Finite Group Databasebeta

← #29689463

Subgroups — #29689463

Order 15616.

1130 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840387C211yesmaximal, characteristic
2#21448416C211yesmaximal, characteristic
3#21448414C211yesmaximal, characteristic
4#21446805C211yesmaximal, characteristic, complemented
5#21446804C211yesmaximal, characteristic, complemented
6#21446803C211yesmaximal, characteristic, complemented
7#21446802C214yesmaximal, complemented
8#21445892C214yesmaximal
9C61×#1996C211yesmaximal, characteristic, complemented
10#16840387—22no—
11#16840387—22no—
12#16840387—22no—
13#16840387—22no—
14C2×#14519757C412yescomplemented
15C2×#14519756C2211yescharacteristic
16C2×#14519755C412yescomplemented
17C2×#14519755C412yescomplemented
18C2×#14519754C2212yes—
19C2×#14519753C412yescomplemented
20C22×#11758198C2211yescharacteristic
21C2×C22⋊C4×D61C2212yes—
22#16839654—22no—
23#16839653—22no—
24#16839652—22no—
25#16839651—22no—
26#16839622C2211yescharacteristic
27#16839620C2212yes—
28#16839617C2212yes—
29#16839617C2212yes—
30C22⋊C4×Dic61C2212yes—
31C4×C22⋊Dic61C2211yescharacteristic
32#16839529C2214yes—
33#16839528C2214yes—
34#16839528C2214yes—
35C2×C22⋊Q8×C61C2211yescharacteristic
36C61×#380C2211yescharacteristic
37C4×C22⋊C4×C61C2211yescharacteristic
38C61×#342C2214yes—
39C24×D61C2311yescharacteristic
40C22×Dic122C2×C412yes—
41C22×Dic122C2×C411yescharacteristic
42C23×Dic61C2311yescharacteristic
43C22×C4×D61C2×C412yes—
44C22×C4×D61C2×C412yes—
45#14519757—24no—
46#14519757—24no—
47#14519756—21no—
48#14519756—21no—
49#14519756—21no—
50#14519756—21no—
51#14519755—24no—
52#14519755—24no—
53#14519755—24no—
54#14519755—24no—
55#14519754—24no—
56#14519754—24no—
57#14519753—24no—
58#14519753—24no—
59C2×C4⋊Dic61—21no—
60C2×C4⋊Dic61C2×C412yes—
61C2×C4⋊Dic61C2×C411yescharacteristic
62C2×#11758196—22no—
63C2×#11758196C2×C412yes—
64C2×#11758196C2×C412yes—
65C2×#11758196C2×C412yes—
66C2×C22⋊Dic61C2312yes—
67C2×C22⋊Dic61C2312yes—
68C2×C22⋊Dic61C2312yes—
69C2×#11758198—22no—
70C2×#11758198—22no—
71C2×#11758198—22no—
72C2×#11758198—22no—
73C2×#11758198C2×C412yes—
74C2×#11758198C2×C412yes—
75C2×#11758198C2×C412yes—
76C2×#11758198C2×C412yes—
77C2×#11758198D412yes—
78C2×#11758198D412yes—
79C2×#11758198D412yes—
80C2×#11758198D412yes—
81C22⋊C4×D61—24no—
82C22⋊C4×D61—24no—
83C2×C4×Dic61—21no—
84C2×C4×Dic61—22no—
85#14519684—21no—
86#14519684—21no—
87#14519684—22no—
88#14519684—22no—
89C22×Q8×C61C2×C411yescharacteristic
90C23×C4×C61C2311yescharacteristic
91C22⋊Q8×C61—21no—
92C22⋊Q8×C61—21no—
93C22⋊Q8×C61—21no—
94C22⋊Q8×C61—21no—
95C2×C4⋊C4×C61—21no—
96C2×C4⋊C4×C61C2×C411yescharacteristic
97C2×C4⋊C4×C61C2×C412yes—
98C2×C22⋊C4×C61C2312yes—
99C2×C22⋊C4×C61C2312yes—
100C2×C22⋊C4×C61C2312yes—
101C2×C42×C61—21no—
102#95×C61—21no—
103#95×C61—21no—
104C24×C61C2411yescharacteristic
105C23×D61—22no—
106C23×D61—22no—
107C23×D61—81no—
108C23×D61—81no—
109C23×D61C22×C412yes—
110C23×D61C2×D412yes—
111C23×D61C2×D412yes—
112C2×Q8×C61—21no—
113C2×Q8×C61—21no—
114C2×Q8×C61—81no—
115C22×C4×C61—22no—
116C22×C4×C61—22no—
117C22×C4×C61—81no—
118C22×C4×C61C22×C412yes—
119C22×C4×C61C22×C412yes—
120C22×C4×C61C22×C411yescharacteristic
121C22×C4×C61C22×C411yescharacteristic
122C22×C4×C61C2×D411yescharacteristic
123C22×C4×C61C2×D411yescharacteristic
124C22×C4×C61C2×D411yescharacteristic
125C22×C4×C61C2×D411yescharacteristic
126C22×Dic61—22no—
127C22×Dic61—22no—
128C22×Dic61—81no—
129C22×Dic61C22×C412yes—
130C22×Dic61C22×C412yes—
131C22×Dic61C22×C412yes—
132C22×Dic61C2×D412yes—
133C22×Dic61C2×D412yes—
134C2×C4×D61—24no—
135C2×C4×D61—24no—
136C2×C4×D61—82no—
137C2×C4×D61—82no—
138C2×Dic122—21no—
139C2×Dic122—21no—
140C2×Dic122—24no—
141C2×Dic122—82no—
142C2×Dic122—81no—
143C4⋊C4×C61—21no—
144C4⋊C4×C61—21no—
145C4⋊C4×C61—22no—
146C4⋊C4×C61—22no—
147C4⋊C4×C61—81no—
148C22⋊C4×C61—22no—
149C22⋊C4×C61—22no—
150C22⋊C4×C61—22no—
151C22⋊C4×C61—24no—
152C22⋊C4×C61—22no—
153C42×C61—81no—
154C22⋊Dic61—22no—
155C22⋊Dic61—22no—
156C22⋊Dic61—22no—
157C22⋊Dic61—22no—
158C22⋊Dic61—24no—
159#11758198—24no—
160#11758198—24no—
161#11758198—82no—
162#11758198—82no—
163#11758198—82no—
164#11758198—82no—
165#11758198C22⋊C414yes—
166#11758198C22⋊C414yes—
167#11758198C22⋊C414yes—
168#11758198C22⋊C414yes—
169C4⋊Dic61—21no—
170C4⋊Dic61—21no—
171C4⋊Dic61—24no—
172C4⋊Dic61—81no—
173#11758196—22no—
174#11758196—22no—
175#11758196—24no—
176#11758196—24no—
177#11758196—82no—
178C4×Dic61—82no—
179C4×Dic61—81no—
180C23×C61—41no—
181C23×C61—41no—
182C23×C61C23×C411yescharacteristic
183C23×C61C22×D411yescharacteristic
184C23×C61C22×D411yescharacteristic
185C23×C612−1+412yes—
186C23×C612−1+412yes—
187C22×D61—24no—
188C22×D61—22no—
189C22×D61—24no—
190C22×D61—22no—
191C22×D61—42no—
192C22×D61—42no—
193C22×D61—42no—
194C22×D61—42no—
195C22×D61—42no—
196C22×D61—42no—
197C22×D61—84no—
198C22×D61—84no—
199C22×D61C2×C22⋊C414yes—
200Q8×C61—81no—
201Q8×C61—81no—
202C2×C4×C61—22no—
203C2×C4×C61—22no—
204C2×C4×C61—41no—
205C2×C4×C61—41no—
206C2×C4×C61—41no—
207C2×C4×C61—41no—
208C2×C4×C61—41no—
209C2×C4×C61—41no—
210C2×C4×C61—42no—
211C2×C4×C61—42no—
212C2×C4×C61—42no—
213C2×C4×C61—42no—
214C2×C4×C61—42no—
215C2×C4×C61—42no—
216C2×C4×C61—42no—
217C2×C4×C61—42no—
218C2×C4×C61C2×C22⋊C411yescharacteristic
219C2×C4×C61C2×C22⋊C411yescharacteristic
220C2×C4×C61C2×C22⋊C411yescharacteristic
221C2×C4×C61C2×C22⋊C411yescharacteristic
222C2×Dic61—22no—
223C2×Dic61—22no—
224C2×Dic61—42no—
225C2×Dic61—42no—
226C2×Dic61—42no—
227C2×Dic61—42no—
228C2×Dic61—42no—
229C2×Dic61—42no—
230C2×Dic61—42no—
231C2×Dic61—42no—
232C2×Dic61—42no—
233C2×Dic61—42no—
234C2×Dic61—42no—
235C2×Dic61C2×C22⋊C414yes—
236C4×D61—84no—
237C4×D61—84no—
238Dic122—84no—
239Dic122—81no—
240Dic122—81no—
241#21827—611nomaximal, Sylow
242C22×C61—21no—
243C22×C61—21no—
244C22×C61—21no—
245C22×C61—21no—
246C22×C61—21no—
247C22×C61—21no—
248C22×C61—42no—
249C22×C61—42no—
250C22×C61C22×C22⋊C411yescharacteristic
251C22×C61#51912yes—
252C22×C61#53611yescharacteristic
253C22×C61#53612yes—
254C22×C61#53611yescharacteristic
255C2×D61—24no—
256C2×D61—24no—
257C2×D61—24no—
258C2×D61—48no—
259C2×D61—82no—
260C2×D61—84no—
261C2×D61—82no—
262C4×C61—21no—
263C4×C61—21no—
264C4×C61—21no—
265C4×C61—21no—
266C4×C61—42no—
267C4×C61—42no—
268C4×C61—82no—
269C4×C61—82no—
270Dic61—24no—
271Dic61—42no—
272Dic61—42no—
273Dic61—82no—
274Dic61—82no—
275C2×#543—611no—
276#2312—611no—
277#2312—611no—
278#1996—611no—
279#1996—611no—
280#1996—614no—
281#1996—611no—
282#1996D6111yescharacteristic, complemented
283#1819—614no—
284C2×C61—21no—
285C2×C61—22no—
286C2×C61—21no—
287C2×C61—22no—
288C2×C61#231011yescharacteristic
289C2×C61#288311yescharacteristic
290C2×C61#288311yescharacteristic
291D61—44no—
292#543—1222no—
293#543—1222no—
294#543—1222no—
295#543—1222no—
296C2×C22⋊Q8—612no—
297C2×C22⋊Q8—611no—
298C2×C22⋊Q8—612no—
299C2×C22⋊Q8—612no—
300C2×C22⋊Q8—612no—
301C2×C22⋊Q8—612no—
302C2×C22⋊Q8C2×D6111yescharacteristic
303C22×C22⋊C4—611no—
304C22×C22⋊C4—612no—
305#388—1222no—
306#388—1222no—
307#388—1222no—
308#388—1222no—
309#380—611no—
310#380—612no—
311#380C2×D6111yescharacteristic
312#379—612no—
313#379—612no—
314C4×C22⋊C4—611no—
315C4×C22⋊C4—612no—
316C4×C22⋊C4C2×D6111yescharacteristic
317#342—614no—
318#342—614no—
319#342—614no—
320#342C2×D6114yes—
321C61#2182711yescharacteristic, complemented, Sylow
322C25—611no—
323C22×Q8—612no—
324C22×Q8—611no—
325C22×Q8C4×D6111yescharacteristic
326C23×C4—611no—
327C23×C4—612no—
328C23×C4—612no—
329C23×C4C22×D6111yescharacteristic
330C22⋊Q8—21no—
331C22⋊Q8—21no—
332C22⋊Q8—21no—
333C22⋊Q8—21no—
334C22⋊Q8—1224no—
335C22⋊Q8—1224no—
336C22⋊Q8—1224no—
337C22⋊Q8—1224no—
338C22⋊Q8—1224no—
339C22⋊Q8—1224no—
340C22⋊Q8—1224no—
341C22⋊Q8—1224no—
342C22⋊Q8—1224no—
343C22⋊Q8—1224no—
344C22⋊Q8—1221no—
345C22⋊Q8—1221no—
346C22⋊Q8—1221no—
347C22⋊Q8—1221no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—612no—
350C2×C4⋊C4—612no—
351C2×C4⋊C4—612no—
352C2×C4⋊C4—612no—
353C2×C4⋊C4—611no—
354C2×C4⋊C4—1222no—
355C2×C4⋊C4—1221no—
356C2×C4⋊C4C4×D6112yes—
357C2×C4⋊C4C4×D6111yescharacteristic
358C2×C22⋊C4—612no—
359C2×C22⋊C4—612no—
360C2×C22⋊C4—612no—
361C2×C22⋊C4—612no—
362C2×C22⋊C4—612no—
363C2×C22⋊C4—612no—
364C2×C22⋊C4—612no—
365C2×C22⋊C4—612no—
366C2×C22⋊C4—612no—
367C2×C22⋊C4—612no—
368C2×C22⋊C4—612no—
369C2×C22⋊C4—1224no—
370C2×C22⋊C4—1224no—
371C2×C22⋊C4—1222no—
372C2×C22⋊C4—1222no—
373C2×C22⋊C4—1222no—
374C2×C22⋊C4—1222no—
375C2×C22⋊C4C22×D6112yes—
376C2×C22⋊C4C22×D6112yes—
377C2×C22⋊C4C22×D6112yes—
378C2×C42—21no—
379C2×C42—1222no—
380C2×C42—1221no—
381#95—21no—
382#95—21no—
383#95—1222no—
384#95—1222no—
385#95—1221no—
386#95—1221no—
387C24—612no—
388C24—612no—
389C24—612no—
390C24—1222no—
391C24—1222no—
392C24—4881no—
393C24—4881no—
394C24C23×D6111yescharacteristic
395C2×Q8—21no—
396C2×Q8—21no—
397C2×Q8—81no—
398C2×Q8—1224no—
399C2×Q8—1221no—
400C2×Q8—1221no—
401C2×Q8—4881no—
402C2×Q8—4882no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—81no—
406C22×C4—612no—
407C22×C4—612no—
408C22×C4—612no—
409C22×C4—612no—
410C22×C4—612no—
411C22×C4—1224no—
412C22×C4—1224no—
413C22×C4—1222no—
414C22×C4—1222no—
415C22×C4—4882no—
416C22×C4—4882no—
417C22×C4—4881no—
418C22×C4C2×C4×D6112yes—
419C22×C4C2×C4×D6112yes—
420C22×C4C2×C4×D6111yescharacteristic
421C22×C4C2×C4×D6111yescharacteristic
422C22×C4D4×D6111yescharacteristic
423C22×C4D4×D6111yescharacteristic
424C22×C4D4×D6111yescharacteristic
425C22×C4D4×D6111yescharacteristic
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—81no—
431C4⋊C4—1224no—
432C4⋊C4—1224no—
433C4⋊C4—1224no—
434C4⋊C4—1222no—
435C4⋊C4—1222no—
436C4⋊C4—1221no—
437C4⋊C4—1221no—
438C4⋊C4—4882no—
439C4⋊C4—4881no—
440C22⋊C4—22no—
441C22⋊C4—22no—
442C22⋊C4—24no—
443C22⋊C4—22no—
444C22⋊C4—22no—
445C22⋊C4—614no—
446C22⋊C4—614no—
447C22⋊C4—614no—
448C22⋊C4—614no—
449C22⋊C4—1224no—
450C22⋊C4—1224no—
451C22⋊C4—1222no—
452C22⋊C4—1222no—
453C22⋊C4—1222no—
454C22⋊C4—1222no—
455C22⋊C4—1224no—
456C22⋊C4—4882no—
457C22⋊C4—4882no—
458C22⋊C4—4882no—
459C22⋊C4—4882no—
460C42—81no—
461C42—4881no—
462C42—4882no—
463C23—41no—
464C23—41no—
465C23—614no—
466C23—1222no—
467C23—1224no—
468C23—1222no—
469C23—1224no—
470C23—4884no—
471C23—4884no—
472C23—2442no—
473C23—2442no—
474C23—2442no—
475C23—2442no—
476C23—2442no—
477C23—2442no—
478C23C22×C4×D6111yescharacteristic
479C23C2×D4×D6111yescharacteristic
480C23C2×D4×D6111yescharacteristic
481C23#1451982412yes—
482C23#1451982512yes—
483Q8—81no—
484Q8—81no—
485Q8—4881no—
486Q8—4881no—
487Q8—4884no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—614no—
505C2×C4—1222no—
506C2×C4—1222no—
507C2×C4—4884no—
508C2×C4—4884no—
509C2×C4—2442no—
510C2×C4—2442no—
511C2×C4—2442no—
512C2×C4—2442no—
513C2×C4—2442no—
514C2×C4—2442no—
515C2×C4—2442no—
516C2×C4—2442no—
517C2×C4—2442no—
518C2×C4—2442no—
519C2×C4—2442no—
520C2×C4C22⋊C4×D6111yescharacteristic
521C2×C4C22⋊C4×D6111yescharacteristic
522C2×C4C22⋊C4×D6111yescharacteristic
523C2×C4C22⋊C4×D6111yescharacteristic
524C22—21no—
525C22—21no—
526C22—21no—
527C22—21no—
528C22—21no—
529C22—21no—
530C22—42no—
531C22—42no—
532C22—1224no—
533C22—1224no—
534C22—1224no—
535C22—4882no—
536C22—4884no—
537C22—4882no—
538C22—2448no—
539C22C2×C22⋊C4×D6111yescharacteristic
540C22#1684023812yes—
541C22#1684033211yescharacteristic
542C22#1684033512yes—
543C22#1684033711yescharacteristic
544C4—21no—
545C4—21no—
546C4—21no—
547C4—21no—
548C4—42no—
549C4—42no—
550C4—82no—
551C4—82no—
552C4—1224no—
553C4—4882no—
554C4—4882no—
555C4—2442no—
556C4—2442no—
557C2—21no—
558C2—22no—
559C2—21no—
560C2—22no—
561C2—2444no—
562C2#2144840211yescharacteristic
563C2#2145427711yescharacteristic
564C2#2145427811yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC23×C61C23×C4
FrattiniC24C23×D61
FittingC61×#1996C2
socleC22×C61C22×C22⋊C4
prime radicalC24×D61C23
cosocleC24×C61C24
hypercenter#1996D61
hypocenterC61#21827
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1996

Complements

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

NH ≅ G/Ntype
4. #21446805C2semidirect
5. #21446804C2semidirect
6. #21446803C2semidirect
7. #21446802C2semidirect
9. C61×#1996C2semidirect
14. C2×#14519757C4semidirect
16. C2×#14519755C4semidirect
17. C2×#14519755C4semidirect
19. C2×#14519753C4semidirect
282. #1996D61semidirect
321. C61#21827semidirect

Maximal subgroups

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

subgroupindex
C2×#168403872
#214484162
#214484142
#214468052
#214468042
#214468032
#214468022
#214458922
C61×#19962
#2182761