Finite Group Databasebeta

← #29689922

Subgroups — #29689922

Order 15616.

1398 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840309C211yesmaximal, characteristic
2C2×C4⋊C8×D61C211yesmaximal, characteristic, complemented
3#21448212C211yesmaximal, characteristic, complemented
4#21446765C212yesmaximal, complemented
5#21446744C212yesmaximal, complemented
6#21446246C212yesmaximal, complemented
7#21446244C212yesmaximal, complemented
8#21445830C211yesmaximal, characteristic, complemented
9#21445829C211yesmaximal, characteristic, complemented
10#21445826C211yesmaximal, characteristic, complemented
11C61×#1804C211yesmaximal, characteristic, complemented
12C22×#11758222—21no—
13C22×C8×D61C2212yes—
14#16840309—24no—
15#16840309C418yescomplemented
16C2×#14519776C2211yescharacteristic
17C2×#14519773C2211yescharacteristic
18C2×#14519763—21no—
19C2×#14519752C2211yescharacteristic
20C2×#14519750—21no—
21C2×#14519748C2211yescharacteristic
22C2×C4×D244C2211yescharacteristic
23C2×C4×C22⋊D61—21no—
24C2×C42×D61C2211yescharacteristic
25C2×C4⋊#92418C2211yescharacteristic
26C2×#14519706C2212yes—
27C4⋊C8×D61—24no—
28C2×#14519699C2212yes—
29C2×#14519698C2212yes—
30C2×#14519697C2212yes—
31#16839629—21no—
32#16839628C2211yescharacteristic
33#16839624—21no—
34#16839619C2212yes—
35C4⋊C4×Dic61C2212yes—
36#16839609—21no—
37#16839607—21no—
38#16839525C2212yes—
39#16839524C2212yes—
40#16839524C2212yes—
41C2×C4×D4×C61C2211yescharacteristic
42C2×C4⋊C8×C61C2211yescharacteristic
43C2×#102×C61C2212yes—
44C61×#382C2211yescharacteristic
45C61×#340C2212yes—
46C2×#11758222—28no—
47C2×#11758222—48no—
48C22×Dic122—21no—
49C22×D244—21no—
50C22×D244C2311yescharacteristic
51C22×C22⋊D61—21no—
52C22×C22⋊D61—21no—
53C22×C22⋊D61—21no—
54C22×C4×D61—21no—
55C22×C4×D61D411yescharacteristic
56C22×C4×D61D411yescharacteristic
57C22×C4×D61C2311yescharacteristic
58C22×#92418C2312yes—
59C2×C8×D61—24no—
60C2×C8×D61—42no—
61C2×C8×D61—42no—
62#14519776—22no—
63#14519776C2×C414yes—
64#14519773—22no—
65#14519773C2×C414yes—
66#14519763—28no—
67#14519752—22no—
68#14519752C2×C414yes—
69#14519750—28no—
70#14519748—22no—
71#14519748C2×C414yes—
72C4×D244—22no—
73C4×D244C2×C414yes—
74C4×C22⋊D61—28no—
75C2×C4⋊Dic61C2312yes—
76C2×C4⋊Dic61C2311yescharacteristic
77C2×#11758196—21no—
78C2×C22⋊Dic61—21no—
79C2×#11758198—21no—
80C2×#11758198—21no—
81C2×#11758198—22no—
82C2×C4×Dic61—22no—
83C2×C4×Dic61D411yescharacteristic
84C2×C4×Dic61D411yescharacteristic
85C2×C4×Dic61C2311yescharacteristic
86C42×D61C2×C414yes—
87C42×D61D414yes—
88C4⋊#92418—22no—
89#14519706—24no—
90#14519699—42no—
91#14519698—42no—
92#14519697—42no—
93#14519684—21no—
94#14519684—21no—
95#14519684—22no—
96#14519684—22no—
97C22×D4×C61C2311yescharacteristic
98C23×C4×C61—21no—
99C22×C8×C61C2312yes—
100C4×D4×C61—22no—
101C4×D4×C61C2×C414yes—
102C2×C4⋊C4×C61C2312yes—
103C2×C4⋊C4×C61C2311yescharacteristic
104C2×C22⋊C4×C61—21no—
105C2×C42×C61C2311yescharacteristic
106C4⋊C8×C61—22no—
107#102×C61—42no—
108#95×C61—21no—
109#95×C61—21no—
110C24×C61—21no—
111C23×D61—21no—
112C23×D61C2×D411yescharacteristic
113C2×D4×C61—21no—
114C2×D4×C61—21no—
115C2×D4×C61—21no—
116C2×D4×C61—21no—
117C2×D4×C61—21no—
118C2×D4×C61C22×C412yes—
119C22×C4×C61—21no—
120C22×C4×C61—24no—
121C22×C4×C61—22no—
122C22×C4×C61—44no—
123C22×C4×C61C2×D411yescharacteristic
124C22×C4×C61C2×D411yescharacteristic
125C22×C4×C61C2411yescharacteristic
126C2×C22⋊D61—24no—
127C2×C22⋊D61—24no—
128C2×C22⋊D61—24no—
129C2×C22⋊D61—44no—
130C2×C22⋊D61—44no—
131C2×C22⋊D61—44no—
132C22×Dic61—21no—
133C22×Dic61—22no—
134C22×Dic61C2×D411yescharacteristic
135C22×Dic61C2×D411yescharacteristic
136C22×Dic61C2×D411yescharacteristic
137#11758222—432no—
138C2×D244—21no—
139C2×D244—21no—
140C2×D244—21no—
141C2×D244—21no—
142C2×D244—24no—
143C2×D244—21no—
144C2×D244—44no—
145C2×D244C22×C412yes—
146C2×C4×D61—24no—
147C2×C4×D61—21no—
148C2×C4×D61—21no—
149C2×C4×D61—21no—
150C2×C4×D61—21no—
151C2×C4×D61—24no—
152C2×C4×D61—24no—
153C2×C4×D61—44no—
154C2×C4×D61C22⋊C414yes—
155C2×C4×D61C22⋊C414yes—
156C2×C4×D61C22×C412yes—
157C2×C4×D61C2×D412yes—
158C2×Dic122—24no—
159C2×Dic122—44no—
160C2×C8×C61—22no—
161C2×C8×C61—22no—
162C2×C8×C61—22no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—22no—
165C4⋊C4×C61—22no—
166C4⋊C4×C61C22×C412yes—
167C22⋊C4×C61—24no—
168C42×C61C22×C412yes—
169C42×C61C2×D412yes—
170C22⋊Dic61—24no—
171#11758198—24no—
172#11758198—24no—
173#11758198—44no—
174C4⋊Dic61—21no—
175C4⋊Dic61—22no—
176C4⋊Dic61—22no—
177C4⋊Dic61C22×C412yes—
178#11758196—24no—
179C4×Dic61—44no—
180C4×Dic61C22⋊C414yes—
181C4×Dic61C22⋊C414yes—
182C4×Dic61C22×C412yes—
183C4×Dic61C2×D412yes—
184C2×#92418—22no—
185C2×#92418—22no—
186C2×#92418—22no—
187C8×D61—48no—
188C23×C61—22no—
189C23×C61—41no—
190C23×C61—41no—
191C23×C61—42no—
192C23×C61—42no—
193C23×C61C22×D411yescharacteristic
194C22×D61—21no—
195C22×D61—21no—
196C22×D61—22no—
197C22×D61—22no—
198C22×D61—41no—
199C22×D61—41no—
200C22×D61—42no—
201C22×D61—42no—
202C22×D61C2×C22⋊C412yes—
203C22×D61AGL(1,ℤ/8)14yes—
204D4×C61—22no—
205D4×C61—22no—
206D4×C61—41no—
207D4×C61—41no—
208C2×C4×C61—22no—
209C2×C4×C61—22no—
210C2×C4×C61—22no—
211C2×C4×C61—48no—
212C2×C4×C61—42no—
213C2×C4×C61—42no—
214C2×C4×C61—42no—
215C2×C4×C61—42no—
216C2×C4×C61C2×C22⋊C412yes—
217C2×C4×C61C2×C22⋊C412yes—
218C2×C4×C61C23×C411yescharacteristic
219C2×C4×C61C22×D411yescharacteristic
220C2×C4×C61H(5,𝔽2)11yescharacteristic
221C2×C4×C61H(5,𝔽2)11yescharacteristic
222C2×C4×C612−1+411yescharacteristic
223C2×C4×C612−1+411yescharacteristic
224C22⋊D61—48no—
225C22⋊D61—48no—
226C22⋊D61—48no—
227C2×Dic61—21no—
228C2×Dic61—21no—
229C2×Dic61—22no—
230C2×Dic61—22no—
231C2×Dic61—42no—
232C2×Dic61—42no—
233C2×Dic61—42no—
234C2×Dic61—42no—
235C2×Dic61C2×C22⋊C412yes—
236C2×Dic61C2×C22⋊C412yes—
237C2×Dic61C2×C22⋊C412yes—
238C2×Dic61#13514yes—
239D244—22no—
240D244—22no—
241D244—48no—
242D244—41no—
243D244—41no—
244C4×D61—28no—
245C4×D61—22no—
246C4×D61—22no—
247C4×D61—22no—
248C4×D61—22no—
249C4×D61—28no—
250C4×D61—48no—
251Dic122—48no—
252C8×C61—42no—
253#92418—42no—
254#21888—611nomaximal, Sylow
255C22×C61—42no—
256C22×C61—42no—
257C22×C61—42no—
258C22×C61—82no—
259C22×C61—82no—
260C22×C61C22×C22⋊C411yescharacteristic
261C22×C61#53711yescharacteristic
262C22×C61#53711yescharacteristic
263C22×C61C2×#13512yes—
264C22×C61C2×AGL(1,ℤ/8)12yes—
265C2×D61—22no—
266C2×D61—22no—
267C2×D61—24no—
268C2×D61—24no—
269C2×D61—42no—
270C2×D61—42no—
271C2×D61—42no—
272C2×D61—82no—
273C2×D61—82no—
274C2×D61#41814yes—
275C4×C61—22no—
276C4×C61—22no—
277C4×C61—42no—
278C4×C61—82no—
279C4×C61#52011yescharacteristic
280C4×C61#52011yescharacteristic
281C4×C61#53511yescharacteristic
282C4×C61#53611yescharacteristic
283Dic61—22no—
284Dic61—22no—
285Dic61—42no—
286Dic61—82no—
287Dic61#41614yes—
288C2×#532—611no—
289C22×C4⋊C8—611no—
290#2283—611no—
291#1987—612no—
292#1982—612no—
293#1893—612no—
294#1893—612no—
295#1804—611no—
296#1804—611no—
297#1804—611no—
298#1804D6111yescharacteristic, complemented
299C2×C61—82no—
300C2×C61#231111yescharacteristic
301C2×C61C2×#41611yescharacteristic
302C2×C61C2×#41811yescharacteristic
303C2×C61#324712yes—
304C2×C61#324712yes—
305D61—24no—
306D61—82no—
307C22×#41—1221no—
308C23×C8—612no—
309#532—618no—
310#532—1224no—
311C2×C4⋊Q8—611no—
312C2×#127—611no—
313C2×#124—1221no—
314C2×#121—611no—
315C2×#121—611no—
316C2×#121—1221no—
317C2×C4×D4—611no—
318C2×C4×D4—1221no—
319C2×C4×D4C2×D6111yescharacteristic
320C22×C42—611no—
321C2×C4⋊C8—612no—
322C2×C4⋊C8—611no—
323C2×C4⋊C8—1224no—
324C2×C4⋊C8C2×D6111yescharacteristic
325C2×#102—612no—
326C2×#102—612no—
327C2×#102—612no—
328C2×#102C2×D6112yes—
329#383—1221no—
330#382—611no—
331#382C2×D6111yescharacteristic
332#381—1221no—
333#380—612no—
334C4×C4⋊C4—612no—
335C4×C22⋊C4—1221no—
336#376—1221no—
337#340—612no—
338#340—612no—
339#340—612no—
340#340C2×D6112yes—
341C61#2188811yescharacteristic, complemented, Sylow
342C2×#41—1228no—
343C2×#41—2448no—
344C22×Q8—1221no—
345C22×D4—611no—
346C22×D4—1221no—
347C22×D4—1221no—
348C22×D4—1221no—
349C22×D4—1221no—
350C22×D4C22×D6111yescharacteristic
351C23×C4—21no—
352C23×C4—611no—
353C23×C4—611no—
354C23×C4—611no—
355C23×C4—1221no—
356C22×C8—612no—
357C22×C8—1224no—
358C22×C8—2442no—
359C22×C8—2442no—
360C22×C8C22×D6112yes—
361C4⋊Q8—614no—
362C4⋊Q8—1222no—
363#127—614no—
364#127—1222no—
365#124—1228no—
366#121—614no—
367#121—614no—
368#121—1228no—
369#121—1222no—
370#121—1222no—
371C4×D4—22no—
372C4×D4—614no—
373C4×D4—1228no—
374C4×D4—1222no—
375C4×D4C4×D6114yes—
376C2×C4⋊C4—611no—
377C2×C4⋊C4—612no—
378C2×C4⋊C4—1221no—
379C2×C4⋊C4C22×D6112yes—
380C2×C4⋊C4C22×D6111yescharacteristic
381C2×C22⋊C4—21no—
382C2×C22⋊C4—1221no—
383C2×C22⋊C4—1221no—
384C2×C22⋊C4—1221no—
385C2×C22⋊C4—1222no—
386C2×C42—611no—
387C2×C42—611no—
388C2×C42—611no—
389C2×C42—614no—
390C2×C42—614no—
391C2×C42—1222no—
392C2×C42C22×D6111yescharacteristic
393C4⋊C8—22no—
394C4⋊C8—1224no—
395C4⋊C8—1222no—
396#102—42no—
397#102—2442no—
398#102—2442no—
399#102—2442no—
400#95—21no—
401#95—21no—
402#95—1222no—
403#95—1222no—
404#95—1221no—
405#95—1221no—
406C24—21no—
407C24—611no—
408C24—1221no—
409#41—24432no—
410C2×Q8—1224no—
411C2×Q8—2444no—
412C2×D4—21no—
413C2×D4—21no—
414C2×D4—21no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4—612no—
418C2×D4—1224no—
419C2×D4—1224no—
420C2×D4—1224no—
421C2×D4—1224no—
422C2×D4—1221no—
423C2×D4—1221no—
424C2×D4—1221no—
425C2×D4—1221no—
426C2×D4—1221no—
427C2×D4—2444no—
428C2×D4—2444no—
429C2×D4—2444no—
430C2×D4—2444no—
431C2×D4C2×C4×D6112yes—
432C22×C4—21no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—44no—
436C22×C4—611no—
437C22×C4—611no—
438C22×C4—611no—
439C22×C4—612no—
440C22×C4—612no—
441C22×C4—614no—
442C22×C4—614no—
443C22×C4—1224no—
444C22×C4—1224no—
445C22×C4—1221no—
446C22×C4—1221no—
447C22×C4—1221no—
448C22×C4—1221no—
449C22×C4—1221no—
450C22×C4—1224no—
451C22×C4—1222no—
452C22×C4—2444no—
453C22×C4D4×D6111yescharacteristic
454C22×C4D4×D6111yescharacteristic
455C22×C4C23×D6111yescharacteristic
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—1222no—
460C2×C8—1222no—
461C2×C8—1222no—
462C2×C8—2448no—
463C4⋊C4—21no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—612no—
467C4⋊C4—1224no—
468C4⋊C4—1221no—
469C4⋊C4—1222no—
470C4⋊C4—1222no—
471C4⋊C4C2×C4×D6112yes—
472C22⋊C4—24no—
473C22⋊C4—1224no—
474C22⋊C4—1224no—
475C22⋊C4—1224no—
476C22⋊C4—2444no—
477C42—612no—
478C42—612no—
479C42—614no—
480C42—614no—
481C42—2444no—
482C42C2×C4×D6112yes—
483C42D4×D6112yes—
484C23—22no—
485C23—42no—
486C23—42no—
487C23—41no—
488C23—41no—
489C23—612no—
490C23—614no—
491C23—1222no—
492C23—1221no—
493C23—1221no—
494C23—1222no—
495C23—2442no—
496C23—2442no—
497C23—2441no—
498C23—2441no—
499C23C2×D4×D6111yescharacteristic
500Q8—2448no—
501D4—22no—
502D4—22no—
503D4—41no—
504D4—41no—
505D4—1222no—
506D4—1222no—
507D4—2448no—
508D4—2448no—
509D4—2448no—
510D4—2448no—
511D4—2441no—
512D4—2441no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—48no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—612no—
522C2×C4—614no—
523C2×C4—612no—
524C2×C4—612no—
525C2×C4—1228no—
526C2×C4—1228no—
527C2×C4—1222no—
528C2×C4—1221no—
529C2×C4—1221no—
530C2×C4—1222no—
531C2×C4—1222no—
532C2×C4—1222no—
533C2×C4—1222no—
534C2×C4—1222no—
535C2×C4—2448no—
536C2×C4—2442no—
537C2×C4—2442no—
538C2×C4—2442no—
539C2×C4—2442no—
540C2×C4C22⋊C4×D6112yes—
541C2×C4C22⋊C4×D6112yes—
542C2×C4C22×C4×D6111yescharacteristic
543C2×C4C2×D4×D6111yescharacteristic
544C2×C4#1451982211yescharacteristic
545C2×C4#1451982311yescharacteristic
546C2×C4#1451982411yescharacteristic
547C2×C4#1451982511yescharacteristic
548C8—42no—
549C8—2442no—
550C22—42no—
551C22—42no—
552C22—42no—
553C22—82no—
554C22—82no—
555C22—614no—
556C22—1222no—
557C22—1222no—
558C22—1224no—
559C22—1224no—
560C22—4882no—
561C22—4882no—
562C22—2442no—
563C22—2442no—
564C22—2442no—
565C22C2×C22⋊C4×D6111yescharacteristic
566C22#1684034111yescharacteristic
567C22#1684034211yescharacteristic
568C22#135×D6112yes—
569C22AGL(1,ℤ/8)×D6112yes—
570C4—22no—
571C4—22no—
572C4—42no—
573C4—82no—
574C4—614no—
575C4—1222no—
576C4—1222no—
577C4—4882no—
578C4—2442no—
579C4#1684024211yescharacteristic
580C4#1684024311yescharacteristic
581C4#1684032811yescharacteristic
582C4#1684033511yescharacteristic
583C2—82no—
584C2—1224no—
585C2—4882no—
586C2#2144840811yescharacteristic
587C2#416×D6111yescharacteristic
588C2#418×D6111yescharacteristic
589C2#2145714212yes—
590C2#2145714412yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1804C2
socleC23×C61C22×D4
prime radicalC2×#16840309C2
cosocleC22×C4×C61C24
hypercenter#1804D61
hypocenterC61#21888
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1804

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C8×D61C2semidirect
3. #21448212C2semidirect
4. #21446765C2semidirect
5. #21446744C2semidirect
6. #21446246C2semidirect
7. #21446244C2semidirect
8. #21445830C2semidirect
9. #21445829C2semidirect
10. #21445826C2semidirect
11. C61×#1804C2semidirect
15. #16840309C4semidirect
298. #1804D61semidirect
341. C61#21888semidirect

Maximal subgroups

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

subgroupindex
C2×#168403092
C2×C4⋊C8×D612
#214482122
#214467652
#214467442
#214462462
#214462442
#214458302
#214458292
#214458262
C61×#18042
#2188861