Finite Group Databasebeta

← #29722465

Subgroups — #29722465

Order 15616.

894 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840032C211yesmaximal, characteristic, complemented
2#21455467C211yesmaximal, characteristic, complemented
3#21455356C212yesmaximal, complemented
4#21454419C211yesmaximal, characteristic, complemented
5#21447174C212yesmaximal, complemented
6#21447028C212yesmaximal, complemented
7#21446685C211yesmaximal, characteristic, complemented
8#21446567C211yesmaximal, characteristic, complemented
9#21446193C212yesmaximal, complemented
10#21445883C211yesmaximal, characteristic, complemented
11C2×C61×#443C211yesmaximal, characteristic, complemented
12C2×#14519802C2212yes—
13C22×D488C2211yescharacteristic
14#16840326C2212yes—
15#16840245C2212yes—
16#16840032—24no—
17#16840015—22no—
18#16839998—21no—
19#16839998—21no—
20#16839986—22no—
21#16839979—22no—
22#16839960—22no—
23#16839955—22no—
24C2×#14519713C2211yescharacteristic
25#16839804—22no—
26#16839804C2212yes—
27#16839800—21no—
28#16839800C2211yescharacteristic
29C2×#14519711—21no—
30C2×#14519711C2211yescharacteristic
31#16839788C2212yes—
32#16839768C2212yes—
33#16839763C2212yes—
34C2×#14519699C2212yes—
35C2×#14519693C2212yes—
36#16839664C2212yes—
37#16839530C2212yes—
38#16839522C2212yes—
39C2×#135×C61C2211yescharacteristic
40C2×#131×C61C2211yescharacteristic
41C2×#131×C61C2211yescharacteristic
42C61×#443—24no—
43C2×#108×C61C2211yescharacteristic
44C2×#104×C61C2211yescharacteristic
45C2×#104×C61C2211yescharacteristic
46C2×C4×C8×C61C2211yescharacteristic
47C22×D244C2312yes—
48#14519802—24no—
49#14519802—24no—
50C2×#11758201—22no—
51C2×D488—21no—
52C2×D488—21no—
53C2×D488—22no—
54C2×D488—22no—
55C2×#11758199—22no—
56C2×#11758194C2312yes—
57#14519773—22no—
58#14519766—22no—
59#14519751—22no—
60#14519751—22no—
61#14519745—42no—
62Q8×Dic61—22no—
63D4×Dic61—22no—
64D4×Dic61—42no—
65#14519736—22no—
66#14519736C2312yes—
67C2×C4⋊Dic61—22no—
68C2×C4⋊Dic61C2312yes—
69C2×#11758198—42no—
70#14519713—22no—
71#14519711—21no—
72#14519711—21no—
73#14519711—22no—
74#14519711D411yescharacteristic
75#14519711D411yescharacteristic
76#14519709—22no—
77#14519709D412yes—
78#14519706—22no—
79#14519706—22no—
80#14519701—22no—
81#14519699—22no—
82#14519699—22no—
83#14519699—22no—
84#14519699—22no—
85#14519698—22no—
86#14519693—24no—
87C2×#41×C61C2311yescharacteristic
88C2×#41×C61C2311yescharacteristic
89#135×C61—22no—
90#135×C61—22no—
91C2×Q16×C61D411yescharacteristic
92C2×SD16×C61D411yescharacteristic
93C2×SD16×C61D411yescharacteristic
94C2×D8×C61D411yescharacteristic
95#131×C61—22no—
96#131×C61—22no—
97#131×C61—22no—
98#131×C61D414yescomplemented
99C2×M16×C61—21no—
100C2×M16×C61—21no—
101C2×M16×C61C2311yescharacteristic
102C2×M16×C61C2311yescharacteristic
103C22×C8×C61—21no—
104C22×C8×C61—21no—
105C22×C8×C61C2311yescharacteristic
106C22×C8×C61C2311yescharacteristic
107C2×C42×C61C2311yescharacteristic
108#108×C61—22no—
109#104×C61—22no—
110#104×C61—22no—
111C4×C8×C61—22no—
112C23×D61—22no—
113#41×C61—21no—
114#41×C61—21no—
115#41×C61—21no—
116#41×C61—21no—
117#41×C61—21no—
118#41×C61—21no—
119#41×C61C2×D412yes—
120#41×C61#4112yes—
121C2×Q8×C61C2×D411yescharacteristic
122C2×Q8×C61C2×D411yescharacteristic
123C2×D4×C61—21no—
124C2×D4×C61—21no—
125C2×D4×C61C2×D411yescharacteristic
126C2×D4×C61C2×D411yescharacteristic
127C22×C4×C61—21no—
128C22×C4×C61—21no—
129C22×C4×C61—21no—
130C22×C4×C61C2411yescharacteristic
131C2×C22⋊D61—42no—
132C2×C22⋊D61—42no—
133C22×Dic61—22no—
134C22×Dic61—42no—
135C2×D244—22no—
136C2×D244—22no—
137C2×D244—22no—
138C2×D244—22no—
139C2×D244—22no—
140C2×D244—22no—
141Q16×C61—22no—
142SD16×C61—22no—
143SD16×C61—22no—
144D8×C61—22no—
145M16×C61—21no—
146M16×C61—24no—
147M16×C61—21no—
148M16×C61—21no—
149M16×C61—42no—
150M16×C61C2×D412yes—
151C2×C8×C61—21no—
152C2×C8×C61—21no—
153C2×C8×C61—21no—
154C2×C8×C61—24no—
155C2×C8×C61—21no—
156C2×C8×C61—42no—
157C2×C8×C61C2×D411yescharacteristic
158C2×C8×C61C2×D411yescharacteristic
159C2×C8×C61C2×D411yescharacteristic
160C2×C8×C61C2×D411yescharacteristic
161C2×C8×C61C2×D412yes—
162C2×C8×C61#4112yes—
163C42×C61—21no—
164C42×C61—21no—
165C22⋊Dic61—22no—
166C22⋊Dic61—42no—
167#11758201—44no—
168#11758199—44no—
169#11758198—42no—
170#11758198—42no—
171#11758198—84no—
172C4⋊Dic61—22no—
173C4⋊Dic61—22no—
174C4⋊Dic61—42no—
175C4⋊Dic61C2×D412yes—
176C4⋊Dic61C2×D412yes—
177C4×Dic61—22no—
178C4×Dic61—42no—
179#11758194—22no—
180#11758194—22no—
181C2×#92418—22no—
182D488—42no—
183D488—42no—
184C23×C61—21no—
185C23×C61—21no—
186C23×C61H(5,𝔽2)11yescharacteristic
187C22×D61—42no—
188C22×D61—42no—
189C22×D61—42no—
190C22×D61—82no—
191C22×D61—82no—
192Q8×C61—21no—
193Q8×C61—21no—
194Q8×C61—21no—
195Q8×C61#12112yes—
196D4×C61—22no—
197D4×C61—22no—
198D4×C61—21no—
199D4×C61—21no—
200D4×C61—21no—
201D4×C61—41no—
202D4×C61—41no—
203D4×C61#12112yes—
204C2×C4×C61—21no—
205C2×C4×C61—22no—
206C2×C4×C61—22no—
207C2×C4×C61—22no—
208C2×C4×C61—21no—
209C2×C4×C61—41no—
210C2×C4×C61—41no—
211C2×C4×C61—41no—
212C2×C4×C61—41no—
213C2×C4×C61C22×D411yescharacteristic
214C2×C4×C61C22×D411yescharacteristic
215C2×C4×C61C22×D411yescharacteristic
216C2×C4×C61C2×#4111yescharacteristic
217C2×C4×C61C2×#4111yescharacteristic
218C2×C4×C612−1+411yescharacteristic
219C22⋊D61—162no—
220C22⋊D61—162no—
221C2×Dic61—22no—
222C2×Dic61—22no—
223C2×Dic61—42no—
224C2×Dic61—82no—
225C2×Dic61—162no—
226D244—42no—
227D244—42no—
228D244—42no—
229D244—42no—
230C8×C61—22no—
231C8×C61—21no—
232C8×C61—21no—
233C8×C61—41no—
234C8×C61#12112yes—
235C8×C61#12112yes—
236#92418—42no—
237#24797—611nomaximal, Sylow
238C22×C61—21no—
239C22×C61—21no—
240C22×C61—22no—
241C22×C61—41no—
242C22×C61—41no—
243C22×C61—41no—
244C22×C61—41no—
245C22×C61—41no—
246C22×C61#53711yescharacteristic
247C22×C61D4211yescharacteristic
248C22×C61#55311yescharacteristic
249C2×D61—84no—
250C2×D61—82no—
251C2×D61—82no—
252C2×D61—82no—
253C2×D61—162no—
254C4×C61—22no—
255C4×C61—42no—
256C4×C61—41no—
257C4×C61C2×#12111yescharacteristic
258C4×C61C2×#12111yescharacteristic
259C4×C61#54711yescharacteristic
260C4×C61#54811yescharacteristic
261Dic61—42no—
262Dic61—42no—
263Dic61—82no—
264C2×#472—611no—
265#3058—611no—
266#3045—612no—
267C2×#443D6111yescharacteristic, complemented
268#2913—611no—
269#2063—612no—
270#2037—612no—
271#1970—611no—
272#1945—611no—
273#1883—612no—
274#1816—611no—
275C2×C61—21no—
276C2×C61—21no—
277C2×C61—42no—
278C2×C61—81no—
279C2×C61#241611yescharacteristic
280C2×C61#329011yescharacteristic
281C2×C61#329211yescharacteristic
282D61—162no—
283C2×AGL(1,ℤ/8)—612no—
284C2×#135C2×D6111yescharacteristic
285C22×D8—611no—
286C2×#131C2×D6111yescharacteristic
287C2×#131C2×D6111yescharacteristic
288#535—612no—
289#520—612no—
290#472—1224no—
291#469—1222no—
292#466—1221no—
293#466—1221no—
294#464—1222no—
295#463—1222no—
296#460—1222no—
297#459—1222no—
298#443—24no—
299C2×#108—611no—
300C2×#108C2×D6111yescharacteristic
301#428—612no—
302#428—1222no—
303#427—611no—
304#427—1221no—
305C2×#107—611no—
306C2×#107—1221no—
307#424—612no—
308C2×#104C2×D6111yescharacteristic
309C2×#104C2×D6111yescharacteristic
310#418—612no—
311#417—612no—
312C2×#102—612no—
313C2×#100—612no—
314C2×C4×C8C2×D6111yescharacteristic
315#390—612no—
316#343—612no—
317#339—612no—
318C61#2479711yescharacteristic, complemented, Sylow
319C2×#41C22×D6111yescharacteristic
320C2×#41C22×D6111yescharacteristic
321C22×D4—612no—
322AGL(1,ℤ/8)—1224no—
323AGL(1,ℤ/8)—1224no—
324#135—22no—
325#135—22no—
326C2×Q16D24411yescharacteristic
327C2×SD16—1222no—
328C2×SD16D24411yescharacteristic
329C2×SD16D24411yescharacteristic
330C2×D8—1222no—
331C2×D8—1221no—
332C2×D8—1221no—
333C2×D8—1222no—
334C2×D8—1222no—
335C2×D8D24411yescharacteristic
336#131—22no—
337#131—22no—
338#131—22no—
339#131C22⋊D6114yescomplemented
340C2×M16—21no—
341C2×M16—21no—
342C2×M16—612no—
343C2×M16C22×D6111yescharacteristic
344C2×M16C22×D6111yescharacteristic
345C22×C8—21no—
346C22×C8—21no—
347C22×C8C22×D6111yescharacteristic
348C22×C8C22×D6111yescharacteristic
349#127—1222no—
350#124—1222no—
351#121—1222no—
352#121—1222no—
353#120—2442no—
354C4×Q8—1222no—
355C4×D4—1222no—
356C4×D4—2442no—
357#117—612no—
358#117—1222no—
359C2×C4⋊C4—612no—
360C2×C4⋊C4—1222no—
361C2×C22⋊C4—2442no—
362C2×C42C22×D6111yescharacteristic
363#108—22no—
364#108—1222no—
365#107—611no—
366#107—611no—
367#107—1222no—
368#107—1221no—
369#107—1221no—
370#106—612no—
371#106—1222no—
372C4⋊C8—1222no—
373C4⋊C8—1222no—
374#104—22no—
375#104—22no—
376#103—1222no—
377#102—1222no—
378#102—1222no—
379#102—1222no—
380#102—1222no—
381#102—1222no—
382#100—1224no—
383C4×C8—22no—
384C24—1222no—
385#41—21no—
386#41—21no—
387#41—21no—
388#41—21no—
389#41—21no—
390#41—21no—
391#41#1175822212yes—
392#41C2×C22⋊D6112yes—
393C2×Q8C2×D24411yescharacteristic
394C2×Q8C2×D24411yescharacteristic
395C2×D4—21no—
396C2×D4—21no—
397C2×D4—1222no—
398C2×D4—1222no—
399C2×D4—1222no—
400C2×D4—1222no—
401C2×D4—1222no—
402C2×D4—1222no—
403C2×D4—2442no—
404C2×D4—2442no—
405C2×D4C2×D24411yescharacteristic
406C2×D4C2×D24411yescharacteristic
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—1222no—
411C22×C4—2442no—
412C22×C4C23×D6111yescharacteristic
413Q16—22no—
414SD16—22no—
415SD16—22no—
416SD16—2444no—
417D8—22no—
418D8—2444no—
419D8—2442no—
420D8—2442no—
421M16—21no—
422M16—24no—
423M16—21no—
424M16—21no—
425M16—42no—
426M16—1222no—
427M16—1222no—
428M16C2×C22⋊D6112yes—
429C2×C8—21no—
430C2×C8—21no—
431C2×C8—21no—
432C2×C8—24no—
433C2×C8—21no—
434C2×C8—42no—
435C2×C8—1222no—
436C2×C8C2×D24411yescharacteristic
437C2×C8C2×D24411yescharacteristic
438C2×C8D4×D6111yescharacteristic
439C2×C8D4×D6111yescharacteristic
440C2×C8#1175822412yes—
441C2×C8C2×C22⋊D6112yes—
442C4⋊C4—612no—
443C4⋊C4—612no—
444C4⋊C4—1222no—
445C4⋊C4—1222no—
446C4⋊C4—2442no—
447C22⋊C4—1222no—
448C22⋊C4—4884no—
449C22⋊C4—2442no—
450C22⋊C4—2442no—
451C22⋊C4—2442no—
452C42—21no—
453C42—21no—
454C42—1222no—
455C42—2442no—
456C23—21no—
457C23—21no—
458C23—4882no—
459C23—4882no—
460C23—2442no—
461C23—2442no—
462C23—2442no—
463C23#1451982311yescharacteristic
464Q8—21no—
465Q8—21no—
466Q8—21no—
467Q8#1451975112yes—
468D4—22no—
469D4—22no—
470D4—21no—
471D4—21no—
472D4—21no—
473D4—41no—
474D4—41no—
475D4—9762no—
476D4—9762no—
477D4—2442no—
478D4—2442no—
479D4—2442no—
480D4—2442no—
481D4#1451975112yes—
482C2×C4—21no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—21no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—1222no—
492C2×C4—1222no—
493C2×C4—9762no—
494C2×C4—4882no—
495C2×C4—2442no—
496C2×C4C2×D4×D6111yescharacteristic
497C2×C4C22×C22⋊D6111yescharacteristic
498C2×C4C22×D24411yescharacteristic
499C2×C4C2×#1175822411yescharacteristic
500C2×C4C2×#1175822211yescharacteristic
501C2×C4#1451982411yescharacteristic
502C8—22no—
503C8—21no—
504C8—21no—
505C8—41no—
506C8—2442no—
507C8#1451974912yes—
508C8#1451974912yes—
509C22—21no—
510C22—21no—
511C22—22no—
512C22—41no—
513C22—41no—
514C22—41no—
515C22—41no—
516C22—41no—
517C22—9762no—
518C22—4884no—
519C22—4882no—
520C22—4882no—
521C22—4882no—
522C22#1684034011yescharacteristic
523C22D4×D24411yescharacteristic
524C22#1684045811yescharacteristic
525C4—22no—
526C4—42no—
527C4—41no—
528C4—4882no—
529C4—2442no—
530C4—2442no—
531C4C2×#1451974911yescharacteristic
532C4C2×#1451975111yescharacteristic
533C4#1684041611yescharacteristic
534C4#1684042111yescharacteristic
535C2—21no—
536C2—21no—
537C2—42no—
538C2—81no—
539C2—9762no—
540C2#2144925911yescharacteristic
541C2#2145760611yescharacteristic
542C2#2145761911yescharacteristic
subgroupG/H
centerC22D4×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#443C2
socleC22×C61D42
hypercenterC2×#443D61
hypocenterC61#24797
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#443

Complements

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

NH ≅ G/Ntype
1. C2×#16840032C2semidirect
2. #21455467C2semidirect
3. #21455356C2semidirect
4. #21454419C2semidirect
5. #21447174C2semidirect
6. #21447028C2semidirect
7. #21446685C2semidirect
8. #21446567C2semidirect
9. #21446193C2semidirect
10. #21445883C2semidirect
11. C2×C61×#443C2semidirect
98. #131×C61D4semidirect
267. C2×#443D61semidirect
318. C61#24797semidirect
339. #131C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168400322
#214554672
#214553562
#214544192
#214471742
#214470282
#214466852
#214465672
#214461932
#214458832
C2×C61×#4432
#2479761