Finite Group Databasebeta

← #29716098

Subgroups — #29716098

Order 15616.

1294 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4⋊Q8×D61C211yesmaximal, characteristic
2C2×#16840045C212yesmaximal, complemented
3#21448357C211yesmaximal, characteristic
4#21447153C211yesmaximal, characteristic, complemented
5#21447149C211yesmaximal, characteristic, complemented
6#21446960C211yesmaximal, characteristic, complemented
7#21446769C212yesmaximal, complemented
8#21446123C212yesmaximal
9#21446013C212yesmaximal
10#21445798C211yesmaximal, characteristic
11C61×#2024C211yesmaximal, characteristic, complemented
12C22×Q8×D61—21no—
13C2×#14519777C2211yescharacteristic
14C2×#14519776—21no—
15C2×#14519776C2212yes—
16C2×#14519775—21no—
17C4⋊Q8×D61—28no—
18C2×#14519753C2211yescharacteristic
19C2×Q8×Dic61C2211yescharacteristic
20C2×C4⋊C4×D61—21no—
21C2×C4⋊C4×D61C2212yes—
22C2×C42×D61C2211yescharacteristic
23#16840045—28no—
24C2×#14519709C2212yes—
25C2×#14519708C2212yes—
26C2×#14519706C2212yes—
27C2×#14519701C2212yes—
28C2×#14519689C2212yes—
29#16839660C2211yescharacteristic
30#16839649—21no—
31#16839636—21no—
32#16839619C2211yescharacteristic
33C4⋊C4×Dic61C2211yescharacteristic
34#16839613—21no—
35#16839609—21no—
36#16839525C2212yes—
37#16839524C2212yes—
38#16839524C2212yes—
39#16839517C2211yescharacteristic
40C2×C4⋊Q8×C61C2211yescharacteristic
41C2×#103×C61C2212yes—
42C61×#391C2211yescharacteristic
43C61×#340C2212yes—
44C61×#336C2211yescharacteristic
45C22×Dic122—21no—
46C22×Dic122—22no—
47C2×Q8×D61—28no—
48C2×Q8×D61—44no—
49C2×Q8×D61—44no—
50C22×C4×D61—22no—
51C22×C4×D61D411yescharacteristic
52C22×C4×D61D411yescharacteristic
53C22×C4×D61C2311yescharacteristic
54C22×#92418C2312yes—
55#14519777—24no—
56#14519776—28no—
57#14519776—24no—
58#14519776D418yes—
59#14519775—28no—
60#14519753—22no—
61#14519753Q814yescomplemented
62Q8×Dic61—24no—
63C2×C4⋊Dic61—21no—
64C2×C4⋊Dic61—21no—
65C2×C4⋊Dic61C2311yescharacteristic
66C2×C4⋊Dic61C2312yes—
67C2×C4⋊Dic61C2311yescharacteristic
68C2×#11758196—22no—
69C2×#11758196—22no—
70C2×#11758196—21no—
71C2×#11758196—21no—
72C4⋊C4×D61—28no—
73C4⋊C4×D61—24no—
74C4⋊C4×D61—24no—
75C2×#11758198—21no—
76C2×#11758198—21no—
77C2×C4×Dic61—21no—
78C2×C4×Dic61—21no—
79C2×C4×Dic61D411yescharacteristic
80C2×C4×Dic61D411yescharacteristic
81C2×C4×Dic61C2311yescharacteristic
82C42×D61—24no—
83#14519709—24no—
84#14519708—24no—
85#14519706—24no—
86#14519701—24no—
87#14519689—24no—
88#14519684—21no—
89#14519684—21no—
90#14519684—21no—
91C22×Q8×C61—21no—
92C22×Q8×C61C2311yescharacteristic
93C22×C8×C61C2312yes—
94C4⋊Q8×C61—24no—
95C2×C4⋊C4×C61—21no—
96C2×C4⋊C4×C61C2311yescharacteristic
97C2×C4⋊C4×C61C2312yes—
98C2×C42×C61C2311yescharacteristic
99#103×C61—24no—
100#95×C61—21no—
101#95×C61—21no—
102C23×D61C2×D411yescharacteristic
103C2×Q8×C61—24no—
104C2×Q8×C61—21no—
105C2×Q8×C61—22no—
106C2×Q8×C61—22no—
107C2×Q8×C61—42no—
108C2×Q8×C61—42no—
109C2×Q8×C61C2×Q812yescomplemented
110C22×C4×C61—22no—
111C22×C4×C61—21no—
112C22×C4×C61—21no—
113C22×C4×C61C2×D411yescharacteristic
114C22×C4×C61C2×D411yescharacteristic
115C22×C4×C61C2411yescharacteristic
116C22×Dic61—22no—
117C22×Dic61—21no—
118C22×Dic61—21no—
119C22×Dic61C2×D411yescharacteristic
120C22×Dic61C2×D411yescharacteristic
121C22×Dic61C2×D411yescharacteristic
122Q8×D61—432no—
123C2×C4×D61—22no—
124C2×C4×D61—22no—
125C2×C4×D61—22no—
126C2×C4×D61—22no—
127C2×C4×D61—22no—
128C2×C4×D61—22no—
129C2×C4×D61—22no—
130C2×C4×D61—28no—
131C2×C4×D61—44no—
132C2×C4×D61—44no—
133C2×C4×D61SD1614yes—
134C2×C4×D61C2×Q812yes—
135C2×C4×D61#4112yes—
136C2×Dic122—24no—
137C2×Dic122—28no—
138C2×Dic122—42no—
139C2×Dic122—42no—
140C2×Dic122—44no—
141C2×Dic122—44no—
142C2×C8×C61—22no—
143C2×C8×C61—22no—
144C2×C8×C61—22no—
145C4⋊C4×C61—24no—
146C4⋊C4×C61—22no—
147C4⋊C4×C61—22no—
148C4⋊C4×C61C2×D414yes—
149C42×C61—22no—
150#11758198—24no—
151#11758198—42no—
152C4⋊Dic61—24no—
153C4⋊Dic61—22no—
154C4⋊Dic61—21no—
155C4⋊Dic61—22no—
156C4⋊Dic61—42no—
157C4⋊Dic61C2×D414yes—
158C4⋊Dic61C2×Q812yescomplemented
159#11758196—28no—
160#11758196—28no—
161#11758196—24no—
162#11758196—42no—
163C4×Dic61—22no—
164C4×Dic61—22no—
165C4×Dic61—42no—
166C4×Dic61—42no—
167C4×Dic61C2×D414yes—
168C2×#92418—22no—
169C2×#92418—22no—
170C2×#92418—22no—
171C23×C61C22×D411yescharacteristic
172C22×D61—21no—
173C22×D61—21no—
174C22×D61—21no—
175C22×D61—21no—
176C22×D61C22⋊Q812yes—
177C22×D61C2×SD1612yes—
178C22×D61#13712yes—
179Q8×C61—24no—
180Q8×C61—48no—
181Q8×C61—42no—
182C2×C4×C61—21no—
183C2×C4×C61—21no—
184C2×C4×C61—21no—
185C2×C4×C61—21no—
186C2×C4×C61—24no—
187C2×C4×C61—22no—
188C2×C4×C61—41no—
189C2×C4×C61—41no—
190C2×C4×C61—42no—
191C2×C4×C61—42no—
192C2×C4×C61—41no—
193C2×C4×C61—41no—
194C2×C4×C61—41no—
195C2×C4×C61C2×SD1612yes—
196C2×C4×C61#13712yes—
197C2×C4×C61C22×D412yes—
198C2×C4×C61C22×Q811yescharacteristic
199C2×C4×C61C2×#4111yescharacteristic
200C2×C4×C612−1+412yes—
201C2×Dic61—21no—
202C2×Dic61—21no—
203C2×Dic61—21no—
204C2×Dic61—21no—
205C2×Dic61—24no—
206C2×Dic61—22no—
207C2×Dic61—42no—
208C2×Dic61—41no—
209C2×Dic61—42no—
210C2×Dic61—41no—
211C2×Dic61—41no—
212C2×Dic61—41no—
213C2×Dic61—41no—
214C2×Dic61#12012yes—
215C2×Dic61#12012yes—
216C2×Dic61C22⋊Q812yes—
217C2×Dic61C2×SD1612yes—
218C2×Dic61AGL(1,ℤ/8)12yes—
219C4×D61—28no—
220C4×D61—24no—
221C4×D61—24no—
222C4×D61—28no—
223C4×D61—416no—
224Dic122—48no—
225Dic122—416no—
226C8×C61—42no—
227#92418—42no—
228#24301—611nomaximal, Sylow
229C22×C61C2×#12011yescharacteristic
230C22×C61C2×C22⋊Q811yescharacteristic
231C22×C61#53611yescharacteristic
232C22×C61C22×SD1611yescharacteristic
233C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
234C22×C61C2×#13711yescharacteristic
235C22×C61C2×#13711yescharacteristic
236C2×D61—22no—
237C2×D61—22no—
238C2×D61—22no—
239C2×D61—22no—
240C2×D61—22no—
241C2×D61—22no—
242C2×D61#47514yescomplemented
243C4×C61—22no—
244C4×C61—22no—
245C4×C61—42no—
246C4×C61—44no—
247C4×C61—81no—
248C4×C61#54812yes—
249C4×C61D4×Q812yes—
250Dic61—22no—
251Dic61—22no—
252Dic61—42no—
253Dic61—44no—
254Dic61—81no—
255Dic61#45014yes—
256C22×C4⋊Q8—611no—
257C2×#475—612no—
258C4×C22⋊Q8—611no—
259#2058—611no—
260#2058—611no—
261#2024—611no—
262#2024D6111yescharacteristic, complemented
263#1988—612no—
264#1872—612no—
265#1853—612no—
266#1797—611no—
267C2×C61#240911yescharacteristic
268C2×C61C2×#45011yescharacteristic
269C2×C61#300411yescharacteristic
270C2×C61C2×#47511yescharacteristic
271C2×C61#307911yescharacteristic
272C2×C61#322911yescharacteristic
273C2×C61#323911yescharacteristic
274D61—24no—
275C23×Q8—1221no—
276C2×C4⋊Q8—612no—
277C2×C4⋊Q8—611no—
278C2×C4⋊Q8—1221no—
279C2×C4⋊Q8—1221no—
280C2×C4⋊Q8—1228no—
281C2×C4⋊Q8C2×D6111yescharacteristic
282C2×C22⋊Q8—611no—
283C2×C4×Q8—611no—
284C22×C4⋊C4—612no—
285C22×C4⋊C4—1221no—
286C22×C42—611no—
287#475—1228no—
288C2×#106—612no—
289C2×#106—612no—
290C2×C4⋊C8—612no—
291C2×#103—612no—
292C2×#103C2×D6112yes—
293C2×C22⋊C8—612no—
294#391C2×D6111yescharacteristic
295#389—611no—
296#387—1221no—
297#385—1221no—
298#380—611no—
299C4×C4⋊C4—611no—
300C4×C4⋊C4—1221no—
301C4×C22⋊C4—1221no—
302#340—612no—
303#340—612no—
304#340—612no—
305#340C2×D6112yes—
306#336—611no—
307#336C2×D6111yescharacteristic
308C61#2430111yescharacteristic, complemented, Sylow
309C22×Q8—21no—
310C22×Q8—1222no—
311C22×Q8—1228no—
312C22×Q8—1221no—
313C22×Q8—2444no—
314C22×Q8—2444no—
315C22×Q8C22×D6111yescharacteristic
316C23×C4—611no—
317C23×C4—611no—
318C23×C4—611no—
319C23×C4—1222no—
320C22×C8—612no—
321C22×C8C22×D6112yes—
322C4⋊Q8—24no—
323C4⋊Q8—618no—
324C4⋊Q8—1224no—
325C4⋊Q8—1224no—
326C4⋊Q8—1228no—
327C4⋊Q8—1228no—
328C22⋊Q8—614no—
329C22⋊Q8—1222no—
330C4×Q8—1224no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—612no—
333C2×C4⋊C4—611no—
334C2×C4⋊C4—611no—
335C2×C4⋊C4—1224no—
336C2×C4⋊C4—1228no—
337C2×C4⋊C4—1222no—
338C2×C4⋊C4—1222no—
339C2×C4⋊C4—1221no—
340C2×C4⋊C4—1221no—
341C2×C4⋊C4—1224no—
342C2×C4⋊C4—1221no—
343C2×C4⋊C4—1221no—
344C2×C4⋊C4C22×D6111yescharacteristic
345C2×C4⋊C4C22×D6112yes—
346C2×C22⋊C4—1221no—
347C2×C22⋊C4—1221no—
348C2×C42—611no—
349C2×C42—611no—
350C2×C42—611no—
351C2×C42—1224no—
352C2×C42—1221no—
353C2×C42—1221no—
354C2×C42C22×D6111yescharacteristic
355#106—1224no—
356#106—1224no—
357C4⋊C8—1224no—
358#103—24no—
359#103—1224no—
360C22⋊C8—1224no—
361#95—21no—
362#95—21no—
363#95—1221no—
364#95—1221no—
365#95—1221no—
366C24—611no—
367C2×Q8—24no—
368C2×Q8—21no—
369C2×Q8—22no—
370C2×Q8—22no—
371C2×Q8—42no—
372C2×Q8—42no—
373C2×Q8—1228no—
374C2×Q8—1224no—
375C2×Q8—2442no—
376C2×Q8—2444no—
377C2×Q8—2444no—
378C2×Q8—24432no—
379C2×Q8—2442no—
380C2×Q8Q8×D6112yescomplemented
381C22×C4—21no—
382C22×C4—22no—
383C22×C4—21no—
384C22×C4—611no—
385C22×C4—611no—
386C22×C4—611no—
387C22×C4—612no—
388C22×C4—612no—
389C22×C4—614no—
390C22×C4—1222no—
391C22×C4—1222no—
392C22×C4—1222no—
393C22×C4—1222no—
394C22×C4—1228no—
395C22×C4—1222no—
396C22×C4—1222no—
397C22×C4—1222no—
398C22×C4—1221no—
399C22×C4—1222no—
400C22×C4—1221no—
401C22×C4—2444no—
402C22×C4—2444no—
403C22×C4D4×D6111yescharacteristic
404C22×C4D4×D6111yescharacteristic
405C22×C4C23×D6111yescharacteristic
406C2×C8—22no—
407C2×C8—22no—
408C2×C8—22no—
409C2×C8—1222no—
410C2×C8—1222no—
411C2×C8—1222no—
412C4⋊C4—24no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—614no—
416C4⋊C4—612no—
417C4⋊C4—1228no—
418C4⋊C4—1224no—
419C4⋊C4—1228no—
420C4⋊C4—1224no—
421C4⋊C4—1222no—
422C4⋊C4—1221no—
423C4⋊C4—1222no—
424C4⋊C4—2442no—
425C4⋊C4—2442no—
426C4⋊C4D4×D6114yes—
427C22⋊C4—1224no—
428C22⋊C4—2442no—
429C42—22no—
430C42—614no—
431C42—1222no—
432C42—1222no—
433C42—2442no—
434C42—2442no—
435C23—612no—
436C23—612no—
437C23—612no—
438C23—1221no—
439C23—1221no—
440C23—1221no—
441C23—1221no—
442C23C2×D4×D6111yescharacteristic
443Q8—24no—
444Q8—48no—
445Q8—42no—
446Q8—2448no—
447Q8—24416no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—24no—
453C2×C4—22no—
454C2×C4—41no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—612no—
462C2×C4—612no—
463C2×C4—612no—
464C2×C4—612no—
465C2×C4—612no—
466C2×C4—1228no—
467C2×C4—1228no—
468C2×C4—1221no—
469C2×C4—1221no—
470C2×C4—1221no—
471C2×C4—1221no—
472C2×C4—1224no—
473C2×C4—1224no—
474C2×C4—1224no—
475C2×C4—1222no—
476C2×C4—24416no—
477C2×C4—2441no—
478C2×C4—2441no—
479C2×C4—2441no—
480C2×C4—2442no—
481C2×C4—2441no—
482C2×C4—2442no—
483C2×C4—2441no—
484C2×C4SD16×D6112yes—
485C2×C4#1451980912yes—
486C2×C4C2×D4×D6112yes—
487C2×C4C2×Q8×D6111yescharacteristic
488C2×C4#41×D6111yescharacteristic
489C2×C4#1451982412yes—
490C8—42no—
491C8—2442no—
492C22—614no—
493C22—1222no—
494C22—1222no—
495C22—1222no—
496C22—1222no—
497C22—1222no—
498C22—1222no—
499C22#120×D6111yescharacteristic
500C22C22⋊Q8×D6111yescharacteristic
501C22#1684033211yescharacteristic
502C22C2×SD16×D6111yescharacteristic
503C22AGL(1,ℤ/8)×D6111yescharacteristic
504C22#137×D6111yescharacteristic
505C22C2×#1451980911yescharacteristic
506C4—22no—
507C4—22no—
508C4—44no—
509C4—42no—
510C4—81no—
511C4—614no—
512C4—1222no—
513C4—1222no—
514C4—4881no—
515C4—2442no—
516C4—2444no—
517C4#1684042012yes—
518C4#1684042512yes—
519C2—1224no—
520C2#2144917711yescharacteristic
521C2#450×D6111yescharacteristic
522C2#2145505411yescharacteristic
523C2#475×D6111yescharacteristic
524C2#2145563311yescharacteristic
525C2#2145695911yescharacteristic
526C2#2145704511yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2024C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
hypercenter#2024D61
hypocenterC61#24301
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2024

Complements

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

NH ≅ G/Ntype
2. C2×#16840045C2semidirect
4. #21447153C2semidirect
5. #21447149C2semidirect
6. #21446960C2semidirect
7. #21446769C2semidirect
11. C61×#2024C2semidirect
61. #14519753Q8semidirect
109. C2×Q8×C61C2×Q8semidirect
158. C4⋊Dic61C2×Q8semidirect
242. C2×D61#475semidirect
262. #2024D61semidirect
308. C61#24301semidirect
380. C2×Q8Q8×D61semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊Q8×D612
C2×#168400452
#214483572
#214471532
#214471492
#214469602
#214467692
#214461232
#214460132
#214457982
C61×#20242
#2430161