Finite Group Databasebeta

← #26109021

Subgroups — #26109021

Order 12032.

698 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20243368C211yesmaximal, characteristic, complemented
2C2×#15688857C212yesmaximal
3#20241615C211yesmaximal, characteristic, complemented
4#20241266C212yesmaximal, complemented
5C4×#121×C47C211yesmaximal, characteristic
6C2×C4⋊(C47⋊C8)—21no—
7C2×C22⋊(C47⋊C8)C2212yes—
8C2×C47⋊#95—21no—
9C2×C47⋊#95—21no—
10C2×C47⋊#95C2212yes—
11#15688878—21no—
12#15688857—24no—
13C2×#121×C47C2211yescharacteristic
14C2×C4×D4×C47—21no—
15C2×C4×D4×C47C411yescharacteristic, complemented
16C22×C42×C47C2211yescharacteristic
17C47×#390C411yescharacteristic, complemented
18C47×#389C411yescharacteristic, complemented
19C47×#388—21no—
20C47×#386—21no—
21C47×#380C2211yescharacteristic
22C4×C4⋊C4×C47C411yescharacteristic, complemented
23C4×C22⋊C4×C47—21no—
24C23×Dic47—22no—
25C22×C47⋊C8—22no—
26C4⋊(C47⋊C8)—42no—
27C22⋊(C47⋊C8)—22no—
28C22⋊(C47⋊C8)—22no—
29C47⋊#95—24no—
30C47⋊#95—22no—
31C47⋊#95—22no—
32C47⋊#95—42no—
33C22×D4×C47—21no—
34C22×D4×C47C2×C411yescharacteristic
35C23×C4×C47—21no—
36C23×C4×C47D411yescharacteristic
37C23×C4×C47D411yescharacteristic
38C23×C4×C47C2311yescharacteristic
39#121×C47—21no—
40#121×C47—21no—
41#121×C47D412yes—
42#121×C47Q812yes—
43C4×D4×C47—42no—
44C4×D4×C47—44no—
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×C42×C47—21no—
55C2×C42×C47—42no—
56C2×C42×C47C2×C411yescharacteristic
57C2×C42×C47C2×C411yescharacteristic
58C2×C42×C47D411yescharacteristic
59C2×C42×C47D411yescharacteristic
60#95×C47—21no—
61#95×C47—21no—
62#95×C47—21no—
63C24×C47—21no—
64C24×C47C2×D411yescharacteristic
65C2×D4×C47—22no—
66C2×D4×C47—22no—
67C2×D4×C47—21no—
68C2×D4×C47—41no—
69C2×D4×C47—41no—
70C2×D4×C47—42no—
71C2×D4×C47—42no—
72C2×D4×C47C4⋊C412yes—
73C22×C4×C47—21no—
74C22×C4×C47—22no—
75C22×C4×C47—22no—
76C22×C4×C47—22no—
77C22×C4×C47—22no—
78C22×C4×C47—21no—
79C22×C4×C47—21no—
80C22×C4×C47—41no—
81C22×C4×C47—41no—
82C22×C4×C47—41no—
83C22×C4×C47—41no—
84C22×C4×C47—42no—
85C22×C4×C47—42no—
86C22×C4×C47—42no—
87C22×C4×C47C22⋊C411yescharacteristic
88C22×C4×C47C22⋊C411yescharacteristic
89C22×C4×C47C22⋊C411yescharacteristic
90C22×C4×C47C22⋊C411yescharacteristic
91C22×C4×C47C22×C411yescharacteristic
92C22×C4×C47C2×D411yescharacteristic
93C22×C4×C47C2×D411yescharacteristic
94C22×C4×C47C2×Q811yescharacteristic
95C22×C4×C47#4111yescharacteristic
96C22×C4×C47#4111yescharacteristic
97C22×Dic47—24no—
98C22×Dic47—42no—
99C22×Dic47—42no—
100C22×Dic47—42no—
101C22×Dic47—42no—
102C22×Dic47—42no—
103C22×Dic47—42no—
104C4⋊C4×C47—21no—
105C4⋊C4×C47—42no—
106C4⋊C4×C47—41no—
107C4⋊C4×C47—41no—
108C4⋊C4×C47C4⋊C412yes—
109C22⋊C4×C47—22no—
110C22⋊C4×C47—22no—
111C22⋊C4×C47—42no—
112C22⋊C4×C47—42no—
113C22⋊C4×C47—42no—
114C22⋊C4×C47—42no—
115C42×C47—22no—
116C42×C47—22no—
117C42×C47—41no—
118C42×C47—41no—
119C42×C47—42no—
120C2×C47⋊C8—42no—
121C2×C47⋊C8—42no—
122C2×C47⋊C8—42no—
123C23×C47—21no—
124C23×C47—21no—
125C23×C47—21no—
126C23×C47—21no—
127C23×C47—22no—
128C23×C47—41no—
129C23×C47—41no—
130C23×C47—41no—
131C23×C47—41no—
132C23×C47—41no—
133C23×C47—41no—
134C23×C47C2×C22⋊C411yescharacteristic
135C23×C47#12111yescharacteristic
136C23×C47C22⋊Q811yescharacteristic
137C23×C47#13512yes—
138C23×C47AGL(1,ℤ/8)12yes—
139D4×C47—42no—
140D4×C47—42no—
141D4×C47—44no—
142D4×C47—44no—
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—22no—
152C2×C4×C47—21no—
153C2×C4×C47—21no—
154C2×C4×C47—21no—
155C2×C4×C47—21no—
156C2×C4×C47—21no—
157C2×C4×C47—21no—
158C2×C4×C47—41no—
159C2×C4×C47—41no—
160C2×C4×C47—41no—
161C2×C4×C47—41no—
162C2×C4×C47—44no—
163C2×C4×C47—44no—
164C2×C4×C47—41no—
165C2×C4×C47—41no—
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—84no—
173C2×C4×C47#9912yes—
174C2×C4×C47#10412yes—
175C2×C4×C47C2×C4⋊C411yescharacteristic
176C2×C4×C47#11711yescharacteristic
177C2×C4×C47#12111yescharacteristic
178C2×C4×C47C22⋊Q811yescharacteristic
179C2×Dic47—44no—
180C2×Dic47—44no—
181C2×Dic47—44no—
182C2×Dic47—82no—
183C2×Dic47—82no—
184C2×Dic47—82no—
185C2×Dic47—82no—
186C47⋊C8—82no—
187#8575—471nomaximal, Sylow
188C22×C47—21no—
189C22×C47—21no—
190C22×C47—21no—
191C22×C47—21no—
192C22×C47—22no—
193C22×C47—22no—
194C22×C47—22no—
195C22×C47—22no—
196C22×C47—42no—
197C22×C47—42no—
198C22×C47—42no—
199C22×C47—81no—
200C22×C47—81no—
201C22×C47—81no—
202C22×C47—81no—
203C22×C47#38011yescharacteristic
204C22×C47C2×#9911yescharacteristic
205C22×C47#41011yescharacteristic
206C22×C47#41611yescharacteristic
207C22×C47#41811yescharacteristic
208C22×C47C2×#10411yescharacteristic
209C22×C47#42111yescharacteristic
210C22×C47#46312yes—
211C22×C47#47812yes—
212C4×C47—21no—
213C4×C47—21no—
214C4×C47—21no—
215C4×C47—21no—
216C4×C47—22no—
217C4×C47—22no—
218C4×C47—42no—
219C4×C47—41no—
220C4×C47—41no—
221C4×C47—81no—
222C4×C47—81no—
223Dic47—84no—
224C4×#121D4711yescharacteristic
225#1797—471no—
226C2×#327—472no—
227#1434—471no—
228#1294—472no—
229C2×C47—22no—
230C2×C47—22no—
231C2×C47—82no—
232C2×C47#150111yescharacteristic
233C2×C47#150511yescharacteristic
234C2×C47#151011yescharacteristic
235C2×C47#152111yescharacteristic
236C2×C47#179911yescharacteristic
237C2×C47#180411yescharacteristic
238C2×C47#180611yescharacteristic
239C2×#121C2×D4711yescharacteristic
240C2×C4×D4—21no—
241C2×C4×D4Dic4711yescharacteristic, complemented
242C22×C42C2×D4711yescharacteristic
243C2×C4⋊C8—941no—
244C2×C22⋊C8—472no—
245#390Dic4711yescharacteristic, complemented
246#389Dic4711yescharacteristic, complemented
247#388—21no—
248#386—21no—
249#380C2×D4711yescharacteristic
250C4×C4⋊C4Dic4711yescharacteristic, complemented
251C4×C22⋊C4—21no—
252C2×#95—472no—
253C2×#95—941no—
254C2×#95—941no—
255#336—941no—
256#327—944no—
257C47#857511yescharacteristic, complemented, Sylow
258C22×D4—21no—
259C22×D4C2×Dic4711yescharacteristic
260C23×C4—21no—
261C23×C4—942no—
262C23×C4C22⋊D4711yescharacteristic
263C23×C4C22⋊D4711yescharacteristic
264C23×C4C22×D4711yescharacteristic
265C22×C8—942no—
266#121—21no—
267#121—21no—
268#121Dic9412yes—
269#121D18812yes—
270C4×D4—42no—
271C4×D4—44no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4C2×Dic4711yescharacteristic
274C2×C4⋊C4C2×Dic4711yescharacteristic
275C2×C4⋊C4C2×Dic4711yescharacteristic
276C2×C22⋊C4—21no—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—21no—
280C2×C22⋊C4—21no—
281C2×C42—21no—
282C2×C42—42no—
283C2×C42C2×Dic4711yescharacteristic
284C2×C42C2×Dic4711yescharacteristic
285C2×C42C22⋊D4711yescharacteristic
286C2×C42C22⋊D4711yescharacteristic
287C4⋊C8—1882no—
288C22⋊C8—942no—
289C22⋊C8—942no—
290#95—21no—
291#95—21no—
292#95—21no—
293#95—944no—
294#95—942no—
295#95—942no—
296#95—1882no—
297C24—21no—
298C24C2×C22⋊D4711yescharacteristic
299C2×D4—22no—
300C2×D4—22no—
301C2×D4—21no—
302C2×D4—41no—
303C2×D4—41no—
304C2×D4—42no—
305C2×D4—42no—
306C2×D4C4⋊Dic4712yes—
307C22×C4—21no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—944no—
322C22×C4—1882no—
323C22×C4—1882no—
324C22×C4—1882no—
325C22×C4—1882no—
326C22×C4—1882no—
327C22×C4—1882no—
328C22×C4C22⋊Dic4711yescharacteristic
329C22×C4C22⋊Dic4711yescharacteristic
330C22×C4C22⋊Dic4711yescharacteristic
331C22×C4C22⋊Dic4711yescharacteristic
332C22×C4C2×Dic9411yescharacteristic
333C22×C4C2×D18811yescharacteristic
334C22×C4#1062535011yescharacteristic
335C22×C4#1062535011yescharacteristic
336C22×C4C22×Dic4711yescharacteristic
337C22×C4C2×C22⋊D4711yescharacteristic
338C2×C8—1882no—
339C2×C8—1882no—
340C2×C8—1882no—
341C4⋊C4—21no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C4⋊C4—42no—
345C4⋊C4C4⋊Dic4712yes—
346C22⋊C4—22no—
347C22⋊C4—22no—
348C22⋊C4—42no—
349C22⋊C4—42no—
350C22⋊C4—42no—
351C22⋊C4—42no—
352C42—22no—
353C42—22no—
354C42—41no—
355C42—41no—
356C42—42no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—21no—
361C23—22no—
362C23—41no—
363C23—41no—
364C23—41no—
365C23—41no—
366C23—41no—
367C23—41no—
368C23C2×C22⋊Dic4711yescharacteristic
369C23#1309399811yescharacteristic
370C23#1309400311yescharacteristic
371C23#1309404412yes—
372C23#1309405012yes—
373D4—42no—
374D4—42no—
375D4—44no—
376D4—44no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—22no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—44no—
397C2×C4—44no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—84no—
407C2×C4—3762no—
408C2×C4—3762no—
409C2×C4—3762no—
410C2×C4—3762no—
411C2×C4—1884no—
412C2×C4—1884no—
413C2×C4—1884no—
414C2×C4#1309393912yes—
415C2×C4#1309395112yes—
416C2×C4C2×C4⋊Dic4711yescharacteristic
417C2×C4#1309398311yescharacteristic
418C2×C4#1309399811yescharacteristic
419C2×C4#1309400311yescharacteristic
420C8—3762no—
421C22—22no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—22no—
427C22—22no—
428C22—22no—
429C22—42no—
430C22—42no—
431C22—42no—
432C22—81no—
433C22—81no—
434C22—81no—
435C22—81no—
436C22#1568898311yescharacteristic
437C22C2×#1309393911yescharacteristic
438C22#1568909711yescharacteristic
439C22#1568911911yescharacteristic
440C22#1568912811yescharacteristic
441C22C2×#1309395111yescharacteristic
442C22#1568914011yescharacteristic
443C22#1568933812yes—
444C22#1568942312yes—
445C4—21no—
446C4—21no—
447C4—21no—
448C4—21no—
449C4—22no—
450C4—22no—
451C4—42no—
452C4—41no—
453C4—41no—
454C4—81no—
455C4—81no—
456C4—3764no—
457C2—22no—
458C2—22no—
459C2—82no—
460C2#2024187911yescharacteristic
461C2#2024189611yescharacteristic
462C2#2024191711yescharacteristic
463C2#2024196511yescharacteristic
464C2#2024337611yescharacteristic
465C2#2024339811yescharacteristic
466C2#2024340811yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic47
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC4×#121×C47C2
socleC23×C47C2×C22⋊C4
prime radicalC2×#121×C47C22
cosocleC23×C4×C47C23
hypercenterC4×#121D47
hypocenterC47#8575
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×#121

Complements

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

NH ≅ G/Ntype
1. #20243368C2semidirect
3. #20241615C2semidirect
4. #20241266C2semidirect
15. C2×C4×D4×C47C4semidirect
17. C47×#390C4semidirect
18. C47×#389C4semidirect
22. C4×C4⋊C4×C47C4semidirect
241. C2×C4×D4Dic47semidirect
245. #390Dic47semidirect
246. #389Dic47semidirect
250. C4×C4⋊C4Dic47semidirect
257. C47#8575semidirect

Maximal subgroups

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

subgroupindex
#202433682
C2×#156888572
#202416152
#202412662
C4×#121×C472
#857547