Finite Group Databasebeta

← #29657395

Subgroups — #29657395

Order 15616.

1024 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456734C211yesmaximal, characteristic, complemented
2C2×#16840110C211yesmaximal, characteristic, complemented
3#21455299C211yesmaximal, characteristic, complemented
4#21444929C214yesmaximal, complemented
5#21444908C214yesmaximal, complemented
6#21444504C211yesmaximal, characteristic, complemented
7#21444270C211yesmaximal, characteristic, complemented
8#21444201C211yesmaximal, characteristic, complemented
9C61×#3511C211yesmaximal, characteristic, complemented
10C2×#14519802—21no—
11C2×#14519774C2211yescharacteristic
12#16840138—21no—
13#16840110—21no—
14#16840110—22no—
15#16840110C2214yes—
16#16839962—21no—
17#16839962C2212yes—
18#16839957—21no—
19#16839957C2212yes—
20#16839957C2214yes—
21#16839895—22no—
22#16839815C2211yescharacteristic
23#16839786C2211yescharacteristic
24#16839766—21no—
25C2×#14519697—22no—
26C2×C4×#92418C2211yescharacteristic
27#16839508C2214yescomplemented
28Q8⋊#92418C2212yescomplemented
29D4⋊#92418C2212yescomplemented
30D4⋊#92418C2214yescomplemented
31C61×#556C2212yescomplemented
32C61×#555—21no—
33C61×#554—21no—
34C61×#553—22no—
35C61×#548—22no—
36C61×#547C2212yescomplemented
37C61×#542—22no—
38C61×#537—21no—
39C61×#533—21no—
40C61×#532—21no—
41C61×#532C2211yescharacteristic
42C2×C4⋊Q8×C61C2211yescharacteristic
43C61×#520—21no—
44C4×#41×C61C2211yescharacteristic
45C22×D244—21no—
46C22×D244—21no—
47C22×D244—21no—
48#14519802—44no—
49C2×#11758201—22no—
50C2×#11758199—22no—
51C2×#11758199—24no—
52C2×#11758194—21no—
53C22×#92418—21no—
54#14519774—21no—
55#14519774—21no—
56#14519774—21no—
57#14519774C2312yes—
58C4⋊#92418—21no—
59C4⋊#92418C2312yes—
60C4⋊#92418C2312yes—
61#14519697—22no—
62#14519697—22no—
63#14519697—24no—
64#14519697—28no—
65C22⋊#92418—21no—
66C4×#92418—21no—
67C4×#92418—21no—
68C4×#92418C2312yes—
69C2×#41×C61—22no—
70C2×#41×C61—21no—
71C2×#41×C61D411yescharacteristic
72C22×Q8×C61—21no—
73C4⋊Q8×C61—21no—
74C4⋊Q8×C61—21no—
75C4⋊Q8×C61D412yes—
76C4⋊Q8×C61D411yescharacteristic
77C4⋊Q8×C61C2312yes—
78#127×C61D411yescharacteristic
79#126×C61—24no—
80#125×C61—22no—
81#124×C61—24no—
82#124×C61—22no—
83#124×C61—21no—
84#123×C61—24no—
85C22⋊Q8×C61—22no—
86C22⋊Q8×C61—24no—
87C22⋊Q8×C61—22no—
88C22⋊Q8×C61—22no—
89#121×C61—22no—
90#121×C61—22no—
91#121×C61—24no—
92#121×C61—22no—
93#121×C61—21no—
94#121×C61D412yes—
95C4×Q8×C61—22no—
96C4×Q8×C61—22no—
97C4×Q8×C61D411yescharacteristic
98C4×Q8×C61C2311yescharacteristic
99C4×D4×C61—22no—
100C4×D4×C61—22no—
101C4×D4×C61—24no—
102C4×D4×C61—21no—
103C4×D4×C61—21no—
104C4×D4×C61—21no—
105C4×D4×C61D411yescharacteristic
106C4×D4×C61C2312yes—
107C4×D4×C61C2311yescharacteristic
108#117×C61—22no—
109#117×C61—22no—
110#117×C61—21no—
111#117×C61D411yescharacteristic
112C2×C4⋊C4×C61—21no—
113C2×C4⋊C4×C61—22no—
114C2×C4⋊C4×C61D412yes—
115C2×C42×C61—21no—
116C2×C42×C61C2311yescharacteristic
117C23×D61—41no—
118#41×C61—22no—
119#41×C61—22no—
120#41×C61—42no—
121#41×C61—42no—
122#41×C61—84no—
123C2×Q8×C61—21no—
124C2×Q8×C61—22no—
125C2×Q8×C61—22no—
126C2×Q8×C61—82no—
127C2×Q8×C61C2×D411yescharacteristic
128C2×D4×C61—21no—
129C2×D4×C61—22no—
130C2×D4×C61—24no—
131C2×D4×C61—21no—
132C2×D4×C61—21no—
133C2×D4×C61C2×D411yescharacteristic
134C2×D4×C61C2×D412yes—
135C22×C4×C61—24no—
136C22×C4×C61—22no—
137C22×C4×C61—21no—
138C22×C4×C61—21no—
139C22×C4×C61—21no—
140C22×C4×C61—22no—
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61C2×D411yescharacteristic
144C2×D244—22no—
145C2×D244—22no—
146C2×D244—22no—
147C2×D244—44no—
148C2×D244—42no—
149C2×D244—42no—
150C2×D244—41no—
151C2×D244—41no—
152C2×D244—41no—
153C2×D244—82no—
154C4⋊C4×C61—24no—
155C4⋊C4×C61—24no—
156C4⋊C4×C61—22no—
157C4⋊C4×C61—22no—
158C4⋊C4×C61—21no—
159C4⋊C4×C61—21no—
160C4⋊C4×C61—22no—
161C4⋊C4×C61—21no—
162C4⋊C4×C61C2×D411yescharacteristic
163C4⋊C4×C61C2×D414yes—
164C4⋊C4×C61C2×D411yescharacteristic
165C4⋊C4×C61C2×D412yes—
166C22⋊C4×C61—24no—
167C22⋊C4×C61—24no—
168C22⋊C4×C61—24no—
169C22⋊C4×C61—21no—
170C22⋊C4×C61—21no—
171C22⋊C4×C61—22no—
172C22⋊C4×C61—21no—
173C22⋊C4×C61—22no—
174C42×C61—24no—
175C42×C61—21no—
176C42×C61—21no—
177C42×C61—21no—
178C42×C61C2×D411yescharacteristic
179C42×C61C2×D411yescharacteristic
180C42×C61C2×D411yescharacteristic
181C42×C61C2411yescharacteristic
182#11758201—44no—
183#11758199—44no—
184#11758199—84no—
185#11758194—42no—
186C2×#92418—22no—
187C2×#92418—22no—
188C2×#92418—22no—
189C2×#92418—42no—
190C23×C61—21no—
191C23×C61—22no—
192C23×C61#12011yescharacteristic
193C22×D61—42no—
194C22×D61—82no—
195C22×D61—82no—
196C22×D61—82no—
197Q8×C61—21no—
198Q8×C61—21no—
199Q8×C61—44no—
200Q8×C61—42no—
201Q8×C61—82no—
202D4×C61—21no—
203D4×C61—21no—
204D4×C61—24no—
205D4×C61—42no—
206D4×C61—44no—
207D4×C61—42no—
208D4×C61—42no—
209D4×C61—84no—
210C2×C4×C61—22no—
211C2×C4×C61—22no—
212C2×C4×C61—21no—
213C2×C4×C61—21no—
214C2×C4×C61—22no—
215C2×C4×C61—21no—
216C2×C4×C61—24no—
217C2×C4×C61—42no—
218C2×C4×C61—42no—
219C2×C4×C61—42no—
220C2×C4×C61—42no—
221C2×C4×C61—42no—
222C2×C4×C61—82no—
223C2×C4×C61—84no—
224C2×C4×C61#12011yescharacteristic
225C2×C4×C61#12011yescharacteristic
226C2×C4×C61#12011yescharacteristic
227C2×C4×C61#13512yes—
228C2×C4×C61AGL(1,ℤ/8)12yes—
229C2×C4×C61C22×D411yescharacteristic
230C2×C4×C61C22×D411yescharacteristic
231C2×C4×C61C22×D411yescharacteristic
232D244—44no—
233D244—42no—
234D244—42no—
235D244—44no—
236D244—82no—
237D244—82no—
238D244—82no—
239#92418—42no—
240#92418—42no—
241#92418—42no—
242#18646—611nomaximal, Sylow
243C22×C61—21no—
244C22×C61—21no—
245C22×C61—41no—
246C22×C61—42no—
247C22×C61—41no—
248C22×C61—42no—
249C22×C61—41no—
250C22×C61—42no—
251C22×C61#44912yes—
252C22×C61C2×#12011yescharacteristic
253C2×D61—82no—
254C2×D61—82no—
255C2×D61—82no—
256C2×D61—164no—
257C4×C61—21no—
258C4×C61—21no—
259C4×C61—41no—
260C4×C61—42no—
261C4×C61—41no—
262C4×C61—44no—
263C4×C61—41no—
264C4×C61#44912yes—
265C4×C61#45412yes—
266C4×C61C2×#13511yescharacteristic
267C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
268C4×C61#57612yes—
269#3511D6111yescharacteristic, complemented
270#3205—611no—
271C2×#486—611no—
272#3038—611no—
273#1628—614no—
274#1625—614no—
275#1549—611no—
276#1492—611no—
277#1476—611no—
278C2×C61—22no—
279C2×C61—81no—
280C2×C61—82no—
281C2×C61C2×#44911yescharacteristic
282C2×C61#300811yescharacteristic
283C2×C61C2×#45411yescharacteristic
284D61—162no—
285C2×AGL(1,ℤ/8)—1221no—
286#556C2×D6112yescomplemented
287#555—21no—
288#554—21no—
289#553—22no—
290#548—22no—
291#547C2×D6112yescomplemented
292#542—22no—
293#537—21no—
294#533—21no—
295#532—21no—
296#532C2×D6111yescharacteristic
297C2×C4⋊Q8C2×D6111yescharacteristic
298C2×#127—611no—
299#520—21no—
300C4×#41C2×D6111yescharacteristic
301#493—1221no—
302#486—614no—
303#486—1222no—
304#486—1221no—
305#460—612no—
306#460—1221no—
307#459—612no—
308#459—614no—
309#459—1221no—
310#447—1222no—
311#431—611no—
312#423—611no—
313#418—1221no—
314C2×#102—1222no—
315C2×C4×C8—611no—
316#332—614no—
317Q8⋊C8—612no—
318D4⋊C8—612no—
319D4⋊C8—614no—
320C61#1864611yescharacteristic, complemented, Sylow
321C2×#41—22no—
322C2×#41—21no—
323C2×#41C22⋊D6111yescharacteristic
324C22×Q8—21no—
325C22×D4—1221no—
326C22×D4—1221no—
327C22×D4—1221no—
328AGL(1,ℤ/8)—2444no—
329C2×SD16—1222no—
330C2×D8—1224no—
331C2×D8—1222no—
332C2×M16—1221no—
333C22×C8—1221no—
334C4⋊Q8—21no—
335C4⋊Q8—21no—
336C4⋊Q8C22⋊D6112yes—
337C4⋊Q8C22⋊D6111yescharacteristic
338C4⋊Q8C22×D6112yes—
339#127—612no—
340#127—1221no—
341#127—1221no—
342#127—1221no—
343#127C22⋊D6111yescharacteristic
344#126—24no—
345#125—22no—
346#124—24no—
347#124—22no—
348#124—21no—
349#123—24no—
350C22⋊Q8—22no—
351C22⋊Q8—24no—
352C22⋊Q8—22no—
353C22⋊Q8—22no—
354#121—22no—
355#121—22no—
356#121—24no—
357#121—22no—
358#121—21no—
359#121C22⋊D6112yes—
360C4×Q8—22no—
361C4×Q8—22no—
362C4×Q8C22⋊D6111yescharacteristic
363C4×Q8C22×D6111yescharacteristic
364C4×D4—22no—
365C4×D4—24no—
366C4×D4—22no—
367C4×D4—21no—
368C4×D4—21no—
369C4×D4—21no—
370C4×D4C22⋊D6111yescharacteristic
371C4×D4C22×D6111yescharacteristic
372C4×D4C22×D6112yes—
373#117—22no—
374#117—22no—
375#117—21no—
376#117C22⋊D6111yescharacteristic
377C2×C4⋊C4—21no—
378C2×C4⋊C4—22no—
379C2×C4⋊C4C22⋊D6112yes—
380C2×C42—21no—
381C2×C42C22×D6111yescharacteristic
382C4⋊C8—612no—
383C4⋊C8—612no—
384C4⋊C8—1221no—
385#102—1228no—
386#102—1222no—
387#102—1222no—
388#102—1224no—
389C22⋊C8—1221no—
390C4×C8—612no—
391C4×C8—1221no—
392C4×C8—1221no—
393C24—2441no—
394#41—22no—
395#41—22no—
396#41—42no—
397#41—42no—
398#41—84no—
399C2×Q8—22no—
400C2×Q8—21no—
401C2×Q8—22no—
402C2×Q8—82no—
403C2×Q8C2×C22⋊D6111yescharacteristic
404C2×D4—21no—
405C2×D4—22no—
406C2×D4—24no—
407C2×D4—21no—
408C2×D4—21no—
409C2×D4—1222no—
410C2×D4—1222no—
411C2×D4—1222no—
412C2×D4—4882no—
413C2×D4—2444no—
414C2×D4—2441no—
415C2×D4—2441no—
416C2×D4—2441no—
417C2×D4—2442no—
418C2×D4—2442no—
419C2×D4C2×C22⋊D6111yescharacteristic
420C2×D4C2×C22⋊D6112yes—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4C2×C22⋊D6111yescharacteristic
428C22×C4C2×C22⋊D6111yescharacteristic
429C22×C4C2×C22⋊D6111yescharacteristic
430SD16—2444no—
431D8—4884no—
432D8—2444no—
433M16—2442no—
434C2×C8—1222no—
435C2×C8—1222no—
436C2×C8—1222no—
437C2×C8—2442no—
438C4⋊C4—24no—
439C4⋊C4—24no—
440C4⋊C4—22no—
441C4⋊C4—21no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4C2×C22⋊D6111yescharacteristic
447C4⋊C4C2×C22⋊D6114yes—
448C4⋊C4C2×C22⋊D6111yescharacteristic
449C4⋊C4C2×C22⋊D6112yes—
450C22⋊C4—24no—
451C22⋊C4—24no—
452C22⋊C4—24no—
453C22⋊C4—21no—
454C22⋊C4—21no—
455C22⋊C4—22no—
456C22⋊C4—21no—
457C22⋊C4—22no—
458C42—24no—
459C42—21no—
460C42—21no—
461C42—21no—
462C42C2×C22⋊D6111yescharacteristic
463C42C2×C22⋊D6111yescharacteristic
464C42C2×C22⋊D6111yescharacteristic
465C42C23×D6111yescharacteristic
466C23—21no—
467C23—22no—
468C23—4882no—
469C23—4882no—
470C23—4882no—
471C23—2442no—
472C23#1451974611yescharacteristic
473Q8—21no—
474Q8—21no—
475Q8—44no—
476Q8—42no—
477Q8—82no—
478D4—21no—
479D4—21no—
480D4—24no—
481D4—42no—
482D4—44no—
483D4—42no—
484D4—42no—
485D4—84no—
486D4—4882no—
487D4—4882no—
488D4—4882no—
489D4—2444no—
490D4—2442no—
491D4—2442no—
492D4—2444no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—22no—
498C2×C4—21no—
499C2×C4—24no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—82no—
506C2×C4—84no—
507C2×C4#1451974611yescharacteristic
508C2×C4#1451974611yescharacteristic
509C2×C4#1451974611yescharacteristic
510C2×C4#1451979712yes—
511C2×C4#1451980212yes—
512C2×C4C22×C22⋊D6111yescharacteristic
513C2×C4C22×C22⋊D6111yescharacteristic
514C2×C4C22×C22⋊D6111yescharacteristic
515C8—2442no—
516C8—2442no—
517C8—2442no—
518C22—21no—
519C22—21no—
520C22—41no—
521C22—42no—
522C22—41no—
523C22—42no—
524C22—41no—
525C22—42no—
526C22—9764no—
527C22—4882no—
528C22—4882no—
529C22—4882no—
530C22#1683990712yes—
531C22C2×#1451974611yescharacteristic
532C4—21no—
533C4—21no—
534C4—41no—
535C4—41no—
536C4—42no—
537C4—41no—
538C4—44no—
539C4#1683990712yes—
540C4#1683993512yes—
541C4C2×#1451979711yescharacteristic
542C4C2×#1451980211yescharacteristic
543C4#1684056512yes—
544C2—22no—
545C2—81no—
546C2—82no—
547C2—9762no—
548C2C2×#1683990711yescharacteristic
549C2#2145508711yescharacteristic
550C2C2×#1683993511yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3511C2
socleC22×C61C2×#120
hypercenter#3511D61
hypocenterC61#18646
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3511

Complements

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

NH ≅ G/Ntype
1. #21456734C2semidirect
2. C2×#16840110C2semidirect
3. #21455299C2semidirect
4. #21444929C2semidirect
5. #21444908C2semidirect
6. #21444504C2semidirect
7. #21444270C2semidirect
8. #21444201C2semidirect
9. C61×#3511C2semidirect
27. #16839508C22semidirect
28. Q8⋊#92418C22semidirect
29. D4⋊#92418C22semidirect
30. D4⋊#92418C22semidirect
31. C61×#556C22semidirect
36. C61×#547C22semidirect
269. #3511D61semidirect
286. #556C2×D61semidirect
291. #547C2×D61semidirect
320. C61#18646semidirect

Maximal subgroups

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

subgroupindex
#214567342
C2×#168401102
#214552992
#214449292
#214449082
#214445042
#214442702
#214442012
C61×#35112
#1864661