Finite Group Databasebeta

← #29684074

Subgroups — #29684074

Order 15616.

1650 conjugacy classes of subgroups.

The subgroup lattice has 698 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840308C211yesmaximal, characteristic
2C22×#14519706C211yesmaximal, characteristic, complemented
3C2×#16839759C211yesmaximal, characteristic, complemented
4C2×#16839758C211yesmaximal, characteristic, complemented
5C2×#16839756C211yesmaximal, characteristic, complemented
6#117×Dic61C211yesmaximal, characteristic, complemented
7#21446750C212yesmaximal, complemented
8#21446748C212yesmaximal, complemented
9#21446744C212yesmaximal, complemented
10#21446740C212yesmaximal, complemented
11C2×C61×#416C211yesmaximal, characteristic, complemented
12C22×#11758222C2211yescharacteristic
13C23×#92418C2211yescharacteristic
14#16840308—24no—
15#16840308C418yescomplemented
16C2×#14519775C2211yescharacteristic
17C2×#14519773C2211yescharacteristic
18C2×#14519766C2211yescharacteristic
19C2×#14519764C2211yescharacteristic
20C2×#14519749—21no—
21C2×#14519749—21no—
22C2×C4×C22⋊D61—21no—
23C2×C4×C22⋊D61—21no—
24C2×#14519736C2211yescharacteristic
25C22×C4×Dic61C2211yescharacteristic
26C2×#14519706—24no—
27C2×#14519706C2212yes—
28C2×#14519706C2212yes—
29#16839759—24no—
30#16839758—24no—
31#16839756—24no—
32C2×#14519702C2212yes—
33C2×#14519701C2212yes—
34C2×#14519700C2212yes—
35C2×#14519699C2212yes—
36C2×#14519698C2212yes—
37C2×#14519697C2212yes—
38C4⋊C4×Dic61C2212yes—
39C4⋊C4×Dic61C2212yes—
40C22⋊C4×Dic61—21no—
41C22⋊C4×Dic61—21no—
42#16839608—21no—
43C42×Dic61—21no—
44C22×#41×C61C2211yescharacteristic
45C23×C8×C61C2211yescharacteristic
46C2×#117×C61C2211yescharacteristic
47C61×#416—24no—
48C2×#103×C61C2212yes—
49C2×#102×C61C2212yes—
50C2×#11758222—22no—
51C2×#11758222—22no—
52C2×#11758222—22no—
53C2×#11758222—22no—
54C2×#11758222—22no—
55C2×#11758222C2×C414yes—
56C22×Dic122C2311yescharacteristic
57C22×D244C2311yescharacteristic
58C22×C22⋊D61—21no—
59C22×C22⋊D61—21no—
60C22×C22⋊D61—21no—
61C23×Dic61D411yescharacteristic
62C23×Dic61D411yescharacteristic
63C22×C4×D61—21no—
64C22×#92418—22no—
65C22×#92418—22no—
66C22×#92418—22no—
67C22×#92418C2312yes—
68#14519775—22no—
69#14519775C2×C414yes—
70#14519773—22no—
71#14519773C2×C414yes—
72#14519766—22no—
73#14519766C2×C414yes—
74#14519764—22no—
75#14519764C2×C414yes—
76#14519749—28no—
77#14519749—28no—
78C4×C22⋊D61—28no—
79C4×C22⋊D61—28no—
80#14519736—22no—
81#14519736D414yescomplemented
82C2×C4⋊Dic61C2312yes—
83C2×#11758196—21no—
84C2×#11758196—21no—
85C2×C22⋊Dic61—21no—
86C2×C22⋊Dic61—21no—
87C2×C22⋊Dic61—21no—
88C2×#11758198—21no—
89C2×#11758198—21no—
90C2×C4×Dic61—21no—
91C2×C4×Dic61—22no—
92C2×C4×Dic61—22no—
93C2×C4×Dic61C2×C414yes—
94C2×C4×Dic61D411yescharacteristic
95C2×C4×Dic61D411yescharacteristic
96C2×C4×Dic61D414yes—
97C2×C4×Dic61C2311yescharacteristic
98C2×C4×Dic61C2311yescharacteristic
99#14519706—24no—
100#14519706—24no—
101#14519702—24no—
102#14519701—24no—
103#14519700—24no—
104#14519699—24no—
105#14519698—24no—
106#14519697—24no—
107#14519684—22no—
108#14519684—22no—
109C2×#41×C61—22no—
110C2×#41×C61—22no—
111C2×#41×C61—22no—
112C2×#41×C61—22no—
113C2×#41×C61—22no—
114C2×#41×C61C2×C414yes—
115C22×Q8×C61C2311yescharacteristic
116C22×D4×C61—21no—
117C22×D4×C61C2311yescharacteristic
118C23×C4×C61—21no—
119C23×C4×C61C2311yescharacteristic
120C22×C8×C61—22no—
121C22×C8×C61—22no—
122C22×C8×C61—22no—
123C22×C8×C61C2312yes—
124#117×C61—22no—
125#117×C61D414yescomplemented
126C2×C4⋊C4×C61C2312yes—
127C2×C22⋊C4×C61—21no—
128C2×C42×C61—21no—
129#103×C61—24no—
130#102×C61—24no—
131C24×C61—21no—
132C24×C61C2×D411yescharacteristic
133C23×D61—21no—
134#41×C61—28no—
135#41×C61—28no—
136#41×C61—44no—
137#41×C61—44no—
138C2×Q8×C61—21no—
139C2×Q8×C61—21no—
140C2×Q8×C61—21no—
141C2×Q8×C61—21no—
142C2×Q8×C61—21no—
143C2×Q8×C61C22×C412yes—
144C2×D4×C61—24no—
145C2×D4×C61—21no—
146C2×D4×C61—21no—
147C2×D4×C61—21no—
148C2×D4×C61—21no—
149C2×D4×C61—21no—
150C2×D4×C61—42no—
151C2×D4×C61—42no—
152C2×D4×C61C22×C412yes—
153C22×C4×C61—22no—
154C22×C4×C61—21no—
155C22×C4×C61—24no—
156C22×C4×C61—42no—
157C22×C4×C61—42no—
158C22×C4×C61C22×C412yes—
159C22×C4×C61C2×D411yescharacteristic
160C22×C4×C61C2×D412yes—
161C22×C4×C61C2×D412yes—
162C22×C4×C61C2×D412yes—
163C22×C4×C61#4112yes—
164C22×C4×C61#4112yes—
165C22×C4×C61C2411yescharacteristic
166C2×C22⋊D61—24no—
167C2×C22⋊D61—24no—
168C2×C22⋊D61—24no—
169C2×C22⋊D61—44no—
170C2×C22⋊D61—42no—
171C2×C22⋊D61—42no—
172C2×C22⋊D61—44no—
173C22×Dic61—24no—
174C22×Dic61—21no—
175C22×Dic61—22no—
176C22×Dic61—42no—
177C22×Dic61C22⋊C414yes—
178C22×Dic61C22⋊C414yes—
179C22×Dic61C2×D411yescharacteristic
180C22×Dic61C2×D411yescharacteristic
181C22×Dic61C2×D411yescharacteristic
182C22×Dic61C2×D411yescharacteristic
183#11758222—28no—
184#11758222—28no—
185#11758222—44no—
186#11758222—44no—
187C2×D244—21no—
188C2×D244—21no—
189C2×D244—21no—
190C2×D244—21no—
191C2×D244—21no—
192C2×D244C22×C412yes—
193C2×C4×D61—24no—
194C2×C4×D61—42no—
195C2×C4×D61—42no—
196C2×Dic122—21no—
197C2×Dic122—21no—
198C2×Dic122—21no—
199C2×Dic122—21no—
200C2×Dic122—21no—
201C2×Dic122C22×C412yes—
202C2×C8×C61—22no—
203C2×C8×C61—22no—
204C2×C8×C61—22no—
205C2×C8×C61—44no—
206C4⋊C4×C61—22no—
207C4⋊C4×C61C2×D414yes—
208C22⋊C4×C61—24no—
209C42×C61—24no—
210C22⋊Dic61—24no—
211C22⋊Dic61—24no—
212C22⋊Dic61—24no—
213#11758198—24no—
214#11758198—24no—
215C4⋊Dic61—22no—
216C4⋊Dic61C2×D414yes—
217#11758196—24no—
218#11758196—24no—
219C4×Dic61—24no—
220C4×Dic61—44no—
221C4×Dic61—44no—
222C4×Dic61C22⋊C414yes—
223C4×Dic61C22⋊C414yes—
224C4×Dic61C22×C412yes—
225C4×Dic61C22×C412yes—
226C4×Dic61C2×D412yes—
227C4×Dic61C2×D412yes—
228C2×#92418—22no—
229C2×#92418—22no—
230C2×#92418—22no—
231C2×#92418—44no—
232C23×C61—22no—
233C23×C61—22no—
234C23×C61—22no—
235C23×C61—41no—
236C23×C61—41no—
237C23×C61—42no—
238C23×C61—42no—
239C23×C61C2×C22⋊C412yes—
240C23×C61#12112yes—
241C23×C61#12112yes—
242C23×C61C22×D411yescharacteristic
243C22×D61—22no—
244C22×D61—41no—
245C22×D61—41no—
246C22×D61—42no—
247C22×D61—42no—
248Q8×C61—22no—
249Q8×C61—22no—
250Q8×C61—41no—
251Q8×C61—41no—
252D4×C61—22no—
253D4×C61—22no—
254D4×C61—48no—
255D4×C61—41no—
256D4×C61—41no—
257C2×C4×C61—22no—
258C2×C4×C61—24no—
259C2×C4×C61—41no—
260C2×C4×C61—41no—
261C2×C4×C61—42no—
262C2×C4×C61—42no—
263C2×C4×C61—48no—
264C2×C4×C61C2×C22⋊C412yes—
265C2×C4×C61C4×D414yes—
266C2×C4×C61C4×D414yes—
267C2×C4×C61#12112yes—
268C2×C4×C61#12112yes—
269C2×C4×C61#12414yes—
270C2×C4×C61#12714yes—
271C2×C4×C61C23×C411yescharacteristic
272C2×C4×C61C22×D411yescharacteristic
273C2×C4×C61C22×D411yescharacteristic
274C2×C4×C61C22×D411yescharacteristic
275C2×C4×C61C2×#4111yescharacteristic
276C2×C4×C61C2×#4111yescharacteristic
277C22⋊D61—48no—
278C22⋊D61—48no—
279C22⋊D61—48no—
280C2×Dic61—22no—
281C2×Dic61—22no—
282C2×Dic61—22no—
283C2×Dic61—28no—
284C2×Dic61—22no—
285C2×Dic61—24no—
286C2×Dic61—44no—
287C2×Dic61—41no—
288C2×Dic61—41no—
289C2×Dic61—42no—
290C2×Dic61—42no—
291C2×Dic61C2×C22⋊C412yes—
292C2×Dic61C2×C22⋊C412yes—
293C2×Dic61C2×C22⋊C412yes—
294C2×Dic61C2×C22⋊C412yes—
295C2×Dic61#13514yes—
296D244—22no—
297D244—22no—
298D244—41no—
299D244—41no—
300C4×D61—48no—
301Dic122—22no—
302Dic122—22no—
303Dic122—41no—
304Dic122—41no—
305C8×C61—42no—
306#92418—42no—
307#21257—611nomaximal, Sylow
308C22×C61—22no—
309C22×C61—22no—
310C22×C61—24no—
311C22×C61—24no—
312C22×C61—42no—
313C22×C61—42no—
314C22×C61—42no—
315C22×C61—82no—
316C22×C61—82no—
317C22×C61#39014yes—
318C22×C61C22×C22⋊C411yescharacteristic
319C22×C61C2×#12111yescharacteristic
320C22×C61C2×#12111yescharacteristic
321C22×C61C2×#13512yes—
322C22×C61#57512yes—
323C2×D61—42no—
324C2×D61—42no—
325C2×D61—42no—
326C2×D61—82no—
327C2×D61—82no—
328C4×C61—44no—
329C4×C61—42no—
330C4×C61#39014yes—
331C4×C61C2×C4×D411yescharacteristic
332C4×C61C2×C4×D411yescharacteristic
333C4×C61C2×#12411yescharacteristic
334C4×C61C2×#12711yescharacteristic
335Dic61—22no—
336Dic61—22no—
337Dic61—22no—
338Dic61—44no—
339Dic61—42no—
340Dic61#41614yescomplemented
341C2×#532—611no—
342C22×C4⋊C8—611no—
343C2×#416—611no—
344C2×#416—611no—
345C2×#416—611no—
346C2×#416D6111yescharacteristic, complemented
347C4×#117—611no—
348#1983—612no—
349#1983—612no—
350#1982—612no—
351#1981—612no—
352C2×C61—24no—
353C2×C61—82no—
354C2×C61C2×#39011yescharacteristic
355C2×C61C2×#41611yescharacteristic
356C2×C61#289611yescharacteristic
357C2×C61#303512yes—
358C2×C61#303512yes—
359D61—82no—
360C22×#41—611no—
361C22×#41C2×D6111yescharacteristic
362C23×C8—611no—
363C23×C8C2×D6111yescharacteristic
364#532—618no—
365#532—1224no—
366C2×C4⋊Q8—611no—
367C2×#127—611no—
368C2×#124—611no—
369C2×#124—611no—
370C2×#121—1221no—
371C2×#121—1221no—
372C2×C4×D4—1221no—
373C2×C4×D4—1221no—
374C2×#117—611no—
375C2×#117C2×D6111yescharacteristic
376C22×C42—611no—
377C2×C4⋊C8—612no—
378C2×C4⋊C8—612no—
379C2×C4⋊C8—1224no—
380#416—24no—
381#416—1224no—
382#416—1224no—
383#416—1224no—
384C2×#103—612no—
385C2×#103—612no—
386C2×#103—612no—
387C2×#103C2×D6112yes—
388C2×#102—612no—
389C2×#102—612no—
390C2×#102—612no—
391C2×#102C2×D6112yes—
392C4×C4⋊C4—612no—
393C4×C4⋊C4—612no—
394C4×C22⋊C4—1221no—
395C4×C22⋊C4—1221no—
396#376—1221no—
397C43—1221no—
398C61#2125711yescharacteristic, complemented, Sylow
399C2×#41—22no—
400C2×#41—22no—
401C2×#41—22no—
402C2×#41—22no—
403C2×#41—22no—
404C2×#41—614no—
405C2×#41—1222no—
406C2×#41—1222no—
407C2×#41—1222no—
408C2×#41—1222no—
409C2×#41—1222no—
410C2×#41C4×D6114yes—
411C22×Q8—611no—
412C22×Q8C22×D6111yescharacteristic
413C22×D4—21no—
414C22×D4—611no—
415C22×D4—1221no—
416C22×D4—1221no—
417C22×D4—1221no—
418C22×D4C22×D6111yescharacteristic
419C23×C4—21no—
420C23×C4—611no—
421C23×C4—611no—
422C23×C4—1221no—
423C23×C4C22×D6111yescharacteristic
424C22×C8—22no—
425C22×C8—22no—
426C22×C8—22no—
427C22×C8—612no—
428C22×C8—1222no—
429C22×C8—1222no—
430C22×C8—1222no—
431C22×C8C22×D6112yes—
432C4⋊Q8—614no—
433C4⋊Q8—1222no—
434#127—614no—
435#127—1222no—
436#124—614no—
437#124—614no—
438#124—1222no—
439#124—1222no—
440#121—1228no—
441#121—1228no—
442C4×D4—1228no—
443C4×D4—1228no—
444#117—22no—
445#117—614no—
446#117—1222no—
447#117C22⋊D6114yescomplemented
448C2×C4⋊C4—612no—
449C2×C4⋊C4—1221no—
450C2×C4⋊C4—1221no—
451C2×C4⋊C4C22×D6112yes—
452C2×C22⋊C4—21no—
453C2×C22⋊C4—1221no—
454C2×C22⋊C4—1221no—
455C2×C22⋊C4—1221no—
456C2×C22⋊C4—1221no—
457C2×C22⋊C4—1221no—
458C2×C42—21no—
459C2×C42—614no—
460C2×C42—614no—
461C2×C42—611no—
462C2×C42—611no—
463C2×C42—611no—
464C2×C42—611no—
465C2×C42—1221no—
466C2×C42—1222no—
467C2×C42—1222no—
468C4⋊C8—1224no—
469C4⋊C8—1224no—
470#103—24no—
471#103—1224no—
472#103—1224no—
473#103—1224no—
474#102—24no—
475#102—1224no—
476#102—1224no—
477#102—1224no—
478#95—1222no—
479#95—1222no—
480C24—21no—
481C24—1221no—
482C24D4×D6111yescharacteristic
483#41—28no—
484#41—28no—
485#41—44no—
486#41—44no—
487#41—1228no—
488#41—1228no—
489#41—2444no—
490#41—2444no—
491C2×Q8—21no—
492C2×Q8—21no—
493C2×Q8—21no—
494C2×Q8—21no—
495C2×Q8—21no—
496C2×Q8—612no—
497C2×Q8—1221no—
498C2×Q8—1221no—
499C2×Q8—1221no—
500C2×Q8—1221no—
501C2×Q8—1221no—
502C2×Q8C2×C4×D6112yes—
503C2×D4—24no—
504C2×D4—21no—
505C2×D4—21no—
506C2×D4—21no—
507C2×D4—21no—
508C2×D4—21no—
509C2×D4—42no—
510C2×D4—42no—
511C2×D4—612no—
512C2×D4—1224no—
513C2×D4—1224no—
514C2×D4—1224no—
515C2×D4—1221no—
516C2×D4—1221no—
517C2×D4—1221no—
518C2×D4—1221no—
519C2×D4—1221no—
520C2×D4—2444no—
521C2×D4—2444no—
522C2×D4—2442no—
523C2×D4—2442no—
524C2×D4C2×C4×D6112yes—
525C22×C4—22no—
526C22×C4—21no—
527C22×C4—24no—
528C22×C4—42no—
529C22×C4—42no—
530C22×C4—611no—
531C22×C4—611no—
532C22×C4—611no—
533C22×C4—611no—
534C22×C4—614no—
535C22×C4—614no—
536C22×C4—1224no—
537C22×C4—1222no—
538C22×C4—1221no—
539C22×C4—1224no—
540C22×C4—2442no—
541C22×C4—2442no—
542C22×C4—2442no—
543C22×C4C2×C4×D6112yes—
544C22×C4#1175822212yes—
545C22×C4D4×D6112yes—
546C22×C4D4×D6112yes—
547C22×C4D4×D6111yescharacteristic
548C22×C4#1175822412yes—
549C22×C4C2×C22⋊D6112yes—
550C22×C4C23×D6111yescharacteristic
551C2×C8—22no—
552C2×C8—22no—
553C2×C8—22no—
554C2×C8—44no—
555C2×C8—1222no—
556C2×C8—1222no—
557C2×C8—1222no—
558C2×C8—2444no—
559C4⋊C4—22no—
560C4⋊C4—614no—
561C4⋊C4—1224no—
562C4⋊C4—1224no—
563C4⋊C4—1222no—
564C4⋊C4C2×C22⋊D6114yes—
565C22⋊C4—24no—
566C22⋊C4—1224no—
567C22⋊C4—1224no—
568C22⋊C4—1224no—
569C22⋊C4—1224no—
570C22⋊C4—1224no—
571C42—24no—
572C42—612no—
573C42—612no—
574C42—612no—
575C42—612no—
576C42—614no—
577C42—614no—
578C42—1224no—
579C42—2444no—
580C42—2444no—
581C23—22no—
582C23—22no—
583C23—22no—
584C23—41no—
585C23—41no—
586C23—42no—
587C23—42no—
588C23—1222no—
589C23—2441no—
590C23—2441no—
591C23—2442no—
592C23—2442no—
593C23C22⋊C4×D6112yes—
594C23#1451974912yes—
595C23#1451975012yes—
596C23C2×D4×D6111yescharacteristic
597Q8—22no—
598Q8—22no—
599Q8—41no—
600Q8—41no—
601Q8—1222no—
602Q8—1222no—
603Q8—2441no—
604Q8—2441no—
605D4—22no—
606D4—22no—
607D4—48no—
608D4—41no—
609D4—41no—
610D4—1222no—
611D4—1222no—
612D4—2448no—
613D4—2448no—
614D4—2448no—
615D4—2441no—
616D4—2441no—
617C2×C4—22no—
618C2×C4—24no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—48no—
624C2×C4—612no—
625C2×C4—614no—
626C2×C4—612no—
627C2×C4—612no—
628C2×C4—612no—
629C2×C4—1228no—
630C2×C4—1222no—
631C2×C4—1222no—
632C2×C4—1222no—
633C2×C4—1222no—
634C2×C4—1224no—
635C2×C4—2444no—
636C2×C4—2441no—
637C2×C4—2441no—
638C2×C4—2442no—
639C2×C4—2442no—
640C2×C4—2448no—
641C2×C4C22⋊C4×D6112yes—
642C2×C4#1451973714yes—
643C2×C4C4×C22⋊D6114yes—
644C2×C4#1451974912yes—
645C2×C4#1451975012yes—
646C2×C4#1451976314yes—
647C2×C4#1451977314yes—
648C2×C4C22×C4×D6111yescharacteristic
649C2×C4C2×D4×D6111yescharacteristic
650C2×C4C2×D4×D6111yescharacteristic
651C2×C4C22×C22⋊D6111yescharacteristic
652C2×C4C2×#1175822411yescharacteristic
653C2×C4C2×#1175822211yescharacteristic
654C8—42no—
655C8—2442no—
656C22—24no—
657C22—24no—
658C22—22no—
659C22—22no—
660C22—42no—
661C22—42no—
662C22—42no—
663C22—82no—
664C22—82no—
665C22—4882no—
666C22—4882no—
667C22—2442no—
668C22—2442no—
669C22—2442no—
670C22#1683966214yes—
671C22C2×C22⋊C4×D6111yescharacteristic
672C22C2×#1451974911yescharacteristic
673C22C2×#1451975011yescharacteristic
674C22#135×D6112yes—
675C22#1684055912yes—
676C4—42no—
677C4—44no—
678C4—614no—
679C4—1222no—
680C4—1222no—
681C4—1222no—
682C4—2442no—
683C4—2444no—
684C4#1683966214yes—
685C4C2×#1451973711yescharacteristic
686C4C2×C4×C22⋊D6111yescharacteristic
687C4C2×#1451976311yescharacteristic
688C4C2×#1451977311yescharacteristic
689C2—24no—
690C2—82no—
691C2—4882no—
692C2C2×#1683966211yescharacteristic
693C2#416×D6111yescharacteristic
694C2#2145433411yescharacteristic
695C2#2145527012yes—
696C2#2145527212yes—
subgroupG/H
centerC22×C4D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC2×C61×#416C2
socleC23×C61C22×D4
prime radicalC2×#16840308C2
cosocleC22×C4×C61C24
hypercenterC2×#416D61
hypocenterC61#21257
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22×C4 ◁ C23×C4 ◁ C2×#416

Complements

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

NH ≅ G/Ntype
2. C22×#14519706C2semidirect
3. C2×#16839759C2semidirect
4. C2×#16839758C2semidirect
5. C2×#16839756C2semidirect
6. #117×Dic61C2semidirect
7. #21446750C2semidirect
8. #21446748C2semidirect
9. #21446744C2semidirect
10. #21446740C2semidirect
11. C2×C61×#416C2semidirect
15. #16840308C4semidirect
81. #14519736D4semidirect
125. #117×C61D4semidirect
340. Dic61#416semidirect
346. C2×#416D61semidirect
398. C61#21257semidirect
447. #117C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168403082
C22×#145197062
C2×#168397592
C2×#168397582
C2×#168397562
#117×Dic612
#214467502
#214467482
#214467442
#214467402
C2×C61×#4162
#2125761