Finite Group Databasebeta

← #29716616

Subgroups — #29716616

Order 15616.

1064 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840489C211yesmaximal, characteristic, complemented
2C2×#16839862C211yesmaximal, characteristic
3#21450741C211yesmaximal, characteristic
4#21447159C211yesmaximal, characteristic, complemented
5#21447145C211yesmaximal, characteristic, complemented
6#21446963C211yesmaximal, characteristic
7#21446953C211yesmaximal, characteristic
8#21446789C211yesmaximal, characteristic, complemented
9#21446758C211yesmaximal, characteristic, complemented
10#21446171C211yesmaximal, characteristic
11#21446171C211yesmaximal, characteristic
12#21446084C211yesmaximal, characteristic, complemented
13#21446070C211yesmaximal, characteristic, complemented
14#21445825C211yesmaximal, characteristic, complemented
15C2×C61×#478C211yesmaximal, characteristic
16C22×#11758200C2211yescharacteristic
17#16840489—24no—
18#16840489—24no—
19C2×#14519776—21no—
20C2×#14519776C2211yescharacteristic
21C2×#14519768C2211yescharacteristic
22C2×#14519767—21no—
23C2×#14519767—21no—
24C2×#14519756—21no—
25C2×C4×Dic122C2211yescharacteristic
26C2×C4×Dic122C2211yescharacteristic
27C2×#14519742—21no—
28C2×#14519742C2211yescharacteristic
29C2×#14519742C2211yescharacteristic
30#16839862—24no—
31C2×#14519710C2211yescharacteristic
32C2×#14519702C2211yescharacteristic
33C2×#14519702C2211yescharacteristic
34C2×#14519700C2211yescharacteristic
35C2×#14519698C2211yescharacteristic
36C2×#14519688C2211yescharacteristic
37#16839685—21no—
38#16839685—21no—
39#16839665C2211yescharacteristic
40#16839665C2211yescharacteristic
41#16839658C2211yescharacteristic
42#16839658C2211yescharacteristic
43#16839632C2211yescharacteristic
44#16839622—21no—
45#16839525C2211yescharacteristic
46#16839525C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839516C2211yescharacteristic
52#16839516C2211yescharacteristic
53#16839516C2211yescharacteristic
54#16839516C2211yescharacteristic
55C2×#125×C61C2211yescharacteristic
56C2×C4×D4×C61C2211yescharacteristic
57C61×#478—24no—
58C2×#107×C61C2211yescharacteristic
59C2×#106×C61C2211yescharacteristic
60C2×C4⋊C8×C61C2211yescharacteristic
61C2×#102×C61C2211yescharacteristic
62C2×#102×C61C2211yescharacteristic
63C22×Dic122—21no—
64C22×Dic122D411yescharacteristic
65C22×Dic122C2311yescharacteristic
66C2×#11758200—22no—
67C2×#11758200—22no—
68C2×#11758200Q814yescomplemented
69C22×#92418C2311yescharacteristic
70C22×#92418C2311yescharacteristic
71#14519776—28no—
72#14519776—22no—
73#14519776—22no—
74#14519768—22no—
75#14519768—22no—
76#14519767—28no—
77#14519767—28no—
78#14519756—44no—
79C4×Dic122—22no—
80C4×Dic122—22no—
81C4×Dic122—22no—
82C4×Dic122D414yes—
83#14519742—28no—
84#14519742—22no—
85#14519742—22no—
86#14519742—22no—
87#14519742—22no—
88C2×C4⋊Dic61—21no—
89C2×C4⋊Dic61—21no—
90C2×C4⋊Dic61—21no—
91C2×C4⋊Dic61D411yescharacteristic
92C2×C4⋊Dic61C2311yescharacteristic
93C2×C4⋊Dic61C2311yescharacteristic
94C2×C4⋊Dic61C2311yescharacteristic
95C2×#11758196—21no—
96C2×#11758196—21no—
97C2×#11758196—21no—
98C2×#11758196—21no—
99C2×#11758196—21no—
100C2×#11758196—21no—
101C2×#11758196—21no—
102C2×#11758196D411yescharacteristic
103C2×#11758196D411yescharacteristic
104C2×C22⋊Dic61—21no—
105C2×C22⋊Dic61—21no—
106C2×C22⋊Dic61—21no—
107C2×C22⋊Dic61—21no—
108C2×C4×Dic61—21no—
109C2×C4×Dic61—21no—
110C2×C4×Dic61—21no—
111C2×C4×Dic61C2311yescharacteristic
112C2×C4×Dic61C2311yescharacteristic
113#14519710—22no—
114#14519702—22no—
115#14519702—22no—
116#14519700—22no—
117#14519698—22no—
118#14519688—22no—
119#14519684—21no—
120#14519684—21no—
121C22×D4×C61C2311yescharacteristic
122C23×C4×C61—21no—
123C22×C8×C61C2311yescharacteristic
124C22×C8×C61C2311yescharacteristic
125#125×C61—22no—
126#125×C61D414yes—
127C4×D4×C61—22no—
128C4×D4×C61—22no—
129C2×C4⋊C4×C61—21no—
130C2×C4⋊C4×C61—21no—
131C2×C4⋊C4×C61C2311yescharacteristic
132C2×C4⋊C4×C61C2311yescharacteristic
133C2×C4⋊C4×C61C2311yescharacteristic
134C2×C22⋊C4×C61—21no—
135C2×C42×C61C2311yescharacteristic
136#107×C61—22no—
137#106×C61—22no—
138C4⋊C8×C61—22no—
139#102×C61—22no—
140#102×C61—22no—
141C24×C61—21no—
142C2×D4×C61—21no—
143C2×D4×C61—21no—
144C2×D4×C61—21no—
145C2×D4×C61—21no—
146C2×D4×C61—21no—
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—42no—
154C22×C4×C61C2×D411yescharacteristic
155C22×C4×C61C2×D411yescharacteristic
156C22×C4×C61C2411yescharacteristic
157C22×Dic61—21no—
158C22×Dic61—21no—
159C22×Dic61—21no—
160C22×Dic61—21no—
161C22×Dic61C2×D411yescharacteristic
162C22×Dic61C2×D411yescharacteristic
163C22×Dic61C2×D411yescharacteristic
164C22×Dic61C2×D411yescharacteristic
165C2×Dic122—21no—
166C2×Dic122—21no—
167C2×Dic122—21no—
168C2×Dic122—24no—
169C2×Dic122—22no—
170C2×Dic122—22no—
171C2×Dic122—41no—
172C2×Dic122—42no—
173C2×Dic122—42no—
174C2×Dic122C2×D412yes—
175C2×Dic122C2×Q812yes—
176C2×Dic122#4112yes—
177C2×C8×C61—21no—
178C2×C8×C61—21no—
179C2×C8×C61—21no—
180C2×C8×C61—21no—
181C2×C8×C61—21no—
182C2×C8×C61—21no—
183C4⋊C4×C61—24no—
184C4⋊C4×C61—24no—
185C4⋊C4×C61—21no—
186C4⋊C4×C61—21no—
187C4⋊C4×C61—21no—
188C4⋊C4×C61—21no—
189C4⋊C4×C61C2×D412yes—
190C4⋊C4×C61C2×D412yes—
191C22⋊C4×C61—42no—
192C42×C61C2×D412yes—
193C42×C61C2×D412yes—
194C22⋊Dic61—42no—
195C22⋊Dic61—42no—
196C22⋊Dic61—81no—
197C22⋊Dic61—81no—
198#11758200—28no—
199C4⋊Dic61—24no—
200C4⋊Dic61—22no—
201C4⋊Dic61—21no—
202C4⋊Dic61—21no—
203C4⋊Dic61—21no—
204C4⋊Dic61—21no—
205C4⋊Dic61—21no—
206C4⋊Dic61—42no—
207C4⋊Dic61—81no—
208C4⋊Dic61C2×D412yes—
209#11758196—24no—
210#11758196—24no—
211#11758196—24no—
212#11758196—24no—
213#11758196—24no—
214#11758196—22no—
215#11758196—22no—
216#11758196—42no—
217#11758196—42no—
218C4×Dic61—24no—
219C4×Dic61—24no—
220C4×Dic61—24no—
221C4×Dic61—21no—
222C4×Dic61—21no—
223C4×Dic61—21no—
224C4×Dic61—21no—
225C2×#92418—21no—
226C2×#92418—21no—
227C2×#92418—21no—
228C2×#92418—21no—
229C2×#92418—21no—
230C2×#92418C2×Q812yes—
231C23×C61—22no—
232C23×C61—41no—
233C23×C61—41no—
234C23×C61—41no—
235C23×C61—41no—
236C23×C61—41no—
237C23×C61—41no—
238C23×C61C22×D411yescharacteristic
239D4×C61—22no—
240D4×C61—22no—
241D4×C61—41no—
242D4×C61—41no—
243C2×C4×C61—21no—
244C2×C4×C61—21no—
245C2×C4×C61—21no—
246C2×C4×C61—21no—
247C2×C4×C61—22no—
248C2×C4×C61—22no—
249C2×C4×C61—41no—
250C2×C4×C61—41no—
251C2×C4×C61—41no—
252C2×C4×C61—41no—
253C2×C4×C61—41no—
254C2×C4×C61—41no—
255C2×C4×C61—41no—
256C2×C4×C61—84no—
257C2×C4×C61#12012yes—
258C2×C4×C61#12012yes—
259C2×C4×C61C22×D411yescharacteristic
260C2×C4×C61C22×D411yescharacteristic
261C2×C4×C61C22×Q811yescharacteristic
262C2×C4×C61C2×#4111yescharacteristic
263C2×C4×C612−1+411yescharacteristic
264C2×C4×C612−1+411yescharacteristic
265C2×Dic61—22no—
266C2×Dic61—21no—
267C2×Dic61—22no—
268C2×Dic61—22no—
269C2×Dic61—21no—
270C2×Dic61—21no—
271C2×Dic61—21no—
272C2×Dic61—21no—
273C2×Dic61—22no—
274C2×Dic61—21no—
275C2×Dic61—21no—
276C2×Dic61—21no—
277C2×Dic61—21no—
278C2×Dic61—21no—
279C2×Dic61—21no—
280C2×Dic61—41no—
281C2×Dic61—41no—
282C2×Dic61—41no—
283C2×Dic61—41no—
284C2×Dic61—41no—
285C2×Dic61—41no—
286C2×Dic61—41no—
287C2×Dic61—41no—
288C2×Dic61AGL(1,ℤ/8)12yes—
289Dic122—22no—
290Dic122—22no—
291Dic122—22no—
292Dic122—44no—
293Dic122—48no—
294Dic122C22⋊Q814yescomplemented
295C8×C61—41no—
296C8×C61—41no—
297#92418—22no—
298#92418—41no—
299#24346—611nomaximal, Sylow
300C22×C61—42no—
301C22×C61—42no—
302C22×C61—42no—
303C22×C61—81no—
304C22×C61—81no—
305C22×C61—81no—
306C22×C61—81no—
307C22×C61C2×#12011yescharacteristic
308C22×C61#54811yescharacteristic
309C22×C61D4×Q811yescharacteristic
310C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
311C22×C61#57511yescharacteristic
312C22×C61#57711yescharacteristic
313C22×C61#57811yescharacteristic
314C4×C61—22no—
315C4×C61—22no—
316C4×C61—42no—
317C4×C61—42no—
318C4×C61—81no—
319C4×C61C2×C22⋊Q811yescharacteristic
320C4×C61#53611yescharacteristic
321C4×C61#54811yescharacteristic
322C4×C61D4×Q811yescharacteristic
323Dic61—24no—
324Dic61—22no—
325Dic61—41no—
326Dic61—42no—
327Dic61—41no—
328Dic61—41no—
329Dic61—81no—
330Dic61—81no—
331C2×#557—611no—
332C2×#478D6111yescharacteristic
333C2×#440—611no—
334#2538—611no—
335#2059—611no—
336#2057—611no—
337#2025—611no—
338#2023—611no—
339#1992—611no—
340#1985—611no—
341#1880—611no—
342#1880—611no—
343#1865—611no—
344#1863—611no—
345#1803—611no—
346C2×C61—82no—
347C2×C61#240911yescharacteristic
348C2×C61#300611yescharacteristic
349C2×C61#301311yescharacteristic
350C2×C61#336811yescharacteristic
351C2×C61#337711yescharacteristic
352C2×C61#338611yescharacteristic
353C2×C61#338811yescharacteristic
354C22×SD16—611no—
355#557—1224no—
356#557—1224no—
357C2×C4⋊Q8—611no—
358C2×C4⋊Q8—1221no—
359C2×#125—611no—
360C2×#125—1221no—
361C2×#125—1221no—
362C2×#125C2×D6111yescharacteristic
363C2×C22⋊Q8—1221no—
364C2×C4×Q8—611no—
365C2×C4×Q8—611no—
366C2×C4×Q8—611no—
367C2×C4×Q8—611no—
368C2×C4×Q8—1221no—
369C2×C4×D4C2×D6111yescharacteristic
370#478—24no—
371#440—1224no—
372C2×#107—611no—
373C2×#107C2×D6111yescharacteristic
374C2×#106C2×D6111yescharacteristic
375C2×C4⋊C8C2×D6111yescharacteristic
376C2×#103—611no—
377C2×#103—611no—
378C2×#103—611no—
379C2×#102—611no—
380C2×#102C2×D6111yescharacteristic
381C2×#102C2×D6111yescharacteristic
382C2×#97—611no—
383#396—1221no—
384#396—1221no—
385#390—611no—
386#390—611no—
387#389—611no—
388#389—611no—
389#384—611no—
390#380—1221no—
391#340—611no—
392#340—611no—
393#340—611no—
394#340—611no—
395#340—611no—
396#340—611no—
397#336—611no—
398#336—611no—
399#336—611no—
400#336—611no—
401C61#2434611yescharacteristic, complemented, Sylow
402C22×Q8—611no—
403C22×Q8—611no—
404C22×Q8—1221no—
405C22×D4C22×D6111yescharacteristic
406C23×C4—21no—
407C2×SD16—614no—
408C2×SD16—1222no—
409C2×SD16—1222no—
410C22×C8—611no—
411C22×C8—611no—
412C22×C8C22×D6111yescharacteristic
413C22×C8C22×D6111yescharacteristic
414C4⋊Q8—1222no—
415C4⋊Q8—1228no—
416C4⋊Q8—1222no—
417#125—22no—
418#125—1222no—
419#125—1228no—
420#125—1228no—
421#125—1222no—
422#125C22⋊D6114yes—
423C22⋊Q8—2444no—
424C4×Q8—614no—
425C4×Q8—1222no—
426C4×Q8—1222no—
427C4×Q8—1228no—
428C4×Q8—1222no—
429C4×Q8—1222no—
430C4×Q8—1222no—
431C4×Q8—1222no—
432C4×Q8—1222no—
433C4×D4—22no—
434C4×D4—22no—
435C2×C4⋊C4—21no—
436C2×C4⋊C4—21no—
437C2×C4⋊C4—611no—
438C2×C4⋊C4—611no—
439C2×C4⋊C4—611no—
440C2×C4⋊C4—611no—
441C2×C4⋊C4—611no—
442C2×C4⋊C4—611no—
443C2×C4⋊C4—1221no—
444C2×C4⋊C4—1221no—
445C2×C4⋊C4—1221no—
446C2×C4⋊C4—1221no—
447C2×C4⋊C4—1221no—
448C2×C4⋊C4—1221no—
449C2×C4⋊C4—1221no—
450C2×C4⋊C4—1221no—
451C2×C4⋊C4—1221no—
452C2×C4⋊C4—1221no—
453C2×C4⋊C4C22×D6111yescharacteristic
454C2×C4⋊C4C22×D6111yescharacteristic
455C2×C4⋊C4C22×D6111yescharacteristic
456C2×C22⋊C4—21no—
457C2×C22⋊C4—1221no—
458C2×C22⋊C4—1221no—
459C2×C22⋊C4—1221no—
460C2×C22⋊C4—1221no—
461C2×C42—611no—
462C2×C42—611no—
463C2×C42—1221no—
464C2×C42—1221no—
465C2×C42—1221no—
466C2×C42C22×D6111yescharacteristic
467#107—22no—
468#107—1222no—
469#106—22no—
470C4⋊C8—22no—
471#103—1222no—
472#103—1222no—
473#103—1222no—
474#102—22no—
475#102—22no—
476#102—1222no—
477#97—1222no—
478#95—1221no—
479#95—1221no—
480C24—21no—
481C2×Q8—612no—
482C2×Q8—612no—
483C2×Q8—612no—
484C2×Q8—1222no—
485C2×Q8—1224no—
486C2×Q8—1221no—
487C2×Q8—1221no—
488C2×Q8—1221no—
489C2×Q8—1222no—
490C2×Q8—2441no—
491C2×Q8—2442no—
492C2×Q8—2442no—
493C2×D4—21no—
494C2×D4—21no—
495C2×D4—21no—
496C2×D4—21no—
497C2×D4—21no—
498C2×D4Q8×D6112yes—
499C22×C4—21no—
500C22×C4—21no—
501C22×C4—21no—
502C22×C4—42no—
503C22×C4—42no—
504C22×C4—42no—
505C22×C4—611no—
506C22×C4—611no—
507C22×C4—611no—
508C22×C4—611no—
509C22×C4—1221no—
510C22×C4—1221no—
511C22×C4—1221no—
512C22×C4—1221no—
513C22×C4D4×D6111yescharacteristic
514C22×C4D4×D6111yescharacteristic
515C22×C4C23×D6111yescharacteristic
516SD16—1228no—
517C2×C8—21no—
518C2×C8—21no—
519C2×C8—21no—
520C2×C8—21no—
521C2×C8—21no—
522C2×C8—21no—
523C2×C8—612no—
524C2×C8—1221no—
525C2×C8—1221no—
526C2×C8—1221no—
527C2×C8—1221no—
528C2×C8—1221no—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—24no—
532C4⋊C4—24no—
533C4⋊C4—21no—
534C4⋊C4—21no—
535C4⋊C4—612no—
536C4⋊C4—1224no—
537C4⋊C4—1224no—
538C4⋊C4—1224no—
539C4⋊C4—1224no—
540C4⋊C4—1224no—
541C4⋊C4—1221no—
542C4⋊C4—1222no—
543C4⋊C4—1222no—
544C4⋊C4—1222no—
545C4⋊C4—1224no—
546C4⋊C4—1221no—
547C4⋊C4—1221no—
548C4⋊C4—1221no—
549C4⋊C4—1221no—
550C4⋊C4—4881no—
551C4⋊C4—2442no—
552C4⋊C4—2442no—
553C4⋊C4—2442no—
554C4⋊C4C2×C22⋊D6112yes—
555C4⋊C4C2×C22⋊D6112yes—
556C22⋊C4—42no—
557C22⋊C4—4881no—
558C22⋊C4—4881no—
559C22⋊C4—2442no—
560C22⋊C4—2442no—
561C42—1224no—
562C42—1224no—
563C42—1221no—
564C42—1221no—
565C42—1221no—
566C42—1221no—
567C42—1224no—
568C42D4×D6112yes—
569C42C2×C22⋊D6112yes—
570C23—22no—
571C23—41no—
572C23—41no—
573C23—41no—
574C23—41no—
575C23—41no—
576C23—41no—
577C23C2×D4×D6111yescharacteristic
578Q8—614no—
579Q8—1222no—
580Q8—1222no—
581Q8—1222no—
582Q8—2444no—
583Q8—2448no—
584D4—22no—
585D4—22no—
586D4—41no—
587D4—41no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—84no—
602C2×C4—612no—
603C2×C4—1222no—
604C2×C4—1221no—
605C2×C4—1221no—
606C2×C4—1222no—
607C2×C4—1222no—
608C2×C4—1221no—
609C2×C4—1222no—
610C2×C4—1221no—
611C2×C4—1221no—
612C2×C4—1221no—
613C2×C4—1221no—
614C2×C4—1221no—
615C2×C4—1221no—
616C2×C4—1221no—
617C2×C4—1221no—
618C2×C4—2441no—
619C2×C4—2441no—
620C2×C4—2441no—
621C2×C4—2441no—
622C2×C4—2441no—
623C2×C4—2441no—
624C2×C4—2441no—
625C2×C4—2441no—
626C2×C4#1451974512yes—
627C2×C4#1451974512yes—
628C2×C4C2×D4×D6111yescharacteristic
629C2×C4C22×C22⋊D6111yescharacteristic
630C2×C4C2×Q8×D6111yescharacteristic
631C2×C4#41×D6111yescharacteristic
632C2×C4#1451982411yescharacteristic
633C2×C4#1451982511yescharacteristic
634C8—41no—
635C8—41no—
636C8—1222no—
637C8—2441no—
638C22—42no—
639C22—42no—
640C22—42no—
641C22—81no—
642C22—81no—
643C22—81no—
644C22—81no—
645C22C2×#1451974511yescharacteristic
646C22#1684042211yescharacteristic
647C22#1684042511yescharacteristic
648C22AGL(1,ℤ/8)×D6111yescharacteristic
649C22#1684055811yescharacteristic
650C22#1684056911yescharacteristic
651C22#1684057611yescharacteristic
652C4—22no—
653C4—22no—
654C4—42no—
655C4—42no—
656C4—81no—
657C4—1222no—
658C4—1224no—
659C4—4881no—
660C4—4881no—
661C4—2441no—
662C4—2441no—
663C4—2441no—
664C4—2442no—
665C4C22⋊Q8×D6111yescharacteristic
666C4#1684033511yescharacteristic
667C4#1684041911yescharacteristic
668C4Q8×C22⋊D6111yescharacteristic
669C2—82no—
670C2#2144918011yescharacteristic
671C2#2145506711yescharacteristic
672C2#2145513011yescharacteristic
673C2#2145855111yescharacteristic
674C2#2145866411yescharacteristic
675C2#2145873511yescharacteristic
676C2#2145876111yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#478C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
hypercenterC2×#478D61
hypocenterC61#24346
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#478

Complements

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

NH ≅ G/Ntype
1. C2×#16840489C2semidirect
4. #21447159C2semidirect
5. #21447145C2semidirect
8. #21446789C2semidirect
9. #21446758C2semidirect
12. #21446084C2semidirect
13. #21446070C2semidirect
14. #21445825C2semidirect
68. C2×#11758200Q8semidirect
294. Dic122C22⋊Q8semidirect
401. C61#24346semidirect

Maximal subgroups

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

subgroupindex
C2×#168404892
C2×#168398622
#214507412
#214471592
#214471452
#214469632
#214469532
#214467892
#214467582
#214461712
#214460842
#214460702
#214458252
C2×C61×#4782
#2434661