Finite Group Databasebeta

← #29713973

Subgroups — #29713973

Order 15616.

1134 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840472C211yesmaximal, characteristic
2C2×#16840058C211yesmaximal, characteristic, complemented
3C2×C4×#11758202C211yesmaximal, characteristic, complemented
4#21450742C211yesmaximal, characteristic
5#21447021C212yesmaximal, complemented
6#21446828C212yesmaximal, complemented
7#21446779C211yesmaximal, characteristic, complemented
8#21446612C211yesmaximal, characteristic
9#21446125C212yesmaximal, complemented
10#21446008C212yesmaximal, complemented
11C61×#1803C211yesmaximal, characteristic, complemented
12C22×#11758202C2211yescharacteristic
13#16840472—24no—
14#16840472—24no—
15C2×#14519777C2211yescharacteristic
16C2×#14519769—21no—
17C2×#14519757C2212yes—
18C2×#14519755—22no—
19C2×#14519753C2212yes—
20C2×C4×Dic122C2211yescharacteristic
21C2×C4×D244—21no—
22C2×C4×D244C2211yescharacteristic
23C2×C4⋊C4×D61C2212yes—
24#16840058—24no—
25C4×#11758202—24no—
26C2×#14519711C2212yes—
27C2×#14519710C2211yescharacteristic
28C2×#14519702C2212yes—
29C2×#14519701C2211yescharacteristic
30C2×#14519700C2211yescharacteristic
31C2×#14519689C2212yes—
32C2×#14519689C2212yes—
33C2×C4×#92418C2211yescharacteristic
34#16839684—22no—
35#16839663—21no—
36#16839661—21no—
37#16839619C2212yes—
38#16839524C2212yes—
39#16839524C2212yes—
40C2×C4×Q8×C61C2211yescharacteristic
41C2×C4⋊C8×C61C2211yescharacteristic
42C2×#103×C61C2212yes—
43C61×#384C2211yescharacteristic
44C61×#340C2212yes—
45C22×Dic122—21no—
46C22×Dic122C2311yescharacteristic
47C22×D244D411yescharacteristic
48C22×C4×D61—22no—
49C22×C4×D61C2312yes—
50C2×#11758202—22no—
51C2×#11758202—22no—
52C2×#11758202—22no—
53C22×#92418C2311yescharacteristic
54C22×#92418C2311yescharacteristic
55#14519777—22no—
56#14519777D414yes—
57#14519769—28no—
58#14519757—24no—
59#14519757—42no—
60#14519755—216no—
61#14519753—24no—
62#14519753—24no—
63C4×Dic122—22no—
64C4×Dic122—22no—
65C4×D244—28no—
66C4×D244—22no—
67C4×D244—22no—
68C2×C4⋊Dic61—21no—
69C2×C4⋊Dic61—22no—
70C2×C4⋊Dic61D411yescharacteristic
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C4⋊Dic61C2312yes—
73C2×#11758196—22no—
74C2×#11758196—22no—
75C2×#11758196—21no—
76C4⋊C4×D61—24no—
77C4⋊C4×D61—24no—
78C2×#11758198—22no—
79C2×#11758198—22no—
80C2×#11758198—22no—
81C2×#11758198D412yes—
82C2×C4×Dic61—21no—
83#14519711—24no—
84#14519710—22no—
85#14519702—24no—
86#14519701—22no—
87#14519700—22no—
88#14519689—24no—
89#14519689—42no—
90C4×#92418—22no—
91#14519684—22no—
92C22×Q8×C61C2311yescharacteristic
93C22×C8×C61C2312yes—
94C4×Q8×C61—22no—
95C4×Q8×C61D414yes—
96C2×C4⋊C4×C61—21no—
97C2×C4⋊C4×C61—21no—
98C2×C4⋊C4×C61—21no—
99C2×C4⋊C4×C61C2312yes—
100C2×C4⋊C4×C61C2311yescharacteristic
101C2×C42×C61—21no—
102C2×C42×C61C2311yescharacteristic
103C4⋊C8×C61—22no—
104#103×C61—42no—
105C23×D61C2×D412yes—
106C2×Q8×C61—21no—
107C2×Q8×C61—21no—
108C2×Q8×C61—21no—
109C2×Q8×C61—21no—
110C2×Q8×C61—21no—
111C2×Q8×C61C2×D412yes—
112C22×C4×C61—21no—
113C22×C4×C61—21no—
114C22×C4×C61—22no—
115C22×C4×C61C2×D411yescharacteristic
116C22×C4×C61C2×D411yescharacteristic
117C22×C4×C61C2411yescharacteristic
118C22×Dic61—22no—
119C22×Dic61—21no—
120C22×Dic61—21no—
121C22×Dic61C2×D412yes—
122C2×D244—22no—
123C2×D244—22no—
124C2×D244Q1614yescomplemented
125C2×C4×D61—22no—
126C2×C4×D61—22no—
127C2×C4×D61—22no—
128C2×C4×D61—22no—
129C2×C4×D61—22no—
130C2×C4×D61—22no—
131C2×C4×D61—28no—
132C2×C4×D61—82no—
133C2×C4×D61—82no—
134C2×Dic122—21no—
135C2×Dic122—21no—
136C2×Dic122—21no—
137C2×Dic122—21no—
138C2×Dic122—21no—
139C2×Dic122—24no—
140C2×Dic122—81no—
141C2×Dic122—81no—
142C2×Dic122#4112yes—
143C2×C8×C61—22no—
144C2×C8×C61—22no—
145C2×C8×C61—22no—
146C4⋊C4×C61—24no—
147C4⋊C4×C61—21no—
148C4⋊C4×C61—22no—
149C4⋊C4×C61—22no—
150C4⋊C4×C61—42no—
151C4⋊C4×C61—81no—
152C4⋊C4×C61C2×D412yes—
153C42×C61—24no—
154C42×C61C2×D412yes—
155C42×C61C2×D412yes—
156#11758202—28no—
157#11758198—28no—
158#11758198—24no—
159#11758198—44no—
160#11758198—82no—
161C4⋊Dic61—24no—
162C4⋊Dic61—22no—
163C4⋊Dic61—22no—
164C4⋊Dic61—21no—
165C4⋊Dic61—21no—
166C4⋊Dic61—44no—
167C4⋊Dic61C2×D414yes—
168#11758196—28no—
169#11758196—28no—
170#11758196—42no—
171C4×Dic61—42no—
172C2×#92418—21no—
173C2×#92418—21no—
174C2×#92418—21no—
175C2×#92418—21no—
176C2×#92418—21no—
177C2×#92418#4112yes—
178C23×C61C22×D411yescharacteristic
179C22×D61—22no—
180C22×D61—22no—
181C22×D61—22no—
182C22×D61—22no—
183C22×D61—22no—
184C22×D61—22no—
185C22×D61C2×Q1614yes—
186Q8×C61—22no—
187Q8×C61—22no—
188Q8×C61—41no—
189Q8×C61—41no—
190C2×C4×C61—21no—
191C2×C4×C61—21no—
192C2×C4×C61—22no—
193C2×C4×C61—22no—
194C2×C4×C61—41no—
195C2×C4×C61—41no—
196C2×C4×C61—41no—
197C2×C4×C61—41no—
198C2×C4×C61—42no—
199C2×C4×C61—42no—
200C2×C4×C61—42no—
201C2×C4×C61#12012yes—
202C2×C4×C61#12012yes—
203C2×C4×C61C2×Q1612yes—
204C2×C4×C61#13712yes—
205C2×C4×C61C22×D411yescharacteristic
206C2×C4×C61C22×D411yescharacteristic
207C2×C4×C61C2×#4111yescharacteristic
208C2×C4×C61C2×#4111yescharacteristic
209C2×C4×C61H(5,𝔽2)11yescharacteristic
210C2×C4×C612−1+411yescharacteristic
211C2×Dic61—22no—
212C2×Dic61—24no—
213C2×Dic61—22no—
214C2×Dic61—22no—
215C2×Dic61—22no—
216C2×Dic61—41no—
217C2×Dic61—41no—
218C2×Dic61—42no—
219C2×Dic61—42no—
220C2×Dic61—41no—
221C2×Dic61—41no—
222C2×Dic61—41no—
223D244—28no—
224C4×D61—42no—
225C4×D61—42no—
226C4×D61—42no—
227C4×D61—42no—
228C4×D61—88no—
229Dic122—22no—
230Dic122—22no—
231Dic122—41no—
232Dic122—41no—
233Dic122—84no—
234C8×C61—42no—
235#92418—22no—
236#92418—41no—
237#24122—611nomaximal, Sylow
238C22×C61C2×#12011yescharacteristic
239C22×C61#54611yescharacteristic
240C22×C61#54711yescharacteristic
241C22×C61C22×Q1611yescharacteristic
242C22×C61C2×#13711yescharacteristic
243C22×C61#57611yescharacteristic
244C22×C61#57711yescharacteristic
245C2×D61—24no—
246C2×D61—24no—
247C2×D61—24no—
248C2×D61—42no—
249C2×D61—42no—
250C2×D61—42no—
251C2×D61—42no—
252C4×C61—22no—
253C4×C61—42no—
254C4×C61—42no—
255C4×C61—82no—
256C4×C61#45114yes—
257C4×C61#52511yescharacteristic
258C4×C61#53711yescharacteristic
259C4×C61#54811yescharacteristic
260C4×C61#55011yescharacteristic
261Dic61—42no—
262Dic61—42no—
263Dic61—44no—
264Dic61—81no—
265C2×#554—611no—
266C2×#477—611no—
267C2×C4×Q16—611no—
268#2538—611no—
269#2036—612no—
270#2001—612no—
271#1990—611no—
272#1954—611no—
273#1872—612no—
274#1852—612no—
275#1803D6111yescharacteristic, complemented
276C2×C61#240711yescharacteristic
277C2×C61C2×#45111yescharacteristic
278C2×C61#301211yescharacteristic
279C2×C61#329911yescharacteristic
280C2×C61#333311yescharacteristic
281C2×C61#333811yescharacteristic
282C2×C61#335611yescharacteristic
283D61—44no—
284C22×Q16—611no—
285#554—1224no—
286#554—1224no—
287C2×C4⋊Q8—611no—
288C2×#125—1221no—
289C2×C22⋊Q8—612no—
290C2×C22⋊Q8—612no—
291C2×C22⋊Q8—1222no—
292C2×C4×Q8—611no—
293C2×C4×Q8C2×D6111yescharacteristic
294C2×C4×D4—611no—
295C2×C4×D4—1221no—
296C22×C4⋊C4—612no—
297#477—1224no—
298C4×Q16—1224no—
299C2×#107—612no—
300C2×#107—611no—
301C2×C4⋊C8C2×D6111yescharacteristic
302C2×#103—611no—
303C2×#103—612no—
304C2×#103—611no—
305C2×#103C2×D6112yes—
306C2×C22⋊C8—612no—
307C2×C22⋊C8—612no—
308C2×C4×C8—611no—
309#396—1222no—
310#390—1221no—
311#389—1221no—
312#384C2×D6111yescharacteristic
313#380—612no—
314#340—612no—
315#340—612no—
316#340C2×D6112yes—
317C61#2412211yescharacteristic, complemented, Sylow
318C22×Q8—611no—
319C22×Q8—1221no—
320C22×Q8C22×D6111yescharacteristic
321C22×D4—611no—
322C23×C4—612no—
323C23×C4—1222no—
324C2×Q16—1222no—
325C2×Q16—1222no—
326C2×Q16—1222no—
327C22×C8—611no—
328C22×C8—611no—
329C22×C8C22×D6112yes—
330C4⋊Q8—614no—
331C4⋊Q8—1222no—
332#125—1228no—
333C22⋊Q8—12216no—
334C22⋊Q8—1224no—
335C22⋊Q8—1224no—
336C22⋊Q8—1224no—
337C22⋊Q8—2442no—
338C4×Q8—22no—
339C4×Q8—1222no—
340C4×Q8—1222no—
341C4×Q8D24414yes—
342C4×D4—1228no—
343C4×D4—1222no—
344C4×D4—1222no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—612no—
349C2×C4⋊C4—611no—
350C2×C4⋊C4—611no—
351C2×C4⋊C4—1224no—
352C2×C4⋊C4—1222no—
353C2×C4⋊C4—1222no—
354C2×C4⋊C4—1221no—
355C2×C4⋊C4—1224no—
356C2×C4⋊C4—1222no—
357C2×C4⋊C4—1221no—
358C2×C4⋊C4C22×D6112yes—
359C2×C4⋊C4C22×D6111yescharacteristic
360C2×C22⋊C4—612no—
361C2×C22⋊C4—1222no—
362C2×C22⋊C4—1222no—
363C2×C22⋊C4—1222no—
364C2×C42—21no—
365C2×C42—1221no—
366C2×C42C22×D6111yescharacteristic
367#107—1224no—
368#107—1222no—
369C4⋊C8—22no—
370#103—42no—
371#103—1224no—
372#103—1222no—
373#103—1222no—
374C22⋊C8—1224no—
375C22⋊C8—2442no—
376C4×C8—1222no—
377#95—1222no—
378C24—612no—
379C2×Q8—21no—
380C2×Q8—21no—
381C2×Q8—21no—
382C2×Q8—21no—
383C2×Q8—21no—
384C2×Q8—612no—
385C2×Q8—1221no—
386C2×Q8—1224no—
387C2×Q8—1221no—
388C2×Q8—1221no—
389C2×Q8—1221no—
390C2×Q8—1221no—
391C2×Q8—4881no—
392C2×Q8—4881no—
393C2×Q8C2×D24412yes—
394C2×D4—614no—
395C2×D4—1222no—
396C2×D4—1222no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—612no—
401C22×C4—1228no—
402C22×C4—1221no—
403C22×C4—1222no—
404C22×C4—1222no—
405C22×C4—1222no—
406C22×C4—1222no—
407C22×C4—1222no—
408C22×C4—1222no—
409C22×C4—1222no—
410C22×C4—1221no—
411C22×C4—4882no—
412C22×C4—4882no—
413C22×C4D4×D6111yescharacteristic
414C22×C4D4×D6111yescharacteristic
415C22×C4C23×D6111yescharacteristic
416Q16—1228no—
417C2×C8—22no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—612no—
421C2×C8—1221no—
422C2×C8—1221no—
423C2×C8—1221no—
424C2×C8—1221no—
425C2×C8—1221no—
426C4⋊C4—21no—
427C4⋊C4—24no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—42no—
431C4⋊C4—81no—
432C4⋊C4—614no—
433C4⋊C4—1228no—
434C4⋊C4—1228no—
435C4⋊C4—1222no—
436C4⋊C4—1222no—
437C4⋊C4—1224no—
438C4⋊C4—1221no—
439C4⋊C4—1221no—
440C4⋊C4—2444no—
441C4⋊C4—2442no—
442C4⋊C4C2×D24412yes—
443C22⋊C4—1228no—
444C22⋊C4—1224no—
445C22⋊C4—4882no—
446C22⋊C4—2444no—
447C42—24no—
448C42—2442no—
449C42C2×D24412yes—
450C42D4×D6112yes—
451C23—614no—
452C23—1222no—
453C23—1222no—
454C23—1222no—
455C23—1222no—
456C23—1222no—
457C23—1222no—
458C23C2×D4×D6111yescharacteristic
459Q8—22no—
460Q8—22no—
461Q8—41no—
462Q8—41no—
463Q8—1222no—
464Q8—1222no—
465Q8—4884no—
466Q8—2441no—
467Q8—2441no—
468D4—1228no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—1224no—
481C2×C4—1222no—
482C2×C4—1222no—
483C2×C4—1222no—
484C2×C4—1222no—
485C2×C4—4888no—
486C2×C4—2441no—
487C2×C4—2441no—
488C2×C4—2441no—
489C2×C4—2441no—
490C2×C4—2442no—
491C2×C4—2442no—
492C2×C4—2441no—
493C2×C4—2442no—
494C2×C4—2442no—
495C2×C4—2442no—
496C2×C4—2442no—
497C2×C4#1451974712yes—
498C2×C4#1451974712yes—
499C2×C4Q16×D6112yes—
500C2×C4#1451980612yes—
501C2×C4C2×D4×D6111yescharacteristic
502C2×C4C22×D24411yescharacteristic
503C2×C4#41×D6111yescharacteristic
504C2×C4#41×D6111yescharacteristic
505C2×C4#1451982211yescharacteristic
506C2×C4#1451982511yescharacteristic
507C8—42no—
508C8—1222no—
509C8—2441no—
510C22—1224no—
511C22—1224no—
512C22—1224no—
513C22—2442no—
514C22—2442no—
515C22—2442no—
516C22—2442no—
517C22C2×#1451974711yescharacteristic
518C22#1684040911yescharacteristic
519C22#1684041411yescharacteristic
520C22C2×Q16×D6111yescharacteristic
521C22C2×#1451980611yescharacteristic
522C22#1684056611yescharacteristic
523C22#1684057111yescharacteristic
524C4—22no—
525C4—42no—
526C4—42no—
527C4—82no—
528C4—4881no—
529C4—2444no—
530C4—2442no—
531C4—2442no—
532C4#1683991914yes—
533C4#1684027411yescharacteristic
534C4#1684034311yescharacteristic
535C4#1684042211yescharacteristic
536C4#1684043011yescharacteristic
537C2—2444no—
538C2#2144916511yescharacteristic
539C2C2×#1683991911yescharacteristic
540C2#2145512511yescharacteristic
541C2#2145769811yescharacteristic
542C2#2145816011yescharacteristic
543C2#2145821011yescharacteristic
544C2#2145843111yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1803C2
socleC23×C61C22×D4
prime radicalC22×D244D4
hypercenter#1803D61
hypocenterC61#24122
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1803

Complements

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

NH ≅ G/Ntype
2. C2×#16840058C2semidirect
3. C2×C4×#11758202C2semidirect
5. #21447021C2semidirect
6. #21446828C2semidirect
7. #21446779C2semidirect
9. #21446125C2semidirect
10. #21446008C2semidirect
11. C61×#1803C2semidirect
124. C2×D244Q16semidirect
275. #1803D61semidirect
317. C61#24122semidirect

Maximal subgroups

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

subgroupindex
C2×#168404722
C2×#168400582
C2×C4×#117582022
#214507422
#214470212
#214468282
#214467792
#214466122
#214461252
#214460082
C61×#18032
#2412261