Finite Group Databasebeta

← #29687699

Subgroups — #29687699

Order 15616.

936 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840065C211yesmaximal, characteristic
2C4×#14519768C211yesmaximal, characteristic, complemented
3C4×#14519739C211yesmaximal, characteristic
4#21446773C211yesmaximal, characteristic, complemented
5#21446588C211yesmaximal, characteristic, complemented
6#21446402C211yesmaximal, characteristic, complemented
7#21446392C211yesmaximal, characteristic, complemented
8#21446077C211yesmaximal, characteristic
9#21446067C211yesmaximal, characteristic
10#21445828C211yesmaximal, characteristic
11#21445717C211yesmaximal, characteristic, complemented
12#21445673C211yesmaximal, characteristic, complemented
13#21445647C211yesmaximal, characteristic
14#21445645C211yesmaximal, characteristic
15C61×#1776C211yesmaximal, characteristic, complemented
16C2×#14519768C2211yescharacteristic
17C2×C4×D244C2211yescharacteristic
18C2×#14519739C2211yescharacteristic
19C2×#14519739C2211yescharacteristic
20C2×C42×D61—21no—
21#16840065C418yescomplemented
22C2×#14519710C2211yescharacteristic
23C2×#14519708C2211yescharacteristic
24C2×#14519706C2211yescharacteristic
25C2×#14519706C2211yescharacteristic
26C2×#14519699C2211yescharacteristic
27C2×#14519699C2211yescharacteristic
28C2×#14519697C2211yescharacteristic
29C2×#14519697C2211yescharacteristic
30C2×#14519688C2211yescharacteristic
31#16839661C2211yescharacteristic
32#16839658C2211yescharacteristic
33#16839649—21no—
34#16839645—21no—
35#16839627C2211yescharacteristic
36C4⋊C4×Dic61C2211yescharacteristic
37C4⋊C4×Dic61C2211yescharacteristic
38C4×#11758196—21no—
39C4×C4⋊Dic61—21no—
40C4×C4⋊Dic61C2211yescharacteristic
41C4×#11758198—21no—
42#16839609—21no—
43#16839609—21no—
44C42×Dic61C2211yescharacteristic
45#16839525C2211yescharacteristic
46#16839525C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839516C2211yescharacteristic
52#16839516C2211yescharacteristic
53C2×#107×C61C2211yescharacteristic
54C2×#106×C61C2211yescharacteristic
55C2×#97×C61C2211yescharacteristic
56C4×C4⋊C4×C61C2211yescharacteristic
57C4×C4⋊C4×C61C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C61×#340C2211yescharacteristic
60C22×D244C2311yescharacteristic
61C22×C4×D61—21no—
62C22×C4×D61—21no—
63C22×C4×D61—21no—
64C22×#92418C2311yescharacteristic
65C22×#92418C2311yescharacteristic
66#14519768—22no—
67#14519768C2×C414yescomplemented
68C4×D244—22no—
69C4×D244Q814yes—
70#14519739—22no—
71#14519739—22no—
72#14519739—22no—
73#14519739C2×C414yes—
74C2×C4⋊Dic61—21no—
75C2×C4⋊Dic61—21no—
76C2×C4⋊Dic61C2311yescharacteristic
77C2×C4⋊Dic61C2311yescharacteristic
78C2×C4⋊Dic61C2311yescharacteristic
79C2×#11758196—21no—
80C2×#11758196—21no—
81C2×#11758198—21no—
82C2×#11758198—21no—
83C2×#11758198—21no—
84C2×C4×Dic61—21no—
85C2×C4×Dic61—21no—
86C2×C4×Dic61—21no—
87C2×C4×Dic61D411yescharacteristic
88C2×C4×Dic61D411yescharacteristic
89C2×C4×Dic61D411yescharacteristic
90C2×C4×Dic61D411yescharacteristic
91C2×C4×Dic61C2311yescharacteristic
92C2×C4×Dic61C2311yescharacteristic
93C42×D61—44no—
94#14519710—22no—
95#14519708C2×C414yescomplemented
96#14519706—22no—
97#14519706C2×C414yescomplemented
98#14519699—22no—
99#14519699C2×C414yes—
100#14519697—22no—
101#14519697C2×C414yes—
102#14519688—22no—
103#14519684—21no—
104#14519684—21no—
105#14519684—21no—
106#14519684—21no—
107#14519684—21no—
108#14519684—21no—
109#14519684—21no—
110C22×C8×C61C2311yescharacteristic
111C22×C8×C61C2311yescharacteristic
112C2×C4⋊C4×C61C2311yescharacteristic
113C2×C4⋊C4×C61C2311yescharacteristic
114C2×C4⋊C4×C61C2311yescharacteristic
115C2×C4⋊C4×C61C2311yescharacteristic
116C2×C42×C61—21no—
117C2×C42×C61—21no—
118C2×C42×C61C2311yescharacteristic
119#107×C61C2×C414yescomplemented
120#106×C61—22no—
121#97×C61—22no—
122#95×C61—21no—
123#95×C61—21no—
124C23×D61—21no—
125C22×C4×C61—21no—
126C22×C4×C61—21no—
127C22×C4×C61—21no—
128C22×C4×C61—21no—
129C22×C4×C61C2×D411yescharacteristic
130C22×C4×C61C2×D411yescharacteristic
131C22×C4×C61C2411yescharacteristic
132C22×Dic61—21no—
133C22×Dic61—21no—
134C22×Dic61—21no—
135C22×Dic61C2×D411yescharacteristic
136C22×Dic61C2×D411yescharacteristic
137C22×Dic61C2×D411yescharacteristic
138C22×Dic61C2×D411yescharacteristic
139C2×D244—21no—
140C2×D244—21no—
141C2×D244—21no—
142C2×D244C22×C412yes—
143C2×D244C2×Q812yes—
144C2×D244#4112yes—
145C2×C4×D61—24no—
146C2×C4×D61—24no—
147C2×C4×D61—24no—
148C2×C4×D61—42no—
149C2×C4×D61—42no—
150C2×C4×D61—42no—
151C2×C4×D61—42no—
152C2×C4×D61—42no—
153C2×C4×D61—42no—
154C2×C8×C61—21no—
155C2×C8×C61—21no—
156C2×C8×C61—21no—
157C2×C8×C61—21no—
158C2×C8×C61—21no—
159C2×C8×C61C22×C412yes—
160C4⋊C4×C61—21no—
161C4⋊C4×C61—21no—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61C22×C412yes—
166C4⋊C4×C61C22×C412yes—
167C4⋊C4×C61#4112yes—
168C42×C61—42no—
169C42×C61—42no—
170C42×C61C2×Q812yes—
171C42×C61#4112yes—
172#11758198—24no—
173#11758198—24no—
174#11758198—24no—
175C4⋊Dic61—21no—
176C4⋊Dic61—21no—
177C4⋊Dic61—24no—
178C4⋊Dic61—21no—
179C4⋊Dic61—21no—
180C4⋊Dic61—42no—
181C4⋊Dic61C22×C412yes—
182C4⋊Dic61C2×Q812yes—
183#11758196—24no—
184#11758196—42no—
185C4×Dic61—21no—
186C4×Dic61—21no—
187C4×Dic61—22no—
188C4×Dic61—41no—
189C4×Dic61—41no—
190C4×Dic61—41no—
191C4×Dic61—42no—
192C4×Dic61—42no—
193C4×Dic61—42no—
194C4×Dic61C22×C412yes—
195C4×Dic61#4112yes—
196C2×#92418—21no—
197C2×#92418—21no—
198C2×#92418—21no—
199C2×#92418—21no—
200C2×#92418—21no—
201C2×#92418C22×C412yes—
202C23×C61C22×D411yescharacteristic
203C22×D61—22no—
204C22×D61—22no—
205C22×D61—22no—
206C22×D61—41no—
207C22×D61—41no—
208C22×D61—41no—
209C22×D61—41no—
210C2×C4×C61—21no—
211C2×C4×C61—21no—
212C2×C4×C61—21no—
213C2×C4×C61—21no—
214C2×C4×C61—22no—
215C2×C4×C61—22no—
216C2×C4×C61—22no—
217C2×C4×C61—41no—
218C2×C4×C61—41no—
219C2×C4×C61—41no—
220C2×C4×C61—41no—
221C2×C4×C61—41no—
222C2×C4×C61—41no—
223C2×C4×C61—41no—
224C2×C4×C61—41no—
225C2×C4×C61—41no—
226C2×C4×C61C22⋊Q812yes—
227C2×C4×C61C22⋊Q812yes—
228C2×C4×C61C23×C411yescharacteristic
229C2×C4×C61C22×Q811yescharacteristic
230C2×C4×C61C2×#4111yescharacteristic
231C2×C4×C61C2×#4111yescharacteristic
232C2×C4×C61C2×#4111yescharacteristic
233C2×C4×C61C2×#4111yescharacteristic
234C2×Dic61—21no—
235C2×Dic61—21no—
236C2×Dic61—21no—
237C2×Dic61—21no—
238C2×Dic61—21no—
239C2×Dic61—21no—
240C2×Dic61—22no—
241C2×Dic61—22no—
242C2×Dic61—21no—
243C2×Dic61—21no—
244C2×Dic61—21no—
245C2×Dic61—41no—
246C2×Dic61—41no—
247C2×Dic61—41no—
248C2×Dic61—41no—
249C2×Dic61—41no—
250C2×Dic61—41no—
251C2×Dic61—41no—
252C2×Dic61C4×D412yes—
253C2×Dic61C4×D412yes—
254C2×Dic61AGL(1,ℤ/8)12yes—
255D244—22no—
256D244—22no—
257D244—22no—
258D244C4×Q814yes—
259C4×D61—48no—
260C4×D61—48no—
261C4×D61—48no—
262C8×C61—22no—
263C8×C61—41no—
264#92418—22no—
265#92418—41no—
266#21617—611nomaximal, Sylow
267C22×C61C2×C4×D411yescharacteristic
268C22×C61C2×C22⋊Q811yescharacteristic
269C22×C61#52511yescharacteristic
270C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
271C22×C61#57511yescharacteristic
272C22×C61#57511yescharacteristic
273C22×C61#57511yescharacteristic
274C2×D61—24no—
275C2×D61—42no—
276C2×D61—42no—
277C2×D61—42no—
278C2×D61—42no—
279C2×D61—42no—
280C2×D61—42no—
281C4×C61—22no—
282C4×C61—22no—
283C4×C61—42no—
284C4×C61—42no—
285C4×C61—42no—
286C4×C61—81no—
287C4×C61C2×C4×Q811yescharacteristic
288C4×C61C4×#4111yescharacteristic
289C4×C61#52911yescharacteristic
290C4×C61#53311yescharacteristic
291Dic61—22no—
292Dic61—22no—
293Dic61—22no—
294Dic61—42no—
295Dic61—41no—
296Dic61—42no—
297Dic61—81no—
298C2×#478—611no—
299C4×#125—611no—
300C42×D4—611no—
301#1989—611no—
302#1949—611no—
303#1920—611no—
304#1918—611no—
305#1864—611no—
306#1862—611no—
307#1804—611no—
308#1776—611no—
309#1776D6111yescharacteristic, complemented
310#1767—611no—
311#1762—611no—
312#1762—611no—
313C2×C61C4×C22⋊Q811yescharacteristic
314C2×C61#294311yescharacteristic
315C2×C61C4×AGL(1,ℤ/8)11yescharacteristic
316C2×C61#307711yescharacteristic
317C2×C61#308011yescharacteristic
318C2×C61#311111yescharacteristic
319C2×C61#311411yescharacteristic
320D61—44no—
321C2×#125—611no—
322C2×C4×D4—611no—
323C2×C4×D4—611no—
324C2×C4×D4—611no—
325C22×C42—1221no—
326#478—618no—
327C2×#107—611no—
328C2×#107C2×D6111yescharacteristic
329C2×#106—611no—
330C2×#106C2×D6111yescharacteristic
331C2×C4⋊C8—611no—
332C2×C4⋊C8—611no—
333C2×#102—611no—
334C2×#102—611no—
335C2×#102—611no—
336C2×#102—611no—
337C2×#97—611no—
338C2×#97C2×D6111yescharacteristic
339#389—611no—
340#389—611no—
341#387—1221no—
342#387—1221no—
343#382—611no—
344C4×C4⋊C4—611no—
345C4×C4⋊C4—611no—
346C4×C4⋊C4—611no—
347C4×C4⋊C4—1221no—
348C4×C4⋊C4—1221no—
349C4×C4⋊C4C2×D6111yescharacteristic
350C4×C4⋊C4C2×D6111yescharacteristic
351C4×C22⋊C4—1221no—
352C4×C22⋊C4—1221no—
353C4×C22⋊C4—1221no—
354C43—611no—
355#340—611no—
356#340—611no—
357#340—611no—
358#340—611no—
359#340—611no—
360#340—611no—
361#340C2×D6111yescharacteristic
362#340C2×D6111yescharacteristic
363#336—611no—
364#336—611no—
365C61#2161711yescharacteristic, complemented, Sylow
366C22×D4—611no—
367C23×C4—1221no—
368C23×C4—1221no—
369C23×C4—1221no—
370C22×C8—611no—
371C22×C8—611no—
372C22×C8C22×D6111yescharacteristic
373C22×C8C22×D6111yescharacteristic
374#125—614no—
375#125—1222no—
376C4×D4—614no—
377C4×D4—614no—
378C4×D4—1222no—
379C4×D4—1222no—
380C4×D4—1222no—
381C4×D4—1222no—
382C2×C4⋊C4—611no—
383C2×C4⋊C4—611no—
384C2×C4⋊C4—611no—
385C2×C4⋊C4—1221no—
386C2×C4⋊C4—1221no—
387C2×C4⋊C4—1221no—
388C2×C4⋊C4—1221no—
389C2×C4⋊C4C22×D6111yescharacteristic
390C2×C4⋊C4C22×D6111yescharacteristic
391C2×C4⋊C4C22×D6111yescharacteristic
392C2×C4⋊C4C22×D6111yescharacteristic
393C2×C22⋊C4—1221no—
394C2×C22⋊C4—1221no—
395C2×C22⋊C4—1221no—
396C2×C42—21no—
397C2×C42—21no—
398C2×C42—611no—
399C2×C42—611no—
400C2×C42—611no—
401C2×C42—611no—
402C2×C42—611no—
403C2×C42—611no—
404C2×C42—1221no—
405C2×C42—1221no—
406C2×C42—1221no—
407C2×C42—2444no—
408C2×C42C22×D6111yescharacteristic
409#107—1222no—
410#107C4×D6114yescomplemented
411#106—22no—
412#106—614no—
413C4⋊C8—614no—
414C4⋊C8—1222no—
415#102—614no—
416#102—614no—
417#102—1222no—
418#102—1222no—
419#97—22no—
420#97—1222no—
421#95—21no—
422#95—21no—
423#95—1221no—
424#95—1221no—
425#95—1221no—
426#95—1221no—
427#95—1221no—
428#95—1221no—
429#95—1221no—
430C24—1221no—
431C2×D4—612no—
432C2×D4—612no—
433C2×D4—612no—
434C2×D4—1221no—
435C2×D4—1221no—
436C2×D4—1221no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—611no—
442C22×C4—611no—
443C22×C4—611no—
444C22×C4—611no—
445C22×C4—1224no—
446C22×C4—1224no—
447C22×C4—1221no—
448C22×C4—1221no—
449C22×C4—1224no—
450C22×C4—1221no—
451C22×C4—2442no—
452C22×C4—2442no—
453C22×C4—2442no—
454C22×C4—2442no—
455C22×C4—2442no—
456C22×C4—2442no—
457C22×C4D4×D6111yescharacteristic
458C22×C4D4×D6111yescharacteristic
459C22×C4C23×D6111yescharacteristic
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—612no—
466C2×C8—1221no—
467C2×C8—1221no—
468C2×C8—1221no—
469C2×C8—1221no—
470C2×C8—1221no—
471C2×C8C2×C4×D6112yes—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—612no—
478C4⋊C4—612no—
479C4⋊C4—1224no—
480C4⋊C4—1224no—
481C4⋊C4—1221no—
482C4⋊C4—1221no—
483C4⋊C4—1221no—
484C4⋊C4—1221no—
485C4⋊C4—2442no—
486C4⋊C4—2442no—
487C4⋊C4C2×C4×D6112yes—
488C4⋊C4C2×C4×D6112yes—
489C4⋊C4#1175822212yes—
490C22⋊C4—1224no—
491C22⋊C4—1224no—
492C22⋊C4—1224no—
493C42—42no—
494C42—42no—
495C42—612no—
496C42—612no—
497C42—1222no—
498C42—1221no—
499C42—1221no—
500C42—2441no—
501C42—2441no—
502C42—2441no—
503C42—2442no—
504C42—2442no—
505C42—2442no—
506C42#1175822212yes—
507C42Q8×D6112yes—
508C23—1222no—
509C23—1222no—
510C23—1222no—
511C23—2441no—
512C23—2441no—
513C23—2441no—
514C23—2441no—
515C23C2×D4×D6111yescharacteristic
516D4—614no—
517D4—1222no—
518D4—1222no—
519D4—1222no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—612no—
537C2×C4—612no—
538C2×C4—612no—
539C2×C4—1221no—
540C2×C4—1221no—
541C2×C4—1221no—
542C2×C4—1221no—
543C2×C4—1221no—
544C2×C4—1221no—
545C2×C4—1222no—
546C2×C4—1222no—
547C2×C4—1221no—
548C2×C4—1221no—
549C2×C4—1221no—
550C2×C4—2448no—
551C2×C4—2448no—
552C2×C4—2448no—
553C2×C4—2441no—
554C2×C4—2441no—
555C2×C4—2441no—
556C2×C4—2441no—
557C2×C4—2441no—
558C2×C4—2441no—
559C2×C4—2441no—
560C2×C4#1451975512yes—
561C2×C4#1451975512yes—
562C2×C4C22×C4×D6111yescharacteristic
563C2×C4C2×Q8×D6111yescharacteristic
564C2×C4#41×D6111yescharacteristic
565C2×C4#41×D6111yescharacteristic
566C2×C4C2×#1175822211yescharacteristic
567C2×C4C2×#1175822211yescharacteristic
568C8—22no—
569C8—41no—
570C8—1222no—
571C8—2441no—
572C22—1224no—
573C22—2442no—
574C22—2442no—
575C22—2442no—
576C22—2442no—
577C22—2442no—
578C22—2442no—
579C22C4×D4×D6111yescharacteristic
580C22C2×#1451975511yescharacteristic
581C22#1684027111yescharacteristic
582C22AGL(1,ℤ/8)×D6111yescharacteristic
583C22#1684055811yescharacteristic
584C22#1684056111yescharacteristic
585C22#1684056111yescharacteristic
586C4—22no—
587C4—22no—
588C4—42no—
589C4—42no—
590C4—42no—
591C4—81no—
592C4—1222no—
593C4—1222no—
594C4—1222no—
595C4—4881no—
596C4—2442no—
597C4—2441no—
598C4—2442no—
599C4C4×Q8×D6111yescharacteristic
600C4C4×#1175822211yescharacteristic
601C4#1684029011yescharacteristic
602C4#1684031611yescharacteristic
603C2—2444no—
604C2C4×#1451975511yescharacteristic
605C2#2145461811yescharacteristic
606C2#2145463011yescharacteristic
607C2#2145561511yescharacteristic
608C2#2145564011yescharacteristic
609C2#2145592611yescharacteristic
610C2#2145596111yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1776C2
socleC23×C61C22×D4
prime radicalC22×D244C23
cosocleC22×C4×C61C24
hypercenter#1776D61
hypocenterC61#21617
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1776

Complements

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

NH ≅ G/Ntype
2. C4×#14519768C2semidirect
4. #21446773C2semidirect
5. #21446588C2semidirect
6. #21446402C2semidirect
7. #21446392C2semidirect
11. #21445717C2semidirect
12. #21445673C2semidirect
15. C61×#1776C2semidirect
21. #16840065C4semidirect
67. #14519768C2×C4semidirect
95. #14519708C2×C4semidirect
97. #14519706C2×C4semidirect
119. #107×C61C2×C4semidirect
309. #1776D61semidirect
365. C61#21617semidirect
410. #107C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168400652
C4×#145197682
C4×#145197392
#214467732
#214465882
#214464022
#214463922
#214460772
#214460672
#214458282
#214457172
#214456732
#214456472
#214456452
C61×#17762
#2161761