Finite Group Databasebeta

← #29714100

Subgroups — #29714100

Order 15616.

1336 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840415C211yesmaximal, characteristic
2C2×#16839860C211yesmaximal, characteristic, complemented
3#21450724C211yesmaximal, characteristic, complemented
4#21447003C212yesmaximal, complemented
5#21446821C212yesmaximal, complemented
6#21446793C211yesmaximal, characteristic, complemented
7#21446616C211yesmaximal, characteristic, complemented
8#21446123C212yesmaximal, complemented
9#21446005C212yesmaximal, complemented
10#21445830C211yesmaximal, characteristic, complemented
11C2×C61×#477C211yesmaximal, characteristic, complemented
12C22×#11758226—21no—
13C22×#11758201C2211yescharacteristic
14#16840415—24no—
15#16840415—24no—
16C2×#14519762—22no—
17C2×#14519753—21no—
18C2×#14519753C2212yes—
19C2×#14519752C2212yes—
20C2×C4×D244C2211yescharacteristic
21C2×C4×D244C2211yescharacteristic
22C2×C4⋊C4×D61C2212yes—
23#16839860—24no—
24C2×#14519710C2211yescharacteristic
25C2×#14519701C2211yescharacteristic
26C2×#14519699C2212yes—
27C2×#14519697C2211yescharacteristic
28C2×#14519689C2212yes—
29C2×#14519689C2212yes—
30C2×#14519688C2211yescharacteristic
31#16839692—21no—
32#16839679—21no—
33#16839653—22no—
34#16839628C2211yescharacteristic
35#16839619C2212yes—
36#16839525C2212yes—
37#16839524C2212yes—
38#16839524C2212yes—
39C2×C4⋊Q8×C61C2211yescharacteristic
40C2×C4×Q8×C61C2211yescharacteristic
41C61×#477—24no—
42C2×#107×C61C2212yes—
43C2×C4⋊C8×C61C2211yescharacteristic
44C2×#103×C61C2212yes—
45C2×#11758226—28no—
46C2×#11758226—48no—
47C22×D244—22no—
48C22×D244D411yescharacteristic
49C22×D244C2311yescharacteristic
50C22×C4×D61—22no—
51C22×C4×D61—21no—
52C22×C4×D61C2312yes—
53C2×#11758201—22no—
54C2×#11758201—22no—
55C2×#11758201—22no—
56C22×#92418C2311yescharacteristic
57C22×#92418C2311yescharacteristic
58#14519762—216no—
59#14519753—28no—
60#14519753—24no—
61#14519753—42no—
62#14519752—24no—
63#14519752—24no—
64C4×D244—22no—
65C4×D244—22no—
66C4×D244—22no—
67C4×D244D414yes—
68C2×C4⋊Dic61D411yescharacteristic
69C2×C4⋊Dic61C2312yes—
70C2×C4⋊Dic61C2311yescharacteristic
71C2×#11758196—22no—
72C2×#11758196—22no—
73C4⋊C4×D61—24no—
74C4⋊C4×D61—24no—
75C2×#11758198—21no—
76C2×#11758198—22no—
77C2×#11758198—22no—
78C2×#11758198—22no—
79C2×#11758198D412yes—
80#14519710—22no—
81#14519701—22no—
82#14519699—42no—
83#14519697—22no—
84#14519689—24no—
85#14519689—42no—
86#14519688—22no—
87#14519684—21no—
88#14519684—21no—
89#14519684—22no—
90C22×Q8×C61—21no—
91C22×Q8×C61C2311yescharacteristic
92C22×C8×C61C2312yes—
93C4⋊Q8×C61—22no—
94C4⋊Q8×C61D414yes—
95C4×Q8×C61—22no—
96C4×Q8×C61—22no—
97C2×C4⋊C4×C61—21no—
98C2×C4⋊C4×C61—21no—
99C2×C4⋊C4×C61C2311yescharacteristic
100C2×C4⋊C4×C61C2312yes—
101C2×C42×C61—21no—
102C2×C42×C61C2311yescharacteristic
103#107×C61—24no—
104C4⋊C8×C61—22no—
105#103×C61—24no—
106C23×D61—21no—
107C23×D61C2×D412yes—
108C2×Q8×C61—24no—
109C2×Q8×C61—21no—
110C2×Q8×C61—21no—
111C2×Q8×C61—21no—
112C2×Q8×C61—21no—
113C2×Q8×C61—21no—
114C2×Q8×C61—44no—
115C2×Q8×C61#4112yescomplemented
116C22×C4×C61—22no—
117C22×C4×C61—21no—
118C22×C4×C61—21no—
119C22×C4×C61C2×D411yescharacteristic
120C22×C4×C61C2×D411yescharacteristic
121C22×C4×C61C2411yescharacteristic
122C22×Dic61—21no—
123C22×Dic61—22no—
124C22×Dic61C2×D412yes—
125#11758226—432no—
126C2×D244—24no—
127C2×D244—22no—
128C2×D244—21no—
129C2×D244—21no—
130C2×D244—21no—
131C2×D244—21no—
132C2×D244—28no—
133C2×D244—21no—
134C2×D244—44no—
135C2×D244—44no—
136C2×D244C2×D412yes—
137C2×C4×D61—28no—
138C2×C4×D61—24no—
139C2×C4×D61—22no—
140C2×C4×D61—22no—
141C2×C4×D61—22no—
142C2×C4×D61—22no—
143C2×C4×D61—22no—
144C2×C4×D61—22no—
145C2×C4×D61—44no—
146C2×C4×D61—44no—
147C2×C4×D61—44no—
148C2×C8×C61—22no—
149C2×C8×C61—22no—
150C2×C8×C61—22no—
151C4⋊C4×C61—24no—
152C4⋊C4×C61—22no—
153C4⋊C4×C61—21no—
154C4⋊C4×C61—21no—
155C4⋊C4×C61—42no—
156C4⋊C4×C61C2×D414yes—
157C42×C61—42no—
158C42×C61C2×D412yes—
159C42×C61C2×D412yes—
160#11758201—28no—
161#11758198—28no—
162#11758198—24no—
163#11758198—24no—
164#11758198—44no—
165#11758198—44no—
166C4⋊Dic61—22no—
167C4⋊Dic61—21no—
168C4⋊Dic61—22no—
169C4⋊Dic61—22no—
170C4⋊Dic61C2×D412yes—
171#11758196—28no—
172#11758196—44no—
173C2×#92418—21no—
174C2×#92418—21no—
175C2×#92418—21no—
176C2×#92418—21no—
177C2×#92418—21no—
178C2×#92418#4112yescomplemented
179C23×C61C22×D411yescharacteristic
180C22×D61—22no—
181C22×D61—22no—
182C22×D61—22no—
183C22×D61—22no—
184C22×D61—22no—
185C22×D61—22no—
186C22×D61—22no—
187C22×D61—41no—
188C22×D61—41no—
189C22×D61—42no—
190C22×D61—42no—
191C22×D61AGL(1,ℤ/8)14yes—
192Q8×C61—22no—
193Q8×C61—22no—
194Q8×C61—48no—
195Q8×C61—41no—
196Q8×C61—41no—
197C2×C4×C61—22no—
198C2×C4×C61—22no—
199C2×C4×C61—24no—
200C2×C4×C61—22no—
201C2×C4×C61—41no—
202C2×C4×C61—41no—
203C2×C4×C61—42no—
204C2×C4×C61—42no—
205C2×C4×C61—41no—
206C2×C4×C61—42no—
207C2×C4×C61#12012yes—
208C2×C4×C61#12012yes—
209C2×C4×C61C22×D411yescharacteristic
210C2×C4×C61C22×D411yescharacteristic
211C2×C4×C61C2×#4111yescharacteristic
212C2×C4×C61C2×#4111yescharacteristic
213C2×C4×C61H(5,𝔽2)11yescharacteristic
214C2×C4×C612−1+411yescharacteristic
215C2×Dic61—22no—
216C2×Dic61—22no—
217C2×Dic61—22no—
218C2×Dic61—22no—
219C2×Dic61—42no—
220C2×Dic61—42no—
221C2×Dic61—42no—
222C2×Dic61—42no—
223D244—28no—
224D244—22no—
225D244—22no—
226D244—416no—
227D244—41no—
228D244—41no—
229C4×D61—48no—
230C4×D61—416no—
231C4×D61—42no—
232C4×D61—42no—
233C4×D61—42no—
234C4×D61—42no—
235C8×C61—42no—
236#92418—22no—
237#92418—41no—
238#24132—611nomaximal, Sylow
239C22×C61C2×#12011yescharacteristic
240C22×C61#54611yescharacteristic
241C22×C61#54711yescharacteristic
242C22×C61C2×AGL(1,ℤ/8)12yes—
243C22×C61#57812yes—
244C2×D61—24no—
245C2×D61—24no—
246C2×D61—24no—
247C2×D61—42no—
248C2×D61—42no—
249C2×D61—42no—
250C2×D61—42no—
251C2×D61—42no—
252C2×D61—42no—
253C2×D61—42no—
254C2×D61—82no—
255C2×D61—82no—
256C4×C61—22no—
257C4×C61—22no—
258C4×C61—42no—
259C4×C61—44no—
260C4×C61—81no—
261C4×C61#52511yescharacteristic
262C4×C61#53711yescharacteristic
263C4×C61#54811yescharacteristic
264C4×C61#55011yescharacteristic
265Dic61—42no—
266Dic61—42no—
267Dic61—82no—
268C2×#547—611no—
269C2×#477D6111yescharacteristic, complemented
270C2×#440—611no—
271#2536—611no—
272#2033—612no—
273#2000—612no—
274#1993—611no—
275#1955—611no—
276#1872—612no—
277#1852—612no—
278#1804—611no—
279C2×C61#240711yescharacteristic
280C2×C61#300311yescharacteristic
281C2×C61#301111yescharacteristic
282C2×C61#330812yes—
283C2×C61#334312yes—
284D61—44no—
285D61—82no—
286C22×#41—1221no—
287C22×SD16—611no—
288#547—1224no—
289#547—1224no—
290C2×C4⋊Q8C2×D6111yescharacteristic
291C2×#123—1222no—
292C2×C22⋊Q8—612no—
293C2×C22⋊Q8—1221no—
294C2×#121—612no—
295C2×C4×Q8C2×D6111yescharacteristic
296C2×C4×D4—611no—
297C2×C4×D4—611no—
298C22×C4⋊C4—612no—
299#477—24no—
300#440—1224no—
301C2×#107—611no—
302C2×#107C2×D6112yes—
303C2×C4⋊C8C2×D6111yescharacteristic
304C2×#103—611no—
305C2×#103C2×D6112yes—
306C2×#102—611no—
307C2×#102—612no—
308C2×C22⋊C8—612no—
309C2×C22⋊C8—612no—
310C2×#97—611no—
311#398—1221no—
312#394—1221no—
313#388—1222no—
314#382—611no—
315#380—612no—
316#340—612no—
317#340—612no—
318#340—612no—
319C61#2413211yescharacteristic, complemented, Sylow
320C2×#41—1228no—
321C2×#41—2448no—
322C22×Q8—21no—
323C22×Q8C22×D6111yescharacteristic
324C22×D4—611no—
325C22×D4—611no—
326C22×D4—1222no—
327C23×C4—612no—
328C23×C4—1222no—
329C23×C4—1221no—
330C2×SD16—1222no—
331C2×SD16—1222no—
332C2×SD16—1222no—
333C22×C8—611no—
334C22×C8—611no—
335C22×C8C22×D6112yes—
336C4⋊Q8—22no—
337C4⋊Q8C22⋊D6114yes—
338#123—12216no—
339C22⋊Q8—1228no—
340C22⋊Q8—1224no—
341C22⋊Q8—2442no—
342#121—1224no—
343#121—1224no—
344C4×Q8—22no—
345C4×Q8—22no—
346C4×D4—614no—
347C4×D4—1222no—
348C4×D4—1222no—
349C4×D4—1222no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—611no—
353C2×C4⋊C4—611no—
354C2×C4⋊C4—612no—
355C2×C4⋊C4—1224no—
356C2×C4⋊C4—1222no—
357C2×C4⋊C4—1224no—
358C2×C4⋊C4—1222no—
359C2×C4⋊C4C22×D6111yescharacteristic
360C2×C4⋊C4C22×D6112yes—
361C2×C22⋊C4—612no—
362C2×C22⋊C4—1222no—
363C2×C22⋊C4—1221no—
364C2×C22⋊C4—1222no—
365C2×C22⋊C4—1222no—
366C2×C42—21no—
367C2×C42C22×D6111yescharacteristic
368#107—24no—
369#107—1222no—
370C4⋊C8—22no—
371#103—24no—
372#103—1222no—
373#102—1222no—
374#102—2442no—
375C22⋊C8—1224no—
376C22⋊C8—2442no—
377#97—1222no—
378#95—1222no—
379#95—1221no—
380#95—1221no—
381C24—612no—
382C24—1221no—
383#41—24432no—
384C2×Q8—21no—
385C2×Q8—24no—
386C2×Q8—21no—
387C2×Q8—21no—
388C2×Q8—21no—
389C2×Q8—21no—
390C2×Q8—44no—
391C2×Q8#1175822612yescomplemented
392C2×D4—612no—
393C2×D4—1224no—
394C2×D4—1228no—
395C2×D4—1221no—
396C2×D4—1221no—
397C2×D4—1221no—
398C2×D4—1221no—
399C2×D4—1221no—
400C2×D4—1222no—
401C2×D4—2444no—
402C2×D4—2444no—
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—21no—
406C22×C4—612no—
407C22×C4—1228no—
408C22×C4—1221no—
409C22×C4—1222no—
410C22×C4—1222no—
411C22×C4—1222no—
412C22×C4—1222no—
413C22×C4—1222no—
414C22×C4—1222no—
415C22×C4—1224no—
416C22×C4—1222no—
417C22×C4—2444no—
418C22×C4—2444no—
419C22×C4—2444no—
420C22×C4D4×D6111yescharacteristic
421C22×C4D4×D6111yescharacteristic
422C22×C4C23×D6111yescharacteristic
423SD16—1228no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—612no—
428C2×C8—1221no—
429C2×C8—1221no—
430C2×C8—1221no—
431C2×C8—1221no—
432C2×C8—1221no—
433C4⋊C4—22no—
434C4⋊C4—24no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—42no—
438C4⋊C4—612no—
439C4⋊C4—1228no—
440C4⋊C4—1221no—
441C4⋊C4—1222no—
442C4⋊C4—1222no—
443C4⋊C4—1222no—
444C4⋊C4—2444no—
445C4⋊C4C2×C22⋊D6114yes—
446C22⋊C4—1228no—
447C22⋊C4—1224no—
448C22⋊C4—1224no—
449C22⋊C4—2444no—
450C22⋊C4—2444no—
451C42—42no—
452C42D4×D6112yes—
453C42C2×C22⋊D6112yes—
454C23—614no—
455C23—1222no—
456C23—1222no—
457C23—1222no—
458C23—1222no—
459C23—1222no—
460C23—1222no—
461C23—1222no—
462C23—2442no—
463C23—2442no—
464C23—2441no—
465C23—2441no—
466C23C2×D4×D6111yescharacteristic
467Q8—22no—
468Q8—22no—
469Q8—48no—
470Q8—41no—
471Q8—41no—
472D4—1228no—
473D4—1222no—
474D4—1222no—
475D4—24416no—
476D4—2441no—
477D4—2441no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—24no—
481C2×C4—22no—
482C2×C4—41no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—42no—
488C2×C4—1222no—
489C2×C4—1222no—
490C2×C4—1222no—
491C2×C4—1222no—
492C2×C4—2448no—
493C2×C4—24416no—
494C2×C4—2442no—
495C2×C4—2442no—
496C2×C4—2442no—
497C2×C4—2442no—
498C2×C4—2442no—
499C2×C4—2442no—
500C2×C4—2442no—
501C2×C4—2442no—
502C2×C4#1451974512yes—
503C2×C4#1451974512yes—
504C2×C4C2×D4×D6111yescharacteristic
505C2×C4C22×C22⋊D6111yescharacteristic
506C2×C4#41×D6111yescharacteristic
507C2×C4C2×#1175822611yescharacteristic
508C2×C4#1451982311yescharacteristic
509C2×C4#1451982511yescharacteristic
510C8—42no—
511C8—1222no—
512C8—2441no—
513C22—1224no—
514C22—1224no—
515C22—1224no—
516C22—4882no—
517C22—4882no—
518C22—2442no—
519C22—2442no—
520C22—2442no—
521C22—2442no—
522C22—2442no—
523C22—2442no—
524C22—2442no—
525C22C2×#1451974511yescharacteristic
526C22#1684040311yescharacteristic
527C22#1684041411yescharacteristic
528C22AGL(1,ℤ/8)×D6112yes—
529C22#1684057512yes—
530C4—22no—
531C4—22no—
532C4—44no—
533C4—42no—
534C4—81no—
535C4—4882no—
536C4—2442no—
537C4—2442no—
538C4#1684027511yescharacteristic
539C4#1684034211yescharacteristic
540C4#1684042311yescharacteristic
541C4#1684043111yescharacteristic
542C2—4882no—
543C2—2444no—
544C2#2144916611yescharacteristic
545C2#2145504411yescharacteristic
546C2#2145511011yescharacteristic
547C2#2145780712yes—
548C2#2145825612yes—
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#477C2
socleC23×C61C22×D4
prime radicalC2×#16840415C2
hypercenterC2×#477D61
hypocenterC61#24132
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#477

Complements

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

NH ≅ G/Ntype
2. C2×#16839860C2semidirect
3. #21450724C2semidirect
4. #21447003C2semidirect
5. #21446821C2semidirect
6. #21446793C2semidirect
7. #21446616C2semidirect
8. #21446123C2semidirect
9. #21446005C2semidirect
10. #21445830C2semidirect
11. C2×C61×#477C2semidirect
115. C2×Q8×C61#41semidirect
178. C2×#92418#41semidirect
269. C2×#477D61semidirect
319. C61#24132semidirect
391. C2×Q8#11758226semidirect

Maximal subgroups

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

subgroupindex
C2×#168404152
C2×#168398602
#214507242
#214470032
#214468212
#214467932
#214466162
#214461232
#214460052
#214458302
C2×C61×#4772
#2413261