Finite Group Databasebeta

← #29721590

Subgroups — #29721590

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×#16839968C211yesmaximal, characteristic
2C2×#16839966C212yesmaximal
3C2×C4×#11758201C211yesmaximal, characteristic
4#21450804C211yesmaximal, characteristic, complemented
5#21448719C211yesmaximal, characteristic
6#21447201C212yesmaximal, complemented
7#21447063C212yesmaximal, complemented
8#21446620C211yesmaximal, characteristic, complemented
9#21446535C211yesmaximal, characteristic, complemented
10#21446154C212yesmaximal
11C61×#1803C211yesmaximal, characteristic, complemented
12C22×#11758190C2212yes—
13C22×#11758201C2211yescharacteristic
14C2×#14519777C2211yescharacteristic
15C2×#14519775C2212yes—
16C2×C4×D244C2211yescharacteristic
17C2×#14519739C2212yes—
18C2×C4⋊C4×D61—21no—
19#16839968—24no—
20#16839966—28no—
21C4×#11758201—24no—
22C2×#14519708C2211yescharacteristic
23C2×#14519706C2212yes—
24C2×#14519702C2212yes—
25C2×#14519701C2211yescharacteristic
26C2×#14519700C2212yes—
27C2×#14519700C2212yes—
28C2×#14519697C2211yescharacteristic
29C2×C4×#92418C2211yescharacteristic
30#16839692—22no—
31#16839667C2212yes—
32#16839660—21no—
33#16839660—21no—
34#16839658C2212yes—
35#16839633—22no—
36#16839619—21no—
37C4×C4⋊Dic61C2211yescharacteristic
38#16839516C2212yes—
39C2×C4×Q8×C61C2211yescharacteristic
40C2×C4⋊C8×C61C2211yescharacteristic
41C2×#103×C61C2212yes—
42C61×#384C2211yescharacteristic
43C61×#340C2212yes—
44C22×Dic122C2312yes—
45C22×D244C2311yescharacteristic
46C22×C4×D61—21no—
47C22×C4×D61—22no—
48C2×#11758190—24no—
49C2×#11758190—24no—
50C2×#11758190—24no—
51C2×#11758201—22no—
52C2×#11758201—22no—
53C2×#11758201D414yes—
54C22×#92418C2311yescharacteristic
55C22×#92418C2311yescharacteristic
56#14519777—22no—
57#14519777—22no—
58#14519775—24no—
59#14519775—42no—
60C4×D244—22no—
61C4×D244D414yes—
62#14519739—24no—
63#14519739—42no—
64C2×C4⋊Dic61—21no—
65C2×C4⋊Dic61—21no—
66C2×C4⋊Dic61D411yescharacteristic
67C2×C4⋊Dic61D411yescharacteristic
68C2×C4⋊Dic61C2311yescharacteristic
69C2×#11758196—22no—
70C2×#11758196—22no—
71C2×#11758196—22no—
72C4⋊C4×D61—44no—
73C2×#11758198—21no—
74C2×#11758198—22no—
75C2×C4×Dic61C2312yes—
76#14519708—22no—
77#14519706—24no—
78#14519702—24no—
79#14519701—22no—
80#14519700—24no—
81#14519700—42no—
82#14519697—22no—
83C4×#92418—22no—
84#14519684—22no—
85#14519684—22no—
86#14519684—22no—
87#14519684D412yes—
88C22×Q8×C61C2311yescharacteristic
89C22×C8×C61C2312yes—
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×C61C2312yes—
96C2×C4⋊C4×C61C2311yescharacteristic
97C2×C42×C61—21no—
98C2×C42×C61C2311yescharacteristic
99C4⋊C8×C61—22no—
100#103×C61—24no—
101C23×D61—21no—
102C2×Q8×C61—21no—
103C2×Q8×C61—21no—
104C2×Q8×C61—21no—
105C2×Q8×C61—21no—
106C2×Q8×C61—21no—
107C2×Q8×C61C2×D412yes—
108C22×C4×C61—21no—
109C22×C4×C61—22no—
110C22×C4×C61—21no—
111C22×C4×C61C2×D411yescharacteristic
112C22×C4×C61C2×D411yescharacteristic
113C22×C4×C61C2411yescharacteristic
114C22×Dic61—21no—
115C22×Dic61—22no—
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—22no—
131C2×Dic122—22no—
132C2×Dic122—22no—
133C2×Dic122—22no—
134C2×Dic122—22no—
135C2×Dic122—22no—
136C2×C8×C61—22no—
137C2×C8×C61—22no—
138C2×C8×C61—22no—
139C4⋊C4×C61—22no—
140C4⋊C4×C61—21no—
141C4⋊C4×C61—21no—
142C4⋊C4×C61—22no—
143C4⋊C4×C61—42no—
144C4⋊C4×C61—42no—
145C4⋊C4×C61—81no—
146C42×C61—42no—
147C42×C61C2×D412yes—
148C42×C61#4112yes—
149#11758201—28no—
150#11758198—24no—
151#11758198—44no—
152C4⋊Dic61—22no—
153C4⋊Dic61—21no—
154C4⋊Dic61—42no—
155C4⋊Dic61—42no—
156C4⋊Dic61SD1614yescomplemented
157C4⋊Dic61C2×D412yes—
158#11758196—44no—
159#11758196—44no—
160#11758196—44no—
161C4×Dic61—22no—
162C4×Dic61—22no—
163C2×#92418—21no—
164C2×#92418—21no—
165C2×#92418—21no—
166C2×#92418—21no—
167C2×#92418—21no—
168C2×#92418C2×D412yes—
169#11758190—48no—
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—22no—
179Q8×C61—22no—
180Q8×C61—41no—
181Q8×C61—41no—
182C2×C4×C61—21no—
183C2×C4×C61—21no—
184C2×C4×C61—22no—
185C2×C4×C61—41no—
186C2×C4×C61—41no—
187C2×C4×C61—41no—
188C2×C4×C61—42no—
189C2×C4×C61—42no—
190C2×C4×C61—42no—
191C2×C4×C61—41no—
192C2×C4×C61—41no—
193C2×C4×C61#12112yes—
194C2×C4×C61#12112yes—
195C2×C4×C61C2×SD1612yes—
196C2×C4×C61#13712yes—
197C2×C4×C61C22×D411yescharacteristic
198C2×C4×C61C22×D411yescharacteristic
199C2×C4×C61C2×#4111yescharacteristic
200C2×C4×C61C2×#4111yescharacteristic
201C2×C4×C61H(5,𝔽2)11yescharacteristic
202C2×C4×C612−1+411yescharacteristic
203C2×Dic61—22no—
204C2×Dic61—22no—
205C2×Dic61—22no—
206C2×Dic61—22no—
207C2×Dic61—22no—
208C2×Dic61—22no—
209C2×Dic61—41no—
210C2×Dic61—41no—
211C2×Dic61—42no—
212C2×Dic61—42no—
213C2×Dic61—42no—
214C2×Dic61C2×SD1614yes—
215D244—22no—
216D244—22no—
217D244—22no—
218D244#12114yes—
219C4×D61—48no—
220C4×D61—88no—
221Dic122—42no—
222Dic122—42no—
223Dic122—42no—
224Dic122—42no—
225C8×C61—42no—
226#92418—22no—
227#92418—41no—
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—42no—
245C4×C61—82no—
246C4×C61—81no—
247C4×C61#46114yescomplemented
248C4×C61C2×#12111yescharacteristic
249C4×C61#53711yescharacteristic
250C4×C61#54711yescharacteristic
251C4×C61#55311yescharacteristic
252Dic61—24no—
253Dic61—42no—
254Dic61—42no—
255Dic61—82no—
256C2×#461—611no—
257C2×#461—612no—
258C2×C4×SD16—611no—
259#2546—611no—
260D4×C4⋊C4—611no—
261#2066—612no—
262#2043—612no—
263#1956—611no—
264#1939—611no—
265#1877—612no—
266#1803D6111yescharacteristic, complemented
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—611no—
276C22×SD16—612no—
277C2×C4⋊Q8—612no—
278C2×C4⋊Q8—611no—
279C2×C4×Q8C2×D6111yescharacteristic
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⋊C8C2×D6111yescharacteristic
289C2×#103—611no—
290C2×#103—612no—
291C2×#103—612no—
292C2×#103—612no—
293C2×#103C2×D6112yes—
294C2×#102—611no—
295C2×C4×C8—611no—
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#340C2×D6112yes—
306#336—612no—
307C61#2473011yescharacteristic, complemented, Sylow
308C22×Q8—612no—
309C22×Q8C22×D6111yescharacteristic
310C22×D4—611no—
311C23×C4—1222no—
312C23×C4—1221no—
313C2×SD16—614no—
314C2×SD16—1224no—
315C2×SD16—1224no—
316C2×SD16—1224no—
317C2×SD16—1222no—
318C2×SD16—1222no—
319C22×C8—611no—
320C22×C8—611no—
321C22×C8C22×D6112yes—
322C4⋊Q8—1224no—
323C4⋊Q8—1222no—
324C4⋊Q8—1222no—
325C4⋊Q8—2442no—
326C4×Q8—22no—
327C4×Q8—22no—
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—611no—
336C2×C4⋊C4—611no—
337C2×C4⋊C4—611no—
338C2×C4⋊C4—1222no—
339C2×C4⋊C4—1222no—
340C2×C4⋊C4—1222no—
341C2×C4⋊C4—1221no—
342C2×C4⋊C4—1221no—
343C2×C4⋊C4—2444no—
344C2×C4⋊C4C22×D6112yes—
345C2×C4⋊C4C22×D6111yescharacteristic
346C2×C22⋊C4—1221no—
347C2×C22⋊C4—1222no—
348C2×C42—21no—
349C2×C42—612no—
350C2×C42C22×D6111yescharacteristic
351#106—1222no—
352C4⋊C8—22no—
353C4⋊C8—1224no—
354#103—24no—
355#103—1224no—
356#103—1224no—
357#103—1222no—
358#103—2442no—
359#102—1222no—
360C4×C8—1222no—
361#95—612no—
362#95—1222no—
363#95—1222no—
364#95—1222no—
365C24—1221no—
366C2×Q8—21no—
367C2×Q8—21no—
368C2×Q8—21no—
369C2×Q8—21no—
370C2×Q8—21no—
371C2×Q8—1222no—
372C2×Q8—1222no—
373C2×Q8—1222no—
374C2×Q8—1222no—
375C2×Q8—1222no—
376C2×Q8—1222no—
377C2×Q8D4×D6112yes—
378C2×D4—612no—
379C2×D4—612no—
380C2×D4—612no—
381C2×D4—1221no—
382C2×D4—1221no—
383C2×D4—1221no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—612no—
388C22×C4—612no—
389C22×C4—1221no—
390C22×C4—1224no—
391C22×C4—1222no—
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—22no—
403C2×C8—22no—
404C2×C8—22no—
405C2×C8—612no—
406C2×C8—1221no—
407C2×C8—1221no—
408C2×C8—1221no—
409C2×C8—1221no—
410C2×C8—1221no—
411C4⋊C4—21no—
412C4⋊C4—22no—
413C4⋊C4—21no—
414C4⋊C4—22no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C4⋊C4—81no—
418C4⋊C4—614no—
419C4⋊C4—612no—
420C4⋊C4—1222no—
421C4⋊C4—1221no—
422C4⋊C4—2444no—
423C4⋊C4—2444no—
424C4⋊C4—2444no—
425C4⋊C4—2442no—
426C4⋊C4—2442no—
427C22⋊C4—1224no—
428C22⋊C4—2444no—
429C42—42no—
430C42—1222no—
431C42—1222no—
432C42#1175822212yes—
433C42D4×D6112yes—
434C23—1222no—
435C23—1222no—
436C23—1222no—
437C23—2441no—
438C23—2441no—
439C23—2441no—
440C23—2441no—
441C23C2×D4×D6111yescharacteristic
442Q8—22no—
443Q8—22no—
444Q8—41no—
445Q8—41no—
446Q8—2442no—
447Q8—2442no—
448Q8—2442no—
449Q8—2442no—
450D4—614no—
451D4—1222no—
452D4—1222no—
453D4—1222no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—22no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—42no—
465C2×C4—614no—
466C2×C4—1222no—
467C2×C4—1222no—
468C2×C4—1222no—
469C2×C4—1222no—
470C2×C4—1222no—
471C2×C4—1222no—
472C2×C4—4888no—
473C2×C4—2442no—
474C2×C4—2442no—
475C2×C4—2442no—
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#1451982311yescharacteristic
488C2×C4#1451982511yescharacteristic
489C8—42no—
490C8—1222no—
491C8—2441no—
492C22—1224no—
493C22—2442no—
494C22—2442no—
495C22—2442no—
496C22—2442no—
497C22—2442no—
498C22—2442no—
499C22C2×#1451975011yescharacteristic
500C22#1684039611yescharacteristic
501C22#1684042211yescharacteristic
502C22C2×SD16×D6111yescharacteristic
503C22C2×#1451980911yescharacteristic
504C22#1684057211yescharacteristic
505C22#1684057811yescharacteristic
506C4—22no—
507C4—42no—
508C4—81no—
509C4—82no—
510C4—1224no—
511C4—4882no—
512C4—2442no—
513C4—2442no—
514C4#1683996614yescomplemented
515C4#121×D6111yescharacteristic
516C4#1684034211yescharacteristic
517C4#1684041311yescharacteristic
518C4#1684045611yescharacteristic
519C2—2444no—
520C2#2144926111yescharacteristic
521C2C2×#1683996611yescharacteristic
522C2#2145536811yescharacteristic
523C2#2145746011yescharacteristic
524C2#2145751011yescharacteristic
525C2#2145846011yescharacteristic
526C2#2145859911yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1803C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenter#1803D61
hypocenterC61#24730
LCSG ▷ C22×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. #21450804C2semidirect
6. #21447201C2semidirect
7. #21447063C2semidirect
8. #21446620C2semidirect
9. #21446535C2semidirect
11. C61×#1803C2semidirect
156. C4⋊Dic61SD16semidirect
247. C4×C61#461semidirect
266. #1803D61semidirect
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×#168399682
C2×#168399662
C2×C4×#117582012
#214508042
#214487192
#214472012
#214470632
#214466202
#214465352
#214461542
C61×#18032
#2473061