Finite Group Databasebeta

← #29713946

Subgroups — #29713946

Order 15616.

1350 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840416C211yesmaximal, characteristic
2C2×#16840047C211yesmaximal, characteristic, complemented
3C2×C4×#11758201C211yesmaximal, characteristic, complemented
4#21447008C212yesmaximal, complemented
5#21446820C212yesmaximal, complemented
6#21446779C211yesmaximal, characteristic, complemented
7#21446614C211yesmaximal, characteristic, complemented
8#21446127C212yesmaximal, complemented
9#21446011C212yesmaximal, complemented
10#21445816C211yesmaximal, characteristic, complemented
11C61×#2538C211yesmaximal, characteristic, complemented
12C22×#11758224—21no—
13C22×#11758201C2211yescharacteristic
14#16840416—24no—
15#16840416—24no—
16C2×#14519777C2211yescharacteristic
17C2×#14519761—22no—
18C2×#14519757—21no—
19C2×#14519751C2212yes—
20C2×C4×D244C2211yescharacteristic
21C22×C4⋊Dic61C2212yes—
22#16840047—24no—
23C4×#11758201—24no—
24C2×#14519708C2212yes—
25C2×#14519708C2211yescharacteristic
26C2×C4⋊#92418C2211yescharacteristic
27C2×#14519701C2212yes—
28C2×#14519701C2211yescharacteristic
29C2×#14519697C2211yescharacteristic
30C2×#14519697C2212yes—
31C2×C22⋊#92418C2212yes—
32C2×C22⋊#92418C2212yes—
33C2×C4×#92418C2211yescharacteristic
34#16839524C2212yes—
35#16839524C2212yes—
36#16839524C2212yes—
37C2×C22⋊Q8×C61C2212yes—
38C2×C4×Q8×C61C2211yescharacteristic
39C2×C4×D4×C61C2211yescharacteristic
40C61×#396—22no—
41C61×#390—21no—
42C61×#389—21no—
43C61×#384C2211yescharacteristic
44C61×#380C2212yes—
45C2×#11758224—28no—
46C2×#11758224—44no—
47C2×#11758224—44no—
48C22×Dic122—21no—
49C22×D244C2311yescharacteristic
50C22×C22⋊D61—22no—
51C23×Dic61—22no—
52C22×C4×D61—21no—
53C2×#11758201—22no—
54C2×#11758201—22no—
55C2×#11758201—22no—
56C22×#92418C2311yescharacteristic
57C22×#92418C2312yes—
58C22×#92418C2311yescharacteristic
59#14519777—22no—
60#14519777D414yes—
61#14519761—216no—
62#14519757—28no—
63#14519751—24no—
64#14519751—24no—
65C4×D244—22no—
66C4×D244D414yes—
67C2×C4⋊Dic61—22no—
68C2×C4⋊Dic61—21no—
69C2×C4⋊Dic61—24no—
70C2×C4⋊Dic61—24no—
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C4⋊Dic61C2312yes—
73C2×#11758198—21no—
74C2×#11758198—22no—
75#14519708—24no—
76#14519708—22no—
77C4⋊#92418—22no—
78#14519701—24no—
79#14519701—22no—
80#14519697—22no—
81#14519697—42no—
82C22⋊#92418—24no—
83C22⋊#92418—42no—
84C4×#92418—22no—
85C22×Q8×C61C2311yescharacteristic
86C22×D4×C61D411yescharacteristic
87C23×C4×C61C2312yes—
88C22⋊Q8×C61—24no—
89C22⋊Q8×C61—42no—
90C4×Q8×C61—22no—
91C4×Q8×C61—22no—
92C4×D4×C61—22no—
93C4×D4×C61—22no—
94C2×C4⋊C4×C61—21no—
95C2×C4⋊C4×C61—22no—
96C2×C4⋊C4×C61—21no—
97C2×C4⋊C4×C61—21no—
98C2×C4⋊C4×C61D411yescharacteristic
99C2×C4⋊C4×C61C2312yes—
100C2×C4⋊C4×C61C2311yescharacteristic
101C2×C22⋊C4×C61—22no—
102C2×C22⋊C4×C61—22no—
103C2×C22⋊C4×C61D412yes—
104C2×C42×C61—21no—
105C2×C42×C61C2311yescharacteristic
106#95×C61—22no—
107C24×C61C2×D412yes—
108C23×D61—21no—
109C2×Q8×C61—21no—
110C2×Q8×C61—21no—
111C2×Q8×C61—21no—
112C2×Q8×C61—21no—
113C2×Q8×C61—21no—
114C2×Q8×C61#4112yescomplemented
115C2×D4×C61—22no—
116C2×D4×C61—22no—
117C2×D4×C61SD1614yescomplemented
118C22×C4×C61—21no—
119C22×C4×C61—22no—
120C22×C4×C61—22no—
121C22×C4×C61—22no—
122C22×C4×C61—22no—
123C22×C4×C61—22no—
124C22×C4×C61—22no—
125C22×C4×C61—21no—
126C22×C4×C61—22no—
127C22×C4×C61C2×D412yes—
128C22×C4×C61C2×D411yescharacteristic
129C22×C4×C61C2×D411yescharacteristic
130C22×C4×C61C2411yescharacteristic
131C2×C22⋊D61—28no—
132C2×C22⋊D61—44no—
133C2×C22⋊D61—44no—
134C22×Dic61—28no—
135C22×Dic61—21no—
136C22×Dic61—22no—
137C22×Dic61—44no—
138C22×Dic61—44no—
139#11758224—432no—
140C2×D244—21no—
141C2×D244—21no—
142C2×D244—21no—
143C2×D244—21no—
144C2×D244—21no—
145C2×D244C2×D412yes—
146C2×C4×D61—24no—
147C2×C4×D61—42no—
148C2×C4×D61—42no—
149C2×Dic122—24no—
150C2×Dic122—42no—
151C2×Dic122—42no—
152C4⋊C4×C61—22no—
153C4⋊C4×C61—21no—
154C4⋊C4×C61—21no—
155C4⋊C4×C61—22no—
156C4⋊C4×C61—22no—
157C4⋊C4×C61—44no—
158C4⋊C4×C61—42no—
159C4⋊C4×C61—42no—
160C4⋊C4×C61—81no—
161C22⋊C4×C61—24no—
162C22⋊C4×C61—44no—
163C22⋊C4×C61—82no—
164C42×C61—42no—
165C42×C61C2×D412yes—
166C42×C61C2×D412yes—
167#11758201—28no—
168#11758198—28no—
169#11758198—24no—
170C4⋊Dic61—28no—
171C4⋊Dic61—24no—
172C4⋊Dic61—21no—
173C4⋊Dic61—22no—
174C4⋊Dic61C2×D412yes—
175C4⋊Dic61C2×D414yes—
176C2×#92418—21no—
177C2×#92418—21no—
178C2×#92418—21no—
179C2×#92418—22no—
180C2×#92418—22no—
181C2×#92418—21no—
182C2×#92418—22no—
183C2×#92418—21no—
184C2×#92418#4112yescomplemented
185C23×C61—22no—
186C23×C61—22no—
187C23×C61—22no—
188C23×C61—22no—
189C23×C61—22no—
190C23×C61—22no—
191C23×C61C2×SD1614yes—
192C23×C61C22×D411yescharacteristic
193C22×D61—22no—
194C22×D61—41no—
195C22×D61—41no—
196C22×D61—41no—
197C22×D61—41no—
198C22×D61—41no—
199C22×D61—41no—
200Q8×C61—22no—
201Q8×C61—22no—
202Q8×C61—41no—
203Q8×C61—41no—
204D4×C61—28no—
205C2×C4×C61—21no—
206C2×C4×C61—21no—
207C2×C4×C61—22no—
208C2×C4×C61—22no—
209C2×C4×C61—22no—
210C2×C4×C61—22no—
211C2×C4×C61—41no—
212C2×C4×C61—41no—
213C2×C4×C61—42no—
214C2×C4×C61—42no—
215C2×C4×C61—42no—
216C2×C4×C61—42no—
217C2×C4×C61—41no—
218C2×C4×C61—42no—
219C2×C4×C61—42no—
220C2×C4×C61—41no—
221C2×C4×C61—42no—
222C2×C4×C61—41no—
223C2×C4×C61#12012yes—
224C2×C4×C61#12012yes—
225C2×C4×C61C2×SD1612yes—
226C2×C4×C61AGL(1,ℤ/8)12yes—
227C2×C4×C61C22×D411yescharacteristic
228C2×C4×C61C22×D411yescharacteristic
229C2×C4×C61C2×#4111yescharacteristic
230C2×C4×C61C2×#4111yescharacteristic
231C2×C4×C61H(5,𝔽2)11yescharacteristic
232C2×C4×C612−1+411yescharacteristic
233C22⋊D61—416no—
234C2×Dic61—24no—
235C2×Dic61—22no—
236C2×Dic61—416no—
237C2×Dic61—42no—
238C2×Dic61—41no—
239C2×Dic61—42no—
240C2×Dic61—41no—
241D244—22no—
242D244—22no—
243D244—41no—
244D244—41no—
245C4×D61—48no—
246Dic122—48no—
247#92418—22no—
248#92418—41no—
249#92418—42no—
250#24120—611nomaximal, Sylow
251C22×C61—24no—
252C22×C61—24no—
253C22×C61—24no—
254C22×C61—42no—
255C22×C61—42no—
256C22×C61—42no—
257C22×C61—42no—
258C22×C61C2×#12011yescharacteristic
259C22×C61#54611yescharacteristic
260C22×C61#54711yescharacteristic
261C22×C61C22×SD1611yescharacteristic
262C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
263C22×C61#57711yescharacteristic
264C22×C61#57811yescharacteristic
265C2×D61—42no—
266C2×D61—42no—
267C2×D61—42no—
268C2×D61—81no—
269C2×D61—81no—
270C2×D61—81no—
271C2×D61—81no—
272C4×C61—22no—
273C4×C61—42no—
274C4×C61—42no—
275C4×C61—82no—
276C4×C61—81no—
277C4×C61#45014yes—
278C4×C61#52511yescharacteristic
279C4×C61#53711yescharacteristic
280C4×C61#54811yescharacteristic
281C4×C61#55011yescharacteristic
282Dic61—44no—
283Dic61—42no—
284C2×#547—611no—
285C2×#475—611no—
286C2×C4×SD16—611no—
287#2538D6111yescharacteristic, complemented
288#2034—612no—
289#2000—612no—
290#1990—611no—
291#1954—611no—
292#1872—612no—
293#1853—612no—
294#1802—611no—
295C2×C61—44no—
296C2×C61#240711yescharacteristic
297C2×C61C2×#45011yescharacteristic
298C2×C61#301311yescharacteristic
299C2×C61#329811yescharacteristic
300C2×C61#333211yescharacteristic
301C2×C61#333511yescharacteristic
302C2×C61#335311yescharacteristic
303D61—82no—
304C22×#41—1221no—
305C22×SD16—611no—
306#547—1224no—
307#547—1224no—
308C2×C4⋊Q8—611no—
309C2×#123—1222no—
310C2×C22⋊Q8—1221no—
311C2×C22⋊Q8C2×D6112yes—
312C2×#121—612no—
313C2×C4×Q8C2×D6111yescharacteristic
314C2×C4×D4—611no—
315C2×C4×D4C2×D6111yescharacteristic
316C22×C4⋊C4—612no—
317#475—1224no—
318C4×SD16—1224no—
319C2×#106—612no—
320C2×#106—611no—
321C2×C4⋊C8—611no—
322C2×#103—612no—
323C2×#103—611no—
324C2×#102—611no—
325C2×#102—612no—
326C2×C22⋊C8—612no—
327C2×C22⋊C8—612no—
328C2×C4×C8—611no—
329#396—22no—
330#390—21no—
331#389—21no—
332#384C2×D6111yescharacteristic
333#380C2×D6112yes—
334#340—612no—
335#340—612no—
336#340—612no—
337C61#2412011yescharacteristic, complemented, Sylow
338C2×#41—1228no—
339C2×#41—2444no—
340C2×#41—2444no—
341C22×Q8—1221no—
342C22×Q8C22×D6111yescharacteristic
343C22×D4—611no—
344C22×D4—1222no—
345C22×D4C22⋊D6111yescharacteristic
346C23×C4—1222no—
347C23×C4—1221no—
348C23×C4C22×D6112yes—
349C2×SD16—1222no—
350C2×SD16—1222no—
351C2×SD16—1222no—
352C22×C8—611no—
353C22×C8—611no—
354C22×C8—612no—
355C4⋊Q8—614no—
356C4⋊Q8—1222no—
357#123—12216no—
358C22⋊Q8—24no—
359C22⋊Q8—42no—
360C22⋊Q8—1228no—
361#121—1224no—
362#121—1224no—
363C4×Q8—22no—
364C4×Q8—22no—
365C4×D4—22no—
366C4×D4—22no—
367C4×D4—614no—
368C4×D4—1222no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—22no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—611no—
374C2×C4⋊C4—612no—
375C2×C4⋊C4—1224no—
376C2×C4⋊C4—1224no—
377C2×C4⋊C4—1222no—
378C2×C4⋊C4—1221no—
379C2×C4⋊C4C22⋊D6111yescharacteristic
380C2×C4⋊C4C22×D6111yescharacteristic
381C2×C4⋊C4C22×D6112yes—
382C2×C22⋊C4—22no—
383C2×C22⋊C4—22no—
384C2×C22⋊C4—1221no—
385C2×C22⋊C4—1222no—
386C2×C22⋊C4C22⋊D6112yes—
387C2×C42—21no—
388C2×C42C22×D6111yescharacteristic
389#106—1222no—
390#106—1224no—
391C4⋊C8—1222no—
392#103—1222no—
393#103—1224no—
394#102—1222no—
395#102—2442no—
396C22⋊C8—1224no—
397C22⋊C8—2442no—
398C4×C8—1222no—
399#95—22no—
400C24—1221no—
401C24C2×C22⋊D6112yes—
402#41—24432no—
403C2×Q8—21no—
404C2×Q8—21no—
405C2×Q8—21no—
406C2×Q8—21no—
407C2×Q8—21no—
408C2×Q8—1224no—
409C2×Q8—2442no—
410C2×Q8—2442no—
411C2×Q8#1175822412yescomplemented
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—612no—
415C2×D4—1228no—
416C2×D4—1221no—
417C2×D4—1221no—
418C2×D4—1221no—
419C2×D4—1221no—
420C2×D4—1221no—
421C2×D4—2444no—
422C2×D4—2444no—
423C2×D4#1175820114yescomplemented
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—1228no—
434C22×C4—1221no—
435C22×C4—1222no—
436C22×C4—1224no—
437C22×C4—2444no—
438C22×C4—2444no—
439C22×C4—2442no—
440C22×C4—2442no—
441C22×C4C2×C22⋊D6112yes—
442C22×C4C2×C22⋊D6111yescharacteristic
443C22×C4C2×C22⋊D6111yescharacteristic
444C22×C4C23×D6111yescharacteristic
445SD16—1228no—
446C2×C8—612no—
447C2×C8—1221no—
448C2×C8—1221no—
449C2×C8—1221no—
450C2×C8—1222no—
451C2×C8—1222no—
452C2×C8—1221no—
453C2×C8—1222no—
454C2×C8—1221no—
455C4⋊C4—22no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—44no—
461C4⋊C4—42no—
462C4⋊C4—42no—
463C4⋊C4—81no—
464C4⋊C4—612no—
465C4⋊C4—614no—
466C4⋊C4—1228no—
467C4⋊C4—1221no—
468C4⋊C4—1222no—
469C4⋊C4—1224no—
470C22⋊C4—24no—
471C22⋊C4—44no—
472C22⋊C4—82no—
473C22⋊C4—1228no—
474C22⋊C4—1224no—
475C42—42no—
476C42D4×D6112yes—
477C42D4×D6112yes—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—22no—
483C23—22no—
484C23—1222no—
485C23—2441no—
486C23—2441no—
487C23—2441no—
488C23—2441no—
489C23—2441no—
490C23—2441no—
491C23C2×#1175820114yes—
492C23C22×C22⋊D6111yescharacteristic
493Q8—22no—
494Q8—22no—
495Q8—41no—
496Q8—41no—
497Q8—2448no—
498D4—28no—
499D4—1222no—
500D4—1222no—
501D4—24416no—
502D4—2441no—
503D4—2441no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—41no—
517C2×C4—42no—
518C2×C4—41no—
519C2×C4—42no—
520C2×C4—41no—
521C2×C4—42no—
522C2×C4—1222no—
523C2×C4—1224no—
524C2×C4—24416no—
525C2×C4—2441no—
526C2×C4—2442no—
527C2×C4—2441no—
528C2×C4—2442no—
529C2×C4—2448no—
530C2×C4#1451974512yes—
531C2×C4#1451974512yes—
532C2×C4C2×#1175820112yes—
533C2×C4#1451980212yes—
534C2×C4C2×D4×D6111yescharacteristic
535C2×C4C2×D4×D6111yescharacteristic
536C2×C4#41×D6111yescharacteristic
537C2×C4C2×#1175822411yescharacteristic
538C2×C4#1451982211yescharacteristic
539C2×C4#1451982411yescharacteristic
540C8—1222no—
541C8—2441no—
542C8—2442no—
543C22—24no—
544C22—24no—
545C22—24no—
546C22—42no—
547C22—42no—
548C22—42no—
549C22—42no—
550C22—4881no—
551C22—4881no—
552C22—4881no—
553C22—4881no—
554C22—2442no—
555C22—2442no—
556C22—2442no—
557C22C2×#1451974511yescharacteristic
558C22#1684040611yescharacteristic
559C22#1684041111yescharacteristic
560C22C22×#1175820111yescharacteristic
561C22C2×#1451980211yescharacteristic
562C22#1684057311yescharacteristic
563C22#1684057711yescharacteristic
564C4—22no—
565C4—42no—
566C4—42no—
567C4—81no—
568C4—82no—
569C4—2442no—
570C4—2444no—
571C4#1683991214yes—
572C4#1684027211yescharacteristic
573C4#1684034111yescharacteristic
574C4#1684041811yescharacteristic
575C4#1684043011yescharacteristic
576C2—44no—
577C2—4882no—
578C2#2144916011yescharacteristic
579C2C2×#1683991211yescharacteristic
580C2#2145513111yescharacteristic
581C2#2145768411yescharacteristic
582C2#2145814611yescharacteristic
583C2#2145817011yescharacteristic
584C2#2145839711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2538C2
socleC23×C61C22×D4
prime radicalC2×#16840416C2
hypercenter#2538D61
hypocenterC61#24120
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2538

Complements

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

NH ≅ G/Ntype
2. C2×#16840047C2semidirect
3. C2×C4×#11758201C2semidirect
4. #21447008C2semidirect
5. #21446820C2semidirect
6. #21446779C2semidirect
7. #21446614C2semidirect
8. #21446127C2semidirect
9. #21446011C2semidirect
10. #21445816C2semidirect
11. C61×#2538C2semidirect
114. C2×Q8×C61#41semidirect
117. C2×D4×C61SD16semidirect
184. C2×#92418#41semidirect
287. #2538D61semidirect
337. C61#24120semidirect
411. C2×Q8#11758224semidirect
423. C2×D4#11758201semidirect

Maximal subgroups

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

subgroupindex
C2×#168404162
C2×#168400472
C2×C4×#117582012
#214470082
#214468202
#214467792
#214466142
#214461272
#214460112
#214458162
C61×#25382
#2412061