Finite Group Databasebeta

← #29714320

Subgroups — #29714320

Order 15616.

1478 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840312C211yesmaximal, characteristic, complemented
2C2×#16840042C211yesmaximal, characteristic, complemented
3C2×#16839957C211yesmaximal, characteristic, complemented
4#21447001C212yesmaximal, complemented
5#21446815C212yesmaximal, complemented
6#21446771C211yesmaximal, characteristic, complemented
7#21446746C211yesmaximal, characteristic, complemented
8#21446170C212yesmaximal, complemented
9#21446065C212yesmaximal, complemented
10#21445799C211yesmaximal, characteristic, complemented
11C4×#121×C61C211yesmaximal, characteristic, complemented
12C22×#11758222—21no—
13C22×#11758199C2212yes—
14#16840312—24no—
15#16840312—24no—
16C2×#14519777C2211yescharacteristic
17C2×#14519774C2211yescharacteristic
18C2×#14519765—21no—
19C2×#14519751—21no—
20C2×#14519751C2212yes—
21C2×C4×D244—21no—
22C2×C4×D244C2212yes—
23#16840042—24no—
24#16839957—24no—
25C2×#14519710C2212yes—
26C2×C4⋊#92418C2211yescharacteristic
27C2×C4⋊#92418C2211yescharacteristic
28C2×#14519698C2212yes—
29C2×#14519697C2212yes—
30C2×#14519697C2212yes—
31C2×#14519697C2212yes—
32C2×C22⋊#92418C2212yes—
33#16839524C2212yes—
34#16839524C2212yes—
35#16839517C2212yes—
36C2×#121×C61C2211yescharacteristic
37C2×C4×D4×C61—21no—
38C2×C4×D4×C61C2211yescharacteristic
39C22×C42×C61C2211yescharacteristic
40C61×#390C2211yescharacteristic
41C61×#389C2211yescharacteristic
42C61×#388—21no—
43C61×#386—21no—
44C61×#380C2211yescharacteristic
45C4×C4⋊C4×C61C2211yescharacteristic
46C4×C22⋊C4×C61—21no—
47C2×#11758222—28no—
48C2×#11758222—44no—
49C2×#11758222—44no—
50C22×Dic122—21no—
51C22×D244—21no—
52C22×D244—21no—
53C22×D244C2312yes—
54C22×C22⋊D61—22no—
55C22×C4×D61—22no—
56C2×#11758199—24no—
57C2×#11758199—24no—
58C2×#11758199—42no—
59C22×#92418C2312yes—
60C22×#92418C2312yes—
61#14519777—22no—
62#14519777D414yescomplemented
63#14519774—22no—
64#14519774D414yescomplemented
65#14519765—28no—
66#14519751—28no—
67#14519751—24no—
68#14519751—24no—
69C4×D244—28no—
70C4×D244—24no—
71C4×D244—24no—
72C2×C4⋊Dic61—21no—
73C2×C4⋊Dic61C2312yes—
74C2×#11758198—22no—
75C2×#11758198—22no—
76#14519710—24no—
77C4⋊#92418—22no—
78C4⋊#92418—22no—
79#14519698—24no—
80#14519697—24no—
81#14519697—24no—
82#14519697—42no—
83C22⋊#92418—42no—
84C22×D4×C61—21no—
85C22×D4×C61C2311yescharacteristic
86C23×C4×C61—21no—
87C23×C4×C61D411yescharacteristic
88C23×C4×C61D411yescharacteristic
89C23×C4×C61C2311yescharacteristic
90#121×C61—22no—
91#121×C61—22no—
92C4×D4×C61—22no—
93C4×D4×C61—22no—
94C4×D4×C61—44no—
95C2×C4⋊C4×C61—21no—
96C2×C4⋊C4×C61C2311yescharacteristic
97C2×C4⋊C4×C61C2311yescharacteristic
98C2×C4⋊C4×C61C2311yescharacteristic
99C2×C22⋊C4×C61—21no—
100C2×C22⋊C4×C61—21no—
101C2×C22⋊C4×C61—21no—
102C2×C22⋊C4×C61—21no—
103C2×C22⋊C4×C61—21no—
104C2×C42×C61—22no—
105C2×C42×C61—22no—
106C2×C42×C61—21no—
107C2×C42×C61D411yescharacteristic
108C2×C42×C61D411yescharacteristic
109C2×C42×C61C2311yescharacteristic
110C2×C42×C61C2311yescharacteristic
111#95×C61—21no—
112#95×C61—21no—
113#95×C61—21no—
114C24×C61—21no—
115C24×C61C2×D411yescharacteristic
116C23×D61—22no—
117C2×D4×C61—24no—
118C2×D4×C61—21no—
119C2×D4×C61—21no—
120C2×D4×C61—21no—
121C2×D4×C61—21no—
122C2×D4×C61—21no—
123C2×D4×C61—42no—
124C2×D4×C61—42no—
125C2×D4×C61#4112yescomplemented
126C22×C4×C61—22no—
127C22×C4×C61—22no—
128C22×C4×C61—21no—
129C22×C4×C61—22no—
130C22×C4×C61—21no—
131C22×C4×C61—21no—
132C22×C4×C61—21no—
133C22×C4×C61—21no—
134C22×C4×C61—21no—
135C22×C4×C61—21no—
136C22×C4×C61—21no—
137C22×C4×C61—21no—
138C22×C4×C61—41no—
139C22×C4×C61—41no—
140C22×C4×C61—42no—
141C22×C4×C61—42no—
142C22×C4×C61—42no—
143C22×C4×C61D814yescomplemented
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2411yescharacteristic
150C2×C22⋊D61—28no—
151C2×C22⋊D61—44no—
152C2×C22⋊D61—44no—
153C22×Dic61—22no—
154#11758222—432no—
155C2×D244—22no—
156C2×D244—22no—
157C2×D244—22no—
158C2×D244—22no—
159C2×D244—24no—
160C2×D244—24no—
161C2×D244—22no—
162C2×D244—42no—
163C2×D244—42no—
164C2×D244—42no—
165C2×D244—42no—
166C2×D244C2×D414yes—
167C2×C4×D61—28no—
168C2×C4×D61—44no—
169C2×C4×D61—44no—
170C2×Dic122—24no—
171C2×Dic122—42no—
172C2×Dic122—42no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—42no—
179C4⋊C4×C61#4112yescomplemented
180C22⋊C4×C61—24no—
181C22⋊C4×C61—42no—
182C22⋊C4×C61—42no—
183C22⋊C4×C61—42no—
184C22⋊C4×C61—42no—
185C42×C61—22no—
186C42×C61—22no—
187C42×C61—21no—
188C42×C61—21no—
189C42×C61—42no—
190C42×C61C2×D412yes—
191C42×C61C2×D412yes—
192#11758199—48no—
193#11758198—28no—
194#11758198—28no—
195C4⋊Dic61—24no—
196C4⋊Dic61—22no—
197C4⋊Dic61C2×D414yes—
198C2×#92418—22no—
199C2×#92418—22no—
200C2×#92418—22no—
201C2×#92418—22no—
202C2×#92418—22no—
203C2×#92418—22no—
204C23×C61—21no—
205C23×C61—21no—
206C23×C61—21no—
207C23×C61—21no—
208C23×C61—21no—
209C23×C61—21no—
210C23×C61—22no—
211C23×C61—41no—
212C23×C61—41no—
213C23×C61—41no—
214C23×C61—41no—
215C23×C61—41no—
216C23×C61—41no—
217C23×C61C2×D812yes—
218C23×C61C22×D411yescharacteristic
219C22×D61—24no—
220C22×D61—42no—
221C22×D61—42no—
222C22×D61—42no—
223C22×D61—42no—
224C22×D61—42no—
225C22×D61—42no—
226D4×C61—22no—
227D4×C61—22no—
228D4×C61—48no—
229D4×C61—41no—
230D4×C61—41no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—28no—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61—21no—
239C2×C4×C61—22no—
240C2×C4×C61—21no—
241C2×C4×C61—21no—
242C2×C4×C61—21no—
243C2×C4×C61—44no—
244C2×C4×C61—41no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61—41no—
248C2×C4×C61—41no—
249C2×C4×C61—41no—
250C2×C4×C61—41no—
251C2×C4×C61—41no—
252C2×C4×C61—41no—
253C2×C4×C61—41no—
254C2×C4×C61—41no—
255C2×C4×C61—41no—
256C2×C4×C61—84no—
257C2×C4×C61#12012yes—
258C2×C4×C61#12012yes—
259C2×C4×C61C2×D812yes—
260C2×C4×C61C2×D812yes—
261C2×C4×C61AGL(1,ℤ/8)12yes—
262C2×C4×C61C22×D411yescharacteristic
263C2×C4×C61C22×D411yescharacteristic
264C2×C4×C61C2×#4111yescharacteristic
265C2×C4×C61C2×#4111yescharacteristic
266C2×C4×C61H(5,𝔽2)11yescharacteristic
267C2×C4×C612−1+411yescharacteristic
268C22⋊D61—416no—
269C2×Dic61—24no—
270C2×Dic61—42no—
271C2×Dic61—42no—
272D244—24no—
273D244—24no—
274D244—48no—
275D244—42no—
276D244—42no—
277D244—48no—
278C4×D61—416no—
279Dic122—48no—
280#92418—42no—
281#92418—42no—
282#24150—611nomaximal, Sylow
283C22×C61—22no—
284C22×C61—22no—
285C22×C61—22no—
286C22×C61—41no—
287C22×C61—41no—
288C22×C61—41no—
289C22×C61—41no—
290C22×C61—42no—
291C22×C61—42no—
292C22×C61—42no—
293C22×C61—81no—
294C22×C61—81no—
295C22×C61—81no—
296C22×C61—81no—
297C22×C61C2×#12011yescharacteristic
298C22×C61#52511yescharacteristic
299C22×C61#53711yescharacteristic
300C22×C61C22×D811yescharacteristic
301C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
302C22×C61#57511yescharacteristic
303C22×C61#57511yescharacteristic
304C2×D61—44no—
305C2×D61—44no—
306C2×D61—44no—
307C2×D61—82no—
308C2×D61—82no—
309C2×D61—82no—
310C2×D61—82no—
311C4×C61—22no—
312C4×C61—22no—
313C4×C61—42no—
314C4×C61—41no—
315C4×C61—41no—
316C4×C61—81no—
317C4×C61—81no—
318C4×C61#44714yes—
319C4×C61#54611yescharacteristic
320C4×C61#54711yescharacteristic
321C4×C61#54811yescharacteristic
322C4×C61#55011yescharacteristic
323Dic61—44no—
324C2×#532—611no—
325C2×#474—611no—
326C2×#459—611no—
327C4×#121D6111yescharacteristic, complemented
328#2033—612no—
329#1999—612no—
330#1988—611no—
331#1982—611no—
332#1879—612no—
333#1862—612no—
334#1797—611no—
335C2×C61—42no—
336C2×C61—82no—
337C2×C61#240711yescharacteristic
338C2×C61C2×#44711yescharacteristic
339C2×C61#300511yescharacteristic
340C2×C61#310011yescharacteristic
341C2×C61#311511yescharacteristic
342C2×C61#324511yescharacteristic
343C2×C61#325111yescharacteristic
344D61—84no—
345C22×#41—1221no—
346C22×D8—612no—
347#532—1224no—
348#532—1224no—
349C2×C4⋊Q8—611no—
350C2×#127—611no—
351C2×#124—1221no—
352C2×#121—612no—
353C2×#121—1221no—
354C2×#121C2×D6111yescharacteristic
355C2×C4×D4—21no—
356C2×C4×D4—612no—
357C2×C4×D4—1221no—
358C2×C4×D4C2×D6111yescharacteristic
359C22×C42C2×D6111yescharacteristic
360#474—1224no—
361#459—1224no—
362C2×#107—612no—
363C2×C4⋊C8—611no—
364C2×C4⋊C8—611no—
365C2×#102—612no—
366C2×#102—612no—
367C2×#102—612no—
368C2×#102—612no—
369C2×C22⋊C8—612no—
370#390C2×D6111yescharacteristic
371#389C2×D6111yescharacteristic
372#388—21no—
373#386—21no—
374#380C2×D6111yescharacteristic
375C4×C4⋊C4C2×D6111yescharacteristic
376C4×C22⋊C4—21no—
377#340—612no—
378#340—612no—
379#336—612no—
380C61#2415011yescharacteristic, complemented, Sylow
381C2×#41—1228no—
382C2×#41—2444no—
383C2×#41—2444no—
384C22×Q8—1221no—
385C22×D4—21no—
386C22×D4—612no—
387C22×D4—1222no—
388C22×D4—1221no—
389C22×D4—1221no—
390C22×D4C22×D6111yescharacteristic
391C23×C4—21no—
392C23×C4—1222no—
393C23×C4C22⋊D6111yescharacteristic
394C23×C4C22⋊D6111yescharacteristic
395C23×C4C22×D6111yescharacteristic
396C2×D8—1224no—
397C2×D8—1224no—
398C2×D8—2442no—
399C22×C8—612no—
400C22×C8—612no—
401C4⋊Q8—614no—
402C4⋊Q8—1222no—
403#127—614no—
404#127—1222no—
405#124—1228no—
406#121—22no—
407#121—22no—
408#121—1224no—
409#121—1224no—
410#121—1228no—
411C4×D4—22no—
412C4×D4—22no—
413C4×D4—44no—
414C4×D4—1224no—
415C4×D4—1224no—
416C4×D4—1228no—
417C2×C4⋊C4—21no—
418C2×C4⋊C4—612no—
419C2×C4⋊C4—1221no—
420C2×C4⋊C4C22×D6111yescharacteristic
421C2×C4⋊C4C22×D6111yescharacteristic
422C2×C4⋊C4C22×D6111yescharacteristic
423C2×C22⋊C4—21no—
424C2×C22⋊C4—21no—
425C2×C22⋊C4—21no—
426C2×C22⋊C4—21no—
427C2×C22⋊C4—21no—
428C2×C22⋊C4—1222no—
429C2×C22⋊C4—1222no—
430C2×C42—22no—
431C2×C42—22no—
432C2×C42—21no—
433C2×C42C22⋊D6111yescharacteristic
434C2×C42C22⋊D6111yescharacteristic
435C2×C42C22×D6111yescharacteristic
436C2×C42C22×D6111yescharacteristic
437#107—1224no—
438C4⋊C8—1222no—
439C4⋊C8—1222no—
440#102—1224no—
441#102—1224no—
442#102—1224no—
443#102—2442no—
444C22⋊C8—2442no—
445#95—21no—
446#95—21no—
447#95—21no—
448C24—21no—
449C24—1222no—
450C24C2×C22⋊D6111yescharacteristic
451#41—24432no—
452C2×Q8—1224no—
453C2×Q8—2442no—
454C2×Q8—2442no—
455C2×D4—24no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4—21no—
461C2×D4—42no—
462C2×D4—42no—
463C2×D4—614no—
464C2×D4—1228no—
465C2×D4—1222no—
466C2×D4—1222no—
467C2×D4—1222no—
468C2×D4—1222no—
469C2×D4—1222no—
470C2×D4—1224no—
471C2×D4—1224no—
472C2×D4—2442no—
473C2×D4—2442no—
474C2×D4—2444no—
475C2×D4—2444no—
476C2×D4—2442no—
477C2×D4—2442no—
478C2×D4#1175822212yescomplemented
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—22no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—41no—
492C22×C4—41no—
493C22×C4—42no—
494C22×C4—42no—
495C22×C4—42no—
496C22×C4—1228no—
497C22×C4—1222no—
498C22×C4—2444no—
499C22×C4—2444no—
500C22×C4#1175819914yescomplemented
501C22×C4C2×C22⋊D6111yescharacteristic
502C22×C4C2×C22⋊D6111yescharacteristic
503C22×C4C2×C22⋊D6111yescharacteristic
504C22×C4C2×C22⋊D6111yescharacteristic
505C22×C4C2×C22⋊D6111yescharacteristic
506C22×C4C23×D6111yescharacteristic
507D8—2448no—
508C2×C8—1222no—
509C2×C8—1222no—
510C2×C8—1222no—
511C2×C8—1222no—
512C2×C8—1222no—
513C2×C8—1222no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—42no—
520C4⋊C4—614no—
521C4⋊C4—1222no—
522C4⋊C4—1224no—
523C4⋊C4#1175822212yescomplemented
524C22⋊C4—24no—
525C22⋊C4—42no—
526C22⋊C4—42no—
527C22⋊C4—42no—
528C22⋊C4—42no—
529C22⋊C4—1228no—
530C22⋊C4—1228no—
531C42—21no—
532C42—21no—
533C42—22no—
534C42—22no—
535C42—42no—
536C42D4×D6112yes—
537C42D4×D6112yes—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—22no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23—41no—
550C23—41no—
551C23—1224no—
552C23—2442no—
553C23—2442no—
554C23—2442no—
555C23—2442no—
556C23—2442no—
557C23—2442no—
558C23C2×#1175819912yes—
559C23C22×C22⋊D6111yescharacteristic
560Q8—2448no—
561D4—22no—
562D4—22no—
563D4—48no—
564D4—41no—
565D4—41no—
566D4—1224no—
567D4—1224no—
568D4—2448no—
569D4—24416no—
570D4—2448no—
571D4—2442no—
572D4—2442no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—28no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—44no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—84no—
599C2×C4—1224no—
600C2×C4—24416no—
601C2×C4—2442no—
602C2×C4—2442no—
603C2×C4#1451974512yes—
604C2×C4#1451974512yes—
605C2×C4C2×#1175819912yes—
606C2×C4C2×#1175819912yes—
607C2×C4#1451980212yes—
608C2×C4C2×D4×D6111yescharacteristic
609C2×C4C2×D4×D6111yescharacteristic
610C2×C4C2×#1175822211yescharacteristic
611C2×C4C2×#1175822211yescharacteristic
612C2×C4#1451982311yescharacteristic
613C2×C4#1451982411yescharacteristic
614C8—2442no—
615C8—2442no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—41no—
620C22—41no—
621C22—41no—
622C22—41no—
623C22—42no—
624C22—42no—
625C22—42no—
626C22—81no—
627C22—81no—
628C22—81no—
629C22—81no—
630C22—4882no—
631C22—4882no—
632C22—4882no—
633C22—4882no—
634C22—2444no—
635C22—2444no—
636C22—2444no—
637C22C2×#1451974511yescharacteristic
638C22#1684027311yescharacteristic
639C22#1684034011yescharacteristic
640C22C22×#1175819911yescharacteristic
641C22C2×#1451980211yescharacteristic
642C22#1684056011yescharacteristic
643C22#1684056011yescharacteristic
644C4—22no—
645C4—22no—
646C4—42no—
647C4—41no—
648C4—41no—
649C4—81no—
650C4—81no—
651C4—2444no—
652C4#1683989514yes—
653C4#1684040811yescharacteristic
654C4#1684041711yescharacteristic
655C4#1684042111yescharacteristic
656C4#1684043311yescharacteristic
657C2—42no—
658C2—82no—
659C2—4884no—
660C2#2144916411yescharacteristic
661C2C2×#1683989511yescharacteristic
662C2#2145505911yescharacteristic
663C2#2145581911yescharacteristic
664C2#2145597011yescharacteristic
665C2#2145712611yescharacteristic
666C2#2145718111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC4×#121×C61C2
socleC23×C61C22×D4
hypercenterC4×#121D61
hypocenterC61#24150
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×#121

Complements

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

NH ≅ G/Ntype
1. C2×#16840312C2semidirect
2. C2×#16840042C2semidirect
3. C2×#16839957C2semidirect
4. #21447001C2semidirect
5. #21446815C2semidirect
6. #21446771C2semidirect
7. #21446746C2semidirect
8. #21446170C2semidirect
9. #21446065C2semidirect
10. #21445799C2semidirect
11. C4×#121×C61C2semidirect
62. #14519777D4semidirect
64. #14519774D4semidirect
125. C2×D4×C61#41semidirect
143. C22×C4×C61D8semidirect
179. C4⋊C4×C61#41semidirect
327. C4×#121D61semidirect
380. C61#24150semidirect
478. C2×D4#11758222semidirect
500. C22×C4#11758199semidirect
523. C4⋊C4#11758222semidirect

Maximal subgroups

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

subgroupindex
C2×#168403122
C2×#168400422
C2×#168399572
#214470012
#214468152
#214467712
#214467462
#214461702
#214460652
#214457992
C4×#121×C612
#2415061