Finite Group Databasebeta

← C2×#21444166 #29645956

Subgroups — C2×#21444166 #29645956

Order 15616.

1490 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519762C211yesmaximal, characteristic
2C2×#16839724C211yesmaximal, characteristic, complemented
3C2×C22⋊C8×D61C211yesmaximal, characteristic, complemented
4#21444166C218yesmaximal, direct product factor
5C2×#16839488C211yesmaximal, characteristic, complemented
6C2×#16839485C211yesmaximal, characteristic, complemented
7C2×#16839485C211yesmaximal, characteristic, complemented
8C2×C61×#324C211yesmaximal, characteristic, complemented
9C23×D244C411yescharacteristic
10C23×C4×D61C2211yescharacteristic
11C22×#11758189—21no—
12C22×C8×D61—21no—
13C2×#14519762—21no—
14C2×#14519762—21no—
15C2×#14519762—22no—
16C2×#14519762—21no—
17C2×#14519762—21no—
18C2×#14519762—22no—
19C2×#14519762C414yes—
20C2×#14519762C2214yes—
21C22×#11758196C411yescharacteristic
22C22×#11758198C411yescharacteristic
23C22×#11758198C411yescharacteristic
24C22×#11758198C2211yescharacteristic
25#16839724—22no—
26#16839724C2214yescomplemented
27C2×C22⋊#92418C2211yescharacteristic
28C2×C22⋊#92418C2211yescharacteristic
29C2×#14519689—21no—
30C2×#14519689—21no—
31C22⋊C8×D61—22no—
32C22⋊C8×D61C2214yescomplemented
33#16839488C2214yescomplemented
34#16839485C2214yescomplemented
35#16839485C2214yescomplemented
36C22×C4⋊C4×C61C2211yescharacteristic
37C2×C22⋊C8×C61C2211yescharacteristic
38C2×C22⋊C8×C61C2211yescharacteristic
39C61×#324C2214yescomplemented
40C24×D61C2×C411yescharacteristic
41C24×D61C2×C411yescharacteristic
42C22×D244—21no—
43C22×D244—22no—
44C22×D244—21no—
45C22×D244—44no—
46C22×D244C2×C414yes—
47C23×Dic61C2×C411yescharacteristic
48C23×Dic61C2×C411yescharacteristic
49C22×C4×D61—44no—
50C22×C4×D61C2×C412yes—
51C22×C4×D61D411yescharacteristic
52C22×C4×D61D411yescharacteristic
53C22×C4×D61D412yes—
54C22×C4×D61D411yescharacteristic
55C22×C4×D61D411yescharacteristic
56C22×C4×D61D412yes—
57C22×C4×D61C2312yes—
58C2×#11758189—24no—
59C2×#11758189—44no—
60C22×#92418—21no—
61C22×#92418—21no—
62C2×C8×D61—24no—
63C2×C8×D61—44no—
64#14519762—24no—
65#14519762—24no—
66#14519762—24no—
67#14519762—24no—
68#14519762—24no—
69#14519762—24no—
70#14519762—24no—
71#14519762—24no—
72C2×#11758196—21no—
73C2×#11758196—21no—
74C2×#11758196—22no—
75C2×#11758196C2×C414yes—
76C2×#11758198—21no—
77C2×#11758198—22no—
78C2×#11758198—21no—
79C2×#11758198—21no—
80C2×#11758198—21no—
81C2×#11758198—21no—
82C2×#11758198—21no—
83C2×#11758198—21no—
84C2×#11758198—21no—
85C2×#11758198—22no—
86C2×#11758198C2×C414yes—
87C2×#11758198C2×C414yes—
88C2×#11758198C2×C412yes—
89C2×#11758198C2312yes—
90C22⋊#92418—21no—
91C22⋊#92418—21no—
92C22⋊#92418C2312yes—
93C22⋊#92418C2312yes—
94#14519689—24no—
95#14519689—24no—
96C23×C4×C61C2×C411yescharacteristic
97C23×C4×C61C2×C411yescharacteristic
98C23×C4×C61C2311yescharacteristic
99C22×C8×C61—21no—
100C22×C8×C61—21no—
101C2×C4⋊C4×C61—21no—
102C2×C4⋊C4×C61—21no—
103C2×C4⋊C4×C61—21no—
104C2×C4⋊C4×C61—21no—
105C2×C4⋊C4×C61C2×C412yes—
106C2×C4⋊C4×C61C2312yes—
107C22⋊C8×C61—21no—
108C22⋊C8×C61—21no—
109C22⋊C8×C61C2312yes—
110C22⋊C8×C61C2312yes—
111C24×C61C22×C411yescharacteristic
112C23×D61—22no—
113C23×D61—22no—
114C23×D61—24no—
115C23×D61—21no—
116C23×D61—21no—
117C23×D61—22no—
118C23×D61—22no—
119C23×D61—42no—
120C23×D61—42no—
121C23×D61—42no—
122C23×D61C22⋊C411yescharacteristic
123C23×D61C22⋊C412yes—
124C23×D61C22⋊C411yescharacteristic
125C23×D61C22×C412yes—
126C23×D61C22×C412yes—
127C22×C4×C61—21no—
128C22×C4×C61—21no—
129C22×C4×C61—21no—
130C22×C4×C61—21no—
131C22×C4×C61—22no—
132C22×C4×C61—22no—
133C22×C4×C61—42no—
134C22×C4×C61—42no—
135C22×C4×C61C22×C412yes—
136C22×C4×C61C22×C412yes—
137C22×C4×C61C22×C411yescharacteristic
138C22×C4×C61C2×D411yescharacteristic
139C22×C4×C61C2×D411yescharacteristic
140C22×C4×C61C2×D411yescharacteristic
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61C2411yescharacteristic
143C22×Dic61—21no—
144C22×Dic61—21no—
145C22×Dic61—42no—
146C22×Dic61—42no—
147C22×Dic61C22⋊C411yescharacteristic
148C22×Dic61C22⋊C411yescharacteristic
149C22×Dic61C22⋊C412yes—
150C22×Dic61C22×C412yes—
151C22×Dic61C22×C412yes—
152C2×D244—24no—
153C2×D244—24no—
154C2×D244—44no—
155C2×D244—48no—
156C2×D244—44no—
157C2×D244—48no—
158C2×C4×D61—24no—
159C2×C4×D61—24no—
160C2×C4×D61—44no—
161C2×C4×D61—44no—
162C2×C4×D61—44no—
163C2×C4×D61—44no—
164C2×C4×D61—42no—
165C2×C4×D61—42no—
166C2×C4×D61C22⋊C412yes—
167C2×C4×D61C22⋊C412yes—
168C2×C4×D61C22⋊C412yes—
169C2×C4×D61C22⋊C412yes—
170C2×C4×D61C2×D412yes—
171C2×C4×D61C2×D412yes—
172C2×C4×D61C2×D412yes—
173C2×C4×D61C2×D412yes—
174C2×C8×C61—22no—
175C2×C8×C61—22no—
176C2×C8×C61—42no—
177C2×C8×C61—42no—
178C4⋊C4×C61—22no—
179C4⋊C4×C61—22no—
180C4⋊C4×C61—22no—
181C4⋊C4×C61—22no—
182#11758198—24no—
183#11758198—24no—
184#11758198—22no—
185#11758198—22no—
186#11758198—22no—
187#11758198—22no—
188#11758198—24no—
189#11758198—24no—
190#11758196—24no—
191#11758196—24no—
192C2×#92418—22no—
193C2×#92418—22no—
194C2×#92418—42no—
195C2×#92418—42no—
196#11758189—48no—
197C8×D61—48no—
198C23×C61—22no—
199C23×C61—22no—
200C23×C61C2×C22⋊C411yescharacteristic
201C23×C61C2×C22⋊C411yescharacteristic
202C23×C61#13112yes—
203C23×C61#13112yes—
204C23×C61C23×C411yescharacteristic
205C22×D61—22no—
206C22×D61—22no—
207C22×D61—22no—
208C22×D61—22no—
209C22×D61—24no—
210C22×D61—22no—
211C22×D61—22no—
212C22×D61—22no—
213C22×D61—24no—
214C22×D61—24no—
215C22×D61—22no—
216C22×D61—22no—
217C22×D61—22no—
218C22×D61—42no—
219C22×D61—42no—
220C22×D61—48no—
221C22×D61—48no—
222C22×D61—42no—
223C22×D61—42no—
224C22×D61—42no—
225C22×D61—42no—
226C22×D61—42no—
227C22×D61—42no—
228C22×D61—42no—
229C22×D61—42no—
230C22×D61—84no—
231C22×D61—84no—
232C22×D61#9914yes—
233C22×D61C2×C22⋊C412yes—
234C22×D61C2×C22⋊C412yes—
235C2×C4×C61—22no—
236C2×C4×C61—22no—
237C2×C4×C61—22no—
238C2×C4×C61—22no—
239C2×C4×C61—22no—
240C2×C4×C61—22no—
241C2×C4×C61—41no—
242C2×C4×C61—41no—
243C2×C4×C61—41no—
244C2×C4×C61—41no—
245C2×C4×C61—42no—
246C2×C4×C61—42no—
247C2×C4×C61—42no—
248C2×C4×C61—42no—
249C2×C4×C61—42no—
250C2×C4×C61C2×C22⋊C411yescharacteristic
251C2×C4×C61C2×C22⋊C411yescharacteristic
252C2×C4×C61C22×D411yescharacteristic
253C2×C4×C61C22×D411yescharacteristic
254C2×Dic61—22no—
255C2×Dic61—22no—
256C2×Dic61—22no—
257C2×Dic61—42no—
258C2×Dic61—42no—
259C2×Dic61—42no—
260C2×Dic61—42no—
261C2×Dic61—42no—
262C2×Dic61C2×C22⋊C412yes—
263C2×Dic61C2×C22⋊C412yes—
264D244—48no—
265D244—48no—
266C4×D61—24no—
267C4×D61—24no—
268C4×D61—24no—
269C4×D61—24no—
270C4×D61—42no—
271C4×D61—42no—
272C4×D61—42no—
273C4×D61—42no—
274C8×C61—42no—
275C8×C61—42no—
276#92418—42no—
277#92418—42no—
278C2×#1468—611nomaximal, Sylow
279C22×C61—21no—
280C22×C61—21no—
281C22×C61—21no—
282C22×C61—21no—
283C22×C61—24no—
284C22×C61—24no—
285C22×C61#40812yes—
286C22×C61#40812yes—
287C22×C61C2×#9912yes—
288C22×C61#41312yes—
289C22×C61C22×C22⋊C411yescharacteristic
290C22×C61C2×#13111yescharacteristic
291C22×C61C2×#13111yescharacteristic
292C2×D61—22no—
293C2×D61—22no—
294C2×D61—28no—
295C2×D61—22no—
296C2×D61—22no—
297C2×D61—22no—
298C2×D61—44no—
299C2×D61—44no—
300C2×D61—48no—
301C2×D61—44no—
302C2×D61—42no—
303C2×D61—42no—
304C2×D61—42no—
305C2×D61—84no—
306C2×D61—84no—
307C2×D61—82no—
308C2×D61—82no—
309C2×D61C2×#9914yes—
310C4×C61—21no—
311C4×C61—21no—
312C4×C61—21no—
313C4×C61—21no—
314C4×C61—42no—
315C4×C61—42no—
316Dic61—22no—
317Dic61—22no—
318Dic61—42no—
319C22×#123—611no—
320C2×#407—611no—
321C22×C22⋊C8—611no—
322#1468—618no—
323C2×#324—611no—
324C2×#324D6111yescharacteristic, complemented
325C2×#323—611no—
326C2×#323—611no—
327C2×C61—21no—
328C2×C61—21no—
329C2×C61—22no—
330C2×C61#146814yescomplemented
331C2×C61C2×#40811yescharacteristic
332C2×C61C22×#9911yescharacteristic
333C2×C61C2×#41311yescharacteristic
334D61—24no—
335D61—42no—
336D61—82no—
337C23×D4—611no—
338C24×C4—611no—
339C22×M16—1221no—
340C23×C8—1221no—
341C2×#123—614no—
342C2×#123—614no—
343C2×#123—1221no—
344C2×#123—1221no—
345C2×#123—1222no—
346C2×#123—1221no—
347C2×#123—1221no—
348C2×#123—1222no—
349C22×C4⋊C4—611no—
350C22×C4⋊C4C2×D6111yescharacteristic
351C22×C22⋊C4—611no—
352C22×C22⋊C4—611no—
353C22×C22⋊C4—611no—
354#407—614no—
355#407—1222no—
356C2×C22⋊C8—611no—
357C2×C22⋊C8—611no—
358C2×C22⋊C8—614no—
359C2×C22⋊C8—1222no—
360C2×C22⋊C8—1221no—
361C2×C22⋊C8—1221no—
362C2×C22⋊C8C2×D6111yescharacteristic
363C2×C22⋊C8C2×D6111yescharacteristic
364#324—614no—
365#324C2×D6114yescomplemented
366#323—614no—
367#323—614no—
368C61C2×#146811yescharacteristic, complemented, Sylow
369C25—611no—
370C25—611no—
371C22×D4—614no—
372C22×D4—1222no—
373C22×D4—1221no—
374C22×D4—1221no—
375C22×D4—2444no—
376C23×C4—611no—
377C23×C4—611no—
378C23×C4—611no—
379C23×C4—612no—
380C23×C4—611no—
381C23×C4—611no—
382C23×C4—612no—
383C23×C4—612no—
384C23×C4—612no—
385C23×C4—611no—
386C23×C4—2444no—
387C23×C4C4×D6111yescharacteristic
388C23×C4C4×D6111yescharacteristic
389C23×C4C22×D6111yescharacteristic
390C2×M16—1224no—
391C2×M16—2444no—
392C22×C8—21no—
393C22×C8—21no—
394C22×C8—1224no—
395C22×C8—1221no—
396C22×C8—1221no—
397C22×C8—2444no—
398#123—1224no—
399#123—1224no—
400#123—1224no—
401#123—1224no—
402#123—1224no—
403#123—1224no—
404#123—1224no—
405#123—1224no—
406C2×C4⋊C4—21no—
407C2×C4⋊C4—21no—
408C2×C4⋊C4—21no—
409C2×C4⋊C4—21no—
410C2×C4⋊C4—614no—
411C2×C4⋊C4—1222no—
412C2×C4⋊C4—1221no—
413C2×C4⋊C4—1221no—
414C2×C4⋊C4C4×D6112yes—
415C2×C4⋊C4C22×D6112yes—
416C2×C22⋊C4—614no—
417C2×C22⋊C4—614no—
418C2×C22⋊C4—612no—
419C2×C22⋊C4—612no—
420C2×C22⋊C4—1222no—
421C2×C22⋊C4—1221no—
422C2×C22⋊C4—1221no—
423C2×C22⋊C4—1222no—
424C2×C22⋊C4—1221no—
425C2×C22⋊C4—1221no—
426C2×C22⋊C4—1221no—
427C2×C22⋊C4—1221no—
428C2×C22⋊C4—1221no—
429C2×C22⋊C4—1221no—
430C22⋊C8—21no—
431C22⋊C8—21no—
432C22⋊C8—612no—
433C22⋊C8—612no—
434C22⋊C8—1224no—
435C22⋊C8—1224no—
436C22⋊C8—1221no—
437C22⋊C8—1221no—
438C22⋊C8C22×D6112yes—
439C22⋊C8C22×D6112yes—
440C24—611no—
441C24—611no—
442C24—612no—
443C24—612no—
444C24—612no—
445C24—1224no—
446C24—1222no—
447C24—1222no—
448C24—1222no—
449C24—1222no—
450C24—1221no—
451C24—1221no—
452C24—2442no—
453C24—2442no—
454C24—2442no—
455C24C2×C4×D6111yescharacteristic
456C2×D4—1224no—
457C2×D4—1224no—
458C2×D4—2444no—
459C2×D4—2448no—
460C2×D4—2444no—
461C2×D4—2448no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—611no—
471C22×C4—611no—
472C22×C4—612no—
473C22×C4—612no—
474C22×C4—612no—
475C22×C4—612no—
476C22×C4—612no—
477C22×C4—612no—
478C22×C4—612no—
479C22×C4—612no—
480C22×C4—612no—
481C22×C4—612no—
482C22×C4—612no—
483C22×C4—1224no—
484C22×C4—1224no—
485C22×C4—1221no—
486C22×C4—1221no—
487C22×C4—2442no—
488C22×C4—2442no—
489C22×C4—2444no—
490C22×C4—2444no—
491C22×C4—2444no—
492C22×C4—2444no—
493C22×C4—2442no—
494C22×C4—2442no—
495C22×C4C2×C4×D6112yes—
496C22×C4C2×C4×D6112yes—
497C22×C4C2×C4×D6111yescharacteristic
498C22×C4D4×D6111yescharacteristic
499C22×C4D4×D6111yescharacteristic
500C22×C4D4×D6111yescharacteristic
501C22×C4D4×D6111yescharacteristic
502C22×C4C23×D6111yescharacteristic
503M16—2448no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—42no—
507C2×C8—42no—
508C2×C8—1222no—
509C2×C8—1222no—
510C2×C8—2448no—
511C2×C8—2442no—
512C2×C8—2442no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—1224no—
518C4⋊C4—1224no—
519C22⋊C4—1224no—
520C22⋊C4—1224no—
521C22⋊C4—1224no—
522C22⋊C4—1224no—
523C22⋊C4—1222no—
524C22⋊C4—1222no—
525C22⋊C4—1222no—
526C22⋊C4—1222no—
527C23—22no—
528C23—22no—
529C23—612no—
530C23—614no—
531C23—612no—
532C23—1224no—
533C23—1224no—
534C23—1222no—
535C23—1222no—
536C23—1224no—
537C23—1222no—
538C23—1222no—
539C23—1222no—
540C23—1222no—
541C23—1222no—
542C23—1222no—
543C23—1222no—
544C23—1222no—
545C23—4884no—
546C23—4884no—
547C23—2442no—
548C23—2442no—
549C23—2448no—
550C23—2448no—
551C23—2442no—
552C23—2442no—
553C23—2442no—
554C23—2442no—
555C23—2442no—
556C23—2442no—
557C23—2442no—
558C23—2442no—
559C23C22⋊C4×D6111yescharacteristic
560C23C22⋊C4×D6111yescharacteristic
561C23#1451978312yes—
562C23#1451978512yes—
563C23C22×C4×D6111yescharacteristic
564D4—2448no—
565D4—2448no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—612no—
582C2×C4—612no—
583C2×C4—1224no—
584C2×C4—1224no—
585C2×C4—1224no—
586C2×C4—1224no—
587C2×C4—1222no—
588C2×C4—1222no—
589C2×C4—1222no—
590C2×C4—2442no—
591C2×C4—2442no—
592C2×C4—2442no—
593C2×C4—2442no—
594C2×C4—2442no—
595C2×C4—2442no—
596C2×C4—2442no—
597C2×C4—2442no—
598C2×C4—2442no—
599C2×C4C22⋊C4×D6111yescharacteristic
600C2×C4C22⋊C4×D6111yescharacteristic
601C2×C4C2×D4×D6111yescharacteristic
602C2×C4C2×D4×D6111yescharacteristic
603C8—42no—
604C8—42no—
605C8—2442no—
606C8—2442no—
607C22—24no—
608C22—24no—
609C22—21no—
610C22—21no—
611C22—21no—
612C22—21no—
613C22—614no—
614C22—1228no—
615C22—1222no—
616C22—1222no—
617C22—1222no—
618C22—1222no—
619C22—1222no—
620C22—4882no—
621C22—4882no—
622C22—4884no—
623C22—4884no—
624C22—2448no—
625C22—2444no—
626C22—2444no—
627C22—2442no—
628C22—2442no—
629C22—2442no—
630C22—2444no—
631C22#1683972712yes—
632C22#1683972712yes—
633C22#99×D6112yes—
634C22#1683974512yes—
635C22C2×C22⋊C4×D6111yescharacteristic
636C22C2×#1451978311yescharacteristic
637C22C2×#1451978511yescharacteristic
638C4—21no—
639C4—21no—
640C4—21no—
641C4—21no—
642C4—42no—
643C4—42no—
644C4—1222no—
645C4—1222no—
646C4—2442no—
647C2—22no—
648C2—21no—
649C2—21no—
650C2—1224no—
651C2—4882no—
652C2—2442no—
653C2#2144416614yesdirect product factor
654C2C2×#1683972711yescharacteristic
655C2C2×#99×D6111yescharacteristic
656C2C2×#1683974511yescharacteristic
subgroupG/H
centerC23C22⋊C4×D61
commutatorC23×C61C23×C4
FrattiniC22×C4C23×D61
FittingC2×C61×#324C2
socleC23×C61C2×C22⋊C4
prime radicalC23×D244C4
cosocleC22×C4×C61C24
hypercenterC2×#324D61
hypocenterC61C2×#1468
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#324

Complements

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

NH ≅ G/Ntype
2. C2×#16839724C2semidirect
3. C2×C22⋊C8×D61C2semidirect
4. #21444166C2direct (N × H)
5. C2×#16839488C2semidirect
6. C2×#16839485C2semidirect
7. C2×#16839485C2semidirect
8. C2×C61×#324C2semidirect
26. #16839724C22semidirect
32. C22⋊C8×D61C22semidirect
33. #16839488C22semidirect
34. #16839485C22semidirect
35. #16839485C22semidirect
39. C61×#324C22semidirect
324. C2×#324D61semidirect
330. C2×C61#1468semidirect
365. #324C2×D61semidirect
368. C61C2×#1468semidirect
653. C2#21444166direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145197622
C2×#168397242
C2×C22⋊C8×D612
#214441662
C2×#168394882
C2×#168394852
C2×C61×#3242
C2×#146861