Finite Group Databasebeta

← #29740011

Subgroups — #29740011

Order 15616.

910 conjugacy classes of subgroups.

The subgroup lattice has 692 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839960C211yesmaximal, characteristic
2C2×#16839847C211yesmaximal, characteristic, complemented
3#21449587C211yesmaximal, characteristic
4#21448740C211yesmaximal, characteristic, complemented
5#21447340C211yesmaximal, characteristic, complemented
6#21447323C211yesmaximal, characteristic, complemented
7#21447234C211yesmaximal, characteristic, complemented
8#21447052C211yesmaximal, characteristic
9#21446504C211yesmaximal, characteristic, complemented
10#21446375C211yesmaximal, characteristic, complemented
11#21446176C211yesmaximal, characteristic, complemented
12#21446174C211yesmaximal, characteristic, complemented
13#21446071C211yesmaximal, characteristic, complemented
14#21446067C211yesmaximal, characteristic, complemented
15C61×#1880C211yesmaximal, characteristic, complemented
16C22×#11758201C2211yescharacteristic
17C2×#14519773C2211yescharacteristic
18C2×#14519759—21no—
19C2×Q8×Dic61C2211yescharacteristic
20C2×Q8×Dic61C2211yescharacteristic
21C2×#14519739C2211yescharacteristic
22#16839960—24no—
23#16839847—24no—
24C2×#14519710C2211yescharacteristic
25C2×#14519708C2211yescharacteristic
26C2×#14519706C2211yescharacteristic
27C2×#14519701C2211yescharacteristic
28C2×#14519699C2211yescharacteristic
29C2×#14519699C2211yescharacteristic
30C2×#14519698C2211yescharacteristic
31C2×#14519698C2211yescharacteristic
32C2×#14519697C2211yescharacteristic
33C2×C8×Dic61C2211yescharacteristic
34#16839678—21no—
35#16839677—21no—
36#16839665C2211yescharacteristic
37#16839664C2211yescharacteristic
38#16839661—21no—
39#16839661—21no—
40#16839661C2211yescharacteristic
41#16839661C2211yescharacteristic
42#16839658C2211yescharacteristic
43#16839638—21no—
44#16839628—21no—
45#16839628C2211yescharacteristic
46C4×C4⋊Dic61—21no—
47C4×C4⋊Dic61C2211yescharacteristic
48#16839525C2211yescharacteristic
49#16839525C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839517C2211yescharacteristic
52#16839516C2211yescharacteristic
53#16839516C2211yescharacteristic
54C22×SD16×C61C2211yescharacteristic
55C2×C4×Q8×C61C2211yescharacteristic
56C2×#103×C61C2211yescharacteristic
57C2×#102×C61C2211yescharacteristic
58C61×#390C2211yescharacteristic
59C61×#340C2211yescharacteristic
60C61×#336C2211yescharacteristic
61C22×D244C2311yescharacteristic
62C22×C22⋊D61—21no—
63C22×C22⋊D61—21no—
64C22×C4×D61—21no—
65C2×#11758201—22no—
66C2×#11758201—22no—
67C2×#11758201D414yes—
68C22×#92418C2311yescharacteristic
69C22×#92418C2311yescharacteristic
70#14519773—22no—
71#14519773—41no—
72#14519759—44no—
73Q8×Dic61—22no—
74Q8×Dic61—22no—
75Q8×Dic61—41no—
76Q8×Dic61—41no—
77#14519739—22no—
78#14519739—41no—
79C2×C4⋊Dic61—21no—
80C2×C4⋊Dic61—21no—
81C2×C4⋊Dic61—21no—
82C2×C4⋊Dic61—21no—
83C2×C4⋊Dic61D411yescharacteristic
84C2×C4⋊Dic61D411yescharacteristic
85C2×C4⋊Dic61C2311yescharacteristic
86C2×C4⋊Dic61C2311yescharacteristic
87C2×C4⋊Dic61C2311yescharacteristic
88C2×#11758196—21no—
89C2×C22⋊Dic61—21no—
90C2×C22⋊Dic61—21no—
91C2×#11758198—21no—
92C2×#11758198—21no—
93C2×#11758198—21no—
94C2×C4×Dic61—21no—
95C2×C4×Dic61—21no—
96C2×C4×Dic61C2311yescharacteristic
97C2×C4×Dic61C2311yescharacteristic
98#14519710—41no—
99#14519708—22no—
100#14519706—22no—
101#14519701—22no—
102#14519699—22no—
103#14519699—22no—
104#14519698—22no—
105#14519698—41no—
106#14519697—22no—
107C8×Dic61—22no—
108#14519684—21no—
109#14519684—21no—
110#14519684—21no—
111#14519684—21no—
112#14519684—21no—
113#14519684D411yescharacteristic
114#14519684D411yescharacteristic
115C22×Q8×C61C2311yescharacteristic
116C22×D4×C61C2311yescharacteristic
117C2×SD16×C61—22no—
118C2×SD16×C61—22no—
119C2×SD16×C61—22no—
120C22×C8×C61C2311yescharacteristic
121C22×C8×C61C2311yescharacteristic
122C4×Q8×C61—22no—
123C4×Q8×C61—22no—
124C2×C4⋊C4×C61—21no—
125C2×C4⋊C4×C61C2311yescharacteristic
126C2×C4⋊C4×C61C2311yescharacteristic
127C2×C22⋊C4×C61—21no—
128C2×C22⋊C4×C61—21no—
129C2×C42×C61—21no—
130C2×C42×C61C2311yescharacteristic
131#103×C61—22no—
132#102×C61—22no—
133C24×C61—21no—
134C23×D61—21no—
135C2×Q8×C61—21no—
136C2×Q8×C61—21no—
137C2×Q8×C61—21no—
138C2×Q8×C61C2×D412yes—
139C2×Q8×C61#4112yes—
140C2×Q8×C61#4112yes—
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—21no—
148C22×C4×C61—21no—
149C22×C4×C61—21no—
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61C2411yescharacteristic
153C2×C22⋊D61—42no—
154C2×C22⋊D61—42no—
155C2×C22⋊D61—42no—
156C2×C22⋊D61—42no—
157C2×C22⋊D61—81no—
158C2×C22⋊D61—81no—
159C22×Dic61—21no—
160C22×Dic61—21no—
161C22×Dic61—21no—
162C22×Dic61C2×D411yescharacteristic
163C22×Dic61C2×D411yescharacteristic
164C22×Dic61C2×D411yescharacteristic
165C22×Dic61C2×D411yescharacteristic
166C2×D244—21no—
167C2×D244—21no—
168C2×D244—21no—
169C2×D244—21no—
170C2×D244—21no—
171C2×D244C2×D412yes—
172C2×C4×D61—42no—
173C2×C4×D61—42no—
174C2×C4×D61—81no—
175SD16×C61—44no—
176C2×C8×C61—21no—
177C2×C8×C61—21no—
178C2×C8×C61—21no—
179C2×C8×C61—21no—
180C2×C8×C61—21no—
181C2×C8×C61—21no—
182C4⋊C4×C61—24no—
183C4⋊C4×C61—21no—
184C4⋊C4×C61—21no—
185C4⋊C4×C61—21no—
186C4⋊C4×C61#4112yes—
187C22⋊C4×C61—42no—
188C22⋊C4×C61—81no—
189C42×C61—24no—
190C42×C61—21no—
191C42×C61—21no—
192C22⋊Dic61—42no—
193C22⋊Dic61—81no—
194#11758201—28no—
195#11758198—42no—
196#11758198—42no—
197#11758198—81no—
198C4⋊Dic61—22no—
199C4⋊Dic61—22no—
200C4⋊Dic61—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61—21no—
203C4⋊Dic61—21no—
204C4⋊Dic61—21no—
205C4⋊Dic61—21no—
206C4⋊Dic61—42no—
207C4⋊Dic61—42no—
208C4⋊Dic61—42no—
209C4⋊Dic61—42no—
210#11758196—42no—
211C4×Dic61—21no—
212C4×Dic61—21no—
213C4×Dic61—21no—
214C4×Dic61—21no—
215C4×Dic61—42no—
216C4×Dic61—42no—
217C2×#92418—21no—
218C2×#92418—21no—
219C2×#92418—21no—
220C2×#92418—21no—
221C2×#92418—21no—
222C2×#92418C2×D412yes—
223C23×C61—41no—
224C23×C61—41no—
225C23×C61—41no—
226C23×C61—41no—
227C23×C61—41no—
228C23×C61—41no—
229C23×C61—41no—
230C23×C61C22×D411yescharacteristic
231C22×D61—22no—
232C22×D61—41no—
233C22×D61—41no—
234C22×D61—41no—
235C22×D61—41no—
236C22×D61—41no—
237C22×D61—41no—
238Q8×C61—22no—
239Q8×C61—22no—
240Q8×C61—22no—
241Q8×C61#12314yescomplemented
242D4×C61—41no—
243D4×C61—41no—
244D4×C61—41no—
245D4×C61—41no—
246C2×C4×C61—22no—
247C2×C4×C61—22no—
248C2×C4×C61—22no—
249C2×C4×C61—21no—
250C2×C4×C61—21no—
251C2×C4×C61—22no—
252C2×C4×C61—21no—
253C2×C4×C61—21no—
254C2×C4×C61—21no—
255C2×C4×C61—21no—
256C2×C4×C61—41no—
257C2×C4×C61—41no—
258C2×C4×C61—41no—
259C2×C4×C61—41no—
260C2×C4×C61—41no—
261C2×C4×C61C22×D411yescharacteristic
262C2×C4×C61C2×#4111yescharacteristic
263C2×C4×C61C2×#4111yescharacteristic
264C2×C4×C61C2×#4111yescharacteristic
265C2×C4×C61H(5,𝔽2)11yescharacteristic
266C2×C4×C612−1+411yescharacteristic
267C22⋊D61—84no—
268C22⋊D61—84no—
269C2×Dic61—21no—
270C2×Dic61—21no—
271C2×Dic61—21no—
272C2×Dic61—21no—
273C2×Dic61—21no—
274C2×Dic61—21no—
275C2×Dic61—21no—
276C2×Dic61—21no—
277C2×Dic61—21no—
278C2×Dic61—21no—
279C2×Dic61—41no—
280C2×Dic61—41no—
281C2×Dic61—41no—
282C2×Dic61—41no—
283C2×Dic61—41no—
284C2×Dic61—41no—
285C2×Dic61—41no—
286C2×Dic61—41no—
287C2×Dic61—41no—
288C2×Dic61#13512yes—
289C2×Dic61AGL(1,ℤ/8)12yes—
290D244—22no—
291D244—22no—
292D244—41no—
293D244—41no—
294C4×D61—84no—
295C8×C61—41no—
296C8×C61—41no—
297#92418—22no—
298#92418—41no—
299#26159—611nomaximal, Sylow
300C22×C61—81no—
301C22×C61—81no—
302C22×C61—81no—
303C22×C61—81no—
304C22×C61—81no—
305C22×C61—81no—
306C22×C61—81no—
307C22×C61#52511yescharacteristic
308C22×C61D4211yescharacteristic
309C22×C61#54811yescharacteristic
310C22×C61C2×#13511yescharacteristic
311C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
312C22×C61#57711yescharacteristic
313C22×C61#57711yescharacteristic
314C2×D61—42no—
315C2×D61—42no—
316C2×D61—42no—
317C2×D61—81no—
318C2×D61—81no—
319C2×D61—81no—
320C2×D61—81no—
321C4×C61—24no—
322C4×C61—41no—
323C4×C61—42no—
324C4×C61—41no—
325C4×C61—81no—
326C4×C61C2×#12311yescharacteristic
327C4×C61#53911yescharacteristic
328C4×C61#54711yescharacteristic
329C4×C61#55311yescharacteristic
330Dic61—22no—
331Dic61—22no—
332Dic61—41no—
333Dic61—41no—
334Dic61—81no—
335Dic61—81no—
336Dic61—81no—
337C2×#460—611no—
338C2×C4×SD16—611no—
339#2445—611no—
340#2351—611no—
341#2092—611no—
342#2088—611no—
343#2073—611no—
344#2041—611no—
345#1934—611no—
346#1916—611no—
347#1880—611no—
348#1880—611no—
349#1880D6111yescharacteristic, complemented
350#1863—611no—
351#1862—611no—
352C2×C61—161no—
353C2×C61#244311yescharacteristic
354C2×C61#310611yescharacteristic
355C2×C61#312611yescharacteristic
356C2×C61#326511yescharacteristic
357C2×C61#328211yescharacteristic
358C2×C61#336511yescharacteristic
359C2×C61#336811yescharacteristic
360D61—82no—
361C22×SD16—611no—
362C22×SD16C2×D6111yescharacteristic
363C2×#127—611no—
364C2×#123—1221no—
365C2×C4×Q8—611no—
366C2×C4×Q8—611no—
367C2×C4×Q8C2×D6111yescharacteristic
368C2×C4×D4—611no—
369#460—1224no—
370C4×SD16—1224no—
371C2×#107—611no—
372C2×#106—611no—
373C2×C4⋊C8—611no—
374C2×#103—611no—
375C2×#103C2×D6111yescharacteristic
376C2×#102—611no—
377C2×#102—611no—
378C2×#102—611no—
379C2×#102—611no—
380C2×#102—611no—
381C2×#102C2×D6111yescharacteristic
382C2×C4×C8—611no—
383#394—1221no—
384#394—1221no—
385#390—611no—
386#390—611no—
387#390C2×D6111yescharacteristic
388#389—611no—
389#389—611no—
390#389—611no—
391#389—1221no—
392#389—1221no—
393#386—1221no—
394#382—611no—
395#382—1221no—
396C4×C4⋊C4—611no—
397C4×C4⋊C4—1221no—
398#340—611no—
399#340—611no—
400#340—611no—
401#340C2×D6111yescharacteristic
402#336—611no—
403#336—611no—
404#336—611no—
405#336C2×D6111yescharacteristic
406C61#2615911yescharacteristic, complemented, Sylow
407C22×Q8C22×D6111yescharacteristic
408C22×D4—611no—
409C22×D4—1221no—
410C22×D4—1221no—
411C22×D4C22×D6111yescharacteristic
412C23×C4—1221no—
413C2×SD16—22no—
414C2×SD16—22no—
415C2×SD16—22no—
416C2×SD16—614no—
417C2×SD16—1222no—
418C2×SD16—1222no—
419C22×C8—611no—
420C22×C8—611no—
421C22×C8C22×D6111yescharacteristic
422C22×C8C22×D6111yescharacteristic
423#127—1222no—
424#127—2441no—
425#123—2444no—
426C4×Q8—22no—
427C4×Q8—22no—
428C4×Q8—1222no—
429C4×Q8—1222no—
430C4×Q8—2441no—
431C4×Q8—2441no—
432C4×D4—1222no—
433C4×D4—2441no—
434C2×C4⋊C4—21no—
435C2×C4⋊C4—611no—
436C2×C4⋊C4—611no—
437C2×C4⋊C4—611no—
438C2×C4⋊C4—611no—
439C2×C4⋊C4—611no—
440C2×C4⋊C4—1221no—
441C2×C4⋊C4—1221no—
442C2×C4⋊C4—1221no—
443C2×C4⋊C4—1221no—
444C2×C4⋊C4—1221no—
445C2×C4⋊C4C22×D6111yescharacteristic
446C2×C4⋊C4C22×D6111yescharacteristic
447C2×C22⋊C4—21no—
448C2×C22⋊C4—21no—
449C2×C22⋊C4—1221no—
450C2×C22⋊C4—1221no—
451C2×C22⋊C4—1221no—
452C2×C22⋊C4—1221no—
453C2×C22⋊C4—1221no—
454C2×C42—21no—
455C2×C42—611no—
456C2×C42—611no—
457C2×C42—1221no—
458C2×C42—1221no—
459C2×C42C22×D6111yescharacteristic
460#107—2441no—
461#106—1222no—
462C4⋊C8—1222no—
463#103—22no—
464#103—1222no—
465#102—22no—
466#102—1222no—
467#102—1222no—
468#102—1222no—
469#102—1222no—
470#102—2441no—
471C4×C8—1222no—
472#95—611no—
473#95—611no—
474#95—1221no—
475#95—1221no—
476#95—1221no—
477#95—1221no—
478#95—1221no—
479C24—21no—
480C24—1221no—
481C2×Q8—21no—
482C2×Q8—21no—
483C2×Q8—21no—
484C2×Q8#1175822212yes—
485C2×Q8D4×D6112yes—
486C2×Q8#1175822412yes—
487C2×D4—21no—
488C2×D4—21no—
489C2×D4—21no—
490C2×D4—21no—
491C2×D4—21no—
492C2×D4—21no—
493C2×D4—612no—
494C2×D4—1221no—
495C2×D4—1221no—
496C2×D4—1221no—
497C2×D4—1221no—
498C2×D4—1221no—
499C2×D4—4881no—
500C2×D4—4881no—
501C2×D4—2442no—
502C2×D4—2442no—
503C2×D4—2442no—
504C2×D4—2442no—
505C22×C4—21no—
506C22×C4—21no—
507C22×C4—21no—
508C22×C4—611no—
509C22×C4—611no—
510C22×C4—611no—
511C22×C4—611no—
512C22×C4—1221no—
513C22×C4—1221no—
514C22×C4—1221no—
515C22×C4—4881no—
516C22×C4—2442no—
517C22×C4—2442no—
518C22×C4D4×D6111yescharacteristic
519C22×C4D4×D6111yescharacteristic
520C22×C4C23×D6111yescharacteristic
521SD16—44no—
522SD16—1228no—
523C2×C8—21no—
524C2×C8—21no—
525C2×C8—21no—
526C2×C8—21no—
527C2×C8—21no—
528C2×C8—21no—
529C2×C8—612no—
530C2×C8—1221no—
531C2×C8—1221no—
532C2×C8—1221no—
533C2×C8—1221no—
534C2×C8—1221no—
535C4⋊C4—24no—
536C4⋊C4—21no—
537C4⋊C4—21no—
538C4⋊C4—21no—
539C4⋊C4—1222no—
540C4⋊C4—1222no—
541C4⋊C4—1221no—
542C4⋊C4—1221no—
543C4⋊C4—1221no—
544C4⋊C4—1221no—
545C4⋊C4—1221no—
546C4⋊C4—1221no—
547C4⋊C4—2442no—
548C4⋊C4—2442no—
549C4⋊C4—2442no—
550C4⋊C4—2442no—
551C4⋊C4—2442no—
552C4⋊C4#1175822412yes—
553C22⋊C4—42no—
554C22⋊C4—81no—
555C22⋊C4—4881no—
556C22⋊C4—4881no—
557C22⋊C4—2442no—
558C22⋊C4—2442no—
559C22⋊C4—2442no—
560C42—24no—
561C42—21no—
562C42—21no—
563C42—1221no—
564C42—1221no—
565C42—1221no—
566C42—1221no—
567C42—2442no—
568C42—2442no—
569C23—41no—
570C23—41no—
571C23—41no—
572C23—41no—
573C23—41no—
574C23—41no—
575C23—41no—
576C23—1222no—
577C23—2441no—
578C23—2441no—
579C23—2441no—
580C23—2441no—
581C23—2441no—
582C23—2441no—
583C23C2×D4×D6111yescharacteristic
584Q8—22no—
585Q8—22no—
586Q8—22no—
587Q8#1451975914yescomplemented
588D4—41no—
589D4—41no—
590D4—41no—
591D4—41no—
592D4—1222no—
593D4—1222no—
594D4—4884no—
595D4—4884no—
596D4—2441no—
597D4—2441no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—22no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—612no—
614C2×C4—612no—
615C2×C4—1221no—
616C2×C4—1221no—
617C2×C4—1221no—
618C2×C4—1221no—
619C2×C4—1221no—
620C2×C4—1221no—
621C2×C4—1221no—
622C2×C4—1221no—
623C2×C4—1221no—
624C2×C4—1221no—
625C2×C4—4884no—
626C2×C4—2441no—
627C2×C4—2441no—
628C2×C4—2441no—
629C2×C4—2441no—
630C2×C4—2441no—
631C2×C4—2441no—
632C2×C4—2441no—
633C2×C4—2441no—
634C2×C4—2441no—
635C2×C4C2×D4×D6111yescharacteristic
636C2×C4C2×#1175822411yescharacteristic
637C2×C4C2×#1175822411yescharacteristic
638C2×C4C2×#1175822211yescharacteristic
639C2×C4#1451982311yescharacteristic
640C2×C4#1451982411yescharacteristic
641C8—41no—
642C8—41no—
643C8—1222no—
644C8—2441no—
645C22—81no—
646C22—81no—
647C22—81no—
648C22—81no—
649C22—81no—
650C22—81no—
651C22—81no—
652C22—4881no—
653C22—4881no—
654C22—4881no—
655C22—4881no—
656C22—2442no—
657C22—2442no—
658C22—2442no—
659C22#1684026811yescharacteristic
660C22#1684039611yescharacteristic
661C22#1684042111yescharacteristic
662C22#135×D6111yescharacteristic
663C22AGL(1,ℤ/8)×D6111yescharacteristic
664C22#1684056811yescharacteristic
665C22#1684057111yescharacteristic
666C4—24no—
667C4—41no—
668C4—42no—
669C4—41no—
670C4—81no—
671C4—1222no—
672C4—1222no—
673C4—4881no—
674C4—4881no—
675C4—4881no—
676C4—2441no—
677C4—2441no—
678C4C2×#1451975911yescharacteristic
679C4#1684035611yescharacteristic
680C4#1684041211yescharacteristic
681C4#1684045811yescharacteristic
682C2—161no—
683C2—4882no—
684C2#2144956011yescharacteristic
685C2#2145587211yescharacteristic
686C2#2145610411yescharacteristic
687C2#2145730911yescharacteristic
688C2#2145752111yescharacteristic
689C2#2145851611yescharacteristic
690C2#2145854911yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1880C2
socleC23×C61C22×D4
prime radicalC2×#14519773C22
hypercenter#1880D61
hypocenterC61#26159
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1880

Complements

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

NH ≅ G/Ntype
2. C2×#16839847C2semidirect
4. #21448740C2semidirect
5. #21447340C2semidirect
6. #21447323C2semidirect
7. #21447234C2semidirect
9. #21446504C2semidirect
10. #21446375C2semidirect
11. #21446176C2semidirect
12. #21446174C2semidirect
13. #21446071C2semidirect
14. #21446067C2semidirect
15. C61×#1880C2semidirect
241. Q8×C61#123semidirect
349. #1880D61semidirect
406. C61#26159semidirect
587. Q8#14519759semidirect

Maximal subgroups

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

subgroupindex
C2×#168399602
C2×#168398472
#214495872
#214487402
#214473402
#214473232
#214472342
#214470522
#214465042
#214463752
#214461762
#214461742
#214460712
#214460672
C61×#18802
#2615961