Finite Group Databasebeta

← #29681962

Subgroups — #29681962

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×#16839764C211yesmaximal, characteristic, complemented
3C2×#16839763C211yesmaximal, characteristic, complemented
4C2×#16839744C211yesmaximal, characteristic, complemented
5#21446334C214yesmaximal, complemented
6#21446327C214yesmaximal, complemented
7C2×#16839530C211yesmaximal, characteristic, complemented
8#21445546C211yesmaximal, characteristic, complemented
9C2×C61×#568C211yesmaximal, characteristic, complemented
10C22×#11758222C2211yescharacteristic
11#16840560—24no—
12#16840560C418yescomplemented
13C2×#14519808—21no—
14C2×#14519807C2212yes—
15C2×#14519803—21no—
16C2×#14519802C2212yes—
17C2×#14519797—22no—
18C22×#11758194C2211yescharacteristic
19C22×#11758194C2211yescharacteristic
20C22×C4⋊Dic61C2211yescharacteristic
21#16839764—22no—
22#16839764—22no—
23#16839763—22no—
24#16839763C2214yes—
25C2×#14519702C2212yescomplemented
26C2×#14519701—21no—
27C2×#14519699C2212yescomplemented
28C2×#14519698—21no—
29#16839744—24no—
30C2×#14519695C2212yescomplemented
31C2×#14519693C2212yescomplemented
32#16839530C2214yescomplemented
33#16839527C2214yescomplemented
34C22×#41×C61C2211yescharacteristic
35C61×#568—24no—
36C61×#568—24no—
37C2×#131×C61—22no—
38C2×#131×C61C2214yes—
39C22×M16×C61—22no—
40C22×M16×C61C2211yescharacteristic
41C22×M16×C61C2211yescharacteristic
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—
53C23×Dic61—21no—
54C22×C4×D61—21no—
55#14519808—28no—
56#14519807—24no—
57#14519807C2×C418yescomplemented
58#14519803—28no—
59#14519802—24no—
60#14519802C2×C418yescomplemented
61#14519797—216no—
62C2×#11758202—22no—
63C2×#11758200—22no—
64C2×#11758201—22no—
65C2×#11758199—22no—
66C2×#11758194—21no—
67C2×#11758194—22no—
68C2×#11758194—21no—
69C2×#11758194—22no—
70C2×#11758194—22no—
71C2×#11758194C2×C414yes—
72C2×#11758194C2312yes—
73C2×#11758194C2312yes—
74C22×#92418—21no—
75C22×#92418—21no—
76C2×C4⋊Dic61—21no—
77C2×C4⋊Dic61—21no—
78C2×C4⋊Dic61—21no—
79C2×C4⋊Dic61—21no—
80C2×C4⋊Dic61—21no—
81C2×C4⋊Dic61C2312yes—
82#14519702—24no—
83#14519701—24no—
84#14519699—24no—
85#14519698—24no—
86#14519695—24no—
87#14519693—24no—
88C2×#41×C61—21no—
89C2×#41×C61—22no—
90C2×#41×C61—22no—
91C2×#41×C61—21no—
92C2×#41×C61C2×C414yes—
93C2×#41×C61D412yes—
94C2×#41×C61D414yes—
95C2×#41×C61C2312yes—
96C22×Q8×C61D411yescharacteristic
97C22×D4×C61—21no—
98C22×D4×C61D411yescharacteristic
99C23×C4×C61—21no—
100C23×C4×C61C2311yescharacteristic
101#131×C61—216no—
102#131×C61—28no—
103#131×C61—28no—
104C2×M16×C61—216no—
105C2×M16×C61—21no—
106C2×M16×C61—22no—
107C2×M16×C61—22no—
108C2×M16×C61—22no—
109C2×M16×C61—24no—
110C2×M16×C61D414yes—
111C2×M16×C61D412yes—
112C2×M16×C61D412yes—
113C2×M16×C61C2312yes—
114C22×C8×C61—21no—
115C22×C8×C61—24no—
116C22×C8×C61C2312yes—
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×C61C22×C412yes—
148C22×C4×C61C2×D412yes—
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61C2×D411yescharacteristic
153C22×C4×C61C2×D412yes—
154C22×C4×C61#4111yescharacteristic
155C22×C4×C61#4112yes—
156C22×C4×C61C2411yescharacteristic
157C2×C22⋊D61—24no—
158C2×C22⋊D61—41no—
159C2×C22⋊D61—41no—
160C2×C22⋊D61—42no—
161C22×Dic61—21no—
162C22×Dic61—22no—
163C22×Dic61—41no—
164C22×Dic61—41no—
165C22×Dic61—82no—
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—216no—
191M16×C61—24no—
192M16×C61—24no—
193M16×C61—24no—
194M16×C61—22no—
195M16×C61C2×D414yes—
196C2×C8×C61—216no—
197C2×C8×C61—24no—
198C2×C8×C61—22no—
199C2×C8×C61—22no—
200C2×C8×C61—22no—
201C2×C8×C61C2×D414yes—
202#11758202—28no—
203#11758201—28no—
204#11758200—28no—
205#11758199—28no—
206C4⋊Dic61—22no—
207C4⋊Dic61—22no—
208C4⋊Dic61—41no—
209C4⋊Dic61—41no—
210#11758194—24no—
211#11758194—22no—
212#11758194—22no—
213#11758194—22no—
214#11758194—42no—
215#11758194C22×C414yes—
216C2×#92418—24no—
217C2×#92418—22no—
218C2×#92418—22no—
219C2×#92418—42no—
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×C61C2×C22⋊C412yes—
251C2×C4×C61C2×C22⋊C412yes—
252C2×C4×C61C4×D412yes—
253C2×C4×C61C4×D412yes—
254C2×C4×C61C4×D412yes—
255C2×C4×C61#12012yes—
256C2×C4×C61#12012yes—
257C2×C4×C61#12112yes—
258C2×C4×C61#12112yes—
259C2×C4×C61#12111yescharacteristic
260C2×C4×C61#12111yescharacteristic
261C2×C4×C61#12312yes—
262C2×C4×C61C23×C411yescharacteristic
263C2×C4×C61C22×D411yescharacteristic
264C2×C4×C61C22×D411yescharacteristic
265C2×C4×C61C22×D411yescharacteristic
266C2×C4×C61C22×D411yescharacteristic
267C2×C4×C61C2×#4111yescharacteristic
268C22⋊D61—44no—
269C22⋊D61—44no—
270C2×Dic61—22no—
271C2×Dic61—41no—
272C2×Dic61—41no—
273C2×Dic61—42no—
274C2×Dic61—82no—
275C2×Dic61—82no—
276C2×Dic61—82no—
277D244—22no—
278D244—22no—
279D244—41no—
280D244—41no—
281C4×D61—44no—
282C4×D61—44no—
283Dic122—22no—
284Dic122—22no—
285Dic122—41no—
286Dic122—41no—
287C8×C61—24no—
288C8×C61—24no—
289#92418—24no—
290#92418—42no—
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#38612yes—
316C4×C61#38612yes—
317C4×C61C22×C22⋊C411yescharacteristic
318C4×C61C2×C4×D411yescharacteristic
319C4×C61C2×#12011yescharacteristic
320C4×C61C2×#12111yescharacteristic
321Dic61—42no—
322Dic61—82no—
323C2×#575—611no—
324C2×#568D6111yescharacteristic, complemented
325C2×#417—611no—
326C2×#417—611no—
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—24no—
351#568—24no—
352C2×#131—22no—
353C2×#131C2×D6114yes—
354C22×M16—22no—
355C22×M16—611no—
356C22×M16—611no—
357C22×M16C2×D6111yescharacteristic
358C22×M16C2×D6111yescharacteristic
359C22×C4⋊C4—611no—
360#417—614no—
361#417—1222no—
362#417—1222no—
363#417—1222no—
364C2×#103—612no—
365C2×#103—1221no—
366C2×#102—612no—
367C2×#102—1221no—
368#413—1224no—
369C2×#101—612no—
370C2×#100—612no—
371#343—614no—
372#341—614no—
373C61#2100411yescharacteristic, complemented, Sylow
374C2×#41—22no—
375C2×#41—21no—
376C2×#41—21no—
377C2×#41—22no—
378C2×#41—614no—
379C2×#41—1222no—
380C2×#41—1221no—
381C2×#41—1221no—
382C2×#41—1222no—
383C2×#41—1222no—
384C2×#41—1221no—
385C2×#41—1221no—
386C2×#41C4×D6114yes—
387C2×#41D24412yes—
388C2×#41C22⋊D6114yes—
389C2×#41C22×D6112yes—
390C22×Q8—611no—
391C22×Q8D24411yescharacteristic
392C22×D4—21no—
393C22×D4—611no—
394C22×D4—1221no—
395C22×D4D24411yescharacteristic
396C23×C4—21no—
397C23×C4—1221no—
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—216no—
412#131—28no—
413#131—28no—
414C2×M16—216no—
415C2×M16—21no—
416C2×M16—22no—
417C2×M16—22no—
418C2×M16—22no—
419C2×M16—24no—
420C2×M16—614no—
421C2×M16—612no—
422C2×M16—612no—
423C2×M16—1221no—
424C2×M16—1222no—
425C2×M16—1221no—
426C2×M16—1222no—
427C2×M16—1222no—
428C2×M16C22⋊D6114yes—
429C2×M16C22⋊D6112yes—
430C2×M16C22⋊D6112yes—
431C2×M16C22×D6112yes—
432C22×C8—21no—
433C22×C8—24no—
434C22×C8—1221no—
435C22×C8—1221no—
436C22×C8C22×D6112yes—
437C2×C4⋊C4—612no—
438C2×C4⋊C4—1221no—
439C2×C4⋊C4—1221no—
440C2×C4⋊C4—1221no—
441C2×C4⋊C4—1221no—
442C2×C4⋊C4—1221no—
443#103—1224no—
444#103—1224no—
445#102—1224no—
446#102—1224no—
447#101—1224no—
448#100—1224no—
449C24—21no—
450C24—1221no—
451C24#1175822211yescharacteristic
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×C22⋊D6114yes—
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×D24412yes—
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×D24412yes—
498C22×C4—21no—
499C22×C4—22no—
500C22×C4—24no—
501C22×C4—24no—
502C22×C4—22no—
503C22×C4—1221no—
504C22×C4—1224no—
505C22×C4—1222no—
506C22×C4—4882no—
507C22×C4—2441no—
508C22×C4—2441no—
509C22×C4—2441no—
510C22×C4—2441no—
511C22×C4—2442no—
512C22×C4C2×C4×D6112yes—
513C22×C4C2×D24411yescharacteristic
514C22×C4C2×D24411yescharacteristic
515C22×C4#1175822212yes—
516C22×C4#1175822211yescharacteristic
517C22×C4C2×C22⋊D6112yes—
518C22×C4C2×C22⋊D6111yescharacteristic
519C22×C4C2×C22⋊D6111yescharacteristic
520C22×C4C2×C22⋊D6112yes—
521C22×C4C23×D6111yescharacteristic
522Q16—1228no—
523SD16—1228no—
524SD16—1228no—
525D8—1228no—
526M16—216no—
527M16—22no—
528M16—24no—
529M16—24no—
530M16—24no—
531M16—614no—
532M16—1222no—
533M16—1224no—
534M16—1222no—
535M16—1222no—
536M16—2442no—
537M16C2×C22⋊D6114yes—
538C2×C8—216no—
539C2×C8—22no—
540C2×C8—22no—
541C2×C8—22no—
542C2×C8—24no—
543C2×C8—1222no—
544C2×C8—1224no—
545C2×C8—1222no—
546C2×C8—2442no—
547C2×C8C2×C22⋊D6114yes—
548C4⋊C4—1222no—
549C4⋊C4—1222no—
550C4⋊C4—2441no—
551C4⋊C4—2441no—
552C23—22no—
553C23—24no—
554C23—41no—
555C23—41no—
556C23—81no—
557C23—81no—
558C23—1222no—
559C23—4881no—
560C23—4881no—
561C23—2441no—
562C23—2441no—
563C23C4×C22⋊D6112yes—
564C23#1451975111yescharacteristic
565C23#1451975111yescharacteristic
566C23#1451976012yes—
567C23C2×#1175822211yescharacteristic
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—1222no—
593C2×C4—4882no—
594C2×C4—4882no—
595C2×C4—4882no—
596C2×C4—2442no—
597C2×C4—2441no—
598C2×C4—2441no—
599C2×C4—2444no—
600C2×C4—2444no—
601C2×C4C2×#1175819812yes—
602C2×C4C2×#1175819812yes—
603C2×C4C4×C22⋊D6112yes—
604C2×C4C4×C22⋊D6112yes—
605C2×C4C4×C22⋊D6112yes—
606C2×C4#1451974612yes—
607C2×C4#1451974612yes—
608C2×C4#1451975112yes—
609C2×C4#1451975112yes—
610C2×C4#1451975111yescharacteristic
611C2×C4#1451975111yescharacteristic
612C2×C4#1451976012yes—
613C2×C4C22×C4×D6111yescharacteristic
614C2×C4C22×C22⋊D6111yescharacteristic
615C2×C4C22×C22⋊D6111yescharacteristic
616C2×C4C22×C22⋊D6111yescharacteristic
617C2×C4C22×D24411yescharacteristic
618C2×C4C2×#1175822211yescharacteristic
619C8—24no—
620C8—24no—
621C8—1224no—
622C8—2442no—
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#1683964112yes—
641C22#1683964112yes—
642C22C2×C4×C22⋊D6111yescharacteristic
643C22C2×#1451975111yescharacteristic
644C22C2×#1451976011yescharacteristic
645C4—24no—
646C4—4882no—
647C4—2442no—
648C4#1683964112yes—
649C4#1683964112yes—
650C4C22×#1175819811yescharacteristic
651C4C2×C4×C22⋊D6111yescharacteristic
652C4C2×#1451974611yescharacteristic
653C4C2×#1451975111yescharacteristic
654C2—22no—
655C2—21no—
656C2—21no—
657C2—22no—
658C2—44no—
659C2—4882no—
660C2C2×#1683964111yescharacteristic
661C2#2145554711yescharacteristic
662C2#2145555611yescharacteristic
subgroupG/H
centerC22C2×#14519751
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC2×C61×#568C2
socleC22×C61C2×#121
prime radicalC2×#16840560C2
cosocleC22×C4×C61C24
hypercenterC2×#568D61
hypocenterC61#21004
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#568

Complements

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

NH ≅ G/Ntype
2. C2×#16839764C2semidirect
3. C2×#16839763C2semidirect
4. C2×#16839744C2semidirect
5. #21446334C2semidirect
6. #21446327C2semidirect
7. C2×#16839530C2semidirect
8. #21445546C2semidirect
9. C2×C61×#568C2semidirect
12. #16840560C4semidirect
25. C2×#14519702C22semidirect
27. C2×#14519699C22semidirect
30. C2×#14519695C22semidirect
31. C2×#14519693C22semidirect
32. #16839530C22semidirect
33. #16839527C22semidirect
57. #14519807C2×C4semidirect
60. #14519802C2×C4semidirect
324. C2×#568D61semidirect
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×#168397642
C2×#168397632
C2×#168397442
#214463342
#214463272
C2×#168395302
#214455462
C2×C61×#5682
#2100461