Finite Group Databasebeta

← #29721586

Subgroups — #29721586

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×#16839968C212yesmaximal
2C2×#16839968C211yesmaximal, characteristic
3C2×C4×#11758190C211yesmaximal, characteristic
4#21450807C211yesmaximal, characteristic, complemented
5C4⋊C4×D244C211yesmaximal, characteristic
6#21447199C212yesmaximal, complemented
7#21447066C212yesmaximal, complemented
8#21446621C211yesmaximal, characteristic, complemented
9#21446152C212yesmaximal
10#21445825C211yesmaximal, characteristic, complemented
11C61×#1939C211yesmaximal, characteristic, complemented
12C22×#11758190C2212yes—
13C22×#11758190C2211yescharacteristic
14C2×#14519777C2211yescharacteristic
15C2×#14519777C2212yes—
16C2×C4×Dic122C2211yescharacteristic
17C2×C4×D244C2212yes—
18C2×C4×D244C2211yescharacteristic
19C2×C4⋊C4×D61—21no—
20#16839968—28no—
21#16839968—24no—
22C4×#11758190—24no—
23C2×#14519709C2211yescharacteristic
24C2×#14519702C2212yes—
25C2×#14519702C2212yes—
26C2×#14519702C2212yes—
27C2×#14519702C2211yescharacteristic
28C2×#14519702C2212yes—
29C2×#14519699C2211yescharacteristic
30#16839694—22no—
31#16839668C2212yes—
32#16839661—21no—
33#16839661—21no—
34#16839660C2212yes—
35#16839636—22no—
36#16839632C2211yescharacteristic
37#16839619—21no—
38#16839525C2212yes—
39C2×C4⋊C8×C61C2212yes—
40C2×C4⋊C8×C61C2211yescharacteristic
41C2×C4×C8×C61C2211yescharacteristic
42C4×C4⋊C4×C61C2211yescharacteristic
43C61×#336C2212yes—
44C22×Dic122C2312yes—
45C22×Dic122C2311yescharacteristic
46C22×D244C2311yescharacteristic
47C22×C4×D61—21no—
48C22×C4×D61—22no—
49C2×#11758190—22no—
50C2×#11758190—22no—
51C2×#11758190—24no—
52C2×#11758190—24no—
53C2×#11758190—24no—
54C2×#11758190D414yes—
55#14519777—24no—
56#14519777—22no—
57#14519777—22no—
58#14519777—42no—
59C4×Dic122—22no—
60C4×Dic122—22no—
61C4×D244—22no—
62C4×D244—24no—
63C4×D244—42no—
64C4×D244D414yes—
65C2×C4⋊Dic61—21no—
66C2×C4⋊Dic61—22no—
67C2×C4⋊Dic61—22no—
68C2×C4⋊Dic61—21no—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61C2312yes—
72C2×C4⋊Dic61C2311yescharacteristic
73C2×C4⋊Dic61C2311yescharacteristic
74C2×#11758196—21no—
75C2×#11758196—22no—
76C4⋊C4×D61—44no—
77C2×#11758198—21no—
78C2×#11758198—22no—
79C2×C4×Dic61—21no—
80#14519709—22no—
81#14519702—24no—
82#14519702—24no—
83#14519702—24no—
84#14519702—22no—
85#14519702—42no—
86#14519699—22no—
87#14519684—22no—
88#14519684—22no—
89#14519684—22no—
90C22×C8×C61C2311yescharacteristic
91C22×C8×C61C2312yes—
92C22×C8×C61C2311yescharacteristic
93C2×C4⋊C4×C61D411yescharacteristic
94C2×C4⋊C4×C61D411yescharacteristic
95C2×C42×C61C2312yes—
96C2×C42×C61C2311yescharacteristic
97C4⋊C8×C61—24no—
98C4⋊C8×C61—22no—
99C4×C8×C61—22no—
100#95×C61D412yes—
101C23×D61—21no—
102C22×C4×C61C2×D412yes—
103C22×C4×C61C2×D411yescharacteristic
104C22×C4×C61C2×D412yes—
105C22×C4×C61C2×D411yescharacteristic
106C22×C4×C61C2411yescharacteristic
107C22×Dic61—21no—
108C22×Dic61—21no—
109C22×Dic61—22no—
110C22×Dic61—22no—
111C22×Dic61—21no—
112C2×D244—21no—
113C2×D244—21no—
114C2×D244—21no—
115C2×D244C2×D412yes—
116C2×D244C2×D412yes—
117C2×D244#4112yes—
118C2×C4×D61—24no—
119C2×C4×D61—42no—
120C2×C4×D61—42no—
121C2×C4×D61—44no—
122C2×C4×D61—44no—
123C2×C4×D61—82no—
124C2×Dic122—21no—
125C2×Dic122—21no—
126C2×Dic122—21no—
127C2×Dic122—21no—
128C2×Dic122—21no—
129C2×Dic122—22no—
130C2×Dic122—22no—
131C2×Dic122—22no—
132C2×Dic122—22no—
133C2×Dic122—22no—
134C2×Dic122—22no—
135C2×Dic122C2×D412yes—
136C2×C8×C61—21no—
137C2×C8×C61—21no—
138C2×C8×C61—21no—
139C2×C8×C61—22no—
140C2×C8×C61—22no—
141C2×C8×C61—22no—
142C2×C8×C61—21no—
143C2×C8×C61—21no—
144C2×C8×C61C2×D412yes—
145C4⋊C4×C61—22no—
146C4⋊C4×C61SD1614yescomplemented
147C42×C61—22no—
148C42×C61—22no—
149C42×C61C2×D412yes—
150C42×C61#4112yes—
151#11758198—24no—
152#11758198—44no—
153C4⋊Dic61—21no—
154C4⋊Dic61—21no—
155C4⋊Dic61—22no—
156C4⋊Dic61—21no—
157C4⋊Dic61—22no—
158C4⋊Dic61—44no—
159C4⋊Dic61—44no—
160C4⋊Dic61—42no—
161C4⋊Dic61—42no—
162C4⋊Dic61—42no—
163C4⋊Dic61—81no—
164C4⋊Dic61C2×D412yes—
165#11758196—42no—
166#11758196—44no—
167C4×Dic61—42no—
168#11758190—28no—
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—
178C2×C4×C61—22no—
179C2×C4×C61—21no—
180C2×C4×C61—22no—
181C2×C4×C61—21no—
182C2×C4×C61—22no—
183C2×C4×C61—22no—
184C2×C4×C61—22no—
185C2×C4×C61#12112yes—
186C2×C4×C61#12112yes—
187C2×C4×C61C2×SD1614yes—
188C2×C4×C61C2×SD1612yes—
189C2×C4×C61#13712yes—
190C2×C4×C61C22×D411yescharacteristic
191C2×C4×C61C22×D411yescharacteristic
192C2×C4×C61C2×#4111yescharacteristic
193C2×C4×C61C2×#4111yescharacteristic
194C2×C4×C61H(5,𝔽2)11yescharacteristic
195C2×C4×C612−1+411yescharacteristic
196C2×Dic61—22no—
197C2×Dic61—22no—
198C2×Dic61—41no—
199C2×Dic61—41no—
200C2×Dic61—41no—
201C2×Dic61—41no—
202C2×Dic61—41no—
203C2×Dic61—42no—
204C2×Dic61—42no—
205C2×Dic61—42no—
206C2×Dic61—42no—
207C2×Dic61—42no—
208C2×Dic61—42no—
209C2×Dic61—41no—
210C2×Dic61—41no—
211D244—22no—
212D244—22no—
213D244—22no—
214D244#12114yes—
215C4×D61—48no—
216C4×D61—88no—
217Dic122—22no—
218Dic122—22no—
219Dic122—41no—
220Dic122—41no—
221Dic122—42no—
222Dic122—42no—
223Dic122—42no—
224Dic122—42no—
225C8×C61—22no—
226C8×C61—42no—
227C8×C61—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—24no—
244C4×C61—22no—
245C4×C61—42no—
246C4×C61#46114yescomplemented
247C4×C61C2×#12111yescharacteristic
248C4×C61#53711yescharacteristic
249C4×C61#54711yescharacteristic
250C4×C61#55311yescharacteristic
251Dic61—42no—
252Dic61—42no—
253Dic61—82no—
254Dic61—82no—
255Dic61—81no—
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#1939D6111yescharacteristic, complemented
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—611no—
276C22×SD16—612no—
277C2×C4⋊Q8—611no—
278C2×C4⋊Q8—612no—
279C2×C4×Q8—611no—
280C2×C4×D4—611no—
281C2×C4×D4—612no—
282C22×C4⋊C4—1221no—
283#461—1224no—
284#461—1228no—
285C4×SD16—1224no—
286C2×#106—611no—
287C2×C4⋊C8C2×D6111yescharacteristic
288C2×C4⋊C8C2×D6112yes—
289C2×#103—612no—
290C2×#103—612no—
291C2×#103—612no—
292C2×#103—611no—
293C2×#103—612no—
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#384—611no—
303#380—1221no—
304C4×C4⋊C4C2×D6111yescharacteristic
305#340—612no—
306#336C2×D6112yes—
307C61#2473011yescharacteristic, complemented, Sylow
308C22×Q8—611no—
309C22×Q8—612no—
310C22×D4—611no—
311C23×C4—1221no—
312C23×C4—1222no—
313C2×SD16—614no—
314C2×SD16—1222no—
315C2×SD16—1222no—
316C2×SD16—1224no—
317C2×SD16—1224no—
318C2×SD16—1224no—
319C22×C8C22×D6112yes—
320C22×C8C22×D6111yescharacteristic
321C22×C8C22×D6111yescharacteristic
322C4⋊Q8—1224no—
323C4⋊Q8—1222no—
324C4⋊Q8—1222no—
325C4⋊Q8—2442no—
326C4×Q8—1222no—
327C4×Q8—1222no—
328C4×D4—614no—
329C4×D4—1224no—
330C4×D4—1222no—
331C4×D4—2442no—
332C2×C4⋊C4—611no—
333C2×C4⋊C4—612no—
334C2×C4⋊C4—611no—
335C2×C4⋊C4—1221no—
336C2×C4⋊C4—1221no—
337C2×C4⋊C4—1222no—
338C2×C4⋊C4—1221no—
339C2×C4⋊C4—1221no—
340C2×C4⋊C4—1222no—
341C2×C4⋊C4—1222no—
342C2×C4⋊C4—1221no—
343C2×C4⋊C4—2444no—
344C2×C4⋊C4D24411yescharacteristic
345C2×C4⋊C4D24411yescharacteristic
346C2×C22⋊C4—1221no—
347C2×C22⋊C4—1222no—
348C2×C42—1221no—
349C2×C42C22×D6112yes—
350C2×C42C22×D6111yescharacteristic
351#106—1222no—
352C4⋊C8—24no—
353C4⋊C8—22no—
354#103—1222no—
355#103—1224no—
356#103—1224no—
357#103—1224no—
358#103—2442no—
359#102—1222no—
360C4×C8—22no—
361#95—1222no—
362#95—1222no—
363#95—1222no—
364#95D24412yes—
365C24—1221no—
366C2×Q8—612no—
367C2×Q8—1221no—
368C2×Q8—1221no—
369C2×Q8—1222no—
370C2×Q8—1222no—
371C2×Q8—1221no—
372C2×Q8—1222no—
373C2×Q8—1221no—
374C2×Q8—1222no—
375C2×Q8—1222no—
376C2×Q8—1221no—
377C2×Q8—1222no—
378C2×D4—612no—
379C2×D4—612no—
380C2×D4—612no—
381C2×D4—1221no—
382C2×D4—1221no—
383C2×D4—1221no—
384C22×C4—1221no—
385C22×C4—1221no—
386C22×C4—1224no—
387C22×C4—1221no—
388C22×C4—1222no—
389C22×C4—1222no—
390C22×C4—4882no—
391C22×C4—2444no—
392C22×C4—2444no—
393C22×C4—2442no—
394C22×C4—2442no—
395C22×C4C2×D24412yes—
396C22×C4C2×D24411yescharacteristic
397C22×C4C2×D24412yes—
398C22×C4C2×D24411yescharacteristic
399C22×C4C23×D6111yescharacteristic
400SD16—1228no—
401SD16—2448no—
402C2×C8—21no—
403C2×C8—21no—
404C2×C8—21no—
405C2×C8—22no—
406C2×C8—22no—
407C2×C8—22no—
408C2×C8—21no—
409C2×C8—21no—
410C2×C8D4×D6112yes—
411C4⋊C4—22no—
412C4⋊C4—612no—
413C4⋊C4—1221no—
414C4⋊C4—1221no—
415C4⋊C4—1222no—
416C4⋊C4—1221no—
417C4⋊C4—1222no—
418C4⋊C4—4881no—
419C4⋊C4—2444no—
420C4⋊C4—2444no—
421C4⋊C4—2442no—
422C4⋊C4—2442no—
423C4⋊C4—2442no—
424C4⋊C4—2444no—
425C4⋊C4—2442no—
426C4⋊C4#1175819014yescomplemented
427C22⋊C4—1224no—
428C22⋊C4—2444no—
429C42—22no—
430C42—22no—
431C42—2442no—
432C42D4×D6112yes—
433C42#1175822612yes—
434C23—1222no—
435C23—1222no—
436C23—1222no—
437C23—2441no—
438C23—2441no—
439C23—2441no—
440C23—2441no—
441C23C22×D24411yescharacteristic
442Q8—1222no—
443Q8—1222no—
444Q8—2442no—
445Q8—2442no—
446Q8—2442no—
447Q8—2442no—
448Q8—2441no—
449Q8—2441no—
450D4—614no—
451D4—1222no—
452D4—1222no—
453D4—1222no—
454C2×C4—21no—
455C2×C4—22no—
456C2×C4—21no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—1222no—
462C2×C4—1222no—
463C2×C4—4888no—
464C2×C4—2441no—
465C2×C4—2442no—
466C2×C4—2442no—
467C2×C4—2442no—
468C2×C4—2441no—
469C2×C4—2441no—
470C2×C4—2442no—
471C2×C4—2442no—
472C2×C4—2442no—
473C2×C4—2441no—
474C2×C4—2441no—
475C2×C4—2441no—
476C2×C4—2441no—
477C2×C4—2448no—
478C2×C4#1451975212yes—
479C2×C4#1451975212yes—
480C2×C4C2×#1175819014yes—
481C2×C4C2×#1175819012yes—
482C2×C4#1451981012yes—
483C2×C4C2×D4×D6111yescharacteristic
484C2×C4C2×D4×D6111yescharacteristic
485C2×C4#41×D6111yescharacteristic
486C2×C4C2×#1175822611yescharacteristic
487C2×C4#1451982311yescharacteristic
488C2×C4#1451982511yescharacteristic
489C8—22no—
490C8—42no—
491C8—41no—
492C22—1224no—
493C22—2442no—
494C22—2442no—
495C22—2442no—
496C22—2442no—
497C22—2442no—
498C22—2442no—
499C22C2×#1451975211yescharacteristic
500C22D4×D24411yescharacteristic
501C22#1684042411yescharacteristic
502C22C22×#1175819011yescharacteristic
503C22C2×#1451981011yescharacteristic
504C22#1684057411yescharacteristic
505C22#1684057911yescharacteristic
506C4—22no—
507C4—24no—
508C4—42no—
509C4—4881no—
510C4—4882no—
511C4—4882no—
512C4—2442no—
513C4—2442no—
514C4#1683996814yescomplemented
515C4#121×D6111yescharacteristic
516C4#1684034211yescharacteristic
517C4#1684041511yescharacteristic
518C4#1684046511yescharacteristic
519C2—2444no—
520C2#2144925711yescharacteristic
521C2C2×#1683996811yescharacteristic
522C2#2145537111yescharacteristic
523C2#2145746711yescharacteristic
524C2D4×#1175819011yescharacteristic
525C2#2145846211yescharacteristic
526C2#2145860211yescharacteristic
subgroupG/H
centerC23C22×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1939C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenter#1939D61
hypocenterC61#24730
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #1939

Complements

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

NH ≅ G/Ntype
4. #21450807C2semidirect
6. #21447199C2semidirect
7. #21447066C2semidirect
8. #21446621C2semidirect
10. #21445825C2semidirect
11. C61×#1939C2semidirect
146. C4⋊C4×C61SD16semidirect
246. C4×C61#461semidirect
264. #1939D61semidirect
307. C61#24730semidirect
426. C4⋊C4#11758190semidirect
514. C4#16839968semidirect

Maximal subgroups

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

subgroupindex
C2×#168399682
C2×C4×#117581902
#214508072
C4⋊C4×D2442
#214471992
#214470662
#214466212
#214461522
#214458252
C61×#19392
#2473061