Finite Group Databasebeta

← #29732221

Subgroups — #29732221

Order 15616.

1194 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840319C211yesmaximal, characteristic, complemented
2C2×#16839991C212yesmaximal
3#21454150C211yesmaximal, characteristic
4#21447153C211yesmaximal, characteristic, complemented
5#21447066C211yesmaximal, characteristic, complemented
6#21446956C211yesmaximal, characteristic, complemented
7#21446904C211yesmaximal, characteristic, complemented
8#21446242C212yesmaximal, complemented
9#21446173C212yesmaximal, complemented
10#21446090C212yesmaximal, complemented
11C2×C61×#336C211yesmaximal, characteristic, complemented
12C22×#11758190C2212yes—
13#16840319—28no—
14C2×#14519777C2211yescharacteristic
15C2×#14519777C2211yescharacteristic
16C2×#14519769—21no—
17C2×#14519760—21no—
18C2×#14519756—21no—
19C2×#14519756C2212yes—
20C2×#14519751C2211yescharacteristic
21C2×C4×Dic122—21no—
22C2×C4×Dic122C2212yes—
23C2×#14519736—21no—
24#16839991—28no—
25C2×#14519709C2212yes—
26C2×#14519702C2212yes—
27C2×#14519702C2212yes—
28C2×#14519702C2212yes—
29C2×#14519699C2212yes—
30#16839664C2211yescharacteristic
31#16839664C2211yescharacteristic
32#16839660C2211yescharacteristic
33#16839660C2211yescharacteristic
34#16839653—21no—
35#16839653—21no—
36#16839641—21no—
37#16839622C2211yescharacteristic
38#16839525C2212yes—
39#16839525C2212yes—
40C23×C8×C61C2212yes—
41C22×C42×C61C2211yescharacteristic
42C61×#336C2212yes—
43C61×#336C2211yescharacteristic
44C61×#336C2211yescharacteristic
45C22×Dic122—21no—
46C22×Dic122C2312yes—
47C22×D244C2311yescharacteristic
48C22×C22⋊D61—21no—
49C2×#11758190—24no—
50C2×#11758190—24no—
51C2×#11758190—42no—
52#14519777—24no—
53#14519777—24no—
54#14519769—28no—
55#14519760—44no—
56#14519756—28no—
57#14519756—24no—
58#14519756D418yes—
59#14519751—24no—
60C4×Dic122—28no—
61C4×Dic122—24no—
62C4×Dic122—24no—
63#14519736—28no—
64C2×C4⋊Dic61—21no—
65C2×C4⋊Dic61—21no—
66C2×C4⋊Dic61—21no—
67C2×C4⋊Dic61C2311yescharacteristic
68C2×C4⋊Dic61C2311yescharacteristic
69C2×C4⋊Dic61C2311yescharacteristic
70C2×C4⋊Dic61C2312yes—
71C2×#11758196—22no—
72C2×#11758196—22no—
73C2×#11758196—21no—
74C2×#11758196—21no—
75C2×C22⋊Dic61—22no—
76C2×C22⋊Dic61—21no—
77C2×#11758198—21no—
78C2×#11758198—21no—
79C2×#11758198—21no—
80C2×#11758198—21no—
81C2×#11758198—21no—
82C2×C4×Dic61—22no—
83#14519709—24no—
84#14519702—24no—
85#14519702—24no—
86#14519702—42no—
87#14519699—24no—
88#14519684—21no—
89#14519684—21no—
90#14519684—21no—
91C23×C4×C61D411yescharacteristic
92C23×C4×C61D411yescharacteristic
93C23×C4×C61C2311yescharacteristic
94C22×C8×C61—24no—
95C22×C8×C61—24no—
96C22×C8×C61—42no—
97C22×C8×C61C2312yes—
98C22×C8×C61C2312yes—
99C2×C42×C61—24no—
100C2×C42×C61D411yescharacteristic
101C2×C42×C61D411yescharacteristic
102C2×C42×C61C2311yescharacteristic
103C2×C42×C61C2311yescharacteristic
104C24×C61C2×D411yescharacteristic
105C23×D61—21no—
106C22×C4×C61—22no—
107C22×C4×C61—22no—
108C22×C4×C61—22no—
109C22×C4×C61—21no—
110C22×C4×C61—21no—
111C22×C4×C61—22no—
112C22×C4×C61—22no—
113C22×C4×C61—41no—
114C22×C4×C61SD1614yescomplemented
115C22×C4×C61C2×D411yescharacteristic
116C22×C4×C61C2×D411yescharacteristic
117C22×C4×C61C2×D414yes—
118C22×C4×C61C2×D411yescharacteristic
119C22×C4×C61C2×D411yescharacteristic
120C22×C4×C61C2×D411yescharacteristic
121C22×C4×C61C2411yescharacteristic
122C2×C22⋊D61—42no—
123C2×C22⋊D61—42no—
124C2×C22⋊D61—42no—
125C22×Dic61—22no—
126C22×Dic61—22no—
127C22×Dic61—21no—
128C22×Dic61—21no—
129C22×Dic61—21no—
130C2×D244—22no—
131C2×D244—22no—
132C2×D244—21no—
133C2×D244—21no—
134C2×Dic122—22no—
135C2×Dic122—22no—
136C2×Dic122—22no—
137C2×Dic122—22no—
138C2×Dic122—24no—
139C2×Dic122—22no—
140C2×Dic122—42no—
141C2×Dic122—42no—
142C2×Dic122C2×D414yes—
143C2×C8×C61—22no—
144C2×C8×C61—22no—
145C2×C8×C61—22no—
146C2×C8×C61—22no—
147C2×C8×C61—22no—
148C2×C8×C61—48no—
149C2×C8×C61#4114yes—
150C42×C61—22no—
151C42×C61—22no—
152C42×C61—22no—
153C42×C61—22no—
154C22⋊Dic61—28no—
155C22⋊Dic61—42no—
156#11758198—42no—
157#11758198—42no—
158#11758198—42no—
159#11758198—42no—
160#11758198—42no—
161C4⋊Dic61—24no—
162C4⋊Dic61—24no—
163C4⋊Dic61—24no—
164C4⋊Dic61—22no—
165C4⋊Dic61—22no—
166C4⋊Dic61—22no—
167C4⋊Dic61—21no—
168C4⋊Dic61—21no—
169C4⋊Dic61C2×D414yes—
170#11758196—28no—
171#11758196—28no—
172#11758196—42no—
173#11758196—42no—
174C4×Dic61—28no—
175#11758190—48no—
176C23×C61—21no—
177C23×C61—21no—
178C23×C61—21no—
179C23×C61—21no—
180C23×C61#12012yes—
181C23×C61C2×SD1612yes—
182C23×C61#13712yes—
183C23×C61C22×D411yescharacteristic
184C22×D61—41no—
185C22×D61—41no—
186C22×D61—41no—
187C22×D61—41no—
188C22×D61—41no—
189C22×D61—41no—
190C22×D61—41no—
191C2×C4×C61—21no—
192C2×C4×C61—21no—
193C2×C4×C61—21no—
194C2×C4×C61—21no—
195C2×C4×C61—21no—
196C2×C4×C61—21no—
197C2×C4×C61—21no—
198C2×C4×C61—21no—
199C2×C4×C61—28no—
200C2×C4×C61—24no—
201C2×C4×C61—24no—
202C2×C4×C61—21no—
203C2×C4×C61—21no—
204C2×C4×C61—21no—
205C2×C4×C61—44no—
206C2×C4×C61#12012yes—
207C2×C4×C61C2×SD1612yes—
208C2×C4×C61C2×SD1612yes—
209C2×C4×C61#13512yes—
210C2×C4×C61C22×D412yes—
211C2×C4×C61C2×#4112yes—
212C2×C4×C61H(5,𝔽2)11yescharacteristic
213C2×C4×C612−1+411yescharacteristic
214C22⋊D61—84no—
215C2×Dic61—24no—
216C2×Dic61—24no—
217C2×Dic61—42no—
218C2×Dic61—42no—
219C2×Dic61—42no—
220C2×Dic61—42no—
221C2×Dic61—41no—
222C2×Dic61—41no—
223C2×Dic61—41no—
224C2×Dic61—41no—
225C2×Dic61—41no—
226C2×Dic61—41no—
227C2×Dic61—41no—
228C2×Dic61—41no—
229C2×Dic61—41no—
230D244—42no—
231D244—42no—
232Dic122—24no—
233Dic122—24no—
234Dic122—48no—
235Dic122—42no—
236Dic122—42no—
237C8×C61—24no—
238C8×C61—42no—
239#25497—611nomaximal, Sylow
240C22×C61—22no—
241C22×C61—22no—
242C22×C61—22no—
243C22×C61—22no—
244C22×C61—22no—
245C22×C61—22no—
246C22×C61#44814yes—
247C22×C61C2×#12011yescharacteristic
248C22×C61#53211yescharacteristic
249C22×C61#53711yescharacteristic
250C22×C61C22×SD1611yescharacteristic
251C22×C61C2×#13511yescharacteristic
252C22×C61C2×#13711yescharacteristic
253C22×C61#57511yescharacteristic
254C2×D61—81no—
255C2×D61—81no—
256C2×D61—81no—
257C2×D61—81no—
258C2×D61—81no—
259C2×D61—81no—
260C2×D61—81no—
261C4×C61—22no—
262C4×C61—22no—
263C4×C61—22no—
264C4×C61—22no—
265C4×C61—41no—
266C4×C61#54612yes—
267C4×C61#54712yes—
268Dic61—44no—
269Dic61—44no—
270Dic61—81no—
271Dic61—81no—
272Dic61—81no—
273C2×#533—611no—
274C2×#465—612no—
275#2854—611no—
276#2058—611no—
277#2043—611no—
278#2023—611no—
279#2013—611no—
280#1892—612no—
281#1880—612no—
282#1865—612no—
283C2×#336D6111yescharacteristic, complemented
284C2×C61—24no—
285C2×C61#243011yescharacteristic
286C2×C61C2×#44811yescharacteristic
287C2×C61#300211yescharacteristic
288C2×C61#316711yescharacteristic
289C2×C61#318111yescharacteristic
290C2×C61#324611yescharacteristic
291C2×C61#324811yescharacteristic
292D61—161no—
293C22×SD16—612no—
294C23×C8C2×D6112yes—
295#533—1228no—
296C2×C4⋊Q8—611no—
297C2×C4⋊Q8—611no—
298C2×#125—1221no—
299C2×#123—1221no—
300C2×C22⋊Q8—612no—
301C2×C22⋊Q8—1221no—
302C2×#121—611no—
303C2×C4×Q8—612no—
304C2×C4×Q8—1221no—
305C2×#117—1221no—
306C22×C42C2×D6111yescharacteristic
307#465—1228no—
308C2×#106—612no—
309C2×#103—612no—
310C2×#103—612no—
311C2×#103—612no—
312C2×#102—612no—
313#390—611no—
314#390—611no—
315#389—611no—
316#389—611no—
317#388—1221no—
318#388—1221no—
319#386—1221no—
320#380—611no—
321#340—612no—
322#340—612no—
323#336C2×D6111yescharacteristic
324#336C2×D6112yes—
325#336C2×D6111yescharacteristic
326C61#2549711yescharacteristic, complemented, Sylow
327C22×Q8—612no—
328C22×Q8—1221no—
329C22×D4—611no—
330C22×D4—1221no—
331C23×C4D24411yescharacteristic
332C23×C4D24411yescharacteristic
333C23×C4C22×D6111yescharacteristic
334C2×SD16—1224no—
335C2×SD16—1224no—
336C2×SD16—2442no—
337C22×C8—24no—
338C22×C8—24no—
339C22×C8—42no—
340C22×C8C22×D6112yes—
341C22×C8C22×D6112yes—
342C4⋊Q8—1224no—
343C4⋊Q8—1224no—
344#125—1228no—
345#123—2444no—
346C22⋊Q8—618no—
347C22⋊Q8—1228no—
348C22⋊Q8—1224no—
349#121—1224no—
350C4×Q8—1224no—
351C4×Q8—1224no—
352C4×Q8—1228no—
353#117—1228no—
354C2×C4⋊C4—612no—
355C2×C4⋊C4—611no—
356C2×C4⋊C4—611no—
357C2×C4⋊C4—611no—
358C2×C4⋊C4—1221no—
359C2×C4⋊C4—1221no—
360C2×C4⋊C4—1222no—
361C2×C4⋊C4—1222no—
362C2×C4⋊C4—1221no—
363C2×C4⋊C4—1221no—
364C2×C4⋊C4—1221no—
365C2×C22⋊C4—1222no—
366C2×C22⋊C4—1221no—
367C2×C22⋊C4—1221no—
368C2×C22⋊C4—1221no—
369C2×C22⋊C4—1221no—
370C2×C22⋊C4—1221no—
371C2×C22⋊C4—1221no—
372C2×C42—24no—
373C2×C42—1222no—
374C2×C42D24411yescharacteristic
375C2×C42D24411yescharacteristic
376C2×C42C22×D6111yescharacteristic
377C2×C42C22×D6111yescharacteristic
378#106—1224no—
379#103—1224no—
380#103—1224no—
381#103—2442no—
382#102—1224no—
383#95—1221no—
384#95—1221no—
385#95—1221no—
386C24—1221no—
387C24C2×D24411yescharacteristic
388C2×Q8—614no—
389C2×Q8—1224no—
390C2×Q8—1222no—
391C2×Q8—1222no—
392C2×Q8—1222no—
393C2×Q8—1222no—
394C2×Q8—1222no—
395C2×Q8—2442no—
396C2×Q8—2442no—
397C2×D4—1222no—
398C2×D4—1222no—
399C2×D4—1221no—
400C2×D4—1221no—
401C2×D4—2442no—
402C2×D4—2442no—
403C2×D4—2442no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—41no—
412C22×C4—1222no—
413C22×C4—1222no—
414C22×C4—1221no—
415C22×C4—1221no—
416C22×C4—1221no—
417C22×C4#1175819014yescomplemented
418C22×C4C2×D24411yescharacteristic
419C22×C4C2×D24411yescharacteristic
420C22×C4C2×D24411yescharacteristic
421C22×C4C2×D24411yescharacteristic
422C22×C4C2×D24411yescharacteristic
423C22×C4D4×D6114yes—
424C22×C4C23×D6111yescharacteristic
425SD16—2448no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—48no—
432C2×C8#1175822214yes—
433C4⋊C4—614no—
434C4⋊C4—1228no—
435C4⋊C4—1224no—
436C4⋊C4—1224no—
437C4⋊C4—1224no—
438C4⋊C4—1222no—
439C4⋊C4—1228no—
440C4⋊C4—1222no—
441C4⋊C4—1222no—
442C4⋊C4—1221no—
443C4⋊C4—1221no—
444C4⋊C4—2442no—
445C4⋊C4—2442no—
446C22⋊C4—1228no—
447C22⋊C4—2442no—
448C22⋊C4—2442no—
449C22⋊C4—2442no—
450C22⋊C4—2442no—
451C22⋊C4—2442no—
452C22⋊C4—2442no—
453C42—22no—
454C42—22no—
455C42—22no—
456C42—22no—
457C42—1228no—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—2441no—
463C23—2441no—
464C23—2441no—
465C23—2441no—
466C23—2441no—
467C23—2441no—
468C23—2441no—
469C23#1451974712yes—
470C23C2×#1175819012yes—
471C23#1451981012yes—
472C23C22×D24411yescharacteristic
473Q8—1224no—
474Q8—1224no—
475Q8—2448no—
476Q8—2442no—
477Q8—2442no—
478D4—4884no—
479D4—2442no—
480D4—2442no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—28no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—44no—
496C2×C4—1224no—
497C2×C4—1224no—
498C2×C4—2441no—
499C2×C4—2441no—
500C2×C4—2441no—
501C2×C4—2441no—
502C2×C4—2441no—
503C2×C4—2441no—
504C2×C4—2441no—
505C2×C4—2441no—
506C2×C4—2441no—
507C2×C4—2442no—
508C2×C4—2442no—
509C2×C4—2442no—
510C2×C4—2442no—
511C2×C4#1451974712yes—
512C2×C4C2×#1175819012yes—
513C2×C4C2×#1175819012yes—
514C2×C4#1451979912yes—
515C2×C4C2×D4×D6112yes—
516C2×C4C2×#1175822212yes—
517C2×C4#1451982211yescharacteristic
518C2×C4#1451982511yescharacteristic
519C8—24no—
520C8—42no—
521C22—22no—
522C22—22no—
523C22—22no—
524C22—22no—
525C22—22no—
526C22—22no—
527C22—4881no—
528C22—4881no—
529C22—4881no—
530C22—4881no—
531C22—4881no—
532C22—4881no—
533C22—4881no—
534C22#1683990314yes—
535C22C2×#1451974711yescharacteristic
536C22#1684031211yescharacteristic
537C22#1684034311yescharacteristic
538C22C22×#1175819011yescharacteristic
539C22C2×#1451979911yescharacteristic
540C22C2×#1451981011yescharacteristic
541C22#1684056211yescharacteristic
542C4—22no—
543C4—22no—
544C4—22no—
545C4—22no—
546C4—41no—
547C4—4881no—
548C4—4881no—
549C4—4881no—
550C4—2444no—
551C4—2444no—
552C4#1684040212yes—
553C4#1684041312yes—
554C2—24no—
555C2—9761no—
556C2#2144942911yescharacteristic
557C2C2×#1683990311yescharacteristic
558C2#2145504111yescharacteristic
559C2#2145637811yescharacteristic
560C2#2145652811yescharacteristic
561C2#2145713611yescharacteristic
562C2#2145715711yescharacteristic
subgroupG/H
centerC23C22×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#336C2
socleC23×C61C22×D4
prime radicalC2×#14519751C22
hypercenterC2×#336D61
hypocenterC61#25497
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#336

Complements

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

NH ≅ G/Ntype
1. C2×#16840319C2semidirect
4. #21447153C2semidirect
5. #21447066C2semidirect
6. #21446956C2semidirect
7. #21446904C2semidirect
8. #21446242C2semidirect
9. #21446173C2semidirect
10. #21446090C2semidirect
11. C2×C61×#336C2semidirect
114. C22×C4×C61SD16semidirect
283. C2×#336D61semidirect
326. C61#25497semidirect
417. C22×C4#11758190semidirect

Maximal subgroups

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

subgroupindex
C2×#168403192
C2×#168399912
#214541502
#214471532
#214470662
#214469562
#214469042
#214462422
#214461732
#214460902
C2×C61×#3362
#2549761