Finite Group Databasebeta

← #29658600

Subgroups — #29658600

Order 15616.

1192 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
1C2×D4×Dic122C211yesmaximal, characteristic, complemented
2#21456944C211yesmaximal, characteristic, complemented
3#21455846C211yesmaximal, characteristic, complemented
4#21445080C214yesmaximal, complemented
5#21444921C214yesmaximal, complemented
6#21444505C211yesmaximal, characteristic, complemented
7#21444247C211yesmaximal, characteristic, complemented
8#21444210C211yesmaximal, characteristic, complemented
9C61×#3036C211yesmaximal, characteristic
10C23×Dic122—21no—
11D4×Dic122—22no—
12D4×Dic122—22no—
13D4×Dic122—22no—
14D4×Dic122C2214yes—
15C2×#14519777C2211yescharacteristic
16C2×#14519776—21no—
17C2×#14519756—21no—
18C2×#14519754—22no—
19C2×C4×Dic122C2211yescharacteristic
20C2×D4×Dic61—21no—
21#16840158C2212yescomplemented
22#16840108C2212yescomplemented
23#16840078—22no—
24#16840078—21no—
25#16840052—22no—
26#16840044C2212yescomplemented
27#16839959C2212yescomplemented
28#16839900—21no—
29C4×#11758200—22no—
30#16839815C2211yescharacteristic
31#16839815C2211yescharacteristic
32#16839507C2214yescomplemented
33#16839492C2214yescomplemented
34D4⋊#92418C2214yes—
35C2×AGL(1,ℤ/8)×C61—21no—
36C61×#532C2211yescharacteristic
37C2×C4×D4×C61C2211yescharacteristic
38C61×#463—22no—
39C61×#461C2212yes—
40C61×#459C2212yes—
41C61×#423C2211yescharacteristic
42C61×#417—21no—
43C22×Dic122—22no—
44C22×Dic122—24no—
45C22×Dic122—21no—
46C22×Dic122—24no—
47C22×Dic122—48no—
48C22×Dic122D411yescharacteristic
49C23×Dic61—22no—
50C2×#11758200—22no—
51#14519777—21no—
52#14519777—21no—
53#14519777D412yes—
54#14519777C2312yes—
55#14519776—24no—
56#14519776—24no—
57#14519756—24no—
58#14519756—24no—
59#14519754—28no—
60#14519754—28no—
61C4×Dic122—21no—
62C4×Dic122—21no—
63C4×Dic122D412yes—
64C4×Dic122C2312yes—
65D4×Dic61—24no—
66D4×Dic61—24no—
67C2×C4⋊Dic61—21no—
68C2×C4⋊Dic61D411yescharacteristic
69C2×C4⋊Dic61D412yes—
70C2×#11758196—22no—
71C2×#11758196—21no—
72C2×C22⋊Dic61—22no—
73C2×C4×Dic61—21no—
74#14519708—22no—
75#14519708—24no—
76C4⋊#92418—21no—
77C4⋊#92418C2312yes—
78#14519700—22no—
79#14519698—22no—
80#14519698—24no—
81C22⋊#92418—21no—
82C22⋊#92418—21no—
83C4×#92418—21no—
84C4×#92418C2312yes—
85C2×#41×C61—21no—
86C22×D4×C61D411yescharacteristic
87C23×C4×C61—21no—
88AGL(1,ℤ/8)×C61—44no—
89C2×SD16×C61—22no—
90C2×D8×C61—22no—
91C2×M16×C61—21no—
92C4⋊Q8×C61C2311yescharacteristic
93#127×C61C2311yescharacteristic
94#124×C61—21no—
95#121×C61—21no—
96#121×C61—21no—
97C4×D4×C61—21no—
98C4×D4×C61—21no—
99C4×D4×C61—21no—
100C4×D4×C61—21no—
101C4×D4×C61D412yes—
102C4×D4×C61C2312yes—
103C2×C4⋊C4×C61D411yescharacteristic
104C2×C22⋊C4×C61—21no—
105C2×C42×C61C2311yescharacteristic
106C4⋊C8×C61—21no—
107C4⋊C8×C61C2312yes—
108#103×C61—22no—
109#102×C61—22no—
110C24×C61—21no—
111#41×C61—42no—
112#41×C61—42no—
113C2×Q8×C61—21no—
114C2×D4×C61—21no—
115C2×D4×C61—22no—
116C2×D4×C61—21no—
117C2×D4×C61—21no—
118C2×D4×C61—21no—
119C2×D4×C61—41no—
120C2×D4×C61SD1614yescomplemented
121C2×D4×C61C2×D412yes—
122C22×C4×C61—21no—
123C22×C4×C61—22no—
124C22×C4×C61—22no—
125C22×C4×C61—44no—
126C22×C4×C61—41no—
127C22×C4×C61C2×D411yescharacteristic
128C22×C4×C61C2×D411yescharacteristic
129C22×C4×C61C2×D411yescharacteristic
130C22×Dic61—21no—
131C22×Dic61—28no—
132C22×Dic61—22no—
133C22×Dic61—21no—
134C22×Dic61—48no—
135C2×Dic122—21no—
136C2×Dic122—24no—
137C2×Dic122—28no—
138C2×Dic122—22no—
139C2×Dic122—22no—
140C2×Dic122—416no—
141C2×Dic122—48no—
142C2×Dic122—48no—
143C2×Dic122—44no—
144C2×Dic122—48no—
145C2×Dic122C2×D412yes—
146SD16×C61—44no—
147D8×C61—44no—
148M16×C61—42no—
149C2×C8×C61—22no—
150C4⋊C4×C61—21no—
151C4⋊C4×C61—21no—
152C4⋊C4×C61—21no—
153C4⋊C4×C61C2×D412yes—
154C22⋊C4×C61—22no—
155C22⋊C4×C61—22no—
156C22⋊C4×C61—41no—
157C22⋊C4×C61—41no—
158C42×C61C2×D411yescharacteristic
159C42×C61C2×D411yescharacteristic
160C42×C61C2×D411yescharacteristic
161C42×C61C2411yescharacteristic
162C22⋊Dic61—28no—
163#11758200—44no—
164C4⋊Dic61—22no—
165C4⋊Dic61—21no—
166C4⋊Dic61—22no—
167C4⋊Dic61—22no—
168C4⋊Dic61C2×D412yes—
169C4⋊Dic61C2×D414yes—
170#11758196—28no—
171#11758196—22no—
172#11758196—22no—
173C4×Dic61—22no—
174C4×Dic61—22no—
175C2×#92418—22no—
176C2×#92418—22no—
177C23×C61—22no—
178C23×C61—24no—
179C23×C61—42no—
180C23×C61—42no—
181C23×C61—41no—
182C23×C61#12011yescharacteristic
183Q8×C61—41no—
184Q8×C61—41no—
185D4×C61—22no—
186D4×C61—24no—
187D4×C61—42no—
188D4×C61—41no—
189D4×C61—42no—
190D4×C61—41no—
191D4×C61—82no—
192D4×C61C2×SD1614yes—
193C2×C4×C61—22no—
194C2×C4×C61—22no—
195C2×C4×C61—41no—
196C2×C4×C61—44no—
197C2×C4×C61—42no—
198C2×C4×C61—44no—
199C2×C4×C61—41no—
200C2×C4×C61—82no—
201C2×C4×C61#12011yescharacteristic
202C2×C4×C61#12011yescharacteristic
203C2×C4×C61#12011yescharacteristic
204C2×C4×C61C2×SD1612yes—
205C2×C4×C61#13712yes—
206C2×C4×C61C22×D411yescharacteristic
207C2×C4×C61C22×D411yescharacteristic
208C2×C4×C61C22×D411yescharacteristic
209C2×Dic61—24no—
210C2×Dic61—22no—
211C2×Dic61—22no—
212C2×Dic61—24no—
213C2×Dic61—416no—
214C2×Dic61—44no—
215C2×Dic61—42no—
216Dic122—24no—
217Dic122—22no—
218Dic122—416no—
219Dic122—44no—
220Dic122—44no—
221Dic122AGL(1,ℤ/8)14yescomplemented
222C8×C61—42no—
223#92418—42no—
224#92418—42no—
225#18758—611nomaximal, Sylow
226C22×C61—21no—
227C22×C61—21no—
228C22×C61—24no—
229C22×C61—48no—
230C22×C61—42no—
231C22×C61—42no—
232C22×C61—81no—
233C22×C61—81no—
234C22×C61—81no—
235C22×C61#44812yes—
236C22×C61C2×#12011yescharacteristic
237C4×C61—21no—
238C4×C61—21no—
239C4×C61—21no—
240C4×C61—42no—
241C4×C61—81no—
242C4×C61#44812yes—
243C4×C61C22×SD1611yescharacteristic
244C4×C61C2×AGL(1,ℤ/8)12yes—
245C4×C61C2×#13711yescharacteristic
246Dic61—24no—
247Dic61—44no—
248Dic61—42no—
249C2×D4×Q8—611no—
250#3227—611no—
251#3103—611no—
252#3036D6111yescharacteristic
253#1651—614no—
254#1627—614no—
255#1549—611no—
256#1487—611no—
257#1478—611no—
258C2×C61—22no—
259C2×C61—44no—
260C2×C61—161no—
261C2×C61C2×#44811yescharacteristic
262C2×C61#300311yescharacteristic
263C2×C61#302011yescharacteristic
264C23×Q8—1221no—
265C2×AGL(1,ℤ/8)—21no—
266D4×Q8—614no—
267D4×Q8—1222no—
268D4×Q8—1222no—
269D4×Q8—1222no—
270#532C2×D6111yescharacteristic
271C2×C4⋊Q8—611no—
272C2×C4⋊Q8—1221no—
273C2×C22⋊Q8—1222no—
274C2×C22⋊Q8—1221no—
275C2×C4×Q8—611no—
276C2×C4×D4—1221no—
277C2×C4×D4C2×D6111yescharacteristic
278#498—612no—
279#486—612no—
280#481—1221no—
281#481—1222no—
282#476—1222no—
283#475—612no—
284#463—22no—
285#461C2×D6112yes—
286#460—612no—
287#459C2×D6112yes—
288#448—1221no—
289C4×SD16—1222no—
290#431—611no—
291#431—611no—
292#423C2×D6111yescharacteristic
293#417—21no—
294#332—614no—
295Q8⋊C8—614no—
296D4⋊C8—614no—
297C61#1875811yescharacteristic, complemented, Sylow
298C2×#41—21no—
299C22×Q8—611no—
300C22×Q8—1224no—
301C22×Q8—1222no—
302C22×Q8—1224no—
303C22×Q8—1221no—
304C22×Q8—2448no—
305C22×D4C22⋊D6111yescharacteristic
306C23×C4—21no—
307C23×C4—1222no—
308AGL(1,ℤ/8)—44no—
309C2×SD16—22no—
310C2×SD16—1222no—
311C2×D8—22no—
312C2×M16—21no—
313C4⋊Q8—612no—
314C4⋊Q8—612no—
315C4⋊Q8—1224no—
316C4⋊Q8—1224no—
317C4⋊Q8—1221no—
318C4⋊Q8—1221no—
319C4⋊Q8C22×D6111yescharacteristic
320#127C22×D6111yescharacteristic
321#124—21no—
322C22⋊Q8—1228no—
323C22⋊Q8—1228no—
324C22⋊Q8—1224no—
325C22⋊Q8—1224no—
326#121—21no—
327#121—21no—
328C4×Q8—612no—
329C4×Q8—612no—
330C4×Q8—1221no—
331C4×Q8—1221no—
332C4×D4—21no—
333C4×D4—21no—
334C4×D4—21no—
335C4×D4—21no—
336C4×D4—1224no—
337C4×D4—1224no—
338C4×D4C22⋊D6112yes—
339C4×D4C22×D6112yes—
340C2×C4⋊C4—611no—
341C2×C4⋊C4—612no—
342C2×C4⋊C4—1222no—
343C2×C4⋊C4—1221no—
344C2×C4⋊C4—1221no—
345C2×C4⋊C4C22⋊D6111yescharacteristic
346C2×C22⋊C4—21no—
347C2×C22⋊C4—1222no—
348C2×C42—1221no—
349C2×C42C22×D6111yescharacteristic
350#106—1222no—
351#106—1224no—
352C4⋊C8—21no—
353C4⋊C8—612no—
354C4⋊C8—1221no—
355C4⋊C8C22×D6112yes—
356#103—22no—
357#103—1222no—
358#102—22no—
359#102—1222no—
360#102—1224no—
361C22⋊C8—1221no—
362C22⋊C8—1221no—
363C4×C8—612no—
364C4×C8—1221no—
365C24—21no—
366#41—42no—
367#41—42no—
368C2×Q8—21no—
369C2×Q8—612no—
370C2×Q8—1224no—
371C2×Q8—1228no—
372C2×Q8—1222no—
373C2×Q8—1221no—
374C2×Q8—1222no—
375C2×Q8—2444no—
376C2×Q8—2448no—
377C2×Q8—2448no—
378C2×Q8—24416no—
379C2×Q8—2448no—
380C2×D4—21no—
381C2×D4—21no—
382C2×D4—21no—
383C2×D4—21no—
384C2×D4—22no—
385C2×D4—41no—
386C2×D4#1175820014yescomplemented
387C2×D4C2×C22⋊D6112yes—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—44no—
392C22×C4—41no—
393C22×C4—1221no—
394C22×C4—1228no—
395C22×C4—1221no—
396C22×C4—1222no—
397C22×C4—2448no—
398C22×C4D4×D6111yescharacteristic
399C22×C4D4×D6111yescharacteristic
400C22×C4C2×C22⋊D6111yescharacteristic
401SD16—44no—
402SD16—2444no—
403D8—44no—
404M16—42no—
405C2×C8—22no—
406C2×C8—1222no—
407C2×C8—1222no—
408C4⋊C4—21no—
409C4⋊C4—21no—
410C4⋊C4—21no—
411C4⋊C4—612no—
412C4⋊C4—614no—
413C4⋊C4—1228no—
414C4⋊C4—1222no—
415C4⋊C4—1222no—
416C4⋊C4—1221no—
417C4⋊C4—1222no—
418C4⋊C4—1222no—
419C4⋊C4—1222no—
420C4⋊C4C2×C22⋊D6112yes—
421C22⋊C4—22no—
422C22⋊C4—22no—
423C22⋊C4—41no—
424C22⋊C4—41no—
425C22⋊C4—1228no—
426C42—1222no—
427C42—1222no—
428C42D4×D6111yescharacteristic
429C42D4×D6111yescharacteristic
430C42C2×C22⋊D6111yescharacteristic
431C42C23×D6111yescharacteristic
432C23—22no—
433C23—24no—
434C23—42no—
435C23—42no—
436C23—41no—
437C23#1451974511yescharacteristic
438Q8—41no—
439Q8—41no—
440Q8—614no—
441Q8—1224no—
442Q8—1222no—
443Q8—2444no—
444Q8—24416no—
445Q8—2444no—
446D4—22no—
447D4—24no—
448D4—42no—
449D4—41no—
450D4—42no—
451D4—41no—
452D4—82no—
453D4C2×#1175820014yes—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—41no—
457C2×C4—44no—
458C2×C4—42no—
459C2×C4—44no—
460C2×C4—41no—
461C2×C4—82no—
462C2×C4—1224no—
463C2×C4—1222no—
464C2×C4—1222no—
465C2×C4—1224no—
466C2×C4—24416no—
467C2×C4—2442no—
468C2×C4—2444no—
469C2×C4#1451974511yescharacteristic
470C2×C4#1451974511yescharacteristic
471C2×C4#1451974511yescharacteristic
472C2×C4C2×#1175820012yes—
473C2×C4#1451980712yes—
474C2×C4C2×D4×D6111yescharacteristic
475C2×C4C2×D4×D6111yescharacteristic
476C2×C4C22×C22⋊D6111yescharacteristic
477C8—42no—
478C8—2442no—
479C8—2442no—
480C22—21no—
481C22—21no—
482C22—24no—
483C22—48no—
484C22—42no—
485C22—42no—
486C22—81no—
487C22—81no—
488C22—81no—
489C22#1683990012yes—
490C22C2×#1451974511yescharacteristic
491C4—21no—
492C4—21no—
493C4—21no—
494C4—42no—
495C4—81no—
496C4—1224no—
497C4—2442no—
498C4—2444no—
499C4#1683990012yes—
500C4C22×#1175820011yescharacteristic
501C4AGL(1,ℤ/8)×D6112yes—
502C4C2×#1451980711yescharacteristic
503C2—22no—
504C2—44no—
505C2—161no—
506C2C2×#1683990011yescharacteristic
507C2#2145504411yescharacteristic
508C2#2145518111yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC61×#3036C2
socleC22×C61C2×#120
hypercenter#3036D61
hypocenterC61#18758
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3036

Complements

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

NH ≅ G/Ntype
1. C2×D4×Dic122C2semidirect
2. #21456944C2semidirect
3. #21455846C2semidirect
4. #21445080C2semidirect
5. #21444921C2semidirect
6. #21444505C2semidirect
7. #21444247C2semidirect
8. #21444210C2semidirect
21. #16840158C22semidirect
22. #16840108C22semidirect
26. #16840044C22semidirect
27. #16839959C22semidirect
32. #16839507C22semidirect
33. #16839492C22semidirect
120. C2×D4×C61SD16semidirect
221. Dic122AGL(1,ℤ/8)semidirect
297. C61#18758semidirect
386. C2×D4#11758200semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Dic1222
#214569442
#214558462
#214450802
#214449212
#214445052
#214442472
#214442102
C61×#30362
#1875861