Finite Group Databasebeta

← #29687263

Subgroups — #29687263

Order 15616.

900 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×#16840080C211yesmaximal, characteristic
2#21448508C211yesmaximal, characteristic
3#21448176C211yesmaximal, characteristic, complemented
4#21446444C211yesmaximal, characteristic, complemented
5#21446437C211yesmaximal, characteristic, complemented
6#21446402C211yesmaximal, characteristic, complemented
7#21446395C211yesmaximal, characteristic, complemented
8#21446139C211yesmaximal, characteristic
9#21446016C211yesmaximal, characteristic, complemented
10#21445715C211yesmaximal, characteristic, complemented
11#21445707C211yesmaximal, characteristic, complemented
12#21445643C211yesmaximal, characteristic, complemented
13#21445633C211yesmaximal, characteristic, complemented
14#21445587C211yesmaximal, characteristic, complemented
15C61×#1823C211yesmaximal, characteristic, complemented
16C2×#14519751C2211yescharacteristic
17C2×#14519739C2211yescharacteristic
18C2×#14519736C2211yescharacteristic
19C2×#14519736C2211yescharacteristic
20#16840080C418yescomplemented
21C2×#14519710C2211yescharacteristic
22C2×#14519708C2211yescharacteristic
23C2×#14519708C2211yescharacteristic
24C2×#14519708C2211yescharacteristic
25C2×#14519699C2211yescharacteristic
26C2×#14519699C2211yescharacteristic
27C2×#14519697C2211yescharacteristic
28C2×#14519697C2211yescharacteristic
29C2×C22⋊#92418C2211yescharacteristic
30C2×#14519687C2211yescharacteristic
31C2×C8×Dic61C2211yescharacteristic
32#16839663C2211yescharacteristic
33#16839662—21no—
34#16839658C2211yescharacteristic
35#16839654—21no—
36#16839638—21no—
37#16839636—21no—
38C4⋊C4×Dic61—21no—
39C4⋊C4×Dic61—21no—
40C4⋊C4×Dic61C2211yescharacteristic
41C4⋊C4×Dic61C2211yescharacteristic
42C4⋊C4×Dic61C2211yescharacteristic
43C4⋊C4×Dic61C2211yescharacteristic
44C4⋊C4×Dic61C2211yescharacteristic
45C4×C22⋊Dic61—21no—
46C4×C22⋊Dic61—21no—
47#16839525C2211yescharacteristic
48#16839525C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839516C2211yescharacteristic
52#16839516C2211yescharacteristic
53C22×C4⋊C4×C61C2211yescharacteristic
54C2×C22⋊C8×C61C2211yescharacteristic
55C61×#380C2211yescharacteristic
56C61×#340C2211yescharacteristic
57C61×#340C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C61×#340C2211yescharacteristic
60C22×D244C2311yescharacteristic
61C22×C22⋊D61—21no—
62C22×C4×D61—21no—
63C22×#92418C2311yescharacteristic
64C22×#92418C2311yescharacteristic
65#14519751—22no—
66#14519751C2×C414yes—
67#14519739—22no—
68#14519739—22no—
69#14519736—22no—
70#14519736—22no—
71#14519736—22no—
72#14519736—22no—
73C2×C4⋊Dic61D411yescharacteristic
74C2×C4⋊Dic61D411yescharacteristic
75C2×C4⋊Dic61C2311yescharacteristic
76C2×C4⋊Dic61C2311yescharacteristic
77C2×C4⋊Dic61C2311yescharacteristic
78C2×#11758196—21no—
79C2×C22⋊Dic61—21no—
80C2×C22⋊Dic61D411yescharacteristic
81C2×C22⋊Dic61D411yescharacteristic
82C2×#11758198—21no—
83C2×#11758198—21no—
84C2×#11758198—21no—
85C2×#11758198—21no—
86C2×C4×Dic61—21no—
87C2×C4×Dic61—21no—
88C2×C4×Dic61—21no—
89C2×C4×Dic61—21no—
90C2×C4×Dic61—21no—
91C2×C4×Dic61—21no—
92C2×C4×Dic61C2311yescharacteristic
93C2×C4×Dic61C2311yescharacteristic
94#14519710—22no—
95#14519708—22no—
96#14519708C2×C414yescomplemented
97#14519708C2×C414yescomplemented
98#14519699—22no—
99#14519699—22no—
100#14519697C2×C414yes—
101#14519697C2×C414yes—
102C22⋊#92418C2×C414yescomplemented
103#14519687—22no—
104C8×Dic61—22no—
105#14519684—21no—
106#14519684—21no—
107#14519684—21no—
108#14519684—21no—
109#14519684—21no—
110#14519684—21no—
111C23×C4×C61—21no—
112C23×C4×C61C2311yescharacteristic
113C22×C8×C61C2311yescharacteristic
114C22×C8×C61C2311yescharacteristic
115C2×C4⋊C4×C61—21no—
116C2×C4⋊C4×C61—22no—
117C2×C4⋊C4×C61C2×C414yescomplemented
118C2×C4⋊C4×C61C2311yescharacteristic
119C2×C4⋊C4×C61C2311yescharacteristic
120C2×C4⋊C4×C61C2311yescharacteristic
121C2×C4⋊C4×C61C2311yescharacteristic
122C2×C22⋊C4×C61—21no—
123C22⋊C8×C61—22no—
124#95×C61—21no—
125C24×C61C2×D411yescharacteristic
126C23×D61—21no—
127C22×C4×C61—21no—
128C22×C4×C61—21no—
129C22×C4×C61—21no—
130C22×C4×C61—21no—
131C22×C4×C61—21no—
132C22×C4×C61—21no—
133C22×C4×C61—24no—
134C22×C4×C61—21no—
135C22×C4×C61—21no—
136C22×C4×C61—81no—
137C22×C4×C61—81no—
138C22×C4×C61C22×C412yes—
139C22×C4×C61C2×D411yescharacteristic
140C22×C4×C61#4112yes—
141C22×C4×C61C2411yescharacteristic
142C2×C22⋊D61—24no—
143C2×C22⋊D61—81no—
144C2×C22⋊D61—81no—
145C22×Dic61—21no—
146C22×Dic61—21no—
147C22×Dic61—21no—
148C22×Dic61C2×D411yescharacteristic
149C22×Dic61C2×D411yescharacteristic
150C22×Dic61C2×D411yescharacteristic
151C22×Dic61C2×D411yescharacteristic
152C2×D244—21no—
153C2×D244—21no—
154C2×D244—21no—
155C2×D244—21no—
156C2×D244—21no—
157C2×D244C22×C412yes—
158C2×C4×D61—42no—
159C2×C4×D61—81no—
160C2×C4×D61—81no—
161C2×C8×C61—21no—
162C2×C8×C61—21no—
163C2×C8×C61—21no—
164C2×C8×C61—21no—
165C2×C8×C61—21no—
166C2×C8×C61—21no—
167C4⋊C4×C61—24no—
168C4⋊C4×C61—21no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61C22×C412yes—
175C4⋊C4×C61C22×C412yes—
176C22⋊C4×C61—81no—
177C22⋊Dic61—24no—
178C22⋊Dic61—22no—
179C22⋊Dic61—22no—
180#11758198—24no—
181#11758198—42no—
182#11758198—81no—
183#11758198—81no—
184C4⋊Dic61—21no—
185C4⋊Dic61—21no—
186C4⋊Dic61—21no—
187C4⋊Dic61—22no—
188C4⋊Dic61—22no—
189C4⋊Dic61C22×C412yes—
190C4⋊Dic61#4112yes—
191C4⋊Dic61#4112yes—
192#11758196—81no—
193C4×Dic61—24no—
194C4×Dic61—21no—
195C4×Dic61—21no—
196C4×Dic61—81no—
197C4×Dic61—81no—
198C4×Dic61—81no—
199C4×Dic61—81no—
200C4×Dic61—81no—
201C4×Dic61#4112yes—
202C4×Dic61#4112yes—
203C2×#92418—21no—
204C2×#92418—21no—
205C2×#92418—21no—
206C2×#92418—21no—
207C2×#92418C22×C412yes—
208C2×#92418C22×C412yes—
209C23×C61—21no—
210C23×C61—21no—
211C23×C61—21no—
212C23×C61—21no—
213C23×C61—21no—
214C23×C61—21no—
215C23×C61C4×D412yes—
216C23×C61C22×D411yescharacteristic
217C22×D61—22no—
218C22×D61—41no—
219C22×D61—41no—
220C22×D61—41no—
221C22×D61—41no—
222C22×D61—41no—
223C22×D61—41no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—22no—
227C2×C4×C61—22no—
228C2×C4×C61—22no—
229C2×C4×C61—22no—
230C2×C4×C61—22no—
231C2×C4×C61—22no—
232C2×C4×C61—41no—
233C2×C4×C61—41no—
234C2×C4×C61—41no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61—41no—
238C2×C4×C61—41no—
239C2×C4×C61—41no—
240C2×C4×C61—41no—
241C2×C4×C61—41no—
242C2×C4×C61—84no—
243C2×C4×C61C4×D412yes—
244C2×C4×C61C23×C411yescharacteristic
245C2×C4×C61C2×#4111yescharacteristic
246C2×C4×C61C2×#4111yescharacteristic
247C2×C4×C61C2×#4111yescharacteristic
248C2×C4×C61C2×#4111yescharacteristic
249C2×C4×C61C2×#4111yescharacteristic
250C22⋊D61—84no—
251C2×Dic61—21no—
252C2×Dic61—21no—
253C2×Dic61—21no—
254C2×Dic61—21no—
255C2×Dic61—22no—
256C2×Dic61—21no—
257C2×Dic61—21no—
258C2×Dic61—22no—
259C2×Dic61—22no—
260C2×Dic61—21no—
261C2×Dic61—21no—
262C2×Dic61—21no—
263C2×Dic61—21no—
264C2×Dic61—41no—
265C2×Dic61—41no—
266C2×Dic61—41no—
267C2×Dic61—41no—
268C2×Dic61—41no—
269C2×Dic61—41no—
270C2×Dic61#12312yes—
271C2×Dic61#12312yes—
272D244—22no—
273D244—22no—
274D244—41no—
275D244—41no—
276C4×D61—84no—
277C8×C61—41no—
278C8×C61—41no—
279#92418—22no—
280#92418—22no—
281#21584—611nomaximal, Sylow
282C22×C61—22no—
283C22×C61—22no—
284C22×C61—22no—
285C22×C61—41no—
286C22×C61—41no—
287C22×C61—41no—
288C22×C61—41no—
289C22×C61C2×C4×D411yescharacteristic
290C22×C61C2×#12311yescharacteristic
291C22×C61#52511yescharacteristic
292C22×C61#57611yescharacteristic
293C22×C61#57711yescharacteristic
294C22×C61#57711yescharacteristic
295C22×C61#57711yescharacteristic
296C2×D61—42no—
297C2×D61—42no—
298C2×D61—42no—
299C2×D61—81no—
300C2×D61—81no—
301C2×D61—81no—
302C2×D61—81no—
303C4×C61—22no—
304C4×C61—42no—
305C4×C61—42no—
306C4×C61—81no—
307C4×C61—81no—
308C4×C61C4×#4111yescharacteristic
309C4×C61C4×#4111yescharacteristic
310C4×C61#52911yescharacteristic
311C4×C61#52911yescharacteristic
312Dic61—22no—
313Dic61—22no—
314Dic61—41no—
315Dic61—41no—
316Dic61—42no—
317Dic61—42no—
318Dic61—42no—
319C2×#481—611no—
320#2324—611no—
321#2274—611no—
322#1926—611no—
323#1925—611no—
324#1920—611no—
325#1919—611no—
326#1875—611no—
327#1854—611no—
328#1823D6111yescharacteristic, complemented
329#1776—611no—
330C4×#106—611no—
331#1762—611no—
332C4×#102—611no—
333#1750—611no—
334C2×C61—42no—
335C2×C61C4×#12311yescharacteristic
336C2×C61#294611yescharacteristic
337C2×C61#294911yescharacteristic
338C2×C61#309611yescharacteristic
339C2×C61#309911yescharacteristic
340C2×C61#311911yescharacteristic
341C2×C61#312511yescharacteristic
342D61—82no—
343C2×#121—611no—
344C2×C4×D4—611no—
345C2×#117—611no—
346C2×#117—611no—
347C22×C4⋊C4C2×D6111yescharacteristic
348#481—618no—
349C2×#107—611no—
350C2×#106—611no—
351C2×#106—611no—
352C2×#106—611no—
353C2×#102—611no—
354C2×#102—611no—
355C2×#102—611no—
356C2×#102—611no—
357C2×C22⋊C8—611no—
358C2×C22⋊C8C2×D6111yescharacteristic
359C2×#97—611no—
360C2×C4×C8—611no—
361#390—611no—
362#390—1221no—
363#389—611no—
364#388—1221no—
365#386—1221no—
366#385—1221no—
367#380C2×D6111yescharacteristic
368C4×C4⋊C4—611no—
369C4×C4⋊C4—611no—
370C4×C4⋊C4—611no—
371C4×C4⋊C4—611no—
372C4×C4⋊C4—611no—
373C4×C4⋊C4—1221no—
374C4×C4⋊C4—1221no—
375C4×C22⋊C4—1221no—
376C4×C22⋊C4—1221no—
377#340—611no—
378#340—611no—
379#340—611no—
380#340—611no—
381#340C2×D6111yescharacteristic
382#340C2×D6111yescharacteristic
383#340C2×D6111yescharacteristic
384#340C2×D6111yescharacteristic
385#336—611no—
386#336—611no—
387C61#2158411yescharacteristic, complemented, Sylow
388C22×D4—611no—
389C22×D4—1221no—
390C23×C4—21no—
391C23×C4—1221no—
392C23×C4C22×D6111yescharacteristic
393C22×C8—611no—
394C22×C8—611no—
395C22×C8C22×D6111yescharacteristic
396C22×C8C22×D6111yescharacteristic
397#121—614no—
398#121—1222no—
399C4×D4—1222no—
400C4×D4—1222no—
401#117—1222no—
402#117—1222no—
403#117—1222no—
404#117—1222no—
405C2×C4⋊C4—21no—
406C2×C4⋊C4—22no—
407C2×C4⋊C4—611no—
408C2×C4⋊C4—611no—
409C2×C4⋊C4—611no—
410C2×C4⋊C4—611no—
411C2×C4⋊C4—611no—
412C2×C4⋊C4—1221no—
413C2×C4⋊C4C4×D6114yescomplemented
414C2×C4⋊C4C22×D6111yescharacteristic
415C2×C4⋊C4C22×D6111yescharacteristic
416C2×C4⋊C4C22×D6111yescharacteristic
417C2×C4⋊C4C22×D6111yescharacteristic
418C2×C22⋊C4—21no—
419C2×C22⋊C4—611no—
420C2×C22⋊C4—611no—
421C2×C22⋊C4—1221no—
422C2×C22⋊C4—1221no—
423C2×C22⋊C4—1221no—
424C2×C22⋊C4—1221no—
425C2×C22⋊C4—1221no—
426C2×C42—611no—
427C2×C42—611no—
428C2×C42—1221no—
429C2×C42—1221no—
430C2×C42—1221no—
431C2×C42—1221no—
432C2×C42—1221no—
433C2×C42—1221no—
434#107—1222no—
435#106—614no—
436#106—614no—
437#106—1222no—
438#102—614no—
439#102—614no—
440#102—1222no—
441#102—1222no—
442C22⋊C8—22no—
443C22⋊C8—614no—
444#97—1222no—
445C4×C8—1222no—
446#95—21no—
447#95—1221no—
448#95—1221no—
449#95—1221no—
450#95—1221no—
451#95—1221no—
452#95—1221no—
453C24—1221no—
454C24D4×D6111yescharacteristic
455C2×D4—612no—
456C2×D4—1224no—
457C2×D4—1221no—
458C2×D4—1221no—
459C2×D4—1221no—
460C2×D4—1221no—
461C2×D4—1221no—
462C2×D4—4881no—
463C2×D4—4881no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—24no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—81no—
474C22×C4—81no—
475C22×C4—611no—
476C22×C4—611no—
477C22×C4—611no—
478C22×C4—611no—
479C22×C4—1221no—
480C22×C4—1221no—
481C22×C4—1221no—
482C22×C4—4881no—
483C22×C4—4881no—
484C22×C4—2442no—
485C22×C4C2×C4×D6112yes—
486C22×C4D4×D6111yescharacteristic
487C22×C4#1175822412yes—
488C22×C4C23×D6111yescharacteristic
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—612no—
496C2×C8—612no—
497C2×C8—1221no—
498C2×C8—1221no—
499C2×C8—1221no—
500C2×C8—1221no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—24no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—612no—
509C4⋊C4—612no—
510C4⋊C4—612no—
511C4⋊C4—1222no—
512C4⋊C4—1222no—
513C4⋊C4—1221no—
514C4⋊C4—1221no—
515C4⋊C4—1221no—
516C4⋊C4—4881no—
517C4⋊C4C2×C4×D6112yes—
518C4⋊C4C2×C4×D6112yes—
519C22⋊C4—81no—
520C22⋊C4—1222no—
521C22⋊C4—1222no—
522C22⋊C4—1224no—
523C22⋊C4—1224no—
524C22⋊C4—4881no—
525C22⋊C4—4881no—
526C22⋊C4—2442no—
527C42—612no—
528C42—612no—
529C42—1224no—
530C42—1221no—
531C42—1221no—
532C42—4881no—
533C42—4881no—
534C42—4881no—
535C42—4881no—
536C42—4881no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—1222no—
544C23—2441no—
545C23—2441no—
546C23—2441no—
547C23—2441no—
548C23—2441no—
549C23—2441no—
550C23#1451973712yes—
551C23C2×D4×D6111yescharacteristic
552D4—1222no—
553D4—1222no—
554D4—4884no—
555D4—2441no—
556D4—2441no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—84no—
576C2×C4—612no—
577C2×C4—612no—
578C2×C4—1221no—
579C2×C4—1221no—
580C2×C4—1221no—
581C2×C4—1221no—
582C2×C4—1222no—
583C2×C4—1221no—
584C2×C4—1221no—
585C2×C4—1222no—
586C2×C4—1222no—
587C2×C4—1221no—
588C2×C4—1221no—
589C2×C4—1221no—
590C2×C4—1221no—
591C2×C4—4884no—
592C2×C4—2441no—
593C2×C4—2441no—
594C2×C4—2441no—
595C2×C4—2441no—
596C2×C4—2441no—
597C2×C4—2441no—
598C2×C4#1451973712yes—
599C2×C4C22×C4×D6111yescharacteristic
600C2×C4#41×D6111yescharacteristic
601C2×C4#41×D6111yescharacteristic
602C2×C4#41×D6111yescharacteristic
603C2×C4#41×D6111yescharacteristic
604C2×C4C2×#1175822411yescharacteristic
605C8—41no—
606C8—41no—
607C8—1222no—
608C8—1222no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—41no—
613C22—41no—
614C22—41no—
615C22—41no—
616C22—4881no—
617C22—4881no—
618C22—4881no—
619C22—4881no—
620C22—2442no—
621C22—2442no—
622C22—2442no—
623C22C2×#1451973711yescharacteristic
624C22#123×D6111yescharacteristic
625C22#1684027011yescharacteristic
626C22#1684056311yescharacteristic
627C22#1684056811yescharacteristic
628C22#1684057111yescharacteristic
629C22#1684057111yescharacteristic
630C4—22no—
631C4—42no—
632C4—42no—
633C4—81no—
634C4—81no—
635C4—1222no—
636C4—1222no—
637C4—2442no—
638C4—2442no—
639C4—2441no—
640C4—2441no—
641C4—2442no—
642C4#1684023011yescharacteristic
643C4#1684023011yescharacteristic
644C4#1684028911yescharacteristic
645C4#1684028911yescharacteristic
646C2—42no—
647C2—4882no—
648C2#2144833611yescharacteristic
649C2#2145464511yescharacteristic
650C2#2145467411yescharacteristic
651C2#2145577011yescharacteristic
652C2#2145580311yescharacteristic
653C2#2145602111yescharacteristic
654C2#2145609111yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1823C2
socleC23×C61C22×D4
prime radicalC2×#14519751C22
cosocleC22×C4×C61C24
hypercenter#1823D61
hypocenterC61#21584
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1823

Complements

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

NH ≅ G/Ntype
3. #21448176C2semidirect
4. #21446444C2semidirect
5. #21446437C2semidirect
6. #21446402C2semidirect
7. #21446395C2semidirect
9. #21446016C2semidirect
10. #21445715C2semidirect
11. #21445707C2semidirect
12. #21445643C2semidirect
13. #21445633C2semidirect
14. #21445587C2semidirect
15. C61×#1823C2semidirect
20. #16840080C4semidirect
96. #14519708C2×C4semidirect
97. #14519708C2×C4semidirect
102. C22⋊#92418C2×C4semidirect
117. C2×C4⋊C4×C61C2×C4semidirect
328. #1823D61semidirect
387. C61#21584semidirect
413. C2×C4⋊C4C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168400802
#214485082
#214481762
#214464442
#214464372
#214464022
#214463952
#214461392
#214460162
#214457152
#214457072
#214456432
#214456332
#214455872
C61×#18232
#2158461