Finite Group Databasebeta

← #29696644

Subgroups — #29696644

Order 15616.

1250 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D4×Dic122C211yesmaximal, characteristic
2C2×#16839891C211yesmaximal, characteristic, complemented
3#21448771C211yesmaximal, characteristic, complemented
4#21446791C211yesmaximal, characteristic, complemented
5#21446790C211yesmaximal, characteristic, complemented
6#21446778C211yesmaximal, characteristic, complemented
7#21446769C211yesmaximal, characteristic, complemented
8#21446766C211yesmaximal, characteristic, complemented
9#21446765C211yesmaximal, characteristic, complemented
10#21445827C211yesmaximal, characteristic
11#21445817C211yesmaximal, characteristic
12#21445646C211yesmaximal, characteristic
13#21445644C211yesmaximal, characteristic
14#102×Dic61C211yesmaximal, characteristic
15C61×#1804C211yesmaximal, characteristic
16C23×Dic122—21no—
17D4×Dic122—24no—
18D4×Dic122C418yescomplemented
19C2×#14519777C2211yescharacteristic
20C2×#14519776C2211yescharacteristic
21C2×#14519776C2211yescharacteristic
22C2×#14519756—21no—
23C2×#14519754—21no—
24C2×#14519754—21no—
25C2×C4×Dic122C2211yescharacteristic
26C2×D4×Dic61C2211yescharacteristic
27C2×D4×Dic61C2211yescharacteristic
28#16839891—24no—
29C2×#14519706C2211yescharacteristic
30C2×#14519706C2211yescharacteristic
31C2×#14519698C2211yescharacteristic
32C2×#14519698C2211yescharacteristic
33C2×#14519688C2211yescharacteristic
34C2×#14519687C2211yescharacteristic
35C2×C8×Dic61C2211yescharacteristic
36#16839666—21no—
37#16839647—21no—
38#16839644—21no—
39#16839631C2211yescharacteristic
40#16839627C2211yescharacteristic
41C4⋊C4×Dic61C2211yescharacteristic
42C4⋊C4×Dic61C2211yescharacteristic
43C4×C4⋊Dic61C2211yescharacteristic
44#16839613—21no—
45#16839525C2211yescharacteristic
46#16839525C2211yescharacteristic
47#16839525C2211yescharacteristic
48#16839525C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839524C2211yescharacteristic
53#16839524C2211yescharacteristic
54#16839524C2211yescharacteristic
55C2×C4×D4×C61C2211yescharacteristic
56C2×C4⋊C8×C61C2211yescharacteristic
57C2×#102×C61C2211yescharacteristic
58C2×#102×C61C2211yescharacteristic
59C61×#382C2211yescharacteristic
60C61×#340C2211yescharacteristic
61C61×#340C2211yescharacteristic
62C22×Dic122—21no—
63C22×Dic122—21no—
64C22×Dic122—21no—
65C22×Dic122—28no—
66C22×Dic122—44no—
67C22×Dic122—44no—
68C22×Dic122D411yescharacteristic
69C23×Dic61—21no—
70C23×Dic61—21no—
71C22×#92418C2311yescharacteristic
72C22×#92418C2311yescharacteristic
73#14519777—22no—
74#14519777C2×C414yes—
75#14519776—22no—
76#14519776—22no—
77#14519776C2×C414yes—
78#14519776C2×C414yes—
79#14519756—28no—
80#14519754—28no—
81#14519754—28no—
82C4×Dic122C2×C414yes—
83C4×Dic122D414yes—
84D4×Dic61—22no—
85D4×Dic61—22no—
86D4×Dic61C2×C414yes—
87D4×Dic61C2×C414yes—
88C2×C4⋊Dic61—21no—
89C2×C4⋊Dic61D411yescharacteristic
90C2×C4⋊Dic61C2311yescharacteristic
91C2×C4⋊Dic61C2311yescharacteristic
92C2×C4⋊Dic61C2311yescharacteristic
93C2×C4⋊Dic61C2311yescharacteristic
94C2×#11758196—21no—
95C2×#11758196—21no—
96C2×#11758196—21no—
97C2×#11758196—21no—
98C2×#11758196D411yescharacteristic
99C2×#11758196D411yescharacteristic
100C2×C22⋊Dic61—21no—
101C2×C22⋊Dic61—21no—
102C2×C4×Dic61—21no—
103C2×C4×Dic61—21no—
104C2×C4×Dic61—21no—
105C2×C4×Dic61C2311yescharacteristic
106C2×C4×Dic61C2311yescharacteristic
107#14519706—22no—
108#14519706—22no—
109#14519698—22no—
110#14519698Q814yescomplemented
111#14519688—22no—
112#14519687—22no—
113C8×Dic61—22no—
114#14519684—21no—
115#14519684—21no—
116#14519684—21no—
117#14519684—21no—
118#14519684—21no—
119C22×D4×C61C2311yescharacteristic
120C23×C4×C61—21no—
121C22×C8×C61C2311yescharacteristic
122C22×C8×C61C2311yescharacteristic
123C4×D4×C61—22no—
124C4×D4×C61C2×C414yes—
125C2×C4⋊C4×C61C2311yescharacteristic
126C2×C4⋊C4×C61C2311yescharacteristic
127C2×C4⋊C4×C61C2311yescharacteristic
128C2×C22⋊C4×C61—21no—
129C2×C42×C61C2311yescharacteristic
130C4⋊C8×C61—22no—
131#102×C61—22no—
132#102×C61—22no—
133#95×C61—21no—
134#95×C61—21no—
135C24×C61—21no—
136C2×D4×C61—21no—
137C2×D4×C61—21no—
138C2×D4×C61—21no—
139C2×D4×C61C22×C412yes—
140C2×D4×C61C2×Q812yes—
141C2×D4×C61#4112yes—
142C22×C4×C61—21no—
143C22×C4×C61—24no—
144C22×C4×C61—21no—
145C22×C4×C61—21no—
146C22×C4×C61—42no—
147C22×C4×C61—42no—
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2411yescharacteristic
151C22×Dic61—21no—
152C22×Dic61—21no—
153C22×Dic61—21no—
154C22×Dic61—24no—
155C22×Dic61—24no—
156C22×Dic61—21no—
157C22×Dic61—42no—
158C22×Dic61—42no—
159C22×Dic61—42no—
160C22×Dic61—42no—
161C22×Dic61C2×D411yescharacteristic
162C22×Dic61C2×D411yescharacteristic
163C22×Dic61C2×D411yescharacteristic
164C22×Dic61C2×D411yescharacteristic
165C2×Dic122—22no—
166C2×Dic122—22no—
167C2×Dic122—24no—
168C2×Dic122—24no—
169C2×Dic122—24no—
170C2×Dic122—432no—
171C2×Dic122—42no—
172C2×Dic122—42no—
173C2×Dic122—42no—
174C2×Dic122—42no—
175C2×Dic122—42no—
176C2×Dic122—42no—
177C2×Dic122C22⋊C414yes—
178C2×C8×C61—21no—
179C2×C8×C61—21no—
180C2×C8×C61—21no—
181C2×C8×C61—21no—
182C2×C8×C61—21no—
183C2×C8×C61—21no—
184C4⋊C4×C61—21no—
185C4⋊C4×C61—21no—
186C4⋊C4×C61—21no—
187C4⋊C4×C61—21no—
188C4⋊C4×C61—21no—
189C4⋊C4×C61C22×C412yes—
190C22⋊C4×C61—24no—
191C42×C61C22×C412yes—
192C42×C61C2×D412yes—
193C22⋊Dic61—24no—
194C22⋊Dic61—24no—
195C4⋊Dic61—24no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201C4⋊Dic61C22⋊C414yes—
202C4⋊Dic61C22×C412yes—
203C4⋊Dic61C22×C412yes—
204C4⋊Dic61C2×Q812yescomplemented
205#11758196—24no—
206#11758196—24no—
207#11758196—42no—
208#11758196—81no—
209#11758196C22⋊C414yes—
210#11758196C22⋊C414yes—
211C4×Dic61—42no—
212C4×Dic61—81no—
213C4×Dic61—81no—
214C4×Dic61C22×C412yes—
215C4×Dic61C22×C412yes—
216C4×Dic61C2×D412yes—
217C4×Dic61C2×D412yes—
218C2×#92418—21no—
219C2×#92418—21no—
220C2×#92418—21no—
221C2×#92418—21no—
222C2×#92418—21no—
223C2×#92418C2×Q812yescomplemented
224C23×C61—22no—
225C23×C61—22no—
226C23×C61—22no—
227C23×C61—41no—
228C23×C61—41no—
229C23×C61—41no—
230C23×C61—41no—
231C23×C61C22×D411yescharacteristic
232D4×C61—22no—
233D4×C61—22no—
234D4×C61—22no—
235D4×C61C4×Q814yes—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61—21no—
239C2×C4×C61—21no—
240C2×C4×C61—22no—
241C2×C4×C61—48no—
242C2×C4×C61—41no—
243C2×C4×C61—41no—
244C2×C4×C61—41no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61—41no—
248C2×C4×C61—41no—
249C2×C4×C61—41no—
250C2×C4×C61C2×C22⋊C412yes—
251C2×C4×C61C2×C22⋊C412yes—
252C2×C4×C61C23×C411yescharacteristic
253C2×C4×C61C22×D411yescharacteristic
254C2×C4×C61C22×Q811yescharacteristic
255C2×C4×C61C2×#4111yescharacteristic
256C2×C4×C612−1+411yescharacteristic
257C2×C4×C612−1+411yescharacteristic
258C2×Dic61—21no—
259C2×Dic61—21no—
260C2×Dic61—21no—
261C2×Dic61—21no—
262C2×Dic61—21no—
263C2×Dic61—21no—
264C2×Dic61—22no—
265C2×Dic61—22no—
266C2×Dic61—22no—
267C2×Dic61—48no—
268C2×Dic61—41no—
269C2×Dic61—41no—
270C2×Dic61—41no—
271C2×Dic61—41no—
272C2×Dic61—41no—
273C2×Dic61—41no—
274C2×Dic61—48no—
275C2×Dic61—41no—
276C2×Dic61—41no—
277C2×Dic61—41no—
278C2×Dic61C2×C22⋊C412yes—
279C2×Dic61C2×C22⋊C412yes—
280C2×Dic61C2×C22⋊C412yes—
281C2×Dic61C2×C22⋊C412yes—
282C2×Dic61AGL(1,ℤ/8)12yes—
283C2×Dic61AGL(1,ℤ/8)12yes—
284Dic122—28no—
285Dic122—48no—
286Dic122—48no—
287Dic122—48no—
288C8×C61—41no—
289C8×C61—41no—
290#92418—22no—
291#92418—41no—
292#22523—611nomaximal, Sylow
293C22×C61—24no—
294C22×C61—42no—
295C22×C61—42no—
296C22×C61—42no—
297C22×C61—42no—
298C22×C61—42no—
299C22×C61—42no—
300C22×C61C22×C22⋊C411yescharacteristic
301C22×C61#54811yescharacteristic
302C22×C61D4×Q811yescharacteristic
303C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
304C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
305C22×C61#57711yescharacteristic
306C22×C61#57811yescharacteristic
307C4×C61—22no—
308C4×C61—22no—
309C4×C61—42no—
310C4×C61—81no—
311C4×C61—81no—
312C4×C61C2×C4×Q811yescharacteristic
313C4×C61#51911yescharacteristic
314C4×C61#54811yescharacteristic
315C4×C61D4×Q811yescharacteristic
316Dic61—22no—
317Dic61—22no—
318Dic61—22no—
319Dic61—42no—
320Dic61—42no—
321Dic61—42no—
322Dic61—81no—
323Dic61#41814yes—
324C2×D4×Q8—611no—
325C2×#446—611no—
326#2355—611no—
327#1993—611no—
328#1993—611no—
329#1990—611no—
330#1988—611no—
331#1987—611no—
332#1987—611no—
333#1804—611no—
334#1804D6111yescharacteristic
335#1802—611no—
336#1762—611no—
337#1762—611no—
338C4×#102—611no—
339C2×C61—44no—
340C2×C61Q8×C22⋊C411yescharacteristic
341C2×C61C2×#41811yescharacteristic
342C2×C61#289811yescharacteristic
343C2×C61#336711yescharacteristic
344C2×C61#337311yescharacteristic
345C2×C61#338411yescharacteristic
346C2×C61#338611yescharacteristic
347C23×Q8—1221no—
348D4×Q8—618no—
349D4×Q8—1224no—
350C2×C4⋊Q8—611no—
351C2×C4⋊Q8—611no—
352C2×C4⋊Q8—611no—
353C2×C22⋊Q8—1221no—
354C2×C22⋊Q8—1221no—
355C2×C22⋊Q8—1221no—
356C2×C4×Q8—611no—
357C2×C4×D4—611no—
358C2×C4×D4—611no—
359C2×C4×D4C2×D6111yescharacteristic
360#446—1224no—
361C2×C4⋊C8—611no—
362C2×C4⋊C8—611no—
363C2×C4⋊C8C2×D6111yescharacteristic
364C2×#102—611no—
365C2×#102—611no—
366C2×#102C2×D6111yescharacteristic
367C2×#102C2×D6111yescharacteristic
368C2×#97—611no—
369C2×#97—611no—
370C2×C4×C8—611no—
371#391—1221no—
372#387—1221no—
373#387—1221no—
374#384—611no—
375#382—611no—
376#382C2×D6111yescharacteristic
377C4×C4⋊C4—611no—
378C4×C4⋊C4—611no—
379C4×C4⋊C4—611no—
380C4×C4⋊C4—1221no—
381#340—611no—
382#340—611no—
383#340—611no—
384#340—611no—
385#340—611no—
386#340—611no—
387#340—611no—
388#340—611no—
389#340—611no—
390#340—611no—
391#340C2×D6111yescharacteristic
392#340C2×D6111yescharacteristic
393C61#2252311yescharacteristic, complemented, Sylow
394C22×Q8—611no—
395C22×Q8—1221no—
396C22×Q8—1221no—
397C22×Q8—1228no—
398C22×Q8—1221no—
399C22×Q8—2444no—
400C22×Q8—2444no—
401C22×D4C22×D6111yescharacteristic
402C23×C4—21no—
403C23×C4—1221no—
404C23×C4—1221no—
405C22×C8—611no—
406C22×C8—611no—
407C22×C8C22×D6111yescharacteristic
408C22×C8C22×D6111yescharacteristic
409C4⋊Q8—614no—
410C4⋊Q8—614no—
411C4⋊Q8—614no—
412C4⋊Q8—1222no—
413C4⋊Q8—1222no—
414C4⋊Q8—1222no—
415C22⋊Q8—1228no—
416C22⋊Q8—1228no—
417C22⋊Q8—1228no—
418C4×Q8—614no—
419C4×Q8—614no—
420C4×D4—22no—
421C4×D4—614no—
422C4×D4—614no—
423C4×D4—1222no—
424C4×D4—1222no—
425C4×D4C4×D6114yes—
426C2×C4⋊C4—611no—
427C2×C4⋊C4—611no—
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⋊C4C22×D6111yescharacteristic
439C2×C4⋊C4C22×D6111yescharacteristic
440C2×C4⋊C4C22×D6111yescharacteristic
441C2×C22⋊C4—21no—
442C2×C22⋊C4—1221no—
443C2×C22⋊C4—1221no—
444C2×C42—611no—
445C2×C42—611no—
446C2×C42—1221no—
447C2×C42—1221no—
448C2×C42—1221no—
449C2×C42C22×D6111yescharacteristic
450C4⋊C8—22no—
451C4⋊C8—1222no—
452C4⋊C8—1222no—
453#102—22no—
454#102—22no—
455#102—614no—
456#102—1222no—
457#97—1222no—
458#97—1222no—
459C4×C8—1222no—
460#95—21no—
461#95—21no—
462#95—1221no—
463#95—1221no—
464#95—1221no—
465#95—1221no—
466#95—1221no—
467C24—21no—
468C2×Q8—614no—
469C2×Q8—1222no—
470C2×Q8—1222no—
471C2×Q8—1224no—
472C2×Q8—1224no—
473C2×Q8—1224no—
474C2×Q8—2442no—
475C2×Q8—2442no—
476C2×Q8—2442no—
477C2×Q8—2442no—
478C2×Q8—2442no—
479C2×Q8—2442no—
480C2×Q8—24432no—
481C2×D4—21no—
482C2×D4—21no—
483C2×D4—21no—
484C2×D4C2×C4×D6112yes—
485C2×D4#1175822412yes—
486C2×D4Q8×D6112yes—
487C22×C4—21no—
488C22×C4—24no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—42no—
492C22×C4—42no—
493C22×C4—611no—
494C22×C4—611no—
495C22×C4—611no—
496C22×C4—611no—
497C22×C4—1224no—
498C22×C4—1224no—
499C22×C4—1221no—
500C22×C4—1221no—
501C22×C4—1221no—
502C22×C4—1221no—
503C22×C4—2442no—
504C22×C4—2442no—
505C22×C4—2442no—
506C22×C4—2442no—
507C22×C4D4×D6111yescharacteristic
508C22×C4D4×D6111yescharacteristic
509C22×C4C23×D6111yescharacteristic
510C2×C8—21no—
511C2×C8—21no—
512C2×C8—21no—
513C2×C8—21no—
514C2×C8—21no—
515C2×C8—21no—
516C2×C8—612no—
517C2×C8—1221no—
518C2×C8—1221no—
519C2×C8—1221no—
520C2×C8—1221no—
521C2×C8—1221no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—614no—
528C4⋊C4—614no—
529C4⋊C4—614no—
530C4⋊C4—612no—
531C4⋊C4—612no—
532C4⋊C4—612no—
533C4⋊C4—1224no—
534C4⋊C4—1224no—
535C4⋊C4—1221no—
536C4⋊C4—1221no—
537C4⋊C4—1221no—
538C4⋊C4—1224no—
539C4⋊C4—1221no—
540C4⋊C4—1221no—
541C4⋊C4—4881no—
542C4⋊C4—2442no—
543C4⋊C4C2×C4×D6112yes—
544C22⋊C4—24no—
545C22⋊C4—1224no—
546C22⋊C4—1224no—
547C42—612no—
548C42—612no—
549C42—612no—
550C42—612no—
551C42—4881no—
552C42—4881no—
553C42—2442no—
554C42C2×C4×D6112yes—
555C42D4×D6112yes—
556C23—22no—
557C23—22no—
558C23—22no—
559C23—41no—
560C23—41no—
561C23—41no—
562C23—41no—
563C23C2×D4×D6111yescharacteristic
564Q8—1228no—
565Q8—2448no—
566Q8—2448no—
567Q8—2448no—
568D4—22no—
569D4—22no—
570D4—22no—
571D4#1451974214yes—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—48no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—612no—
587C2×C4—612no—
588C2×C4—612no—
589C2×C4—612no—
590C2×C4—612no—
591C2×C4—612no—
592C2×C4—1221no—
593C2×C4—1221no—
594C2×C4—1221no—
595C2×C4—1221no—
596C2×C4—1221no—
597C2×C4—1221no—
598C2×C4—1222no—
599C2×C4—1222no—
600C2×C4—1222no—
601C2×C4—2448no—
602C2×C4—2448no—
603C2×C4—2441no—
604C2×C4—2441no—
605C2×C4—2441no—
606C2×C4—2441no—
607C2×C4—2441no—
608C2×C4—2441no—
609C2×C4—2441no—
610C2×C4—2441no—
611C2×C4—2441no—
612C2×C4C22⋊C4×D6112yes—
613C2×C4C22⋊C4×D6112yes—
614C2×C4C22×C4×D6111yescharacteristic
615C2×C4C2×D4×D6111yescharacteristic
616C2×C4C2×Q8×D6111yescharacteristic
617C2×C4C2×#1175822411yescharacteristic
618C2×C4#1451982411yescharacteristic
619C2×C4#1451982411yescharacteristic
620C8—41no—
621C8—41no—
622C8—1222no—
623C8—2441no—
624C22—24no—
625C22—42no—
626C22—42no—
627C22—42no—
628C22—42no—
629C22—42no—
630C22—42no—
631C22C2×C22⋊C4×D6111yescharacteristic
632C22#1684041811yescharacteristic
633C22#1684042511yescharacteristic
634C22AGL(1,ℤ/8)×D6111yescharacteristic
635C22AGL(1,ℤ/8)×D6111yescharacteristic
636C22#1684056911yescharacteristic
637C22#1684057611yescharacteristic
638C4—22no—
639C4—22no—
640C4—42no—
641C4—81no—
642C4—81no—
643C4—614no—
644C4—1222no—
645C4—1222no—
646C4—1222no—
647C4—4881no—
648C4—2442no—
649C4—2442no—
650C4—2442no—
651C4C2×#1451974211yescharacteristic
652C4#1684023911yescharacteristic
653C4#1684041811yescharacteristic
654C4#1684042511yescharacteristic
655C2—44no—
656C2#2144865111yescharacteristic
657C2#418×D6111yescharacteristic
658C2#2145434811yescharacteristic
659C2#2145853211yescharacteristic
660C2#2145860611yescharacteristic
661C2#2145872111yescharacteristic
662C2#2145873511yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1804C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
cosocleC22×C4×C61C24
hypercenter#1804D61
hypocenterC61#22523
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1804

Complements

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

NH ≅ G/Ntype
2. C2×#16839891C2semidirect
3. #21448771C2semidirect
4. #21446791C2semidirect
5. #21446790C2semidirect
6. #21446778C2semidirect
7. #21446769C2semidirect
8. #21446766C2semidirect
9. #21446765C2semidirect
18. D4×Dic122C4semidirect
110. #14519698Q8semidirect
204. C4⋊Dic61C2×Q8semidirect
223. C2×#92418C2×Q8semidirect
393. C61#22523semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Dic1222
C2×#168398912
#214487712
#214467912
#214467902
#214467782
#214467692
#214467662
#214467652
#214458272
#214458172
#214456462
#214456442
#102×Dic612
C61×#18042
#2252361