
43C4×D4×C47—22no—
44C4×D4×C47—22no—
45C2×C4⋊C4×C47—21no—
46C2×C4⋊C4×C47—21no—
47C2×C4⋊C4×C47D411yescharacteristic
48C2×C22⋊C4×C47—21no—
49C2×C22⋊C4×C47—21no—
50C2×C22⋊C4×C47—21no—
51C2×C22⋊C4×C47—22no—
52C2×C22⋊C4×C47—21no—
53C2×C22⋊C4×C47—21no—
54C2×C22⋊C4×C47—22no—
55C2×C22⋊C4×C47—21no—
56C2×C22⋊C4×C47C2×C411yescharacteristic
57C2×C22⋊C4×C47C2×C411yescharacteristic
58C2×C22⋊C4×C47C2×C411yescharacteristic
59C2×C22⋊C4×C47D411yescharacteristic
60C2×C42×C47D411yescharacteristic
61#95×C47—21no—
62#95×C47—21no—
63#95×C47—21no—
64C24×C47—21no—
65C24×C47—22no—
66C24×C47—42no—
67C24×C47—41no—
68C24×C47—41no—
69C24×C47—81no—
70C24×C47C22⋊C411yescharacteristic
71C24×C47C22×C411yescharacteristic
72C2×D4×C47—22no—
73C2×D4×C47—22no—
74C2×D4×C47—41no—
75C2×D4×C47—42no—
76C2×D4×C47—42no—
77C2×D4×C47—81no—
78C22×C4×C47—21no—
79C22×C4×C47—21no—
80C22×C4×C47—21no—
81C22×C4×C47—21no—
82C22×C4×C47—21no—
83C22×C4×C47—22no—
84C22×C4×C47—41no—
85C22×C4×C47—41no—
86C22×C4×C47—81no—
87C22×C4×C47—81no—
88C22×C4×C47C22⋊C411yescharacteristic
89C22×C4×C47C22⋊C411yescharacteristic
90C22×C4×C47C22⋊C411yescharacteristic
91C22×C4×C47C2×D412yes—
92C22×C4×C47C2×D411yescharacteristic
93C22×C4×C47C2×D411yescharacteristic
94C22×C4×C47#4112yes—
95C4⋊C4×C47—22no—
96C4⋊C4×C47—42no—
97C4⋊C4×C47—81no—
98C22⋊C4×C47—22no—
99C22⋊C4×C47—22no—
100C22⋊C4×C47—42no—
101C22⋊C4×C47—42no—
102C22⋊C4×C47—42no—
103C22⋊C4×C47—41no—
104C22⋊C4×C47—42no—
105C22⋊C4×C47—41no—
106C22⋊C4×C47—41no—
107C22⋊C4×C47—41no—
108C22⋊C4×C47—81no—
109C42×C47—21no—
110C42×C47—21no—
111#10625329—82no—
112#10625329—82no—
113C2×C47⋊C8—24no—
114C2×C47⋊C8—42no—
115C2×C47⋊C8—82no—
116C23×C47—21no—
117C23×C47—21no—
118C23×C47—21no—
119C23×C47—21no—
120C23×C47—22no—
121C23×C47—24no—
122C23×C47—41no—
123C23×C47—41no—
124C23×C47—41no—
125C23×C47—41no—
126C23×C47—41no—
127C23×C47—41no—
128C23×C47—41no—
129C23×C47—42no—
130C23×C47—41no—
131C23×C47—41no—
132C23×C47—41no—
133C23×C47—82no—
134C23×C47—82no—
135C23×C47—84no—
136C23×C47—84no—
137C23×C47#10012yes—
138C23×C47C2×C22⋊C411yescharacteristic
139C23×C47C4×D412yes—
140D4×C47—44no—
141D4×C47—84no—
142C2×C4×C47—21no—
143C2×C4×C47—21no—
144C2×C4×C47—21no—
145C2×C4×C47—21no—
146C2×C4×C47—21no—
147C2×C4×C47—42no—
148C2×C4×C47—42no—
149C2×C4×C47—41no—
150C2×C4×C47—41no—
151C2×C4×C47—42no—
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—41no—
161C2×C4×C47—41no—
162C2×C4×C47—41no—
163C2×C4×C47—81no—
164C2×C4×C47—82no—
165C2×C4×C47#12011yescharacteristic
166C2×C4×C47#12112yes—
167C2×C4×C47#12311yescharacteristic
168C47⋊C8—44no—
169C47⋊C8—82no—
170#9515—471nomaximal, Sylow
171C22×C47—21no—
172C22×C47—21no—
173C22×C47—22no—
174C22×C47—22no—
175C22×C47—22no—
176C22×C47—22no—
177C22×C47—42no—
178C22×C47—42no—
179C22×C47—42no—
180C22×C47—42no—
181C22×C47—42no—
182C22×C47—41no—
183C22×C47—41no—
184C22×C47—42no—
185C22×C47—81no—
186C22×C47—81no—
187C22×C47—81no—
188C22×C47—84no—
189C22×C47—84no—
190C22×C47—81no—
191C22×C47—81no—
192C22×C47#38611yescharacteristic
193C22×C47C2×#10011yescharacteristic
194C22×C47#41311yescharacteristic
195C4×C47—21no—
196C4×C47—22no—
197C4×C47—21no—
198C4×C47—41no—
199C4×C47—41no—
200C4×C47—41no—
201C4×C47—81no—
202C4×C47—81no—
203C4×C47—81no—
204#2973—471no—
205#2534D4711yescharacteristic
206#1465—471no—
207#1462—472no—
208#1305—472no—
209C2×C47—21no—
210C2×C47—21no—
211C2×C47—42no—
212C2×C47—42no—
213C2×C47—82no—
214C2×C47#189111yescharacteristic
215C2×C47#190011yescharacteristic
216C2×C47#191311yescharacteristic
217C2×#123Dic4711yescharacteristic
218C2×#120Dic4711yescharacteristic
219C2×C4×D4C2×D4711yescharacteristic
220C22×C22⋊C4C2×D4711yescharacteristic
221#435—944no—
222#432—941no—
223#407—472no—
224#407—472no—
225C2×C22⋊C8—941no—
226#396—21no—
227#394—21no—
228#388—21no—
229#386—21no—
230#381Dic4711yescharacteristic
231#380C2×D4711yescharacteristic
232#379Dic4711yescharacteristic
233#324—941no—
234#323—944no—
235#323—941no—
236C47#951511yescharacteristic, complemented, Sylow
237C25C2×Dic4711yescharacteristic
238C22×D4—21no—
239C22×D4C2×Dic4711yescharacteristic
240C23×C4C2×Dic4711yescharacteristic
241C23×C4C22⋊D4711yescharacteristic
242C23×C4C22×D4711yescharacteristic
243C2×M16—942no—
244C2×M16—942no—
245C22×C8—942no—
246#123—42no—
247#120—42no—
248C4×D4—22no—
249C4×D4—22no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4C22⋊D4711yescharacteristic
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—21no—
259C2×C22⋊C4—21no—
260C2×C22⋊C4—21no—
261C2×C22⋊C4C2×Dic4711yescharacteristic
262C2×C22⋊C4C2×Dic4711yescharacteristic
263C2×C22⋊C4C2×Dic4711yescharacteristic
264C2×C22⋊C4C22⋊D4711yescharacteristic
265C2×C42C22⋊D4711yescharacteristic
266C4⋊C8—942no—
267C22⋊C8—474no—
268C22⋊C8—942no—
269C22⋊C8—942no—
270C22⋊C8—942no—
271C22⋊C8—942no—
272#97—942no—
273#95—21no—
274#95—21no—
275#95—21no—
276C24—21no—
277C24—22no—
278C24—42no—
279C24—41no—
280C24—41no—
281C24—81no—
282C24C22⋊Dic4711yescharacteristic
283C24C22×Dic4711yescharacteristic
284C2×D4—22no—
285C2×D4—22no—
286C2×D4—41no—
287C2×D4—42no—
288C2×D4—42no—
289C2×D4—81no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—22no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—81no—
299C22×C4—81no—
300C22×C4C22⋊Dic4711yescharacteristic
301C22×C4C22⋊Dic4711yescharacteristic
302C22×C4C22⋊Dic4711yescharacteristic
303C22×C4D4×D4712yes—
304C22×C4#1062535212yes—
305C22×C4C2×C22⋊D4711yescharacteristic
306C22×C4C2×C22⋊D4711yescharacteristic
307M16—3762no—
308M16—3762no—
309C2×C8—944no—
310C2×C8—3762no—
311C2×C8—1882no—
312C4⋊C4—22no—
313C4⋊C4—42no—
314C4⋊C4—81no—
315C22⋊C4—22no—
316C22⋊C4—22no—
317C22⋊C4—42no—
318C22⋊C4—42no—
319C22⋊C4—42no—
320C22⋊C4—41no—
321C22⋊C4—42no—
322C22⋊C4—41no—
323C22⋊C4—41no—
324C22⋊C4—41no—
325C22⋊C4—81no—
326C42—21no—
327C42—21no—
328C23—21no—
329C23—21no—
330C23—21no—
331C23—21no—
332C23—22no—
333C23—24no—
334C23—41no—
335C23—41no—
336C23—41no—
337C23—41no—
338C23—41no—
339C23—41no—
340C23—41no—
341C23—42no—
342C23—41no—
343C23—41no—
344C23—41no—
345C23—82no—
346C23—82no—
347C23—84no—
348C23—84no—
349C23#1309394112yes—
350C23C2×C22⋊Dic4711yescharacteristic
351C23D4×Dic4712yes—
352D4—44no—
353D4—84no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—41no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—41no—
363C2×C4—42no—
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—82no—
377C2×C4#1309399211yescharacteristic
378C2×C4#1309399612yes—
379C2×C4#1309400511yescharacteristic
380C8—3762no—
381C8—1884no—
382C22—21no—
383C22—21no—
384C22—22no—
385C22—22no—
386C22—22no—
387C22—22no—
388C22—42no—
389C22—42no—
390C22—42no—
391C22—42no—
392C22—42no—
393C22—41no—
394C22—41no—
395C22—42no—
396C22—81no—
397C22—81no—
398C22—81no—
399C22—84no—
400C22—84no—
401C22—81no—
402C22—81no—
403C22#1568900311yescharacteristic
404C22C2×#1309394111yescharacteristic
405C22#1568910711yescharacteristic
406C4—22no—
407C4—21no—
408C4—21no—
409C4—41no—
410C4—41no—
411C4—41no—
412C4—81no—
413C4—81no—
414C4—81no—
415C2—21no—
416C2—21no—
417C2—42no—
418C2—42no—
419C2—82no—
420C2#2024380711yescharacteristic
421C2#2024385211yescharacteristic
422C2#2024393111yescharacteristic
subgroupG/H
centerC22#15689003
commutatorC24×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#2534C2
socleC22×C47#386
prime radicalC2×#120×C47C4
cosocleC23×C4×C47C23
hypercenter#2534D47
hypocenterC47#9515
LCSG ▷ C24×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2534

Complements

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

NH ≅ G/Ntype
1. #20252417C2semidirect
2. #20241726C2semidirect
3. #20241715C2semidirect
4. #20241294C2semidirect
33. C22⋊(C47⋊C8)D4semidirect
236. C47#9515semidirect

Maximal subgroups

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

subgroupindex
#202524172
#202417262
#202417152
#202412942
C47×#25342
#951547