Finite Group Databasebeta

← #29698429

Subgroups — #29698429

Order 15616.

986 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×SD16×Dic61C211yesmaximal, characteristic
2C2×SD16×Dic61C211yesmaximal, characteristic
3#21448737C211yesmaximal, characteristic, complemented
4#21448724C211yesmaximal, characteristic
5#21446547C211yesmaximal, characteristic, complemented
6#21446545C211yesmaximal, characteristic, complemented
7#21446073C211yesmaximal, characteristic, complemented
8#21446071C211yesmaximal, characteristic, complemented
9#21446066C211yesmaximal, characteristic
10#21446064C211yesmaximal, characteristic
11#21445980C211yesmaximal, characteristic, complemented
12#21445832C211yesmaximal, characteristic, complemented
13#21445827C211yesmaximal, characteristic
14C8×#11758196C211yesmaximal, characteristic, complemented
15C2×C4×SD16×C61C211yesmaximal, characteristic
16C2×Q8×Dic61C2211yescharacteristic
17C2×Q8×Dic61C2211yescharacteristic
18C2×D4×Dic61C2211yescharacteristic
19C2×D4×Dic61C2211yescharacteristic
20SD16×Dic61—24no—
21SD16×Dic61C418yescomplemented
22C2×#14519710C2211yescharacteristic
23C2×#14519710C2211yescharacteristic
24C2×#14519709C2211yescharacteristic
25C2×#14519709C2211yescharacteristic
26C2×#14519706C2211yescharacteristic
27C2×#14519701C2211yescharacteristic
28C2×#14519701C2211yescharacteristic
29C2×#14519698C2211yescharacteristic
30C2×#14519698C2211yescharacteristic
31C2×C8×Dic61C2211yescharacteristic
32C2×C8×Dic61C2211yescharacteristic
33#16839660—21no—
34#16839660C2211yescharacteristic
35#16839660C2211yescharacteristic
36#16839658—21no—
37#16839658C2211yescharacteristic
38#16839658C2211yescharacteristic
39#16839635—21no—
40#16839634—21no—
41#16839627—21no—
42#16839627C2211yescharacteristic
43#16839627C2211yescharacteristic
44#16839620—21no—
45C4×#11758196—21no—
46C4×#11758196C2211yescharacteristic
47C2×#14519684—21no—
48#16839525C2211yescharacteristic
49#16839525C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839517C2211yescharacteristic
53#16839516C2211yescharacteristic
54C22×SD16×C61C2211yescharacteristic
55C2×C4×Q8×C61C2211yescharacteristic
56C2×C4×D4×C61C2211yescharacteristic
57C4×SD16×C61—24no—
58C2×#106×C61C2211yescharacteristic
59C2×#103×C61C2211yescharacteristic
60C2×#102×C61C2211yescharacteristic
61C2×C4×C8×C61C2211yescharacteristic
62C23×Dic61—21no—
63C23×Dic61—21no—
64C22×#92418C2311yescharacteristic
65C22×#92418C2311yescharacteristic
66Q8×Dic61—22no—
67Q8×Dic61—22no—
68Q8×Dic61—41no—
69Q8×Dic61C2×C414yescomplemented
70D4×Dic61—22no—
71D4×Dic61—22no—
72D4×Dic61—41no—
73D4×Dic61C2×C414yes—
74C2×C4⋊Dic61—21no—
75C2×C4⋊Dic61—21no—
76C2×C4⋊Dic61C2311yescharacteristic
77C2×C4⋊Dic61C2311yescharacteristic
78C2×C4⋊Dic61C2311yescharacteristic
79C2×C4⋊Dic61C2311yescharacteristic
80C2×#11758196—21no—
81C2×#11758196—21no—
82C2×#11758196—21no—
83C2×#11758196—21no—
84C2×#11758196D411yescharacteristic
85C2×#11758196D411yescharacteristic
86C2×C22⋊Dic61—21no—
87C2×C22⋊Dic61—21no—
88C2×C4×Dic61—21no—
89C2×C4×Dic61—21no—
90C2×C4×Dic61C2311yescharacteristic
91C2×C4×Dic61C2311yescharacteristic
92#14519710—41no—
93#14519710—41no—
94#14519709—22no—
95#14519709C2×C414yescomplemented
96#14519706—22no—
97#14519701—22no—
98#14519701C2×C414yescomplemented
99#14519698—22no—
100#14519698C2×C414yes—
101C8×Dic61—22no—
102C8×Dic61C2×C414yescomplemented
103#14519684—21no—
104#14519684—21no—
105#14519684—21no—
106#14519684—21no—
107#14519684—21no—
108#14519684—21no—
109#14519684—21no—
110#14519684—21no—
111#14519684D411yescharacteristic
112#14519684D411yescharacteristic
113C22×Q8×C61C2311yescharacteristic
114C22×D4×C61C2311yescharacteristic
115C23×C4×C61—21no—
116C2×SD16×C61C2×C414yes—
117C2×SD16×C61D414yes—
118C2×SD16×C61Q814yes—
119C22×C8×C61C2311yescharacteristic
120C22×C8×C61C2311yescharacteristic
121C4×Q8×C61—22no—
122C4×Q8×C61—41no—
123C4×D4×C61—22no—
124C4×D4×C61—41no—
125C2×C4⋊C4×C61—21no—
126C2×C4⋊C4×C61C2311yescharacteristic
127C2×C4⋊C4×C61C2311yescharacteristic
128C2×C22⋊C4×C61—21no—
129C2×C42×C61—21no—
130C2×C42×C61C2311yescharacteristic
131#106×C61—22no—
132#103×C61—22no—
133#102×C61—22no—
134C4×C8×C61—22no—
135C24×C61—21no—
136C2×Q8×C61—21no—
137C2×Q8×C61—21no—
138C2×Q8×C61—21no—
139C2×Q8×C61C22×C412yes—
140C2×Q8×C61C2×D412yes—
141C2×Q8×C61C2×Q812yes—
142C2×D4×C61—21no—
143C2×D4×C61—21no—
144C2×D4×C61—21no—
145C2×D4×C61C22×C412yes—
146C2×D4×C61C2×D412yes—
147C2×D4×C61C2×Q812yes—
148C22×C4×C61—21no—
149C22×C4×C61—21no—
150C22×C4×C61—21no—
151C22×C4×C61—42no—
152C22×C4×C61—42no—
153C22×C4×C61—81no—
154C22×C4×C61C2×D411yescharacteristic
155C22×C4×C61C2×D411yescharacteristic
156C22×C4×C61C2411yescharacteristic
157C22×Dic61—21no—
158C22×Dic61—21no—
159C22×Dic61—24no—
160C22×Dic61—21no—
161C22×Dic61—21no—
162C22×Dic61—42no—
163C22×Dic61—42no—
164C22×Dic61—81no—
165C22×Dic61—81no—
166C22×Dic61—81no—
167C22×Dic61C2×D411yescharacteristic
168C22×Dic61C2×D411yescharacteristic
169C22×Dic61C2×D411yescharacteristic
170C22×Dic61C2×D411yescharacteristic
171SD16×C61C4⋊C4116yescomplemented
172C2×C8×C61—21no—
173C2×C8×C61—21no—
174C2×C8×C61—21no—
175C2×C8×C61C22×C412yes—
176C2×C8×C61C2×D412yes—
177C2×C8×C61C2×Q812yes—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61—42no—
183C22⋊C4×C61—42no—
184C42×C61—21no—
185C42×C61—21no—
186C42×C61—42no—
187C22⋊Dic61—24no—
188C22⋊Dic61—42no—
189C4⋊Dic61—21no—
190C4⋊Dic61—21no—
191C4⋊Dic61—24no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—42no—
197C4⋊Dic61C22×C412yes—
198C4⋊Dic61C22×C412yes—
199#11758196—22no—
200#11758196—22no—
201#11758196—42no—
202#11758196—42no—
203#11758196—81no—
204#11758196—81no—
205C4×Dic61—21no—
206C4×Dic61—21no—
207C4×Dic61—24no—
208C4×Dic61—42no—
209C4×Dic61C22×C412yes—
210C4×Dic61#4112yes—
211C2×#92418—21no—
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418—21no—
215C2×#92418—21no—
216C2×#92418C22×C412yes—
217C23×C61—22no—
218C23×C61—22no—
219C23×C61—22no—
220C23×C61—41no—
221C23×C61—41no—
222C23×C61—41no—
223C23×C61—41no—
224C23×C61C22×D411yescharacteristic
225Q8×C61—22no—
226Q8×C61—22no—
227Q8×C61—22no—
228Q8×C61C2×C4⋊C414yes—
229D4×C61—22no—
230D4×C61—22no—
231D4×C61—22no—
232D4×C61C2×C4⋊C414yes—
233C2×C4×C61—22no—
234C2×C4×C61—22no—
235C2×C4×C61—21no—
236C2×C4×C61—22no—
237C2×C4×C61—21no—
238C2×C4×C61—21no—
239C2×C4×C61—21no—
240C2×C4×C61—21no—
241C2×C4×C61—41no—
242C2×C4×C61—41no—
243C2×C4×C61—41no—
244C2×C4×C61—41no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61—84no—
248C2×C4×C61#13512yes—
249C2×C4×C61C23×C411yescharacteristic
250C2×C4×C61C22×D411yescharacteristic
251C2×C4×C61C22×Q811yescharacteristic
252C2×C4×C61C2×#4111yescharacteristic
253C2×C4×C61H(5,𝔽2)11yescharacteristic
254C2×C4×C612−1+411yescharacteristic
255C2×Dic61—21no—
256C2×Dic61—21no—
257C2×Dic61—21no—
258C2×Dic61—21no—
259C2×Dic61—21no—
260C2×Dic61—21no—
261C2×Dic61—22no—
262C2×Dic61—22no—
263C2×Dic61—21no—
264C2×Dic61—21no—
265C2×Dic61—21no—
266C2×Dic61—41no—
267C2×Dic61—41no—
268C2×Dic61—41no—
269C2×Dic61—41no—
270C2×Dic61—41no—
271C2×Dic61—41no—
272C2×Dic61—41no—
273C2×Dic61—41no—
274C2×Dic61—41no—
275C2×Dic61—41no—
276C2×Dic61—84no—
277C2×Dic61—84no—
278C2×Dic61C4×D412yes—
279C2×Dic61C4×D412yes—
280C2×Dic61#13512yes—
281C8×C61—41no—
282C8×C61C2×C4⋊C414yes—
283#92418—22no—
284#92418—41no—
285#22683—611nomaximal, Sylow
286C22×C61—24no—
287C22×C61—42no—
288C22×C61—42no—
289C22×C61—42no—
290C22×C61—42no—
291C22×C61—42no—
292C22×C61—42no—
293C22×C61C2×C4×D411yescharacteristic
294C22×C61D4211yescharacteristic
295C22×C61D4×Q811yescharacteristic
296C22×C61C2×#13511yescharacteristic
297C22×C61C2×#13511yescharacteristic
298C22×C61#57711yescharacteristic
299C22×C61#57711yescharacteristic
300C4×C61—24no—
301C4×C61—22no—
302C4×C61—41no—
303C4×C61—81no—
304C4×C61—81no—
305C4×C61C22×C4⋊C411yescharacteristic
306C4×C61#52011yescharacteristic
307C4×C61#54711yescharacteristic
308C4×C61#55411yescharacteristic
309Dic61—22no—
310Dic61—22no—
311Dic61—22no—
312Dic61—42no—
313Dic61—42no—
314Dic61—41no—
315Dic61—81no—
316Dic61—81no—
317C2×C4×SD16—611no—
318C2×C4×SD16—611no—
319C2×C4×SD16D6111yescharacteristic
320#2351—611no—
321#2349—611no—
322#1941—611no—
323#1941—611no—
324#1863—611no—
325#1863—611no—
326#1862—611no—
327#1862—611no—
328#1846—611no—
329#1805—611no—
330#1804—611no—
331C8×C4⋊C4—611no—
332C2×C61—44no—
333C2×C61D4×C4⋊C411yescharacteristic
334C2×C61C4×#13511yescharacteristic
335C2×C61#294911yescharacteristic
336C2×C61#328211yescharacteristic
337C2×C61#328211yescharacteristic
338C2×C61#338111yescharacteristic
339C2×C61#338111yescharacteristic
340C22×SD16C2×D6111yescharacteristic
341C2×C4×Q8—611no—
342C2×C4×Q8—611no—
343C2×C4×Q8C2×D6111yescharacteristic
344C2×C4×D4—611no—
345C2×C4×D4—611no—
346C2×C4×D4C2×D6111yescharacteristic
347C4×SD16—24no—
348C4×SD16—618no—
349C4×SD16—1224no—
350C2×#107—611no—
351C2×#107—611no—
352C2×#106—611no—
353C2×#106—611no—
354C2×#106C2×D6111yescharacteristic
355C2×C4⋊C8—611no—
356C2×#103—611no—
357C2×#103—611no—
358C2×#103C2×D6111yescharacteristic
359C2×#102—611no—
360C2×#102—611no—
361C2×#102C2×D6111yescharacteristic
362C2×C4×C8—611no—
363C2×C4×C8—611no—
364C2×C4×C8C2×D6111yescharacteristic
365#389—611no—
366#389—611no—
367#389—611no—
368#389—611no—
369#389—1221no—
370#389—1221no—
371#385—1221no—
372#385—1221no—
373#382—611no—
374#382—611no—
375#382—1221no—
376#380—1221no—
377C4×C4⋊C4—611no—
378C4×C4⋊C4—1221no—
379C2×#95—1221no—
380#340—611no—
381#340—611no—
382#340—611no—
383#340—611no—
384#336—611no—
385#336—611no—
386C61#2268311yescharacteristic, complemented, Sylow
387C22×Q8C22×D6111yescharacteristic
388C22×D4C22×D6111yescharacteristic
389C23×C4—21no—
390C23×C4—1221no—
391C23×C4—1221no—
392C2×SD16Dic12214yes—
393C2×SD16C4×D6114yes—
394C2×SD16C22⋊D6114yes—
395C22×C8—611no—
396C22×C8—611no—
397C22×C8C22×D6111yescharacteristic
398C22×C8C22×D6111yescharacteristic
399C4×Q8—22no—
400C4×Q8—41no—
401C4×Q8—614no—
402C4×Q8—1222no—
403C4×Q8—1222no—
404C4×Q8—2441no—
405C4×D4—22no—
406C4×D4—41no—
407C4×D4—614no—
408C4×D4—1222no—
409C4×D4—1222no—
410C4×D4—2441no—
411C2×C4⋊C4—21no—
412C2×C4⋊C4—611no—
413C2×C4⋊C4—611no—
414C2×C4⋊C4—611no—
415C2×C4⋊C4—611no—
416C2×C4⋊C4—611no—
417C2×C4⋊C4—611no—
418C2×C4⋊C4—1221no—
419C2×C4⋊C4—1221no—
420C2×C4⋊C4—1221no—
421C2×C4⋊C4—1221no—
422C2×C4⋊C4—1221no—
423C2×C4⋊C4—1221no—
424C2×C4⋊C4C22×D6111yescharacteristic
425C2×C4⋊C4C22×D6111yescharacteristic
426C2×C22⋊C4—21no—
427C2×C22⋊C4—1221no—
428C2×C22⋊C4—1221no—
429C2×C42—21no—
430C2×C42—611no—
431C2×C42—611no—
432C2×C42—1221no—
433C2×C42—1221no—
434C2×C42C22×D6111yescharacteristic
435#107—2441no—
436#107—2441no—
437#106—22no—
438#106—614no—
439#106—1222no—
440C4⋊C8—1222no—
441#103—22no—
442#103—614no—
443#103—1222no—
444#102—22no—
445#102—614no—
446#102—1222no—
447C4×C8—22no—
448C4×C8—614no—
449C4×C8—1222no—
450#95—611no—
451#95—611no—
452#95—1221no—
453#95—1221no—
454#95—1221no—
455#95—1221no—
456#95—1221no—
457#95—1221no—
458#95—1221no—
459#95—1221no—
460C24—21no—
461C2×Q8—21no—
462C2×Q8—21no—
463C2×Q8—21no—
464C2×Q8C2×Dic12212yes—
465C2×Q8C2×C4×D6112yes—
466C2×Q8C2×C22⋊D6112yes—
467C2×D4—21no—
468C2×D4—21no—
469C2×D4—21no—
470C2×D4C2×Dic12212yes—
471C2×D4C2×C4×D6112yes—
472C2×D4C2×C22⋊D6112yes—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—81no—
479C22×C4—611no—
480C22×C4—611no—
481C22×C4—611no—
482C22×C4—611no—
483C22×C4—1224no—
484C22×C4—1221no—
485C22×C4—1221no—
486C22×C4—1221no—
487C22×C4—1221no—
488C22×C4—4881no—
489C22×C4—4881no—
490C22×C4—4881no—
491C22×C4—2442no—
492C22×C4—2442no—
493C22×C4D4×D6111yescharacteristic
494C22×C4D4×D6111yescharacteristic
495C22×C4C23×D6111yescharacteristic
496SD16#11758196116yescomplemented
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—612no—
501C2×C8—1221no—
502C2×C8—1221no—
503C2×C8—1221no—
504C2×C8—1221no—
505C2×C8—1221no—
506C2×C8C2×Dic12212yes—
507C2×C8C2×C4×D6112yes—
508C2×C8C2×C22⋊D6112yes—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—42no—
514C4⋊C4—612no—
515C4⋊C4—612no—
516C4⋊C4—1222no—
517C4⋊C4—1224no—
518C4⋊C4—1221no—
519C4⋊C4—1221no—
520C4⋊C4—1222no—
521C4⋊C4—1221no—
522C4⋊C4—1221no—
523C4⋊C4—1221no—
524C4⋊C4—1221no—
525C4⋊C4—4881no—
526C4⋊C4—4881no—
527C4⋊C4—2442no—
528C4⋊C4—2442no—
529C4⋊C4—2442no—
530C22⋊C4—42no—
531C22⋊C4—1224no—
532C22⋊C4—2442no—
533C42—21no—
534C42—21no—
535C42—42no—
536C42—612no—
537C42—612no—
538C42—1224no—
539C42—1221no—
540C42—1221no—
541C42—2442no—
542C23—22no—
543C23—22no—
544C23—22no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23C2×D4×D6111yescharacteristic
550Q8—22no—
551Q8—22no—
552Q8—22no—
553Q8C2×#1175819614yes—
554D4—22no—
555D4—22no—
556D4—22no—
557D4C2×#1175819614yes—
558C2×C4—22no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—84no—
573C2×C4—612no—
574C2×C4—612no—
575C2×C4—612no—
576C2×C4—1221no—
577C2×C4—1221no—
578C2×C4—1221no—
579C2×C4—1221no—
580C2×C4—1221no—
581C2×C4—1221no—
582C2×C4—1222no—
583C2×C4—1222no—
584C2×C4—1221no—
585C2×C4—1221no—
586C2×C4—1221no—
587C2×C4—4884no—
588C2×C4—4884no—
589C2×C4—2441no—
590C2×C4—2441no—
591C2×C4—2441no—
592C2×C4—2441no—
593C2×C4—2441no—
594C2×C4—2441no—
595C2×C4—2441no—
596C2×C4—2441no—
597C2×C4—2441no—
598C2×C4—2441no—
599C2×C4#1451979812yes—
600C2×C4C22×C4×D6111yescharacteristic
601C2×C4C22×C22⋊D6111yescharacteristic
602C2×C4C22×Dic12211yescharacteristic
603C2×C4#41×D6111yescharacteristic
604C2×C4#1451982211yescharacteristic
605C2×C4#1451982411yescharacteristic
606C8—41no—
607C8—1222no—
608C8—2441no—
609C8C2×#1175819614yes—
610C22—24no—
611C22—42no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22—42no—
616C22—42no—
617C22C4×D4×D6111yescharacteristic
618C22D4×C22⋊D6111yescharacteristic
619C22D4×Dic12211yescharacteristic
620C22#135×D6111yescharacteristic
621C22C2×#1451979811yescharacteristic
622C22#1684056811yescharacteristic
623C22#1684056911yescharacteristic
624C4—24no—
625C4—22no—
626C4—41no—
627C4—81no—
628C4—81no—
629C4—1222no—
630C4—1222no—
631C4—1222no—
632C4—4881no—
633C4—4881no—
634C4—2442no—
635C4—2441no—
636C4—2442no—
637C4C22×#1175819611yescharacteristic
638C4#1684024111yescharacteristic
639C4#1684041111yescharacteristic
640C4#1684047111yescharacteristic
641C2—44no—
642C2D4×#1175819611yescharacteristic
643C2C4×#1451979811yescharacteristic
644C2#2145467311yescharacteristic
645C2#2145752011yescharacteristic
646C2#2145752611yescharacteristic
647C2#2145869211yescharacteristic
648C2#2145869511yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC2×C4×SD16×C61C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
cosocleC22×C4×C61C24
hypercenterC2×C4×SD16D61
hypocenterC61#22683
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×SD16

Complements

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

NH ≅ G/Ntype
3. #21448737C2semidirect
5. #21446547C2semidirect
6. #21446545C2semidirect
7. #21446073C2semidirect
8. #21446071C2semidirect
11. #21445980C2semidirect
12. #21445832C2semidirect
14. C8×#11758196C2semidirect
21. SD16×Dic61C4semidirect
69. Q8×Dic61C2×C4semidirect
95. #14519709C2×C4semidirect
98. #14519701C2×C4semidirect
102. C8×Dic61C2×C4semidirect
171. SD16×C61C4⋊C4semidirect
386. C61#22683semidirect
496. SD16#11758196semidirect

Maximal subgroups

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

subgroupindex
C2×SD16×Dic612
#214487372
#214487242
#214465472
#214465452
#214460732
#214460712
#214460662
#214460642
#214459802
#214458322
#214458272
C8×#117581962
C2×C4×SD16×C612
#2268361