Finite Group Databasebeta

← C2×#21446838 #29706902

Subgroups — C2×#21446838 #29706902

Order 15616.

2126 conjugacy classes of subgroups.

The subgroup lattice has 704 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840333C211yesmaximal, characteristic, complemented
2C22×#14519764C211yesmaximal, characteristic, complemented
3C22×#14519704C211yesmaximal, characteristic, complemented
4C2×#16839767C211yesmaximal, characteristic, complemented
5C22×#14519695C211yesmaximal, characteristic, complemented
6#21446838C218yesmaximal, direct product factor
7C2×#16839523C211yesmaximal, characteristic, complemented
8C22×AGL(1,ℤ/8)×C61C211yesmaximal, characteristic
9C23×Dic122C2211yescharacteristic
10C22×#11758194C2211yescharacteristic
11#16840333—22no—
12#16840333—22no—
13#16840333—22no—
14#16840333C2214yescomplemented
15C2×#14519775—21no—
16C2×#14519764—24no—
17C2×#14519764—24no—
18C2×#14519764—21no—
19C2×#14519764—22no—
20C2×#14519764C2214yescomplemented
21C2×#14519758—22no—
22C2×#14519756—22no—
23C2×#14519736C2211yescharacteristic
24C22×C22⋊Dic61—22no—
25C22×C4×Dic61C2211yescharacteristic
26C2×#14519704—24no—
27C2×#14519704C2214yescomplemented
28#16839767—22no—
29#16839767C2214yescomplemented
30C2×#14519698—22no—
31C2×#14519695—24no—
32C2×#14519695C2214yescomplemented
33#16839523C2214yescomplemented
34C22×#41×C61C2211yescharacteristic
35C23×D4×C61C2211yescharacteristic
36C2×AGL(1,ℤ/8)×C61—22no—
37C2×AGL(1,ℤ/8)×C61—24no—
38C2×AGL(1,ℤ/8)×C61—24no—
39C2×AGL(1,ℤ/8)×C61C2214yes—
40C22×SD16×C61—21no—
41C22×D8×C61—21no—
42C22×M16×C61C2211yescharacteristic
43C22×Dic122—22no—
44C22×Dic122—21no—
45C22×Dic122—22no—
46C22×Dic122—22no—
47C22×Dic122—21no—
48C22×Dic122—21no—
49C22×Dic122—21no—
50C22×Dic122—22no—
51C22×Dic122D412yes—
52C22×Dic122C2312yes—
53C23×Dic61—21no—
54C23×Dic61D412yes—
55C2×#11758194—21no—
56C2×#11758194—22no—
57C2×#11758194—22no—
58C2×#11758194C2312yes—
59C22×#92418—21no—
60#14519775—24no—
61#14519775—24no—
62#14519764—216no—
63#14519764—24no—
64#14519764—28no—
65#14519764—24no—
66#14519764—28no—
67#14519758—28no—
68#14519758—28no—
69#14519756—28no—
70#14519756—28no—
71#14519736—21no—
72#14519736—21no—
73#14519736D412yes—
74#14519736C2312yes—
75C2×C4⋊Dic61D412yes—
76C2×#11758196—24no—
77C2×C22⋊Dic61—216no—
78C2×C22⋊Dic61—24no—
79C2×C22⋊Dic61—21no—
80C2×C22⋊Dic61—28no—
81C2×C4×Dic61—21no—
82C2×C4×Dic61—22no—
83C2×C4×Dic61—21no—
84C2×C4×Dic61D414yes—
85C2×C4×Dic61C2312yes—
86#14519704—28no—
87#14519698—28no—
88#14519695—28no—
89C25×C61—21no—
90C2×#41×C61—21no—
91C2×#41×C61—21no—
92C2×#41×C61—22no—
93C2×#41×C61—22no—
94C2×#41×C61—22no—
95C2×#41×C61—22no—
96C2×#41×C61D412yes—
97C2×#41×C61C2312yes—
98C22×Q8×C61D411yescharacteristic
99C22×D4×C61—21no—
100C22×D4×C61—21no—
101C22×D4×C61—21no—
102C22×D4×C61—22no—
103C22×D4×C61—21no—
104C22×D4×C61—21no—
105C22×D4×C61—21no—
106C22×D4×C61—22no—
107C22×D4×C61—22no—
108C22×D4×C61D411yescharacteristic
109C22×D4×C61D414yes—
110C22×D4×C61C2312yes—
111C23×C4×C61—21no—
112C23×C4×C61C2311yescharacteristic
113AGL(1,ℤ/8)×C61—216no—
114AGL(1,ℤ/8)×C61—28no—
115AGL(1,ℤ/8)×C61—28no—
116C2×SD16×C61—28no—
117C2×SD16×C61—24no—
118C2×D8×C61—28no—
119C2×D8×C61—24no—
120C2×M16×C61—22no—
121C2×M16×C61—21no—
122C2×M16×C61D414yes—
123C2×M16×C61C2312yes—
124C22×C8×C61—21no—
125C24×C61—21no—
126C24×C61—24no—
127C24×C61—22no—
128C24×C61—48no—
129C24×C61—41no—
130C24×C61—41no—
131C24×C61—42no—
132C24×C61C2×D411yescharacteristic
133#41×C61—24no—
134#41×C61—24no—
135#41×C61—22no—
136#41×C61—22no—
137#41×C61—22no—
138#41×C61—22no—
139#41×C61—42no—
140#41×C61—42no—
141#41×C61—42no—
142#41×C61—42no—
143C2×Q8×C61—21no—
144C2×Q8×C61—22no—
145C2×Q8×C61—22no—
146C2×Q8×C61C2×D412yes—
147C2×D4×C61—22no—
148C2×D4×C61—22no—
149C2×D4×C61—22no—
150C2×D4×C61—24no—
151C2×D4×C61—24no—
152C2×D4×C61—24no—
153C2×D4×C61—21no—
154C2×D4×C61—24no—
155C2×D4×C61—24no—
156C2×D4×C61—22no—
157C2×D4×C61—22no—
158C2×D4×C61—22no—
159C2×D4×C61—22no—
160C2×D4×C61—22no—
161C2×D4×C61—22no—
162C2×D4×C61—42no—
163C2×D4×C61—42no—
164C2×D4×C61—41no—
165C2×D4×C61—41no—
166C2×D4×C61—42no—
167C2×D4×C61—42no—
168C2×D4×C61—42no—
169C2×D4×C61—42no—
170C2×D4×C61—42no—
171C2×D4×C61C2×D412yes—
172C2×D4×C61C2×D414yes—
173C22×C4×C61—21no—
174C22×C4×C61—22no—
175C22×C4×C61—22no—
176C22×C4×C61—42no—
177C22×C4×C61—42no—
178C22×C4×C61C2×D411yescharacteristic
179C22×C4×C61C2×D411yescharacteristic
180C22×C4×C61C2×D411yescharacteristic
181C22×C4×C61C2×D412yes—
182C22×C4×C61C2×D412yes—
183C22×C4×C61C2×D411yescharacteristic
184C22×C4×C61C2×D412yes—
185C22×C4×C61C2×D411yescharacteristic
186C22×C4×C61#4112yes—
187C22×C4×C61C2411yescharacteristic
188C22×Dic61—24no—
189C22×Dic61—24no—
190C22×Dic61—22no—
191C22×Dic61—22no—
192C22×Dic61—22no—
193C22×Dic61—24no—
194C22×Dic61—22no—
195C22×Dic61—44no—
196C22×Dic61C2×D414yes—
197C2×Dic122—22no—
198C2×Dic122—24no—
199C2×Dic122—22no—
200C2×Dic122—24no—
201C2×Dic122—22no—
202C2×Dic122—22no—
203C2×Dic122—22no—
204C2×Dic122—22no—
205C2×Dic122—22no—
206C2×Dic122—22no—
207C2×Dic122—42no—
208C2×Dic122—42no—
209C2×Dic122—42no—
210C2×Dic122—42no—
211C2×Dic122—42no—
212C2×Dic122—42no—
213C2×Dic122—42no—
214C2×Dic122—42no—
215SD16×C61—216no—
216D8×C61—216no—
217M16×C61—24no—
218M16×C61—22no—
219M16×C61C2×D414yes—
220C2×C8×C61—24no—
221C2×C8×C61—22no—
222C22⋊Dic61—232no—
223C22⋊Dic61—216no—
224C22⋊Dic61—22no—
225C22⋊Dic61—22no—
226C4⋊Dic61—22no—
227C4⋊Dic61C2×D414yes—
228#11758196—216no—
229C4×Dic61—24no—
230C4×Dic61—22no—
231C4×Dic61—22no—
232C4×Dic61—22no—
233C4×Dic61C2×D414yes—
234#11758194—22no—
235#11758194—22no—
236#11758194—42no—
237C2×#92418—22no—
238C2×#92418—42no—
239C23×C61—24no—
240C23×C61—22no—
241C23×C61—24no—
242C23×C61—48no—
243C23×C61—416no—
244C23×C61—42no—
245C23×C61—42no—
246C23×C61—42no—
247C23×C61—44no—
248C23×C61—44no—
249C23×C61—44no—
250C23×C61—84no—
251C23×C61—88no—
252C23×C61—84no—
253C23×C61#12011yescharacteristic
254C23×C61#12012yes—
255C23×C61#12011yescharacteristic
256C23×C61#12112yes—
257C23×C61C22×D411yescharacteristic
258Q8×C61—22no—
259Q8×C61—22no—
260Q8×C61—42no—
261D4×C61—22no—
262D4×C61—24no—
263D4×C61—24no—
264D4×C61—22no—
265D4×C61—42no—
266D4×C61—42no—
267D4×C61—44no—
268D4×C61—42no—
269D4×C61—42no—
270D4×C61—44no—
271D4×C61—44no—
272D4×C61—42no—
273C2×C4×C61—22no—
274C2×C4×C61—42no—
275C2×C4×C61—42no—
276C2×C4×C61—42no—
277C2×C4×C61—44no—
278C2×C4×C61#12012yes—
279C2×C4×C61#12012yes—
280C2×C4×C61#12012yes—
281C2×C4×C61#12011yescharacteristic
282C2×C4×C61#12011yescharacteristic
283C2×C4×C61#12112yes—
284C2×C4×C61#12112yes—
285C2×C4×C61#12112yes—
286C2×C4×C61#12112yes—
287C2×C4×C61#12112yes—
288C2×C4×C61#12712yes—
289C2×C4×C61#12712yes—
290C2×C4×C61C22×D411yescharacteristic
291C2×C4×C61C22×D411yescharacteristic
292C2×C4×C61C22×D411yescharacteristic
293C2×C4×C61C22×D411yescharacteristic
294C2×C4×C61C22×D411yescharacteristic
295C2×C4×C61C2×#4111yescharacteristic
296C2×Dic61—24no—
297C2×Dic61—24no—
298C2×Dic61—24no—
299C2×Dic61—24no—
300C2×Dic61—24no—
301C2×Dic61—44no—
302C2×Dic61—48no—
303C2×Dic61—44no—
304C2×Dic61—44no—
305C2×Dic61—44no—
306Dic122—42no—
307Dic122—42no—
308Dic122—42no—
309Dic122—42no—
310Dic122—44no—
311Dic122—44no—
312C8×C61—24no—
313#92418—42no—
314C2×#2003—611nomaximal, Sylow
315C22×C61—22no—
316C22×C61—22no—
317C22×C61—24no—
318C22×C61—24no—
319C22×C61—48no—
320C22×C61—42no—
321C22×C61—44no—
322C22×C61—44no—
323C22×C61—84no—
324C22×C61—88no—
325C22×C61—84no—
326C22×C61—84no—
327C22×C61—88no—
328C22×C61#39212yes—
329C22×C61#39212yes—
330C22×C61#45414yes—
331C22×C61C2×#12011yescharacteristic
332C22×C61C2×#12011yescharacteristic
333C22×C61C2×#12111yescharacteristic
334C4×C61—42no—
335C4×C61#39212yes—
336C4×C61#39212yes—
337C4×C61C2×#12011yescharacteristic
338C4×C61C2×#12111yescharacteristic
339C4×C61C2×#12111yescharacteristic
340C4×C61C2×#12711yescharacteristic
341Dic61—44no—
342Dic61—44no—
343Dic61—44no—
344C22×AGL(1,ℤ/8)D6111yescharacteristic
345C2×#536—611no—
346C22×#124—611no—
347C22×#104—611no—
348C2×#418—611no—
349C22×#101—611no—
350#2003—618no—
351C2×#339—611no—
352C2×C61—21no—
353C2×C61—21no—
354C2×C61—22no—
355C2×C61—82no—
356C2×C61—84no—
357C2×C61#200314yescomplemented
358C2×C61C2×#39211yescharacteristic
359C2×C61C2×#45412yes—
360C22×#41C2×D6111yescharacteristic
361C23×Q8—611no—
362C23×D4C2×D6111yescharacteristic
363C2×AGL(1,ℤ/8)—22no—
364C2×AGL(1,ℤ/8)—24no—
365C2×AGL(1,ℤ/8)—24no—
366C2×AGL(1,ℤ/8)C2×D6114yes—
367C22×SD16—21no—
368C22×D8—21no—
369C22×M16—611no—
370C22×M16C2×D6111yescharacteristic
371#536—614no—
372#536—1222no—
373#536—1222no—
374#536—1222no—
375C2×C4⋊Q8—1221no—
376C2×#124—614no—
377C2×#124—1224no—
378C2×#124—1222no—
379C2×#124—1224no—
380C2×#124—1221no—
381C2×#123—1222no—
382C2×C22⋊Q8—1222no—
383C2×#117—611no—
384C22×C22⋊C4—1222no—
385C22×C42—611no—
386C2×#104—614no—
387C2×#104—1224no—
388#418—614no—
389#418—1222no—
390C2×#102—1222no—
391C2×#101—614no—
392C2×#101—1224no—
393#339—614no—
394C61C2×#200311yescharacteristic, complemented, Sylow
395C25—21no—
396C2×#41—21no—
397C2×#41—21no—
398C2×#41—22no—
399C2×#41—22no—
400C2×#41—22no—
401C2×#41—22no—
402C2×#41C22⋊D6112yes—
403C2×#41C22×D6112yes—
404C22×Q8—612no—
405C22×Q8—612no—
406C22×Q8—1221no—
407C22×Q8—1222no—
408C22×Q8—1221no—
409C22×Q8—1222no—
410C22×Q8—1222no—
411C22×Q8—1221no—
412C22×Q8—1221no—
413C22×Q8—1222no—
414C22×Q8C22⋊D6111yescharacteristic
415C22×D4—21no—
416C22×D4—21no—
417C22×D4—21no—
418C22×D4—22no—
419C22×D4—21no—
420C22×D4—22no—
421C22×D4—21no—
422C22×D4—21no—
423C22×D4—22no—
424C22×D4C22⋊D6111yescharacteristic
425C22×D4C22⋊D6114yes—
426C22×D4C22×D6112yes—
427C23×C4—21no—
428C23×C4—612no—
429C23×C4—1221no—
430C23×C4C22×D6111yescharacteristic
431AGL(1,ℤ/8)—216no—
432AGL(1,ℤ/8)—28no—
433AGL(1,ℤ/8)—28no—
434C2×SD16—28no—
435C2×SD16—24no—
436C2×D8—28no—
437C2×D8—24no—
438C2×M16—21no—
439C2×M16—22no—
440C2×M16—612no—
441C2×M16—1221no—
442C2×M16—1222no—
443C2×M16—1222no—
444C2×M16C22⋊D6114yes—
445C2×M16C22×D6112yes—
446C22×C8—21no—
447C22×C8—1221no—
448C4⋊Q8—1224no—
449C4⋊Q8—1224no—
450#124—1228no—
451#124—12216no—
452#124—1224no—
453#124—1228no—
454#124—1224no—
455#123—1228no—
456#123—1228no—
457C22⋊Q8—1228no—
458C22⋊Q8—1228no—
459#117—612no—
460#117—612no—
461#117—1221no—
462#117—1221no—
463C2×C4⋊C4—612no—
464C2×C4⋊C4—1224no—
465C2×C22⋊C4—1228no—
466C2×C22⋊C4—12216no—
467C2×C22⋊C4—1224no—
468C2×C22⋊C4—1221no—
469C2×C42—614no—
470C2×C42—612no—
471C2×C42—1221no—
472C2×C42—1221no—
473C2×C42—1222no—
474#104—1228no—
475#102—1228no—
476#101—1228no—
477C24—22no—
478C24—24no—
479C24—21no—
480C24—48no—
481C24—41no—
482C24—41no—
483C24—42no—
484C24D4×D6111yescharacteristic
485#41—22no—
486#41—22no—
487#41—22no—
488#41—22no—
489#41—24no—
490#41—24no—
491#41—42no—
492#41—42no—
493#41—42no—
494#41—42no—
495C2×Q8—21no—
496C2×Q8—22no—
497C2×Q8—22no—
498C2×Q8—1222no—
499C2×Q8—1222no—
500C2×Q8—1222no—
501C2×Q8—1222no—
502C2×Q8—1222no—
503C2×Q8—1222no—
504C2×Q8—1222no—
505C2×Q8—1224no—
506C2×Q8—1222no—
507C2×Q8—1224no—
508C2×Q8—2442no—
509C2×Q8—2442no—
510C2×Q8—2442no—
511C2×Q8—2442no—
512C2×Q8—2442no—
513C2×Q8—2442no—
514C2×Q8—2442no—
515C2×Q8—2442no—
516C2×Q8C2×C22⋊D6112yes—
517C2×D4—22no—
518C2×D4—22no—
519C2×D4—22no—
520C2×D4—22no—
521C2×D4—22no—
522C2×D4—22no—
523C2×D4—22no—
524C2×D4—24no—
525C2×D4—21no—
526C2×D4—22no—
527C2×D4—24no—
528C2×D4—24no—
529C2×D4—24no—
530C2×D4—24no—
531C2×D4—22no—
532C2×D4—42no—
533C2×D4—42no—
534C2×D4—41no—
535C2×D4—41no—
536C2×D4—42no—
537C2×D4—42no—
538C2×D4—42no—
539C2×D4—42no—
540C2×D4—42no—
541C2×D4C2×C22⋊D6112yes—
542C2×D4C2×C22⋊D6114yes—
543C22×C4—21no—
544C22×C4—22no—
545C22×C4—22no—
546C22×C4—42no—
547C22×C4—42no—
548C22×C4—614no—
549C22×C4—1222no—
550C22×C4—1224no—
551C22×C4—1224no—
552C22×C4—1222no—
553C22×C4—1222no—
554C22×C4—1222no—
555C22×C4—1224no—
556C22×C4—2444no—
557C22×C4D4×D6112yes—
558C22×C4D4×D6111yescharacteristic
559C22×C4D4×D6111yescharacteristic
560C22×C4D4×D6111yescharacteristic
561C22×C4#1175822412yes—
562C22×C4C2×C22⋊D6112yes—
563C22×C4C2×C22⋊D6111yescharacteristic
564C22×C4C2×C22⋊D6111yescharacteristic
565C22×C4C2×C22⋊D6112yes—
566C22×C4C23×D6111yescharacteristic
567SD16—216no—
568D8—216no—
569M16—22no—
570M16—24no—
571M16—1222no—
572M16—1222no—
573M16—2442no—
574M16C2×C22⋊D6114yes—
575C2×C8—22no—
576C2×C8—24no—
577C2×C8—1222no—
578C2×C8—2442no—
579C4⋊C4—614no—
580C4⋊C4—12216no—
581C4⋊C4—1222no—
582C22⋊C4—12232no—
583C22⋊C4—12216no—
584C22⋊C4—1222no—
585C22⋊C4—1222no—
586C42—614no—
587C42—1222no—
588C42—1222no—
589C42—1224no—
590C42—1222no—
591C23—24no—
592C23—22no—
593C23—24no—
594C23—48no—
595C23—416no—
596C23—42no—
597C23—42no—
598C23—42no—
599C23—44no—
600C23—44no—
601C23—44no—
602C23—84no—
603C23—88no—
604C23—84no—
605C23#1451974511yescharacteristic
606C23#1451974512yes—
607C23#1451974511yescharacteristic
608C23#1451974812yes—
609C23C2×D4×D6111yescharacteristic
610Q8—22no—
611Q8—22no—
612Q8—42no—
613Q8—2442no—
614Q8—2442no—
615Q8—2442no—
616Q8—2442no—
617Q8—2444no—
618Q8—2444no—
619D4—22no—
620D4—24no—
621D4—24no—
622D4—22no—
623D4—42no—
624D4—42no—
625D4—44no—
626D4—42no—
627D4—42no—
628D4—44no—
629D4—44no—
630D4—42no—
631C2×C4—22no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—44no—
636C2×C4—1224no—
637C2×C4—1224no—
638C2×C4—1224no—
639C2×C4—1224no—
640C2×C4—1224no—
641C2×C4—2444no—
642C2×C4—2448no—
643C2×C4—2444no—
644C2×C4—2444no—
645C2×C4—2444no—
646C2×C4#1451974512yes—
647C2×C4#1451974511yescharacteristic
648C2×C4#1451974511yescharacteristic
649C2×C4#1451974612yes—
650C2×C4#1451974612yes—
651C2×C4#1451974812yes—
652C2×C4#1451974912yes—
653C2×C4#1451974912yes—
654C2×C4#1451974912yes—
655C2×C4#1451974912yes—
656C2×C4#1451977312yes—
657C2×C4#1451977312yes—
658C2×C4C2×D4×D6111yescharacteristic
659C2×C4C2×D4×D6111yescharacteristic
660C2×C4C22×C22⋊D6111yescharacteristic
661C2×C4C22×C22⋊D6111yescharacteristic
662C2×C4C22×C22⋊D6111yescharacteristic
663C2×C4C2×#1175822411yescharacteristic
664C8—24no—
665C8—2442no—
666C22—22no—
667C22—22no—
668C22—24no—
669C22—24no—
670C22—48no—
671C22—44no—
672C22—42no—
673C22—44no—
674C22—84no—
675C22—88no—
676C22—84no—
677C22—84no—
678C22—88no—
679C22#1683967012yes—
680C22#1683967012yes—
681C22#1683993314yes—
682C22C2×#1451974511yescharacteristic
683C22C2×#1451974511yescharacteristic
684C22C2×#1451974811yescharacteristic
685C4—42no—
686C4—2444no—
687C4—2444no—
688C4—2444no—
689C4#1683967012yes—
690C4#1683967012yes—
691C4C2×#1451974611yescharacteristic
692C4C2×#1451974911yescharacteristic
693C4C2×#1451974911yescharacteristic
694C4C2×#1451977311yescharacteristic
695C2—21no—
696C2—21no—
697C2—22no—
698C2—84no—
699C2—82no—
700C2#2144683814yesdirect product factor
701C2C2×#1683967011yescharacteristic
702C2C2×#1683993312yes—
subgroupG/H
centerC23#14519745
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
Fitting, prime radicalC22×AGL(1,ℤ/8)×C61C2
socleC23×C61#120
hypercenterC22×AGL(1,ℤ/8)D61
hypocenterC61C2×#2003
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#16840333C2semidirect
2. C22×#14519764C2semidirect
3. C22×#14519704C2semidirect
4. C2×#16839767C2semidirect
5. C22×#14519695C2semidirect
6. #21446838C2direct (N × H)
7. C2×#16839523C2semidirect
14. #16840333C22semidirect
20. C2×#14519764C22semidirect
27. C2×#14519704C22semidirect
29. #16839767C22semidirect
32. C2×#14519695C22semidirect
33. #16839523C22semidirect
357. C2×C61#2003semidirect
394. C61C2×#2003semidirect
700. C2#21446838direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168403332
C22×#145197642
C22×#145197042
C2×#168397672
C22×#145196952
#214468382
C2×#168395232
C22×AGL(1,ℤ/8)×C612
C2×#200361