Finite Group Databasebeta

← #29713068

Subgroups — #29713068

Order 15616.

1338 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D4×Dic122C211yesmaximal, characteristic
2C2×#16840093C211yesmaximal, characteristic, complemented
3C22×#14519702C211yesmaximal, characteristic, complemented
4#21450127C211yesmaximal, characteristic
5#21447066C212yesmaximal, complemented
6#21447016C212yesmaximal, complemented
7#21446242C211yesmaximal, characteristic, complemented
8#21446090C212yesmaximal, complemented
9#21445755C212yesmaximal, complemented
10#21445755C211yesmaximal, characteristic, complemented
11C61×#1869C211yesmaximal, characteristic
12C23×Dic122C2211yescharacteristic
13C23×Dic122C2211yescharacteristic
14D4×Dic122—24no—
15D4×Dic122—24no—
16C2×#14519777C2211yescharacteristic
17C2×#14519776—21no—
18C2×#14519758—21no—
19C2×#14519756C2211yescharacteristic
20C2×#14519756C2211yescharacteristic
21C2×#14519754—21no—
22C2×#14519754—21no—
23C2×C4×Dic122C2211yescharacteristic
24C2×D4×Dic61—21no—
25C22×C4⋊Dic61C2211yescharacteristic
26#16840093—24no—
27C2×#14519709C2212yes—
28C2×#14519702—24no—
29C2×#14519702C2212yes—
30C2×#14519702C2212yes—
31C2×#14519702C2212yes—
32C2×#14519702C2212yes—
33C2×#14519702C2212yes—
34C2×#14519702C2212yes—
35#16839692—21no—
36#16839687—21no—
37#16839660C2212yes—
38#16839623—21no—
39#16839622C2211yescharacteristic
40#16839622C2212yes—
41#16839525C2212yes—
42C23×C8×C61C2211yescharacteristic
43C2×C4×D4×C61C2211yescharacteristic
44C2×C22⋊C8×C61C2211yescharacteristic
45C2×C22⋊C8×C61C2212yes—
46C61×#336C2212yes—
47C22×Dic122—24no—
48C22×Dic122—21no—
49C22×Dic122—21no—
50C22×Dic122—22no—
51C22×Dic122—22no—
52C22×Dic122—22no—
53C22×Dic122—24no—
54C22×Dic122—42no—
55C22×Dic122—42no—
56C22×Dic122D414yes—
57C22×Dic122C2311yescharacteristic
58C22×Dic122C2311yescharacteristic
59C22×Dic122C2311yescharacteristic
60C23×Dic61—21no—
61C23×Dic61—21no—
62C23×Dic61—21no—
63#14519777—22no—
64#14519777—22no—
65#14519776—28no—
66#14519758—44no—
67#14519756—22no—
68#14519756—22no—
69#14519756—22no—
70#14519756D414yes—
71#14519754—28no—
72#14519754—28no—
73C4×Dic122—22no—
74C4×Dic122—22no—
75D4×Dic61—28no—
76C2×C4⋊Dic61—21no—
77C2×C4⋊Dic61—21no—
78C2×C4⋊Dic61—22no—
79C2×C4⋊Dic61—22no—
80C2×C4⋊Dic61—21no—
81C2×C4⋊Dic61C2312yes—
82C2×C4⋊Dic61C2312yes—
83C2×C4⋊Dic61C2311yescharacteristic
84C2×#11758196—21no—
85C2×#11758196—21no—
86C2×#11758196—21no—
87C2×#11758196—22no—
88C2×C22⋊Dic61—21no—
89C2×C22⋊Dic61—21no—
90C2×C22⋊Dic61—22no—
91C2×C22⋊Dic61—21no—
92C2×C4×Dic61—21no—
93#14519709—24no—
94#14519702—24no—
95#14519702—24no—
96#14519702—24no—
97#14519702—24no—
98#14519702—42no—
99#14519702—42no—
100#14519684—21no—
101#14519684—22no—
102#14519684—22no—
103C22×D4×C61D411yescharacteristic
104C23×C4×C61C2311yescharacteristic
105C23×C4×C61C2311yescharacteristic
106C22×C8×C61—22no—
107C22×C8×C61—42no—
108C22×C8×C61C2312yes—
109C22×C8×C61C2312yes—
110C4×D4×C61—22no—
111C4×D4×C61—22no—
112C2×C4⋊C4×C61D411yescharacteristic
113C2×C22⋊C4×C61D411yescharacteristic
114C2×C22⋊C4×C61D411yescharacteristic
115C2×C42×C61C2311yescharacteristic
116C22⋊C8×C61—22no—
117C22⋊C8×C61—42no—
118C24×C61C2×D411yescharacteristic
119C24×C61C2×D411yescharacteristic
120C2×D4×C61—22no—
121C2×D4×C61—22no—
122C2×D4×C61SD1614yes—
123C22×C4×C61—22no—
124C22×C4×C61—22no—
125C22×C4×C61—21no—
126C22×C4×C61—21no—
127C22×C4×C61—22no—
128C22×C4×C61—22no—
129C22×C4×C61C2×D411yescharacteristic
130C22×C4×C61C2×D411yescharacteristic
131C22×C4×C61C2×D411yescharacteristic
132C22×C4×C61C2×D412yes—
133C22×C4×C61C2×D412yes—
134C22×C4×C61C2×D411yescharacteristic
135C22×C4×C61C2411yescharacteristic
136C22×Dic61—21no—
137C22×Dic61—24no—
138C22×Dic61—24no—
139C22×Dic61—21no—
140C22×Dic61—21no—
141C22×Dic61—21no—
142C22×Dic61—22no—
143C22×Dic61—22no—
144C22×Dic61—44no—
145C22×Dic61—44no—
146C22×Dic61—42no—
147C22×Dic61—42no—
148C22×Dic61—42no—
149C2×Dic122—21no—
150C2×Dic122—22no—
151C2×Dic122—22no—
152C2×Dic122—22no—
153C2×Dic122—22no—
154C2×Dic122—22no—
155C2×Dic122—28no—
156C2×Dic122—24no—
157C2×Dic122—24no—
158C2×Dic122—21no—
159C2×Dic122—21no—
160C2×Dic122—44no—
161C2×Dic122—48no—
162C2×Dic122—48no—
163C2×Dic122—44no—
164C2×Dic122—44no—
165C2×Dic122—44no—
166C2×Dic122—44no—
167C2×Dic122C2×D412yes—
168C2×Dic122C2×D412yes—
169C2×Dic122C2×D412yes—
170C2×Dic122#4114yes—
171C2×C8×C61—22no—
172C2×C8×C61—22no—
173C2×C8×C61—22no—
174C2×C8×C61—22no—
175C2×C8×C61—22no—
176C2×C8×C61—22no—
177C2×C8×C61—44no—
178C4⋊C4×C61—22no—
179C22⋊C4×C61—22no—
180C22⋊C4×C61—22no—
181C42×C61—21no—
182C42×C61—21no—
183C22⋊Dic61—24no—
184C22⋊Dic61—24no—
185C22⋊Dic61—42no—
186C22⋊Dic61—44no—
187C4⋊Dic61—24no—
188C4⋊Dic61—24no—
189C4⋊Dic61—21no—
190C4⋊Dic61—22no—
191C4⋊Dic61—22no—
192C4⋊Dic61—22no—
193C4⋊Dic61—22no—
194C4⋊Dic61—42no—
195C4⋊Dic61C2×D412yes—
196#11758196—24no—
197#11758196—24no—
198#11758196—24no—
199#11758196—44no—
200C4×Dic61—24no—
201C23×C61—22no—
202C23×C61—22no—
203C23×C61—21no—
204C23×C61—22no—
205C23×C61—21no—
206C23×C61—22no—
207C23×C61#12012yes—
208C23×C61C2×SD1612yes—
209C23×C61C2×SD1612yes—
210C23×C61#13712yes—
211C23×C61C22×D411yescharacteristic
212D4×C61—28no—
213C2×C4×C61—21no—
214C2×C4×C61—22no—
215C2×C4×C61—21no—
216C2×C4×C61—22no—
217C2×C4×C61—22no—
218C2×C4×C61—24no—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61—22no—
222C2×C4×C61—22no—
223C2×C4×C61—41no—
224C2×C4×C61—41no—
225C2×C4×C61—42no—
226C2×C4×C61#12012yes—
227C2×C4×C61C2×SD1612yes—
228C2×C4×C61C22×D411yescharacteristic
229C2×C4×C61C22×D411yescharacteristic
230C2×C4×C61C2×#4112yes—
231C2×C4×C61H(5,𝔽2)12yes—
232C2×Dic61—22no—
233C2×Dic61—24no—
234C2×Dic61—22no—
235C2×Dic61—22no—
236C2×Dic61—22no—
237C2×Dic61—48no—
238C2×Dic61—48no—
239C2×Dic61—42no—
240C2×Dic61—42no—
241C2×Dic61—42no—
242C2×Dic61—42no—
243C2×Dic61—42no—
244C2×Dic61—42no—
245C2×Dic61—42no—
246C2×Dic61—42no—
247C2×Dic61—42no—
248C2×Dic61—84no—
249Dic122—22no—
250Dic122—24no—
251Dic122—22no—
252Dic122—22no—
253Dic122—24no—
254Dic122—48no—
255Dic122—48no—
256Dic122—41no—
257Dic122—41no—
258Dic122—42no—
259Dic122#12314yescomplemented
260C8×C61—42no—
261C8×C61—42no—
262#24041—611nomaximal, Sylow
263C22×C61—22no—
264C22×C61—22no—
265C22×C61—24no—
266C22×C61—24no—
267C22×C61—22no—
268C22×C61—24no—
269C22×C61—41no—
270C22×C61—41no—
271C22×C61—42no—
272C22×C61#44814yes—
273C22×C61C2×#12011yescharacteristic
274C22×C61#54612yes—
275C22×C61C22×SD1611yescharacteristic
276C22×C61C2×#13711yescharacteristic
277C22×C61#57812yes—
278C4×C61—22no—
279C4×C61—22no—
280C4×C61—41no—
281C4×C61—41no—
282C4×C61C2×#12311yescharacteristic
283C4×C61#53411yescharacteristic
284C4×C61#54612yes—
285Dic61—24no—
286Dic61—42no—
287Dic61—42no—
288Dic61—42no—
289Dic61—82no—
290Dic61—82no—
291C2×D4×Q8—611no—
292C2×#483—611no—
293C22×#103—611no—
294#2486—611no—
295#2043—612no—
296#2035—612no—
297#1892—611no—
298#1869D6111yescharacteristic
299#1865—612no—
300#1787—611no—
301#1787—612no—
302C2×C61—24no—
303C2×C61—42no—
304C2×C61#240511yescharacteristic
305C2×C61C2×#44811yescharacteristic
306C2×C61#301111yescharacteristic
307C2×C61#329812yes—
308C2×C61#331212yes—
309C23×Q8—611no—
310C23×Q8—611no—
311C23×C8C2×D6111yescharacteristic
312D4×Q8—1224no—
313D4×Q8—1224no—
314C2×C4⋊Q8—611no—
315C2×C4⋊Q8—1221no—
316C2×#123—1221no—
317C2×C22⋊Q8—611no—
318C2×C22⋊Q8—611no—
319C2×C22⋊Q8—1221no—
320C2×C22⋊Q8—1221no—
321C2×C4×Q8—611no—
322C2×C4×D4—1221no—
323C2×C4×D4C2×D6111yescharacteristic
324C22×C4⋊C4—611no—
325#483—1224no—
326C2×#106—612no—
327C2×#103—612no—
328C2×#103—612no—
329C2×#103—612no—
330C2×#103—612no—
331C2×#103—612no—
332C2×#103—612no—
333C2×#103—1224no—
334C2×C22⋊C8C2×D6111yescharacteristic
335C2×C22⋊C8C2×D6112yes—
336#398—1221no—
337#397—1221no—
338#389—612no—
339#381—1221no—
340#380—611no—
341#380—612no—
342#340—612no—
343#336C2×D6112yes—
344C61#2404111yescharacteristic, complemented, Sylow
345C22×Q8—614no—
346C22×Q8—611no—
347C22×Q8—611no—
348C22×Q8—611no—
349C22×Q8—1222no—
350C22×Q8—1222no—
351C22×Q8—1224no—
352C22×Q8—1224no—
353C22×Q8—1221no—
354C22×Q8—1221no—
355C22×Q8—1222no—
356C22×Q8—2442no—
357C22×Q8—2442no—
358C22×D4D24411yescharacteristic
359C23×C4—1221no—
360C23×C4—1221no—
361C23×C4—1221no—
362C23×C4C22×D6111yescharacteristic
363C23×C4C22×D6111yescharacteristic
364C22×C8—22no—
365C22×C8—42no—
366C22×C8C22×D6112yes—
367C22×C8C22×D6112yes—
368C4⋊Q8—1222no—
369C4⋊Q8—1222no—
370C4⋊Q8—1228no—
371#123—2444no—
372C22⋊Q8—614no—
373C22⋊Q8—1228no—
374C22⋊Q8—1228no—
375C22⋊Q8—1222no—
376C22⋊Q8—1222no—
377C22⋊Q8—1222no—
378C4×Q8—1222no—
379C4×Q8—1222no—
380C4×D4—22no—
381C4×D4—22no—
382C4×D4—1228no—
383C2×C4⋊C4—611no—
384C2×C4⋊C4—612no—
385C2×C4⋊C4—612no—
386C2×C4⋊C4—1221no—
387C2×C4⋊C4—1221no—
388C2×C4⋊C4—1221no—
389C2×C4⋊C4—1221no—
390C2×C4⋊C4—1221no—
391C2×C4⋊C4—1222no—
392C2×C4⋊C4—1222no—
393C2×C4⋊C4—1221no—
394C2×C4⋊C4—1222no—
395C2×C4⋊C4D24411yescharacteristic
396C2×C22⋊C4—1221no—
397C2×C22⋊C4—1221no—
398C2×C22⋊C4—1221no—
399C2×C22⋊C4—1222no—
400C2×C22⋊C4D24411yescharacteristic
401C2×C22⋊C4D24411yescharacteristic
402C2×C42—1221no—
403C2×C42C22×D6111yescharacteristic
404#106—1224no—
405#103—1224no—
406#103—1224no—
407#103—1224no—
408#103—1224no—
409#103—2442no—
410#103—2442no—
411C22⋊C8—22no—
412C22⋊C8—42no—
413#95—1222no—
414#95—1221no—
415#95—1222no—
416C24C2×D24411yescharacteristic
417C24C2×D24411yescharacteristic
418C2×Q8—612no—
419C2×Q8—612no—
420C2×Q8—612no—
421C2×Q8—614no—
422C2×Q8—1224no—
423C2×Q8—1228no—
424C2×Q8—1221no—
425C2×Q8—1221no—
426C2×Q8—1221no—
427C2×Q8—1222no—
428C2×Q8—1224no—
429C2×Q8—1222no—
430C2×Q8—1222no—
431C2×Q8—1222no—
432C2×Q8—1222no—
433C2×Q8—2444no—
434C2×Q8—2444no—
435C2×Q8—2444no—
436C2×Q8—2448no—
437C2×Q8—2444no—
438C2×Q8—2448no—
439C2×Q8—2444no—
440C2×D4—22no—
441C2×D4—22no—
442C2×D4#1175819014yes—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—1221no—
450C22×C4—1224no—
451C22×C4—1221no—
452C22×C4—1221no—
453C22×C4—1221no—
454C22×C4—1224no—
455C22×C4—1222no—
456C22×C4—1222no—
457C22×C4—2444no—
458C22×C4—2444no—
459C22×C4—2442no—
460C22×C4—2442no—
461C22×C4—2442no—
462C22×C4C2×D24411yescharacteristic
463C22×C4C2×D24411yescharacteristic
464C22×C4C2×D24411yescharacteristic
465C22×C4C2×D24411yescharacteristic
466C22×C4D4×D6112yes—
467C22×C4D4×D6112yes—
468C22×C4C23×D6111yescharacteristic
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—44no—
476C4⋊C4—22no—
477C4⋊C4—612no—
478C4⋊C4—1224no—
479C4⋊C4—1224no—
480C4⋊C4—1224no—
481C4⋊C4—1224no—
482C4⋊C4—1221no—
483C4⋊C4—1224no—
484C4⋊C4—1222no—
485C4⋊C4—1222no—
486C4⋊C4—1222no—
487C4⋊C4—1222no—
488C4⋊C4—2442no—
489C4⋊C4—2444no—
490C22⋊C4—22no—
491C22⋊C4—22no—
492C22⋊C4—1224no—
493C22⋊C4—1224no—
494C22⋊C4—2442no—
495C22⋊C4—2444no—
496C42—21no—
497C42—21no—
498C42—1224no—
499C23—22no—
500C23—22no—
501C23—21no—
502C23—21no—
503C23—22no—
504C23—22no—
505C23#1451974712yes—
506C23C2×#1175819012yes—
507C23C2×#1175819012yes—
508C23#1451981012yes—
509C23C22×D24411yescharacteristic
510Q8—614no—
511Q8—1222no—
512Q8—1224no—
513Q8—1222no—
514Q8—1222no—
515Q8—1224no—
516Q8—2448no—
517Q8—2448no—
518Q8—2441no—
519Q8—2441no—
520Q8—2442no—
521D4—28no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—24no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—42no—
535C2×C4—1222no—
536C2×C4—1224no—
537C2×C4—1222no—
538C2×C4—1222no—
539C2×C4—1222no—
540C2×C4—4884no—
541C2×C4—2448no—
542C2×C4—2448no—
543C2×C4—2442no—
544C2×C4—2442no—
545C2×C4—2442no—
546C2×C4—2442no—
547C2×C4—2442no—
548C2×C4—2442no—
549C2×C4—2442no—
550C2×C4—2442no—
551C2×C4—2442no—
552C2×C4#1451974712yes—
553C2×C4C2×#1175819012yes—
554C2×C4C2×D4×D6111yescharacteristic
555C2×C4C2×D4×D6111yescharacteristic
556C2×C4#41×D6112yes—
557C2×C4#1451982212yes—
558C8—42no—
559C8—42no—
560C22—22no—
561C22—24no—
562C22—22no—
563C22—24no—
564C22—24no—
565C22—22no—
566C22—41no—
567C22—41no—
568C22—42no—
569C22#1683990314yes—
570C22C2×#1451974711yescharacteristic
571C22#1684040712yes—
572C22C22×#1175819011yescharacteristic
573C22C2×#1451981011yescharacteristic
574C22#1684057912yes—
575C4—22no—
576C4—22no—
577C4—41no—
578C4—41no—
579C4—1224no—
580C4—4882no—
581C4—4882no—
582C4—2442no—
583C4—2442no—
584C4—2442no—
585C4#123×D6111yescharacteristic
586C4#1684032211yescharacteristic
587C4#1684040112yes—
588C2—24no—
589C2—42no—
590C2#2144914511yescharacteristic
591C2C2×#1683990311yescharacteristic
592C2#2145511611yescharacteristic
593C2#2145768512yes—
594C2#2145787212yes—
subgroupG/H
centerC23C22×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1869C2
socleC23×C61C22×D4
prime radicalC22×D4×C61D4
hypercenter#1869D61
hypocenterC61#24041
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #1869

Complements

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

NH ≅ G/Ntype
2. C2×#16840093C2semidirect
3. C22×#14519702C2semidirect
5. #21447066C2semidirect
6. #21447016C2semidirect
7. #21446242C2semidirect
8. #21446090C2semidirect
9. #21445755C2semidirect
10. #21445755C2semidirect
259. Dic122#123semidirect
344. C61#24041semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Dic1222
C2×#168400932
C22×#145197022
#214501272
#214470662
#214470162
#214462422
#214460902
#214457552
C61×#18692
#2404161