Finite Group Databasebeta

← #26111848

Subgroups — #26111848

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
1#20251719C211yesmaximal, characteristic
2C2×#15688858C212yesmaximal
3#20241615C211yesmaximal, characteristic, complemented
4#20241268C212yesmaximal, complemented
5C2×C47×#431C211yesmaximal, characteristic, complemented
6C2×#13094007—21no—
7C2×#13093998C2211yescharacteristic
8#15689025C411yescharacteristic, complemented
9#15689025C411yescharacteristic, complemented
10#15689021C411yescharacteristic, complemented
11#15689021C411yescharacteristic, complemented
12#15689014—21no—
13#15689014—21no—
14#15689002—21no—
15#15688983C2211yescharacteristic
16C2×C47⋊#95—21no—
17C2×C47⋊#95—21no—
18C2×C47⋊#95C2212yes—
19#15688858—24no—
20C22×C42×C47C2211yescharacteristic
21C47×#431—24no—
22C2×C4⋊C8×C47—21no—
23C2×C22⋊C8×C47C2212yes—
24C2×C4×C8×C47—21no—
25C22×D188C2×C411yescharacteristic
26C22×C22⋊D47—21no—
27C23×Dic47—22no—
28#13094007—42no—
29#13094007—42no—
30#13093998—22no—
31#13093998—22no—
32C2×C4⋊Dic47C2×C411yescharacteristic
33C2×C4⋊Dic47C2×C411yescharacteristic
34C2×C4⋊Dic47C2×C411yescharacteristic
35C2×Dic47⋊C4—21no—
36C2×Dic47⋊C4—21no—
37C2×C22⋊Dic47—21no—
38C2×(C2×D47)⋊C4—21no—
39C2×(C2×D47)⋊C4—21no—
40C2×(C2×D47)⋊C4—21no—
41C2×(C2×D47)⋊C4—21no—
42C2×(C2×D47)⋊C4—21no—
43C47⋊#95—24no—
44C47⋊#95—21no—
45C47⋊#95—21no—
46C47⋊#95—21no—
47C47⋊#95—22no—
48C47⋊#95—22no—
49C47⋊#95—42no—
50C23×C4×C47D411yescharacteristic
51C23×C4×C47D411yescharacteristic
52C23×C4×C47C2311yescharacteristic
53C22×C8×C47—22no—
54C2×C42×C47—22no—
55C2×C42×C47—22no—
56C2×C42×C47C2×C411yescharacteristic
57C2×C42×C47C2×C411yescharacteristic
58C2×C42×C47D411yescharacteristic
59C2×C42×C47D411yescharacteristic
60C4⋊C8×C47—24no—
61C22⋊C8×C47—22no—
62C22⋊C8×C47D414yes—
63C4×C8×C47—24no—
64C24×C47C2×D411yescharacteristic
65C23×D47—21no—
66C22×C4×C47—22no—
67C22×C4×C47—22no—
68C22×C4×C47—22no—
69C22×C4×C47—22no—
70C22×C4×C47—41no—
71C22×C4×C47—41no—
72C22×C4×C47—41no—
73C22×C4×C47C22⋊C411yescharacteristic
74C22×C4×C47C22⋊C411yescharacteristic
75C22×C4×C47C22⋊C411yescharacteristic
76C22×C4×C47C22⋊C411yescharacteristic
77C22×C4×C47D812yes—
78C22×C4×C47SD1612yes—
79C22×C4×C47C22×C411yescharacteristic
80C22×C4×C47C2×D411yescharacteristic
81C22×C4×C47C2×D412yes—
82C22×C4×C47#4112yes—
83C2×C22⋊D47—42no—
84C2×C22⋊D47—41no—
85C2×C22⋊D47—41no—
86C2×C22⋊D47—42no—
87C22×Dic47—24no—
88C22×Dic47—21no—
89C22×Dic47—21no—
90C22×Dic47—21no—
91C22×Dic47—42no—
92C22×Dic47—42no—
93C22×Dic47—42no—
94C22×Dic47—42no—
95C22×Dic47—42no—
96C22×Dic47—42no—
97C2×D188—22no—
98C2×D188—41no—
99C2×D188—41no—
100C2×C8×C47—24no—
101C2×C8×C47—42no—
102C2×C8×C47—42no—
103C42×C47—22no—
104C42×C47—22no—
105C42×C47—21no—
106C42×C47—21no—
107C42×C47—21no—
108C42×C47—21no—
109C22⋊Dic47—41no—
110C22⋊Dic47—41no—
111(C2×D47)⋊C4—42no—
112(C2×D47)⋊C4—42no—
113(C2×D47)⋊C4—42no—
114(C2×D47)⋊C4—42no—
115(C2×D47)⋊C4—41no—
116(C2×D47)⋊C4—41no—
117C4⋊Dic47—22no—
118C4⋊Dic47—41no—
119C4⋊Dic47—41no—
120Dic47⋊C4—42no—
121Dic47⋊C4—42no—
122C23×C47—21no—
123C23×C47—21no—
124C23×C47—21no—
125C23×C47—21no—
126C23×C47C2×C22⋊C411yescharacteristic
127C23×C47#12112yes—
128C23×C47C2×D811yescharacteristic
129C23×C47C2×SD1611yescharacteristic
130C23×C47#13512yes—
131C22×D47—41no—
132C22×D47—41no—
133C22×D47—41no—
134C22×D47—41no—
135C22×D47—41no—
136C22×D47—41no—
137C22×D47—41no—
138C2×C4×C47—21no—
139C2×C4×C47—21no—
140C2×C4×C47—21no—
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—24no—
149C2×C4×C47—24no—
150C2×C4×C47—21no—
151C2×C4×C47—21no—
152C2×C4×C47—21no—
153C2×C4×C47—42no—
154C2×C4×C47—42no—
155C2×C4×C47—44no—
156C2×C4×C47#10212yes—
157C2×C4×C47#10212yes—
158C2×C4×C47#10412yes—
159C2×C4×C47C4×D412yes—
160C2×C4×C47#12011yescharacteristic
161C2×C4×C47#12311yescharacteristic
162C22⋊D47—82no—
163C22⋊D47—82no—
164C2×Dic47—44no—
165C2×Dic47—44no—
166C2×Dic47—44no—
167C2×Dic47—41no—
168C2×Dic47—41no—
169C2×Dic47—41no—
170C2×Dic47—41no—
171C2×Dic47—41no—
172C2×Dic47—41no—
173C2×Dic47—41no—
174C2×Dic47—41no—
175C2×Dic47—41no—
176C2×Dic47—82no—
177C2×Dic47—82no—
178C2×Dic47—82no—
179C2×Dic47—82no—
180D188—82no—
181C8×C47—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#38611yescharacteristic
192C22×C47#41011yescharacteristic
193C22×C47#41011yescharacteristic
194C22×C47C2×#10211yescharacteristic
195C22×C47#41611yescharacteristic
196C22×C47C2×#10411yescharacteristic
197C22×C47#42111yescharacteristic
198C22×C47#46512yes—
199C22×C47#46612yes—
200C2×D47—81no—
201C2×D47—81no—
202C2×D47—81no—
203C2×D47—81no—
204C2×D47—81no—
205C2×D47—81no—
206C2×D47—81no—
207C4×C47—22no—
208C4×C47—22no—
209C4×C47—22no—
210C4×C47—21no—
211C4×C47—22no—
212C4×C47—21no—
213C4×C47—41no—
214Dic47—84no—
215Dic47—81no—
216Dic47—81no—
217Dic47—81no—
218C2×#431D4711yescharacteristic, complemented
219#2854—471no—
220C2×#327—472no—
221#1434—471no—
222#1294—472no—
223C2×C47—22no—
224C2×C47—22no—
225C2×C47#150011yescharacteristic
226C2×C47#150211yescharacteristic
227C2×C47#151711yescharacteristic
228C2×C47#152111yescharacteristic
229C2×C47#187111yescharacteristic
230C2×C47#189311yescharacteristic
231C2×C47#189911yescharacteristic
232D47—161no—
233C2×#123—941no—
234C2×#121—471no—
235C22×C42C2×D4711yescharacteristic
236#431—24no—
237C2×C4⋊C8—21no—
238C2×C22⋊C8C2×D4712yes—
239C2×C4×C8—21no—
240#390—471no—
241#390—471no—
242#389—471no—
243#389—471no—
244#388—941no—
245#388—941no—
246#386—941no—
247#380—471no—
248C2×#95—472no—
249C2×#95—941no—
250C2×#95—941no—
251#327—944no—
252C47#911411yescharacteristic, complemented, Sylow
253C22×D4—471no—
254C22×D4—941no—
255C23×C4—942no—
256C23×C4D18811yescharacteristic
257C23×C4D18811yescharacteristic
258C23×C4C22×D4711yescharacteristic
259C22×C8—22no—
260#123—1882no—
261#123—1882no—
262#121—942no—
263#121—942no—
264C2×C4⋊C4—471no—
265C2×C4⋊C4—471no—
266C2×C4⋊C4—471no—
267C2×C4⋊C4—941no—
268C2×C4⋊C4—941no—
269C2×C22⋊C4—941no—
270C2×C22⋊C4—941no—
271C2×C22⋊C4—941no—
272C2×C22⋊C4—941no—
273C2×C22⋊C4—941no—
274C2×C22⋊C4—941no—
275C2×C42—22no—
276C2×C42—22no—
277C2×C42C4×D4711yescharacteristic
278C2×C42C4×D4711yescharacteristic
279C2×C42C22⋊D4711yescharacteristic
280C2×C42C22⋊D4711yescharacteristic
281C4⋊C8—24no—
282C22⋊C8—22no—
283C22⋊C8C22⋊D4714yes—
284C4×C8—24no—
285#95—944no—
286#95—941no—
287#95—941no—
288#95—941no—
289#95—942no—
290#95—942no—
291#95—1882no—
292C24—941no—
293C24C2×D18811yescharacteristic
294C2×D4—942no—
295C2×D4—1881no—
296C2×D4—1881no—
297C2×D4—1882no—
298C2×D4—1881no—
299C2×D4—1881no—
300C2×D4—1882no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—944no—
309C22×C4—941no—
310C22×C4—941no—
311C22×C4—941no—
312C22×C4—1882no—
313C22×C4—1882no—
314C22×C4—1882no—
315C22×C4—1882no—
316C22×C4—1882no—
317C22×C4—1882no—
318C22×C4#1062532512yes—
319C22×C4D37612yes—
320C22×C4(C2×D47)⋊C411yescharacteristic
321C22×C4(C2×D47)⋊C411yescharacteristic
322C22×C4(C2×D47)⋊C411yescharacteristic
323C22×C4(C2×D47)⋊C411yescharacteristic
324C22×C4C2×C4×D4711yescharacteristic
325C22×C4#1062535012yes—
326C22×C4C2×C22⋊D4711yescharacteristic
327C22×C4C2×C22⋊D4712yes—
328C2×C8—24no—
329C2×C8—42no—
330C2×C8—42no—
331C4⋊C4—942no—
332C4⋊C4—1881no—
333C4⋊C4—1881no—
334C4⋊C4—1882no—
335C4⋊C4—1882no—
336C22⋊C4—1882no—
337C22⋊C4—1881no—
338C22⋊C4—1881no—
339C22⋊C4—1882no—
340C22⋊C4—1882no—
341C22⋊C4—1882no—
342C22⋊C4—1881no—
343C22⋊C4—1881no—
344C42—22no—
345C42—22no—
346C42—21no—
347C42—21no—
348C42—21no—
349C42—21no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—1881no—
355C23—1881no—
356C23—1881no—
357C23—1881no—
358C23—1881no—
359C23—1881no—
360C23—1881no—
361C23C2×(C2×D47)⋊C411yescharacteristic
362C23#1309399812yes—
363C23C2×D37611yescharacteristic
364C23C2×#1062532511yescharacteristic
365C23#1309404612yes—
366D4—3762no—
367D4—3762no—
368D4—3762no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—24no—
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—3762no—
388C2×C4—3762no—
389C2×C4—3762no—
390C2×C4—3762no—
391C2×C4—1884no—
392C2×C4—1884no—
393C2×C4—1884no—
394C2×C4—1881no—
395C2×C4—1881no—
396C2×C4—1881no—
397C2×C4—1881no—
398C2×C4—1881no—
399C2×C4—1881no—
400C2×C4—1881no—
401C2×C4—1881no—
402C2×C4—1881no—
403C2×C4#1309394612yes—
404C2×C4#1309394612yes—
405C2×C4#1309395012yes—
406C2×C4C4×C22⋊D4712yes—
407C2×C4#1309399311yescharacteristic
408C2×C4#1309400711yescharacteristic
409C8—44no—
410C22—21no—
411C22—21no—
412C22—22no—
413C22—22no—
414C22—22no—
415C22—21no—
416C22—21no—
417C22—22no—
418C22—3761no—
419C22—3761no—
420C22—3761no—
421C22—3761no—
422C22—3761no—
423C22—3761no—
424C22—3761no—
425C22#1568900211yescharacteristic
426C22#1568909611yescharacteristic
427C22#1568909611yescharacteristic
428C22C2×#1309394611yescharacteristic
429C22#1568912011yescharacteristic
430C22C2×#1309395011yescharacteristic
431C22#1568913911yescharacteristic
432C22#1568935212yes—
433C22#1568935912yes—
434C4—22no—
435C4—22no—
436C4—21no—
437C4—22no—
438C4—21no—
439C4—22no—
440C4—41no—
441C4—3764no—
442C4—3761no—
443C4—3761no—
444C4—3761no—
445C2—22no—
446C2—22no—
447C2—7521no—
448C2#2024187511yescharacteristic
449C2#2024188411yescharacteristic
450C2#2024194911yescharacteristic
451C2#2024196611yescharacteristic
452C2#2024369011yescharacteristic
453C2#2024381611yescharacteristic
454C2#2024384611yescharacteristic
subgroupG/H
centerC23C2×(C2×D47)⋊C4
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC2×C47×#431C2
socleC23×C47C2×C22⋊C4
prime radicalC2×#13093998C22
cosocleC23×C4×C47C23
hypercenterC2×#431D47
hypocenterC47#9114
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#431

Complements

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

NH ≅ G/Ntype
3. #20241615C2semidirect
4. #20241268C2semidirect
5. C2×C47×#431C2semidirect
8. #15689025C4semidirect
9. #15689025C4semidirect
10. #15689021C4semidirect
11. #15689021C4semidirect
218. C2×#431D47semidirect
252. C47#9114semidirect

Maximal subgroups

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

subgroupindex
#202517192
C2×#156888582
#202416152
#202412682
C2×C47×#4312
#911447