Finite Group Databasebeta

← #30600405

Subgroups — #30600405

Order 15616.

1270 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×(C2×#6923)⋊C4C211yesmaximal, characteristic
2#21461642C211yesmaximal, characteristic
3C2×#16840616C212yesmaximal, complemented
4#21460692C212yesmaximal, complemented
5#21454103C211yesmaximal, characteristic
6C24×#6923—21no—
7C2×#14519834—21no—
8C2×#14519834C2212yes—
9C2×C22⋊#92420—21no—
10C2×(C2×#6923)⋊C4—24no—
11C2×(C2×#6923)⋊C4—21no—
12C2×(C2×#6923)⋊C4C2212yes—
13#16840616—24no—
14C23×C4×D61C2211yescharacteristic
15C2×#14519753C411yescharacteristic, complemented
16C2×#14519753C2211yescharacteristic
17#16839634—21no—
18#16839633—21no—
19#16839633—21no—
20#16839622C411yescharacteristic, complemented
21#16839619C411yescharacteristic, complemented
22#16839619C2211yescharacteristic
23#16839617—21no—
24C2×C22⋊Q8×C61C411yescharacteristic, complemented
25C23×#6923—24no—
26C23×#6923—24no—
27C23×#6923—24no—
28C23×#6923—22no—
29C23×#6923—42no—
30C23×#6923—42no—
31C23×#6923—42no—
32C23×#6923—42no—
33C22×#92420—22no—
34#14519834—22no—
35#14519834—22no—
36#14519834—42no—
37C22⋊#92420—42no—
38(C2×#6923)⋊C4—24no—
39(C2×#6923)⋊C4—22no—
40(C2×#6923)⋊C4D414yes—
41C24×D61D411yescharacteristic
42C23×Dic61D411yescharacteristic
43C22×C4×D61—22no—
44C22×C4×D61—22no—
45C22×C4×D61—42no—
46C22×C4×D61—42no—
47C22×C4×D61C2×C411yescharacteristic
48C22×C4×D61D411yescharacteristic
49C22×C4×D61D411yescharacteristic
50C22×C4×D61C2311yescharacteristic
51#14519753—22no—
52#14519753—22no—
53#14519753—42no—
54C2×C4⋊Dic61C2×C411yescharacteristic, complemented
55C2×C4⋊Dic61C2×C411yescharacteristic, complemented
56C2×#11758196—21no—
57C2×#11758196—21no—
58C2×C22⋊Dic61—21no—
59C2×#11758198—21no—
60C2×#11758198—21no—
61C2×#11758198—21no—
62C2×#11758198—21no—
63#14519684—21no—
64#14519684—21no—
65#14519684—21no—
66C22×Q8×C61C2×C411yescharacteristic, complemented
67C23×C4×C61C2×C411yescharacteristic
68C22⋊Q8×C61—42no—
69C2×C4⋊C4×C61—21no—
70C2×C4⋊C4×C61C2×C411yescharacteristic, complemented
71C2×C22⋊C4×C61—21no—
72C24×C61C22⋊C411yescharacteristic
73C23×D61—22no—
74C23×D61—22no—
75C23×D61—22no—
76C23×D61—22no—
77C23×D61—21no—
78C23×D61—21no—
79C23×D61—21no—
80C23×D61—21no—
81C23×D61C22⋊C411yescharacteristic
82C23×D61SD1612yes—
83C23×D61Q1612yes—
84C23×D61C2×D411yescharacteristic
85C22×#6923—216no—
86C22×#6923—24no—
87C22×#6923—24no—
88C22×#6923—24no—
89C22×#6923—48no—
90C22×#6923—48no—
91C22×#6923—48no—
92C22×#6923—48no—
93C22×#6923—42no—
94C22×#6923—42no—
95C22×#6923—48no—
96C22×#6923—48no—
97C22×#6923—42no—
98C22×#6923—42no—
99C2×Q8×C61—22no—
100C2×Q8×C61—41no—
101C2×Q8×C61—41no—
102C22×C4×C61—22no—
103C22×C4×C61—22no—
104C22×C4×C61—21no—
105C22×C4×C61—21no—
106C22×C4×C61—41no—
107C22×C4×C61C22⋊C411yescharacteristic
108C22×C4×C61C22×C411yescharacteristic
109C22×Dic61—22no—
110C22×Dic61—22no—
111C22×Dic61—21no—
112C22×Dic61—21no—
113C22×Dic61—41no—
114C22×Dic61C22⋊C411yescharacteristic
115C22×Dic61C2×D411yescharacteristic
116C2×C4×D61—22no—
117C2×C4×D61—22no—
118C2×C4×D61—21no—
119C2×C4×D61—21no—
120C2×C4×D61—22no—
121C2×C4×D61—22no—
122C2×C4×D61—41no—
123C2×C4×D61—41no—
124C2×C4×D61—41no—
125C2×C4×D61—41no—
126C2×C4×D61—41no—
127C2×C4×D61—41no—
128C2×C4×D61—48no—
129C2×C4×D61—48no—
130C2×C4×D61C2×D412yes—
131C2×C4×D61#4112yes—
132C2×#92420—42no—
133C2×#92420—42no—
134C2×#92420—42no—
135C4⋊C4×C61—42no—
136C4⋊C4×C61—41no—
137C22⋊C4×C61—42no—
138C22⋊Dic61—42no—
139#11758198—42no—
140#11758198—42no—
141#11758198—41no—
142#11758198—41no—
143#11758198—41no—
144#11758198—41no—
145C4⋊Dic61—22no—
146C4⋊Dic61—41no—
147#11758196—42no—
148#11758196—42no—
149C23×C61—21no—
150C23×C61—21no—
151C23×C61—21no—
152C23×C61—21no—
153C23×C61#9912yes—
154C23×C61#10312yes—
155C23×C61#10412yes—
156C23×C61C2×C22⋊C411yescharacteristic
157C22×D61—24no—
158C22×D61—24no—
159C22×D61—24no—
160C22×D61—24no—
161C22×D61—24no—
162C22×D61—24no—
163C22×D61—24no—
164C22×D61—24no—
165C22×D61—24no—
166C22×D61—24no—
167C22×D61—21no—
168C22×D61—21no—
169C22×D61—21no—
170C22×D61—21no—
171C22×D61—21no—
172C22×D61—21no—
173C22×D61—21no—
174C22×D61—21no—
175C22×D61—24no—
176C22×D61—24no—
177C22×D61—22no—
178C22×D61—22no—
179C22×D61—22no—
180C22×D61—22no—
181C22×D61#9912yes—
182C22×D61#10312yes—
183C22×D61#10412yes—
184C22×D61#12011yescharacteristic
185C22×D61#12311yescharacteristic
186C22×D61C2×SD1611yescharacteristic
187C22×D61C2×Q1611yescharacteristic
188C22×D61AGL(1,ℤ/8)11yescharacteristic
189C22×D61#13711yescharacteristic
190C2×#6923—28no—
191C2×#6923—432no—
192C2×#6923—44no—
193C2×#6923—44no—
194C2×#6923—44no—
195C2×#6923—44no—
196C2×#6923—44no—
197C2×#6923—44no—
198Q8×C61—82no—
199C2×C4×C61—21no—
200C2×C4×C61—21no—
201C2×C4×C61—21no—
202C2×C4×C61—21no—
203C2×C4×C61—21no—
204C2×C4×C61—44no—
205C2×C4×C61—41no—
206C2×C4×C61—41no—
207C2×C4×C61—41no—
208C2×C4×C61—41no—
209C2×C4×C61—41no—
210C2×C4×C61—41no—
211C2×C4×C61C4×D412yes—
212C2×Dic61—21no—
213C2×Dic61—21no—
214C2×Dic61—21no—
215C2×Dic61—21no—
216C2×Dic61—21no—
217C2×Dic61—44no—
218C2×Dic61—41no—
219C2×Dic61—41no—
220C2×Dic61—41no—
221C2×Dic61—41no—
222C2×Dic61—41no—
223C2×Dic61—41no—
224C2×Dic61#12112yes—
225C4×D61—42no—
226C4×D61—42no—
227C4×D61—44no—
228C4×D61—44no—
229C4×D61—42no—
230C4×D61—42no—
231#92420—82no—
232#9143—611nomaximal, Sylow
233C22×C61—22no—
234C22×C61—22no—
235C22×C61—22no—
236C22×C61—22no—
237C22×C61—22no—
238C22×C61—22no—
239C22×C61#32814yescomplemented
240C22×C61#38611yescharacteristic
241C22×C61C2×#9911yescharacteristic
242C22×C61C2×#9911yescharacteristic
243C22×C61C2×#10311yescharacteristic
244C22×C61#41711yescharacteristic
245C22×C61C2×#10411yescharacteristic
246C22×C61C2×#10411yescharacteristic
247C2×D61—28no—
248C2×D61—28no—
249C2×D61—28no—
250C2×D61—28no—
251C2×D61—22no—
252C2×D61—21no—
253C2×D61—22no—
254C2×D61—21no—
255C2×D61—21no—
256C2×D61—22no—
257C2×D61—21no—
258C2×D61—21no—
259C2×D61—21no—
260C2×D61—22no—
261C2×D61—21no—
262C2×D61—21no—
263C2×D61—21no—
264C2×D61—21no—
265C2×D61—22no—
266C2×D61—22no—
267C2×D61—21no—
268C2×D61—21no—
269C2×D61#32814yescomplemented
270C2×D61#45011yescharacteristic
271C2×D61#45111yescharacteristic
272C2×D61#48311yescharacteristic
273C2×D61#48411yescharacteristic
274#6923—48no—
275C4×C61—22no—
276C4×C61—41no—
277C4×C61—81no—
278C4×C61—81no—
279Dic61—22no—
280Dic61—41no—
281Dic61—81no—
282Dic61—81no—
283#2848—611no—
284C22×#95—611no—
285#1779—611no—
286C2×#328—612no—
287#1294—612no—
288C2×C61—24no—
289C2×C61C2×#32811yescharacteristic
290C2×C61C2×#32811yescharacteristic
291C2×C61#150411yescharacteristic
292C2×C61#150411yescharacteristic
293C2×C61#187211yescharacteristic
294C2×C61#189611yescharacteristic
295C2×C61#189811yescharacteristic
296D61—21no—
297D61—24no—
298D61—21no—
299D61—21no—
300D61—21no—
301C24×C4—611no—
302C24×C4—1221no—
303C2×C22⋊Q8—611no—
304C2×C22⋊Q8—611no—
305C2×C22⋊Q8#692311yescharacteristic, complemented
306C2×C22⋊C8—612no—
307C2×C22⋊C8—1221no—
308C2×C22⋊C8—1221no—
309#385—1221no—
310#385—1221no—
311#385—1221no—
312#380—611no—
313#380—611no—
314#380—611no—
315#379—1221no—
316C2×#95—612no—
317C2×#95—1224no—
318C2×#95—1221no—
319#328—1224no—
320C61#914311yescharacteristic, complemented, Sylow
321C25—611no—
322C22×Q8C2×#692311yescharacteristic, complemented
323C23×C4—611no—
324C23×C4—611no—
325C23×C4—611no—
326C23×C4—611no—
327C23×C4—611no—
328C23×C4—1222no—
329C23×C4—1222no—
330C23×C4—1222no—
331C23×C4—1224no—
332C23×C4—1224no—
333C23×C4—1224no—
334C23×C4—2442no—
335C23×C4—2442no—
336C23×C4—2442no—
337C23×C4—2442no—
338C23×C4—2442no—
339C23×C4—2442no—
340C23×C4C2×#692311yescharacteristic
341C22×C8—1222no—
342C22⋊Q8—42no—
343C22⋊Q8—1222no—
344C22⋊Q8—1222no—
345C22⋊Q8—2442no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—611no—
348C2×C4⋊C4—611no—
349C2×C4⋊C4—1221no—
350C2×C4⋊C4—1221no—
351C2×C4⋊C4C2×#692311yescharacteristic, complemented
352C2×C22⋊C4—21no—
353C2×C22⋊C4—1221no—
354C2×C22⋊C4—1221no—
355C2×C22⋊C4—1221no—
356C2×C22⋊C4—1221no—
357C2×C22⋊C4—1221no—
358C22⋊C8—1222no—
359C22⋊C8—1222no—
360C22⋊C8—2442no—
361C22⋊C8—2442no—
362#95—614no—
363#95—1221no—
364#95—1221no—
365#95—1222no—
366#95—1224no—
367#95—1221no—
368C24—611no—
369C24—611no—
370C24—612no—
371C24—612no—
372C24—1221no—
373C24—1221no—
374C24—1222no—
375C24—1222no—
376C24—1221no—
377C24—1221no—
378C24—1222no—
379C24—1222no—
380C24C22⋊#692311yescharacteristic
381C2×Q8—22no—
382C2×Q8—41no—
383C2×Q8—41no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—41no—
389C22×C4—611no—
390C22×C4—612no—
391C22×C4—612no—
392C22×C4—611no—
393C22×C4—1222no—
394C22×C4—1221no—
395C22×C4—1222no—
396C22×C4—1222no—
397C22×C4—1221no—
398C22×C4—1222no—
399C22×C4—1224no—
400C22×C4—12216no—
401C22×C4—1224no—
402C22×C4—1224no—
403C22×C4—1222no—
404C22×C4—1222no—
405C22×C4—1221no—
406C22×C4—1221no—
407C22×C4—2441no—
408C22×C4—2441no—
409C22×C4—2441no—
410C22×C4—2441no—
411C22×C4—2441no—
412C22×C4—2441no—
413C22×C4—2448no—
414C22×C4—2448no—
415C22×C4—2441no—
416C22×C4—2442no—
417C22×C4—2442no—
418C22×C4—2448no—
419C22×C4—2448no—
420C22×C4—2448no—
421C22×C4—2448no—
422C22×C4—2448no—
423C22×C4—2442no—
424C22×C4—2442no—
425C22×C4—2448no—
426C22×C4C22⋊#692311yescharacteristic
427C22×C4C22×#692311yescharacteristic
428C2×C8—2442no—
429C2×C8—2442no—
430C2×C8—2442no—
431C4⋊C4—41no—
432C4⋊C4—42no—
433C4⋊C4—1222no—
434C4⋊C4—2441no—
435C4⋊C4—2442no—
436C4⋊C4—2442no—
437C22⋊C4—42no—
438C22⋊C4—2441no—
439C22⋊C4—2441no—
440C22⋊C4—2442no—
441C22⋊C4—2441no—
442C22⋊C4—2441no—
443C22⋊C4—2442no—
444C22⋊C4—2442no—
445C23—21no—
446C23—21no—
447C23—21no—
448C23—21no—
449C23—611no—
450C23—611no—
451C23—612no—
452C23—612no—
453C23—611no—
454C23—611no—
455C23—611no—
456C23—611no—
457C23—612no—
458C23—1222no—
459C23—1224no—
460C23—1222no—
461C23—1222no—
462C23—1222no—
463C23—1224no—
464C23—1224no—
465C23—1224no—
466C23—1224no—
467C23—1224no—
468C23—1224no—
469C23—1224no—
470C23—1224no—
471C23—1224no—
472C23—1221no—
473C23—1221no—
474C23—1221no—
475C23—1221no—
476C23—1221no—
477C23—1221no—
478C23—1221no—
479C23—1221no—
480C23—1224no—
481C23—1224no—
482C23#1451983512yes—
483C23#1451984012yes—
484C23#1451984212yes—
485C23C2×C22⋊#692311yescharacteristic
486Q8—82no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—44no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—612no—
500C2×C4—1221no—
501C2×C4—1221no—
502C2×C4—1228no—
503C2×C4—1221no—
504C2×C4—1221no—
505C2×C4—1221no—
506C2×C4—2442no—
507C2×C4—2442no—
508C2×C4—2444no—
509C2×C4—2444no—
510C2×C4—2442no—
511C2×C4—2442no—
512C2×C4—2444no—
513C2×C4—2441no—
514C2×C4—2441no—
515C2×C4—2441no—
516C2×C4—2441no—
517C2×C4—2441no—
518C2×C4—2441no—
519C2×C4—2444no—
520C2×C4—24432no—
521C2×C4—2444no—
522C2×C4—2444no—
523C2×C4—2444no—
524C2×C4—2444no—
525C2×C4—2444no—
526C2×C4D4×#692312yes—
527C8—4882no—
528C22—22no—
529C22—22no—
530C22—22no—
531C22—22no—
532C22—22no—
533C22—22no—
534C22—611no—
535C22—611no—
536C22—611no—
537C22—611no—
538C22—614no—
539C22—1228no—
540C22—1228no—
541C22—1228no—
542C22—1228no—
543C22—1222no—
544C22—1221no—
545C22—1222no—
546C22—1221no—
547C22—1221no—
548C22—1221no—
549C22—1222no—
550C22—1221no—
551C22—1221no—
552C22—1221no—
553C22—1222no—
554C22—1221no—
555C22—1221no—
556C22—1222no—
557C22—1221no—
558C22—1221no—
559C22—1222no—
560C22—1221no—
561C22#1684061614yescomplemented
562C22#1684071011yescharacteristic
563C22C2×#1451983511yescharacteristic
564C22D61⋊#9911yescharacteristic
565C22C2×#1451984011yescharacteristic
566C22#1684075311yescharacteristic
567C22D61⋊#10411yescharacteristic
568C22C2×#1451984211yescharacteristic
569C4—22no—
570C4—41no—
571C4—81no—
572C4—81no—
573C4—1222no—
574C4—4881no—
575C4—4881no—
576C4—2448no—
577C4—2441no—
578C2—24no—
579C2—1221no—
580C2—1221no—
581C2—1221no—
582C2—1221no—
583C2—1224no—
584C2D61⋊#32811yescharacteristic
585C2C2×#1684061611yescharacteristic
586C2#2146120511yescharacteristic
587C2#2146120611yescharacteristic
588C2#2146180411yescharacteristic
589C2#2146183611yescharacteristic
590C2#2146184011yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC2×C22⋊Q8×C61C4
socleC23×C61C2×C22⋊C4
prime radicalC24×D61D4
cosocleC22×C4×D61C23
hypercenterC2×C22⋊Q8#6923
hypocenterC61#9143
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. C2×#16840616C2semidirect
4. #21460692C2semidirect
15. C2×#14519753C4semidirect
20. #16839622C4semidirect
21. #16839619C4semidirect
24. C2×C22⋊Q8×C61C4semidirect
54. C2×C4⋊Dic61C2×C4semidirect
55. C2×C4⋊Dic61C2×C4semidirect
66. C22×Q8×C61C2×C4semidirect
70. C2×C4⋊C4×C61C2×C4semidirect
239. C22×C61#328semidirect
269. C2×D61#328semidirect
305. C2×C22⋊Q8#6923semidirect
320. C61#9143semidirect
322. C22×Q8C2×#6923semidirect
351. C2×C4⋊C4C2×#6923semidirect
561. C22#16840616semidirect

Maximal subgroups

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

subgroupindex
C22×(C2×#6923)⋊C42
#214616422
C2×#168406162
#214606922
#214541032
#914361