Finite Group Databasebeta

← #29696250

Subgroups — #29696250

Order 15616.

1258 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840472C211yesmaximal, characteristic
2C2×#16839827C211yesmaximal, characteristic
3#21448719C211yesmaximal, characteristic
4#21446779C211yesmaximal, characteristic, complemented
5#21446766C211yesmaximal, characteristic, complemented
6#21446239C212yesmaximal, complemented
7#21446123C212yesmaximal, complemented
8#21445908C212yesmaximal, complemented
9C4×#14519701C211yesmaximal, characteristic, complemented
10#340×D61C212yesmaximal, complemented
11C61×#1803C211yesmaximal, characteristic, complemented
12C22×C8×D61C2212yes—
13#16840472—24no—
14#16840472C418yescomplemented
15C2×#14519777C2211yescharacteristic
16C2×#14519769—21no—
17C2×#14519755—22no—
18C2×#14519753C2212yes—
19C2×C4×D244—21no—
20C2×C4×D244C2211yescharacteristic
21C2×#14519739—21no—
22C2×C4⋊C4×D61C2212yes—
23C2×C4⋊C4×D61C2212yes—
24#16839827—24no—
25C2×#14519701C2211yescharacteristic
26C2×#14519701C2211yescharacteristic
27C2×#14519689C2212yes—
28C2×C4×#92418C2211yescharacteristic
29#16839658—21no—
30#16839633—22no—
31#16839619C2212yes—
32C4×C4⋊Dic61C2211yescharacteristic
33#16839525C2212yes—
34#16839525C2212yes—
35#16839524C2211yescharacteristic
36#16839524C2211yescharacteristic
37#16839524C2212yes—
38#16839524C2212yes—
39C2×C4×Q8×C61C2211yescharacteristic
40C2×C4⋊C8×C61C2211yescharacteristic
41C2×#103×C61C2212yes—
42C61×#384C2211yescharacteristic
43C61×#340C2212yes—
44C22×Dic122—21no—
45C22×D244D411yescharacteristic
46C22×C4×D61—22no—
47C22×C4×D61—22no—
48C22×C4×D61C2312yes—
49C22×#92418C2311yescharacteristic
50C22×#92418C2311yescharacteristic
51C2×C8×D61—24no—
52C2×C8×D61—24no—
53C2×C8×D61—42no—
54#14519777—22no—
55#14519777C2×C414yescomplemented
56#14519769—28no—
57#14519755—216no—
58#14519753—24no—
59#14519753C2×C418yescomplemented
60C4×D244—28no—
61C4×D244C2×C414yes—
62C4×D244D414yes—
63#14519739—44no—
64C2×C4⋊Dic61—21no—
65C2×C4⋊Dic61D411yescharacteristic
66C2×C4⋊Dic61C2311yescharacteristic
67C2×C4⋊Dic61C2312yes—
68C2×C4⋊Dic61C2311yescharacteristic
69C2×#11758196—22no—
70C2×#11758196—22no—
71C2×#11758196—22no—
72C4⋊C4×D61—24no—
73C4⋊C4×D61—24no—
74C4⋊C4×D61—42no—
75C4⋊C4×D61C2×C418yescomplemented
76C2×#11758198—22no—
77C2×#11758198—22no—
78C2×#11758198D412yes—
79C2×C4×Dic61—21no—
80#14519701—22no—
81#14519701—22no—
82#14519689—24no—
83C4×#92418—22no—
84#14519684—21no—
85#14519684—22no—
86C22×Q8×C61C2311yescharacteristic
87C22×C8×C61C2312yes—
88C4×Q8×C61—22no—
89C4×Q8×C61C2×C414yescomplemented
90C2×C4⋊C4×C61—21no—
91C2×C4⋊C4×C61—21no—
92C2×C4⋊C4×C61—21no—
93C2×C4⋊C4×C61C2312yes—
94C2×C4⋊C4×C61C2311yescharacteristic
95C2×C42×C61—21no—
96C2×C42×C61C2311yescharacteristic
97C4⋊C8×C61—22no—
98#103×C61—42no—
99C23×D61C2×D412yes—
100C2×Q8×C61—21no—
101C2×Q8×C61—21no—
102C2×Q8×C61—21no—
103C2×Q8×C61—21no—
104C2×Q8×C61—21no—
105C2×Q8×C61C22×C412yes—
106C22×C4×C61—21no—
107C22×C4×C61—21no—
108C22×C4×C61—22no—
109C22×C4×C61C2×D411yescharacteristic
110C22×C4×C61C2×D411yescharacteristic
111C22×C4×C61C2411yescharacteristic
112C22×Dic61—21no—
113C22×Dic61—22no—
114C22×Dic61—21no—
115C22×Dic61C2×D412yes—
116C2×D244C22⋊C414yes—
117C2×D244SD1614yes—
118C2×D244Q1614yes—
119C2×C4×D61—22no—
120C2×C4×D61—22no—
121C2×C4×D61—22no—
122C2×C4×D61—22no—
123C2×C4×D61—28no—
124C2×C4×D61—44no—
125C2×C4×D61—44no—
126C2×C4×D61—82no—
127C2×C4×D61—82no—
128C2×C4×D61—82no—
129C2×C4×D61C22×C414yes—
130C2×C4×D61C2×D414yes—
131C2×Dic122—24no—
132C2×Dic122—81no—
133C2×Dic122—81no—
134C2×C8×C61—22no—
135C2×C8×C61—22no—
136C2×C8×C61—22no—
137C4⋊C4×C61—24no—
138C4⋊C4×C61—21no—
139C4⋊C4×C61—22no—
140C4⋊C4×C61—22no—
141C4⋊C4×C61—42no—
142C4⋊C4×C61—81no—
143C4⋊C4×C61C22×C412yes—
144C42×C61—24no—
145C42×C61C22×C412yes—
146C42×C61C2×D412yes—
147#11758198—28no—
148#11758198—44no—
149#11758198C22⋊C418yes—
150C4⋊Dic61—24no—
151C4⋊Dic61—22no—
152C4⋊Dic61—21no—
153C4⋊Dic61—21no—
154C4⋊Dic61—21no—
155C4⋊Dic61C22⋊C414yes—
156C4⋊Dic61C22×C414yes—
157C4⋊Dic61#4112yes—
158#11758196—28no—
159#11758196—28no—
160#11758196—44no—
161C4×Dic61—42no—
162C2×#92418—21no—
163C2×#92418—21no—
164C2×#92418—21no—
165C2×#92418—21no—
166C2×#92418—21no—
167C2×#92418#4112yes—
168C8×D61—48no—
169C23×C61C22×D411yescharacteristic
170C22×D61—22no—
171C22×D61—22no—
172C22×D61—22no—
173C22×D61—22no—
174C22×D61C2×C22⋊C414yes—
175C22×D61C2×SD1614yes—
176C22×D61C2×Q1614yes—
177Q8×C61—22no—
178Q8×C61—22no—
179Q8×C61—41no—
180Q8×C61—41no—
181C2×C4×C61—21no—
182C2×C4×C61—21no—
183C2×C4×C61—22no—
184C2×C4×C61—22no—
185C2×C4×C61—41no—
186C2×C4×C61—41no—
187C2×C4×C61—41no—
188C2×C4×C61—41no—
189C2×C4×C61—42no—
190C2×C4×C61—42no—
191C2×C4×C61—42no—
192C2×C4×C61C2×C22⋊C412yes—
193C2×C4×C61C2×C22⋊C412yes—
194C2×C4×C61C2×SD1612yes—
195C2×C4×C61C2×Q1612yes—
196C2×C4×C61C23×C411yescharacteristic
197C2×C4×C61C22×D411yescharacteristic
198C2×C4×C61C2×#4111yescharacteristic
199C2×C4×C61C2×#4111yescharacteristic
200C2×C4×C61H(5,𝔽2)11yescharacteristic
201C2×C4×C612−1+411yescharacteristic
202C2×Dic61—22no—
203C2×Dic61—22no—
204C2×Dic61—24no—
205C2×Dic61—22no—
206C2×Dic61—41no—
207C2×Dic61—42no—
208C2×Dic61—41no—
209C2×Dic61—42no—
210C2×Dic61—41no—
211C2×Dic61—41no—
212C2×Dic61—41no—
213C2×Dic61C2×C22⋊C414yes—
214D244#103116yescomplemented
215C4×D61—24no—
216C4×D61—24no—
217C4×D61—24no—
218C4×D61—24no—
219C4×D61—88no—
220C4×D61—88no—
221Dic122—84no—
222C8×C61—42no—
223#92418—22no—
224#92418—41no—
225#22487—611nomaximal, Sylow
226C22×C61C22×C22⋊C411yescharacteristic
227C22×C61#54611yescharacteristic
228C22×C61#54711yescharacteristic
229C22×C61C22×SD1611yescharacteristic
230C22×C61C22×Q1611yescharacteristic
231C22×C61#57611yescharacteristic
232C22×C61#57711yescharacteristic
233C2×D61—24no—
234C2×D61—24no—
235C2×D61—24no—
236C2×D61—24no—
237C2×D61—24no—
238C2×D61—24no—
239C2×D61C2×#10318yes—
240C4×C61—22no—
241C4×C61—42no—
242C4×C61—42no—
243C4×C61—82no—
244C4×C61C2×#10314yes—
245C4×C61C4×#4111yescharacteristic
246C4×C61#52011yescharacteristic
247C4×C61#54611yescharacteristic
248C4×C61#54711yescharacteristic
249Dic61—24no—
250Dic61—44no—
251Dic61—42no—
252Dic61—81no—
253C2×#554—611no—
254C2×C8×D4—611no—
255D4×C4⋊C4—611no—
256#1990—611no—
257#1987—611no—
258#1892—612no—
259#1872—612no—
260#1824—612no—
261#1803D6111yescharacteristic, complemented
262C4×#103—611no—
263C2×#340—612no—
264C2×C61#233911yescharacteristic
265C2×C61C22×#10311yescharacteristic
266C2×C61#289711yescharacteristic
267C2×C61#329511yescharacteristic
268C2×C61#329911yescharacteristic
269C2×C61#333511yescharacteristic
270C2×C61#333811yescharacteristic
271D61—28no—
272C23×C8—612no—
273#554—618no—
274#554—1224no—
275C2×C4⋊Q8—611no—
276C2×#125—1221no—
277C2×C22⋊Q8—612no—
278C2×C22⋊Q8—1222no—
279C2×C4×Q8C2×D6111yescharacteristic
280C2×C4×D4—611no—
281C2×C4×D4—1221no—
282C2×C4×D4—1221no—
283C22×C4⋊C4—612no—
284C22×C4⋊C4—612no—
285C8×D4—1224no—
286C2×C4⋊C8C2×D6111yescharacteristic
287C2×#103—611no—
288C2×#103—611no—
289C2×#103C2×D6112yes—
290C2×C22⋊C8—612no—
291C2×C4×C8—611no—
292#389—1221no—
293#385—1222no—
294#384C2×D6111yescharacteristic
295#380—612no—
296C4×C4⋊C4—611no—
297#340—612no—
298#340—612no—
299#340—611no—
300#340—612no—
301#340—612no—
302#340—611no—
303#340C2×D6112yes—
304C61#2248711yescharacteristic, complemented, Sylow
305C22×Q8—1221no—
306C22×Q8C22×D6111yescharacteristic
307C22×D4—611no—
308C23×C4—612no—
309C23×C4—1222no—
310C23×C4—1222no—
311C22×C8—611no—
312C22×C8—611no—
313C22×C8—1224no—
314C22×C8—1224no—
315C22×C8—2442no—
316C22×C8C22×D6112yes—
317C4⋊Q8—614no—
318C4⋊Q8—1222no—
319#125—1228no—
320C22⋊Q8—618no—
321C22⋊Q8—12216no—
322C22⋊Q8—1224no—
323C4×Q8—22no—
324C4×Q8C4×D6114yescomplemented
325C4×D4—614no—
326C4×D4—614no—
327C4×D4—1228no—
328C4×D4—2444no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—611no—
333C2×C4⋊C4—611no—
334C2×C4⋊C4—612no—
335C2×C4⋊C4—618no—
336C2×C4⋊C4—611no—
337C2×C4⋊C4—1224no—
338C2×C4⋊C4—1224no—
339C2×C4⋊C4—1222no—
340C2×C4⋊C4—1222no—
341C2×C4⋊C4—1221no—
342C2×C4⋊C4—1222no—
343C2×C4⋊C4—2442no—
344C2×C4⋊C4C22×D6112yes—
345C2×C4⋊C4C22×D6111yescharacteristic
346C2×C22⋊C4—612no—
347C2×C22⋊C4—1222no—
348C2×C22⋊C4—1222no—
349C2×C42—21no—
350C2×C42—1221no—
351C2×C42C22×D6111yescharacteristic
352C4⋊C8—22no—
353#103—42no—
354#103—1222no—
355#103—1222no—
356C22⋊C8—1224no—
357C4×C8—1222no—
358#95—1222no—
359#95—1221no—
360C24—612no—
361C2×Q8—21no—
362C2×Q8—21no—
363C2×Q8—21no—
364C2×Q8—21no—
365C2×Q8—21no—
366C2×Q8—1224no—
367C2×Q8—4881no—
368C2×Q8—4881no—
369C2×Q8C2×C4×D6112yes—
370C2×D4—614no—
371C2×D4—614no—
372C2×D4—614no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—612no—
377C22×C4—614no—
378C22×C4—614no—
379C22×C4—1228no—
380C22×C4—1222no—
381C22×C4—1222no—
382C22×C4—1222no—
383C22×C4—1222no—
384C22×C4—1221no—
385C22×C4—1222no—
386C22×C4—1221no—
387C22×C4—4882no—
388C22×C4—4882no—
389C22×C4—4882no—
390C22×C4—2444no—
391C22×C4—2444no—
392C22×C4D4×D6111yescharacteristic
393C22×C4D4×D6111yescharacteristic
394C22×C4C23×D6111yescharacteristic
395C2×C8—22no—
396C2×C8—22no—
397C2×C8—22no—
398C2×C8—612no—
399C2×C8—1221no—
400C2×C8—1221no—
401C2×C8—1221no—
402C2×C8—1221no—
403C2×C8—1221no—
404C2×C8—2448no—
405C4⋊C4—21no—
406C4⋊C4—24no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—42no—
410C4⋊C4—81no—
411C4⋊C4—614no—
412C4⋊C4—614no—
413C4⋊C4—612no—
414C4⋊C4—1228no—
415C4⋊C4—1228no—
416C4⋊C4—1222no—
417C4⋊C4—1221no—
418C4⋊C4—1224no—
419C4⋊C4—1221no—
420C4⋊C4—1221no—
421C4⋊C4—2444no—
422C4⋊C4C2×C4×D6112yes—
423C22⋊C4—618no—
424C22⋊C4—1228no—
425C22⋊C4—2444no—
426C42—24no—
427C42—2442no—
428C42C2×C4×D6112yes—
429C42D4×D6112yes—
430C23—614no—
431C23—614no—
432C23—614no—
433C23—1222no—
434C23—1222no—
435C23—1222no—
436C23—1222no—
437C23C2×D4×D6111yescharacteristic
438Q8—22no—
439Q8—22no—
440Q8—41no—
441Q8—41no—
442Q8—4884no—
443D4—6116no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—614no—
456C2×C4—1222no—
457C2×C4—1222no—
458C2×C4—1224no—
459C2×C4—1224no—
460C2×C4—1224no—
461C2×C4—1224no—
462C2×C4—1224no—
463C2×C4—1222no—
464C2×C4—4888no—
465C2×C4—4888no—
466C2×C4—2441no—
467C2×C4—2441no—
468C2×C4—2441no—
469C2×C4—2441no—
470C2×C4—2441no—
471C2×C4—2442no—
472C2×C4—2442no—
473C2×C4C22⋊C4×D6112yes—
474C2×C4C22⋊C4×D6112yes—
475C2×C4SD16×D6112yes—
476C2×C4Q16×D6112yes—
477C2×C4C22×C4×D6111yescharacteristic
478C2×C4C2×D4×D6111yescharacteristic
479C2×C4#41×D6111yescharacteristic
480C2×C4#41×D6111yescharacteristic
481C2×C4#1451982211yescharacteristic
482C2×C4#1451982511yescharacteristic
483C8—42no—
484C8—1222no—
485C8—2441no—
486C22—618no—
487C22—1224no—
488C22—1224no—
489C22—1224no—
490C22—1224no—
491C22—1224no—
492C22—1224no—
493C22C2×C22⋊C4×D6111yescharacteristic
494C22#1684040911yescharacteristic
495C22#1684041411yescharacteristic
496C22C2×SD16×D6111yescharacteristic
497C22C2×Q16×D6111yescharacteristic
498C22#1684056611yescharacteristic
499C22#1684057111yescharacteristic
500C4—22no—
501C4—42no—
502C4—42no—
503C4—82no—
504C4—1224no—
505C4—4881no—
506C4—2444no—
507C4—2442no—
508C4#103×D6114yes—
509C4#1684023011yescharacteristic
510C4#1684024311yescharacteristic
511C4#1684040911yescharacteristic
512C4#1684041411yescharacteristic
513C2—1228no—
514C2#2144864711yescharacteristic
515C2C2×#103×D6111yescharacteristic
516C2#2145434311yescharacteristic
517C2#2145764611yescharacteristic
518C2#2145769811yescharacteristic
519C2#2145817911yescharacteristic
520C2#2145821011yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1803C2
socleC23×C61C22×D4
prime radicalC22×D244D4
cosocleC22×C4×C61C24
hypercenter#1803D61
hypocenterC61#22487
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1803

Complements

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

NH ≅ G/Ntype
4. #21446779C2semidirect
5. #21446766C2semidirect
6. #21446239C2semidirect
7. #21446123C2semidirect
8. #21445908C2semidirect
9. C4×#14519701C2semidirect
10. #340×D61C2semidirect
11. C61×#1803C2semidirect
14. #16840472C4semidirect
55. #14519777C2×C4semidirect
59. #14519753C2×C4semidirect
75. C4⋊C4×D61C2×C4semidirect
89. C4×Q8×C61C2×C4semidirect
214. D244#103semidirect
261. #1803D61semidirect
304. C61#22487semidirect
324. C4×Q8C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168404722
C2×#168398272
#214487192
#214467792
#214467662
#214462392
#214461232
#214459082
C4×#145197012
#340×D612
C61×#18032
#2248761