Finite Group Databasebeta

← #29706883

Subgroups — #29706883

Order 15616.

2014 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519810C211yesmaximal, characteristic, complemented
2C2×#16840340C211yesmaximal, characteristic
3C2×#16840333C211yesmaximal, characteristic, complemented
4C2×#16839759C211yesmaximal, characteristic, complemented
5C22×#14519702C211yesmaximal, characteristic, complemented
6#21446829C214yesmaximal, complemented
7#21446824C214yesmaximal, complemented
8#21445539C211yesmaximal, characteristic, complemented
9C2×C61×#408C211yesmaximal, characteristic, complemented
10C22×#11758222C2211yescharacteristic
11C23×Dic122C2211yescharacteristic
12C2×#14519810—24no—
13C2×#14519810—24no—
14C2×#14519810—24no—
15C22×Dic244C2212yes—
16C22×#11758190C2212yes—
17#16840340—24no—
18#16840340—24no—
19#16840333—24no—
20#16840333—24no—
21C2×#14519775—21no—
22C2×#14519764—21no—
23C2×#14519758—22no—
24C2×#14519756C2214yes—
25C2×#14519756C2212yes—
26C2×#14519753—21no—
27C2×#14519751C2212yes—
28C2×#14519749—22no—
29C2×#14519748—21no—
30C2×#14519736C2211yescharacteristic
31C22×C4⋊Dic61C2211yescharacteristic
32#16839759—24no—
33C2×#14519702—24no—
34C2×#14519702C2214yes—
35C2×#14519702C2212yes—
36C2×#14519699C2212yes—
37#16839525C2214yes—
38C22×#41×C61C2211yescharacteristic
39C22×M16×C61C2211yescharacteristic
40C23×C8×C61C2211yescharacteristic
41C61×#408—24no—
42C2×C22⋊C8×C61C2214yes—
43C2×#11758222—22no—
44C2×#11758222—22no—
45C2×#11758222—22no—
46C2×#11758222—22no—
47C2×#11758222—22no—
48C2×#11758222D414yes—
49C22×Dic122—22no—
50C22×Dic122—22no—
51C22×Dic122—22no—
52C22×Dic122—22no—
53C22×Dic122—21no—
54C22×Dic122—22no—
55C22×Dic122D414yescomplemented
56C22×Dic122C2312yes—
57C22×Dic122C2311yescharacteristic
58C22×D244C2311yescharacteristic
59C22×C22⋊D61—21no—
60C22×C22⋊D61—22no—
61C23×Dic61—21no—
62C23×Dic61—21no—
63C22×C4×D61—21no—
64#14519810—232no—
65C2×Dic244—24no—
66C2×Dic244—24no—
67C2×Dic244—24no—
68C2×#11758190—24no—
69C2×#11758190—24no—
70C2×#11758190—24no—
71#14519775—28no—
72#14519764—28no—
73#14519758—216no—
74#14519756—28no—
75#14519756—24no—
76#14519756—28no—
77#14519756—24no—
78#14519753—28no—
79#14519751—24no—
80#14519751—24no—
81#14519749—216no—
82#14519748—28no—
83#14519736—22no—
84#14519736D414yescomplemented
85C2×C4⋊Dic61—21no—
86C2×C4⋊Dic61—22no—
87C2×C4⋊Dic61D414yes—
88C2×C4⋊Dic61C2312yes—
89C2×C4⋊Dic61C2312yes—
90C2×#11758196—22no—
91C2×#11758196—24no—
92C2×C22⋊Dic61—21no—
93C2×C22⋊Dic61—22no—
94C2×C22⋊Dic61—24no—
95C2×#11758198—22no—
96C2×C4×Dic61—21no—
97#14519702—28no—
98#14519702—24no—
99#14519699—24no—
100C2×#41×C61—22no—
101C2×#41×C61—22no—
102C2×#41×C61—22no—
103C2×#41×C61—22no—
104C2×#41×C61—22no—
105C2×#41×C61—22no—
106C22×Q8×C61D411yescharacteristic
107C22×D4×C61D411yescharacteristic
108C22×D4×C61D412yes—
109C23×C4×C61C2312yes—
110C23×C4×C61C2311yescharacteristic
111C2×M16×C61—22no—
112C2×M16×C61—22no—
113C2×M16×C61D414yescomplemented
114C22×C8×C61—22no—
115C22×C8×C61—22no—
116C22×C8×C61—22no—
117C22×C8×C61C2312yes—
118C22×C8×C61C2312yes—
119C22⋊C8×C61—28no—
120C24×C61C2×D412yes—
121C24×C61C2×D411yescharacteristic
122C23×D61—21no—
123#41×C61—44no—
124#41×C61—44no—
125#41×C61—44no—
126#41×C61—44no—
127C2×Q8×C61—22no—
128C2×Q8×C61—22no—
129C2×Q8×C61—22no—
130C2×D4×C61—24no—
131C2×D4×C61—24no—
132C2×D4×C61—24no—
133C2×D4×C61—22no—
134C2×D4×C61—22no—
135C2×D4×C61—22no—
136C22×C4×C61—22no—
137C22×C4×C61—22no—
138C22×C4×C61—22no—
139C22×C4×C61—22no—
140C22×C4×C61—22no—
141C22×C4×C61—22no—
142C22×C4×C61C2×D412yes—
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2×D412yes—
145C22×C4×C61C2×D412yes—
146C22×C4×C61C2×D412yes—
147C22×C4×C61C2×D412yes—
148C22×C4×C61C2×D412yes—
149C22×C4×C61#4112yes—
150C22×C4×C61C2411yescharacteristic
151C2×C22⋊D61—24no—
152C2×C22⋊D61—28no—
153C2×C22⋊D61—42no—
154C2×C22⋊D61—42no—
155C2×C22⋊D61—84no—
156C22×Dic61—24no—
157C22×Dic61—21no—
158C22×Dic61—22no—
159C22×Dic61—22no—
160C22×Dic61—22no—
161C22×Dic61—24no—
162C22×Dic61—42no—
163C22×Dic61—42no—
164C22×Dic61—82no—
165#11758222—28no—
166#11758222—28no—
167#11758222—44no—
168#11758222—44no—
169C2×D244—21no—
170C2×D244—21no—
171C2×D244—21no—
172C2×D244—21no—
173C2×D244—21no—
174C2×D244C2×D412yes—
175C2×C4×D61—24no—
176C2×C4×D61—42no—
177C2×C4×D61—42no—
178C2×Dic122—28no—
179C2×Dic122—28no—
180C2×Dic122—22no—
181C2×Dic122—22no—
182C2×Dic122—22no—
183C2×Dic122—21no—
184C2×Dic122—21no—
185C2×Dic122—21no—
186C2×Dic122—22no—
187C2×Dic122—21no—
188C2×Dic122—24no—
189C2×Dic122—22no—
190C2×Dic122—21no—
191C2×Dic122—42no—
192C2×Dic122—44no—
193C2×Dic122—42no—
194C2×Dic122—44no—
195C2×Dic122C2×D414yes—
196C2×Dic122C2×D412yes—
197M16×C61—28no—
198C2×C8×C61—22no—
199C2×C8×C61—22no—
200C2×C8×C61—22no—
201C2×C8×C61—22no—
202C2×C8×C61—22no—
203C2×C8×C61—44no—
204C2×C8×C61C2×D414yes—
205C22⋊Dic61—28no—
206C22⋊Dic61—216no—
207C22⋊Dic61—24no—
208#11758198—28no—
209C4⋊Dic61—24no—
210C4⋊Dic61—22no—
211C4⋊Dic61—22no—
212C4⋊Dic61C2×D414yes—
213C4⋊Dic61C2×D414yes—
214#11758196—28no—
215#11758196—216no—
216C4×Dic61—24no—
217Dic244—216no—
218#11758190—216no—
219C23×C61—22no—
220C23×C61—22no—
221C23×C61—22no—
222C23×C61—22no—
223C23×C61—22no—
224C23×C61—24no—
225C23×C61—24no—
226C23×C61#12012yes—
227C23×C61#12012yes—
228C23×C61#12112yes—
229C23×C61C22×D411yescharacteristic
230C22×D61—22no—
231C22×D61—41no—
232C22×D61—41no—
233C22×D61—42no—
234C22×D61—42no—
235Q8×C61—44no—
236D4×C61—48no—
237D4×C61—44no—
238C2×C4×C61—22no—
239C2×C4×C61—22no—
240C2×C4×C61—22no—
241C2×C4×C61—42no—
242C2×C4×C61—42no—
243C2×C4×C61—42no—
244C2×C4×C61—42no—
245C2×C4×C61#12014yes—
246C2×C4×C61#12012yes—
247C2×C4×C61#12012yes—
248C2×C4×C61#12114yes—
249C2×C4×C61#12114yes—
250C2×C4×C61#12112yes—
251C2×C4×C61#12714yes—
252C2×C4×C61C22×D411yescharacteristic
253C2×C4×C61C22×D411yescharacteristic
254C2×C4×C61C22×D411yescharacteristic
255C2×C4×C61C22×D411yescharacteristic
256C2×C4×C61C22×D411yescharacteristic
257C2×C4×C61C2×#4111yescharacteristic
258C22⋊D61—48no—
259C22⋊D61—88no—
260C2×Dic61—24no—
261C2×Dic61—24no—
262C2×Dic61—22no—
263C2×Dic61—24no—
264C2×Dic61—42no—
265C2×Dic61—41no—
266C2×Dic61—42no—
267C2×Dic61—41no—
268C2×Dic61—42no—
269C2×Dic61—44no—
270C2×Dic61—42no—
271C2×Dic61—48no—
272C2×Dic61—84no—
273D244—22no—
274D244—22no—
275D244—41no—
276D244—41no—
277C4×D61—48no—
278Dic122—24no—
279Dic122—22no—
280Dic122—24no—
281Dic122—22no—
282Dic122—42no—
283Dic122—41no—
284Dic122—42no—
285Dic122—41no—
286Dic122—48no—
287C8×C61—24no—
288C8×C61—42no—
289#23394—611nomaximal, Sylow
290C22×C61—22no—
291C22×C61—22no—
292C22×C61—24no—
293C22×C61—24no—
294C22×C61—42no—
295C22×C61—42no—
296C22×C61—42no—
297C22×C61—44no—
298C22×C61—44no—
299C22×C61#39214yes—
300C22×C61C2×#12011yescharacteristic
301C22×C61C2×#12011yescharacteristic
302C22×C61C2×#12111yescharacteristic
303C22×C61#57712yes—
304C22×C61#57812yes—
305C2×D61—42no—
306C2×D61—42no—
307C2×D61—42no—
308C2×D61—82no—
309C2×D61—82no—
310C4×C61—42no—
311C4×C61#39214yes—
312C4×C61C2×#12011yescharacteristic
313C4×C61C2×#12111yescharacteristic
314C4×C61C2×#12111yescharacteristic
315C4×C61C2×#12711yescharacteristic
316Dic61—44no—
317Dic61—42no—
318Dic61—44no—
319Dic61—44no—
320C22×#137—611no—
321C2×#537—611no—
322C2×#536—611no—
323C2×#416—611no—
324C22×#103—611no—
325C2×#408D6111yescharacteristic, complemented
326#2001—614no—
327#2000—614no—
328#1736—611no—
329C2×C61—24no—
330C2×C61—82no—
331C2×C61C2×#39211yescharacteristic
332C2×C61#300911yescharacteristic
333C2×C61#301111yescharacteristic
334C2×C61#301312yes—
335C2×C61#306312yes—
336D61—82no—
337C22×#41—611no—
338C22×#41C2×D6111yescharacteristic
339C23×Q8—611no—
340C2×#137—1224no—
341C2×#137—1224no—
342C2×#137—1224no—
343C22×Q16—612no—
344C22×SD16—612no—
345C22×M16C2×D6111yescharacteristic
346C23×C8C2×D6111yescharacteristic
347#537—1224no—
348#537—1224no—
349#536—1224no—
350#536—1224no—
351C2×C4⋊Q8—1221no—
352C2×#124—1221no—
353C2×#123—1222no—
354C2×C22⋊Q8—614no—
355C2×C22⋊Q8—612no—
356C2×C22⋊Q8—1221no—
357C2×#121—612no—
358C2×#121—1221no—
359C2×#121—1222no—
360C2×#117—611no—
361C22×C4⋊C4—611no—
362#416—1224no—
363C2×#103—614no—
364C2×#103—612no—
365C2×#103—1224no—
366C2×#102—612no—
367#408—24no—
368C2×C22⋊C8C2×D6114yes—
369#340—614no—
370C61#2339411yescharacteristic, complemented, Sylow
371C2×#41—22no—
372C2×#41—22no—
373C2×#41—22no—
374C2×#41—22no—
375C2×#41—22no—
376C2×#41—22no—
377C2×#41—614no—
378C2×#41—1222no—
379C2×#41—1222no—
380C2×#41—1222no—
381C2×#41—1222no—
382C2×#41—1222no—
383C22×Q8—614no—
384C22×Q8—612no—
385C22×Q8—611no—
386C22×Q8—1221no—
387C22×Q8—1222no—
388C22×Q8—1222no—
389C22×Q8—1222no—
390C22×Q8—1222no—
391C22×Q8—1222no—
392C22×Q8D24411yescharacteristic
393C22×D4—611no—
394C22×D4—1222no—
395C22×D4—1221no—
396C22×D4D24412yes—
397C22×D4D24411yescharacteristic
398C23×C4—1221no—
399C23×C4—1221no—
400C23×C4—1221no—
401C23×C4C22×D6112yes—
402C23×C4C22×D6111yescharacteristic
403#137—12232no—
404C2×Q16—1224no—
405C2×Q16—1224no—
406C2×Q16—1224no—
407C2×SD16—1224no—
408C2×SD16—1224no—
409C2×SD16—1224no—
410C2×M16—22no—
411C2×M16—22no—
412C2×M16C22⋊D6114yescomplemented
413C22×C8—22no—
414C22×C8—22no—
415C22×C8—22no—
416C22×C8C22×D6112yes—
417C22×C8C22×D6112yes—
418C4⋊Q8—1228no—
419#124—1228no—
420#123—12216no—
421C22⋊Q8—1228no—
422C22⋊Q8—1224no—
423C22⋊Q8—1228no—
424C22⋊Q8—1224no—
425C22⋊Q8—1228no—
426#121—1224no—
427#121—1224no—
428#121—1228no—
429#121—12216no—
430#117—614no—
431#117—1222no—
432C2×C4⋊C4—614no—
433C2×C4⋊C4—612no—
434C2×C4⋊C4—612no—
435C2×C4⋊C4—1221no—
436C2×C4⋊C4—1222no—
437C2×C4⋊C4—1224no—
438C2×C4⋊C4—1222no—
439C2×C22⋊C4—1221no—
440C2×C22⋊C4—1222no—
441C2×C22⋊C4—1224no—
442C2×C22⋊C4—1222no—
443C2×C42—1221no—
444#103—1228no—
445#103—1224no—
446#102—1224no—
447C22⋊C8—28no—
448C24—1221no—
449C24C2×D24412yes—
450C24C2×D24411yescharacteristic
451#41—44no—
452#41—44no—
453#41—44no—
454#41—44no—
455#41—1228no—
456#41—1228no—
457#41—2444no—
458#41—2444no—
459C2×Q8—22no—
460C2×Q8—22no—
461C2×Q8—22no—
462C2×Q8—614no—
463C2×Q8—612no—
464C2×Q8—1222no—
465C2×Q8—1221no—
466C2×Q8—1224no—
467C2×Q8—1222no—
468C2×Q8—1221no—
469C2×Q8—1222no—
470C2×Q8—1221no—
471C2×Q8—1222no—
472C2×Q8—1221no—
473C2×Q8—1222no—
474C2×Q8—1221no—
475C2×Q8—1228no—
476C2×Q8—1228no—
477C2×Q8—2442no—
478C2×Q8—2444no—
479C2×Q8—2442no—
480C2×Q8—2444no—
481C2×D4—24no—
482C2×D4—24no—
483C2×D4—24no—
484C2×D4—22no—
485C2×D4—22no—
486C2×D4—22no—
487C2×D4—612no—
488C2×D4—1228no—
489C2×D4—1224no—
490C2×D4—1221no—
491C2×D4—1221no—
492C2×D4—1221no—
493C2×D4—1221no—
494C2×D4—1221no—
495C2×D4—4884no—
496C2×D4—2442no—
497C2×D4—2442no—
498C22×C4—22no—
499C22×C4—22no—
500C22×C4—22no—
501C22×C4—22no—
502C22×C4—22no—
503C22×C4—22no—
504C22×C4—1224no—
505C22×C4—1222no—
506C22×C4—1222no—
507C22×C4—1221no—
508C22×C4—1222no—
509C22×C4—1224no—
510C22×C4—1224no—
511C22×C4—4882no—
512C22×C4—2442no—
513C22×C4—2442no—
514C22×C4—2442no—
515C22×C4—2442no—
516C22×C4C2×D24412yes—
517C22×C4C2×D24411yescharacteristic
518C22×C4#1175822212yes—
519C22×C4D4×D6112yes—
520C22×C4D4×D6112yes—
521C22×C4D4×D6112yes—
522C22×C4D4×D6112yes—
523C22×C4C2×C22⋊D6112yes—
524C22×C4C23×D6111yescharacteristic
525Q16—12216no—
526SD16—12216no—
527M16—28no—
528C2×C8—22no—
529C2×C8—22no—
530C2×C8—22no—
531C2×C8—22no—
532C2×C8—22no—
533C2×C8—44no—
534C2×C8C2×C22⋊D6114yes—
535C4⋊C4—614no—
536C4⋊C4—614no—
537C4⋊C4—1228no—
538C4⋊C4—12216no—
539C4⋊C4—1222no—
540C4⋊C4—1222no—
541C4⋊C4—1224no—
542C22⋊C4—1228no—
543C22⋊C4—1228no—
544C22⋊C4—12216no—
545C22⋊C4—1224no—
546C42—1224no—
547C23—22no—
548C23—22no—
549C23—22no—
550C23—22no—
551C23—22no—
552C23—24no—
553C23—24no—
554C23—1222no—
555C23—2441no—
556C23—2441no—
557C23—2442no—
558C23—2442no—
559C23#1451974712yes—
560C23#1451974712yes—
561C23#1451975112yes—
562C23C22×D24411yescharacteristic
563Q8—44no—
564Q8—1224no—
565Q8—1222no—
566Q8—1224no—
567Q8—1222no—
568Q8—2442no—
569Q8—2441no—
570Q8—2442no—
571Q8—2441no—
572Q8—2448no—
573D4—48no—
574D4—44no—
575D4—1222no—
576D4—1222no—
577D4—4888no—
578D4—2448no—
579D4—2441no—
580D4—2441no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—1224no—
589C2×C4—1224no—
590C2×C4—1222no—
591C2×C4—1224no—
592C2×C4—4884no—
593C2×C4—2441no—
594C2×C4—2442no—
595C2×C4—2441no—
596C2×C4—2442no—
597C2×C4—2442no—
598C2×C4—2444no—
599C2×C4—2442no—
600C2×C4—2448no—
601C2×C4—2448no—
602C2×C4#1451974514yes—
603C2×C4#1451974712yes—
604C2×C4#1451974712yes—
605C2×C4#1451975014yes—
606C2×C4#1451975014yes—
607C2×C4#1451975112yes—
608C2×C4#1451977314yes—
609C2×C4C2×D4×D6111yescharacteristic
610C2×C4C2×D4×D6111yescharacteristic
611C2×C4C2×D4×D6111yescharacteristic
612C2×C4C2×D4×D6111yescharacteristic
613C2×C4C22×C22⋊D6111yescharacteristic
614C2×C4C2×#1175822211yescharacteristic
615C8—24no—
616C8—42no—
617C22—22no—
618C22—24no—
619C22—24no—
620C22—22no—
621C22—42no—
622C22—44no—
623C22—42no—
624C22—44no—
625C22—42no—
626C22—4882no—
627C22—4882no—
628C22—2442no—
629C22—2442no—
630C22—2442no—
631C22#1683967114yes—
632C22C2×#1451974711yescharacteristic
633C22C2×#1451974711yescharacteristic
634C22C2×#1451975111yescharacteristic
635C22#1684057412yes—
636C22#1684057912yes—
637C4—42no—
638C4—2444no—
639C4—2442no—
640C4—2444no—
641C4—2444no—
642C4#1683967114yes—
643C4C2×#1451974511yescharacteristic
644C4C2×#1451975011yescharacteristic
645C4C2×#1451975011yescharacteristic
646C4C2×#1451977311yescharacteristic
647C2—24no—
648C2—82no—
649C2—4882no—
650C2C2×#1683967111yescharacteristic
651C2#2145509811yescharacteristic
652C2#2145511611yescharacteristic
653C2#2145513712yes—
654C2#2145551012yes—
subgroupG/H
centerC23C22×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#408C2
socleC23×C61C22×D4
prime radicalC2×#16840340C2
hypercenterC2×#408D61
hypocenterC61#23394
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×#41 ◁ C2×#408

Complements

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

NH ≅ G/Ntype
1. C22×#14519810C2semidirect
3. C2×#16840333C2semidirect
4. C2×#16839759C2semidirect
5. C22×#14519702C2semidirect
6. #21446829C2semidirect
7. #21446824C2semidirect
8. #21445539C2semidirect
9. C2×C61×#408C2semidirect
55. C22×Dic122D4semidirect
84. #14519736D4semidirect
113. C2×M16×C61D4semidirect
325. C2×#408D61semidirect
370. C61#23394semidirect
412. C2×M16C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
C22×#145198102
C2×#168403402
C2×#168403332
C2×#168397592
C22×#145197022
#214468292
#214468242
#214455392
C2×C61×#4082
#2339461