Finite Group Databasebeta

← #26182193

Subgroups — #26182193

Order 12032.

1210 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20252417C212yesmaximal, complemented
2#20241732C212yesmaximal, complemented
3#20241726C212yesmaximal, complemented
4#20241722C212yesmaximal, complemented
5#20241718C212yesmaximal, complemented
6#20241706C212yesmaximal, complemented
7C2×#15688849C212yesmaximal, complemented
8C2×C47×#520C211yesmaximal, characteristic
9#15689195—28no—
10#15689180C2214yes—
11#15689087C2214yes—
12C2×C22⋊(C47⋊C8)C2214yes—
13#15688849C2214yes—
14#15688849C2214yes—
15#15688849C2214yes—
16#15688847C2214yes—
17C22×#41×C47C411yescharacteristic
18C47×#520—28no—
19C2×C4×Q8×C47C411yescharacteristic
20C2×C4×Q8×C47C2211yescharacteristic
21C2×C4×D4×C47C411yescharacteristic
22C2×C4×D4×C47C411yescharacteristic
23C2×C4×D4×C47C411yescharacteristic
24C2×C4×D4×C47C2211yescharacteristic
25C2×C4×D4×C47C2211yescharacteristic
26C2×C4×D4×C47C2211yescharacteristic
27C2×#117×C47C411yescharacteristic
28C2×#117×C47C411yescharacteristic
29C2×#117×C47C2211yescharacteristic
30C22×C4⋊C4×C47C411yescharacteristic
31C22×C4⋊C4×C47C2211yescharacteristic
32C22×C4⋊C4×C47C2211yescharacteristic
33C2×#10625329—24no—
34C22×C47⋊C8—24no—
35C4⋊(C47⋊C8)—24no—
36C22⋊(C47⋊C8)—24no—
37C22⋊(C47⋊C8)—24no—
38C22⋊(C47⋊C8)C2318yes—
39#13093935—24no—
40C2×#41×C47—24no—
41C2×#41×C47—24no—
42C2×#41×C47—24no—
43C22×Q8×C47C2×C411yescharacteristic
44C22×D4×C47C2×C411yescharacteristic
45C22×D4×C47C2×C411yescharacteristic
46C22×D4×C47C2×C411yescharacteristic
47C23×C4×C47C2×C411yescharacteristic
48C23×C4×C47C2×C411yescharacteristic
49C23×C4×C47C2×C411yescharacteristic
50C23×C4×C47C2×C411yescharacteristic
51C23×C4×C47C2×C411yescharacteristic
52C23×C4×C47C2×C411yescharacteristic
53C23×C4×C47C2×C411yescharacteristic
54C23×C4×C47C2311yescharacteristic
55C23×C4×C47C2311yescharacteristic
56C4×Q8×C47—24no—
57C4×Q8×C47—24no—
58C4×D4×C47—24no—
59C4×D4×C47—24no—
60C4×D4×C47—24no—
61C4×D4×C47—24no—
62C4×D4×C47—24no—
63C4×D4×C47D418yes—
64#117×C47—24no—
65#117×C47—24no—
66#117×C47—24no—
67C2×C4⋊C4×C47—24no—
68C2×C4⋊C4×C47—24no—
69C2×C4⋊C4×C47—24no—
70C2×C4⋊C4×C47C2×C411yescharacteristic
71C2×C4⋊C4×C47C2×C411yescharacteristic
72C2×C4⋊C4×C47C2×C411yescharacteristic
73C2×C4⋊C4×C47C2×C411yescharacteristic
74C2×C4⋊C4×C47C2×C411yescharacteristic
75C2×C4⋊C4×C47C2×C411yescharacteristic
76C2×C4⋊C4×C47C2×C411yescharacteristic
77C2×C4⋊C4×C47C2311yescharacteristic
78C2×C4⋊C4×C47C2311yescharacteristic
79C2×C4⋊C4×C47C2311yescharacteristic
80C2×C22⋊C4×C47C2×C411yescharacteristic
81C2×C22⋊C4×C47C2×C411yescharacteristic
82C2×C22⋊C4×C47C2×C411yescharacteristic
83C2×C22⋊C4×C47C2×C411yescharacteristic
84C2×C22⋊C4×C47C2×C411yescharacteristic
85C2×C22⋊C4×C47C2311yescharacteristic
86C2×C42×C47C2×C411yescharacteristic
87C2×C42×C47C2×C411yescharacteristic
88C2×C42×C47C2×C411yescharacteristic
89C2×C42×C47C2×C411yescharacteristic
90C2×C42×C47C2×C411yescharacteristic
91C2×C42×C47C2311yescharacteristic
92C24×C47C22×C411yescharacteristic
93C24×C47C22×C411yescharacteristic
94C24×C47C22×C411yescharacteristic
95#41×C47—232no—
96C2×Q8×C47—22no—
97C2×Q8×C47—22no—
98C2×Q8×C47—22no—
99C2×D4×C47—22no—
100C2×D4×C47—22no—
101C2×D4×C47—22no—
102C2×D4×C47—22no—
103C2×D4×C47—22no—
104C2×D4×C47—22no—
105C2×D4×C47—22no—
106C2×D4×C47C22⋊C414yes—
107C2×D4×C47M1614yescomplemented
108C22×C4×C47—22no—
109C22×C4×C47—22no—
110C22×C4×C47—22no—
111C22×C4×C47—22no—
112C22×C4×C47—22no—
113C22×C4×C47—22no—
114C22×C4×C47—22no—
115C22×C4×C47—22no—
116C22×C4×C47—22no—
117C22×C4×C47—22no—
118C22×C4×C47—22no—
119C22×C4×C47—22no—
120C22×C4×C47—81no—
121C22×C4×C47—81no—
122C22×C4×C47—81no—
123C22×C4×C47—81no—
124C22×C4×C47—81no—
125C22×C4×C47—81no—
126C22×C4×C47M1614yescomplemented
127C22×C4×C47C22×C411yescharacteristic
128C22×C4×C47C22×C411yescharacteristic
129C22×C4×C47C22×C411yescharacteristic
130C22×C4×C47C22×C411yescharacteristic
131C22×C4×C47C22×C411yescharacteristic
132C22×C4×C47C22×C411yescharacteristic
133C22×C4×C47C22×C411yescharacteristic
134C22×C4×C47C22×C411yescharacteristic
135C22×C4×C47C22×C411yescharacteristic
136C22×C4×C47C22×C411yescharacteristic
137C22×C4×C47C22×C411yescharacteristic
138C22×C4×C47C2×D414yes—
139C22×C4×C47C2×D414yes—
140C22×C4×C47C2411yescharacteristic
141C4⋊C4×C47—22no—
142C4⋊C4×C47—22no—
143C4⋊C4×C47—22no—
144C4⋊C4×C47—22no—
145C4⋊C4×C47—22no—
146C4⋊C4×C47—22no—
147C4⋊C4×C47—22no—
148C4⋊C4×C47—22no—
149C4⋊C4×C47—22no—
150C4⋊C4×C47C22⋊C414yes—
151C22⋊C4×C47—22no—
152C22⋊C4×C47—22no—
153C22⋊C4×C47—22no—
154C22⋊C4×C47—22no—
155C22⋊C4×C47C22⋊C414yes—
156C22⋊C4×C47C22⋊C414yes—
157C42×C47—22no—
158C42×C47—22no—
159C42×C47—22no—
160C42×C47—22no—
161C42×C47—22no—
162C42×C47C2×D414yes—
163#10625329—84no—
164C2×C47⋊C8—28no—
165C2×C47⋊C8—84no—
166C23×C47—21no—
167C23×C47—21no—
168C23×C47—21no—
169C23×C47—21no—
170C23×C47—41no—
171C23×C47—41no—
172C23×C47—41no—
173C23×C47—41no—
174C23×C47—41no—
175C23×C47—41no—
176C23×C47C2×C22⋊C412yes—
177C23×C47C2×C22⋊C412yes—
178C23×C47C2×M1612yes—
179C23×C47C2×M1612yes—
180C23×C47C2×M1612yes—
181C23×C47C23×C411yescharacteristic
182Q8×C47—28no—
183D4×C47—28no—
184D4×C47—28no—
185D4×C47—28no—
186C2×C4×C47—28no—
187C2×C4×C47—28no—
188C2×C4×C47—21no—
189C2×C4×C47—21no—
190C2×C4×C47—21no—
191C2×C4×C47—21no—
192C2×C4×C47—21no—
193C2×C4×C47—21no—
194C2×C4×C47—21no—
195C2×C4×C47—21no—
196C2×C4×C47—28no—
197C2×C4×C47—21no—
198C2×C4×C47—21no—
199C2×C4×C47—21no—
200C2×C4×C47—21no—
201C2×C4×C47—44no—
202C2×C4×C47—44no—
203C2×C4×C47—41no—
204C2×C4×C47—41no—
205C2×C4×C47—41no—
206C2×C4×C47—41no—
207C2×C4×C47—41no—
208C2×C4×C47—41no—
209C2×C4×C47—41no—
210C2×C4×C47—41no—
211C2×C4×C47—82no—
212C2×C4×C47—82no—
213C2×C4×C47—82no—
214C2×C4×C47—82no—
215C2×C4×C47C2×C22⋊C412yes—
216C2×C4×C47C2×C22⋊C412yes—
217C2×C4×C47C2×C22⋊C412yes—
218C2×C4×C47C2×C22⋊C412yes—
219C2×C4×C47C2×M1612yes—
220C2×C4×C47C2×M1612yes—
221C2×C4×C47C2×M1612yes—
222C2×C4×C47C22×D412yes—
223C47⋊C8—48no—
224#17368—471nomaximal, Sylow
225C22×C47—21no—
226C22×C47—22no—
227C22×C47—21no—
228C22×C47—22no—
229C22×C47—22no—
230C22×C47—22no—
231C22×C47—22no—
232C22×C47—22no—
233C22×C47—42no—
234C22×C47—42no—
235C22×C47—42no—
236C22×C47—42no—
237C22×C47—42no—
238C22×C47—42no—
239C22×C47—41no—
240C22×C47—41no—
241C22×C47#40714yes—
242C22×C47C22×C22⋊C411yescharacteristic
243C22×C47C22×M1611yescharacteristic
244C22×C47C22×M1611yescharacteristic
245C4×C47—22no—
246C4×C47—22no—
247C4×C47—22no—
248C4×C47—22no—
249C4×C47—22no—
250C4×C47—22no—
251C4×C47—22no—
252C4×C47—22no—
253C4×C47—41no—
254C4×C47—41no—
255C4×C47—41no—
256C4×C47—41no—
257C2×#520D4711yescharacteristic
258#2973—472no—
259#1467—472no—
260#1465—472no—
261#1464—472no—
262#1463—472no—
263#1459—472no—
264C2×#324—472no—
265C2×C47—21no—
266C2×C47—21no—
267C2×C47—24no—
268C2×C47—42no—
269C2×C47—42no—
270C2×C47C2×#40711yescharacteristic
271C2×C47#288311yescharacteristic
272C2×C47#288911yescharacteristic
273C22×#41Dic4711yescharacteristic
274#520—28no—
275C2×C4×Q8Dic4711yescharacteristic
276C2×C4×Q8C2×D4711yescharacteristic
277C2×C4×D4Dic4711yescharacteristic
278C2×C4×D4Dic4711yescharacteristic
279C2×C4×D4Dic4711yescharacteristic
280C2×C4×D4C2×D4711yescharacteristic
281C2×C4×D4C2×D4711yescharacteristic
282C2×C4×D4C2×D4711yescharacteristic
283C2×#117Dic4711yescharacteristic
284C2×#117Dic4711yescharacteristic
285C2×#117C2×D4711yescharacteristic
286C22×C4⋊C4Dic4711yescharacteristic
287C22×C4⋊C4C2×D4711yescharacteristic
288C22×C4⋊C4C2×D4711yescharacteristic
289#435—948no—
290#432—474no—
291#407—474no—
292C2×C22⋊C8—474no—
293#324—474no—
294#324—474no—
295#324—474no—
296#323—474no—
297C47#1736811yescharacteristic, complemented, Sylow
298C2×#41—24no—
299C2×#41—24no—
300C2×#41—24no—
301C22×Q8C2×Dic4711yescharacteristic
302C22×D4C2×Dic4711yescharacteristic
303C22×D4C2×Dic4711yescharacteristic
304C22×D4C2×Dic4711yescharacteristic
305C23×C4C2×Dic4711yescharacteristic
306C23×C4C2×Dic4711yescharacteristic
307C23×C4C2×Dic4711yescharacteristic
308C23×C4C2×Dic4711yescharacteristic
309C23×C4C2×Dic4711yescharacteristic
310C23×C4C2×Dic4711yescharacteristic
311C23×C4C2×Dic4711yescharacteristic
312C23×C4C22×D4711yescharacteristic
313C23×C4C22×D4711yescharacteristic
314C2×M16—944no—
315C22×C8—944no—
316C4×Q8—24no—
317C4×Q8—24no—
318C4×D4—24no—
319C4×D4—24no—
320C4×D4—24no—
321C4×D4—24no—
322C4×D4—24no—
323C4×D4C22⋊D4718yes—
324#117—24no—
325#117—24no—
326#117—24no—
327C2×C4⋊C4—24no—
328C2×C4⋊C4—24no—
329C2×C4⋊C4—24no—
330C2×C4⋊C4C2×Dic4711yescharacteristic
331C2×C4⋊C4C2×Dic4711yescharacteristic
332C2×C4⋊C4C2×Dic4711yescharacteristic
333C2×C4⋊C4C2×Dic4711yescharacteristic
334C2×C4⋊C4C2×Dic4711yescharacteristic
335C2×C4⋊C4C2×Dic4711yescharacteristic
336C2×C4⋊C4C2×Dic4711yescharacteristic
337C2×C4⋊C4C22×D4711yescharacteristic
338C2×C4⋊C4C22×D4711yescharacteristic
339C2×C4⋊C4C22×D4711yescharacteristic
340C2×C22⋊C4C2×Dic4711yescharacteristic
341C2×C22⋊C4C2×Dic4711yescharacteristic
342C2×C22⋊C4C2×Dic4711yescharacteristic
343C2×C22⋊C4C2×Dic4711yescharacteristic
344C2×C22⋊C4C2×Dic4711yescharacteristic
345C2×C22⋊C4C22×D4711yescharacteristic
346C2×C42C2×Dic4711yescharacteristic
347C2×C42C2×Dic4711yescharacteristic
348C2×C42C2×Dic4711yescharacteristic
349C2×C42C2×Dic4711yescharacteristic
350C2×C42C2×Dic4711yescharacteristic
351C2×C42C22×D4711yescharacteristic
352C4⋊C8—944no—
353C22⋊C8—478no—
354C22⋊C8—944no—
355C22⋊C8—944no—
356#97—944no—
357C24C22×Dic4711yescharacteristic
358C24C22×Dic4711yescharacteristic
359C24C22×Dic4711yescharacteristic
360#41—232no—
361C2×Q8—22no—
362C2×Q8—22no—
363C2×Q8—22no—
364C2×D4—22no—
365C2×D4—22no—
366C2×D4—22no—
367C2×D4—22no—
368C2×D4—22no—
369C2×D4—22no—
370C2×D4—22no—
371C2×D4#1062532914yescomplemented
372C2×D4C22⋊Dic4714yes—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—81no—
386C22×C4—81no—
387C22×C4—81no—
388C22×C4—81no—
389C22×C4—81no—
390C22×C4—81no—
391C22×C4#1062532914yescomplemented
392C22×C4C22×Dic4711yescharacteristic
393C22×C4C22×Dic4711yescharacteristic
394C22×C4C22×Dic4711yescharacteristic
395C22×C4C22×Dic4711yescharacteristic
396C22×C4C22×Dic4711yescharacteristic
397C22×C4C22×Dic4711yescharacteristic
398C22×C4C22×Dic4711yescharacteristic
399C22×C4C22×Dic4711yescharacteristic
400C22×C4C22×Dic4711yescharacteristic
401C22×C4C22×Dic4711yescharacteristic
402C22×C4C22×Dic4711yescharacteristic
403C22×C4C2×C22⋊D4714yes—
404C22×C4C2×C22⋊D4714yes—
405C22×C4C23×D4711yescharacteristic
406M16—3764no—
407C2×C8—948no—
408C2×C8—3764no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4C22⋊Dic4714yes—
419C22⋊C4—22no—
420C22⋊C4—22no—
421C22⋊C4—22no—
422C22⋊C4—22no—
423C22⋊C4C22⋊Dic4714yes—
424C22⋊C4C22⋊Dic4714yes—
425C42—22no—
426C42—22no—
427C42—22no—
428C42—22no—
429C42—22no—
430C42C2×C22⋊D4714yes—
431C23—21no—
432C23—21no—
433C23—21no—
434C23—21no—
435C23—41no—
436C23—41no—
437C23—41no—
438C23—41no—
439C23—41no—
440C23—41no—
441C23C2×C22⋊Dic4712yes—
442C23C2×C22⋊Dic4712yes—
443C23C2×#1062532912yes—
444C23C2×#1062532912yes—
445C23C2×#1062532912yes—
446C23C23×Dic4711yescharacteristic
447Q8—28no—
448D4—28no—
449D4—28no—
450D4—28no—
451C2×C4—28no—
452C2×C4—28no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—28no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4C2×C22⋊Dic4712yes—
481C2×C4C2×C22⋊Dic4712yes—
482C2×C4C2×C22⋊Dic4712yes—
483C2×C4C2×C22⋊Dic4712yes—
484C2×C4C2×#1062532912yes—
485C2×C4C2×#1062532912yes—
486C2×C4C2×#1062532912yes—
487C2×C4C22×C22⋊D4712yes—
488C8—1888no—
489C22—21no—
490C22—22no—
491C22—21no—
492C22—22no—
493C22—22no—
494C22—22no—
495C22—22no—
496C22—22no—
497C22—42no—
498C22—42no—
499C22—42no—
500C22—42no—
501C22—42no—
502C22—42no—
503C22—41no—
504C22—41no—
505C22#1568908714yes—
506C22C22×C22⋊Dic4711yescharacteristic
507C22C22×#1062532911yescharacteristic
508C22C22×#1062532911yescharacteristic
509C4—22no—
510C4—22no—
511C4—22no—
512C4—22no—
513C4—22no—
514C4—22no—
515C4—22no—
516C4—22no—
517C4—41no—
518C4—41no—
519C4—41no—
520C4—41no—
521C2—21no—
522C2—21no—
523C2—24no—
524C2—42no—
525C2—42no—
526C2C2×#1568908711yescharacteristic
527C2#2025184911yescharacteristic
528C2#2025187111yescharacteristic
subgroupG/H
centerC22C22×C22⋊Dic47
commutatorC23×C47C23×C4
FrattiniC22×C4C23×D47
FittingC2×C47×#520C2
socleC22×C47C22×C22⋊C4
prime radicalC22×#41×C47C4
cosocleC22×C4×C47C24
hypercenterC2×#520D47
hypocenterC47#17368
LCSG ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ C2×#520

Complements

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

NH ≅ G/Ntype
1. #20252417C2semidirect
2. #20241732C2semidirect
3. #20241726C2semidirect
4. #20241722C2semidirect
5. #20241718C2semidirect
6. #20241706C2semidirect
7. C2×#15688849C2semidirect
107. C2×D4×C47M16semidirect
126. C22×C4×C47M16semidirect
297. C47#17368semidirect
371. C2×D4#10625329semidirect
391. C22×C4#10625329semidirect

Maximal subgroups

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

subgroupindex
#202524172
#202417322
#202417262
#202417222
#202417182
#202417062
C2×#156888492
C2×C47×#5202
#1736847