Finite Group Databasebeta

← #29721591

Subgroups — #29721591

Order 15616.

1018 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×#16839966C212yesmaximal
2C2×#16839964C211yesmaximal, characteristic
3C2×C4×#11758190C211yesmaximal, characteristic
4#21450808C211yesmaximal, characteristic, complemented
5#21448719C211yesmaximal, characteristic
6#21447198C212yesmaximal, complemented
7#21447063C212yesmaximal, complemented
8#21446537C211yesmaximal, characteristic, complemented
9#21446150C212yesmaximal
10#21445821C211yesmaximal, characteristic, complemented
11C61×#1956C211yesmaximal, characteristic, complemented
12C22×#11758190C2211yescharacteristic
13C22×#11758201C2212yes—
14C2×#14519775C2212yes—
15C2×C4×Dic122C2211yescharacteristic
16C2×C4×D244C2211yescharacteristic
17C2×#14519739C2212yes—
18C2×C4⋊C4×D61—21no—
19#16839966—28no—
20#16839964—24no—
21C4×#11758190—24no—
22C2×#14519709C2211yescharacteristic
23C2×C4⋊#92418C2211yescharacteristic
24C2×#14519706C2212yes—
25C2×#14519702C2211yescharacteristic
26C2×#14519701C2212yes—
27C2×#14519700C2212yes—
28C2×#14519699C2211yescharacteristic
29#16839693—22no—
30#16839669C2212yes—
31#16839658—21no—
32#16839658—21no—
33#16839658C2212yes—
34#16839633—22no—
35#16839619—21no—
36C4×C4⋊Dic61C2211yescharacteristic
37#16839524C2212yes—
38#16839516C2212yes—
39C2×C4⋊Q8×C61C2211yescharacteristic
40C2×#103×C61C2212yes—
41C2×#103×C61C2212yes—
42C2×C4×C8×C61C2211yescharacteristic
43C61×#384C2211yescharacteristic
44C22×Dic122C2311yescharacteristic
45C22×D244C2311yescharacteristic
46C22×C4×D61—21no—
47C22×C4×D61—22no—
48C2×#11758190—22no—
49C2×#11758190—22no—
50C2×#11758190D414yes—
51C2×#11758201—24no—
52C2×#11758201—24no—
53C2×#11758201—24no—
54C22×#92418C2312yes—
55#14519775—24no—
56#14519775—42no—
57C4×Dic122—22no—
58C4×Dic122—22no—
59C4×D244—22no—
60C4×D244D414yes—
61#14519739—24no—
62#14519739—42no—
63C2×C4⋊Dic61D411yescharacteristic
64C2×C4⋊Dic61D411yescharacteristic
65C2×C4⋊Dic61C2311yescharacteristic
66C2×C4⋊Dic61C2311yescharacteristic
67C2×#11758196—21no—
68C2×#11758196—22no—
69C2×#11758196—22no—
70C2×#11758196—22no—
71C4⋊C4×D61—44no—
72C2×#11758198—21no—
73C2×#11758198—22no—
74C2×C4×Dic61—21no—
75C2×C4×Dic61C2312yes—
76#14519709—22no—
77C4⋊#92418—22no—
78#14519706—24no—
79#14519702—22no—
80#14519701—24no—
81#14519700—24no—
82#14519699—22no—
83#14519684—22no—
84#14519684—22no—
85#14519684—22no—
86#14519684D412yes—
87C22×Q8×C61C2312yes—
88C22×C8×C61C2311yescharacteristic
89C22×C8×C61C2311yescharacteristic
90C4⋊Q8×C61—22no—
91C4⋊Q8×C61—22no—
92C2×C4⋊C4×C61—21no—
93C2×C4⋊C4×C61—21no—
94C2×C4⋊C4×C61—21no—
95C2×C4⋊C4×C61—21no—
96C2×C4⋊C4×C61C2312yes—
97C2×C42×C61C2311yescharacteristic
98#103×C61—24no—
99#103×C61—42no—
100C4×C8×C61—22no—
101C23×D61—21no—
102C2×Q8×C61—22no—
103C2×Q8×C61—22no—
104C2×Q8×C61—22no—
105C2×Q8×C61—22no—
106C2×Q8×C61—22no—
107C2×Q8×C61—22no—
108C22×C4×C61—22no—
109C22×C4×C61—22no—
110C22×C4×C61C2×D411yescharacteristic
111C22×C4×C61C2×D411yescharacteristic
112C22×C4×C61C2411yescharacteristic
113C22×Dic61—21no—
114C22×Dic61—21no—
115C22×Dic61—21no—
116C22×Dic61C2×D412yes—
117C22×Dic61C2×D412yes—
118C2×D244—21no—
119C2×D244—21no—
120C2×D244—21no—
121C2×D244C2×D412yes—
122C2×D244C2×D412yes—
123C2×D244#4112yes—
124C2×C4×D61—24no—
125C2×C4×D61—42no—
126C2×C4×D61—42no—
127C2×C4×D61—44no—
128C2×C4×D61—44no—
129C2×C4×D61—82no—
130C2×Dic122—21no—
131C2×Dic122—21no—
132C2×Dic122—21no—
133C2×Dic122—21no—
134C2×Dic122—21no—
135C2×Dic122C2×D412yes—
136C2×C8×C61—21no—
137C2×C8×C61—21no—
138C2×C8×C61—21no—
139C2×C8×C61—21no—
140C2×C8×C61—21no—
141C2×C8×C61C2×D412yes—
142C4⋊C4×C61—22no—
143C4⋊C4×C61—22no—
144C4⋊C4×C61—42no—
145C4⋊C4×C61—42no—
146C4⋊C4×C61—42no—
147C4⋊C4×C61—81no—
148C42×C61C2×D412yes—
149C42×C61#4112yes—
150#11758201—48no—
151#11758198—24no—
152#11758198—44no—
153C4⋊Dic61—22no—
154C4⋊Dic61—21no—
155C4⋊Dic61—21no—
156C4⋊Dic61—21no—
157C4⋊Dic61SD1614yescomplemented
158C4⋊Dic61C2×D412yes—
159#11758196—44no—
160#11758196—44no—
161#11758196—42no—
162#11758196—44no—
163C4×Dic61—22no—
164C4×Dic61—22no—
165C4×Dic61—42no—
166C2×#92418—22no—
167C2×#92418—22no—
168C2×#92418—22no—
169#11758190—28no—
170C23×C61C22×D411yescharacteristic
171C22×D61—22no—
172C22×D61—22no—
173C22×D61—22no—
174C22×D61—41no—
175C22×D61—41no—
176C22×D61—41no—
177C22×D61—41no—
178Q8×C61—42no—
179Q8×C61—42no—
180Q8×C61—42no—
181Q8×C61—42no—
182C2×C4×C61—21no—
183C2×C4×C61—21no—
184C2×C4×C61—42no—
185C2×C4×C61—42no—
186C2×C4×C61—42no—
187C2×C4×C61—42no—
188C2×C4×C61—42no—
189C2×C4×C61—42no—
190C2×C4×C61#12112yes—
191C2×C4×C61#12112yes—
192C2×C4×C61C2×SD1612yes—
193C2×C4×C61#13712yes—
194C2×C4×C61C22×D411yescharacteristic
195C2×C4×C61C22×D411yescharacteristic
196C2×C4×C61C2×#4111yescharacteristic
197C2×C4×C61C2×#4111yescharacteristic
198C2×C4×C61H(5,𝔽2)11yescharacteristic
199C2×C4×C612−1+411yescharacteristic
200C2×Dic61—22no—
201C2×Dic61—22no—
202C2×Dic61—22no—
203C2×Dic61—22no—
204C2×Dic61—22no—
205C2×Dic61—22no—
206C2×Dic61—22no—
207C2×Dic61—41no—
208C2×Dic61—41no—
209C2×Dic61—41no—
210C2×Dic61—41no—
211C2×Dic61—41no—
212C2×Dic61—41no—
213C2×Dic61—41no—
214C2×Dic61C2×SD1614yes—
215D244—22no—
216D244—22no—
217D244—22no—
218D244#12114yes—
219C4×D61—48no—
220C4×D61—88no—
221Dic122—22no—
222Dic122—22no—
223Dic122—41no—
224Dic122—41no—
225C8×C61—22no—
226C8×C61—41no—
227#92418—42no—
228#24730—611nomaximal, Sylow
229C22×C61C2×#12111yescharacteristic
230C22×C61D4211yescharacteristic
231C22×C61#54811yescharacteristic
232C22×C61C22×SD1611yescharacteristic
233C22×C61C2×#13711yescharacteristic
234C22×C61#57711yescharacteristic
235C22×C61#57811yescharacteristic
236C2×D61—24no—
237C2×D61—42no—
238C2×D61—42no—
239C2×D61—42no—
240C2×D61—42no—
241C2×D61—42no—
242C2×D61—42no—
243C4×C61—22no—
244C4×C61—82no—
245C4×C61—82no—
246C4×C61#46114yescomplemented
247C4×C61C2×#12111yescharacteristic
248C4×C61#53711yescharacteristic
249C4×C61#54711yescharacteristic
250C4×C61#55311yescharacteristic
251Dic61—24no—
252Dic61—42no—
253Dic61—42no—
254Dic61—42no—
255Dic61—81no—
256C2×#461—612no—
257C2×#461—611no—
258C2×C4×SD16—611no—
259#2546—611no—
260D4×C4⋊C4—611no—
261#2066—612no—
262#2043—612no—
263#1956D6111yescharacteristic, complemented
264#1939—611no—
265#1877—612no—
266#1803—611no—
267C2×C61#241611yescharacteristic
268C2×C61C2×#46111yescharacteristic
269C2×C61#304611yescharacteristic
270C2×C61#327611yescharacteristic
271C2×C61D4×SD1611yescharacteristic
272C2×C61#335911yescharacteristic
273C2×C61#337211yescharacteristic
274D61—44no—
275C22×SD16—612no—
276C22×SD16—611no—
277C2×C4⋊Q8—612no—
278C2×C4⋊Q8C2×D6111yescharacteristic
279C2×C4×Q8—611no—
280C2×C4×D4—612no—
281C2×C4×D4—611no—
282C22×C4⋊C4—1221no—
283#461—1228no—
284#461—1224no—
285C4×SD16—1224no—
286C2×#106—611no—
287C2×C4⋊C8—612no—
288C2×C4⋊C8—611no—
289C2×#103—612no—
290C2×#103—611no—
291C2×#103—612no—
292C2×#103C2×D6112yes—
293C2×#103C2×D6112yes—
294C2×#102—611no—
295C2×C4×C8C2×D6111yescharacteristic
296#398—1222no—
297#391—612no—
298#389—612no—
299#389—1221no—
300#389—1221no—
301#385—1222no—
302#384C2×D6111yescharacteristic
303#380—1221no—
304C4×C4⋊C4—611no—
305#340—612no—
306#336—612no—
307C61#2473011yescharacteristic, complemented, Sylow
308C22×Q8—611no—
309C22×Q8C22×D6112yes—
310C22×D4—611no—
311C23×C4—1222no—
312C23×C4—1221no—
313C2×SD16—614no—
314C2×SD16—1222no—
315C2×SD16—1222no—
316C2×SD16—1224no—
317C2×SD16—1224no—
318C2×SD16—1224no—
319C22×C8—612no—
320C22×C8C22×D6111yescharacteristic
321C22×C8C22×D6111yescharacteristic
322C4⋊Q8—22no—
323C4⋊Q8—22no—
324C4⋊Q8—1224no—
325C4⋊Q8—2442no—
326C4×Q8—1222no—
327C4×Q8—1222no—
328C4×D4—614no—
329C4×D4—1224no—
330C4×D4—1222no—
331C4×D4—2442no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—611no—
337C2×C4⋊C4—611no—
338C2×C4⋊C4—611no—
339C2×C4⋊C4—611no—
340C2×C4⋊C4—1222no—
341C2×C4⋊C4—1222no—
342C2×C4⋊C4—1222no—
343C2×C4⋊C4—1221no—
344C2×C4⋊C4—2444no—
345C2×C4⋊C4C22×D6112yes—
346C2×C22⋊C4—1221no—
347C2×C22⋊C4—1222no—
348C2×C42—612no—
349C2×C42—1221no—
350C2×C42C22×D6111yescharacteristic
351#106—1222no—
352C4⋊C8—1224no—
353C4⋊C8—1222no—
354#103—24no—
355#103—42no—
356#103—1224no—
357#103—1222no—
358#103—1224no—
359#102—1222no—
360C4×C8—22no—
361#95—612no—
362#95—1222no—
363#95—1222no—
364#95—1222no—
365C24—1221no—
366C2×Q8—22no—
367C2×Q8—22no—
368C2×Q8—22no—
369C2×Q8—22no—
370C2×Q8—22no—
371C2×Q8—22no—
372C2×Q8—612no—
373C2×Q8—1221no—
374C2×Q8—1221no—
375C2×Q8—1221no—
376C2×Q8—1221no—
377C2×Q8—1221no—
378C2×D4—612no—
379C2×D4—612no—
380C2×D4—612no—
381C2×D4—1221no—
382C2×D4—1221no—
383C2×D4—1221no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—612no—
387C22×C4—612no—
388C22×C4—1221no—
389C22×C4—1221no—
390C22×C4—1224no—
391C22×C4—1221no—
392C22×C4—4882no—
393C22×C4—2444no—
394C22×C4—2444no—
395C22×C4—2442no—
396C22×C4—2442no—
397C22×C4D4×D6111yescharacteristic
398C22×C4D4×D6111yescharacteristic
399C22×C4C23×D6111yescharacteristic
400SD16—1228no—
401SD16—2448no—
402C2×C8—21no—
403C2×C8—21no—
404C2×C8—21no—
405C2×C8—21no—
406C2×C8—21no—
407C2×C8—1222no—
408C2×C8—1222no—
409C2×C8—1222no—
410C2×C8D4×D6112yes—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—81no—
417C4⋊C4—614no—
418C4⋊C4—612no—
419C4⋊C4—1222no—
420C4⋊C4—1221no—
421C4⋊C4—1221no—
422C4⋊C4—1221no—
423C4⋊C4—2444no—
424C4⋊C4—2444no—
425C4⋊C4—2444no—
426C4⋊C4—2442no—
427C22⋊C4—1224no—
428C22⋊C4—2444no—
429C42—1222no—
430C42—1222no—
431C42—2442no—
432C42#1175822212yes—
433C42D4×D6112yes—
434C23—1222no—
435C23—1222no—
436C23—1222no—
437C23—2441no—
438C23—2441no—
439C23—2441no—
440C23—2441no—
441C23C2×D4×D6111yescharacteristic
442Q8—42no—
443Q8—42no—
444Q8—42no—
445Q8—42no—
446Q8—1222no—
447Q8—1222no—
448Q8—2441no—
449Q8—2441no—
450D4—614no—
451D4—1222no—
452D4—1222no—
453D4—1222no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—614no—
463C2×C4—1222no—
464C2×C4—1222no—
465C2×C4—1222no—
466C2×C4—1222no—
467C2×C4—1222no—
468C2×C4—1222no—
469C2×C4—1222no—
470C2×C4—4888no—
471C2×C4—2441no—
472C2×C4—2441no—
473C2×C4—2441no—
474C2×C4—2441no—
475C2×C4—2441no—
476C2×C4—2441no—
477C2×C4—2441no—
478C2×C4—2448no—
479C2×C4#1451975012yes—
480C2×C4#1451975012yes—
481C2×C4SD16×D6112yes—
482C2×C4#1451980912yes—
483C2×C4C2×D4×D6111yescharacteristic
484C2×C4C2×D4×D6111yescharacteristic
485C2×C4#41×D6111yescharacteristic
486C2×C4C2×#1175822211yescharacteristic
487C2×C4#1451982211yescharacteristic
488C2×C4#1451982411yescharacteristic
489C8—22no—
490C8—41no—
491C8—2442no—
492C22—1224no—
493C22—2442no—
494C22—2442no—
495C22—2442no—
496C22—2442no—
497C22—2442no—
498C22—2442no—
499C22C2×#1451975011yescharacteristic
500C22#1684039711yescharacteristic
501C22#1684042011yescharacteristic
502C22C2×SD16×D6111yescharacteristic
503C22C2×#1451980911yescharacteristic
504C22#1684056811yescharacteristic
505C22#1684057511yescharacteristic
506C4—22no—
507C4—82no—
508C4—82no—
509C4—1224no—
510C4—4881no—
511C4—2442no—
512C4—2442no—
513C4—2442no—
514C4#1683996614yescomplemented
515C4#121×D6111yescharacteristic
516C4#1684034111yescharacteristic
517C4#1684041711yescharacteristic
518C4#1684046211yescharacteristic
519C2—2444no—
520C2#2144926211yescharacteristic
521C2C2×#1683996611yescharacteristic
522C2#2145537411yescharacteristic
523C2#2145746411yescharacteristic
524C2#2145751411yescharacteristic
525C2#2145845811yescharacteristic
526C2#2145860511yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1956C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenter#1956D61
hypocenterC61#24730
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1956

Complements

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

NH ≅ G/Ntype
4. #21450808C2semidirect
6. #21447198C2semidirect
7. #21447063C2semidirect
8. #21446537C2semidirect
10. #21445821C2semidirect
11. C61×#1956C2semidirect
157. C4⋊Dic61SD16semidirect
246. C4×C61#461semidirect
263. #1956D61semidirect
307. C61#24730semidirect
514. C4#16839966semidirect

Maximal subgroups

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

subgroupindex
C2×#168399662
C2×#168399642
C2×C4×#117581902
#214508082
#214487192
#214471982
#214470632
#214465372
#214461502
#214458212
C61×#19562
#2473061