Finite Group Databasebeta

← #29698361

Subgroups — #29698361

Order 15616.

1010 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839862C211yesmaximal, characteristic
2C2×#16839859C211yesmaximal, characteristic
3C2×#16839853C211yesmaximal, characteristic
4#21448738C211yesmaximal, characteristic, complemented
5D4×#11758196C211yesmaximal, characteristic
6#21446554C211yesmaximal, characteristic, complemented
7#21446551C211yesmaximal, characteristic, complemented
8#21446084C211yesmaximal, characteristic, complemented
9#21446076C211yesmaximal, characteristic, complemented
10#21446068C211yesmaximal, characteristic
11#21445984C211yesmaximal, characteristic, complemented
12#21445827C211yesmaximal, characteristic
13#21445823C211yesmaximal, characteristic, complemented
14#21445692C211yesmaximal, characteristic, complemented
15C61×#1864C211yesmaximal, characteristic
16C22×#11758200C2211yescharacteristic
17C2×C4×Dic122C2211yescharacteristic
18C2×#14519742C2211yescharacteristic
19C2×#14519742C2211yescharacteristic
20C2×D4×Dic61C2211yescharacteristic
21C2×D4×Dic61C2211yescharacteristic
22C22×#11758196—21no—
23#16839862—24no—
24#16839859C418yescomplemented
25#16839853—24no—
26C2×#14519711C2211yescharacteristic
27C2×#14519710C2211yescharacteristic
28C2×#14519710C2211yescharacteristic
29C2×#14519708C2211yescharacteristic
30C2×#14519706C2211yescharacteristic
31C2×#14519702C2211yescharacteristic
32C2×#14519702C2211yescharacteristic
33C2×#14519700C2211yescharacteristic
34C2×#14519698C2211yescharacteristic
35C2×#14519688C2211yescharacteristic
36C2×#14519687C2211yescharacteristic
37C2×C8×Dic61C2211yescharacteristic
38#16839661C2211yescharacteristic
39#16839660C2211yescharacteristic
40#16839659—21no—
41#16839658C2211yescharacteristic
42#16839657—21no—
43#16839635—21no—
44#16839634—21no—
45#16839631C2211yescharacteristic
46#16839627C2211yescharacteristic
47#16839626—21no—
48#16839620—21no—
49C4⋊C4×Dic61—21no—
50C4×#11758196C2211yescharacteristic
51#16839525C2211yescharacteristic
52#16839524C2211yescharacteristic
53#16839524C2211yescharacteristic
54#16839516C2211yescharacteristic
55C2×C4×D4×C61C2211yescharacteristic
56C2×#106×C61C2211yescharacteristic
57C2×#102×C61C2211yescharacteristic
58C2×#102×C61C2211yescharacteristic
59C61×#389C2211yescharacteristic
60C61×#340C2211yescharacteristic
61C61×#336C2211yescharacteristic
62C22×Dic122C2311yescharacteristic
63C23×Dic61—21no—
64C23×Dic61—21no—
65C2×#11758200C2×C414yes—
66C2×#11758200D414yes—
67C2×#11758200Q814yes—
68C22×#92418C2311yescharacteristic
69C22×#92418C2311yescharacteristic
70C4×Dic122—22no—
71C4×Dic122—41no—
72#14519742—22no—
73#14519742—22no—
74#14519742—41no—
75#14519742C2×C414yescomplemented
76D4×Dic61—22no—
77D4×Dic61—22no—
78D4×Dic61—41no—
79D4×Dic61C2×C414yes—
80C2×C4⋊Dic61—21no—
81C2×C4⋊Dic61C2311yescharacteristic
82C2×C4⋊Dic61C2311yescharacteristic
83C2×C4⋊Dic61C2311yescharacteristic
84C2×#11758196—21no—
85C2×#11758196—21no—
86C2×#11758196—21no—
87C2×#11758196—21no—
88C2×#11758196—21no—
89C2×#11758196—21no—
90C2×#11758196—21no—
91C2×#11758196—44no—
92C2×#11758196D411yescharacteristic
93C2×#11758196D411yescharacteristic
94C2×C22⋊Dic61—21no—
95C2×C22⋊Dic61—21no—
96C2×C4×Dic61—21no—
97C2×C4×Dic61—21no—
98C2×C4×Dic61—21no—
99C2×C4×Dic61C2311yescharacteristic
100C2×C4×Dic61C2311yescharacteristic
101#14519711—41no—
102#14519710—22no—
103#14519710C2×C414yescomplemented
104#14519708—22no—
105#14519706—22no—
106#14519702—22no—
107#14519702C2×C414yescomplemented
108#14519700—22no—
109#14519698—22no—
110#14519688—22no—
111#14519687C2×C414yescomplemented
112C8×Dic61—22no—
113#14519684—21no—
114#14519684—21no—
115#14519684—21no—
116#14519684—21no—
117#14519684—21no—
118#14519684D411yescharacteristic
119#14519684D411yescharacteristic
120C22×D4×C61C2311yescharacteristic
121C23×C4×C61—21no—
122C22×C8×C61C2311yescharacteristic
123C22×C8×C61C2311yescharacteristic
124C4×D4×C61—22no—
125C4×D4×C61—41no—
126C2×C4⋊C4×C61—21no—
127C2×C4⋊C4×C61C2311yescharacteristic
128C2×C4⋊C4×C61C2311yescharacteristic
129C2×C4⋊C4×C61C2311yescharacteristic
130C2×C22⋊C4×C61—21no—
131C2×C42×C61C2311yescharacteristic
132#106×C61—41no—
133#102×C61—22no—
134#102×C61C2×C414yes—
135#95×C61—21no—
136C24×C61—21no—
137C2×D4×C61—21no—
138C2×D4×C61—21no—
139C2×D4×C61—21no—
140C2×D4×C61C22×C412yes—
141C2×D4×C61C2×D412yes—
142C2×D4×C61C2×Q812yes—
143C22×C4×C61—21no—
144C22×C4×C61—21no—
145C22×C4×C61—21no—
146C22×C4×C61—42no—
147C22×C4×C61—42no—
148C22×C4×C61—81no—
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2411yescharacteristic
152C22×Dic61—21no—
153C22×Dic61—21no—
154C22×Dic61—24no—
155C22×Dic61—21no—
156C22×Dic61—21no—
157C22×Dic61—42no—
158C22×Dic61—42no—
159C22×Dic61—42no—
160C22×Dic61—42no—
161C22×Dic61—81no—
162C22×Dic61C2×D411yescharacteristic
163C22×Dic61C2×D411yescharacteristic
164C22×Dic61C2×D411yescharacteristic
165C22×Dic61C2×D411yescharacteristic
166C2×Dic122—21no—
167C2×Dic122—21no—
168C2×Dic122—21no—
169C2×Dic122C22×C412yes—
170C2×Dic122C2×D412yes—
171C2×Dic122C2×Q812yes—
172C2×C8×C61—21no—
173C2×C8×C61—21no—
174C2×C8×C61—21no—
175C2×C8×C61—21no—
176C2×C8×C61—21no—
177C2×C8×C61C22×C412yes—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61—21no—
183C4⋊C4×C61—42no—
184C4⋊C4×C61C22×C412yes—
185C22⋊C4×C61—42no—
186C42×C61—21no—
187C42×C61—21no—
188C22⋊Dic61—24no—
189C22⋊Dic61—42no—
190#11758200C4⋊C4116yescomplemented
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—42no—
197C4⋊Dic61C22×C412yes—
198#11758196—24no—
199#11758196—22no—
200#11758196—41no—
201#11758196—42no—
202#11758196—42no—
203#11758196—42no—
204#11758196—42no—
205#11758196—42no—
206#11758196—81no—
207C4×Dic61—21no—
208C4×Dic61—21no—
209C4×Dic61—24no—
210C4×Dic61—42no—
211C4×Dic61—42no—
212C4×Dic61C22×C412yes—
213C4×Dic61#4112yes—
214C2×#92418—21no—
215C2×#92418—21no—
216C2×#92418—21no—
217C2×#92418C22×C412yes—
218C2×#92418C2×D412yes—
219C2×#92418C2×Q812yes—
220C23×C61—22no—
221C23×C61—22no—
222C23×C61—22no—
223C23×C61—41no—
224C23×C61—41no—
225C23×C61—41no—
226C23×C61—41no—
227C23×C61C22×D411yescharacteristic
228D4×C61—22no—
229D4×C61—22no—
230D4×C61—22no—
231D4×C61C2×C4⋊C414yes—
232C2×C4×C61—22no—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61—21no—
239C2×C4×C61—41no—
240C2×C4×C61—41no—
241C2×C4×C61—41no—
242C2×C4×C61—41no—
243C2×C4×C61—41no—
244C2×C4×C61—41no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61—84no—
248C2×C4×C61C23×C411yescharacteristic
249C2×C4×C61C22×D411yescharacteristic
250C2×C4×C61C22×Q811yescharacteristic
251C2×C4×C61C2×#4111yescharacteristic
252C2×C4×C61H(5,𝔽2)11yescharacteristic
253C2×C4×C612−1+411yescharacteristic
254C2×Dic61—22no—
255C2×Dic61—21no—
256C2×Dic61—22no—
257C2×Dic61—22no—
258C2×Dic61—21no—
259C2×Dic61—21no—
260C2×Dic61—21no—
261C2×Dic61—21no—
262C2×Dic61—21no—
263C2×Dic61—22no—
264C2×Dic61—21no—
265C2×Dic61—21no—
266C2×Dic61—21no—
267C2×Dic61—48no—
268C2×Dic61—41no—
269C2×Dic61—41no—
270C2×Dic61—41no—
271C2×Dic61—41no—
272C2×Dic61—41no—
273C2×Dic61—41no—
274C2×Dic61—41no—
275C2×Dic61—41no—
276C2×Dic61—84no—
277C2×Dic61C4×D412yes—
278C2×Dic61C4×D412yes—
279C2×Dic61AGL(1,ℤ/8)12yes—
280Dic122—22no—
281Dic122—22no—
282Dic122—22no—
283Dic122C2×C4⋊C414yescomplemented
284C8×C61—22no—
285C8×C61—41no—
286#92418—41no—
287#92418C2×C4⋊C414yescomplemented
288#22677—611nomaximal, Sylow
289C22×C61—24no—
290C22×C61—42no—
291C22×C61—42no—
292C22×C61—42no—
293C22×C61—42no—
294C22×C61—42no—
295C22×C61—42no—
296C22×C61C2×C4×D411yescharacteristic
297C22×C61D4211yescharacteristic
298C22×C61D4×Q811yescharacteristic
299C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
300C22×C61#57511yescharacteristic
301C22×C61#57711yescharacteristic
302C22×C61#57811yescharacteristic
303C4×C61—42no—
304C4×C61—41no—
305C4×C61—41no—
306C4×C61—81no—
307C4×C61—81no—
308C4×C61C22×C4⋊C411yescharacteristic
309C4×C61#52011yescharacteristic
310C4×C61#54711yescharacteristic
311C4×C61#55411yescharacteristic
312Dic61—24no—
313Dic61—22no—
314Dic61—22no—
315Dic61—22no—
316Dic61—42no—
317Dic61—41no—
318Dic61—81no—
319Dic61—81no—
320C2×#440—611no—
321C2×#440—611no—
322C2×C4×SD16—611no—
323#2351—611no—
324D4×C4⋊C4—611no—
325#1943—611no—
326#1942—611no—
327#1865—611no—
328#1864D6111yescharacteristic
329#1863—611no—
330#1862—611no—
331#1847—611no—
332#1804—611no—
333#1803—611no—
334#1771—611no—
335C2×C61—44no—
336C2×C61D4×C4⋊C411yescharacteristic
337C2×C61C4×AGL(1,ℤ/8)11yescharacteristic
338C2×C61#294711yescharacteristic
339C2×C61#326511yescharacteristic
340C2×C61#326911yescharacteristic
341C2×C61#338611yescharacteristic
342C2×C61#338811yescharacteristic
343C22×SD16—611no—
344C2×C4×Q8—611no—
345C2×C4×Q8—611no—
346C2×C4×Q8—611no—
347C2×C4×D4—611no—
348C2×C4×D4—611no—
349C2×C4×D4C2×D6111yescharacteristic
350C22×C4⋊C4—1221no—
351#440—618no—
352#440—1224no—
353C4×SD16—1224no—
354C2×#107—611no—
355C2×#107—611no—
356C2×#107—611no—
357C2×#106—611no—
358C2×#106C2×D6111yescharacteristic
359C2×C4⋊C8—611no—
360C2×#103—611no—
361C2×#103—611no—
362C2×#103—611no—
363C2×#102—611no—
364C2×#102C2×D6111yescharacteristic
365C2×#102C2×D6111yescharacteristic
366C2×#97—611no—
367C2×#97—611no—
368C2×C4×C8—611no—
369#389—611no—
370#389—611no—
371#389—611no—
372#389—1221no—
373#389—1221no—
374#389C2×D6111yescharacteristic
375#385—1221no—
376#385—1221no—
377#384—611no—
378#382—611no—
379#382—1221no—
380#380—1221no—
381C4×C4⋊C4—611no—
382C4×C4⋊C4—1221no—
383#340—611no—
384#340—611no—
385#340—611no—
386#340C2×D6111yescharacteristic
387#336—611no—
388#336C2×D6111yescharacteristic
389C61#2267711yescharacteristic, complemented, Sylow
390C22×Q8—611no—
391C22×D4C22×D6111yescharacteristic
392C23×C4—21no—
393C23×C4—1221no—
394C23×C4—1221no—
395C2×SD16—614no—
396C2×SD16—614no—
397C2×SD16—614no—
398C22×C8—611no—
399C22×C8—611no—
400C22×C8C22×D6111yescharacteristic
401C22×C8C22×D6111yescharacteristic
402C4×Q8—614no—
403C4×Q8—1222no—
404C4×Q8—1222no—
405C4×Q8—1222no—
406C4×Q8—2441no—
407C4×Q8—2441no—
408C4×D4—22no—
409C4×D4—41no—
410C4×D4—614no—
411C4×D4—1222no—
412C4×D4—1222no—
413C4×D4—2441no—
414C2×C4⋊C4—21no—
415C2×C4⋊C4—611no—
416C2×C4⋊C4—611no—
417C2×C4⋊C4—611no—
418C2×C4⋊C4—611no—
419C2×C4⋊C4—611no—
420C2×C4⋊C4—1221no—
421C2×C4⋊C4—1221no—
422C2×C4⋊C4—1221no—
423C2×C4⋊C4—1221no—
424C2×C4⋊C4—1221no—
425C2×C4⋊C4—1221no—
426C2×C4⋊C4—1221no—
427C2×C4⋊C4—1221no—
428C2×C4⋊C4—2444no—
429C2×C4⋊C4C22×D6111yescharacteristic
430C2×C4⋊C4C22×D6111yescharacteristic
431C2×C4⋊C4C22×D6111yescharacteristic
432C2×C22⋊C4—21no—
433C2×C22⋊C4—1221no—
434C2×C22⋊C4—1221no—
435C2×C42—611no—
436C2×C42—611no—
437C2×C42—1221no—
438C2×C42—1221no—
439C2×C42—1221no—
440C2×C42C22×D6111yescharacteristic
441#107—614no—
442#107—1222no—
443#107—2441no—
444#106—41no—
445#106—1222no—
446C4⋊C8—1222no—
447#103—614no—
448#103—1222no—
449#103—1222no—
450#102—22no—
451#102—1222no—
452#102C4×D6114yes—
453#97—614no—
454#97—1222no—
455C4×C8—1222no—
456#95—21no—
457#95—611no—
458#95—611no—
459#95—1221no—
460#95—1221no—
461#95—1221no—
462#95—1221no—
463#95—1221no—
464C24—21no—
465C2×Q8—612no—
466C2×Q8—612no—
467C2×Q8—612no—
468C2×Q8—1221no—
469C2×Q8—1221no—
470C2×Q8—1221no—
471C2×D4—21no—
472C2×D4—21no—
473C2×D4—21no—
474C2×D4C2×C4×D6112yes—
475C2×D4D4×D6112yes—
476C2×D4Q8×D6112yes—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—81no—
483C22×C4—611no—
484C22×C4—611no—
485C22×C4—611no—
486C22×C4—611no—
487C22×C4—1224no—
488C22×C4—1221no—
489C22×C4—1221no—
490C22×C4—1221no—
491C22×C4—1221no—
492C22×C4—4881no—
493C22×C4—2442no—
494C22×C4—2442no—
495C22×C4—2442no—
496C22×C4—2442no—
497C22×C4D4×D6111yescharacteristic
498C22×C4D4×D6111yescharacteristic
499C22×C4C23×D6111yescharacteristic
500SD16—6116no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—612no—
507C2×C8—612no—
508C2×C8—612no—
509C2×C8—1221no—
510C2×C8—1221no—
511C2×C8—1221no—
512C2×C8C2×C4×D6112yes—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—42no—
519C4⋊C4—612no—
520C4⋊C4—1222no—
521C4⋊C4—1224no—
522C4⋊C4—1221no—
523C4⋊C4—1221no—
524C4⋊C4—1221no—
525C4⋊C4—1221no—
526C4⋊C4—1221no—
527C4⋊C4—4881no—
528C4⋊C4—2441no—
529C4⋊C4—2442no—
530C4⋊C4—2442no—
531C4⋊C4—2442no—
532C4⋊C4—2442no—
533C4⋊C4—2442no—
534C4⋊C4—2442no—
535C4⋊C4C2×C4×D6112yes—
536C22⋊C4—42no—
537C22⋊C4—1224no—
538C22⋊C4—2442no—
539C42—21no—
540C42—21no—
541C42—612no—
542C42—612no—
543C42—1224no—
544C42—1221no—
545C42—1221no—
546C42—2442no—
547C42—2442no—
548C23—22no—
549C23—22no—
550C23—22no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23C2×D4×D6111yescharacteristic
556Q8—614no—
557Q8—1222no—
558Q8—1222no—
559Q8—1222no—
560D4—22no—
561D4—22no—
562D4—22no—
563D4C4⋊C4×D6114yes—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—84no—
580C2×C4—612no—
581C2×C4—612no—
582C2×C4—612no—
583C2×C4—1222no—
584C2×C4—1221no—
585C2×C4—1221no—
586C2×C4—1221no—
587C2×C4—1221no—
588C2×C4—1221no—
589C2×C4—1221no—
590C2×C4—1222no—
591C2×C4—1222no—
592C2×C4—1222no—
593C2×C4—1221no—
594C2×C4—1221no—
595C2×C4—1221no—
596C2×C4—4884no—
597C2×C4—2448no—
598C2×C4—2441no—
599C2×C4—2441no—
600C2×C4—2441no—
601C2×C4—2441no—
602C2×C4—2441no—
603C2×C4—2441no—
604C2×C4—2441no—
605C2×C4—2441no—
606C2×C4C22×C4×D6111yescharacteristic
607C2×C4C2×D4×D6111yescharacteristic
608C2×C4C2×Q8×D6111yescharacteristic
609C2×C4#41×D6111yescharacteristic
610C2×C4#1451982211yescharacteristic
611C2×C4#1451982411yescharacteristic
612C8—22no—
613C8—41no—
614C8—614no—
615C8—2441no—
616C22—24no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—42no—
622C22—42no—
623C22C4×D4×D6111yescharacteristic
624C22#1684039411yescharacteristic
625C22#1684042511yescharacteristic
626C22AGL(1,ℤ/8)×D6111yescharacteristic
627C22#1684055811yescharacteristic
628C22#1684056811yescharacteristic
629C22#1684057611yescharacteristic
630C4—42no—
631C4—41no—
632C4—41no—
633C4—81no—
634C4—81no—
635C4—1224no—
636C4—1222no—
637C4—1222no—
638C4—1222no—
639C4—4881no—
640C4—4881no—
641C4—2442no—
642C4—2441no—
643C4C2×C4⋊C4×D6111yescharacteristic
644C4#1684024111yescharacteristic
645C4#1684041011yescharacteristic
646C4#1684046611yescharacteristic
647C2—44no—
648C2#2144871711yescharacteristic
649C2#2145463011yescharacteristic
650C2#2145465511yescharacteristic
651C2#2145731211yescharacteristic
652C2#2145737211yescharacteristic
653C2#2145873511yescharacteristic
654C2#2145875711yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1864C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
cosocleC22×C4×C61C24
hypercenter#1864D61
hypocenterC61#22677
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1864

Complements

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

NH ≅ G/Ntype
4. #21448738C2semidirect
6. #21446554C2semidirect
7. #21446551C2semidirect
8. #21446084C2semidirect
9. #21446076C2semidirect
11. #21445984C2semidirect
13. #21445823C2semidirect
14. #21445692C2semidirect
24. #16839859C4semidirect
75. #14519742C2×C4semidirect
103. #14519710C2×C4semidirect
107. #14519702C2×C4semidirect
111. #14519687C2×C4semidirect
190. #11758200C4⋊C4semidirect
283. Dic122C2×C4⋊C4semidirect
287. #92418C2×C4⋊C4semidirect
389. C61#22677semidirect

Maximal subgroups

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

subgroupindex
C2×#168398622
C2×#168398592
C2×#168398532
#214487382
D4×#117581962
#214465542
#214465512
#214460842
#214460762
#214460682
#214459842
#214458272
#214458232
#214456922
C61×#18642
#2267761