Finite Group Databasebeta

← #29730135

Subgroups — #29730135

Order 15616.

958 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840052C211yesmaximal, characteristic, complemented
2C2×C8×D244C211yesmaximal, characteristic, complemented
3#21450988C211yesmaximal, characteristic, complemented
4#21450732C211yesmaximal, characteristic
5#21447257C212yesmaximal, complemented
6#21447006C212yesmaximal, complemented
7#21446780C211yesmaximal, characteristic, complemented
8#21446582C211yesmaximal, characteristic, complemented
9#21446244C212yesmaximal, complemented
10#21445843C212yesmaximal, complemented
11C4×#102×C61C211yesmaximal, characteristic, complemented
12C22×C8×D61C2212yes—
13C2×#14519777C2211yescharacteristic
14C2×#14519751—21no—
15C2×#14519748C2212yes—
16C2×C4×D244C2211yescharacteristic
17#16840052—24no—
18C8×D244—24no—
19C2×#14519708C2212yes—
20C2×#14519708C2212yes—
21C2×C4⋊#92418C2211yescharacteristic
22C2×#14519698C2212yes—
23C2×#14519689C2212yes—
24#16839678—22no—
25#16839675—22no—
26#16839664—21no—
27#16839632C2211yescharacteristic
28#16839628—21no—
29#16839622—21no—
30#16839619C2212yes—
31#16839619C2212yes—
32#16839619C2212yes—
33#16839525C2212yes—
34#16839525C2212yes—
35#16839524C2212yes—
36C2×C4×D4×C61C2211yescharacteristic
37C2×#102×C61C2211yescharacteristic
38C2×#102×C61C2211yescharacteristic
39C2×C4×C8×C61C2211yescharacteristic
40C4×C4⋊C4×C61C2211yescharacteristic
41C61×#340C2211yescharacteristic
42C61×#340C2211yescharacteristic
43C22×Dic122—21no—
44C22×D244D411yescharacteristic
45C22×C22⋊D61—22no—
46C22×C4×D61C2312yes—
47C22×#92418C2312yes—
48C2×C8×D61—24no—
49C2×C8×D61—24no—
50C2×C8×D61—42no—
51#14519777—22no—
52#14519777—22no—
53#14519751—44no—
54#14519748—24no—
55#14519748—42no—
56C4×D244—22no—
57C4×D244—22no—
58C2×C4⋊Dic61—21no—
59C2×C4⋊Dic61—21no—
60C2×C4⋊Dic61—21no—
61C2×C4⋊Dic61D411yescharacteristic
62C2×C4⋊Dic61C2312yes—
63C2×C4⋊Dic61C2312yes—
64C2×C22⋊Dic61—22no—
65C2×#11758198—22no—
66C2×#11758198—22no—
67C2×#11758198—22no—
68C2×#11758198D412yes—
69#14519708—24no—
70#14519708—42no—
71C4⋊#92418D414yescomplemented
72#14519698—24no—
73#14519689—24no—
74#14519684—22no—
75#14519684—22no—
76#14519684—22no—
77C22×D4×C61C2311yescharacteristic
78C23×C4×C61—21no—
79C22×C8×C61C2311yescharacteristic
80C22×C8×C61C2311yescharacteristic
81C4×D4×C61—22no—
82C4×D4×C61—22no—
83C2×C4⋊C4×C61C2311yescharacteristic
84C2×C4⋊C4×C61C2311yescharacteristic
85C2×C4⋊C4×C61C2311yescharacteristic
86C2×C22⋊C4×C61—21no—
87C2×C42×C61—21no—
88C2×C42×C61C2311yescharacteristic
89#102×C61—22no—
90#102×C61—22no—
91C4×C8×C61—22no—
92#95×C61—21no—
93C24×C61—21no—
94C23×D61C2×D412yes—
95C2×D4×C61—21no—
96C2×D4×C61—21no—
97C2×D4×C61—21no—
98C2×D4×C61—21no—
99C2×D4×C61—21no—
100C2×D4×C61#4112yes—
101C22×C4×C61—21no—
102C22×C4×C61—21no—
103C22×C4×C61—21no—
104C22×C4×C61—42no—
105C22×C4×C61—42no—
106C22×C4×C61—42no—
107C22×C4×C61C2×D411yescharacteristic
108C22×C4×C61C2×D411yescharacteristic
109C22×C4×C61C2411yescharacteristic
110C2×C22⋊D61—44no—
111C2×C22⋊D61—44no—
112C2×C22⋊D61—82no—
113C22×Dic61—22no—
114C22×Dic61—22no—
115C22×Dic61C2×D412yes—
116C2×D244—22no—
117C2×D244—22no—
118C2×D244SD1614yes—
119C2×C4×D61—22no—
120C2×C4×D61—22no—
121C2×C4×D61—22no—
122C2×C4×D61—22no—
123C2×C4×D61—22no—
124C2×C4×D61—22no—
125C2×Dic122—42no—
126C2×Dic122—81no—
127C2×Dic122—81no—
128C2×C8×C61—21no—
129C2×C8×C61—21no—
130C2×C8×C61—21no—
131C2×C8×C61—21no—
132C2×C8×C61—21no—
133C2×C8×C61#4112yes—
134C4⋊C4×C61—21no—
135C4⋊C4×C61—21no—
136C4⋊C4×C61—21no—
137C4⋊C4×C61—21no—
138C4⋊C4×C61—21no—
139C4⋊C4×C61—21no—
140C22⋊C4×C61—42no—
141C42×C61—42no—
142C42×C61C2×D412yes—
143C42×C61#4112yes—
144C22⋊Dic61—44no—
145#11758198—24no—
146#11758198—44no—
147#11758198—82no—
148#11758198—82no—
149C4⋊Dic61—22no—
150C4⋊Dic61—22no—
151C4⋊Dic61—22no—
152C4⋊Dic61—22no—
153C4⋊Dic61—22no—
154C4⋊Dic61—42no—
155C4⋊Dic61—42no—
156C4⋊Dic61—81no—
157C2×#92418—22no—
158C2×#92418—22no—
159C2×#92418C2×D414yes—
160C8×D61—48no—
161C23×C61—22no—
162C23×C61—41no—
163C23×C61—41no—
164C23×C61—41no—
165C23×C61—41no—
166C23×C61—41no—
167C23×C61—41no—
168C23×C61C22×D411yescharacteristic
169C22×D61—22no—
170C22×D61—22no—
171C22×D61—22no—
172C22×D61—22no—
173C22×D61—22no—
174C22×D61—22no—
175C22×D61C2×SD1614yes—
176D4×C61—22no—
177D4×C61—22no—
178D4×C61—41no—
179D4×C61—41no—
180C2×C4×C61—21no—
181C2×C4×C61—21no—
182C2×C4×C61—41no—
183C2×C4×C61—41no—
184C2×C4×C61—41no—
185C2×C4×C61—41no—
186C2×C4×C61—41no—
187C2×C4×C61—41no—
188C2×C4×C61—41no—
189C2×C4×C61—41no—
190C2×C4×C61—41no—
191C2×C4×C61—84no—
192C2×C4×C61#12112yes—
193C2×C4×C61#12112yes—
194C2×C4×C61C2×SD1612yes—
195C2×C4×C61#13512yes—
196C2×C4×C61C22×D411yescharacteristic
197C2×C4×C61C2×#4111yescharacteristic
198C2×C4×C61C2×#4111yescharacteristic
199C2×C4×C61C2×#4111yescharacteristic
200C2×C4×C61H(5,𝔽2)11yescharacteristic
201C2×C4×C612−1+411yescharacteristic
202C22⋊D61—88no—
203C2×Dic61—22no—
204C2×Dic61—22no—
205C2×Dic61—22no—
206C2×Dic61—42no—
207C2×Dic61—42no—
208C2×Dic61—42no—
209C2×Dic61—42no—
210C2×Dic61—42no—
211C2×Dic61—42no—
212D244—28no—
213C4×D61—42no—
214C4×D61—42no—
215C4×D61—42no—
216C4×D61—42no—
217Dic122—84no—
218C8×C61—22no—
219C8×C61—41no—
220#92418—24no—
221#25339—611nomaximal, Sylow
222C22×C61—42no—
223C22×C61—42no—
224C22×C61—42no—
225C22×C61—81no—
226C22×C61—81no—
227C22×C61—81no—
228C22×C61—81no—
229C22×C61C2×#12111yescharacteristic
230C22×C61#54611yescharacteristic
231C22×C61#54711yescharacteristic
232C22×C61C22×SD1611yescharacteristic
233C22×C61C2×#13511yescharacteristic
234C22×C61#57611yescharacteristic
235C22×C61#57711yescharacteristic
236C2×D61—24no—
237C2×D61—24no—
238C2×D61—24no—
239C2×D61—42no—
240C2×D61—42no—
241C2×D61—42no—
242C2×D61—42no—
243C4×C61—22no—
244C4×C61—81no—
245C4×C61—81no—
246C4×C61—81no—
247C4×C61#46514yes—
248C4×C61#52911yescharacteristic
249C4×C61#54211yescharacteristic
250C4×C61#54611yescharacteristic
251C4×C61#54811yescharacteristic
252Dic61—42no—
253Dic61—82no—
254Dic61—82no—
255C2×#476—611no—
256C2×C8×D4—611no—
257#2563—611no—
258#2537—611no—
259#2077—612no—
260#2034—612no—
261#1990—611no—
262#1948—611no—
263#1893—612no—
264#1807—612no—
265C4×#102D6111yescharacteristic, complemented
266C2×C61—82no—
267C2×C61#242711yescharacteristic
268C2×C61C2×#46511yescharacteristic
269C2×C61#306211yescharacteristic
270C2×C61#329511yescharacteristic
271C2×C61#330311yescharacteristic
272C2×C61#333511yescharacteristic
273C2×C61#334111yescharacteristic
274D61—44no—
275C23×C8—612no—
276C2×C4⋊Q8—611no—
277C2×#121—612no—
278C2×#121—1221no—
279C2×C4×D4—611no—
280C2×C4×D4C2×D6111yescharacteristic
281#476—1224no—
282C8×D4—1224no—
283C2×#106—612no—
284C2×#106—612no—
285C2×C4⋊C8—611no—
286C2×#102—612no—
287C2×#102C2×D6111yescharacteristic
288C2×#102C2×D6111yescharacteristic
289C2×C22⋊C8—612no—
290C2×C4×C8C2×D6111yescharacteristic
291#394—1222no—
292#393—1222no—
293#390—1221no—
294#384—611no—
295#382—1221no—
296#380—612no—
297#380—612no—
298#380—612no—
299#380—1221no—
300C4×C4⋊C4C2×D6111yescharacteristic
301#340—612no—
302#340—612no—
303#340—612no—
304#340C2×D6111yescharacteristic
305#340C2×D6111yescharacteristic
306C61#2533911yescharacteristic, complemented, Sylow
307C22×Q8—1221no—
308C22×D4—611no—
309C22×D4—1222no—
310C22×D4C22×D6111yescharacteristic
311C23×C4—21no—
312C23×C4—612no—
313C22×C8—612no—
314C22×C8—1224no—
315C22×C8—1224no—
316C22×C8—2442no—
317C22×C8C22×D6111yescharacteristic
318C22×C8C22×D6111yescharacteristic
319C4⋊Q8—1222no—
320C4⋊Q8—1222no—
321#121—1224no—
322#121—2442no—
323#121—2444no—
324C4×D4—22no—
325C4×D4—22no—
326C4×D4—1222no—
327C4×D4—1222no—
328C2×C4⋊C4—611no—
329C2×C4⋊C4—612no—
330C2×C4⋊C4—612no—
331C2×C4⋊C4—1221no—
332C2×C4⋊C4—1221no—
333C2×C4⋊C4—1221no—
334C2×C4⋊C4C22×D6111yescharacteristic
335C2×C4⋊C4C22×D6111yescharacteristic
336C2×C4⋊C4C22×D6111yescharacteristic
337C2×C22⋊C4—21no—
338C2×C22⋊C4—612no—
339C2×C22⋊C4—1222no—
340C2×C22⋊C4—1222no—
341C2×C22⋊C4—1222no—
342C2×C22⋊C4—1222no—
343C2×C42—21no—
344C2×C42C22×D6111yescharacteristic
345#106—1224no—
346#106—2442no—
347C4⋊C8—614no—
348#102—22no—
349#102—22no—
350#102—1224no—
351C22⋊C8—1224no—
352C4×C8—22no—
353#95—21no—
354#95—1222no—
355#95—1222no—
356#95—1222no—
357C24—21no—
358C24—612no—
359C2×Q8—4881no—
360C2×Q8—4881no—
361C2×Q8—2442no—
362C2×D4—21no—
363C2×D4—21no—
364C2×D4—21no—
365C2×D4—21no—
366C2×D4—21no—
367C2×D4—614no—
368C2×D4—1222no—
369C2×D4—1222no—
370C2×D4—4882no—
371C2×D4—2444no—
372C2×D4—2444no—
373C2×D4#1175822212yes—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—612no—
381C22×C4—1222no—
382C22×C4—1222no—
383C22×C4—1222no—
384C22×C4—1222no—
385C22×C4—1222no—
386C22×C4—1222no—
387C22×C4—1222no—
388C22×C4—1222no—
389C22×C4D4×D6111yescharacteristic
390C22×C4D4×D6111yescharacteristic
391C22×C4C23×D6111yescharacteristic
392C2×C8—21no—
393C2×C8—21no—
394C2×C8—21no—
395C2×C8—21no—
396C2×C8—21no—
397C2×C8—614no—
398C2×C8—1222no—
399C2×C8—1222no—
400C2×C8—2448no—
401C2×C8#1175822212yes—
402C4⋊C4—21no—
403C4⋊C4—21no—
404C4⋊C4—21no—
405C4⋊C4—21no—
406C4⋊C4—21no—
407C4⋊C4—21no—
408C4⋊C4—1222no—
409C4⋊C4—1222no—
410C4⋊C4—1222no—
411C4⋊C4—1222no—
412C4⋊C4—1222no—
413C4⋊C4—4881no—
414C4⋊C4—2442no—
415C4⋊C4—2442no—
416C22⋊C4—42no—
417C22⋊C4—1224no—
418C22⋊C4—4882no—
419C22⋊C4—4882no—
420C22⋊C4—2444no—
421C22⋊C4—2444no—
422C42—42no—
423C42#1175822212yes—
424C42D4×D6112yes—
425C23—22no—
426C23—41no—
427C23—41no—
428C23—41no—
429C23—41no—
430C23—41no—
431C23—41no—
432C23—614no—
433C23—1222no—
434C23—1222no—
435C23—1222no—
436C23—1222no—
437C23—1222no—
438C23—1222no—
439C23C2×D4×D6111yescharacteristic
440Q8—4884no—
441D4—22no—
442D4—22no—
443D4—41no—
444D4—41no—
445D4—1228no—
446D4—4888no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—84no—
459C2×C4—1222no—
460C2×C4—1222no—
461C2×C4—1222no—
462C2×C4—2442no—
463C2×C4—2442no—
464C2×C4—2442no—
465C2×C4—2442no—
466C2×C4—2442no—
467C2×C4—2442no—
468C2×C4—2442no—
469C2×C4—2442no—
470C2×C4—2442no—
471C2×C4—2442no—
472C2×C4#1451975012yes—
473C2×C4#1451975012yes—
474C2×C4SD16×D6112yes—
475C2×C4#1451979612yes—
476C2×C4C2×D4×D6111yescharacteristic
477C2×C4C2×#1175822211yescharacteristic
478C2×C4C2×#1175822211yescharacteristic
479C2×C4C2×#1175822211yescharacteristic
480C2×C4#1451982311yescharacteristic
481C2×C4#1451982411yescharacteristic
482C8—22no—
483C8—41no—
484C8—1224no—
485C22—42no—
486C22—42no—
487C22—42no—
488C22—81no—
489C22—81no—
490C22—81no—
491C22—81no—
492C22—1224no—
493C22—1224no—
494C22—1224no—
495C22—2442no—
496C22—2442no—
497C22—2442no—
498C22—2442no—
499C22C2×#1451975011yescharacteristic
500C22#1684040811yescharacteristic
501C22#1684041711yescharacteristic
502C22C2×SD16×D6111yescharacteristic
503C22C2×#1451979611yescharacteristic
504C22#1684056411yescharacteristic
505C22#1684056911yescharacteristic
506C4—22no—
507C4—81no—
508C4—81no—
509C4—81no—
510C4—4882no—
511C4—4882no—
512C4—2442no—
513C4#1683999014yes—
514C4#1684029611yescharacteristic
515C4#1684038311yescharacteristic
516C4#1684040811yescharacteristic
517C4#1684042111yescharacteristic
518C2—82no—
519C2—2444no—
520C2#2144939611yescharacteristic
521C2C2×#1683999011yescharacteristic
522C2#2145550211yescharacteristic
523C2#2145764511yescharacteristic
524C2#2145774911yescharacteristic
525C2#2145817811yescharacteristic
526C2#2145824311yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC4×#102×C61C2
socleC23×C61C22×D4
prime radical#21450732C2
hypercenterC4×#102D61
hypocenterC61#25339
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C4×#102

Complements

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

NH ≅ G/Ntype
1. C2×#16840052C2semidirect
2. C2×C8×D244C2semidirect
3. #21450988C2semidirect
5. #21447257C2semidirect
6. #21447006C2semidirect
7. #21446780C2semidirect
8. #21446582C2semidirect
9. #21446244C2semidirect
10. #21445843C2semidirect
11. C4×#102×C61C2semidirect
71. C4⋊#92418D4semidirect
265. C4×#102D61semidirect
306. C61#25339semidirect

Maximal subgroups

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

subgroupindex
C2×#168400522
C2×C8×D2442
#214509882
#214507322
#214472572
#214470062
#214467802
#214465822
#214462442
#214458432
C4×#102×C612
#2533961