Finite Group Databasebeta

← C2×#20241518 #26131270

Subgroups — C2×#20241518 #26131270

Order 12032.

766 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689097C211yesmaximal, characteristic
2C2×#15689090C211yesmaximal, characteristic
3#20241518C214yesmaximal, direct product factor
4C2×C47×#413C211yesmaximal, characteristic
5C22×#10625329C411yescharacteristic, complemented
6C23×C47⋊C8C411yescharacteristic, complemented
7C2×#13093983—21no—
8#15689097—21no—
9#15689097—21no—
10#15689097C412yes—
11#15689097C2212yes—
12C2×#13093939—21no—
13C2×#13093939C412yes—
14#15689090—21no—
15#15689090—21no—
16#15689090C412yescomplemented
17#15689090C2212yes—
18C2×C22⋊(C47⋊C8)—21no—
19C2×C22⋊(C47⋊C8)—21no—
20C22×#41×C47C2211yescharacteristic
21C22×M16×C47—21no—
22C47×#413—21no—
23C47×#413—21no—
24C47×#413C412yescomplemented
25C47×#413C2212yes—
26C2×#101×C47C412yescomplemented
27C2×#100×C47—21no—
28C2×#10625329—21no—
29C2×#10625329—41no—
30C2×#10625329—41no—
31C2×#10625329C2×C412yescomplemented
32C22×C47⋊C8—21no—
33C22×C47⋊C8—21no—
34C22×C47⋊C8—21no—
35C22×C47⋊C8—41no—
36C22×C47⋊C8—41no—
37C22×C47⋊C8C2×C412yescomplemented
38#13093983—22no—
39#13093983—22no—
40#13093983—41no—
41#13093983—41no—
42C2×C4⋊Dic47—41no—
43C2×C22⋊Dic47—22no—
44C2×C4×Dic47—41no—
45#13093939—22no—
46#13093939—22no—
47#13093939—22no—
48#13093939C2×C414yes—
49C22⋊(C47⋊C8)—22no—
50C22⋊(C47⋊C8)—22no—
51C22⋊(C47⋊C8)—22no—
52C22⋊(C47⋊C8)—22no—
53C2×#41×C47—41no—
54C2×#41×C47—41no—
55C2×#41×C47—41no—
56C2×#41×C47—41no—
57C2×#41×C47C2×C411yescharacteristic
58C2×#41×C47C2×C411yescharacteristic
59C2×#41×C47C2×C411yescharacteristic
60C2×#41×C47D411yescharacteristic
61C2×#41×C47D411yescharacteristic
62C2×#41×C47D411yescharacteristic
63C2×#41×C47Q811yescharacteristic
64C2×#41×C47C2311yescharacteristic
65C22×Q8×C47C2×C411yescharacteristic
66C22×D4×C47C2×C411yescharacteristic
67C22×D4×C47D411yescharacteristic
68C22×D4×C47Q811yescharacteristic
69C23×C4×C47C2×C411yescharacteristic
70C23×C4×C47D411yescharacteristic
71C23×C4×C47D411yescharacteristic
72C2×M16×C47—22no—
73C2×M16×C47—22no—
74C2×M16×C47—22no—
75C2×M16×C47—41no—
76C2×M16×C47—41no—
77C2×M16×C47—41no—
78C22×C8×C47—41no—
79#101×C47—22no—
80#101×C47C2×C414yescomplemented
81#100×C47—22no—
82#100×C47—22no—
83C24×C47C22⋊C411yescharacteristic
84C24×C47C4⋊C411yescharacteristic
85C24×C47C4⋊C411yescharacteristic
86#41×C47—41no—
87#41×C47—41no—
88#41×C47—41no—
89#41×C47—41no—
90#41×C47—41no—
91#41×C47—41no—
92#41×C47—41no—
93#41×C47—41no—
94#41×C47—41no—
95#41×C47—41no—
96#41×C47—41no—
97#41×C47—41no—
98#41×C47—41no—
99#41×C47—41no—
100#41×C47—41no—
101#41×C47—41no—
102C2×Q8×C47—41no—
103C2×Q8×C47—41no—
104C2×Q8×C47C4211yescharacteristic
105C2×Q8×C47C22⋊C411yescharacteristic
106C2×Q8×C47C4⋊C411yescharacteristic
107C2×Q8×C47C22×C411yescharacteristic
108C2×D4×C47—41no—
109C2×D4×C47—41no—
110C2×D4×C47—41no—
111C2×D4×C47—41no—
112C2×D4×C47—41no—
113C2×D4×C47—41no—
114C2×D4×C47C4211yescharacteristic
115C2×D4×C47C22⋊C411yescharacteristic
116C2×D4×C47C22⋊C411yescharacteristic
117C2×D4×C47C22⋊C411yescharacteristic
118C2×D4×C47C4⋊C411yescharacteristic
119C2×D4×C47C4⋊C411yescharacteristic
120C2×D4×C47C4⋊C411yescharacteristic
121C2×D4×C47C4⋊C411yescharacteristic
122C2×D4×C47C4⋊C411yescharacteristic
123C2×D4×C47C22×C411yescharacteristic
124C2×D4×C47C2×D411yescharacteristic
125C2×D4×C47C2×Q811yescharacteristic
126C22×C4×C47—21no—
127C22×C4×C47—21no—
128C22×C4×C47—21no—
129C22×C4×C47—21no—
130C22×C4×C47—21no—
131C22×C4×C47—21no—
132C22×C4×C47—21no—
133C22×C4×C47—21no—
134C22×C4×C47—21no—
135C22×C4×C47—21no—
136C22×C4×C47—21no—
137C22×C4×C47—21no—
138C22×C4×C47C4211yescharacteristic
139C22×C4×C47C4211yescharacteristic
140C22×C4×C47C22⋊C411yescharacteristic
141C22×C4×C47C22⋊C411yescharacteristic
142C22×C4×C47C22⋊C411yescharacteristic
143C22×C4×C47C22⋊C411yescharacteristic
144C22×C4×C47C22⋊C411yescharacteristic
145C22×C4×C47C22⋊C411yescharacteristic
146C22×C4×C47C22⋊C411yescharacteristic
147C22×C4×C47C4⋊C411yescharacteristic
148C22×C4×C47C4⋊C411yescharacteristic
149C22×C4×C47C4⋊C411yescharacteristic
150C22×C4×C47C4⋊C411yescharacteristic
151C22×C4×C47C22×C411yescharacteristic
152C22×C4×C47C2×D411yescharacteristic
153C22×C4×C47C2×D411yescharacteristic
154C22×Dic47—42no—
155M16×C47—24no—
156M16×C47—42no—
157M16×C47—42no—
158M16×C47—42no—
159C2×C8×C47—42no—
160C2×C8×C47—42no—
161C2×C8×C47—42no—
162C22⋊Dic47—24no—
163C22⋊Dic47—42no—
164C4⋊Dic47—42no—
165C4⋊Dic47—42no—
166C4×Dic47—42no—
167C4×Dic47—42no—
168#10625329—42no—
169#10625329—42no—
170C2×C47⋊C8—22no—
171C2×C47⋊C8—22no—
172C2×C47⋊C8—42no—
173C2×C47⋊C8—41no—
174C2×C47⋊C8—41no—
175C2×C47⋊C8—41no—
176C2×C47⋊C8—41no—
177C2×C47⋊C8—42no—
178C23×C47—21no—
179C23×C47—21no—
180C23×C47—21no—
181C23×C47—21no—
182C23×C47—21no—
183C23×C47—21no—
184C23×C47—42no—
185C23×C47—42no—
186C23×C47—42no—
187C23×C47#9511yescharacteristic
188C23×C47#9511yescharacteristic
189C23×C47#9511yescharacteristic
190C23×C47#9511yescharacteristic
191C23×C47C2×C22⋊C411yescharacteristic
192C23×C47C2×C4⋊C411yescharacteristic
193C23×C47C2×C4⋊C411yescharacteristic
194Q8×C47—41no—
195Q8×C47—41no—
196Q8×C47—41no—
197Q8×C47—41no—
198D4×C47—41no—
199D4×C47—41no—
200D4×C47—41no—
201D4×C47—41no—
202D4×C47—41no—
203D4×C47—41no—
204D4×C47—41no—
205D4×C47—41no—
206D4×C47—41no—
207D4×C47—41no—
208D4×C47—41no—
209D4×C47—41no—
210C2×C4×C47—21no—
211C2×C4×C47—21no—
212C2×C4×C47—21no—
213C2×C4×C47—21no—
214C2×C4×C47—21no—
215C2×C4×C47—21no—
216C2×C4×C47—21no—
217C2×C4×C47—21no—
218C2×C4×C47—21no—
219C2×C4×C47—21no—
220C2×C4×C47—21no—
221C2×C4×C47—21no—
222C2×C4×C47—21no—
223C2×C4×C47—21no—
224C2×C4×C47—21no—
225C2×C4×C47—21no—
226C2×C4×C47—21no—
227C2×C4×C47—21no—
228C2×C4×C47—21no—
229C2×C4×C47—21no—
230C2×C4×C47—21no—
231C2×C4×C47—21no—
232C2×C4×C47—21no—
233C2×C4×C47—21no—
234C2×C4×C47—21no—
235C2×C4×C47—21no—
236C2×C4×C47—21no—
237C2×C4×C47—21no—
238C2×C4×C47—21no—
239C2×C4×C47—21no—
240C2×C4×C47#9511yescharacteristic
241C2×C4×C47#9511yescharacteristic
242C2×C4×C47#9511yescharacteristic
243C2×C4×C47#9511yescharacteristic
244C2×C4×C47#9911yescharacteristic
245C2×C4×C47#9911yescharacteristic
246C2×C4×C47#9911yescharacteristic
247C2×C4×C47#9911yescharacteristic
248C2×C4×C47C2×C4211yescharacteristic
249C2×C4×C47C2×C22⋊C411yescharacteristic
250C2×C4×C47C2×C22⋊C411yescharacteristic
251C2×C4×C47C2×C4⋊C411yescharacteristic
252C2×Dic47—44no—
253C2×Dic47—84no—
254C8×C47—44no—
255C47⋊C8—42no—
256C47⋊C8—42no—
257C2×#1395—471nomaximal, Sylow
258C22×C47—22no—
259C22×C47—21no—
260C22×C47—21no—
261C22×C47—21no—
262C22×C47—21no—
263C22×C47—21no—
264C22×C47—21no—
265C22×C47—41no—
266C22×C47—41no—
267C22×C47—41no—
268C22×C47—41no—
269C22×C47—41no—
270C22×C47—41no—
271C22×C47—42no—
272C22×C47—42no—
273C22×C47—42no—
274C22×C47—42no—
275C22×C47—81no—
276C22×C47—81no—
277C22×C47#34211yescharacteristic
278C22×C47#34211yescharacteristic
279C22×C47C2×#9511yescharacteristic
280C4×C47—21no—
281C4×C47—21no—
282C4×C47—21no—
283C4×C47—21no—
284C4×C47—21no—
285C4×C47—21no—
286C4×C47#34211yescharacteristic
287C4×C47#34211yescharacteristic
288C4×C47C2×#9911yescharacteristic
289C4×C47C2×#9911yescharacteristic
290Dic47—84no—
291C2×#413D4711yescharacteristic
292C2×#410—471no—
293C2×#408—471no—
294#1395—474no—
295C2×C47—21no—
296C2×C47—21no—
297C2×C47—41no—
298C2×C47—41no—
299C2×C47—41no—
300C2×C47—41no—
301C2×C47—41no—
302C2×C47—41no—
303C2×C47#139512yescomplemented
304C2×C47C2×#34211yescharacteristic
305C22×#41C2×D4711yescharacteristic
306C22×M16—21no—
307C22×M16—471no—
308C23×C8—471no—
309C2×#117—941no—
310#413—21no—
311#413—21no—
312#413Dic4712yescomplemented
313#413C2×D4712yes—
314C2×#101Dic4712yescomplemented
315C2×#100—21no—
316#410—472no—
317#410—472no—
318#410—941no—
319#410—941no—
320C2×#99—472no—
321C2×#99—941no—
322#408—472no—
323#408—472no—
324#408—941no—
325#408—941no—
326C2×C22⋊C8—941no—
327C2×C22⋊C8—941no—
328C47C2×#139511yescharacteristic, complemented, Sylow
329C2×#41—41no—
330C2×#41—41no—
331C2×#41—41no—
332C2×#41—41no—
333C2×#41Dic9411yescharacteristic
334C2×#41C4×D4711yescharacteristic
335C2×#41C4×D4711yescharacteristic
336C2×#41D18811yescharacteristic
337C2×#41C2×Dic4711yescharacteristic
338C2×#41C22⋊D4711yescharacteristic
339C2×#41C22⋊D4711yescharacteristic
340C2×#41C22×D4711yescharacteristic
341C22×Q8C2×Dic4711yescharacteristic
342C22×D4Dic9411yescharacteristic
343C22×D4C4×D4711yescharacteristic
344C22×D4C22⋊D4711yescharacteristic
345C23×C4C4×D4711yescharacteristic
346C23×C4D18811yescharacteristic
347C23×C4C22⋊D4711yescharacteristic
348C2×M16—22no—
349C2×M16—22no—
350C2×M16—22no—
351C2×M16—41no—
352C2×M16—41no—
353C2×M16—41no—
354C2×M16—472no—
355C2×M16—941no—
356C2×M16—1881no—
357C2×M16—1881no—
358C22×C8—41no—
359C22×C8—472no—
360C22×C8—941no—
361C22×C8—941no—
362C22×C8—941no—
363C22×C8—1881no—
364C22×C8—1881no—
365#117—942no—
366#117—942no—
367#117—1881no—
368#117—1881no—
369C2×C4⋊C4—1881no—
370C2×C22⋊C4—942no—
371C2×C42—1881no—
372#101—22no—
373#101C2×Dic4714yescomplemented
374#100—22no—
375#100—22no—
376#99—474no—
377#99—942no—
378#99—942no—
379#99—942no—
380C22⋊C8—942no—
381C22⋊C8—942no—
382C22⋊C8—942no—
383C22⋊C8—942no—
384C24Dic47⋊C411yescharacteristic
385C24Dic47⋊C411yescharacteristic
386C24(C2×D47)⋊C411yescharacteristic
387#41—41no—
388#41—41no—
389#41—41no—
390#41—41no—
391#41—41no—
392#41—41no—
393#41—41no—
394#41—41no—
395#41—41no—
396#41—41no—
397#41—41no—
398#41—41no—
399#41—41no—
400#41—41no—
401#41—41no—
402#41—41no—
403C2×Q8—41no—
404C2×Q8—41no—
405C2×Q8C4×Dic4711yescharacteristic
406C2×Q8C4⋊Dic4711yescharacteristic
407C2×Q8C22⋊Dic4711yescharacteristic
408C2×Q8C22×Dic4711yescharacteristic
409C2×D4—41no—
410C2×D4—41no—
411C2×D4—41no—
412C2×D4—41no—
413C2×D4—41no—
414C2×D4—41no—
415C2×D4C4×Dic4711yescharacteristic
416C2×D4Dic47⋊C411yescharacteristic
417C2×D4Dic47⋊C411yescharacteristic
418C2×D4Dic47⋊C411yescharacteristic
419C2×D4Dic47⋊C411yescharacteristic
420C2×D4C4⋊Dic4711yescharacteristic
421C2×D4(C2×D47)⋊C411yescharacteristic
422C2×D4(C2×D47)⋊C411yescharacteristic
423C2×D4C22⋊Dic4711yescharacteristic
424C2×D4C2×Dic9411yescharacteristic
425C2×D4C2×C4×D4711yescharacteristic
426C2×D4C2×C22⋊D4711yescharacteristic
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—1882no—
440C22×C4C4×Dic4711yescharacteristic
441C22×C4C4×Dic4711yescharacteristic
442C22×C4Dic47⋊C411yescharacteristic
443C22×C4Dic47⋊C411yescharacteristic
444C22×C4C4⋊Dic4711yescharacteristic
445C22×C4C4⋊Dic4711yescharacteristic
446C22×C4(C2×D47)⋊C411yescharacteristic
447C22×C4(C2×D47)⋊C411yescharacteristic
448C22×C4(C2×D47)⋊C411yescharacteristic
449C22×C4(C2×D47)⋊C411yescharacteristic
450C22×C4(C2×D47)⋊C411yescharacteristic
451C22×C4C22⋊Dic4711yescharacteristic
452C22×C4C22⋊Dic4711yescharacteristic
453C22×C4C2×C4×D4711yescharacteristic
454C22×C4C2×D18811yescharacteristic
455C22×C4C2×C22⋊D4711yescharacteristic
456M16—24no—
457M16—42no—
458M16—42no—
459M16—42no—
460M16—1882no—
461M16—1882no—
462C2×C8—42no—
463C2×C8—42no—
464C2×C8—42no—
465C2×C8—942no—
466C2×C8—942no—
467C2×C8—1882no—
468C2×C8—1881no—
469C2×C8—1881no—
470C2×C8—1881no—
471C2×C8—1881no—
472C2×C8—1882no—
473C4⋊C4—1882no—
474C4⋊C4—1882no—
475C22⋊C4—944no—
476C22⋊C4—1882no—
477C42—1882no—
478C42—1882no—
479C23—21no—
480C23—21no—
481C23—21no—
482C23—21no—
483C23—21no—
484C23—21no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23C47⋊#9511yescharacteristic
489C23C47⋊#9511yescharacteristic
490C23C47⋊#9511yescharacteristic
491C23C47⋊#9511yescharacteristic
492C23C2×(C2×D47)⋊C411yescharacteristic
493C23C2×Dic47⋊C411yescharacteristic
494C23C2×Dic47⋊C411yescharacteristic
495Q8—41no—
496Q8—41no—
497Q8—41no—
498Q8—41no—
499D4—41no—
500D4—41no—
501D4—41no—
502D4—41no—
503D4—41no—
504D4—41no—
505D4—41no—
506D4—41no—
507D4—41no—
508D4—41no—
509D4—41no—
510D4—41no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—3764no—
542C2×C4—1884no—
543C2×C4C47⋊#9511yescharacteristic
544C2×C4C47⋊#9511yescharacteristic
545C2×C4C47⋊#9511yescharacteristic
546C2×C4C47⋊#9511yescharacteristic
547C2×C4#1309393811yescharacteristic
548C2×C4#1309393811yescharacteristic
549C2×C4#1309393811yescharacteristic
550C2×C4#1309393811yescharacteristic
551C2×C4C2×C4×Dic4711yescharacteristic
552C2×C4C2×(C2×D47)⋊C411yescharacteristic
553C2×C4C2×C22⋊Dic4711yescharacteristic
554C2×C4C2×C4⋊Dic4711yescharacteristic
555C8—44no—
556C8—1882no—
557C8—1882no—
558C22—22no—
559C22—21no—
560C22—21no—
561C22—21no—
562C22—21no—
563C22—21no—
564C22—21no—
565C22—41no—
566C22—41no—
567C22—41no—
568C22—41no—
569C22—41no—
570C22—41no—
571C22—42no—
572C22—42no—
573C22—42no—
574C22—42no—
575C22—81no—
576C22—81no—
577C22#1568888911yescharacteristic
578C22#1568888911yescharacteristic
579C22C2×C47⋊#9511yescharacteristic
580C4—21no—
581C4—21no—
582C4—21no—
583C4—21no—
584C4—21no—
585C4—21no—
586C4—3764no—
587C4#1568888911yescharacteristic
588C4#1568888911yescharacteristic
589C4C2×#1309393811yescharacteristic
590C4C2×#1309393811yescharacteristic
591C2—21no—
592C2—21no—
593C2—41no—
594C2—41no—
595C2—41no—
596C2—41no—
597C2—41no—
598C2—41no—
599C2#2024151812yesdirect product factor
600C2C2×#1568888911yescharacteristic
subgroupG/H
centerC2×C4#13093938
commutatorC2×C4×C47C2×C42
FrattiniC2×#41C22×D47
FittingC2×C47×#413C2
socleC22×C47#342
prime radicalC22×#41×C47C22
cosocleC2×#41×C47C23
hypercenterC2×#413D47
hypocenterC47C2×#1395
LCSG ▷ C2×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C22×#41 ◁ C2×#413

Complements

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

NH ≅ G/Ntype
3. #20241518C2direct (N × H)
5. C22×#10625329C4semidirect
6. C23×C47⋊C8C4semidirect
16. #15689090C4semidirect
24. C47×#413C4semidirect
26. C2×#101×C47C4semidirect
31. C2×#10625329C2×C4semidirect
37. C22×C47⋊C8C2×C4semidirect
80. #101×C47C2×C4semidirect
303. C2×C47#1395semidirect
312. #413Dic47semidirect
314. C2×#101Dic47semidirect
328. C47C2×#1395semidirect
373. #101C2×Dic47semidirect
599. C2#20241518direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156890972
C2×#156890902
#202415182
C2×C47×#4132
C2×#139547