Finite Group Databasebeta

← #29702331

Subgroups — #29702331

Order 15616.

1018 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839965C211yesmaximal, characteristic
2C2×#16839863C211yesmaximal, characteristic
3C2×#16839862C211yesmaximal, characteristic
4#21448876C211yesmaximal, characteristic, complemented
5D4×#11758196C211yesmaximal, characteristic
6#21446619C211yesmaximal, characteristic, complemented
7#21446576C211yesmaximal, characteristic, complemented
8#21446512C211yesmaximal, characteristic, complemented
9#21446507C211yesmaximal, characteristic, complemented
10#21446151C211yesmaximal, characteristic
11#21446076C211yesmaximal, characteristic, complemented
12#21446071C211yesmaximal, characteristic, complemented
13#21445685C211yesmaximal, characteristic, complemented
14#102×Dic61C211yesmaximal, characteristic
15C61×#1877C211yesmaximal, characteristic
16C22×#11758200C2211yescharacteristic
17C2×#14519775C2211yescharacteristic
18C2×Q8×Dic61C2211yescharacteristic
19C2×C4×Dic122C2211yescharacteristic
20C2×D4×Dic61C2211yescharacteristic
21C2×D4×Dic61C2211yescharacteristic
22C22×#11758196—21no—
23#16839965C418yescomplemented
24#16839863—24no—
25#16839862—24no—
26C2×#14519711C2211yescharacteristic
27C2×#14519710C2211yescharacteristic
28C2×#14519706C2211yescharacteristic
29C2×#14519702C2211yescharacteristic
30C2×#14519702C2211yescharacteristic
31C2×#14519701C2211yescharacteristic
32C2×#14519701C2211yescharacteristic
33C2×#14519700C2211yescharacteristic
34C2×#14519698C2211yescharacteristic
35C2×#14519688C2211yescharacteristic
36C2×#14519687C2211yescharacteristic
37C2×C8×Dic61C2211yescharacteristic
38#16839667C2211yescharacteristic
39#16839660C2211yescharacteristic
40#16839659—21no—
41#16839658C2211yescharacteristic
42#16839649—21no—
43#16839644—21no—
44#16839635—21no—
45#16839634—21no—
46#16839631C2211yescharacteristic
47#16839620—21no—
48C4⋊C4×Dic61C2211yescharacteristic
49C4×#11758196C2211yescharacteristic
50#16839613—21no—
51#16839525C2211yescharacteristic
52#16839524C2211yescharacteristic
53#16839524C2211yescharacteristic
54#16839516C2211yescharacteristic
55C22×SD16×C61C2211yescharacteristic
56C2×C4×D4×C61C2211yescharacteristic
57C2×#103×C61C2211yescharacteristic
58C2×#102×C61C2211yescharacteristic
59C61×#391C2211yescharacteristic
60C61×#340C2211yescharacteristic
61C61×#336C2211yescharacteristic
62C22×Dic122C2311yescharacteristic
63C23×Dic61—21no—
64C23×Dic61—21no—
65C2×#11758200—22no—
66C2×#11758200—22no—
67C2×#11758200C2×C414yes—
68C22×#92418C2311yescharacteristic
69C22×#92418C2311yescharacteristic
70#14519775—22no—
71#14519775C2×C414yescomplemented
72Q8×Dic61—22no—
73Q8×Dic61—41no—
74C4×Dic122—22no—
75C4×Dic122—41no—
76D4×Dic61—22no—
77D4×Dic61—22no—
78D4×Dic61—41no—
79D4×Dic61C2×C414yes—
80C2×C4⋊Dic61—21no—
81C2×C4⋊Dic61C2311yescharacteristic
82C2×C4⋊Dic61C2311yescharacteristic
83C2×C4⋊Dic61C2311yescharacteristic
84C2×#11758196—21no—
85C2×#11758196—21no—
86C2×#11758196—21no—
87C2×#11758196—21no—
88C2×#11758196—21no—
89C2×#11758196—21no—
90C2×#11758196—21no—
91C2×#11758196—44no—
92C2×#11758196D411yescharacteristic
93C2×#11758196D411yescharacteristic
94C2×C22⋊Dic61—21no—
95C2×C22⋊Dic61—21no—
96C2×C4×Dic61—21no—
97C2×C4×Dic61—21no—
98C2×C4×Dic61—21no—
99C2×C4×Dic61C2311yescharacteristic
100C2×C4×Dic61C2311yescharacteristic
101#14519711—22no—
102#14519710—22no—
103#14519706C2×C414yescomplemented
104#14519702—41no—
105#14519702C2×C414yescomplemented
106#14519701—22no—
107#14519701C2×C414yescomplemented
108#14519700—22no—
109#14519698D414yes—
110#14519688—22no—
111#14519687—22no—
112C8×Dic61—22no—
113#14519684—21no—
114#14519684—21no—
115#14519684—21no—
116#14519684—21no—
117#14519684—21no—
118#14519684D411yescharacteristic
119#14519684D411yescharacteristic
120C22×Q8×C61C2311yescharacteristic
121C22×D4×C61C2311yescharacteristic
122C23×C4×C61—21no—
123C2×SD16×C61—22no—
124C2×SD16×C61—22no—
125C2×SD16×C61C2×C414yes—
126C22×C8×C61C2311yescharacteristic
127C22×C8×C61C2311yescharacteristic
128C4×D4×C61—22no—
129C4×D4×C61—41no—
130C2×C4⋊C4×C61C2311yescharacteristic
131C2×C4⋊C4×C61C2311yescharacteristic
132C2×C22⋊C4×C61—21no—
133C2×C42×C61C2311yescharacteristic
134#103×C61—41no—
135#102×C61—22no—
136#95×C61—21no—
137#95×C61—21no—
138C24×C61—21no—
139C2×Q8×C61—21no—
140C2×Q8×C61—21no—
141C2×Q8×C61—21no—
142C2×Q8×C61—21no—
143C2×Q8×C61—21no—
144C2×Q8×C61C22×C412yes—
145C2×D4×C61—21no—
146C2×D4×C61—21no—
147C2×D4×C61—21no—
148C2×D4×C61C22×C412yes—
149C2×D4×C61C2×D412yes—
150C2×D4×C61#4112yes—
151C22×C4×C61—21no—
152C22×C4×C61—21no—
153C22×C4×C61—21no—
154C22×C4×C61—42no—
155C22×C4×C61—42no—
156C22×C4×C61—81no—
157C22×C4×C61C2×D411yescharacteristic
158C22×C4×C61C2×D411yescharacteristic
159C22×C4×C61C2411yescharacteristic
160C22×Dic61—21no—
161C22×Dic61—21no—
162C22×Dic61—21no—
163C22×Dic61—24no—
164C22×Dic61—21no—
165C22×Dic61—42no—
166C22×Dic61—42no—
167C22×Dic61—42no—
168C22×Dic61—42no—
169C22×Dic61—81no—
170C22×Dic61C2×D411yescharacteristic
171C22×Dic61C2×D411yescharacteristic
172C22×Dic61C2×D411yescharacteristic
173C22×Dic61C2×D411yescharacteristic
174C2×Dic122—21no—
175C2×Dic122—21no—
176C2×Dic122—21no—
177C2×Dic122—21no—
178C2×Dic122—21no—
179C2×Dic122C22×C412yes—
180SD16×C61—28no—
181C2×C8×C61—21no—
182C2×C8×C61—21no—
183C2×C8×C61—21no—
184C2×C8×C61—21no—
185C2×C8×C61—21no—
186C2×C8×C61C22×C412yes—
187C4⋊C4×C61—21no—
188C4⋊C4×C61—21no—
189C4⋊C4×C61—21no—
190C4⋊C4×C61—21no—
191C22⋊C4×C61—42no—
192C42×C61—21no—
193C42×C61—21no—
194C22⋊Dic61—24no—
195C22⋊Dic61—42no—
196#11758200—28no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201C4⋊Dic61—42no—
202C4⋊Dic61C22×C412yes—
203C4⋊Dic61C2×D412yes—
204#11758196—24no—
205#11758196—24no—
206#11758196—22no—
207#11758196—22no—
208#11758196—42no—
209#11758196—42no—
210#11758196—42no—
211#11758196—42no—
212#11758196—81no—
213C4×Dic61—21no—
214C4×Dic61—21no—
215C4×Dic61—42no—
216C4×Dic61—42no—
217C4×Dic61—42no—
218C4×Dic61C22×C412yes—
219C4×Dic61#4112yes—
220C2×#92418—21no—
221C2×#92418—21no—
222C2×#92418—21no—
223C2×#92418—21no—
224C2×#92418C22×C412yes—
225C2×#92418C2×D412yes—
226C23×C61—22no—
227C23×C61—22no—
228C23×C61—22no—
229C23×C61—41no—
230C23×C61—41no—
231C23×C61—41no—
232C23×C61—41no—
233C23×C61C22×D411yescharacteristic
234Q8×C61—22no—
235Q8×C61—22no—
236Q8×C61—41no—
237Q8×C61—41no—
238D4×C61—22no—
239D4×C61—22no—
240D4×C61—22no—
241D4×C61C4×D414yes—
242C2×C4×C61—22no—
243C2×C4×C61—21no—
244C2×C4×C61—21no—
245C2×C4×C61—21no—
246C2×C4×C61—21no—
247C2×C4×C61—21no—
248C2×C4×C61—21no—
249C2×C4×C61—41no—
250C2×C4×C61—41no—
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—84no—
258C2×C4×C61C23×C411yescharacteristic
259C2×C4×C61C22×D411yescharacteristic
260C2×C4×C61C2×#4111yescharacteristic
261C2×C4×C61C2×#4111yescharacteristic
262C2×C4×C61H(5,𝔽2)11yescharacteristic
263C2×C4×C612−1+411yescharacteristic
264C2×Dic61—21no—
265C2×Dic61—21no—
266C2×Dic61—22no—
267C2×Dic61—22no—
268C2×Dic61—21no—
269C2×Dic61—21no—
270C2×Dic61—22no—
271C2×Dic61—21no—
272C2×Dic61—21no—
273C2×Dic61—21no—
274C2×Dic61—21no—
275C2×Dic61—41no—
276C2×Dic61—41no—
277C2×Dic61—41no—
278C2×Dic61—41no—
279C2×Dic61—41no—
280C2×Dic61—41no—
281C2×Dic61—48no—
282C2×Dic61—41no—
283C2×Dic61—41no—
284C2×Dic61—41no—
285C2×Dic61—84no—
286C2×Dic61C4×D412yes—
287C2×Dic61C4×D412yes—
288C2×Dic61AGL(1,ℤ/8)12yes—
289C2×Dic61#13712yes—
290Dic122—22no—
291Dic122—22no—
292Dic122—41no—
293Dic122—41no—
294C8×C61—22no—
295C8×C61—41no—
296#92418—22no—
297#92418—22no—
298#23002—611nomaximal, Sylow
299C22×C61—24no—
300C22×C61—42no—
301C22×C61—42no—
302C22×C61—42no—
303C22×C61—42no—
304C22×C61—42no—
305C22×C61—42no—
306C22×C61C2×C4×D411yescharacteristic
307C22×C61D4211yescharacteristic
308C22×C61#54811yescharacteristic
309C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
310C22×C61C2×#13711yescharacteristic
311C22×C61#57711yescharacteristic
312C22×C61#57811yescharacteristic
313C4×C61—42no—
314C4×C61—41no—
315C4×C61—41no—
316C4×C61—81no—
317C4×C61—81no—
318C4×C61C2×C4×D411yescharacteristic
319C4×C61#52011yescharacteristic
320C4×C61#54711yescharacteristic
321C4×C61#55311yescharacteristic
322Dic61—22no—
323Dic61—42no—
324Dic61—42no—
325Dic61—41no—
326Dic61—42no—
327Dic61—41no—
328Dic61—81no—
329Dic61#44014yescomplemented
330C2×#461—611no—
331C2×#440—611no—
332C2×#440—611no—
333#2365—611no—
334D4×C4⋊C4—611no—
335#1956—611no—
336#1947—611no—
337#1935—611no—
338#1935—611no—
339#1877—611no—
340#1877D6111yescharacteristic
341#1863—611no—
342#1863—611no—
343#1770—611no—
344C4×#102—611no—
345C2×C61—44no—
346C2×C61#236111yescharacteristic
347C2×C61C2×#44011yescharacteristic
348C2×C61#294711yescharacteristic
349C2×C61#327211yescharacteristic
350C2×C61#327611yescharacteristic
351C2×C61#336711yescharacteristic
352C2×C61#337211yescharacteristic
353C22×SD16—611no—
354C22×SD16C2×D6111yescharacteristic
355C2×C4⋊Q8—611no—
356C2×C4×Q8—611no—
357C2×C4×Q8—611no—
358C2×C4×D4—611no—
359C2×C4×D4—611no—
360C2×C4×D4C2×D6111yescharacteristic
361C22×C4⋊C4—1221no—
362#461—618no—
363#440—1224no—
364#440—1224no—
365C2×#107—611no—
366C2×#107—611no—
367C2×C4⋊C8—611no—
368C2×#103—611no—
369C2×#103—611no—
370C2×#103—611no—
371C2×#103—611no—
372C2×#103—611no—
373C2×#103C2×D6111yescharacteristic
374C2×#102—611no—
375C2×#102C2×D6111yescharacteristic
376C2×#97—611no—
377C2×#97—611no—
378C2×C4×C8—611no—
379#391—611no—
380#391C2×D6111yescharacteristic
381#389—611no—
382#389—611no—
383#389—1221no—
384#387—1221no—
385#387—1221no—
386#385—1221no—
387#385—1221no—
388#384—611no—
389#380—1221no—
390C4×C4⋊C4—611no—
391C4×C4⋊C4—611no—
392C4×C4⋊C4—1221no—
393#340—611no—
394#340—611no—
395#340—611no—
396#340C2×D6111yescharacteristic
397#336—611no—
398#336C2×D6111yescharacteristic
399C61#2300211yescharacteristic, complemented, Sylow
400C22×Q8—611no—
401C22×Q8C22×D6111yescharacteristic
402C22×D4C22×D6111yescharacteristic
403C23×C4—21no—
404C23×C4—1221no—
405C23×C4—1221no—
406C2×SD16—22no—
407C2×SD16—22no—
408C2×SD16—614no—
409C2×SD16—1222no—
410C2×SD16—1222no—
411C2×SD16C4×D6114yes—
412C22×C8—611no—
413C22×C8—611no—
414C22×C8C22×D6111yescharacteristic
415C22×C8C22×D6111yescharacteristic
416C4⋊Q8—614no—
417C4⋊Q8—1222no—
418C4×Q8—1222no—
419C4×Q8—1222no—
420C4×Q8—2441no—
421C4×Q8—2441no—
422C4×D4—22no—
423C4×D4—41no—
424C4×D4—614no—
425C4×D4—1222no—
426C4×D4—1222no—
427C4×D4—2441no—
428C2×C4⋊C4—611no—
429C2×C4⋊C4—611no—
430C2×C4⋊C4—611no—
431C2×C4⋊C4—611no—
432C2×C4⋊C4—611no—
433C2×C4⋊C4—1221no—
434C2×C4⋊C4—1221no—
435C2×C4⋊C4—1221no—
436C2×C4⋊C4—1221no—
437C2×C4⋊C4—1221no—
438C2×C4⋊C4—1221no—
439C2×C4⋊C4—1221no—
440C2×C4⋊C4—1221no—
441C2×C4⋊C4—2444no—
442C2×C4⋊C4C22×D6111yescharacteristic
443C2×C4⋊C4C22×D6111yescharacteristic
444C2×C22⋊C4—21no—
445C2×C22⋊C4—1221no—
446C2×C22⋊C4—1221no—
447C2×C42—611no—
448C2×C42—611no—
449C2×C42—1221no—
450C2×C42—1221no—
451C2×C42—1221no—
452C2×C42C22×D6111yescharacteristic
453#107—1222no—
454#107—1222no—
455C4⋊C8—614no—
456#103—41no—
457#103—614no—
458#103—614no—
459#103—1222no—
460#103—1222no—
461#103—2441no—
462#102—22no—
463#102—614no—
464#97—1222no—
465#97—1222no—
466C4×C8—1222no—
467#95—21no—
468#95—21no—
469#95—611no—
470#95—611no—
471#95—1221no—
472#95—1221no—
473#95—1221no—
474#95—1221no—
475#95—1221no—
476C24—21no—
477C2×Q8—21no—
478C2×Q8—21no—
479C2×Q8—21no—
480C2×Q8—21no—
481C2×Q8—21no—
482C2×Q8—612no—
483C2×Q8—1221no—
484C2×Q8—1221no—
485C2×Q8—1221no—
486C2×Q8—1221no—
487C2×Q8—1221no—
488C2×Q8C2×C4×D6112yes—
489C2×D4—21no—
490C2×D4—21no—
491C2×D4—21no—
492C2×D4C2×C4×D6112yes—
493C2×D4D4×D6112yes—
494C2×D4#1175822412yes—
495C22×C4—21no—
496C22×C4—21no—
497C22×C4—21no—
498C22×C4—42no—
499C22×C4—42no—
500C22×C4—81no—
501C22×C4—611no—
502C22×C4—611no—
503C22×C4—611no—
504C22×C4—611no—
505C22×C4—1224no—
506C22×C4—1221no—
507C22×C4—1221no—
508C22×C4—1221no—
509C22×C4—1221no—
510C22×C4—4881no—
511C22×C4—2442no—
512C22×C4—2442no—
513C22×C4—2442no—
514C22×C4—2442no—
515C22×C4D4×D6111yescharacteristic
516C22×C4D4×D6111yescharacteristic
517C22×C4C23×D6111yescharacteristic
518SD16—28no—
519SD16—1228no—
520C2×C8—21no—
521C2×C8—21no—
522C2×C8—21no—
523C2×C8—21no—
524C2×C8—21no—
525C2×C8—612no—
526C2×C8—612no—
527C2×C8—1221no—
528C2×C8—1221no—
529C2×C8—1221no—
530C2×C8—1221no—
531C2×C8C2×C4×D6112yes—
532C4⋊C4—21no—
533C4⋊C4—21no—
534C4⋊C4—21no—
535C4⋊C4—21no—
536C4⋊C4—612no—
537C4⋊C4—612no—
538C4⋊C4—1222no—
539C4⋊C4—1222no—
540C4⋊C4—1224no—
541C4⋊C4—1224no—
542C4⋊C4—1221no—
543C4⋊C4—1221no—
544C4⋊C4—1221no—
545C4⋊C4—1221no—
546C4⋊C4—4881no—
547C4⋊C4—2442no—
548C4⋊C4—2442no—
549C4⋊C4—2442no—
550C4⋊C4—2442no—
551C4⋊C4—2442no—
552C22⋊C4—42no—
553C22⋊C4—1224no—
554C22⋊C4—2442no—
555C42—21no—
556C42—21no—
557C42—612no—
558C42—612no—
559C42—1221no—
560C42—1221no—
561C42—2442no—
562C42—2442no—
563C42—2442no—
564C23—22no—
565C23—22no—
566C23—22no—
567C23—41no—
568C23—41no—
569C23—41no—
570C23—41no—
571C23C2×D4×D6111yescharacteristic
572Q8—22no—
573Q8—22no—
574Q8—41no—
575Q8—41no—
576Q8—1222no—
577Q8—1222no—
578Q8—2441no—
579Q8—2441no—
580D4—22no—
581D4—22no—
582D4—22no—
583D4#1451973714yes—
584C2×C4—22no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—84no—
600C2×C4—612no—
601C2×C4—612no—
602C2×C4—612no—
603C2×C4—612no—
604C2×C4—1221no—
605C2×C4—1221no—
606C2×C4—1222no—
607C2×C4—1222no—
608C2×C4—1221no—
609C2×C4—1221no—
610C2×C4—1221no—
611C2×C4—1221no—
612C2×C4—1221no—
613C2×C4—1221no—
614C2×C4—1222no—
615C2×C4—4884no—
616C2×C4—2448no—
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—2441no—
626C2×C4C22×C4×D6111yescharacteristic
627C2×C4C2×D4×D6111yescharacteristic
628C2×C4#41×D6111yescharacteristic
629C2×C4C2×#1175822411yescharacteristic
630C2×C4#1451982211yescharacteristic
631C2×C4#1451982411yescharacteristic
632C8—22no—
633C8—41no—
634C8—1222no—
635C8—1222no—
636C22—24no—
637C22—42no—
638C22—42no—
639C22—42no—
640C22—42no—
641C22—42no—
642C22—42no—
643C22C4×D4×D6111yescharacteristic
644C22#1684039411yescharacteristic
645C22#1684041811yescharacteristic
646C22AGL(1,ℤ/8)×D6111yescharacteristic
647C22#137×D6111yescharacteristic
648C22#1684056811yescharacteristic
649C22#1684057611yescharacteristic
650C4—41no—
651C4—41no—
652C4—42no—
653C4—81no—
654C4—81no—
655C4—614no—
656C4—1222no—
657C4—4881no—
658C4—2442no—
659C4—2441no—
660C4—2442no—
661C4—2441no—
662C4—2442no—
663C4C2×#1451973711yescharacteristic
664C4#1684024111yescharacteristic
665C4#1684041011yescharacteristic
666C4#1684045511yescharacteristic
667C2—44no—
668C2#2144883011yescharacteristic
669C2#440×D6111yescharacteristic
670C2#2145465511yescharacteristic
671C2#2145741711yescharacteristic
672C2#2145746311yescharacteristic
673C2#2145853211yescharacteristic
674C2#2145859511yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1877C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
cosocleC22×C4×C61C24
hypercenter#1877D61
hypocenterC61#23002
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1877

Complements

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

NH ≅ G/Ntype
4. #21448876C2semidirect
6. #21446619C2semidirect
7. #21446576C2semidirect
8. #21446512C2semidirect
9. #21446507C2semidirect
11. #21446076C2semidirect
12. #21446071C2semidirect
13. #21445685C2semidirect
23. #16839965C4semidirect
71. #14519775C2×C4semidirect
103. #14519706C2×C4semidirect
105. #14519702C2×C4semidirect
107. #14519701C2×C4semidirect
329. Dic61#440semidirect
399. C61#23002semidirect

Maximal subgroups

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

subgroupindex
C2×#168399652
C2×#168398632
C2×#168398622
#214488762
D4×#117581962
#214466192
#214465762
#214465122
#214465072
#214461512
#214460762
#214460712
#214456852
#102×Dic612
C61×#18772
#2300261