Finite Group Databasebeta

← #26111849

Subgroups — #26111849

Order 12032.

694 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689176C211yesmaximal, characteristic, complemented
2C2×#15688857C212yesmaximal
3#20241615C211yesmaximal, characteristic, complemented
4#20241266C212yesmaximal, complemented
5C47×#2854C211yesmaximal, characteristic
6#15689176—24no—
7C2×C4⋊(C47⋊C8)—21no—
8C2×C22⋊(C47⋊C8)C2212yes—
9C2×C4×C47⋊C8—21no—
10C2×C47⋊#95—21no—
11C2×C47⋊#95—21no—
12C2×C47⋊#95C2212yes—
13#15688857—24no—
14C2×#123×C47—21no—
15C2×#121×C47C2211yescharacteristic
16C22×C42×C47C2211yescharacteristic
17C47×#390C411yescharacteristic, complemented
18C47×#390C411yescharacteristic, complemented
19C47×#389C411yescharacteristic, complemented
20C47×#389C411yescharacteristic, complemented
21C47×#388—21no—
22C47×#388—21no—
23C47×#386—21no—
24C47×#380C2211yescharacteristic
25C23×Dic47—22no—
26C22×C47⋊C8—22no—
27C4⋊(C47⋊C8)—24no—
28C22⋊(C47⋊C8)—22no—
29C22⋊(C47⋊C8)D414yes—
30C4×C47⋊C8—24no—
31C47⋊#95—24no—
32C47⋊#95—22no—
33C47⋊#95—22no—
34C47⋊#95—42no—
35C22×D4×C47—21no—
36C22×D4×C47C2×C411yescharacteristic
37C23×C4×C47D411yescharacteristic
38C23×C4×C47D411yescharacteristic
39C23×C4×C47C2311yescharacteristic
40#123×C47—42no—
41#123×C47—42no—
42#121×C47—22no—
43#121×C47—22no—
44C2×C4⋊C4×C47—21no—
45C2×C4⋊C4×C47—21no—
46C2×C4⋊C4×C47C2×C411yescharacteristic
47C2×C4⋊C4×C47C2×C411yescharacteristic
48C2×C4⋊C4×C47C2×C411yescharacteristic
49C2×C22⋊C4×C47—21no—
50C2×C22⋊C4×C47—21no—
51C2×C22⋊C4×C47—21no—
52C2×C22⋊C4×C47—21no—
53C2×C22⋊C4×C47—21no—
54C2×C22⋊C4×C47—21no—
55C2×C42×C47—22no—
56C2×C42×C47—22no—
57C2×C42×C47C2×C411yescharacteristic
58C2×C42×C47C2×C411yescharacteristic
59C2×C42×C47D411yescharacteristic
60C2×C42×C47D411yescharacteristic
61#95×C47—21no—
62#95×C47—21no—
63#95×C47—21no—
64C24×C47—21no—
65C24×C47C2×D411yescharacteristic
66C2×D4×C47—22no—
67C2×D4×C47—41no—
68C2×D4×C47—41no—
69C2×D4×C47—42no—
70C2×D4×C47—41no—
71C2×D4×C47—41no—
72C2×D4×C47—42no—
73C22×C4×C47—22no—
74C22×C4×C47—22no—
75C22×C4×C47—22no—
76C22×C4×C47—22no—
77C22×C4×C47—21no—
78C22×C4×C47—21no—
79C22×C4×C47—21no—
80C22×C4×C47—41no—
81C22×C4×C47—41no—
82C22×C4×C47—41no—
83C22×C4×C47C22⋊C411yescharacteristic
84C22×C4×C47C22⋊C411yescharacteristic
85C22×C4×C47C22⋊C411yescharacteristic
86C22×C4×C47C22⋊C411yescharacteristic
87C22×C4×C47D812yes—
88C22×C4×C47SD1612yes—
89C22×C4×C47C22×C411yescharacteristic
90C22×C4×C47C2×D411yescharacteristic
91C22×C4×C47C2×D412yes—
92C22×C4×C47#4112yes—
93C22×Dic47—24no—
94C22×Dic47—42no—
95C22×Dic47—42no—
96C22×Dic47—42no—
97C22×Dic47—42no—
98C22×Dic47—42no—
99C22×Dic47—42no—
100C4⋊C4×C47—22no—
101C4⋊C4×C47—41no—
102C4⋊C4×C47—41no—
103C4⋊C4×C47—42no—
104C4⋊C4×C47—42no—
105C22⋊C4×C47—42no—
106C22⋊C4×C47—41no—
107C22⋊C4×C47—41no—
108C22⋊C4×C47—42no—
109C22⋊C4×C47—42no—
110C22⋊C4×C47—42no—
111C22⋊C4×C47—41no—
112C22⋊C4×C47—41no—
113C42×C47—22no—
114C42×C47—22no—
115C42×C47—21no—
116C42×C47—21no—
117C42×C47—21no—
118C42×C47—21no—
119C2×C47⋊C8—24no—
120C2×C47⋊C8—42no—
121C2×C47⋊C8—42no—
122C23×C47—21no—
123C23×C47—21no—
124C23×C47—21no—
125C23×C47—21no—
126C23×C47—41no—
127C23×C47—41no—
128C23×C47—41no—
129C23×C47—41no—
130C23×C47—41no—
131C23×C47—41no—
132C23×C47—41no—
133C23×C47C2×C22⋊C411yescharacteristic
134C23×C47#12112yes—
135C23×C47C2×D811yescharacteristic
136C23×C47C2×SD1611yescharacteristic
137C23×C47#13512yes—
138D4×C47—82no—
139D4×C47—82no—
140D4×C47—82no—
141C2×C4×C47—21no—
142C2×C4×C47—21no—
143C2×C4×C47—21no—
144C2×C4×C47—21no—
145C2×C4×C47—21no—
146C2×C4×C47—21no—
147C2×C4×C47—21no—
148C2×C4×C47—21no—
149C2×C4×C47—21no—
150C2×C4×C47—21no—
151C2×C4×C47—24no—
152C2×C4×C47—24no—
153C2×C4×C47—21no—
154C2×C4×C47—21no—
155C2×C4×C47—21no—
156C2×C4×C47—42no—
157C2×C4×C47—42no—
158C2×C4×C47—44no—
159C2×C4×C47—41no—
160C2×C4×C47—41no—
161C2×C4×C47—41no—
162C2×C4×C47—41no—
163C2×C4×C47—41no—
164C2×C4×C47—41no—
165C2×C4×C47—41no—
166C2×C4×C47—41no—
167C2×C4×C47—41no—
168C2×C4×C47#10212yes—
169C2×C4×C47#10212yes—
170C2×C4×C47#10412yes—
171C2×C4×C47C4×D412yes—
172C2×C4×C47#12011yescharacteristic
173C2×C4×C47#12311yescharacteristic
174C2×Dic47—44no—
175C2×Dic47—44no—
176C2×Dic47—44no—
177C2×Dic47—82no—
178C2×Dic47—82no—
179C2×Dic47—82no—
180C2×Dic47—82no—
181C47⋊C8—44no—
182#9114—471nomaximal, Sylow
183C22×C47—22no—
184C22×C47—22no—
185C22×C47—22no—
186C22×C47—21no—
187C22×C47—21no—
188C22×C47—21no—
189C22×C47—21no—
190C22×C47—22no—
191C22×C47—81no—
192C22×C47—81no—
193C22×C47—81no—
194C22×C47—81no—
195C22×C47—81no—
196C22×C47—81no—
197C22×C47—81no—
198C22×C47#38611yescharacteristic
199C22×C47#41011yescharacteristic
200C22×C47#41011yescharacteristic
201C22×C47C2×#10211yescharacteristic
202C22×C47#41611yescharacteristic
203C22×C47C2×#10411yescharacteristic
204C22×C47#42111yescharacteristic
205C22×C47#46512yes—
206C22×C47#46612yes—
207C4×C47—22no—
208C4×C47—22no—
209C4×C47—22no—
210C4×C47—21no—
211C4×C47—22no—
212C4×C47—21no—
213C4×C47—41no—
214C4×C47—81no—
215C4×C47—81no—
216C4×C47—81no—
217Dic47—84no—
218C2×#431—471no—
219#2854D4711yescharacteristic
220C2×#327—472no—
221#1434—471no—
222#1294—472no—
223C2×C47—22no—
224C2×C47—22no—
225C2×C47—161no—
226C2×C47#150011yescharacteristic
227C2×C47#150211yescharacteristic
228C2×C47#151711yescharacteristic
229C2×C47#152111yescharacteristic
230C2×C47#187111yescharacteristic
231C2×C47#189311yescharacteristic
232C2×C47#189911yescharacteristic
233C2×#123—21no—
234C2×#121C2×D4711yescharacteristic
235C22×C42C2×D4711yescharacteristic
236#431—944no—
237C2×C4⋊C8—941no—
238C2×C22⋊C8—472no—
239C2×C4×C8—941no—
240#390Dic4711yescharacteristic, complemented
241#390Dic4711yescharacteristic, complemented
242#389Dic4711yescharacteristic, complemented
243#389Dic4711yescharacteristic, complemented
244#388—21no—
245#388—21no—
246#386—21no—
247#380C2×D4711yescharacteristic
248C2×#95—472no—
249C2×#95—941no—
250C2×#95—941no—
251#327—944no—
252C47#911411yescharacteristic, complemented, Sylow
253C22×D4—21no—
254C22×D4C2×Dic4711yescharacteristic
255C23×C4—942no—
256C23×C4C22⋊D4711yescharacteristic
257C23×C4C22⋊D4711yescharacteristic
258C23×C4C22×D4711yescharacteristic
259C22×C8—942no—
260#123—42no—
261#123—42no—
262#121—22no—
263#121—22no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4C2×Dic4711yescharacteristic
267C2×C4⋊C4C2×Dic4711yescharacteristic
268C2×C4⋊C4C2×Dic4711yescharacteristic
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—21no—
275C2×C42—22no—
276C2×C42—22no—
277C2×C42C2×Dic4711yescharacteristic
278C2×C42C2×Dic4711yescharacteristic
279C2×C42C22⋊D4711yescharacteristic
280C2×C42C22⋊D4711yescharacteristic
281C4⋊C8—944no—
282C22⋊C8—474no—
283C22⋊C8—942no—
284C4×C8—944no—
285#95—21no—
286#95—21no—
287#95—21no—
288#95—944no—
289#95—942no—
290#95—942no—
291#95—1882no—
292C24—21no—
293C24C2×C22⋊D4711yescharacteristic
294C2×D4—22no—
295C2×D4—41no—
296C2×D4—41no—
297C2×D4—42no—
298C2×D4—41no—
299C2×D4—41no—
300C2×D4—42no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—944no—
312C22×C4—1882no—
313C22×C4—1882no—
314C22×C4—1882no—
315C22×C4—1882no—
316C22×C4—1882no—
317C22×C4—1882no—
318C22×C4#1062533412yes—
319C22×C4#1062533512yes—
320C22×C4C22⋊Dic4711yescharacteristic
321C22×C4C22⋊Dic4711yescharacteristic
322C22×C4C22⋊Dic4711yescharacteristic
323C22×C4C22⋊Dic4711yescharacteristic
324C22×C4D4×D4712yes—
325C22×C4#1062535212yes—
326C22×C4C22×Dic4711yescharacteristic
327C22×C4C2×C22⋊D4711yescharacteristic
328C2×C8—944no—
329C2×C8—1882no—
330C2×C8—1882no—
331C4⋊C4—22no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C22⋊C4—42no—
337C22⋊C4—41no—
338C22⋊C4—41no—
339C22⋊C4—42no—
340C22⋊C4—42no—
341C22⋊C4—42no—
342C22⋊C4—41no—
343C22⋊C4—41no—
344C42—22no—
345C42—22no—
346C42—21no—
347C42—21no—
348C42—21no—
349C42—21no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—41no—
355C23—41no—
356C23—41no—
357C23—41no—
358C23—41no—
359C23—41no—
360C23—41no—
361C23C2×C22⋊Dic4711yescharacteristic
362C23#1309399612yes—
363C23C2×#1062533411yescharacteristic
364C23C2×#1062533511yescharacteristic
365C23#1309404412yes—
366D4—82no—
367D4—82no—
368D4—82no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—24no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—24no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—44no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—3762no—
397C2×C4—3762no—
398C2×C4—3762no—
399C2×C4—3762no—
400C2×C4—1884no—
401C2×C4—1884no—
402C2×C4—1884no—
403C2×C4#1309394512yes—
404C2×C4#1309394512yes—
405C2×C4#1309395112yes—
406C2×C4D4×Dic4712yes—
407C2×C4#1309399211yescharacteristic
408C2×C4#1309400511yescharacteristic
409C8—1884no—
410C22—21no—
411C22—21no—
412C22—22no—
413C22—22no—
414C22—22no—
415C22—21no—
416C22—21no—
417C22—22no—
418C22—81no—
419C22—81no—
420C22—81no—
421C22—81no—
422C22—81no—
423C22—81no—
424C22—81no—
425C22#1568900311yescharacteristic
426C22#1568909711yescharacteristic
427C22#1568909711yescharacteristic
428C22C2×#1309394511yescharacteristic
429C22#1568911911yescharacteristic
430C22C2×#1309395111yescharacteristic
431C22#1568914011yescharacteristic
432C22#1568935012yes—
433C22#1568935712yes—
434C4—22no—
435C4—22no—
436C4—21no—
437C4—22no—
438C4—21no—
439C4—22no—
440C4—41no—
441C4—81no—
442C4—81no—
443C4—81no—
444C4—3764no—
445C2—22no—
446C2—22no—
447C2—161no—
448C2#2024187411yescharacteristic
449C2#2024188311yescharacteristic
450C2#2024194811yescharacteristic
451C2#2024196511yescharacteristic
452C2#2024369111yescharacteristic
453C2#2024381711yescharacteristic
454C2#2024384711yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic47
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#2854C2
socleC23×C47C2×C22⋊C4
prime radicalC2×#121×C47C22
cosocleC23×C4×C47C23
hypercenter#2854D47
hypocenterC47#9114
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2854

Complements

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

NH ≅ G/Ntype
1. C2×#15689176C2semidirect
3. #20241615C2semidirect
4. #20241266C2semidirect
17. C47×#390C4semidirect
18. C47×#390C4semidirect
19. C47×#389C4semidirect
20. C47×#389C4semidirect
240. #390Dic47semidirect
241. #390Dic47semidirect
242. #389Dic47semidirect
243. #389Dic47semidirect
252. C47#9114semidirect

Maximal subgroups

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

subgroupindex
C2×#156891762
C2×#156888572
#202416152
#202412662
C47×#28542
#911447