Finite Group Databasebeta

← #29736147

Subgroups — #29736147

Order 15616.

1076 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840471C211yesmaximal, characteristic
2C2×#16840169C211yesmaximal, characteristic, complemented
3C2×#16840050C211yesmaximal, characteristic
4#21450806C211yesmaximal, characteristic, complemented
5#21447158C211yesmaximal, characteristic, complemented
6#21447157C211yesmaximal, characteristic, complemented
7#21447151C211yesmaximal, characteristic, complemented
8#21447150C211yesmaximal, characteristic, complemented
9#21446775C211yesmaximal, characteristic, complemented
10#21446757C211yesmaximal, characteristic, complemented
11#21446080C211yesmaximal, characteristic
12#21446068C211yesmaximal, characteristic
13#21446064C211yesmaximal, characteristic
14#21445646C211yesmaximal, characteristic
15C61×#1864C211yesmaximal, characteristic
16#16840471—24no—
17#16840471—24no—
18C2×#14519776—21no—
19C2×#14519776C2211yescharacteristic
20C2×#14519776C2211yescharacteristic
21C2×#14519769C2211yescharacteristic
22C2×#14519768C2211yescharacteristic
23C2×#14519756—21no—
24C2×#14519754—21no—
25C2×#14519754—21no—
26C2×C4×Dic122C2211yescharacteristic
27C2×D4×Dic61C2211yescharacteristic
28C2×D4×Dic61C2211yescharacteristic
29C22×#11758196—21no—
30#16840169—24no—
31#16840050—24no—
32C2×#14519711C2211yescharacteristic
33C2×#14519710C2211yescharacteristic
34C2×#14519709C2211yescharacteristic
35C2×#14519708C2211yescharacteristic
36C2×#14519708C2211yescharacteristic
37C2×#14519706C2211yescharacteristic
38C2×#14519698C2211yescharacteristic
39C2×#14519698C2211yescharacteristic
40C2×#14519687C2211yescharacteristic
41#16839696—21no—
42#16839695—21no—
43#16839666—21no—
44#16839661C2211yescharacteristic
45#16839660C2211yescharacteristic
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×#106×C61C2211yescharacteristic
58C2×#102×C61C2211yescharacteristic
59C2×#102×C61C2211yescharacteristic
60C61×#389C2211yescharacteristic
61C61×#340C2211yescharacteristic
62C61×#336C2211yescharacteristic
63C22×Dic122—21no—
64C22×Dic122—21no—
65C22×Dic122D411yescharacteristic
66C23×Dic61—21no—
67C23×Dic61—21no—
68C22×#92418C2311yescharacteristic
69C22×#92418C2311yescharacteristic
70#14519776—22no—
71#14519776—22no—
72#14519776—22no—
73#14519776—22no—
74#14519776—44no—
75#14519769—22no—
76#14519769—22no—
77#14519768—22no—
78#14519768D414yes—
79#14519756—28no—
80#14519754—28no—
81#14519754—28no—
82C4×Dic122—22no—
83C4×Dic122—22no—
84D4×Dic61—22no—
85D4×Dic61—22no—
86D4×Dic61—22no—
87D4×Dic61D414yes—
88C2×C4⋊Dic61—21no—
89C2×C4⋊Dic61—21no—
90C2×C4⋊Dic61D411yescharacteristic
91C2×C4⋊Dic61C2311yescharacteristic
92C2×C4⋊Dic61C2311yescharacteristic
93C2×C4⋊Dic61C2311yescharacteristic
94C2×C4⋊Dic61C2311yescharacteristic
95C2×#11758196—28no—
96C2×#11758196—21no—
97C2×#11758196—21no—
98C2×#11758196—21no—
99C2×#11758196—21no—
100C2×#11758196—21no—
101C2×#11758196—21no—
102C2×#11758196—21no—
103C2×#11758196D411yescharacteristic
104C2×#11758196D411yescharacteristic
105C2×C22⋊Dic61—21no—
106C2×C22⋊Dic61—21no—
107C2×C4×Dic61—21no—
108C2×C4×Dic61C2311yescharacteristic
109C2×C4×Dic61C2311yescharacteristic
110#14519711—22no—
111#14519710—22no—
112#14519709—22no—
113#14519708—22no—
114#14519708—22no—
115#14519706—22no—
116#14519698—22no—
117#14519698Q814yescomplemented
118#14519687—22no—
119#14519684—21no—
120#14519684—21no—
121#14519684—21no—
122#14519684—21no—
123C22×D4×C61C2311yescharacteristic
124C23×C4×C61—21no—
125C22×C8×C61C2311yescharacteristic
126C22×C8×C61C2311yescharacteristic
127C4×D4×C61—22no—
128C4×D4×C61—22no—
129C2×C4⋊C4×C61—21no—
130C2×C4⋊C4×C61C2311yescharacteristic
131C2×C4⋊C4×C61C2311yescharacteristic
132C2×C4⋊C4×C61C2311yescharacteristic
133C2×C22⋊C4×C61—21no—
134C2×C42×C61C2311yescharacteristic
135#106×C61—22no—
136#102×C61—22no—
137#102×C61Q814yescomplemented
138#95×C61—21no—
139C24×C61—21no—
140C2×D4×C61—21no—
141C2×D4×C61—21no—
142C2×D4×C61—21no—
143C2×D4×C61C2×D412yes—
144C2×D4×C61C2×Q812yes—
145C2×D4×C61C2×Q812yes—
146C22×C4×C61—21no—
147C22×C4×C61—21no—
148C22×C4×C61—24no—
149C22×C4×C61—21no—
150C22×C4×C61—81no—
151C22×C4×C61—81no—
152C22×C4×C61C2×D411yescharacteristic
153C22×C4×C61C2×D411yescharacteristic
154C22×C4×C61C2411yescharacteristic
155C22×Dic61—24no—
156C22×Dic61—21no—
157C22×Dic61—21no—
158C22×Dic61—21no—
159C22×Dic61—24no—
160C22×Dic61—21no—
161C22×Dic61—42no—
162C22×Dic61—42no—
163C22×Dic61—81no—
164C22×Dic61—81no—
165C22×Dic61C2×D411yescharacteristic
166C22×Dic61C2×D411yescharacteristic
167C22×Dic61C2×D411yescharacteristic
168C22×Dic61C2×D411yescharacteristic
169C2×Dic122—22no—
170C2×Dic122—22no—
171C2×Dic122—24no—
172C2×Dic122—41no—
173C2×Dic122—42no—
174C2×Dic122—42no—
175C2×Dic122—42no—
176C2×Dic122—81no—
177C2×Dic122—81no—
178C2×C8×C61—21no—
179C2×C8×C61—21no—
180C2×C8×C61—21no—
181C2×C8×C61—21no—
182C2×C8×C61—21no—
183C2×C8×C61C2×Q812yes—
184C4⋊C4×C61—21no—
185C4⋊C4×C61—21no—
186C4⋊C4×C61—21no—
187C4⋊C4×C61—21no—
188C4⋊C4×C61—42no—
189C4⋊C4×C61C2×D412yes—
190C4⋊C4×C61C2×Q812yes—
191C22⋊C4×C61—24no—
192C42×C61—21no—
193C42×C61—21no—
194C22⋊Dic61—24no—
195C22⋊Dic61—24no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61—24no—
200C4⋊Dic61—22no—
201C4⋊Dic61—21no—
202C4⋊Dic61—21no—
203C4⋊Dic61—42no—
204C4⋊Dic61C2×D412yes—
205C4⋊Dic61C2×D412yes—
206C4⋊Dic61C2×Q812yes—
207#11758196—24no—
208#11758196—24no—
209#11758196—24no—
210#11758196—22no—
211#11758196—24no—
212#11758196—22no—
213#11758196—42no—
214#11758196—42no—
215#11758196—42no—
216C4×Dic61—21no—
217C4×Dic61—21no—
218C4×Dic61—42no—
219C4×Dic61C2×D412yes—
220C4×Dic61C2×D412yes—
221C2×#92418—21no—
222C2×#92418—21no—
223C2×#92418—21no—
224C2×#92418—21no—
225C2×#92418—21no—
226C2×#92418C2×Q812yes—
227C23×C61—22no—
228C23×C61—22no—
229C23×C61—22no—
230C23×C61—41no—
231C23×C61—41no—
232C23×C61—41no—
233C23×C61—41no—
234C23×C61C22×D411yescharacteristic
235D4×C61—22no—
236D4×C61—22no—
237D4×C61—22no—
238D4×C61C4⋊Q814yescomplemented
239C2×C4×C61—21no—
240C2×C4×C61—21no—
241C2×C4×C61—22no—
242C2×C4×C61—22no—
243C2×C4×C61—21no—
244C2×C4×C61—21no—
245C2×C4×C61—21no—
246C2×C4×C61—21no—
247C2×C4×C61—41no—
248C2×C4×C61—41no—
249C2×C4×C61—41no—
250C2×C4×C61—41no—
251C2×C4×C61—41no—
252C2×C4×C61—41no—
253C2×C4×C61—41no—
254C2×C4×C61—84no—
255C2×C4×C61C22×D411yescharacteristic
256C2×C4×C61C22×D411yescharacteristic
257C2×C4×C61C22×Q811yescharacteristic
258C2×C4×C61C22×Q811yescharacteristic
259C2×C4×C612−1+411yescharacteristic
260C2×C4×C612−1+411yescharacteristic
261C2×Dic61—21no—
262C2×Dic61—21no—
263C2×Dic61—21no—
264C2×Dic61—21no—
265C2×Dic61—21no—
266C2×Dic61—21no—
267C2×Dic61—21no—
268C2×Dic61—22no—
269C2×Dic61—22no—
270C2×Dic61—22no—
271C2×Dic61—21no—
272C2×Dic61—21no—
273C2×Dic61—48no—
274C2×Dic61—41no—
275C2×Dic61—41no—
276C2×Dic61—41no—
277C2×Dic61—41no—
278C2×Dic61—41no—
279C2×Dic61—41no—
280C2×Dic61—41no—
281C2×Dic61—41no—
282C2×Dic61—41no—
283C2×Dic61—84no—
284C2×Dic61#12012yes—
285C2×Dic61#12012yes—
286C2×Dic61AGL(1,ℤ/8)12yes—
287Dic122—44no—
288Dic122—84no—
289Dic122—84no—
290C8×C61—22no—
291C8×C61—41no—
292#92418—22no—
293#92418—41no—
294#25851—611nomaximal, Sylow
295C22×C61—24no—
296C22×C61—42no—
297C22×C61—42no—
298C22×C61—42no—
299C22×C61—42no—
300C22×C61—42no—
301C22×C61—42no—
302C22×C61C2×#12011yescharacteristic
303C22×C61D4×Q811yescharacteristic
304C22×C61D4×Q811yescharacteristic
305C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
306C22×C61#57511yescharacteristic
307C22×C61#57711yescharacteristic
308C22×C61#57711yescharacteristic
309C4×C61—42no—
310C4×C61—41no—
311C4×C61—42no—
312C4×C61—41no—
313C4×C61—81no—
314C4×C61C2×C4⋊Q811yescharacteristic
315C4×C61#53611yescharacteristic
316C4×C61D4×Q811yescharacteristic
317C4×C61D4×Q811yescharacteristic
318Dic61—22no—
319Dic61—22no—
320Dic61—22no—
321Dic61—42no—
322Dic61—42no—
323Dic61—41no—
324Dic61—42no—
325Dic61—81no—
326C2×#554—611no—
327C2×#501—611no—
328C2×#476—611no—
329#2546—611no—
330#2059—611no—
331#2059—611no—
332#2058—611no—
333#2058—611no—
334#1989—611no—
335#1985—611no—
336#1864—611no—
337#1864D6111yescharacteristic
338#1862—611no—
339#1862—611no—
340#1762—611no—
341C2×C61—44no—
342C2×C61#243411yescharacteristic
343C2×C61#300611yescharacteristic
344C2×C61#300911yescharacteristic
345C2×C61#338411yescharacteristic
346C2×C61#338411yescharacteristic
347C2×C61#338811yescharacteristic
348C2×C61#338811yescharacteristic
349#554—1224no—
350#554—1224no—
351C2×C4⋊Q8—611no—
352C2×C4⋊Q8—611no—
353C2×C4⋊Q8—1221no—
354C2×#125—611no—
355C2×#125—611no—
356C2×C22⋊Q8—1221no—
357C2×C22⋊Q8—1221no—
358C2×C22⋊Q8—1221no—
359C2×C4×Q8—611no—
360C2×C4×D4—611no—
361C2×C4×D4—611no—
362C2×C4×D4C2×D6111yescharacteristic
363C22×C4⋊C4—1221no—
364#501—1224no—
365#476—1224no—
366C2×#107—611no—
367C2×#107—611no—
368C2×#106—611no—
369C2×#106—611no—
370C2×#106—611no—
371C2×#106C2×D6111yescharacteristic
372C2×C4⋊C8—611no—
373C2×#102—611no—
374C2×#102—611no—
375C2×#102C2×D6111yescharacteristic
376C2×#102C2×D6111yescharacteristic
377C2×#97—611no—
378#398—1221no—
379#398—1221no—
380#391—1221no—
381#389—611no—
382#389—611no—
383#389—611no—
384#389C2×D6111yescharacteristic
385C4×C4⋊C4—611no—
386#340—611no—
387#340—611no—
388#340—611no—
389#340—611no—
390#340—611no—
391#340C2×D6111yescharacteristic
392#336—611no—
393#336—611no—
394#336—611no—
395#336C2×D6111yescharacteristic
396C61#2585111yescharacteristic, complemented, Sylow
397C22×Q8—611no—
398C22×Q8—1221no—
399C22×Q8—1221no—
400C22×D4C22×D6111yescharacteristic
401C23×C4—21no—
402C23×C4—1221no—
403C23×C4—1221no—
404C22×C8—611no—
405C22×C8—611no—
406C22×C8C22×D6111yescharacteristic
407C22×C8C22×D6111yescharacteristic
408C4⋊Q8—1222no—
409C4⋊Q8—1222no—
410C4⋊Q8—1222no—
411C4⋊Q8—1222no—
412C4⋊Q8—2444no—
413#125—614no—
414#125—1222no—
415#125—1222no—
416#125—1222no—
417C22⋊Q8—1228no—
418C22⋊Q8—1228no—
419C22⋊Q8—1228no—
420C4×Q8—1222no—
421C4×Q8—1222no—
422C4×D4—22no—
423C4×D4—22no—
424C4×D4—614no—
425C4×D4—1222no—
426C4×D4—1222no—
427C4×D4—1222no—
428C2×C4⋊C4—21no—
429C2×C4⋊C4—611no—
430C2×C4⋊C4—611no—
431C2×C4⋊C4—611no—
432C2×C4⋊C4—611no—
433C2×C4⋊C4—611no—
434C2×C4⋊C4—611no—
435C2×C4⋊C4—611no—
436C2×C4⋊C4—1221no—
437C2×C4⋊C4—1221no—
438C2×C4⋊C4—1221no—
439C2×C4⋊C4—1228no—
440C2×C4⋊C4—1221no—
441C2×C4⋊C4—1221no—
442C2×C4⋊C4—1221no—
443C2×C4⋊C4—1221no—
444C2×C4⋊C4—1221no—
445C2×C4⋊C4—1221no—
446C2×C4⋊C4C22×D6111yescharacteristic
447C2×C4⋊C4C22×D6111yescharacteristic
448C2×C4⋊C4C22×D6111yescharacteristic
449C2×C22⋊C4—21no—
450C2×C22⋊C4—1221no—
451C2×C22⋊C4—1221no—
452C2×C42—611no—
453C2×C42—611no—
454C2×C42—1221no—
455C2×C42C22×D6111yescharacteristic
456#107—1222no—
457#107—1222no—
458#106—22no—
459#106—1222no—
460#106—1222no—
461#106—1222no—
462C4⋊C8—1222no—
463#102—22no—
464#102—614no—
465#102—1222no—
466#102Dic12214yescomplemented
467#97—1222no—
468#95—21no—
469#95—1221no—
470#95—1221no—
471#95—1221no—
472#95—1221no—
473C24—21no—
474C2×Q8—1222no—
475C2×Q8—1222no—
476C2×Q8—1224no—
477C2×Q8—4881no—
478C2×Q8—4881no—
479C2×Q8—2441no—
480C2×Q8—2442no—
481C2×Q8—2442no—
482C2×Q8—2442no—
483C2×D4—21no—
484C2×D4—21no—
485C2×D4—21no—
486C2×D4C2×Dic12212yes—
487C2×D4D4×D6112yes—
488C2×D4Q8×D6112yes—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—24no—
492C22×C4—21no—
493C22×C4—81no—
494C22×C4—81no—
495C22×C4—611no—
496C22×C4—611no—
497C22×C4—611no—
498C22×C4—611no—
499C22×C4—1224no—
500C22×C4—1224no—
501C22×C4—1221no—
502C22×C4—1221no—
503C22×C4—1221no—
504C22×C4—1221no—
505C22×C4—4881no—
506C22×C4—4881no—
507C22×C4—2442no—
508C22×C4—2442no—
509C22×C4D4×D6111yescharacteristic
510C22×C4D4×D6111yescharacteristic
511C22×C4C23×D6111yescharacteristic
512C2×C8—21no—
513C2×C8—21no—
514C2×C8—21no—
515C2×C8—21no—
516C2×C8—21no—
517C2×C8—612no—
518C2×C8—1221no—
519C2×C8—1221no—
520C2×C8—1221no—
521C2×C8—1221no—
522C2×C8—1221no—
523C2×C8C2×Dic12212yes—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—42no—
529C4⋊C4—612no—
530C4⋊C4—612no—
531C4⋊C4—612no—
532C4⋊C4—1222no—
533C4⋊C4—1222no—
534C4⋊C4—1222no—
535C4⋊C4—1224no—
536C4⋊C4—1224no—
537C4⋊C4—1224no—
538C4⋊C4—1224no—
539C4⋊C4—1224no—
540C4⋊C4—1221no—
541C4⋊C4—1221no—
542C4⋊C4—1221no—
543C4⋊C4—1221no—
544C4⋊C4—1221no—
545C4⋊C4—2442no—
546C4⋊C4—2442no—
547C4⋊C4—2442no—
548C4⋊C4—2442no—
549C4⋊C4C2×Dic12212yes—
550C4⋊C4D4×D6112yes—
551C22⋊C4—24no—
552C22⋊C4—1224no—
553C22⋊C4—1224no—
554C42—21no—
555C42—21no—
556C42—612no—
557C42—612no—
558C42—1221no—
559C42—1221no—
560C42—2442no—
561C23—22no—
562C23—22no—
563C23—22no—
564C23—41no—
565C23—41no—
566C23—41no—
567C23—41no—
568C23C2×D4×D6111yescharacteristic
569Q8—4884no—
570Q8—4884no—
571Q8—2444no—
572D4—22no—
573D4—22no—
574D4—22no—
575D4#1451977614yescomplemented
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—22no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—84no—
592C2×C4—612no—
593C2×C4—612no—
594C2×C4—612no—
595C2×C4—1221no—
596C2×C4—1221no—
597C2×C4—1221no—
598C2×C4—1221no—
599C2×C4—1221no—
600C2×C4—1221no—
601C2×C4—1221no—
602C2×C4—1222no—
603C2×C4—1222no—
604C2×C4—1221no—
605C2×C4—1221no—
606C2×C4—1222no—
607C2×C4—4884no—
608C2×C4—2448no—
609C2×C4—2441no—
610C2×C4—2441no—
611C2×C4—2441no—
612C2×C4—2441no—
613C2×C4—2441no—
614C2×C4—2441no—
615C2×C4—2441no—
616C2×C4—2441no—
617C2×C4—2441no—
618C2×C4C2×D4×D6111yescharacteristic
619C2×C4C2×D4×D6111yescharacteristic
620C2×C4C2×Q8×D6111yescharacteristic
621C2×C4C22×Dic12211yescharacteristic
622C2×C4#1451982411yescharacteristic
623C2×C4#1451982411yescharacteristic
624C8—22no—
625C8—41no—
626C8—1222no—
627C8—2441no—
628C22—24no—
629C22—42no—
630C22—42no—
631C22—42no—
632C22—42no—
633C22—42no—
634C22—42no—
635C22#120×D6111yescharacteristic
636C22#1684042511yescharacteristic
637C22D4×Dic12211yescharacteristic
638C22AGL(1,ℤ/8)×D6111yescharacteristic
639C22#1684055911yescharacteristic
640C22#1684056811yescharacteristic
641C22#1684057211yescharacteristic
642C4—42no—
643C4—41no—
644C4—41no—
645C4—42no—
646C4—81no—
647C4—1222no—
648C4—1222no—
649C4—1222no—
650C4—4881no—
651C4—2442no—
652C4—2442no—
653C4—2441no—
654C4—2442no—
655C4C2×#1451977611yescharacteristic
656C4#1684033411yescharacteristic
657C4#1684042511yescharacteristic
658C4D4×Dic12211yescharacteristic
659C2—44no—
660C2#2144946111yescharacteristic
661C2#2145506811yescharacteristic
662C2#2145509111yescharacteristic
663C2#2145871311yescharacteristic
664C2#2145871911yescharacteristic
665C2#2145876011yescharacteristic
666C2#2145876811yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1864C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
hypercenter#1864D61
hypocenterC61#25851
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1864

Complements

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

NH ≅ G/Ntype
2. C2×#16840169C2semidirect
4. #21450806C2semidirect
5. #21447158C2semidirect
6. #21447157C2semidirect
7. #21447151C2semidirect
8. #21447150C2semidirect
9. #21446775C2semidirect
10. #21446757C2semidirect
117. #14519698Q8semidirect
137. #102×C61Q8semidirect
238. D4×C61C4⋊Q8semidirect
396. C61#25851semidirect
466. #102Dic122semidirect
575. D4#14519776semidirect

Maximal subgroups

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

subgroupindex
C2×#168404712
C2×#168401692
C2×#168400502
#214508062
#214471582
#214471572
#214471512
#214471502
#214467752
#214467572
#214460802
#214460682
#214460642
#214456462
C61×#18642
#2585161