Finite Group Databasebeta

← #29721589

Subgroups — #29721589

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×#16839965C212yesmaximal
2C2×C4×#11758200C211yesmaximal, characteristic
3#21450808C211yesmaximal, characteristic, complemented
4D4×C4⋊Dic61C211yesmaximal, characteristic
5#21447196C212yesmaximal, complemented
6#21447064C212yesmaximal, complemented
7#21446622C211yesmaximal, characteristic, complemented
8#21446535C211yesmaximal, characteristic, complemented
9#21446155C212yesmaximal
10#21445825C211yesmaximal, characteristic, complemented
11C2×C61×#461C211yesmaximal, characteristic
12C22×#11758200C2211yescharacteristic
13C2×#14519775C2212yes—
14C2×C4×Dic122C2211yescharacteristic
15C2×D4×Dic61C2212yes—
16C22×C4⋊Dic61—21no—
17#16839965—28no—
18C4×#11758200—24no—
19C2×#14519708C2211yescharacteristic
20C2×#14519706C2212yes—
21C2×#14519702C2212yes—
22C2×#14519701C2212yes—
23C2×#14519701C2212yes—
24C2×#14519700C2211yescharacteristic
25C2×#14519698C2211yescharacteristic
26C2×C4×#92418C2211yescharacteristic
27#16839693—22no—
28#16839669C2212yes—
29#16839661C2212yes—
30#16839658—21no—
31#16839658—21no—
32#16839637—22no—
33#16839632C2211yescharacteristic
34#16839622—21no—
35C4×C4⋊Dic61C2211yescharacteristic
36#16839525C2212yes—
37#16839516C2212yes—
38C22×SD16×C61C2212yes—
39C2×C4⋊Q8×C61C2211yescharacteristic
40C2×C4×D4×C61C2211yescharacteristic
41C61×#461—24no—
42C2×C4⋊C8×C61C2211yescharacteristic
43C2×#103×C61C2212yes—
44C22×Dic122C2311yescharacteristic
45C23×Dic61—22no—
46C2×#11758200—22no—
47C2×#11758200—22no—
48C2×#11758200D414yes—
49C22×#92418C2311yescharacteristic
50C22×#92418C2311yescharacteristic
51#14519775—24no—
52#14519775—42no—
53C4×Dic122—22no—
54C4×Dic122—22no—
55D4×Dic61—24no—
56D4×Dic61—42no—
57C2×C4⋊Dic61—21no—
58C2×C4⋊Dic61—22no—
59C2×C4⋊Dic61—21no—
60C2×C4⋊Dic61—44no—
61C2×C4⋊Dic61D411yescharacteristic
62C2×C4⋊Dic61D411yescharacteristic
63C2×C4⋊Dic61C2312yes—
64C2×C4⋊Dic61C2311yescharacteristic
65C2×#11758196—21no—
66C2×#11758196—22no—
67C2×#11758196—22no—
68C2×C22⋊Dic61—22no—
69C2×C4×Dic61—21no—
70C2×C4×Dic61C2312yes—
71#14519708—22no—
72#14519706—24no—
73#14519702—24no—
74#14519701—24no—
75#14519701—42no—
76#14519700—22no—
77#14519698—22no—
78C4×#92418—22no—
79#14519684—22no—
80#14519684—22no—
81#14519684—22no—
82#14519684D412yes—
83C22×Q8×C61C2312yes—
84C22×D4×C61C2311yescharacteristic
85C23×C4×C61—21no—
86C2×SD16×C61—24no—
87C2×SD16×C61—24no—
88C2×SD16×C61—24no—
89C22×C8×C61C2312yes—
90C4⋊Q8×C61—22no—
91C4⋊Q8×C61—22no—
92C4×D4×C61—22no—
93C4×D4×C61D414yes—
94C2×C4⋊C4×C61—21no—
95C2×C4⋊C4×C61—21no—
96C2×C4⋊C4×C61C2311yescharacteristic
97C2×C22⋊C4×C61—21no—
98C2×C42×C61C2311yescharacteristic
99C4⋊C8×C61—22no—
100#103×C61—24no—
101C24×C61—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—
108C2×D4×C61—21no—
109C2×D4×C61—21no—
110C2×D4×C61—21no—
111C2×D4×C61C2×D412yes—
112C2×D4×C61C2×D412yes—
113C2×D4×C61#4112yes—
114C22×C4×C61—21no—
115C22×C4×C61—24no—
116C22×C4×C61—22no—
117C22×C4×C61—42no—
118C22×C4×C61—42no—
119C22×C4×C61C2×D411yescharacteristic
120C22×C4×C61C2×D411yescharacteristic
121C22×C4×C61C2411yescharacteristic
122C22×Dic61—21no—
123C22×Dic61—22no—
124C22×Dic61—21no—
125C22×Dic61—44no—
126C22×Dic61—44no—
127C22×Dic61—82no—
128C22×Dic61C2×D412yes—
129C22×Dic61C2×D412yes—
130C2×Dic122—21no—
131C2×Dic122—21no—
132C2×Dic122—21no—
133C2×Dic122—21no—
134C2×Dic122—21no—
135C2×Dic122C2×D412yes—
136SD16×C61—48no—
137C2×C8×C61—22no—
138C2×C8×C61—22no—
139C2×C8×C61—22no—
140C4⋊C4×C61—21no—
141C4⋊C4×C61—42no—
142C4⋊C4×C61—42no—
143C4⋊C4×C61C2×D412yes—
144C22⋊C4×C61—24no—
145C42×C61C2×D412yes—
146C42×C61#4112yes—
147C22⋊Dic61—44no—
148#11758200—28no—
149C4⋊Dic61—22no—
150C4⋊Dic61—21no—
151C4⋊Dic61—22no—
152C4⋊Dic61—21no—
153C4⋊Dic61—22no—
154C4⋊Dic61—42no—
155C4⋊Dic61—44no—
156C4⋊Dic61—81no—
157C4⋊Dic61SD1614yescomplemented
158#11758196—44no—
159#11758196—44no—
160#11758196—42no—
161C4×Dic61—22no—
162C4×Dic61—22no—
163C4×Dic61—42no—
164C2×#92418—21no—
165C2×#92418—21no—
166C2×#92418—21no—
167C2×#92418—21no—
168C2×#92418—21no—
169C2×#92418C2×D412yes—
170C23×C61—22no—
171C23×C61—22no—
172C23×C61—22no—
173C23×C61—41no—
174C23×C61—41no—
175C23×C61—41no—
176C23×C61—41no—
177C23×C61C22×D411yescharacteristic
178Q8×C61—42no—
179Q8×C61—42no—
180Q8×C61—42no—
181Q8×C61—42no—
182D4×C61—22no—
183D4×C61—22no—
184D4×C61—22no—
185D4×C61#12114yes—
186C2×C4×C61—21no—
187C2×C4×C61—21no—
188C2×C4×C61—22no—
189C2×C4×C61—41no—
190C2×C4×C61—41no—
191C2×C4×C61—48no—
192C2×C4×C61—42no—
193C2×C4×C61—42no—
194C2×C4×C61—42no—
195C2×C4×C61#12112yes—
196C2×C4×C61#12112yes—
197C2×C4×C61C2×SD1612yes—
198C2×C4×C61#13712yes—
199C2×C4×C61C22×D411yescharacteristic
200C2×C4×C61C22×D411yescharacteristic
201C2×C4×C61C2×#4111yescharacteristic
202C2×C4×C61C2×#4111yescharacteristic
203C2×C4×C61H(5,𝔽2)11yescharacteristic
204C2×C4×C612−1+411yescharacteristic
205C2×Dic61—22no—
206C2×Dic61—22no—
207C2×Dic61—22no—
208C2×Dic61—22no—
209C2×Dic61—22no—
210C2×Dic61—22no—
211C2×Dic61—41no—
212C2×Dic61—41no—
213C2×Dic61—41no—
214C2×Dic61—42no—
215C2×Dic61—42no—
216C2×Dic61—42no—
217C2×Dic61—41no—
218C2×Dic61—41no—
219C2×Dic61—88no—
220C2×Dic61C2×SD1614yes—
221Dic122—22no—
222Dic122—22no—
223Dic122—41no—
224Dic122—41no—
225C8×C61—42no—
226#92418—22no—
227#92418—41no—
228#24730—611nomaximal, Sylow
229C22×C61—24no—
230C22×C61—42no—
231C22×C61—42no—
232C22×C61—42no—
233C22×C61—42no—
234C22×C61—42no—
235C22×C61—42no—
236C22×C61C2×#12111yescharacteristic
237C22×C61D4211yescharacteristic
238C22×C61#54811yescharacteristic
239C22×C61C22×SD1611yescharacteristic
240C22×C61C2×#13711yescharacteristic
241C22×C61#57711yescharacteristic
242C22×C61#57811yescharacteristic
243C4×C61—22no—
244C4×C61—42no—
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—82no—
255Dic61—81no—
256C2×#461—612no—
257C2×#461D6111yescharacteristic
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#1803—611no—
267C2×C61—44no—
268C2×C61#241611yescharacteristic
269C2×C61C2×#46111yescharacteristic
270C2×C61#304611yescharacteristic
271C2×C61#327611yescharacteristic
272C2×C61D4×SD1611yescharacteristic
273C2×C61#335911yescharacteristic
274C2×C61#337211yescharacteristic
275C22×SD16—611no—
276C22×SD16C2×D6112yes—
277C2×C4⋊Q8—612no—
278C2×C4⋊Q8C2×D6111yescharacteristic
279C2×C4×Q8—611no—
280C2×C4×D4—612no—
281C2×C4×D4C2×D6111yescharacteristic
282C22×C4⋊C4—1221no—
283#461—24no—
284#461—1228no—
285C4×SD16—1224no—
286C2×#106—611no—
287C2×C4⋊C8—612no—
288C2×C4⋊C8C2×D6111yescharacteristic
289C2×#103—612no—
290C2×#103—612no—
291C2×#103—611no—
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#384—611no—
303#380—1221no—
304C4×C4⋊C4—611no—
305#340—612no—
306#336—612no—
307C61#2473011yescharacteristic, complemented, Sylow
308C22×Q8—611no—
309C22×Q8C22×D6112yes—
310C22×D4C22×D6111yescharacteristic
311C23×C4—21no—
312C23×C4—1222no—
313C2×SD16—24no—
314C2×SD16—24no—
315C2×SD16—24no—
316C2×SD16—614no—
317C2×SD16—1222no—
318C2×SD16—1222no—
319C22×C8—611no—
320C22×C8—611no—
321C22×C8C22×D6112yes—
322C4⋊Q8—22no—
323C4⋊Q8—22no—
324C4⋊Q8—1224no—
325C4⋊Q8—2442no—
326C4×Q8—1222no—
327C4×Q8—1222no—
328C4×D4—22no—
329C4×D4—1224no—
330C4×D4—2442no—
331C4×D4C22⋊D6114yes—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—611no—
335C2×C4⋊C4—611no—
336C2×C4⋊C4—612no—
337C2×C4⋊C4—611no—
338C2×C4⋊C4—1221no—
339C2×C4⋊C4—1222no—
340C2×C4⋊C4—1222no—
341C2×C4⋊C4—1222no—
342C2×C4⋊C4—1221no—
343C2×C4⋊C4—1221no—
344C2×C4⋊C4—2444no—
345C2×C4⋊C4C22×D6111yescharacteristic
346C2×C22⋊C4—21no—
347C2×C22⋊C4—1222no—
348C2×C42—612no—
349C2×C42—1221no—
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—21no—
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—21no—
379C2×D4—21no—
380C2×D4—21no—
381C2×D4D4×D6112yes—
382C2×D4#1175822412yes—
383C2×D4C2×C22⋊D6112yes—
384C22×C4—21no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—612no—
390C22×C4—612no—
391C22×C4—1221no—
392C22×C4—1221no—
393C22×C4—1222no—
394C22×C4—4882no—
395C22×C4—2444no—
396C22×C4—2444no—
397C22×C4D4×D6111yescharacteristic
398C22×C4D4×D6111yescharacteristic
399C22×C4C23×D6111yescharacteristic
400SD16—48no—
401SD16—1228no—
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—42no—
413C4⋊C4—42no—
414C4⋊C4—614no—
415C4⋊C4—1222no—
416C4⋊C4—1222no—
417C4⋊C4—1221no—
418C4⋊C4—1221no—
419C4⋊C4—1222no—
420C4⋊C4—4881no—
421C4⋊C4—2444no—
422C4⋊C4—2444no—
423C4⋊C4—2444no—
424C4⋊C4—2442no—
425C4⋊C4—2442no—
426C4⋊C4C2×C22⋊D6112yes—
427C22⋊C4—24no—
428C22⋊C4—2444no—
429C42—1222no—
430C42—1222no—
431C42—2442no—
432C42#1175822412yes—
433C42C2×C22⋊D6112yes—
434C23—22no—
435C23—22no—
436C23—22no—
437C23—41no—
438C23—41no—
439C23—41no—
440C23—41no—
441C23C2×D4×D6111yescharacteristic
442Q8—42no—
443Q8—42no—
444Q8—42no—
445Q8—42no—
446Q8—1222no—
447Q8—1222no—
448Q8—2441no—
449Q8—2441no—
450D4—22no—
451D4—22no—
452D4—22no—
453D4#1451974814yes—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—22no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—48no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—614no—
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—2442no—
473C2×C4—2442no—
474C2×C4—2441no—
475C2×C4—2441no—
476C2×C4—2442no—
477C2×C4—2441no—
478C2×C4—2441no—
479C2×C4#1451974912yes—
480C2×C4#1451974912yes—
481C2×C4SD16×D6112yes—
482C2×C4#1451980612yes—
483C2×C4C2×D4×D6111yescharacteristic
484C2×C4C22×C22⋊D6111yescharacteristic
485C2×C4C2×#1175822411yescharacteristic
486C2×C4C2×#1175822411yescharacteristic
487C2×C4#1451982311yescharacteristic
488C2×C4#1451982411yescharacteristic
489C8—42no—
490C8—1222no—
491C8—2441no—
492C22—24no—
493C22—42no—
494C22—42no—
495C22—42no—
496C22—42no—
497C22—42no—
498C22—42no—
499C22C2×#1451974911yescharacteristic
500C22#1684039611yescharacteristic
501C22#1684041811yescharacteristic
502C22C2×SD16×D6111yescharacteristic
503C22C2×#1451980611yescharacteristic
504C22#1684057111yescharacteristic
505C22#1684057811yescharacteristic
506C4—22no—
507C4—42no—
508C4—82no—
509C4—1224no—
510C4—4881no—
511C4—4882no—
512C4—2442no—
513C4—2442no—
514C4#1683996514yescomplemented
515C4C2×#1451974811yescharacteristic
516C4#1684034011yescharacteristic
517C4#1684041211yescharacteristic
518C4#1684046411yescharacteristic
519C2—44no—
520C2#2144926011yescharacteristic
521C2C2×#1683996511yescharacteristic
522C2#2145537311yescharacteristic
523C2#2145746811yescharacteristic
524C2#2145751811yescharacteristic
525C2#2145845611yescharacteristic
526C2#2145860411yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#461C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
hypercenterC2×#461D61
hypocenterC61#24730
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#461

Complements

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

NH ≅ G/Ntype
3. #21450808C2semidirect
5. #21447196C2semidirect
6. #21447064C2semidirect
7. #21446622C2semidirect
8. #21446535C2semidirect
10. #21445825C2semidirect
157. C4⋊Dic61SD16semidirect
246. C4×C61#461semidirect
307. C61#24730semidirect
514. C4#16839965semidirect

Maximal subgroups

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

subgroupindex
C2×#168399652
C2×C4×#117582002
#214508082
D4×C4⋊Dic612
#214471962
#214470642
#214466222
#214465352
#214461552
#214458252
C2×C61×#4612
#2473061