Finite Group Databasebeta

← #29580223

Subgroups — #29580223

Order 15616.

968 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839901C211yesmaximal, characteristic, complemented
2#21451553C211yesmaximal, characteristic
3#21445754C211yesmaximal, characteristic, complemented
4C2×#16839495C211yesmaximal, characteristic, complemented
5C2×#16839486C211yesmaximal, characteristic, complemented
6#21443698C211yesmaximal, characteristic, complemented
7C2×C61×#393C211yesmaximal, characteristic, complemented
8C22×#11758201—21no—
9C2×#14519751C2211yescharacteristic
10C2×#14519746—21no—
11#16839901—22no—
12#16839901—22no—
13C2×#14519701—21no—
14C2×#14519701—22no—
15C2×C22⋊#92418C2211yescharacteristic
16C2×C22⋊#92418C2211yescharacteristic
17#16839687—21no—
18#16839678—22no—
19#16839671—21no—
20#16839641—21no—
21#16839622C2211yescharacteristic
22#16839495—22no—
23#16839486—22no—
24C23×Q8×C61C2211yescharacteristic
25C22×C22⋊C4×C61—21no—
26C22×C22⋊C4×C61C2211yescharacteristic
27C61×#393—22no—
28C61×#393—22no—
29C2×#95×C61—21no—
30C2×#95×C61C2211yescharacteristic
31C22×D244D411yescharacteristic
32C22×C22⋊D61—21no—
33C22×C22⋊D61—22no—
34C2×#11758201—24no—
35C2×#11758201—44no—
36C22×#92418—21no—
37C22×#92418—21no—
38#14519751—21no—
39#14519751—21no—
40#14519751—21no—
41#14519751—21no—
42#14519746—42no—
43#14519746—42no—
44C2×C4⋊Dic61D411yescharacteristic
45C2×C4⋊Dic61D412yes—
46C2×C22⋊Dic61—22no—
47C2×C22⋊Dic61—21no—
48C2×#11758198—21no—
49C2×#11758198—22no—
50#14519701—24no—
51#14519701—44no—
52C22⋊#92418—21no—
53C22⋊#92418—21no—
54C22⋊#92418—21no—
55C22⋊#92418D412yescomplemented
56#14519684—22no—
57#14519684—21no—
58C25×C61D411yescharacteristic
59C22×Q8×C61—21no—
60C22×Q8×C61—21no—
61C22×Q8×C61—21no—
62C22×Q8×C61—21no—
63C22×Q8×C61—21no—
64C22×Q8×C61—44no—
65C22×Q8×C61D412yescomplemented
66C23×C4×C61—21no—
67C23×C4×C61—21no—
68C23×C4×C61D411yescharacteristic
69C23×C4×C61C2311yescharacteristic
70C2×C22⋊C4×C61—24no—
71C2×C22⋊C4×C61—21no—
72C2×C22⋊C4×C61—21no—
73C2×C22⋊C4×C61—21no—
74C2×C22⋊C4×C61—21no—
75C2×C22⋊C4×C61—21no—
76C2×C22⋊C4×C61—21no—
77C2×C22⋊C4×C61—44no—
78#95×C61—24no—
79#95×C61—21no—
80#95×C61D412yescomplemented
81C24×C61—21no—
82C24×C61—21no—
83C24×C61—21no—
84C24×C61—21no—
85C24×C61—21no—
86C24×C61—21no—
87C24×C61—42no—
88C24×C61—42no—
89C24×C61SD1612yescomplemented
90C24×C61C2×D411yescharacteristic
91C23×D61—21no—
92C2×Q8×C61—24no—
93C2×Q8×C61—24no—
94C2×Q8×C61—22no—
95C2×Q8×C61—22no—
96C2×Q8×C61—44no—
97C2×Q8×C61—42no—
98C2×Q8×C61—42no—
99C2×Q8×C61—44no—
100C2×Q8×C61—44no—
101C2×Q8×C61—44no—
102C2×Q8×C61—41no—
103C2×Q8×C61—41no—
104C22×C4×C61—24no—
105C22×C4×C61—22no—
106C22×C4×C61—21no—
107C22×C4×C61—22no—
108C22×C4×C61—21no—
109C22×C4×C61—21no—
110C22×C4×C61—42no—
111C22×C4×C61—42no—
112C22×C4×C61—41no—
113C22×C4×C61—41no—
114C22×C4×C61—42no—
115C22×C4×C61—42no—
116C22×C4×C61—42no—
117C22×C4×C61—41no—
118C22×C4×C61—41no—
119C22×C4×C61C2×D411yescharacteristic
120C22×C4×C61C2×D411yescharacteristic
121C22×C4×C61C2×D411yescharacteristic
122C22×C4×C61C2×D411yescharacteristic
123C22×C4×C61C2×D411yescharacteristic
124C22×C4×C61#4111yescharacteristic
125C2×C22⋊D61—44no—
126C2×C22⋊D61—41no—
127C2×C22⋊D61—41no—
128C2×C22⋊D61—41no—
129C2×C22⋊D61—41no—
130C2×C22⋊D61—41no—
131C2×C22⋊D61—41no—
132C2×C22⋊D61—82no—
133C2×C22⋊D61—82no—
134C22×Dic61—21no—
135C22×Dic61—22no—
136C2×D244—21no—
137C2×D244—21no—
138C2×D244—42no—
139C22⋊C4×C61—48no—
140C22⋊C4×C61—41no—
141C22⋊C4×C61—41no—
142C22⋊C4×C61—41no—
143C22⋊C4×C61—41no—
144C22⋊Dic61—44no—
145C22⋊Dic61—41no—
146C22⋊Dic61—41no—
147#11758201—48no—
148#11758198—41no—
149#11758198—41no—
150#11758198—82no—
151C4⋊Dic61—21no—
152C4⋊Dic61—21no—
153C4⋊Dic61—42no—
154C2×#92418—22no—
155C2×#92418—42no—
156C2×#92418—42no—
157C2×#92418—41no—
158C23×C61—22no—
159C23×C61—22no—
160C23×C61—22no—
161C23×C61—22no—
162C23×C61—22no—
163C23×C61—44no—
164C23×C61—42no—
165C23×C61—44no—
166C23×C61—42no—
167C23×C61—41no—
168C23×C61—41no—
169C23×C61—41no—
170C23×C61—42no—
171C23×C61—42no—
172C23×C61—41no—
173C23×C61—82no—
174C23×C61—82no—
175C23×C61—81no—
176C23×C61—81no—
177C23×C61#12011yescharacteristic
178C23×C61#12011yescharacteristic
179C23×C61#12111yescharacteristic
180C23×C61C2×SD1611yescharacteristic
181C23×C61#13511yescharacteristic
182C23×C61AGL(1,ℤ/8)11yescharacteristic
183C23×C61#13711yescharacteristic
184C22×D61—41no—
185C22×D61—41no—
186C22×D61—41no—
187C22×D61—42no—
188C22×D61—42no—
189Q8×C61—24no—
190Q8×C61—24no—
191Q8×C61—42no—
192Q8×C61—44no—
193Q8×C61—44no—
194Q8×C61—42no—
195C2×C4×C61—22no—
196C2×C4×C61—22no—
197C2×C4×C61—24no—
198C2×C4×C61—41no—
199C2×C4×C61—41no—
200C2×C4×C61—41no—
201C2×C4×C61—41no—
202C2×C4×C61—42no—
203C2×C4×C61—44no—
204C2×C4×C61—42no—
205C2×C4×C61—44no—
206C2×C4×C61—42no—
207C2×C4×C61—42no—
208C2×C4×C61—42no—
209C2×C4×C61—41no—
210C2×C4×C61—42no—
211C2×C4×C61—42no—
212C2×C4×C61—82no—
213C2×C4×C61—82no—
214C2×C4×C61—81no—
215C2×C4×C61#12011yescharacteristic
216C2×C4×C61#12111yescharacteristic
217C2×C4×C61#12111yescharacteristic
218C2×C4×C61#12711yescharacteristic
219C22⋊D61—88no—
220C22⋊D61—81no—
221C22⋊D61—81no—
222C22⋊D61—81no—
223C22⋊D61—81no—
224C2×Dic61—41no—
225C2×Dic61—42no—
226C2×Dic61—41no—
227C2×Dic61—41no—
228C2×Dic61—42no—
229C2×Dic61—42no—
230D244—42no—
231D244—81no—
232#92418—42no—
233#92418—81no—
234#9916—611nomaximal, Sylow
235C22×C61—22no—
236C22×C61—22no—
237C22×C61—22no—
238C22×C61—22no—
239C22×C61—21no—
240C22×C61—21no—
241C22×C61—21no—
242C22×C61—21no—
243C22×C61—44no—
244C22×C61—44no—
245C22×C61—42no—
246C22×C61—82no—
247C22×C61—82no—
248C22×C61—82no—
249C22×C61—82no—
250C22×C61—81no—
251C22×C61—81no—
252C22×C61#39211yescharacteristic
253C22×C61#44811yescharacteristic
254C22×C61#44911yescharacteristic
255C22×C61#45011yescharacteristic
256C22×C61#45211yescharacteristic
257C22×C61#45512yes—
258C22×C61B(4,𝔽2)12yes—
259C22×C61#46511yescharacteristic
260C22×C61#46811yescharacteristic
261C2×D61—82no—
262C2×D61—82no—
263C2×D61—81no—
264C2×D61—81no—
265C2×D61—81no—
266C4×C61—21no—
267C4×C61—21no—
268C4×C61—21no—
269C4×C61—21no—
270C4×C61—44no—
271C4×C61—82no—
272C4×C61—81no—
273Dic61—82no—
274Dic61—81no—
275C2×#448—611no—
276#2617—611no—
277C2×#393D6111yescharacteristic, complemented
278#1787—611no—
279C2×#327—611no—
280C2×#323—611no—
281#1294—611no—
282C2×C61—21no—
283C2×C61—21no—
284C2×C61—21no—
285C2×C61—21no—
286C2×C61—82no—
287C2×C61#160112yes—
288C2×C61#160612yes—
289C2×C61#200011yescharacteristic
290C2×C61#200411yescharacteristic
291C2×C61#202111yescharacteristic
292D61—161no—
293C23×Q8C2×D6111yescharacteristic
294C22×SD16—1221no—
295C2×#121—611no—
296C2×#120—1221no—
297C22×C22⋊C4—21no—
298C22×C22⋊C4C2×D6111yescharacteristic
299#448—1222no—
300#448—1222no—
301C2×#103—1221no—
302C2×#103—1222no—
303C2×C22⋊C8—611no—
304C2×C22⋊C8—611no—
305#397—1221no—
306#394—1222no—
307#393—22no—
308#393—22no—
309#392—1221no—
310#386—1221no—
311#380—611no—
312C2×#95—21no—
313C2×#95C2×D6111yescharacteristic
314#327—1222no—
315#323—1222no—
316C61#991611yescharacteristic, complemented, Sylow
317C25C22⋊D6111yescharacteristic
318C22×Q8—21no—
319C22×Q8—21no—
320C22×Q8—21no—
321C22×Q8—21no—
322C22×Q8—21no—
323C22×Q8—44no—
324C22×Q8C22⋊D6112yescomplemented
325C22×D4—611no—
326C22×D4—1222no—
327C22×D4—1221no—
328C23×C4—21no—
329C23×C4—21no—
330C23×C4C22⋊D6111yescharacteristic
331C23×C4C22×D6111yescharacteristic
332C2×SD16—1224no—
333C2×SD16—2444no—
334C22×C8—1221no—
335C22×C8—1221no—
336#121—1221no—
337#121—1221no—
338#121—1221no—
339#121—1221no—
340#120—2442no—
341#120—2442no—
342C2×C4⋊C4—611no—
343C2×C4⋊C4—612no—
344C2×C22⋊C4—24no—
345C2×C22⋊C4—21no—
346C2×C22⋊C4—21no—
347C2×C22⋊C4—21no—
348C2×C22⋊C4—21no—
349C2×C22⋊C4—21no—
350C2×C22⋊C4—21no—
351C2×C22⋊C4—44no—
352C2×C22⋊C4—1222no—
353C2×C22⋊C4—1222no—
354C2×C22⋊C4—1221no—
355C2×C22⋊C4—1221no—
356#103—1224no—
357#103—2444no—
358C22⋊C8—612no—
359C22⋊C8—1221no—
360C22⋊C8—1221no—
361C22⋊C8—1221no—
362#95—24no—
363#95—21no—
364#95—1222no—
365#95—1221no—
366#95C22⋊D6112yescomplemented
367C24—21no—
368C24—21no—
369C24—21no—
370C24—21no—
371C24—21no—
372C24—21no—
373C24—42no—
374C24—42no—
375C24—1221no—
376C24#1175820112yescomplemented
377C24C2×C22⋊D6111yescharacteristic
378C2×Q8—24no—
379C2×Q8—24no—
380C2×Q8—22no—
381C2×Q8—22no—
382C2×Q8—44no—
383C2×Q8—42no—
384C2×Q8—42no—
385C2×Q8—44no—
386C2×Q8—44no—
387C2×Q8—44no—
388C2×Q8—41no—
389C2×Q8—41no—
390C2×D4—1221no—
391C2×D4—1221no—
392C2×D4—4882no—
393C2×D4—4882no—
394C2×D4—2442no—
395C2×D4—2444no—
396C2×D4—2441no—
397C2×D4—2441no—
398C2×D4—2441no—
399C2×D4—2441no—
400C2×D4—2441no—
401C2×D4—2441no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—41no—
416C22×C4—41no—
417C22×C4—1222no—
418C22×C4—1221no—
419C22×C4D4×D6111yescharacteristic
420C22×C4D4×D6111yescharacteristic
421C22×C4D4×D6111yescharacteristic
422C22×C4#1175822411yescharacteristic
423C22×C4C2×C22⋊D6111yescharacteristic
424C22×C4C2×C22⋊D6111yescharacteristic
425SD16—2448no—
426C2×C8—1222no—
427C2×C8—2442no—
428C2×C8—2441no—
429C2×C8—2442no—
430C4⋊C4—1221no—
431C4⋊C4—1221no—
432C4⋊C4—2442no—
433C22⋊C4—48no—
434C22⋊C4—41no—
435C22⋊C4—41no—
436C22⋊C4—41no—
437C22⋊C4—41no—
438C22⋊C4—4882no—
439C22⋊C4—2444no—
440C22⋊C4—2441no—
441C22⋊C4—2441no—
442C22⋊C4—2441no—
443C22⋊C4—2441no—
444C23—22no—
445C23—22no—
446C23—22no—
447C23—22no—
448C23—22no—
449C23—44no—
450C23—42no—
451C23—44no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23—41no—
456C23—41no—
457C23—41no—
458C23—41no—
459C23—82no—
460C23—81no—
461C23—82no—
462C23—81no—
463C23—2442no—
464C23—2442no—
465C23—2441no—
466C23—2441no—
467C23—2441no—
468C23#1451974511yescharacteristic
469C23#1451974611yescharacteristic
470C23#1451974911yescharacteristic
471C23C2×#1175820111yescharacteristic
472C23#1451979711yescharacteristic
473C23#1451980211yescharacteristic
474C23#1451980811yescharacteristic
475Q8—24no—
476Q8—24no—
477Q8—44no—
478Q8—42no—
479Q8—44no—
480Q8—42no—
481D4—4881no—
482D4—4888no—
483D4—4881no—
484D4—4881no—
485D4—4881no—
486D4—4881no—
487D4—2442no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—24no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—42no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—41no—
505C2×C4—82no—
506C2×C4—82no—
507C2×C4—81no—
508C2×C4—2442no—
509C2×C4—2442no—
510C2×C4—2442no—
511C2×C4—2441no—
512C2×C4—2441no—
513C2×C4—2441no—
514C2×C4#1451974511yescharacteristic
515C2×C4#1451974811yescharacteristic
516C2×C4#1451974911yescharacteristic
517C2×C4#1451977311yescharacteristic
518C8—4881no—
519C8—2442no—
520C22—21no—
521C22—21no—
522C22—22no—
523C22—22no—
524C22—22no—
525C22—22no—
526C22—21no—
527C22—21no—
528C22—44no—
529C22—44no—
530C22—42no—
531C22—82no—
532C22—82no—
533C22—82no—
534C22—81no—
535C22—82no—
536C22—81no—
537C22—4881no—
538C22—4882no—
539C22—4881no—
540C22—4882no—
541C22—4881no—
542C22#1683967011yescharacteristic
543C22#1683990111yescharacteristic
544C22#1683990711yescharacteristic
545C22#1683991211yescharacteristic
546C22#1683992211yescharacteristic
547C22#1683993912yes—
548C22#1683994812yes—
549C22#1683999311yescharacteristic
550C22#1684001011yescharacteristic
551C4—21no—
552C4—21no—
553C4—21no—
554C4—21no—
555C4—44no—
556C4—82no—
557C4—81no—
558C4—4882no—
559C4—4881no—
560C2—21no—
561C2—21no—
562C2—21no—
563C2—21no—
564C2—82no—
565C2—9761no—
566C2#2144477012yes—
567C2#2144479712yes—
568C2#2144682011yescharacteristic
569C2#2144684811yescharacteristic
570C2#2144694211yescharacteristic
subgroupG/H
centerC23#14519745
commutator, cosocleC23×C4×C61C23
FrattiniC23×C4C22×D61
FittingC2×C61×#393C2
socleC23×C61#120
prime radical#21451553C2
hypercenterC2×#393D61
hypocenterC61#9916
LCSG ▷ C23×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#393

Complements

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

NH ≅ G/Ntype
1. C2×#16839901C2semidirect
3. #21445754C2semidirect
4. C2×#16839495C2semidirect
5. C2×#16839486C2semidirect
6. #21443698C2semidirect
7. C2×C61×#393C2semidirect
55. C22⋊#92418D4semidirect
65. C22×Q8×C61D4semidirect
80. #95×C61D4semidirect
89. C24×C61SD16semidirect
277. C2×#393D61semidirect
316. C61#9916semidirect
324. C22×Q8C22⋊D61semidirect
366. #95C22⋊D61semidirect
376. C24#11758201semidirect

Maximal subgroups

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

subgroupindex
C2×#168399012
#214515532
#214457542
C2×#168394952
C2×#168394862
#214436982
C2×C61×#3932
#991661