Finite Group Databasebeta

← #29749868

Subgroups — #29749868

Order 15616.

882 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840120C211yesmaximal, characteristic, complemented
2#21449663C211yesmaximal, characteristic, complemented
3#21449600C211yesmaximal, characteristic, complemented
4#21447195C211yesmaximal, characteristic, complemented
5#21447150C211yesmaximal, characteristic, complemented
6#21447127C211yesmaximal, characteristic
7#21446894C211yesmaximal, characteristic
8#21446674C211yesmaximal, characteristic, complemented
9#21446513C211yesmaximal, characteristic, complemented
10#21446509C211yesmaximal, characteristic, complemented
11#21446504C211yesmaximal, characteristic, complemented
12#21446153C211yesmaximal, characteristic, complemented
13#21446080C211yesmaximal, characteristic, complemented
14#21445687C211yesmaximal, characteristic, complemented
15C2×C61×#442C211yesmaximal, characteristic
16C22×#11758200C2211yescharacteristic
17C2×#14519776C2211yescharacteristic
18C2×#14519764C2211yescharacteristic
19C2×#14519756—21no—
20#16840120—24no—
21C2×#14519708C2211yescharacteristic
22C2×#14519706C2211yescharacteristic
23C2×#14519702C2211yescharacteristic
24C2×#14519702C2211yescharacteristic
25C2×#14519700C2211yescharacteristic
26C2×#14519698C2211yescharacteristic
27C2×#14519698C2211yescharacteristic
28C2×#14519698C2211yescharacteristic
29C2×#14519687C2211yescharacteristic
30C2×C8×Dic61C2211yescharacteristic
31#16839690—21no—
32#16839690—21no—
33#16839685—21no—
34#16839674—21no—
35#16839668C2211yescharacteristic
36#16839667C2211yescharacteristic
37#16839665C2211yescharacteristic
38#16839665C2211yescharacteristic
39#16839665C2211yescharacteristic
40#16839661C2211yescharacteristic
41#16839660C2211yescharacteristic
42#16839658C2211yescharacteristic
43#16839635—21no—
44#16839635—21no—
45#16839625—21no—
46C4×#11758196C2211yescharacteristic
47#16839525C2211yescharacteristic
48#16839525C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839517C2211yescharacteristic
52#16839516C2211yescharacteristic
53C22×D8×C61C2211yescharacteristic
54C2×C4×D4×C61C2211yescharacteristic
55C2×C4×D4×C61C2211yescharacteristic
56C61×#442—24no—
57C2×#106×C61C2211yescharacteristic
58C2×#102×C61C2211yescharacteristic
59C2×#102×C61C2211yescharacteristic
60C2×#97×C61C2211yescharacteristic
61C22×Dic122—21no—
62C22×Dic122C2311yescharacteristic
63C2×#11758200—22no—
64C2×#11758200—22no—
65C2×#11758200—41no—
66C22×#92418C2311yescharacteristic
67C22×#92418C2311yescharacteristic
68#14519776—22no—
69#14519776—41no—
70#14519764—22no—
71#14519764—41no—
72#14519756—44no—
73C2×C4⋊Dic61—21no—
74C2×C4⋊Dic61C2311yescharacteristic
75C2×C4⋊Dic61C2311yescharacteristic
76C2×C4⋊Dic61C2311yescharacteristic
77C2×#11758196—21no—
78C2×#11758196—21no—
79C2×#11758196—21no—
80C2×#11758196D411yescharacteristic
81C2×#11758196D411yescharacteristic
82C2×C22⋊Dic61—21no—
83C2×C22⋊Dic61—21no—
84C2×C22⋊Dic61—21no—
85C2×C22⋊Dic61—21no—
86C2×C22⋊Dic61—21no—
87C2×C22⋊Dic61—21no—
88C2×C22⋊Dic61—21no—
89C2×C22⋊Dic61—21no—
90C2×C4×Dic61C2311yescharacteristic
91C2×C4×Dic61C2311yescharacteristic
92#14519708—41no—
93#14519706—22no—
94#14519702—22no—
95#14519702—22no—
96#14519700—22no—
97#14519698—22no—
98#14519698—22no—
99#14519698—22no—
100#14519687—22no—
101C8×Dic61—22no—
102#14519684—21no—
103#14519684—21no—
104#14519684—21no—
105#14519684—21no—
106#14519684—21no—
107#14519684—21no—
108#14519684—21no—
109#14519684D411yescharacteristic
110#14519684D411yescharacteristic
111C22×D4×C61C2311yescharacteristic
112C22×D4×C61C2311yescharacteristic
113C23×C4×C61—21no—
114C23×C4×C61—21no—
115C2×D8×C61—22no—
116C2×D8×C61—22no—
117C2×D8×C61—22no—
118C22×C8×C61C2311yescharacteristic
119C22×C8×C61C2311yescharacteristic
120C4×D4×C61—22no—
121C4×D4×C61—22no—
122C4×D4×C61—41no—
123C4×D4×C61—41no—
124C2×C4⋊C4×C61C2311yescharacteristic
125C2×C4⋊C4×C61C2311yescharacteristic
126C2×C22⋊C4×C61—21no—
127C2×C22⋊C4×C61—21no—
128C2×C42×C61C2311yescharacteristic
129#106×C61—22no—
130#102×C61—22no—
131#102×C61D414yes—
132#97×C61—22no—
133C24×C61—21no—
134C24×C61—21no—
135C2×D4×C61—21no—
136C2×D4×C61—21no—
137C2×D4×C61—21no—
138C2×D4×C61—21no—
139C2×D4×C61—21no—
140C2×D4×C61—21no—
141C2×D4×C61—21no—
142C2×D4×C61—21no—
143C2×D4×C61—21no—
144C2×D4×C61—21no—
145C2×D4×C61C2×D412yes—
146C2×D4×C61#4112yes—
147C22×C4×C61—21no—
148C22×C4×C61—21no—
149C22×C4×C61—42no—
150C22×C4×C61—42no—
151C22×C4×C61—42no—
152C22×C4×C61—42no—
153C22×C4×C61—81no—
154C22×C4×C61—81no—
155C22×C4×C61C2×D411yescharacteristic
156C22×C4×C61C2×D411yescharacteristic
157C22×C4×C61C2411yescharacteristic
158C22×Dic61—21no—
159C22×Dic61—21no—
160C22×Dic61—21no—
161C22×Dic61—21no—
162C22×Dic61C2×D411yescharacteristic
163C22×Dic61C2×D411yescharacteristic
164C22×Dic61C2×D411yescharacteristic
165C22×Dic61C2×D411yescharacteristic
166C2×Dic122—21no—
167C2×Dic122—21no—
168C2×Dic122—21no—
169C2×Dic122—21no—
170C2×Dic122—21no—
171C2×Dic122—21no—
172C2×Dic122—42no—
173C2×Dic122—42no—
174C2×Dic122—81no—
175D8×C61—28no—
176C2×C8×C61—21no—
177C2×C8×C61—21no—
178C2×C8×C61—21no—
179C2×C8×C61—21no—
180C2×C8×C61C2×D412yes—
181C2×C8×C61#4112yes—
182C4⋊C4×C61—21no—
183C4⋊C4×C61—21no—
184C4⋊C4×C61—21no—
185C4⋊C4×C61C2×D412yes—
186C22⋊C4×C61—42no—
187C22⋊C4×C61—42no—
188C42×C61—21no—
189C42×C61—21no—
190C22⋊Dic61—42no—
191C22⋊Dic61—42no—
192C22⋊Dic61—42no—
193C22⋊Dic61—42no—
194C22⋊Dic61—42no—
195C22⋊Dic61—42no—
196C22⋊Dic61—81no—
197C22⋊Dic61—81no—
198#11758200—44no—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61—21no—
203C4⋊Dic61—21no—
204C4⋊Dic61—21no—
205C4⋊Dic61—42no—
206#11758196—22no—
207#11758196—41no—
208#11758196—42no—
209#11758196—42no—
210#11758196—81no—
211C4×Dic61—21no—
212C4×Dic61—21no—
213C4×Dic61—21no—
214C4×Dic61—21no—
215C2×#92418—21no—
216C2×#92418—21no—
217C2×#92418—21no—
218C2×#92418—21no—
219C2×#92418—21no—
220C2×#92418#4112yes—
221C23×C61—22no—
222C23×C61—22no—
223C23×C61—41no—
224C23×C61—41no—
225C23×C61—41no—
226C23×C61—41no—
227C23×C61—41no—
228C23×C61—41no—
229C23×C61—41no—
230C23×C61—41no—
231C23×C61—41no—
232C23×C61—41no—
233C23×C61—41no—
234C23×C61—41no—
235C23×C61C22×D411yescharacteristic
236D4×C61—22no—
237D4×C61—22no—
238D4×C61—22no—
239D4×C61—22no—
240D4×C61—41no—
241D4×C61—41no—
242D4×C61—41no—
243D4×C61—41no—
244C2×C4×C61—22no—
245C2×C4×C61—21no—
246C2×C4×C61—21no—
247C2×C4×C61—21no—
248C2×C4×C61—21no—
249C2×C4×C61—21no—
250C2×C4×C61—21no—
251C2×C4×C61—41no—
252C2×C4×C61—41no—
253C2×C4×C61—41no—
254C2×C4×C61—41no—
255C2×C4×C61—41no—
256C2×C4×C61—84no—
257C2×C4×C61—84no—
258C2×C4×C61C22×D411yescharacteristic
259C2×C4×C61C2×#4111yescharacteristic
260C2×C4×C61C2×#4111yescharacteristic
261C2×C4×C61C2×#4111yescharacteristic
262C2×C4×C61H(5,𝔽2)11yescharacteristic
263C2×C4×C612−1+411yescharacteristic
264C2×Dic61—21no—
265C2×Dic61—21no—
266C2×Dic61—21no—
267C2×Dic61—21no—
268C2×Dic61—21no—
269C2×Dic61—21no—
270C2×Dic61—21no—
271C2×Dic61—21no—
272C2×Dic61—21no—
273C2×Dic61—21no—
274C2×Dic61—21no—
275C2×Dic61—41no—
276C2×Dic61—41no—
277C2×Dic61—41no—
278C2×Dic61—41no—
279C2×Dic61—41no—
280C2×Dic61—41no—
281C2×Dic61—41no—
282C2×Dic61—41no—
283C2×Dic61—41no—
284C2×Dic61—41no—
285C2×Dic61—41no—
286C2×Dic61—41no—
287C2×Dic61AGL(1,ℤ/8)12yes—
288Dic122—41no—
289Dic122—41no—
290Dic122—41no—
291Dic122—41no—
292Dic122—84no—
293C8×C61—22no—
294C8×C61—22no—
295#92418—22no—
296#92418—41no—
297#26842—611nomaximal, Sylow
298C22×C61—42no—
299C22×C61—42no—
300C22×C61—42no—
301C22×C61—42no—
302C22×C61—42no—
303C22×C61—42no—
304C22×C61—81no—
305C22×C61—81no—
306C22×C61—81no—
307C22×C61—81no—
308C22×C61—81no—
309C22×C61—81no—
310C22×C61—81no—
311C22×C61—81no—
312C22×C61#53211yescharacteristic
313C22×C61D4211yescharacteristic
314C22×C61#54811yescharacteristic
315C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
316C22×C61#57511yescharacteristic
317C22×C61#57711yescharacteristic
318C22×C61#57711yescharacteristic
319C4×C61—42no—
320C4×C61—41no—
321C4×C61—41no—
322C4×C61—81no—
323C4×C61#52511yescharacteristic
324C4×C61#54211yescharacteristic
325C4×C61#54711yescharacteristic
326C4×C61#55111yescharacteristic
327Dic61—22no—
328Dic61—41no—
329Dic61—41no—
330Dic61—41no—
331Dic61—81no—
332Dic61—81no—
333Dic61—81no—
334Dic61—81no—
335C2×#489—611no—
336C2×#442D6111yescharacteristic
337#2452—611no—
338#2447—611no—
339#2066—611no—
340#2058—611no—
341#2054—611no—
342#2012—611no—
343#1968—611no—
344#1936—611no—
345#1935—611no—
346#1934—611no—
347#1877—611no—
348#1864—611no—
349#1770—611no—
350C2×C61—82no—
351C2×C61—82no—
352C2×C61#246111yescharacteristic
353C2×C61#317311yescharacteristic
354C2×C61#318011yescharacteristic
355C2×C61#327211yescharacteristic
356C2×C61#327511yescharacteristic
357C2×C61#337311yescharacteristic
358C2×C61#337611yescharacteristic
359C22×SD16—611no—
360C22×D8C2×D6111yescharacteristic
361C2×C4⋊Q8—611no—
362C2×#124—611no—
363C2×C22⋊Q8—1221no—
364C2×C4×D4C2×D6111yescharacteristic
365C2×C4×D4C2×D6111yescharacteristic
366#489—1224no—
367#442—24no—
368C2×#106—611no—
369C2×#106C2×D6111yescharacteristic
370C2×C4⋊C8—611no—
371C2×#103—611no—
372C2×#103—611no—
373C2×#103—611no—
374C2×#102—611no—
375C2×#102—611no—
376C2×#102—611no—
377C2×#102C2×D6111yescharacteristic
378C2×#102C2×D6111yescharacteristic
379C2×#97—611no—
380C2×#97C2×D6111yescharacteristic
381C2×C4×C8—611no—
382#397—1221no—
383#397—1221no—
384#396—1221no—
385#393—1221no—
386#391—611no—
387#391—611no—
388#390—611no—
389#390—611no—
390#390—611no—
391#389—611no—
392#389—611no—
393#389—611no—
394#385—1221no—
395#385—1221no—
396#381—1221no—
397C4×C4⋊C4—611no—
398#340—611no—
399#340—611no—
400#340—611no—
401#340—611no—
402#336—611no—
403#336—611no—
404C61#2684211yescharacteristic, complemented, Sylow
405C22×Q8—611no—
406C22×Q8—1221no—
407C22×D4C22×D6111yescharacteristic
408C22×D4C22×D6111yescharacteristic
409C23×C4—21no—
410C23×C4—21no—
411C2×SD16—1222no—
412C2×SD16—1222no—
413C2×SD16—2441no—
414C2×D8—22no—
415C2×D8—22no—
416C2×D8—22no—
417C22×C8—611no—
418C22×C8—611no—
419C22×C8C22×D6111yescharacteristic
420C22×C8C22×D6111yescharacteristic
421C4⋊Q8—1222no—
422C4⋊Q8—2441no—
423#124—1222no—
424#124—2441no—
425C22⋊Q8—2444no—
426C4×D4—22no—
427C4×D4—22no—
428C4×D4—41no—
429C4×D4—41no—
430C2×C4⋊C4—611no—
431C2×C4⋊C4—611no—
432C2×C4⋊C4—611no—
433C2×C4⋊C4—611no—
434C2×C4⋊C4—611no—
435C2×C4⋊C4—1221no—
436C2×C4⋊C4—1221no—
437C2×C4⋊C4—1221no—
438C2×C4⋊C4—1221no—
439C2×C4⋊C4C22×D6111yescharacteristic
440C2×C4⋊C4C22×D6111yescharacteristic
441C2×C22⋊C4—21no—
442C2×C22⋊C4—21no—
443C2×C22⋊C4—1221no—
444C2×C22⋊C4—1221no—
445C2×C22⋊C4—1221no—
446C2×C22⋊C4—1221no—
447C2×C22⋊C4—1221no—
448C2×C22⋊C4—1221no—
449C2×C22⋊C4—1221no—
450C2×C22⋊C4—1221no—
451C2×C42—611no—
452C2×C42—611no—
453C2×C42C22×D6111yescharacteristic
454#106—22no—
455#106—2441no—
456C4⋊C8—1222no—
457#103—1222no—
458#103—1222no—
459#103—1222no—
460#102—22no—
461#102—1222no—
462#102—1222no—
463#102—1222no—
464#102C22⋊D6114yes—
465#97—22no—
466#97—1222no—
467C4×C8—1222no—
468#95—611no—
469#95—611no—
470#95—1221no—
471#95—1221no—
472#95—1221no—
473#95—1221no—
474#95—1221no—
475#95—1221no—
476#95—1221no—
477C24—21no—
478C24—21no—
479C2×Q8—1221no—
480C2×Q8—1221no—
481C2×Q8—1221no—
482C2×Q8—1221no—
483C2×Q8—1221no—
484C2×Q8—1221no—
485C2×Q8—4881no—
486C2×Q8—2442no—
487C2×Q8—2442no—
488C2×D4—21no—
489C2×D4—21no—
490C2×D4—21no—
491C2×D4—21no—
492C2×D4—21no—
493C2×D4—21no—
494C2×D4—21no—
495C2×D4—21no—
496C2×D4—21no—
497C2×D4—21no—
498C2×D4#1175822212yes—
499C2×D4C2×C22⋊D6112yes—
500C22×C4—21no—
501C22×C4—21no—
502C22×C4—42no—
503C22×C4—42no—
504C22×C4—42no—
505C22×C4—42no—
506C22×C4—81no—
507C22×C4—81no—
508C22×C4—611no—
509C22×C4—611no—
510C22×C4—611no—
511C22×C4—611no—
512C22×C4—1221no—
513C22×C4—1221no—
514C22×C4—1221no—
515C22×C4—1221no—
516C22×C4D4×D6111yescharacteristic
517C22×C4D4×D6111yescharacteristic
518C22×C4C23×D6111yescharacteristic
519SD16—2444no—
520D8—28no—
521C2×C8—21no—
522C2×C8—21no—
523C2×C8—21no—
524C2×C8—21no—
525C2×C8—612no—
526C2×C8—1221no—
527C2×C8—1221no—
528C2×C8—1221no—
529C2×C8—1221no—
530C2×C8—1221no—
531C2×C8#1175822212yes—
532C2×C8C2×C22⋊D6112yes—
533C4⋊C4—21no—
534C4⋊C4—21no—
535C4⋊C4—21no—
536C4⋊C4—1222no—
537C4⋊C4—1221no—
538C4⋊C4—1221no—
539C4⋊C4—1221no—
540C4⋊C4—1221no—
541C4⋊C4—1221no—
542C4⋊C4—1221no—
543C4⋊C4—4881no—
544C4⋊C4—2441no—
545C4⋊C4—2442no—
546C4⋊C4—2442no—
547C4⋊C4—2442no—
548C4⋊C4C2×C22⋊D6112yes—
549C22⋊C4—42no—
550C22⋊C4—42no—
551C22⋊C4—4881no—
552C22⋊C4—4881no—
553C22⋊C4—2442no—
554C22⋊C4—2442no—
555C22⋊C4—2442no—
556C22⋊C4—2442no—
557C22⋊C4—2442no—
558C22⋊C4—2442no—
559C42—21no—
560C42—21no—
561C42—1221no—
562C42—1221no—
563C42—1221no—
564C42—1221no—
565C23—22no—
566C23—22no—
567C23—41no—
568C23—41no—
569C23—41no—
570C23—41no—
571C23—41no—
572C23—41no—
573C23—41no—
574C23—41no—
575C23—41no—
576C23—41no—
577C23—41no—
578C23—41no—
579C23C2×D4×D6111yescharacteristic
580Q8—4884no—
581Q8—2441no—
582Q8—2441no—
583Q8—2441no—
584Q8—2441no—
585D4—22no—
586D4—22no—
587D4—22no—
588D4—22no—
589D4—41no—
590D4—41no—
591D4—41no—
592D4—41no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—22no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—84no—
606C2×C4—84no—
607C2×C4—612no—
608C2×C4—1221no—
609C2×C4—1221no—
610C2×C4—1221no—
611C2×C4—1221no—
612C2×C4—1221no—
613C2×C4—1221no—
614C2×C4—1221no—
615C2×C4—1221no—
616C2×C4—1221no—
617C2×C4—1221no—
618C2×C4—1221no—
619C2×C4—2441no—
620C2×C4—2441no—
621C2×C4—2441no—
622C2×C4—2441no—
623C2×C4—2441no—
624C2×C4—2441no—
625C2×C4—2441no—
626C2×C4—2441no—
627C2×C4—2441no—
628C2×C4—2441no—
629C2×C4—2441no—
630C2×C4—2441no—
631C2×C4C22×C22⋊D6111yescharacteristic
632C2×C4#41×D6111yescharacteristic
633C2×C4C2×#1175822211yescharacteristic
634C2×C4C2×#1175822211yescharacteristic
635C2×C4#1451982211yescharacteristic
636C2×C4#1451982411yescharacteristic
637C8—22no—
638C8—22no—
639C8—1222no—
640C8—2441no—
641C22—42no—
642C22—42no—
643C22—42no—
644C22—42no—
645C22—42no—
646C22—42no—
647C22—81no—
648C22—81no—
649C22—81no—
650C22—81no—
651C22—81no—
652C22—81no—
653C22—81no—
654C22—81no—
655C22#1684030911yescharacteristic
656C22D4×C22⋊D6111yescharacteristic
657C22#1684042111yescharacteristic
658C22AGL(1,ℤ/8)×D6111yescharacteristic
659C22#1684055811yescharacteristic
660C22#1684057211yescharacteristic
661C22#1684057211yescharacteristic
662C4—41no—
663C4—42no—
664C4—41no—
665C4—81no—
666C4—1222no—
667C4—4881no—
668C4—4881no—
669C4—4881no—
670C4—4881no—
671C4—2441no—
672C4—2441no—
673C4—2441no—
674C4#1684027311yescharacteristic
675C4#1684037711yescharacteristic
676C4#1684041111yescharacteristic
677C4#1684044111yescharacteristic
678C2—82no—
679C2—82no—
680C2#2144978011yescharacteristic
681C2#2145643411yescharacteristic
682C2#2145651511yescharacteristic
683C2#2145741311yescharacteristic
684C2#2145745011yescharacteristic
685C2#2145861211yescharacteristic
686C2#2145864711yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#442C2
socleC23×C61C22×D4
prime radicalC22×D8×C61C22
hypercenterC2×#442D61
hypocenterC61#26842
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#442

Complements

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

NH ≅ G/Ntype
1. C2×#16840120C2semidirect
2. #21449663C2semidirect
3. #21449600C2semidirect
4. #21447195C2semidirect
5. #21447150C2semidirect
8. #21446674C2semidirect
9. #21446513C2semidirect
10. #21446509C2semidirect
11. #21446504C2semidirect
12. #21446153C2semidirect
13. #21446080C2semidirect
14. #21445687C2semidirect
404. C61#26842semidirect

Maximal subgroups

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

subgroupindex
C2×#168401202
#214496632
#214496002
#214471952
#214471502
#214471272
#214468942
#214466742
#214465132
#214465092
#214465042
#214461532
#214460802
#214456872
C2×C61×#4422
#2684261