Finite Group Databasebeta

← #29716386

Subgroups — #29716386

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×Q8×D244C211yesmaximal, characteristic, complemented
2C2×#16840042C211yesmaximal, characteristic, complemented
3C2×C4×#11758199C211yesmaximal, characteristic
4#21447147C212yesmaximal, complemented
5#21446955C212yesmaximal, complemented
6#21446779C211yesmaximal, characteristic, complemented
7#21446756C211yesmaximal, characteristic, complemented
8#21446170C212yesmaximal
9#21446065C212yesmaximal, complemented
10#21445816C211yesmaximal, characteristic, complemented
11C61×#2538C211yesmaximal, characteristic, complemented
12C22×Q8×D61—21no—
13C22×#11758199C2211yescharacteristic
14Q8×D244—24no—
15Q8×D244—24no—
16C2×#14519777C2212yes—
17C2×#14519777C2211yescharacteristic
18C2×#14519757—21no—
19C2×#14519757—22no—
20C2×C4×D244C2212yes—
21C2×C4×D244C2211yescharacteristic
22#16840042—24no—
23C4×#11758199—24no—
24C2×#14519710C2212yes—
25C2×#14519710C2211yescharacteristic
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—22no—
46C22×Dic122—21no—
47C2×Q8×D61—28no—
48C2×Q8×D61—44no—
49C2×Q8×D61—44no—
50C22×D244C2311yescharacteristic
51C22×C4×D61—22no—
52C22×C4×D61—21no—
53C2×#11758199—22no—
54C2×#11758199—22no—
55C2×#11758199Q814yescomplemented
56C22×#92418C2311yescharacteristic
57C22×#92418C2312yes—
58C22×#92418C2311yescharacteristic
59#14519777—22no—
60#14519777—24no—
61#14519777—24no—
62#14519777D414yes—
63#14519757—28no—
64#14519757—216no—
65C4×D244—24no—
66C4×D244—22no—
67C4×D244—24no—
68C4×D244D414yes—
69C2×C4⋊Dic61—22no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61C2312yes—
72C2×C4⋊Dic61C2311yescharacteristic
73C2×#11758198—22no—
74C2×#11758198—21no—
75#14519710—24no—
76#14519710—22no—
77C4⋊#92418—22no—
78#14519698—24no—
79#14519698—22no—
80#14519697—22no—
81#14519697—24no—
82C4×#92418—22no—
83C22×Q8×C61D411yescharacteristic
84C22×D4×C61C2311yescharacteristic
85C23×C4×C61—21no—
86C22⋊Q8×C61—44no—
87C4×Q8×C61—22no—
88C4×Q8×C61—22no—
89C4×D4×C61—22no—
90C4×D4×C61—22no—
91C2×C4⋊C4×C61—21no—
92C2×C4⋊C4×C61—22no—
93C2×C4⋊C4×C61—21no—
94C2×C4⋊C4×C61D411yescharacteristic
95C2×C4⋊C4×C61D412yes—
96C2×C4⋊C4×C61C2312yes—
97C2×C4⋊C4×C61C2311yescharacteristic
98C2×C22⋊C4×C61—22no—
99C2×C22⋊C4×C61—21no—
100C2×C22⋊C4×C61—22no—
101C2×C42×C61C2312yes—
102C2×C42×C61C2311yescharacteristic
103#95×C61—22no—
104C24×C61—21no—
105C23×D61—21no—
106C2×Q8×C61—22no—
107C2×Q8×C61—22no—
108C2×Q8×C61D814yescomplemented
109C2×D4×C61—21no—
110C2×D4×C61—21no—
111C2×D4×C61—21no—
112C2×D4×C61—21no—
113C2×D4×C61—21no—
114C2×D4×C61C2×Q812yescomplemented
115C22×C4×C61—21no—
116C22×C4×C61—22no—
117C22×C4×C61—42no—
118C22×C4×C61—42no—
119C22×C4×C61—42no—
120C22×C4×C61C2×D412yes—
121C22×C4×C61C2×D411yescharacteristic
122C22×C4×C61C2×D412yes—
123C22×C4×C61C2×D411yescharacteristic
124C22×C4×C61C2411yescharacteristic
125C22×Dic61—21no—
126C22×Dic61—22no—
127Q8×D61—432no—
128C2×D244—21no—
129C2×D244—21no—
130C2×D244—21no—
131C2×D244C2×D412yes—
132C2×D244C2×Q812yes—
133C2×D244#4112yes—
134C2×C4×D61—28no—
135C2×C4×D61—24no—
136C2×C4×D61—44no—
137C2×C4×D61—44no—
138C2×C4×D61—42no—
139C2×C4×D61—42no—
140C2×Dic122—28no—
141C2×Dic122—24no—
142C2×Dic122—44no—
143C2×Dic122—44no—
144C2×Dic122—42no—
145C2×Dic122—42no—
146C4⋊C4×C61—24no—
147C4⋊C4×C61—22no—
148C4⋊C4×C61—21no—
149C4⋊C4×C61—21no—
150C4⋊C4×C61—22no—
151C4⋊C4×C61—22no—
152C4⋊C4×C61—42no—
153C4⋊C4×C61—44no—
154C4⋊C4×C61—81no—
155C22⋊C4×C61—42no—
156C22⋊C4×C61—44no—
157C22⋊C4×C61—82no—
158C42×C61—22no—
159C42×C61—22no—
160C42×C61C2×D412yes—
161C42×C61C2×D412yes—
162#11758199—28no—
163#11758198—28no—
164#11758198—24no—
165C4⋊Dic61—28no—
166C4⋊Dic61—24no—
167C4⋊Dic61—22no—
168C4⋊Dic61—21no—
169C4⋊Dic61C2×D414yes—
170C4⋊Dic61C2×D412yes—
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—41no—
182C23×C61—41no—
183C23×C61—41no—
184C23×C61—41no—
185C23×C61—41no—
186C23×C61—41no—
187C23×C61C22×D411yescharacteristic
188C22×D61—22no—
189C22×D61—22no—
190C22×D61—22no—
191C22×D61—41no—
192C22×D61—41no—
193C22×D61—41no—
194C22×D61—41no—
195Q8×C61—28no—
196D4×C61—22no—
197D4×C61—22no—
198D4×C61—41no—
199D4×C61—41no—
200C2×C4×C61—22no—
201C2×C4×C61—22no—
202C2×C4×C61—21no—
203C2×C4×C61—21no—
204C2×C4×C61—22no—
205C2×C4×C61—22no—
206C2×C4×C61—22no—
207C2×C4×C61—41no—
208C2×C4×C61—42no—
209C2×C4×C61—42no—
210C2×C4×C61—41no—
211C2×C4×C61—41no—
212C2×C4×C61—42no—
213C2×C4×C61—84no—
214C2×C4×C61#12012yes—
215C2×C4×C61#12012yes—
216C2×C4×C61C2×D814yes—
217C2×C4×C61C2×D812yes—
218C2×C4×C61AGL(1,ℤ/8)12yes—
219C2×C4×C61C22×D411yescharacteristic
220C2×C4×C61C22×D411yescharacteristic
221C2×C4×C61C22×Q811yescharacteristic
222C2×C4×C61C2×#4111yescharacteristic
223C2×C4×C612−1+411yescharacteristic
224C2×C4×C612−1+411yescharacteristic
225C2×Dic61—22no—
226C2×Dic61—24no—
227C2×Dic61—41no—
228C2×Dic61—42no—
229C2×Dic61—41no—
230C2×Dic61—42no—
231D244—22no—
232D244—22no—
233D244—22no—
234D244C22⋊Q814yescomplemented
235C4×D61—416no—
236C4×D61—48no—
237Dic122—416no—
238Dic122—48no—
239#92418—22no—
240#92418—41no—
241#92418—42no—
242#24326—611nomaximal, Sylow
243C22×C61—42no—
244C22×C61—42no—
245C22×C61—42no—
246C22×C61—81no—
247C22×C61—81no—
248C22×C61—81no—
249C22×C61—81no—
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—24no—
258C2×D61—42no—
259C2×D61—42no—
260C2×D61—42no—
261C2×D61—42no—
262C2×D61—42no—
263C2×D61—42no—
264C4×C61—24no—
265C4×C61—22no—
266C4×C61—42no—
267C4×C61—82no—
268C4×C61—81no—
269C4×C61#44714yes—
270C4×C61C2×C22⋊Q811yescharacteristic
271C4×C61#53611yescharacteristic
272C4×C61#54811yescharacteristic
273C4×C61D4×Q811yescharacteristic
274Dic61—42no—
275Dic61—44no—
276C2×D4×Q8—611no—
277C2×#474—611no—
278C2×C4×D8—611no—
279#2538D6111yescharacteristic, complemented
280#2057—612no—
281#2023—612no—
282#1990—611no—
283#1984—611no—
284#1879—612no—
285#1862—612no—
286#1802—611no—
287C2×C61—82no—
288C2×C61#240911yescharacteristic
289C2×C61C2×#44711yescharacteristic
290C2×C61#301311yescharacteristic
291C2×C61#335811yescharacteristic
292C2×C61#336711yescharacteristic
293C2×C61Q8×D811yescharacteristic
294C2×C61#338411yescharacteristic
295D61—44no—
296C23×Q8—1221no—
297C22×D8—611no—
298D4×Q8—1224no—
299D4×Q8—1224no—
300C2×C4⋊Q8—611no—
301C2×C4⋊Q8—612no—
302C2×C22⋊Q8—21no—
303C2×C22⋊Q8—1221no—
304C2×C22⋊Q8—1222no—
305C2×C4×Q8C2×D6111yescharacteristic
306C2×C4×D4—611no—
307C2×C4×D4—612no—
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—22no—
320#390C2×D6112yes—
321#389C2×D6112yes—
322#384C2×D6111yescharacteristic
323#380—21no—
324#340—612no—
325#340—612no—
326#340—612no—
327#336—612no—
328#336—612no—
329C61#2432611yescharacteristic, complemented, Sylow
330C22×Q8—1228no—
331C22×Q8—1222no—
332C22×Q8—1221no—
333C22×Q8—2444no—
334C22×Q8—2444no—
335C22×Q8C22⋊D6111yescharacteristic
336C22×D4—611no—
337C22×D4C22×D6111yescharacteristic
338C23×C4—21no—
339C23×C4—1222no—
340C23×C4—1221no—
341C2×D8—614no—
342C2×D8—1222no—
343C2×D8—1222no—
344C22×C8—611no—
345C22×C8—611no—
346C22×C8—612no—
347C4⋊Q8—614no—
348C4⋊Q8—1224no—
349C4⋊Q8—1224no—
350C4⋊Q8—1222no—
351C22⋊Q8—44no—
352C22⋊Q8—1228no—
353C22⋊Q8—12216no—
354C4×Q8—22no—
355C4×Q8—22no—
356C4×D4—22no—
357C4×D4—22no—
358C4×D4—614no—
359C4×D4—1224no—
360C4×D4—1224no—
361C4×D4—1222no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—22no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—611no—
366C2×C4⋊C4—612no—
367C2×C4⋊C4—1222no—
368C2×C4⋊C4—1221no—
369C2×C4⋊C4C22⋊D6111yescharacteristic
370C2×C4⋊C4C22⋊D6112yes—
371C2×C4⋊C4C22×D6111yescharacteristic
372C2×C4⋊C4C22×D6112yes—
373C2×C22⋊C4—21no—
374C2×C22⋊C4—22no—
375C2×C22⋊C4—22no—
376C2×C22⋊C4—1222no—
377C2×C22⋊C4—1221no—
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—22no—
389C24—21no—
390C24—1221no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—1228no—
394C2×Q8—1224no—
395C2×Q8—2444no—
396C2×Q8—24432no—
397C2×Q8—2444no—
398C2×Q8—2442no—
399C2×Q8—2442no—
400C2×Q8#1175819914yescomplemented
401C2×D4—21no—
402C2×D4—21no—
403C2×D4—21no—
404C2×D4—21no—
405C2×D4—21no—
406C2×D4—612no—
407C2×D4—612no—
408C2×D4—612no—
409C2×D4—1221no—
410C2×D4—1221no—
411C2×D4—1221no—
412C2×D4Q8×D6112yescomplemented
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—1228no—
419C22×C4—1222no—
420C22×C4—1221no—
421C22×C4—1224no—
422C22×C4—2444no—
423C22×C4—2444no—
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—24no—
442C4⋊C4—22no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—42no—
448C4⋊C4—44no—
449C4⋊C4—81no—
450C4⋊C4—612no—
451C4⋊C4—614no—
452C4⋊C4—1228no—
453C4⋊C4—1222no—
454C4⋊C4—1221no—
455C4⋊C4—1224no—
456C22⋊C4—42no—
457C22⋊C4—44no—
458C22⋊C4—82no—
459C22⋊C4—1228no—
460C22⋊C4—1224no—
461C42—22no—
462C42—22no—
463C42D4×D6112yes—
464C42D4×D6112yes—
465C23—22no—
466C23—41no—
467C23—41no—
468C23—41no—
469C23—41no—
470C23—41no—
471C23—41no—
472C23—1222no—
473C23—1222no—
474C23—1222no—
475C23—2441no—
476C23—2441no—
477C23—2441no—
478C23—2441no—
479C23C22×C22⋊D6111yescharacteristic
480Q8—28no—
481Q8—24416no—
482Q8—2448no—
483D4—22no—
484D4—22no—
485D4—41no—
486D4—41no—
487D4—614no—
488D4—1222no—
489D4—1222no—
490D4—1222no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—41no—
499C2×C4—42no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—84no—
505C2×C4—1224no—
506C2×C4—1222no—
507C2×C4—24416no—
508C2×C4—2442no—
509C2×C4—2441no—
510C2×C4—2442no—
511C2×C4—2441no—
512C2×C4—2448no—
513C2×C4#1451974512yes—
514C2×C4#1451974512yes—
515C2×C4C2×#1175819914yes—
516C2×C4C2×#1175819912yes—
517C2×C4#1451980212yes—
518C2×C4C2×D4×D6111yescharacteristic
519C2×C4C2×D4×D6111yescharacteristic
520C2×C4C2×Q8×D6111yescharacteristic
521C2×C4#41×D6111yescharacteristic
522C2×C4#1451982411yescharacteristic
523C2×C4#1451982511yescharacteristic
524C8—1222no—
525C8—2441no—
526C8—2442no—
527C22—42no—
528C22—42no—
529C22—42no—
530C22—81no—
531C22—81no—
532C22—81no—
533C22—81no—
534C22—1224no—
535C22—2442no—
536C22—2442no—
537C22—2442no—
538C22—2442no—
539C22—2442no—
540C22—2442no—
541C22C2×#1451974511yescharacteristic
542C22#1684041911yescharacteristic
543C22Q8×C22⋊D6111yescharacteristic
544C22C22×#1175819911yescharacteristic
545C22C2×#1451980211yescharacteristic
546C22#1684057311yescharacteristic
547C22#1684057711yescharacteristic
548C4—22no—
549C4—24no—
550C4—42no—
551C4—82no—
552C4—81no—
553C4—2444no—
554C4—2442no—
555C4#1683989514yes—
556C4C22⋊Q8×D6111yescharacteristic
557C4#1684033511yescharacteristic
558C4#1684042211yescharacteristic
559C4#1684042511yescharacteristic
560C2—82no—
561C2—2444no—
562C2#2144918011yescharacteristic
563C2C2×#1683989511yescharacteristic
564C2#2145513111yescharacteristic
565C2#2145845011yescharacteristic
566C2#2145853311yescharacteristic
567C2Q8×#1175819911yescharacteristic
568C2#2145871811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2538C2
socleC23×C61C22×D4
prime radicalC22×#11758199C22
hypercenter#2538D61
hypocenterC61#24326
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×Q8×D244C2semidirect
2. C2×#16840042C2semidirect
4. #21447147C2semidirect
5. #21446955C2semidirect
6. #21446779C2semidirect
7. #21446756C2semidirect
9. #21446065C2semidirect
10. #21445816C2semidirect
11. C61×#2538C2semidirect
55. C2×#11758199Q8semidirect
108. C2×Q8×C61D8semidirect
114. C2×D4×C61C2×Q8semidirect
179. C2×#92418C2×Q8semidirect
234. D244C22⋊Q8semidirect
279. #2538D61semidirect
329. C61#24326semidirect
400. C2×Q8#11758199semidirect
412. C2×D4Q8×D61semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×D2442
C2×#168400422
C2×C4×#117581992
#214471472
#214469552
#214467792
#214467562
#214461702
#214460652
#214458162
C61×#25382
#2432661