Finite Group Databasebeta

← #29730750

Subgroups — #29730750

Order 15616.

1018 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840003C212yesmaximal, complemented
2C2×#16839973C211yesmaximal, characteristic, complemented
3#21450772C211yesmaximal, characteristic
4#21450741C211yesmaximal, characteristic
5#21447023C212yesmaximal, complemented
6#21446829C212yesmaximal, complemented
7#21446621C211yesmaximal, characteristic, complemented
8#21446578C211yesmaximal, characteristic, complemented
9#21446242C212yesmaximal, complemented
10C4×#14519702C211yesmaximal, characteristic, complemented
11C2×C8×D4×C61C211yesmaximal, characteristic
12C22×Dic244C2212yes—
13C2×#14519777C2211yescharacteristic
14C2×#14519756C2212yes—
15C2×#14519756C2212yes—
16C2×#14519756C2212yes—
17C2×C4×Dic122C2211yescharacteristic
18#16840003—28no—
19#16839973—24no—
20C2×#14519711C2212yes—
21C2×#14519702C2211yescharacteristic
22C2×#14519702C2212yes—
23C2×#14519702C2212yes—
24C2×#14519702C2212yes—
25C2×#14519702C2211yescharacteristic
26#16839698—22no—
27#16839685—22no—
28#16839665—21no—
29#16839665—21no—
30#16839658—21no—
31#16839632C2211yescharacteristic
32#16839631—21no—
33#16839622C2212yes—
34C4×C4⋊Dic61C2211yescharacteristic
35#16839525C2211yescharacteristic
36#16839525C2211yescharacteristic
37#16839525C2212yes—
38C23×C8×C61C2212yes—
39C2×C4×D4×C61C2211yescharacteristic
40C8×D4×C61—24no—
41C2×C4⋊C8×C61C2211yescharacteristic
42C2×C22⋊C8×C61C2212yes—
43C2×C4×C8×C61C2211yescharacteristic
44C22×Dic122C2312yes—
45C22×Dic122C2311yescharacteristic
46C2×Dic244—24no—
47C2×Dic244—24no—
48C2×Dic244—42no—
49#14519777—22no—
50#14519777—22no—
51#14519756—24no—
52#14519756—24no—
53#14519756—24no—
54#14519756—42no—
55#14519756—42no—
56#14519756—42no—
57C4×Dic122—22no—
58C4×Dic122—22no—
59C2×C4⋊Dic61—21no—
60C2×C4⋊Dic61—21no—
61C2×C4⋊Dic61—21no—
62C2×C4⋊Dic61—21no—
63C2×C4⋊Dic61C2311yescharacteristic
64C2×C4⋊Dic61C2312yes—
65C2×C4⋊Dic61C2311yescharacteristic
66C2×C4⋊Dic61C2311yescharacteristic
67C2×#11758196—21no—
68C2×#11758196—22no—
69C2×#11758196—22no—
70C2×#11758196—22no—
71C2×C22⋊Dic61—22no—
72C2×C22⋊Dic61—22no—
73C2×C22⋊Dic61—22no—
74C2×C22⋊Dic61—22no—
75C2×C4×Dic61—21no—
76C2×C4×Dic61—21no—
77#14519711—24no—
78#14519702—24no—
79#14519702—24no—
80#14519702—22no—
81#14519702—42no—
82#14519702D414yescomplemented
83#14519684—22no—
84#14519684—21no—
85C22×D4×C61D411yescharacteristic
86C23×C4×C61C2312yes—
87C22×C8×C61—24no—
88C22×C8×C61—24no—
89C22×C8×C61—42no—
90C22×C8×C61C2311yescharacteristic
91C22×C8×C61C2311yescharacteristic
92C22×C8×C61C2312yes—
93C4×D4×C61—22no—
94C4×D4×C61—22no—
95C2×C4⋊C4×C61D411yescharacteristic
96C2×C22⋊C4×C61D412yes—
97C2×C42×C61C2311yescharacteristic
98C4⋊C8×C61D414yes—
99C22⋊C8×C61—24no—
100C4×C8×C61—22no—
101C24×C61C2×D412yes—
102C2×D4×C61—22no—
103C2×D4×C61—22no—
104C2×D4×C61Q1614yescomplemented
105C22×C4×C61—22no—
106C22×C4×C61—22no—
107C22×C4×C61—22no—
108C22×C4×C61—22no—
109C22×C4×C61—22no—
110C22×C4×C61—22no—
111C22×C4×C61C2×D412yes—
112C22×C4×C61C2×D411yescharacteristic
113C22×C4×C61C2×D411yescharacteristic
114C22×C4×C61C2411yescharacteristic
115C22×Dic61—21no—
116C22×Dic61—21no—
117C22×Dic61—21no—
118C22×Dic61—22no—
119C22×Dic61—21no—
120C22×Dic61—22no—
121C2×Dic122—21no—
122C2×Dic122—21no—
123C2×Dic122—21no—
124C2×Dic122—21no—
125C2×Dic122—21no—
126C2×Dic122—22no—
127C2×Dic122—22no—
128C2×Dic122—22no—
129C2×Dic122—22no—
130C2×Dic122—22no—
131C2×Dic122—22no—
132C2×Dic122C2×D412yes—
133C2×C8×C61—21no—
134C2×C8×C61—22no—
135C2×C8×C61—21no—
136C2×C8×C61—22no—
137C2×C8×C61—21no—
138C2×C8×C61—21no—
139C2×C8×C61—21no—
140C2×C8×C61—48no—
141C2×C8×C61C2×D412yes—
142C2×C8×C61C2×D414yes—
143C4⋊C4×C61—22no—
144C22⋊C4×C61—24no—
145C42×C61C2×D412yes—
146C42×C61#4112yes—
147C22⋊Dic61—44no—
148C22⋊Dic61—44no—
149C22⋊Dic61—44no—
150C22⋊Dic61—82no—
151C4⋊Dic61—21no—
152C4⋊Dic61—21no—
153C4⋊Dic61—21no—
154C4⋊Dic61—22no—
155C4⋊Dic61—21no—
156C4⋊Dic61—22no—
157C4⋊Dic61—42no—
158C4⋊Dic61—42no—
159C4⋊Dic61—42no—
160C4⋊Dic61—81no—
161C4⋊Dic61C2×D412yes—
162C4⋊Dic61#4112yes—
163#11758196—44no—
164#11758196—44no—
165#11758196—44no—
166#11758196—42no—
167C4×Dic61—42no—
168C4×Dic61—42no—
169Dic244—48no—
170C23×C61—22no—
171C23×C61—22no—
172C23×C61—22no—
173C23×C61—22no—
174C23×C61—22no—
175C23×C61—22no—
176C23×C61C2×Q1614yes—
177C23×C61C22×D411yescharacteristic
178D4×C61—28no—
179C2×C4×C61—21no—
180C2×C4×C61—21no—
181C2×C4×C61—22no—
182C2×C4×C61—22no—
183C2×C4×C61—22no—
184C2×C4×C61—42no—
185C2×C4×C61—42no—
186C2×C4×C61—42no—
187C2×C4×C61—42no—
188C2×C4×C61#12112yes—
189C2×C4×C61#12112yes—
190C2×C4×C61C2×Q1612yes—
191C2×C4×C61#13512yes—
192C2×C4×C61C22×D411yescharacteristic
193C2×C4×C61C22×D411yescharacteristic
194C2×C4×C61C2×#4111yescharacteristic
195C2×C4×C61C2×#4111yescharacteristic
196C2×C4×C61H(5,𝔽2)11yescharacteristic
197C2×C4×C61H(5,𝔽2)11yescharacteristic
198C2×Dic61—22no—
199C2×Dic61—22no—
200C2×Dic61—22no—
201C2×Dic61—41no—
202C2×Dic61—41no—
203C2×Dic61—41no—
204C2×Dic61—41no—
205C2×Dic61—41no—
206C2×Dic61—41no—
207C2×Dic61—41no—
208C2×Dic61—42no—
209C2×Dic61—42no—
210C2×Dic61—42no—
211C2×Dic61—42no—
212C2×Dic61—42no—
213C2×Dic61—42no—
214C2×Dic61—41no—
215C2×Dic61—41no—
216Dic122—22no—
217Dic122—22no—
218Dic122—41no—
219Dic122—41no—
220Dic122—42no—
221Dic122—42no—
222Dic122—42no—
223Dic122—42no—
224C8×C61—24no—
225C8×C61—22no—
226C8×C61—41no—
227#25386—611nomaximal, Sylow
228C22×C61—24no—
229C22×C61—24no—
230C22×C61—24no—
231C22×C61—42no—
232C22×C61—42no—
233C22×C61—42no—
234C22×C61—42no—
235C22×C61C2×#12111yescharacteristic
236C22×C61D4211yescharacteristic
237C22×C61#54611yescharacteristic
238C22×C61C22×Q1611yescharacteristic
239C22×C61C2×#13511yescharacteristic
240C22×C61#57711yescharacteristic
241C22×C61#57811yescharacteristic
242C4×C61—22no—
243C4×C61—42no—
244C4×C61#46714yescomplemented
245C4×C61#53211yescharacteristic
246C4×C61#54011yescharacteristic
247C4×C61D4211yescharacteristic
248C4×C61#55011yescharacteristic
249Dic61—42no—
250Dic61—42no—
251Dic61—42no—
252Dic61—82no—
253Dic61—82no—
254Dic61—81no—
255C2×#467—612no—
256C2×#462—611no—
257C2×C8×D4D6111yescharacteristic
258#2542—611no—
259#2538—611no—
260#2036—612no—
261#2001—612no—
262#1956—611no—
263#1947—611no—
264#1892—612no—
265C4×#103—611no—
266C2×C61—44no—
267C2×C61#242811yescharacteristic
268C2×C61C2×#46711yescharacteristic
269C2×C61#306311yescharacteristic
270C2×C61#328411yescharacteristic
271C2×C61D4×Q1611yescharacteristic
272C2×C61#329911yescharacteristic
273C2×C61#330211yescharacteristic
274C22×Q16—612no—
275C23×C8C2×D6112yes—
276C2×C4⋊Q8—611no—
277C2×C22⋊Q8—612no—
278C2×C22⋊Q8—612no—
279C2×C22⋊Q8—612no—
280C2×C4×Q8—611no—
281C2×C4×D4C2×D6111yescharacteristic
282#467—1228no—
283#462—1224no—
284C8×D4—24no—
285C2×#107—612no—
286C2×C4⋊C8C2×D6111yescharacteristic
287C2×#103—611no—
288C2×#103—612no—
289C2×#103—612no—
290C2×#103—611no—
291C2×#103—612no—
292C2×C22⋊C8C2×D6112yes—
293C2×C4×C8C2×D6111yescharacteristic
294#399—1222no—
295#396—1222no—
296#390—1221no—
297#390—1221no—
298#389—1221no—
299#384—611no—
300#384—1221no—
301#380—612no—
302C4×C4⋊C4—611no—
303#340—611no—
304#340—612no—
305#340—611no—
306C61#2538611yescharacteristic, complemented, Sylow
307C22×Q8—611no—
308C22×Q8—612no—
309C22×D4D24411yescharacteristic
310C23×C4C22×D6112yes—
311C2×Q16—1224no—
312C2×Q16—1224no—
313C2×Q16—2442no—
314C22×C8—24no—
315C22×C8—24no—
316C22×C8—42no—
317C22×C8C22×D6111yescharacteristic
318C22×C8C22×D6111yescharacteristic
319C22×C8C22×D6112yes—
320C4⋊Q8—1222no—
321C4⋊Q8—1222no—
322C22⋊Q8—1224no—
323C22⋊Q8—1224no—
324C22⋊Q8—1224no—
325C22⋊Q8—2442no—
326C22⋊Q8—2442no—
327C22⋊Q8—2442no—
328C4×Q8—1222no—
329C4×Q8—1222no—
330C4×D4—22no—
331C4×D4—22no—
332C2×C4⋊C4—611no—
333C2×C4⋊C4—611no—
334C2×C4⋊C4—611no—
335C2×C4⋊C4—612no—
336C2×C4⋊C4—1221no—
337C2×C4⋊C4—1221no—
338C2×C4⋊C4—1222no—
339C2×C4⋊C4—1222no—
340C2×C4⋊C4—1222no—
341C2×C4⋊C4—1221no—
342C2×C4⋊C4—1221no—
343C2×C4⋊C4—1221no—
344C2×C4⋊C4D24411yescharacteristic
345C2×C22⋊C4—1222no—
346C2×C22⋊C4—1222no—
347C2×C22⋊C4—1222no—
348C2×C22⋊C4—1222no—
349C2×C22⋊C4D24412yes—
350C2×C42—1221no—
351C2×C42—1221no—
352C2×C42C22×D6111yescharacteristic
353#107—1224no—
354C4⋊C8C22⋊D6114yes—
355#103—614no—
356#103—1222no—
357#103—1224no—
358#103—1224no—
359#103—2442no—
360C22⋊C8—24no—
361C4×C8—22no—
362#95—1221no—
363#95—1222no—
364C24C2×D24412yes—
365C2×Q8—612no—
366C2×Q8—1221no—
367C2×Q8—1222no—
368C2×Q8—1221no—
369C2×Q8—1222no—
370C2×Q8—1221no—
371C2×Q8—1222no—
372C2×Q8—1221no—
373C2×Q8—1222no—
374C2×Q8—1221no—
375C2×Q8—1222no—
376C2×Q8—1222no—
377C2×D4—22no—
378C2×D4—22no—
379C2×D4Dic24414yescomplemented
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—1221no—
387C22×C4—1221no—
388C22×C4—1221no—
389C22×C4—1221no—
390C22×C4—1222no—
391C22×C4—1222no—
392C22×C4C2×D24412yes—
393C22×C4C2×D24411yescharacteristic
394C22×C4C2×D24411yescharacteristic
395C22×C4C23×D6111yescharacteristic
396Q16—2448no—
397C2×C8—22no—
398C2×C8—21no—
399C2×C8—21no—
400C2×C8—22no—
401C2×C8—21no—
402C2×C8—21no—
403C2×C8—21no—
404C2×C8—48no—
405C2×C8D4×D6112yes—
406C2×C8C2×C22⋊D6114yes—
407C4⋊C4—22no—
408C4⋊C4—612no—
409C4⋊C4—612no—
410C4⋊C4—1221no—
411C4⋊C4—1221no—
412C4⋊C4—1222no—
413C4⋊C4—1221no—
414C4⋊C4—1221no—
415C4⋊C4—1222no—
416C4⋊C4—4881no—
417C4⋊C4—2444no—
418C4⋊C4—2444no—
419C4⋊C4—2444no—
420C4⋊C4—2442no—
421C4⋊C4—2442no—
422C4⋊C4—2442no—
423C4⋊C4—2442no—
424C22⋊C4—24no—
425C22⋊C4—4882no—
426C22⋊C4—2444no—
427C22⋊C4—2444no—
428C22⋊C4—2444no—
429C42—2442no—
430C42—2442no—
431C42#1175822212yes—
432C42C2×C22⋊D6112yes—
433C23—22no—
434C23—22no—
435C23—22no—
436C23—22no—
437C23—22no—
438C23—22no—
439C23C2×Dic24414yes—
440C23C22×D24411yescharacteristic
441Q8—1222no—
442Q8—1222no—
443Q8—2442no—
444Q8—2442no—
445Q8—2442no—
446Q8—2442no—
447Q8—2441no—
448Q8—2441no—
449D4—28no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—1222no—
460C2×C4—1222no—
461C2×C4—1222no—
462C2×C4—2441no—
463C2×C4—2442no—
464C2×C4—2442no—
465C2×C4—2441no—
466C2×C4—2442no—
467C2×C4—2441no—
468C2×C4—2442no—
469C2×C4—2442no—
470C2×C4—2442no—
471C2×C4—2441no—
472C2×C4—2441no—
473C2×C4—2441no—
474C2×C4—2441no—
475C2×C4—2441no—
476C2×C4—2441no—
477C2×C4#1451975112yes—
478C2×C4#1451975112yes—
479C2×C4C2×Dic24412yes—
480C2×C4#1451979912yes—
481C2×C4C2×D4×D6111yescharacteristic
482C2×C4C22×C22⋊D6111yescharacteristic
483C2×C4#41×D6111yescharacteristic
484C2×C4C2×#1175822211yescharacteristic
485C2×C4#1451982211yescharacteristic
486C2×C4#1451982311yescharacteristic
487C8—22no—
488C8—24no—
489C8—41no—
490C22—24no—
491C22—24no—
492C22—24no—
493C22—42no—
494C22—42no—
495C22—42no—
496C22—42no—
497C22C2×#1451975111yescharacteristic
498C22D4×D24411yescharacteristic
499C22#1684040711yescharacteristic
500C22C22×Dic24411yescharacteristic
501C22C2×#1451979911yescharacteristic
502C22#1684057411yescharacteristic
503C22#1684057911yescharacteristic
504C4—22no—
505C4—42no—
506C4—4882no—
507C4—4881no—
508C4—4882no—
509C4—2442no—
510C4—2442no—
511C4—2442no—
512C4#1684000314yescomplemented
513C4#1684030911yescharacteristic
514C4#1684036311yescharacteristic
515C4D4×C22⋊D6111yescharacteristic
516C4#1684043111yescharacteristic
517C2—44no—
518C2#2144940111yescharacteristic
519C2C2×#1684000311yescharacteristic
520C2#2145551011yescharacteristic
521C2#2145755111yescharacteristic
522C2D4×Dic24411yescharacteristic
523C2#2145769611yescharacteristic
524C2#2145773711yescharacteristic
subgroupG/H
centerC23C22×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C8×D4×C61C2
socleC23×C61C22×D4
prime radicalC22×D4×C61D4
hypercenterC2×C8×D4D61
hypocenterC61#25386
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×C8×D4

Complements

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

NH ≅ G/Ntype
1. C2×#16840003C2semidirect
2. C2×#16839973C2semidirect
5. #21447023C2semidirect
6. #21446829C2semidirect
7. #21446621C2semidirect
8. #21446578C2semidirect
9. #21446242C2semidirect
10. C4×#14519702C2semidirect
82. #14519702D4semidirect
104. C2×D4×C61Q16semidirect
244. C4×C61#467semidirect
306. C61#25386semidirect
379. C2×D4Dic244semidirect
512. C4#16840003semidirect

Maximal subgroups

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

subgroupindex
C2×#168400032
C2×#168399732
#214507722
#214507412
#214470232
#214468292
#214466212
#214465782
#214462422
C4×#145197022
C2×C8×D4×C612
#2538661