Finite Group Databasebeta

← #29707138

Subgroups — #29707138

Order 15616.

1986 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
1C22×#14519797C211yesmaximal, characteristic
2C2×#16840336C211yesmaximal, characteristic, complemented
3C2×#16839764C211yesmaximal, characteristic, complemented
4C22×#14519701C211yesmaximal, characteristic, complemented
5#21448411C211yesmaximal, characteristic, complemented
6#21446917C212yesmaximal, complemented
7#21446910C212yesmaximal, complemented
8#21446900C212yesmaximal, complemented
9#21446893C212yesmaximal, complemented
10#21445517C211yesmaximal, characteristic, complemented
11C22×#137×C61C211yesmaximal, characteristic, complemented
12C22×#11758222C2211yescharacteristic
13C2×#14519797—24no—
14C2×#14519797—24no—
15C2×#14519797—24no—
16C22×#11758202C2211yescharacteristic
17C22×#11758200C2211yescharacteristic
18C22×#11758201C2211yescharacteristic
19C22×#11758199C2211yescharacteristic
20C23×#92418C2211yescharacteristic
21#16840336—24no—
22#16840336—24no—
23C2×#14519775C2212yes—
24C2×#14519766C2212yes—
25C2×#14519760—22no—
26C2×#14519753—22no—
27C2×#14519736C2211yescharacteristic
28C22×C4⋊Dic61C2211yescharacteristic
29#16839764—24no—
30C2×#14519702C2212yes—
31C2×#14519701—24no—
32C2×#14519701C2212yes—
33C2×#14519701C2212yes—
34C2×#14519699C2212yes—
35#16839669C2212yes—
36#16839665C2212yes—
37#16839642—22no—
38#16839622—22no—
39#16839516C2212yes—
40#16839516C2212yes—
41C22×#41×C61C2211yescharacteristic
42C23×Q8×C61C2211yescharacteristic
43C2×#137×C61—24no—
44C2×#137×C61—24no—
45C2×#137×C61—24no—
46C22×Q16×C61C2212yes—
47C22×SD16×C61C2212yes—
48C22×M16×C61C2211yescharacteristic
49C2×#11758222—22no—
50C2×#11758222—22no—
51C2×#11758222—22no—
52C2×#11758222—22no—
53C2×#11758222—22no—
54C2×#11758222D414yes—
55C22×Dic122C2311yescharacteristic
56C22×D244C2311yescharacteristic
57C22×C22⋊D61—21no—
58C23×Dic61—21no—
59C22×C4×D61—21no—
60#14519797—232no—
61C2×#11758202—22no—
62C2×#11758202—22no—
63C2×#11758202—22no—
64C2×#11758200—22no—
65C2×#11758200—22no—
66C2×#11758200—22no—
67C2×#11758201—22no—
68C2×#11758201—22no—
69C2×#11758201—22no—
70C2×#11758199—22no—
71C2×#11758199—22no—
72C2×#11758199—22no—
73C22×#92418—22no—
74C22×#92418—22no—
75C22×#92418D414yes—
76C22×#92418C2311yescharacteristic
77C22×#92418C2311yescharacteristic
78#14519775—24no—
79#14519775—24no—
80#14519766—24no—
81#14519766—24no—
82#14519760—216no—
83#14519753—216no—
84#14519736—22no—
85#14519736—22no—
86C2×C4⋊Dic61—22no—
87C2×C4⋊Dic61—22no—
88C2×C4⋊Dic61—21no—
89C2×C4⋊Dic61D412yes—
90C2×C4⋊Dic61C2312yes—
91C2×#11758196—24no—
92C2×C22⋊Dic61—24no—
93C2×C22⋊Dic61D412yes—
94C2×#11758198—24no—
95C2×C4×Dic61C2312yes—
96#14519702—24no—
97#14519701—24no—
98#14519701—24no—
99#14519699—24no—
100#14519684—24no—
101C2×#41×C61—22no—
102C2×#41×C61—22no—
103C2×#41×C61—22no—
104C2×#41×C61—22no—
105C2×#41×C61—22no—
106C2×#41×C61D414yes—
107C22×Q8×C61—21no—
108C22×Q8×C61—22no—
109C22×Q8×C61—22no—
110C22×Q8×C61—22no—
111C22×Q8×C61—22no—
112C22×Q8×C61—22no—
113C22×Q8×C61D414yes—
114C22×Q8×C61C2312yes—
115C22×Q8×C61C2311yescharacteristic
116C22×D4×C61—21no—
117C22×D4×C61C2311yescharacteristic
118C23×C4×C61—21no—
119C23×C4×C61—21no—
120C23×C4×C61C2311yescharacteristic
121#137×C61—232no—
122C2×Q16×C61—24no—
123C2×Q16×C61—24no—
124C2×Q16×C61—24no—
125C2×SD16×C61—24no—
126C2×SD16×C61—24no—
127C2×SD16×C61—24no—
128C2×M16×C61—22no—
129C2×M16×C61—22no—
130C2×M16×C61D414yes—
131C22×C8×C61C2312yes—
132C24×C61—21no—
133C24×C61C2×D411yescharacteristic
134C23×D61—21no—
135#41×C61—28no—
136#41×C61—28no—
137#41×C61—44no—
138#41×C61—44no—
139C2×Q8×C61—24no—
140C2×Q8×C61—22no—
141C2×Q8×C61—21no—
142C2×Q8×C61—28no—
143C2×Q8×C61—28no—
144C2×Q8×C61—22no—
145C2×Q8×C61—22no—
146C2×Q8×C61—21no—
147C2×Q8×C61—21no—
148C2×Q8×C61—22no—
149C2×Q8×C61—21no—
150C2×Q8×C61—21no—
151C2×Q8×C61—22no—
152C2×Q8×C61—44no—
153C2×Q8×C61—42no—
154C2×Q8×C61—44no—
155C2×Q8×C61—42no—
156C2×Q8×C61C2×D414yes—
157C2×Q8×C61C2×D412yes—
158C2×D4×C61—24no—
159C2×D4×C61—21no—
160C2×D4×C61—21no—
161C2×D4×C61—21no—
162C2×D4×C61—21no—
163C2×D4×C61—21no—
164C2×D4×C61—42no—
165C2×D4×C61—42no—
166C2×D4×C61C2×D412yes—
167C22×C4×C61—24no—
168C22×C4×C61—21no—
169C22×C4×C61—22no—
170C22×C4×C61—24no—
171C22×C4×C61—42no—
172C22×C4×C61—42no—
173C22×C4×C61—42no—
174C22×C4×C61—42no—
175C22×C4×C61C2×D412yes—
176C22×C4×C61C2×D412yes—
177C22×C4×C61C2×D412yes—
178C22×C4×C61C2×D411yescharacteristic
179C22×C4×C61C2×D412yes—
180C22×C4×C61C2×D412yes—
181C22×C4×C61#4112yes—
182C22×C4×C61C2411yescharacteristic
183C2×C22⋊D61—24no—
184C2×C22⋊D61—42no—
185C2×C22⋊D61—42no—
186C22×Dic61—21no—
187C22×Dic61—22no—
188C22×Dic61—42no—
189C22×Dic61—82no—
190C22×Dic61C2×D414yes—
191#11758222—28no—
192#11758222—28no—
193#11758222—44no—
194#11758222—44no—
195C2×D244—21no—
196C2×D244—21no—
197C2×D244—21no—
198C2×D244—21no—
199C2×D244—21no—
200C2×D244C2×D412yes—
201C2×C4×D61—24no—
202C2×C4×D61—42no—
203C2×C4×D61—42no—
204C2×Dic122—21no—
205C2×Dic122—21no—
206C2×Dic122—21no—
207C2×Dic122—21no—
208C2×Dic122—21no—
209C2×Dic122C2×D412yes—
210Q16×C61—216no—
211SD16×C61—216no—
212M16×C61—28no—
213C2×C8×C61—22no—
214C2×C8×C61—22no—
215C2×C8×C61C2×D414yes—
216C22⋊Dic61—24no—
217C22⋊Dic61—84no—
218#11758202—28no—
219#11758201—28no—
220#11758200—28no—
221#11758199—28no—
222#11758198—216no—
223C4⋊Dic61—24no—
224C4⋊Dic61—22no—
225C4⋊Dic61—22no—
226C4⋊Dic61—42no—
227#11758196—216no—
228C4×Dic61—22no—
229C4×Dic61—22no—
230C2×#92418—28no—
231C2×#92418—21no—
232C2×#92418—21no—
233C2×#92418—21no—
234C2×#92418—21no—
235C2×#92418C2×D412yes—
236C2×#92418C2×D412yes—
237C23×C61—22no—
238C23×C61—22no—
239C23×C61—22no—
240C23×C61—41no—
241C23×C61—41no—
242C23×C61—42no—
243C23×C61—42no—
244C23×C61#12012yes—
245C23×C61#12112yes—
246C23×C61#12712yes—
247C23×C61C22×D411yescharacteristic
248C22×D61—22no—
249C22×D61—41no—
250C22×D61—41no—
251C22×D61—42no—
252C22×D61—42no—
253Q8×C61—22no—
254Q8×C61—24no—
255Q8×C61—22no—
256Q8×C61—24no—
257Q8×C61—42no—
258Q8×C61—42no—
259Q8×C61—41no—
260Q8×C61—41no—
261Q8×C61—48no—
262D4×C61—22no—
263D4×C61—22no—
264D4×C61—48no—
265D4×C61—41no—
266D4×C61—41no—
267C2×C4×C61—24no—
268C2×C4×C61—22no—
269C2×C4×C61—41no—
270C2×C4×C61—42no—
271C2×C4×C61—41no—
272C2×C4×C61—42no—
273C2×C4×C61—48no—
274C2×C4×C61—48no—
275C2×C4×C61#12014yes—
276C2×C4×C61#12014yes—
277C2×C4×C61#12012yes—
278C2×C4×C61#12114yes—
279C2×C4×C61#12114yes—
280C2×C4×C61#12112yes—
281C2×C4×C61#12712yes—
282C2×C4×C61C22×D411yescharacteristic
283C2×C4×C61C22×D411yescharacteristic
284C2×C4×C61C22×D411yescharacteristic
285C2×C4×C61C22×D411yescharacteristic
286C2×C4×C61C22×D411yescharacteristic
287C2×C4×C61C2×#4111yescharacteristic
288C22⋊D61—48no—
289C2×Dic61—24no—
290C2×Dic61—22no—
291C2×Dic61—24no—
292C2×Dic61—24no—
293C2×Dic61—41no—
294C2×Dic61—41no—
295C2×Dic61—42no—
296C2×Dic61—44no—
297C2×Dic61—84no—
298D244—22no—
299D244—22no—
300D244—41no—
301D244—41no—
302C4×D61—48no—
303Dic122—22no—
304Dic122—22no—
305Dic122—41no—
306Dic122—41no—
307C8×C61—24no—
308#92418—22no—
309#92418—22no—
310#23427—611nomaximal, Sylow
311C22×C61—22no—
312C22×C61—22no—
313C22×C61—24no—
314C22×C61—24no—
315C22×C61—42no—
316C22×C61—44no—
317C22×C61—81no—
318C22×C61—81no—
319C22×C61—82no—
320C22×C61#39214yes—
321C22×C61C2×#12011yescharacteristic
322C22×C61C2×#12111yescharacteristic
323C22×C61C2×#12711yescharacteristic
324C22×C61#57712yes—
325C22×C61#57812yes—
326C2×D61—42no—
327C2×D61—44no—
328C2×D61—81no—
329C2×D61—81no—
330C2×D61—82no—
331C4×C61—42no—
332C4×C61—44no—
333C4×C61#39214yes—
334C4×C61C2×#12011yescharacteristic
335C4×C61C2×#12011yescharacteristic
336C4×C61C2×#12111yescharacteristic
337C4×C61C2×#12111yescharacteristic
338Dic61—42no—
339Dic61—44no—
340Dic61—82no—
341C22×#137D6111yescharacteristic, complemented
342C22×#135—611no—
343C2×#536—611no—
344C2×#417—611no—
345C22×#103—611no—
346#2311—611no—
347#2016—612no—
348#2015—612no—
349#2013—612no—
350#2012—612no—
351#1729—611no—
352C2×C61—24no—
353C2×C61—82no—
354C2×C61C2×#39211yescharacteristic
355C2×C61#301312yes—
356C2×C61#304612yes—
357C2×C61#315411yescharacteristic
358C2×C61#315611yescharacteristic
359D61—82no—
360C22×#41—611no—
361C22×#41C2×D6111yescharacteristic
362C23×Q8C2×D6111yescharacteristic
363C2×#137—24no—
364C2×#137—24no—
365C2×#137—24no—
366C2×#135—1224no—
367C2×#135—1224no—
368C2×#135—1224no—
369C22×Q16—611no—
370C22×Q16C2×D6112yes—
371C22×SD16—611no—
372C22×SD16—611no—
373C22×SD16C2×D6112yes—
374C22×D8—611no—
375C22×M16C2×D6111yescharacteristic
376C23×C8—611no—
377#536—1224no—
378#536—1224no—
379C2×C4⋊Q8—612no—
380C2×#124—612no—
381C2×#123—1222no—
382C2×C22⋊Q8—1222no—
383C2×#117—611no—
384C22×C4⋊C4—611no—
385#417—1224no—
386C2×#103—612no—
387C2×#103—612no—
388C2×#103—612no—
389C2×#103—1224no—
390C2×#102—612no—
391#391—612no—
392#390—612no—
393#386—1222no—
394#380—1222no—
395#336—612no—
396#336—612no—
397C61#2342711yescharacteristic, complemented, Sylow
398C2×#41—22no—
399C2×#41—22no—
400C2×#41—22no—
401C2×#41—22no—
402C2×#41—22no—
403C2×#41—614no—
404C2×#41—1222no—
405C2×#41—1222no—
406C2×#41—1222no—
407C2×#41—1222no—
408C2×#41—1222no—
409C2×#41C22⋊D6114yes—
410C22×Q8—21no—
411C22×Q8—22no—
412C22×Q8—22no—
413C22×Q8—22no—
414C22×Q8—22no—
415C22×Q8—22no—
416C22×Q8—611no—
417C22×Q8C22⋊D6114yes—
418C22×Q8C22×D6112yes—
419C22×Q8C22×D6111yescharacteristic
420C22×D4—21no—
421C22×D4—611no—
422C22×D4—1221no—
423C22×D4C22×D6111yescharacteristic
424C23×C4—21no—
425C23×C4—21no—
426C23×C4—1221no—
427C23×C4—1221no—
428C23×C4C22×D6111yescharacteristic
429#137—232no—
430#135—12232no—
431C2×Q16—24no—
432C2×Q16—24no—
433C2×Q16—24no—
434C2×Q16—1222no—
435C2×Q16—1222no—
436C2×Q16—1222no—
437C2×SD16—24no—
438C2×SD16—24no—
439C2×SD16—24no—
440C2×SD16—1222no—
441C2×SD16—1222no—
442C2×SD16—1222no—
443C2×SD16—1222no—
444C2×SD16—1222no—
445C2×SD16—1222no—
446C2×D8—1222no—
447C2×D8—1222no—
448C2×D8—1222no—
449C2×M16—22no—
450C2×M16—22no—
451C2×M16C22⋊D6114yes—
452C22×C8—614no—
453C22×C8—611no—
454C22×C8—611no—
455C22×C8—1222no—
456C22×C8—1222no—
457C22×C8C22×D6112yes—
458C4⋊Q8—1224no—
459C4⋊Q8—1224no—
460#124—1224no—
461#124—1224no—
462#123—12216no—
463C22⋊Q8—12216no—
464#117—1222no—
465#117—1222no—
466C2×C4⋊C4—612no—
467C2×C4⋊C4—612no—
468C2×C4⋊C4—1224no—
469C2×C4⋊C4—1222no—
470C2×C4⋊C4—1222no—
471C2×C4⋊C4—1221no—
472C2×C22⋊C4—612no—
473C2×C22⋊C4—1224no—
474C2×C22⋊C4—1224no—
475C2×C42—612no—
476#103—1224no—
477#103—1224no—
478#103—1224no—
479#102—1224no—
480#95—1224no—
481C24—21no—
482C24—1221no—
483C24D4×D6111yescharacteristic
484#41—28no—
485#41—28no—
486#41—44no—
487#41—44no—
488#41—1228no—
489#41—1228no—
490#41—2444no—
491#41—2444no—
492C2×Q8—22no—
493C2×Q8—21no—
494C2×Q8—24no—
495C2×Q8—22no—
496C2×Q8—21no—
497C2×Q8—21no—
498C2×Q8—22no—
499C2×Q8—22no—
500C2×Q8—21no—
501C2×Q8—21no—
502C2×Q8—22no—
503C2×Q8—28no—
504C2×Q8—28no—
505C2×Q8—44no—
506C2×Q8—44no—
507C2×Q8—42no—
508C2×Q8—42no—
509C2×Q8—612no—
510C2×Q8—1221no—
511C2×Q8—1221no—
512C2×Q8—1221no—
513C2×Q8—1221no—
514C2×Q8—1221no—
515C2×Q8C2×C22⋊D6114yes—
516C2×Q8C2×C22⋊D6112yes—
517C2×D4—24no—
518C2×D4—21no—
519C2×D4—21no—
520C2×D4—21no—
521C2×D4—21no—
522C2×D4—21no—
523C2×D4—42no—
524C2×D4—42no—
525C2×D4—612no—
526C2×D4—1224no—
527C2×D4—1221no—
528C2×D4—1221no—
529C2×D4—1221no—
530C2×D4—1221no—
531C2×D4—1221no—
532C2×D4—2442no—
533C2×D4—2442no—
534C2×D4C2×C22⋊D6112yes—
535C22×C4—21no—
536C22×C4—22no—
537C22×C4—24no—
538C22×C4—24no—
539C22×C4—42no—
540C22×C4—42no—
541C22×C4—42no—
542C22×C4—42no—
543C22×C4—614no—
544C22×C4—1221no—
545C22×C4—1224no—
546C22×C4—1222no—
547C22×C4—4882no—
548C22×C4—2442no—
549C22×C4—2442no—
550C22×C4—2442no—
551C22×C4D4×D6112yes—
552C22×C4D4×D6112yes—
553C22×C4D4×D6111yescharacteristic
554C22×C4#1175822412yes—
555C22×C4C2×C22⋊D6112yes—
556C22×C4C2×C22⋊D6112yes—
557C22×C4C2×C22⋊D6112yes—
558C22×C4C23×D6111yescharacteristic
559Q16—216no—
560Q16—1228no—
561SD16—216no—
562SD16—1228no—
563SD16—1228no—
564D8—1228no—
565M16—28no—
566C2×C8—22no—
567C2×C8—22no—
568C2×C8—612no—
569C2×C8—612no—
570C2×C8—1221no—
571C2×C8—1221no—
572C2×C8—1221no—
573C2×C8—1221no—
574C2×C8—1228no—
575C2×C8C2×C22⋊D6114yes—
576C4⋊C4—12216no—
577C4⋊C4—1224no—
578C4⋊C4—1222no—
579C4⋊C4—1222no—
580C4⋊C4—2442no—
581C22⋊C4—12216no—
582C22⋊C4—1224no—
583C22⋊C4—4884no—
584C42—1222no—
585C42—1222no—
586C23—22no—
587C23—22no—
588C23—22no—
589C23—41no—
590C23—41no—
591C23—42no—
592C23—42no—
593C23—1222no—
594C23—2441no—
595C23—2441no—
596C23—2442no—
597C23—2442no—
598C23#1451974512yes—
599C23#1451974912yes—
600C23#1451977312yes—
601C23C2×D4×D6111yescharacteristic
602Q8—22no—
603Q8—24no—
604Q8—22no—
605Q8—24no—
606Q8—42no—
607Q8—42no—
608Q8—41no—
609Q8—41no—
610Q8—48no—
611Q8—1222no—
612Q8—1222no—
613Q8—2441no—
614Q8—2441no—
615D4—22no—
616D4—22no—
617D4—48no—
618D4—41no—
619D4—41no—
620D4—1222no—
621D4—1222no—
622D4—2448no—
623D4—2441no—
624D4—2441no—
625C2×C4—24no—
626C2×C4—22no—
627C2×C4—41no—
628C2×C4—42no—
629C2×C4—41no—
630C2×C4—42no—
631C2×C4—48no—
632C2×C4—48no—
633C2×C4—1224no—
634C2×C4—1222no—
635C2×C4—1224no—
636C2×C4—1224no—
637C2×C4—4884no—
638C2×C4—2442no—
639C2×C4—2444no—
640C2×C4—2441no—
641C2×C4—2441no—
642C2×C4—2448no—
643C2×C4#1451974514yes—
644C2×C4#1451974512yes—
645C2×C4#1451974614yes—
646C2×C4#1451974814yes—
647C2×C4#1451974914yes—
648C2×C4#1451974912yes—
649C2×C4#1451977312yes—
650C2×C4C2×D4×D6111yescharacteristic
651C2×C4C2×D4×D6111yescharacteristic
652C2×C4C22×C22⋊D6111yescharacteristic
653C2×C4C22×C22⋊D6111yescharacteristic
654C2×C4C22×C22⋊D6111yescharacteristic
655C2×C4C2×#1175822411yescharacteristic
656C8—24no—
657C8—1222no—
658C8—1222no—
659C22—24no—
660C22—24no—
661C22—22no—
662C22—22no—
663C22—44no—
664C22—42no—
665C22—82no—
666C22—81no—
667C22—81no—
668C22—4882no—
669C22—4881no—
670C22—4881no—
671C22—2444no—
672C22—2442no—
673C22#1683967014yes—
674C22C2×#1451974511yescharacteristic
675C22C2×#1451974911yescharacteristic
676C22C2×#1451977311yescharacteristic
677C22#1684057112yes—
678C22#1684057812yes—
679C4—42no—
680C4—44no—
681C4—4882no—
682C4—2444no—
683C4—2442no—
684C4#1683967014yes—
685C4C2×#1451974511yescharacteristic
686C4C2×#1451974611yescharacteristic
687C4C2×#1451974811yescharacteristic
688C4C2×#1451974911yescharacteristic
689C2—24no—
690C2—82no—
691C2—4882no—
692C2C2×#1683967011yescharacteristic
693C2#2145513312yes—
694C2#2145537312yes—
695C2#2145629811yescharacteristic
696C2#2145631211yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC22×#137×C61C2
socleC23×C61C22×D4
prime radicalC22×#14519797C2
hypercenterC22×#137D61
hypocenterC61#23427
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#137

Complements

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

NH ≅ G/Ntype
2. C2×#16840336C2semidirect
3. C2×#16839764C2semidirect
4. C22×#14519701C2semidirect
5. #21448411C2semidirect
6. #21446917C2semidirect
7. #21446910C2semidirect
8. #21446900C2semidirect
9. #21446893C2semidirect
10. #21445517C2semidirect
11. C22×#137×C61C2semidirect
341. C22×#137D61semidirect
397. C61#23427semidirect

Maximal subgroups

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

subgroupindex
C22×#145197972
C2×#168403362
C2×#168397642
C22×#145197012
#214484112
#214469172
#214469102
#214469002
#214468932
#214455172
C22×#137×C612
#2342761