Finite Group Databasebeta

← #29740241

Subgroups — #29740241

Order 15616.

904 conjugacy classes of subgroups.

The subgroup lattice has 692 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839863C211yesmaximal, characteristic, complemented
2#21449580C211yesmaximal, characteristic
3#21448737C211yesmaximal, characteristic, complemented
4#21447338C211yesmaximal, characteristic, complemented
5#21447307C211yesmaximal, characteristic, complemented
6#21447235C211yesmaximal, characteristic, complemented
7#21447050C211yesmaximal, characteristic
8#21446513C211yesmaximal, characteristic, complemented
9#21446380C211yesmaximal, characteristic, complemented
10#21446176C211yesmaximal, characteristic, complemented
11#21446176C211yesmaximal, characteristic, complemented
12#21446176C211yesmaximal, characteristic, complemented
13#21446087C211yesmaximal, characteristic, complemented
14#21446078C211yesmaximal, characteristic, complemented
15C2×C61×#460C211yesmaximal, characteristic
16C2×#14519758—21no—
17C2×Q8×Dic61C2211yescharacteristic
18C2×Q8×Dic61C2211yescharacteristic
19C2×D4×Dic61C2211yescharacteristic
20#16839863—24no—
21C2×#14519711C2211yescharacteristic
22C2×#14519708C2211yescharacteristic
23C2×#14519701C2211yescharacteristic
24C2×#14519701C2211yescharacteristic
25C2×#14519698C2211yescharacteristic
26C2×#14519698C2211yescharacteristic
27C2×#14519698C2211yescharacteristic
28C2×#14519698C2211yescharacteristic
29C2×#14519687C2211yescharacteristic
30#16839676—21no—
31#16839676—21no—
32#16839665C2211yescharacteristic
33#16839665C2211yescharacteristic
34#16839665C2211yescharacteristic
35#16839660—21no—
36#16839660C2211yescharacteristic
37#16839660C2211yescharacteristic
38#16839658—21no—
39#16839658C2211yescharacteristic
40#16839642—21no—
41#16839627—21no—
42#16839627C2211yescharacteristic
43C4×#11758196—21no—
44C4×#11758196C2211yescharacteristic
45#16839525C2211yescharacteristic
46#16839525C2211yescharacteristic
47#16839525C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839517C2211yescharacteristic
50#16839516C2211yescharacteristic
51#16839516C2211yescharacteristic
52#16839516C2211yescharacteristic
53C22×SD16×C61C2211yescharacteristic
54C22×SD16×C61C2211yescharacteristic
55C2×#127×C61C2211yescharacteristic
56C2×C4×Q8×C61C2211yescharacteristic
57C61×#460—24no—
58C2×C4⋊C8×C61C2211yescharacteristic
59C2×#102×C61C2211yescharacteristic
60C2×#102×C61C2211yescharacteristic
61C23×Dic61—21no—
62C22×#92418C2311yescharacteristic
63C22×#92418C2311yescharacteristic
64#14519758—44no—
65Q8×Dic61—22no—
66Q8×Dic61—22no—
67Q8×Dic61—22no—
68Q8×Dic61—41no—
69D4×Dic61—22no—
70D4×Dic61—41no—
71C2×C4⋊Dic61—21no—
72C2×C4⋊Dic61—21no—
73C2×C4⋊Dic61C2311yescharacteristic
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×C4×Dic61—21no—
90C2×C4×Dic61—21no—
91C2×C4×Dic61C2311yescharacteristic
92C2×C4×Dic61C2311yescharacteristic
93#14519711—22no—
94#14519708—41no—
95#14519701—22no—
96#14519701—22no—
97#14519698—22no—
98#14519698—22no—
99#14519698—22no—
100#14519698—41no—
101#14519687—22no—
102#14519684—21no—
103#14519684—21no—
104#14519684—21no—
105#14519684—21no—
106#14519684—21no—
107#14519684D411yescharacteristic
108#14519684D411yescharacteristic
109C22×Q8×C61C2311yescharacteristic
110C22×D4×C61—21no—
111C22×D4×C61—21no—
112C22×D4×C61C2311yescharacteristic
113C22×D4×C61C2311yescharacteristic
114C2×SD16×C61—22no—
115C2×SD16×C61—22no—
116C2×SD16×C61—22no—
117C2×SD16×C61—22no—
118C2×SD16×C61—22no—
119C2×SD16×C61D414yes—
120C22×C8×C61C2311yescharacteristic
121C22×C8×C61C2311yescharacteristic
122#127×C61—22no—
123#127×C61—41no—
124C4×Q8×C61—22no—
125C4×Q8×C61—41no—
126C2×C4⋊C4×C61—21no—
127C2×C4⋊C4×C61C2311yescharacteristic
128C2×C42×C61—21no—
129C2×C42×C61C2311yescharacteristic
130C4⋊C8×C61—22no—
131#102×C61—22no—
132#102×C61—22no—
133C24×C61—21no—
134C24×C61—21no—
135C2×Q8×C61—21no—
136C2×Q8×C61—21no—
137C2×Q8×C61—21no—
138C2×Q8×C61C2×D412yes—
139C2×Q8×C61#4112yes—
140C2×Q8×C61#4112yes—
141C2×D4×C61—21no—
142C2×D4×C61—21no—
143C2×D4×C61—21no—
144C2×D4×C61—21no—
145C2×D4×C61—21no—
146C2×D4×C61—21no—
147C2×D4×C61—21no—
148C2×D4×C61—21no—
149C2×D4×C61—21no—
150C2×D4×C61—21no—
151C2×D4×C61—21no—
152C2×D4×C61—42no—
153C2×D4×C61—42no—
154C2×D4×C61—42no—
155C2×D4×C61—42no—
156C2×D4×C61—81no—
157C2×D4×C61—81no—
158C2×D4×C61C2×D412yes—
159C22×C4×C61—21no—
160C22×C4×C61—21no—
161C22×C4×C61C2×D411yescharacteristic
162C22×C4×C61C2×D411yescharacteristic
163C22×C4×C61C2411yescharacteristic
164C22×Dic61—21no—
165C22×Dic61—21no—
166C22×Dic61—21no—
167C22×Dic61—21no—
168C22×Dic61—42no—
169C22×Dic61—42no—
170C22×Dic61—81no—
171C22×Dic61C2×D411yescharacteristic
172C22×Dic61C2×D411yescharacteristic
173C22×Dic61C2×D411yescharacteristic
174C22×Dic61C2×D411yescharacteristic
175SD16×C61—28no—
176SD16×C61—44no—
177C2×C8×C61—21no—
178C2×C8×C61—21no—
179C2×C8×C61—21no—
180C2×C8×C61—21no—
181C2×C8×C61—21no—
182C2×C8×C61C2×D412yes—
183C4⋊C4×C61—21no—
184C4⋊C4×C61—21no—
185C4⋊C4×C61—42no—
186C42×C61—21no—
187C42×C61—21no—
188C42×C61—42no—
189C22⋊Dic61—42no—
190C22⋊Dic61—42no—
191C22⋊Dic61—42no—
192C22⋊Dic61—42no—
193C22⋊Dic61—81no—
194C22⋊Dic61—81no—
195C22⋊Dic61—81no—
196C4⋊Dic61—24no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61—21no—
203C4⋊Dic61—21no—
204C4⋊Dic61—42no—
205C4⋊Dic61#4112yes—
206#11758196—22no—
207#11758196—41no—
208#11758196—42no—
209#11758196—42no—
210#11758196—42no—
211C4×Dic61—24no—
212C4×Dic61—21no—
213C4×Dic61—21no—
214C4×Dic61—21no—
215C4×Dic61—21no—
216C4×Dic61—42no—
217C2×#92418—21no—
218C2×#92418—21no—
219C2×#92418—21no—
220C2×#92418—21no—
221C2×#92418—21no—
222C2×#92418—21no—
223C23×C61—22no—
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×C61—41no—
236C23×C61—41no—
237C23×C61C22×D411yescharacteristic
238Q8×C61—22no—
239Q8×C61—22no—
240Q8×C61—22no—
241Q8×C61#12314yescomplemented
242D4×C61—22no—
243D4×C61—22no—
244D4×C61—41no—
245D4×C61—41no—
246D4×C61—41no—
247D4×C61—41no—
248D4×C61—41no—
249D4×C61—41no—
250D4×C61—84no—
251D4×C61—84no—
252C2×C4×C61—22no—
253C2×C4×C61—22no—
254C2×C4×C61—21no—
255C2×C4×C61—22no—
256C2×C4×C61—21no—
257C2×C4×C61—21no—
258C2×C4×C61—21no—
259C2×C4×C61—21no—
260C2×C4×C61—41no—
261C2×C4×C61—41no—
262C2×C4×C61—41no—
263C2×C4×C61AGL(1,ℤ/8)12yes—
264C2×C4×C61C22×D411yescharacteristic
265C2×C4×C61C2×#4111yescharacteristic
266C2×C4×C61C2×#4111yescharacteristic
267C2×C4×C61C2×#4111yescharacteristic
268C2×C4×C61H(5,𝔽2)11yescharacteristic
269C2×C4×C612−1+411yescharacteristic
270C2×Dic61—21no—
271C2×Dic61—21no—
272C2×Dic61—22no—
273C2×Dic61—21no—
274C2×Dic61—21no—
275C2×Dic61—21no—
276C2×Dic61—21no—
277C2×Dic61—21no—
278C2×Dic61—21no—
279C2×Dic61—21no—
280C2×Dic61—21no—
281C2×Dic61—21no—
282C2×Dic61—21no—
283C2×Dic61—41no—
284C2×Dic61—41no—
285C2×Dic61—41no—
286C2×Dic61—41no—
287C2×Dic61—41no—
288C2×Dic61—41no—
289C2×Dic61—41no—
290C2×Dic61—41no—
291C2×Dic61—41no—
292C2×Dic61—41no—
293C2×Dic61—41no—
294C2×Dic61—84no—
295C8×C61—22no—
296C8×C61—41no—
297#92418—41no—
298#92418—41no—
299#26175—611nomaximal, Sylow
300C22×C61—42no—
301C22×C61—42no—
302C22×C61—42no—
303C22×C61—81no—
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—81no—
313C22×C61—81no—
314C22×C61#52511yescharacteristic
315C22×C61D4211yescharacteristic
316C22×C61#54811yescharacteristic
317C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
318C22×C61#57511yescharacteristic
319C22×C61#57711yescharacteristic
320C22×C61#57711yescharacteristic
321C4×C61—24no—
322C4×C61—22no—
323C4×C61—41no—
324C4×C61—81no—
325C4×C61C2×#12311yescharacteristic
326C4×C61#53911yescharacteristic
327C4×C61#54711yescharacteristic
328C4×C61#55311yescharacteristic
329Dic61—41no—
330Dic61—42no—
331Dic61—41no—
332Dic61—41no—
333Dic61—41no—
334Dic61—81no—
335Dic61—81no—
336Dic61—81no—
337C2×#460D6111yescharacteristic
338C2×#440—611no—
339#2445—611no—
340#2351—611no—
341#2092—611no—
342#2086—611no—
343#2073—611no—
344#2041—611no—
345#1936—611no—
346#1917—611no—
347#1880—611no—
348#1880—611no—
349#1880—611no—
350#1865—611no—
351#1864—611no—
352C2×C61—82no—
353C2×C61—161no—
354C2×C61#244311yescharacteristic
355C2×C61#311511yescharacteristic
356C2×C61#312611yescharacteristic
357C2×C61#326511yescharacteristic
358C2×C61#327511yescharacteristic
359C2×C61#336811yescharacteristic
360C2×C61#337111yescharacteristic
361C22×SD16C2×D6111yescharacteristic
362C22×SD16C2×D6111yescharacteristic
363C2×#127C2×D6111yescharacteristic
364C2×#123—1221no—
365C2×C4×Q8—611no—
366C2×C4×Q8—611no—
367C2×C4×Q8C2×D6111yescharacteristic
368C2×C4×D4—611no—
369#460—24no—
370#440—1224no—
371C2×#107—611no—
372C2×#106—611no—
373C2×C4⋊C8C2×D6111yescharacteristic
374C2×#103—611no—
375C2×#103—611no—
376C2×#102—611no—
377C2×#102—611no—
378C2×#102—611no—
379C2×#102—611no—
380C2×#102C2×D6111yescharacteristic
381C2×#102C2×D6111yescharacteristic
382C2×#97—611no—
383#394—1221no—
384#394—1221no—
385#390—611no—
386#390—611no—
387#390—611no—
388#389—611no—
389#389—611no—
390#389—611no—
391#389—1221no—
392#389—1221no—
393#386—1221no—
394#382—611no—
395#382—1221no—
396C4×C4⋊C4—611no—
397C4×C4⋊C4—1221no—
398#340—611no—
399#340—611no—
400#340—611no—
401#340—611no—
402#336—611no—
403#336—611no—
404#336—611no—
405#336—611no—
406C61#2617511yescharacteristic, complemented, Sylow
407C22×Q8C22×D6111yescharacteristic
408C22×D4—21no—
409C22×D4—21no—
410C22×D4C22×D6111yescharacteristic
411C22×D4C22×D6111yescharacteristic
412C23×C4—1221no—
413C2×SD16—22no—
414C2×SD16—22no—
415C2×SD16—22no—
416C2×SD16—22no—
417C2×SD16—22no—
418C2×SD16C22⋊D6114yes—
419C22×C8—611no—
420C22×C8—611no—
421C22×C8C22×D6111yescharacteristic
422C22×C8C22×D6111yescharacteristic
423#127—22no—
424#127—41no—
425#123—2444no—
426C4×Q8—22no—
427C4×Q8—41no—
428C4×Q8—1222no—
429C4×Q8—1222no—
430C4×Q8—1222no—
431C4×Q8—2441no—
432C4×D4—1222no—
433C4×D4—2441no—
434C2×C4⋊C4—21no—
435C2×C4⋊C4—611no—
436C2×C4⋊C4—611no—
437C2×C4⋊C4—611no—
438C2×C4⋊C4—611no—
439C2×C4⋊C4—611no—
440C2×C4⋊C4—611no—
441C2×C4⋊C4—1221no—
442C2×C4⋊C4—1221no—
443C2×C4⋊C4—1221no—
444C2×C4⋊C4—1221no—
445C2×C4⋊C4—1221no—
446C2×C4⋊C4C22×D6111yescharacteristic
447C2×C22⋊C4—1221no—
448C2×C22⋊C4—1221no—
449C2×C22⋊C4—1221no—
450C2×C22⋊C4—1221no—
451C2×C22⋊C4—1221no—
452C2×C22⋊C4—1221no—
453C2×C22⋊C4—1221no—
454C2×C42—21no—
455C2×C42—611no—
456C2×C42—611no—
457C2×C42—1221no—
458C2×C42—1221no—
459C2×C42C22×D6111yescharacteristic
460#107—1222no—
461#106—2441no—
462C4⋊C8—22no—
463#103—1222no—
464#103—1222no—
465#102—22no—
466#102—22no—
467#102—1222no—
468#102—1222no—
469#102—1222no—
470#102—2441no—
471#97—1222no—
472#95—611no—
473#95—611no—
474#95—1221no—
475#95—1221no—
476#95—1221no—
477#95—1221no—
478#95—1221no—
479C24—21no—
480C24—21no—
481C2×Q8—21no—
482C2×Q8—21no—
483C2×Q8—21no—
484C2×Q8#1175822412yes—
485C2×Q8#1175822412yes—
486C2×Q8C2×C22⋊D6112yes—
487C2×D4—21no—
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—42no—
499C2×D4—42no—
500C2×D4—42no—
501C2×D4—42no—
502C2×D4—81no—
503C2×D4—81no—
504C2×D4C2×C22⋊D6112yes—
505C22×C4—21no—
506C22×C4—21no—
507C22×C4—611no—
508C22×C4—611no—
509C22×C4—611no—
510C22×C4—611no—
511C22×C4—1221no—
512C22×C4—1221no—
513C22×C4—1221no—
514C22×C4—1221no—
515C22×C4—4881no—
516C22×C4—2442no—
517C22×C4—2442no—
518C22×C4D4×D6111yescharacteristic
519C22×C4D4×D6111yescharacteristic
520C22×C4C23×D6111yescharacteristic
521SD16—28no—
522SD16—44no—
523C2×C8—21no—
524C2×C8—21no—
525C2×C8—21no—
526C2×C8—21no—
527C2×C8—21no—
528C2×C8—1221no—
529C2×C8—1221no—
530C2×C8—1221no—
531C2×C8—1221no—
532C2×C8—1221no—
533C2×C8—1221no—
534C2×C8C2×C22⋊D6112yes—
535C4⋊C4—21no—
536C4⋊C4—21no—
537C4⋊C4—42no—
538C4⋊C4—612no—
539C4⋊C4—1224no—
540C4⋊C4—1221no—
541C4⋊C4—1222no—
542C4⋊C4—1221no—
543C4⋊C4—1221no—
544C4⋊C4—1221no—
545C4⋊C4—1221no—
546C4⋊C4—1221no—
547C4⋊C4—1221no—
548C4⋊C4—2441no—
549C4⋊C4—2442no—
550C4⋊C4—2442no—
551C4⋊C4—2442no—
552C4⋊C4—2442no—
553C22⋊C4—4881no—
554C22⋊C4—4881no—
555C22⋊C4—4881no—
556C22⋊C4—2442no—
557C22⋊C4—2442no—
558C22⋊C4—2442no—
559C22⋊C4—2442no—
560C42—21no—
561C42—21no—
562C42—42no—
563C42—1224no—
564C42—1221no—
565C42—1221no—
566C42—1221no—
567C42—1221no—
568C42—2442no—
569C23—22no—
570C23—41no—
571C23—41no—
572C23—41no—
573C23—41no—
574C23—41no—
575C23—41no—
576C23—41no—
577C23—41no—
578C23—41no—
579C23—41no—
580C23—41no—
581C23—41no—
582C23—41no—
583C23C2×D4×D6111yescharacteristic
584Q8—22no—
585Q8—22no—
586Q8—22no—
587Q8#1451975814yescomplemented
588D4—22no—
589D4—22no—
590D4—41no—
591D4—41no—
592D4—41no—
593D4—41no—
594D4—41no—
595D4—41no—
596D4—84no—
597D4—84no—
598C2×C4—22no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—1221no—
610C2×C4—1222no—
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—1221no—
620C2×C4—1221no—
621C2×C4—1221no—
622C2×C4—4884no—
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×C4—2441no—
632C2×C4—2441no—
633C2×C4—2441no—
634C2×C4#1451980412yes—
635C2×C4C22×C22⋊D6111yescharacteristic
636C2×C4#41×D6111yescharacteristic
637C2×C4C2×#1175822411yescharacteristic
638C2×C4C2×#1175822411yescharacteristic
639C2×C4#1451982211yescharacteristic
640C2×C4#1451982411yescharacteristic
641C8—22no—
642C8—41no—
643C8—2441no—
644C8—2441no—
645C22—42no—
646C22—42no—
647C22—42no—
648C22—81no—
649C22—81no—
650C22—81no—
651C22—81no—
652C22—81no—
653C22—81no—
654C22—81no—
655C22—81no—
656C22—81no—
657C22—81no—
658C22—81no—
659C22#1684027211yescharacteristic
660C22D4×C22⋊D6111yescharacteristic
661C22#1684041811yescharacteristic
662C22C2×#1451980411yescharacteristic
663C22#1684055811yescharacteristic
664C22#1684056811yescharacteristic
665C22#1684057211yescharacteristic
666C4—24no—
667C4—22no—
668C4—41no—
669C4—81no—
670C4—4881no—
671C4—4881no—
672C4—4881no—
673C4—2441no—
674C4—2441no—
675C4—2442no—
676C4—2441no—
677C4—2441no—
678C4C2×#1451975811yescharacteristic
679C4#1684035711yescharacteristic
680C4#1684041111yescharacteristic
681C4#1684045511yescharacteristic
682C2—82no—
683C2—161no—
684C2#2144955011yescharacteristic
685C2#2145596911yescharacteristic
686C2#2145610811yescharacteristic
687C2#2145731411yescharacteristic
688C2#2145745011yescharacteristic
689C2#2145855211yescharacteristic
690C2#2145858011yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#460C2
socleC23×C61C22×D4
prime radicalC2×#127×C61C22
hypercenterC2×#460D61
hypocenterC61#26175
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#460

Complements

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

NH ≅ G/Ntype
1. C2×#16839863C2semidirect
3. #21448737C2semidirect
4. #21447338C2semidirect
5. #21447307C2semidirect
6. #21447235C2semidirect
8. #21446513C2semidirect
9. #21446380C2semidirect
10. #21446176C2semidirect
11. #21446176C2semidirect
12. #21446176C2semidirect
13. #21446087C2semidirect
14. #21446078C2semidirect
241. Q8×C61#123semidirect
406. C61#26175semidirect
587. Q8#14519758semidirect

Maximal subgroups

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

subgroupindex
C2×#168398632
#214495802
#214487372
#214473382
#214473072
#214472352
#214470502
#214465132
#214463802
#214461762
#214460872
#214460782
C2×C61×#4602
#2617561