Finite Group Databasebeta

← #29730748

Subgroups — #29730748

Order 15616.

1102 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839993C212yesmaximal, complemented
2C2×#16839964C211yesmaximal, characteristic, complemented
3C2×D4×#92418C211yesmaximal, characteristic, complemented
4#21450732C211yesmaximal, characteristic
5#21447012C212yesmaximal, complemented
6#21446820C212yesmaximal, complemented
7#21446622C211yesmaximal, characteristic, complemented
8#21446582C211yesmaximal, characteristic, complemented
9#21446243C212yesmaximal, complemented
10C4×#14519697C211yesmaximal, characteristic, complemented
11C61×#2542C211yesmaximal, characteristic, complemented
12C22×#11758201C2212yes—
13C23×#92418C2212yes—
14C2×#14519751C2212yes—
15C2×#14519748—21no—
16C2×C4×D244C2211yescharacteristic
17#16839993—28no—
18#16839964—24no—
19D4×#92418—24no—
20C2×#14519708C2212yes—
21C2×C4⋊#92418C2211yescharacteristic
22C2×#14519701C2212yes—
23C2×#14519701C2212yes—
24C2×#14519701C2212yes—
25C2×#14519697C2211yescharacteristic
26C2×#14519697C2211yescharacteristic
27C2×C22⋊#92418C2212yes—
28C2×C4×#92418C2211yescharacteristic
29#16839678—22no—
30#16839632C2211yescharacteristic
31#16839622C2212yes—
32#16839619—21no—
33#16839524C2211yescharacteristic
34#16839524C2211yescharacteristic
35#16839524C2212yes—
36C2×C4⋊Q8×C61C2211yescharacteristic
37C2×C22⋊Q8×C61C2212yes—
38C2×C22⋊Q8×C61C2212yes—
39C2×C4×D4×C61C2211yescharacteristic
40C61×#399—22no—
41C61×#390—21no—
42C61×#384—21no—
43C4×C4⋊C4×C61C2211yescharacteristic
44C22×D244C2311yescharacteristic
45C22×C22⋊D61—22no—
46C22×C4×D61—21no—
47C2×#11758201—24no—
48C2×#11758201—24no—
49C2×#11758201—42no—
50C22×#92418—24no—
51C22×#92418—24no—
52C22×#92418—42no—
53C22×#92418C2311yescharacteristic
54C22×#92418C2311yescharacteristic
55C22×#92418C2312yes—
56#14519751—24no—
57#14519751—42no—
58#14519748—44no—
59C4×D244—22no—
60C4×D244—22no—
61C2×C4⋊Dic61—21no—
62C2×C4⋊Dic61—21no—
63C2×C4⋊Dic61C2311yescharacteristic
64C2×C4⋊Dic61C2312yes—
65C2×C22⋊Dic61—22no—
66C2×#11758198—21no—
67C2×#11758198—22no—
68#14519708—24no—
69C4⋊#92418D414yescomplemented
70#14519701—24no—
71#14519701—24no—
72#14519701—42no—
73#14519697—22no—
74#14519697D414yescomplemented
75C22⋊#92418—24no—
76C4×#92418—22no—
77#14519684—22no—
78C22×Q8×C61C2312yes—
79C22×D4×C61D411yescharacteristic
80C23×C4×C61C2312yes—
81C4⋊Q8×C61—22no—
82C4⋊Q8×C61—22no—
83C22⋊Q8×C61—24no—
84C22⋊Q8×C61—24no—
85C22⋊Q8×C61—42no—
86C22⋊Q8×C61—42no—
87C4×D4×C61—22no—
88C4×D4×C61—22no—
89C2×C4⋊C4×C61—21no—
90C2×C4⋊C4×C61—21no—
91C2×C4⋊C4×C61—22no—
92C2×C4⋊C4×C61—22no—
93C2×C4⋊C4×C61D411yescharacteristic
94C2×C4⋊C4×C61C2311yescharacteristic
95C2×C4⋊C4×C61C2311yescharacteristic
96C2×C22⋊C4×C61—22no—
97C2×C22⋊C4×C61—22no—
98C2×C22⋊C4×C61D412yes—
99C2×C42×C61—21no—
100C2×C42×C61C2311yescharacteristic
101#95×C61—21no—
102C24×C61C2×D412yes—
103C23×D61—21no—
104C2×Q8×C61—22no—
105C2×Q8×C61—22no—
106C2×Q8×C61—22no—
107C2×Q8×C61—22no—
108C2×Q8×C61—22no—
109C2×Q8×C61—22no—
110C2×D4×C61—22no—
111C2×D4×C61—22no—
112C2×D4×C61SD1614yescomplemented
113C22×C4×C61—21no—
114C22×C4×C61—21no—
115C22×C4×C61—22no—
116C22×C4×C61—22no—
117C22×C4×C61—22no—
118C22×C4×C61—22no—
119C22×C4×C61—22no—
120C22×C4×C61—22no—
121C22×C4×C61—22no—
122C22×C4×C61C2×D412yes—
123C22×C4×C61C2×D411yescharacteristic
124C22×C4×C61C2×D411yescharacteristic
125C22×C4×C61C2411yescharacteristic
126C2×C22⋊D61—44no—
127C2×C22⋊D61—44no—
128C2×C22⋊D61—82no—
129C22×Dic61—21no—
130C22×Dic61—22no—
131C2×D244—21no—
132C2×D244—21no—
133C2×D244—21no—
134C2×D244—21no—
135C2×D244—21no—
136C2×D244C2×D412yes—
137C2×C4×D61—42no—
138C2×C4×D61—42no—
139C2×C4×D61—42no—
140C4⋊C4×C61—22no—
141C4⋊C4×C61—21no—
142C4⋊C4×C61—21no—
143C4⋊C4×C61—44no—
144C4⋊C4×C61—44no—
145C4⋊C4×C61—42no—
146C4⋊C4×C61—42no—
147C4⋊C4×C61C2×D412yes—
148C4⋊C4×C61#4112yes—
149C22⋊C4×C61—24no—
150C22⋊C4×C61—44no—
151C22⋊C4×C61—44no—
152C42×C61—42no—
153C42×C61C2×D412yes—
154C42×C61#4112yes—
155C22⋊Dic61—44no—
156#11758201—48no—
157#11758198—44no—
158#11758198—42no—
159C4⋊Dic61—21no—
160C4⋊Dic61—22no—
161C4⋊Dic61—21no—
162C4⋊Dic61—22no—
163C4⋊Dic61—42no—
164C4⋊Dic61—81no—
165C2×#92418—21no—
166C2×#92418—22no—
167C2×#92418—21no—
168C2×#92418—22no—
169C2×#92418—21no—
170C2×#92418—21no—
171C2×#92418—21no—
172C2×#92418—48no—
173C2×#92418C2×D414yes—
174C2×#92418C2×D412yes—
175C23×C61—22no—
176C23×C61—22no—
177C23×C61—22no—
178C23×C61—22no—
179C23×C61—22no—
180C23×C61—22no—
181C23×C61C2×SD1614yes—
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—42no—
191Q8×C61—42no—
192Q8×C61—42no—
193Q8×C61—42no—
194D4×C61—28no—
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—41no—
203C2×C4×C61—41no—
204C2×C4×C61—41no—
205C2×C4×C61—41no—
206C2×C4×C61—42no—
207C2×C4×C61—42no—
208C2×C4×C61—42no—
209C2×C4×C61—42no—
210C2×C4×C61—42no—
211C2×C4×C61—42no—
212C2×C4×C61—42no—
213C2×C4×C61#12112yes—
214C2×C4×C61#12112yes—
215C2×C4×C61C2×SD1612yes—
216C2×C4×C61#13512yes—
217C2×C4×C61C22×D411yescharacteristic
218C2×C4×C61C22×D411yescharacteristic
219C2×C4×C61C2×#4111yescharacteristic
220C2×C4×C61C2×#4111yescharacteristic
221C2×C4×C61H(5,𝔽2)11yescharacteristic
222C2×C4×C61H(5,𝔽2)11yescharacteristic
223C22⋊D61—88no—
224C2×Dic61—41no—
225C2×Dic61—42no—
226C2×Dic61—42no—
227C2×Dic61—41no—
228C2×Dic61—41no—
229C2×Dic61—42no—
230D244—22no—
231D244—22no—
232D244—41no—
233D244—41no—
234C4×D61—84no—
235#92418—22no—
236#92418—24no—
237#92418—41no—
238#25385—611nomaximal, Sylow
239C22×C61—24no—
240C22×C61—24no—
241C22×C61—24no—
242C22×C61—42no—
243C22×C61—42no—
244C22×C61—42no—
245C22×C61—42no—
246C22×C61C2×#12111yescharacteristic
247C22×C61D4211yescharacteristic
248C22×C61#54611yescharacteristic
249C22×C61C22×SD1611yescharacteristic
250C22×C61C2×#13511yescharacteristic
251C22×C61#57711yescharacteristic
252C22×C61#57811yescharacteristic
253C2×D61—42no—
254C2×D61—42no—
255C2×D61—42no—
256C2×D61—81no—
257C2×D61—81no—
258C2×D61—81no—
259C2×D61—81no—
260C4×C61—22no—
261C4×C61—42no—
262C4×C61—42no—
263C4×C61—42no—
264C4×C61—82no—
265C4×C61#46514yescomplemented
266C4×C61#53211yescharacteristic
267C4×C61#54011yescharacteristic
268C4×C61D4211yescharacteristic
269C4×C61#55011yescharacteristic
270Dic61—82no—
271Dic61—81no—
272C2×#465—612no—
273C2×#461—611no—
274C2×C8×D4—611no—
275#2542D6111yescharacteristic, complemented
276#2537—611no—
277#2035—612no—
278#2000—612no—
279#1956—611no—
280#1948—611no—
281#1892—612no—
282C4×#102—611no—
283C2×C61—44no—
284C2×C61#242811yescharacteristic
285C2×C61C2×#46511yescharacteristic
286C2×C61#306311yescharacteristic
287C2×C61D4×SD1611yescharacteristic
288C2×C61#328311yescharacteristic
289C2×C61#329811yescharacteristic
290C2×C61#330311yescharacteristic
291D61—82no—
292C22×SD16—612no—
293C23×C8—612no—
294C2×C4⋊Q8C2×D6111yescharacteristic
295C2×C22⋊Q8C2×D6112yes—
296C2×C22⋊Q8C2×D6112yes—
297C2×#121—612no—
298C2×#121—1221no—
299C2×C4×D4—611no—
300C2×C4×D4C2×D6111yescharacteristic
301#465—1228no—
302#461—1224no—
303C8×D4—1224no—
304C2×#106—612no—
305C2×C4⋊C8—611no—
306C2×#103—612no—
307C2×#103—612no—
308C2×#103—612no—
309C2×#102—611no—
310C2×#102—611no—
311C2×C22⋊C8—612no—
312C2×C4×C8—611no—
313#399—22no—
314#394—1222no—
315#390—21no—
316#384—21no—
317#384—611no—
318#380—612no—
319#380—1221no—
320C4×C4⋊C4C2×D6111yescharacteristic
321#340—611no—
322#340—612no—
323#340—611no—
324C61#2538511yescharacteristic, complemented, Sylow
325C22×Q8C22×D6112yes—
326C22×D4—611no—
327C22×D4—1222no—
328C22×D4C22⋊D6111yescharacteristic
329C23×C4—1221no—
330C23×C4C22×D6112yes—
331C2×SD16—1224no—
332C2×SD16—1224no—
333C2×SD16—2442no—
334C22×C8—611no—
335C22×C8—612no—
336C22×C8—611no—
337C22×C8—1224no—
338C22×C8—1224no—
339C22×C8—2442no—
340C4⋊Q8—22no—
341C4⋊Q8—22no—
342C22⋊Q8—24no—
343C22⋊Q8—24no—
344C22⋊Q8—42no—
345C22⋊Q8—42no—
346#121—1224no—
347#121—2442no—
348#121—2444no—
349C4×D4—22no—
350C4×D4—22no—
351C4×D4—1222no—
352C4×D4—1222no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—21no—
357C2×C4⋊C4—611no—
358C2×C4⋊C4—612no—
359C2×C4⋊C4—1221no—
360C2×C4⋊C4—1221no—
361C2×C4⋊C4C22⋊D6111yescharacteristic
362C2×C4⋊C4C22×D6111yescharacteristic
363C2×C4⋊C4C22×D6111yescharacteristic
364C2×C22⋊C4—22no—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—1222no—
367C2×C22⋊C4—1221no—
368C2×C22⋊C4—1222no—
369C2×C22⋊C4C22⋊D6112yes—
370C2×C42—21no—
371C2×C42C22×D6111yescharacteristic
372#106—1224no—
373C4⋊C8—614no—
374#103—1224no—
375#103—1224no—
376#103—2442no—
377#102—614no—
378#102—1222no—
379C22⋊C8—1224no—
380C4×C8—1222no—
381#95—21no—
382#95—1222no—
383C24—1221no—
384C24C2×C22⋊D6112yes—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—612no—
394C2×D4—1221no—
395C2×D4—1221no—
396C2×D4—1221no—
397C2×D4—1221no—
398C2×D4—1221no—
399C2×D4—4882no—
400C2×D4—2444no—
401C2×D4—2444no—
402C2×D4#1175820114yescomplemented
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—1221no—
413C22×C4—1222no—
414C22×C4—2442no—
415C22×C4—2442no—
416C22×C4—2442no—
417C22×C4C2×C22⋊D6112yes—
418C22×C4C2×C22⋊D6111yescharacteristic
419C22×C4C2×C22⋊D6111yescharacteristic
420C22×C4C23×D6111yescharacteristic
421SD16—2448no—
422C2×C8—614no—
423C2×C8—612no—
424C2×C8—1221no—
425C2×C8—1222no—
426C2×C8—1221no—
427C2×C8—1222no—
428C2×C8—1221no—
429C2×C8—1221no—
430C2×C8—1221no—
431C2×C8—2448no—
432C4⋊C4—21no—
433C4⋊C4—21no—
434C4⋊C4—22no—
435C4⋊C4—44no—
436C4⋊C4—44no—
437C4⋊C4—42no—
438C4⋊C4—42no—
439C4⋊C4—1221no—
440C4⋊C4—1222no—
441C4⋊C4—1221no—
442C4⋊C4—1222no—
443C4⋊C4—4881no—
444C4⋊C4—2442no—
445C4⋊C4D4×D6112yes—
446C4⋊C4#1175822412yes—
447C22⋊C4—24no—
448C22⋊C4—44no—
449C22⋊C4—44no—
450C22⋊C4—2444no—
451C22⋊C4—2444no—
452C22⋊C4—2442no—
453C42—42no—
454C42D4×D6112yes—
455C42#1175822412yes—
456C23—22no—
457C23—22no—
458C23—22no—
459C23—22no—
460C23—22no—
461C23—22no—
462C23—1222no—
463C23—2441no—
464C23—2441no—
465C23—2441no—
466C23—2441no—
467C23—2441no—
468C23—2441no—
469C23C2×#1175820114yes—
470C23C22×C22⋊D6111yescharacteristic
471Q8—42no—
472Q8—42no—
473Q8—42no—
474Q8—42no—
475D4—28no—
476D4—1222no—
477D4—1222no—
478D4—4888no—
479D4—2441no—
480D4—2441no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—4884no—
500C2×C4—2441no—
501C2×C4—2441no—
502C2×C4—2442no—
503C2×C4—2441no—
504C2×C4—2442no—
505C2×C4—2442no—
506C2×C4#1451974912yes—
507C2×C4#1451974912yes—
508C2×C4C2×#1175820112yes—
509C2×C4#1451979712yes—
510C2×C4C2×D4×D6111yescharacteristic
511C2×C4C2×D4×D6111yescharacteristic
512C2×C4C2×#1175822411yescharacteristic
513C2×C4C2×#1175822411yescharacteristic
514C2×C4#1451982211yescharacteristic
515C2×C4#1451982211yescharacteristic
516C8—1222no—
517C8—1224no—
518C8—2441no—
519C22—24no—
520C22—24no—
521C22—24no—
522C22—42no—
523C22—42no—
524C22—42no—
525C22—42no—
526C22—4881no—
527C22—4881no—
528C22—4881no—
529C22—4881no—
530C22—2442no—
531C22—2442no—
532C22—2442no—
533C22C2×#1451974911yescharacteristic
534C22D4×C22⋊D6111yescharacteristic
535C22#1684040611yescharacteristic
536C22C22×#1175820111yescharacteristic
537C22C2×#1451979711yescharacteristic
538C22#1684057011yescharacteristic
539C22#1684057711yescharacteristic
540C4—22no—
541C4—42no—
542C4—42no—
543C4—42no—
544C4—82no—
545C4—4881no—
546C4—4882no—
547C4#1683999314yescomplemented
548C4#1684031011yescharacteristic
549C4#1684036611yescharacteristic
550C4#1684039711yescharacteristic
551C4#1684043211yescharacteristic
552C2—44no—
553C2—4882no—
554C2#2144941011yescharacteristic
555C2C2×#1683999311yescharacteristic
556C2#2145550611yescharacteristic
557C2D4×#1175820111yescharacteristic
558C2#2145753711yescharacteristic
559C2#2145768411yescharacteristic
560C2#2145774711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2542C2
socleC23×C61C22×D4
prime radical#21450732C2
hypercenter#2542D61
hypocenterC61#25385
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2542

Complements

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

NH ≅ G/Ntype
1. C2×#16839993C2semidirect
2. C2×#16839964C2semidirect
3. C2×D4×#92418C2semidirect
5. #21447012C2semidirect
6. #21446820C2semidirect
7. #21446622C2semidirect
8. #21446582C2semidirect
9. #21446243C2semidirect
10. C4×#14519697C2semidirect
11. C61×#2542C2semidirect
69. C4⋊#92418D4semidirect
74. #14519697D4semidirect
112. C2×D4×C61SD16semidirect
265. C4×C61#465semidirect
275. #2542D61semidirect
324. C61#25385semidirect
402. C2×D4#11758201semidirect
547. C4#16839993semidirect

Maximal subgroups

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

subgroupindex
C2×#168399932
C2×#168399642
C2×D4×#924182
#214507322
#214470122
#214468202
#214466222
#214465822
#214462432
C4×#145196972
C61×#25422
#2538561