Finite Group Databasebeta

← C2×#20241855 #26182895

Subgroups — C2×#20241855 #26182895

Order 12032.

1270 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689670C211yesmaximal, characteristic
2C2×#15689179C211yesmaximal, characteristic, complemented
3C2×#15689174C211yesmaximal, characteristic, complemented
4#20241855C218yesmaximal, direct product factor
5C2×#15688852C211yesmaximal, characteristic, complemented
6C2×D4⋊(C47⋊C8)C211yesmaximal, characteristic, complemented
7C2×#15688850C211yesmaximal, characteristic, complemented
8C2×C47×D4⋊C8C211yesmaximal, characteristic, complemented
9C22×#10625350—21no—
10#15689670—22no—
11#15689670—22no—
12#15689670C414yes—
13#15689670C2214yes—
14C2×#13094023C411yescharacteristic
15C2×#13094020C411yescharacteristic
16C2×#13094010—21no—
17C2×#13093999C411yescharacteristic
18C2×#13093997—21no—
19C2×#13093995C411yescharacteristic
20C2×C4×D188C2211yescharacteristic
21C2×C4×C22⋊D47—21no—
22C2×C42×D47C2211yescharacteristic
23#15689179—22no—
24#15689179C2214yescomplemented
25#15689174—22no—
26#15689174C2214yescomplemented
27C2×C4⋊(C47⋊C8)C2211yescharacteristic
28C2×Dic47⋊C8—21no—
29C2×(C2×D47)⋊C8—21no—
30C2×(C2×D47)⋊C8—21no—
31C2×#13093934—21no—
32C2×C4×C47⋊C8C2211yescharacteristic
33#15688852C2214yescomplemented
34D4⋊(C47⋊C8)C2214yescomplemented
35#15688850C2214yescomplemented
36C2×C4×D4×C47C2211yescharacteristic
37C2×C4⋊C8×C47C2211yescharacteristic
38C2×C4×C8×C47C2211yescharacteristic
39C47×D4⋊C8C2214yescomplemented
40C2×#10625350—24no—
41C2×#10625350—24no—
42C2×#10625350—44no—
43C2×#10625350—44no—
44C22×Dic94—21no—
45C22×D188—21no—
46C22×D188C2×C411yescharacteristic
47C22×C22⋊D47—21no—
48C22×C22⋊D47—21no—
49C22×C22⋊D47—21no—
50C22×C4×D47—21no—
51C22×C4×D47C2×C411yescharacteristic
52C22×C4×D47D411yescharacteristic
53C22×C4×D47D411yescharacteristic
54C22×C47⋊C8—21no—
55C22×C47⋊C8—21no—
56#13094023—22no—
57#13094023C2×C414yes—
58#13094020—22no—
59#13094020C2×C414yes—
60#13094010—24no—
61#13094010—24no—
62#13093999—22no—
63#13093999C2×C414yes—
64#13093997—24no—
65#13093997—24no—
66#13093995—22no—
67#13093995C2×C414yes—
68C4×D188—21no—
69C4×D188—21no—
70C4×D188C2×C412yes—
71C4×D188C2312yes—
72C4×C22⋊D47—24no—
73C4×C22⋊D47—24no—
74C2×C4⋊Dic47C2×C411yescharacteristic
75C2×Dic47⋊C4—21no—
76C2×C22⋊Dic47—21no—
77C2×(C2×D47)⋊C4—21no—
78C2×(C2×D47)⋊C4—21no—
79C2×C4×Dic47C2×C411yescharacteristic
80C2×C4×Dic47D411yescharacteristic
81C2×C4×Dic47D411yescharacteristic
82C42×D47C2×C412yes—
83C42×D47D412yes—
84C42×D47D412yes—
85C42×D47C2312yes—
86C4⋊(C47⋊C8)—21no—
87C4⋊(C47⋊C8)C2312yes—
88Dic47⋊C8—24no—
89(C2×D47)⋊C8—24no—
90(C2×D47)⋊C8—24no—
91#13093934—24no—
92C4×C47⋊C8—21no—
93C4×C47⋊C8C2312yes—
94C22×D4×C47C2×C411yescharacteristic
95C23×C4×C47—21no—
96C22×C8×C47—21no—
97C22×C8×C47—21no—
98C4×D4×C47—21no—
99C4×D4×C47—21no—
100C4×D4×C47C2×C412yes—
101C4×D4×C47C2312yes—
102C2×C4⋊C4×C47C2×C411yescharacteristic
103C2×C22⋊C4×C47—21no—
104C2×C42×C47C2311yescharacteristic
105C4⋊C8×C47—21no—
106C4⋊C8×C47C2312yes—
107C4×C8×C47—21no—
108C4×C8×C47C2312yes—
109C24×C47—21no—
110C23×D47—21no—
111C23×D47C22⋊C411yescharacteristic
112C2×D4×C47—21no—
113C2×D4×C47—22no—
114C2×D4×C47—22no—
115C2×D4×C47C22×C412yes—
116C22×C4×C47—21no—
117C22×C4×C47—22no—
118C22×C4×C47—22no—
119C22×C4×C47—42no—
120C22×C4×C47—42no—
121C22×C4×C47C22×C411yescharacteristic
122C22×C4×C47C2×D411yescharacteristic
123C22×C4×C47C2×D411yescharacteristic
124C2×C22⋊D47—24no—
125C2×C22⋊D47—24no—
126C2×C22⋊D47—24no—
127C2×C22⋊D47—44no—
128C2×C22⋊D47—44no—
129C2×C22⋊D47—44no—
130C22×Dic47—21no—
131C22×Dic47C22⋊C411yescharacteristic
132C22×Dic47C22⋊C411yescharacteristic
133C22×Dic47C22⋊C411yescharacteristic
134#10625350—48no—
135#10625350—48no—
136#10625350—416no—
137C2×D188—22no—
138C2×D188—22no—
139C2×D188—24no—
140C2×D188—21no—
141C2×D188—44no—
142C2×D188C22×C412yes—
143C2×C4×D47—22no—
144C2×C4×D47—22no—
145C2×C4×D47—22no—
146C2×C4×D47—22no—
147C2×C4×D47—22no—
148C2×C4×D47—22no—
149C2×C4×D47—42no—
150C2×C4×D47—42no—
151C2×C4×D47—42no—
152C2×C4×D47C22⋊C412yes—
153C2×C4×D47C22⋊C412yes—
154C2×C4×D47C22⋊C412yes—
155C2×C4×D47C22×C412yes—
156C2×C4×D47C2×D412yes—
157C2×C4×D47C2×D412yes—
158C2×Dic94—24no—
159C2×Dic94—44no—
160C2×C8×C47—22no—
161C2×C8×C47—22no—
162C2×C8×C47—42no—
163C2×C8×C47—42no—
164C4⋊C4×C47—21no—
165C4⋊C4×C47C22×C412yes—
166C22⋊C4×C47—22no—
167C22⋊C4×C47—22no—
168C42×C47C22×C411yescharacteristic
169C42×C47C2×D411yescharacteristic
170C42×C47C2×D411yescharacteristic
171C42×C47C2411yescharacteristic
172C22⋊Dic47—24no—
173(C2×D47)⋊C4—22no—
174(C2×D47)⋊C4—22no—
175(C2×D47)⋊C4—24no—
176C4⋊Dic47—21no—
177C4⋊Dic47C22×C412yes—
178Dic47⋊C4—24no—
179C4×Dic47C22⋊C412yes—
180C4×Dic47C22⋊C412yes—
181C4×Dic47C22⋊C412yes—
182C4×Dic47C22×C412yes—
183C4×Dic47C2×D412yes—
184C4×Dic47C2×D412yes—
185C2×C47⋊C8—22no—
186C2×C47⋊C8—22no—
187C2×C47⋊C8—42no—
188C2×C47⋊C8—42no—
189C23×C47—22no—
190C23×C47—42no—
191C23×C47—42no—
192C23×C47—42no—
193C23×C47C2×C22⋊C411yescharacteristic
194C22×D47—22no—
195C22×D47—22no—
196C22×D47—22no—
197C22×D47—22no—
198C22×D47—42no—
199C22×D47—42no—
200C22×D47—42no—
201C22×D47C2×C22⋊C412yes—
202D4×C47—22no—
203D4×C47—22no—
204D4×C47—42no—
205C2×C4×C47—22no—
206C2×C4×C47—22no—
207C2×C4×C47—42no—
208C2×C4×C47—42no—
209C2×C4×C47—42no—
210C2×C4×C47—44no—
211C2×C4×C47C2×C22⋊C411yescharacteristic
212C2×C4×C47C2×C22⋊C411yescharacteristic
213C2×C4×C47C2×C22⋊C411yescharacteristic
214C2×C4×C47#13112yes—
215C2×C4×C47#13112yes—
216C2×C4×C47AGL(1,ℤ/8)12yes—
217C2×C4×C47AGL(1,ℤ/8)12yes—
218C2×C4×C47C23×C411yescharacteristic
219C2×C4×C47C22×D411yescharacteristic
220C2×C4×C47C22×D411yescharacteristic
221C22⋊D47—48no—
222C22⋊D47—48no—
223C22⋊D47—48no—
224C2×Dic47—22no—
225C2×Dic47—22no—
226C2×Dic47—22no—
227C2×Dic47—22no—
228C2×Dic47—42no—
229C2×Dic47C2×C22⋊C412yes—
230C2×Dic47C2×C22⋊C412yes—
231C2×Dic47C2×C22⋊C412yes—
232D188—22no—
233D188—22no—
234D188—48no—
235D188—42no—
236C4×D47—22no—
237C4×D47—22no—
238C4×D47—22no—
239C4×D47—22no—
240C4×D47—24no—
241C4×D47—24no—
242C4×D47—42no—
243C4×D47—42no—
244C4×D47—42no—
245C4×D47—42no—
246C4×D47—44no—
247Dic94—48no—
248C8×C47—42no—
249C8×C47—42no—
250C47⋊C8—42no—
251C47⋊C8—42no—
252C2×#1496—471nomaximal, Sylow
253C22×C47—42no—
254C22×C47—42no—
255C22×C47—42no—
256C22×C47—84no—
257C22×C47#40812yes—
258C22×C47#41812yes—
259C22×C47#42112yes—
260C22×C47C22×C22⋊C411yescharacteristic
261C2×D47—22no—
262C2×D47—22no—
263C2×D47—22no—
264C2×D47—44no—
265C2×D47—42no—
266C2×D47—42no—
267C2×D47—42no—
268C2×D47—84no—
269C4×C47—21no—
270C4×C47—21no—
271C4×C47—42no—
272C4×C47#40812yes—
273C4×C47#41812yes—
274C4×C47C2×#13111yescharacteristic
275C4×C47C2×#13111yescharacteristic
276C4×C47C2×AGL(1,ℤ/8)11yescharacteristic
277C4×C47C2×AGL(1,ℤ/8)11yescharacteristic
278Dic47—22no—
279Dic47—22no—
280Dic47—22no—
281Dic47—42no—
282C2×#532—471no—
283C2×#432—471no—
284C2×#431—471no—
285#1496—478no—
286C2×D4⋊C8—471no—
287C2×D4⋊C8—471no—
288C2×D4⋊C8—471no—
289C2×D4⋊C8D4711yescharacteristic, complemented
290C2×C47—82no—
291C2×C47#149614yescomplemented
292C2×C47C2×#40811yescharacteristic
293C2×C47C2×#41811yescharacteristic
294C2×C47C2×#42111yescharacteristic
295D47—42no—
296D47—82no—
297C22×#41—941no—
298#532—474no—
299#532—474no—
300#532—942no—
301#532—942no—
302C2×C4⋊Q8—471no—
303C2×#127—471no—
304C2×#124—941no—
305C2×#121—471no—
306C2×#121—471no—
307C2×#121—941no—
308C2×C4×D4—471no—
309C2×C4×D4—941no—
310C2×C4×D4C2×D4711yescharacteristic
311C22×C42—471no—
312#432—474no—
313#432—942no—
314#431—474no—
315#431—942no—
316C2×C4⋊C8—471no—
317C2×C4⋊C8—941no—
318C2×C4⋊C8C2×D4711yescharacteristic
319C2×C22⋊C8—941no—
320C2×C22⋊C8—941no—
321C2×#97—941no—
322C2×C4×C8—471no—
323C2×C4×C8C2×D4711yescharacteristic
324D4⋊C8—474no—
325D4⋊C8—474no—
326D4⋊C8—474no—
327D4⋊C8C2×D4714yescomplemented
328C47C2×#149611yescharacteristic, complemented, Sylow
329C2×#41—944no—
330C2×#41—944no—
331C2×#41—1884no—
332C2×#41—1884no—
333C22×Q8—941no—
334C22×D4—471no—
335C22×D4—941no—
336C22×D4—941no—
337C22×D4—941no—
338C22×D4—941no—
339C22×D4C4×D4711yescharacteristic
340C23×C4—21no—
341C23×C4—471no—
342C23×C4—471no—
343C23×C4—471no—
344C23×C4—941no—
345C22×C8—21no—
346C22×C8—21no—
347C22×C8—941no—
348C22×C8—941no—
349C4⋊Q8—474no—
350C4⋊Q8—942no—
351#127—474no—
352#127—942no—
353#124—944no—
354#124—944no—
355#121—474no—
356#121—474no—
357#121—944no—
358#121—944no—
359#121—942no—
360#121—942no—
361C4×D4—21no—
362C4×D4—21no—
363C4×D4—472no—
364C4×D4—472no—
365C4×D4—944no—
366C4×D4—944no—
367C4×D4—941no—
368C4×D4—941no—
369C4×D4C4×D4712yes—
370C4×D4C22×D4712yes—
371C2×C4⋊C4—471no—
372C2×C4⋊C4—941no—
373C2×C4⋊C4C4×D4711yescharacteristic
374C2×C22⋊C4—21no—
375C2×C22⋊C4—941no—
376C2×C22⋊C4—941no—
377C2×C22⋊C4—941no—
378C2×C42—471no—
379C2×C42—471no—
380C2×C42—471no—
381C2×C42—472no—
382C2×C42—472no—
383C2×C42—472no—
384C2×C42—472no—
385C2×C42C22×D4711yescharacteristic
386C4⋊C8—21no—
387C4⋊C8—472no—
388C4⋊C8—944no—
389C4⋊C8—941no—
390C4⋊C8C22×D4712yes—
391C22⋊C8—944no—
392C22⋊C8—944no—
393#97—944no—
394C4×C8—21no—
395C4×C8—472no—
396C4×C8—941no—
397C4×C8C22×D4712yes—
398C24—21no—
399C24—471no—
400C24—941no—
401#41—18816no—
402#41—1888no—
403#41—1888no—
404C2×Q8—944no—
405C2×Q8—1884no—
406C2×D4—21no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—472no—
410C2×D4—944no—
411C2×D4—944no—
412C2×D4—944no—
413C2×D4—944no—
414C2×D4—941no—
415C2×D4—942no—
416C2×D4—942no—
417C2×D4—1884no—
418C2×D4—1884no—
419C2×D4—1884no—
420C2×D4—1884no—
421C2×D4C2×C4×D4712yes—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—471no—
428C22×C4—471no—
429C22×C4—471no—
430C22×C4—472no—
431C22×C4—472no—
432C22×C4—472no—
433C22×C4—472no—
434C22×C4—472no—
435C22×C4—472no—
436C22×C4—942no—
437C22×C4—942no—
438C22×C4—942no—
439C22×C4—942no—
440C22×C4—941no—
441C22×C4—942no—
442C22×C4—942no—
443C22×C4—1882no—
444C22×C4—1882no—
445C22×C4—1882no—
446C22×C4C2×C4×D4711yescharacteristic
447C22×C4D4×D4711yescharacteristic
448C22×C4D4×D4711yescharacteristic
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—42no—
452C2×C8—42no—
453C2×C8—942no—
454C2×C8—942no—
455C2×C8—1882no—
456C2×C8—1882no—
457C4⋊C4—21no—
458C4⋊C4—472no—
459C4⋊C4—944no—
460C4⋊C4—941no—
461C4⋊C4C2×C4×D4712yes—
462C22⋊C4—22no—
463C22⋊C4—22no—
464C22⋊C4—944no—
465C22⋊C4—944no—
466C22⋊C4—942no—
467C22⋊C4—942no—
468C42—472no—
469C42—472no—
470C42—472no—
471C42—472no—
472C42—472no—
473C42—472no—
474C42C2×C4×D4711yescharacteristic
475C42D4×D4711yescharacteristic
476C42D4×D4711yescharacteristic
477C42C23×D4711yescharacteristic
478C23—22no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—472no—
483C23—942no—
484C23—942no—
485C23—942no—
486C23—942no—
487C23—1882no—
488C23—1882no—
489C23—1882no—
490C23C22⋊C4×D4711yescharacteristic
491Q8—1888no—
492D4—22no—
493D4—22no—
494D4—42no—
495D4—942no—
496D4—942no—
497D4—1888no—
498D4—1888no—
499D4—1888no—
500D4—1888no—
501D4—1882no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—42no—
505C2×C4—44no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—472no—
509C2×C4—472no—
510C2×C4—472no—
511C2×C4—942no—
512C2×C4—942no—
513C2×C4—942no—
514C2×C4—944no—
515C2×C4—942no—
516C2×C4—942no—
517C2×C4—944no—
518C2×C4—942no—
519C2×C4—942no—
520C2×C4—942no—
521C2×C4—1882no—
522C2×C4—1882no—
523C2×C4—1882no—
524C2×C4—1884no—
525C2×C4—1882no—
526C2×C4—1882no—
527C2×C4C22⋊C4×D4711yescharacteristic
528C2×C4C22⋊C4×D4711yescharacteristic
529C2×C4C22⋊C4×D4711yescharacteristic
530C2×C4#1309403012yes—
531C2×C4#1309403212yes—
532C2×C4#1309404812yes—
533C2×C4#1309405112yes—
534C2×C4C22×C4×D4711yescharacteristic
535C2×C4C2×D4×D4711yescharacteristic
536C2×C4C2×D4×D4711yescharacteristic
537C8—42no—
538C8—42no—
539C8—1882no—
540C8—1882no—
541C22—42no—
542C22—42no—
543C22—42no—
544C22—84no—
545C22—942no—
546C22—942no—
547C22—942no—
548C22—3764no—
549C22—1882no—
550C22—1882no—
551C22—1882no—
552C22—1884no—
553C22#1568908812yes—
554C22#1568912612yes—
555C22#1568913812yes—
556C22C2×C22⋊C4×D4711yescharacteristic
557C4—21no—
558C4—21no—
559C4—42no—
560C4—942no—
561C4—942no—
562C4—942no—
563C4—1882no—
564C4#1568908812yes—
565C4#1568912612yes—
566C4C2×#1309403011yescharacteristic
567C4C2×#1309403211yescharacteristic
568C4C2×#1309404811yescharacteristic
569C4C2×#1309405111yescharacteristic
570C2—82no—
571C2—3762no—
572C2—1882no—
573C2#2024185514yesdirect product factor
574C2C2×#1568908811yescharacteristic
575C2C2×#1568912611yescharacteristic
576C2C2×#1568913811yescharacteristic
subgroupG/H
centerC22×C4D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC42C23×D47
FittingC2×C47×D4⋊C8C2
socleC23×C47C2×C22⋊C4
prime radicalC2×#15689670C2
cosocleC42×C47C24
hypercenterC2×D4⋊C8D47
hypocenterC47C2×#1496
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22×C4 ◁ C2×C42 ◁ C2×D4⋊C8

Complements

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

NH ≅ G/Ntype
2. C2×#15689179C2semidirect
3. C2×#15689174C2semidirect
4. #20241855C2direct (N × H)
5. C2×#15688852C2semidirect
6. C2×D4⋊(C47⋊C8)C2semidirect
7. C2×#15688850C2semidirect
8. C2×C47×D4⋊C8C2semidirect
24. #15689179C22semidirect
26. #15689174C22semidirect
33. #15688852C22semidirect
34. D4⋊(C47⋊C8)C22semidirect
35. #15688850C22semidirect
39. C47×D4⋊C8C22semidirect
289. C2×D4⋊C8D47semidirect
291. C2×C47#1496semidirect
327. D4⋊C8C2×D47semidirect
328. C47C2×#1496semidirect
573. C2#20241855direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156896702
C2×#156891792
C2×#156891742
#202418552
C2×#156888522
C2×D4⋊(C47⋊C8)2
C2×#156888502
C2×C47×D4⋊C82
C2×#149647