Finite Group Databasebeta

← #29716103

Subgroups — #29716103

Order 15616.

1294 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×C4⋊Q8×D61C211yesmaximal, characteristic
2C2×#16840048C211yesmaximal, characteristic, complemented
3C2×#16840044C211yesmaximal, characteristic, complemented
4C4×#14519757C211yesmaximal, characteristic
5#21447151C212yesmaximal, complemented
6#21446959C212yesmaximal, complemented
7#21446771C211yesmaximal, characteristic, complemented
8#21446125C212yesmaximal
9#21446013C212yesmaximal
10#21445796C211yesmaximal, characteristic
11C61×#1988C211yesmaximal, characteristic, complemented
12C22×Q8×D61—21no—
13C2×#14519777C2211yescharacteristic
14C2×#14519776—21no—
15C2×#14519776C2212yes—
16C2×#14519775—21no—
17C4⋊Q8×D61—24no—
18C4⋊Q8×D61—24no—
19C2×#14519757C2211yescharacteristic
20C2×C4×Dic122C2211yescharacteristic
21C2×C4⋊C4×D61—21no—
22C2×C4⋊C4×D61C2212yes—
23C2×C42×D61C2211yescharacteristic
24#16840048—24no—
25#16840044—24no—
26C2×#14519709C2212yes—
27C2×#14519708C2212yes—
28C2×C4⋊#92418C2211yescharacteristic
29C2×#14519702C2212yes—
30C2×#14519700C2212yes—
31C2×#14519689C2212yes—
32#16839667C2211yescharacteristic
33#16839661C2211yescharacteristic
34#16839647—21no—
35#16839636—21no—
36#16839619C2211yescharacteristic
37C4×C4⋊Dic61—21no—
38C4×#11758198—21no—
39#16839524C2212yes—
40#16839524C2212yes—
41#16839516C2212yes—
42C2×C4⋊Q8×C61C2211yescharacteristic
43C2×C4⋊C8×C61C2211yescharacteristic
44C4×C4⋊C4×C61C2211yescharacteristic
45C61×#340C2212yes—
46C61×#340C2212yes—
47C22×Dic122—21no—
48C22×Dic122—22no—
49C22×Dic122C2311yescharacteristic
50C2×Q8×D61—28no—
51C2×Q8×D61—44no—
52C2×Q8×D61—44no—
53C22×C4×D61—22no—
54C22×C4×D61D411yescharacteristic
55C22×C4×D61D411yescharacteristic
56C22×C4×D61C2311yescharacteristic
57C22×#92418C2312yes—
58#14519777—22no—
59#14519777D414yes—
60#14519776—28no—
61#14519776—24no—
62#14519776—24no—
63#14519775—28no—
64#14519757—22no—
65#14519757Q814yescomplemented
66C4×Dic122—22no—
67C4×Dic122—22no—
68C2×C4⋊Dic61—21no—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C4⋊Dic61C2312yes—
73C2×#11758196—22no—
74C2×#11758196—22no—
75C2×#11758196—21no—
76C4⋊C4×D61—28no—
77C4⋊C4×D61—24no—
78C4⋊C4×D61—24no—
79C2×#11758198—21no—
80C2×#11758198—21no—
81C2×C4×Dic61—21no—
82C2×C4×Dic61D411yescharacteristic
83C2×C4×Dic61D411yescharacteristic
84C2×C4×Dic61C2311yescharacteristic
85C42×D61—22no—
86C42×D61—22no—
87#14519709—24no—
88#14519708—24no—
89C4⋊#92418—22no—
90#14519702—24no—
91#14519700—24no—
92#14519689—24no—
93#14519684—21no—
94#14519684—21no—
95#14519684—21no—
96#14519684—21no—
97C22×Q8×C61—21no—
98C22×C8×C61C2312yes—
99C4⋊Q8×C61—22no—
100C4⋊Q8×C61D414yes—
101C2×C4⋊C4×C61—21no—
102C2×C4⋊C4×C61C2311yescharacteristic
103C2×C4⋊C4×C61C2312yes—
104C2×C4⋊C4×C61C2311yescharacteristic
105C2×C42×C61—21no—
106C2×C42×C61C2311yescharacteristic
107C4⋊C8×C61—22no—
108#95×C61—21no—
109C23×D61C2×D411yescharacteristic
110C2×Q8×C61—24no—
111C2×Q8×C61—42no—
112C2×Q8×C61—42no—
113C22×C4×C61—22no—
114C22×C4×C61—21no—
115C22×C4×C61—21no—
116C22×C4×C61C2×D411yescharacteristic
117C22×C4×C61C2×D411yescharacteristic
118C22×C4×C61C2411yescharacteristic
119C22×Dic61—22no—
120C22×Dic61—21no—
121C22×Dic61—21no—
122C22×Dic61C2×D411yescharacteristic
123C22×Dic61C2×D411yescharacteristic
124C22×Dic61C2×D411yescharacteristic
125Q8×D61—432no—
126C2×C4×D61—22no—
127C2×C4×D61—22no—
128C2×C4×D61—22no—
129C2×C4×D61—22no—
130C2×C4×D61—21no—
131C2×C4×D61—21no—
132C2×C4×D61—21no—
133C2×C4×D61—21no—
134C2×C4×D61—22no—
135C2×C4×D61—28no—
136C2×C4×D61—44no—
137C2×C4×D61—44no—
138C2×C4×D61SD1614yes—
139C2×C4×D61C2×Q812yes—
140C2×C4×D61#4112yes—
141C2×Dic122—21no—
142C2×Dic122—21no—
143C2×Dic122—21no—
144C2×Dic122—21no—
145C2×Dic122—28no—
146C2×Dic122—24no—
147C2×Dic122—21no—
148C2×Dic122—44no—
149C2×Dic122—42no—
150C2×Dic122—42no—
151C2×Dic122—44no—
152C2×Dic122C2×Q812yescomplemented
153C2×C8×C61—22no—
154C2×C8×C61—22no—
155C2×C8×C61—22no—
156C4⋊C4×C61—24no—
157C4⋊C4×C61—22no—
158C4⋊C4×C61—21no—
159C4⋊C4×C61—21no—
160C4⋊C4×C61—21no—
161C4⋊C4×C61C2×D414yes—
162C4⋊C4×C61C2×Q812yescomplemented
163C42×C61—42no—
164C42×C61C2×D412yes—
165C42×C61C2×D412yes—
166#11758198—24no—
167#11758198—42no—
168C4⋊Dic61—24no—
169C4⋊Dic61—22no—
170C4⋊Dic61—24no—
171C4⋊Dic61—21no—
172C4⋊Dic61—21no—
173C4⋊Dic61—42no—
174C4⋊Dic61C2×D414yes—
175#11758196—28no—
176#11758196—28no—
177#11758196—42no—
178C4×Dic61—22no—
179C4×Dic61—22no—
180C4×Dic61—21no—
181C4×Dic61—21no—
182C4×Dic61—42no—
183C2×#92418—22no—
184C2×#92418—22no—
185C2×#92418—22no—
186C23×C61C22×D411yescharacteristic
187C22×D61—21no—
188C22×D61—21no—
189C22×D61—21no—
190C22×D61—21no—
191C22×D61C22⋊Q812yes—
192C22×D61C2×SD1612yes—
193C22×D61#13712yes—
194Q8×C61—48no—
195C2×C4×C61—21no—
196C2×C4×C61—21no—
197C2×C4×C61—24no—
198C2×C4×C61—22no—
199C2×C4×C61—42no—
200C2×C4×C61—41no—
201C2×C4×C61—42no—
202C2×C4×C61—41no—
203C2×C4×C61—41no—
204C2×C4×C61—41no—
205C2×C4×C61—41no—
206C2×C4×C61#12012yes—
207C2×C4×C61#12012yes—
208C2×C4×C61C2×SD1612yes—
209C2×C4×C61AGL(1,ℤ/8)12yes—
210C2×C4×C61C22×D411yescharacteristic
211C2×C4×C61C22×D411yescharacteristic
212C2×C4×C61C22×Q811yescharacteristic
213C2×C4×C61C2×#4111yescharacteristic
214C2×C4×C612−1+411yescharacteristic
215C2×C4×C612−1+411yescharacteristic
216C2×Dic61—21no—
217C2×Dic61—21no—
218C2×Dic61—21no—
219C2×Dic61—21no—
220C2×Dic61—21no—
221C2×Dic61—21no—
222C2×Dic61—24no—
223C2×Dic61—22no—
224C2×Dic61—41no—
225C2×Dic61—41no—
226C2×Dic61—42no—
227C2×Dic61—42no—
228C2×Dic61—41no—
229C2×Dic61—41no—
230C2×Dic61—41no—
231C2×Dic61C22⋊Q812yes—
232C2×Dic61C2×SD1612yes—
233C2×Dic61#13712yes—
234C4×D61—28no—
235C4×D61—22no—
236C4×D61—22no—
237C4×D61—22no—
238C4×D61—22no—
239C4×D61—28no—
240C4×D61—416no—
241Dic122—22no—
242Dic122—22no—
243Dic122—416no—
244Dic122—41no—
245Dic122—41no—
246Dic122—48no—
247C8×C61—42no—
248#92418—42no—
249#24301—611nomaximal, Sylow
250C22×C61C2×#12011yescharacteristic
251C22×C61C2×C22⋊Q811yescharacteristic
252C22×C61#53611yescharacteristic
253C22×C61C22×SD1611yescharacteristic
254C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
255C22×C61C2×#13711yescharacteristic
256C22×C61C2×#13711yescharacteristic
257C2×D61—22no—
258C2×D61—22no—
259C2×D61—22no—
260C2×D61—22no—
261C2×D61—22no—
262C2×D61—22no—
263C2×D61#47514yescomplemented
264C4×C61—22no—
265C4×C61—42no—
266C4×C61—44no—
267C4×C61—81no—
268C4×C61#45014yes—
269C4×C61#54811yescharacteristic
270C4×C61#54811yescharacteristic
271C4×C61D4×Q811yescharacteristic
272C4×C61D4×Q811yescharacteristic
273Dic61—22no—
274Dic61—22no—
275Dic61—22no—
276Dic61—42no—
277Dic61—44no—
278Dic61—81no—
279C22×C4⋊Q8—611no—
280C2×#475—611no—
281C2×#475—611no—
282C4×C22⋊Q8—611no—
283#2058—612no—
284#2024—612no—
285#1988—611no—
286#1988D6111yescharacteristic, complemented
287#1872—612no—
288#1853—612no—
289#1797—611no—
290C2×C61#240911yescharacteristic
291C2×C61C2×#45011yescharacteristic
292C2×C61#300411yescharacteristic
293C2×C61C2×#47511yescharacteristic
294C2×C61#307911yescharacteristic
295C2×C61#322911yescharacteristic
296C2×C61#323911yescharacteristic
297D61—24no—
298C23×Q8—1221no—
299C2×C4⋊Q8—612no—
300C2×C4⋊Q8—611no—
301C2×C4⋊Q8—1221no—
302C2×C4⋊Q8—1221no—
303C2×C4⋊Q8—1224no—
304C2×C4⋊Q8—1224no—
305C2×C4⋊Q8C2×D6111yescharacteristic
306C2×C22⋊Q8—611no—
307C2×C4×Q8—611no—
308C22×C4⋊C4—612no—
309C22×C4⋊C4—1221no—
310C22×C42—611no—
311#475—1224no—
312#475—1224no—
313C2×#106—612no—
314C2×#106—612no—
315C2×C4⋊C8—611no—
316C2×C4⋊C8C2×D6111yescharacteristic
317C2×#103—612no—
318C2×#103—612no—
319C2×C22⋊C8—612no—
320#391—611no—
321#389—611no—
322#387—1221no—
323#385—1221no—
324#380—611no—
325C4×C4⋊C4—1221no—
326C4×C4⋊C4C2×D6111yescharacteristic
327C4×C22⋊C4—1221no—
328#340—612no—
329#340—612no—
330#340C2×D6112yes—
331#340C2×D6112yes—
332#336—612no—
333C61#2430111yescharacteristic, complemented, Sylow
334C22×Q8—21no—
335C22×Q8—611no—
336C22×Q8—1222no—
337C22×Q8—1228no—
338C22×Q8—1221no—
339C22×Q8—2444no—
340C22×Q8—2444no—
341C23×C4—611no—
342C23×C4—611no—
343C23×C4—611no—
344C23×C4—1222no—
345C22×C8—612no—
346C22×C8C22×D6112yes—
347C4⋊Q8—22no—
348C4⋊Q8—614no—
349C4⋊Q8—1224no—
350C4⋊Q8—1228no—
351C4⋊Q8—1228no—
352C4⋊Q8—1224no—
353C4⋊Q8—1222no—
354C4⋊Q8D24414yes—
355C22⋊Q8—614no—
356C22⋊Q8—1222no—
357C4×Q8—1222no—
358C4×Q8—1222no—
359C2×C4⋊C4—21no—
360C2×C4⋊C4—612no—
361C2×C4⋊C4—611no—
362C2×C4⋊C4—1224no—
363C2×C4⋊C4—1228no—
364C2×C4⋊C4—1222no—
365C2×C4⋊C4—1222no—
366C2×C4⋊C4—1221no—
367C2×C4⋊C4—1221no—
368C2×C4⋊C4—1224no—
369C2×C4⋊C4—1221no—
370C2×C4⋊C4—1221no—
371C2×C4⋊C4C22×D6111yescharacteristic
372C2×C4⋊C4C22×D6112yes—
373C2×C4⋊C4C22×D6111yescharacteristic
374C2×C22⋊C4—1221no—
375C2×C22⋊C4—1221no—
376C2×C42—21no—
377C2×C42—611no—
378C2×C42—611no—
379C2×C42—611no—
380C2×C42—1222no—
381C2×C42—1222no—
382C2×C42—1221no—
383C2×C42C22×D6111yescharacteristic
384#106—1224no—
385#106—1224no—
386C4⋊C8—22no—
387C4⋊C8—1222no—
388#103—1224no—
389#103—1224no—
390C22⋊C8—1224no—
391#95—21no—
392#95—1221no—
393#95—1221no—
394#95—1221no—
395#95—1221no—
396C24—611no—
397C2×Q8—24no—
398C2×Q8—42no—
399C2×Q8—42no—
400C2×Q8—612no—
401C2×Q8—1228no—
402C2×Q8—1221no—
403C2×Q8—1224no—
404C2×Q8—1221no—
405C2×Q8—1221no—
406C2×Q8—1221no—
407C2×Q8—1221no—
408C2×Q8—2444no—
409C2×Q8—2444no—
410C2×Q8—24432no—
411C2×Q8—2442no—
412C2×Q8—2442no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—611no—
417C22×C4—611no—
418C22×C4—611no—
419C22×C4—612no—
420C22×C4—612no—
421C22×C4—614no—
422C22×C4—1222no—
423C22×C4—1222no—
424C22×C4—1228no—
425C22×C4—1221no—
426C22×C4—1221no—
427C22×C4—1221no—
428C22×C4—1221no—
429C22×C4—1222no—
430C22×C4—1222no—
431C22×C4—1222no—
432C22×C4—1221no—
433C22×C4—1222no—
434C22×C4—1221no—
435C22×C4—2444no—
436C22×C4—2444no—
437C22×C4D4×D6111yescharacteristic
438C22×C4D4×D6111yescharacteristic
439C22×C4C23×D6111yescharacteristic
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—1222no—
444C2×C8—1222no—
445C2×C8—1222no—
446C4⋊C4—22no—
447C4⋊C4—21no—
448C4⋊C4—24no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—614no—
452C4⋊C4—1224no—
453C4⋊C4—1228no—
454C4⋊C4—1228no—
455C4⋊C4—1222no—
456C4⋊C4—1224no—
457C4⋊C4—1221no—
458C4⋊C4—1221no—
459C4⋊C4—2442no—
460C4⋊C4—2442no—
461C4⋊C4C2×D24414yes—
462C4⋊C4Q8×D6112yescomplemented
463C22⋊C4—1224no—
464C22⋊C4—2442no—
465C42—42no—
466C42—1221no—
467C42—1221no—
468C42—1222no—
469C42—1222no—
470C42—2442no—
471C42C2×D24412yes—
472C42D4×D6112yes—
473C23—612no—
474C23—612no—
475C23—612no—
476C23—1221no—
477C23—1221no—
478C23—1221no—
479C23—1221no—
480C23C2×D4×D6111yescharacteristic
481Q8—48no—
482Q8—1222no—
483Q8—1222no—
484Q8—24416no—
485Q8—2441no—
486Q8—2441no—
487Q8—2448no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—24no—
491C2×C4—22no—
492C2×C4—42no—
493C2×C4—41no—
494C2×C4—42no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—612no—
500C2×C4—612no—
501C2×C4—612no—
502C2×C4—1228no—
503C2×C4—1221no—
504C2×C4—1221no—
505C2×C4—1221no—
506C2×C4—1221no—
507C2×C4—1221no—
508C2×C4—1221no—
509C2×C4—1222no—
510C2×C4—1222no—
511C2×C4—1228no—
512C2×C4—1222no—
513C2×C4—1222no—
514C2×C4—1224no—
515C2×C4—1222no—
516C2×C4—24416no—
517C2×C4—2441no—
518C2×C4—2441no—
519C2×C4—2441no—
520C2×C4—2441no—
521C2×C4—2442no—
522C2×C4—2442no—
523C2×C4—2441no—
524C2×C4#1451974712yes—
525C2×C4#1451974712yes—
526C2×C4SD16×D6112yes—
527C2×C4#1451980112yes—
528C2×C4C2×D4×D6111yescharacteristic
529C2×C4C22×D24411yescharacteristic
530C2×C4C2×Q8×D6111yescharacteristic
531C2×C4#41×D6111yescharacteristic
532C2×C4#1451982411yescharacteristic
533C2×C4#1451982511yescharacteristic
534C8—42no—
535C8—2442no—
536C22—614no—
537C22—1222no—
538C22—1222no—
539C22—1222no—
540C22—1222no—
541C22—1222no—
542C22—1222no—
543C22C2×#1451974711yescharacteristic
544C22C22⋊Q8×D6111yescharacteristic
545C22#1684033511yescharacteristic
546C22C2×SD16×D6111yescharacteristic
547C22C2×#1451980111yescharacteristic
548C22#137×D6111yescharacteristic
549C22#137×D6111yescharacteristic
550C4—22no—
551C4—42no—
552C4—44no—
553C4—81no—
554C4—1222no—
555C4—1222no—
556C4—1222no—
557C4—4881no—
558C4—2444no—
559C4—2442no—
560C4#1683991414yes—
561C4#1684042011yescharacteristic
562C4#1684042411yescharacteristic
563C4#1684042711yescharacteristic
564C4Q8×D24411yescharacteristic
565C2—1224no—
566C2#2144918211yescharacteristic
567C2C2×#1683991411yescharacteristic
568C2#2145505511yescharacteristic
569C2#475×D6111yescharacteristic
570C2#2145562711yescharacteristic
571C2#2145696211yescharacteristic
572C2#2145704811yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1988C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
hypercenter#1988D61
hypocenterC61#24301
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1988

Complements

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

NH ≅ G/Ntype
2. C2×#16840048C2semidirect
3. C2×#16840044C2semidirect
5. #21447151C2semidirect
6. #21446959C2semidirect
7. #21446771C2semidirect
11. C61×#1988C2semidirect
65. #14519757Q8semidirect
152. C2×Dic122C2×Q8semidirect
162. C4⋊C4×C61C2×Q8semidirect
263. C2×D61#475semidirect
286. #1988D61semidirect
333. C61#24301semidirect
462. C4⋊C4Q8×D61semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊Q8×D612
C2×#168400482
C2×#168400442
C4×#145197572
#214471512
#214469592
#214467712
#214461252
#214460132
#214457962
C61×#19882
#2430161