Finite Group Databasebeta

← #29716222

Subgroups — #29716222

Order 15616.

1012 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840390C211yesmaximal, characteristic
2C2×#16840072C211yesmaximal, characteristic, complemented
3C2×#16840056C211yesmaximal, characteristic, complemented
4#21448497C211yesmaximal, characteristic
5#21447159C211yesmaximal, characteristic, complemented
6#21447145C211yesmaximal, characteristic, complemented
7#21446969C211yesmaximal, characteristic, complemented
8#21446959C211yesmaximal, characteristic, complemented
9#21446775C211yesmaximal, characteristic, complemented
10#21446769C211yesmaximal, characteristic, complemented
11#21446131C211yesmaximal, characteristic
12#21446124C211yesmaximal, characteristic
13#21446006C211yesmaximal, characteristic
14#21445801C211yesmaximal, characteristic
15C61×#1854C211yesmaximal, characteristic
16#16840390—24no—
17#16840390—24no—
18C2×#14519777—21no—
19C2×#14519776C2211yescharacteristic
20C2×#14519776C2211yescharacteristic
21C2×#14519769—21no—
22C2×#14519768C2211yescharacteristic
23C2×#14519768C2211yescharacteristic
24C2×#14519756C2211yescharacteristic
25C2×#14519754—21no—
26C2×#14519754—21no—
27C2×#14519742C2211yescharacteristic
28C2×#14519736C2211yescharacteristic
29C22×C4⋊Dic61C2211yescharacteristic
30#16840072—24no—
31#16840056—24no—
32C2×#14519710C2211yescharacteristic
33C2×#14519710C2211yescharacteristic
34C2×#14519706C2211yescharacteristic
35C2×#14519706C2211yescharacteristic
36C2×#14519702C2211yescharacteristic
37C2×#14519702C2211yescharacteristic
38C2×#14519700C2211yescharacteristic
39C2×#14519700C2211yescharacteristic
40C2×C22⋊#92418C2211yescharacteristic
41#16839667C2211yescharacteristic
42#16839659—21no—
43#16839658C2211yescharacteristic
44#16839656—21no—
45#16839650—21no—
46#16839649—21no—
47C4⋊C4×Dic61C2211yescharacteristic
48#16839525C2211yescharacteristic
49#16839525C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839524C2211yescharacteristic
53#16839524C2211yescharacteristic
54#16839516C2211yescharacteristic
55#16839516C2211yescharacteristic
56C2×#117×C61C2211yescharacteristic
57C2×#107×C61C2211yescharacteristic
58C2×#106×C61C2211yescharacteristic
59C2×C22⋊C8×C61C2211yescharacteristic
60C61×#380C2211yescharacteristic
61C61×#340C2211yescharacteristic
62C61×#340C2211yescharacteristic
63C22×Dic122—21no—
64C22×Dic122—21no—
65C22×Dic122C2311yescharacteristic
66C23×Dic61—21no—
67C22×#92418C2311yescharacteristic
68C22×#92418C2311yescharacteristic
69#14519777—28no—
70#14519776—22no—
71#14519776—22no—
72#14519776—22no—
73#14519776D414yes—
74#14519769—28no—
75#14519768—22no—
76#14519768—22no—
77#14519768—22no—
78#14519768D414yes—
79#14519756—22no—
80#14519756Q814yescomplemented
81#14519754—28no—
82#14519754—28no—
83#14519742—22no—
84#14519742—22no—
85#14519736—22no—
86#14519736—22no—
87C2×C4⋊Dic61—21no—
88C2×C4⋊Dic61—21no—
89C2×C4⋊Dic61—21no—
90C2×C4⋊Dic61—22no—
91C2×C4⋊Dic61—22no—
92C2×C4⋊Dic61D411yescharacteristic
93C2×C4⋊Dic61D411yescharacteristic
94C2×C4⋊Dic61C2311yescharacteristic
95C2×C4⋊Dic61C2311yescharacteristic
96C2×C4⋊Dic61C2311yescharacteristic
97C2×#11758196—21no—
98C2×#11758196—21no—
99C2×#11758196—21no—
100C2×#11758196—21no—
101C2×#11758196—21no—
102C2×#11758196—21no—
103C2×#11758196—21no—
104C2×C22⋊Dic61—21no—
105C2×C22⋊Dic61D411yescharacteristic
106C2×C22⋊Dic61D411yescharacteristic
107C2×C4×Dic61—21no—
108C2×C4×Dic61—21no—
109C2×C4×Dic61C2311yescharacteristic
110C2×C4×Dic61C2311yescharacteristic
111#14519710—22no—
112#14519710—22no—
113#14519706—22no—
114#14519706—22no—
115#14519702—22no—
116#14519702—22no—
117#14519700—22no—
118#14519700—22no—
119C22⋊#92418—22no—
120#14519684—21no—
121#14519684—21no—
122#14519684—21no—
123#14519684—21no—
124C23×C4×C61C2311yescharacteristic
125C22×C8×C61C2311yescharacteristic
126C22×C8×C61C2311yescharacteristic
127#117×C61—22no—
128#117×C61—22no—
129C2×C4⋊C4×C61C2311yescharacteristic
130C2×C4⋊C4×C61C2311yescharacteristic
131C2×C4⋊C4×C61C2311yescharacteristic
132C2×C4⋊C4×C61C2311yescharacteristic
133C2×C22⋊C4×C61—21no—
134C2×C22⋊C4×C61—21no—
135C2×C42×C61—21no—
136#107×C61—22no—
137#106×C61—22no—
138C22⋊C8×C61—22no—
139#95×C61—21no—
140C24×C61C2×D411yescharacteristic
141C22×C4×C61—21no—
142C22×C4×C61—21no—
143C22×C4×C61—21no—
144C22×C4×C61—21no—
145C22×C4×C61—21no—
146C22×C4×C61—21no—
147C22×C4×C61—21no—
148C22×C4×C61—21no—
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2×Q812yes—
151C22×C4×C61#4112yes—
152C22×C4×C61C2411yescharacteristic
153C22×Dic61—21no—
154C22×Dic61—21no—
155C22×Dic61—21no—
156C22×Dic61—21no—
157C22×Dic61—24no—
158C22×Dic61—81no—
159C22×Dic61—81no—
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×Dic122—24no—
170C2×Dic122—24no—
171C2×Dic122—81no—
172C2×Dic122—81no—
173C2×Dic122—81no—
174C2×Dic122—81no—
175C2×Dic122C2×Q812yes—
176C2×C8×C61—21no—
177C2×C8×C61—21no—
178C2×C8×C61—21no—
179C2×C8×C61—21no—
180C2×C8×C61—21no—
181C2×C8×C61—21no—
182C4⋊C4×C61—21no—
183C4⋊C4×C61—21no—
184C4⋊C4×C61—21no—
185C4⋊C4×C61—21no—
186C4⋊C4×C61—21no—
187C4⋊C4×C61—21no—
188C4⋊C4×C61C2×D412yes—
189C4⋊C4×C61C2×D412yes—
190C22⋊C4×C61—24no—
191C22⋊C4×C61—81no—
192C42×C61—24no—
193C22⋊Dic61—24no—
194C22⋊Dic61—22no—
195C22⋊Dic61—22no—
196C4⋊Dic61—24no—
197C4⋊Dic61—24no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201C4⋊Dic61—24no—
202C4⋊Dic61—22no—
203C4⋊Dic61—22no—
204C4⋊Dic61C2×D412yes—
205C4⋊Dic61C2×D412yes—
206C4⋊Dic61C2×Q812yes—
207#11758196—24no—
208#11758196—24no—
209#11758196—24no—
210#11758196—24no—
211#11758196—24no—
212#11758196—42no—
213#11758196—81no—
214C4×Dic61—21no—
215C4×Dic61—21no—
216C4×Dic61—42no—
217C4×Dic61—81no—
218C4×Dic61C2×D412yes—
219C4×Dic61C2×D412yes—
220C2×#92418—21no—
221C2×#92418—21no—
222C2×#92418—21no—
223C2×#92418—21no—
224C2×#92418—21no—
225C2×#92418—21no—
226C23×C61—21no—
227C23×C61—21no—
228C23×C61—21no—
229C23×C61—21no—
230C23×C61—21no—
231C23×C61—21no—
232C23×C61C22⋊Q812yes—
233C23×C61C22×D411yescharacteristic
234C2×C4×C61—21no—
235C2×C4×C61—21no—
236C2×C4×C61—22no—
237C2×C4×C61—22no—
238C2×C4×C61—22no—
239C2×C4×C61—22no—
240C2×C4×C61—22no—
241C2×C4×C61—22no—
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×C61—41no—
251C2×C4×C61—41no—
252C2×C4×C61C22⋊Q812yes—
253C2×C4×C61C22×D411yescharacteristic
254C2×C4×C61C22×D411yescharacteristic
255C2×C4×C61C22×Q811yescharacteristic
256C2×C4×C61C2×#4111yescharacteristic
257C2×C4×C612−1+411yescharacteristic
258C2×C4×C612−1+411yescharacteristic
259C2×Dic61—21no—
260C2×Dic61—21no—
261C2×Dic61—21no—
262C2×Dic61—21no—
263C2×Dic61—21no—
264C2×Dic61—21no—
265C2×Dic61—22no—
266C2×Dic61—22no—
267C2×Dic61—22no—
268C2×Dic61—22no—
269C2×Dic61—21no—
270C2×Dic61—21no—
271C2×Dic61—21no—
272C2×Dic61—21no—
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—84no—
282C2×Dic61#12012yes—
283C2×Dic61#12012yes—
284Dic122—22no—
285Dic122—22no—
286Dic122—41no—
287Dic122—41no—
288Dic122—84no—
289Dic122—84no—
290C8×C61—41no—
291C8×C61—41no—
292#92418—41no—
293#92418—41no—
294#24311—611nomaximal, Sylow
295C22×C61—22no—
296C22×C61—22no—
297C22×C61—22no—
298C22×C61—41no—
299C22×C61—41no—
300C22×C61—41no—
301C22×C61—41no—
302C22×C61C2×#12011yescharacteristic
303C22×C61C2×C22⋊Q811yescharacteristic
304C22×C61#53611yescharacteristic
305C22×C61#57611yescharacteristic
306C22×C61#57711yescharacteristic
307C22×C61#57811yescharacteristic
308C22×C61#57811yescharacteristic
309C4×C61—22no—
310C4×C61—42no—
311C4×C61—42no—
312C4×C61—81no—
313C4×C61—81no—
314C4×C61#54811yescharacteristic
315C4×C61#54811yescharacteristic
316C4×C61D4×Q811yescharacteristic
317C4×C61D4×Q811yescharacteristic
318Dic61—22no—
319Dic61—22no—
320Dic61—42no—
321Dic61—42no—
322Dic61—41no—
323Dic61—41no—
324Dic61—42no—
325Dic61—42no—
326C2×#544—611no—
327C2×#479—611no—
328C2×#477—611no—
329#2323—611no—
330#2059—611no—
331#2057—611no—
332#2026—611no—
333#2024—611no—
334#1989—611no—
335#1988—611no—
336#1873—611no—
337#1872—611no—
338#1854D6111yescharacteristic
339#1852—611no—
340#1798—611no—
341C2×C61—42no—
342C2×C61#240911yescharacteristic
343C2×C61#301111yescharacteristic
344C2×C61#301211yescharacteristic
345C2×C61#308211yescharacteristic
346C2×C61#308311yescharacteristic
347C2×C61#324211yescharacteristic
348C2×C61#324311yescharacteristic
349#544—1224no—
350#544—1224no—
351C2×C4⋊Q8—611no—
352C2×C4⋊Q8—611no—
353C2×C4⋊Q8—1221no—
354C2×#125—611no—
355C2×#125—611no—
356C2×#125—1221no—
357C2×C22⋊Q8—611no—
358C2×C22⋊Q8—1221no—
359C2×C22⋊Q8—1221no—
360C2×C4×Q8—611no—
361C2×#117—611no—
362C2×#117C2×D6111yescharacteristic
363C22×C4⋊C4—611no—
364#479—1224no—
365#477—1224no—
366C2×#107—611no—
367C2×#107—611no—
368C2×#107C2×D6111yescharacteristic
369C2×#106C2×D6111yescharacteristic
370C2×C4⋊C8—611no—
371C2×C4⋊C8—611no—
372C2×#103—611no—
373C2×#103—611no—
374C2×#103—611no—
375C2×#103—611no—
376C2×C22⋊C8—611no—
377C2×C22⋊C8C2×D6111yescharacteristic
378#391—611no—
379#389—611no—
380#389—1221no—
381#388—1221no—
382#388—1221no—
383#387—1221no—
384#380C2×D6111yescharacteristic
385C4×C4⋊C4—611no—
386#340—611no—
387#340—611no—
388#340—611no—
389#340—611no—
390#340—611no—
391#340—611no—
392#340C2×D6111yescharacteristic
393#340C2×D6111yescharacteristic
394#336—611no—
395#336—611no—
396C61#2431111yescharacteristic, complemented, Sylow
397C22×Q8—611no—
398C22×Q8—1221no—
399C22×Q8—1221no—
400C23×C4—1221no—
401C23×C4C22×D6111yescharacteristic
402C22×C8—611no—
403C22×C8—611no—
404C22×C8C22×D6111yescharacteristic
405C22×C8C22×D6111yescharacteristic
406C4⋊Q8—614no—
407C4⋊Q8—1228no—
408C4⋊Q8—1222no—
409C4⋊Q8—1222no—
410C4⋊Q8—1222no—
411#125—614no—
412#125—1228no—
413#125—1222no—
414#125—1222no—
415#125—1222no—
416C22⋊Q8—614no—
417C22⋊Q8—1228no—
418C22⋊Q8—1228no—
419C22⋊Q8—1222no—
420C4×Q8—1222no—
421C4×Q8—1222no—
422#117—22no—
423#117—22no—
424#117—1222no—
425#117—1222no—
426C2×C4⋊C4—611no—
427C2×C4⋊C4—611no—
428C2×C4⋊C4—611no—
429C2×C4⋊C4—611no—
430C2×C4⋊C4—611no—
431C2×C4⋊C4—1222no—
432C2×C4⋊C4—1222no—
433C2×C4⋊C4—1221no—
434C2×C4⋊C4—1221no—
435C2×C4⋊C4—1221no—
436C2×C4⋊C4—1221no—
437C2×C4⋊C4—1221no—
438C2×C4⋊C4—1221no—
439C2×C4⋊C4—1221no—
440C2×C4⋊C4—1221no—
441C2×C4⋊C4—1221no—
442C2×C4⋊C4—1221no—
443C2×C4⋊C4C22×D6111yescharacteristic
444C2×C4⋊C4C22×D6111yescharacteristic
445C2×C4⋊C4C22×D6111yescharacteristic
446C2×C4⋊C4C22×D6111yescharacteristic
447C2×C22⋊C4—21no—
448C2×C22⋊C4—21no—
449C2×C22⋊C4—611no—
450C2×C22⋊C4—611no—
451C2×C22⋊C4—1221no—
452C2×C42—21no—
453C2×C42—611no—
454C2×C42—611no—
455C2×C42—1221no—
456C2×C42—1221no—
457#107—22no—
458#107—1222no—
459#107—1222no—
460#106—22no—
461C4⋊C8—1222no—
462C4⋊C8—1222no—
463#103—1222no—
464#103—1222no—
465#103—1222no—
466#103—1222no—
467C22⋊C8—22no—
468C22⋊C8—1222no—
469#95—21no—
470#95—1221no—
471#95—1221no—
472#95—1221no—
473#95—1221no—
474C24D4×D6111yescharacteristic
475C2×Q8—612no—
476C2×Q8—1224no—
477C2×Q8—1224no—
478C2×Q8—1221no—
479C2×Q8—1221no—
480C2×Q8—1221no—
481C2×Q8—1221no—
482C2×Q8—1221no—
483C2×Q8—4881no—
484C2×Q8—4881no—
485C2×Q8—4881no—
486C2×Q8—4881no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—611no—
496C22×C4—611no—
497C22×C4—611no—
498C22×C4—611no—
499C22×C4—1224no—
500C22×C4—1221no—
501C22×C4—1221no—
502C22×C4—1221no—
503C22×C4—1221no—
504C22×C4—4881no—
505C22×C4—4881no—
506C22×C4#1175822212yes—
507C22×C4D4×D6111yescharacteristic
508C22×C4Q8×D6112yes—
509C22×C4C23×D6111yescharacteristic
510C2×C8—21no—
511C2×C8—21no—
512C2×C8—21no—
513C2×C8—21no—
514C2×C8—21no—
515C2×C8—21no—
516C2×C8—1221no—
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—21no—
528C4⋊C4—612no—
529C4⋊C4—612no—
530C4⋊C4—612no—
531C4⋊C4—1222no—
532C4⋊C4—1222no—
533C4⋊C4—1224no—
534C4⋊C4—1224no—
535C4⋊C4—1224no—
536C4⋊C4—1224no—
537C4⋊C4—1224no—
538C4⋊C4—1224no—
539C4⋊C4—1224no—
540C4⋊C4—1221no—
541C4⋊C4—1221no—
542C4⋊C4—1221no—
543C4⋊C4—1224no—
544C4⋊C4—4881no—
545C4⋊C4—2442no—
546C4⋊C4D4×D6112yes—
547C4⋊C4D4×D6112yes—
548C22⋊C4—24no—
549C22⋊C4—81no—
550C22⋊C4—1222no—
551C22⋊C4—1222no—
552C22⋊C4—1224no—
553C42—24no—
554C42—612no—
555C42—612no—
556C42—1221no—
557C42—1221no—
558C42—4881no—
559C42—2442no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23#1451975512yes—
567C23C2×D4×D6111yescharacteristic
568Q8—1222no—
569Q8—1222no—
570Q8—4884no—
571Q8—4884no—
572Q8—2441no—
573Q8—2441no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—612no—
593C2×C4—612no—
594C2×C4—1221no—
595C2×C4—1221no—
596C2×C4—1221no—
597C2×C4—1221no—
598C2×C4—1221no—
599C2×C4—1221no—
600C2×C4—1222no—
601C2×C4—1222no—
602C2×C4—1222no—
603C2×C4—1222no—
604C2×C4—1221no—
605C2×C4—1221no—
606C2×C4—1221no—
607C2×C4—1221no—
608C2×C4—4884no—
609C2×C4—2441no—
610C2×C4—2441no—
611C2×C4—2441no—
612C2×C4—2441no—
613C2×C4—2441no—
614C2×C4—2441no—
615C2×C4—2441no—
616C2×C4—2441no—
617C2×C4#1451975512yes—
618C2×C4C2×D4×D6111yescharacteristic
619C2×C4C2×D4×D6111yescharacteristic
620C2×C4C2×Q8×D6111yescharacteristic
621C2×C4C2×#1175822211yescharacteristic
622C2×C4#1451982411yescharacteristic
623C2×C4#1451982411yescharacteristic
624C8—41no—
625C8—41no—
626C8—2441no—
627C8—2441no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—41no—
632C22—41no—
633C22—41no—
634C22—41no—
635C22#120×D6111yescharacteristic
636C22C2×#1451975511yescharacteristic
637C22#1684033411yescharacteristic
638C22#1684056611yescharacteristic
639C22#1684056911yescharacteristic
640C22#1684057611yescharacteristic
641C22#1684057811yescharacteristic
642C4—22no—
643C4—42no—
644C4—42no—
645C4—81no—
646C4—81no—
647C4—1222no—
648C4—1222no—
649C4—2441no—
650C4—2441no—
651C4—2442no—
652C4—2442no—
653C4—2442no—
654C4—2442no—
655C4#1684042111yescharacteristic
656C4#1684042111yescharacteristic
657C4#1684042511yescharacteristic
658C4#1684042511yescharacteristic
659C2—42no—
660C2#2144918311yescharacteristic
661C2#2145511111yescharacteristic
662C2#2145512111yescharacteristic
663C2#2145566411yescharacteristic
664C2#2145566911yescharacteristic
665C2#2145709311yescharacteristic
666C2#2145710011yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1854C2
socleC23×C61C22×D4
prime radicalC24×C61C2×D4
hypercenter#1854D61
hypocenterC61#24311
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1854

Complements

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

NH ≅ G/Ntype
2. C2×#16840072C2semidirect
3. C2×#16840056C2semidirect
5. #21447159C2semidirect
6. #21447145C2semidirect
7. #21446969C2semidirect
8. #21446959C2semidirect
9. #21446775C2semidirect
10. #21446769C2semidirect
80. #14519756Q8semidirect
396. C61#24311semidirect

Maximal subgroups

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

subgroupindex
C2×#168403902
C2×#168400722
C2×#168400562
#214484972
#214471592
#214471452
#214469692
#214469592
#214467752
#214467692
#214461312
#214461242
#214460062
#214458012
C61×#18542
#2431161