Finite Group Databasebeta

← #29658518

Subgroups — #29658518

Order 15616.

1040 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456737C211yesmaximal, characteristic
2#21455850C211yesmaximal, characteristic, complemented
3#21455291C211yesmaximal, characteristic, complemented
4#21445005C214yesmaximal, complemented
5#21444878C214yesmaximal, complemented
6#21444487C211yesmaximal, characteristic, complemented
7#21444248C211yesmaximal, characteristic, complemented
8#21444214C211yesmaximal, characteristic, complemented
9C2×C61×#559C211yesmaximal, characteristic, complemented
10C2×#14519808—21no—
11#16840312C2211yescharacteristic
12#16840138C2212yes—
13#16840108C2212yes—
14#16840080—21no—
15#16840047C2212yescomplemented
16#16839984—22no—
17#16839972C2212yescomplemented
18#16839962C2212yescomplemented
19#16839962C2212yescomplemented
20#16839954—22no—
21#16839901—21no—
22#16839896—22no—
23C4×#11758201—22no—
24#16839815C2211yescharacteristic
25#16839815C2211yescharacteristic
26#16839786C2211yescharacteristic
27#16839761—21no—
28#16839506C2214yes—
29Q8⋊#92418C2214yescomplemented
30Q8⋊#92418C2214yescomplemented
31C61×#559—22no—
32C61×#559—22no—
33C61×#559—22no—
34C61×#559C2214yes—
35C2×#127×C61C2211yescharacteristic
36C2×#124×C61—21no—
37C2×#124×C61—21no—
38C2×#123×C61—22no—
39C2×C4×Q8×C61C2211yescharacteristic
40C2×C4×Q8×C61C2211yescharacteristic
41C2×C4×D4×C61—21no—
42C2×C4×D4×C61—21no—
43C2×#11758222—21no—
44#14519808—44no—
45C2×#11758202—22no—
46C2×#11758201—22no—
47C2×#11758201—22no—
48C2×#11758199—24no—
49C2×#11758194—21no—
50#14519777C2311yescharacteristic
51#14519774C2311yescharacteristic
52#14519765—21no—
53#14519751—21no—
54#14519751—21no—
55C4×D244—21no—
56C4×D244—21no—
57#14519708—22no—
58C4⋊#92418—21no—
59C4⋊#92418—21no—
60C4⋊#92418C2312yes—
61C4⋊#92418C2312yes—
62#14519701—22no—
63#14519700—22no—
64#14519698—24no—
65#14519697—22no—
66#14519697—22no—
67C22⋊#92418—21no—
68C22⋊#92418—21no—
69C4×#92418—21no—
70C4×#92418C2312yes—
71C22×Q8×C61D411yescharacteristic
72C22×Q8×C61D411yescharacteristic
73C22×D4×C61—21no—
74C22×D4×C61—21no—
75C22×D4×C61D412yes—
76C23×C4×C61—22no—
77#127×C61—21no—
78#127×C61—21no—
79#127×C61D412yes—
80#127×C61C2312yes—
81#124×C61—24no—
82#124×C61—24no—
83#124×C61—24no—
84#124×C61—24no—
85#123×C61—28no—
86#123×C61—28no—
87C4×Q8×C61—21no—
88C4×Q8×C61—21no—
89C4×Q8×C61—21no—
90C4×Q8×C61—21no—
91C4×Q8×C61D412yes—
92C4×Q8×C61D412yes—
93C4×Q8×C61C2312yes—
94C4×Q8×C61C2312yes—
95C4×D4×C61—24no—
96C4×D4×C61—24no—
97C4×D4×C61—24no—
98C4×D4×C61—24no—
99C2×C4⋊C4×C61—21no—
100C2×C4⋊C4×C61—21no—
101C2×C4⋊C4×C61D411yescharacteristic
102C2×C4⋊C4×C61D411yescharacteristic
103C2×C22⋊C4×C61—22no—
104C2×C22⋊C4×C61—22no—
105C2×C22⋊C4×C61—22no—
106C2×C42×C61—21no—
107C2×C42×C61—21no—
108C2×C42×C61C2311yescharacteristic
109C24×C61—22no—
110C2×Q8×C61—21no—
111C2×Q8×C61—21no—
112C2×Q8×C61—22no—
113C2×Q8×C61—22no—
114C2×Q8×C61—42no—
115C2×Q8×C61C2×D412yes—
116C2×Q8×C61C2×D412yes—
117C2×D4×C61—22no—
118C2×D4×C61—22no—
119C2×D4×C61—22no—
120C2×D4×C61—22no—
121C2×D4×C61—22no—
122C2×D4×C61—44no—
123C2×D4×C61—42no—
124C2×D4×C61—42no—
125C2×D4×C61—82no—
126C2×D4×C61C2×D414yes—
127C22×C4×C61—21no—
128C22×C4×C61—21no—
129C22×C4×C61—21no—
130C22×C4×C61—21no—
131C22×C4×C61—28no—
132C22×C4×C61—84no—
133C22×C4×C61C2×D411yescharacteristic
134C22×C4×C61C2×D411yescharacteristic
135C22×C4×C61C2×D411yescharacteristic
136C2×C22⋊D61—41no—
137#11758222—42no—
138#11758222—42no—
139C2×D244—21no—
140C2×D244—21no—
141C2×D244—21no—
142C2×C4×D61—41no—
143C2×Dic122—21no—
144C4⋊C4×C61—22no—
145C4⋊C4×C61—22no—
146C4⋊C4×C61—21no—
147C4⋊C4×C61—21no—
148C4⋊C4×C61—22no—
149C4⋊C4×C61—22no—
150C4⋊C4×C61C2×D412yes—
151C4⋊C4×C61C2×D412yes—
152C22⋊C4×C61—28no—
153C22⋊C4×C61—28no—
154C22⋊C4×C61—28no—
155C42×C61—22no—
156C42×C61—22no—
157C42×C61—22no—
158C42×C61—22no—
159C42×C61C2×D411yescharacteristic
160C42×C61C2×D411yescharacteristic
161C42×C61C2×D411yescharacteristic
162C42×C61C2411yescharacteristic
163#11758202—44no—
164#11758201—44no—
165#11758201—44no—
166#11758199—84no—
167#11758198—41no—
168#11758198—41no—
169C4⋊Dic61—21no—
170C4⋊Dic61—21no—
171#11758194—42no—
172C2×#92418—22no—
173C2×#92418—22no—
174C2×#92418—22no—
175C23×C61—24no—
176C23×C61—44no—
177C23×C61—44no—
178C23×C61—44no—
179C23×C61#12011yescharacteristic
180C22×D61—41no—
181Q8×C61—22no—
182Q8×C61—22no—
183Q8×C61—22no—
184Q8×C61—22no—
185Q8×C61—42no—
186Q8×C61—42no—
187D4×C61—44no—
188D4×C61—44no—
189D4×C61—42no—
190D4×C61—42no—
191D4×C61—44no—
192D4×C61—44no—
193D4×C61—82no—
194C2×C4×C61—22no—
195C2×C4×C61—22no—
196C2×C4×C61—22no—
197C2×C4×C61—22no—
198C2×C4×C61—22no—
199C2×C4×C61—41no—
200C2×C4×C61—42no—
201C2×C4×C61—42no—
202C2×C4×C61—42no—
203C2×C4×C61—42no—
204C2×C4×C61—88no—
205C2×C4×C61#12011yescharacteristic
206C2×C4×C61#12011yescharacteristic
207C2×C4×C61#12011yescharacteristic
208C2×C4×C61#13512yes—
209C2×C4×C61AGL(1,ℤ/8)12yes—
210C2×C4×C61C22×D411yescharacteristic
211C2×C4×C61C22×D411yescharacteristic
212C2×C4×C61C22×D411yescharacteristic
213C22⋊D61—82no—
214C2×Dic61—41no—
215D244—42no—
216D244—41no—
217D244—41no—
218D244—42no—
219C4×D61—82no—
220Dic122—41no—
221Dic122—41no—
222#92418—42no—
223#92418—42no—
224#92418—42no—
225#18750—611nomaximal, Sylow
226C22×C61—21no—
227C22×C61—21no—
228C22×C61—44no—
229C22×C61—44no—
230C22×C61—44no—
231C22×C61—88no—
232C22×C61#44912yes—
233C22×C61C2×#12011yescharacteristic
234C2×D61—81no—
235C2×D61—81no—
236C2×D61—81no—
237C4×C61—21no—
238C4×C61—21no—
239C4×C61—21no—
240C4×C61—42no—
241C4×C61—42no—
242C4×C61—42no—
243C4×C61—42no—
244C4×C61#44912yes—
245C4×C61C2×#13511yescharacteristic
246C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
247C4×C61#57512yes—
248Dic61—81no—
249C2×#559D6111yescharacteristic, complemented
250#3205—611no—
251#3103—611no—
252#3037—611no—
253#1640—614no—
254#1620—614no—
255#1545—611no—
256#1487—611no—
257#1479—611no—
258C2×C61—22no—
259C2×C61—84no—
260C2×C61C2×#44911yescharacteristic
261C2×C61#300511yescharacteristic
262C2×C61#301911yescharacteristic
263D61—161no—
264C2×#137—1221no—
265#559—22no—
266#559—22no—
267#559—22no—
268#559C2×D6114yes—
269#532—611no—
270C2×#127C2×D6111yescharacteristic
271C2×#124—21no—
272C2×#124—21no—
273C2×#123—22no—
274C2×C4×Q8C2×D6111yescharacteristic
275C2×C4×Q8C2×D6111yescharacteristic
276C2×C4×D4—21no—
277C2×C4×D4—21no—
278#493—612no—
279#486—612no—
280#481—1221no—
281#475—612no—
282#464—1222no—
283#462—612no—
284#460—612no—
285#460—612no—
286#459—1222no—
287#448—1221no—
288#447—1222no—
289C4×SD16—1222no—
290#431—611no—
291#431—611no—
292#423—611no—
293#417—1221no—
294#331—614no—
295Q8⋊C8—614no—
296Q8⋊C8—614no—
297C61#1875011yescharacteristic, complemented, Sylow
298C2×#41—1221no—
299C22×Q8C22⋊D6111yescharacteristic
300C22×Q8C22⋊D6111yescharacteristic
301C22×D4—21no—
302C22×D4—21no—
303C22×D4C22⋊D6112yes—
304C23×C4—22no—
305#137—2444no—
306C2×Q16—1222no—
307C2×SD16—1222no—
308C2×SD16—1222no—
309C2×D8—1224no—
310C2×M16—1221no—
311C4⋊Q8—611no—
312#127—21no—
313#127—21no—
314#127—611no—
315#127C22⋊D6112yes—
316#127C22×D6112yes—
317#124—24no—
318#124—24no—
319#124—24no—
320#124—24no—
321#124—1221no—
322#123—28no—
323#123—28no—
324#121—1221no—
325#121—1221no—
326C4×Q8—21no—
327C4×Q8—21no—
328C4×Q8—21no—
329C4×Q8—21no—
330C4×Q8C22⋊D6112yes—
331C4×Q8C22⋊D6112yes—
332C4×Q8C22×D6112yes—
333C4×Q8C22×D6112yes—
334C4×D4—24no—
335C4×D4—24no—
336C4×D4—24no—
337C4×D4—24no—
338C4×D4—1221no—
339C4×D4—1221no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4C22⋊D6111yescharacteristic
343C2×C4⋊C4C22⋊D6111yescharacteristic
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C42—21no—
348C2×C42—21no—
349C2×C42C22×D6111yescharacteristic
350#106—1222no—
351C4⋊C8—612no—
352C4⋊C8—612no—
353C4⋊C8—1221no—
354C4⋊C8—1221no—
355#103—1222no—
356#103—1222no—
357#102—1224no—
358#102—1222no—
359#102—1222no—
360C22⋊C8—1221no—
361C22⋊C8—1221no—
362C4×C8—612no—
363C4×C8—1221no—
364C24—22no—
365#41—2442no—
366#41—2442no—
367C2×Q8—21no—
368C2×Q8—21no—
369C2×Q8—22no—
370C2×Q8—22no—
371C2×Q8—42no—
372C2×Q8—1221no—
373C2×Q8C2×C22⋊D6112yes—
374C2×Q8C2×C22⋊D6112yes—
375C2×D4—22no—
376C2×D4—22no—
377C2×D4—22no—
378C2×D4—22no—
379C2×D4—22no—
380C2×D4—44no—
381C2×D4—42no—
382C2×D4—42no—
383C2×D4—82no—
384C2×D4—1221no—
385C2×D4—1221no—
386C2×D4—1221no—
387C2×D4—2441no—
388C2×D4C2×C22⋊D6114yes—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—28no—
394C22×C4—84no—
395C22×C4—2441no—
396C22×C4C2×C22⋊D6111yescharacteristic
397C22×C4C2×C22⋊D6111yescharacteristic
398C22×C4C2×C22⋊D6111yescharacteristic
399Q16—2444no—
400SD16—2444no—
401SD16—2444no—
402D8—4884no—
403M16—2442no—
404C2×C8—1222no—
405C2×C8—1222no—
406C2×C8—1222no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—21no—
412C4⋊C4—21no—
413C4⋊C4—1221no—
414C4⋊C4—1221no—
415C4⋊C4C2×C22⋊D6112yes—
416C4⋊C4C2×C22⋊D6112yes—
417C22⋊C4—28no—
418C22⋊C4—28no—
419C22⋊C4—28no—
420C22⋊C4—2441no—
421C22⋊C4—2441no—
422C42—22no—
423C42—22no—
424C42—22no—
425C42—22no—
426C42C2×C22⋊D6111yescharacteristic
427C42C2×C22⋊D6111yescharacteristic
428C42C2×C22⋊D6111yescharacteristic
429C42C23×D6111yescharacteristic
430C23—24no—
431C23—44no—
432C23—44no—
433C23—44no—
434C23—2441no—
435C23#1451974611yescharacteristic
436Q8—22no—
437Q8—22no—
438Q8—22no—
439Q8—22no—
440Q8—42no—
441Q8—42no—
442Q8—2441no—
443Q8—2441no—
444D4—44no—
445D4—44no—
446D4—42no—
447D4—42no—
448D4—44no—
449D4—44no—
450D4—82no—
451D4—4882no—
452D4—2442no—
453D4—2441no—
454D4—2441no—
455D4—2442no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—41no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—88no—
467C2×C4—4882no—
468C2×C4—2441no—
469C2×C4#1451974611yescharacteristic
470C2×C4#1451974611yescharacteristic
471C2×C4#1451974611yescharacteristic
472C2×C4#1451979712yes—
473C2×C4#1451980212yes—
474C2×C4C22×C22⋊D6111yescharacteristic
475C2×C4C22×C22⋊D6111yescharacteristic
476C2×C4C22×C22⋊D6111yescharacteristic
477C8—2442no—
478C8—2442no—
479C8—2442no—
480C22—21no—
481C22—21no—
482C22—44no—
483C22—44no—
484C22—44no—
485C22—88no—
486C22—4881no—
487C22—4881no—
488C22—4881no—
489C22#1683990712yes—
490C22C2×#1451974611yescharacteristic
491C4—21no—
492C4—21no—
493C4—21no—
494C4—42no—
495C4—42no—
496C4—42no—
497C4—42no—
498C4—4881no—
499C4#1683990712yes—
500C4C2×#1451979711yescharacteristic
501C4C2×#1451980211yescharacteristic
502C4#1684056012yes—
503C2—22no—
504C2—84no—
505C2—9761no—
506C2C2×#1683990711yescharacteristic
507C2#2145506111yescharacteristic
508C2#2145517611yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC2×C61×#559C2
socleC22×C61C2×#120
prime radical#21456737C2
hypercenterC2×#559D61
hypocenterC61#18750
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#559

Complements

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

NH ≅ G/Ntype
2. #21455850C2semidirect
3. #21455291C2semidirect
4. #21445005C2semidirect
5. #21444878C2semidirect
6. #21444487C2semidirect
7. #21444248C2semidirect
8. #21444214C2semidirect
9. C2×C61×#559C2semidirect
15. #16840047C22semidirect
17. #16839972C22semidirect
18. #16839962C22semidirect
19. #16839962C22semidirect
29. Q8⋊#92418C22semidirect
30. Q8⋊#92418C22semidirect
249. C2×#559D61semidirect
297. C61#18750semidirect

Maximal subgroups

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

subgroupindex
#214567372
#214558502
#214552912
#214450052
#214448782
#214444872
#214442482
#214442142
C2×C61×#5592
#1875061