Finite Group Databasebeta

← #29681959

Subgroups — #29681959

Order 15616.

1686 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
1C2×#16840560C211yesmaximal, characteristic
2C2×#16840529C211yesmaximal, characteristic, complemented
3C2×#16839761C211yesmaximal, characteristic, complemented
4C2×#16839744C211yesmaximal, characteristic, complemented
5#21446332C214yesmaximal, complemented
6#21446325C214yesmaximal, complemented
7C2×#16839531C211yesmaximal, characteristic, complemented
8#21445545C211yesmaximal, characteristic, complemented
9C2×C61×#417C211yesmaximal, characteristic, complemented
10C22×#11758222C2211yescharacteristic
11#16840560—24no—
12#16840560C418yescomplemented
13C2×#14519808—21no—
14C2×#14519807C2212yes—
15C2×#14519803—21no—
16C2×#14519802C2212yes—
17C2×#14519797—22no—
18#16840529—24no—
19#16840529—24no—
20C2×#14519784—22no—
21C2×#14519784C2214yes—
22C22×#11758194—22no—
23C22×#11758194C2211yescharacteristic
24C22×#11758194C2211yescharacteristic
25C22×#11758194C2211yescharacteristic
26#16839761—22no—
27#16839761—22no—
28C2×#14519700C2212yescomplemented
29C2×#14519697C2212yescomplemented
30#16839744—24no—
31C2×#14519695C2212yescomplemented
32C2×#14519693C2212yescomplemented
33#16839531C2214yescomplemented
34#16839526C2214yescomplemented
35C22×#41×C61C2211yescharacteristic
36C22×M16×C61C2211yescharacteristic
37C22×C4⋊C4×C61C2211yescharacteristic
38C61×#417—22no—
39C61×#417C2214yes—
40C2×#103×C61—21no—
41C2×#102×C61—21no—
42C2×#11758222—22no—
43C2×#11758222—21no—
44C2×#11758222—21no—
45C2×#11758222—21no—
46C2×#11758222—21no—
47C2×#11758222—22no—
48C2×#11758222—22no—
49C2×#11758222C2×C414yes—
50C22×Dic122C2311yescharacteristic
51C22×D244C2311yescharacteristic
52C22×C22⋊D61—21no—
53C22×C4×D61—21no—
54#14519808—28no—
55#14519807—24no—
56#14519807C2×C418yescomplemented
57#14519803—28no—
58#14519802—24no—
59#14519802C2×C418yescomplemented
60#14519797—216no—
61C2×#11758202—22no—
62C2×#11758200—22no—
63C2×#11758201—22no—
64C2×#11758199—22no—
65#14519784—216no—
66#14519784—28no—
67#14519784—28no—
68C2×#11758194—216no—
69C2×#11758194—21no—
70C2×#11758194—22no—
71C2×#11758194—22no—
72C2×#11758194—22no—
73C2×#11758194—21no—
74C2×#11758194—22no—
75C2×#11758194—24no—
76C2×#11758194C2×C414yes—
77C2×#11758194D414yes—
78C2×#11758194D412yes—
79C2×#11758194D412yes—
80C2×#11758194C2312yes—
81C2×#11758194C2312yes—
82C22×#92418—21no—
83C22×#92418—21no—
84C22×#92418—24no—
85C22×#92418C2312yes—
86#14519700—24no—
87#14519697—24no—
88#14519695—24no—
89#14519693—24no—
90C2×#41×C61—21no—
91C2×#41×C61—22no—
92C2×#41×C61—22no—
93C2×#41×C61—21no—
94C2×#41×C61C2×C414yes—
95C2×#41×C61D414yes—
96C2×#41×C61D412yes—
97C2×#41×C61C2312yes—
98C22×Q8×C61D411yescharacteristic
99C22×D4×C61—21no—
100C22×D4×C61D411yescharacteristic
101C23×C4×C61—21no—
102C23×C4×C61—21no—
103C23×C4×C61C2311yescharacteristic
104C2×M16×C61—21no—
105C2×M16×C61—22no—
106C2×M16×C61—22no—
107C2×M16×C61C2312yes—
108C22×C8×C61—21no—
109C2×C4⋊C4×C61—21no—
110C2×C4⋊C4×C61—21no—
111C2×C4⋊C4×C61—21no—
112C2×C4⋊C4×C61—21no—
113C2×C4⋊C4×C61—21no—
114C2×C4⋊C4×C61C2312yes—
115#103×C61—24no—
116#102×C61—24no—
117C24×C61—21no—
118C24×C61#4111yescharacteristic
119C23×D61—21no—
120#41×C61—24no—
121#41×C61—24no—
122#41×C61—24no—
123#41×C61—24no—
124#41×C61—24no—
125#41×C61—22no—
126#41×C61—22no—
127#41×C61C22⋊C418yes—
128#41×C61C22×C414yes—
129#41×C61C2×D414yes—
130C2×Q8×C61—21no—
131C2×Q8×C61—22no—
132C2×Q8×C61C22⋊C414yes—
133C2×Q8×C61C2×D412yes—
134C2×D4×C61—24no—
135C2×D4×C61—22no—
136C2×D4×C61—24no—
137C2×D4×C61—22no—
138C2×D4×C61—22no—
139C2×D4×C61—21no—
140C2×D4×C61C22⋊C414yes—
141C2×D4×C61C2×D412yes—
142C22×C4×C61—21no—
143C22×C4×C61—24no—
144C22×C4×C61—24no—
145C22×C4×C61—22no—
146C22×C4×C61—22no—
147C22×C4×C61—22no—
148C22×C4×C61—41no—
149C22×C4×C61—41no—
150C22×C4×C61—82no—
151C22×C4×C61C22×C412yes—
152C22×C4×C61C2×D412yes—
153C22×C4×C61C2×D411yescharacteristic
154C22×C4×C61C2×D411yescharacteristic
155C22×C4×C61C2×D411yescharacteristic
156C22×C4×C61C2×D411yescharacteristic
157C22×C4×C61C2×D412yes—
158C22×C4×C61#4111yescharacteristic
159C22×C4×C61#4112yes—
160C22×C4×C61C2411yescharacteristic
161C2×C22⋊D61—24no—
162C2×C22⋊D61—41no—
163C2×C22⋊D61—41no—
164C2×C22⋊D61—42no—
165C22×Dic61—21no—
166#11758222—24no—
167#11758222—24no—
168#11758222—24no—
169#11758222—24no—
170#11758222—42no—
171#11758222—42no—
172#11758222—42no—
173#11758222—42no—
174C2×D244—21no—
175C2×D244—21no—
176C2×D244—21no—
177C2×D244—21no—
178C2×D244—21no—
179C2×D244C22×C412yes—
180C2×C4×D61—24no—
181C2×C4×D61—41no—
182C2×C4×D61—41no—
183C2×C4×D61—42no—
184C2×Dic122—21no—
185C2×Dic122—21no—
186C2×Dic122—21no—
187C2×Dic122—21no—
188C2×Dic122—21no—
189C2×Dic122C22×C412yes—
190M16×C61—22no—
191M16×C61—22no—
192M16×C61—42no—
193C2×C8×C61—22no—
194C2×C8×C61—42no—
195C4⋊C4×C61—22no—
196C4⋊C4×C61—22no—
197C4⋊C4×C61—41no—
198C4⋊C4×C61—41no—
199#11758202—28no—
200#11758201—28no—
201#11758200—28no—
202#11758199—28no—
203#11758194—216no—
204#11758194—24no—
205#11758194—24no—
206#11758194—24no—
207#11758194—24no—
208#11758194—22no—
209#11758194—22no—
210#11758194C22×C414yes—
211#11758194C2×D414yes—
212C2×#92418—216no—
213C2×#92418—24no—
214C2×#92418—24no—
215C2×#92418—22no—
216C2×#92418—22no—
217C2×#92418—22no—
218C2×#92418—22no—
219C2×#92418C2×D414yes—
220C23×C61—22no—
221C23×C61—24no—
222C23×C61—41no—
223C23×C61—41no—
224C23×C61—81no—
225C23×C61—81no—
226C23×C61C4×D412yes—
227C23×C61#12111yescharacteristic
228C23×C61#12111yescharacteristic
229C23×C61#12312yes—
230C23×C61C2×#4111yescharacteristic
231C22×D61—22no—
232C22×D61—41no—
233C22×D61—41no—
234C22×D61—81no—
235C22×D61—81no—
236Q8×C61—22no—
237Q8×C61—24no—
238Q8×C61C2×C22⋊C414yes—
239D4×C61—24no—
240D4×C61—24no—
241D4×C61—28no—
242D4×C61—22no—
243D4×C61—24no—
244D4×C61C2×C22⋊C414yes—
245C2×C4×C61—22no—
246C2×C4×C61—24no—
247C2×C4×C61—24no—
248C2×C4×C61—24no—
249C2×C4×C61—28no—
250C2×C4×C61—42no—
251C2×C4×C61—82no—
252C2×C4×C61—82no—
253C2×C4×C61—82no—
254C2×C4×C61C2×C22⋊C412yes—
255C2×C4×C61C2×C22⋊C412yes—
256C2×C4×C61C4×D412yes—
257C2×C4×C61C4×D412yes—
258C2×C4×C61C4×D412yes—
259C2×C4×C61#12012yes—
260C2×C4×C61#12012yes—
261C2×C4×C61#12112yes—
262C2×C4×C61#12112yes—
263C2×C4×C61#12111yescharacteristic
264C2×C4×C61#12111yescharacteristic
265C2×C4×C61#12312yes—
266C2×C4×C61C23×C411yescharacteristic
267C2×C4×C61C22×D411yescharacteristic
268C2×C4×C61C22×D411yescharacteristic
269C2×C4×C61C22×D411yescharacteristic
270C2×C4×C61C22×D411yescharacteristic
271C2×C4×C61C2×#4111yescharacteristic
272C22⋊D61—44no—
273C22⋊D61—44no—
274C2×Dic61—22no—
275C2×Dic61—41no—
276C2×Dic61—41no—
277D244—22no—
278D244—22no—
279D244—41no—
280D244—41no—
281C4×D61—44no—
282C4×D61—44no—
283Dic122—22no—
284Dic122—22no—
285Dic122—41no—
286Dic122—41no—
287C8×C61—42no—
288#92418—24no—
289#92418—24no—
290#92418—24no—
291#21004—611nomaximal, Sylow
292C22×C61—21no—
293C22×C61—21no—
294C22×C61—21no—
295C22×C61—21no—
296C22×C61—21no—
297C22×C61—21no—
298C22×C61—24no—
299C22×C61—42no—
300C22×C61—42no—
301C22×C61—42no—
302C22×C61—42no—
303C22×C61—84no—
304C22×C61#38612yes—
305C22×C61#38612yes—
306C22×C61C2×C4×D411yescharacteristic
307C22×C61C2×#12111yescharacteristic
308C22×C61C2×#12311yescharacteristic
309C2×D61—42no—
310C2×D61—82no—
311C2×D61—81no—
312C2×D61—81no—
313C2×D61—82no—
314C4×C61—24no—
315C4×C61—82no—
316C4×C61#38612yes—
317C4×C61#38612yes—
318C4×C61C22×C22⋊C411yescharacteristic
319C4×C61C2×C4×D411yescharacteristic
320C4×C61C2×#12011yescharacteristic
321C4×C61C2×#12111yescharacteristic
322Dic61—42no—
323C2×#575—611no—
324C2×#568—611no—
325C2×#417—611no—
326C2×#417D6111yescharacteristic, complemented
327C2×#413—611no—
328#1908—614no—
329#1907—614no—
330C2×#343—611no—
331#1738—611no—
332C2×C61—21no—
333C2×C61—21no—
334C2×C61—22no—
335C2×C61—22no—
336C2×C61—44no—
337C2×C61C2×#38611yescharacteristic
338C2×C61#306811yescharacteristic
339C2×C61#306911yescharacteristic
340D61—82no—
341C22×#41—611no—
342C22×#41C2×D6111yescharacteristic
343#575—618no—
344#575—1224no—
345C2×#137—612no—
346C2×#137—1221no—
347C2×AGL(1,ℤ/8)—612no—
348C2×AGL(1,ℤ/8)—1221no—
349C2×#135—1222no—
350#568—1224no—
351#568—1224no—
352C2×#131—614no—
353C2×#131—1222no—
354C22×M16—611no—
355C22×M16—611no—
356C22×M16—611no—
357C22×M16—1222no—
358C22×M16C2×D6111yescharacteristic
359C22×C4⋊C4C2×D6111yescharacteristic
360#417—22no—
361#417—1222no—
362#417—1222no—
363#417C2×D6114yes—
364C2×#103—21no—
365C2×#103—612no—
366C2×#102—21no—
367C2×#102—612no—
368#413—1224no—
369C2×#101—612no—
370C2×#100—612no—
371#343—614no—
372#341—614no—
373C61#2100411yescharacteristic, complemented, Sylow
374C2×#41—21no—
375C2×#41—22no—
376C2×#41—22no—
377C2×#41—21no—
378C2×#41—614no—
379C2×#41—1222no—
380C2×#41—1222no—
381C2×#41—1221no—
382C2×#41—1221no—
383C2×#41—1222no—
384C2×#41—1221no—
385C2×#41—1221no—
386C2×#41C4×D6114yes—
387C2×#41D24414yes—
388C2×#41C22⋊D6112yes—
389C2×#41C22×D6112yes—
390C22×Q8—611no—
391C22×Q8C22⋊D6111yescharacteristic
392C22×D4—21no—
393C22×D4—611no—
394C22×D4—1221no—
395C22×D4C22⋊D6111yescharacteristic
396C23×C4—21no—
397C23×C4—21no—
398C23×C4—1221no—
399C23×C4C22×D6111yescharacteristic
400#137—618no—
401#137—1224no—
402#137—1228no—
403AGL(1,ℤ/8)—618no—
404AGL(1,ℤ/8)—1224no—
405AGL(1,ℤ/8)—1228no—
406#135—12216no—
407C2×Q16—1222no—
408C2×SD16—1222no—
409C2×SD16—1222no—
410C2×D8—1222no—
411#131—1228no—
412#131—12216no—
413#131—1228no—
414C2×M16—21no—
415C2×M16—22no—
416C2×M16—22no—
417C2×M16—612no—
418C2×M16—612no—
419C2×M16—614no—
420C2×M16—614no—
421C2×M16—612no—
422C2×M16—612no—
423C2×M16—1224no—
424C2×M16—12216no—
425C2×M16—1222no—
426C2×M16—1222no—
427C2×M16—1222no—
428C2×M16—1221no—
429C2×M16—1222no—
430C2×M16—1221no—
431C2×M16C22×D6112yes—
432C22×C8—21no—
433C22×C8—612no—
434C22×C8—1224no—
435C22×C8—1221no—
436C22×C8—1221no—
437C2×C4⋊C4—21no—
438C2×C4⋊C4—21no—
439C2×C4⋊C4—21no—
440C2×C4⋊C4—21no—
441C2×C4⋊C4—21no—
442C2×C4⋊C4C22×D6112yes—
443#103—24no—
444#103—1224no—
445#102—24no—
446#102—1224no—
447#101—1224no—
448#100—1224no—
449C24—21no—
450C24—1221no—
451C24#1175822411yescharacteristic
452#41—22no—
453#41—22no—
454#41—24no—
455#41—24no—
456#41—24no—
457#41—24no—
458#41—24no—
459#41—1224no—
460#41—1224no—
461#41—1224no—
462#41—1224no—
463#41—2442no—
464#41—2442no—
465#41—2442no—
466#41—2442no—
467#41#1175819818yes—
468#41C2×C4×D6114yes—
469#41C2×D24414yes—
470C2×Q8—21no—
471C2×Q8—22no—
472C2×Q8—612no—
473C2×Q8—1221no—
474C2×Q8—1221no—
475C2×Q8—1221no—
476C2×Q8—1221no—
477C2×Q8—1221no—
478C2×Q8#1175819814yes—
479C2×Q8C2×C22⋊D6112yes—
480C2×D4—22no—
481C2×D4—24no—
482C2×D4—22no—
483C2×D4—24no—
484C2×D4—22no—
485C2×D4—21no—
486C2×D4—612no—
487C2×D4—1224no—
488C2×D4—1221no—
489C2×D4—1221no—
490C2×D4—1221no—
491C2×D4—1221no—
492C2×D4—1221no—
493C2×D4—2441no—
494C2×D4—2441no—
495C2×D4—2442no—
496C2×D4#1175819814yes—
497C2×D4C2×C22⋊D6112yes—
498C22×C4—21no—
499C22×C4—22no—
500C22×C4—24no—
501C22×C4—24no—
502C22×C4—22no—
503C22×C4—22no—
504C22×C4—41no—
505C22×C4—41no—
506C22×C4—82no—
507C22×C4—1221no—
508C22×C4—1224no—
509C22×C4—2441no—
510C22×C4—2441no—
511C22×C4—2442no—
512C22×C4C2×C4×D6112yes—
513C22×C4C2×D24412yes—
514C22×C4D4×D6111yescharacteristic
515C22×C4D4×D6111yescharacteristic
516C22×C4D4×D6112yes—
517C22×C4#1175822412yes—
518C22×C4#1175822411yescharacteristic
519C22×C4C2×C22⋊D6111yescharacteristic
520C22×C4C2×C22⋊D6111yescharacteristic
521C22×C4C23×D6111yescharacteristic
522Q16—1228no—
523SD16—1228no—
524SD16—1228no—
525D8—1228no—
526M16—22no—
527M16—22no—
528M16—42no—
529M16—614no—
530M16—614no—
531M16—12216no—
532M16—1222no—
533M16—1222no—
534M16—1224no—
535M16—1224no—
536M16—1224no—
537M16—1224no—
538C2×C8—22no—
539C2×C8—42no—
540C2×C8—614no—
541C2×C8—12216no—
542C2×C8—1222no—
543C2×C8—1222no—
544C2×C8—1222no—
545C2×C8—1222no—
546C2×C8—1224no—
547C2×C8—1224no—
548C4⋊C4—22no—
549C4⋊C4—22no—
550C4⋊C4—41no—
551C4⋊C4—41no—
552C23—22no—
553C23—24no—
554C23—41no—
555C23—41no—
556C23—81no—
557C23—81no—
558C23—1222no—
559C23—4881no—
560C23—4881no—
561C23—2441no—
562C23—2441no—
563C23#1451973712yes—
564C23#1451974911yescharacteristic
565C23#1451974911yescharacteristic
566C23#1451976112yes—
567C23C2×#1175822411yescharacteristic
568Q8—22no—
569Q8—24no—
570Q8—1222no—
571Q8—1222no—
572Q8—2441no—
573Q8—2441no—
574Q8C2×#1175819814yes—
575D4—24no—
576D4—24no—
577D4—28no—
578D4—22no—
579D4—24no—
580D4—1222no—
581D4—1222no—
582D4—2444no—
583D4—2444no—
584D4—2441no—
585D4—2441no—
586D4C2×#1175819814yes—
587C2×C4—22no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—28no—
592C2×C4—42no—
593C2×C4—82no—
594C2×C4—82no—
595C2×C4—82no—
596C2×C4—1222no—
597C2×C4—2441no—
598C2×C4—2441no—
599C2×C4—2444no—
600C2×C4—2444no—
601C2×C4C2×#1175819812yes—
602C2×C4C2×#1175819812yes—
603C2×C4#1451973712yes—
604C2×C4#1451973712yes—
605C2×C4#1451973712yes—
606C2×C4#1451974712yes—
607C2×C4#1451974712yes—
608C2×C4#1451974912yes—
609C2×C4#1451974912yes—
610C2×C4#1451974911yescharacteristic
611C2×C4#1451974911yescharacteristic
612C2×C4#1451976112yes—
613C2×C4C22×C4×D6111yescharacteristic
614C2×C4C2×D4×D6111yescharacteristic
615C2×C4C2×D4×D6111yescharacteristic
616C2×C4C22×C22⋊D6111yescharacteristic
617C2×C4C22×D24411yescharacteristic
618C2×C4C2×#1175822411yescharacteristic
619C8—42no—
620C8—1224no—
621C8—1224no—
622C8—1224no—
623C22—21no—
624C22—21no—
625C22—21no—
626C22—21no—
627C22—21no—
628C22—21no—
629C22—24no—
630C22—42no—
631C22—42no—
632C22—42no—
633C22—42no—
634C22—84no—
635C22—4882no—
636C22—4882no—
637C22—4881no—
638C22—4881no—
639C22—2442no—
640C22#1683963912yes—
641C22#1683963912yes—
642C22C2×#1451973711yescharacteristic
643C22C2×#1451974911yescharacteristic
644C22C2×#1451976111yescharacteristic
645C4—24no—
646C4—82no—
647C4—2442no—
648C4#1683963912yes—
649C4#1683963912yes—
650C4C22×#1175819811yescharacteristic
651C4C2×#1451973711yescharacteristic
652C4C2×#1451974711yescharacteristic
653C4C2×#1451974911yescharacteristic
654C2—22no—
655C2—21no—
656C2—21no—
657C2—22no—
658C2—44no—
659C2—4882no—
660C2C2×#1683963911yescharacteristic
661C2#2145554411yescharacteristic
662C2#2145555311yescharacteristic
subgroupG/H
centerC22C2×#14519749
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC2×C61×#417C2
socleC22×C61C2×#121
prime radicalC2×#16840560C2
cosocleC22×C4×C61C24
hypercenterC2×#417D61
hypocenterC61#21004
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#417

Complements

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

NH ≅ G/Ntype
2. C2×#16840529C2semidirect
3. C2×#16839761C2semidirect
4. C2×#16839744C2semidirect
5. #21446332C2semidirect
6. #21446325C2semidirect
7. C2×#16839531C2semidirect
8. #21445545C2semidirect
9. C2×C61×#417C2semidirect
12. #16840560C4semidirect
28. C2×#14519700C22semidirect
29. C2×#14519697C22semidirect
31. C2×#14519695C22semidirect
32. C2×#14519693C22semidirect
33. #16839531C22semidirect
34. #16839526C22semidirect
56. #14519807C2×C4semidirect
59. #14519802C2×C4semidirect
326. C2×#417D61semidirect
373. C61#21004semidirect

Maximal subgroups

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

subgroupindex
C2×#168405602
C2×#168405292
C2×#168397612
C2×#168397442
#214463322
#214463252
C2×#168395312
#214455452
C2×C61×#4172
#2100461