Finite Group Databasebeta

← #29740106

Subgroups — #29740106

Order 15616.

904 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×SD16×Dic61C211yesmaximal, characteristic
2#21449591C211yesmaximal, characteristic, complemented
3D4×#11758196C211yesmaximal, characteristic
4#21447360C211yesmaximal, characteristic, complemented
5#21447317C211yesmaximal, characteristic, complemented
6#21447230C211yesmaximal, characteristic, complemented
7#21447062C211yesmaximal, characteristic, complemented
8#21446492C211yesmaximal, characteristic, complemented
9#21446373C211yesmaximal, characteristic, complemented
10#21446155C211yesmaximal, characteristic
11#21446155C211yesmaximal, characteristic
12#21446155C211yesmaximal, characteristic
13#21446085C211yesmaximal, characteristic, complemented
14#21446080C211yesmaximal, characteristic
15C2×C61×#461C211yesmaximal, characteristic
16C2×Q8×Dic61C2211yescharacteristic
17C2×D4×Dic61C2211yescharacteristic
18C2×D4×Dic61C2211yescharacteristic
19C22×#11758196—21no—
20SD16×Dic61—24no—
21C2×#14519709C2211yescharacteristic
22C2×#14519708C2211yescharacteristic
23C2×#14519701C2211yescharacteristic
24C2×#14519701C2211yescharacteristic
25C2×#14519701C2211yescharacteristic
26C2×#14519701C2211yescharacteristic
27C2×#14519698C2211yescharacteristic
28C2×#14519698C2211yescharacteristic
29C2×C8×Dic61C2211yescharacteristic
30#16839701—21no—
31#16839692—21no—
32#16839669C2211yescharacteristic
33#16839669C2211yescharacteristic
34#16839669C2211yescharacteristic
35#16839660C2211yescharacteristic
36#16839660C2211yescharacteristic
37#16839658C2211yescharacteristic
38#16839643—21no—
39#16839643—21no—
40#16839635—21no—
41#16839634—21no—
42#16839631C2211yescharacteristic
43#16839620—21no—
44C4×#11758196C2211yescharacteristic
45#16839525C2211yescharacteristic
46#16839525C2211yescharacteristic
47#16839525C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839517C2211yescharacteristic
50#16839516C2211yescharacteristic
51#16839516C2211yescharacteristic
52#16839516C2211yescharacteristic
53C22×SD16×C61C2211yescharacteristic
54C22×SD16×C61C2211yescharacteristic
55C2×C4⋊Q8×C61C2211yescharacteristic
56C2×C4×D4×C61C2211yescharacteristic
57C61×#461—24no—
58C2×C4⋊C8×C61C2211yescharacteristic
59C2×#103×C61C2211yescharacteristic
60C2×#103×C61C2211yescharacteristic
61C23×Dic61—21no—
62C23×Dic61—21no—
63C22×#92418C2311yescharacteristic
64C22×#92418C2311yescharacteristic
65Q8×Dic61—22no—
66Q8×Dic61—41no—
67D4×Dic61—22no—
68D4×Dic61—22no—
69D4×Dic61—22no—
70D4×Dic61—41no—
71C2×C4⋊Dic61—21no—
72C2×C4⋊Dic61C2311yescharacteristic
73C2×C4⋊Dic61C2311yescharacteristic
74C2×C4⋊Dic61C2311yescharacteristic
75C2×C4⋊Dic61C2311yescharacteristic
76C2×#11758196—21no—
77C2×#11758196—21no—
78C2×#11758196—21no—
79C2×#11758196—21no—
80C2×#11758196—21no—
81C2×#11758196—44no—
82C2×#11758196D411yescharacteristic
83C2×#11758196D411yescharacteristic
84C2×C22⋊Dic61—21no—
85C2×C22⋊Dic61—21no—
86C2×C4×Dic61—21no—
87C2×C4×Dic61C2311yescharacteristic
88C2×C4×Dic61C2311yescharacteristic
89#14519709—22no—
90#14519708—41no—
91#14519701—22no—
92#14519701—22no—
93#14519701—22no—
94#14519701—41no—
95#14519698—22no—
96#14519698—22no—
97C8×Dic61—22no—
98#14519684—21no—
99#14519684—21no—
100#14519684—21no—
101#14519684—21no—
102#14519684—21no—
103#14519684—21no—
104#14519684—21no—
105#14519684—21no—
106#14519684—21no—
107#14519684D411yescharacteristic
108#14519684D411yescharacteristic
109C22×Q8×C61C2311yescharacteristic
110C22×Q8×C61C2311yescharacteristic
111C22×D4×C61C2311yescharacteristic
112C23×C4×C61—21no—
113C2×SD16×C61—22no—
114C2×SD16×C61—22no—
115C2×SD16×C61—22no—
116C2×SD16×C61—22no—
117C2×SD16×C61—22no—
118C2×SD16×C61D414yes—
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×C61—21no—
127C2×C4⋊C4×C61C2311yescharacteristic
128C2×C22⋊C4×C61—21no—
129C2×C42×C61C2311yescharacteristic
130C4⋊C8×C61—22no—
131#103×C61—22no—
132#103×C61—22no—
133C24×C61—21no—
134C2×Q8×C61—21no—
135C2×Q8×C61—21no—
136C2×Q8×C61—21no—
137C2×Q8×C61—21no—
138C2×Q8×C61—21no—
139C2×Q8×C61—21no—
140C2×Q8×C61—21no—
141C2×Q8×C61—21no—
142C2×Q8×C61—21no—
143C2×Q8×C61—21no—
144C2×Q8×C61—21no—
145C2×Q8×C61C2×D412yes—
146C2×D4×C61—21no—
147C2×D4×C61—21no—
148C2×D4×C61—21no—
149C2×D4×C61C2×D412yes—
150C2×D4×C61#4112yes—
151C2×D4×C61#4112yes—
152C22×C4×C61—21no—
153C22×C4×C61—21no—
154C22×C4×C61—21no—
155C22×C4×C61—42no—
156C22×C4×C61—42no—
157C22×C4×C61—81no—
158C22×C4×C61C2×D411yescharacteristic
159C22×C4×C61C2×D411yescharacteristic
160C22×C4×C61C2411yescharacteristic
161C22×Dic61—24no—
162C22×Dic61—21no—
163C22×Dic61—21no—
164C22×Dic61—21no—
165C22×Dic61—21no—
166C22×Dic61—42no—
167C22×Dic61—42no—
168C22×Dic61—42no—
169C22×Dic61—42no—
170C22×Dic61—81no—
171C22×Dic61C2×D411yescharacteristic
172C22×Dic61C2×D411yescharacteristic
173C22×Dic61C2×D411yescharacteristic
174C22×Dic61C2×D411yescharacteristic
175SD16×C61—28no—
176SD16×C61—44no—
177C2×C8×C61—21no—
178C2×C8×C61—21no—
179C2×C8×C61—21no—
180C2×C8×C61—21no—
181C2×C8×C61—21no—
182C2×C8×C61C2×D412yes—
183C4⋊C4×C61—21no—
184C4⋊C4×C61—21no—
185C4⋊C4×C61—42no—
186C4⋊C4×C61—42no—
187C22⋊C4×C61—42no—
188C42×C61—21no—
189C42×C61—21no—
190C22⋊Dic61—24no—
191C22⋊Dic61—42no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61—42no—
200C4⋊Dic61#4112yes—
201#11758196—41no—
202#11758196—42no—
203#11758196—42no—
204#11758196—42no—
205#11758196—42no—
206#11758196—81no—
207#11758196SD1614yescomplemented
208C4×Dic61—21no—
209C4×Dic61—21no—
210C4×Dic61—21no—
211C4×Dic61—21no—
212C4×Dic61—42no—
213C2×#92418—21no—
214C2×#92418—21no—
215C2×#92418—21no—
216C2×#92418—21no—
217C2×#92418—21no—
218C2×#92418—21no—
219C23×C61—22no—
220C23×C61—22no—
221C23×C61—22no—
222C23×C61—41no—
223C23×C61—41no—
224C23×C61—41no—
225C23×C61—41no—
226C23×C61C22×D411yescharacteristic
227Q8×C61—22no—
228Q8×C61—22no—
229Q8×C61—41no—
230Q8×C61—41no—
231Q8×C61—41no—
232Q8×C61—41no—
233Q8×C61—41no—
234Q8×C61—41no—
235D4×C61—22no—
236D4×C61—22no—
237D4×C61—22no—
238D4×C61#12314yes—
239C2×C4×C61—21no—
240C2×C4×C61—21no—
241C2×C4×C61—22no—
242C2×C4×C61—21no—
243C2×C4×C61—21no—
244C2×C4×C61—21no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
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—84no—
254C2×C4×C61C2×SD1612yes—
255C2×C4×C61C22×D411yescharacteristic
256C2×C4×C61C2×#4111yescharacteristic
257C2×C4×C61C2×#4111yescharacteristic
258C2×C4×C61C2×#4111yescharacteristic
259C2×C4×C61H(5,𝔽2)11yescharacteristic
260C2×C4×C612−1+411yescharacteristic
261C2×Dic61—21no—
262C2×Dic61—21no—
263C2×Dic61—21no—
264C2×Dic61—21no—
265C2×Dic61—21no—
266C2×Dic61—22no—
267C2×Dic61—21no—
268C2×Dic61—21no—
269C2×Dic61—21no—
270C2×Dic61—21no—
271C2×Dic61—21no—
272C2×Dic61—48no—
273C2×Dic61—41no—
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—41no—
284C2×Dic61—84no—
285C2×Dic61C2×SD1612yes—
286C2×Dic61C2×SD1612yes—
287C8×C61—22no—
288C8×C61—41no—
289#92418—41no—
290#92418—41no—
291#26166—611nomaximal, Sylow
292C22×C61—24no—
293C22×C61—42no—
294C22×C61—42no—
295C22×C61—42no—
296C22×C61—42no—
297C22×C61—42no—
298C22×C61—42no—
299C22×C61#52511yescharacteristic
300C22×C61D4211yescharacteristic
301C22×C61#54811yescharacteristic
302C22×C61C22×SD1611yescharacteristic
303C22×C61#57511yescharacteristic
304C22×C61#57711yescharacteristic
305C22×C61#57811yescharacteristic
306C4×C61—22no—
307C4×C61—42no—
308C4×C61—41no—
309C4×C61—81no—
310C4×C61—81no—
311C4×C61C2×#12311yescharacteristic
312C4×C61#53911yescharacteristic
313C4×C61#54711yescharacteristic
314C4×C61#55311yescharacteristic
315Dic61—22no—
316Dic61—22no—
317Dic61—42no—
318Dic61—41no—
319Dic61—41no—
320Dic61—81no—
321Dic61—81no—
322Dic61—81no—
323C2×#461D6111yescharacteristic
324C2×C4×SD16—611no—
325#2446—611no—
326D4×C4⋊C4—611no—
327#2096—611no—
328#2088—611no—
329#2072—611no—
330#2043—611no—
331#1933—611no—
332#1916—611no—
333#1877—611no—
334#1877—611no—
335#1877—611no—
336#1865—611no—
337#1864—611no—
338C2×C61—44no—
339C2×C61#244311yescharacteristic
340C2×C61#310111yescharacteristic
341C2×C61#312011yescharacteristic
342C2×C61#326911yescharacteristic
343C2×C61D4×SD1611yescharacteristic
344C2×C61#335911yescharacteristic
345C2×C61#337611yescharacteristic
346C22×SD16C2×D6111yescharacteristic
347C22×SD16C2×D6111yescharacteristic
348C2×C4⋊Q8C2×D6111yescharacteristic
349C2×C4×Q8—611no—
350C2×C4×D4—611no—
351C2×C4×D4—611no—
352C2×C4×D4C2×D6111yescharacteristic
353C22×C4⋊C4—1221no—
354#461—24no—
355C4×SD16—1224no—
356C2×#106—611no—
357C2×#106—611no—
358C2×C4⋊C8C2×D6111yescharacteristic
359C2×#103—611no—
360C2×#103—611no—
361C2×#103—611no—
362C2×#103—611no—
363C2×#103C2×D6111yescharacteristic
364C2×#103C2×D6111yescharacteristic
365C2×#102—611no—
366C2×#102—611no—
367C2×C4×C8—611no—
368#400—1221no—
369#398—1221no—
370#391—611no—
371#391—611no—
372#391—611no—
373#389—611no—
374#389—611no—
375#389—611no—
376#387—1221no—
377#387—1221no—
378#385—1221no—
379#385—1221no—
380#384—611no—
381#380—1221no—
382C4×C4⋊C4—611no—
383#340—611no—
384#340—611no—
385#340—611no—
386#340—611no—
387#336—611no—
388#336—611no—
389#336—611no—
390#336—611no—
391C61#2616611yescharacteristic, complemented, Sylow
392C22×Q8C22×D6111yescharacteristic
393C22×Q8C22×D6111yescharacteristic
394C22×D4C22×D6111yescharacteristic
395C23×C4—21no—
396C23×C4—1221no—
397C23×C4—1221no—
398C2×SD16—22no—
399C2×SD16—22no—
400C2×SD16—22no—
401C2×SD16—22no—
402C2×SD16—22no—
403C2×SD16C22⋊D6114yes—
404C22×C8—611no—
405C22×C8—611no—
406C22×C8C22×D6111yescharacteristic
407C22×C8C22×D6111yescharacteristic
408C4⋊Q8—22no—
409C4⋊Q8—41no—
410C4×Q8—1222no—
411C4×Q8—2441no—
412C4×D4—22no—
413C4×D4—41no—
414C4×D4—1222no—
415C4×D4—1222no—
416C4×D4—1222no—
417C4×D4—2441no—
418C2×C4⋊C4—21no—
419C2×C4⋊C4—21no—
420C2×C4⋊C4—611no—
421C2×C4⋊C4—611no—
422C2×C4⋊C4—611no—
423C2×C4⋊C4—611no—
424C2×C4⋊C4—611no—
425C2×C4⋊C4—611no—
426C2×C4⋊C4—1221no—
427C2×C4⋊C4—1221no—
428C2×C4⋊C4—1221no—
429C2×C4⋊C4—1221no—
430C2×C4⋊C4—1221no—
431C2×C4⋊C4—1221no—
432C2×C4⋊C4—2444no—
433C2×C4⋊C4C22×D6111yescharacteristic
434C2×C22⋊C4—21no—
435C2×C22⋊C4—1221no—
436C2×C22⋊C4—1221no—
437C2×C42—611no—
438C2×C42—611no—
439C2×C42—1221no—
440C2×C42C22×D6111yescharacteristic
441#106—1222no—
442#106—2441no—
443C4⋊C8—22no—
444#103—22no—
445#103—22no—
446#103—1222no—
447#103—1222no—
448#103—1222no—
449#103—2441no—
450#102—1222no—
451#102—1222no—
452C4×C8—1222no—
453#95—611no—
454#95—611no—
455#95—1221no—
456#95—1221no—
457#95—1221no—
458#95—1221no—
459#95—1221no—
460#95—1221no—
461#95—1221no—
462#95—1221no—
463#95—1221no—
464C24—21no—
465C2×Q8—21no—
466C2×Q8—21no—
467C2×Q8—21no—
468C2×Q8—21no—
469C2×Q8—21no—
470C2×Q8—21no—
471C2×Q8—21no—
472C2×Q8—21no—
473C2×Q8—21no—
474C2×Q8—21no—
475C2×Q8—21no—
476C2×Q8C2×C22⋊D6112yes—
477C2×D4—21no—
478C2×D4—21no—
479C2×D4—21no—
480C2×D4#1175822412yes—
481C2×D4#1175822412yes—
482C2×D4C2×C22⋊D6112yes—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—42no—
487C22×C4—42no—
488C22×C4—81no—
489C22×C4—611no—
490C22×C4—611no—
491C22×C4—611no—
492C22×C4—611no—
493C22×C4—1224no—
494C22×C4—1221no—
495C22×C4—1221no—
496C22×C4—1221no—
497C22×C4—1221no—
498C22×C4—4881no—
499C22×C4—2442no—
500C22×C4—2442no—
501C22×C4—2442no—
502C22×C4—2442no—
503C22×C4D4×D6111yescharacteristic
504C22×C4D4×D6111yescharacteristic
505C22×C4C23×D6111yescharacteristic
506SD16—28no—
507SD16—44no—
508C2×C8—21no—
509C2×C8—21no—
510C2×C8—21no—
511C2×C8—21no—
512C2×C8—21no—
513C2×C8—1221no—
514C2×C8—1221no—
515C2×C8—1221no—
516C2×C8—1221no—
517C2×C8—1221no—
518C2×C8—1221no—
519C2×C8C2×C22⋊D6112yes—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—42no—
523C4⋊C4—42no—
524C4⋊C4—614no—
525C4⋊C4—612no—
526C4⋊C4—1221no—
527C4⋊C4—1221no—
528C4⋊C4—1221no—
529C4⋊C4—1221no—
530C4⋊C4—1221no—
531C4⋊C4—1221no—
532C4⋊C4—1221no—
533C4⋊C4—4881no—
534C4⋊C4—2441no—
535C4⋊C4—2442no—
536C4⋊C4—2442no—
537C4⋊C4—2442no—
538C4⋊C4—2442no—
539C4⋊C4—2442no—
540C22⋊C4—42no—
541C22⋊C4—1224no—
542C22⋊C4—2442no—
543C42—21no—
544C42—21no—
545C42—1221no—
546C42—1221no—
547C42—1221no—
548C42—1221no—
549C42—2442no—
550C23—22no—
551C23—22no—
552C23—22no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23C2×D4×D6111yescharacteristic
558Q8—22no—
559Q8—22no—
560Q8—41no—
561Q8—41no—
562Q8—41no—
563Q8—41no—
564Q8—41no—
565Q8—41no—
566D4—22no—
567D4—22no—
568D4—22no—
569D4#1451975814yes—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—84no—
585C2×C4—612no—
586C2×C4—612no—
587C2×C4—1221no—
588C2×C4—1221no—
589C2×C4—1221no—
590C2×C4—1221no—
591C2×C4—1221no—
592C2×C4—1222no—
593C2×C4—1221no—
594C2×C4—1221no—
595C2×C4—1221no—
596C2×C4—1221no—
597C2×C4—1221no—
598C2×C4—4884no—
599C2×C4—2448no—
600C2×C4—2441no—
601C2×C4—2441no—
602C2×C4—2441no—
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×C4SD16×D6112yes—
612C2×C4C22×C22⋊D6111yescharacteristic
613C2×C4#41×D6111yescharacteristic
614C2×C4C2×#1175822411yescharacteristic
615C2×C4C2×#1175822411yescharacteristic
616C2×C4#1451982211yescharacteristic
617C2×C4#1451982411yescharacteristic
618C8—22no—
619C8—41no—
620C8—2441no—
621C8—2441no—
622C22—24no—
623C22—42no—
624C22—42no—
625C22—42no—
626C22—42no—
627C22—42no—
628C22—42no—
629C22#1684027211yescharacteristic
630C22D4×C22⋊D6111yescharacteristic
631C22#1684041811yescharacteristic
632C22C2×SD16×D6111yescharacteristic
633C22#1684056111yescharacteristic
634C22#1684056811yescharacteristic
635C22#1684057811yescharacteristic
636C4—22no—
637C4—42no—
638C4—41no—
639C4—81no—
640C4—81no—
641C4—1222no—
642C4—1222no—
643C4—4881no—
644C4—4881no—
645C4—4881no—
646C4—2441no—
647C4—2442no—
648C4—2441no—
649C4C2×#1451975811yescharacteristic
650C4#1684035711yescharacteristic
651C4#1684041111yescharacteristic
652C4#1684045511yescharacteristic
653C2—44no—
654C2#2144955011yescharacteristic
655C2#2145582611yescharacteristic
656C2#2145603011yescharacteristic
657C2#2145736811yescharacteristic
658C2SD16×C22⋊D6111yescharacteristic
659C2#2145845611yescharacteristic
660C2#2145863911yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#461C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
hypercenterC2×#461D61
hypocenterC61#26166
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#461

Complements

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

NH ≅ G/Ntype
2. #21449591C2semidirect
4. #21447360C2semidirect
5. #21447317C2semidirect
6. #21447230C2semidirect
7. #21447062C2semidirect
8. #21446492C2semidirect
9. #21446373C2semidirect
13. #21446085C2semidirect
207. #11758196SD16semidirect
391. C61#26166semidirect

Maximal subgroups

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

subgroupindex
C2×SD16×Dic612
#214495912
D4×#117581962
#214473602
#214473172
#214472302
#214470622
#214464922
#214463732
#214461552
#214460852
#214460802
C2×C61×#4612
#2616661