Finite Group Databasebeta

← #29743403

Subgroups — #29743403

Order 15616.

858 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840057C211yesmaximal, characteristic, complemented
2C2×#16839863C211yesmaximal, characteristic
3#21449671C211yesmaximal, characteristic
4Q8×#11758196C211yesmaximal, characteristic, complemented
5#21447345C211yesmaximal, characteristic
6#21447294C211yesmaximal, characteristic, complemented
7#21447193C211yesmaximal, characteristic
8#21447148C211yesmaximal, characteristic, complemented
9#21446561C211yesmaximal, characteristic, complemented
10#21446386C211yesmaximal, characteristic, complemented
11#21446176C211yesmaximal, characteristic
12#21446085C211yesmaximal, characteristic, complemented
13#21446075C211yesmaximal, characteristic, complemented
14#21446064C211yesmaximal, characteristic
15C61×#1880C211yesmaximal, characteristic
16C2×#14519777C2211yescharacteristic
17C2×#14519754—21no—
18C2×Q8×Dic61C2211yescharacteristic
19C2×Q8×Dic61C2211yescharacteristic
20C2×D4×Dic61C2211yescharacteristic
21#16840057—24no—
22#16839863—24no—
23C2×#14519711C2211yescharacteristic
24C2×#14519711C2211yescharacteristic
25C2×#14519710C2211yescharacteristic
26C2×#14519710C2211yescharacteristic
27C2×#14519708C2211yescharacteristic
28C2×C4⋊#92418C2211yescharacteristic
29C2×#14519701C2211yescharacteristic
30C2×#14519701C2211yescharacteristic
31C2×#14519698C2211yescharacteristic
32C2×#14519698C2211yescharacteristic
33C2×#14519687C2211yescharacteristic
34#16839685—21no—
35#16839674—21no—
36#16839665C2211yescharacteristic
37#16839661C2211yescharacteristic
38#16839660—21no—
39#16839660C2211yescharacteristic
40#16839658—21no—
41#16839658C2211yescharacteristic
42#16839658C2211yescharacteristic
43#16839637—21no—
44#16839631—21no—
45#16839631C2211yescharacteristic
46C4×#11758196—21no—
47C4×#11758196C2211yescharacteristic
48#16839525C2211yescharacteristic
49#16839525C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839516C2211yescharacteristic
52#16839516C2211yescharacteristic
53#16839516C2211yescharacteristic
54C22×SD16×C61C2211yescharacteristic
55C2×C4×Q8×C61C2211yescharacteristic
56C2×#103×C61C2211yescharacteristic
57C2×#102×C61C2211yescharacteristic
58C61×#390C2211yescharacteristic
59C61×#340C2211yescharacteristic
60C61×#336C2211yescharacteristic
61C22×Dic122—21no—
62C23×Dic61—21no—
63C22×#92418C2311yescharacteristic
64C22×#92418C2311yescharacteristic
65#14519777—22no—
66#14519777—41no—
67#14519754—44no—
68Q8×Dic61—22no—
69Q8×Dic61—22no—
70Q8×Dic61—22no—
71Q8×Dic61—41no—
72D4×Dic61—22no—
73D4×Dic61—41no—
74C2×C4⋊Dic61—21no—
75C2×C4⋊Dic61—21no—
76C2×C4⋊Dic61—21no—
77C2×C4⋊Dic61—21no—
78C2×C4⋊Dic61C2311yescharacteristic
79C2×C4⋊Dic61C2311yescharacteristic
80C2×C4⋊Dic61C2311yescharacteristic
81C2×C4⋊Dic61C2311yescharacteristic
82C2×#11758196—21no—
83C2×#11758196—21no—
84C2×#11758196—21no—
85C2×#11758196—21no—
86C2×#11758196—21no—
87C2×#11758196D411yescharacteristic
88C2×#11758196D411yescharacteristic
89C2×C22⋊Dic61—21no—
90C2×C22⋊Dic61—21no—
91C2×C22⋊Dic61—21no—
92C2×C4×Dic61—21no—
93C2×C4×Dic61—21no—
94C2×C4×Dic61C2311yescharacteristic
95C2×C4×Dic61C2311yescharacteristic
96#14519711—22no—
97#14519711—41no—
98#14519710—22no—
99#14519710—22no—
100#14519708—22no—
101C4⋊#92418—22no—
102#14519701—22no—
103#14519701D414yes—
104#14519698—22no—
105#14519698—22no—
106#14519687—22no—
107#14519684—21no—
108#14519684—21no—
109#14519684—21no—
110#14519684—21no—
111#14519684D411yescharacteristic
112#14519684D411yescharacteristic
113C22×Q8×C61C2311yescharacteristic
114C22×D4×C61C2311yescharacteristic
115C2×SD16×C61—22no—
116C2×SD16×C61—22no—
117C2×SD16×C61Q814yescomplemented
118C22×C8×C61C2311yescharacteristic
119C22×C8×C61C2311yescharacteristic
120C4×Q8×C61—22no—
121C4×Q8×C61—41no—
122C2×C4⋊C4×C61—21no—
123C2×C4⋊C4×C61C2311yescharacteristic
124C2×C4⋊C4×C61C2311yescharacteristic
125C2×C22⋊C4×C61—21no—
126C2×C22⋊C4×C61—21no—
127C2×C42×C61—21no—
128C2×C42×C61C2311yescharacteristic
129#103×C61—22no—
130#102×C61—41no—
131C24×C61—21no—
132C2×Q8×C61—21no—
133C2×Q8×C61—21no—
134C2×Q8×C61—21no—
135C2×Q8×C61C2×D412yes—
136C2×Q8×C61C2×Q812yes—
137C2×Q8×C61#4112yes—
138C2×D4×C61—21no—
139C2×D4×C61—21no—
140C2×D4×C61—21no—
141C2×D4×C61—21no—
142C2×D4×C61—21no—
143C2×D4×C61C2×Q812yes—
144C22×C4×C61—21no—
145C22×C4×C61—21no—
146C22×C4×C61—21no—
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2411yescharacteristic
150C22×Dic61—21no—
151C22×Dic61—21no—
152C22×Dic61—21no—
153C22×Dic61—21no—
154C22×Dic61—42no—
155C22×Dic61—42no—
156C22×Dic61—81no—
157C22×Dic61C2×D411yescharacteristic
158C22×Dic61C2×D411yescharacteristic
159C22×Dic61C2×D411yescharacteristic
160C22×Dic61C2×D411yescharacteristic
161C2×Dic122—42no—
162C2×Dic122—42no—
163C2×Dic122—81no—
164SD16×C61—28no—
165C2×C8×C61—21no—
166C2×C8×C61—21no—
167C2×C8×C61—21no—
168C2×C8×C61—21no—
169C2×C8×C61—21no—
170C2×C8×C61C2×Q812yes—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—42no—
176C22⋊C4×C61—42no—
177C22⋊C4×C61—42no—
178C42×C61—21no—
179C42×C61—21no—
180C42×C61—42no—
181C22⋊Dic61—42no—
182C22⋊Dic61—42no—
183C22⋊Dic61—81no—
184C4⋊Dic61—24no—
185C4⋊Dic61—21no—
186C4⋊Dic61—21no—
187C4⋊Dic61—21no—
188C4⋊Dic61—21no—
189C4⋊Dic61—21no—
190C4⋊Dic61—21no—
191C4⋊Dic61—42no—
192C4⋊Dic61—42no—
193C4⋊Dic61—42no—
194C4⋊Dic61C2×D412yes—
195C4⋊Dic61#4112yes—
196#11758196—22no—
197#11758196—22no—
198#11758196—42no—
199#11758196—42no—
200#11758196—81no—
201#11758196—81no—
202#11758196—81no—
203C4×Dic61—24no—
204C4×Dic61—21no—
205C4×Dic61—21no—
206C4×Dic61—21no—
207C4×Dic61—21no—
208C4×Dic61—42no—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418C2×D412yes—
215C23×C61—22no—
216C23×C61—41no—
217C23×C61—41no—
218C23×C61—41no—
219C23×C61—41no—
220C23×C61—41no—
221C23×C61—41no—
222C23×C61C22×D411yescharacteristic
223Q8×C61—22no—
224Q8×C61—22no—
225Q8×C61—22no—
226Q8×C61C22⋊Q814yescomplemented
227D4×C61—22no—
228D4×C61—22no—
229D4×C61—41no—
230D4×C61—41no—
231C2×C4×C61—22no—
232C2×C4×C61—22no—
233C2×C4×C61—21no—
234C2×C4×C61—22no—
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#13712yes—
246C2×C4×C61C22×D411yescharacteristic
247C2×C4×C61C22×Q811yescharacteristic
248C2×C4×C61C2×#4111yescharacteristic
249C2×C4×C61C2×#4111yescharacteristic
250C2×C4×C61H(5,𝔽2)11yescharacteristic
251C2×C4×C612−1+411yescharacteristic
252C2×Dic61—21no—
253C2×Dic61—21no—
254C2×Dic61—21no—
255C2×Dic61—22no—
256C2×Dic61—21no—
257C2×Dic61—21no—
258C2×Dic61—22no—
259C2×Dic61—21no—
260C2×Dic61—21no—
261C2×Dic61—21no—
262C2×Dic61—21no—
263C2×Dic61—21no—
264C2×Dic61—21no—
265C2×Dic61—41no—
266C2×Dic61—41no—
267C2×Dic61—41no—
268C2×Dic61—41no—
269C2×Dic61—41no—
270C2×Dic61—41no—
271C2×Dic61—41no—
272C2×Dic61—41no—
273C2×Dic61—41no—
274C2×Dic61—41no—
275C2×Dic61—84no—
276C2×Dic61AGL(1,ℤ/8)12yes—
277Dic122—84no—
278C8×C61—22no—
279C8×C61—41no—
280#92418—22no—
281#92418—41no—
282#26397—611nomaximal, Sylow
283C22×C61—42no—
284C22×C61—42no—
285C22×C61—42no—
286C22×C61—81no—
287C22×C61—81no—
288C22×C61—81no—
289C22×C61—81no—
290C22×C61#52511yescharacteristic
291C22×C61D4211yescharacteristic
292C22×C61D4×Q811yescharacteristic
293C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
294C22×C61C2×#13711yescharacteristic
295C22×C61#57611yescharacteristic
296C22×C61#57811yescharacteristic
297C4×C61—24no—
298C4×C61—22no—
299C4×C61—41no—
300C4×C61—81no—
301C4×C61—81no—
302C4×C61C2×C22⋊Q811yescharacteristic
303C4×C61#53911yescharacteristic
304C4×C61#54711yescharacteristic
305C4×C61#55411yescharacteristic
306Dic61—22no—
307Dic61—41no—
308Dic61—42no—
309Dic61—41no—
310Dic61—42no—
311Dic61—41no—
312Dic61—81no—
313Dic61—81no—
314C2×#477—611no—
315C2×#440—611no—
316#2452—611no—
317Q8×C4⋊C4—611no—
318#2093—611no—
319#2083—611no—
320#2065—611no—
321#2057—611no—
322#1944—611no—
323#1917—611no—
324#1880—611no—
325#1880D6111yescharacteristic
326#1865—611no—
327#1863—611no—
328#1862—611no—
329C2×C61—82no—
330C2×C61#244911yescharacteristic
331C2×C61#311211yescharacteristic
332C2×C61#312411yescharacteristic
333C2×C61#327011yescharacteristic
334C2×C61#327411yescharacteristic
335C2×C61#338511yescharacteristic
336C2×C61#338611yescharacteristic
337C22×SD16C2×D6111yescharacteristic
338C2×C4⋊Q8—611no—
339C2×C22⋊Q8—1221no—
340C2×C4×Q8—611no—
341C2×C4×Q8—611no—
342C2×C4×Q8C2×D6111yescharacteristic
343C2×C4×D4—611no—
344#477—1224no—
345#440—1224no—
346C2×#107—611no—
347C2×#107—611no—
348C2×#107—611no—
349C2×#107—611no—
350C2×#106—611no—
351C2×C4⋊C8—611no—
352C2×#103—611no—
353C2×#103—611no—
354C2×#103C2×D6111yescharacteristic
355C2×#102—611no—
356C2×#102—611no—
357C2×#102C2×D6111yescharacteristic
358C2×#97—611no—
359#396—1221no—
360#393—1221no—
361#390—611no—
362#390C2×D6111yescharacteristic
363#389—611no—
364#389—611no—
365#389—611no—
366#389—611no—
367#389—1221no—
368#389—1221no—
369#385—1221no—
370#384—611no—
371#384—1221no—
372C4×C4⋊C4—611no—
373C4×C4⋊C4—1221no—
374#340—611no—
375#340—611no—
376#340—611no—
377#340C2×D6111yescharacteristic
378#336—611no—
379#336—611no—
380#336—611no—
381#336C2×D6111yescharacteristic
382C61#2639711yescharacteristic, complemented, Sylow
383C22×Q8—1221no—
384C22×Q8C22×D6111yescharacteristic
385C22×D4C22×D6111yescharacteristic
386C23×C4—1221no—
387C2×SD16—22no—
388C2×SD16—22no—
389C2×SD16Dic12214yescomplemented
390C22×C8—611no—
391C22×C8—611no—
392C22×C8C22×D6111yescharacteristic
393C22×C8C22×D6111yescharacteristic
394C4⋊Q8—1222no—
395C4⋊Q8—2441no—
396C22⋊Q8—2444no—
397C4×Q8—22no—
398C4×Q8—41no—
399C4×Q8—1222no—
400C4×Q8—1222no—
401C4×Q8—1222no—
402C4×Q8—2441no—
403C4×D4—1222no—
404C4×D4—2441no—
405C2×C4⋊C4—21no—
406C2×C4⋊C4—611no—
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⋊C4—1221no—
414C2×C4⋊C4—1221no—
415C2×C4⋊C4—1221no—
416C2×C4⋊C4—1221no—
417C2×C4⋊C4—1221no—
418C2×C4⋊C4—1221no—
419C2×C4⋊C4—1221no—
420C2×C4⋊C4—1221no—
421C2×C4⋊C4C22×D6111yescharacteristic
422C2×C4⋊C4C22×D6111yescharacteristic
423C2×C22⋊C4—21no—
424C2×C22⋊C4—21no—
425C2×C22⋊C4—1221no—
426C2×C22⋊C4—1221no—
427C2×C22⋊C4—1221no—
428C2×C42—21no—
429C2×C42—611no—
430C2×C42—611no—
431C2×C42—1221no—
432C2×C42—1221no—
433C2×C42C22×D6111yescharacteristic
434#107—1222no—
435#107—1222no—
436#107—1222no—
437#107—2441no—
438#106—1222no—
439C4⋊C8—1222no—
440#103—22no—
441#103—614no—
442#103—1222no—
443#102—41no—
444#102—1222no—
445#102—1222no—
446#97—1222no—
447#95—611no—
448#95—611no—
449#95—1221no—
450#95—1221no—
451#95—1221no—
452#95—1221no—
453C24—21no—
454C2×Q8—21no—
455C2×Q8—21no—
456C2×Q8—21no—
457C2×Q8—4881no—
458C2×Q8—2442no—
459C2×Q8—2442no—
460C2×Q8C2×Dic12212yes—
461C2×Q8D4×D6112yes—
462C2×Q8#1175822412yes—
463C2×D4—21no—
464C2×D4—21no—
465C2×D4—21no—
466C2×D4—21no—
467C2×D4—21no—
468C2×D4C2×Dic12212yes—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—611no—
473C22×C4—611no—
474C22×C4—611no—
475C22×C4—611no—
476C22×C4—1221no—
477C22×C4—1221no—
478C22×C4—1221no—
479C22×C4—1221no—
480C22×C4—4881no—
481C22×C4—2442no—
482C22×C4—2442no—
483C22×C4D4×D6111yescharacteristic
484C22×C4D4×D6111yescharacteristic
485C22×C4C23×D6111yescharacteristic
486SD16—28no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—612no—
493C2×C8—1221no—
494C2×C8—1221no—
495C2×C8—1221no—
496C2×C8—1221no—
497C2×C8—1221no—
498C2×C8C2×Dic12212yes—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—42no—
504C4⋊C4—612no—
505C4⋊C4—612no—
506C4⋊C4—1222no—
507C4⋊C4—1224no—
508C4⋊C4—1221no—
509C4⋊C4—1221no—
510C4⋊C4—1222no—
511C4⋊C4—1221no—
512C4⋊C4—1221no—
513C4⋊C4—1221no—
514C4⋊C4—1221no—
515C4⋊C4—4881no—
516C4⋊C4—4881no—
517C4⋊C4—4881no—
518C4⋊C4—2442no—
519C4⋊C4—2442no—
520C4⋊C4—2442no—
521C4⋊C4—2442no—
522C4⋊C4—2442no—
523C22⋊C4—42no—
524C22⋊C4—42no—
525C22⋊C4—4881no—
526C22⋊C4—2442no—
527C22⋊C4—2442no—
528C42—21no—
529C42—21no—
530C42—42no—
531C42—1224no—
532C42—1221no—
533C42—1221no—
534C42—1221no—
535C42—1221no—
536C42—2442no—
537C23—22no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23C2×D4×D6111yescharacteristic
545Q8—22no—
546Q8—22no—
547Q8—22no—
548Q8—4884no—
549Q8#1451975414yescomplemented
550D4—22no—
551D4—22no—
552D4—41no—
553D4—41no—
554C2×C4—22no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—612no—
569C2×C4—1221no—
570C2×C4—1221no—
571C2×C4—1222no—
572C2×C4—1221no—
573C2×C4—1221no—
574C2×C4—1221no—
575C2×C4—1221no—
576C2×C4—1221no—
577C2×C4—1221no—
578C2×C4—1221no—
579C2×C4—1221no—
580C2×C4—1221no—
581C2×C4—1222no—
582C2×C4—4884no—
583C2×C4—2441no—
584C2×C4—2441no—
585C2×C4—2441no—
586C2×C4—2441no—
587C2×C4—2441no—
588C2×C4—2441no—
589C2×C4—2441no—
590C2×C4—2441no—
591C2×C4—2441no—
592C2×C4—2441no—
593C2×C4#1451980612yes—
594C2×C4C2×D4×D6111yescharacteristic
595C2×C4C22×Dic12211yescharacteristic
596C2×C4#41×D6111yescharacteristic
597C2×C4C2×#1175822411yescharacteristic
598C2×C4#1451982211yescharacteristic
599C2×C4#1451982411yescharacteristic
600C8—22no—
601C8—41no—
602C8—1222no—
603C8—2441no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—81no—
608C22—81no—
609C22—81no—
610C22—81no—
611C22#1684027211yescharacteristic
612C22#1684039411yescharacteristic
613C22D4×Dic12211yescharacteristic
614C22AGL(1,ℤ/8)×D6111yescharacteristic
615C22C2×#1451980611yescharacteristic
616C22#1684056411yescharacteristic
617C22#1684057811yescharacteristic
618C4—24no—
619C4—22no—
620C4—41no—
621C4—81no—
622C4—81no—
623C4—1222no—
624C4—4881no—
625C4—4881no—
626C4—2441no—
627C4—2442no—
628C4—2441no—
629C4—2442no—
630C4—2441no—
631C4C2×#1451975411yescharacteristic
632C4#1684035711yescharacteristic
633C4#1684041011yescharacteristic
634C4#1684047111yescharacteristic
635C2—82no—
636C2#2144963211yescharacteristic
637C2#2145594011yescharacteristic
638C2#2145608011yescharacteristic
639C2#2145738211yescharacteristic
640C2#2145743211yescharacteristic
641C2#2145872711yescharacteristic
642C2#2145874111yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1880C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
hypercenter#1880D61
hypocenterC61#26397
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1880

Complements

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

NH ≅ G/Ntype
1. C2×#16840057C2semidirect
4. Q8×#11758196C2semidirect
6. #21447294C2semidirect
8. #21447148C2semidirect
9. #21446561C2semidirect
10. #21446386C2semidirect
12. #21446085C2semidirect
13. #21446075C2semidirect
117. C2×SD16×C61Q8semidirect
226. Q8×C61C22⋊Q8semidirect
382. C61#26397semidirect
389. C2×SD16Dic122semidirect
549. Q8#14519754semidirect

Maximal subgroups

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

subgroupindex
C2×#168400572
C2×#168398632
#214496712
Q8×#117581962
#214473452
#214472942
#214471932
#214471482
#214465612
#214463862
#214461762
#214460852
#214460752
#214460642
C61×#18802
#2639761