Finite Group Databasebeta

← #29716389

Subgroups — #29716389

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×#16840049C211yesmaximal, characteristic, complemented
2C2×C4×#11758199C211yesmaximal, characteristic
3#21450737C211yesmaximal, characteristic, complemented
4#21447154C212yesmaximal, complemented
5#21446962C212yesmaximal, complemented
6#21446781C211yesmaximal, characteristic, complemented
7#21446756C211yesmaximal, characteristic, complemented
8#21446167C212yesmaximal
9#21446080C212yesmaximal, complemented
10#21445819C211yesmaximal, characteristic, complemented
11C2×C61×#554C211yesmaximal, characteristic, complemented
12C22×#11758199C2211yescharacteristic
13C2×#14519753—21no—
14C2×C4×D244C2211yescharacteristic
15#16840049—24no—
16C4×#11758199—24no—
17C2×#14519710C2211yescharacteristic
18C2×#14519708C2212yes—
19C2×C4⋊#92418C2211yescharacteristic
20C2×#14519698C2212yes—
21C2×#14519698C2211yescharacteristic
22C2×#14519697C2212yes—
23C2×#14519697C2211yescharacteristic
24C2×C4×#92418C2211yescharacteristic
25#16839684—22no—
26#16839664C2212yes—
27#16839660C2212yes—
28#16839632C2211yescharacteristic
29#16839619—21no—
30#16839524C2212yes—
31#16839524C2212yes—
32#16839524C2212yes—
33#16839517C2212yes—
34#16839517C2212yes—
35C61×#554—24no—
36C61×#554—24no—
37C2×C4⋊Q8×C61C2211yescharacteristic
38C2×#125×C61C2212yes—
39C2×C22⋊Q8×C61—21no—
40C2×C22⋊Q8×C61—22no—
41C2×C4×Q8×C61C2211yescharacteristic
42C2×C4×D4×C61C2212yes—
43C2×C4×D4×C61C2211yescharacteristic
44C22×C4⋊C4×C61—21no—
45C22×D244C2311yescharacteristic
46C22×C4×D61—21no—
47C2×#11758199—22no—
48C2×#11758199—22no—
49C2×#11758199Q814yescomplemented
50C22×#92418C2311yescharacteristic
51C22×#92418C2312yes—
52C22×#92418C2311yescharacteristic
53#14519753—44no—
54C4×D244—22no—
55C4×D244—22no—
56C2×C4⋊Dic61—21no—
57C2×C4⋊Dic61—21no—
58C2×C4⋊Dic61C2311yescharacteristic
59C2×C4⋊Dic61C2312yes—
60C2×#11758196—22no—
61C2×#11758198—21no—
62C2×#11758198—22no—
63C2×#11758198—22no—
64#14519710—22no—
65#14519708—24no—
66C4⋊#92418—22no—
67#14519698—22no—
68#14519698—24no—
69#14519697—24no—
70#14519697—22no—
71C4×#92418—22no—
72#14519684—22no—
73C22×Q8×C61—21no—
74C22×Q8×C61D411yescharacteristic
75C22×D4×C61C2311yescharacteristic
76C23×C4×C61—21no—
77C23×C4×C61—22no—
78C4⋊Q8×C61—22no—
79C4⋊Q8×C61D414yes—
80#125×C61—24no—
81#125×C61—24no—
82C22⋊Q8×C61—28no—
83C22⋊Q8×C61—216no—
84C4×Q8×C61—22no—
85C4×Q8×C61—22no—
86C4×D4×C61—24no—
87C4×D4×C61—24no—
88C4×D4×C61—22no—
89C4×D4×C61D414yes—
90C2×C4⋊C4×C61—28no—
91C2×C4⋊C4×C61—21no—
92C2×C4⋊C4×C61—22no—
93C2×C4⋊C4×C61—22no—
94C2×C4⋊C4×C61D412yes—
95C2×C4⋊C4×C61D411yescharacteristic
96C2×C4⋊C4×C61C2312yes—
97C2×C4⋊C4×C61C2311yescharacteristic
98C2×C22⋊C4×C61—22no—
99C2×C22⋊C4×C61—21no—
100C2×C42×C61C2312yes—
101C2×C42×C61C2311yescharacteristic
102C24×C61—21no—
103C23×D61—21no—
104C2×Q8×C61—22no—
105C2×Q8×C61—24no—
106C2×Q8×C61—22no—
107C2×Q8×C61—22no—
108C2×Q8×C61—81no—
109C2×Q8×C61—81no—
110C2×D4×C61—21no—
111C2×D4×C61—21no—
112C2×D4×C61—21no—
113C2×D4×C61C2×D412yes—
114C2×D4×C61C2×Q812yes—
115C2×D4×C61#4112yes—
116C22×C4×C61—21no—
117C22×C4×C61—22no—
118C22×C4×C61—28no—
119C22×C4×C61—24no—
120C22×C4×C61—42no—
121C22×C4×C61—42no—
122C22×C4×C61—82no—
123C22×C4×C61—82no—
124C22×C4×C61C2×D412yes—
125C22×C4×C61C2×D412yes—
126C22×C4×C61C2×D411yescharacteristic
127C22×C4×C61C2×D411yescharacteristic
128C22×C4×C61C2411yescharacteristic
129C22×Dic61—21no—
130C22×Dic61—22no—
131C2×D244—21no—
132C2×D244—21no—
133C2×D244—21no—
134C2×D244—21no—
135C2×D244—21no—
136C2×D244C2×Q812yes—
137C2×C4×D61—42no—
138C2×C4×D61—42no—
139C2×C4×D61—42no—
140C4⋊C4×C61—28no—
141C4⋊C4×C61—28no—
142C4⋊C4×C61—22no—
143C4⋊C4×C61—24no—
144C4⋊C4×C61—24no—
145C4⋊C4×C61—21no—
146C4⋊C4×C61—22no—
147C4⋊C4×C61C2×D414yes—
148C4⋊C4×C61C2×D412yes—
149C22⋊C4×C61—28no—
150C22⋊C4×C61—24no—
151C42×C61—22no—
152C42×C61—22no—
153C42×C61C2×D412yes—
154C42×C61C2×D412yes—
155#11758199—28no—
156#11758198—44no—
157#11758198—42no—
158#11758198—82no—
159C4⋊Dic61—21no—
160C4⋊Dic61—21no—
161C4⋊Dic61—22no—
162C4⋊Dic61—22no—
163C4⋊Dic61—42no—
164C4⋊Dic61—81no—
165#11758196—44no—
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—22no—
177C23×C61—22no—
178C23×C61—41no—
179C23×C61—41no—
180C23×C61—41no—
181C23×C61—41no—
182C23×C61C22×D411yescharacteristic
183C22×D61—22no—
184C22×D61—41no—
185C22×D61—41no—
186C22×D61—41no—
187C22×D61—41no—
188C22×D61—41no—
189C22×D61—41no—
190Q8×C61—44no—
191Q8×C61—84no—
192D4×C61—22no—
193D4×C61—22no—
194D4×C61—22no—
195D4×C61C22⋊Q814yescomplemented
196C2×C4×C61—21no—
197C2×C4×C61—21no—
198C2×C4×C61—22no—
199C2×C4×C61—22no—
200C2×C4×C61—22no—
201C2×C4×C61—24no—
202C2×C4×C61—22no—
203C2×C4×C61—22no—
204C2×C4×C61—22no—
205C2×C4×C61—22no—
206C2×C4×C61—41no—
207C2×C4×C61—41no—
208C2×C4×C61—42no—
209C2×C4×C61—42no—
210C2×C4×C61—48no—
211C2×C4×C61—88no—
212C2×C4×C61#12012yes—
213C2×C4×C61#12012yes—
214C2×C4×C61#13512yes—
215C2×C4×C61AGL(1,ℤ/8)12yes—
216C2×C4×C61C22×D411yescharacteristic
217C2×C4×C61C22×D411yescharacteristic
218C2×C4×C61C22×Q811yescharacteristic
219C2×C4×C61C2×#4111yescharacteristic
220C2×C4×C612−1+411yescharacteristic
221C2×C4×C612−1+411yescharacteristic
222C2×Dic61—41no—
223C2×Dic61—42no—
224C2×Dic61—41no—
225C2×Dic61—41no—
226C2×Dic61—42no—
227C2×Dic61—42no—
228D244—22no—
229D244—22no—
230D244—41no—
231D244—41no—
232C4×D61—84no—
233#92418—22no—
234#92418—41no—
235#92418—42no—
236#24327—611nomaximal, Sylow
237C22×C61—24no—
238C22×C61—42no—
239C22×C61—42no—
240C22×C61—42no—
241C22×C61—42no—
242C22×C61—42no—
243C22×C61—42no—
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—42no—
252C2×D61—42no—
253C2×D61—42no—
254C2×D61—81no—
255C2×D61—81no—
256C2×D61—81no—
257C2×D61—81no—
258C4×C61—22no—
259C4×C61—42no—
260C4×C61—42no—
261C4×C61—42no—
262C4×C61—44no—
263C4×C61#44914yes—
264C4×C61C2×C22⋊Q811yescharacteristic
265C4×C61#53611yescharacteristic
266C4×C61#54811yescharacteristic
267C4×C61D4×Q811yescharacteristic
268Dic61—82no—
269Dic61—81no—
270C2×#554D6111yescharacteristic, complemented
271C2×#476—611no—
272C2×C4×D8—611no—
273#2538—611no—
274#2059—612no—
275#2025—612no—
276#1990—611no—
277#1984—611no—
278#1879—612no—
279#1864—612no—
280#1802—611no—
281C2×C61—44no—
282C2×C61#240911yescharacteristic
283C2×C61C2×#44911yescharacteristic
284C2×C61#301211yescharacteristic
285C2×C61#336011yescharacteristic
286C2×C61#336611yescharacteristic
287C2×C61#338011yescharacteristic
288C2×C61#338411yescharacteristic
289D61—82no—
290C22×D8—611no—
291#554—24no—
292#554—24no—
293C2×C4⋊Q8C2×D6111yescharacteristic
294C2×#125C2×D6112yes—
295C2×C22⋊Q8—21no—
296C2×C22⋊Q8—22no—
297C2×C22⋊Q8—1221no—
298C2×C4×Q8C2×D6111yescharacteristic
299C2×C4×D4—611no—
300C2×C4×D4C2×D6112yes—
301C2×C4×D4C2×D6111yescharacteristic
302C22×C4⋊C4—21no—
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—1222no—
314#390—612no—
315#389—612no—
316#384—611no—
317#380—1221no—
318#340—612no—
319#340—612no—
320#340—612no—
321#336—612no—
322#336—612no—
323C61#2432711yescharacteristic, complemented, Sylow
324C22×Q8—21no—
325C22×Q8C22⋊D6111yescharacteristic
326C22×D4—611no—
327C22×D4C22×D6111yescharacteristic
328C23×C4—21no—
329C23×C4—22no—
330C23×C4—1221no—
331C2×D8—614no—
332C2×D8—1222no—
333C2×D8—1222no—
334C22×C8—611no—
335C22×C8—611no—
336C22×C8—612no—
337C4⋊Q8—22no—
338C4⋊Q8C22⋊D6114yes—
339#125—24no—
340#125—24no—
341C22⋊Q8—28no—
342C22⋊Q8—216no—
343C22⋊Q8—2444no—
344C4×Q8—22no—
345C4×Q8—22no—
346C4×D4—24no—
347C4×D4—24no—
348C4×D4—22no—
349C4×D4—1222no—
350C4×D4—1222no—
351C4×D4C22⋊D6114yes—
352C2×C4⋊C4—28no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—611no—
357C2×C4⋊C4—612no—
358C2×C4⋊C4—1221no—
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—22no—
366C2×C22⋊C4—21no—
367C2×C22⋊C4—1222no—
368C2×C22⋊C4—1221no—
369C2×C22⋊C4—1222no—
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—1222no—
381C24—21no—
382C24—1221no—
383C2×Q8—22no—
384C2×Q8—24no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—81no—
388C2×Q8—81no—
389C2×D4—21no—
390C2×D4—21no—
391C2×D4—21no—
392C2×D4—612no—
393C2×D4—1221no—
394C2×D4—1221no—
395C2×D4—1221no—
396C2×D4—1221no—
397C2×D4—1221no—
398C2×D4Q8×D6112yes—
399C2×D4#1175822612yes—
400C2×D4C2×C22⋊D6112yes—
401C22×C4—21no—
402C22×C4—22no—
403C22×C4—28no—
404C22×C4—24no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—82no—
408C22×C4—82no—
409C22×C4—1221no—
410C22×C4—1222no—
411C22×C4—2442no—
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—28no—
430C4⋊C4—28no—
431C4⋊C4—21no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—24no—
435C4⋊C4—24no—
436C4⋊C4—1221no—
437C4⋊C4—1221no—
438C4⋊C4—1222no—
439C4⋊C4—1222no—
440C4⋊C4—4881no—
441C4⋊C4—2444no—
442C4⋊C4—2442no—
443C4⋊C4C2×C22⋊D6114yes—
444C4⋊C4C2×C22⋊D6112yes—
445C22⋊C4—28no—
446C22⋊C4—24no—
447C22⋊C4—4882no—
448C22⋊C4—2444no—
449C22⋊C4—2442no—
450C42—22no—
451C42—22no—
452C42C2×C22⋊D6112yes—
453C42C2×C22⋊D6112yes—
454C23—22no—
455C23—22no—
456C23—22no—
457C23—41no—
458C23—41no—
459C23—41no—
460C23—41no—
461C23—1222no—
462C23—2441no—
463C23—2441no—
464C23—2441no—
465C23—2441no—
466C23—2441no—
467C23—2441no—
468C23C22×C22⋊D6111yescharacteristic
469Q8—44no—
470Q8—84no—
471D4—22no—
472D4—22no—
473D4—22no—
474D4—1222no—
475D4—1222no—
476D4—2441no—
477D4—2441no—
478D4#1451975314yescomplemented
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—24no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—48no—
494C2×C4—88no—
495C2×C4—4884no—
496C2×C4—2441no—
497C2×C4—2441no—
498C2×C4—2442no—
499C2×C4—2441no—
500C2×C4—2442no—
501C2×C4—2442no—
502C2×C4#1451974612yes—
503C2×C4#1451974612yes—
504C2×C4#1451979712yes—
505C2×C4#1451980212yes—
506C2×C4C22×C22⋊D6111yescharacteristic
507C2×C4C22×C22⋊D6111yescharacteristic
508C2×C4C2×Q8×D6111yescharacteristic
509C2×C4C2×#1175822611yescharacteristic
510C2×C4#1451982511yescharacteristic
511C2×C4#1451982511yescharacteristic
512C8—1222no—
513C8—2441no—
514C8—2442no—
515C22—24no—
516C22—42no—
517C22—42no—
518C22—42no—
519C22—42no—
520C22—42no—
521C22—42no—
522C22—4881no—
523C22—4881no—
524C22—4881no—
525C22—4881no—
526C22—2442no—
527C22—2442no—
528C22—2442no—
529C22C2×#1451974611yescharacteristic
530C22#1684042311yescharacteristic
531C22Q8×C22⋊D6111yescharacteristic
532C22C2×#1451979711yescharacteristic
533C22C2×#1451980211yescharacteristic
534C22#1684056511yescharacteristic
535C22#1684057311yescharacteristic
536C4—22no—
537C4—42no—
538C4—44no—
539C4—42no—
540C4—42no—
541C4—4882no—
542C4—4881no—
543C4#1683990714yes—
544C4C2×#1451975311yescharacteristic
545C4#1684033611yescharacteristic
546C4#1684042311yescharacteristic
547C4Q8×C22⋊D6111yescharacteristic
548C2—44no—
549C2—4882no—
550C2#2144917811yescharacteristic
551C2C2×#1683990711yescharacteristic
552C2#2145512311yescharacteristic
553C2#2145846811yescharacteristic
554C2#2145852611yescharacteristic
555C2#2145868211yescharacteristic
556C2#2145871811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#554C2
socleC23×C61C22×D4
prime radicalC22×#11758199C22
hypercenterC2×#554D61
hypocenterC61#24327
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#554

Complements

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

NH ≅ G/Ntype
1. C2×#16840049C2semidirect
3. #21450737C2semidirect
4. #21447154C2semidirect
5. #21446962C2semidirect
6. #21446781C2semidirect
7. #21446756C2semidirect
9. #21446080C2semidirect
10. #21445819C2semidirect
11. C2×C61×#554C2semidirect
49. C2×#11758199Q8semidirect
195. D4×C61C22⋊Q8semidirect
270. C2×#554D61semidirect
323. C61#24327semidirect
478. D4#14519753semidirect

Maximal subgroups

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

subgroupindex
C2×#168400492
C2×C4×#117581992
#214507372
#214471542
#214469622
#214467812
#214467562
#214461672
#214460802
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C2×C61×#5542
#2432761