Finite Group Databasebeta

← #29697170

Subgroups — #29697170

Order 15616.

1006 conjugacy classes of subgroups.

The subgroup lattice has 690 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839755C211yesmaximal, characteristic
2#21448700C211yesmaximal, characteristic, complemented
3#21448644C211yesmaximal, characteristic
4#21446495C211yesmaximal, characteristic, complemented
5#21446493C211yesmaximal, characteristic, complemented
6#21446453C211yesmaximal, characteristic, complemented
7#21446451C211yesmaximal, characteristic, complemented
8#21446080C211yesmaximal, characteristic, complemented
9#21446078C211yesmaximal, characteristic, complemented
10#21445912C211yesmaximal, characteristic, complemented
11#21445843C211yesmaximal, characteristic, complemented
12#21445756C211yesmaximal, characteristic
13#102×Dic61C211yesmaximal, characteristic, complemented
14C8×#11758198C211yesmaximal, characteristic, complemented
15C4×#102×C61C211yesmaximal, characteristic, complemented
16C22×#11758224C2211yescharacteristic
17C22×C8×D61C2211yescharacteristic
18C2×D4×Dic61C2211yescharacteristic
19C2×#14519735C2211yescharacteristic
20C2×#14519708C2211yescharacteristic
21C2×#14519708C2211yescharacteristic
22#16839755C418yescomplemented
23C2×#14519702C2211yescharacteristic
24C2×#14519702C2211yescharacteristic
25C2×#14519700C2211yescharacteristic
26C2×#14519700C2211yescharacteristic
27C2×#14519698C2211yescharacteristic
28C2×#14519698C2211yescharacteristic
29C2×#14519689C2211yescharacteristic
30C2×C8×Dic61C2211yescharacteristic
31#16839668C2211yescharacteristic
32#16839667C2211yescharacteristic
33#16839660C2211yescharacteristic
34#16839658C2211yescharacteristic
35#16839653—21no—
36#16839651—21no—
37#16839641—21no—
38#16839639—21no—
39#16839626—21no—
40#16839623—21no—
41#16839619C2211yescharacteristic
42#16839619C2211yescharacteristic
43C4⋊C4×Dic61C2211yescharacteristic
44C4×#11758198C2211yescharacteristic
45#16839608—21no—
46C2×#14519684—21no—
47#16839525C2211yescharacteristic
48#16839525C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839517C2211yescharacteristic
52#16839516C2211yescharacteristic
53C2×C4×D4×C61C2211yescharacteristic
54C2×#102×C61C2211yescharacteristic
55C2×#102×C61C2211yescharacteristic
56C2×C4×C8×C61C2211yescharacteristic
57C4×C4⋊C4×C61C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C61×#340C2211yescharacteristic
60C2×#11758224—22no—
61C2×#11758224—22no—
62C2×#11758224—22no—
63C2×#11758224—41no—
64C2×#11758224—41no—
65C2×#11758224C2×C414yes—
66C22×Dic122C2311yescharacteristic
67C22×C22⋊D61—21no—
68C23×Dic61—21no—
69C23×Dic61—21no—
70C22×C4×D61C2311yescharacteristic
71C22×#92418C2311yescharacteristic
72C22×#92418C2311yescharacteristic
73C2×C8×D61—41no—
74C2×C8×D61—41no—
75C2×C8×D61C2×C414yescomplemented
76D4×Dic61—22no—
77D4×Dic61—41no—
78#14519735—22no—
79#14519735C2×C414yescomplemented
80C2×C4⋊Dic61C2311yescharacteristic
81C2×C4⋊Dic61C2311yescharacteristic
82C2×C4⋊Dic61C2311yescharacteristic
83C2×#11758196—21no—
84C2×#11758196—21no—
85C2×C22⋊Dic61—21no—
86C2×#11758198—21no—
87C2×#11758198—21no—
88C2×#11758198—21no—
89C2×#11758198D411yescharacteristic
90C2×#11758198D411yescharacteristic
91C2×C4×Dic61—21no—
92C2×C4×Dic61—21no—
93C2×C4×Dic61C2311yescharacteristic
94#14519708—41no—
95#14519708—41no—
96#14519702—41no—
97#14519702C2×C414yescomplemented
98#14519700—41no—
99#14519700C2×C414yescomplemented
100#14519698—22no—
101#14519698C2×C414yes—
102#14519689—22no—
103C8×Dic61—41no—
104#14519684—21no—
105#14519684—21no—
106#14519684—21no—
107#14519684—21no—
108#14519684—21no—
109#14519684—21no—
110#14519684—21no—
111#14519684—21no—
112#14519684—21no—
113#14519684D411yescharacteristic
114#14519684D411yescharacteristic
115C22×D4×C61C2311yescharacteristic
116C23×C4×C61—21no—
117C22×C8×C61C2311yescharacteristic
118C22×C8×C61C2311yescharacteristic
119C4×D4×C61—22no—
120C4×D4×C61—41no—
121C2×C4⋊C4×C61C2311yescharacteristic
122C2×C4⋊C4×C61C2311yescharacteristic
123C2×C4⋊C4×C61C2311yescharacteristic
124C2×C22⋊C4×C61—21no—
125C2×C42×C61—21no—
126C2×C42×C61C2311yescharacteristic
127#102×C61—22no—
128#102×C61C2×C414yes—
129C4×C8×C61—41no—
130#95×C61—21no—
131C24×C61—21no—
132C23×D61C2×D411yescharacteristic
133C2×D4×C61—21no—
134C2×D4×C61—21no—
135C2×D4×C61—21no—
136C2×D4×C61C22×C412yes—
137C2×D4×C61#4112yes—
138C2×D4×C61#4112yes—
139C22×C4×C61—21no—
140C22×C4×C61—21no—
141C22×C4×C61—21no—
142C22×C4×C61—42no—
143C22×C4×C61—42no—
144C22×C4×C61—81no—
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2411yescharacteristic
148C2×C22⋊D61—24no—
149C2×C22⋊D61—42no—
150C2×C22⋊D61—42no—
151C22×Dic61—21no—
152C22×Dic61—21no—
153C22×Dic61—24no—
154C22×Dic61—21no—
155C22×Dic61—21no—
156C22×Dic61—42no—
157C22×Dic61—42no—
158C22×Dic61—42no—
159C22×Dic61—42no—
160C22×Dic61—81no—
161C22×Dic61C2×D411yescharacteristic
162C22×Dic61C2×D411yescharacteristic
163C22×Dic61C2×D411yescharacteristic
164#11758224—28no—
165#11758224—44no—
166#11758224—44no—
167#11758224—44no—
168C2×C4×D61—21no—
169C2×C4×D61—21no—
170C2×C4×D61—21no—
171C2×C4×D61—21no—
172C2×C4×D61C22×C412yes—
173C2×C4×D61#4112yes—
174C2×Dic122—21no—
175C2×Dic122—21no—
176C2×Dic122—21no—
177C2×Dic122—21no—
178C2×Dic122—21no—
179C2×Dic122C22×C412yes—
180C2×C8×C61—21no—
181C2×C8×C61—21no—
182C2×C8×C61—21no—
183C2×C8×C61—21no—
184C2×C8×C61—21no—
185C2×C8×C61C22×C412yes—
186C4⋊C4×C61—21no—
187C4⋊C4×C61—21no—
188C4⋊C4×C61—21no—
189C4⋊C4×C61—21no—
190C4⋊C4×C61—21no—
191C4⋊C4×C61C22×C412yes—
192C22⋊C4×C61—42no—
193C42×C61—21no—
194C42×C61—21no—
195C42×C61—42no—
196C22⋊Dic61—42no—
197#11758198—24no—
198#11758198—22no—
199#11758198—22no—
200#11758198—81no—
201#11758198—81no—
202C4⋊Dic61—21no—
203C4⋊Dic61—21no—
204C4⋊Dic61—21no—
205C4⋊Dic61—21no—
206C4⋊Dic61—21no—
207C4⋊Dic61C22×C412yes—
208#11758196—81no—
209#11758196—81no—
210C4×Dic61—21no—
211C4×Dic61—21no—
212C4×Dic61—24no—
213C4×Dic61—42no—
214C2×#92418—21no—
215C2×#92418—21no—
216C2×#92418—21no—
217C2×#92418—21no—
218C2×#92418—21no—
219C2×#92418C22×C412yes—
220C8×D61—44no—
221C23×C61—22no—
222C23×C61—22no—
223C23×C61—22no—
224C23×C61—41no—
225C23×C61—41no—
226C23×C61—41no—
227C23×C61—41no—
228C23×C61C22×D411yescharacteristic
229C22×D61—21no—
230C22×D61—21no—
231C22×D61—21no—
232C22×D61—21no—
233C22×D61—21no—
234C22×D61—21no—
235C22×D61C4×D412yes—
236D4×C61—22no—
237D4×C61—22no—
238D4×C61—22no—
239D4×C61#11714yes—
240C2×C4×C61—21no—
241C2×C4×C61—21no—
242C2×C4×C61—22no—
243C2×C4×C61—21no—
244C2×C4×C61—21no—
245C2×C4×C61—21no—
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—41no—
254C2×C4×C61—84no—
255C2×C4×C61#13512yes—
256C2×C4×C61C23×C411yescharacteristic
257C2×C4×C61C2×#4111yescharacteristic
258C2×C4×C61C2×#4111yescharacteristic
259C2×C4×C61C2×#4111yescharacteristic
260C2×C4×C61H(5,𝔽2)11yescharacteristic
261C2×C4×C61H(5,𝔽2)11yescharacteristic
262C22⋊D61—48no—
263C2×Dic61—21no—
264C2×Dic61—21no—
265C2×Dic61—21no—
266C2×Dic61—21no—
267C2×Dic61—22no—
268C2×Dic61—22no—
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—84no—
284C2×Dic61C4×D412yes—
285C2×Dic61#13512yes—
286C4×D61—22no—
287C4×D61—22no—
288C4×D61—22no—
289C4×D61—22no—
290Dic122—22no—
291Dic122—22no—
292Dic122—41no—
293Dic122—41no—
294C8×C61—22no—
295C8×C61—41no—
296#92418—22no—
297#92418—41no—
298#22572—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×C61#54611yescharacteristic
308C22×C61#54611yescharacteristic
309C22×C61C2×#13511yescharacteristic
310C22×C61C2×#13511yescharacteristic
311C22×C61#57711yescharacteristic
312C22×C61#57711yescharacteristic
313C2×D61—22no—
314C2×D61—22no—
315C2×D61—22no—
316C2×D61—41no—
317C2×D61—41no—
318C2×D61—41no—
319C2×D61—41no—
320C4×C61—22no—
321C4×C61—42no—
322C4×C61—41no—
323C4×C61—81no—
324C4×C61—81no—
325C4×C61C2×#11711yescharacteristic
326C4×C61#51811yescharacteristic
327C4×C61#55111yescharacteristic
328C4×C61#55111yescharacteristic
329Dic61—22no—
330Dic61—22no—
331Dic61—42no—
332Dic61—42no—
333Dic61—41no—
334Dic61—81no—
335Dic61—81no—
336C2×#416—611no—
337#2346—611no—
338#2339—611no—
339#1933—611no—
340#1933—611no—
341#1927—611no—
342#1927—611no—
343#1864—611no—
344#1864—611no—
345#1825—611no—
346#1807—611no—
347#1788—611no—
348C4×#102—611no—
349C4×#102D6111yescharacteristic, complemented
350C8×C22⋊C4—611no—
351C2×C61—44no—
352C2×C61#234211yescharacteristic
353C2×C61C4×#13511yescharacteristic
354C2×C61#295211yescharacteristic
355C2×C61#329611yescharacteristic
356C2×C61#329611yescharacteristic
357C2×C61#330311yescharacteristic
358C2×C61#330311yescharacteristic
359D61—42no—
360C22×#41—611no—
361C23×C8—611no—
362C2×C4×D4—611no—
363C2×C4×D4C2×D6111yescharacteristic
364C2×#117—611no—
365C2×#106—611no—
366C2×#106—611no—
367#416—618no—
368C2×#103—611no—
369C2×#103—611no—
370C2×#103—611no—
371C2×#103—611no—
372C2×#102—611no—
373C2×#102—611no—
374C2×#102C2×D6111yescharacteristic
375C2×#102C2×D6111yescharacteristic
376C2×C22⋊C8—611no—
377C2×C4×C8—611no—
378C2×C4×C8C2×D6111yescharacteristic
379#391—611no—
380#391—611no—
381#389—611no—
382#389—611no—
383#388—1221no—
384#388—1221no—
385#386—1221no—
386#386—1221no—
387#382—1221no—
388#381—1221no—
389#380—611no—
390#380—611no—
391C4×C4⋊C4—611no—
392C4×C4⋊C4C2×D6111yescharacteristic
393C4×C22⋊C4—611no—
394#376—1221no—
395C2×#95—1221no—
396#340—611no—
397#340—611no—
398#340—611no—
399#340—611no—
400#340C2×D6111yescharacteristic
401#340C2×D6111yescharacteristic
402#336—611no—
403#336—611no—
404C61#2257211yescharacteristic, complemented, Sylow
405C2×#41—614no—
406C2×#41—1222no—
407C2×#41—1222no—
408C2×#41—1222no—
409C2×#41—2441no—
410C2×#41—2441no—
411C22×Q8—611no—
412C22×D4—1221no—
413C22×D4C22×D6111yescharacteristic
414C23×C4—21no—
415C23×C4—611no—
416C23×C4—1221no—
417C23×C4—1221no—
418C22×C8—611no—
419C22×C8—614no—
420C22×C8—611no—
421C22×C8—2441no—
422C22×C8—2441no—
423C22×C8C22×D6111yescharacteristic
424C22×C8C22×D6111yescharacteristic
425C4×D4—22no—
426C4×D4—41no—
427C4×D4—1222no—
428C4×D4—2441no—
429#117—614no—
430#117—1222no—
431C2×C4⋊C4—611no—
432C2×C4⋊C4—611no—
433C2×C4⋊C4—611no—
434C2×C4⋊C4—1221no—
435C2×C4⋊C4—1221no—
436C2×C4⋊C4C22×D6111yescharacteristic
437C2×C4⋊C4C22×D6111yescharacteristic
438C2×C4⋊C4C22×D6111yescharacteristic
439C2×C22⋊C4—21no—
440C2×C22⋊C4—611no—
441C2×C22⋊C4—611no—
442C2×C22⋊C4—1221no—
443C2×C22⋊C4—1221no—
444C2×C22⋊C4—1221no—
445C2×C22⋊C4—1221no—
446C2×C42—21no—
447C2×C42—611no—
448C2×C42—1221no—
449C2×C42—1221no—
450C2×C42C22×D6111yescharacteristic
451#106—2441no—
452#106—2441no—
453#103—614no—
454#103—614no—
455#103—2441no—
456#103—2441no—
457#102—22no—
458#102—614no—
459#102—1222no—
460#102C4×D6114yes—
461C22⋊C8—1222no—
462C4×C8—41no—
463C4×C8—2441no—
464#95—21no—
465#95—611no—
466#95—611no—
467#95—1221no—
468#95—1221no—
469#95—1221no—
470#95—1221no—
471#95—1221no—
472#95—1221no—
473#95—1221no—
474#95—1221no—
475#95—1221no—
476C24—21no—
477C24—611no—
478#41—1228no—
479#41—2444no—
480#41—2444no—
481#41—2444no—
482C2×Q8—612no—
483C2×Q8—1221no—
484C2×Q8—1221no—
485C2×Q8—1221no—
486C2×Q8—1221no—
487C2×Q8—1221no—
488C2×D4—21no—
489C2×D4—21no—
490C2×D4—21no—
491C2×D4—1224no—
492C2×D4—2442no—
493C2×D4—2442no—
494C2×D4C2×C4×D6112yes—
495C2×D4#1175822212yes—
496C2×D4#1175822212yes—
497C22×C4—21no—
498C22×C4—21no—
499C22×C4—21no—
500C22×C4—42no—
501C22×C4—42no—
502C22×C4—81no—
503C22×C4—611no—
504C22×C4—611no—
505C22×C4—612no—
506C22×C4—612no—
507C22×C4—611no—
508C22×C4—1224no—
509C22×C4—1221no—
510C22×C4—1221no—
511C22×C4—1221no—
512C22×C4—1221no—
513C22×C4—1221no—
514C22×C4—1221no—
515C22×C4—1221no—
516C22×C4—1221no—
517C22×C4—4881no—
518C22×C4—2442no—
519C22×C4—2442no—
520C22×C4—2442no—
521C22×C4—2442no—
522C22×C4D4×D6111yescharacteristic
523C22×C4D4×D6111yescharacteristic
524C22×C4C23×D6111yescharacteristic
525C2×C8—21no—
526C2×C8—21no—
527C2×C8—21no—
528C2×C8—21no—
529C2×C8—21no—
530C2×C8—612no—
531C2×C8—1221no—
532C2×C8—1221no—
533C2×C8—1221no—
534C2×C8—1221no—
535C2×C8—1221no—
536C2×C8—2444no—
537C2×C8C2×C4×D6112yes—
538C4⋊C4—21no—
539C4⋊C4—21no—
540C4⋊C4—21no—
541C4⋊C4—21no—
542C4⋊C4—21no—
543C4⋊C4—612no—
544C4⋊C4—1221no—
545C4⋊C4—1221no—
546C4⋊C4—1221no—
547C4⋊C4—1221no—
548C4⋊C4—1221no—
549C4⋊C4—4881no—
550C4⋊C4—4881no—
551C4⋊C4C2×C4×D6112yes—
552C22⋊C4—42no—
553C22⋊C4—1222no—
554C22⋊C4—1224no—
555C22⋊C4—1222no—
556C22⋊C4—4881no—
557C22⋊C4—4881no—
558C22⋊C4—2442no—
559C42—21no—
560C42—21no—
561C42—42no—
562C42—1224no—
563C42—1221no—
564C42—1221no—
565C42—2442no—
566C23—22no—
567C23—22no—
568C23—22no—
569C23—41no—
570C23—41no—
571C23—41no—
572C23—41no—
573C23—612no—
574C23—1221no—
575C23—1221no—
576C23—1221no—
577C23—1221no—
578C23—1221no—
579C23—1221no—
580C23C2×D4×D6111yescharacteristic
581Q8—1222no—
582Q8—1222no—
583Q8—2441no—
584Q8—2441no—
585D4—22no—
586D4—22no—
587D4—22no—
588D4—2448no—
589D4#1451973414yes—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—22no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—84no—
605C2×C4—612no—
606C2×C4—612no—
607C2×C4—1221no—
608C2×C4—1221no—
609C2×C4—1221no—
610C2×C4—1221no—
611C2×C4—1222no—
612C2×C4—1222no—
613C2×C4—1222no—
614C2×C4—1222no—
615C2×C4—1222no—
616C2×C4—1222no—
617C2×C4—1221no—
618C2×C4—1221no—
619C2×C4—1221no—
620C2×C4—4884no—
621C2×C4—2448no—
622C2×C4—2441no—
623C2×C4—2441no—
624C2×C4—2441no—
625C2×C4—2441no—
626C2×C4—2441no—
627C2×C4—2441no—
628C2×C4—2441no—
629C2×C4—2441no—
630C2×C4—2441no—
631C2×C4—2441no—
632C2×C4#1451979612yes—
633C2×C4C22×C4×D6111yescharacteristic
634C2×C4#41×D6111yescharacteristic
635C2×C4C2×#1175822211yescharacteristic
636C2×C4C2×#1175822211yescharacteristic
637C2×C4#1451982211yescharacteristic
638C2×C4#1451982311yescharacteristic
639C8—22no—
640C8—41no—
641C8—1222no—
642C8—2441no—
643C22—24no—
644C22—42no—
645C22—42no—
646C22—42no—
647C22—42no—
648C22—42no—
649C22—42no—
650C22—1222no—
651C22—1222no—
652C22—1222no—
653C22—2441no—
654C22—2441no—
655C22—2441no—
656C22—2441no—
657C22C4×D4×D6111yescharacteristic
658C22#1684040211yescharacteristic
659C22#1684040811yescharacteristic
660C22#135×D6111yescharacteristic
661C22C2×#1451979611yescharacteristic
662C22#1684056811yescharacteristic
663C22#1684056911yescharacteristic
664C4—22no—
665C4—41no—
666C4—42no—
667C4—81no—
668C4—81no—
669C4—1222no—
670C4—1222no—
671C4—4881no—
672C4—4881no—
673C4—2441no—
674C4—2442no—
675C4—2442no—
676C4C2×#1451973411yescharacteristic
677C4#1684023411yescharacteristic
678C4#1684043611yescharacteristic
679C4#1684044111yescharacteristic
680C2—44no—
681C2—2442no—
682C2#2144866211yescharacteristic
683C2C4×#1451979611yescharacteristic
684C2#2145469711yescharacteristic
685C2#2145765011yescharacteristic
686C2#2145765911yescharacteristic
687C2#2145774311yescharacteristic
688C2#2145774911yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC4×#102×C61C2
socleC23×C61C22×D4
prime radicalC22×#11758224C22
cosocleC22×C4×C61C24
hypercenterC4×#102D61
hypocenterC61#22572
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C4×#102

Complements

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

NH ≅ G/Ntype
2. #21448700C2semidirect
4. #21446495C2semidirect
5. #21446493C2semidirect
6. #21446453C2semidirect
7. #21446451C2semidirect
8. #21446080C2semidirect
9. #21446078C2semidirect
10. #21445912C2semidirect
11. #21445843C2semidirect
13. #102×Dic61C2semidirect
14. C8×#11758198C2semidirect
15. C4×#102×C61C2semidirect
22. #16839755C4semidirect
75. C2×C8×D61C2×C4semidirect
79. #14519735C2×C4semidirect
97. #14519702C2×C4semidirect
99. #14519700C2×C4semidirect
349. C4×#102D61semidirect
404. C61#22572semidirect

Maximal subgroups

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

subgroupindex
C2×#168397552
#214487002
#214486442
#214464952
#214464932
#214464532
#214464512
#214460802
#214460782
#214459122
#214458432
#214457562
#102×Dic612
C8×#117581982
C4×#102×C612
#2257261