Finite Group Databasebeta

← #26106788

Subgroups — #26106788

Order 12032.

914 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20243150C212yesmaximal, complemented
2C2×#15688847C212yesmaximal
3#20241231C212yesmaximal, complemented
4C2×C4×C22⋊C4×C47C211yesmaximal, characteristic
5C2×C22⋊(C47⋊C8)C2214yes—
6C2×#13093935—22no—
7#15688878—22no—
8#15688847C418yescomplemented
9C24×C4×C47C411yescharacteristic
10C22×C22⋊C4×C47C411yescharacteristic
11C22×C22⋊C4×C47C2211yescharacteristic
12C22×C42×C47C411yescharacteristic
13C22×C42×C47C2211yescharacteristic
14C4×C22⋊C4×C47—22no—
15C4×C22⋊C4×C47—22no—
16C2×#95×C47C411yescharacteristic
17C2×#95×C47C2211yescharacteristic
18C22×C47⋊C8—24no—
19C22⋊(C47⋊C8)—24no—
20C22⋊(C47⋊C8)C2×C418yescomplemented
21#13093935—44no—
22C25×C47C2×C411yescharacteristic
23C23×C4×C47—22no—
24C23×C4×C47—21no—
25C23×C4×C47—21no—
26C23×C4×C47—22no—
27C23×C4×C47—42no—
28C23×C4×C47—42no—
29C23×C4×C47C2×C411yescharacteristic
30C23×C4×C47C2×C411yescharacteristic
31C23×C4×C47C2×C411yescharacteristic
32C23×C4×C47C2×C411yescharacteristic
33C23×C4×C47C2×C411yescharacteristic
34C23×C4×C47C2311yescharacteristic
35C2×C22⋊C4×C47—22no—
36C2×C22⋊C4×C47—22no—
37C2×C22⋊C4×C47—22no—
38C2×C22⋊C4×C47—22no—
39C2×C22⋊C4×C47—22no—
40C2×C22⋊C4×C47C2×C414yes—
41C2×C42×C47—21no—
42C2×C42×C47—21no—
43C2×C42×C47—42no—
44C2×C42×C47—42no—
45C2×C42×C47D412yes—
46C2×C42×C47D412yes—
47#95×C47—22no—
48#95×C47—22no—
49C24×C47—21no—
50C24×C47—21no—
51C24×C47—21no—
52C24×C47—21no—
53C24×C47—41no—
54C24×C47—41no—
55C24×C47—41no—
56C24×C47—41no—
57C24×C47C4212yes—
58C24×C47M1612yes—
59C24×C47M1612yes—
60C24×C47C22×C411yescharacteristic
61C22×C4×C47—21no—
62C22×C4×C47—21no—
63C22×C4×C47—22no—
64C22×C4×C47—22no—
65C22×C4×C47—24no—
66C22×C4×C47—24no—
67C22×C4×C47—21no—
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—41no—
76C22×C4×C47—41no—
77C22×C4×C47—41no—
78C22×C4×C47—41no—
79C22×C4×C47—48no—
80C22×C4×C47—48no—
81C22×C4×C47—42no—
82C22×C4×C47—42no—
83C22×C4×C47—42no—
84C22×C4×C47—42no—
85C22×C4×C47—41no—
86C22×C4×C47—41no—
87C22×C4×C47—41no—
88C22×C4×C47—41no—
89C22×C4×C47—41no—
90C22×C4×C47—41no—
91C22×C4×C47C4212yes—
92C22×C4×C47C22⋊C412yes—
93C22×C4×C47C22⋊C411yescharacteristic
94C22×C4×C47C22⋊C411yescharacteristic
95C22×C4×C47M1612yes—
96C22×C4×C47M1612yes—
97C22×C4×C47C22×C412yes—
98C22×C4×C47C2×D412yes—
99C22×C4×C47#4112yes—
100C22⋊C4×C47—28no—
101C22⋊C4×C47—28no—
102C42×C47—24no—
103C42×C47—42no—
104C42×C47—42no—
105C42×C47—42no—
106C2×C47⋊C8—28no—
107C2×C47⋊C8—44no—
108C2×C47⋊C8—44no—
109C23×C47—21no—
110C23×C47—21no—
111C23×C47—21no—
112C23×C47—21no—
113C23×C47—22no—
114C23×C47—22no—
115C23×C47—22no—
116C23×C47—22no—
117C23×C47—22no—
118C23×C47—22no—
119C23×C47—42no—
120C23×C47—42no—
121C23×C47—42no—
122C23×C47—42no—
123C23×C47—42no—
124C23×C47—42no—
125C23×C47—42no—
126C23×C47—42no—
127C23×C47—42no—
128C23×C47—42no—
129C23×C47—42no—
130C23×C47—42no—
131C23×C47—41no—
132C23×C47—41no—
133C23×C47—41no—
134C23×C47—41no—
135C23×C47#9714yescomplemented
136C23×C47C2×C4211yescharacteristic
137C23×C47C2×C22⋊C411yescharacteristic
138C23×C47#11711yescharacteristic
139C23×C47C2×M1611yescharacteristic
140C23×C47C2×M1611yescharacteristic
141C23×C47C2×M1611yescharacteristic
142C23×C47C2×M1611yescharacteristic
143C2×C4×C47—24no—
144C2×C4×C47—24no—
145C2×C4×C47—21no—
146C2×C4×C47—21no—
147C2×C4×C47—22no—
148C2×C4×C47—21no—
149C2×C4×C47—22no—
150C2×C4×C47—21no—
151C2×C4×C47—22no—
152C2×C4×C47—22no—
153C2×C4×C47—22no—
154C2×C4×C47—22no—
155C2×C4×C47—22no—
156C2×C4×C47—22no—
157C2×C4×C47—42no—
158C2×C4×C47—42no—
159C2×C4×C47—48no—
160C2×C4×C47—48no—
161C2×C4×C47—42no—
162C2×C4×C47—42no—
163C2×C4×C47—42no—
164C2×C4×C47—42no—
165C2×C4×C47—42no—
166C2×C4×C47—41no—
167C2×C4×C47—41no—
168C2×C4×C47—41no—
169C2×C4×C47—41no—
170C2×C4×C47—41no—
171C2×C4×C47—41no—
172C2×C4×C47—42no—
173C2×C4×C47—42no—
174C2×C4×C47—42no—
175C2×C4×C47#9912yes—
176C2×C4×C47#10012yes—
177C2×C4×C47C4×D414yes—
178C47⋊C8—48no—
179#8152—471nomaximal, Sylow
180C22×C47—21no—
181C22×C47—21no—
182C22×C47—21no—
183C22×C47—21no—
184C22×C47—21no—
185C22×C47—21no—
186C22×C47—21no—
187C22×C47—21no—
188C22×C47—21no—
189C22×C47—21no—
190C22×C47—21no—
191C22×C47—21no—
192C22×C47—24no—
193C22×C47—44no—
194C22×C47—44no—
195C22×C47—44no—
196C22×C47—44no—
197C22×C47—42no—
198C22×C47—42no—
199C22×C47—42no—
200C22×C47—42no—
201C22×C47—42no—
202C22×C47—42no—
203C22×C47C4×C22⋊C411yescharacteristic
204C22×C47C2×#9711yescharacteristic
205C22×C47C4×M1611yescharacteristic
206C22×C47#40711yescharacteristic
207C22×C47#40711yescharacteristic
208C22×C47C2×#9911yescharacteristic
209C22×C47#41011yescharacteristic
210C22×C47C2×#10011yescharacteristic
211C22×C47#41311yescharacteristic
212C22×C47#43211yescharacteristic
213C22×C47#43211yescharacteristic
214C4×C47—22no—
215C4×C47—22no—
216C4×C47—24no—
217C4×C47—42no—
218C4×C47—42no—
219C4×C47—42no—
220C4×C47—42no—
221C2×C4×C22⋊C4D4711yescharacteristic
222#1749—472no—
223C2×#323—472no—
224#1280—472no—
225C2×C47—21no—
226C2×C47—21no—
227C2×C47—21no—
228C2×C47—21no—
229C2×C47—44no—
230C2×C47#146211yescharacteristic
231C2×C47#146411yescharacteristic
232C2×C47#146511yescharacteristic
233C2×C47#146611yescharacteristic
234C2×C47#174911yescharacteristic
235C2×C47C4×#9911yescharacteristic
236C2×C47C4×#10011yescharacteristic
237C24×C4Dic4711yescharacteristic
238C22×C22⋊C4Dic4711yescharacteristic
239C22×C22⋊C4C2×D4711yescharacteristic
240C22×C42Dic4711yescharacteristic
241C22×C42C2×D4711yescharacteristic
242C2×C22⋊C8—474no—
243C2×#97—942no—
244C4×C22⋊C4—22no—
245C4×C22⋊C4—22no—
246C2×#95Dic4711yescharacteristic
247C2×#95C2×D4711yescharacteristic
248#336—942no—
249#323—478no—
250C47#815211yescharacteristic, complemented, Sylow
251C25C2×Dic4711yescharacteristic
252C23×C4—22no—
253C23×C4—22no—
254C23×C4—21no—
255C23×C4—21no—
256C23×C4—42no—
257C23×C4—42no—
258C23×C4C2×Dic4711yescharacteristic
259C23×C4C2×Dic4711yescharacteristic
260C23×C4C2×Dic4711yescharacteristic
261C23×C4C2×Dic4711yescharacteristic
262C23×C4C2×Dic4711yescharacteristic
263C23×C4C22×D4711yescharacteristic
264C22×C8—944no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4C4×D4714yes—
271C2×C42—21no—
272C2×C42—21no—
273C2×C42—42no—
274C2×C42—42no—
275C2×C42C22⋊D4712yes—
276C2×C42C22⋊D4712yes—
277C22⋊C8—478no—
278C22⋊C8—944no—
279#97—1884no—
280#95—22no—
281#95—22no—
282C24—21no—
283C24—21no—
284C24—21no—
285C24—21no—
286C24—41no—
287C24—41no—
288C24—41no—
289C24—41no—
290C24#1062532912yes—
291C24#1062532912yes—
292C24C4×Dic4712yes—
293C24C22×Dic4711yescharacteristic
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—48no—
313C22×C4—48no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4#1062532912yes—
325C22×C4#1062532912yes—
326C22×C4C4×Dic4712yes—
327C22×C4C22⋊Dic4712yes—
328C22×C4C22⋊Dic4711yescharacteristic
329C22×C4C22⋊Dic4711yescharacteristic
330C22×C4C2×C4×D4712yes—
331C22×C4#1062535012yes—
332C22×C4C2×C22⋊D4712yes—
333C2×C8—948no—
334C2×C8—1884no—
335C2×C8—1884no—
336C22⋊C4—28no—
337C22⋊C4—28no—
338C42—24no—
339C42—42no—
340C42—42no—
341C42—42no—
342C23—21no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—22no—
347C23—22no—
348C23—22no—
349C23—22no—
350C23—22no—
351C23—22no—
352C23—42no—
353C23—42no—
354C23—42no—
355C23—42no—
356C23—42no—
357C23—42no—
358C23—42no—
359C23—42no—
360C23—42no—
361C23—42no—
362C23—42no—
363C23—42no—
364C23—41no—
365C23—41no—
366C23—41no—
367C23—41no—
368C23#1309393514yescomplemented
369C23C2×C4×Dic4711yescharacteristic
370C23C2×C22⋊Dic4711yescharacteristic
371C23#1309398311yescharacteristic
372C23C2×#1062532911yescharacteristic
373C23C2×#1062532911yescharacteristic
374C23C2×#1062532911yescharacteristic
375C23C2×#1062532911yescharacteristic
376C2×C4—24no—
377C2×C4—24no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—22no—
381C2×C4—21no—
382C2×C4—22no—
383C2×C4—21no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—48no—
393C2×C4—48no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4#1309393912yes—
409C2×C4#1309394112yes—
410C2×C4C4×C22⋊D4714yes—
411C8—1888no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—24no—
425C22—44no—
426C22—44no—
427C22—44no—
428C22—44no—
429C22—42no—
430C22—42no—
431C22—42no—
432C22—42no—
433C22—42no—
434C22—42no—
435C22C4×C22⋊Dic4711yescharacteristic
436C22C2×#1309393511yescharacteristic
437C22C4×#1062532911yescharacteristic
438C22#1568908711yescharacteristic
439C22#1568908711yescharacteristic
440C22C2×#1309393911yescharacteristic
441C22#1568909711yescharacteristic
442C22C2×#1309394111yescharacteristic
443C22#1568910711yescharacteristic
444C22#1568918011yescharacteristic
445C22#1568918011yescharacteristic
446C4—22no—
447C4—22no—
448C4—24no—
449C4—42no—
450C4—42no—
451C4—42no—
452C4—42no—
453C2—21no—
454C2—21no—
455C2—21no—
456C2—21no—
457C2—44no—
458C2#2024171511yescharacteristic
459C2#2024172211yescharacteristic
460C2#2024172611yescharacteristic
461C2#2024172911yescharacteristic
462C2#2024315011yescharacteristic
463C2C4×#1309393911yescharacteristic
464C2C4×#1309394111yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic47
commutatorC23×C47C2×C42
FrattiniC23×C4C22×D47
FittingC2×C4×C22⋊C4×C47C2
socleC23×C47C2×C22⋊C4
prime radicalC25×C47C2×C4
cosocleC23×C4×C47C23
hypercenterC2×C4×C22⋊C4D47
hypocenterC47#8152
LCSG ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. #20243150C2semidirect
3. #20241231C2semidirect
8. #15688847C4semidirect
20. C22⋊(C47⋊C8)C2×C4semidirect
135. C23×C47#97semidirect
250. C47#8152semidirect
368. C23#13093935semidirect

Maximal subgroups

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

subgroupindex
#202431502
C2×#156888472
#202412312
C2×C4×C22⋊C4×C472
#815247