Finite Group Databasebeta

← #29688576

Subgroups — #29688576

Order 15616.

1262 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840150C211yesmaximal, characteristic
2C2×#16839874C211yesmaximal, characteristic, complemented
3C2×#16839871C211yesmaximal, characteristic, complemented
4C2×#16839861C211yesmaximal, characteristic, complemented
5C2×#16839859C211yesmaximal, characteristic, complemented
6#21448383C211yesmaximal, characteristic, complemented
7#21448366C211yesmaximal, characteristic, complemented
8#21446684C211yesmaximal, characteristic, complemented
9#21446673C211yesmaximal, characteristic, complemented
10#21446672C211yesmaximal, characteristic, complemented
11#21446641C211yesmaximal, characteristic, complemented
12#21446634C211yesmaximal, characteristic, complemented
13#21445986C211yesmaximal, characteristic, complemented
14C4×#14519687C211yesmaximal, characteristic, complemented
15C61×#1970C211yesmaximal, characteristic, complemented
16C22×#11758190C2211yescharacteristic
17C22×#11758200C2211yescharacteristic
18C22×#11758199C2211yescharacteristic
19C2×#14519773C2211yescharacteristic
20C2×#14519764C2211yescharacteristic
21C2×#14519742C2211yescharacteristic
22C2×D4×Dic61C2211yescharacteristic
23C2×D4×Dic61C2211yescharacteristic
24C2×#14519739C2211yescharacteristic
25C2×#14519737—22no—
26#16840150C418yescomplemented
27#16839874—24no—
28#16839871—24no—
29#16839861—24no—
30#16839859—24no—
31C2×#14519710C2211yescharacteristic
32C2×#14519709C2211yescharacteristic
33C2×#14519708C2211yescharacteristic
34C2×#14519702C2211yescharacteristic
35C2×#14519700C2211yescharacteristic
36C2×#14519699C2211yescharacteristic
37C2×#14519698C2211yescharacteristic
38C2×#14519698C2211yescharacteristic
39C2×#14519697C2211yescharacteristic
40C2×#14519687C2211yescharacteristic
41C2×#14519687C2211yescharacteristic
42C2×#14519687C2211yescharacteristic
43C2×#14519687C2211yescharacteristic
44C2×C4×#92418C2211yescharacteristic
45#16839668C2211yescharacteristic
46#16839664C2211yescharacteristic
47#16839662—22no—
48#16839650—22no—
49#16839631C2211yescharacteristic
50#16839627C2211yescharacteristic
51C22⋊C4×Dic61—22no—
52C42×Dic61C2211yescharacteristic
53C22×D8×C61C2211yescharacteristic
54C2×#107×C61C2211yescharacteristic
55C2×#102×C61C2211yescharacteristic
56C2×#102×C61C2211yescharacteristic
57C2×C4×C8×C61C2211yescharacteristic
58C61×#390C2211yescharacteristic
59C61×#390C2211yescharacteristic
60C22×Dic122C2311yescharacteristic
61C22×D244C2311yescharacteristic
62C22×C22⋊D61—22no—
63C23×Dic61—21no—
64C23×Dic61—21no—
65C22×C4×D61—21no—
66C2×#11758190—22no—
67C2×#11758190—22no—
68C2×#11758190C2×C414yescomplemented
69C2×#11758200—22no—
70C2×#11758200—22no—
71C2×#11758200C2×C414yescomplemented
72C2×#11758199—22no—
73C2×#11758199—22no—
74C2×#11758199C2×C414yescomplemented
75C22×#92418C2311yescharacteristic
76C22×#92418C2311yescharacteristic
77#14519773—22no—
78#14519773C2×C414yescomplemented
79#14519764—22no—
80#14519764C2×C414yescomplemented
81#14519742—22no—
82#14519742—22no—
83D4×Dic61—22no—
84D4×Dic61—22no—
85D4×Dic61—22no—
86D4×Dic61—22no—
87#14519739—22no—
88#14519739—22no—
89#14519737—48no—
90C2×C4⋊Dic61C2311yescharacteristic
91C2×C4⋊Dic61C2311yescharacteristic
92C2×#11758196—22no—
93C2×#11758196—21no—
94C2×C22⋊Dic61—22no—
95C2×C22⋊Dic61—21no—
96C2×C22⋊Dic61—21no—
97C2×#11758198—22no—
98C2×#11758198—21no—
99C2×C4×Dic61—21no—
100C2×C4×Dic61D412yes—
101C2×C4×Dic61D412yes—
102C2×C4×Dic61C2311yescharacteristic
103C2×C4×Dic61C2311yescharacteristic
104#14519710—22no—
105#14519709—22no—
106#14519708—22no—
107#14519702—22no—
108#14519700—22no—
109#14519699—22no—
110#14519698—22no—
111#14519698—22no—
112#14519697—22no—
113#14519687—22no—
114#14519687C2×C414yescomplemented
115#14519687D414yescomplemented
116#14519687D414yescomplemented
117C4×#92418—22no—
118#14519684—22no—
119#14519684—22no—
120C22×D4×C61C2311yescharacteristic
121C22×D4×C61C2311yescharacteristic
122C2×D8×C61—22no—
123C2×D8×C61—22no—
124C2×D8×C61C2×C414yescomplemented
125C22×C8×C61C2311yescharacteristic
126C22×C8×C61C2311yescharacteristic
127C2×C4⋊C4×C61C2311yescharacteristic
128C2×C4⋊C4×C61C2311yescharacteristic
129C2×C22⋊C4×C61—22no—
130C2×C22⋊C4×C61—22no—
131C2×C42×C61C2311yescharacteristic
132#107×C61—22no—
133#102×C61—22no—
134#102×C61—22no—
135C4×C8×C61—22no—
136C24×C61—21no—
137C24×C61—21no—
138C23×D61—21no—
139C2×D4×C61—21no—
140C2×D4×C61—21no—
141C2×D4×C61—21no—
142C2×D4×C61—21no—
143C2×D4×C61—21no—
144C2×D4×C61—21no—
145C2×D4×C61—21no—
146C2×D4×C61—21no—
147C2×D4×C61—21no—
148C2×D4×C61—21no—
149C2×D4×C61C22×C412yes—
150C2×D4×C61C22×C412yes—
151C22×C4×C61—21no—
152C22×C4×C61—21no—
153C22×C4×C61C2×D412yes—
154C22×C4×C61C2411yescharacteristic
155C2×C22⋊D61—28no—
156C2×C22⋊D61—44no—
157C2×C22⋊D61—44no—
158C22×Dic61—21no—
159C22×Dic61—21no—
160C22×Dic61—21no—
161C22×Dic61—44no—
162C22×Dic61—44no—
163C22×Dic61—42no—
164C22×Dic61—42no—
165C22×Dic61C2×D412yes—
166C22×Dic61C2×D411yescharacteristic
167C22×Dic61C2×D411yescharacteristic
168C2×D244—21no—
169C2×D244—21no—
170C2×D244—21no—
171C2×D244—21no—
172C2×D244—21no—
173C2×D244C22×C412yes—
174C2×C4×D61—44no—
175C2×C4×D61—42no—
176C2×Dic122—21no—
177C2×Dic122—21no—
178C2×Dic122—21no—
179C2×Dic122—21no—
180C2×Dic122—21no—
181C2×Dic122C22×C412yes—
182D8×C61—28no—
183C2×C8×C61—21no—
184C2×C8×C61—21no—
185C2×C8×C61C22×C412yes—
186C2×C8×C61C2×D412yes—
187C2×C8×C61C2×D412yes—
188C2×C8×C61#4112yes—
189C4⋊C4×C61—21no—
190C4⋊C4×C61—21no—
191C4⋊C4×C61—21no—
192C4⋊C4×C61—21no—
193C22⋊C4×C61—44no—
194C22⋊C4×C61—44no—
195C42×C61—21no—
196C42×C61—21no—
197C22⋊Dic61—28no—
198C22⋊Dic61—42no—
199C22⋊Dic61—42no—
200#11758200—28no—
201#11758199—28no—
202#11758198—44no—
203#11758198—42no—
204C4⋊Dic61—21no—
205C4⋊Dic61—21no—
206C4⋊Dic61—21no—
207C4⋊Dic61—21no—
208#11758196—44no—
209#11758196—42no—
210C4×Dic61—24no—
211C4×Dic61—42no—
212C4×Dic61—42no—
213C4×Dic61C22×C412yes—
214C4×Dic61C2×D412yes—
215C4×Dic61C2×D412yes—
216C4×Dic61#4112yes—
217C2×#92418—21no—
218C2×#92418—21no—
219C2×#92418C22×C412yes—
220C2×#92418C2×D412yes—
221C2×#92418C2×D412yes—
222C2×#92418#4112yes—
223#11758190—28no—
224C23×C61—22no—
225C23×C61—22no—
226C23×C61—41no—
227C23×C61—41no—
228C23×C61—41no—
229C23×C61—41no—
230C23×C61—42no—
231C23×C61—42no—
232C23×C61—42no—
233C23×C61—42no—
234C23×C61C22×D411yescharacteristic
235C22×D61—22no—
236C22×D61—41no—
237C22×D61—41no—
238C22×D61—42no—
239C22×D61—42no—
240D4×C61—22no—
241D4×C61—22no—
242D4×C61—22no—
243D4×C61—22no—
244D4×C61—41no—
245D4×C61—41no—
246D4×C61—41no—
247D4×C61—41no—
248C2×C4×C61—22no—
249C2×C4×C61—22no—
250C2×C4×C61—41no—
251C2×C4×C61—41no—
252C2×C4×C61—42no—
253C2×C4×C61—42no—
254C2×C4×C61AGL(1,ℤ/8)14yes—
255C2×C4×C61C23×C411yescharacteristic
256C2×C4×C61C22×D411yescharacteristic
257C2×C4×C61C22×D411yescharacteristic
258C2×C4×C61C2×#4111yescharacteristic
259C2×C4×C61C2×#4111yescharacteristic
260C2×C4×C61C2×#4111yescharacteristic
261C22⋊D61—416no—
262C2×Dic61—22no—
263C2×Dic61—22no—
264C2×Dic61—22no—
265C2×Dic61—22no—
266C2×Dic61—22no—
267C2×Dic61—41no—
268C2×Dic61—41no—
269C2×Dic61—41no—
270C2×Dic61—41no—
271C2×Dic61—42no—
272C2×Dic61—42no—
273C2×Dic61—84no—
274C2×Dic61—84no—
275C2×Dic61C4×D414yes—
276C2×Dic61#12712yes—
277C2×Dic61#12712yes—
278D244—22no—
279D244—22no—
280D244—41no—
281D244—41no—
282C4×D61—84no—
283Dic122—22no—
284Dic122—22no—
285Dic122—41no—
286Dic122—41no—
287C8×C61—22no—
288C8×C61C4×D414yescomplemented
289#92418—22no—
290#92418C4×D414yescomplemented
291#21701—611nomaximal, Sylow
292C22×C61—42no—
293C22×C61—42no—
294C22×C61—44no—
295C22×C61—44no—
296C22×C61—81no—
297C22×C61—81no—
298C22×C61—81no—
299C22×C61—81no—
300C22×C61—82no—
301C22×C61—82no—
302C22×C61C2×C4×D411yescharacteristic
303C22×C61C2×#12711yescharacteristic
304C22×C61#53211yescharacteristic
305C22×C61C2×AGL(1,ℤ/8)12yes—
306C22×C61#57512yes—
307C2×D61—42no—
308C2×D61—44no—
309C2×D61—81no—
310C2×D61—81no—
311C2×D61—82no—
312C4×C61—24no—
313C4×C61—81no—
314C4×C61—81no—
315C4×C61C2×C4×D411yescharacteristic
316C4×C61C2×C4×D411yescharacteristic
317C4×C61#53211yescharacteristic
318C4×C61#53211yescharacteristic
319Dic61—22no—
320Dic61—22no—
321Dic61—24no—
322Dic61—42no—
323Dic61—81no—
324Dic61—81no—
325C2×#496—611no—
326C2×#442—611no—
327C2×#442—611no—
328C2×#440—611no—
329C2×#440—611no—
330C4×#127—611no—
331C4×#124—611no—
332#1970—611no—
333#1970D6111yescharacteristic, complemented
334#1968—611no—
335#1968—611no—
336#1961—611no—
337#1959—611no—
338#1847—611no—
339C4×#97—611no—
340C2×C61—82no—
341C2×C61—82no—
342C2×C61C4×#12711yescharacteristic
343C2×C61C4×AGL(1,ℤ/8)12yes—
344C2×C61#315111yescharacteristic
345C2×C61#315311yescharacteristic
346C2×C61#318012yes—
347D61—82no—
348C22×SD16—611no—
349C22×SD16—611no—
350C22×D8—611no—
351C22×D8C2×D6111yescharacteristic
352C2×#127—611no—
353C2×#124—611no—
354C2×C4×Q8—611no—
355C2×C4×D4—611no—
356C2×C4×D4—611no—
357C2×C4×D4—611no—
358C2×C4×D4—1222no—
359#496—618no—
360#442—1224no—
361#442—1224no—
362#440—1224no—
363#440—1224no—
364C2×#107—611no—
365C2×#107C2×D6111yescharacteristic
366C2×#106—611no—
367C2×#106—611no—
368C2×#103—611no—
369C2×#103—611no—
370C2×#102—611no—
371C2×#102—611no—
372C2×#102—611no—
373C2×#102—611no—
374C2×#102C2×D6111yescharacteristic
375C2×#102C2×D6111yescharacteristic
376C2×#97—611no—
377C2×#97—611no—
378C2×#97—611no—
379C2×#97—611no—
380C2×C4×C8—611no—
381C2×C4×C8C2×D6111yescharacteristic
382#391—611no—
383#390—611no—
384#390—1222no—
385#390C2×D6111yescharacteristic
386#390C2×D6111yescharacteristic
387#388—1222no—
388#384—611no—
389#382—611no—
390C4×C22⋊C4—1222no—
391C43—611no—
392C61#2170111yescharacteristic, complemented, Sylow
393C22×Q8—611no—
394C22×D4—611no—
395C22×D4—1222no—
396C22×D4C22×D6111yescharacteristic
397C22×D4C22×D6111yescharacteristic
398C23×C4—1221no—
399C23×C4—1221no—
400C23×C4—1221no—
401C2×SD16—614no—
402C2×SD16—614no—
403C2×SD16—1222no—
404C2×SD16—1222no—
405C2×SD16—1222no—
406C2×SD16—1222no—
407C2×D8—22no—
408C2×D8—22no—
409C2×D8—614no—
410C2×D8—1222no—
411C2×D8—1222no—
412C2×D8C4×D6114yescomplemented
413C22×C8—611no—
414C22×C8—611no—
415C22×C8C22×D6111yescharacteristic
416C22×C8C22×D6111yescharacteristic
417#127—614no—
418#127—1222no—
419#124—614no—
420#124—1222no—
421C4×Q8—1222no—
422C4×Q8—1222no—
423C4×D4—1222no—
424C4×D4—1222no—
425C4×D4—1222no—
426C4×D4—1222no—
427C4×D4—1222no—
428C4×D4—1222no—
429C4×D4—2448no—
430C2×C4⋊C4—611no—
431C2×C4⋊C4—611no—
432C2×C4⋊C4—1222no—
433C2×C4⋊C4—1221no—
434C2×C4⋊C4C22×D6111yescharacteristic
435C2×C4⋊C4C22×D6111yescharacteristic
436C2×C22⋊C4—22no—
437C2×C22⋊C4—22no—
438C2×C22⋊C4—1222no—
439C2×C22⋊C4—1221no—
440C2×C22⋊C4—1221no—
441C2×C22⋊C4—1221no—
442C2×C22⋊C4—1222no—
443C2×C42—611no—
444C2×C42—611no—
445C2×C42—612no—
446C2×C42—612no—
447C2×C42—1221no—
448C2×C42C22×D6111yescharacteristic
449#107—22no—
450#107—1222no—
451#106—1222no—
452#106—1222no—
453#103—1222no—
454#103—1222no—
455#102—22no—
456#102—22no—
457#102—1222no—
458#102—1222no—
459#102—1222no—
460#102—1222no—
461#97—614no—
462#97—614no—
463#97—614no—
464#97—1222no—
465C4×C8—22no—
466C4×C8—1222no—
467#95—1222no—
468#95—1222no—
469C24—21no—
470C24—21no—
471C24—1221no—
472C2×Q8—612no—
473C2×Q8—1221no—
474C2×Q8—1221no—
475C2×Q8—1221no—
476C2×Q8—1221no—
477C2×Q8—1221no—
478C2×D4—21no—
479C2×D4—21no—
480C2×D4—21no—
481C2×D4—21no—
482C2×D4—21no—
483C2×D4—21no—
484C2×D4—21no—
485C2×D4—21no—
486C2×D4—21no—
487C2×D4—21no—
488C2×D4—612no—
489C2×D4—1228no—
490C2×D4—1221no—
491C2×D4—1221no—
492C2×D4—1221no—
493C2×D4—1221no—
494C2×D4—1221no—
495C2×D4—2444no—
496C2×D4—2444no—
497C2×D4C2×C4×D6112yes—
498C2×D4C2×C4×D6112yes—
499C22×C4—21no—
500C22×C4—21no—
501C22×C4—612no—
502C22×C4—611no—
503C22×C4—611no—
504C22×C4—1221no—
505C22×C4—1221no—
506C22×C4—1221no—
507C22×C4—2442no—
508C22×C4—2444no—
509C22×C4—2442no—
510C22×C4—2442no—
511C22×C4—2444no—
512C22×C4—2444no—
513C22×C4D4×D6112yes—
514C22×C4C23×D6111yescharacteristic
515SD16—1228no—
516SD16—1228no—
517D8—28no—
518D8—1228no—
519C2×C8—21no—
520C2×C8—21no—
521C2×C8—612no—
522C2×C8—612no—
523C2×C8—612no—
524C2×C8—612no—
525C2×C8—1221no—
526C2×C8—1221no—
527C2×C8C2×C4×D6112yes—
528C2×C8D4×D6112yes—
529C2×C8D4×D6112yes—
530C2×C8#1175822412yes—
531C4⋊C4—21no—
532C4⋊C4—21no—
533C4⋊C4—21no—
534C4⋊C4—21no—
535C4⋊C4—1221no—
536C4⋊C4—1221no—
537C4⋊C4—1221no—
538C4⋊C4—1221no—
539C4⋊C4—2442no—
540C4⋊C4—2444no—
541C22⋊C4—44no—
542C22⋊C4—44no—
543C22⋊C4—1228no—
544C22⋊C4—2442no—
545C22⋊C4—2442no—
546C22⋊C4—2442no—
547C22⋊C4—2444no—
548C42—21no—
549C42—21no—
550C42—612no—
551C42—612no—
552C42—612no—
553C42—612no—
554C42—1224no—
555C42—2442no—
556C42—2442no—
557C23—22no—
558C23—22no—
559C23—42no—
560C23—42no—
561C23—41no—
562C23—41no—
563C23—42no—
564C23—42no—
565C23—41no—
566C23—41no—
567C23—1222no—
568C23—2442no—
569C23—2441no—
570C23—2442no—
571C23—2441no—
572C23C2×D4×D6111yescharacteristic
573Q8—1222no—
574Q8—1222no—
575Q8—2441no—
576Q8—2441no—
577D4—22no—
578D4—22no—
579D4—22no—
580D4—22no—
581D4—41no—
582D4—41no—
583D4—41no—
584D4—41no—
585D4—1222no—
586D4—1222no—
587D4—24416no—
588D4—2441no—
589D4—2441no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—614no—
597C2×C4—612no—
598C2×C4—612no—
599C2×C4—1222no—
600C2×C4—1222no—
601C2×C4—1222no—
602C2×C4—1222no—
603C2×C4—1222no—
604C2×C4—4884no—
605C2×C4—4884no—
606C2×C4—4884no—
607C2×C4—2442no—
608C2×C4—2441no—
609C2×C4—2441no—
610C2×C4—2442no—
611C2×C4—2441no—
612C2×C4—2441no—
613C2×C4#1451980114yes—
614C2×C4C22×C4×D6111yescharacteristic
615C2×C4C2×D4×D6111yescharacteristic
616C2×C4C2×D4×D6111yescharacteristic
617C2×C4#41×D6111yescharacteristic
618C2×C4#41×D6111yescharacteristic
619C2×C4C2×#1175822411yescharacteristic
620C8—22no—
621C8—614no—
622C8—1222no—
623C8#1451973714yescomplemented
624C22—44no—
625C22—44no—
626C22—42no—
627C22—42no—
628C22—82no—
629C22—82no—
630C22—81no—
631C22—81no—
632C22—81no—
633C22—81no—
634C22—4882no—
635C22—4881no—
636C22—4881no—
637C22—2444no—
638C22—2442no—
639C22C4×D4×D6111yescharacteristic
640C22#127×D6111yescharacteristic
641C22#1684030711yescharacteristic
642C22C2×#1451980112yes—
643C22#1684055912yes—
644C4—24no—
645C4—81no—
646C4—81no—
647C4—1224no—
648C4—1222no—
649C4—1222no—
650C4—4881no—
651C4—4881no—
652C4—2442no—
653C4C4×D4×D6111yescharacteristic
654C4C2×#1451973711yescharacteristic
655C4#1684030711yescharacteristic
656C4#1684031111yescharacteristic
657C2—82no—
658C2—82no—
659C2—4882no—
660C2#2144838311yescharacteristic
661C2C4×#1451980112yes—
662C2#2145627011yescharacteristic
663C2#2145628011yescharacteristic
664C2#2145651212yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1970C2
socleC23×C61C22×D4
prime radicalC2×#16840150C2
cosocleC22×C4×C61C24
hypercenter#1970D61
hypocenterC61#21701
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1970

Complements

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

NH ≅ G/Ntype
2. C2×#16839874C2semidirect
3. C2×#16839871C2semidirect
4. C2×#16839861C2semidirect
5. C2×#16839859C2semidirect
6. #21448383C2semidirect
7. #21448366C2semidirect
8. #21446684C2semidirect
9. #21446673C2semidirect
10. #21446672C2semidirect
11. #21446641C2semidirect
12. #21446634C2semidirect
13. #21445986C2semidirect
14. C4×#14519687C2semidirect
15. C61×#1970C2semidirect
26. #16840150C4semidirect
68. C2×#11758190C2×C4semidirect
71. C2×#11758200C2×C4semidirect
74. C2×#11758199C2×C4semidirect
78. #14519773C2×C4semidirect
80. #14519764C2×C4semidirect
114. #14519687C2×C4semidirect
115. #14519687D4semidirect
116. #14519687D4semidirect
124. C2×D8×C61C2×C4semidirect
288. C8×C61C4×D4semidirect
290. #92418C4×D4semidirect
333. #1970D61semidirect
392. C61#21701semidirect
412. C2×D8C4×D61semidirect
623. C8#14519737semidirect

Maximal subgroups

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

subgroupindex
C2×#168401502
C2×#168398742
C2×#168398712
C2×#168398612
C2×#168398592
#214483832
#214483662
#214466842
#214466732
#214466722
#214466412
#214466342
#214459862
C4×#145196872
C61×#19702
#2170161