Finite Group Databasebeta

← #29714078

Subgroups — #29714078

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×#16840473C211yesmaximal, characteristic, complemented
2C2×#16839862C211yesmaximal, characteristic, complemented
3#21450716C211yesmaximal, characteristic
4#21447024C212yesmaximal, complemented
5#21446823C212yesmaximal, complemented
6#21446793C211yesmaximal, characteristic, complemented
7#21446616C211yesmaximal, characteristic, complemented
8#21446133C212yesmaximal, complemented
9#21446005C212yesmaximal, complemented
10#21445835C211yesmaximal, characteristic, complemented
11C2×C61×#474C211yesmaximal, characteristic, complemented
12C22×#11758200C2211yescharacteristic
13#16840473—24no—
14#16840473—24no—
15C2×#14519768—21no—
16C2×#14519757C2212yes—
17C2×#14519753—22no—
18C2×#14519748C2212yes—
19C2×C4×Dic122C2211yescharacteristic
20C2×C4×D244C2211yescharacteristic
21C2×#14519739—21no—
22C2×C4⋊C4×D61C2212yes—
23#16839862—24no—
24C2×#14519710C2211yescharacteristic
25C2×#14519702C2212yes—
26C2×#14519700C2211yescharacteristic
27C2×#14519698C2211yescharacteristic
28C2×#14519689C2212yes—
29C2×#14519689C2212yes—
30C2×#14519688C2211yescharacteristic
31#16839687—21no—
32#16839671—21no—
33#16839641—22no—
34#16839628C2211yescharacteristic
35#16839619C2212yes—
36#16839525C2212yes—
37#16839524C2212yes—
38#16839524C2212yes—
39C2×C4⋊Q8×C61C2211yescharacteristic
40C2×C4×D4×C61C2211yescharacteristic
41C61×#474—24no—
42C2×#107×C61C2212yes—
43C2×C4⋊C8×C61C2211yescharacteristic
44C2×#102×C61C2212yes—
45C22×Dic122C2311yescharacteristic
46C22×D244D411yescharacteristic
47C22×C22⋊D61—22no—
48C22×C4×D61—22no—
49C22×C4×D61C2312yes—
50C2×#11758200—22no—
51C2×#11758200—22no—
52C2×#11758200—22no—
53C22×#92418C2311yescharacteristic
54C22×#92418C2311yescharacteristic
55#14519768—28no—
56#14519757—24no—
57#14519757—24no—
58#14519753—216no—
59#14519748—24no—
60#14519748—42no—
61C4×Dic122—22no—
62C4×Dic122D414yescomplemented
63C4×D244—22no—
64C4×D244—22no—
65#14519739—28no—
66C2×C4⋊Dic61—22no—
67C2×C4⋊Dic61D411yescharacteristic
68C2×C4⋊Dic61C2312yes—
69C2×C4⋊Dic61C2311yescharacteristic
70C2×#11758196—21no—
71C2×#11758196—22no—
72C4⋊C4×D61—24no—
73C4⋊C4×D61—24no—
74C2×C22⋊Dic61—22no—
75C2×#11758198—22no—
76C2×#11758198—22no—
77C2×#11758198D412yes—
78C2×C4×Dic61—21no—
79#14519710—22no—
80#14519702—42no—
81#14519700—22no—
82#14519698—22no—
83#14519689—24no—
84#14519689—42no—
85#14519688—22no—
86#14519684—21no—
87#14519684—21no—
88#14519684—22no—
89C22×Q8×C61—21no—
90C22×D4×C61C2311yescharacteristic
91C23×C4×C61—21no—
92C22×C8×C61C2312yes—
93C4⋊Q8×C61—22no—
94C4⋊Q8×C61D414yescomplemented
95C4×D4×C61—22no—
96C4×D4×C61—22no—
97C2×C4⋊C4×C61—21no—
98C2×C4⋊C4×C61C2311yescharacteristic
99C2×C4⋊C4×C61C2312yes—
100C2×C22⋊C4×C61—21no—
101C2×C42×C61C2311yescharacteristic
102#107×C61—24no—
103C4⋊C8×C61—22no—
104#102×C61—24no—
105C24×C61—21no—
106C23×D61C2×D412yes—
107C2×Q8×C61—24no—
108C2×Q8×C61—82no—
109C2×D4×C61—21no—
110C2×D4×C61—21no—
111C2×D4×C61—21no—
112C2×D4×C61—21no—
113C2×D4×C61—21no—
114C2×D4×C61#4112yes—
115C22×C4×C61—22no—
116C22×C4×C61—21no—
117C22×C4×C61—42no—
118C22×C4×C61—82no—
119C22×C4×C61C2×D411yescharacteristic
120C22×C4×C61C2×D411yescharacteristic
121C22×C4×C61C2411yescharacteristic
122C2×C22⋊D61—44no—
123C2×C22⋊D61—44no—
124C2×C22⋊D61—82no—
125C22×Dic61—21no—
126C22×Dic61—21no—
127C22×Dic61—22no—
128C22×Dic61C2×D412yes—
129C2×D244—24no—
130C2×D244—22no—
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—21no—
141C2×Dic122—21no—
142C2×Dic122—21no—
143C2×Dic122—21no—
144C2×Dic122—21no—
145C2×Dic122C2×D412yes—
146C2×C8×C61—22no—
147C2×C8×C61—22no—
148C2×C8×C61—22no—
149C4⋊C4×C61—24no—
150C4⋊C4×C61—22no—
151C4⋊C4×C61—21no—
152C4⋊C4×C61—21no—
153C4⋊C4×C61C2×D414yes—
154C22⋊C4×C61—42no—
155C42×C61C2×D412yes—
156C42×C61C2×D412yes—
157C22⋊Dic61—44no—
158#11758200—28no—
159#11758198—28no—
160#11758198—24no—
161#11758198—82no—
162C4⋊Dic61—28no—
163C4⋊Dic61—22no—
164C4⋊Dic61—21no—
165C4⋊Dic61—22no—
166C4⋊Dic61—22no—
167C4⋊Dic61C2×D412yes—
168#11758196—28no—
169#11758196—24no—
170C4×Dic61—24no—
171C2×#92418—21no—
172C2×#92418—21no—
173C2×#92418—21no—
174C2×#92418—21no—
175C2×#92418—21no—
176C2×#92418#4112yes—
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—84no—
191D4×C61—22no—
192D4×C61—22no—
193D4×C61—41no—
194D4×C61—41no—
195C2×C4×C61—22no—
196C2×C4×C61—22no—
197C2×C4×C61—24no—
198C2×C4×C61—42no—
199C2×C4×C61—42no—
200C2×C4×C61—41no—
201C2×C4×C61—42no—
202C2×C4×C61—84no—
203C2×C4×C61#12012yes—
204C2×C4×C61#12012yes—
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
211C22⋊D61—88no—
212C2×Dic61—22no—
213C2×Dic61—22no—
214C2×Dic61—22no—
215C2×Dic61—22no—
216C2×Dic61—22no—
217C2×Dic61—41no—
218C2×Dic61—41no—
219C2×Dic61—42no—
220C2×Dic61—42no—
221C2×Dic61—42no—
222C2×Dic61—42no—
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—
233C8×C61—42no—
234#92418—22no—
235#92418—41no—
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—44no—
257C4×C61—81no—
258C4×C61#52511yescharacteristic
259C4×C61#53711yescharacteristic
260C4×C61#54811yescharacteristic
261C4×C61#55011yescharacteristic
262Dic61—42no—
263Dic61—42no—
264Dic61—42no—
265Dic61—82no—
266C2×#554—611no—
267C2×#474D6111yescharacteristic, complemented
268C2×#440—611no—
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×SD16—611no—
285#554—1224no—
286#554—1224no—
287C2×C4⋊Q8C2×D6111yescharacteristic
288C2×#125—1221no—
289C2×C22⋊Q8—612no—
290C2×C22⋊Q8—1222no—
291C2×#121—612no—
292C2×C4×Q8—611no—
293C2×C4×D4—611no—
294C2×C4×D4—1221no—
295C2×C4×D4C2×D6111yescharacteristic
296C22×C4⋊C4—612no—
297#474—24no—
298#440—1224no—
299C2×#107—611no—
300C2×#107C2×D6112yes—
301C2×C4⋊C8C2×D6111yescharacteristic
302C2×#103—611no—
303C2×#103—612no—
304C2×#102—611no—
305C2×#102C2×D6112yes—
306C2×C22⋊C8—612no—
307C2×C22⋊C8—612no—
308C2×#97—611no—
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—21no—
319C22×Q8—611no—
320C22×D4—611no—
321C22×D4—1222no—
322C22×D4C22×D6111yescharacteristic
323C23×C4—21no—
324C23×C4—612no—
325C23×C4—1222no—
326C2×SD16—1222no—
327C2×SD16—1222no—
328C2×SD16—1222no—
329C22×C8—611no—
330C22×C8—611no—
331C22×C8C22×D6112yes—
332C4⋊Q8—22no—
333C4⋊Q8C22⋊D6114yescomplemented
334#125—1228no—
335C22⋊Q8—1224no—
336C22⋊Q8—12216no—
337C22⋊Q8—1224no—
338#121—1224no—
339#121—2442no—
340C4×Q8—614no—
341C4×Q8—1222no—
342C4×D4—22no—
343C4×D4—22no—
344C4×D4—1228no—
345C4×D4—1222no—
346C4×D4—1222no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—611no—
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×D6112yes—
358C2×C22⋊C4—21no—
359C2×C22⋊C4—612no—
360C2×C22⋊C4—1222no—
361C2×C22⋊C4—1222no—
362C2×C22⋊C4—1222no—
363C2×C42—1221no—
364C2×C42C22×D6111yescharacteristic
365#107—24no—
366#107—1222no—
367C4⋊C8—22no—
368#103—1222no—
369#103—2442no—
370#102—24no—
371#102—1222no—
372C22⋊C8—1224no—
373C22⋊C8—2442no—
374#97—1222no—
375#95—1222no—
376#95—1221no—
377#95—1221no—
378C24—21no—
379C24—612no—
380C2×Q8—24no—
381C2×Q8—82no—
382C2×Q8—612no—
383C2×Q8—1221no—
384C2×Q8—1221no—
385C2×Q8—1221no—
386C2×Q8—1221no—
387C2×Q8—1221no—
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#1175822612yes—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—42no—
402C22×C4—82no—
403C22×C4—612no—
404C22×C4—1228no—
405C22×C4—1221no—
406C22×C4—1221no—
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—1228no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—612no—
424C2×C8—1221no—
425C2×C8—1221no—
426C2×C8—1221no—
427C2×C8—1221no—
428C2×C8—1221no—
429C4⋊C4—22no—
430C4⋊C4—24no—
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—612no—
434C4⋊C4—1228no—
435C4⋊C4—1228no—
436C4⋊C4—1221no—
437C4⋊C4—1222no—
438C4⋊C4—1224no—
439C4⋊C4—1222no—
440C4⋊C4—1222no—
441C4⋊C4C2×C22⋊D6114yes—
442C22⋊C4—42no—
443C22⋊C4—1228no—
444C22⋊C4—1224no—
445C22⋊C4—4882no—
446C22⋊C4—2444no—
447C42—1224no—
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—84no—
464Q8—1222no—
465Q8—1222no—
466Q8—2441no—
467Q8—2441no—
468D4—22no—
469D4—22no—
470D4—41no—
471D4—41no—
472D4—1228no—
473D4—4888no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—24no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—41no—
480C2×C4—42no—
481C2×C4—84no—
482C2×C4—1222no—
483C2×C4—1222no—
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—2441no—
493C2×C4—2441no—
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×C4#41×D6111yescharacteristic
503C2×C4C2×#1175822611yescharacteristic
504C2×C4#1451982311yescharacteristic
505C2×C4#1451982511yescharacteristic
506C8—42no—
507C8—1222no—
508C8—2441no—
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#1684040311yescharacteristic
523C22#1684041411yescharacteristic
524C22#137×D6112yes—
525C22#1684056312yes—
526C4—22no—
527C4—22no—
528C4—44no—
529C4—81no—
530C4—4882no—
531C4—2442no—
532C4—2442no—
533C4—2442no—
534C4#1684027511yescharacteristic
535C4#1684034211yescharacteristic
536C4#1684042311yescharacteristic
537C4#1684043111yescharacteristic
538C2—82no—
539C2—2444no—
540C2#2144916611yescharacteristic
541C2#2145505111yescharacteristic
542C2#2145508311yescharacteristic
543C2#2145789712yes—
544C2#2145830112yes—
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#474C2
socleC23×C61C22×D4
prime radical#21450716C2
hypercenterC2×#474D61
hypocenterC61#24130
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#474

Complements

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

NH ≅ G/Ntype
1. C2×#16840473C2semidirect
2. C2×#16839862C2semidirect
4. #21447024C2semidirect
5. #21446823C2semidirect
6. #21446793C2semidirect
7. #21446616C2semidirect
8. #21446133C2semidirect
9. #21446005C2semidirect
10. #21445835C2semidirect
11. C2×C61×#474C2semidirect
62. C4×Dic122D4semidirect
94. C4⋊Q8×C61D4semidirect
267. C2×#474D61semidirect
317. C61#24130semidirect
333. C4⋊Q8C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168404732
C2×#168398622
#214507162
#214470242
#214468232
#214467932
#214466162
#214461332
#214460052
#214458352
C2×C61×#4742
#2413061