Finite Group Databasebeta

← #29707616

Subgroups — #29707616

Order 15616.

1450 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459148C211yesmaximal, characteristic
2#21454409C211yesmaximal, characteristic, complemented
3C22×#14519703C211yesmaximal, characteristic, complemented
4#21454339C211yesmaximal, characteristic, complemented
5#21448310C211yesmaximal, characteristic, complemented
6#21447037C214yesmaximal, complemented
7#21447030C214yesmaximal, complemented
8C2×#16839523C211yesmaximal, characteristic, complemented
9C2×C61×#413C211yesmaximal, characteristic, complemented
10C22×#11758222C2211yescharacteristic
11#16840340—22no—
12#16840336C2212yes—
13#16840333—21no—
14#16840329C2212yes—
15#16840326—21no—
16#16840321—22no—
17#16840308—22no—
18#16840273—22no—
19#16840269—22no—
20C2×#14519736C2211yescharacteristic
21C2×#14519736C2211yescharacteristic
22C22×C4×Dic61C2211yescharacteristic
23#16839804—22no—
24#16839804C2214yescomplemented
25#16839800—22no—
26C2×#14519703—24no—
27C2×#14519703C2214yes—
28#16839772—21no—
29#16839768—21no—
30#16839764C2214yes—
31#16839759—22no—
32#16839669C2212yescomplemented
33#16839665C2212yescomplemented
34#16839642—22no—
35C4×C22⋊Dic61—22no—
36#16839523C2214yescomplemented
37C22×#41×C61C2211yescharacteristic
38C22×M16×C61C2211yescharacteristic
39C22×M16×C61C2211yescharacteristic
40C61×#413—24no—
41C2×#101×C61C2212yescomplemented
42C2×#100×C61C2212yescomplemented
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—
54C23×Dic61—21no—
55C22×C4×D61—21no—
56#14519775—22no—
57#14519773—22no—
58#14519766—22no—
59#14519764—22no—
60#14519760—24no—
61#14519758—24no—
62#14519756—24no—
63#14519753—24no—
64#14519751—24no—
65#14519749—28no—
66#14519748—24no—
67#14519746—24no—
68#14519745—24no—
69C4×C22⋊D61—28no—
70#14519736—21no—
71#14519736—21no—
72#14519736—21no—
73#14519736—21no—
74#14519736—21no—
75#14519736—21no—
76#14519736D412yes—
77#14519736C2312yes—
78C2×C4⋊Dic61D412yes—
79C2×C4⋊Dic61C2312yes—
80C2×C22⋊Dic61—21no—
81C2×C22⋊Dic61—21no—
82C2×C22⋊Dic61—24no—
83C2×C4×Dic61—21no—
84C2×C4×Dic61—22no—
85C2×C4×Dic61—22no—
86C2×C4×Dic61—21no—
87C2×C4×Dic61—21no—
88C2×C4×Dic61C2312yes—
89#14519711—24no—
90#14519709—24no—
91#14519703—28no—
92#14519702—24no—
93#14519699—24no—
94#14519684—24no—
95C2×#41×C61—21no—
96C2×#41×C61—21no—
97C2×#41×C61—21no—
98C2×#41×C61—22no—
99C2×#41×C61—21no—
100C2×#41×C61—22no—
101C2×#41×C61—22no—
102C2×#41×C61—22no—
103C22×Q8×C61C2311yescharacteristic
104C22×D4×C61D412yes—
105C22×D4×C61C2311yescharacteristic
106C23×C4×C61D412yes—
107C23×C4×C61C2311yescharacteristic
108C2×M16×C61—21no—
109C2×M16×C61—22no—
110C2×M16×C61—21no—
111C2×M16×C61—22no—
112C2×M16×C61—22no—
113C2×M16×C61D414yescomplemented
114C2×M16×C61C2312yes—
115C2×M16×C61C2312yes—
116C22×C8×C61—21no—
117C22×C8×C61—21no—
118#101×C61—24no—
119#100×C61—24no—
120C24×C61C2×D412yes—
121C24×C61C2×D411yescharacteristic
122C23×D61—21no—
123#41×C61—42no—
124#41×C61—42no—
125#41×C61—42no—
126#41×C61—42no—
127#41×C61—42no—
128#41×C61—42no—
129#41×C61—42no—
130#41×C61—42no—
131C2×Q8×C61—21no—
132C2×Q8×C61—21no—
133C2×Q8×C61—21no—
134C2×Q8×C61—21no—
135C2×Q8×C61—21no—
136C2×Q8×C61—21no—
137C2×D4×C61—22no—
138C2×D4×C61—22no—
139C2×D4×C61—24no—
140C2×D4×C61—24no—
141C2×D4×C61—21no—
142C2×D4×C61—21no—
143C2×D4×C61—21no—
144C2×D4×C61—21no—
145C2×D4×C61—21no—
146C2×D4×C61—21no—
147C22×C4×C61—22no—
148C22×C4×C61—22no—
149C22×C4×C61—24no—
150C22×C4×C61—24no—
151C22×C4×C61C2×D412yes—
152C22×C4×C61C2×D411yescharacteristic
153C22×C4×C61C2×D411yescharacteristic
154C22×C4×C61C2×D411yescharacteristic
155C22×C4×C61C2×D411yescharacteristic
156C22×C4×C61C2×D411yescharacteristic
157C22×C4×C61C2×D412yes—
158C22×C4×C61#4112yes—
159C22×C4×C61#4112yes—
160C22×C4×C61#4112yes—
161C22×C4×C61C2411yescharacteristic
162C2×C22⋊D61—22no—
163C2×C22⋊D61—22no—
164C2×C22⋊D61—28no—
165C2×C22⋊D61—42no—
166C2×C22⋊D61—42no—
167C22×Dic61—21no—
168C22×Dic61—22no—
169C22×Dic61—22no—
170C22×Dic61—22no—
171C22×Dic61—44no—
172C22×Dic61—42no—
173#11758222—24no—
174#11758222—24no—
175#11758222—22no—
176#11758222—22no—
177#11758222—22no—
178#11758222—22no—
179#11758222—42no—
180#11758222—42no—
181#11758222—42no—
182#11758222—42no—
183C2×D244—22no—
184C2×D244—22no—
185C2×D244—21no—
186C2×D244C2×D412yes—
187C2×C4×D61—22no—
188C2×C4×D61—22no—
189C2×C4×D61—42no—
190C2×C4×D61—42no—
191C2×Dic122—22no—
192C2×Dic122—22no—
193C2×Dic122—21no—
194C2×Dic122C2×D412yes—
195M16×C61—24no—
196M16×C61—22no—
197M16×C61—22no—
198M16×C61—22no—
199M16×C61—42no—
200M16×C61C2×D414yes—
201C2×C8×C61—24no—
202C2×C8×C61—22no—
203C2×C8×C61—22no—
204C2×C8×C61—42no—
205C22⋊Dic61—28no—
206C22⋊Dic61—22no—
207C22⋊Dic61—22no—
208C22⋊Dic61—48no—
209C22⋊Dic61—41no—
210C22⋊Dic61—41no—
211#11758198—28no—
212C4⋊Dic61—22no—
213C4⋊Dic61—22no—
214C4⋊Dic61—22no—
215C4⋊Dic61C2×D414yes—
216#11758196—28no—
217C4×Dic61—22no—
218C4×Dic61—22no—
219C4×Dic61—22no—
220C4×Dic61—22no—
221C4×Dic61—41no—
222C4×Dic61—41no—
223C4×Dic61—42no—
224C23×C61—22no—
225C23×C61—22no—
226C23×C61—22no—
227C23×C61—42no—
228C23×C61—44no—
229C23×C61#12011yescharacteristic
230C23×C61#12011yescharacteristic
231C23×C61#12112yes—
232C23×C61#12312yes—
233C23×C61C22×D411yescharacteristic
234C22×D61—22no—
235C22×D61—42no—
236C22×D61—82no—
237Q8×C61—41no—
238Q8×C61—41no—
239Q8×C61—41no—
240Q8×C61—41no—
241D4×C61—44no—
242D4×C61—44no—
243D4×C61—41no—
244D4×C61—41no—
245D4×C61—41no—
246D4×C61—41no—
247C2×C4×C61—22no—
248C2×C4×C61—22no—
249C2×C4×C61—22no—
250C2×C4×C61—44no—
251C2×C4×C61—44no—
252C2×C4×C61#12011yescharacteristic
253C2×C4×C61#12011yescharacteristic
254C2×C4×C61#12112yes—
255C2×C4×C61#12112yes—
256C2×C4×C61#12112yes—
257C2×C4×C61#12112yes—
258C2×C4×C61#12112yes—
259C2×C4×C61#12312yes—
260C2×C4×C61#12312yes—
261C2×C4×C61#12312yes—
262C2×C4×C61#12412yes—
263C2×C4×C61#12412yes—
264C2×C4×C61C22×D411yescharacteristic
265C2×C4×C61C22×D411yescharacteristic
266C2×C4×C61C22×D411yescharacteristic
267C2×C4×C61C2×#4111yescharacteristic
268C2×C4×C61C2×#4111yescharacteristic
269C2×C4×C61C2×#4111yescharacteristic
270C22⋊D61—42no—
271C22⋊D61—42no—
272C22⋊D61—44no—
273C22⋊D61—88no—
274C2×Dic61—24no—
275C2×Dic61—24no—
276C2×Dic61—22no—
277C2×Dic61—42no—
278C2×Dic61—48no—
279C2×Dic61—42no—
280C2×Dic61—42no—
281C2×Dic61—44no—
282C2×Dic61—42no—
283C2×Dic61—82no—
284D244—22no—
285D244—22no—
286D244—42no—
287C4×D61—42no—
288C4×D61—42no—
289C4×D61—44no—
290Dic122—22no—
291Dic122—22no—
292Dic122—42no—
293C8×C61—24no—
294C8×C61—42no—
295#23492—611nomaximal, Sylow
296C22×C61—22no—
297C22×C61—22no—
298C22×C61—22no—
299C22×C61—42no—
300C22×C61—42no—
301C22×C61—44no—
302C22×C61—42no—
303C22×C61—44no—
304C22×C61—82no—
305C22×C61#39412yes—
306C22×C61#39412yes—
307C22×C61C2×#12011yescharacteristic
308C22×C61C2×#12111yescharacteristic
309C22×C61C2×#12311yescharacteristic
310C2×D61—42no—
311C2×D61—44no—
312C2×D61—84no—
313C4×C61—42no—
314C4×C61#39412yes—
315C4×C61#39412yes—
316C4×C61C2×#12111yescharacteristic
317C4×C61C2×#12111yescharacteristic
318C4×C61C2×#12311yescharacteristic
319C4×C61C2×#12411yescharacteristic
320Dic61—44no—
321Dic61—42no—
322Dic61—44no—
323Dic61—82no—
324#3449—611no—
325#2911—611no—
326C22×#104—611no—
327#2896—611no—
328C2×#413D6111yescharacteristic, complemented
329#2297—611no—
330#2038—614no—
331#2037—614no—
332C2×#339—611no—
333C2×C61—21no—
334C2×C61—22no—
335C2×C61—21no—
336C2×C61—22no—
337C2×C61—82no—
338C2×C61C2×#39411yescharacteristic
339C2×C61#301812yes—
340D61—82no—
341C22×#41—611no—
342C22×#41C2×D6111yescharacteristic
343C22×M16C2×D6111yescharacteristic
344C22×M16C2×D6111yescharacteristic
345#537—1222no—
346#536—612no—
347#536—1221no—
348#535—612no—
349#535—1221no—
350#534—1222no—
351#532—1222no—
352#525—1222no—
353#525—1222no—
354C2×#117—611no—
355C2×#117—611no—
356C22×C42—611no—
357#428—614no—
358#428—1222no—
359#427—1222no—
360C2×#104—614no—
361C2×#104—1224no—
362#419—1221no—
363#418—1221no—
364#417—614no—
365#416—1222no—
366#413—24no—
367C2×#101C2×D6112yescomplemented
368C2×#100C2×D6112yescomplemented
369#391—612no—
370#390—612no—
371#386—1222no—
372C4×C22⋊C4—1222no—
373#339—614no—
374C61#2349211yescharacteristic, complemented, Sylow
375C2×#41—22no—
376C2×#41—21no—
377C2×#41—21no—
378C2×#41—21no—
379C2×#41—21no—
380C2×#41—22no—
381C2×#41—22no—
382C2×#41—22no—
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—611no—
392C22×Q8C22×D6111yescharacteristic
393C22×D4—611no—
394C22×D4—1221no—
395C22×D4D24412yes—
396C22×D4C22×D6111yescharacteristic
397C23×C4—1221no—
398C23×C4—1221no—
399C23×C4D24412yes—
400C23×C4C22×D6111yescharacteristic
401C2×M16—21no—
402C2×M16—22no—
403C2×M16—21no—
404C2×M16—22no—
405C2×M16—22no—
406C2×M16C22⋊D6114yescomplemented
407C2×M16C22×D6112yes—
408C2×M16C22×D6112yes—
409C22×C8—21no—
410C22×C8—21no—
411C4⋊Q8—1222no—
412#127—1222no—
413#124—1222no—
414#124—1222no—
415#123—1224no—
416#123—1224no—
417C22⋊Q8—1224no—
418C22⋊Q8—1224no—
419#121—1224no—
420#121—1224no—
421#121—1228no—
422#120—1224no—
423#120—1224no—
424C4×D4—1228no—
425#117—612no—
426#117—612no—
427#117—1221no—
428#117—1221no—
429#117—1221no—
430#117—1221no—
431#117—1221no—
432#117—1221no—
433C2×C4⋊C4—612no—
434C2×C4⋊C4—612no—
435C2×C22⋊C4—1221no—
436C2×C22⋊C4—1221no—
437C2×C22⋊C4—1224no—
438C2×C42—612no—
439C2×C42—1221no—
440C2×C42—1222no—
441C2×C42—1222no—
442C2×C42—1221no—
443C2×C42—1221no—
444#107—1224no—
445#106—1224no—
446#104—1228no—
447#103—1224no—
448#102—1224no—
449#101—24no—
450#100—24no—
451#95—1224no—
452C24—1221no—
453C24C2×D24412yes—
454C24C2×D24411yescharacteristic
455#41—42no—
456#41—42no—
457#41—42no—
458#41—42no—
459#41—42no—
460#41—42no—
461#41—42no—
462#41—42no—
463#41—1222no—
464#41—1222no—
465#41—1222no—
466#41—1222no—
467#41—1224no—
468#41—1224no—
469#41—2442no—
470#41—2442no—
471#41—2442no—
472#41—2442no—
473C2×Q8—21no—
474C2×Q8—21no—
475C2×Q8—21no—
476C2×Q8—21no—
477C2×Q8—21no—
478C2×Q8—21no—
479C2×Q8—612no—
480C2×Q8—1221no—
481C2×Q8—1222no—
482C2×Q8—1222no—
483C2×D4—22no—
484C2×D4—22no—
485C2×D4—24no—
486C2×D4—24no—
487C2×D4—21no—
488C2×D4—21no—
489C2×D4—21no—
490C2×D4—21no—
491C2×D4—21no—
492C2×D4—21no—
493C2×D4—612no—
494C2×D4—1228no—
495C2×D4—1222no—
496C2×D4—1222no—
497C2×D4—1221no—
498C2×D4—1222no—
499C2×D4—1222no—
500C2×D4—2442no—
501C2×D4—2442no—
502C22×C4—22no—
503C22×C4—22no—
504C22×C4—24no—
505C22×C4—24no—
506C22×C4—1222no—
507C22×C4—1222no—
508C22×C4—1221no—
509C22×C4—1222no—
510C22×C4—1222no—
511C22×C4—1222no—
512C22×C4—2444no—
513C22×C4—2442no—
514C22×C4—2442no—
515C22×C4—2442no—
516C22×C4C2×D24412yes—
517C22×C4C2×D24411yescharacteristic
518C22×C4#1175822212yes—
519C22×C4D4×D6111yescharacteristic
520C22×C4D4×D6111yescharacteristic
521C22×C4D4×D6111yescharacteristic
522C22×C4D4×D6111yescharacteristic
523C22×C4#1175822412yes—
524C22×C4#1175822412yes—
525C22×C4C2×C22⋊D6112yes—
526C22×C4C23×D6111yescharacteristic
527M16—22no—
528M16—24no—
529M16—22no—
530M16—22no—
531M16—42no—
532M16C2×C22⋊D6114yes—
533C2×C8—22no—
534C2×C8—24no—
535C2×C8—22no—
536C2×C8—42no—
537C4⋊C4—614no—
538C4⋊C4—1228no—
539C4⋊C4—1222no—
540C4⋊C4—1222no—
541C4⋊C4—1222no—
542C22⋊C4—1228no—
543C22⋊C4—1228no—
544C22⋊C4—1222no—
545C22⋊C4—1222no—
546C22⋊C4—2448no—
547C22⋊C4—2441no—
548C22⋊C4—2441no—
549C42—1222no—
550C42—1222no—
551C42—1222no—
552C42—1222no—
553C42—2441no—
554C42—2441no—
555C42—2442no—
556C23—22no—
557C23—22no—
558C23—22no—
559C23—42no—
560C23—44no—
561C23—1222no—
562C23—4882no—
563C23—2442no—
564C23#1451974711yescharacteristic
565C23#1451974711yescharacteristic
566C23#1451975112yes—
567C23#1451976112yes—
568C23C22×D24411yescharacteristic
569Q8—41no—
570Q8—41no—
571Q8—41no—
572Q8—41no—
573Q8—1222no—
574Q8—1222no—
575Q8—2442no—
576D4—44no—
577D4—44no—
578D4—41no—
579D4—41no—
580D4—41no—
581D4—41no—
582D4—1222no—
583D4—1222no—
584D4—4888no—
585D4—2442no—
586D4—2442no—
587D4—2444no—
588D4—2442no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—1224no—
595C2×C4—1224no—
596C2×C4—1222no—
597C2×C4—4882no—
598C2×C4—2444no—
599C2×C4—2442no—
600C2×C4—2442no—
601C2×C4—2448no—
602C2×C4—2442no—
603C2×C4—2442no—
604C2×C4—2442no—
605C2×C4—2442no—
606C2×C4—2444no—
607C2×C4#1451974711yescharacteristic
608C2×C4#1451974711yescharacteristic
609C2×C4#1451974812yes—
610C2×C4#1451974812yes—
611C2×C4#1451974912yes—
612C2×C4#1451974912yes—
613C2×C4#1451975112yes—
614C2×C4#1451975912yes—
615C2×C4#1451975912yes—
616C2×C4#1451976112yes—
617C2×C4#1451976312yes—
618C2×C4#1451976312yes—
619C2×C4C2×D4×D6111yescharacteristic
620C2×C4C2×D4×D6111yescharacteristic
621C2×C4C22×C22⋊D6111yescharacteristic
622C2×C4C2×#1175822411yescharacteristic
623C2×C4C2×#1175822411yescharacteristic
624C2×C4C2×#1175822211yescharacteristic
625C8—24no—
626C8—42no—
627C22—22no—
628C22—22no—
629C22—22no—
630C22—42no—
631C22—42no—
632C22—44no—
633C22—42no—
634C22—44no—
635C22—82no—
636C22—4884no—
637C22—2442no—
638C22—2444no—
639C22#1683967812yes—
640C22#1683967812yes—
641C22C2×#1451974711yescharacteristic
642C22C2×#1451975111yescharacteristic
643C22C2×#1451976111yescharacteristic
644C4—42no—
645C4—4882no—
646C4—2444no—
647C4—2442no—
648C4—2444no—
649C4#1683967812yes—
650C4#1683967812yes—
651C4C2×#1451974811yescharacteristic
652C4C2×#1451974911yescharacteristic
653C4C2×#1451975911yescharacteristic
654C4C2×#1451976311yescharacteristic
655C2—21no—
656C2—21no—
657C2—22no—
658C2—22no—
659C2—82no—
660C2—4882no—
661C2C2×#1683967811yescharacteristic
662C2#2145517212yes—
subgroupG/H
centerC2×C4#14519747
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#413C2
socleC22×C61C2×#120
prime radical#21459148C2
hypercenterC2×#413D61
hypocenterC61#23492
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#413

Complements

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

NH ≅ G/Ntype
2. #21454409C2semidirect
3. C22×#14519703C2semidirect
4. #21454339C2semidirect
5. #21448310C2semidirect
6. #21447037C2semidirect
7. #21447030C2semidirect
8. C2×#16839523C2semidirect
9. C2×C61×#413C2semidirect
24. #16839804C22semidirect
32. #16839669C22semidirect
33. #16839665C22semidirect
36. #16839523C22semidirect
41. C2×#101×C61C22semidirect
42. C2×#100×C61C22semidirect
113. C2×M16×C61D4semidirect
328. C2×#413D61semidirect
367. C2×#101C2×D61semidirect
368. C2×#100C2×D61semidirect
374. C61#23492semidirect
406. C2×M16C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
#214591482
#214544092
C22×#145197032
#214543392
#214483102
#214470372
#214470302
C2×#168395232
C2×C61×#4132
#2349261