Finite Group Databasebeta

← #29721560

Subgroups — #29721560

Order 15616.

1118 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839962C212yesmaximal, complemented
2C2×#16839962C211yesmaximal, characteristic, complemented
3C2×C4×#11758201C211yesmaximal, characteristic, complemented
4#21450761C211yesmaximal, characteristic
5#21447190C212yesmaximal, complemented
6#21447053C212yesmaximal, complemented
7#21446624C211yesmaximal, characteristic, complemented
8#21446538C211yesmaximal, characteristic, complemented
9#21446174C212yesmaximal, complemented
10#21445816C211yesmaximal, characteristic, complemented
11Q8×C4⋊C4×C61C211yesmaximal, characteristic, complemented
12C22×#11758201C2212yes—
13C22×#11758201C2211yescharacteristic
14C2×#14519774C2211yescharacteristic
15C2×#14519774C2212yes—
16C2×#14519752—21no—
17C2×#14519752—21no—
18C2×C4×D244C2211yescharacteristic
19#16839962—28no—
20#16839962—24no—
21C4×#11758201—24no—
22C2×#14519708C2211yescharacteristic
23C2×C4⋊#92418C2212yes—
24C2×C4⋊#92418C2211yescharacteristic
25C2×#14519701C2211yescharacteristic
26C2×#14519697C2212yes—
27C2×#14519697C2212yes—
28C2×#14519697C2211yescharacteristic
29C2×#14519697C2212yes—
30C2×#14519697C2212yes—
31C2×C4×#92418C2211yescharacteristic
32#16839672—22no—
33#16839664C2212yes—
34#16839524C2212yes—
35#16839517C2212yes—
36C2×C4×Q8×C61C2212yes—
37C2×C4×Q8×C61C2211yescharacteristic
38C61×#389—22no—
39C61×#389C2212yes—
40C61×#384—21no—
41C61×#384C2211yescharacteristic
42C4×C4⋊C4×C61—21no—
43C4×C4⋊C4×C61C2211yescharacteristic
44C22×D244—21no—
45C22×D244—22no—
46C22×D244—22no—
47C22×D244—21no—
48C22×D244C2312yes—
49C22×D244C2311yescharacteristic
50C22×C4×D61—21no—
51C2×#11758201—22no—
52C2×#11758201—22no—
53C2×#11758201—24no—
54C2×#11758201—24no—
55C2×#11758201—24no—
56C2×#11758201D414yescomplemented
57C22×#92418C2311yescharacteristic
58C22×#92418C2312yes—
59C22×#92418C2311yescharacteristic
60#14519774—24no—
61#14519774—22no—
62#14519774—22no—
63#14519774—42no—
64#14519752—44no—
65#14519752—44no—
66C4×D244—22no—
67C4×D244—22no—
68C2×C4⋊Dic61C2311yescharacteristic
69C2×#11758198—21no—
70C2×#11758198—22no—
71C2×#11758198—22no—
72#14519708—22no—
73C4⋊#92418—22no—
74C4⋊#92418—24no—
75#14519701—22no—
76#14519697—24no—
77#14519697—24no—
78#14519697—22no—
79#14519697—24no—
80#14519697—42no—
81C4×#92418—22no—
82C22×Q8×C61C2311yescharacteristic
83C4×Q8×C61—22no—
84C4×Q8×C61—24no—
85C4×Q8×C61—42no—
86C4×Q8×C61D414yescomplemented
87C2×C4⋊C4×C61—21no—
88C2×C4⋊C4×C61—21no—
89C2×C4⋊C4×C61—22no—
90C2×C4⋊C4×C61—22no—
91C2×C4⋊C4×C61—21no—
92C2×C4⋊C4×C61D411yescharacteristic
93C2×C4⋊C4×C61D411yescharacteristic
94C2×C4⋊C4×C61C2312yes—
95C2×C4⋊C4×C61C2311yescharacteristic
96C2×C42×C61—21no—
97C2×C42×C61—22no—
98C2×C42×C61C2312yes—
99C2×C42×C61C2311yescharacteristic
100#95×C61—22no—
101#95×C61D412yes—
102C23×D61—21no—
103C23×D61—22no—
104C2×Q8×C61—21no—
105C2×Q8×C61—21no—
106C2×Q8×C61—21no—
107C2×Q8×C61C2×D412yes—
108C2×Q8×C61C2×D412yes—
109C2×Q8×C61#4112yes—
110C22×C4×C61—21no—
111C22×C4×C61—21no—
112C22×C4×C61—22no—
113C22×C4×C61C2×D412yes—
114C22×C4×C61C2×D411yescharacteristic
115C22×C4×C61C2×D412yes—
116C22×C4×C61C2×D411yescharacteristic
117C22×C4×C61C2411yescharacteristic
118C22×Dic61—21no—
119C2×D244—21no—
120C2×D244—21no—
121C2×D244—21no—
122C2×D244—21no—
123C2×D244—22no—
124C2×D244—22no—
125C2×D244—22no—
126C2×D244—22no—
127C2×D244—21no—
128C2×D244—22no—
129C2×D244—22no—
130C2×D244—44no—
131C2×D244—44no—
132C2×D244—44no—
133C2×D244—42no—
134C2×D244—42no—
135C2×D244—44no—
136C2×D244—42no—
137C2×D244—42no—
138C2×D244—82no—
139C2×D244—81no—
140C2×D244—81no—
141C2×D244—82no—
142C2×D244C2×D412yes—
143C2×C4×D61—42no—
144C2×C4×D61—42no—
145C2×C4×D61—42no—
146C4⋊C4×C61—24no—
147C4⋊C4×C61—22no—
148C4⋊C4×C61—21no—
149C4⋊C4×C61—22no—
150C4⋊C4×C61—22no—
151C4⋊C4×C61—44no—
152C4⋊C4×C61—42no—
153C4⋊C4×C61—81no—
154C4⋊C4×C61—82no—
155C4⋊C4×C61SD1614yescomplemented
156C4⋊C4×C61C2×D412yes—
157C42×C61—24no—
158C42×C61—22no—
159C42×C61—22no—
160C42×C61—44no—
161C42×C61C2×D412yes—
162C42×C61#4112yes—
163#11758201—28no—
164#11758201—48no—
165#11758198—44no—
166#11758198—42no—
167#11758198—44no—
168C4⋊Dic61—21no—
169C4⋊Dic61—21no—
170C2×#92418—21no—
171C2×#92418—21no—
172C2×#92418—21no—
173C2×#92418—22no—
174C2×#92418—22no—
175C2×#92418—22no—
176C2×#92418—21no—
177C2×#92418—21no—
178C2×#92418C2×D412yes—
179C23×C61C22×D411yescharacteristic
180C22×D61—22no—
181C22×D61—41no—
182C22×D61—41no—
183C22×D61—41no—
184C22×D61—41no—
185C22×D61—41no—
186C22×D61—41no—
187C22×D61—42no—
188C22×D61—42no—
189C22×D61—42no—
190C22×D61—42no—
191C22×D61—42no—
192C22×D61—42no—
193C22×D61—42no—
194Q8×C61—22no—
195Q8×C61—22no—
196Q8×C61—22no—
197Q8×C61#12114yescomplemented
198C2×C4×C61—22no—
199C2×C4×C61—22no—
200C2×C4×C61—22no—
201C2×C4×C61—22no—
202C2×C4×C61—21no—
203C2×C4×C61—22no—
204C2×C4×C61—21no—
205C2×C4×C61—22no—
206C2×C4×C61—22no—
207C2×C4×C61—22no—
208C2×C4×C61—22no—
209C2×C4×C61—41no—
210C2×C4×C61—41no—
211C2×C4×C61—42no—
212C2×C4×C61—42no—
213C2×C4×C61—42no—
214C2×C4×C61#12112yes—
215C2×C4×C61#12112yes—
216C2×C4×C61C2×SD1614yes—
217C2×C4×C61C2×SD1612yes—
218C2×C4×C61AGL(1,ℤ/8)12yes—
219C2×C4×C61C22×D411yescharacteristic
220C2×C4×C61C22×D411yescharacteristic
221C2×C4×C61C2×#4111yescharacteristic
222C2×C4×C61C2×#4111yescharacteristic
223C2×C4×C61H(5,𝔽2)11yescharacteristic
224C2×C4×C612−1+411yescharacteristic
225C2×Dic61—41no—
226C2×Dic61—41no—
227C2×Dic61—41no—
228D244—22no—
229D244—22no—
230D244—42no—
231D244—41no—
232D244—42no—
233D244—42no—
234D244—41no—
235D244—42no—
236D244—88no—
237D244—84no—
238D244—88no—
239D244—84no—
240C4×D61—84no—
241#92418—22no—
242#92418—42no—
243#92418—41no—
244#24728—611nomaximal, Sylow
245C22×C61C2×#12111yescharacteristic
246C22×C61D4211yescharacteristic
247C22×C61#54811yescharacteristic
248C22×C61C22×SD1611yescharacteristic
249C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
250C22×C61#57611yescharacteristic
251C22×C61#57711yescharacteristic
252C2×D61—42no—
253C2×D61—42no—
254C2×D61—42no—
255C2×D61—81no—
256C2×D61—81no—
257C2×D61—81no—
258C2×D61—81no—
259C2×D61—82no—
260C2×D61—82no—
261C2×D61—82no—
262C2×D61—82no—
263C2×D61—82no—
264C2×D61—82no—
265C2×D61—82no—
266C4×C61—24no—
267C4×C61—24no—
268C4×C61—22no—
269C4×C61—42no—
270C4×C61—42no—
271C4×C61—82no—
272C4×C61#46014yescomplemented
273C4×C61C2×#12111yescharacteristic
274C4×C61#53711yescharacteristic
275C4×C61#54711yescharacteristic
276C4×C61#55311yescharacteristic
277Dic61—81no—
278C2×#460—612no—
279C2×#460—611no—
280C2×C4×SD16—611no—
281#2541—611no—
282Q8×C4⋊C4D6111yescharacteristic, complemented
283#2065—612no—
284#2041—612no—
285#1957—611no—
286#1939—611no—
287#1880—612no—
288#1802—611no—
289C2×C61#241611yescharacteristic
290C2×C61C2×#46011yescharacteristic
291C2×C61#304511yescharacteristic
292C2×C61#327411yescharacteristic
293C2×C61D4×SD1611yescharacteristic
294C2×C61#336211yescharacteristic
295C2×C61#336811yescharacteristic
296D61—82no—
297D61—162no—
298C22×SD16—611no—
299C22×SD16—612no—
300C2×#127—611no—
301C2×#127—612no—
302C2×#121—1221no—
303C2×#121—1221no—
304C2×C4×Q8C2×D6112yes—
305C2×C4×Q8C2×D6111yescharacteristic
306C2×C4×D4—611no—
307#460—1228no—
308#460—1224no—
309C4×SD16—1224no—
310C2×#106—611no—
311C2×C4⋊C8—611no—
312C2×C4⋊C8—612no—
313C2×#103—611no—
314C2×#102—612no—
315C2×#102—612no—
316C2×#102—612no—
317C2×#102—611no—
318C2×#102—612no—
319C2×C4×C8—611no—
320#392—1222no—
321#390—612no—
322#389—22no—
323#389C2×D6112yes—
324#384—21no—
325#384C2×D6111yescharacteristic
326C4×C4⋊C4—21no—
327C4×C4⋊C4C2×D6111yescharacteristic
328#340—612no—
329#336—612no—
330C61#2472811yescharacteristic, complemented, Sylow
331C22×Q8C22×D6111yescharacteristic
332C22×D4—611no—
333C22×D4—612no—
334C22×D4—1221no—
335C22×D4—1222no—
336C22×D4—1222no—
337C22×D4—1221no—
338C23×C4—1221no—
339C2×SD16—614no—
340C2×SD16—1222no—
341C2×SD16—1222no—
342C2×SD16—1224no—
343C2×SD16—1224no—
344C2×SD16—1224no—
345C22×C8—611no—
346C22×C8—612no—
347C22×C8—611no—
348#127—1224no—
349#127—1222no—
350#127—1222no—
351#127—2442no—
352#121—2444no—
353#121—2444no—
354C4×Q8—22no—
355C4×Q8—24no—
356C4×Q8—42no—
357C4×Q8D24414yescomplemented
358C4×D4—1222no—
359C4×D4—1222no—
360C2×C4⋊C4—21no—
361C2×C4⋊C4—22no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—22no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—611no—
366C2×C4⋊C4C22⋊D6111yescharacteristic
367C2×C4⋊C4C22⋊D6111yescharacteristic
368C2×C4⋊C4C22×D6112yes—
369C2×C4⋊C4C22×D6111yescharacteristic
370C2×C22⋊C4—1221no—
371C2×C22⋊C4—1222no—
372C2×C22⋊C4—1222no—
373C2×C42—21no—
374C2×C42—22no—
375C2×C42C22×D6112yes—
376C2×C42C22×D6111yescharacteristic
377#106—1222no—
378C4⋊C8—1224no—
379C4⋊C8—1222no—
380#103—1222no—
381#102—1224no—
382#102—1224no—
383#102—1222no—
384#102—1224no—
385#102—2442no—
386C4×C8—1222no—
387#95—22no—
388#95C22⋊D6112yes—
389C24—1221no—
390C24—1222no—
391C2×Q8—21no—
392C2×Q8—21no—
393C2×Q8—21no—
394C2×Q8C2×D24412yes—
395C2×Q8D4×D6112yes—
396C2×Q8#1175822612yes—
397C2×D4—612no—
398C2×D4—1222no—
399C2×D4—1221no—
400C2×D4—1222no—
401C2×D4—1221no—
402C2×D4—1222no—
403C2×D4—1221no—
404C2×D4—1222no—
405C2×D4—1221no—
406C2×D4—1222no—
407C2×D4—1221no—
408C2×D4—1222no—
409C2×D4—4881no—
410C2×D4—4881no—
411C2×D4—4882no—
412C2×D4—4882no—
413C2×D4—2444no—
414C2×D4—2444no—
415C2×D4—2444no—
416C2×D4—2442no—
417C2×D4—2442no—
418C2×D4—2444no—
419C2×D4—2442no—
420C2×D4—2442no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—1221no—
425C22×C4—2442no—
426C22×C4—2442no—
427C22×C4—2442no—
428C22×C4C2×C22⋊D6112yes—
429C22×C4C2×C22⋊D6111yescharacteristic
430C22×C4C2×C22⋊D6112yes—
431C22×C4C2×C22⋊D6111yescharacteristic
432C22×C4C23×D6111yescharacteristic
433SD16—1228no—
434SD16—2448no—
435C2×C8—612no—
436C2×C8—1221no—
437C2×C8—1221no—
438C2×C8—1222no—
439C2×C8—1221no—
440C2×C8—1222no—
441C2×C8—1222no—
442C2×C8—1221no—
443C2×C8—1221no—
444C4⋊C4—21no—
445C4⋊C4—24no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—44no—
450C4⋊C4—42no—
451C4⋊C4—81no—
452C4⋊C4—82no—
453C4⋊C4—1221no—
454C4⋊C4—1221no—
455C4⋊C4#1175820114yescomplemented
456C4⋊C4C2×D24412yes—
457C22⋊C4—2444no—
458C22⋊C4—2442no—
459C22⋊C4—2444no—
460C42—22no—
461C42—22no—
462C42—24no—
463C42—44no—
464C42C2×D24412yes—
465C42#1175822212yes—
466C23—1222no—
467C23—2442no—
468C23—2442no—
469C23—2442no—
470C23—2442no—
471C23—2442no—
472C23—2442no—
473C23—2442no—
474C23—2441no—
475C23—2441no—
476C23—2441no—
477C23—2441no—
478C23—2441no—
479C23—2441no—
480C23C22×C22⋊D6111yescharacteristic
481Q8—22no—
482Q8—22no—
483Q8—22no—
484Q8#1451975214yescomplemented
485D4—1222no—
486D4—1222no—
487D4—4884no—
488D4—4888no—
489D4—4888no—
490D4—4884no—
491D4—2442no—
492D4—2441no—
493D4—2442no—
494D4—2442no—
495D4—2441no—
496D4—2442no—
497C2×C4—22no—
498C2×C4—21no—
499C2×C4—22no—
500C2×C4—21no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—4884no—
514C2×C4—2441no—
515C2×C4—2441no—
516C2×C4—2441no—
517C2×C4#1451975112yes—
518C2×C4#1451975112yes—
519C2×C4C2×#1175820114yes—
520C2×C4C2×#1175820112yes—
521C2×C4#1451980212yes—
522C2×C4C2×D4×D6111yescharacteristic
523C2×C4C22×D24411yescharacteristic
524C2×C4C2×#1175822211yescharacteristic
525C2×C4C2×#1175822611yescharacteristic
526C2×C4#1451982211yescharacteristic
527C2×C4#1451982511yescharacteristic
528C8—1222no—
529C8—2442no—
530C8—2441no—
531C22—4882no—
532C22—4882no—
533C22—4882no—
534C22—4882no—
535C22—4882no—
536C22—4882no—
537C22—4882no—
538C22—4881no—
539C22—4881no—
540C22—4881no—
541C22—4881no—
542C22—2442no—
543C22—2442no—
544C22—2442no—
545C22C2×#1451975111yescharacteristic
546C22D4×C22⋊D6111yescharacteristic
547C22#1684042311yescharacteristic
548C22C22×#1175820111yescharacteristic
549C22C2×#1451980211yescharacteristic
550C22#1684056511yescharacteristic
551C22#1684057011yescharacteristic
552C4—24no—
553C4—24no—
554C4—22no—
555C4—42no—
556C4—42no—
557C4—82no—
558C4—4881no—
559C4#1683996214yescomplemented
560C4C2×#1451975211yescharacteristic
561C4#1684034311yescharacteristic
562C4#1684041311yescharacteristic
563C4#1684046011yescharacteristic
564C2—9762no—
565C2—4882no—
566C2#2144925311yescharacteristic
567C2C2×#1683996211yescharacteristic
568C2#2145535811yescharacteristic
569C2#2145743911yescharacteristic
570C2D4×#1175820111yescharacteristic
571C2#2145849011yescharacteristic
572C2#2145854811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingQ8×C4⋊C4×C61C2
socleC23×C61C22×D4
prime radical#21450761C2
hypercenterQ8×C4⋊C4D61
hypocenterC61#24728
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ Q8×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#16839962C2semidirect
2. C2×#16839962C2semidirect
3. C2×C4×#11758201C2semidirect
5. #21447190C2semidirect
6. #21447053C2semidirect
7. #21446624C2semidirect
8. #21446538C2semidirect
9. #21446174C2semidirect
10. #21445816C2semidirect
11. Q8×C4⋊C4×C61C2semidirect
56. C2×#11758201D4semidirect
86. C4×Q8×C61D4semidirect
155. C4⋊C4×C61SD16semidirect
197. Q8×C61#121semidirect
272. C4×C61#460semidirect
282. Q8×C4⋊C4D61semidirect
330. C61#24728semidirect
357. C4×Q8D244semidirect
455. C4⋊C4#11758201semidirect
484. Q8#14519752semidirect
559. C4#16839962semidirect

Maximal subgroups

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

subgroupindex
C2×#168399622
C2×C4×#117582012
#214507612
#214471902
#214470532
#214466242
#214465382
#214461742
#214458162
Q8×C4⋊C4×C612
#2472861