Finite Group Databasebeta

← C2×#21446858 #29706916

Subgroups — C2×#21446858 #29706916

Order 15616.

1998 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519810C211yesmaximal, characteristic
2C2×#16840338C211yesmaximal, characteristic
3C22×#14519777C211yesmaximal, characteristic, complemented
4C22×#14519705C211yesmaximal, characteristic
5C2×#16839770C211yesmaximal, characteristic, complemented
6C22×#14519695C211yesmaximal, characteristic, complemented
7#21446858C218yesmaximal, direct product factor
8C2×C61×#339C211yesmaximal, characteristic, complemented
9C22×#11758222C2211yescharacteristic
10C23×Dic122C2211yescharacteristic
11C23×Dic122C2211yescharacteristic
12C2×#14519810—24no—
13C2×#14519810—22no—
14C2×#14519810—24no—
15C2×#14519810C2214yes—
16C22×Dic244—21no—
17C22×#11758190—21no—
18C22×#11758194C2211yescharacteristic
19#16840338—22no—
20#16840338—22no—
21#16840338—22no—
22#16840338C2214yes—
23C2×#14519777—24no—
24C2×#14519777—22no—
25C2×#14519777—24no—
26C2×#14519777—21no—
27C2×#14519777C2214yescomplemented
28C2×#14519765—21no—
29C2×#14519761—22no—
30C2×#14519757—22no—
31C22×C4⋊Dic61—22no—
32C2×#14519705—24no—
33C2×#14519705C2214yes—
34#16839770—22no—
35#16839770C2214yescomplemented
36C2×#14519700—22no—
37C2×#14519695—24no—
38C2×#14519695C2214yescomplemented
39C22×M16×C61C2211yescharacteristic
40C2×#117×C61C2211yescharacteristic
41C22×C42×C61C2211yescharacteristic
42C61×#339C2214yescomplemented
43C2×#11758222—21no—
44C2×#11758222—21no—
45C2×#11758222—22no—
46C2×#11758222—22no—
47C2×#11758222—22no—
48C2×#11758222—22no—
49C2×#11758222D412yes—
50C2×#11758222C2312yes—
51C22×Dic122—21no—
52C22×Dic122—21no—
53C22×Dic122—21no—
54C22×Dic122—21no—
55C22×Dic122—22no—
56C22×Dic122—22no—
57C22×Dic122—22no—
58C22×Dic122—21no—
59C22×Dic122—21no—
60C22×Dic122—21no—
61C22×Dic122—22no—
62C22×Dic122—22no—
63C22×Dic122—22no—
64C22×Dic122—21no—
65C22×Dic122—21no—
66C22×Dic122—22no—
67C22×Dic122D412yes—
68C22×Dic122D411yescharacteristic
69C22×Dic122D414yes—
70C22×Dic122C2312yes—
71C22×Dic122C2312yes—
72C22×D244D411yescharacteristic
73C22×C22⋊D61—21no—
74C23×Dic61—21no—
75C23×Dic61D412yes—
76C22×C4×D61—21no—
77#14519810—216no—
78#14519810—28no—
79#14519810—28no—
80C2×Dic244—28no—
81C2×Dic244—24no—
82C2×#11758190—28no—
83C2×#11758190—24no—
84C2×#11758194—21no—
85C2×#11758194—22no—
86C2×#11758194—22no—
87C2×#11758194C2312yes—
88C22×#92418—21no—
89#14519777—216no—
90#14519777—24no—
91#14519777—28no—
92#14519777—24no—
93#14519777—28no—
94#14519765—24no—
95#14519765—24no—
96#14519761—28no—
97#14519761—28no—
98#14519757—28no—
99#14519757—28no—
100C2×C4⋊Dic61—216no—
101C2×C4⋊Dic61—24no—
102C2×C4⋊Dic61—28no—
103C2×#11758198—24no—
104#14519705—28no—
105#14519700—28no—
106#14519695—28no—
107C23×C4×C61—21no—
108C23×C4×C61C2311yescharacteristic
109C2×M16×C61—22no—
110C2×M16×C61—21no—
111C2×M16×C61D414yes—
112C2×M16×C61C2312yes—
113C22×C8×C61—21no—
114#117×C61—21no—
115#117×C61—21no—
116#117×C61D412yes—
117#117×C61C2312yes—
118C2×C4⋊C4×C61D412yes—
119C2×C22⋊C4×C61—21no—
120C2×C42×C61—21no—
121C2×C42×C61—21no—
122C2×C42×C61—22no—
123C2×C42×C61D414yes—
124C2×C42×C61C2312yes—
125C24×C61C2×D411yescharacteristic
126C23×D61—21no—
127C22×C4×C61—28no—
128C22×C4×C61—24no—
129C22×C4×C61—22no—
130C22×C4×C61—22no—
131C22×C4×C61C2×D411yescharacteristic
132C22×C4×C61C2×D411yescharacteristic
133C22×C4×C61C2×D411yescharacteristic
134C22×C4×C61C2×D412yes—
135C22×C4×C61C2×D412yes—
136C22×C4×C61C2×D411yescharacteristic
137C22×C4×C61C2×D412yes—
138C22×C4×C61C2×D411yescharacteristic
139C22×C4×C61#4112yes—
140C22×C4×C61C2411yescharacteristic
141C2×C22⋊D61—22no—
142C2×C22⋊D61—22no—
143C2×C22⋊D61—42no—
144C2×C22⋊D61—42no—
145C22×Dic61—21no—
146C22×Dic61—24no—
147C22×Dic61—22no—
148C22×Dic61—24no—
149C22×Dic61—22no—
150C22×Dic61—24no—
151C22×Dic61—22no—
152C22×Dic61—41no—
153C22×Dic61—41no—
154C22×Dic61—42no—
155C22×Dic61C2×D414yes—
156#11758222—24no—
157#11758222—24no—
158#11758222—22no—
159#11758222—22no—
160#11758222—22no—
161#11758222—22no—
162#11758222—42no—
163#11758222—42no—
164#11758222—42no—
165#11758222—42no—
166C2×D244—22no—
167C2×D244—22no—
168C2×D244—21no—
169C2×D244C2×D412yes—
170C2×C4×D61—22no—
171C2×C4×D61—22no—
172C2×C4×D61—42no—
173C2×C4×D61—42no—
174C2×Dic122—24no—
175C2×Dic122—24no—
176C2×Dic122—22no—
177C2×Dic122—24no—
178C2×Dic122—24no—
179C2×Dic122—22no—
180C2×Dic122—22no—
181C2×Dic122—22no—
182C2×Dic122—22no—
183C2×Dic122—22no—
184C2×Dic122—22no—
185C2×Dic122—22no—
186C2×Dic122—24no—
187C2×Dic122—24no—
188C2×Dic122—21no—
189C2×Dic122—24no—
190C2×Dic122—22no—
191C2×Dic122—22no—
192C2×Dic122—22no—
193C2×Dic122—22no—
194C2×Dic122—22no—
195C2×Dic122—22no—
196C2×Dic122—22no—
197C2×Dic122—42no—
198C2×Dic122—42no—
199C2×Dic122—41no—
200C2×Dic122—41no—
201C2×Dic122—42no—
202C2×Dic122—42no—
203C2×Dic122—42no—
204C2×Dic122—42no—
205C2×Dic122—42no—
206C2×Dic122—42no—
207C2×Dic122—42no—
208C2×Dic122—42no—
209C2×Dic122—42no—
210C2×Dic122—42no—
211C2×Dic122—42no—
212C2×Dic122C2×D414yes—
213C2×Dic122C2×D412yes—
214M16×C61—24no—
215M16×C61—22no—
216M16×C61C2×D414yes—
217C2×C8×C61—24no—
218C2×C8×C61—22no—
219C4⋊C4×C61—22no—
220C4⋊C4×C61C2×D414yes—
221C22⋊C4×C61—22no—
222C22⋊C4×C61—22no—
223C42×C61—24no—
224C42×C61—22no—
225C42×C61—22no—
226C42×C61—22no—
227C42×C61C2×D414yes—
228#11758198—216no—
229C4⋊Dic61—232no—
230C4⋊Dic61—216no—
231#11758194—22no—
232#11758194—22no—
233#11758194—42no—
234C2×#92418—22no—
235C2×#92418—42no—
236Dic244—216no—
237#11758190—216no—
238C23×C61—24no—
239C23×C61#12011yescharacteristic
240C23×C61#12012yes—
241C23×C61#12011yescharacteristic
242C23×C61#12112yes—
243C23×C61C22×D411yescharacteristic
244C22×D61—22no—
245C22×D61—42no—
246C22×D61—44no—
247C2×C4×C61—28no—
248C2×C4×C61—216no—
249C2×C4×C61—24no—
250C2×C4×C61—24no—
251C2×C4×C61—44no—
252C2×C4×C61#12012yes—
253C2×C4×C61#12012yes—
254C2×C4×C61#12012yes—
255C2×C4×C61#12011yescharacteristic
256C2×C4×C61#12011yescharacteristic
257C2×C4×C61#12112yes—
258C2×C4×C61#12112yes—
259C2×C4×C61#12112yes—
260C2×C4×C61#12112yes—
261C2×C4×C61#12112yes—
262C2×C4×C61#12712yes—
263C2×C4×C61#12712yes—
264C2×C4×C61C22×D411yescharacteristic
265C2×C4×C61C22×D411yescharacteristic
266C2×C4×C61C22×D411yescharacteristic
267C2×C4×C61C22×D411yescharacteristic
268C2×C4×C61C22×D411yescharacteristic
269C2×C4×C61C2×#4111yescharacteristic
270C22⋊D61—42no—
271C22⋊D61—42no—
272C22⋊D61—44no—
273C2×Dic61—24no—
274C2×Dic61—24no—
275C2×Dic61—22no—
276C2×Dic61—24no—
277C2×Dic61—24no—
278C2×Dic61—42no—
279C2×Dic61—42no—
280C2×Dic61—44no—
281C2×Dic61—44no—
282C2×Dic61—42no—
283C2×Dic61—44no—
284C2×Dic61—44no—
285D244—22no—
286D244—22no—
287D244—42no—
288C4×D61—42no—
289C4×D61—42no—
290C4×D61—44no—
291Dic122—24no—
292Dic122—22no—
293Dic122—24no—
294Dic122—22no—
295Dic122—42no—
296Dic122—42no—
297Dic122—42no—
298Dic122—42no—
299Dic122—42no—
300Dic122—42no—
301Dic122—44no—
302Dic122—44no—
303Dic122—44no—
304Dic122—44no—
305Dic122—42no—
306C8×C61—24no—
307#92418—42no—
308C2×#2005—611nomaximal, Sylow
309C22×C61—22no—
310C22×C61—22no—
311C22×C61—24no—
312C22×C61—24no—
313C22×C61#39212yes—
314C22×C61#39212yes—
315C22×C61#45614yes—
316C22×C61C2×#12011yescharacteristic
317C22×C61C2×#12011yescharacteristic
318C22×C61C2×#12111yescharacteristic
319C2×D61—42no—
320C2×D61—44no—
321C2×D61—84no—
322C4×C61—28no—
323C4×C61—44no—
324C4×C61#39212yes—
325C4×C61#39212yes—
326C4×C61C2×#12011yescharacteristic
327C4×C61C2×#12111yescharacteristic
328C4×C61C2×#12111yescharacteristic
329C4×C61C2×#12711yescharacteristic
330Dic61—44no—
331Dic61—42no—
332Dic61—44no—
333C22×#137—611no—
334C2×#536—611no—
335C22×C4⋊Q8—611no—
336C22×#104—611no—
337C2×#419—611no—
338C22×#101—611no—
339#2005—618no—
340C2×#339D6111yescharacteristic, complemented
341C2×C61—21no—
342C2×C61—21no—
343C2×C61—22no—
344C2×C61#200514yescomplemented
345C2×C61C2×#39211yescharacteristic
346C2×C61C2×#45612yes—
347D61—82no—
348C22×#41—611no—
349C23×Q8—611no—
350C23×Q8—611no—
351C2×#137—614no—
352C2×#137—1224no—
353C2×#137—1222no—
354C2×#137—1224no—
355C22×Q16—1221no—
356C22×SD16—1221no—
357C22×M16—611no—
358C22×M16C2×D6111yescharacteristic
359#536—614no—
360#536—1222no—
361#536—1222no—
362#536—1222no—
363C2×C4⋊Q8—614no—
364C2×C4⋊Q8—1222no—
365C2×C4⋊Q8—1224no—
366C2×C4⋊Q8—1221no—
367C2×C4⋊Q8—1224no—
368C2×#124—1221no—
369C2×#123—1222no—
370C2×C22⋊Q8—1222no—
371C2×#117C2×D6111yescharacteristic
372C22×C4⋊C4—1222no—
373C22×C42C2×D6111yescharacteristic
374C2×#104—614no—
375C2×#104—1224no—
376#419—614no—
377#419—1222no—
378C2×#103—1222no—
379C2×#101—614no—
380C2×#101—1224no—
381#339C2×D6114yescomplemented
382C61C2×#200511yescharacteristic, complemented, Sylow
383C2×#41—612no—
384C2×#41—612no—
385C2×#41—1221no—
386C2×#41—1221no—
387C2×#41—1222no—
388C2×#41—1222no—
389C2×#41—1222no—
390C2×#41—1222no—
391C22×Q8—612no—
392C22×Q8—612no—
393C22×Q8—611no—
394C22×Q8—614no—
395C22×Q8—612no—
396C22×Q8—1221no—
397C22×Q8—1222no—
398C22×Q8—1222no—
399C22×Q8—1221no—
400C22×Q8—1222no—
401C22×Q8—1221no—
402C22×Q8—1221no—
403C22×Q8—1222no—
404C22×Q8—1221no—
405C22×Q8—1222no—
406C22×Q8—1222no—
407C22×Q8—1221no—
408C22×Q8—1221no—
409C22×Q8—1221no—
410C22×Q8—1221no—
411C22×Q8—1222no—
412C22×D4—611no—
413C22×D4—1221no—
414C23×C4—21no—
415C23×C4—612no—
416C23×C4—1221no—
417C23×C4—1221no—
418C23×C4C22×D6111yescharacteristic
419#137—1228no—
420#137—12216no—
421#137—1228no—
422C2×Q16—1224no—
423C2×Q16—1228no—
424C2×SD16—1224no—
425C2×SD16—1228no—
426C2×M16—21no—
427C2×M16—22no—
428C2×M16—612no—
429C2×M16—1221no—
430C2×M16—1222no—
431C2×M16—1222no—
432C2×M16D24414yes—
433C2×M16C22×D6112yes—
434C22×C8—21no—
435C22×C8—1221no—
436C4⋊Q8—1228no—
437C4⋊Q8—1224no—
438C4⋊Q8—12216no—
439C4⋊Q8—1224no—
440C4⋊Q8—1228no—
441#124—1224no—
442#124—1224no—
443#123—1228no—
444#123—1228no—
445C22⋊Q8—1228no—
446C22⋊Q8—1228no—
447#117—21no—
448#117—21no—
449#117D24412yes—
450#117C22×D6112yes—
451C2×C4⋊C4—1228no—
452C2×C4⋊C4—12216no—
453C2×C4⋊C4—1224no—
454C2×C4⋊C4D24412yes—
455C2×C22⋊C4—21no—
456C2×C22⋊C4—1224no—
457C2×C42—21no—
458C2×C42—22no—
459C2×C42—21no—
460C2×C42D24414yes—
461C2×C42C22×D6112yes—
462#104—1228no—
463#103—1228no—
464#101—1228no—
465C24—1221no—
466C24D4×D6111yescharacteristic
467#41—1222no—
468#41—1222no—
469#41—1222no—
470#41—1222no—
471#41—1224no—
472#41—1224no—
473#41—2442no—
474#41—2442no—
475#41—2442no—
476#41—2442no—
477C2×Q8—612no—
478C2×Q8—614no—
479C2×Q8—1222no—
480C2×Q8—1222no—
481C2×Q8—1222no—
482C2×Q8—1222no—
483C2×Q8—1222no—
484C2×Q8—1222no—
485C2×Q8—1222no—
486C2×Q8—1222no—
487C2×Q8—1222no—
488C2×Q8—1224no—
489C2×Q8—1224no—
490C2×Q8—1221no—
491C2×Q8—1224no—
492C2×Q8—1222no—
493C2×Q8—1222no—
494C2×Q8—1222no—
495C2×Q8—1222no—
496C2×Q8—1224no—
497C2×Q8—1222no—
498C2×Q8—1224no—
499C2×Q8—1224no—
500C2×Q8—1224no—
501C2×Q8—1222no—
502C2×Q8—2441no—
503C2×Q8—2441no—
504C2×Q8—2442no—
505C2×Q8—2442no—
506C2×Q8—2442no—
507C2×Q8—2442no—
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×Q8—2442no—
517C2×D4—612no—
518C2×D4—1222no—
519C2×D4—1222no—
520C2×D4—1222no—
521C2×D4—1221no—
522C2×D4—1222no—
523C2×D4—2442no—
524C2×D4—2442no—
525C22×C4—28no—
526C22×C4—22no—
527C22×C4—24no—
528C22×C4—22no—
529C22×C4—614no—
530C22×C4—1222no—
531C22×C4—1224no—
532C22×C4—1224no—
533C22×C4—1222no—
534C22×C4—1222no—
535C22×C4—1221no—
536C22×C4—1224no—
537C22×C4—1222no—
538C22×C4—1222no—
539C22×C4—2441no—
540C22×C4—2441no—
541C22×C4—2442no—
542C22×C4—2442no—
543C22×C4—2442no—
544C22×C4C2×D24412yes—
545C22×C4C2×D24412yes—
546C22×C4C2×D24411yescharacteristic
547C22×C4C2×D24411yescharacteristic
548C22×C4D4×D6112yes—
549C22×C4D4×D6111yescharacteristic
550C22×C4D4×D6111yescharacteristic
551C22×C4D4×D6111yescharacteristic
552C22×C4#1175822612yes—
553C22×C4C23×D6111yescharacteristic
554Q16—12216no—
555SD16—12216no—
556M16—22no—
557M16—24no—
558M16—1222no—
559M16—1222no—
560M16—2442no—
561M16C2×D24414yes—
562C2×C8—22no—
563C2×C8—24no—
564C2×C8—1222no—
565C2×C8—2442no—
566C4⋊C4—22no—
567C4⋊C4—12232no—
568C4⋊C4—12216no—
569C4⋊C4C2×D24414yes—
570C22⋊C4—22no—
571C22⋊C4—22no—
572C22⋊C4—12216no—
573C42—22no—
574C42—22no—
575C42—24no—
576C42—22no—
577C42C2×D24414yes—
578C23—24no—
579C23—1222no—
580C23—2444no—
581C23—2442no—
582C23#1451974711yescharacteristic
583C23#1451974712yes—
584C23#1451974711yescharacteristic
585C23#1451975212yes—
586C23C2×D4×D6111yescharacteristic
587Q8—1224no—
588Q8—1222no—
589Q8—1224no—
590Q8—1222no—
591Q8—2442no—
592Q8—2442no—
593Q8—2442no—
594Q8—2442no—
595Q8—2442no—
596Q8—2442no—
597Q8—2444no—
598Q8—2444no—
599Q8—2444no—
600Q8—2444no—
601Q8—2442no—
602D4—1222no—
603D4—1222no—
604D4—2442no—
605D4—2442no—
606D4—2444no—
607D4—2442no—
608C2×C4—28no—
609C2×C4—216no—
610C2×C4—24no—
611C2×C4—24no—
612C2×C4—44no—
613C2×C4—1224no—
614C2×C4—1224no—
615C2×C4—1222no—
616C2×C4—1224no—
617C2×C4—1224no—
618C2×C4—2442no—
619C2×C4—2442no—
620C2×C4—2444no—
621C2×C4—2444no—
622C2×C4—2442no—
623C2×C4—2444no—
624C2×C4—2444no—
625C2×C4—2442no—
626C2×C4—2442no—
627C2×C4—2444no—
628C2×C4#1451974712yes—
629C2×C4#1451974712yes—
630C2×C4#1451974711yescharacteristic
631C2×C4#1451974712yes—
632C2×C4#1451974711yescharacteristic
633C2×C4#1451975212yes—
634C2×C4#1451975212yes—
635C2×C4#1451975212yes—
636C2×C4#1451975212yes—
637C2×C4#1451975212yes—
638C2×C4#1451977412yes—
639C2×C4#1451977412yes—
640C2×C4C2×D4×D6111yescharacteristic
641C2×C4C2×D4×D6111yescharacteristic
642C2×C4C22×D24411yescharacteristic
643C2×C4C22×D24411yescharacteristic
644C2×C4C22×D24411yescharacteristic
645C2×C4C2×#1175822611yescharacteristic
646C8—24no—
647C8—2442no—
648C22—22no—
649C22—22no—
650C22—24no—
651C22—24no—
652C22—4884no—
653C22—2442no—
654C22—2444no—
655C22#1683967212yes—
656C22#1683967212yes—
657C22#1683994614yes—
658C22C2×#1451974711yescharacteristic
659C22C2×#1451974711yescharacteristic
660C22C2×#1451975211yescharacteristic
661C4—28no—
662C4—44no—
663C4—2444no—
664C4—2442no—
665C4—2444no—
666C4#1683967212yes—
667C4#1683967212yes—
668C4C2×#1451974711yescharacteristic
669C4C2×#1451975211yescharacteristic
670C4C2×#1451975211yescharacteristic
671C4C2×#1451977411yescharacteristic
672C2—21no—
673C2—21no—
674C2—22no—
675C2—4882no—
676C2#2144685814yesdirect product factor
677C2C2×#1683967211yescharacteristic
678C2C2×#1683994612yes—
subgroupG/H
centerC23#14519747
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#339C2
socleC23×C61#120
prime radicalC22×#11758222C22
hypercenterC2×#339D61
hypocenterC61C2×#2005
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#339

Complements

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

NH ≅ G/Ntype
3. C22×#14519777C2semidirect
5. C2×#16839770C2semidirect
6. C22×#14519695C2semidirect
7. #21446858C2direct (N × H)
8. C2×C61×#339C2semidirect
27. C2×#14519777C22semidirect
35. #16839770C22semidirect
38. C2×#14519695C22semidirect
42. C61×#339C22semidirect
340. C2×#339D61semidirect
344. C2×C61#2005semidirect
381. #339C2×D61semidirect
382. C61C2×#2005semidirect
676. C2#21446858direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145198102
C2×#168403382
C22×#145197772
C22×#145197052
C2×#168397702
C22×#145196952
#214468582
C2×C61×#3392
C2×#200561