Finite Group Databasebeta

← #29716378

Subgroups — #29716378

Order 15616.

1330 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840039C211yesmaximal, characteristic, complemented
2C2×C4×#11758199C211yesmaximal, characteristic
3#21450737C211yesmaximal, characteristic, complemented
4#21447144C212yesmaximal, complemented
5#21446952C212yesmaximal, complemented
6#21446781C211yesmaximal, characteristic, complemented
7#21446756C211yesmaximal, characteristic, complemented
8#21446167C212yesmaximal
9#21446066C212yesmaximal, complemented
10#21445819C211yesmaximal, characteristic, complemented
11C2×D4×Q8×C61C211yesmaximal, characteristic, complemented
12C22×#11758199C2211yescharacteristic
13C2×#14519753—21no—
14C2×C4×D244C2211yescharacteristic
15#16840039—24no—
16C4×#11758199—24no—
17C2×#14519710C2212yes—
18C2×#14519710C2211yescharacteristic
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—
35C23×Q8×C61—21no—
36D4×Q8×C61—24no—
37D4×Q8×C61—24no—
38C2×C4⋊Q8×C61C2212yes—
39C2×C4⋊Q8×C61C2211yescharacteristic
40C2×C22⋊Q8×C61—22no—
41C2×C22⋊Q8×C61—21no—
42C2×C4×Q8×C61C2211yescharacteristic
43C2×C4×D4×C61C2212yes—
44C2×C4×D4×C61C2211yescharacteristic
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—24no—
65#14519710—22no—
66C4⋊#92418—22no—
67#14519698—22no—
68#14519698—24no—
69#14519697—24no—
70#14519697—22no—
71C4×#92418—22no—
72#14519684—22no—
73C22×Q8×C61—28no—
74C22×Q8×C61—22no—
75C22×Q8×C61—21no—
76C22×Q8×C61—44no—
77C22×Q8×C61—44no—
78C22×Q8×C61D411yescharacteristic
79C22×D4×C61C2311yescharacteristic
80C23×C4×C61—22no—
81C23×C4×C61—21no—
82C4⋊Q8×C61—24no—
83C4⋊Q8×C61—24no—
84C4⋊Q8×C61—22no—
85C4⋊Q8×C61D414yes—
86C22⋊Q8×C61—28no—
87C22⋊Q8×C61—216no—
88C4×Q8×C61—22no—
89C4×Q8×C61—22no—
90C4×D4×C61—24no—
91C4×D4×C61—24no—
92C4×D4×C61—22no—
93C4×D4×C61D414yes—
94C2×C4⋊C4×C61—22no—
95C2×C4⋊C4×C61—21no—
96C2×C4⋊C4×C61D412yes—
97C2×C4⋊C4×C61D411yescharacteristic
98C2×C4⋊C4×C61C2312yes—
99C2×C4⋊C4×C61C2311yescharacteristic
100C2×C22⋊C4×C61—22no—
101C2×C22⋊C4×C61—21no—
102C2×C42×C61C2312yes—
103C2×C42×C61C2311yescharacteristic
104C24×C61—21no—
105C23×D61—21no—
106C2×Q8×C61—28no—
107C2×Q8×C61—22no—
108C2×Q8×C61—24no—
109C2×Q8×C61—22no—
110C2×Q8×C61—44no—
111C2×Q8×C61—432no—
112C2×Q8×C61—44no—
113C2×Q8×C61—42no—
114C2×Q8×C61—42no—
115C2×Q8×C61D814yescomplemented
116C2×D4×C61—21no—
117C2×D4×C61—21no—
118C2×D4×C61—21no—
119C2×D4×C61C2×D412yes—
120C2×D4×C61C2×Q812yes—
121C2×D4×C61#4112yes—
122C22×C4×C61—28no—
123C22×C4×C61—22no—
124C22×C4×C61—21no—
125C22×C4×C61—24no—
126C22×C4×C61—44no—
127C22×C4×C61—44no—
128C22×C4×C61—42no—
129C22×C4×C61—42no—
130C22×C4×C61C2×D412yes—
131C22×C4×C61C2×D411yescharacteristic
132C22×C4×C61C2×D412yes—
133C22×C4×C61C2×D411yescharacteristic
134C22×C4×C61C2411yescharacteristic
135C22×Dic61—21no—
136C22×Dic61—22no—
137C2×D244—21no—
138C2×D244—21no—
139C2×D244—21no—
140C2×D244—21no—
141C2×D244—21no—
142C2×D244C2×Q812yescomplemented
143C2×C4×D61—42no—
144C2×C4×D61—42no—
145C2×C4×D61—42no—
146C4⋊C4×C61—28no—
147C4⋊C4×C61—24no—
148C4⋊C4×C61—24no—
149C4⋊C4×C61—22no—
150C4⋊C4×C61—22no—
151C4⋊C4×C61—21no—
152C4⋊C4×C61C2×D414yes—
153C4⋊C4×C61C2×D412yes—
154C22⋊C4×C61—28no—
155C22⋊C4×C61—24no—
156C42×C61—22no—
157C42×C61—22no—
158C42×C61C2×D412yes—
159C42×C61C2×D412yes—
160#11758199—28no—
161#11758198—42no—
162#11758198—44no—
163#11758198—82no—
164C4⋊Dic61—21no—
165C4⋊Dic61—21no—
166C4⋊Dic61—22no—
167C4⋊Dic61—22no—
168C4⋊Dic61—42no—
169C4⋊Dic61—81no—
170#11758196—44no—
171C2×#92418—21no—
172C2×#92418—21no—
173C2×#92418—21no—
174C2×#92418—22no—
175C2×#92418—22no—
176C2×#92418—21no—
177C2×#92418—21no—
178C2×#92418—22no—
179C2×#92418C2×Q812yescomplemented
180C23×C61—22no—
181C23×C61—22no—
182C23×C61—22no—
183C23×C61—41no—
184C23×C61—41no—
185C23×C61—41no—
186C23×C61—41no—
187C23×C61C22×D411yescharacteristic
188C22×D61—22no—
189C22×D61—41no—
190C22×D61—41no—
191C22×D61—41no—
192C22×D61—41no—
193C22×D61—41no—
194C22×D61—41no—
195Q8×C61—28no—
196Q8×C61—416no—
197Q8×C61—48no—
198D4×C61—22no—
199D4×C61—22no—
200D4×C61—22no—
201D4×C61C22⋊Q814yescomplemented
202C2×C4×C61—22no—
203C2×C4×C61—22no—
204C2×C4×C61—21no—
205C2×C4×C61—21no—
206C2×C4×C61—22no—
207C2×C4×C61—24no—
208C2×C4×C61—22no—
209C2×C4×C61—22no—
210C2×C4×C61—22no—
211C2×C4×C61—416no—
212C2×C4×C61—42no—
213C2×C4×C61—41no—
214C2×C4×C61—42no—
215C2×C4×C61—41no—
216C2×C4×C61—48no—
217C2×C4×C61#12012yes—
218C2×C4×C61#12012yes—
219C2×C4×C61C2×D814yes—
220C2×C4×C61C2×D812yes—
221C2×C4×C61AGL(1,ℤ/8)12yes—
222C2×C4×C61C22×D411yescharacteristic
223C2×C4×C61C22×D411yescharacteristic
224C2×C4×C61C22×Q811yescharacteristic
225C2×C4×C61C2×#4111yescharacteristic
226C2×C4×C612−1+411yescharacteristic
227C2×C4×C612−1+411yescharacteristic
228C2×Dic61—41no—
229C2×Dic61—42no—
230C2×Dic61—41no—
231C2×Dic61—42no—
232C2×Dic61—41no—
233C2×Dic61—42no—
234D244—22no—
235D244—22no—
236D244—41no—
237D244—41no—
238C4×D61—84no—
239#92418—22no—
240#92418—41no—
241#92418—42no—
242#24326—611nomaximal, Sylow
243C22×C61—24no—
244C22×C61—42no—
245C22×C61—42no—
246C22×C61—42no—
247C22×C61—42no—
248C22×C61—42no—
249C22×C61—42no—
250C22×C61C2×#12011yescharacteristic
251C22×C61#54811yescharacteristic
252C22×C61D4×Q811yescharacteristic
253C22×C61C22×D811yescharacteristic
254C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
255C22×C61#57711yescharacteristic
256C22×C61#57811yescharacteristic
257C2×D61—42no—
258C2×D61—42no—
259C2×D61—42no—
260C2×D61—81no—
261C2×D61—81no—
262C2×D61—81no—
263C2×D61—81no—
264C4×C61—24no—
265C4×C61—22no—
266C4×C61—44no—
267C4×C61—42no—
268C4×C61—42no—
269C4×C61#44714yes—
270C4×C61C2×C22⋊Q811yescharacteristic
271C4×C61#53611yescharacteristic
272C4×C61#54811yescharacteristic
273C4×C61D4×Q811yescharacteristic
274Dic61—82no—
275Dic61—81no—
276C2×D4×Q8D6111yescharacteristic, complemented
277C2×#474—611no—
278C2×C4×D8—611no—
279#2538—611no—
280#2057—612no—
281#2023—612no—
282#1990—611no—
283#1984—611no—
284#1879—612no—
285#1862—612no—
286#1802—611no—
287C2×C61—44no—
288C2×C61#240911yescharacteristic
289C2×C61C2×#44711yescharacteristic
290C2×C61#301311yescharacteristic
291C2×C61#335811yescharacteristic
292C2×C61#336711yescharacteristic
293C2×C61Q8×D811yescharacteristic
294C2×C61#338411yescharacteristic
295D61—82no—
296C23×Q8—21no—
297C22×D8—611no—
298D4×Q8—24no—
299D4×Q8—24no—
300C2×C4⋊Q8C2×D6112yes—
301C2×C4⋊Q8C2×D6111yescharacteristic
302C2×C22⋊Q8—21no—
303C2×C22⋊Q8—22no—
304C2×C22⋊Q8—1221no—
305C2×C4×Q8C2×D6111yescharacteristic
306C2×C4×D4—611no—
307C2×C4×D4C2×D6112yes—
308C2×C4×D4C2×D6111yescharacteristic
309#474—1224no—
310C4×D8—1224no—
311C2×#107—611no—
312C2×#107—612no—
313C2×C4⋊C8—611no—
314C2×#102—612no—
315C2×#102—611no—
316C2×#102—612no—
317C2×#102—611no—
318C2×C4×C8—611no—
319#396—1222no—
320#390—612no—
321#389—612no—
322#384—611no—
323#380—1221no—
324#340—612no—
325#340—612no—
326#340—612no—
327#336—612no—
328#336—612no—
329C61#2432611yescharacteristic, complemented, Sylow
330C22×Q8—28no—
331C22×Q8—22no—
332C22×Q8—21no—
333C22×Q8—44no—
334C22×Q8—44no—
335C22×Q8C22⋊D6111yescharacteristic
336C22×D4—611no—
337C22×D4C22×D6111yescharacteristic
338C23×C4—22no—
339C23×C4—21no—
340C23×C4—1221no—
341C2×D8—614no—
342C2×D8—1222no—
343C2×D8—1222no—
344C22×C8—611no—
345C22×C8—611no—
346C22×C8—612no—
347C4⋊Q8—24no—
348C4⋊Q8—24no—
349C4⋊Q8—22no—
350C4⋊Q8C22⋊D6114yes—
351C22⋊Q8—28no—
352C22⋊Q8—216no—
353C22⋊Q8—2444no—
354C4×Q8—22no—
355C4×Q8—22no—
356C4×D4—24no—
357C4×D4—24no—
358C4×D4—22no—
359C4×D4—1222no—
360C4×D4—1222no—
361C4×D4C22⋊D6114yes—
362C2×C4⋊C4—22no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—611no—
365C2×C4⋊C4—612no—
366C2×C4⋊C4—1221no—
367C2×C4⋊C4—1221no—
368C2×C4⋊C4—1222no—
369C2×C4⋊C4C22⋊D6111yescharacteristic
370C2×C4⋊C4C22⋊D6112yes—
371C2×C4⋊C4C22×D6112yes—
372C2×C4⋊C4C22×D6111yescharacteristic
373C2×C22⋊C4—22no—
374C2×C22⋊C4—21no—
375C2×C22⋊C4—1222no—
376C2×C22⋊C4—1221no—
377C2×C22⋊C4—1222no—
378C2×C42C22×D6112yes—
379C2×C42C22×D6111yescharacteristic
380#107—1222no—
381#107—1224no—
382C4⋊C8—1222no—
383#102—1224no—
384#102—1222no—
385#102—1222no—
386#102—1224no—
387C4×C8—1222no—
388#95—1222no—
389C24—21no—
390C24—1221no—
391C2×Q8—28no—
392C2×Q8—22no—
393C2×Q8—24no—
394C2×Q8—22no—
395C2×Q8—44no—
396C2×Q8—432no—
397C2×Q8—44no—
398C2×Q8—42no—
399C2×Q8—42no—
400C2×Q8#1175819914yescomplemented
401C2×D4—21no—
402C2×D4—21no—
403C2×D4—21no—
404C2×D4—612no—
405C2×D4—1221no—
406C2×D4—1221no—
407C2×D4—1221no—
408C2×D4—1221no—
409C2×D4—1221no—
410C2×D4Q8×D6112yes—
411C2×D4#1175822612yes—
412C2×D4C2×C22⋊D6112yes—
413C22×C4—28no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—24no—
417C22×C4—44no—
418C22×C4—44no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—1221no—
422C22×C4—1222no—
423C22×C4—2442no—
424C22×C4—2442no—
425C22×C4—2442no—
426C22×C4C2×C22⋊D6112yes—
427C22×C4C2×C22⋊D6111yescharacteristic
428C22×C4C2×C22⋊D6112yes—
429C22×C4C2×C22⋊D6111yescharacteristic
430C22×C4C23×D6111yescharacteristic
431D8—1228no—
432C2×C8—612no—
433C2×C8—1221no—
434C2×C8—1221no—
435C2×C8—1221no—
436C2×C8—1222no—
437C2×C8—1222no—
438C2×C8—1221no—
439C2×C8—1221no—
440C2×C8—1222no—
441C4⋊C4—28no—
442C4⋊C4—22no—
443C4⋊C4—21no—
444C4⋊C4—24no—
445C4⋊C4—24no—
446C4⋊C4—22no—
447C4⋊C4—1221no—
448C4⋊C4—1221no—
449C4⋊C4—1222no—
450C4⋊C4—1222no—
451C4⋊C4—4881no—
452C4⋊C4—2444no—
453C4⋊C4—2442no—
454C4⋊C4C2×C22⋊D6114yes—
455C4⋊C4C2×C22⋊D6112yes—
456C22⋊C4—28no—
457C22⋊C4—24no—
458C22⋊C4—4882no—
459C22⋊C4—2442no—
460C22⋊C4—2444no—
461C42—22no—
462C42—22no—
463C42C2×C22⋊D6112yes—
464C42C2×C22⋊D6112yes—
465C23—22no—
466C23—22no—
467C23—22no—
468C23—41no—
469C23—41no—
470C23—41no—
471C23—41no—
472C23—1222no—
473C23—2441no—
474C23—2441no—
475C23—2441no—
476C23—2441no—
477C23—2441no—
478C23—2441no—
479C23C22×C22⋊D6111yescharacteristic
480Q8—28no—
481Q8—416no—
482Q8—48no—
483D4—22no—
484D4—22no—
485D4—22no—
486D4—1222no—
487D4—1222no—
488D4—2441no—
489D4—2441no—
490D4#1451975314yescomplemented
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—22no—
496C2×C4—24no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—416no—
501C2×C4—42no—
502C2×C4—41no—
503C2×C4—42no—
504C2×C4—41no—
505C2×C4—48no—
506C2×C4—4884no—
507C2×C4—2441no—
508C2×C4—2442no—
509C2×C4—2441no—
510C2×C4—2441no—
511C2×C4—2442no—
512C2×C4—2442no—
513C2×C4#1451974612yes—
514C2×C4#1451974612yes—
515C2×C4C2×#1175819914yes—
516C2×C4C2×#1175819912yes—
517C2×C4#1451980312yes—
518C2×C4C22×C22⋊D6111yescharacteristic
519C2×C4C22×C22⋊D6111yescharacteristic
520C2×C4C2×Q8×D6111yescharacteristic
521C2×C4C2×#1175822611yescharacteristic
522C2×C4#1451982511yescharacteristic
523C2×C4#1451982511yescharacteristic
524C8—1222no—
525C8—2441no—
526C8—2442no—
527C22—24no—
528C22—42no—
529C22—42no—
530C22—42no—
531C22—42no—
532C22—42no—
533C22—42no—
534C22—4881no—
535C22—4881no—
536C22—4881no—
537C22—4881no—
538C22—2442no—
539C22—2442no—
540C22—2442no—
541C22C2×#1451974611yescharacteristic
542C22#1684042311yescharacteristic
543C22Q8×C22⋊D6111yescharacteristic
544C22C22×#1175819911yescharacteristic
545C22C2×#1451980311yescharacteristic
546C22#1684057011yescharacteristic
547C22#1684057711yescharacteristic
548C4—22no—
549C4—24no—
550C4—42no—
551C4—44no—
552C4—42no—
553C4—4882no—
554C4—4881no—
555C4#1683989614yes—
556C4C2×#1451975311yescharacteristic
557C4#1684033611yescharacteristic
558C4#1684042311yescharacteristic
559C4Q8×C22⋊D6111yescharacteristic
560C2—44no—
561C2—4882no—
562C2#2144917811yescharacteristic
563C2C2×#1683989611yescharacteristic
564C2#2145513411yescharacteristic
565C2#2145845411yescharacteristic
566C2#2145853711yescharacteristic
567C2Q8×#1175819911yescharacteristic
568C2#2145871411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×D4×Q8×C61C2
socleC23×C61C22×D4
prime radicalC22×#11758199C22
hypercenterC2×D4×Q8D61
hypocenterC61#24326
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×D4×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#16840039C2semidirect
3. #21450737C2semidirect
4. #21447144C2semidirect
5. #21446952C2semidirect
6. #21446781C2semidirect
7. #21446756C2semidirect
9. #21446066C2semidirect
10. #21445819C2semidirect
11. C2×D4×Q8×C61C2semidirect
49. C2×#11758199Q8semidirect
115. C2×Q8×C61D8semidirect
142. C2×D244C2×Q8semidirect
179. C2×#92418C2×Q8semidirect
201. D4×C61C22⋊Q8semidirect
276. C2×D4×Q8D61semidirect
329. C61#24326semidirect
400. C2×Q8#11758199semidirect
490. D4#14519753semidirect

Maximal subgroups

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

subgroupindex
C2×#168400392
C2×C4×#117581992
#214507372
#214471442
#214469522
#214467812
#214467562
#214461672
#214460662
#214458192
C2×D4×Q8×C612
#2432661