Finite Group Databasebeta

← #29714076

Subgroups — #29714076

Order 15616.

1184 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, complemented
2C2×#16840042C211yesmaximal, characteristic, complemented
3#21450716C211yesmaximal, characteristic
4#21447020C212yesmaximal, complemented
5#21446819C212yesmaximal, complemented
6#21446792C211yesmaximal, characteristic, complemented
7#21446617C211yesmaximal, characteristic, complemented
8#21446133C212yesmaximal, complemented
9#21446005C212yesmaximal, complemented
10#21445834C211yesmaximal, characteristic, complemented
11C2×C61×#440C211yesmaximal, characteristic, complemented
12#16840472—24no—
13#16840472—24no—
14C2×#14519777C2211yescharacteristic
15C2×#14519769—21no—
16C2×#14519755—22no—
17C2×#14519753C2212yes—
18C2×#14519748C2212yes—
19C2×C4×D244—21no—
20C2×C4×D244C2211yescharacteristic
21C2×C4⋊C4×D61C2212yes—
22#16840042—24no—
23C2×#14519710C2212yes—
24C2×C4⋊#92418C2211yescharacteristic
25C2×#14519701C2212yes—
26C2×#14519698C2212yes—
27C2×#14519689C2212yes—
28C2×#14519689C2212yes—
29#16839687—21no—
30#16839671—21no—
31#16839641—22no—
32#16839628C2211yescharacteristic
33#16839619C2212yes—
34#16839525C2212yes—
35#16839525C2212yes—
36#16839524C2212yes—
37C22×SD16×C61C2211yescharacteristic
38C2×C4×Q8×C61C2211yescharacteristic
39C2×C4×D4×C61C2211yescharacteristic
40C61×#440—24no—
41C2×#107×C61C2211yescharacteristic
42C2×#103×C61C2211yescharacteristic
43C2×#102×C61C2211yescharacteristic
44C2×#97×C61C2211yescharacteristic
45C22×Dic122—21no—
46C22×D244D411yescharacteristic
47C22×C22⋊D61—22no—
48C22×C4×D61—22no—
49C22×C4×D61C2312yes—
50C22×#92418C2312yes—
51#14519777—22no—
52#14519777D414yescomplemented
53#14519769—28no—
54#14519755—216no—
55#14519753—24no—
56#14519753—24no—
57#14519748—24no—
58#14519748—42no—
59C4×D244—28no—
60C4×D244—22no—
61C4×D244—22no—
62C2×C4⋊Dic61—21no—
63C2×C4⋊Dic61D411yescharacteristic
64C2×C4⋊Dic61C2312yes—
65C2×C4⋊Dic61C2312yes—
66C2×#11758196—22no—
67C2×#11758196—22no—
68C4⋊C4×D61—24no—
69C4⋊C4×D61—24no—
70C2×C22⋊Dic61—22no—
71C2×#11758198—22no—
72C2×#11758198—22no—
73C2×#11758198D412yes—
74#14519710—24no—
75C4⋊#92418—22no—
76#14519701—42no—
77#14519698—24no—
78#14519689—24no—
79#14519689—42no—
80#14519684—21no—
81#14519684—21no—
82#14519684—22no—
83C22×Q8×C61C2311yescharacteristic
84C22×D4×C61C2311yescharacteristic
85C23×C4×C61—21no—
86C2×SD16×C61—22no—
87C2×SD16×C61—22no—
88C2×SD16×C61—22no—
89C22×C8×C61C2311yescharacteristic
90C22×C8×C61C2311yescharacteristic
91C4×Q8×C61—22no—
92C4×Q8×C61D414yescomplemented
93C4×D4×C61—22no—
94C4×D4×C61—22no—
95C2×C4⋊C4×C61—21no—
96C2×C4⋊C4×C61C2311yescharacteristic
97C2×C4⋊C4×C61C2311yescharacteristic
98C2×C22⋊C4×C61—21no—
99C2×C42×C61—21no—
100C2×C42×C61C2311yescharacteristic
101#107×C61—22no—
102#103×C61—22no—
103#102×C61—22no—
104#97×C61—22no—
105C24×C61—21no—
106C23×D61C2×D412yes—
107C2×Q8×C61—21no—
108C2×Q8×C61—21no—
109C2×Q8×C61—21no—
110C2×Q8×C61—21no—
111C2×Q8×C61—21no—
112C2×Q8×C61C2×D412yes—
113C2×D4×C61—21no—
114C2×D4×C61—21no—
115C2×D4×C61—21no—
116C2×D4×C61—21no—
117C2×D4×C61—21no—
118C2×D4×C61#4112yes—
119C22×C4×C61—21no—
120C22×C4×C61—21no—
121C22×C4×C61—21no—
122C22×C4×C61—42no—
123C22×C4×C61—82no—
124C22×C4×C61C2×D411yescharacteristic
125C22×C4×C61C2×D411yescharacteristic
126C22×C4×C61C2411yescharacteristic
127C2×C22⋊D61—44no—
128C2×C22⋊D61—44no—
129C2×C22⋊D61—82no—
130C22×Dic61—22no—
131C22×Dic61—22no—
132C22×Dic61C2×D412yes—
133C2×D244—24no—
134C2×D244—22no—
135C2×C4×D61—22no—
136C2×C4×D61—22no—
137C2×C4×D61—22no—
138C2×C4×D61—22no—
139C2×C4×D61—22no—
140C2×C4×D61—22no—
141C2×C4×D61—28no—
142C2×C4×D61—82no—
143C2×C4×D61—82no—
144C2×Dic122—24no—
145C2×Dic122—82no—
146SD16×C61—28no—
147C2×C8×C61—21no—
148C2×C8×C61—21no—
149C2×C8×C61—21no—
150C2×C8×C61—21no—
151C2×C8×C61—21no—
152C2×C8×C61#4112yes—
153C4⋊C4×C61—24no—
154C4⋊C4×C61—21no—
155C4⋊C4×C61—21no—
156C4⋊C4×C61—21no—
157C4⋊C4×C61C2×D412yes—
158C22⋊C4×C61—42no—
159C42×C61—24no—
160C42×C61C2×D412yes—
161C42×C61C2×D412yes—
162C22⋊Dic61—44no—
163#11758198—28no—
164#11758198—24no—
165#11758198—82no—
166C4⋊Dic61—24no—
167C4⋊Dic61—22no—
168C4⋊Dic61—22no—
169C4⋊Dic61—22no—
170C4⋊Dic61—22no—
171C4⋊Dic61C2×D414yes—
172#11758196—28no—
173#11758196—28no—
174C2×#92418—22no—
175C2×#92418—22no—
176C2×#92418—22no—
177C23×C61—22no—
178C23×C61—41no—
179C23×C61—41no—
180C23×C61—42no—
181C23×C61—42no—
182C23×C61C22×D411yescharacteristic
183C22×D61—22no—
184C22×D61—22no—
185C22×D61—22no—
186C22×D61—22no—
187C22×D61—22no—
188C22×D61—22no—
189C22×D61#13714yes—
190Q8×C61—22no—
191Q8×C61—22no—
192Q8×C61—41no—
193Q8×C61—41no—
194D4×C61—22no—
195D4×C61—22no—
196D4×C61—41no—
197D4×C61—41no—
198C2×C4×C61—22no—
199C2×C4×C61—22no—
200C2×C4×C61—22no—
201C2×C4×C61—22no—
202C2×C4×C61—41no—
203C2×C4×C61—41no—
204C2×C4×C61—42no—
205C2×C4×C61—41no—
206C2×C4×C61—42no—
207C2×C4×C61—84no—
208C2×C4×C61#12012yes—
209C2×C4×C61#12012yes—
210C2×C4×C61C22×D411yescharacteristic
211C2×C4×C61C22×D411yescharacteristic
212C2×C4×C61C2×#4111yescharacteristic
213C2×C4×C61C2×#4111yescharacteristic
214C2×C4×C61H(5,𝔽2)11yescharacteristic
215C2×C4×C612−1+411yescharacteristic
216C22⋊D61—88no—
217C2×Dic61—22no—
218C2×Dic61—24no—
219C2×Dic61—22no—
220C2×Dic61—22no—
221C2×Dic61—42no—
222C2×Dic61—42no—
223C2×Dic61—42no—
224C2×Dic61—42no—
225C2×Dic61—42no—
226D244—28no—
227C4×D61—42no—
228C4×D61—42no—
229C4×D61—42no—
230C4×D61—42no—
231C4×D61—88no—
232Dic122—84no—
233C8×C61—22no—
234C8×C61—41no—
235#92418—42no—
236#24130—611nomaximal, Sylow
237C22×C61—42no—
238C22×C61—44no—
239C22×C61—82no—
240C22×C61—81no—
241C22×C61—81no—
242C22×C61C2×#12011yescharacteristic
243C22×C61#54611yescharacteristic
244C22×C61#54711yescharacteristic
245C22×C61C2×#13712yes—
246C22×C61#57612yes—
247C2×D61—24no—
248C2×D61—24no—
249C2×D61—24no—
250C2×D61—42no—
251C2×D61—42no—
252C2×D61—42no—
253C2×D61—42no—
254C4×C61—22no—
255C4×C61—22no—
256C4×C61—42no—
257C4×C61—42no—
258C4×C61—81no—
259C4×C61#52511yescharacteristic
260C4×C61#53711yescharacteristic
261C4×C61#54811yescharacteristic
262C4×C61#55011yescharacteristic
263Dic61—42no—
264Dic61—44no—
265Dic61—82no—
266C2×#554—611no—
267C2×#474—611no—
268C2×#440D6111yescharacteristic, complemented
269#2535—611no—
270#2036—612no—
271#2000—612no—
272#1993—611no—
273#1955—611no—
274#1874—612no—
275#1852—612no—
276#1805—611no—
277C2×C61—82no—
278C2×C61#240711yescharacteristic
279C2×C61#300411yescharacteristic
280C2×C61#300811yescharacteristic
281C2×C61#331412yes—
282C2×C61#334612yes—
283D61—44no—
284C22×SD16C2×D6111yescharacteristic
285#554—1224no—
286#554—1224no—
287C2×C4⋊Q8—611no—
288C2×#125—1221no—
289C2×C22⋊Q8—612no—
290C2×C22⋊Q8—1222no—
291C2×#121—612no—
292C2×C4×Q8C2×D6111yescharacteristic
293C2×C4×D4—611no—
294C2×C4×D4—1221no—
295C2×C4×D4C2×D6111yescharacteristic
296C22×C4⋊C4—612no—
297#474—1224no—
298#440—24no—
299C2×#107—612no—
300C2×#107C2×D6111yescharacteristic
301C2×C4⋊C8—611no—
302C2×#103—612no—
303C2×#103C2×D6111yescharacteristic
304C2×#102—612no—
305C2×#102C2×D6111yescharacteristic
306C2×C22⋊C8—612no—
307C2×C22⋊C8—612no—
308C2×#97C2×D6111yescharacteristic
309#397—1221no—
310#392—1221no—
311#386—1222no—
312#382—611no—
313#380—612no—
314#340—612no—
315#340—612no—
316#340—612no—
317C61#2413011yescharacteristic, complemented, Sylow
318C22×Q8—1221no—
319C22×Q8C22×D6111yescharacteristic
320C22×D4—611no—
321C22×D4—1222no—
322C22×D4C22×D6111yescharacteristic
323C23×C4—21no—
324C23×C4—612no—
325C23×C4—1222no—
326C2×SD16—22no—
327C2×SD16—22no—
328C2×SD16—22no—
329C22×C8—612no—
330C22×C8C22×D6111yescharacteristic
331C22×C8C22×D6111yescharacteristic
332C4⋊Q8—614no—
333C4⋊Q8—1222no—
334#125—1228no—
335C22⋊Q8—12216no—
336C22⋊Q8—1224no—
337C22⋊Q8—1224no—
338#121—1224no—
339#121—2442no—
340C4×Q8—22no—
341C4×Q8C22⋊D6114yescomplemented
342C4×D4—22no—
343C4×D4—22no—
344C4×D4—1228no—
345C4×D4—1222no—
346C4×D4—1222no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—612no—
349C2×C4⋊C4—611no—
350C2×C4⋊C4—612no—
351C2×C4⋊C4—1224no—
352C2×C4⋊C4—1222no—
353C2×C4⋊C4—1222no—
354C2×C4⋊C4—1221no—
355C2×C4⋊C4—1224no—
356C2×C4⋊C4C22×D6111yescharacteristic
357C2×C4⋊C4C22×D6111yescharacteristic
358C2×C22⋊C4—21no—
359C2×C22⋊C4—612no—
360C2×C22⋊C4—1222no—
361C2×C22⋊C4—1222no—
362C2×C22⋊C4—1222no—
363C2×C42—21no—
364C2×C42C22×D6111yescharacteristic
365#107—22no—
366#107—1224no—
367C4⋊C8—1222no—
368#103—22no—
369#103—2442no—
370#102—22no—
371#102—1224no—
372C22⋊C8—1224no—
373C22⋊C8—2442no—
374#97—22no—
375#95—1222no—
376#95—1221no—
377#95—1221no—
378C24—21no—
379C24—612no—
380C2×Q8—21no—
381C2×Q8—21no—
382C2×Q8—21no—
383C2×Q8—21no—
384C2×Q8—21no—
385C2×Q8—1224no—
386C2×Q8—4882no—
387C2×Q8C2×C22⋊D6112yes—
388C2×D4—21no—
389C2×D4—21no—
390C2×D4—21no—
391C2×D4—21no—
392C2×D4—21no—
393C2×D4—1224no—
394C2×D4—1222no—
395C2×D4—4882no—
396C2×D4—2444no—
397C2×D4—2444no—
398C2×D4#1175822212yes—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—42no—
403C22×C4—82no—
404C22×C4—612no—
405C22×C4—1228no—
406C22×C4—1222no—
407C22×C4—1222no—
408C22×C4—1222no—
409C22×C4—1222no—
410C22×C4—1222no—
411C22×C4—1222no—
412C22×C4—1222no—
413C22×C4—1222no—
414C22×C4—4882no—
415C22×C4—4882no—
416C22×C4D4×D6111yescharacteristic
417C22×C4D4×D6111yescharacteristic
418C22×C4C23×D6111yescharacteristic
419SD16—28no—
420C2×C8—21no—
421C2×C8—21no—
422C2×C8—21no—
423C2×C8—21no—
424C2×C8—21no—
425C2×C8—1222no—
426C2×C8—1222no—
427C2×C8—1222no—
428C2×C8#1175822212yes—
429C4⋊C4—21no—
430C4⋊C4—24no—
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—614no—
434C4⋊C4—1228no—
435C4⋊C4—1228no—
436C4⋊C4—1222no—
437C4⋊C4—1222no—
438C4⋊C4—1224no—
439C4⋊C4—1222no—
440C4⋊C4—1222no—
441C4⋊C4C2×C22⋊D6112yes—
442C22⋊C4—42no—
443C22⋊C4—1228no—
444C22⋊C4—1224no—
445C22⋊C4—4882no—
446C22⋊C4—2444no—
447C42—24no—
448C42D4×D6112yes—
449C42C2×C22⋊D6112yes—
450C23—22no—
451C23—41no—
452C23—42no—
453C23—42no—
454C23—41no—
455C23—614no—
456C23—1222no—
457C23—1222no—
458C23—1222no—
459C23—1222no—
460C23—1222no—
461C23—1222no—
462C23C2×D4×D6111yescharacteristic
463Q8—22no—
464Q8—22no—
465Q8—41no—
466Q8—41no—
467Q8—4884no—
468D4—22no—
469D4—22no—
470D4—41no—
471D4—41no—
472D4—1228no—
473D4—4888no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—42no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—42no—
483C2×C4—84no—
484C2×C4—1224no—
485C2×C4—1222no—
486C2×C4—1222no—
487C2×C4—1222no—
488C2×C4—4888no—
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—2442no—
498C2×C4#1451974512yes—
499C2×C4#1451974512yes—
500C2×C4C2×D4×D6111yescharacteristic
501C2×C4C22×C22⋊D6111yescharacteristic
502C2×C4C2×#1175822211yescharacteristic
503C2×C4C2×#1175822211yescharacteristic
504C2×C4#1451982311yescharacteristic
505C2×C4#1451982411yescharacteristic
506C8—22no—
507C8—41no—
508C8—2442no—
509C22—42no—
510C22—44no—
511C22—82no—
512C22—81no—
513C22—81no—
514C22—1224no—
515C22—1224no—
516C22—1224no—
517C22—2442no—
518C22—2442no—
519C22—2442no—
520C22—2442no—
521C22C2×#1451974511yescharacteristic
522C22#1684040811yescharacteristic
523C22#1684041711yescharacteristic
524C22#137×D6112yes—
525C22#1684056612yes—
526C4—22no—
527C4—22no—
528C4—42no—
529C4—42no—
530C4—81no—
531C4—4882no—
532C4—2442no—
533C4—2444no—
534C4#1684027311yescharacteristic
535C4#1684034011yescharacteristic
536C4#1684042111yescharacteristic
537C4#1684043311yescharacteristic
538C2—82no—
539C2—2444no—
540C2#2144916411yescharacteristic
541C2#2145505111yescharacteristic
542C2#2145508611yescharacteristic
543C2#2145790512yes—
544C2#2145830912yes—
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#440C2
socleC23×C61C22×D4
prime radical#21450716C2
hypercenterC2×#440D61
hypocenterC61#24130
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#440

Complements

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

NH ≅ G/Ntype
1. C2×#16840472C2semidirect
2. C2×#16840042C2semidirect
4. #21447020C2semidirect
5. #21446819C2semidirect
6. #21446792C2semidirect
7. #21446617C2semidirect
8. #21446133C2semidirect
9. #21446005C2semidirect
10. #21445834C2semidirect
11. C2×C61×#440C2semidirect
52. #14519777D4semidirect
92. C4×Q8×C61D4semidirect
268. C2×#440D61semidirect
317. C61#24130semidirect
341. C4×Q8C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168404722
C2×#168400422
#214507162
#214470202
#214468192
#214467922
#214466172
#214461332
#214460052
#214458342
C2×C61×#4402
#2413061