Finite Group Databasebeta

← #29721599

Subgroups — #29721599

Order 15616.

1086 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839979C212yesmaximal, complemented
2C2×#16839957C211yesmaximal, characteristic
3#21450818C211yesmaximal, characteristic, complemented
4D4×C4⋊Dic61C211yesmaximal, characteristic, complemented
5#21447188C212yesmaximal, complemented
6#21447057C212yesmaximal, complemented
7#21446625C211yesmaximal, characteristic, complemented
8#21446537C211yesmaximal, characteristic, complemented
9#21446165C212yesmaximal, complemented
10#21445824C211yesmaximal, characteristic, complemented
11C2×C4×SD16×C61C211yesmaximal, characteristic, complemented
12C22×#11758199C2212yes—
13C2×#14519774C2211yescharacteristic
14C2×#14519766C2212yes—
15C2×D4×Dic61C2212yes—
16C22×C4⋊Dic61—21no—
17#16839979—28no—
18#16839957—24no—
19C2×C4⋊#92418C2211yescharacteristic
20C2×#14519706C2212yes—
21C2×#14519701C2212yes—
22C2×#14519699C2212yes—
23C2×#14519699C2212yes—
24C2×#14519697C2212yes—
25#16839679—22no—
26#16839664—21no—
27#16839664—21no—
28#16839663C2212yes—
29#16839661C2212yes—
30#16839637—22no—
31#16839632C2211yescharacteristic
32#16839622—21no—
33C4×C4⋊Dic61C2211yescharacteristic
34#16839524C2212yes—
35#16839516C2212yes—
36C22×SD16×C61C2211yescharacteristic
37C2×C4×Q8×C61C2211yescharacteristic
38C2×C4×D4×C61C2211yescharacteristic
39C4×SD16×C61—24no—
40C2×#106×C61C2211yescharacteristic
41C2×#103×C61C2211yescharacteristic
42C2×#102×C61C2211yescharacteristic
43C2×C4×C8×C61C2211yescharacteristic
44C22×D244—21no—
45C22×D244—21no—
46C22×D244C2312yes—
47C23×Dic61—22no—
48C2×#11758199—24no—
49C2×#11758199—24no—
50C2×#11758199—24no—
51C22×#92418C2312yes—
52#14519774—22no—
53#14519774—22no—
54#14519766—24no—
55#14519766—42no—
56D4×Dic61—24no—
57D4×Dic61—42no—
58C2×C4⋊Dic61—21no—
59C2×C4⋊Dic61—22no—
60C2×C4⋊Dic61—21no—
61C2×C4⋊Dic61—44no—
62C2×C4⋊Dic61D411yescharacteristic
63C2×C4⋊Dic61D411yescharacteristic
64C2×C4⋊Dic61C2312yes—
65C2×C22⋊Dic61—22no—
66C2×#11758198—22no—
67C2×#11758198—22no—
68C2×#11758198—22no—
69C2×#11758198—22no—
70C2×C4×Dic61C2312yes—
71C4⋊#92418—22no—
72#14519706—24no—
73#14519701—24no—
74#14519699—24no—
75#14519699—42no—
76#14519697—24no—
77#14519684—22no—
78#14519684D412yes—
79C22×Q8×C61C2311yescharacteristic
80C22×D4×C61C2311yescharacteristic
81C23×C4×C61—21no—
82C2×SD16×C61—22no—
83C2×SD16×C61—22no—
84C2×SD16×C61D414yescomplemented
85C22×C8×C61C2311yescharacteristic
86C22×C8×C61C2311yescharacteristic
87C4×Q8×C61—22no—
88C4×Q8×C61—22no—
89C4×D4×C61—22no—
90C4×D4×C61D414yes—
91C2×C4⋊C4×C61—21no—
92C2×C4⋊C4×C61C2311yescharacteristic
93C2×C4⋊C4×C61C2311yescharacteristic
94C2×C22⋊C4×C61—21no—
95C2×C42×C61—21no—
96C2×C42×C61C2311yescharacteristic
97#106×C61—22no—
98#103×C61—22no—
99#102×C61—22no—
100C4×C8×C61—22no—
101C24×C61—21no—
102C23×D61—22no—
103C2×Q8×C61—21no—
104C2×Q8×C61—21no—
105C2×Q8×C61—21no—
106C2×Q8×C61—21no—
107C2×Q8×C61—21no—
108C2×Q8×C61C2×D412yes—
109C2×D4×C61—21no—
110C2×D4×C61—21no—
111C2×D4×C61—21no—
112C2×D4×C61C2×D412yes—
113C2×D4×C61C2×D412yes—
114C2×D4×C61#4112yes—
115C22×C4×C61—21no—
116C22×C4×C61—21no—
117C22×C4×C61—24no—
118C22×C4×C61—21no—
119C22×C4×C61—42no—
120C22×C4×C61—42no—
121C22×C4×C61C2×D411yescharacteristic
122C22×C4×C61C2×D411yescharacteristic
123C22×C4×C61C2411yescharacteristic
124C22×Dic61—22no—
125C22×Dic61—44no—
126C22×Dic61—44no—
127C22×Dic61—82no—
128C22×Dic61C2×D412yes—
129C22×Dic61C2×D412yes—
130C2×D244—22no—
131C2×D244—22no—
132C2×D244—22no—
133C2×D244—22no—
134C2×D244—22no—
135C2×D244—22no—
136C2×D244—42no—
137C2×D244—42no—
138C2×D244—42no—
139C2×D244—42no—
140C2×D244—81no—
141C2×D244—81no—
142SD16×C61—28no—
143C2×C8×C61—21no—
144C2×C8×C61—21no—
145C2×C8×C61—21no—
146C2×C8×C61—21no—
147C2×C8×C61—21no—
148C2×C8×C61C2×D412yes—
149C4⋊C4×C61—21no—
150C4⋊C4×C61—21no—
151C4⋊C4×C61—21no—
152C4⋊C4×C61—42no—
153C4⋊C4×C61C2×D412yes—
154C22⋊C4×C61—24no—
155C42×C61—42no—
156C42×C61C2×D412yes—
157C42×C61#4112yes—
158C22⋊Dic61—44no—
159#11758199—48no—
160#11758198—44no—
161#11758198—44no—
162#11758198—82no—
163#11758198—82no—
164C4⋊Dic61—22no—
165C4⋊Dic61—22no—
166C4⋊Dic61—22no—
167C4⋊Dic61—22no—
168C4⋊Dic61—44no—
169C4⋊Dic61—42no—
170C4⋊Dic61—81no—
171C4×Dic61—22no—
172C4×Dic61—22no—
173C2×#92418—22no—
174C2×#92418—22no—
175C2×#92418—22no—
176C23×C61—22no—
177C23×C61—22no—
178C23×C61—22no—
179C23×C61—41no—
180C23×C61—41no—
181C23×C61—41no—
182C23×C61—41no—
183C23×C61C22×D411yescharacteristic
184C22×D61—42no—
185C22×D61—42no—
186C22×D61—42no—
187C22×D61—42no—
188C22×D61—42no—
189C22×D61—42no—
190C22×D61—42no—
191Q8×C61—22no—
192Q8×C61—22no—
193Q8×C61—41no—
194Q8×C61—41no—
195D4×C61—22no—
196D4×C61—22no—
197D4×C61—22no—
198D4×C61#12114yes—
199C2×C4×C61—21no—
200C2×C4×C61—21no—
201C2×C4×C61—22no—
202C2×C4×C61—22no—
203C2×C4×C61—41no—
204C2×C4×C61—41no—
205C2×C4×C61—41no—
206C2×C4×C61—41no—
207C2×C4×C61—48no—
208C2×C4×C61—41no—
209C2×C4×C61—41no—
210C2×C4×C61—41no—
211C2×C4×C61#12112yes—
212C2×C4×C61#12112yes—
213C2×C4×C61#13512yes—
214C2×C4×C61AGL(1,ℤ/8)12yes—
215C2×C4×C61C22×D411yescharacteristic
216C2×C4×C61C22×D411yescharacteristic
217C2×C4×C61C2×#4111yescharacteristic
218C2×C4×C61C2×#4111yescharacteristic
219C2×C4×C61H(5,𝔽2)11yescharacteristic
220C2×C4×C612−1+411yescharacteristic
221C2×Dic61—22no—
222C2×Dic61—22no—
223C2×Dic61—22no—
224C2×Dic61—22no—
225C2×Dic61—22no—
226C2×Dic61—22no—
227C2×Dic61—42no—
228C2×Dic61—42no—
229C2×Dic61—42no—
230C2×Dic61—88no—
231D244—42no—
232D244—42no—
233D244—42no—
234D244—42no—
235D244—84no—
236D244—84no—
237C8×C61—22no—
238C8×C61—41no—
239#92418—42no—
240#24731—611nomaximal, Sylow
241C22×C61—24no—
242C22×C61—42no—
243C22×C61—42no—
244C22×C61—42no—
245C22×C61—42no—
246C22×C61—42no—
247C22×C61—42no—
248C22×C61C2×#12111yescharacteristic
249C22×C61D4211yescharacteristic
250C22×C61#54811yescharacteristic
251C22×C61C2×#13511yescharacteristic
252C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
253C22×C61#57611yescharacteristic
254C22×C61#57711yescharacteristic
255C2×D61—82no—
256C2×D61—82no—
257C2×D61—82no—
258C2×D61—82no—
259C2×D61—82no—
260C2×D61—82no—
261C2×D61—82no—
262C4×C61—22no—
263C4×C61—42no—
264C4×C61—42no—
265C4×C61—81no—
266C4×C61#46314yescomplemented
267C4×C61C2×#12111yescharacteristic
268C4×C61#53711yescharacteristic
269C4×C61#54711yescharacteristic
270C4×C61#55311yescharacteristic
271Dic61—42no—
272Dic61—42no—
273Dic61—82no—
274C2×#463—612no—
275C2×#459—611no—
276C2×C4×SD16D6111yescharacteristic, complemented
277#2548—611no—
278D4×C4⋊C4—611no—
279#2065—612no—
280#2042—612no—
281#1957—611no—
282#1939—611no—
283#1879—612no—
284#1803—611no—
285C2×C61—44no—
286C2×C61#241611yescharacteristic
287C2×C61C2×#46311yescharacteristic
288C2×C61#304511yescharacteristic
289C2×C61#327411yescharacteristic
290C2×C61#328411yescharacteristic
291C2×C61#336111yescharacteristic
292C2×C61#337311yescharacteristic
293D61—162no—
294C22×SD16C2×D6111yescharacteristic
295C22×D8—612no—
296C2×#127—611no—
297C2×#124—612no—
298C2×C4×Q8C2×D6111yescharacteristic
299C2×C4×D4—612no—
300C2×C4×D4C2×D6111yescharacteristic
301C22×C4⋊C4—1221no—
302#463—1228no—
303#459—1224no—
304C4×SD16—24no—
305C2×#106C2×D6111yescharacteristic
306C2×C4⋊C8—612no—
307C2×C4⋊C8—611no—
308C2×#103—612no—
309C2×#103C2×D6111yescharacteristic
310C2×#102—612no—
311C2×#102—612no—
312C2×#102—612no—
313C2×#102C2×D6111yescharacteristic
314C2×C4×C8C2×D6111yescharacteristic
315#394—1222no—
316#390—612no—
317#390—1221no—
318#390—1221no—
319#389—612no—
320#385—1222no—
321#384—611no—
322#380—1221no—
323C4×C4⋊C4—611no—
324#340—612no—
325#336—612no—
326C61#2473111yescharacteristic, complemented, Sylow
327C22×Q8C22×D6111yescharacteristic
328C22×D4—612no—
329C22×D4—1221no—
330C22×D4—1221no—
331C22×D4C22×D6111yescharacteristic
332C23×C4—21no—
333C23×C4—1222no—
334C2×SD16—22no—
335C2×SD16—22no—
336C2×SD16D24414yescomplemented
337C2×D8—1224no—
338C2×D8—1224no—
339C2×D8—1224no—
340C22×C8—612no—
341C22×C8C22×D6111yescharacteristic
342C22×C8C22×D6111yescharacteristic
343#127—1222no—
344#127—1222no—
345#124—1224no—
346#124—2442no—
347C4×Q8—22no—
348C4×Q8—22no—
349C4×D4—22no—
350C4×D4—1224no—
351C4×D4—2442no—
352C4×D4C22⋊D6114yes—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—611no—
355C2×C4⋊C4—611no—
356C2×C4⋊C4—612no—
357C2×C4⋊C4—1221no—
358C2×C4⋊C4—1222no—
359C2×C4⋊C4—1221no—
360C2×C4⋊C4—2444no—
361C2×C4⋊C4C22×D6111yescharacteristic
362C2×C4⋊C4C22×D6111yescharacteristic
363C2×C22⋊C4—21no—
364C2×C22⋊C4—1222no—
365C2×C22⋊C4—1222no—
366C2×C22⋊C4—1222no—
367C2×C22⋊C4—1222no—
368C2×C22⋊C4—1222no—
369C2×C42—21no—
370C2×C42—612no—
371C2×C42C22×D6111yescharacteristic
372#106—22no—
373C4⋊C8—1224no—
374C4⋊C8—1222no—
375#103—22no—
376#103—1224no—
377#102—22no—
378#102—1224no—
379#102—1224no—
380#102—2442no—
381C4×C8—22no—
382#95—612no—
383#95—1222no—
384C24—21no—
385C24—1222no—
386C2×Q8—21no—
387C2×Q8—21no—
388C2×Q8—21no—
389C2×Q8—21no—
390C2×Q8—21no—
391C2×Q8C2×D24412yes—
392C2×D4—21no—
393C2×D4—21no—
394C2×D4—21no—
395C2×D4—1222no—
396C2×D4—1222no—
397C2×D4—1222no—
398C2×D4—1222no—
399C2×D4—1222no—
400C2×D4—1222no—
401C2×D4—4881no—
402C2×D4—4881no—
403C2×D4—2442no—
404C2×D4—2442no—
405C2×D4—2442no—
406C2×D4—2442no—
407C2×D4C2×D24412yes—
408C2×D4#1175822212yes—
409C2×D4C2×C22⋊D6112yes—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—24no—
413C22×C4—21no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—612no—
417C22×C4—612no—
418C22×C4—1222no—
419C22×C4—4882no—
420C22×C4—2444no—
421C22×C4—2444no—
422C22×C4D4×D6111yescharacteristic
423C22×C4D4×D6111yescharacteristic
424C22×C4C23×D6111yescharacteristic
425SD16—28no—
426D8—2448no—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—21no—
430C2×C8—21no—
431C2×C8—21no—
432C2×C8—1222no—
433C2×C8—1222no—
434C2×C8—1222no—
435C2×C8C2×D24412yes—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—42no—
440C4⋊C4—1222no—
441C4⋊C4—1222no—
442C4⋊C4—1222no—
443C4⋊C4—1222no—
444C4⋊C4—4881no—
445C4⋊C4—2444no—
446C4⋊C4—2442no—
447C4⋊C4C2×C22⋊D6112yes—
448C22⋊C4—24no—
449C22⋊C4—4882no—
450C22⋊C4—4882no—
451C22⋊C4—2444no—
452C22⋊C4—2444no—
453C22⋊C4—2444no—
454C42—42no—
455C42—1222no—
456C42—1222no—
457C42#1175822412yes—
458C42C2×C22⋊D6112yes—
459C23—22no—
460C23—22no—
461C23—22no—
462C23—41no—
463C23—41no—
464C23—41no—
465C23—41no—
466C23—2442no—
467C23—2442no—
468C23—2442no—
469C23—2442no—
470C23—2442no—
471C23—2442no—
472C23—2442no—
473C23C2×D4×D6111yescharacteristic
474Q8—22no—
475Q8—22no—
476Q8—41no—
477Q8—41no—
478D4—22no—
479D4—22no—
480D4—22no—
481D4—4884no—
482D4—4884no—
483D4—2442no—
484D4—2442no—
485D4—2442no—
486D4—2442no—
487D4#1451975114yes—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—48no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—1222no—
501C2×C4—1222no—
502C2×C4—1222no—
503C2×C4—1222no—
504C2×C4—1222no—
505C2×C4—1222no—
506C2×C4—4888no—
507C2×C4—2442no—
508C2×C4—2442no—
509C2×C4—2442no—
510C2×C4#1451974912yes—
511C2×C4#1451974912yes—
512C2×C4#1451979812yes—
513C2×C4#1451980412yes—
514C2×C4C22×C22⋊D6111yescharacteristic
515C2×C4C22×D24411yescharacteristic
516C2×C4C2×#1175822411yescharacteristic
517C2×C4C2×#1175822211yescharacteristic
518C2×C4#1451982311yescharacteristic
519C2×C4#1451982411yescharacteristic
520C8—22no—
521C8—41no—
522C8—2442no—
523C22—24no—
524C22—42no—
525C22—42no—
526C22—42no—
527C22—42no—
528C22—42no—
529C22—42no—
530C22—4882no—
531C22—4882no—
532C22—4882no—
533C22—4882no—
534C22—4882no—
535C22—4882no—
536C22—4882no—
537C22C2×#1451974911yescharacteristic
538C22D4×D24411yescharacteristic
539C22#1684042111yescharacteristic
540C22C2×#1451979811yescharacteristic
541C22C2×#1451980411yescharacteristic
542C22#1684056411yescharacteristic
543C22#1684056911yescharacteristic
544C4—22no—
545C4—42no—
546C4—42no—
547C4—81no—
548C4—4882no—
549C4—2442no—
550C4—2442no—
551C4#1683997914yescomplemented
552C4C2×#1451975111yescharacteristic
553C4#1684034011yescharacteristic
554C4#1684041611yescharacteristic
555C4#1684045811yescharacteristic
556C2—44no—
557C2—9762no—
558C2#2144925911yescharacteristic
559C2C2×#1683997911yescharacteristic
560C2#2145535611yescharacteristic
561C2#2145744011yescharacteristic
562C2#2145755011yescharacteristic
563C2#2145847711yescharacteristic
564C2#2145860911yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C4×SD16×C61C2
socleC23×C61C22×D4
prime radicalC2×#16839957C2
hypercenterC2×C4×SD16D61
hypocenterC61#24731
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×SD16

Complements

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

NH ≅ G/Ntype
1. C2×#16839979C2semidirect
3. #21450818C2semidirect
4. D4×C4⋊Dic61C2semidirect
5. #21447188C2semidirect
6. #21447057C2semidirect
7. #21446625C2semidirect
8. #21446537C2semidirect
9. #21446165C2semidirect
10. #21445824C2semidirect
11. C2×C4×SD16×C61C2semidirect
84. C2×SD16×C61D4semidirect
266. C4×C61#463semidirect
276. C2×C4×SD16D61semidirect
326. C61#24731semidirect
336. C2×SD16D244semidirect
551. C4#16839979semidirect

Maximal subgroups

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

subgroupindex
C2×#168399792
C2×#168399572
#214508182
D4×C4⋊Dic612
#214471882
#214470572
#214466252
#214465372
#214461652
#214458242
C2×C4×SD16×C612
#2473161