Finite Group Databasebeta

← C2×#21444169 #29645960

Subgroups — C2×#21444169 #29645960

Order 15616.

1346 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519763C211yesmaximal, characteristic
2C2×#16839816C211yesmaximal, characteristic, complemented
3C2×#16839812C211yesmaximal, characteristic, complemented
4#21444169C218yesmaximal, direct product factor
5C2×#16839486C211yesmaximal, characteristic, complemented
6C2×#16839485C211yesmaximal, characteristic, complemented
7C2×#16839485C211yesmaximal, characteristic, complemented
8C2×C61×#323C211yesmaximal, characteristic, complemented
9C23×Dic122C411yescharacteristic
10C23×C22⋊D61C411yescharacteristic
11C2×#14519763—21no—
12C2×#14519763—21no—
13C2×#14519763—21no—
14C2×#14519763—21no—
15C2×#14519763—22no—
16C2×#14519763—22no—
17C2×#14519763C414yes—
18C2×#14519763C2214yes—
19C22×C22⋊Dic61C411yescharacteristic
20C22×#11758198C411yescharacteristic
21C22×#11758198C2211yescharacteristic
22C22×C4×Dic61C2211yescharacteristic
23#16839816—22no—
24#16839816C2214yescomplemented
25#16839812—22no—
26#16839812C2214yescomplemented
27C2×#14519706—21no—
28C2×#14519706—21no—
29C2×C22⋊#92418C2211yescharacteristic
30C2×C22⋊#92418C2211yescharacteristic
31C2×#14519687—21no—
32C2×C8×Dic61—21no—
33#16839486C2214yescomplemented
34#16839485C2214yescomplemented
35#16839485C2214yescomplemented
36C22×C22⋊C4×C61C2211yescharacteristic
37C2×C22⋊C8×C61C2211yescharacteristic
38C2×C22⋊C8×C61C2211yescharacteristic
39C61×#323C2214yescomplemented
40C24×D61C2×C411yescharacteristic
41C22×Dic122—21no—
42C22×Dic122—22no—
43C22×Dic122—21no—
44C22×Dic122—44no—
45C22×Dic122C2×C414yes—
46C22×C22⋊D61—22no—
47C22×C22⋊D61—21no—
48C22×C22⋊D61—21no—
49C22×C22⋊D61—44no—
50C22×C22⋊D61C2×C414yes—
51C23×Dic61C2×C411yescharacteristic
52C23×Dic61C2×C411yescharacteristic
53C23×Dic61C2×C411yescharacteristic
54C22×#92418—21no—
55C22×#92418—21no—
56#14519763—24no—
57#14519763—24no—
58#14519763—24no—
59#14519763—24no—
60#14519763—24no—
61#14519763—24no—
62#14519763—24no—
63#14519763—24no—
64C2×C22⋊Dic61—22no—
65C2×C22⋊Dic61—21no—
66C2×C22⋊Dic61—21no—
67C2×C22⋊Dic61C2×C414yes—
68C2×#11758198—21no—
69C2×#11758198—21no—
70C2×#11758198—21no—
71C2×#11758198—21no—
72C2×#11758198—22no—
73C2×#11758198—21no—
74C2×#11758198—21no—
75C2×#11758198C2×C414yes—
76C2×#11758198C2×C412yes—
77C2×#11758198C2312yes—
78C2×C4×Dic61C2×C412yes—
79C2×C4×Dic61D411yescharacteristic
80C2×C4×Dic61D411yescharacteristic
81C2×C4×Dic61D411yescharacteristic
82C2×C4×Dic61D411yescharacteristic
83C2×C4×Dic61D412yes—
84C2×C4×Dic61D412yes—
85C2×C4×Dic61C2312yes—
86#14519706—24no—
87#14519706—24no—
88C22⋊#92418—21no—
89C22⋊#92418—21no—
90C22⋊#92418C2312yes—
91C22⋊#92418C2312yes—
92#14519687—24no—
93C8×Dic61—24no—
94C25×C61C2×C411yescharacteristic
95C23×C4×C61C2×C411yescharacteristic
96C23×C4×C61C2311yescharacteristic
97C22×C8×C61—21no—
98C22×C8×C61—21no—
99C2×C22⋊C4×C61—21no—
100C2×C22⋊C4×C61—21no—
101C2×C22⋊C4×C61—21no—
102C2×C22⋊C4×C61—21no—
103C2×C22⋊C4×C61C2×C412yes—
104C2×C22⋊C4×C61C2312yes—
105C22⋊C8×C61—21no—
106C22⋊C8×C61—21no—
107C22⋊C8×C61C2312yes—
108C22⋊C8×C61C2312yes—
109C24×C61—21no—
110C24×C61—21no—
111C24×C61—22no—
112C24×C61—22no—
113C24×C61—42no—
114C24×C61—42no—
115C24×C61C22×C411yescharacteristic
116C24×C61C22×C412yes—
117C23×D61—21no—
118C23×D61—21no—
119C23×D61—22no—
120C23×D61—22no—
121C23×D61—42no—
122C23×D61—42no—
123C23×D61C22×C412yes—
124C22×C4×C61—21no—
125C22×C4×C61—21no—
126C22×C4×C61—42no—
127C22×C4×C61—42no—
128C22×C4×C61C22×C412yes—
129C22×C4×C61C22×C411yescharacteristic
130C22×C4×C61C2×D411yescharacteristic
131C22×C4×C61C2×D411yescharacteristic
132C22×C4×C61C2×D411yescharacteristic
133C22×C4×C61C2×D411yescharacteristic
134C22×C4×C61C2411yescharacteristic
135C2×C22⋊D61—24no—
136C2×C22⋊D61—24no—
137C2×C22⋊D61—44no—
138C2×C22⋊D61—48no—
139C2×C22⋊D61—48no—
140C2×C22⋊D61—44no—
141C22×Dic61—22no—
142C22×Dic61—22no—
143C22×Dic61—24no—
144C22×Dic61—21no—
145C22×Dic61—21no—
146C22×Dic61—42no—
147C22×Dic61C22⋊C411yescharacteristic
148C22×Dic61C22⋊C412yes—
149C22×Dic61C22⋊C412yes—
150C22×Dic61C22⋊C411yescharacteristic
151C22×Dic61C22⋊C411yescharacteristic
152C22×Dic61C22⋊C411yescharacteristic
153C22×Dic61C22×C412yes—
154C22×Dic61C22×C412yes—
155C22×Dic61C22×C412yes—
156C2×Dic122—24no—
157C2×Dic122—24no—
158C2×Dic122—44no—
159C2×Dic122—48no—
160C2×Dic122—44no—
161C2×Dic122—48no—
162C2×C8×C61—22no—
163C2×C8×C61—22no—
164C2×C8×C61—42no—
165C2×C8×C61—42no—
166C22⋊C4×C61—22no—
167C22⋊C4×C61—22no—
168C22⋊C4×C61—22no—
169C22⋊C4×C61—22no—
170C22⋊Dic61—24no—
171C22⋊Dic61—24no—
172#11758198—22no—
173#11758198—22no—
174#11758198—22no—
175#11758198—22no—
176#11758198—24no—
177#11758198—24no—
178C4×Dic61C22⋊C412yes—
179C4×Dic61C22⋊C412yes—
180C4×Dic61C22⋊C412yes—
181C4×Dic61C22⋊C412yes—
182C4×Dic61C2×D412yes—
183C4×Dic61C2×D412yes—
184C4×Dic61C2×D412yes—
185C4×Dic61C2×D412yes—
186C2×#92418—22no—
187C2×#92418—22no—
188C2×#92418—42no—
189C2×#92418—42no—
190C23×C61—22no—
191C23×C61—22no—
192C23×C61—22no—
193C23×C61—22no—
194C23×C61—22no—
195C23×C61—42no—
196C23×C61—42no—
197C23×C61—42no—
198C23×C61—42no—
199C23×C61—42no—
200C23×C61—42no—
201C23×C61—42no—
202C23×C61—42no—
203C23×C61—84no—
204C23×C61—84no—
205C23×C61C2×C22⋊C411yescharacteristic
206C23×C61C2×C22⋊C411yescharacteristic
207C23×C61#13112yes—
208C23×C61#13112yes—
209C23×C61C23×C411yescharacteristic
210C22×D61—22no—
211C22×D61—22no—
212C22×D61—22no—
213C22×D61—42no—
214C22×D61—42no—
215C22×D61—42no—
216C22×D61—42no—
217C22×D61—42no—
218C22×D61—42no—
219C22×D61—42no—
220C22×D61—42no—
221C22×D61—84no—
222C22×D61—84no—
223C2×C4×C61—22no—
224C2×C4×C61—22no—
225C2×C4×C61—22no—
226C2×C4×C61—41no—
227C2×C4×C61—41no—
228C2×C4×C61—41no—
229C2×C4×C61—41no—
230C2×C4×C61—42no—
231C2×C4×C61—42no—
232C2×C4×C61—42no—
233C2×C4×C61C2×C22⋊C411yescharacteristic
234C2×C4×C61C2×C22⋊C411yescharacteristic
235C2×C4×C61C22×D411yescharacteristic
236C2×C4×C61C22×D411yescharacteristic
237C22⋊D61—48no—
238C22⋊D61—48no—
239C2×Dic61—22no—
240C2×Dic61—22no—
241C2×Dic61—24no—
242C2×Dic61—22no—
243C2×Dic61—22no—
244C2×Dic61—22no—
245C2×Dic61—24no—
246C2×Dic61—24no—
247C2×Dic61—22no—
248C2×Dic61—42no—
249C2×Dic61—42no—
250C2×Dic61—42no—
251C2×Dic61#10014yes—
252C2×Dic61C2×C22⋊C412yes—
253C2×Dic61C2×C22⋊C412yes—
254C2×Dic61C2×C22⋊C412yes—
255C2×Dic61C2×C22⋊C412yes—
256Dic122—48no—
257Dic122—48no—
258C8×C61—42no—
259C8×C61—42no—
260#92418—42no—
261#92418—42no—
262C2×#1469—611nomaximal, Sylow
263C22×C61—21no—
264C22×C61—21no—
265C22×C61—21no—
266C22×C61—21no—
267C22×C61—24no—
268C22×C61—24no—
269C22×C61—42no—
270C22×C61—42no—
271C22×C61—42no—
272C22×C61—82no—
273C22×C61—82no—
274C22×C61—84no—
275C22×C61—84no—
276C22×C61#40812yes—
277C22×C61#40812yes—
278C22×C61#41012yes—
279C22×C61C2×#10012yes—
280C22×C61C22×C22⋊C411yescharacteristic
281C22×C61C2×#13111yescharacteristic
282C22×C61C2×#13111yescharacteristic
283C2×D61—42no—
284C2×D61—42no—
285C2×D61—42no—
286C2×D61—84no—
287C2×D61—84no—
288C2×D61—82no—
289C2×D61—82no—
290C4×C61—21no—
291C4×C61—21no—
292C4×C61—21no—
293C4×C61—21no—
294C4×C61—42no—
295Dic61—22no—
296Dic61—22no—
297Dic61—22no—
298Dic61—42no—
299Dic61C2×#10014yes—
300C22×#124—611no—
301C2×#432—611no—
302C2×#431—611no—
303#1469—618no—
304C2×#323—611no—
305C2×#323—611no—
306C2×#323—611no—
307C2×#323D6111yescharacteristic, complemented
308C2×C61—21no—
309C2×C61—21no—
310C2×C61—22no—
311C2×C61—82no—
312C2×C61#146914yescomplemented
313C2×C61C2×#40811yescharacteristic
314C2×C61C2×#41011yescharacteristic
315C2×C61C22×#10011yescharacteristic
316D61—82no—
317C23×Q8—611no—
318C23×D4—611no—
319C2×#124—614no—
320C2×#124—614no—
321C2×#124—1221no—
322C2×#124—1221no—
323C2×#124—1221no—
324C2×#124—1221no—
325C2×#124—1222no—
326C2×#124—1222no—
327C22×C22⋊C4—611no—
328C22×C22⋊C4—611no—
329C22×C22⋊C4—611no—
330C22×C22⋊C4C2×D6111yescharacteristic
331C22×C42—611no—
332#432—614no—
333#432—1222no—
334#431—614no—
335#431—1222no—
336C2×C4⋊C8—1221no—
337C2×C4⋊C8—1221no—
338C2×C22⋊C8—611no—
339C2×C22⋊C8—611no—
340C2×C22⋊C8C2×D6111yescharacteristic
341C2×C22⋊C8C2×D6111yescharacteristic
342C2×#97—1221no—
343C2×C4×C8—1221no—
344#323—614no—
345#323—614no—
346#323—614no—
347#323C2×D6114yescomplemented
348C61C2×#146911yescharacteristic, complemented, Sylow
349C25—611no—
350C25C4×D6111yescharacteristic
351C22×Q8—614no—
352C22×Q8—1222no—
353C22×Q8—1221no—
354C22×Q8—1221no—
355C22×Q8—2444no—
356C22×D4—614no—
357C22×D4—1221no—
358C22×D4—1222no—
359C22×D4—1221no—
360C22×D4—2444no—
361C23×C4—611no—
362C23×C4—611no—
363C23×C4—611no—
364C23×C4C4×D6111yescharacteristic
365C23×C4C22×D6111yescharacteristic
366C22×C8—21no—
367C22×C8—21no—
368C22×C8—1221no—
369C22×C8—1221no—
370#124—1224no—
371#124—1224no—
372#124—1224no—
373#124—1224no—
374#124—1224no—
375#124—1224no—
376#124—1224no—
377#124—1224no—
378C2×C22⋊C4—21no—
379C2×C22⋊C4—21no—
380C2×C22⋊C4—21no—
381C2×C22⋊C4—21no—
382C2×C22⋊C4—614no—
383C2×C22⋊C4—614no—
384C2×C22⋊C4—612no—
385C2×C22⋊C4—612no—
386C2×C22⋊C4—1222no—
387C2×C22⋊C4—1221no—
388C2×C22⋊C4—1221no—
389C2×C22⋊C4—1222no—
390C2×C22⋊C4—1221no—
391C2×C22⋊C4—1221no—
392C2×C22⋊C4—1221no—
393C2×C22⋊C4—1221no—
394C2×C22⋊C4—1221no—
395C2×C22⋊C4—1221no—
396C2×C22⋊C4C4×D6112yes—
397C2×C22⋊C4C22×D6112yes—
398C2×C42—612no—
399C2×C42—611no—
400C2×C42—611no—
401C2×C42—611no—
402C2×C42—611no—
403C2×C42—612no—
404C2×C42—612no—
405C2×C42—612no—
406C4⋊C8—1224no—
407C4⋊C8—1224no—
408C22⋊C8—21no—
409C22⋊C8—21no—
410C22⋊C8—612no—
411C22⋊C8—612no—
412C22⋊C8—1221no—
413C22⋊C8—1221no—
414C22⋊C8C22×D6112yes—
415C22⋊C8C22×D6112yes—
416#97—1224no—
417C4×C8—1224no—
418C24—21no—
419C24—21no—
420C24—22no—
421C24—22no—
422C24—42no—
423C24—42no—
424C24—612no—
425C24—1222no—
426C24—1222no—
427C24—1221no—
428C24—1221no—
429C24—2442no—
430C24—2442no—
431C24C2×C4×D6111yescharacteristic
432C24C2×C4×D6112yes—
433C2×Q8—1224no—
434C2×Q8—1224no—
435C2×Q8—2444no—
436C2×Q8—2448no—
437C2×Q8—2448no—
438C2×Q8—2444no—
439C2×D4—1224no—
440C2×D4—1224no—
441C2×D4—2444no—
442C2×D4—2448no—
443C2×D4—2444no—
444C2×D4—2448no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—611no—
450C22×C4—612no—
451C22×C4—612no—
452C22×C4—612no—
453C22×C4—612no—
454C22×C4—611no—
455C22×C4—611no—
456C22×C4—611no—
457C22×C4—612no—
458C22×C4—1224no—
459C22×C4—1222no—
460C22×C4—1222no—
461C22×C4—1221no—
462C22×C4—1221no—
463C22×C4—2442no—
464C22×C4C2×C4×D6112yes—
465C22×C4C2×C4×D6111yescharacteristic
466C22×C4D4×D6111yescharacteristic
467C22×C4D4×D6111yescharacteristic
468C22×C4D4×D6111yescharacteristic
469C22×C4D4×D6111yescharacteristic
470C22×C4C23×D6111yescharacteristic
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—42no—
474C2×C8—42no—
475C2×C8—1222no—
476C2×C8—1222no—
477C2×C8—2442no—
478C2×C8—2442no—
479C22⋊C4—22no—
480C22⋊C4—22no—
481C22⋊C4—22no—
482C22⋊C4—22no—
483C22⋊C4—1224no—
484C22⋊C4—1224no—
485C22⋊C4—1224no—
486C22⋊C4—1224no—
487C22⋊C4—1222no—
488C22⋊C4—1222no—
489C22⋊C4—1222no—
490C22⋊C4—1222no—
491C42—612no—
492C42—612no—
493C42—612no—
494C42—612no—
495C42—612no—
496C42—612no—
497C42—612no—
498C42—612no—
499C23—22no—
500C23—22no—
501C23—22no—
502C23—22no—
503C23—22no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—84no—
513C23—84no—
514C23—1222no—
515C23—1222no—
516C23—1222no—
517C23—4884no—
518C23—4884no—
519C23—2442no—
520C23—2442no—
521C23—2442no—
522C23—2442no—
523C23—2442no—
524C23—2442no—
525C23—2442no—
526C23—2442no—
527C23C22⋊C4×D6111yescharacteristic
528C23C22⋊C4×D6111yescharacteristic
529C23#1451978312yes—
530C23#1451978512yes—
531C23C22×C4×D6111yescharacteristic
532Q8—2448no—
533Q8—2448no—
534D4—2448no—
535D4—2448no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—612no—
547C2×C4—614no—
548C2×C4—612no—
549C2×C4—612no—
550C2×C4—612no—
551C2×C4—1224no—
552C2×C4—1224no—
553C2×C4—1224no—
554C2×C4—1222no—
555C2×C4—1222no—
556C2×C4—1222no—
557C2×C4—1222no—
558C2×C4—1222no—
559C2×C4—1222no—
560C2×C4—2442no—
561C2×C4—2442no—
562C2×C4—2442no—
563C2×C4C22⋊C4×D6111yescharacteristic
564C2×C4C22⋊C4×D6111yescharacteristic
565C2×C4C2×D4×D6111yescharacteristic
566C2×C4C2×D4×D6111yescharacteristic
567C8—42no—
568C8—42no—
569C8—2442no—
570C8—2442no—
571C22—24no—
572C22—24no—
573C22—21no—
574C22—21no—
575C22—21no—
576C22—21no—
577C22—42no—
578C22—42no—
579C22—42no—
580C22—82no—
581C22—82no—
582C22—84no—
583C22—84no—
584C22—4882no—
585C22—4882no—
586C22—4884no—
587C22—4884no—
588C22—2442no—
589C22—2442no—
590C22—2442no—
591C22#1683972712yes—
592C22#1683972712yes—
593C22#1683973412yes—
594C22#100×D6112yes—
595C22C2×C22⋊C4×D6111yescharacteristic
596C22C2×#1451978311yescharacteristic
597C22C2×#1451978511yescharacteristic
598C4—21no—
599C4—21no—
600C4—21no—
601C4—21no—
602C4—42no—
603C4—614no—
604C4—1222no—
605C4—1222no—
606C4—1222no—
607C4—2442no—
608C2—22no—
609C2—21no—
610C2—21no—
611C2—82no—
612C2—4882no—
613C2#2144416914yesdirect product factor
614C2C2×#1683972711yescharacteristic
615C2C2×#1683973411yescharacteristic
616C2C2×#100×D6111yescharacteristic
subgroupG/H
centerC23C22⋊C4×D61
commutatorC23×C61C23×C4
FrattiniC22×C4C23×D61
FittingC2×C61×#323C2
socleC23×C61C2×C22⋊C4
prime radicalC23×C22⋊D61C4
cosocleC22×C4×C61C24
hypercenterC2×#323D61
hypocenterC61C2×#1469
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#323

Complements

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

NH ≅ G/Ntype
2. C2×#16839816C2semidirect
3. C2×#16839812C2semidirect
4. #21444169C2direct (N × H)
5. C2×#16839486C2semidirect
6. C2×#16839485C2semidirect
7. C2×#16839485C2semidirect
8. C2×C61×#323C2semidirect
24. #16839816C22semidirect
26. #16839812C22semidirect
33. #16839486C22semidirect
34. #16839485C22semidirect
35. #16839485C22semidirect
39. C61×#323C22semidirect
307. C2×#323D61semidirect
312. C2×C61#1469semidirect
347. #323C2×D61semidirect
348. C61C2×#1469semidirect
613. C2#21444169direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145197632
C2×#168398162
C2×#168398122
#214441692
C2×#168394862
C2×#168394852
C2×C61×#3232
C2×#146961