Finite Group Databasebeta

← #29716397

Subgroups — #29716397

Order 15616.

1146 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840472C211yesmaximal, characteristic, complemented
2C2×#16840052C211yesmaximal, characteristic, complemented
3C2×C4×#11758199C211yesmaximal, characteristic
4#21447157C212yesmaximal, complemented
5#21446965C212yesmaximal, complemented
6#21446779C211yesmaximal, characteristic, complemented
7#21446756C211yesmaximal, characteristic, complemented
8#21446170C212yesmaximal
9#21446079C212yesmaximal, complemented
10#21445816C211yesmaximal, characteristic, complemented
11C61×#2538C211yesmaximal, characteristic, complemented
12C22×#11758199C2211yescharacteristic
13#16840472—24no—
14#16840472—24no—
15C2×#14519777C2211yescharacteristic
16C2×#14519769C2212yes—
17C2×#14519755—22no—
18C2×#14519753—21no—
19C2×C4×D244C2212yes—
20C2×C4×D244C2211yescharacteristic
21C2×C4⋊C4×D61—21no—
22#16840052—24no—
23C4×#11758199—24no—
24C2×#14519710C2211yescharacteristic
25C2×#14519708C2212yes—
26C2×C4⋊#92418C2211yescharacteristic
27C2×#14519698C2212yes—
28C2×#14519698C2211yescharacteristic
29C2×#14519697C2212yes—
30C2×#14519697C2211yescharacteristic
31C2×C4×#92418C2211yescharacteristic
32#16839524C2212yes—
33#16839524C2212yes—
34#16839524C2212yes—
35#16839517C2212yes—
36#16839517C2212yes—
37C2×C22⋊Q8×C61—21no—
38C2×C4×Q8×C61C2211yescharacteristic
39C2×C4×D4×C61C2211yescharacteristic
40C61×#396—22no—
41C61×#390C2212yes—
42C61×#389C2212yes—
43C61×#384C2211yescharacteristic
44C61×#380—21no—
45C22×Dic122—21no—
46C22×D244C2311yescharacteristic
47C22×C4×D61—21no—
48C22×C4×D61—22no—
49C2×#11758199—22no—
50C2×#11758199—22no—
51C2×#11758199Q814yescomplemented
52C22×#92418C2311yescharacteristic
53C22×#92418C2312yes—
54C22×#92418C2311yescharacteristic
55#14519777—22no—
56#14519777D414yes—
57#14519769—24no—
58#14519769—24no—
59#14519755—216no—
60#14519753—28no—
61C4×D244—24no—
62C4×D244—22no—
63C4×D244—24no—
64C4×D244D414yes—
65C2×C4⋊Dic61—21no—
66C2×C4⋊Dic61C2311yescharacteristic
67C2×C4⋊Dic61C2312yes—
68C2×#11758196—22no—
69C2×#11758196—22no—
70C4⋊C4×D61—28no—
71C2×#11758198—22no—
72C2×#11758198—21no—
73#14519710—22no—
74#14519708—24no—
75C4⋊#92418—22no—
76#14519698—24no—
77#14519698—22no—
78#14519697—22no—
79#14519697—24no—
80C4×#92418—22no—
81C22×Q8×C61D411yescharacteristic
82C22×D4×C61C2311yescharacteristic
83C23×C4×C61—21no—
84C22⋊Q8×C61—44no—
85C4×Q8×C61—22no—
86C4×Q8×C61—22no—
87C4×D4×C61—22no—
88C4×D4×C61—22no—
89C2×C4⋊C4×C61—21no—
90C2×C4⋊C4×C61—22no—
91C2×C4⋊C4×C61—21no—
92C2×C4⋊C4×C61D411yescharacteristic
93C2×C4⋊C4×C61D412yes—
94C2×C4⋊C4×C61C2312yes—
95C2×C4⋊C4×C61C2311yescharacteristic
96C2×C22⋊C4×C61—22no—
97C2×C22⋊C4×C61—21no—
98C2×C22⋊C4×C61—22no—
99C2×C42×C61C2312yes—
100C2×C42×C61C2311yescharacteristic
101#95×C61—22no—
102C24×C61—21no—
103C23×D61—21no—
104C2×Q8×C61—22no—
105C2×Q8×C61—22no—
106C2×Q8×C61—22no—
107C2×D4×C61—21no—
108C2×D4×C61—21no—
109C2×D4×C61—21no—
110C2×D4×C61—21no—
111C2×D4×C61—21no—
112C2×D4×C61C2×Q812yes—
113C22×C4×C61—21no—
114C22×C4×C61—22no—
115C22×C4×C61—42no—
116C22×C4×C61—42no—
117C22×C4×C61—42no—
118C22×C4×C61C2×D412yes—
119C22×C4×C61C2×D412yes—
120C22×C4×C61C2×D411yescharacteristic
121C22×C4×C61C2×D411yescharacteristic
122C22×C4×C61C2411yescharacteristic
123C22×Dic61—22no—
124C22×Dic61—21no—
125C2×D244—21no—
126C2×D244—21no—
127C2×D244—21no—
128C2×D244C2×D412yes—
129C2×D244C2×Q812yes—
130C2×D244#4112yes—
131C2×C4×D61—24no—
132C2×C4×D61—28no—
133C2×C4×D61—42no—
134C2×C4×D61—42no—
135C2×C4×D61—82no—
136C2×C4×D61—82no—
137C2×Dic122—24no—
138C2×Dic122—81no—
139C2×Dic122—81no—
140C4⋊C4×C61—22no—
141C4⋊C4×C61—24no—
142C4⋊C4×C61—21no—
143C4⋊C4×C61—21no—
144C4⋊C4×C61—22no—
145C4⋊C4×C61—22no—
146C4⋊C4×C61—44no—
147C4⋊C4×C61—42no—
148C4⋊C4×C61—81no—
149C22⋊C4×C61—44no—
150C22⋊C4×C61—42no—
151C22⋊C4×C61—82no—
152C42×C61—22no—
153C42×C61—22no—
154C42×C61C2×D412yes—
155C42×C61C2×D412yes—
156#11758199—28no—
157#11758198—28no—
158#11758198—24no—
159C4⋊Dic61—24no—
160C4⋊Dic61—21no—
161C4⋊Dic61—22no—
162C4⋊Dic61C2×D414yes—
163C4⋊Dic61C2×D412yes—
164#11758196—28no—
165#11758196—28no—
166C2×#92418—21no—
167C2×#92418—21no—
168C2×#92418—21no—
169C2×#92418—22no—
170C2×#92418—22no—
171C2×#92418—21no—
172C2×#92418—21no—
173C2×#92418—22no—
174C2×#92418C2×Q812yes—
175C23×C61—22no—
176C23×C61—41no—
177C23×C61—41no—
178C23×C61—41no—
179C23×C61—41no—
180C23×C61—41no—
181C23×C61—41no—
182C23×C61C22×D411yescharacteristic
183C22×D61—22no—
184C22×D61—22no—
185C22×D61—22no—
186C22×D61—41no—
187C22×D61—41no—
188C22×D61—41no—
189C22×D61—41no—
190Q8×C61—44no—
191D4×C61—22no—
192D4×C61—22no—
193D4×C61—41no—
194D4×C61—41no—
195C2×C4×C61—21no—
196C2×C4×C61—21no—
197C2×C4×C61—22no—
198C2×C4×C61—22no—
199C2×C4×C61—22no—
200C2×C4×C61—22no—
201C2×C4×C61—22no—
202C2×C4×C61—22no—
203C2×C4×C61—41no—
204C2×C4×C61—42no—
205C2×C4×C61—41no—
206C2×C4×C61—42no—
207C2×C4×C61—41no—
208C2×C4×C61—42no—
209C2×C4×C61—84no—
210C2×C4×C61#12012yes—
211C2×C4×C61#12012yes—
212C2×C4×C61#13512yes—
213C2×C4×C61AGL(1,ℤ/8)12yes—
214C2×C4×C61C22×D411yescharacteristic
215C2×C4×C61C22×D411yescharacteristic
216C2×C4×C61C22×Q811yescharacteristic
217C2×C4×C61C2×#4111yescharacteristic
218C2×C4×C612−1+411yescharacteristic
219C2×C4×C612−1+411yescharacteristic
220C2×Dic61—22no—
221C2×Dic61—24no—
222C2×Dic61—41no—
223C2×Dic61—41no—
224C2×Dic61—42no—
225C2×Dic61—42no—
226D244—22no—
227D244—22no—
228D244—22no—
229D244C22⋊Q814yescomplemented
230C4×D61—48no—
231C4×D61—88no—
232Dic122—84no—
233#92418—22no—
234#92418—41no—
235#92418—42no—
236#24327—611nomaximal, Sylow
237C22×C61—42no—
238C22×C61—42no—
239C22×C61—42no—
240C22×C61—81no—
241C22×C61—81no—
242C22×C61—81no—
243C22×C61—81no—
244C22×C61C2×#12011yescharacteristic
245C22×C61#54811yescharacteristic
246C22×C61D4×Q811yescharacteristic
247C22×C61C2×#13511yescharacteristic
248C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
249C22×C61#57611yescharacteristic
250C22×C61#57711yescharacteristic
251C2×D61—24no—
252C2×D61—42no—
253C2×D61—42no—
254C2×D61—42no—
255C2×D61—42no—
256C2×D61—42no—
257C2×D61—42no—
258C4×C61—22no—
259C4×C61—42no—
260C4×C61—42no—
261C4×C61—82no—
262C4×C61—81no—
263C4×C61#44914yes—
264C4×C61C2×C22⋊Q811yescharacteristic
265C4×C61#53611yescharacteristic
266C4×C61#54811yescharacteristic
267C4×C61D4×Q811yescharacteristic
268Dic61—42no—
269Dic61—44no—
270C2×#554—611no—
271C2×#476—611no—
272C2×C4×D8—611no—
273#2538D6111yescharacteristic, complemented
274#2059—612no—
275#2025—612no—
276#1990—611no—
277#1984—611no—
278#1879—612no—
279#1864—612no—
280#1802—611no—
281C2×C61—82no—
282C2×C61#240911yescharacteristic
283C2×C61C2×#44911yescharacteristic
284C2×C61#301211yescharacteristic
285C2×C61#336011yescharacteristic
286C2×C61#336611yescharacteristic
287C2×C61#338011yescharacteristic
288C2×C61#338411yescharacteristic
289D61—44no—
290C22×D8—611no—
291#554—1224no—
292#554—1224no—
293C2×C4⋊Q8—611no—
294C2×#125—612no—
295C2×C22⋊Q8—21no—
296C2×C22⋊Q8—1221no—
297C2×C22⋊Q8—1222no—
298C2×C4×Q8C2×D6111yescharacteristic
299C2×C4×D4—611no—
300C2×C4×D4—612no—
301C2×C4×D4C2×D6111yescharacteristic
302C22×C4⋊C4—1221no—
303#476—1224no—
304C4×D8—1224no—
305C2×#107—611no—
306C2×#106—612no—
307C2×C4⋊C8—611no—
308C2×#102—612no—
309C2×#102—611no—
310C2×#102—612no—
311C2×#102—611no—
312C2×C4×C8—611no—
313#396—22no—
314#390C2×D6112yes—
315#389C2×D6112yes—
316#384C2×D6111yescharacteristic
317#380—21no—
318#340—612no—
319#340—612no—
320#340—612no—
321#336—612no—
322#336—612no—
323C61#2432711yescharacteristic, complemented, Sylow
324C22×Q8—1221no—
325C22×Q8C22⋊D6111yescharacteristic
326C22×D4—611no—
327C22×D4C22×D6111yescharacteristic
328C23×C4—21no—
329C23×C4—1222no—
330C23×C4—1221no—
331C2×D8—614no—
332C2×D8—1222no—
333C2×D8—1222no—
334C22×C8—611no—
335C22×C8—611no—
336C22×C8—612no—
337C4⋊Q8—614no—
338C4⋊Q8—1222no—
339#125—1224no—
340#125—1224no—
341C22⋊Q8—44no—
342C22⋊Q8—12216no—
343C22⋊Q8—1228no—
344C4×Q8—22no—
345C4×Q8—22no—
346C4×D4—22no—
347C4×D4—22no—
348C4×D4—614no—
349C4×D4—1224no—
350C4×D4—1224no—
351C4×D4—1222no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—611no—
356C2×C4⋊C4—612no—
357C2×C4⋊C4—1228no—
358C2×C4⋊C4—1222no—
359C2×C4⋊C4—1221no—
360C2×C4⋊C4—1222no—
361C2×C4⋊C4C22⋊D6111yescharacteristic
362C2×C4⋊C4C22⋊D6112yes—
363C2×C4⋊C4C22×D6111yescharacteristic
364C2×C4⋊C4C22×D6112yes—
365C2×C22⋊C4—21no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—1222no—
369C2×C22⋊C4—1221no—
370C2×C42C22×D6112yes—
371C2×C42C22×D6111yescharacteristic
372#107—1222no—
373#106—1224no—
374C4⋊C8—1222no—
375#102—1224no—
376#102—1222no—
377#102—1222no—
378#102—1224no—
379C4×C8—1222no—
380#95—22no—
381C24—21no—
382C24—1221no—
383C2×Q8—22no—
384C2×Q8—22no—
385C2×Q8—22no—
386C2×Q8—1224no—
387C2×Q8—4881no—
388C2×Q8—4881no—
389C2×D4—21no—
390C2×D4—21no—
391C2×D4—21no—
392C2×D4—21no—
393C2×D4—21no—
394C2×D4—612no—
395C2×D4—612no—
396C2×D4—612no—
397C2×D4—1221no—
398C2×D4—1221no—
399C2×D4—1221no—
400C2×D4Q8×D6112yes—
401C22×C4—21no—
402C22×C4—22no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—1228no—
407C22×C4—1221no—
408C22×C4—1222no—
409C22×C4—1224no—
410C22×C4—4882no—
411C22×C4—4882no—
412C22×C4—2442no—
413C22×C4—2442no—
414C22×C4C2×C22⋊D6112yes—
415C22×C4C2×C22⋊D6112yes—
416C22×C4C2×C22⋊D6111yescharacteristic
417C22×C4C2×C22⋊D6111yescharacteristic
418C22×C4C23×D6111yescharacteristic
419D8—1228no—
420C2×C8—612no—
421C2×C8—1221no—
422C2×C8—1221no—
423C2×C8—1221no—
424C2×C8—1222no—
425C2×C8—1222no—
426C2×C8—1221no—
427C2×C8—1221no—
428C2×C8—1222no—
429C4⋊C4—22no—
430C4⋊C4—24no—
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—44no—
436C4⋊C4—42no—
437C4⋊C4—81no—
438C4⋊C4—614no—
439C4⋊C4—612no—
440C4⋊C4—1228no—
441C4⋊C4—1228no—
442C4⋊C4—1221no—
443C4⋊C4—1222no—
444C4⋊C4—1224no—
445C22⋊C4—44no—
446C22⋊C4—42no—
447C22⋊C4—82no—
448C22⋊C4—1228no—
449C22⋊C4—1224no—
450C42—22no—
451C42—22no—
452C42D4×D6112yes—
453C42D4×D6112yes—
454C23—22no—
455C23—41no—
456C23—41no—
457C23—41no—
458C23—41no—
459C23—41no—
460C23—41no—
461C23—1222no—
462C23—1222no—
463C23—1222no—
464C23—2441no—
465C23—2441no—
466C23—2441no—
467C23—2441no—
468C23C22×C22⋊D6111yescharacteristic
469Q8—44no—
470Q8—4884no—
471D4—22no—
472D4—22no—
473D4—41no—
474D4—41no—
475D4—614no—
476D4—1222no—
477D4—1222no—
478D4—1222no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—42no—
490C2×C4—41no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—84no—
494C2×C4—1222no—
495C2×C4—1224no—
496C2×C4—4888no—
497C2×C4—2441no—
498C2×C4—2441no—
499C2×C4—2442no—
500C2×C4—2442no—
501C2×C4—2448no—
502C2×C4#1451974512yes—
503C2×C4#1451974512yes—
504C2×C4#1451979712yes—
505C2×C4#1451980312yes—
506C2×C4C2×D4×D6111yescharacteristic
507C2×C4C2×D4×D6111yescharacteristic
508C2×C4C2×Q8×D6111yescharacteristic
509C2×C4#41×D6111yescharacteristic
510C2×C4#1451982411yescharacteristic
511C2×C4#1451982511yescharacteristic
512C8—1222no—
513C8—2441no—
514C8—2442no—
515C22—42no—
516C22—42no—
517C22—42no—
518C22—81no—
519C22—81no—
520C22—81no—
521C22—81no—
522C22—1224no—
523C22—2442no—
524C22—2442no—
525C22—2442no—
526C22—2442no—
527C22—2442no—
528C22—2442no—
529C22C2×#1451974511yescharacteristic
530C22#1684041911yescharacteristic
531C22Q8×C22⋊D6111yescharacteristic
532C22C2×#1451979711yescharacteristic
533C22C2×#1451980311yescharacteristic
534C22#1684056511yescharacteristic
535C22#1684057011yescharacteristic
536C4—22no—
537C4—42no—
538C4—42no—
539C4—82no—
540C4—81no—
541C4—2444no—
542C4—2442no—
543C4#1683990514yes—
544C4C22⋊Q8×D6111yescharacteristic
545C4#1684033511yescharacteristic
546C4#1684042211yescharacteristic
547C4#1684042511yescharacteristic
548C2—82no—
549C2—2444no—
550C2#2144918011yescharacteristic
551C2C2×#1683990511yescharacteristic
552C2#2145512011yescharacteristic
553C2#2145846411yescharacteristic
554C2#2145852211yescharacteristic
555C2#2145868211yescharacteristic
556C2#2145871411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2538C2
socleC23×C61C22×D4
prime radicalC22×#11758199C22
hypercenter#2538D61
hypocenterC61#24327
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2538

Complements

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

NH ≅ G/Ntype
1. C2×#16840472C2semidirect
2. C2×#16840052C2semidirect
4. #21447157C2semidirect
5. #21446965C2semidirect
6. #21446779C2semidirect
7. #21446756C2semidirect
9. #21446079C2semidirect
10. #21445816C2semidirect
11. C61×#2538C2semidirect
51. C2×#11758199Q8semidirect
229. D244C22⋊Q8semidirect
273. #2538D61semidirect
323. C61#24327semidirect

Maximal subgroups

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

subgroupindex
C2×#168404722
C2×#168400522
C2×C4×#117581992
#214471572
#214469652
#214467792
#214467562
#214461702
#214460792
#214458162
C61×#25382
#2432761