Finite Group Databasebeta

← C2×#21446855 #29706913

Subgroups — C2×#21446855 #29706913

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×#14519808C211yesmaximal, characteristic
2C2×#16840336C211yesmaximal, characteristic
3C22×#14519705C211yesmaximal, characteristic
4C2×#16839771C211yesmaximal, characteristic, complemented
5C22×#14519696C211yesmaximal, characteristic, complemented
6#21446855C218yesmaximal, direct product factor
7C2×#16839522C211yesmaximal, characteristic, complemented
8C22×C4⋊Q8×C61C211yesmaximal, characteristic, complemented
9C22×#11758222C2211yescharacteristic
10C2×#14519808—24no—
11C2×#14519808—24no—
12C2×#14519808—22no—
13C2×#14519808C2214yes—
14C22×#11758202—21no—
15C22×#11758201—21no—
16C22×#11758194C2211yescharacteristic
17C22×#11758194C2211yescharacteristic
18#16840336—22no—
19#16840336—22no—
20#16840336—22no—
21#16840336C2214yes—
22C2×#14519775—21no—
23C2×#14519766—21no—
24C2×#14519760—22no—
25C2×#14519753—22no—
26C2×#14519736C2211yescharacteristic
27C2×#14519705—24no—
28C2×#14519705C2214yes—
29#16839771—22no—
30#16839771C2214yescomplemented
31C2×#14519701—22no—
32C2×#14519696—24no—
33C2×#14519696C2214yescomplemented
34#16839522C2214yescomplemented
35C23×Q8×C61C2211yescharacteristic
36C23×Q8×C61C2211yescharacteristic
37C2×C4⋊Q8×C61—24no—
38C2×C4⋊Q8×C61—22no—
39C2×C4⋊Q8×C61—24no—
40C2×C4⋊Q8×C61C2214yescomplemented
41C22×C4⋊C4×C61—22no—
42C22×C42×C61C2211yescharacteristic
43C2×#11758222—21no—
44C2×#11758222—21no—
45C2×#11758222—22no—
46C2×#11758222—22no—
47C2×#11758222—22no—
48C2×#11758222—22no—
49C2×#11758222D412yes—
50C2×#11758222C2312yes—
51C22×Dic122D411yescharacteristic
52C22×D244D411yescharacteristic
53C22×C22⋊D61—21no—
54C22×C4×D61—21no—
55#14519808—216no—
56#14519808—28no—
57#14519808—28no—
58C2×#11758202—28no—
59C2×#11758202—24no—
60C2×#11758201—28no—
61C2×#11758201—24no—
62C2×#11758194—22no—
63C2×#11758194—21no—
64C2×#11758194—21no—
65C2×#11758194—22no—
66C2×#11758194—22no—
67C2×#11758194D414yes—
68C2×#11758194C2312yes—
69C2×#11758194C2312yes—
70C22×#92418—21no—
71C22×#92418—21no—
72#14519775—24no—
73#14519775—24no—
74#14519766—24no—
75#14519766—24no—
76#14519760—28no—
77#14519760—28no—
78#14519753—28no—
79#14519753—28no—
80#14519736—21no—
81#14519736—21no—
82#14519736D412yes—
83#14519736C2312yes—
84C2×C4⋊Dic61D412yes—
85C2×#11758196—24no—
86C2×C22⋊Dic61—21no—
87C2×#11758198—24no—
88C2×C4×Dic61—21no—
89#14519705—28no—
90#14519701—28no—
91#14519696—28no—
92C22×Q8×C61—22no—
93C22×Q8×C61—22no—
94C22×Q8×C61—22no—
95C22×Q8×C61—21no—
96C22×Q8×C61—22no—
97C22×Q8×C61—21no—
98C22×Q8×C61—21no—
99C22×Q8×C61—21no—
100C22×Q8×C61—21no—
101C22×Q8×C61—22no—
102C22×Q8×C61—21no—
103C22×Q8×C61—21no—
104C22×Q8×C61—21no—
105C22×Q8×C61—21no—
106C22×Q8×C61—22no—
107C22×Q8×C61—22no—
108C22×Q8×C61D414yes—
109C22×Q8×C61D412yes—
110C22×Q8×C61C2312yes—
111C22×Q8×C61C2312yes—
112C23×C4×C61—21no—
113C23×C4×C61—21no—
114C23×C4×C61D412yes—
115C23×C4×C61C2311yescharacteristic
116C4⋊Q8×C61—216no—
117C4⋊Q8×C61—28no—
118C4⋊Q8×C61—28no—
119C2×C4⋊C4×C61—216no—
120C2×C4⋊C4×C61—28no—
121C2×C42×C61—21no—
122C2×C42×C61—22no—
123C2×C42×C61D414yes—
124C2×C42×C61C2312yes—
125C24×C61C2×D411yescharacteristic
126C23×D61—21no—
127C2×Q8×C61—24no—
128C2×Q8×C61—24no—
129C2×Q8×C61—22no—
130C2×Q8×C61—22no—
131C2×Q8×C61—24no—
132C2×Q8×C61—22no—
133C2×Q8×C61—22no—
134C2×Q8×C61—22no—
135C2×Q8×C61—22no—
136C2×Q8×C61—22no—
137C2×Q8×C61—22no—
138C2×Q8×C61—22no—
139C2×Q8×C61—24no—
140C2×Q8×C61—24no—
141C2×Q8×C61—24no—
142C2×Q8×C61—24no—
143C2×Q8×C61—22no—
144C2×Q8×C61—22no—
145C2×Q8×C61—22no—
146C2×Q8×C61—22no—
147C2×Q8×C61—41no—
148C2×Q8×C61—41no—
149C2×Q8×C61—42no—
150C2×Q8×C61—42no—
151C2×Q8×C61—42no—
152C2×Q8×C61—42no—
153C2×Q8×C61—42no—
154C2×Q8×C61—42no—
155C2×Q8×C61—42no—
156C2×Q8×C61—42no—
157C2×Q8×C61—42no—
158C2×Q8×C61—42no—
159C2×Q8×C61—42no—
160C2×Q8×C61—42no—
161C2×Q8×C61—42no—
162C2×Q8×C61C2×D414yes—
163C22×C4×C61—28no—
164C22×C4×C61—24no—
165C22×C4×C61—22no—
166C22×C4×C61—24no—
167C22×C4×C61—24no—
168C22×C4×C61—22no—
169C22×C4×C61—22no—
170C22×C4×C61—24no—
171C22×C4×C61—22no—
172C22×C4×C61—41no—
173C22×C4×C61—41no—
174C22×C4×C61—42no—
175C22×C4×C61C2×D414yes—
176C22×C4×C61C2×D411yescharacteristic
177C22×C4×C61C2×D411yescharacteristic
178C22×C4×C61C2×D411yescharacteristic
179C22×C4×C61C2×D412yes—
180C22×C4×C61C2×D412yes—
181C22×C4×C61C2×D411yescharacteristic
182C22×C4×C61C2×D411yescharacteristic
183C22×C4×C61C2×D412yes—
184C22×C4×C61#4112yes—
185C22×C4×C61C2411yescharacteristic
186C2×C22⋊D61—22no—
187C2×C22⋊D61—22no—
188C2×C22⋊D61—42no—
189C2×C22⋊D61—42no—
190C22×Dic61—21no—
191C22×Dic61—22no—
192#11758222—24no—
193#11758222—24no—
194#11758222—22no—
195#11758222—22no—
196#11758222—22no—
197#11758222—22no—
198#11758222—42no—
199#11758222—42no—
200#11758222—42no—
201#11758222—42no—
202C2×D244—22no—
203C2×D244—22no—
204C2×D244—21no—
205C2×D244C2×D412yes—
206C2×C4×D61—22no—
207C2×C4×D61—22no—
208C2×C4×D61—42no—
209C2×C4×D61—42no—
210C2×Dic122—22no—
211C2×Dic122—22no—
212C2×Dic122—21no—
213C2×Dic122C2×D412yes—
214C4⋊C4×C61—232no—
215C42×C61—24no—
216C42×C61—22no—
217C42×C61C2×D414yes—
218C22⋊Dic61—22no—
219C22⋊Dic61—22no—
220#11758202—216no—
221#11758201—216no—
222#11758198—216no—
223C4⋊Dic61—22no—
224C4⋊Dic61C2×D414yes—
225#11758196—216no—
226C4×Dic61—22no—
227C4×Dic61—22no—
228#11758194—24no—
229#11758194—22no—
230#11758194—22no—
231#11758194—22no—
232#11758194—42no—
233#11758194C2×D414yes—
234C2×#92418—24no—
235C2×#92418—22no—
236C2×#92418—22no—
237C2×#92418—42no—
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—
247Q8×C61—24no—
248Q8×C61—24no—
249Q8×C61—42no—
250Q8×C61—42no—
251Q8×C61—42no—
252Q8×C61—42no—
253Q8×C61—42no—
254Q8×C61—42no—
255Q8×C61—44no—
256Q8×C61—44no—
257Q8×C61—44no—
258Q8×C61—44no—
259C2×C4×C61—28no—
260C2×C4×C61—216no—
261C2×C4×C61—24no—
262C2×C4×C61—24no—
263C2×C4×C61—24no—
264C2×C4×C61—24no—
265C2×C4×C61—24no—
266C2×C4×C61—42no—
267C2×C4×C61—42no—
268C2×C4×C61—44no—
269C2×C4×C61—44no—
270C2×C4×C61—44no—
271C2×C4×C61—44no—
272C2×C4×C61#12012yes—
273C2×C4×C61#12012yes—
274C2×C4×C61#12011yescharacteristic
275C2×C4×C61#12011yescharacteristic
276C2×C4×C61#12012yes—
277C2×C4×C61#12112yes—
278C2×C4×C61#12112yes—
279C2×C4×C61#12112yes—
280C2×C4×C61#12112yes—
281C2×C4×C61#12112yes—
282C2×C4×C61#12712yes—
283C2×C4×C61#12712yes—
284C2×C4×C61C22×D411yescharacteristic
285C2×C4×C61C22×D411yescharacteristic
286C2×C4×C61C22×D411yescharacteristic
287C2×C4×C61C22×D411yescharacteristic
288C2×C4×C61C22×D411yescharacteristic
289C2×C4×C61C2×#4111yescharacteristic
290C22⋊D61—42no—
291C22⋊D61—42no—
292C22⋊D61—44no—
293C2×Dic61—24no—
294C2×Dic61—22no—
295C2×Dic61—42no—
296C2×Dic61—44no—
297D244—22no—
298D244—22no—
299D244—42no—
300C4×D61—42no—
301C4×D61—42no—
302C4×D61—44no—
303Dic122—22no—
304Dic122—22no—
305Dic122—42no—
306#92418—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—44no—
325C4×C61#39212yes—
326C4×C61#39212yes—
327C4×C61C2×#12011yescharacteristic
328C4×C61C2×#12111yescharacteristic
329C4×C61C2×#12111yescharacteristic
330C4×C61C2×#12711yescharacteristic
331Dic61—42no—
332Dic61—44no—
333C22×#137—611no—
334C2×#536—611no—
335C22×C4⋊Q8D6111yescharacteristic, complemented
336C22×#104—611no—
337C2×#419—611no—
338C22×#101—611no—
339#2005—618no—
340C2×#339—611no—
341C2×C61—21no—
342C2×C61—21no—
343C2×C61—22no—
344C2×C61#200514yescomplemented
345C2×C61C2×#39211yescharacteristic
346C2×C61C2×#45612yes—
347D61—82no—
348C22×#41—611no—
349C23×Q8C2×D6111yescharacteristic
350C23×Q8C2×D6111yescharacteristic
351C2×#137—614no—
352C2×#137—1224no—
353C2×#137—1222no—
354C2×#137—1224no—
355C22×Q16—1221no—
356C22×SD16—1221no—
357C22×M16—611no—
358C22×M16—611no—
359#536—614no—
360#536—1222no—
361#536—1222no—
362#536—1222no—
363C2×C4⋊Q8—24no—
364C2×C4⋊Q8—22no—
365C2×C4⋊Q8—24no—
366C2×C4⋊Q8—1221no—
367C2×C4⋊Q8C2×D6114yescomplemented
368C2×#124—1221no—
369C2×#123—1222no—
370C2×C22⋊Q8—1222no—
371C2×#117—611no—
372C22×C4⋊C4—22no—
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#339—614no—
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—21no—
392C22×Q8—21no—
393C22×Q8—21no—
394C22×Q8—21no—
395C22×Q8—22no—
396C22×Q8—22no—
397C22×Q8—21no—
398C22×Q8—22no—
399C22×Q8—22no—
400C22×Q8—22no—
401C22×Q8—21no—
402C22×Q8—21no—
403C22×Q8—22no—
404C22×Q8—21no—
405C22×Q8—21no—
406C22×Q8—22no—
407C22×Q8—611no—
408C22×Q8C22⋊D6114yes—
409C22×Q8C22⋊D6112yes—
410C22×Q8C22×D6112yes—
411C22×Q8C22×D6112yes—
412C22×D4—611no—
413C22×D4—1221no—
414C23×C4—21no—
415C23×C4—21no—
416C23×C4—1221no—
417C23×C4C22⋊D6112yes—
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—614no—
427C2×M16—612no—
428C2×M16—612no—
429C2×M16—1221no—
430C2×M16—1222no—
431C2×M16—1221no—
432C2×M16—1222no—
433C2×M16—1222no—
434C22×C8—1221no—
435C22×C8—1221no—
436C4⋊Q8—216no—
437C4⋊Q8—28no—
438C4⋊Q8—28no—
439C4⋊Q8—1224no—
440C4⋊Q8—1224no—
441#124—1224no—
442#124—1224no—
443#123—1228no—
444#123—1228no—
445C22⋊Q8—1228no—
446C22⋊Q8—1228no—
447#117—612no—
448#117—612no—
449#117—1221no—
450#117—1221no—
451C2×C4⋊C4—216no—
452C2×C4⋊C4—28no—
453C2×C4⋊C4—612no—
454C2×C4⋊C4—1224no—
455C2×C22⋊C4—1224no—
456C2×C22⋊C4—1221no—
457C2×C42—22no—
458C2×C42—21no—
459C2×C42—1221no—
460C2×C42C22⋊D6114yes—
461C2×C42C22×D6112yes—
462#104—1228no—
463#103—1228no—
464#101—1228no—
465C24—1221no—
466C24C2×C22⋊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—22no—
478C2×Q8—22no—
479C2×Q8—22no—
480C2×Q8—22no—
481C2×Q8—22no—
482C2×Q8—22no—
483C2×Q8—22no—
484C2×Q8—22no—
485C2×Q8—22no—
486C2×Q8—24no—
487C2×Q8—24no—
488C2×Q8—24no—
489C2×Q8—22no—
490C2×Q8—22no—
491C2×Q8—22no—
492C2×Q8—22no—
493C2×Q8—24no—
494C2×Q8—24no—
495C2×Q8—24no—
496C2×Q8—24no—
497C2×Q8—41no—
498C2×Q8—41no—
499C2×Q8—42no—
500C2×Q8—42no—
501C2×Q8—42no—
502C2×Q8—42no—
503C2×Q8—42no—
504C2×Q8—42no—
505C2×Q8—42no—
506C2×Q8—42no—
507C2×Q8—42no—
508C2×Q8—42no—
509C2×Q8—42no—
510C2×Q8—42no—
511C2×Q8—42no—
512C2×Q8—612no—
513C2×Q8—1221no—
514C2×Q8—1222no—
515C2×Q8—1222no—
516C2×Q8C2×C22⋊D6114yes—
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—22no—
530C22×C4—22no—
531C22×C4—24no—
532C22×C4—24no—
533C22×C4—24no—
534C22×C4—41no—
535C22×C4—41no—
536C22×C4—42no—
537C22×C4—1222no—
538C22×C4—1221no—
539C22×C4—1222no—
540C22×C4—1222no—
541C22×C4—2442no—
542C22×C4—2442no—
543C22×C4D4×D6111yescharacteristic
544C22×C4D4×D6111yescharacteristic
545C22×C4D4×D6112yes—
546C22×C4D4×D6111yescharacteristic
547C22×C4D4×D6111yescharacteristic
548C22×C4#1175822412yes—
549C22×C4C2×C22⋊D6114yes—
550C22×C4C2×C22⋊D6112yes—
551C22×C4C2×C22⋊D6112yes—
552C22×C4C2×C22⋊D6111yescharacteristic
553C22×C4C23×D6111yescharacteristic
554Q16—12216no—
555SD16—12216no—
556M16—614no—
557M16—1222no—
558M16—1224no—
559M16—1222no—
560M16—1222no—
561M16—2442no—
562C2×C8—1222no—
563C2×C8—1224no—
564C2×C8—1222no—
565C2×C8—2442no—
566C4⋊C4—232no—
567C4⋊C4—614no—
568C4⋊C4—12216no—
569C4⋊C4—1222no—
570C22⋊C4—12216no—
571C22⋊C4—1222no—
572C22⋊C4—1222no—
573C42—22no—
574C42—24no—
575C42—1222no—
576C42—1222no—
577C42C2×C22⋊D6114yes—
578C23—24no—
579C23—1222no—
580C23—2444no—
581C23—2442no—
582C23#1451974511yescharacteristic
583C23#1451974511yescharacteristic
584C23#1451974612yes—
585C23#1451974912yes—
586C23C22×C22⋊D6111yescharacteristic
587Q8—24no—
588Q8—24no—
589Q8—42no—
590Q8—42no—
591Q8—42no—
592Q8—42no—
593Q8—42no—
594Q8—42no—
595Q8—44no—
596Q8—44no—
597Q8—44no—
598Q8—44no—
599Q8—1222no—
600Q8—1222no—
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—24no—
613C2×C4—24no—
614C2×C4—24no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—1224no—
622C2×C4—1222no—
623C2×C4—2444no—
624C2×C4—2442no—
625C2×C4—2442no—
626C2×C4—2442no—
627C2×C4—2444no—
628C2×C4#1451974512yes—
629C2×C4#1451974512yes—
630C2×C4#1451974511yescharacteristic
631C2×C4#1451974511yescharacteristic
632C2×C4#1451974612yes—
633C2×C4#1451974812yes—
634C2×C4#1451974812yes—
635C2×C4#1451974912yes—
636C2×C4#1451974912yes—
637C2×C4#1451974912yes—
638C2×C4#1451977312yes—
639C2×C4#1451977312yes—
640C2×C4C2×D4×D6111yescharacteristic
641C2×C4C2×D4×D6111yescharacteristic
642C2×C4C2×D4×D6111yescharacteristic
643C2×C4C22×C22⋊D6111yescharacteristic
644C2×C4C22×C22⋊D6111yescharacteristic
645C2×C4C2×#1175822411yescharacteristic
646C8—1224no—
647C8—2442no—
648C22—22no—
649C22—22no—
650C22—24no—
651C22—24no—
652C22—4884no—
653C22—2442no—
654C22—2444no—
655C22#1683967012yes—
656C22#1683967012yes—
657C22#1683994414yes—
658C22C2×#1451974511yescharacteristic
659C22C2×#1451974611yescharacteristic
660C22C2×#1451974911yescharacteristic
661C4—28no—
662C4—44no—
663C4—44no—
664C4—2442no—
665C4—2444no—
666C4#1683967012yes—
667C4#1683967012yes—
668C4C2×#1451974511yescharacteristic
669C4C2×#1451974811yescharacteristic
670C4C2×#1451974911yescharacteristic
671C4C2×#1451977311yescharacteristic
672C2—21no—
673C2—21no—
674C2—22no—
675C2—4882no—
676C2#2144685514yesdirect product factor
677C2C2×#1683967011yescharacteristic
678C2C2×#1683994412yes—
subgroupG/H
centerC23#14519745
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC22×C4⋊Q8×C61C2
socleC23×C61#120
prime radicalC22×#11758222C22
hypercenterC22×C4⋊Q8D61
hypocenterC61C2×#2005
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4⋊Q8

Complements

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

NH ≅ G/Ntype
4. C2×#16839771C2semidirect
5. C22×#14519696C2semidirect
6. #21446855C2direct (N × H)
7. C2×#16839522C2semidirect
8. C22×C4⋊Q8×C61C2semidirect
30. #16839771C22semidirect
33. C2×#14519696C22semidirect
34. #16839522C22semidirect
40. C2×C4⋊Q8×C61C22semidirect
335. C22×C4⋊Q8D61semidirect
344. C2×C61#2005semidirect
367. C2×C4⋊Q8C2×D61semidirect
382. C61C2×#2005semidirect
676. C2#21446855direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145198082
C2×#168403362
C22×#145197052
C2×#168397712
C22×#145196962
#214468552
C2×#168395222
C22×C4⋊Q8×C612
C2×#200561