Finite Group Databasebeta

← #29736073

Subgroups — #29736073

Order 15616.

1224 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D4×Dic122C211yesmaximal, characteristic
2C2×#16840170C211yesmaximal, characteristic, complemented
3C2×#16840064C211yesmaximal, characteristic
4#21451204C211yesmaximal, characteristic, complemented
5#21447153C211yesmaximal, characteristic, complemented
6#21447151C211yesmaximal, characteristic, complemented
7#21447147C211yesmaximal, characteristic, complemented
8#21447145C211yesmaximal, characteristic, complemented
9#21446765C211yesmaximal, characteristic, complemented
10#21446757C211yesmaximal, characteristic, complemented
11#21446082C211yesmaximal, characteristic
12#21446080C211yesmaximal, characteristic
13#21446064C211yesmaximal, characteristic
14#21445646C211yesmaximal, characteristic
15C61×#1862C211yesmaximal, characteristic
16C23×Dic122—21no—
17D4×Dic122—24no—
18D4×Dic122—24no—
19C2×#14519777C2211yescharacteristic
20C2×#14519776C2211yescharacteristic
21C2×#14519776C2211yescharacteristic
22C2×#14519768C2211yescharacteristic
23C2×#14519756—21no—
24C2×#14519754—21no—
25C2×#14519754—21no—
26C2×C4×Dic122C2211yescharacteristic
27C2×D4×Dic61C2211yescharacteristic
28C2×D4×Dic61C2211yescharacteristic
29#16840170—24no—
30#16840064—24no—
31C2×#14519710C2211yescharacteristic
32C2×#14519710C2211yescharacteristic
33C2×#14519709C2211yescharacteristic
34C2×#14519709C2211yescharacteristic
35C2×#14519708C2211yescharacteristic
36C2×#14519706C2211yescharacteristic
37C2×#14519698C2211yescharacteristic
38C2×#14519698C2211yescharacteristic
39C2×#14519687C2211yescharacteristic
40#16839696—21no—
41#16839682—21no—
42#16839661—21no—
43#16839661C2211yescharacteristic
44#16839660C2211yescharacteristic
45#16839659—21no—
46#16839658C2211yescharacteristic
47C4⋊C4×Dic61C2211yescharacteristic
48#16839525C2211yescharacteristic
49#16839525C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839524C2211yescharacteristic
53#16839517C2211yescharacteristic
54#16839516C2211yescharacteristic
55#16839516C2211yescharacteristic
56C2×C4×D4×C61C2211yescharacteristic
57C2×#107×C61C2211yescharacteristic
58C2×#102×C61C2211yescharacteristic
59C2×#102×C61C2211yescharacteristic
60C61×#389C2211yescharacteristic
61C61×#340C2211yescharacteristic
62C61×#336C2211yescharacteristic
63C22×Dic122—28no—
64C22×Dic122—21no—
65C22×Dic122—21no—
66C22×Dic122—21no—
67C22×Dic122—44no—
68C22×Dic122—44no—
69C22×Dic122D411yescharacteristic
70C23×Dic61—21no—
71C23×Dic61—21no—
72C22×#92418C2311yescharacteristic
73C22×#92418C2311yescharacteristic
74#14519777—22no—
75#14519777—22no—
76#14519776—22no—
77#14519776—22no—
78#14519776—22no—
79#14519776D414yes—
80#14519768—22no—
81#14519768—22no—
82#14519756—28no—
83#14519754—28no—
84#14519754—28no—
85C4×Dic122—22no—
86C4×Dic122—22no—
87D4×Dic61—22no—
88D4×Dic61—22no—
89D4×Dic61—22no—
90D4×Dic61D414yes—
91C2×C4⋊Dic61—21no—
92C2×C4⋊Dic61—21no—
93C2×C4⋊Dic61—21no—
94C2×C4⋊Dic61D411yescharacteristic
95C2×C4⋊Dic61C2311yescharacteristic
96C2×C4⋊Dic61C2311yescharacteristic
97C2×C4⋊Dic61C2311yescharacteristic
98C2×C4⋊Dic61C2311yescharacteristic
99C2×#11758196—21no—
100C2×#11758196—21no—
101C2×#11758196—21no—
102C2×#11758196—21no—
103C2×#11758196—21no—
104C2×#11758196D411yescharacteristic
105C2×#11758196D411yescharacteristic
106C2×C22⋊Dic61—21no—
107C2×C22⋊Dic61—21no—
108C2×C4×Dic61—21no—
109C2×C4×Dic61C2311yescharacteristic
110C2×C4×Dic61C2311yescharacteristic
111#14519710—22no—
112#14519710—22no—
113#14519709—22no—
114#14519709—22no—
115#14519708—22no—
116#14519706—22no—
117#14519698—22no—
118#14519698Q814yescomplemented
119#14519687—22no—
120#14519684—21no—
121#14519684—21no—
122#14519684—21no—
123#14519684—21no—
124C22×D4×C61C2311yescharacteristic
125C23×C4×C61—21no—
126C22×C8×C61C2311yescharacteristic
127C22×C8×C61C2311yescharacteristic
128C4×D4×C61—22no—
129C4×D4×C61—22no—
130C2×C4⋊C4×C61—21no—
131C2×C4⋊C4×C61C2311yescharacteristic
132C2×C4⋊C4×C61C2311yescharacteristic
133C2×C4⋊C4×C61C2311yescharacteristic
134C2×C22⋊C4×C61—21no—
135C2×C42×C61C2311yescharacteristic
136#107×C61—22no—
137#102×C61—22no—
138#102×C61Q814yescomplemented
139#95×C61—21no—
140C24×C61—21no—
141C2×D4×C61—21no—
142C2×D4×C61—21no—
143C2×D4×C61—21no—
144C2×D4×C61C2×D412yes—
145C2×D4×C61C2×Q812yes—
146C2×D4×C61C2×Q812yes—
147C22×C4×C61—24no—
148C22×C4×C61—21no—
149C22×C4×C61—21no—
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—24no—
157C22×Dic61—21no—
158C22×Dic61—21no—
159C22×Dic61—21no—
160C22×Dic61—24no—
161C22×Dic61—21no—
162C22×Dic61—42no—
163C22×Dic61—42no—
164C22×Dic61—42no—
165C22×Dic61—42no—
166C22×Dic61C2×D411yescharacteristic
167C22×Dic61C2×D411yescharacteristic
168C22×Dic61C2×D411yescharacteristic
169C22×Dic61C2×D411yescharacteristic
170C2×Dic122—22no—
171C2×Dic122—22no—
172C2×Dic122—22no—
173C2×Dic122—24no—
174C2×Dic122—24no—
175C2×Dic122—24no—
176C2×Dic122—432no—
177C2×Dic122—42no—
178C2×Dic122—42no—
179C2×Dic122—42no—
180C2×Dic122—42no—
181C2×Dic122—42no—
182C2×Dic122—42no—
183C2×C8×C61—21no—
184C2×C8×C61—21no—
185C2×C8×C61—21no—
186C2×C8×C61—21no—
187C2×C8×C61—21no—
188C2×C8×C61C2×Q812yescomplemented
189C4⋊C4×C61—21no—
190C4⋊C4×C61—21no—
191C4⋊C4×C61—21no—
192C4⋊C4×C61—21no—
193C4⋊C4×C61—42no—
194C4⋊C4×C61C2×D412yes—
195C4⋊C4×C61C2×Q812yescomplemented
196C22⋊C4×C61—24no—
197C42×C61—21no—
198C42×C61—21no—
199C22⋊Dic61—24no—
200C22⋊Dic61—24no—
201C4⋊Dic61—24no—
202C4⋊Dic61—21no—
203C4⋊Dic61—21no—
204C4⋊Dic61—21no—
205C4⋊Dic61—22no—
206C4⋊Dic61—21no—
207C4⋊Dic61—21no—
208C4⋊Dic61—42no—
209C4⋊Dic61—81no—
210C4⋊Dic61C2×D412yes—
211C4⋊Dic61C2×D412yes—
212C4⋊Dic61C2×Q812yescomplemented
213#11758196—24no—
214#11758196—24no—
215#11758196—22no—
216#11758196—22no—
217#11758196—42no—
218#11758196—42no—
219#11758196—81no—
220C4×Dic61—21no—
221C4×Dic61—21no—
222C4×Dic61—81no—
223C4×Dic61C2×D412yes—
224C4×Dic61C2×D412yes—
225C2×#92418—21no—
226C2×#92418—21no—
227C2×#92418—21no—
228C2×#92418—21no—
229C2×#92418—21no—
230C2×#92418C2×Q812yescomplemented
231C23×C61—22no—
232C23×C61—22no—
233C23×C61—22no—
234C23×C61—41no—
235C23×C61—41no—
236C23×C61—41no—
237C23×C61—41no—
238C23×C61C22×D411yescharacteristic
239D4×C61—22no—
240D4×C61—22no—
241D4×C61—22no—
242D4×C61C4⋊Q814yes—
243C2×C4×C61—21no—
244C2×C4×C61—21no—
245C2×C4×C61—21no—
246C2×C4×C61—22no—
247C2×C4×C61—22no—
248C2×C4×C61—21no—
249C2×C4×C61—21no—
250C2×C4×C61—48no—
251C2×C4×C61—41no—
252C2×C4×C61—41no—
253C2×C4×C61—41no—
254C2×C4×C61—41no—
255C2×C4×C61—41no—
256C2×C4×C61—41no—
257C2×C4×C61—41no—
258C2×C4×C61AGL(1,ℤ/8)12yes—
259C2×C4×C61C22×D411yescharacteristic
260C2×C4×C61C22×D411yescharacteristic
261C2×C4×C61C22×Q811yescharacteristic
262C2×C4×C61C22×Q811yescharacteristic
263C2×C4×C612−1+411yescharacteristic
264C2×C4×C612−1+411yescharacteristic
265C2×Dic61—21no—
266C2×Dic61—21no—
267C2×Dic61—21no—
268C2×Dic61—21no—
269C2×Dic61—21no—
270C2×Dic61—21no—
271C2×Dic61—21no—
272C2×Dic61—22no—
273C2×Dic61—22no—
274C2×Dic61—22no—
275C2×Dic61—21no—
276C2×Dic61—21no—
277C2×Dic61—48no—
278C2×Dic61—48no—
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#12012yes—
289C2×Dic61#12012yes—
290C2×Dic61AGL(1,ℤ/8)12yes—
291Dic122—28no—
292Dic122—48no—
293Dic122—48no—
294Dic122—48no—
295C8×C61—22no—
296C8×C61—41no—
297#92418—22no—
298#92418—41no—
299#25844—611nomaximal, Sylow
300C22×C61—24no—
301C22×C61—42no—
302C22×C61—42no—
303C22×C61—42no—
304C22×C61—42no—
305C22×C61—42no—
306C22×C61—42no—
307C22×C61C2×#12011yescharacteristic
308C22×C61D4×Q811yescharacteristic
309C22×C61D4×Q811yescharacteristic
310C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
311C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
312C22×C61#57711yescharacteristic
313C22×C61#57811yescharacteristic
314C4×C61—22no—
315C4×C61—42no—
316C4×C61—42no—
317C4×C61—41no—
318C4×C61—81no—
319C4×C61C2×C4⋊Q811yescharacteristic
320C4×C61#53611yescharacteristic
321C4×C61D4×Q811yescharacteristic
322C4×C61D4×Q811yescharacteristic
323Dic61—22no—
324Dic61—22no—
325Dic61—22no—
326Dic61—42no—
327Dic61—42no—
328Dic61—42no—
329Dic61—41no—
330Dic61—81no—
331C2×D4×Q8—611no—
332C2×#501—611no—
333C2×#478—611no—
334#2584—611no—
335#2058—611no—
336#2058—611no—
337#2057—611no—
338#2057—611no—
339#1987—611no—
340#1985—611no—
341#1864—611no—
342#1864—611no—
343#1862—611no—
344#1862D6111yescharacteristic
345#1762—611no—
346C2×C61—44no—
347C2×C61#243411yescharacteristic
348C2×C61#300311yescharacteristic
349C2×C61#301311yescharacteristic
350C2×C61#338411yescharacteristic
351C2×C61#338411yescharacteristic
352C2×C61#338611yescharacteristic
353C2×C61#338611yescharacteristic
354C23×Q8—1221no—
355D4×Q8—1224no—
356D4×Q8—1224no—
357C2×C4⋊Q8—611no—
358C2×C4⋊Q8—611no—
359C2×C4⋊Q8—611no—
360C2×#125—611no—
361C2×C22⋊Q8—1221no—
362C2×C22⋊Q8—1221no—
363C2×C22⋊Q8—1221no—
364C2×C4×Q8—611no—
365C2×C4×D4—611no—
366C2×C4×D4—611no—
367C2×C4×D4C2×D6111yescharacteristic
368#501—1224no—
369#478—1224no—
370C2×#107—611no—
371C2×#107—611no—
372C2×#107C2×D6111yescharacteristic
373C2×#106—611no—
374C2×#106—611no—
375C2×#106—611no—
376C2×C4⋊C8—611no—
377C2×#102—611no—
378C2×#102—611no—
379C2×#102C2×D6111yescharacteristic
380C2×#102C2×D6111yescharacteristic
381C2×#97—611no—
382#398—1221no—
383#395—1221no—
384#389—611no—
385#389—611no—
386#389—611no—
387#389—1221no—
388#389—1221no—
389#389C2×D6111yescharacteristic
390C4×C4⋊C4—611no—
391#340—611no—
392#340—611no—
393#340—611no—
394#340—611no—
395#340—611no—
396#340C2×D6111yescharacteristic
397#336—611no—
398#336—611no—
399#336—611no—
400#336C2×D6111yescharacteristic
401C61#2584411yescharacteristic, complemented, Sylow
402C22×Q8—611no—
403C22×Q8—1221no—
404C22×Q8—1221no—
405C22×Q8—1228no—
406C22×Q8—1221no—
407C22×Q8—2444no—
408C22×Q8—2444no—
409C22×D4C22×D6111yescharacteristic
410C23×C4—21no—
411C23×C4—1221no—
412C23×C4—1221no—
413C22×C8—611no—
414C22×C8—611no—
415C22×C8C22×D6111yescharacteristic
416C22×C8C22×D6111yescharacteristic
417C4⋊Q8—614no—
418C4⋊Q8—1222no—
419C4⋊Q8—1222no—
420C4⋊Q8—1222no—
421C4⋊Q8—1222no—
422C4⋊Q8—1222no—
423#125—1222no—
424#125—1222no—
425C22⋊Q8—1228no—
426C22⋊Q8—1228no—
427C22⋊Q8—1228no—
428C4×Q8—1222no—
429C4×Q8—1222no—
430C4×D4—22no—
431C4×D4—22no—
432C4×D4—614no—
433C4×D4—1222no—
434C4×D4—1222no—
435C4×D4—1222no—
436C2×C4⋊C4—21no—
437C2×C4⋊C4—611no—
438C2×C4⋊C4—611no—
439C2×C4⋊C4—611no—
440C2×C4⋊C4—611no—
441C2×C4⋊C4—611no—
442C2×C4⋊C4—611no—
443C2×C4⋊C4—611no—
444C2×C4⋊C4—1221no—
445C2×C4⋊C4—1221no—
446C2×C4⋊C4—1221no—
447C2×C4⋊C4—1221no—
448C2×C4⋊C4—1221no—
449C2×C4⋊C4—1221no—
450C2×C4⋊C4—1221no—
451C2×C4⋊C4—1221no—
452C2×C4⋊C4C22×D6111yescharacteristic
453C2×C4⋊C4C22×D6111yescharacteristic
454C2×C4⋊C4C22×D6111yescharacteristic
455C2×C22⋊C4—21no—
456C2×C22⋊C4—1221no—
457C2×C22⋊C4—1221no—
458C2×C42—611no—
459C2×C42—611no—
460C2×C42—1221no—
461C2×C42C22×D6111yescharacteristic
462#107—22no—
463#107—1222no—
464#107—1222no—
465#106—1222no—
466#106—1222no—
467#106—1222no—
468C4⋊C8—1222no—
469#102—22no—
470#102—614no—
471#102—1222no—
472#102Dic12214yescomplemented
473#97—1222no—
474#95—21no—
475#95—1221no—
476#95—1221no—
477#95—1221no—
478#95—1221no—
479C24—21no—
480C2×Q8—1222no—
481C2×Q8—1222no—
482C2×Q8—1224no—
483C2×Q8—1224no—
484C2×Q8—1224no—
485C2×Q8—1222no—
486C2×Q8—2442no—
487C2×Q8—2442no—
488C2×Q8—2442no—
489C2×Q8—2442no—
490C2×Q8—2442no—
491C2×Q8—24432no—
492C2×Q8—2442no—
493C2×D4—21no—
494C2×D4—21no—
495C2×D4—21no—
496C2×D4C2×Dic12212yes—
497C2×D4D4×D6112yes—
498C2×D4Q8×D6112yes—
499C22×C4—24no—
500C22×C4—21no—
501C22×C4—21no—
502C22×C4—21no—
503C22×C4—42no—
504C22×C4—42no—
505C22×C4—611no—
506C22×C4—611no—
507C22×C4—611no—
508C22×C4—611no—
509C22×C4—1224no—
510C22×C4—1224no—
511C22×C4—1221no—
512C22×C4—1221no—
513C22×C4—1221no—
514C22×C4—1221no—
515C22×C4—2442no—
516C22×C4—2442no—
517C22×C4—2442no—
518C22×C4—2442no—
519C22×C4D4×D6111yescharacteristic
520C22×C4D4×D6111yescharacteristic
521C22×C4C23×D6111yescharacteristic
522C2×C8—21no—
523C2×C8—21no—
524C2×C8—21no—
525C2×C8—21no—
526C2×C8—21no—
527C2×C8—612no—
528C2×C8—1221no—
529C2×C8—1221no—
530C2×C8—1221no—
531C2×C8—1221no—
532C2×C8—1221no—
533C2×C8C2×Dic12212yescomplemented
534C4⋊C4—21no—
535C4⋊C4—21no—
536C4⋊C4—21no—
537C4⋊C4—21no—
538C4⋊C4—42no—
539C4⋊C4—612no—
540C4⋊C4—612no—
541C4⋊C4—612no—
542C4⋊C4—1222no—
543C4⋊C4—1222no—
544C4⋊C4—1222no—
545C4⋊C4—1224no—
546C4⋊C4—1224no—
547C4⋊C4—1224no—
548C4⋊C4—1221no—
549C4⋊C4—1221no—
550C4⋊C4—1221no—
551C4⋊C4—1221no—
552C4⋊C4—1221no—
553C4⋊C4—4881no—
554C4⋊C4—4881no—
555C4⋊C4—2442no—
556C4⋊C4—2442no—
557C4⋊C4—2442no—
558C4⋊C4C2×Dic12212yescomplemented
559C4⋊C4D4×D6112yes—
560C22⋊C4—24no—
561C22⋊C4—1224no—
562C22⋊C4—1224no—
563C42—21no—
564C42—21no—
565C42—612no—
566C42—612no—
567C42—1221no—
568C42—1221no—
569C42—4881no—
570C23—22no—
571C23—22no—
572C23—22no—
573C23—41no—
574C23—41no—
575C23—41no—
576C23—41no—
577C23C2×D4×D6111yescharacteristic
578Q8—1228no—
579Q8—2448no—
580Q8—2448no—
581Q8—2448no—
582D4—22no—
583D4—22no—
584D4—22no—
585D4#1451977614yes—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—22no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—22no—
593C2×C4—48no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—612no—
602C2×C4—612no—
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—1222no—
612C2×C4—1222no—
613C2×C4—1221no—
614C2×C4—1221no—
615C2×C4—1222no—
616C2×C4—2448no—
617C2×C4—2448no—
618C2×C4—2441no—
619C2×C4—2441no—
620C2×C4—2441no—
621C2×C4—2441no—
622C2×C4—2441no—
623C2×C4—2441no—
624C2×C4—2441no—
625C2×C4—2441no—
626C2×C4—2441no—
627C2×C4#1451980112yes—
628C2×C4C2×D4×D6111yescharacteristic
629C2×C4C2×D4×D6111yescharacteristic
630C2×C4C2×Q8×D6111yescharacteristic
631C2×C4C22×Dic12211yescharacteristic
632C2×C4#1451982411yescharacteristic
633C2×C4#1451982411yescharacteristic
634C8—22no—
635C8—41no—
636C8—1222no—
637C8—2441no—
638C22—24no—
639C22—42no—
640C22—42no—
641C22—42no—
642C22—42no—
643C22—42no—
644C22—42no—
645C22#120×D6111yescharacteristic
646C22#1684042511yescharacteristic
647C22D4×Dic12211yescharacteristic
648C22AGL(1,ℤ/8)×D6111yescharacteristic
649C22C2×#1451980111yescharacteristic
650C22#1684056911yescharacteristic
651C22#1684057811yescharacteristic
652C4—22no—
653C4—42no—
654C4—41no—
655C4—42no—
656C4—81no—
657C4—1222no—
658C4—1222no—
659C4—1222no—
660C4—4881no—
661C4—2442no—
662C4—2441no—
663C4—2442no—
664C4—2442no—
665C4C2×#1451977611yescharacteristic
666C4#1684033411yescharacteristic
667C4#1684042511yescharacteristic
668C4D4×Dic12211yescharacteristic
669C2—44no—
670C2#2144946111yescharacteristic
671C2#2145504311yescharacteristic
672C2#2145513811yescharacteristic
673C2#2145871611yescharacteristic
674C2#2145872111yescharacteristic
675C2#2145874111yescharacteristic
676C2#2145874411yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1862C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
hypercenter#1862D61
hypocenterC61#25844
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1862

Complements

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

NH ≅ G/Ntype
2. C2×#16840170C2semidirect
4. #21451204C2semidirect
5. #21447153C2semidirect
6. #21447151C2semidirect
7. #21447147C2semidirect
8. #21447145C2semidirect
9. #21446765C2semidirect
10. #21446757C2semidirect
118. #14519698Q8semidirect
138. #102×C61Q8semidirect
188. C2×C8×C61C2×Q8semidirect
195. C4⋊C4×C61C2×Q8semidirect
212. C4⋊Dic61C2×Q8semidirect
230. C2×#92418C2×Q8semidirect
401. C61#25844semidirect
472. #102Dic122semidirect
533. C2×C8C2×Dic122semidirect
558. C4⋊C4C2×Dic122semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Dic1222
C2×#168401702
C2×#168400642
#214512042
#214471532
#214471512
#214471472
#214471452
#214467652
#214467572
#214460822
#214460802
#214460642
#214456462
C61×#18622
#2584461