Finite Group Databasebeta

← #29699338

Subgroups — #29699338

Order 15616.

942 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839871C211yesmaximal, characteristic
2#21448788C211yesmaximal, characteristic, complemented
3D4×#11758196C211yesmaximal, characteristic, complemented
4#21446516C211yesmaximal, characteristic, complemented
5#21446500C211yesmaximal, characteristic, complemented
6#21446404C211yesmaximal, characteristic, complemented
7#21446395C211yesmaximal, characteristic, complemented
8#21446167C211yesmaximal, characteristic
9#21446165C211yesmaximal, characteristic
10#21445951C211yesmaximal, characteristic, complemented
11#21445827C211yesmaximal, characteristic, complemented
12#21445818C211yesmaximal, characteristic, complemented
13#21445695C211yesmaximal, characteristic, complemented
14#102×Dic61C211yesmaximal, characteristic, complemented
15C61×#1762C211yesmaximal, characteristic, complemented
16C22×#11758199C2211yescharacteristic
17C2×D4×Dic61C2211yescharacteristic
18C2×D4×Dic61C2211yescharacteristic
19C2×#14519739C2211yescharacteristic
20C22×#11758196—21no—
21#16839871C418yescomplemented
22C2×#14519708C2211yescharacteristic
23C2×#14519706C2211yescharacteristic
24C2×#14519699C2211yescharacteristic
25C2×#14519699C2211yescharacteristic
26C2×#14519698C2211yescharacteristic
27C2×#14519697C2211yescharacteristic
28C2×#14519687C2211yescharacteristic
29C2×C8×Dic61C2211yescharacteristic
30#16839664C2211yescharacteristic
31#16839663C2211yescharacteristic
32#16839660C2211yescharacteristic
33#16839658C2211yescharacteristic
34#16839653—21no—
35#16839651—21no—
36#16839635—21no—
37#16839634—21no—
38#16839631C2211yescharacteristic
39#16839627C2211yescharacteristic
40#16839624—21no—
41#16839620—21no—
42C4⋊C4×Dic61C2211yescharacteristic
43C4×#11758196C2211yescharacteristic
44C4×#11758198—21no—
45#16839525C2211yescharacteristic
46#16839525C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839517C2211yescharacteristic
52#16839516C2211yescharacteristic
53C2×C4×D4×C61C2211yescharacteristic
54C2×#102×C61C2211yescharacteristic
55C2×#102×C61C2211yescharacteristic
56C2×#97×C61C2211yescharacteristic
57C4×C4⋊C4×C61C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C61×#340C2211yescharacteristic
60C22×D244C2311yescharacteristic
61C23×Dic61—21no—
62C23×Dic61—21no—
63C22×C4×D61—21no—
64C2×#11758199—22no—
65C2×#11758199—22no—
66C2×#11758199C2×C414yes—
67C22×#92418C2311yescharacteristic
68C22×#92418C2311yescharacteristic
69D4×Dic61—22no—
70D4×Dic61—22no—
71D4×Dic61—41no—
72D4×Dic61C2×C414yes—
73#14519739—22no—
74#14519739C2×C414yescomplemented
75C2×C4⋊Dic61C2311yescharacteristic
76C2×C4⋊Dic61C2311yescharacteristic
77C2×C4⋊Dic61C2311yescharacteristic
78C2×#11758196—21no—
79C2×#11758196—21no—
80C2×#11758196—21no—
81C2×#11758196—21no—
82C2×#11758196—44no—
83C2×#11758196D411yescharacteristic
84C2×#11758196D411yescharacteristic
85C2×C22⋊Dic61—21no—
86C2×C22⋊Dic61—21no—
87C2×#11758198—21no—
88C2×#11758198—21no—
89C2×#11758198—21no—
90C2×#11758198—21no—
91C2×#11758198—21no—
92C2×C4×Dic61—21no—
93C2×C4×Dic61C2311yescharacteristic
94C2×C4×Dic61C2311yescharacteristic
95#14519708C2×C414yescomplemented
96#14519706—22no—
97#14519699—41no—
98#14519699C2×C414yescomplemented
99#14519698—22no—
100#14519697—41no—
101#14519687C2×C414yescomplemented
102C8×Dic61—41no—
103#14519684—21no—
104#14519684—21no—
105#14519684—21no—
106#14519684—21no—
107#14519684—21no—
108#14519684D411yescharacteristic
109#14519684D411yescharacteristic
110C22×D4×C61C2311yescharacteristic
111C23×C4×C61—21no—
112C22×C8×C61C2311yescharacteristic
113C22×C8×C61C2311yescharacteristic
114C4×D4×C61—22no—
115C4×D4×C61—41no—
116C2×C4⋊C4×C61C2311yescharacteristic
117C2×C4⋊C4×C61C2311yescharacteristic
118C2×C4⋊C4×C61C2311yescharacteristic
119C2×C22⋊C4×C61—21no—
120C2×C42×C61—21no—
121C2×C42×C61C2311yescharacteristic
122#102×C61—22no—
123#102×C61C2×C414yes—
124#97×C61—41no—
125#95×C61—21no—
126C24×C61—21no—
127C23×D61—21no—
128C2×D4×C61—21no—
129C2×D4×C61—21no—
130C2×D4×C61—21no—
131C2×D4×C61C22×C412yes—
132C2×D4×C61#4112yes—
133C2×D4×C61#4112yes—
134C22×C4×C61—21no—
135C22×C4×C61—21no—
136C22×C4×C61—21no—
137C22×C4×C61—42no—
138C22×C4×C61—42no—
139C22×C4×C61—81no—
140C22×C4×C61C2×D411yescharacteristic
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61C2411yescharacteristic
143C22×Dic61—21no—
144C22×Dic61—24no—
145C22×Dic61—21no—
146C22×Dic61—21no—
147C22×Dic61—42no—
148C22×Dic61—42no—
149C22×Dic61—42no—
150C22×Dic61—42no—
151C22×Dic61—81no—
152C22×Dic61C2×D411yescharacteristic
153C22×Dic61C2×D411yescharacteristic
154C22×Dic61C2×D411yescharacteristic
155C22×Dic61C2×D411yescharacteristic
156C2×D244—21no—
157C2×D244—21no—
158C2×D244—21no—
159C2×D244—21no—
160C2×D244—21no—
161C2×D244C22×C412yes—
162C2×C4×D61—24no—
163C2×C4×D61—81no—
164C2×C4×D61—81no—
165C2×C8×C61—21no—
166C2×C8×C61—21no—
167C2×C8×C61—21no—
168C2×C8×C61—21no—
169C2×C8×C61—21no—
170C2×C8×C61C22×C412yes—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61C22×C412yes—
177C22⋊C4×C61—42no—
178C42×C61—21no—
179C42×C61—21no—
180C42×C61—81no—
181C22⋊Dic61—24no—
182C22⋊Dic61—42no—
183#11758199—28no—
184#11758198—24no—
185#11758198—42no—
186#11758198—42no—
187#11758198—81no—
188#11758198—81no—
189C4⋊Dic61—21no—
190C4⋊Dic61—21no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61C22×C412yes—
195#11758196—22no—
196#11758196—41no—
197#11758196—42no—
198#11758196—42no—
199#11758196—42no—
200#11758196—81no—
201C4×Dic61—21no—
202C4×Dic61—21no—
203C4×Dic61—81no—
204C4×Dic61C22×C412yes—
205C4×Dic61#4112yes—
206C2×#92418—21no—
207C2×#92418—21no—
208C2×#92418—21no—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418C22×C412yes—
212C23×C61—22no—
213C23×C61—22no—
214C23×C61—22no—
215C23×C61—41no—
216C23×C61—41no—
217C23×C61—41no—
218C23×C61—41no—
219C23×C61C22×D411yescharacteristic
220C22×D61—22no—
221C22×D61—41no—
222C22×D61—41no—
223C22×D61—41no—
224C22×D61—41no—
225C22×D61—41no—
226C22×D61—41no—
227D4×C61—22no—
228D4×C61—22no—
229D4×C61—22no—
230D4×C61#11714yes—
231C2×C4×C61—22no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61—41no—
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—84no—
247C2×C4×C61C23×C411yescharacteristic
248C2×C4×C61C2×#4111yescharacteristic
249C2×C4×C61C2×#4111yescharacteristic
250C2×C4×C61C2×#4111yescharacteristic
251C2×C4×C61H(5,𝔽2)11yescharacteristic
252C2×C4×C612−1+411yescharacteristic
253C2×Dic61—21no—
254C2×Dic61—21no—
255C2×Dic61—21no—
256C2×Dic61—21no—
257C2×Dic61—21no—
258C2×Dic61—21no—
259C2×Dic61—22no—
260C2×Dic61—21no—
261C2×Dic61—21no—
262C2×Dic61—21no—
263C2×Dic61—48no—
264C2×Dic61—41no—
265C2×Dic61—41no—
266C2×Dic61—41no—
267C2×Dic61—41no—
268C2×Dic61—41no—
269C2×Dic61—41no—
270C2×Dic61—41no—
271C2×Dic61—41no—
272C2×Dic61—84no—
273C2×Dic61C4×D412yes—
274C2×Dic61C4×D412yes—
275C2×Dic61AGL(1,ℤ/8)12yes—
276D244—22no—
277D244—22no—
278D244—41no—
279D244—41no—
280C4×D61—84no—
281C8×C61—22no—
282C8×C61—41no—
283#92418—22no—
284#92418—41no—
285#22774—611nomaximal, Sylow
286C22×C61—24no—
287C22×C61—42no—
288C22×C61—42no—
289C22×C61—42no—
290C22×C61—42no—
291C22×C61—42no—
292C22×C61—42no—
293C22×C61C2×C4×D411yescharacteristic
294C22×C61#54611yescharacteristic
295C22×C61#54811yescharacteristic
296C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
297C22×C61#57511yescharacteristic
298C22×C61#57611yescharacteristic
299C22×C61#57711yescharacteristic
300C2×D61—42no—
301C2×D61—42no—
302C2×D61—42no—
303C2×D61—81no—
304C2×D61—81no—
305C2×D61—81no—
306C2×D61—81no—
307C4×C61—42no—
308C4×C61—41no—
309C4×C61—41no—
310C4×C61—81no—
311C4×C61—81no—
312C4×C61C2×#11711yescharacteristic
313C4×C61#52011yescharacteristic
314C4×C61#55211yescharacteristic
315C4×C61#55311yescharacteristic
316Dic61—22no—
317Dic61—22no—
318Dic61—22no—
319Dic61—42no—
320Dic61—41no—
321Dic61—81no—
322Dic61—81no—
323C2×#442—611no—
324#2357—611no—
325D4×C4⋊C4—611no—
326#1936—611no—
327#1934—611no—
328#1920—611no—
329#1919—611no—
330#1879—611no—
331#1879—611no—
332#1836—611no—
333#1804—611no—
334#1802—611no—
335#1771—611no—
336#1762D6111yescharacteristic, complemented
337C4×#102—611no—
338C2×C61—44no—
339C2×C61#235211yescharacteristic
340C2×C61C4×AGL(1,ℤ/8)11yescharacteristic
341C2×C61#294611yescharacteristic
342C2×C61#330711yescharacteristic
343C2×C61#331811yescharacteristic
344C2×C61#337011yescharacteristic
345C2×C61#337311yescharacteristic
346D61—82no—
347C22×D8—611no—
348C2×C4×D4—611no—
349C2×C4×D4—611no—
350C2×C4×D4—611no—
351C2×C4×D4C2×D6111yescharacteristic
352C22×C4⋊C4—1221no—
353#442—618no—
354C2×#106—611no—
355C2×C4⋊C8—611no—
356C2×#102—611no—
357C2×#102—611no—
358C2×#102—611no—
359C2×#102—611no—
360C2×#102C2×D6111yescharacteristic
361C2×#102C2×D6111yescharacteristic
362C2×#97—611no—
363C2×#97C2×D6111yescharacteristic
364C2×C4×C8—611no—
365#390—611no—
366#390—611no—
367#389—611no—
368#389—611no—
369#388—1221no—
370#388—1221no—
371#385—1221no—
372#385—1221no—
373#384—611no—
374#382—611no—
375#381—1221no—
376#380—1221no—
377C4×C4⋊C4—611no—
378C4×C4⋊C4—611no—
379C4×C4⋊C4C2×D6111yescharacteristic
380C4×C22⋊C4—1221no—
381#340—611no—
382#340—611no—
383#340—611no—
384#340—611no—
385#340—611no—
386#340—611no—
387#340C2×D6111yescharacteristic
388#340C2×D6111yescharacteristic
389#336—611no—
390#336—611no—
391C61#2277411yescharacteristic, complemented, Sylow
392C22×D4—611no—
393C22×D4C22×D6111yescharacteristic
394C23×C4—21no—
395C23×C4—1221no—
396C23×C4—1221no—
397C23×C4—1221no—
398C2×D8—614no—
399C2×D8—1222no—
400C2×D8—1222no—
401C22×C8—611no—
402C22×C8—611no—
403C22×C8C22×D6111yescharacteristic
404C22×C8C22×D6111yescharacteristic
405C4×D4—22no—
406C4×D4—41no—
407C4×D4—614no—
408C4×D4—614no—
409C4×D4—1222no—
410C4×D4—1222no—
411C4×D4—1222no—
412C4×D4—2441no—
413C2×C4⋊C4—611no—
414C2×C4⋊C4—611no—
415C2×C4⋊C4—611no—
416C2×C4⋊C4—611no—
417C2×C4⋊C4—611no—
418C2×C4⋊C4—1221no—
419C2×C4⋊C4—1221no—
420C2×C4⋊C4—1221no—
421C2×C4⋊C4—1221no—
422C2×C4⋊C4—2444no—
423C2×C4⋊C4C22×D6111yescharacteristic
424C2×C4⋊C4C22×D6111yescharacteristic
425C2×C4⋊C4C22×D6111yescharacteristic
426C2×C22⋊C4—21no—
427C2×C22⋊C4—1221no—
428C2×C22⋊C4—1221no—
429C2×C22⋊C4—1221no—
430C2×C22⋊C4—1221no—
431C2×C22⋊C4—1221no—
432C2×C22⋊C4—1221no—
433C2×C22⋊C4—1221no—
434C2×C42—21no—
435C2×C42—611no—
436C2×C42—611no—
437C2×C42—1221no—
438C2×C42C22×D6111yescharacteristic
439#106—614no—
440C4⋊C8—1222no—
441#102—22no—
442#102—614no—
443#102—1222no—
444#102—2441no—
445#102—2441no—
446#102C4×D6114yes—
447#97—41no—
448#97—614no—
449C4×C8—2441no—
450#95—21no—
451#95—611no—
452#95—611no—
453#95—1221no—
454#95—1221no—
455#95—1221no—
456#95—1221no—
457#95—1221no—
458C24—21no—
459C24—1221no—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4—21no—
463C2×D4—612no—
464C2×D4—1221no—
465C2×D4—1221no—
466C2×D4—1221no—
467C2×D4—1221no—
468C2×D4—1221no—
469C2×D4C2×C4×D6112yes—
470C2×D4#1175822212yes—
471C2×D4#1175822212yes—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4—81no—
478C22×C4—611no—
479C22×C4—611no—
480C22×C4—611no—
481C22×C4—611no—
482C22×C4—1224no—
483C22×C4—1224no—
484C22×C4—1221no—
485C22×C4—1221no—
486C22×C4—1221no—
487C22×C4—4881no—
488C22×C4—4881no—
489C22×C4—4881no—
490C22×C4—2442no—
491C22×C4—2442no—
492C22×C4—2442no—
493C22×C4—2442no—
494C22×C4D4×D6111yescharacteristic
495C22×C4D4×D6111yescharacteristic
496C22×C4C23×D6111yescharacteristic
497D8—1228no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—612no—
504C2×C8—1221no—
505C2×C8—1221no—
506C2×C8—1221no—
507C2×C8—1221no—
508C2×C8—1221no—
509C2×C8C2×C4×D6112yes—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—612no—
516C4⋊C4—1222no—
517C4⋊C4—1221no—
518C4⋊C4—1221no—
519C4⋊C4—1221no—
520C4⋊C4—1221no—
521C4⋊C4—1221no—
522C4⋊C4—4881no—
523C4⋊C4—2441no—
524C4⋊C4—2442no—
525C4⋊C4—2442no—
526C4⋊C4—2442no—
527C4⋊C4C2×C4×D6112yes—
528C22⋊C4—42no—
529C22⋊C4—1224no—
530C22⋊C4—1224no—
531C22⋊C4—4881no—
532C22⋊C4—4881no—
533C22⋊C4—2442no—
534C22⋊C4—2442no—
535C22⋊C4—2442no—
536C42—21no—
537C42—21no—
538C42—81no—
539C42—612no—
540C42—612no—
541C42—1221no—
542C42—1221no—
543C42—4881no—
544C23—22no—
545C23—22no—
546C23—22no—
547C23—41no—
548C23—41no—
549C23—41no—
550C23—41no—
551C23—1222no—
552C23—2441no—
553C23—2441no—
554C23—2441no—
555C23—2441no—
556C23—2441no—
557C23—2441no—
558C23C2×D4×D6111yescharacteristic
559D4—22no—
560D4—22no—
561D4—22no—
562D4—1222no—
563D4—1222no—
564D4—2441no—
565D4—2441no—
566D4#1451973414yes—
567C2×C4—22no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—84no—
583C2×C4—612no—
584C2×C4—612no—
585C2×C4—612no—
586C2×C4—1221no—
587C2×C4—1221no—
588C2×C4—1221no—
589C2×C4—1221no—
590C2×C4—1221no—
591C2×C4—1221no—
592C2×C4—1222no—
593C2×C4—1221no—
594C2×C4—1221no—
595C2×C4—1221no—
596C2×C4—4884no—
597C2×C4—4884no—
598C2×C4—2448no—
599C2×C4—2441no—
600C2×C4—2441no—
601C2×C4—2441no—
602C2×C4—2441no—
603C2×C4—2441no—
604C2×C4—2441no—
605C2×C4—2441no—
606C2×C4—2441no—
607C2×C4C22×C4×D6111yescharacteristic
608C2×C4#41×D6111yescharacteristic
609C2×C4C2×#1175822211yescharacteristic
610C2×C4C2×#1175822211yescharacteristic
611C2×C4#1451982211yescharacteristic
612C2×C4#1451982411yescharacteristic
613C8—22no—
614C8—41no—
615C8—1222no—
616C8—2441no—
617C22—24no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—42no—
622C22—42no—
623C22—42no—
624C22—4881no—
625C22—4881no—
626C22—4881no—
627C22—4881no—
628C22—2442no—
629C22—2442no—
630C22—2442no—
631C22C4×D4×D6111yescharacteristic
632C22#1684040211yescharacteristic
633C22#1684042111yescharacteristic
634C22AGL(1,ℤ/8)×D6111yescharacteristic
635C22#1684056111yescharacteristic
636C22#1684056611yescharacteristic
637C22#1684057211yescharacteristic
638C4—41no—
639C4—42no—
640C4—41no—
641C4—81no—
642C4—81no—
643C4—1222no—
644C4—1222no—
645C4—1222no—
646C4—4881no—
647C4—4881no—
648C4—2442no—
649C4—2441no—
650C4C2×#1451973411yescharacteristic
651C4#1684024111yescharacteristic
652C4#1684044511yescharacteristic
653C4#1684046111yescharacteristic
654C2—44no—
655C2—4882no—
656C2#2144874211yescharacteristic
657C2#2145462411yescharacteristic
658C2#2145464811yescharacteristic
659C2#2145779611yescharacteristic
660C2#2145796111yescharacteristic
661C2#2145857311yescharacteristic
662C2#2145861211yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1762C2
socleC23×C61C22×D4
prime radicalC22×#11758199C22
cosocleC22×C4×C61C24
hypercenter#1762D61
hypocenterC61#22774
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1762

Complements

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

NH ≅ G/Ntype
2. #21448788C2semidirect
3. D4×#11758196C2semidirect
4. #21446516C2semidirect
5. #21446500C2semidirect
6. #21446404C2semidirect
7. #21446395C2semidirect
10. #21445951C2semidirect
11. #21445827C2semidirect
12. #21445818C2semidirect
13. #21445695C2semidirect
14. #102×Dic61C2semidirect
15. C61×#1762C2semidirect
21. #16839871C4semidirect
74. #14519739C2×C4semidirect
95. #14519708C2×C4semidirect
98. #14519699C2×C4semidirect
101. #14519687C2×C4semidirect
336. #1762D61semidirect
391. C61#22774semidirect

Maximal subgroups

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

subgroupindex
C2×#168398712
#214487882
D4×#117581962
#214465162
#214465002
#214464042
#214463952
#214461672
#214461652
#214459512
#214458272
#214458182
#214456952
#102×Dic612
C61×#17622
#2277461