Finite Group Databasebeta

← C2×#20242419 #26188426

Subgroups — C2×#20242419 #26188426

Order 12032.

1016 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689456C211yesmaximal, characteristic, complemented
2C2×#15689439C211yesmaximal, characteristic
3C2×#15689062C211yesmaximal, characteristic, complemented
4#20242419C218yesmaximal, direct product factor
5C2×#15688860C211yesmaximal, characteristic, complemented
6C2×#15688857C211yesmaximal, characteristic
7C2×#15688849C211yesmaximal, characteristic, complemented
8C2×C47×#539C211yesmaximal, characteristic
9C22×C4⋊Dic47C2211yescharacteristic
10#15689456—22no—
11#15689456C2214yescomplemented
12#15689439—22no—
13#15689439C2214yes—
14C2×#13093957—21no—
15C2×#13093955—21no—
16C2×#13093948—21no—
17C2×#13093945—21no—
18C2×C22⋊(C47⋊C8)C2211yescharacteristic
19C2×C22⋊(C47⋊C8)C2211yescharacteristic
20#15689062—22no—
21#15689062C2214yescomplemented
22C2×C47⋊#95—21no—
23C2×C47⋊#95—21no—
24C2×C47⋊#95C2211yescharacteristic
25#15688860C2214yescomplemented
26#15688857C2214yes—
27#15688849C2214yescomplemented
28C47×#539—22no—
29C47×#539—22no—
30C47×#539—22no—
31C47×#539C2214yes—
32C2×#126×C47—21no—
33C2×#125×C47—21no—
34C2×#123×C47—21no—
35C2×#123×C47—21no—
36C2×C22⋊Q8×C47—21no—
37C2×C22⋊Q8×C47C2211yescharacteristic
38C2×#121×C47C2211yescharacteristic
39C2×C4×D4×C47—21no—
40C22×C4⋊C4×C47C2211yescharacteristic
41C23×Dic47—21no—
42C23×Dic47—21no—
43C22×C47⋊C8—21no—
44C22×C47⋊C8—21no—
45C2×C4⋊Dic47—21no—
46C2×C4⋊Dic47—21no—
47C2×C4⋊Dic47—21no—
48C2×C4⋊Dic47—21no—
49C2×C4⋊Dic47—21no—
50C2×C4⋊Dic47C2312yes—
51#13093957—24no—
52#13093955—24no—
53#13093948—24no—
54#13093945—24no—
55C22⋊(C47⋊C8)—21no—
56C22⋊(C47⋊C8)—21no—
57C22⋊(C47⋊C8)C2312yes—
58C22⋊(C47⋊C8)C2312yes—
59C47⋊#95—24no—
60C47⋊#95—24no—
61C47⋊#95—21no—
62C47⋊#95C2312yes—
63C22×Q8×C47D411yescharacteristic
64C22×D4×C47—21no—
65C22×D4×C47D411yescharacteristic
66C23×C4×C47—21no—
67C23×C4×C47D411yescharacteristic
68C23×C4×C47D411yescharacteristic
69C23×C4×C47C2311yescharacteristic
70#126×C47—24no—
71#126×C47—24no—
72#125×C47—24no—
73#125×C47—24no—
74#123×C47—24no—
75#123×C47—24no—
76#123×C47—24no—
77#123×C47—24no—
78C22⋊Q8×C47—24no—
79C22⋊Q8×C47—24no—
80C22⋊Q8×C47—21no—
81C22⋊Q8×C47—21no—
82C22⋊Q8×C47D412yes—
83C22⋊Q8×C47C2312yes—
84#121×C47—21no—
85#121×C47—21no—
86#121×C47D412yes—
87#121×C47C2312yes—
88C4×D4×C47—24no—
89C4×D4×C47—24no—
90C2×C4⋊C4×C47—21no—
91C2×C4⋊C4×C47—21no—
92C2×C4⋊C4×C47—21no—
93C2×C4⋊C4×C47—21no—
94C2×C4⋊C4×C47—21no—
95C2×C4⋊C4×C47—21no—
96C2×C4⋊C4×C47—21no—
97C2×C4⋊C4×C47—21no—
98C2×C4⋊C4×C47D412yes—
99C2×C4⋊C4×C47D411yescharacteristic
100C2×C4⋊C4×C47D411yescharacteristic
101C2×C4⋊C4×C47C2312yes—
102C2×C22⋊C4×C47—21no—
103C2×C22⋊C4×C47—21no—
104C2×C22⋊C4×C47—21no—
105C2×C22⋊C4×C47—21no—
106C2×C22⋊C4×C47—21no—
107C2×C42×C47—21no—
108C24×C47—21no—
109C24×C47C2×D411yescharacteristic
110C2×Q8×C47—21no—
111C2×Q8×C47—42no—
112C2×Q8×C47C2×D412yes—
113C2×D4×C47—22no—
114C2×D4×C47—22no—
115C2×D4×C47—21no—
116C2×D4×C47—42no—
117C2×D4×C47—42no—
118C2×D4×C47—42no—
119C2×D4×C47C2×D412yes—
120C22×C4×C47—21no—
121C22×C4×C47—21no—
122C22×C4×C47—21no—
123C22×C4×C47—21no—
124C22×C4×C47—21no—
125C22×C4×C47—21no—
126C22×C4×C47—22no—
127C22×C4×C47—22no—
128C22×C4×C47—24no—
129C22×C4×C47—21no—
130C22×C4×C47—42no—
131C22×C4×C47—42no—
132C22×C4×C47—82no—
133C22×C4×C47C2×D412yes—
134C22×C4×C47C2×D411yescharacteristic
135C22×C4×C47C2×D411yescharacteristic
136C22×C4×C47C2×D412yes—
137C22×C4×C47C2×D411yescharacteristic
138C22×C4×C47C2×D411yescharacteristic
139C22×C4×C47C2×D411yescharacteristic
140C22×C4×C47C2411yescharacteristic
141C22×Dic47—22no—
142C22×Dic47—22no—
143C22×Dic47—42no—
144C22×Dic47—42no—
145C22×Dic47—44no—
146C22×Dic47—41no—
147C22×Dic47—41no—
148C22×Dic47—42no—
149C4⋊C4×C47—24no—
150C4⋊C4×C47—24no—
151C4⋊C4×C47—24no—
152C4⋊C4×C47—22no—
153C4⋊C4×C47—22no—
154C4⋊C4×C47—22no—
155C4⋊C4×C47—21no—
156C4⋊C4×C47—21no—
157C4⋊C4×C47—22no—
158C4⋊C4×C47—22no—
159C4⋊C4×C47—22no—
160C4⋊C4×C47C2×D412yes—
161C4⋊C4×C47C2×D412yes—
162C22⋊C4×C47—24no—
163C22⋊C4×C47—24no—
164C22⋊C4×C47—24no—
165C22⋊C4×C47—22no—
166C22⋊C4×C47—22no—
167C22⋊C4×C47—22no—
168C22⋊C4×C47—22no—
169C42×C47—24no—
170C4⋊Dic47—22no—
171C4⋊Dic47—22no—
172C4⋊Dic47—41no—
173C4⋊Dic47—41no—
174C2×C47⋊C8—22no—
175C2×C47⋊C8—22no—
176C2×C47⋊C8—42no—
177C2×C47⋊C8—42no—
178C23×C47—22no—
179C23×C47—22no—
180C23×C47—22no—
181C23×C47—42no—
182C23×C47—42no—
183C23×C47—42no—
184C23×C47#12011yescharacteristic
185C23×C47#12011yescharacteristic
186C23×C47#13512yes—
187C23×C47#13712yes—
188C23×C47C22×D411yescharacteristic
189Q8×C47—42no—
190Q8×C47—81no—
191D4×C47—42no—
192D4×C47—42no—
193D4×C47—42no—
194D4×C47—44no—
195D4×C47—81no—
196C2×C4×C47—22no—
197C2×C4×C47—22no—
198C2×C4×C47—22no—
199C2×C4×C47—22no—
200C2×C4×C47—22no—
201C2×C4×C47—22no—
202C2×C4×C47—22no—
203C2×C4×C47—22no—
204C2×C4×C47—22no—
205C2×C4×C47—41no—
206C2×C4×C47—41no—
207C2×C4×C47—41no—
208C2×C4×C47—41no—
209C2×C4×C47—42no—
210C2×C4×C47—42no—
211C2×C4×C47—42no—
212C2×C4×C47—42no—
213C2×C4×C47—42no—
214C2×C4×C47—42no—
215C2×C4×C47—42no—
216C2×C4×C47—81no—
217C2×C4×C47—84no—
218C2×C4×C47#12011yescharacteristic
219C2×C4×C47#12011yescharacteristic
220C2×C4×C47C22×D411yescharacteristic
221C2×C4×C47C22×D411yescharacteristic
222C2×Dic47—42no—
223C2×Dic47—42no—
224C2×Dic47—44no—
225C2×Dic47—42no—
226C2×Dic47—42no—
227C2×Dic47—84no—
228C2×Dic47—82no—
229C2×Dic47—82no—
230C47⋊C8—42no—
231C47⋊C8—42no—
232C2×#1615—471nomaximal, Sylow
233C22×C47—24no—
234C22×C47—24no—
235C22×C47—21no—
236C22×C47—21no—
237C22×C47—21no—
238C22×C47—21no—
239C22×C47—42no—
240C22×C47—42no—
241C22×C47—42no—
242C22×C47—84no—
243C22×C47#45212yes—
244C22×C47#45212yes—
245C22×C47#45412yes—
246C22×C47#45812yes—
247C22×C47C2×#12011yescharacteristic
248C22×C47C2×#13511yescharacteristic
249C22×C47C2×#13711yescharacteristic
250C4×C47—21no—
251C4×C47—21no—
252C4×C47—21no—
253C4×C47—21no—
254C4×C47—42no—
255C4×C47—42no—
256C4×C47—42no—
257C4×C47—42no—
258C4×C47—42no—
259Dic47—82no—
260Dic47—82no—
261C2×#539D4711yescharacteristic
262C2×#484—471no—
263C2×#481—471no—
264C2×#400—471no—
265#1615—478no—
266C2×#328—471no—
267C2×#327—471no—
268C2×#324—471no—
269C2×C47—21no—
270C2×C47—21no—
271C2×C47—22no—
272C2×C47—82no—
273C2×C47#161514yescomplemented
274C2×C47C2×#45211yescharacteristic
275C2×C47C2×#45411yescharacteristic
276C2×C47C2×#45811yescharacteristic
277#539—22no—
278#539—22no—
279#539—22no—
280#539C2×D4714yes—
281C2×#126—21no—
282C2×#125—21no—
283C2×#123—21no—
284C2×#123—21no—
285C2×C22⋊Q8—21no—
286C2×C22⋊Q8C2×D4711yescharacteristic
287C2×#121C2×D4711yescharacteristic
288C2×C4×D4—21no—
289C22×C4⋊C4—471no—
290C22×C4⋊C4C2×D4711yescharacteristic
291#484—474no—
292#484—942no—
293#481—474no—
294#481—942no—
295C2×#107—941no—
296C2×#106—941no—
297C2×#103—941no—
298C2×#102—941no—
299C2×C22⋊C8—471no—
300C2×C22⋊C8—471no—
301#400—474no—
302#400—942no—
303C2×#95—471no—
304C2×#95—941no—
305C2×#95—941no—
306#328—474no—
307#327—474no—
308#324—474no—
309C47C2×#161511yescharacteristic, complemented, Sylow
310C22×Q8C22⋊D4711yescharacteristic
311C22×D4—21no—
312C22×D4C22⋊D4711yescharacteristic
313C23×C4—21no—
314C23×C4—941no—
315C23×C4—941no—
316C23×C4C22⋊D4711yescharacteristic
317C23×C4C22⋊D4711yescharacteristic
318C23×C4C22×D4711yescharacteristic
319C22×C8—941no—
320C22×C8—941no—
321#126—24no—
322#126—24no—
323#125—24no—
324#125—24no—
325#123—24no—
326#123—24no—
327#123—24no—
328#123—24no—
329C22⋊Q8—24no—
330C22⋊Q8—24no—
331C22⋊Q8—21no—
332C22⋊Q8—21no—
333C22⋊Q8C22⋊D4712yes—
334C22⋊Q8C22×D4712yes—
335#121—21no—
336#121—21no—
337#121C22⋊D4712yes—
338#121C22×D4712yes—
339C4×D4—24no—
340C4×D4—24no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—472no—
350C2×C4⋊C4—941no—
351C2×C4⋊C4—941no—
352C2×C4⋊C4—941no—
353C2×C4⋊C4—941no—
354C2×C4⋊C4—941no—
355C2×C4⋊C4C22⋊D4712yes—
356C2×C4⋊C4C22⋊D4711yescharacteristic
357C2×C4⋊C4C22⋊D4711yescharacteristic
358C2×C4⋊C4C22×D4712yes—
359C2×C22⋊C4—21no—
360C2×C22⋊C4—21no—
361C2×C22⋊C4—21no—
362C2×C22⋊C4—21no—
363C2×C22⋊C4—21no—
364C2×C42—21no—
365#107—944no—
366#106—944no—
367#103—944no—
368#102—944no—
369C22⋊C8—472no—
370C22⋊C8—472no—
371C22⋊C8—941no—
372C22⋊C8—941no—
373#95—472no—
374#95—941no—
375#95—944no—
376#95—944no—
377C24—21no—
378C24C2×C22⋊D4711yescharacteristic
379C2×Q8—21no—
380C2×Q8—42no—
381C2×Q8C2×C22⋊D4712yes—
382C2×D4—22no—
383C2×D4—22no—
384C2×D4—21no—
385C2×D4—42no—
386C2×D4—42no—
387C2×D4—42no—
388C2×D4C2×C22⋊D4712yes—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—21no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—82no—
402C22×C4—942no—
403C22×C4—942no—
404C22×C4—1882no—
405C22×C4—1882no—
406C22×C4—1884no—
407C22×C4—1881no—
408C22×C4—1881no—
409C22×C4—1882no—
410C22×C4C2×C22⋊D4712yes—
411C22×C4C2×C22⋊D4711yescharacteristic
412C22×C4C2×C22⋊D4711yescharacteristic
413C22×C4C2×C22⋊D4712yes—
414C22×C4C2×C22⋊D4711yescharacteristic
415C22×C4C2×C22⋊D4711yescharacteristic
416C22×C4C2×C22⋊D4711yescharacteristic
417C22×C4C23×D4711yescharacteristic
418C2×C8—942no—
419C2×C8—942no—
420C2×C8—1882no—
421C2×C8—1882no—
422C4⋊C4—24no—
423C4⋊C4—24no—
424C4⋊C4—24no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—21no—
429C4⋊C4—21no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—942no—
434C4⋊C4—942no—
435C4⋊C4—1881no—
436C4⋊C4—1881no—
437C4⋊C4C2×C22⋊D4712yes—
438C4⋊C4C2×C22⋊D4712yes—
439C22⋊C4—24no—
440C22⋊C4—24no—
441C22⋊C4—24no—
442C22⋊C4—22no—
443C22⋊C4—22no—
444C22⋊C4—22no—
445C22⋊C4—22no—
446C42—24no—
447C23—22no—
448C23—22no—
449C23—22no—
450C23—42no—
451C23—42no—
452C23—42no—
453C23#1309399311yescharacteristic
454C23#1309399311yescharacteristic
455C23#1309404412yes—
456C23#1309405412yes—
457C23C22×C22⋊D4711yescharacteristic
458Q8—42no—
459Q8—81no—
460D4—42no—
461D4—44no—
462D4—42no—
463D4—42no—
464D4—81no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—81no—
486C2×C4—84no—
487C2×C4—3764no—
488C2×C4—3762no—
489C2×C4—3762no—
490C2×C4—1882no—
491C2×C4—1882no—
492C2×C4—1884no—
493C2×C4—1882no—
494C2×C4—1882no—
495C2×C4#1309399311yescharacteristic
496C2×C4#1309399311yescharacteristic
497C2×C4C22×C22⋊D4711yescharacteristic
498C2×C4C22×C22⋊D4711yescharacteristic
499C8—1882no—
500C8—1882no—
501C22—21no—
502C22—21no—
503C22—21no—
504C22—21no—
505C22—24no—
506C22—24no—
507C22—42no—
508C22—42no—
509C22—42no—
510C22—84no—
511C22#1568928512yes—
512C22#1568928512yes—
513C22#1568929612yes—
514C22#1568931312yes—
515C22C2×#1309399311yescharacteristic
516C22C2×#1309404411yescharacteristic
517C22C2×#1309405411yescharacteristic
518C4—21no—
519C4—21no—
520C4—21no—
521C4—21no—
522C4—42no—
523C4—42no—
524C4—42no—
525C4—42no—
526C4—42no—
527C4—3762no—
528C4—3762no—
529C2—21no—
530C2—22no—
531C2—21no—
532C2—82no—
533C2#2024241914yesdirect product factor
534C2C2×#1568928511yescharacteristic
535C2C2×#1568929611yescharacteristic
536C2C2×#1568931311yescharacteristic
subgroupG/H
centerC23#13093993
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC2×C47×#539C2
socleC23×C47#120
prime radicalC2×#121×C47C22
hypercenterC2×#539D47
hypocenterC47C2×#1615
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#539

Complements

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

NH ≅ G/Ntype
1. C2×#15689456C2semidirect
3. C2×#15689062C2semidirect
4. #20242419C2direct (N × H)
5. C2×#15688860C2semidirect
7. C2×#15688849C2semidirect
11. #15689456C22semidirect
21. #15689062C22semidirect
25. #15688860C22semidirect
27. #15688849C22semidirect
273. C2×C47#1615semidirect
309. C47C2×#1615semidirect
533. C2#20242419direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156894562
C2×#156894392
C2×#156890622
#202424192
C2×#156888602
C2×#156888572
C2×#156888492
C2×C47×#5392
C2×#161547