Finite Group Databasebeta

← #29722487

Subgroups — #29722487

Order 15616.

890 conjugacy classes of subgroups.

The subgroup lattice has 534 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#21454414C211yesmaximal, characteristic, complemented
5#21447174C212yesmaximal, complemented
6#21447028C212yesmaximal, complemented
7#21446695C211yesmaximal, characteristic, complemented
8#21446572C211yesmaximal, characteristic, complemented
9#21446193C212yesmaximal, complemented
10#21445878C211yesmaximal, characteristic, complemented
11C2×C61×#444C211yesmaximal, characteristic, complemented
12C2×#14519802C2212yes—
13C22×D488C2211yescharacteristic
14#16840326C2212yes—
15#16840245C2212yes—
16#16840032—24no—
17#16840015—21no—
18#16840015—21no—
19#16839998—22no—
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×#14519709—21no—
30C2×#14519709C2211yescharacteristic
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×#444—24no—
43C2×#108×C61C2211yescharacteristic
44C2×#104×C61C2211yescharacteristic
45C2×#104×C61C2211yescharacteristic
46C2×#97×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—22no—
72#14519711D412yes—
73#14519709—21no—
74#14519709—21no—
75#14519709—22no—
76#14519709D411yescharacteristic
77#14519709D411yescharacteristic
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—
111#97×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—21no—
208C2×C4×C61—41no—
209C2×C4×C61—41no—
210C2×C4×C61—42no—
211C2×C4×C61—41no—
212C2×C4×C61C22×D411yescharacteristic
213C2×C4×C61C22×D411yescharacteristic
214C2×C4×C61C22×D411yescharacteristic
215C2×C4×C61C2×#4111yescharacteristic
216C2×C4×C61C2×#4111yescharacteristic
217C2×C4×C612−1+411yescharacteristic
218C22⋊D61—162no—
219C22⋊D61—162no—
220C2×Dic61—22no—
221C2×Dic61—22no—
222C2×Dic61—42no—
223C2×Dic61—82no—
224C2×Dic61—162no—
225D244—42no—
226D244—42no—
227D244—42no—
228D244—42no—
229C8×C61—22no—
230C8×C61—21no—
231C8×C61—21no—
232C8×C61—41no—
233C8×C61#12112yes—
234C8×C61#12112yes—
235#92418—42no—
236#24799—611nomaximal, Sylow
237C22×C61—22no—
238C22×C61—22no—
239C22×C61—42no—
240C22×C61—41no—
241C22×C61—42no—
242C22×C61#53711yescharacteristic
243C22×C61D4211yescharacteristic
244C22×C61#55311yescharacteristic
245C2×D61—84no—
246C2×D61—82no—
247C2×D61—82no—
248C2×D61—82no—
249C2×D61—162no—
250C4×C61—22no—
251C4×C61—41no—
252C4×C61—81no—
253C4×C61C2×#12111yescharacteristic
254C4×C61C2×#12111yescharacteristic
255C4×C61#54711yescharacteristic
256C4×C61#54811yescharacteristic
257Dic61—42no—
258Dic61—42no—
259Dic61—82no—
260C2×#472—611no—
261#3058—611no—
262#3045—612no—
263C2×#444D6111yescharacteristic, complemented
264#2912—611no—
265#2063—612no—
266#2037—612no—
267#1972—611no—
268#1946—611no—
269#1883—612no—
270#1815—611no—
271C2×C61—21no—
272C2×C61—21no—
273C2×C61—42no—
274C2×C61—81no—
275C2×C61#241611yescharacteristic
276C2×C61#328812yes—
277D61—162no—
278C2×AGL(1,ℤ/8)—612no—
279C2×#135C2×D6111yescharacteristic
280C22×D8—611no—
281C2×#131C2×D6111yescharacteristic
282C2×#131C2×D6111yescharacteristic
283#535—612no—
284#520—612no—
285#472—1224no—
286#469—1221no—
287#469—1221no—
288#466—1222no—
289#464—1222no—
290#463—1222no—
291#460—1222no—
292#459—1222no—
293#444—24no—
294C2×#108—611no—
295C2×#108C2×D6111yescharacteristic
296#428—612no—
297#428—1222no—
298#427—611no—
299#427—1221no—
300C2×#106—611no—
301C2×#106—1221no—
302#424—612no—
303C2×#104C2×D6111yescharacteristic
304C2×#104C2×D6111yescharacteristic
305#418—612no—
306#417—612no—
307C2×#102—612no—
308C2×#100—612no—
309C2×#97C2×D6111yescharacteristic
310#390—612no—
311#343—612no—
312#339—612no—
313C61#2479911yescharacteristic, complemented, Sylow
314C2×#41C22×D6111yescharacteristic
315C2×#41C22×D6111yescharacteristic
316C22×D4—612no—
317AGL(1,ℤ/8)—1224no—
318AGL(1,ℤ/8)—1224no—
319#135—22no—
320#135—22no—
321C2×Q16D24411yescharacteristic
322C2×SD16—1222no—
323C2×SD16D24411yescharacteristic
324C2×SD16D24411yescharacteristic
325C2×D8—1222no—
326C2×D8—1221no—
327C2×D8—1221no—
328C2×D8—1222no—
329C2×D8—1222no—
330C2×D8D24411yescharacteristic
331#131—22no—
332#131—22no—
333#131—22no—
334#131C22⋊D6114yescomplemented
335C2×M16—21no—
336C2×M16—21no—
337C2×M16—612no—
338C2×M16C22×D6111yescharacteristic
339C2×M16C22×D6111yescharacteristic
340C22×C8—21no—
341C22×C8—21no—
342C22×C8C22×D6111yescharacteristic
343C22×C8C22×D6111yescharacteristic
344#127—1222no—
345#124—1222no—
346#121—1222no—
347#121—1222no—
348#120—2442no—
349C4×Q8—1222no—
350C4×D4—1222no—
351C4×D4—2442no—
352#117—612no—
353#117—1222no—
354C2×C4⋊C4—612no—
355C2×C4⋊C4—1222no—
356C2×C22⋊C4—2442no—
357C2×C42C22×D6111yescharacteristic
358#108—22no—
359#108—1222no—
360#107—612no—
361#107—1222no—
362#106—611no—
363#106—611no—
364#106—1222no—
365#106—1221no—
366#106—1221no—
367C4⋊C8—1222no—
368C4⋊C8—1222no—
369#104—22no—
370#104—22no—
371#103—1222no—
372#102—1222no—
373#102—1222no—
374#102—1222no—
375#102—1222no—
376#102—1222no—
377#100—1224no—
378#97—22no—
379C24—1222no—
380#41—21no—
381#41—21no—
382#41—21no—
383#41—21no—
384#41—21no—
385#41—21no—
386#41#1175822212yes—
387#41C2×C22⋊D6112yes—
388C2×Q8C2×D24411yescharacteristic
389C2×Q8C2×D24411yescharacteristic
390C2×D4—21no—
391C2×D4—21no—
392C2×D4—1222no—
393C2×D4—1222no—
394C2×D4—1222no—
395C2×D4—1222no—
396C2×D4—1222no—
397C2×D4—1222no—
398C2×D4—2442no—
399C2×D4—2442no—
400C2×D4C2×D24411yescharacteristic
401C2×D4C2×D24411yescharacteristic
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—1222no—
406C22×C4—2442no—
407C22×C4C23×D6111yescharacteristic
408Q16—22no—
409SD16—22no—
410SD16—22no—
411SD16—2444no—
412D8—22no—
413D8—2444no—
414D8—2442no—
415D8—2442no—
416M16—21no—
417M16—24no—
418M16—21no—
419M16—21no—
420M16—42no—
421M16—1222no—
422M16—1222no—
423M16C2×C22⋊D6112yes—
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—21no—
427C2×C8—24no—
428C2×C8—21no—
429C2×C8—42no—
430C2×C8—1222no—
431C2×C8C2×D24411yescharacteristic
432C2×C8C2×D24411yescharacteristic
433C2×C8D4×D6111yescharacteristic
434C2×C8D4×D6111yescharacteristic
435C2×C8#1175822412yes—
436C2×C8C2×C22⋊D6112yes—
437C4⋊C4—612no—
438C4⋊C4—612no—
439C4⋊C4—1222no—
440C4⋊C4—1222no—
441C4⋊C4—2442no—
442C22⋊C4—1222no—
443C22⋊C4—4884no—
444C22⋊C4—2442no—
445C22⋊C4—2442no—
446C22⋊C4—2442no—
447C42—21no—
448C42—21no—
449C42—1222no—
450C42—2442no—
451C23—21no—
452C23—21no—
453C23—4882no—
454C23—4882no—
455C23—2442no—
456C23—2442no—
457C23—2442no—
458C23#1451982311yescharacteristic
459Q8—21no—
460Q8—21no—
461Q8—21no—
462Q8#1451975112yes—
463D4—22no—
464D4—22no—
465D4—21no—
466D4—21no—
467D4—21no—
468D4—41no—
469D4—41no—
470D4—9762no—
471D4—9762no—
472D4—2442no—
473D4—2442no—
474D4—2442no—
475D4—2442no—
476D4#1451975112yes—
477C2×C4—21no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—21no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—42no—
484C2×C4—41no—
485C2×C4—1222no—
486C2×C4—1222no—
487C2×C4—9762no—
488C2×C4—4882no—
489C2×C4—2442no—
490C2×C4C2×D4×D6111yescharacteristic
491C2×C4C22×C22⋊D6111yescharacteristic
492C2×C4C22×D24411yescharacteristic
493C2×C4C2×#1175822411yescharacteristic
494C2×C4C2×#1175822211yescharacteristic
495C2×C4#1451982411yescharacteristic
496C8—22no—
497C8—21no—
498C8—21no—
499C8—41no—
500C8—2442no—
501C8#1451974912yes—
502C8#1451974912yes—
503C22—22no—
504C22—22no—
505C22—42no—
506C22—41no—
507C22—42no—
508C22—9762no—
509C22—4884no—
510C22—4882no—
511C22—4882no—
512C22—4882no—
513C22#1684034011yescharacteristic
514C22D4×D24411yescharacteristic
515C22#1684045811yescharacteristic
516C4—22no—
517C4—41no—
518C4—81no—
519C4—4882no—
520C4—2442no—
521C4—2442no—
522C4C2×#1451974911yescharacteristic
523C4C2×#1451975111yescharacteristic
524C4#1684041611yescharacteristic
525C4#1684042111yescharacteristic
526C2—21no—
527C2—21no—
528C2—42no—
529C2—81no—
530C2—9762no—
531C2#2144925911yescharacteristic
532C2#2145758612yes—
subgroupG/H
centerC22D4×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#444C2
socleC22×C61D42
hypercenterC2×#444D61
hypocenterC61#24799
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#444

Complements

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

NH ≅ G/Ntype
1. C2×#16840032C2semidirect
2. #21455467C2semidirect
3. #21455356C2semidirect
4. #21454414C2semidirect
5. #21447174C2semidirect
6. #21447028C2semidirect
7. #21446695C2semidirect
8. #21446572C2semidirect
9. #21446193C2semidirect
10. #21445878C2semidirect
11. C2×C61×#444C2semidirect
98. #131×C61D4semidirect
263. C2×#444D61semidirect
313. C61#24799semidirect
334. #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
#214544142
#214471742
#214470282
#214466952
#214465722
#214461932
#214458782
C2×C61×#4442
#2479961