Finite Group Databasebeta

← #29714761

Subgroups — #29714761

Order 15616.

1150 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840482C211yesmaximal, characteristic, complemented
2C2×#16839967C211yesmaximal, characteristic, complemented
3#21450741C211yesmaximal, characteristic, complemented
4#21447058C212yesmaximal, complemented
5#21446894C212yesmaximal
6#21446755C211yesmaximal, characteristic, complemented
7#21446621C211yesmaximal, characteristic, complemented
8#21446171C212yesmaximal, complemented
9#21446070C212yesmaximal, complemented
10#21445819C211yesmaximal, characteristic, complemented
11C2×C4×D8×C61C211yesmaximal, characteristic
12C22×#11758200C2212yes—
13#16840482—24no—
14#16840482—24no—
15C2×#14519777C2211yescharacteristic
16C2×#14519770—22no—
17C2×#14519764C2212yes—
18C2×#14519756—21no—
19C2×#14519756—21no—
20C2×C4×Dic122C2211yescharacteristic
21C2×#14519742C2212yes—
22C2×#14519736—21no—
23#16839967—24no—
24C2×C4⋊#92418C2211yescharacteristic
25C2×#14519702C2212yes—
26C2×#14519702C2212yes—
27C2×#14519700C2212yes—
28C2×#14519698C2212yes—
29#16839685—22no—
30#16839665C2212yes—
31#16839658C2212yes—
32#16839632C2211yescharacteristic
33#16839622—21no—
34#16839524C2212yes—
35#16839516C2212yes—
36#16839516C2212yes—
37C22×D8×C61C2211yescharacteristic
38C2×C4×D4×C61C2211yescharacteristic
39C2×C4×D4×C61C2211yescharacteristic
40C4×D8×C61—24no—
41C2×#107×C61C2211yescharacteristic
42C2×#102×C61C2211yescharacteristic
43C2×#102×C61C2211yescharacteristic
44C2×C4×C8×C61C2211yescharacteristic
45C22×Dic122D411yescharacteristic
46C22×Dic122C2312yes—
47C2×#11758200—24no—
48C2×#11758200—24no—
49C2×#11758200—42no—
50C22×#92418C2312yes—
51#14519777—22no—
52#14519777D414yes—
53#14519770—216no—
54#14519764—24no—
55#14519764—24no—
56#14519756—28no—
57#14519756—44no—
58C4×Dic122—22no—
59C4×Dic122—22no—
60#14519742—24no—
61#14519742—24no—
62#14519736—28no—
63C2×C4⋊Dic61—21no—
64C2×C4⋊Dic61—21no—
65C2×C4⋊Dic61—21no—
66C2×C4⋊Dic61—21no—
67C2×C4⋊Dic61D411yescharacteristic
68C2×C4⋊Dic61C2312yes—
69C2×#11758196—22no—
70C2×#11758196—22no—
71C2×#11758196D412yes—
72C2×C22⋊Dic61—22no—
73C2×C22⋊Dic61—22no—
74C2×C22⋊Dic61—22no—
75C2×C22⋊Dic61—22no—
76C2×C4×Dic61—22no—
77C2×C4×Dic61C2312yes—
78C4⋊#92418—22no—
79#14519702—24no—
80#14519702—42no—
81#14519700—24no—
82#14519698—42no—
83#14519684—22no—
84C22×D4×C61C2311yescharacteristic
85C22×D4×C61C2311yescharacteristic
86C23×C4×C61—21no—
87C23×C4×C61—21no—
88C2×D8×C61—22no—
89C2×D8×C61—22no—
90C2×D8×C61—22no—
91C22×C8×C61C2311yescharacteristic
92C22×C8×C61C2311yescharacteristic
93C4×D4×C61—22no—
94C4×D4×C61—22no—
95C4×D4×C61—22no—
96C4×D4×C61D414yes—
97C2×C4⋊C4×C61C2311yescharacteristic
98C2×C4⋊C4×C61C2311yescharacteristic
99C2×C22⋊C4×C61—21no—
100C2×C22⋊C4×C61—21no—
101C2×C42×C61C2311yescharacteristic
102#107×C61—22no—
103#102×C61—22no—
104#102×C61—22no—
105C4×C8×C61—22no—
106C24×C61—21no—
107C24×C61—21no—
108C2×D4×C61—21no—
109C2×D4×C61—21no—
110C2×D4×C61—21no—
111C2×D4×C61—21no—
112C2×D4×C61—21no—
113C2×D4×C61—21no—
114C2×D4×C61—21no—
115C2×D4×C61—21no—
116C2×D4×C61—21no—
117C2×D4×C61—21no—
118C2×D4×C61C2×D412yes—
119C2×D4×C61#4112yes—
120C22×C4×C61—21no—
121C22×C4×C61—24no—
122C22×C4×C61—21no—
123C22×C4×C61—42no—
124C22×C4×C61—42no—
125C22×C4×C61—42no—
126C22×C4×C61—42no—
127C22×C4×C61—42no—
128C22×C4×C61C2×D411yescharacteristic
129C22×C4×C61C2×D411yescharacteristic
130C22×C4×C61C2411yescharacteristic
131C22×Dic61—22no—
132C22×Dic61—22no—
133C22×Dic61C2×D412yes—
134C22×Dic61C2×D412yes—
135C2×Dic122—22no—
136C2×Dic122—22no—
137C2×Dic122—22no—
138C2×Dic122—22no—
139C2×Dic122—22no—
140C2×Dic122—22no—
141C2×Dic122—22no—
142C2×Dic122—22no—
143C2×Dic122C2×D414yes—
144D8×C61—28no—
145C2×C8×C61—21no—
146C2×C8×C61—21no—
147C2×C8×C61—21no—
148C2×C8×C61—21no—
149C2×C8×C61—21no—
150C2×C8×C61#4112yes—
151C4⋊C4×C61—21no—
152C4⋊C4×C61—21no—
153C4⋊C4×C61—21no—
154C4⋊C4×C61C2×D412yes—
155C22⋊C4×C61—24no—
156C22⋊C4×C61—42no—
157C42×C61C2×D412yes—
158C42×C61C2×D412yes—
159C22⋊Dic61—28no—
160C22⋊Dic61—28no—
161C22⋊Dic61—44no—
162C22⋊Dic61—82no—
163#11758200—48no—
164C4⋊Dic61—24no—
165C4⋊Dic61—24no—
166C4⋊Dic61—22no—
167C4⋊Dic61—22no—
168C4⋊Dic61—22no—
169C4⋊Dic61—42no—
170C4⋊Dic61—81no—
171#11758196—28no—
172#11758196—24no—
173#11758196—44no—
174C4×Dic61—28no—
175C4×Dic61—22no—
176C4×Dic61—22no—
177C2×#92418—22no—
178C2×#92418—22no—
179C2×#92418—22no—
180C23×C61—22no—
181C23×C61—22no—
182C23×C61—41no—
183C23×C61—41no—
184C23×C61—41no—
185C23×C61—41no—
186C23×C61—41no—
187C23×C61—41no—
188C23×C61—41no—
189C23×C61—41no—
190C23×C61—41no—
191C23×C61—41no—
192C23×C61—41no—
193C23×C61—41no—
194C23×C61C22×D411yescharacteristic
195D4×C61—22no—
196D4×C61—22no—
197D4×C61—22no—
198D4×C61—22no—
199D4×C61—41no—
200D4×C61—41no—
201D4×C61—41no—
202D4×C61—41no—
203C2×C4×C61—21no—
204C2×C4×C61—21no—
205C2×C4×C61—22no—
206C2×C4×C61—41no—
207C2×C4×C61—41no—
208C2×C4×C61—48no—
209C2×C4×C61—41no—
210C2×C4×C61—41no—
211C2×C4×C61—41no—
212C2×C4×C61—84no—
213C2×C4×C61#12012yes—
214C2×C4×C61#12012yes—
215C2×C4×C61#13512yes—
216C2×C4×C61AGL(1,ℤ/8)12yes—
217C2×C4×C61C22×D411yescharacteristic
218C2×C4×C61C22×D411yescharacteristic
219C2×C4×C61C2×#4111yescharacteristic
220C2×C4×C61C2×#4111yescharacteristic
221C2×C4×C61H(5,𝔽2)11yescharacteristic
222C2×C4×C612−1+411yescharacteristic
223C2×Dic61—22no—
224C2×Dic61—22no—
225C2×Dic61—24no—
226C2×Dic61—22no—
227C2×Dic61—22no—
228C2×Dic61—22no—
229C2×Dic61—22no—
230C2×Dic61—42no—
231C2×Dic61—42no—
232C2×Dic61—42no—
233C2×Dic61—42no—
234C2×Dic61—42no—
235Dic122—24no—
236Dic122—24no—
237Dic122—42no—
238Dic122—42no—
239Dic122—44no—
240C8×C61—22no—
241C8×C61—41no—
242#92418—42no—
243#24185—611nomaximal, Sylow
244C22×C61—42no—
245C22×C61—42no—
246C22×C61—42no—
247C22×C61—42no—
248C22×C61—42no—
249C22×C61—42no—
250C22×C61—81no—
251C22×C61—81no—
252C22×C61—81no—
253C22×C61—81no—
254C22×C61—81no—
255C22×C61—81no—
256C22×C61—81no—
257C22×C61—81no—
258C22×C61C2×#12011yescharacteristic
259C22×C61#54811yescharacteristic
260C22×C61#55011yescharacteristic
261C22×C61C2×#13511yescharacteristic
262C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
263C22×C61#57711yescharacteristic
264C22×C61#57811yescharacteristic
265C4×C61—22no—
266C4×C61—42no—
267C4×C61—81no—
268C4×C61#44914yes—
269C4×C61#52511yescharacteristic
270C4×C61#53711yescharacteristic
271C4×C61#54611yescharacteristic
272C4×C61#54711yescharacteristic
273Dic61—44no—
274Dic61—42no—
275Dic61—42no—
276Dic61—82no—
277C2×#556—611no—
278C2×#461—611no—
279C2×C4×D8D6111yescharacteristic
280#2538—611no—
281#2042—612no—
282#2012—612no—
283#1984—611no—
284#1956—611no—
285#1880—612no—
286#1863—612no—
287#1802—611no—
288C2×C61—82no—
289C2×C61—82no—
290C2×C61#240711yescharacteristic
291C2×C61C2×#44911yescharacteristic
292C2×C61#301311yescharacteristic
293C2×C61#336011yescharacteristic
294C2×C61#336711yescharacteristic
295C2×C61#339311yescharacteristic
296C2×C61#339711yescharacteristic
297C22×SD16—612no—
298C22×D8C2×D6111yescharacteristic
299#556—1224no—
300#556—1224no—
301C2×C4⋊Q8—611no—
302C2×#126—1222no—
303C2×#124—612no—
304C2×C22⋊Q8—1221no—
305C2×C22⋊Q8—1221no—
306C2×C4×Q8—611no—
307C2×C4×Q8—612no—
308C2×C4×D4C2×D6111yescharacteristic
309C2×C4×D4C2×D6111yescharacteristic
310C2×#117—1221no—
311#461—1224no—
312C4×D8—24no—
313C2×#107C2×D6111yescharacteristic
314C2×C4⋊C8—611no—
315C2×#103—612no—
316C2×#103—612no—
317C2×#103—612no—
318C2×#102—612no—
319C2×#102C2×D6111yescharacteristic
320C2×#102C2×D6111yescharacteristic
321C2×C4×C8C2×D6111yescharacteristic
322#396—1222no—
323#390—612no—
324#389—612no—
325#384—611no—
326#380—1221no—
327#340—612no—
328#336—612no—
329#336—612no—
330C61#2418511yescharacteristic, complemented, Sylow
331C22×Q8—611no—
332C22×Q8—612no—
333C22×D4C22×D6111yescharacteristic
334C22×D4C22×D6111yescharacteristic
335C23×C4—21no—
336C23×C4—21no—
337C2×SD16—1224no—
338C2×SD16—1224no—
339C2×SD16—2442no—
340C2×D8—22no—
341C2×D8—22no—
342C2×D8—22no—
343C22×C8—612no—
344C22×C8C22×D6111yescharacteristic
345C22×C8C22×D6111yescharacteristic
346C4⋊Q8—614no—
347C4⋊Q8—1222no—
348#126—12216no—
349#124—1224no—
350#124—1224no—
351C22⋊Q8—1228no—
352C22⋊Q8—2444no—
353C4×Q8—1224no—
354C4×Q8—1224no—
355C4×Q8—1222no—
356C4×Q8—1222no—
357C4×D4—22no—
358C4×D4—22no—
359C4×D4—22no—
360C4×D4C22⋊D6114yes—
361#117—1228no—
362C2×C4⋊C4—611no—
363C2×C4⋊C4—612no—
364C2×C4⋊C4—612no—
365C2×C4⋊C4—1222no—
366C2×C4⋊C4—1221no—
367C2×C4⋊C4—1221no—
368C2×C4⋊C4—1221no—
369C2×C4⋊C4—1222no—
370C2×C4⋊C4—1221no—
371C2×C4⋊C4C22×D6111yescharacteristic
372C2×C4⋊C4C22×D6111yescharacteristic
373C2×C22⋊C4—21no—
374C2×C22⋊C4—21no—
375C2×C22⋊C4—1222no—
376C2×C22⋊C4—1222no—
377C2×C22⋊C4—1222no—
378C2×C22⋊C4—1222no—
379C2×C42—612no—
380C2×C42—1222no—
381C2×C42C22×D6111yescharacteristic
382#107—22no—
383C4⋊C8—1222no—
384#103—1224no—
385#103—1224no—
386#103—2442no—
387#102—22no—
388#102—22no—
389#102—2442no—
390C4×C8—22no—
391#95—1222no—
392C24—21no—
393C24—21no—
394C2×Q8—614no—
395C2×Q8—1222no—
396C2×Q8—1222no—
397C2×Q8—1222no—
398C2×Q8—1222no—
399C2×Q8—1222no—
400C2×Q8—1222no—
401C2×Q8—1222no—
402C2×Q8—1222no—
403C2×D4—21no—
404C2×D4—21no—
405C2×D4—21no—
406C2×D4—21no—
407C2×D4—21no—
408C2×D4—21no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4#1175822212yes—
414C2×D4C2×C22⋊D6112yes—
415C22×C4—21no—
416C22×C4—24no—
417C22×C4—21no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—612no—
424C22×C4—612no—
425C22×C4—1222no—
426C22×C4—1222no—
427C22×C4D4×D6111yescharacteristic
428C22×C4D4×D6111yescharacteristic
429C22×C4C23×D6111yescharacteristic
430SD16—2448no—
431D8—28no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—1222no—
438C2×C8—1222no—
439C2×C8—1222no—
440C2×C8#1175822212yes—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—1228no—
445C4⋊C4—1224no—
446C4⋊C4—1222no—
447C4⋊C4—1224no—
448C4⋊C4—1224no—
449C4⋊C4—1222no—
450C4⋊C4—1222no—
451C4⋊C4—4881no—
452C4⋊C4—2444no—
453C4⋊C4—2442no—
454C4⋊C4C2×C22⋊D6112yes—
455C22⋊C4—24no—
456C22⋊C4—42no—
457C22⋊C4—1228no—
458C22⋊C4—1228no—
459C22⋊C4—4882no—
460C22⋊C4—2444no—
461C42—1228no—
462C42—1222no—
463C42—1222no—
464C42D4×D6112yes—
465C42C2×C22⋊D6112yes—
466C23—22no—
467C23—22no—
468C23—41no—
469C23—41no—
470C23—41no—
471C23—41no—
472C23—41no—
473C23—41no—
474C23—41no—
475C23—41no—
476C23—41no—
477C23—41no—
478C23—41no—
479C23—41no—
480C23C2×D4×D6111yescharacteristic
481Q8—1224no—
482Q8—1224no—
483Q8—2442no—
484Q8—2442no—
485Q8—2444no—
486D4—22no—
487D4—22no—
488D4—22no—
489D4—22no—
490D4—41no—
491D4—41no—
492D4—41no—
493D4—41no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—22no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—48no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—84no—
504C2×C4—1222no—
505C2×C4—1222no—
506C2×C4—1224no—
507C2×C4—1222no—
508C2×C4—1222no—
509C2×C4—1222no—
510C2×C4—1222no—
511C2×C4—2442no—
512C2×C4—2442no—
513C2×C4—2442no—
514C2×C4—2442no—
515C2×C4—2442no—
516C2×C4#1451974512yes—
517C2×C4#1451974512yes—
518C2×C4#1451979612yes—
519C2×C4#1451980112yes—
520C2×C4C2×D4×D6111yescharacteristic
521C2×C4C22×C22⋊D6111yescharacteristic
522C2×C4C2×#1175822211yescharacteristic
523C2×C4C2×#1175822211yescharacteristic
524C2×C4#1451982311yescharacteristic
525C2×C4#1451982411yescharacteristic
526C8—22no—
527C8—41no—
528C8—2442no—
529C22—42no—
530C22—42no—
531C22—42no—
532C22—42no—
533C22—42no—
534C22—42no—
535C22—81no—
536C22—81no—
537C22—81no—
538C22—81no—
539C22—81no—
540C22—81no—
541C22—81no—
542C22—81no—
543C22C2×#1451974511yescharacteristic
544C22#1684042111yescharacteristic
545C22#1684043311yescharacteristic
546C22C2×#1451979611yescharacteristic
547C22C2×#1451980111yescharacteristic
548C22#1684056911yescharacteristic
549C22#1684057611yescharacteristic
550C4—22no—
551C4—42no—
552C4—81no—
553C4—4882no—
554C4—2442no—
555C4—2442no—
556C4—2444no—
557C4#1683990414yes—
558C4#1684027311yescharacteristic
559C4#1684034011yescharacteristic
560C4#1684040811yescharacteristic
561C4#1684041711yescharacteristic
562C2—82no—
563C2—82no—
564C2#2144916411yescharacteristic
565C2C2×#1683990411yescharacteristic
566C2#2145513011yescharacteristic
567C2#2145846611yescharacteristic
568C2#2145853511yescharacteristic
569C2#2145880511yescharacteristic
570C2#2145883711yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C4×D8×C61C2
socleC23×C61C22×D4
prime radicalC22×D8×C61C22
hypercenterC2×C4×D8D61
hypocenterC61#24185
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×D8

Complements

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

NH ≅ G/Ntype
1. C2×#16840482C2semidirect
2. C2×#16839967C2semidirect
3. #21450741C2semidirect
4. #21447058C2semidirect
6. #21446755C2semidirect
7. #21446621C2semidirect
8. #21446171C2semidirect
9. #21446070C2semidirect
10. #21445819C2semidirect
330. C61#24185semidirect

Maximal subgroups

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

subgroupindex
C2×#168404822
C2×#168399672
#214507412
#214470582
#214468942
#214467552
#214466212
#214461712
#214460702
#214458192
C2×C4×D8×C612
#2418561