Finite Group Databasebeta

← #29713953

Subgroups — #29713953

Order 15616.

1350 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840415C211yesmaximal, characteristic
2C2×#16840044C211yesmaximal, characteristic, complemented
3C2×C4×#11758190C211yesmaximal, characteristic, complemented
4#21450742C211yesmaximal, characteristic, complemented
5#21447011C212yesmaximal, complemented
6#21446825C212yesmaximal, complemented
7#21446613C211yesmaximal, characteristic, complemented
8#21446125C212yesmaximal, complemented
9#21446013C212yesmaximal, complemented
10#21445816C211yesmaximal, characteristic, complemented
11C61×#1990C211yesmaximal, characteristic, complemented
12C22×#11758226—21no—
13C22×#11758190C2211yescharacteristic
14#16840415—24no—
15#16840415—24no—
16C2×#14519762—22no—
17C2×#14519757C2212yes—
18C2×#14519753—21no—
19C2×#14519752C2212yes—
20C2×C4×Dic122C2211yescharacteristic
21C2×C4×D244C2211yescharacteristic
22C2×C4×D244C2211yescharacteristic
23C2×C4⋊C4×D61C2212yes—
24#16840044—24no—
25C4×#11758190—24no—
26C2×#14519709C2211yescharacteristic
27C2×#14519708C2212yes—
28C2×C4⋊#92418C2211yescharacteristic
29C2×#14519702C2211yescharacteristic
30C2×#14519700C2212yes—
31C2×#14519699C2211yescharacteristic
32C2×#14519697C2212yes—
33C2×#14519689C2212yes—
34C2×#14519689C2212yes—
35#16839684—22no—
36#16839663—21no—
37#16839661—21no—
38#16839619C2212yes—
39#16839524C2212yes—
40C2×C4⋊Q8×C61C2211yescharacteristic
41C2×C4×C8×C61C2211yescharacteristic
42C61×#384C2211yescharacteristic
43C61×#340C2212yes—
44C61×#340C2212yes—
45C2×#11758226—28no—
46C2×#11758226—44no—
47C2×#11758226—44no—
48C22×Dic122C2311yescharacteristic
49C22×D244—22no—
50C22×D244D411yescharacteristic
51C22×D244C2311yescharacteristic
52C22×C4×D61—22no—
53C22×C4×D61—21no—
54C22×C4×D61C2312yes—
55C2×#11758190—22no—
56C2×#11758190—22no—
57C2×#11758190—22no—
58C22×#92418C2312yes—
59#14519762—216no—
60#14519757—24no—
61#14519757—42no—
62#14519753—28no—
63#14519752—24no—
64#14519752—24no—
65C4×Dic122—22no—
66C4×Dic122—22no—
67C4×D244—22no—
68C4×D244—22no—
69C4×D244—22no—
70C4×D244D414yes—
71C2×C4⋊Dic61—22no—
72C2×C4⋊Dic61D411yescharacteristic
73C2×C4⋊Dic61C2311yescharacteristic
74C2×C4⋊Dic61C2311yescharacteristic
75C2×#11758196—22no—
76C2×#11758196—21no—
77C4⋊C4×D61—24no—
78C4⋊C4×D61—24no—
79C2×#11758198—21no—
80C2×#11758198—22no—
81C2×#11758198—22no—
82C2×#11758198—22no—
83C2×#11758198D412yes—
84C2×C4×Dic61—21no—
85#14519709—22no—
86#14519708—24no—
87C4⋊#92418—22no—
88#14519702—22no—
89#14519700—24no—
90#14519699—22no—
91#14519697—42no—
92#14519689—24no—
93#14519689—42no—
94#14519684—22no—
95C22×Q8×C61—21no—
96C22×C8×C61C2311yescharacteristic
97C22×C8×C61C2311yescharacteristic
98C4⋊Q8×C61—22no—
99C4⋊Q8×C61D414yes—
100C2×C4⋊C4×C61—21no—
101C2×C4⋊C4×C61—21no—
102C2×C4⋊C4×C61—21no—
103C2×C4⋊C4×C61C2312yes—
104C2×C4⋊C4×C61C2312yes—
105C2×C42×C61C2311yescharacteristic
106C4×C8×C61—22no—
107C23×D61—21no—
108C23×D61C2×D412yes—
109C2×Q8×C61—24no—
110C2×Q8×C61—42no—
111C2×Q8×C61—42no—
112C22×C4×C61—22no—
113C22×C4×C61—22no—
114C22×C4×C61C2×D411yescharacteristic
115C22×C4×C61C2×D411yescharacteristic
116C22×C4×C61C2411yescharacteristic
117C22×Dic61—21no—
118C22×Dic61—21no—
119C22×Dic61—21no—
120C22×Dic61C2×D412yes—
121#11758226—432no—
122C2×D244—22no—
123C2×D244—22no—
124C2×D244—21no—
125C2×D244—21no—
126C2×D244—21no—
127C2×D244—21no—
128C2×D244—28no—
129C2×D244—21no—
130C2×D244—44no—
131C2×D244—44no—
132C2×D244SD1614yescomplemented
133C2×D244C2×D412yes—
134C2×C4×D61—28no—
135C2×C4×D61—24no—
136C2×C4×D61—22no—
137C2×C4×D61—22no—
138C2×C4×D61—22no—
139C2×C4×D61—22no—
140C2×C4×D61—22no—
141C2×C4×D61—22no—
142C2×C4×D61—44no—
143C2×C4×D61—44no—
144C2×C4×D61—42no—
145C2×C4×D61—42no—
146C2×Dic122—21no—
147C2×Dic122—21no—
148C2×Dic122—21no—
149C2×Dic122—21no—
150C2×Dic122—21no—
151C2×Dic122#4112yescomplemented
152C2×C8×C61—21no—
153C2×C8×C61—21no—
154C2×C8×C61—21no—
155C2×C8×C61—21no—
156C2×C8×C61—21no—
157C2×C8×C61#4112yescomplemented
158C4⋊C4×C61—24no—
159C4⋊C4×C61—22no—
160C4⋊C4×C61—22no—
161C4⋊C4×C61—22no—
162C4⋊C4×C61—42no—
163C4⋊C4×C61—81no—
164C4⋊C4×C61C2×D414yes—
165C42×C61C2×D412yes—
166C42×C61C2×D412yes—
167#11758198—28no—
168#11758198—24no—
169#11758198—24no—
170#11758198—44no—
171#11758198—82no—
172C4⋊Dic61—22no—
173C4⋊Dic61—21no—
174C4⋊Dic61—21no—
175C4⋊Dic61—21no—
176C4⋊Dic61—44no—
177C4⋊Dic61C2×D412yes—
178#11758196—28no—
179#11758196—42no—
180C4×Dic61—42no—
181C2×#92418—22no—
182C2×#92418—22no—
183C2×#92418—22no—
184#11758190—28no—
185C23×C61C22×D411yescharacteristic
186C22×D61—22no—
187C22×D61—22no—
188C22×D61—22no—
189C22×D61—22no—
190C22×D61—22no—
191C22×D61—22no—
192C22×D61—22no—
193C22×D61—41no—
194C22×D61—41no—
195C22×D61—41no—
196C22×D61—41no—
197C22×D61—41no—
198C22×D61—41no—
199C22×D61C2×SD1614yes—
200Q8×C61—48no—
201C2×C4×C61—21no—
202C2×C4×C61—21no—
203C2×C4×C61—24no—
204C2×C4×C61—42no—
205C2×C4×C61—42no—
206C2×C4×C61—42no—
207C2×C4×C61—42no—
208C2×C4×C61—42no—
209C2×C4×C61#12012yes—
210C2×C4×C61#12012yes—
211C2×C4×C61C2×SD1612yes—
212C2×C4×C61AGL(1,ℤ/8)12yes—
213C2×C4×C61C22×D411yescharacteristic
214C2×C4×C61C22×D411yescharacteristic
215C2×C4×C61C2×#4111yescharacteristic
216C2×C4×C61C2×#4111yescharacteristic
217C2×C4×C61H(5,𝔽2)11yescharacteristic
218C2×C4×C612−1+411yescharacteristic
219C2×Dic61—22no—
220C2×Dic61—22no—
221C2×Dic61—22no—
222C2×Dic61—22no—
223C2×Dic61—22no—
224C2×Dic61—41no—
225C2×Dic61—41no—
226C2×Dic61—41no—
227C2×Dic61—41no—
228C2×Dic61—41no—
229C2×Dic61—41no—
230C2×Dic61—41no—
231D244—22no—
232D244—22no—
233D244—28no—
234D244—416no—
235D244—41no—
236D244—41no—
237C4×D61—416no—
238C4×D61—42no—
239C4×D61—42no—
240C4×D61—42no—
241C4×D61—42no—
242C4×D61—48no—
243Dic122—22no—
244Dic122—22no—
245Dic122—41no—
246Dic122—41no—
247C8×C61—22no—
248C8×C61—41no—
249#92418—42no—
250#24120—611nomaximal, Sylow
251C22×C61C2×#12011yescharacteristic
252C22×C61#54611yescharacteristic
253C22×C61#54711yescharacteristic
254C22×C61C22×SD1611yescharacteristic
255C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
256C22×C61#57711yescharacteristic
257C22×C61#57811yescharacteristic
258C2×D61—24no—
259C2×D61—24no—
260C2×D61—24no—
261C2×D61—42no—
262C2×D61—42no—
263C2×D61—42no—
264C2×D61—42no—
265C2×D61—42no—
266C2×D61—42no—
267C2×D61—42no—
268C2×D61—81no—
269C2×D61—81no—
270C2×D61—81no—
271C2×D61—81no—
272C4×C61—22no—
273C4×C61—44no—
274C4×C61—82no—
275C4×C61#45014yes—
276C4×C61#52511yescharacteristic
277C4×C61#53711yescharacteristic
278C4×C61#54811yescharacteristic
279C4×C61#55011yescharacteristic
280Dic61—42no—
281Dic61—42no—
282Dic61—42no—
283Dic61—81no—
284C2×#547—611no—
285C2×#475—611no—
286C2×C4×SD16—611no—
287#2538—611no—
288#2034—612no—
289#2000—612no—
290#1990D6111yescharacteristic, complemented
291#1954—611no—
292#1872—612no—
293#1853—612no—
294#1802—611no—
295C2×C61#240711yescharacteristic
296C2×C61C2×#45011yescharacteristic
297C2×C61#301311yescharacteristic
298C2×C61#329811yescharacteristic
299C2×C61#333211yescharacteristic
300C2×C61#333511yescharacteristic
301C2×C61#335311yescharacteristic
302D61—44no—
303D61—82no—
304C22×#41—1221no—
305C22×SD16—611no—
306#547—1224no—
307#547—1224no—
308C2×C4⋊Q8C2×D6111yescharacteristic
309C2×#123—1222no—
310C2×C22⋊Q8—612no—
311C2×C22⋊Q8—1221no—
312C2×#121—612no—
313C2×C4×Q8—611no—
314C2×C4×D4—611no—
315C2×C4×D4—611no—
316C22×C4⋊C4—612no—
317#475—1224no—
318C4×SD16—1224no—
319C2×#106—611no—
320C2×#106—612no—
321C2×C4⋊C8—611no—
322C2×#103—611no—
323C2×#103—612no—
324C2×#102—611no—
325C2×#102—612no—
326C2×C22⋊C8—612no—
327C2×C22⋊C8—612no—
328C2×C4×C8C2×D6111yescharacteristic
329#396—1222no—
330#390—1221no—
331#389—1221no—
332#384C2×D6111yescharacteristic
333#380—612no—
334#340—612no—
335#340C2×D6112yes—
336#340C2×D6112yes—
337C61#2412011yescharacteristic, complemented, Sylow
338C2×#41—1228no—
339C2×#41—2444no—
340C2×#41—2444no—
341C22×Q8—21no—
342C22×Q8—611no—
343C22×D4—611no—
344C22×D4—611no—
345C22×D4—1222no—
346C23×C4—612no—
347C23×C4—1222no—
348C23×C4—1221no—
349C2×SD16—1222no—
350C2×SD16—1222no—
351C2×SD16—1222no—
352C22×C8—612no—
353C22×C8C22×D6111yescharacteristic
354C22×C8C22×D6111yescharacteristic
355C4⋊Q8—22no—
356C4⋊Q8D24414yes—
357#123—12216no—
358C22⋊Q8—1228no—
359C22⋊Q8—1224no—
360C22⋊Q8—2442no—
361#121—1224no—
362#121—1224no—
363C4×Q8—1222no—
364C4×Q8—1222no—
365C4×D4—614no—
366C4×D4—1222no—
367C4×D4—1222no—
368C4×D4—1222no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—611no—
373C2×C4⋊C4—611no—
374C2×C4⋊C4—611no—
375C2×C4⋊C4—1224no—
376C2×C4⋊C4—1222no—
377C2×C4⋊C4—1224no—
378C2×C4⋊C4—1222no—
379C2×C4⋊C4—1221no—
380C2×C4⋊C4C22×D6112yes—
381C2×C4⋊C4C22×D6112yes—
382C2×C22⋊C4—612no—
383C2×C22⋊C4—1222no—
384C2×C22⋊C4—1221no—
385C2×C22⋊C4—1222no—
386C2×C22⋊C4—1222no—
387C2×C42—1221no—
388C2×C42C22×D6111yescharacteristic
389#106—1222no—
390#106—1224no—
391C4⋊C8—1222no—
392#103—1222no—
393#103—1224no—
394#102—1222no—
395#102—2442no—
396C22⋊C8—1224no—
397C22⋊C8—2442no—
398C4×C8—22no—
399#95—1222no—
400C24—612no—
401C24—1221no—
402#41—24432no—
403C2×Q8—24no—
404C2×Q8—42no—
405C2×Q8—42no—
406C2×Q8—612no—
407C2×Q8—1221no—
408C2×Q8—1221no—
409C2×Q8—1221no—
410C2×Q8—1221no—
411C2×Q8—1221no—
412C2×D4—612no—
413C2×D4—614no—
414C2×D4—1228no—
415C2×D4—1221no—
416C2×D4—1221no—
417C2×D4—1221no—
418C2×D4—1221no—
419C2×D4—1221no—
420C2×D4—1222no—
421C2×D4—1222no—
422C2×D4—2444no—
423C2×D4—2444no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—612no—
427C22×C4—1228no—
428C22×C4—1221no—
429C22×C4—1221no—
430C22×C4—1222no—
431C22×C4—1222no—
432C22×C4—1222no—
433C22×C4—1222no—
434C22×C4—1222no—
435C22×C4—1222no—
436C22×C4—1224no—
437C22×C4—1221no—
438C22×C4—2444no—
439C22×C4—2444no—
440C22×C4—2442no—
441C22×C4—2442no—
442C22×C4D4×D6111yescharacteristic
443C22×C4D4×D6111yescharacteristic
444C22×C4C23×D6111yescharacteristic
445SD16—1228no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—1222no—
452C2×C8—1222no—
453C2×C8—1222no—
454C2×C8#1175822612yescomplemented
455C4⋊C4—22no—
456C4⋊C4—24no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—42no—
460C4⋊C4—81no—
461C4⋊C4—612no—
462C4⋊C4—1228no—
463C4⋊C4—1221no—
464C4⋊C4—1222no—
465C4⋊C4—1221no—
466C4⋊C4—1221no—
467C4⋊C4—2444no—
468C4⋊C4—2442no—
469C4⋊C4C2×D24414yes—
470C22⋊C4—1228no—
471C22⋊C4—1224no—
472C22⋊C4—1224no—
473C22⋊C4—4882no—
474C22⋊C4—2444no—
475C42—2442no—
476C42C2×D24412yes—
477C42D4×D6112yes—
478C23—614no—
479C23—1222no—
480C23—1222no—
481C23—1222no—
482C23—1222no—
483C23—1222no—
484C23—1222no—
485C23—1222no—
486C23—2441no—
487C23—2441no—
488C23—2441no—
489C23—2441no—
490C23—2441no—
491C23—2441no—
492C23C2×D4×D6111yescharacteristic
493Q8—48no—
494Q8—1222no—
495Q8—1222no—
496Q8—2441no—
497Q8—2441no—
498D4—1222no—
499D4—1222no—
500D4—1228no—
501D4—24416no—
502D4—2441no—
503D4—2441no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—24no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—1222no—
513C2×C4—1222no—
514C2×C4—1222no—
515C2×C4—1222no—
516C2×C4—1222no—
517C2×C4—24416no—
518C2×C4—2441no—
519C2×C4—2441no—
520C2×C4—2441no—
521C2×C4—2441no—
522C2×C4—2441no—
523C2×C4—2441no—
524C2×C4—2441no—
525C2×C4—2442no—
526C2×C4—2442no—
527C2×C4—2442no—
528C2×C4—2448no—
529C2×C4—2442no—
530C2×C4#1451974712yes—
531C2×C4#1451974712yes—
532C2×C4SD16×D6112yes—
533C2×C4#1451980112yes—
534C2×C4C2×D4×D6111yescharacteristic
535C2×C4C22×D24411yescharacteristic
536C2×C4#41×D6111yescharacteristic
537C2×C4C2×#1175822611yescharacteristic
538C2×C4#1451982311yescharacteristic
539C2×C4#1451982511yescharacteristic
540C8—22no—
541C8—41no—
542C8—2442no—
543C22—1224no—
544C22—1224no—
545C22—1224no—
546C22—4881no—
547C22—4881no—
548C22—4881no—
549C22—4881no—
550C22—2442no—
551C22—2442no—
552C22—2442no—
553C22—2442no—
554C22—2442no—
555C22—2442no—
556C22—2442no—
557C22C2×#1451974711yescharacteristic
558C22#1684040511yescharacteristic
559C22#1684041511yescharacteristic
560C22C2×SD16×D6111yescharacteristic
561C22C2×#1451980111yescharacteristic
562C22#1684056811yescharacteristic
563C22#1684057511yescharacteristic
564C4—22no—
565C4—44no—
566C4—82no—
567C4—4881no—
568C4—2442no—
569C4—2442no—
570C4—2442no—
571C4#1683991414yes—
572C4#1684027511yescharacteristic
573C4#1684034211yescharacteristic
574C4#1684042411yescharacteristic
575C4#1684043411yescharacteristic
576C2—4882no—
577C2—2444no—
578C2#2144916711yescharacteristic
579C2C2×#1683991411yescharacteristic
580C2#2145513911yescharacteristic
581C2#2145768311yescharacteristic
582C2#2145814511yescharacteristic
583C2#2145817511yescharacteristic
584C2#2145840211yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1990C2
socleC23×C61C22×D4
prime radicalC2×#16840415C2
hypercenter#1990D61
hypocenterC61#24120
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1990

Complements

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

NH ≅ G/Ntype
2. C2×#16840044C2semidirect
3. C2×C4×#11758190C2semidirect
4. #21450742C2semidirect
5. #21447011C2semidirect
6. #21446825C2semidirect
7. #21446613C2semidirect
8. #21446125C2semidirect
9. #21446013C2semidirect
10. #21445816C2semidirect
11. C61×#1990C2semidirect
132. C2×D244SD16semidirect
151. C2×Dic122#41semidirect
157. C2×C8×C61#41semidirect
290. #1990D61semidirect
337. C61#24120semidirect
454. C2×C8#11758226semidirect

Maximal subgroups

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

subgroupindex
C2×#168404152
C2×#168400442
C2×C4×#117581902
#214507422
#214470112
#214468252
#214466132
#214461252
#214460132
#214458162
C61×#19902
#2412061