Finite Group Databasebeta

← #29686702

Subgroups — #29686702

Order 15616.

954 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840018C211yesmaximal, characteristic
2C2×#16839854C211yesmaximal, characteristic, complemented
3C2×#16839717C211yesmaximal, characteristic
4#21448503C211yesmaximal, characteristic
5#21448497C211yesmaximal, characteristic
6#21446687C211yesmaximal, characteristic, complemented
7#21446561C211yesmaximal, characteristic, complemented
8#21446496C211yesmaximal, characteristic, complemented
9#21446492C211yesmaximal, characteristic, complemented
10#21446242C211yesmaximal, characteristic
11#21445850C211yesmaximal, characteristic
12#21445707C211yesmaximal, characteristic, complemented
13#21445651C211yesmaximal, characteristic, complemented
14#21445648C211yesmaximal, characteristic, complemented
15C61×#1892C211yesmaximal, characteristic
16C22×#11758202C2211yescharacteristic
17C22×#11758194C2211yescharacteristic
18C2×#14519756C2211yescharacteristic
19C2×Q8×Dic61C2211yescharacteristic
20C2×#14519742C2211yescharacteristic
21C2×#14519736C2211yescharacteristic
22#16840018C418yescomplemented
23#16839854—24no—
24C2×#14519710C2211yescharacteristic
25C2×#14519708C2211yescharacteristic
26C2×#14519708C2211yescharacteristic
27C2×#14519702C2211yescharacteristic
28C2×#14519702C2211yescharacteristic
29C2×#14519701C2211yescharacteristic
30C2×#14519700C2211yescharacteristic
31#16839717—24no—
32C2×#14519687C2211yescharacteristic
33C2×#14519687C2211yescharacteristic
34C2×C8×Dic61C2211yescharacteristic
35C2×C8×Dic61C2211yescharacteristic
36#16839669C2211yescharacteristic
37#16839667C2211yescharacteristic
38#16839661—21no—
39#16839661C2211yescharacteristic
40#16839659—21no—
41#16839658C2211yescharacteristic
42#16839656—21no—
43#16839655—21no—
44#16839655—21no—
45#16839650—21no—
46#16839649—21no—
47#16839648—21no—
48#16839622C2211yescharacteristic
49C4⋊C4×Dic61C2211yescharacteristic
50C4⋊C4×Dic61C2211yescharacteristic
51#16839525C2211yescharacteristic
52#16839525C2211yescharacteristic
53#16839524C2211yescharacteristic
54#16839524C2211yescharacteristic
55C23×C8×C61C2211yescharacteristic
56C2×C22⋊Q8×C61C2211yescharacteristic
57C2×#103×C61C2211yescharacteristic
58C2×#103×C61C2211yescharacteristic
59C61×#380C2211yescharacteristic
60C61×#340C2211yescharacteristic
61C61×#340C2211yescharacteristic
62C22×Dic122C2311yescharacteristic
63C2×#11758202—22no—
64C2×#11758202—22no—
65C2×#11758202C2×C414yes—
66C2×#11758194—22no—
67C2×#11758194—22no—
68C2×#11758194C2×C414yes—
69C22×#92418C2311yescharacteristic
70C22×#92418C2311yescharacteristic
71#14519756—22no—
72#14519756C2×C414yes—
73Q8×Dic61—22no—
74Q8×Dic61—22no—
75#14519742—22no—
76#14519742—22no—
77#14519736—22no—
78#14519736—22no—
79C2×C4⋊Dic61—21no—
80C2×C4⋊Dic61—21no—
81C2×C4⋊Dic61D411yescharacteristic
82C2×C4⋊Dic61D411yescharacteristic
83C2×C4⋊Dic61C2311yescharacteristic
84C2×C4⋊Dic61C2311yescharacteristic
85C2×C4⋊Dic61C2311yescharacteristic
86C2×#11758196—21no—
87C2×#11758196—21no—
88C2×#11758196—21no—
89C2×C22⋊Dic61—21no—
90C2×C22⋊Dic61—21no—
91C2×C22⋊Dic61D411yescharacteristic
92C2×C22⋊Dic61D411yescharacteristic
93C2×C4×Dic61—21no—
94C2×C4×Dic61—21no—
95C2×C4×Dic61—21no—
96C2×C4×Dic61—21no—
97C2×C4×Dic61C2311yescharacteristic
98C2×C4×Dic61C2311yescharacteristic
99#14519710—22no—
100#14519708—22no—
101#14519708C2×C414yes—
102#14519702—22no—
103#14519702—22no—
104#14519701C2×C414yes—
105#14519700C2×C414yes—
106#14519687—22no—
107#14519687—22no—
108C8×Dic61—22no—
109C8×Dic61D414yes—
110#14519684—21no—
111#14519684—21no—
112#14519684—21no—
113#14519684—21no—
114#14519684—21no—
115#14519684—21no—
116#14519684—21no—
117#14519684—21no—
118#14519684—21no—
119C22×Q8×C61C2311yescharacteristic
120C23×C4×C61C2311yescharacteristic
121C22×C8×C61—22no—
122C22×C8×C61—22no—
123C22×C8×C61—22no—
124C22×C8×C61C2311yescharacteristic
125C22×C8×C61C2311yescharacteristic
126C22⋊Q8×C61—22no—
127C22⋊Q8×C61C2×C414yes—
128C2×C4⋊C4×C61—21no—
129C2×C4⋊C4×C61C2311yescharacteristic
130C2×C4⋊C4×C61C2311yescharacteristic
131C2×C4⋊C4×C61C2311yescharacteristic
132C2×C22⋊C4×C61—21no—
133C2×C22⋊C4×C61—21no—
134#103×C61—22no—
135#103×C61—22no—
136#95×C61—21no—
137C24×C61C2×D411yescharacteristic
138C2×Q8×C61—21no—
139C2×Q8×C61—21no—
140C2×Q8×C61—21no—
141C2×Q8×C61—21no—
142C2×Q8×C61—21no—
143C2×Q8×C61C22×C412yes—
144C22×C4×C61—21no—
145C22×C4×C61—21no—
146C22×C4×C61—21no—
147C22×C4×C61—21no—
148C22×C4×C61—21no—
149C22×C4×C61—21no—
150C22×C4×C61—21no—
151C22×C4×C61—21no—
152C22×C4×C61C22×C412yes—
153C22×C4×C61C2×D411yescharacteristic
154C22×C4×C61#4112yes—
155C22×C4×C61C2411yescharacteristic
156C22×Dic61—21no—
157C22×Dic61—21no—
158C22×Dic61—21no—
159C22×Dic61—21no—
160C22×Dic61C2×D411yescharacteristic
161C22×Dic61C2×D411yescharacteristic
162C22×Dic61C2×D411yescharacteristic
163C22×Dic61C2×D411yescharacteristic
164C2×Dic122—21no—
165C2×Dic122—21no—
166C2×Dic122—21no—
167C2×Dic122—21no—
168C2×Dic122—21no—
169C2×Dic122C22×C412yes—
170C2×C8×C61—28no—
171C2×C8×C61—21no—
172C2×C8×C61—21no—
173C2×C8×C61—21no—
174C2×C8×C61—21no—
175C2×C8×C61C2×D412yes—
176C2×C8×C61#4112yes—
177C4⋊C4×C61—24no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61—21no—
183C4⋊C4×C61C22×C412yes—
184C22⋊C4×C61—24no—
185C22⋊C4×C61—81no—
186C22⋊Dic61—24no—
187C22⋊Dic61—22no—
188C22⋊Dic61—22no—
189C22⋊Dic61—81no—
190#11758202—28no—
191C4⋊Dic61—21no—
192C4⋊Dic61—22no—
193C4⋊Dic61—22no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—42no—
199C4⋊Dic61—81no—
200C4⋊Dic61C22×C412yes—
201#11758196—24no—
202#11758196—42no—
203#11758196—81no—
204C4×Dic61—21no—
205C4×Dic61—21no—
206C4×Dic61—42no—
207C4×Dic61—42no—
208C4×Dic61—81no—
209C4×Dic61—81no—
210C4×Dic61C2×D412yes—
211C4×Dic61#4112yes—
212#11758194—28no—
213C2×#92418—21no—
214C2×#92418—21no—
215C2×#92418C22×C412yes—
216C2×#92418C22×C412yes—
217C2×#92418C2×D412yes—
218C2×#92418#4112yes—
219C23×C61—21no—
220C23×C61—21no—
221C23×C61—21no—
222C23×C61—21no—
223C23×C61—21no—
224C23×C61—21no—
225C23×C61C4×D412yes—
226C23×C61C22×D411yescharacteristic
227Q8×C61—22no—
228Q8×C61—22no—
229Q8×C61—41no—
230Q8×C61—41no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—22no—
234C2×C4×C61—22no—
235C2×C4×C61—22no—
236C2×C4×C61—22no—
237C2×C4×C61—22no—
238C2×C4×C61—22no—
239C2×C4×C61—41no—
240C2×C4×C61—41no—
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—41no—
248C2×C4×C61—41no—
249C2×C4×C61C4×D412yes—
250C2×C4×C61C23×C411yescharacteristic
251C2×C4×C61C22×D411yescharacteristic
252C2×C4×C61C2×#4111yescharacteristic
253C2×C4×C61C2×#4111yescharacteristic
254C2×C4×C61C2×#4111yescharacteristic
255C2×C4×C61C2×#4111yescharacteristic
256C2×Dic61—21no—
257C2×Dic61—21no—
258C2×Dic61—21no—
259C2×Dic61—21no—
260C2×Dic61—22no—
261C2×Dic61—22no—
262C2×Dic61—21no—
263C2×Dic61—21no—
264C2×Dic61—21no—
265C2×Dic61—21no—
266C2×Dic61—21no—
267C2×Dic61—21no—
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—41no—
277C2×Dic61—41no—
278C2×Dic61#12112yes—
279C2×Dic61#12112yes—
280Dic122—22no—
281Dic122—22no—
282Dic122—41no—
283Dic122—41no—
284C8×C61—22no—
285C8×C61—22no—
286#92418—22no—
287#92418C4×D414yes—
288#21537—611nomaximal, Sylow
289C22×C61—22no—
290C22×C61—22no—
291C22×C61—22no—
292C22×C61—41no—
293C22×C61—41no—
294C22×C61—41no—
295C22×C61—41no—
296C22×C61C2×C4×D411yescharacteristic
297C22×C61C2×#12111yescharacteristic
298C22×C61#53211yescharacteristic
299C22×C61#57711yescharacteristic
300C22×C61#57711yescharacteristic
301C22×C61#57811yescharacteristic
302C22×C61#57811yescharacteristic
303C4×C61—22no—
304C4×C61—42no—
305C4×C61—42no—
306C4×C61—81no—
307C4×C61—81no—
308C4×C61C2×C4×D411yescharacteristic
309C4×C61C4×#4111yescharacteristic
310C4×C61#52511yescharacteristic
311C4×C61#52911yescharacteristic
312Dic61—22no—
313Dic61—22no—
314Dic61—42no—
315Dic61—41no—
316Dic61—41no—
317Dic61—42no—
318Dic61—81no—
319Dic61—81no—
320C2×#470—611no—
321C2×C4×Q16—611no—
322C2×#405—611no—
323#2323—611no—
324#2323—611no—
325#1971—611no—
326#1944—611no—
327#1933—611no—
328#1933—611no—
329#1892—611no—
330#1892D6111yescharacteristic
331#1809—611no—
332C4×#106—611no—
333#1763—611no—
334#1763—611no—
335C2×C61—42no—
336C2×C61C4×#12111yescharacteristic
337C2×C61#294711yescharacteristic
338C2×C61#294711yescharacteristic
339C2×C61#305911yescharacteristic
340C2×C61#306011yescharacteristic
341C2×C61#317811yescharacteristic
342C2×C61#317811yescharacteristic
343C22×Q16—611no—
344C22×M16—611no—
345C23×C8C2×D6111yescharacteristic
346C2×C22⋊Q8—611no—
347C2×C22⋊Q8C2×D6111yescharacteristic
348C2×C4×Q8—611no—
349C2×C4×Q8—611no—
350C2×#117—611no—
351#470—618no—
352C4×Q16—1224no—
353C2×#107—611no—
354C2×#106—611no—
355C2×#106—611no—
356C2×#103—611no—
357C2×#103—611no—
358C2×#103—611no—
359C2×#103—611no—
360C2×#103C2×D6111yescharacteristic
361C2×#103C2×D6111yescharacteristic
362#405—1224no—
363C2×#97—611no—
364C2×#97—611no—
365C2×C4×C8—611no—
366C2×C4×C8—611no—
367#391—611no—
368#391—611no—
369#389—611no—
370#389—611no—
371#389—1221no—
372#389—1221no—
373#388—1221no—
374#388—1221no—
375#388—1221no—
376#388—1221no—
377#387—1221no—
378#387—1221no—
379#380—611no—
380#380C2×D6111yescharacteristic
381C4×C4⋊C4—611no—
382C4×C4⋊C4—611no—
383#340—611no—
384#340—611no—
385#340—611no—
386#340—611no—
387#340C2×D6111yescharacteristic
388#340C2×D6111yescharacteristic
389C61#2153711yescharacteristic, complemented, Sylow
390C22×Q8—611no—
391C22×Q8C22×D6111yescharacteristic
392C23×C4C22×D6111yescharacteristic
393C2×Q16—614no—
394C2×Q16—1222no—
395C2×Q16—1222no—
396C2×M16—614no—
397C2×M16—1222no—
398C2×M16—1222no—
399C22×C8—22no—
400C22×C8—22no—
401C22×C8—22no—
402C22×C8—611no—
403C22×C8—611no—
404C22×C8C22×D6111yescharacteristic
405C22×C8C22×D6111yescharacteristic
406C22⋊Q8—22no—
407C22⋊Q8—614no—
408C22⋊Q8—1222no—
409C22⋊Q8C4×D6114yes—
410C4×Q8—1222no—
411C4×Q8—1222no—
412C4×Q8—1222no—
413C4×Q8—1222no—
414#117—1222no—
415#117—1222no—
416C2×C4⋊C4—21no—
417C2×C4⋊C4—611no—
418C2×C4⋊C4—611no—
419C2×C4⋊C4—611no—
420C2×C4⋊C4—611no—
421C2×C4⋊C4—611no—
422C2×C4⋊C4—1221no—
423C2×C4⋊C4—1221no—
424C2×C4⋊C4—1221no—
425C2×C4⋊C4—1221no—
426C2×C4⋊C4—1221no—
427C2×C4⋊C4C22×D6111yescharacteristic
428C2×C4⋊C4C22×D6111yescharacteristic
429C2×C4⋊C4C22×D6111yescharacteristic
430C2×C22⋊C4—21no—
431C2×C22⋊C4—21no—
432C2×C22⋊C4—611no—
433C2×C22⋊C4—611no—
434C2×C22⋊C4—1221no—
435C2×C22⋊C4—1221no—
436C2×C42—611no—
437C2×C42—611no—
438C2×C42—1221no—
439C2×C42—1221no—
440C2×C42—1221no—
441C2×C42—1221no—
442#107—1222no—
443#106—614no—
444#106—1222no—
445#103—22no—
446#103—22no—
447#103—614no—
448#103—614no—
449#103—1222no—
450#103—1222no—
451#97—1222no—
452#97—1222no—
453C4×C8—614no—
454C4×C8—1222no—
455#95—21no—
456#95—1221no—
457#95—1221no—
458#95—1221no—
459#95—1221no—
460#95—1221no—
461#95—1221no—
462#95—1221no—
463#95—1221no—
464#95—1221no—
465C24D4×D6111yescharacteristic
466C2×Q8—21no—
467C2×Q8—21no—
468C2×Q8—21no—
469C2×Q8—21no—
470C2×Q8—21no—
471C2×Q8—612no—
472C2×Q8—1221no—
473C2×Q8—1221no—
474C2×Q8—1221no—
475C2×Q8—1221no—
476C2×Q8—1221no—
477C2×Q8C2×C4×D6112yes—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—611no—
487C22×C4—611no—
488C22×C4—611no—
489C22×C4—611no—
490C22×C4—1221no—
491C22×C4—1221no—
492C22×C4—1221no—
493C22×C4—1221no—
494C22×C4C2×C4×D6112yes—
495C22×C4D4×D6111yescharacteristic
496C22×C4#1175822412yes—
497C22×C4C23×D6111yescharacteristic
498Q16—1228no—
499M16—1228no—
500C2×C8—28no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—612no—
506C2×C8—612no—
507C2×C8—612no—
508C2×C8—612no—
509C2×C8—1221no—
510C2×C8—1221no—
511C2×C8D4×D6112yes—
512C2×C8#1175822412yes—
513C4⋊C4—21no—
514C4⋊C4—24no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—612no—
520C4⋊C4—1222no—
521C4⋊C4—1222no—
522C4⋊C4—1221no—
523C4⋊C4—1224no—
524C4⋊C4—1221no—
525C4⋊C4—1221no—
526C4⋊C4—1221no—
527C4⋊C4—1221no—
528C4⋊C4—4881no—
529C4⋊C4—4881no—
530C4⋊C4—2442no—
531C4⋊C4—2442no—
532C4⋊C4C2×C4×D6112yes—
533C22⋊C4—24no—
534C22⋊C4—81no—
535C22⋊C4—1222no—
536C22⋊C4—1222no—
537C22⋊C4—1224no—
538C22⋊C4—4881no—
539C42—612no—
540C42—612no—
541C42—1221no—
542C42—1221no—
543C42—4881no—
544C42—4881no—
545C42—2442no—
546C42—2442no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23#1451973712yes—
554C23C2×D4×D6111yescharacteristic
555Q8—22no—
556Q8—22no—
557Q8—41no—
558Q8—41no—
559Q8—1222no—
560Q8—1222no—
561Q8—2441no—
562Q8—2441no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—612no—
582C2×C4—612no—
583C2×C4—1221no—
584C2×C4—1221no—
585C2×C4—1221no—
586C2×C4—1221no—
587C2×C4—1222no—
588C2×C4—1222no—
589C2×C4—1221no—
590C2×C4—1221no—
591C2×C4—1221no—
592C2×C4—1221no—
593C2×C4—1221no—
594C2×C4—1221no—
595C2×C4—2441no—
596C2×C4—2441no—
597C2×C4—2441no—
598C2×C4—2441no—
599C2×C4—2441no—
600C2×C4—2441no—
601C2×C4—2441no—
602C2×C4—2441no—
603C2×C4—2441no—
604C2×C4—2441no—
605C2×C4#1451973712yes—
606C2×C4C22×C4×D6111yescharacteristic
607C2×C4C2×D4×D6111yescharacteristic
608C2×C4#41×D6111yescharacteristic
609C2×C4#41×D6111yescharacteristic
610C2×C4C2×#1175822411yescharacteristic
611C2×C4C2×#1175822411yescharacteristic
612C8—22no—
613C8—22no—
614C8—614no—
615C8—1222no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—41no—
620C22—41no—
621C22—41no—
622C22—41no—
623C22C2×#1451973711yescharacteristic
624C22#121×D6111yescharacteristic
625C22#1684030711yescharacteristic
626C22#1684056811yescharacteristic
627C22#1684056911yescharacteristic
628C22#1684057511yescharacteristic
629C22#1684057611yescharacteristic
630C4—22no—
631C4—42no—
632C4—42no—
633C4—81no—
634C4—81no—
635C4—1222no—
636C4—1222no—
637C4—4881no—
638C4—4881no—
639C4—2442no—
640C4—2441no—
641C4—2441no—
642C4—2442no—
643C4C4×D4×D6111yescharacteristic
644C4C4×#1175822411yescharacteristic
645C4#1684026811yescharacteristic
646C4#1684028911yescharacteristic
647C2—42no—
648C2#2144832911yescharacteristic
649C2#2145465611yescharacteristic
650C2#2145466511yescharacteristic
651C2#2145546911yescharacteristic
652C2#2145547611yescharacteristic
653C2#2145648211yescharacteristic
654C2#2145649011yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1892C2
socleC23×C61C22×D4
prime radicalC24×C61C2×D4
cosocleC22×C4×C61C24
hypercenter#1892D61
hypocenterC61#21537
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1892

Complements

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

NH ≅ G/Ntype
2. C2×#16839854C2semidirect
6. #21446687C2semidirect
7. #21446561C2semidirect
8. #21446496C2semidirect
9. #21446492C2semidirect
12. #21445707C2semidirect
13. #21445651C2semidirect
14. #21445648C2semidirect
22. #16840018C4semidirect
389. C61#21537semidirect

Maximal subgroups

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

subgroupindex
C2×#168400182
C2×#168398542
C2×#168397172
#214485032
#214484972
#214466872
#214465612
#214464962
#214464922
#214462422
#214458502
#214457072
#214456512
#214456482
C61×#18922
#2153761