Finite Group Databasebeta

← #26232833

Subgroups — #26232833

Order 12032.

1114 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20246265C212yesmaximal, complemented
2#20244374C212yesmaximal, complemented
3#20244374C214yesmaximal, complemented
4#20243376C211yesmaximal, characteristic
5#20243376C212yesmaximal
6C4×#13093939C211yesmaximal, characteristic
7C4×#13093939C212yesmaximal
8C2×C47×#554C211yesmaximal, characteristic
9C2×#13093939C2211yescharacteristic
10C2×#13093939C2212yes—
11C2×#13093939C2211yescharacteristic
12#15689022—22no—
13#15689009—24no—
14#15688983C2212yes—
15#15688983C2214yes—
16C4⋊C4×Dic47—22no—
17C4×C22⋊Dic47C2212yes—
18C4×C22⋊Dic47C2214yes—
19#15688890C2214yes—
20#15688890C2212yes—
21#15688890C2214yes—
22#15688890C2212yes—
23C47×#554—22no—
24C47×#554—24no—
25C47×#554—22no—
26C2×C4⋊Q8×C47C411yescharacteristic, complemented
27C2×#125×C47C412yescomplemented
28C2×C22⋊Q8×C47C412yescomplemented
29C2×C22⋊Q8×C47C411yescharacteristic, complemented
30C2×C22⋊Q8×C47C2212yes—
31C2×C22⋊Q8×C47C2211yescharacteristic
32C2×C4×Q8×C47C411yescharacteristic, complemented
33C2×C4×D4×C47C2212yes—
34C2×C4×D4×C47C2211yescharacteristic
35C22×C4⋊C4×C47C411yescharacteristic, complemented
36C22×C4⋊C4×C47C2211yescharacteristic
37C2×C4⋊Dic47—22no—
38C2×C4⋊Dic47—24no—
39C2×C22⋊Dic47C2312yes—
40C2×C22⋊Dic47C2314yes—
41C2×C22⋊Dic47C2312yes—
42C2×C4×Dic47—22no—
43C2×C4×Dic47—24no—
44#13093939—22no—
45#13093939—24no—
46#13093939Q814yescomplemented
47C47⋊#95—22no—
48C47⋊#95—24no—
49C47⋊#95—24no—
50C47⋊#95—22no—
51C22×Q8×C47C2×C411yescharacteristic
52C22×Q8×C47C2×C412yes—
53C22×D4×C47C2311yescharacteristic
54C23×C4×C47C2×C411yescharacteristic
55C23×C4×C47C2×C412yes—
56C23×C4×C47C2311yescharacteristic
57C23×C4×C47C2312yes—
58C4⋊Q8×C47—22no—
59C4⋊Q8×C47—22no—
60#125×C47—24no—
61#125×C47—24no—
62C22⋊Q8×C47—24no—
63C22⋊Q8×C47—24no—
64C22⋊Q8×C47—22no—
65C22⋊Q8×C47—22no—
66C22⋊Q8×C47—24no—
67C22⋊Q8×C47—24no—
68C22⋊Q8×C47—22no—
69C22⋊Q8×C47—22no—
70C4×Q8×C47—22no—
71C4×Q8×C47—21no—
72C4×Q8×C47—21no—
73C4×D4×C47—24no—
74C4×D4×C47—24no—
75C4×D4×C47—22no—
76C4×D4×C47—22no—
77C2×C4⋊C4×C47—22no—
78C2×C4⋊C4×C47—21no—
79C2×C4⋊C4×C47—21no—
80C2×C4⋊C4×C47C2×C412yes—
81C2×C4⋊C4×C47C2×C411yescharacteristic
82C2×C4⋊C4×C47C2×C411yescharacteristic
83C2×C4⋊C4×C47C2×C412yes—
84C2×C4⋊C4×C47C2×C412yes—
85C2×C4⋊C4×C47C2×C412yes—
86C2×C4⋊C4×C47C2×C412yes—
87C2×C4⋊C4×C47C2×C411yescharacteristic
88C2×C4⋊C4×C47C2×C412yes—
89C2×C4⋊C4×C47C2×C411yescharacteristic
90C2×C4⋊C4×C47D412yes—
91C2×C4⋊C4×C47D414yes—
92C2×C4⋊C4×C47D412yes—
93C2×C22⋊C4×C47C2×C412yes—
94C2×C22⋊C4×C47C2×C411yescharacteristic
95C2×C22⋊C4×C47C2312yes—
96C2×C22⋊C4×C47C2311yescharacteristic
97C2×C42×C47C2×C412yes—
98C2×C42×C47C2×C411yescharacteristic
99C24×C47C22×C411yescharacteristic
100C24×C47C2411yescharacteristic
101C2×Q8×C47—21no—
102C2×Q8×C47—21no—
103C2×Q8×C47—24no—
104C2×Q8×C47—42no—
105C2×Q8×C47—82no—
106C2×D4×C47—22no—
107C2×D4×C47—22no—
108C2×D4×C47C2×Q812yes—
109C2×D4×C47#4112yes—
110C22×C4×C47—21no—
111C22×C4×C47—21no—
112C22×C4×C47—22no—
113C22×C4×C47—22no—
114C22×C4×C47—42no—
115C22×C4×C47—42no—
116C22×C4×C47—82no—
117C22×C4×C47—82no—
118C22×C4×C47C22×C411yescharacteristic
119C22×C4×C47C22×C411yescharacteristic
120C22×C4×C47C22×C412yes—
121C22×C4×C47C22×C411yescharacteristic
122C22×C4×C47C22×C411yescharacteristic
123C22×C4×C47C22×C412yes—
124C22×C4×C47C22×C412yes—
125C22×C4×C47C22×C411yescharacteristic
126C22×C4×C47C22×C412yes—
127C22×C4×C47C2×D412yes—
128C22×C4×C47C2×D412yes—
129C22×C4×C47C2×D414yes—
130C22×C4×C47C2×D414yes—
131C22×Dic47—22no—
132C22×Dic47—22no—
133C22×Dic47—24no—
134C4⋊C4×C47—22no—
135C4⋊C4×C47—22no—
136C4⋊C4×C47—21no—
137C4⋊C4×C47—21no—
138C4⋊C4×C47—22no—
139C4⋊C4×C47—24no—
140C4⋊C4×C47—22no—
141C4⋊C4×C47—22no—
142C4⋊C4×C47—24no—
143C4⋊C4×C47—24no—
144C4⋊C4×C47C22⋊C412yes—
145C4⋊C4×C47C22⋊C411yescharacteristic
146C4⋊C4×C47C22⋊C411yescharacteristic
147C4⋊C4×C47C22⋊C414yes—
148C4⋊C4×C47C22⋊C414yes—
149C4⋊C4×C47C22⋊C412yes—
150C4⋊C4×C47C22⋊C412yes—
151C22⋊C4×C47—24no—
152C22⋊C4×C47—22no—
153C22⋊C4×C47—24no—
154C22⋊C4×C47—22no—
155C42×C47—22no—
156C42×C47—22no—
157C42×C47—21no—
158C42×C47—21no—
159C22⋊Dic47—22no—
160C22⋊Dic47—22no—
161C22⋊Dic47—24no—
162C22⋊Dic47—22no—
163C22⋊Dic47—24no—
164C22⋊Dic47C2×Q814yes—
165C4⋊Dic47—44no—
166C4⋊Dic47—84no—
167C4×Dic47—44no—
168C4×Dic47—84no—
169C23×C47—21no—
170C23×C47—21no—
171C23×C47—41no—
172C23×C47—41no—
173C23×C47—41no—
174C23×C47—41no—
175C23×C47C4×Q812yes—
176C23×C47C23×C411yescharacteristic
177C23×C47C22×D412yes—
178C23×C47C22×Q811yescharacteristic
179C23×C47C2×#4111yescharacteristic
180C23×C472−1+412yes—
181Q8×C47—42no—
182Q8×C47—42no—
183Q8×C47—84no—
184D4×C47—24no—
185D4×C47—24no—
186C2×C4×C47—21no—
187C2×C4×C47—21no—
188C2×C4×C47—24no—
189C2×C4×C47—22no—
190C2×C4×C47—22no—
191C2×C4×C47—21no—
192C2×C4×C47—21no—
193C2×C4×C47—22no—
194C2×C4×C47—44no—
195C2×C4×C47—44no—
196C2×C4×C47—41no—
197C2×C4×C47—42no—
198C2×C4×C47—42no—
199C2×C4×C47—42no—
200C2×C4×C47—42no—
201C2×C4×C47—42no—
202C2×C4×C47—41no—
203C2×C4×C47—42no—
204C2×C4×C47—42no—
205C2×C4×C47—41no—
206C2×C4×C47—82no—
207C2×C4×C47—82no—
208C2×C4×C47C2×C22⋊C412yes—
209C2×C4×C47C2×C22⋊C412yes—
210C2×C4×C47C2×C22⋊C411yescharacteristic
211C2×C4×C47C2×C22⋊C412yes—
212C2×C4×C47C2×C22⋊C414yes—
213C2×C4×C47C2×C22⋊C411yescharacteristic
214C2×C4×C47C4×Q812yes—
215C2×Dic47—24no—
216C2×Dic47—42no—
217C2×Dic47—42no—
218C2×Dic47—42no—
219C2×Dic47—44no—
220C2×Dic47—44no—
221C2×Dic47—42no—
222C2×Dic47—42no—
223C2×Dic47—44no—
224#22513—471nomaximal, Sylow
225C22×C47—22no—
226C22×C47—21no—
227C22×C47—22no—
228C22×C47—21no—
229C22×C47—22no—
230C22×C47—41no—
231C22×C47—41no—
232C22×C47—42no—
233C22×C47—42no—
234C22×C47—42no—
235C22×C47—41no—
236C22×C47—41no—
237C22×C47C22×C22⋊C411yescharacteristic
238C22×C47C2×C4×Q811yescharacteristic
239C22×C47#51911yescharacteristic
240C22×C47#54812yes—
241C22×C47D4×Q812yes—
242C4×C47—22no—
243C4×C47—21no—
244C4×C47—22no—
245C4×C47—22no—
246C4×C47—22no—
247C4×C47—24no—
248C4×C47—41no—
249C4×C47—42no—
250C4×C47—42no—
251C4×C47—41no—
252C4×C47#41012yes—
253Dic47—44no—
254Dic47—82no—
255Dic47—84no—
256C2×#554D4711yescharacteristic
257#2345—472no—
258#1996—474no—
259#1996—472no—
260#1799—472no—
261#1799—471no—
262C4×#99—472no—
263C4×#99—471no—
264C2×C47—22no—
265C2×C47—22no—
266C2×C47—21no—
267C2×C47—21no—
268C2×C47—42no—
269C2×C47Q8×C22⋊C411yescharacteristic
270C2×C47C2×#41011yescharacteristic
271C2×C47#288311yescharacteristic
272#554—22no—
273#554—24no—
274#554—22no—
275C2×C4⋊Q8Dic4711yescharacteristic, complemented
276C2×#125Dic4712yescomplemented
277C2×C22⋊Q8Dic4712yescomplemented
278C2×C22⋊Q8Dic4711yescharacteristic, complemented
279C2×C22⋊Q8C2×D4712yes—
280C2×C22⋊Q8C2×D4711yescharacteristic
281C2×C4×Q8Dic4711yescharacteristic, complemented
282C2×C4×D4C2×D4712yes—
283C2×C4×D4C2×D4711yescharacteristic
284C22×C4⋊C4Dic4711yescharacteristic, complemented
285C22×C4⋊C4C2×D4711yescharacteristic
286C2×#99—471no—
287C2×#99—472no—
288C2×#99—471no—
289#389—942no—
290#387—944no—
291#380—474no—
292#380—472no—
293C4×C4⋊C4—942no—
294C4×C22⋊C4—472no—
295C4×C22⋊C4—474no—
296#342—474no—
297#342—472no—
298#342—474no—
299#342—472no—
300C47#2251311yescharacteristic, complemented, Sylow
301C22×Q8C2×Dic4711yescharacteristic
302C22×Q8C2×Dic4712yes—
303C22×D4C22×D4711yescharacteristic
304C23×C4C2×Dic4711yescharacteristic
305C23×C4C2×Dic4712yes—
306C23×C4C22×D4711yescharacteristic
307C23×C4C22×D4712yes—
308C4⋊Q8—22no—
309C4⋊Q8—22no—
310#125—24no—
311#125—24no—
312C22⋊Q8—24no—
313C22⋊Q8—24no—
314C22⋊Q8—22no—
315C22⋊Q8—22no—
316C22⋊Q8—24no—
317C22⋊Q8—24no—
318C22⋊Q8—22no—
319C22⋊Q8—22no—
320C4×Q8—22no—
321C4×Q8—21no—
322C4×Q8—21no—
323C4×D4—24no—
324C4×D4—24no—
325C4×D4—22no—
326C4×D4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—944no—
331C2×C4⋊C4—942no—
332C2×C4⋊C4C2×Dic4712yes—
333C2×C4⋊C4C2×Dic4711yescharacteristic
334C2×C4⋊C4C2×Dic4711yescharacteristic
335C2×C4⋊C4C2×Dic4712yes—
336C2×C4⋊C4C2×Dic4712yes—
337C2×C4⋊C4C2×Dic4712yes—
338C2×C4⋊C4C2×Dic4712yes—
339C2×C4⋊C4C2×Dic4711yescharacteristic
340C2×C4⋊C4C2×Dic4712yes—
341C2×C4⋊C4C2×Dic4711yescharacteristic
342C2×C4⋊C4C22⋊D4712yes—
343C2×C4⋊C4C22⋊D4714yes—
344C2×C4⋊C4C22⋊D4712yes—
345C2×C22⋊C4—472no—
346C2×C22⋊C4—474no—
347C2×C22⋊C4—472no—
348C2×C22⋊C4C2×Dic4712yes—
349C2×C22⋊C4C2×Dic4711yescharacteristic
350C2×C22⋊C4C22×D4712yes—
351C2×C22⋊C4C22×D4711yescharacteristic
352C2×C42—944no—
353C2×C42—942no—
354C2×C42C2×Dic4712yes—
355C2×C42C2×Dic4711yescharacteristic
356#99—474no—
357#99—944no—
358#99—942no—
359#95—942no—
360#95—944no—
361#95—944no—
362#95—942no—
363C24C22×Dic4711yescharacteristic
364C24C23×D4711yescharacteristic
365C2×Q8—21no—
366C2×Q8—21no—
367C2×Q8—24no—
368C2×Q8—42no—
369C2×Q8—82no—
370C2×D4—22no—
371C2×D4—22no—
372C2×D4Q8×D4712yes—
373C2×D4#1062535412yes—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—82no—
381C22×C4—82no—
382C22×C4—942no—
383C22×C4—944no—
384C22×C4—942no—
385C22×C4C22×Dic4711yescharacteristic
386C22×C4C22×Dic4711yescharacteristic
387C22×C4C22×Dic4712yes—
388C22×C4C22×Dic4711yescharacteristic
389C22×C4C22×Dic4711yescharacteristic
390C22×C4C22×Dic4712yes—
391C22×C4C22×Dic4712yes—
392C22×C4C22×Dic4711yescharacteristic
393C22×C4C22×Dic4712yes—
394C22×C4C2×C22⋊D4712yes—
395C22×C4C2×C22⋊D4712yes—
396C22×C4C2×C22⋊D4714yes—
397C22×C4C2×C22⋊D4714yes—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—21no—
401C4⋊C4—21no—
402C4⋊C4—22no—
403C4⋊C4—24no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—24no—
407C4⋊C4—24no—
408C4⋊C4—3764no—
409C4⋊C4—1884no—
410C4⋊C4C22⋊Dic4712yes—
411C4⋊C4C22⋊Dic4711yescharacteristic
412C4⋊C4C22⋊Dic4711yescharacteristic
413C4⋊C4C22⋊Dic4714yes—
414C4⋊C4C22⋊Dic4714yes—
415C4⋊C4C22⋊Dic4712yes—
416C4⋊C4C22⋊Dic4712yes—
417C22⋊C4—24no—
418C22⋊C4—22no—
419C22⋊C4—24no—
420C22⋊C4—22no—
421C22⋊C4—474no—
422C22⋊C4—942no—
423C22⋊C4—944no—
424C22⋊C4—942no—
425C22⋊C4—944no—
426C22⋊C4—942no—
427C42—22no—
428C42—22no—
429C42—21no—
430C42—21no—
431C42—3764no—
432C42—1884no—
433C23—21no—
434C23—21no—
435C23—41no—
436C23—41no—
437C23—41no—
438C23—41no—
439C23Q8×Dic4712yes—
440C23C23×Dic4711yescharacteristic
441C23C22×C22⋊D4712yes—
442C23C2×Q8×D4711yescharacteristic
443C23C2×#1062535411yescharacteristic
444C23#1309407212yes—
445Q8—42no—
446Q8—42no—
447Q8—84no—
448D4—24no—
449D4—24no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—24no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—22no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—41no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—41no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—41no—
470C2×C4—82no—
471C2×C4—82no—
472C2×C4—944no—
473C2×C4—1884no—
474C2×C4—1882no—
475C2×C4—1882no—
476C2×C4—1884no—
477C2×C4—1884no—
478C2×C4—1882no—
479C2×C4—1882no—
480C2×C4—1882no—
481C2×C4C2×C22⋊Dic4712yes—
482C2×C4C2×C22⋊Dic4712yes—
483C2×C4C2×C22⋊Dic4711yescharacteristic
484C2×C4C2×C22⋊Dic4712yes—
485C2×C4C2×C22⋊Dic4714yes—
486C2×C4C2×C22⋊Dic4711yescharacteristic
487C2×C4Q8×Dic4712yes—
488C22—21no—
489C22—22no—
490C22—21no—
491C22—22no—
492C22—22no—
493C22—41no—
494C22—41no—
495C22—42no—
496C22—42no—
497C22—42no—
498C22—41no—
499C22—41no—
500C22C22×C22⋊Dic4711yescharacteristic
501C22C2×Q8×Dic4711yescharacteristic
502C22#1568960111yescharacteristic
503C22#1568978412yes—
504C22Q8×C22⋊D4712yes—
505C4—21no—
506C4—22no—
507C4—22no—
508C4—22no—
509C4—24no—
510C4—22no—
511C4—41no—
512C4—42no—
513C4—42no—
514C4—41no—
515C4—3764no—
516C4—3762no—
517C4—1884no—
518C4#1568909712yes—
519C2—22no—
520C2—22no—
521C2—21no—
522C2—21no—
523C2—42no—
524C2Q8×C22⋊Dic4711yescharacteristic
525C2C2×#1568909711yescharacteristic
526C2#2025184911yescharacteristic
subgroupG/H
centerC22C22×C22⋊Dic47
commutatorC23×C47C23×C4
FrattiniC24C23×D47
FittingC2×C47×#554C2
socleC22×C47C22×C22⋊C4
prime radicalC22×D4×C47C23
cosocleC24×C47C24
hypercenterC2×#554D47
hypocenterC47#22513
LCSG ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4⋊C4 ◁ C2×#554

Complements

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

NH ≅ G/Ntype
1. #20246265C2semidirect
2. #20244374C2semidirect
3. #20244374C2semidirect
26. C2×C4⋊Q8×C47C4semidirect
27. C2×#125×C47C4semidirect
28. C2×C22⋊Q8×C47C4semidirect
29. C2×C22⋊Q8×C47C4semidirect
32. C2×C4×Q8×C47C4semidirect
35. C22×C4⋊C4×C47C4semidirect
46. #13093939Q8semidirect
275. C2×C4⋊Q8Dic47semidirect
276. C2×#125Dic47semidirect
277. C2×C22⋊Q8Dic47semidirect
278. C2×C22⋊Q8Dic47semidirect
281. C2×C4×Q8Dic47semidirect
284. C22×C4⋊C4Dic47semidirect
300. C47#22513semidirect

Maximal subgroups

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

subgroupindex
#202462652
#202443742
#202433762
C4×#130939392
C2×C47×#5542
#2251347