Finite Group Databasebeta

← #29709201

Subgroups — #29709201

Order 15616.

956 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#11758199C211yesmaximal, characteristic
2C4×#14519766C211yesmaximal, characteristic, complemented
3#21448284C211yesmaximal, characteristic, complemented
4#21447304C211yesmaximal, characteristic, complemented
5#21447235C211yesmaximal, characteristic, complemented
6#21447210C211yesmaximal, characteristic, complemented
7#21447188C211yesmaximal, characteristic, complemented
8#21447132C211yesmaximal, characteristic, complemented
9#21447128C211yesmaximal, characteristic
10#21447057C211yesmaximal, characteristic, complemented
11#21446893C211yesmaximal, characteristic
12C4×#14519706C211yesmaximal, characteristic, complemented
13C4×#14519701C211yesmaximal, characteristic, complemented
14C4×#14519699C211yesmaximal, characteristic, complemented
15C2×C4×SD16×C61C211yesmaximal, characteristic, complemented
16C22×#11758199C2211yescharacteristic
17C2×#14519766C2211yescharacteristic
18C2×C4×D244C2211yescharacteristic
19C2×#14519736—21no—
20C4×#11758199—24no—
21C2×#14519710C2211yescharacteristic
22C2×#14519706C2211yescharacteristic
23C2×#14519706C2211yescharacteristic
24C2×#14519701C2211yescharacteristic
25C2×#14519701C2211yescharacteristic
26C2×#14519699C2211yescharacteristic
27C2×#14519699C2211yescharacteristic
28C2×#14519698C2211yescharacteristic
29C2×#14519697C2211yescharacteristic
30C2×C4×#92418C2211yescharacteristic
31#16839669C2211yescharacteristic
32#16839665C2211yescharacteristic
33#16839665C2211yescharacteristic
34#16839663C2211yescharacteristic
35#16839661C2211yescharacteristic
36#16839661C2211yescharacteristic
37#16839655—21no—
38#16839655—21no—
39#16839652—21no—
40#16839652—21no—
41#16839627C2211yescharacteristic
42C4×C4⋊Dic61C2211yescharacteristic
43C4×#11758198—21no—
44C4×#11758198—21no—
45C4×C22⋊Dic61—21no—
46C42×Dic61C2211yescharacteristic
47#16839525C2211yescharacteristic
48#16839525C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839516C2211yescharacteristic
52#16839516C2211yescharacteristic
53C22×SD16×C61C2211yescharacteristic
54C2×C4×Q8×C61C2211yescharacteristic
55C2×C4×D4×C61C2211yescharacteristic
56C4×SD16×C61—24no—
57C2×#106×C61C2211yescharacteristic
58C2×#103×C61C2211yescharacteristic
59C2×#102×C61C2211yescharacteristic
60C2×C4×C8×C61C2211yescharacteristic
61C22×D244C2311yescharacteristic
62C22×C4×D61—21no—
63C2×#11758199—22no—
64C2×#11758199—22no—
65C2×#11758199—22no—
66C22×#92418C2311yescharacteristic
67C22×#92418C2311yescharacteristic
68#14519766—22no—
69#14519766—22no—
70C4×D244—22no—
71C4×D244—22no—
72#14519736—44no—
73C2×C4⋊Dic61—21no—
74C2×C4⋊Dic61—21no—
75C2×C4⋊Dic61C2311yescharacteristic
76C2×C4⋊Dic61C2311yescharacteristic
77C2×C4⋊Dic61C2311yescharacteristic
78C2×C22⋊Dic61—21no—
79C2×C22⋊Dic61—21no—
80C2×C22⋊Dic61—21no—
81C2×C22⋊Dic61—21no—
82C2×#11758198—21no—
83C2×#11758198—21no—
84C2×#11758198—21no—
85C2×#11758198—21no—
86C2×#11758198—21no—
87C2×C4×Dic61—21no—
88C2×C4×Dic61D411yescharacteristic
89C2×C4×Dic61D411yescharacteristic
90C2×C4×Dic61D411yescharacteristic
91C2×C4×Dic61D411yescharacteristic
92C2×C4×Dic61C2311yescharacteristic
93C2×C4×Dic61C2311yescharacteristic
94#14519710—22no—
95#14519706—22no—
96#14519706—22no—
97#14519701—22no—
98#14519701—22no—
99#14519699—22no—
100#14519699—22no—
101#14519698—22no—
102#14519697—22no—
103C4×#92418—22no—
104#14519684—21no—
105#14519684—21no—
106#14519684—21no—
107#14519684—21no—
108#14519684—21no—
109#14519684—21no—
110#14519684—21no—
111C22×Q8×C61C2311yescharacteristic
112C22×D4×C61C2311yescharacteristic
113C23×C4×C61—21no—
114C2×SD16×C61—22no—
115C2×SD16×C61—22no—
116C2×SD16×C61—22no—
117C22×C8×C61C2311yescharacteristic
118C22×C8×C61C2311yescharacteristic
119C4×Q8×C61—22no—
120C4×Q8×C61—22no—
121C4×D4×C61—22no—
122C4×D4×C61D414yes—
123C2×C4⋊C4×C61—21no—
124C2×C4⋊C4×C61C2311yescharacteristic
125C2×C4⋊C4×C61C2311yescharacteristic
126C2×C22⋊C4×C61—21no—
127C2×C42×C61—21no—
128C2×C42×C61C2311yescharacteristic
129#106×C61—22no—
130#103×C61—22no—
131#102×C61—22no—
132C4×C8×C61—22no—
133C24×C61—21no—
134C23×D61—21no—
135C2×Q8×C61—21no—
136C2×Q8×C61—21no—
137C2×Q8×C61—21no—
138C2×Q8×C61—21no—
139C2×Q8×C61—21no—
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×C61C2×D412yes—
147C22×C4×C61—21no—
148C22×C4×C61—21no—
149C22×C4×C61—24no—
150C22×C4×C61—21no—
151C22×C4×C61—42no—
152C22×C4×C61—42no—
153C22×C4×C61C2×D411yescharacteristic
154C22×C4×C61C2×D411yescharacteristic
155C22×C4×C61C2411yescharacteristic
156C22×Dic61—21no—
157C22×Dic61—21no—
158C22×Dic61—21no—
159C22×Dic61C2×D411yescharacteristic
160C22×Dic61C2×D411yescharacteristic
161C22×Dic61C2×D411yescharacteristic
162C22×Dic61C2×D411yescharacteristic
163C2×D244—21no—
164C2×D244—21no—
165C2×D244—21no—
166C2×D244—21no—
167C2×D244—21no—
168C2×D244#4112yes—
169C2×C4×D61—42no—
170C2×C4×D61—42no—
171C2×C4×D61—42no—
172SD16×C61—28no—
173C2×C8×C61—21no—
174C2×C8×C61—21no—
175C2×C8×C61—21no—
176C2×C8×C61—21no—
177C2×C8×C61—21no—
178C2×C8×C61#4112yes—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61—42no—
183C4⋊C4×C61C2×D412yes—
184C22⋊C4×C61—24no—
185C42×C61—42no—
186C42×C61C2×D412yes—
187C42×C61#4112yes—
188C22⋊Dic61—42no—
189C22⋊Dic61—42no—
190C22⋊Dic61—42no—
191C22⋊Dic61—42no—
192#11758199—28no—
193#11758198—24no—
194#11758198—24no—
195#11758198—42no—
196#11758198—42no—
197#11758198—42no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61—21no—
203C4⋊Dic61—21no—
204C4⋊Dic61—42no—
205C4⋊Dic61—42no—
206C4×Dic61—22no—
207C4×Dic61—22no—
208C4×Dic61—21no—
209C4×Dic61—21no—
210C4×Dic61—21no—
211C4×Dic61—21no—
212C4×Dic61—22no—
213C4×Dic61—41no—
214C4×Dic61—42no—
215C2×#92418—21no—
216C2×#92418—21no—
217C2×#92418—21no—
218C2×#92418—21no—
219C2×#92418—21no—
220C2×#92418#4112yes—
221C23×C61—22no—
222C23×C61—41no—
223C23×C61—41no—
224C23×C61—41no—
225C23×C61—41no—
226C23×C61—41no—
227C23×C61—41no—
228C23×C61C22×D411yescharacteristic
229C22×D61—22no—
230C22×D61—41no—
231C22×D61—41no—
232C22×D61—41no—
233C22×D61—41no—
234C22×D61—41no—
235C22×D61—41no—
236Q8×C61—22no—
237Q8×C61—22no—
238Q8×C61—41no—
239Q8×C61—41no—
240D4×C61—22no—
241D4×C61—22no—
242D4×C61—41no—
243D4×C61—41no—
244C2×C4×C61—21no—
245C2×C4×C61—21no—
246C2×C4×C61—22no—
247C2×C4×C61—22no—
248C2×C4×C61—41no—
249C2×C4×C61—41no—
250C2×C4×C61—41no—
251C2×C4×C61—41no—
252C2×C4×C61—48no—
253C2×C4×C61—41no—
254C2×C4×C61—41no—
255C2×C4×C61—41no—
256C2×C4×C61#12112yes—
257C2×C4×C61#12112yes—
258C2×C4×C61#13512yes—
259C2×C4×C61AGL(1,ℤ/8)12yes—
260C2×C4×C61C22×D411yescharacteristic
261C2×C4×C61C2×#4111yescharacteristic
262C2×C4×C61C2×#4111yescharacteristic
263C2×C4×C61C2×#4111yescharacteristic
264C2×C4×C61C2×#4111yescharacteristic
265C2×C4×C61C2×#4111yescharacteristic
266C2×Dic61—21no—
267C2×Dic61—21no—
268C2×Dic61—21no—
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—41no—
278C2×Dic61—41no—
279C2×Dic61—41no—
280C2×Dic61—41no—
281C2×Dic61—41no—
282C2×Dic61—41no—
283C2×Dic61—41no—
284C2×Dic61—41no—
285C2×Dic61—41no—
286C2×Dic61#13512yes—
287D244—22no—
288D244—22no—
289D244—41no—
290D244—41no—
291C4×D61—84no—
292C8×C61—22no—
293C8×C61—41no—
294#92418—22no—
295#92418—41no—
296#23680—611nomaximal, Sylow
297C22×C61—42no—
298C22×C61—42no—
299C22×C61—42no—
300C22×C61—81no—
301C22×C61—81no—
302C22×C61—81no—
303C22×C61—81no—
304C22×C61C2×#12111yescharacteristic
305C22×C61#52511yescharacteristic
306C22×C61#53211yescharacteristic
307C22×C61C2×#13511yescharacteristic
308C22×C61C2×#13511yescharacteristic
309C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
310C22×C61#57511yescharacteristic
311C2×D61—42no—
312C2×D61—42no—
313C2×D61—42no—
314C2×D61—81no—
315C2×D61—81no—
316C2×D61—81no—
317C2×D61—81no—
318C4×C61—22no—
319C4×C61—42no—
320C4×C61—42no—
321C4×C61—81no—
322C4×C61#46314yes—
323C4×C61#52511yescharacteristic
324C4×C61#52511yescharacteristic
325C4×C61#52911yescharacteristic
326C4×C61#52911yescharacteristic
327Dic61—22no—
328Dic61—41no—
329Dic61—41no—
330Dic61—41no—
331Dic61—81no—
332Dic61—81no—
333Dic61—81no—
334C2×C4×SD16D6111yescharacteristic, complemented
335C2×C4×D8—611no—
336C4×#124—611no—
337#2293—611no—
338#2085—611no—
339#2073—611no—
340#2068—611no—
341#2065—611no—
342#2055—611no—
343#2054—611no—
344#2042—611no—
345#2012—611no—
346C4×C4⋊C8—611no—
347C4×#103—611no—
348C4×#102—611no—
349C2×C61—82no—
350C2×C61#239211yescharacteristic
351C2×C61C2×#46311yescharacteristic
352C2×C61#303511yescharacteristic
353C2×C61#310211yescharacteristic
354C2×C61#310611yescharacteristic
355C2×C61#316811yescharacteristic
356C2×C61#317111yescharacteristic
357D61—82no—
358C22×SD16C2×D6111yescharacteristic
359C22×D8—611no—
360C2×#124—611no—
361C2×C4×Q8C2×D6111yescharacteristic
362C2×C4×D4—611no—
363C2×C4×D4C2×D6111yescharacteristic
364C2×#117—1221no—
365C4×SD16—24no—
366C4×D8—1224no—
367C2×#107—611no—
368C2×#106C2×D6111yescharacteristic
369C2×C4⋊C8—611no—
370C2×C4⋊C8—611no—
371C2×#103—611no—
372C2×#103—611no—
373C2×#103C2×D6111yescharacteristic
374C2×#102—611no—
375C2×#102—611no—
376C2×#102—611no—
377C2×#102—611no—
378C2×#102C2×D6111yescharacteristic
379C2×C4×C8—611no—
380C2×C4×C8C2×D6111yescharacteristic
381#391—611no—
382#390—611no—
383#390—611no—
384#390—611no—
385#389—611no—
386#389—611no—
387#388—1221no—
388#388—1221no—
389#388—1221no—
390#388—1221no—
391#382—611no—
392C4×C4⋊C4—611no—
393C4×C22⋊C4—1221no—
394C4×C22⋊C4—1221no—
395C4×C22⋊C4—1221no—
396C43—611no—
397#340—611no—
398#340—611no—
399#340—611no—
400#340—611no—
401#336—611no—
402#336—611no—
403C61#2368011yescharacteristic, complemented, Sylow
404C22×Q8C22×D6111yescharacteristic
405C22×D4—611no—
406C22×D4C22×D6111yescharacteristic
407C23×C4—21no—
408C23×C4—1221no—
409C2×SD16—22no—
410C2×SD16—22no—
411C2×SD16—22no—
412C2×D8—1222no—
413C2×D8—1222no—
414C2×D8—1222no—
415C22×C8—611no—
416C22×C8—611no—
417C22×C8C22×D6111yescharacteristic
418C22×C8C22×D6111yescharacteristic
419#124—1222no—
420#124—1222no—
421C4×Q8—22no—
422C4×Q8—22no—
423C4×D4—22no—
424C4×D4—1222no—
425C4×D4—1222no—
426C4×D4C22⋊D6114yes—
427#117—2444no—
428C2×C4⋊C4—21no—
429C2×C4⋊C4—611no—
430C2×C4⋊C4—611no—
431C2×C4⋊C4—611no—
432C2×C4⋊C4—1221no—
433C2×C4⋊C4—1221no—
434C2×C4⋊C4C22×D6111yescharacteristic
435C2×C4⋊C4C22×D6111yescharacteristic
436C2×C22⋊C4—21no—
437C2×C22⋊C4—1221no—
438C2×C22⋊C4—1221no—
439C2×C22⋊C4—1221no—
440C2×C22⋊C4—1221no—
441C2×C22⋊C4—1221no—
442C2×C22⋊C4—1221no—
443C2×C22⋊C4—1221no—
444C2×C22⋊C4—1221no—
445C2×C22⋊C4—1221no—
446C2×C42—21no—
447C2×C42—611no—
448C2×C42—611no—
449C2×C42—611no—
450C2×C42—611no—
451C2×C42—611no—
452C2×C42—611no—
453C2×C42—1221no—
454C2×C42C22×D6111yescharacteristic
455#107—1222no—
456#106—22no—
457C4⋊C8—1222no—
458C4⋊C8—1222no—
459#103—22no—
460#103—1222no—
461#103—1222no—
462#102—22no—
463#102—1222no—
464#102—1222no—
465#102—1222no—
466#102—1222no—
467C4×C8—22no—
468C4×C8—1222no—
469#95—1221no—
470#95—1221no—
471#95—1221no—
472#95—1221no—
473#95—1221no—
474#95—1221no—
475#95—1221no—
476C24—21no—
477C24—1221no—
478C2×Q8—21no—
479C2×Q8—21no—
480C2×Q8—21no—
481C2×Q8—21no—
482C2×Q8—21no—
483C2×Q8#1175822212yes—
484C2×D4—21no—
485C2×D4—21no—
486C2×D4—21no—
487C2×D4—21no—
488C2×D4—21no—
489C2×D4—612no—
490C2×D4—1221no—
491C2×D4—1221no—
492C2×D4—1221no—
493C2×D4—1221no—
494C2×D4—1221no—
495C2×D4C2×C22⋊D6112yes—
496C22×C4—21no—
497C22×C4—21no—
498C22×C4—24no—
499C22×C4—21no—
500C22×C4—42no—
501C22×C4—42no—
502C22×C4—611no—
503C22×C4—611no—
504C22×C4—611no—
505C22×C4—611no—
506C22×C4—1221no—
507C22×C4—1221no—
508C22×C4—1221no—
509C22×C4—2442no—
510C22×C4—2442no—
511C22×C4—2442no—
512C22×C4D4×D6111yescharacteristic
513C22×C4D4×D6111yescharacteristic
514C22×C4C23×D6111yescharacteristic
515SD16—28no—
516D8—1228no—
517C2×C8—21no—
518C2×C8—21no—
519C2×C8—21no—
520C2×C8—21no—
521C2×C8—21no—
522C2×C8—612no—
523C2×C8—1221no—
524C2×C8—1221no—
525C2×C8—1221no—
526C2×C8—1221no—
527C2×C8—1221no—
528C2×C8#1175822212yes—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—21no—
532C4⋊C4—42no—
533C4⋊C4—1221no—
534C4⋊C4—1221no—
535C4⋊C4—1221no—
536C4⋊C4—1221no—
537C4⋊C4—1221no—
538C4⋊C4—1221no—
539C4⋊C4—2442no—
540C4⋊C4—2442no—
541C4⋊C4C2×C22⋊D6112yes—
542C22⋊C4—24no—
543C22⋊C4—1224no—
544C22⋊C4—1224no—
545C22⋊C4—2442no—
546C22⋊C4—2442no—
547C22⋊C4—2442no—
548C22⋊C4—2442no—
549C22⋊C4—2442no—
550C22⋊C4—2442no—
551C22⋊C4—2442no—
552C42—42no—
553C42—1222no—
554C42—1221no—
555C42—1221no—
556C42—1221no—
557C42—1221no—
558C42—1222no—
559C42—1222no—
560C42—2441no—
561C42—2442no—
562C42#1175822412yes—
563C42C2×C22⋊D6112yes—
564C23—22no—
565C23—41no—
566C23—41no—
567C23—41no—
568C23—41no—
569C23—41no—
570C23—41no—
571C23—1222no—
572C23—2441no—
573C23—2441no—
574C23—2441no—
575C23—2441no—
576C23—2441no—
577C23—2441no—
578C23C2×D4×D6111yescharacteristic
579Q8—22no—
580Q8—22no—
581Q8—41no—
582Q8—41no—
583D4—22no—
584D4—22no—
585D4—41no—
586D4—41no—
587D4—1222no—
588D4—1222no—
589D4—2441no—
590D4—2441no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—48no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—612no—
604C2×C4—1221no—
605C2×C4—1221no—
606C2×C4—1221no—
607C2×C4—1221no—
608C2×C4—1221no—
609C2×C4—1221no—
610C2×C4—1221no—
611C2×C4—1221no—
612C2×C4—1221no—
613C2×C4—1221no—
614C2×C4—1221no—
615C2×C4—4884no—
616C2×C4—2441no—
617C2×C4—2441no—
618C2×C4—2441no—
619C2×C4—2441no—
620C2×C4—2441no—
621C2×C4—2441no—
622C2×C4—2441no—
623C2×C4—2441no—
624C2×C4—2441no—
625C2×C4#1451974912yes—
626C2×C4#1451974912yes—
627C2×C4#1451979812yes—
628C2×C4#1451980412yes—
629C2×C4C22×C22⋊D6111yescharacteristic
630C2×C4#41×D6111yescharacteristic
631C2×C4#41×D6111yescharacteristic
632C2×C4C2×#1175822411yescharacteristic
633C2×C4C2×#1175822211yescharacteristic
634C2×C4C2×#1175822211yescharacteristic
635C8—22no—
636C8—41no—
637C8—1222no—
638C8—2441no—
639C22—42no—
640C22—42no—
641C22—42no—
642C22—81no—
643C22—81no—
644C22—81no—
645C22—81no—
646C22—4881no—
647C22—4881no—
648C22—4881no—
649C22—4881no—
650C22—2442no—
651C22—2442no—
652C22—2442no—
653C22C2×#1451974911yescharacteristic
654C22#1684027111yescharacteristic
655C22#1684030911yescharacteristic
656C22#135×D6111yescharacteristic
657C22C2×#1451979811yescharacteristic
658C22C2×#1451980411yescharacteristic
659C22#1684055911yescharacteristic
660C4—22no—
661C4—42no—
662C4—42no—
663C4—81no—
664C4—1222no—
665C4—4881no—
666C4—4881no—
667C4—4881no—
668C4—2441no—
669C4—2441no—
670C4—2441no—
671C4#1683997914yes—
672C4#1684026911yescharacteristic
673C4#1684027311yescharacteristic
674C4#1684029211yescharacteristic
675C4#1684029411yescharacteristic
676C2—82no—
677C2—4882no—
678C2#2144906111yescharacteristic
679C2C2×#1683997911yescharacteristic
680C2#2145527011yescharacteristic
681C2#2145583511yescharacteristic
682C2#2145587711yescharacteristic
683C2#2145638511yescharacteristic
684C2#2145641011yescharacteristic
subgroupG/H
centerC22×C4D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C4×SD16×C61C2
socleC23×C61C22×D4
prime radicalC22×#11758199C22
hypercenterC2×C4×SD16D61
hypocenterC61#23680
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22×C4 ◁ C2×C42 ◁ C2×C4×SD16

Complements

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

NH ≅ G/Ntype
2. C4×#14519766C2semidirect
3. #21448284C2semidirect
4. #21447304C2semidirect
5. #21447235C2semidirect
6. #21447210C2semidirect
7. #21447188C2semidirect
8. #21447132C2semidirect
10. #21447057C2semidirect
12. C4×#14519706C2semidirect
13. C4×#14519701C2semidirect
14. C4×#14519699C2semidirect
15. C2×C4×SD16×C61C2semidirect
334. C2×C4×SD16D61semidirect
403. C61#23680semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#117581992
C4×#145197662
#214482842
#214473042
#214472352
#214472102
#214471882
#214471322
#214471282
#214470572
#214468932
C4×#145197062
C4×#145197012
C4×#145196992
C2×C4×SD16×C612
#2368061