Finite Group Databasebeta

← #29658842

Subgroups — #29658842

Order 15616.

1224 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456947C211yesmaximal, characteristic, complemented
2#21455819C211yesmaximal, characteristic, complemented
3#21455283C211yesmaximal, characteristic, complemented
4#21445103C214yesmaximal, complemented
5#21444908C214yesmaximal, complemented
6#21444504C211yesmaximal, characteristic, complemented
7#21444244C211yesmaximal, characteristic, complemented
8#21444210C211yesmaximal, characteristic, complemented
9C2×C61×#547C211yesmaximal, characteristic, complemented
10C2×#14519803—21no—
11#16840312C2211yescharacteristic
12#16840158C2212yescomplemented
13#16840110C2212yescomplemented
14#16840080—22no—
15#16840075—21no—
16#16840049—22no—
17#16840042C2212yescomplemented
18#16839977—22no—
19#16839967C2212yescomplemented
20#16839957C2212yescomplemented
21#16839957C2212yescomplemented
22#16839896—21no—
23C4×#11758199—22no—
24#16839815C2211yescharacteristic
25#16839815C2211yescharacteristic
26#16839786C2211yescharacteristic
27#16839761—21no—
28#16839508C2214yescomplemented
29D4⋊#92418C2214yescomplemented
30D4⋊#92418C2214yescomplemented
31C22×#41×C61—21no—
32C61×#547—22no—
33C61×#547—22no—
34C61×#547—22no—
35C61×#547C2214yes—
36C2×C4⋊Q8×C61C2211yescharacteristic
37C2×#123×C61—22no—
38C2×C22⋊Q8×C61—21no—
39C2×#121×C61—21no—
40C2×C4×D4×C61C2211yescharacteristic
41C2×C4×D4×C61C2211yescharacteristic
42C22×C4⋊C4×C61—21no—
43C2×#11758222—21no—
44#14519803—44no—
45C2×#11758200—22no—
46C2×#11758199—22no—
47C2×#11758199—22no—
48C2×#11758194—21no—
49#14519777C2311yescharacteristic
50#14519774C2311yescharacteristic
51#14519765—21no—
52#14519751—21no—
53#14519751—21no—
54C4×D244—21no—
55C4×D244—21no—
56#14519710—22no—
57#14519708—24no—
58C4⋊#92418—21no—
59C4⋊#92418—21no—
60C4⋊#92418C2312yes—
61C4⋊#92418C2312yes—
62#14519700—22no—
63#14519698—22no—
64#14519697—22no—
65#14519697—22no—
66#14519697—24no—
67C22⋊#92418—21no—
68C22⋊#92418—21no—
69C4×#92418—21no—
70C4×#92418C2312yes—
71C2×#41×C61—24no—
72C2×#41×C61—24no—
73C2×#41×C61—48no—
74C22×Q8×C61—21no—
75C22×D4×C61—22no—
76C22×D4×C61D411yescharacteristic
77C22×D4×C61D411yescharacteristic
78C23×C4×C61—22no—
79C23×C4×C61—21no—
80C23×C4×C61—21no—
81C4⋊Q8×C61—21no—
82C4⋊Q8×C61—21no—
83C4⋊Q8×C61D412yes—
84C4⋊Q8×C61C2312yes—
85#123×C61—28no—
86#123×C61—28no—
87C22⋊Q8×C61—24no—
88C22⋊Q8×C61—24no—
89#121×C61—24no—
90#121×C61—24no—
91C4×D4×C61—21no—
92C4×D4×C61—21no—
93C4×D4×C61—21no—
94C4×D4×C61—21no—
95C4×D4×C61D412yes—
96C4×D4×C61D412yes—
97C4×D4×C61C2312yes—
98C4×D4×C61C2312yes—
99C2×C4⋊C4×C61—24no—
100C2×C4⋊C4×C61—22no—
101C2×C4⋊C4×C61—21no—
102C2×C4⋊C4×C61—24no—
103C2×C4⋊C4×C61D412yes—
104C2×C4⋊C4×C61D411yescharacteristic
105C2×C4⋊C4×C61D411yescharacteristic
106C2×C22⋊C4×C61—22no—
107C2×C22⋊C4×C61—21no—
108C2×C22⋊C4×C61—21no—
109C2×C42×C61C2311yescharacteristic
110C24×C61—21no—
111C24×C61—21no—
112#41×C61—416no—
113#41×C61—48no—
114#41×C61—48no—
115C2×Q8×C61—22no—
116C2×Q8×C61—22no—
117C2×Q8×C61—44no—
118C2×D4×C61—24no—
119C2×D4×C61—28no—
120C2×D4×C61—21no—
121C2×D4×C61—21no—
122C2×D4×C61—22no—
123C2×D4×C61—22no—
124C2×D4×C61—48no—
125C2×D4×C61C2×D412yes—
126C2×D4×C61C2×D412yes—
127C22×C4×C61—28no—
128C22×C4×C61—21no—
129C22×C4×C61—21no—
130C22×C4×C61—22no—
131C22×C4×C61—22no—
132C22×C4×C61—22no—
133C22×C4×C61—22no—
134C22×C4×C61—22no—
135C22×C4×C61—44no—
136C22×C4×C61—48no—
137C22×C4×C61—42no—
138C22×C4×C61—42no—
139C22×C4×C61C2×D411yescharacteristic
140C22×C4×C61C2×D411yescharacteristic
141C22×C4×C61C2×D411yescharacteristic
142C2×C22⋊D61—41no—
143#11758222—42no—
144#11758222—42no—
145C2×D244—21no—
146C2×D244—21no—
147C2×D244—21no—
148C2×C4×D61—41no—
149C2×Dic122—21no—
150C4⋊C4×C61—28no—
151C4⋊C4×C61—21no—
152C4⋊C4×C61—22no—
153C4⋊C4×C61—21no—
154C4⋊C4×C61—22no—
155C4⋊C4×C61—22no—
156C4⋊C4×C61C2×D412yes—
157C4⋊C4×C61C2×D412yes—
158C4⋊C4×C61C2×D414yes—
159C22⋊C4×C61—28no—
160C22⋊C4×C61—22no—
161C22⋊C4×C61—22no—
162C22⋊C4×C61—22no—
163C22⋊C4×C61—22no—
164C42×C61C2×D411yescharacteristic
165C42×C61C2×D411yescharacteristic
166C42×C61C2×D411yescharacteristic
167C42×C61C2411yescharacteristic
168#11758200—44no—
169#11758199—44no—
170#11758199—44no—
171#11758198—41no—
172#11758198—41no—
173C4⋊Dic61—21no—
174C4⋊Dic61—21no—
175#11758194—42no—
176C2×#92418—22no—
177C2×#92418—22no—
178C2×#92418—22no—
179C23×C61—24no—
180C23×C61—22no—
181C23×C61—22no—
182C23×C61—42no—
183C23×C61—42no—
184C23×C61—42no—
185C23×C61—42no—
186C23×C61—42no—
187C23×C61#12011yescharacteristic
188C22×D61—41no—
189Q8×C61—44no—
190Q8×C61—44no—
191D4×C61—24no—
192D4×C61—22no—
193D4×C61—22no—
194D4×C61—22no—
195D4×C61—416no—
196D4×C61—42no—
197D4×C61AGL(1,ℤ/8)14yescomplemented
198C2×C4×C61—22no—
199C2×C4×C61—22no—
200C2×C4×C61—22no—
201C2×C4×C61—24no—
202C2×C4×C61—41no—
203C2×C4×C61—44no—
204C2×C4×C61—42no—
205C2×C4×C61—44no—
206C2×C4×C61—416no—
207C2×C4×C61—42no—
208C2×C4×C61—42no—
209C2×C4×C61—42no—
210C2×C4×C61—44no—
211C2×C4×C61—44no—
212C2×C4×C61#12011yescharacteristic
213C2×C4×C61#12011yescharacteristic
214C2×C4×C61#12011yescharacteristic
215C2×C4×C61#13512yes—
216C2×C4×C61AGL(1,ℤ/8)12yes—
217C2×C4×C61C22×D411yescharacteristic
218C2×C4×C61C22×D411yescharacteristic
219C2×C4×C61C22×D411yescharacteristic
220C22⋊D61—82no—
221C2×Dic61—41no—
222D244—42no—
223D244—41no—
224D244—41no—
225D244—42no—
226C4×D61—82no—
227Dic122—41no—
228Dic122—41no—
229#92418—42no—
230#92418—42no—
231#92418—42no—
232#18782—611nomaximal, Sylow
233C22×C61—21no—
234C22×C61—21no—
235C22×C61—24no—
236C22×C61—48no—
237C22×C61—42no—
238C22×C61—42no—
239C22×C61—42no—
240C22×C61—42no—
241C22×C61—42no—
242C22×C61—84no—
243C22×C61#44912yes—
244C22×C61C2×#12011yescharacteristic
245C2×D61—81no—
246C2×D61—81no—
247C2×D61—81no—
248C4×C61—21no—
249C4×C61—21no—
250C4×C61—21no—
251C4×C61—42no—
252C4×C61—42no—
253C4×C61—44no—
254C4×C61#44912yes—
255C4×C61C2×#13511yescharacteristic
256C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
257C4×C61C2×AGL(1,ℤ/8)12yes—
258Dic61—81no—
259C2×#547D6111yescharacteristic, complemented
260#3227—611no—
261#3100—611no—
262#3036—611no—
263#1654—614no—
264#1625—614no—
265#1549—611no—
266#1486—611no—
267#1478—611no—
268C2×C61—22no—
269C2×C61—44no—
270C2×C61—82no—
271C2×C61C2×#44911yescharacteristic
272C2×C61#300311yescharacteristic
273C2×C61#302011yescharacteristic
274D61—161no—
275C22×#41—21no—
276C2×AGL(1,ℤ/8)—1221no—
277#547—22no—
278#547—22no—
279#547—22no—
280#547C2×D6114yes—
281#532—611no—
282C2×C4⋊Q8C2×D6111yescharacteristic
283C2×#123—22no—
284C2×C22⋊Q8—21no—
285C2×#121—21no—
286C2×C4×D4C2×D6111yescharacteristic
287C2×C4×D4C2×D6111yescharacteristic
288C22×C4⋊C4—21no—
289#498—612no—
290#486—612no—
291#481—1222no—
292#480—1221no—
293#476—1222no—
294#474—612no—
295#463—1222no—
296#461—612no—
297#459—612no—
298#459—612no—
299#447—1221no—
300C4×D8—1222no—
301#431—611no—
302#431—611no—
303#423—611no—
304#417—1221no—
305#332—614no—
306D4⋊C8—614no—
307D4⋊C8—614no—
308C61#1878211yescharacteristic, complemented, Sylow
309C2×#41—24no—
310C2×#41—24no—
311C2×#41—48no—
312C2×#41—1221no—
313C22×Q8—21no—
314C22×D4—22no—
315C22×D4C22⋊D6111yescharacteristic
316C22×D4C22⋊D6111yescharacteristic
317C23×C4—22no—
318C23×C4—21no—
319C23×C4—21no—
320AGL(1,ℤ/8)—2444no—
321C2×SD16—1222no—
322C2×D8—1222no—
323C2×D8—1222no—
324C2×M16—1221no—
325C4⋊Q8—21no—
326C4⋊Q8—21no—
327C4⋊Q8—611no—
328C4⋊Q8C22⋊D6112yes—
329C4⋊Q8C22×D6112yes—
330#127—611no—
331#124—1221no—
332#123—28no—
333#123—28no—
334C22⋊Q8—24no—
335C22⋊Q8—24no—
336#121—24no—
337#121—24no—
338#121—1221no—
339#121—1221no—
340C4×D4—21no—
341C4×D4—21no—
342C4×D4—21no—
343C4×D4—21no—
344C4×D4—1221no—
345C4×D4—1221no—
346C4×D4C22⋊D6112yes—
347C4×D4C22⋊D6112yes—
348C4×D4C22×D6112yes—
349C4×D4C22×D6112yes—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—24no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—24no—
354C2×C4⋊C4C22⋊D6111yescharacteristic
355C2×C4⋊C4C22⋊D6111yescharacteristic
356C2×C4⋊C4C22⋊D6112yes—
357C2×C22⋊C4—22no—
358C2×C22⋊C4—21no—
359C2×C22⋊C4—21no—
360C2×C42C22×D6111yescharacteristic
361#107—1222no—
362#106—1224no—
363C4⋊C8—612no—
364C4⋊C8—612no—
365C4⋊C8—1221no—
366C4⋊C8—1221no—
367#103—1222no—
368#102—1222no—
369#102—1222no—
370#102—1222no—
371#102—1224no—
372C22⋊C8—1221no—
373C22⋊C8—1221no—
374C4×C8—612no—
375C4×C8—1221no—
376C24—21no—
377C24—21no—
378#41—416no—
379#41—48no—
380#41—48no—
381#41—2442no—
382#41—2442no—
383C2×Q8—22no—
384C2×Q8—22no—
385C2×Q8—44no—
386C2×Q8—1221no—
387C2×D4—24no—
388C2×D4—28no—
389C2×D4—21no—
390C2×D4—21no—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—48no—
394C2×D4—1221no—
395C2×D4—1221no—
396C2×D4—1221no—
397C2×D4—2441no—
398C2×D4C2×C22⋊D6112yes—
399C2×D4C2×C22⋊D6112yes—
400C22×C4—28no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—44no—
409C22×C4—48no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—2441no—
413C22×C4C2×C22⋊D6111yescharacteristic
414C22×C4C2×C22⋊D6111yescharacteristic
415C22×C4C2×C22⋊D6111yescharacteristic
416SD16—2444no—
417D8—2444no—
418D8—2444no—
419M16—2442no—
420C2×C8—1222no—
421C2×C8—1222no—
422C2×C8—1222no—
423C4⋊C4—28no—
424C4⋊C4—22no—
425C4⋊C4—21no—
426C4⋊C4—22no—
427C4⋊C4—21no—
428C4⋊C4—22no—
429C4⋊C4—1221no—
430C4⋊C4—1221no—
431C4⋊C4C2×C22⋊D6112yes—
432C4⋊C4C2×C22⋊D6114yes—
433C4⋊C4C2×C22⋊D6112yes—
434C22⋊C4—28no—
435C22⋊C4—22no—
436C22⋊C4—22no—
437C22⋊C4—22no—
438C22⋊C4—22no—
439C22⋊C4—2441no—
440C22⋊C4—2441no—
441C42C2×C22⋊D6111yescharacteristic
442C42C2×C22⋊D6111yescharacteristic
443C42C2×C22⋊D6111yescharacteristic
444C42C23×D6111yescharacteristic
445C23—24no—
446C23—22no—
447C23—22no—
448C23—42no—
449C23—42no—
450C23—42no—
451C23—42no—
452C23—42no—
453C23—2441no—
454C23#1451974611yescharacteristic
455Q8—44no—
456Q8—44no—
457Q8—2441no—
458Q8—2441no—
459D4—24no—
460D4—22no—
461D4—22no—
462D4—22no—
463D4—416no—
464D4—42no—
465D4—4882no—
466D4—2442no—
467D4—2441no—
468D4—2441no—
469D4—2442no—
470D4#1451980314yescomplemented
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—24no—
475C2×C4—41no—
476C2×C4—44no—
477C2×C4—42no—
478C2×C4—44no—
479C2×C4—416no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—4882no—
486C2×C4—2441no—
487C2×C4#1451974611yescharacteristic
488C2×C4#1451974611yescharacteristic
489C2×C4#1451974611yescharacteristic
490C2×C4#1451979712yes—
491C2×C4#1451980212yes—
492C2×C4C22×C22⋊D6111yescharacteristic
493C2×C4C22×C22⋊D6111yescharacteristic
494C2×C4C22×C22⋊D6111yescharacteristic
495C8—2442no—
496C8—2442no—
497C8—2442no—
498C22—21no—
499C22—21no—
500C22—24no—
501C22—48no—
502C22—42no—
503C22—42no—
504C22—42no—
505C22—42no—
506C22—42no—
507C22—84no—
508C22—4881no—
509C22—4881no—
510C22—4881no—
511C22#1683990712yes—
512C22C2×#1451974611yescharacteristic
513C4—21no—
514C4—21no—
515C4—21no—
516C4—42no—
517C4—42no—
518C4—44no—
519C4—4881no—
520C4#1683990712yes—
521C4C2×#1451979711yescharacteristic
522C4C2×#1451980211yescharacteristic
523C4C2×#1451980312yes—
524C2—22no—
525C2—44no—
526C2—82no—
527C2—9761no—
528C2C2×#1683990711yescharacteristic
529C2#2145504611yescharacteristic
530C2#2145518311yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC2×C61×#547C2
socleC22×C61C2×#120
hypercenterC2×#547D61
hypocenterC61#18782
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#547

Complements

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

NH ≅ G/Ntype
1. #21456947C2semidirect
2. #21455819C2semidirect
3. #21455283C2semidirect
4. #21445103C2semidirect
5. #21444908C2semidirect
6. #21444504C2semidirect
7. #21444244C2semidirect
8. #21444210C2semidirect
9. C2×C61×#547C2semidirect
12. #16840158C22semidirect
13. #16840110C22semidirect
17. #16840042C22semidirect
19. #16839967C22semidirect
20. #16839957C22semidirect
21. #16839957C22semidirect
28. #16839508C22semidirect
29. D4⋊#92418C22semidirect
30. D4⋊#92418C22semidirect
197. D4×C61AGL(1,ℤ/8)semidirect
259. C2×#547D61semidirect
308. C61#18782semidirect
470. D4#14519803semidirect

Maximal subgroups

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

subgroupindex
#214569472
#214558192
#214552832
#214451032
#214449082
#214445042
#214442442
#214442102
C2×C61×#5472
#1878261