Finite Group Databasebeta

← C2×#20245043 #26369016

Subgroups — C2×#20245043 #26369016

Order 12032.

1214 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689694C211yesmaximal, characteristic
2C2×#15689097C211yesmaximal, characteristic
3#20245043C218yesmaximal, direct product factor
4C2×#15688914C214yesmaximal, complemented
5C2×C47×#413C211yesmaximal, characteristic
6C23×Dic94C411yescharacteristic, complemented
7#15689694—22no—
8#15689694—22no—
9#15689694C414yescomplemented
10#15689694C2214yes—
11C2×#13094022—21no—
12C2×#13094011—21no—
13C2×#13094005C412yescomplemented
14C2×#13094005C2212yes—
15C2×#13094003—21no—
16C2×#13094003—21no—
17C2×#13093983—21no—
18C2×#13093983C411yescharacteristic, complemented
19#15689097—22no—
20#15689097C2214yes—
21C2×#13093939—21no—
22C2×#13093939C2212yes—
23#15688914C22116yescomplemented
24C22×#41×C47C2211yescharacteristic
25C22×M16×C47—21no—
26C47×#413—22no—
27C47×#413C2214yes—
28C2×#101×C47—21no—
29C2×#100×C47C2212yes—
30C22×Dic94—21no—
31C22×Dic94—21no—
32C22×Dic94—22no—
33C22×Dic94—42no—
34C22×Dic94—41no—
35C22×Dic94—41no—
36C22×Dic94C2×C414yescomplemented
37C23×Dic47C2×C412yes—
38#13094022—24no—
39#13094022—24no—
40#13094011—24no—
41#13094011—24no—
42#13094005—24no—
43#13094005—22no—
44#13094005—22no—
45#13094005C2×C414yes—
46#13094005C2×C418yescomplemented
47#13094005C2314yes—
48#13094003—24no—
49#13094003—24no—
50#13094003—24no—
51#13094003—24no—
52#13093983—22no—
53#13093983—24no—
54#13093983—42no—
55#13093983C2×C414yescomplemented
56C2×C4⋊Dic47—21no—
57C2×C4⋊Dic47—41no—
58C2×Dic47⋊C4—22no—
59C2×Dic47⋊C4—22no—
60C2×C22⋊Dic47—22no—
61C2×C22⋊Dic47—22no—
62C2×C22⋊Dic47—22no—
63C2×C22⋊Dic47C2×C412yes—
64C2×C4×Dic47—21no—
65C2×C4×Dic47—41no—
66#13093939—24no—
67#13093939—22no—
68#13093939C2314yes—
69C2×#41×C47—41no—
70C2×#41×C47—41no—
71C2×#41×C47—42no—
72C2×#41×C47C2×C412yes—
73C2×#41×C47D412yes—
74C2×#41×C47D412yes—
75C2×#41×C47C2312yes—
76C22×Q8×C47D411yescharacteristic
77C22×D4×C47C2×C411yescharacteristic
78C22×D4×C47D411yescharacteristic
79C22×D4×C47C2311yescharacteristic
80C23×C4×C47C2×C411yescharacteristic
81C23×C4×C47D411yescharacteristic
82C23×C4×C47D411yescharacteristic
83C2×M16×C47—22no—
84C2×M16×C47—24no—
85C2×M16×C47—41no—
86C2×M16×C47—42no—
87C22×C8×C47—41no—
88#101×C47—24no—
89#100×C47—22no—
90#100×C47C2314yes—
91C24×C47C22⋊C411yescharacteristic
92C24×C47C22×C411yescharacteristic
93C24×C47C2×D411yescharacteristic
94#41×C47—42no—
95#41×C47—42no—
96#41×C47—42no—
97#41×C47—42no—
98#41×C47—42no—
99#41×C47—42no—
100#41×C47—42no—
101#41×C47—42no—
102C2×Q8×C47—41no—
103C2×Q8×C47—41no—
104C2×Q8×C47C22⋊C412yes—
105C2×Q8×C47C2×D412yes—
106C2×D4×C47—41no—
107C2×D4×C47—41no—
108C2×D4×C47—41no—
109C2×D4×C47—41no—
110C2×D4×C47—41no—
111C2×D4×C47—41no—
112C2×D4×C47C22⋊C412yes—
113C2×D4×C47C22⋊C412yes—
114C2×D4×C47C22×C412yes—
115C2×D4×C47C22×C411yescharacteristic
116C2×D4×C47C2×D412yes—
117C2×D4×C47C2×D411yescharacteristic
118C2×D4×C47C2×D411yescharacteristic
119C2×D4×C47C2411yescharacteristic
120C22×C4×C47—21no—
121C22×C4×C47—21no—
122C22×C4×C47—22no—
123C22×C4×C47—22no—
124C22×C4×C47—21no—
125C22×C4×C47—21no—
126C22×C4×C47—21no—
127C22×C4×C47—21no—
128C22×C4×C47—22no—
129C22×C4×C47C22⋊C411yescharacteristic
130C22×C4×C47C22⋊C412yes—
131C22×C4×C47C22⋊C411yescharacteristic
132C22×C4×C47C22⋊C412yes—
133C22×C4×C47C22⋊C412yes—
134C22×C4×C47C22⋊C411yescharacteristic
135C22×C4×C47C22×C412yes—
136C22×C4×C47C2×D412yes—
137C22×C4×C47C2×D411yescharacteristic
138C22×C4×C47C2×D412yes—
139C22×Dic47—22no—
140C22×Dic47—22no—
141C22×Dic47—22no—
142C22×Dic47—42no—
143C22×Dic47—42no—
144C22×Dic47—42no—
145C22×Dic47C22×C414yes—
146C2×Dic94—24no—
147C2×Dic94—24no—
148C2×Dic94—42no—
149C2×Dic94—42no—
150C2×Dic94—44no—
151C2×Dic94—44no—
152C2×Dic94—42no—
153C2×Dic94—41no—
154C2×Dic94—41no—
155C2×Dic94—44no—
156C2×Dic94—44no—
157M16×C47—24no—
158M16×C47—42no—
159M16×C47—44no—
160C2×C8×C47—42no—
161C2×C8×C47—44no—
162C22⋊Dic47—24no—
163C22⋊Dic47—28no—
164C22⋊Dic47—22no—
165C22⋊Dic47—24no—
166C22⋊Dic47—24no—
167C22⋊Dic47—42no—
168C22⋊Dic47C22×C414yes—
169C4⋊Dic47—24no—
170C4⋊Dic47—44no—
171Dic47⋊C4—24no—
172Dic47⋊C4—28no—
173Dic47⋊C4—24no—
174C4×Dic47—24no—
175C4×Dic47—44no—
176C23×C47—21no—
177C23×C47—21no—
178C23×C47—21no—
179C23×C47—21no—
180C23×C47—21no—
181C23×C47—21no—
182C23×C47—42no—
183C23×C47—42no—
184C23×C47—42no—
185C23×C47C2×C22⋊C411yescharacteristic
186C23×C47C2×C22⋊C411yescharacteristic
187C23×C47C2×C22⋊C412yes—
188C23×C47C2×C22⋊C411yescharacteristic
189C23×C47C23×C411yescharacteristic
190C23×C47C22×D411yescharacteristic
191Q8×C47—42no—
192Q8×C47—42no—
193D4×C47—42no—
194D4×C47—42no—
195D4×C47—42no—
196D4×C47—41no—
197D4×C47—42no—
198D4×C47—41no—
199D4×C47—41no—
200D4×C47—41no—
201C2×C4×C47—21no—
202C2×C4×C47—21no—
203C2×C4×C47—21no—
204C2×C4×C47—21no—
205C2×C4×C47—21no—
206C2×C4×C47—21no—
207C2×C4×C47—22no—
208C2×C4×C47—22no—
209C2×C4×C47—22no—
210C2×C4×C47—22no—
211C2×C4×C47—22no—
212C2×C4×C47—22no—
213C2×C4×C47—22no—
214C2×C4×C47—22no—
215C2×C4×C47—22no—
216C2×C4×C47—22no—
217C2×C4×C47—22no—
218C2×C4×C47—22no—
219C2×C4×C47#9912yes—
220C2×C4×C47#9911yescharacteristic
221C2×C4×C47#9911yescharacteristic
222C2×C4×C47C2×C22⋊C412yes—
223C2×C4×C47C2×C22⋊C412yes—
224C2×C4×C47C2×C22⋊C411yescharacteristic
225C2×C4×C47C2×C22⋊C412yes—
226C2×C4×C47C22×D411yescharacteristic
227C2×Dic47—24no—
228C2×Dic47—24no—
229C2×Dic47—44no—
230C2×Dic47—44no—
231C2×Dic47—42no—
232C2×Dic47—42no—
233C2×Dic47—44no—
234C2×Dic47—44no—
235C2×Dic47—84no—
236Dic94—48no—
237Dic94—44no—
238Dic94—44no—
239C8×C47—44no—
240C2×#2123—471nomaximal, Sylow
241C22×C47—22no—
242C22×C47—21no—
243C22×C47—21no—
244C22×C47—21no—
245C22×C47—22no—
246C22×C47—21no—
247C22×C47—41no—
248C22×C47—42no—
249C22×C47—41no—
250C22×C47—41no—
251C22×C47—41no—
252C22×C47—42no—
253C22×C47—42no—
254C22×C47—42no—
255C22×C47—42no—
256C22×C47—81no—
257C22×C47—81no—
258C22×C47C2×#9911yescharacteristic
259C22×C47C2×#9911yescharacteristic
260C22×C47C22×C22⋊C411yescharacteristic
261C4×C47—21no—
262C4×C47—22no—
263C4×C47—21no—
264C4×C47—22no—
265C4×C47C2×#9912yes—
266C4×C47C2×#9912yes—
267Dic47—44no—
268Dic47—44no—
269Dic47—84no—
270C2×#536—471no—
271C2×#413D4711yescharacteristic
272C2×#410—471no—
273#2123—478no—
274C2×#352—474no—
275C2×C47—21no—
276C2×C47—21no—
277C2×C47—41no—
278C2×C47—41no—
279C2×C47—41no—
280C2×C47—41no—
281C2×C47—42no—
282C2×C47#212312yescomplemented
283C2×C47C22×#9911yescharacteristic
284C22×#41C2×D4711yescharacteristic
285C23×Q8—471no—
286C22×M16—21no—
287#536—474no—
288#536—474no—
289#536—942no—
290#536—942no—
291C2×C4⋊Q8—941no—
292C2×#124—941no—
293C2×#123—472no—
294C2×#123—472no—
295C2×C22⋊Q8—941no—
296C2×C22⋊Q8—941no—
297C2×#117—471no—
298C2×#117—941no—
299#413—22no—
300#413C2×D4714yes—
301C2×#101—21no—
302C2×#100C2×D4712yes—
303#410—474no—
304#410—942no—
305C2×#99—472no—
306C2×#99—941no—
307#352—4716no—
308C47C2×#212311yescharacteristic, complemented, Sylow
309C2×#41—41no—
310C2×#41—41no—
311C2×#41—42no—
312C2×#41C4×D4712yes—
313C2×#41D18812yes—
314C2×#41C22⋊D4712yes—
315C2×#41C22×D4712yes—
316C22×Q8—474no—
317C22×Q8—941no—
318C22×Q8—942no—
319C22×Q8—941no—
320C22×Q8—1882no—
321C22×Q8—1881no—
322C22×Q8—1881no—
323C22×Q8C22⋊D4711yescharacteristic
324C22×D4C4×D4711yescharacteristic
325C22×D4D18811yescharacteristic
326C22×D4C22×D4711yescharacteristic
327C23×C4—472no—
328C23×C4C4×D4711yescharacteristic
329C23×C4D18811yescharacteristic
330C23×C4C22⋊D4711yescharacteristic
331C2×M16—22no—
332C2×M16—24no—
333C2×M16—41no—
334C2×M16—42no—
335C22×C8—41no—
336C4⋊Q8—944no—
337C4⋊Q8—944no—
338#124—944no—
339#124—944no—
340#123—474no—
341#123—478no—
342#123—474no—
343#123—944no—
344#123—942no—
345#123—942no—
346C22⋊Q8—944no—
347C22⋊Q8—944no—
348C22⋊Q8—944no—
349C22⋊Q8—944no—
350#117—474no—
351#117—942no—
352#117—944no—
353#117—1882no—
354C2×C4⋊C4—942no—
355C2×C4⋊C4—941no—
356C2×C4⋊C4—942no—
357C2×C4⋊C4—1881no—
358C2×C22⋊C4—472no—
359C2×C22⋊C4—942no—
360C2×C22⋊C4—942no—
361C2×C22⋊C4—942no—
362C2×C42—941no—
363C2×C42—1881no—
364#101—24no—
365#100—22no—
366#100C22×D4714yes—
367#99—474no—
368#99—942no—
369#99—944no—
370C24(C2×D47)⋊C411yescharacteristic
371C24C2×C4×D4711yescharacteristic
372C24C2×D18811yescharacteristic
373#41—42no—
374#41—42no—
375#41—42no—
376#41—42no—
377#41—42no—
378#41—42no—
379#41—42no—
380#41—42no—
381C2×Q8—41no—
382C2×Q8—41no—
383C2×Q8—944no—
384C2×Q8—944no—
385C2×Q8—1884no—
386C2×Q8—1884no—
387C2×Q8—1882no—
388C2×Q8—1882no—
389C2×Q8—1884no—
390C2×Q8—1884no—
391C2×Q8—1882no—
392C2×Q8—1881no—
393C2×Q8—1881no—
394C2×Q8(C2×D47)⋊C412yes—
395C2×Q8C2×C22⋊D4712yes—
396C2×D4—41no—
397C2×D4—41no—
398C2×D4—41no—
399C2×D4—41no—
400C2×D4—41no—
401C2×D4—41no—
402C2×D4(C2×D47)⋊C412yes—
403C2×D4(C2×D47)⋊C412yes—
404C2×D4C2×C4×D4712yes—
405C2×D4C2×C4×D4711yescharacteristic
406C2×D4C2×D18811yescharacteristic
407C2×D4C2×D18812yes—
408C2×D4C2×C22⋊D4711yescharacteristic
409C2×D4C23×D4711yescharacteristic
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—474no—
420C22×C4—942no—
421C22×C4—942no—
422C22×C4—942no—
423C22×C4—1882no—
424C22×C4—1882no—
425C22×C4—1882no—
426C22×C4(C2×D47)⋊C411yescharacteristic
427C22×C4(C2×D47)⋊C411yescharacteristic
428C22×C4(C2×D47)⋊C412yes—
429C22×C4(C2×D47)⋊C412yes—
430C22×C4(C2×D47)⋊C412yes—
431C22×C4(C2×D47)⋊C411yescharacteristic
432C22×C4C2×C4×D4712yes—
433C22×C4C2×D18812yes—
434C22×C4C2×C22⋊D4711yescharacteristic
435C22×C4C2×C22⋊D4712yes—
436M16—24no—
437M16—42no—
438M16—44no—
439C2×C8—42no—
440C2×C8—44no—
441C4⋊C4—948no—
442C4⋊C4—944no—
443C4⋊C4—944no—
444C4⋊C4—944no—
445C4⋊C4—1884no—
446C22⋊C4—474no—
447C22⋊C4—942no—
448C22⋊C4—944no—
449C22⋊C4—948no—
450C22⋊C4—944no—
451C22⋊C4—944no—
452C22⋊C4—1882no—
453C42—944no—
454C42—1884no—
455C23—21no—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23C2×(C2×D47)⋊C411yescharacteristic
465C23C2×(C2×D47)⋊C411yescharacteristic
466C23C2×(C2×D47)⋊C412yes—
467C23C2×(C2×D47)⋊C411yescharacteristic
468C23C22×C4×D4711yescharacteristic
469C23C22×D18811yescharacteristic
470Q8—42no—
471Q8—42no—
472Q8—1888no—
473Q8—1884no—
474Q8—1884no—
475D4—42no—
476D4—42no—
477D4—42no—
478D4—41no—
479D4—42no—
480D4—41no—
481D4—41no—
482D4—41no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—944no—
502C2×C4—944no—
503C2×C4—3764no—
504C2×C4—1884no—
505C2×C4—1884no—
506C2×C4—1884no—
507C2×C4—1882no—
508C2×C4—1882no—
509C2×C4—1884no—
510C2×C4#1309393812yes—
511C2×C4#1309393811yescharacteristic
512C2×C4#1309393811yescharacteristic
513C2×C4C2×(C2×D47)⋊C412yes—
514C2×C4C2×(C2×D47)⋊C412yes—
515C2×C4C2×(C2×D47)⋊C411yescharacteristic
516C2×C4C2×(C2×D47)⋊C412yes—
517C2×C4C22×C22⋊D4711yescharacteristic
518C8—44no—
519C22—22no—
520C22—21no—
521C22—21no—
522C22—21no—
523C22—22no—
524C22—21no—
525C22—41no—
526C22—42no—
527C22—41no—
528C22—41no—
529C22—41no—
530C22—42no—
531C22—42no—
532C22—42no—
533C22—42no—
534C22—81no—
535C22—81no—
536C22C2×#1309393811yescharacteristic
537C22C2×#1309393811yescharacteristic
538C22C22×(C2×D47)⋊C411yescharacteristic
539C4—21no—
540C4—22no—
541C4—21no—
542C4—22no—
543C4—3764no—
544C4—1884no—
545C4—1884no—
546C4C2×#1309393812yes—
547C4C2×#1309393812yes—
548C2—21no—
549C2—21no—
550C2—41no—
551C2—41no—
552C2—41no—
553C2—41no—
554C2—42no—
555C2#2024504312yesdirect product factor
556C2C22×#1309393811yescharacteristic
subgroupG/H
centerC22C2×#13093938
commutatorC23×C47C23×C4
FrattiniC2×D4C23×D47
FittingC2×C47×#413C2
socleC22×C47C2×#99
prime radicalC22×#41×C47C22
cosocleC2×D4×C47C24
hypercenterC2×#413D47
hypocenterC47C2×#2123
LCSG ▷ C23×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ C2×#413

Complements

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

NH ≅ G/Ntype
3. #20245043C2direct (N × H)
4. C2×#15688914C2semidirect
6. C23×Dic94C4semidirect
9. #15689694C4semidirect
13. C2×#13094005C4semidirect
18. C2×#13093983C4semidirect
23. #15688914C22semidirect
36. C22×Dic94C2×C4semidirect
46. #13094005C2×C4semidirect
55. #13093983C2×C4semidirect
282. C2×C47#2123semidirect
308. C47C2×#2123semidirect
555. C2#20245043direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156896942
C2×#156890972
#202450432
C2×#156889142
C2×C47×#4132
C2×#212347