Finite Group Databasebeta

← #29713972

Subgroups — #29713972

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×#16840057C211yesmaximal, characteristic, complemented
3#21450737C211yesmaximal, characteristic
4#21447020C212yesmaximal, complemented
5#21446826C212yesmaximal, complemented
6#21446780C211yesmaximal, characteristic, complemented
7#21446613C211yesmaximal, characteristic
8#21446123C212yesmaximal, complemented
9#21446005C212yesmaximal, complemented
10#21445824C211yesmaximal, characteristic, complemented
11C2×C4×Q16×C61C211yesmaximal, characteristic, complemented
12#16840472—24no—
13#16840472—24no—
14C2×#14519777C2211yescharacteristic
15C2×#14519769—21no—
16C2×#14519755—22no—
17C2×#14519753C2212yes—
18C2×#14519753C2212yes—
19C2×C4×D244—21no—
20C2×C4×D244C2211yescharacteristic
21C2×C4⋊C4×D61C2212yes—
22#16840057—24no—
23C2×#14519710C2212yes—
24C2×C4⋊#92418C2211yescharacteristic
25C2×#14519701C2212yes—
26C2×#14519701C2212yes—
27C2×#14519689C2212yes—
28C2×#14519689C2212yes—
29#16839684—22no—
30#16839664—21no—
31#16839660—21no—
32#16839632C2211yescharacteristic
33#16839619C2212yes—
34#16839525C2212yes—
35#16839525C2212yes—
36#16839524C2212yes—
37C22×Q16×C61C2211yescharacteristic
38C2×C4×Q8×C61C2211yescharacteristic
39C2×C4×Q8×C61C2211yescharacteristic
40C4×Q16×C61—24no—
41C2×#107×C61C2211yescharacteristic
42C2×#103×C61C2211yescharacteristic
43C2×#103×C61C2211yescharacteristic
44C2×C4×C8×C61C2211yescharacteristic
45C22×Dic122—21no—
46C22×D244D411yescharacteristic
47C22×C4×D61—22no—
48C22×C4×D61C2312yes—
49C22×#92418C2312yes—
50#14519777—22no—
51#14519777D414yes—
52#14519769—28no—
53#14519755—216no—
54#14519753—24no—
55#14519753—24no—
56#14519753—24no—
57#14519753—42no—
58C4×D244—28no—
59C4×D244—22no—
60C4×D244—22no—
61C2×C4⋊Dic61—21no—
62C2×C4⋊Dic61—21no—
63C2×C4⋊Dic61—21no—
64C2×C4⋊Dic61D411yescharacteristic
65C2×C4⋊Dic61C2312yes—
66C2×C4⋊Dic61C2312yes—
67C2×#11758196—22no—
68C2×#11758196—22no—
69C2×#11758196—22no—
70C4⋊C4×D61—24no—
71C4⋊C4×D61—24no—
72C2×#11758198—22no—
73C2×#11758198—22no—
74C2×#11758198—22no—
75C2×#11758198D412yes—
76#14519710—24no—
77C4⋊#92418—22no—
78#14519701—24no—
79#14519701—42no—
80#14519689—24no—
81#14519689—42no—
82#14519684—22no—
83C22×Q8×C61C2311yescharacteristic
84C22×Q8×C61C2311yescharacteristic
85C2×Q16×C61—22no—
86C2×Q16×C61—22no—
87C2×Q16×C61—22no—
88C22×C8×C61C2311yescharacteristic
89C22×C8×C61C2311yescharacteristic
90C4×Q8×C61—22no—
91C4×Q8×C61—22no—
92C4×Q8×C61—22no—
93C4×Q8×C61D414yes—
94C2×C4⋊C4×C61—21no—
95C2×C4⋊C4×C61—21no—
96C2×C4⋊C4×C61C2311yescharacteristic
97C2×C4⋊C4×C61C2311yescharacteristic
98C2×C42×C61—21no—
99C2×C42×C61—21no—
100C2×C42×C61C2311yescharacteristic
101#107×C61—22no—
102#103×C61—22no—
103#103×C61—22no—
104C4×C8×C61—22no—
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×C61—21no—
112C2×Q8×C61—21no—
113C2×Q8×C61—21no—
114C2×Q8×C61—21no—
115C2×Q8×C61—21no—
116C2×Q8×C61C2×D412yes—
117C2×Q8×C61#4112yes—
118C22×C4×C61—21no—
119C22×C4×C61—21no—
120C22×C4×C61—21no—
121C22×C4×C61—21no—
122C22×C4×C61C2×D411yescharacteristic
123C22×C4×C61C2×D411yescharacteristic
124C22×C4×C61C2411yescharacteristic
125C22×Dic61—22no—
126C22×Dic61—22no—
127C22×Dic61C2×D412yes—
128C2×D244—22no—
129C2×D244—22no—
130C2×D244Q1614yescomplemented
131C2×C4×D61—22no—
132C2×C4×D61—22no—
133C2×C4×D61—22no—
134C2×C4×D61—22no—
135C2×C4×D61—22no—
136C2×C4×D61—22no—
137C2×C4×D61—28no—
138C2×C4×D61—82no—
139C2×C4×D61—82no—
140C2×Dic122—24no—
141C2×Dic122—81no—
142C2×Dic122—81no—
143Q16×C61—28no—
144C2×C8×C61—21no—
145C2×C8×C61—21no—
146C2×C8×C61—21no—
147C2×C8×C61—21no—
148C2×C8×C61—21no—
149C2×C8×C61#4112yes—
150C4⋊C4×C61—24no—
151C4⋊C4×C61—21no—
152C4⋊C4×C61—21no—
153C4⋊C4×C61—21no—
154C4⋊C4×C61—42no—
155C4⋊C4×C61C2×D412yes—
156C42×C61—24no—
157C42×C61—42no—
158C42×C61C2×D412yes—
159C42×C61C2×D412yes—
160#11758198—28no—
161#11758198—24no—
162#11758198—44no—
163#11758198—82no—
164C4⋊Dic61—24no—
165C4⋊Dic61—22no—
166C4⋊Dic61—22no—
167C4⋊Dic61—22no—
168C4⋊Dic61—22no—
169C4⋊Dic61—42no—
170C4⋊Dic61—81no—
171C4⋊Dic61C2×D414yes—
172#11758196—28no—
173#11758196—28no—
174#11758196—44no—
175C2×#92418—22no—
176C2×#92418—22no—
177C2×#92418—22no—
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—22no—
189Q8×C61—22no—
190Q8×C61—41no—
191Q8×C61—41no—
192Q8×C61—41no—
193Q8×C61—41no—
194C2×C4×C61—21no—
195C2×C4×C61—21no—
196C2×C4×C61—22no—
197C2×C4×C61—22no—
198C2×C4×C61—22no—
199C2×C4×C61—41no—
200C2×C4×C61—41no—
201C2×C4×C61—41no—
202C2×C4×C61—41no—
203C2×C4×C61—41no—
204C2×C4×C61—41no—
205C2×C4×C61—41no—
206C2×C4×C61—41no—
207C2×C4×C61—41no—
208C2×C4×C61#12012yes—
209C2×C4×C61#12012yes—
210C2×C4×C61C2×Q1612yes—
211C2×C4×C61#13712yes—
212C2×C4×C61C22×D411yescharacteristic
213C2×C4×C61C22×D411yescharacteristic
214C2×C4×C61C2×#4111yescharacteristic
215C2×C4×C61C2×#4111yescharacteristic
216C2×C4×C61H(5,𝔽2)11yescharacteristic
217C2×C4×C612−1+411yescharacteristic
218C2×Dic61—22no—
219C2×Dic61—24no—
220C2×Dic61—22no—
221C2×Dic61—22no—
222C2×Dic61—42no—
223C2×Dic61—42no—
224C2×Dic61—42no—
225C2×Dic61—42no—
226C2×Dic61—42no—
227D244—28no—
228C4×D61—42no—
229C4×D61—42no—
230C4×D61—42no—
231C4×D61—42no—
232C4×D61—88no—
233Dic122—84no—
234C8×C61—22no—
235C8×C61—41no—
236#92418—42no—
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—42no—
256C4×C61—81no—
257C4×C61#45114yes—
258C4×C61#52511yescharacteristic
259C4×C61#53711yescharacteristic
260C4×C61#54811yescharacteristic
261C4×C61#55011yescharacteristic
262Dic61—42no—
263Dic61—44no—
264Dic61—82no—
265C2×#554—611no—
266C2×#477—611no—
267C2×C4×Q16D6111yescharacteristic, complemented
268#2538—611no—
269#2036—612no—
270#2001—612no—
271#1990—611no—
272#1954—611no—
273#1872—612no—
274#1852—612no—
275#1803—611no—
276C2×C61#240711yescharacteristic
277C2×C61C2×#45111yescharacteristic
278C2×C61#301211yescharacteristic
279C2×C61#329911yescharacteristic
280C2×C61#333311yescharacteristic
281C2×C61#333811yescharacteristic
282C2×C61#335611yescharacteristic
283D61—44no—
284C22×Q16C2×D6111yescharacteristic
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×Q8C2×D6111yescharacteristic
293C2×C4×Q8C2×D6111yescharacteristic
294C2×C4×D4—611no—
295C2×C4×D4—1221no—
296C22×C4⋊C4—612no—
297#477—1224no—
298C4×Q16—24no—
299C2×#107—612no—
300C2×#107C2×D6111yescharacteristic
301C2×C4⋊C8—611no—
302C2×#103—612no—
303C2×#103—612no—
304C2×#103C2×D6111yescharacteristic
305C2×#103C2×D6111yescharacteristic
306C2×C22⋊C8—612no—
307C2×C22⋊C8—612no—
308C2×C4×C8C2×D6111yescharacteristic
309#396—1222no—
310#390—1221no—
311#389—1221no—
312#384—611no—
313#380—612no—
314#340—612no—
315#340—612no—
316#340—612no—
317C61#2412211yescharacteristic, complemented, Sylow
318C22×Q8—1221no—
319C22×Q8C22×D6111yescharacteristic
320C22×Q8C22×D6111yescharacteristic
321C22×D4—611no—
322C23×C4—612no—
323C23×C4—1222no—
324C2×Q16—22no—
325C2×Q16—22no—
326C2×Q16—22no—
327C22×C8—612no—
328C22×C8C22×D6111yescharacteristic
329C22×C8C22×D6111yescharacteristic
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—22no—
340C4×Q8—22no—
341C4×Q8C22⋊D6114yes—
342C4×D4—1228no—
343C4×D4—1222no—
344C4×D4—1222no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—612no—
348C2×C4⋊C4—611no—
349C2×C4⋊C4—612no—
350C2×C4⋊C4—1224no—
351C2×C4⋊C4—1222no—
352C2×C4⋊C4—1222no—
353C2×C4⋊C4—1221no—
354C2×C4⋊C4—1224no—
355C2×C4⋊C4—1221no—
356C2×C4⋊C4—1222no—
357C2×C4⋊C4—1221no—
358C2×C4⋊C4C22×D6111yescharacteristic
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—21no—
366C2×C42C22×D6111yescharacteristic
367#107—22no—
368#107—1224no—
369C4⋊C8—1222no—
370#103—22no—
371#103—22no—
372#103—1224no—
373#103—2442no—
374C22⋊C8—1224no—
375C22⋊C8—2442no—
376C4×C8—22no—
377#95—1222no—
378C24—612no—
379C2×Q8—21no—
380C2×Q8—21no—
381C2×Q8—21no—
382C2×Q8—21no—
383C2×Q8—21no—
384C2×Q8—21no—
385C2×Q8—21no—
386C2×Q8—21no—
387C2×Q8—21no—
388C2×Q8—21no—
389C2×Q8—1224no—
390C2×Q8—4881no—
391C2×Q8—4881no—
392C2×Q8#1175822212yes—
393C2×Q8C2×C22⋊D6112yes—
394C2×D4—614no—
395C2×D4—1222no—
396C2×D4—1222no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—612no—
402C22×C4—1228no—
403C22×C4—1222no—
404C22×C4—1222no—
405C22×C4—1222no—
406C22×C4—1222no—
407C22×C4—1222no—
408C22×C4—1222no—
409C22×C4—1222no—
410C22×C4—1222no—
411C22×C4—4882no—
412C22×C4—4882no—
413C22×C4D4×D6111yescharacteristic
414C22×C4D4×D6111yescharacteristic
415C22×C4C23×D6111yescharacteristic
416Q16—28no—
417C2×C8—21no—
418C2×C8—21no—
419C2×C8—21no—
420C2×C8—21no—
421C2×C8—21no—
422C2×C8—1222no—
423C2×C8—1222no—
424C2×C8—1222no—
425C2×C8#1175822212yes—
426C4⋊C4—21no—
427C4⋊C4—24no—
428C4⋊C4—21no—
429C4⋊C4—21no—
430C4⋊C4—42no—
431C4⋊C4—614no—
432C4⋊C4—1228no—
433C4⋊C4—1228no—
434C4⋊C4—1222no—
435C4⋊C4—1222no—
436C4⋊C4—1224no—
437C4⋊C4—1222no—
438C4⋊C4—1222no—
439C4⋊C4—4881no—
440C4⋊C4—2444no—
441C4⋊C4—2442no—
442C4⋊C4C2×C22⋊D6112yes—
443C22⋊C4—1228no—
444C22⋊C4—1224no—
445C22⋊C4—4882no—
446C22⋊C4—2444no—
447C42—24no—
448C42—42no—
449C42D4×D6112yes—
450C42C2×C22⋊D6112yes—
451C23—614no—
452C23—1222no—
453C23—1222no—
454C23—1222no—
455C23—1222no—
456C23—1222no—
457C23—1222no—
458C23C2×D4×D6111yescharacteristic
459Q8—22no—
460Q8—22no—
461Q8—22no—
462Q8—22no—
463Q8—41no—
464Q8—41no—
465Q8—41no—
466Q8—41no—
467Q8—4884no—
468D4—1228no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—1224no—
484C2×C4—1222no—
485C2×C4—1222no—
486C2×C4—1222no—
487C2×C4—4888no—
488C2×C4—2442no—
489C2×C4—2442no—
490C2×C4—2442no—
491C2×C4—2442no—
492C2×C4—2442no—
493C2×C4—2442no—
494C2×C4—2442no—
495C2×C4—2442no—
496C2×C4—2442no—
497C2×C4#1451974512yes—
498C2×C4#1451974512yes—
499C2×C4Q16×D6112yes—
500C2×C4#1451980912yes—
501C2×C4C2×D4×D6111yescharacteristic
502C2×C4C22×C22⋊D6111yescharacteristic
503C2×C4C2×#1175822211yescharacteristic
504C2×C4C2×#1175822211yescharacteristic
505C2×C4#1451982311yescharacteristic
506C2×C4#1451982411yescharacteristic
507C8—22no—
508C8—41no—
509C8—2442no—
510C22—1224no—
511C22—1224no—
512C22—1224no—
513C22—2442no—
514C22—2442no—
515C22—2442no—
516C22—2442no—
517C22C2×#1451974511yescharacteristic
518C22#1684040811yescharacteristic
519C22#1684041711yescharacteristic
520C22C2×Q16×D6111yescharacteristic
521C22C2×#1451980911yescharacteristic
522C22#1684056411yescharacteristic
523C22#1684056911yescharacteristic
524C4—22no—
525C4—42no—
526C4—42no—
527C4—42no—
528C4—81no—
529C4—4882no—
530C4—2444no—
531C4—2442no—
532C4#1683991614yes—
533C4#1684027311yescharacteristic
534C4#1684034011yescharacteristic
535C4#1684042111yescharacteristic
536C4#1684043311yescharacteristic
537C2—2444no—
538C2#2144916411yescharacteristic
539C2C2×#1683991611yescharacteristic
540C2#2145511911yescharacteristic
541C2#2145769711yescharacteristic
542C2#2145815911yescharacteristic
543C2#2145821311yescharacteristic
544C2#2145843611yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C4×Q16×C61C2
socleC23×C61C22×D4
prime radicalC22×D244D4
hypercenterC2×C4×Q16D61
hypocenterC61#24122
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×Q16

Complements

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

NH ≅ G/Ntype
2. C2×#16840057C2semidirect
4. #21447020C2semidirect
5. #21446826C2semidirect
6. #21446780C2semidirect
8. #21446123C2semidirect
9. #21446005C2semidirect
10. #21445824C2semidirect
11. C2×C4×Q16×C61C2semidirect
130. C2×D244Q16semidirect
267. C2×C4×Q16D61semidirect
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×#168400572
#214507372
#214470202
#214468262
#214467802
#214466132
#214461232
#214460052
#214458242
C2×C4×Q16×C612
#2412261