Finite Group Databasebeta

← #29681954

Subgroups — #29681954

Order 15616.

1686 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840562C211yesmaximal, characteristic
2C22×#14519785C211yesmaximal, characteristic, complemented
3#21454336C211yesmaximal, characteristic, complemented
4C2×#16839756C211yesmaximal, characteristic, complemented
5C2×#16839744C211yesmaximal, characteristic, complemented
6#21446335C214yesmaximal, complemented
7#21446328C214yesmaximal, complemented
8C2×#16839526C211yesmaximal, characteristic, complemented
9C61×#1742C211yesmaximal, characteristic, complemented
10C22×#11758222C2211yescharacteristic
11C22×#11758222C2211yescharacteristic
12#16840562—24no—
13#16840562C418yescomplemented
14C2×#14519810—21no—
15C2×#14519810C2212yes—
16C2×#14519805—21no—
17C2×#14519805C2212yes—
18C2×#14519799—22no—
19C2×#14519785—22no—
20C2×#14519785—24no—
21C2×#14519785—24no—
22C2×#14519785C2214yes—
23C22×#11758194C2211yescharacteristic
24C22×#11758189—21no—
25C23×#92418C2211yescharacteristic
26C22×C8×D61—21no—
27#16839770—21no—
28#16839766—21no—
29#16839761C2214yes—
30#16839756—21no—
31#16839756—21no—
32#16839756—22no—
33#16839756—22no—
34C2×#14519700C2212yescomplemented
35C2×#14519697C2212yescomplemented
36#16839744—24no—
37C2×#14519695C2212yescomplemented
38C2×#14519693C2212yescomplemented
39#16839526C2214yescomplemented
40C22×M16×C61C2211yescharacteristic
41C22×M16×C61C2211yescharacteristic
42C2×#117×C61C2211yescharacteristic
43C61×#343C2214yescomplemented
44C2×#11758222—22no—
45C2×#11758222—21no—
46C2×#11758222—22no—
47C2×#11758222—21no—
48C2×#11758222—22no—
49C2×#11758222—21no—
50C2×#11758222—21no—
51C2×#11758222—21no—
52C2×#11758222—21no—
53C2×#11758222—22no—
54C2×#11758222—22no—
55C2×#11758222C2×C414yes—
56C2×#11758222C2×C414yes—
57C2×#11758222D414yes—
58C2×#11758222D412yes—
59C2×#11758222C2312yes—
60C22×Dic122D411yescharacteristic
61C22×Dic122C2311yescharacteristic
62C22×D244D411yescharacteristic
63C22×D244C2311yescharacteristic
64C22×C22⋊D61—21no—
65C22×C22⋊D61—21no—
66C22×C4×D61—21no—
67C22×C4×D61—21no—
68#14519810—24no—
69#14519810—28no—
70#14519810C2×C418yes—
71#14519805—24no—
72#14519805—28no—
73#14519805C2×C418yes—
74#14519799—216no—
75C2×Dic244—22no—
76C2×#11758190—22no—
77C2×#11758190—22no—
78C2×D488—22no—
79#14519785—216no—
80#14519785—28no—
81#14519785—28no—
82C2×#11758194—21no—
83C2×#11758194—22no—
84C2×#11758194—22no—
85C2×#11758194C2312yes—
86C2×#11758189—28no—
87C2×#11758189—24no—
88C22×#92418—22no—
89C22×#92418—22no—
90C22×#92418—21no—
91C22×#92418D411yescharacteristic
92C22×#92418D411yescharacteristic
93C22×#92418D412yes—
94C22×#92418C2312yes—
95C2×C8×D61—28no—
96C2×C8×D61—24no—
97#14519700—24no—
98#14519700—22no—
99#14519700—22no—
100#14519697—24no—
101#14519697—22no—
102#14519697—22no—
103#14519695—24no—
104#14519693—24no—
105C23×C4×C61C2311yescharacteristic
106C2×M16×C61—22no—
107C2×M16×C61—21no—
108C2×M16×C61—21no—
109C2×M16×C61—22no—
110C2×M16×C61C2×C414yes—
111C2×M16×C61D414yes—
112C2×M16×C61C2312yes—
113C2×M16×C61C2312yes—
114C22×C8×C61—21no—
115C22×C8×C61—21no—
116#117×C61—21no—
117#117×C61—21no—
118#117×C61—21no—
119#117×C61—21no—
120C2×C4⋊C4×C61C2312yes—
121C2×C22⋊C4×C61—21no—
122C2×C42×C61—21no—
123C24×C61#4111yescharacteristic
124C23×D61—21no—
125C23×D61—21no—
126C22×C4×C61—22no—
127C22×C4×C61C22×C412yes—
128C22×C4×C61C2×D412yes—
129C22×C4×C61C2×D411yescharacteristic
130C22×C4×C61C2×D411yescharacteristic
131C22×C4×C61C2×D411yescharacteristic
132C22×C4×C61C2×D411yescharacteristic
133C22×C4×C61C2×D412yes—
134C22×C4×C61#4111yescharacteristic
135C22×C4×C61#4112yes—
136C22×C4×C61C2411yescharacteristic
137C2×C22⋊D61—24no—
138C2×C22⋊D61—22no—
139C2×C22⋊D61—24no—
140C2×C22⋊D61—22no—
141C2×C22⋊D61—24no—
142C2×C22⋊D61—41no—
143C2×C22⋊D61—41no—
144C2×C22⋊D61—42no—
145C22×Dic61—21no—
146C22×Dic61—21no—
147#11758222—24no—
148#11758222—24no—
149#11758222—24no—
150#11758222—24no—
151#11758222—24no—
152#11758222—22no—
153#11758222—22no—
154#11758222—24no—
155#11758222—24no—
156#11758222—24no—
157#11758222—24no—
158#11758222—42no—
159#11758222—42no—
160#11758222—42no—
161#11758222—42no—
162#11758222C22⋊C418yes—
163#11758222C22×C414yes—
164#11758222C2×D414yes—
165C2×D244—22no—
166C2×D244—21no—
167C2×D244—21no—
168C2×D244—21no—
169C2×D244—21no—
170C2×D244—21no—
171C2×D244—21no—
172C2×D244C22⋊C414yes—
173C2×D244C22×C412yes—
174C2×D244C2×D412yes—
175C2×C4×D61—24no—
176C2×C4×D61—24no—
177C2×C4×D61—22no—
178C2×C4×D61—22no—
179C2×C4×D61—24no—
180C2×C4×D61—41no—
181C2×C4×D61—41no—
182C2×C4×D61—42no—
183C2×Dic122—22no—
184C2×Dic122—21no—
185C2×Dic122—21no—
186C2×Dic122—21no—
187C2×Dic122—21no—
188C2×Dic122—21no—
189C2×Dic122—21no—
190C2×Dic122C22⋊C414yes—
191C2×Dic122C22×C412yes—
192C2×Dic122C2×D412yes—
193M16×C61—24no—
194M16×C61—24no—
195M16×C61—22no—
196M16×C61—22no—
197M16×C61C22×C414yes—
198M16×C61C2×D414yes—
199C2×C8×C61—24no—
200C2×C8×C61—24no—
201C2×C8×C61—22no—
202C2×C8×C61—22no—
203C4⋊C4×C61—22no—
204C4⋊C4×C61—22no—
205C22⋊C4×C61—41no—
206C22⋊C4×C61—41no—
207C42×C61—41no—
208C42×C61—41no—
209#11758194—22no—
210#11758194—22no—
211#11758194—42no—
212C2×#92418—22no—
213C2×#92418—22no—
214C2×#92418—24no—
215C2×#92418—22no—
216C2×#92418—22no—
217C2×#92418—22no—
218C2×#92418—42no—
219C2×#92418C2×D412yes—
220C2×#92418C2×D412yes—
221Dic244—28no—
222D488—28no—
223#11758190—28no—
224#11758190—28no—
225#11758189—216no—
226C8×D61—216no—
227C23×C61—42no—
228C23×C61C4×D412yes—
229C23×C61#12111yescharacteristic
230C23×C61#12111yescharacteristic
231C23×C61#12312yes—
232C23×C61C2×#4111yescharacteristic
233C22×D61—22no—
234C22×D61—24no—
235C22×D61—22no—
236C22×D61—44no—
237C22×D61—41no—
238C22×D61—41no—
239C22×D61—82no—
240C2×C4×C61—42no—
241C2×C4×C61—82no—
242C2×C4×C61C2×C22⋊C412yes—
243C2×C4×C61C2×C22⋊C412yes—
244C2×C4×C61C4×D412yes—
245C2×C4×C61C4×D412yes—
246C2×C4×C61C4×D412yes—
247C2×C4×C61#12012yes—
248C2×C4×C61#12012yes—
249C2×C4×C61#12112yes—
250C2×C4×C61#12112yes—
251C2×C4×C61#12111yescharacteristic
252C2×C4×C61#12111yescharacteristic
253C2×C4×C61#12312yes—
254C2×C4×C61C23×C411yescharacteristic
255C2×C4×C61C22×D411yescharacteristic
256C2×C4×C61C22×D411yescharacteristic
257C2×C4×C61C22×D411yescharacteristic
258C2×C4×C61C22×D411yescharacteristic
259C2×C4×C61C2×#4111yescharacteristic
260C22⋊D61—24no—
261C22⋊D61—24no—
262C22⋊D61—28no—
263C22⋊D61—44no—
264C22⋊D61—44no—
265C2×Dic61—22no—
266C2×Dic61—24no—
267C2×Dic61—22no—
268C2×Dic61—41no—
269C2×Dic61—41no—
270D244—22no—
271D244—22no—
272D244—22no—
273D244—24no—
274D244—41no—
275D244—41no—
276D244C2×C22⋊C414yes—
277C4×D61—24no—
278C4×D61—24no—
279C4×D61—28no—
280C4×D61—44no—
281C4×D61—44no—
282Dic122—22no—
283Dic122—24no—
284Dic122—22no—
285Dic122—22no—
286Dic122—41no—
287Dic122—41no—
288Dic122C2×C22⋊C414yes—
289C8×C61—24no—
290C8×C61—24no—
291#92418—22no—
292#92418—22no—
293#92418—42no—
294#21003—611nomaximal, Sylow
295C22×C61—22no—
296C22×C61—22no—
297C22×C61—22no—
298C22×C61—42no—
299C22×C61—42no—
300C22×C61#38612yes—
301C22×C61#38612yes—
302C22×C61C2×C4×D411yescharacteristic
303C22×C61C2×#12111yescharacteristic
304C22×C61C2×#12311yescharacteristic
305C2×D61—24no—
306C2×D61—48no—
307C2×D61—44no—
308C2×D61—42no—
309C2×D61—82no—
310C2×D61—82no—
311C2×D61—82no—
312C4×C61—82no—
313C4×C61#38612yes—
314C4×C61#38612yes—
315C4×C61C22×C22⋊C411yescharacteristic
316C4×C61C2×C4×D411yescharacteristic
317C4×C61C2×#12011yescharacteristic
318C4×C61C2×#12111yescharacteristic
319Dic61—24no—
320Dic61—42no—
321C2×#575—611no—
322C22×#131—611no—
323#2896—611no—
324C2×#416—611no—
325C2×#413—611no—
326#1908—614no—
327#1907—614no—
328#1742D6111yescharacteristic, complemented
329C2×#341—611no—
330C2×C61—22no—
331C2×C61—21no—
332C2×C61—21no—
333C2×C61—22no—
334C2×C61C2×#38611yescharacteristic
335C2×C61#306712yes—
336D61—44no—
337D61—82no—
338C22×#41—611no—
339C22×#41—611no—
340#575—618no—
341#575—1224no—
342C2×#137—612no—
343C2×#137—1221no—
344C2×AGL(1,ℤ/8)—612no—
345C2×AGL(1,ℤ/8)—1221no—
346C2×#135—1222no—
347C2×#131—614no—
348C2×#131—1222no—
349C2×#131—1224no—
350C2×#131—1224no—
351C22×M16—611no—
352C22×M16—1221no—
353C22×M16C2×D6111yescharacteristic
354C22×M16C2×D6111yescharacteristic
355C23×C8—611no—
356C23×C8—1221no—
357C2×#117C2×D6111yescharacteristic
358#419—1221no—
359#418—1221no—
360#417—614no—
361#416—1221no—
362#416—1221no—
363#416—1222no—
364#416—1222no—
365C2×#103—612no—
366C2×#102—612no—
367#413—1224no—
368C2×#101—612no—
369C2×#100—612no—
370#343C2×D6114yescomplemented
371#341—614no—
372C61#2100311yescharacteristic, complemented, Sylow
373C2×#41—614no—
374C2×#41—614no—
375C2×#41—614no—
376C2×#41—612no—
377C2×#41—612no—
378C2×#41—1222no—
379C2×#41—1222no—
380C2×#41—1221no—
381C2×#41—1221no—
382C2×#41—1222no—
383C2×#41—1221no—
384C2×#41—1221no—
385C2×#41—1222no—
386C2×#41—1221no—
387C2×#41—1221no—
388C2×#41—1222no—
389C22×Q8—611no—
390C22×Q8—611no—
391C22×D4—611no—
392C22×D4—611no—
393C22×D4—1221no—
394C22×D4—1221no—
395C23×C4—1221no—
396C23×C4—1221no—
397C23×C4C22×D6111yescharacteristic
398#137—618no—
399#137—1228no—
400#137—1224no—
401AGL(1,ℤ/8)—618no—
402AGL(1,ℤ/8)—1228no—
403AGL(1,ℤ/8)—1224no—
404#135—12216no—
405C2×Q16—1222no—
406C2×SD16—1222no—
407C2×SD16—1222no—
408C2×D8—1222no—
409#131—1228no—
410#131—12216no—
411#131—1228no—
412C2×M16—21no—
413C2×M16—22no—
414C2×M16—21no—
415C2×M16—22no—
416C2×M16—612no—
417C2×M16—1224no—
418C2×M16—1228no—
419C2×M16—1221no—
420C2×M16—1222no—
421C2×M16—1222no—
422C2×M16C4×D6114yes—
423C2×M16D24414yes—
424C2×M16C22×D6112yes—
425C2×M16C22×D6112yes—
426C22×C8—21no—
427C22×C8—21no—
428C22×C8—611no—
429C22×C8—611no—
430C22×C8—612no—
431C22×C8—612no—
432C22×C8—1224no—
433C22×C8—1228no—
434C22×C8—1222no—
435C22×C8—1222no—
436C22×C8—1221no—
437#117—21no—
438#117—21no—
439#117—21no—
440#117—21no—
441C2×C4⋊C4C22×D6112yes—
442C2×C22⋊C4—21no—
443C2×C42—21no—
444#103—1222no—
445#103—1222no—
446#103—1224no—
447#102—1222no—
448#102—1222no—
449#102—1224no—
450#101—1224no—
451#100—1224no—
452C24—1221no—
453C24—1221no—
454C24#1175822211yescharacteristic
455#41—614no—
456#41—614no—
457#41—618no—
458#41—1222no—
459#41—1222no—
460#41—1224no—
461#41—1224no—
462#41—1224no—
463#41—1224no—
464#41—1224no—
465#41—1224no—
466#41—1224no—
467#41—1224no—
468#41—1224no—
469#41—2442no—
470#41—2442no—
471#41—2442no—
472#41—2442no—
473C2×Q8—612no—
474C2×Q8—612no—
475C2×Q8—614no—
476C2×Q8—1221no—
477C2×Q8—1221no—
478C2×Q8—1222no—
479C2×Q8—1221no—
480C2×Q8—1221no—
481C2×Q8—1221no—
482C2×Q8—1221no—
483C2×D4—612no—
484C2×D4—612no—
485C2×D4—614no—
486C2×D4—1222no—
487C2×D4—1224no—
488C2×D4—1222no—
489C2×D4—1224no—
490C2×D4—1224no—
491C2×D4—1221no—
492C2×D4—1221no—
493C2×D4—1221no—
494C2×D4—1221no—
495C2×D4—1221no—
496C2×D4—1221no—
497C2×D4—1222no—
498C2×D4—2441no—
499C2×D4—2441no—
500C2×D4—2442no—
501C22×C4—22no—
502C22×C4—1221no—
503C22×C4—1222no—
504C22×C4—1224no—
505C22×C4—1224no—
506C22×C4—1222no—
507C22×C4—1221no—
508C22×C4—1224no—
509C22×C4—2441no—
510C22×C4—2441no—
511C22×C4—2442no—
512C22×C4C2×C4×D6112yes—
513C22×C4C2×D24412yes—
514C22×C4#1175822211yescharacteristic
515C22×C4D4×D6112yes—
516C22×C4D4×D6111yescharacteristic
517C22×C4D4×D6111yescharacteristic
518C22×C4D4×D6111yescharacteristic
519C22×C4D4×D6111yescharacteristic
520C22×C4#1175822612yes—
521C22×C4C23×D6111yescharacteristic
522Q16—1228no—
523SD16—1228no—
524SD16—1228no—
525D8—1228no—
526M16—22no—
527M16—22no—
528M16—24no—
529M16—24no—
530M16—12216no—
531M16—1222no—
532M16—1222no—
533M16—2442no—
534M16C2×C4×D6114yes—
535M16C2×D24414yes—
536C2×C8—22no—
537C2×C8—22no—
538C2×C8—24no—
539C2×C8—24no—
540C2×C8—612no—
541C2×C8—612no—
542C2×C8—12216no—
543C2×C8—1222no—
544C2×C8—1222no—
545C2×C8—1222no—
546C2×C8—1222no—
547C2×C8—1224no—
548C2×C8—1222no—
549C2×C8—2442no—
550C4⋊C4—22no—
551C4⋊C4—22no—
552C22⋊C4—41no—
553C22⋊C4—41no—
554C42—41no—
555C42—41no—
556C23—42no—
557C23—1222no—
558C23—1224no—
559C23—1222no—
560C23—4882no—
561C23—2444no—
562C23—2441no—
563C23—2441no—
564C23C4×D24412yes—
565C23#1451975011yescharacteristic
566C23#1451975011yescharacteristic
567C23#1451976212yes—
568C23C2×#1175822211yescharacteristic
569Q8—614no—
570Q8—1222no—
571Q8—1224no—
572Q8—1222no—
573Q8—1222no—
574Q8—2441no—
575Q8—2441no—
576D4—614no—
577D4—1224no—
578D4—1224no—
579D4—1228no—
580D4—1222no—
581D4—1222no—
582D4—1222no—
583D4—1224no—
584D4—2444no—
585D4—2444no—
586D4—2441no—
587D4—2441no—
588C2×C4—42no—
589C2×C4—82no—
590C2×C4—1222no—
591C2×C4—1224no—
592C2×C4—1224no—
593C2×C4—1224no—
594C2×C4—1228no—
595C2×C4—1222no—
596C2×C4—2441no—
597C2×C4—2441no—
598C2×C4—2444no—
599C2×C4—2444no—
600C2×C4C22⋊C4×D6112yes—
601C2×C4C22⋊C4×D6112yes—
602C2×C4#1451973912yes—
603C2×C4#1451973912yes—
604C2×C4C4×D24412yes—
605C2×C4#1451974712yes—
606C2×C4#1451974712yes—
607C2×C4#1451975011yescharacteristic
608C2×C4#1451975011yescharacteristic
609C2×C4#1451975212yes—
610C2×C4#1451975212yes—
611C2×C4#1451976212yes—
612C2×C4C22×C4×D6111yescharacteristic
613C2×C4C2×D4×D6111yescharacteristic
614C2×C4C2×D4×D6111yescharacteristic
615C2×C4C2×D4×D6111yescharacteristic
616C2×C4C22×D24411yescharacteristic
617C2×C4C2×#1175822611yescharacteristic
618C8—24no—
619C8—24no—
620C8—1222no—
621C8—1222no—
622C8—2442no—
623C22—22no—
624C22—22no—
625C22—22no—
626C22—42no—
627C22—42no—
628C22—1224no—
629C22—4882no—
630C22—4882no—
631C22—4882no—
632C22—2448no—
633C22—2442no—
634C22—2444no—
635C22#1683964012yes—
636C22#1683964012yes—
637C22C2×C4×D24411yescharacteristic
638C22C2×#1451975011yescharacteristic
639C22C2×#1451976211yescharacteristic
640C4—82no—
641C4—1224no—
642C4—2442no—
643C4#1683964012yes—
644C4#1683964012yes—
645C4C2×C22⋊C4×D6111yescharacteristic
646C4C2×#1451973911yescharacteristic
647C4C2×#1451974711yescharacteristic
648C4C2×#1451975211yescharacteristic
649C2—21no—
650C2—21no—
651C2—22no—
652C2—22no—
653C2—4882no—
654C2—2444no—
655C2C2×#1683964011yescharacteristic
656C2#2145553812yes—
subgroupG/H
centerC2×C4#14519750
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1742C2
socleC22×C61C2×#121
prime radicalC2×#16840562C2
cosocleC22×C4×C61C24
hypercenter#1742D61
hypocenterC61#21003
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #1742

Complements

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

NH ≅ G/Ntype
2. C22×#14519785C2semidirect
3. #21454336C2semidirect
4. C2×#16839756C2semidirect
5. C2×#16839744C2semidirect
6. #21446335C2semidirect
7. #21446328C2semidirect
8. C2×#16839526C2semidirect
9. C61×#1742C2semidirect
13. #16840562C4semidirect
34. C2×#14519700C22semidirect
35. C2×#14519697C22semidirect
37. C2×#14519695C22semidirect
38. C2×#14519693C22semidirect
39. #16839526C22semidirect
43. C61×#343C22semidirect
328. #1742D61semidirect
370. #343C2×D61semidirect
372. C61#21003semidirect

Maximal subgroups

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

subgroupindex
C2×#168405622
C22×#145197852
#214543362
C2×#168397562
C2×#168397442
#214463352
#214463282
C2×#168395262
C61×#17422
#2100361