Finite Group Databasebeta

← #29729394

Subgroups — #29729394

Order 15616.

886 conjugacy classes of subgroups.

The subgroup lattice has 690 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840122C211yesmaximal, characteristic, complemented
2C2×#16840052C211yesmaximal, characteristic, complemented
3#21450786C211yesmaximal, characteristic
4#21448875C211yesmaximal, characteristic, complemented
5#21447201C211yesmaximal, characteristic, complemented
6#21447194C211yesmaximal, characteristic, complemented
7#21447058C211yesmaximal, characteristic, complemented
8#21447051C211yesmaximal, characteristic
9#21446751C211yesmaximal, characteristic, complemented
10#21446750C211yesmaximal, characteristic, complemented
11#21446744C211yesmaximal, characteristic
12#21446542C211yesmaximal, characteristic, complemented
13#21446451C211yesmaximal, characteristic, complemented
14#21446444C211yesmaximal, characteristic, complemented
15C61×#1762C211yesmaximal, characteristic, complemented
16C2×#14519777C2211yescharacteristic
17C2×#14519773C2211yescharacteristic
18C2×#14519765C2211yescharacteristic
19C2×#14519764C2211yescharacteristic
20C2×C4×Dic122—21no—
21#16840122—24no—
22#16840052—24no—
23C2×#14519708C2211yescharacteristic
24C2×#14519708C2211yescharacteristic
25C2×C4⋊#92418C2211yescharacteristic
26C2×#14519706C2211yescharacteristic
27C2×#14519706C2211yescharacteristic
28C2×#14519702C2211yescharacteristic
29C2×#14519702C2211yescharacteristic
30C2×#14519700C2211yescharacteristic
31C2×#14519700C2211yescharacteristic
32C2×#14519699C2211yescharacteristic
33C2×#14519699C2211yescharacteristic
34C2×#14519698C2211yescharacteristic
35C2×#14519698C2211yescharacteristic
36C2×#14519697C2211yescharacteristic
37C2×#14519697C2211yescharacteristic
38#16839678—21no—
39#16839678—21no—
40#16839668—21no—
41#16839667C2211yescharacteristic
42#16839663C2211yescharacteristic
43#16839658C2211yescharacteristic
44#16839658C2211yescharacteristic
45#16839645—21no—
46#16839645—21no—
47#16839641—21no—
48#16839641—21no—
49C4⋊C4×Dic61C2211yescharacteristic
50C4⋊C4×Dic61C2211yescharacteristic
51C4×C4⋊Dic61C2211yescharacteristic
52#16839516C2211yescharacteristic
53#16839516C2211yescharacteristic
54C2×C4×D4×C61C2211yescharacteristic
55C2×#102×C61C2211yescharacteristic
56C2×#102×C61C2211yescharacteristic
57C2×#97×C61C2211yescharacteristic
58C4×C4⋊C4×C61C2211yescharacteristic
59C61×#340C2211yescharacteristic
60C61×#340C2211yescharacteristic
61C22×Dic122—21no—
62C22×Dic122C2311yescharacteristic
63C22×D244C2311yescharacteristic
64C22×C22⋊D61—21no—
65C22×C22⋊D61—21no—
66C22×#92418C2311yescharacteristic
67C22×#92418C2311yescharacteristic
68#14519777—22no—
69#14519777D414yescomplemented
70#14519773—22no—
71#14519773—41no—
72#14519765—22no—
73#14519765—22no—
74#14519764—22no—
75#14519764—41no—
76C4×Dic122—44no—
77C2×C4⋊Dic61—21no—
78C2×C4⋊Dic61D411yescharacteristic
79C2×C4⋊Dic61D411yescharacteristic
80C2×C4⋊Dic61C2311yescharacteristic
81C2×C4⋊Dic61C2311yescharacteristic
82C2×#11758196—21no—
83C2×#11758196—21no—
84C2×C22⋊Dic61—21no—
85C2×C22⋊Dic61—21no—
86C2×#11758198—21no—
87C2×#11758198—21no—
88C2×#11758198—21no—
89C2×#11758198—21no—
90C2×C4×Dic61—21no—
91C2×C4×Dic61—21no—
92C2×C4×Dic61C2311yescharacteristic
93C2×C4×Dic61C2311yescharacteristic
94#14519708—22no—
95#14519708—22no—
96C4⋊#92418—22no—
97#14519706—41no—
98#14519706—41no—
99#14519702—22no—
100#14519702—22no—
101#14519700—22no—
102#14519700—41no—
103#14519699—22no—
104#14519699—22no—
105#14519698—22no—
106#14519698—22no—
107#14519697—22no—
108#14519697—41no—
109#14519684—21no—
110#14519684—21no—
111#14519684—21no—
112#14519684—21no—
113#14519684—21no—
114#14519684—21no—
115#14519684D411yescharacteristic
116#14519684D411yescharacteristic
117C22×D4×C61C2311yescharacteristic
118C23×C4×C61—21no—
119C22×C8×C61C2311yescharacteristic
120C22×C8×C61C2311yescharacteristic
121C4×D4×C61—22no—
122C4×D4×C61—22no—
123C2×C4⋊C4×C61C2311yescharacteristic
124C2×C4⋊C4×C61C2311yescharacteristic
125C2×C4⋊C4×C61C2311yescharacteristic
126C2×C22⋊C4×C61—21no—
127C2×C42×C61—21no—
128C2×C42×C61C2311yescharacteristic
129#102×C61—22no—
130#102×C61—22no—
131#97×C61—22no—
132#95×C61—21no—
133C24×C61—21no—
134C23×D61—21no—
135C2×D4×C61—21no—
136C2×D4×C61—21no—
137C2×D4×C61—21no—
138C2×D4×C61—21no—
139C2×D4×C61—21no—
140C2×D4×C61#4112yes—
141C22×C4×C61—21no—
142C22×C4×C61—21no—
143C22×C4×C61—21no—
144C22×C4×C61—42no—
145C22×C4×C61—81no—
146C22×C4×C61—81no—
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2411yescharacteristic
150C2×C22⋊D61—42no—
151C2×C22⋊D61—42no—
152C2×C22⋊D61—42no—
153C2×C22⋊D61—42no—
154C2×C22⋊D61—81no—
155C2×C22⋊D61—81no—
156C22×Dic61—21no—
157C22×Dic61—21no—
158C22×Dic61—21no—
159C22×Dic61C2×D411yescharacteristic
160C22×Dic61C2×D411yescharacteristic
161C22×Dic61C2×D411yescharacteristic
162C22×Dic61C2×D411yescharacteristic
163C2×D244—21no—
164C2×D244—21no—
165C2×D244—21no—
166C2×D244—21no—
167C2×D244—21no—
168C2×D244—21no—
169C2×Dic122—24no—
170C2×Dic122—21no—
171C2×Dic122—21no—
172C2×Dic122—21no—
173C2×Dic122—21no—
174C2×Dic122—21no—
175C2×Dic122—21no—
176C2×Dic122—42no—
177C2×Dic122—42no—
178C2×C8×C61—21no—
179C2×C8×C61—21no—
180C2×C8×C61—21no—
181C2×C8×C61—21no—
182C2×C8×C61—21no—
183C2×C8×C61—21no—
184C4⋊C4×C61—21no—
185C4⋊C4×C61—21no—
186C4⋊C4×C61—21no—
187C4⋊C4×C61—21no—
188C4⋊C4×C61—21no—
189C4⋊C4×C61#4112yes—
190C22⋊C4×C61—42no—
191C42×C61—42no—
192C42×C61C2×D412yes—
193C42×C61#4112yes—
194C22⋊Dic61—42no—
195C22⋊Dic61—42no—
196#11758198—42no—
197#11758198—42no—
198#11758198—81no—
199#11758198—81no—
200C4⋊Dic61—24no—
201C4⋊Dic61—21no—
202C4⋊Dic61—21no—
203C4⋊Dic61—22no—
204C4⋊Dic61—41no—
205C4⋊Dic61C2×D412yes—
206C4⋊Dic61C2×D412yes—
207#11758196—42no—
208#11758196—42no—
209C4×Dic61—21no—
210C4×Dic61—21no—
211C4×Dic61—21no—
212C4×Dic61—21no—
213C4×Dic61—42no—
214C4×Dic61—42no—
215C2×#92418—21no—
216C2×#92418—21no—
217C2×#92418—21no—
218C2×#92418—21no—
219C2×#92418—21no—
220C2×#92418—21no—
221C23×C61—22no—
222C23×C61—41no—
223C23×C61—41no—
224C23×C61—41no—
225C23×C61—41no—
226C23×C61—41no—
227C23×C61—41no—
228C23×C61C22×D411yescharacteristic
229C22×D61—41no—
230C22×D61—41no—
231C22×D61—41no—
232C22×D61—41no—
233C22×D61—41no—
234C22×D61—41no—
235C22×D61—41no—
236D4×C61—22no—
237D4×C61—22no—
238D4×C61—41no—
239D4×C61—41no—
240C2×C4×C61—21no—
241C2×C4×C61—21no—
242C2×C4×C61—21no—
243C2×C4×C61—21no—
244C2×C4×C61—22no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61—41no—
248C2×C4×C61—41no—
249C2×C4×C61—41no—
250C2×C4×C61—41no—
251C2×C4×C61—41no—
252C2×C4×C61—41no—
253C2×C4×C61—84no—
254C2×C4×C61#12112yes—
255C2×C4×C61#12112yes—
256C2×C4×C61C22×D411yescharacteristic
257C2×C4×C61C2×#4111yescharacteristic
258C2×C4×C61C2×#4111yescharacteristic
259C2×C4×C61C2×#4111yescharacteristic
260C2×C4×C61H(5,𝔽2)11yescharacteristic
261C2×C4×C612−1+411yescharacteristic
262C22⋊D61—84no—
263C22⋊D61—84no—
264C2×Dic61—21no—
265C2×Dic61—21no—
266C2×Dic61—22no—
267C2×Dic61—22no—
268C2×Dic61—21no—
269C2×Dic61—21no—
270C2×Dic61—21no—
271C2×Dic61—21no—
272C2×Dic61—21no—
273C2×Dic61—21no—
274C2×Dic61—21no—
275C2×Dic61—21no—
276C2×Dic61—21no—
277C2×Dic61—41no—
278C2×Dic61—41no—
279C2×Dic61—41no—
280C2×Dic61—41no—
281C2×Dic61—41no—
282C2×Dic61—41no—
283C2×Dic61—41no—
284C2×Dic61AGL(1,ℤ/8)12yes—
285D244—41no—
286D244—41no—
287D244—41no—
288D244—41no—
289Dic122—48no—
290Dic122—41no—
291Dic122—41no—
292Dic122—41no—
293Dic122—41no—
294C8×C61—41no—
295C8×C61—41no—
296#92418—41no—
297#92418—41no—
298#25284—611nomaximal, Sylow
299C22×C61—42no—
300C22×C61—42no—
301C22×C61—42no—
302C22×C61—81no—
303C22×C61—81no—
304C22×C61—81no—
305C22×C61—81no—
306C22×C61C2×#12111yescharacteristic
307C22×C61#54611yescharacteristic
308C22×C61#54811yescharacteristic
309C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
310C22×C61#57511yescharacteristic
311C22×C61#57711yescharacteristic
312C22×C61#57711yescharacteristic
313C2×D61—81no—
314C2×D61—81no—
315C2×D61—81no—
316C2×D61—81no—
317C2×D61—81no—
318C2×D61—81no—
319C2×D61—81no—
320C4×C61—22no—
321C4×C61—22no—
322C4×C61—42no—
323C4×C61—81no—
324C4×C61—81no—
325C4×C61#52911yescharacteristic
326C4×C61#54211yescharacteristic
327C4×C61#54811yescharacteristic
328C4×C61#55011yescharacteristic
329Dic61—22no—
330Dic61—42no—
331Dic61—42no—
332Dic61—41no—
333Dic61—41no—
334Dic61—41no—
335Dic61—81no—
336C2×#489—611no—
337C2×#476—611no—
338#2544—611no—
339#2365—611no—
340#2066—611no—
341#2065—611no—
342#2042—611no—
343#2041—611no—
344#1983—611no—
345#1983—611no—
346#1982—611no—
347#1940—611no—
348#1927—611no—
349#1926—611no—
350#1762D6111yescharacteristic, complemented
351C2×C61—82no—
352C2×C61#242611yescharacteristic
353C2×C61#303611yescharacteristic
354C2×C61#304311yescharacteristic
355C2×C61#330611yescharacteristic
356C2×C61#331111yescharacteristic
357C2×C61#337111yescharacteristic
358C2×C61#337311yescharacteristic
359D61—161no—
360C2×C4⋊Q8—611no—
361C2×#127—611no—
362C2×#124—611no—
363C2×#124—611no—
364C2×C4×Q8—1221no—
365C2×C4×D4C2×D6111yescharacteristic
366#489—1224no—
367#476—1224no—
368C2×#106—611no—
369C2×#106—611no—
370C2×C4⋊C8—611no—
371C2×C4⋊C8—611no—
372C2×C4⋊C8—611no—
373C2×#103—611no—
374C2×#103—611no—
375C2×#103—611no—
376C2×#103—611no—
377C2×#102—611no—
378C2×#102—611no—
379C2×#102—611no—
380C2×#102—611no—
381C2×#102—611no—
382C2×#102—611no—
383C2×#102C2×D6111yescharacteristic
384C2×#102C2×D6111yescharacteristic
385C2×#97C2×D6111yescharacteristic
386#394—1221no—
387#394—1221no—
388#391—611no—
389#391—1221no—
390#390—611no—
391#389—611no—
392#389—611no—
393#387—1221no—
394#387—1221no—
395#386—1221no—
396#386—1221no—
397C4×C4⋊C4—611no—
398C4×C4⋊C4—611no—
399C4×C4⋊C4—611no—
400C4×C4⋊C4C2×D6111yescharacteristic
401#340C2×D6111yescharacteristic
402#340C2×D6111yescharacteristic
403#336—611no—
404#336—611no—
405C61#2528411yescharacteristic, complemented, Sylow
406C22×Q8—611no—
407C22×Q8—1221no—
408C22×D4—611no—
409C22×D4—1221no—
410C22×D4—1221no—
411C22×D4C22×D6111yescharacteristic
412C23×C4—21no—
413C22×C8—611no—
414C22×C8—611no—
415C22×C8C22×D6111yescharacteristic
416C22×C8C22×D6111yescharacteristic
417C4⋊Q8—614no—
418C4⋊Q8—1222no—
419#127—1222no—
420#127—2441no—
421#124—1222no—
422#124—1222no—
423#124—1222no—
424#124—2441no—
425C4×Q8—2444no—
426C4×D4—22no—
427C4×D4—22no—
428C2×C4⋊C4—611no—
429C2×C4⋊C4—611no—
430C2×C4⋊C4—611no—
431C2×C4⋊C4—611no—
432C2×C4⋊C4—1221no—
433C2×C4⋊C4—1221no—
434C2×C4⋊C4—1221no—
435C2×C4⋊C4C22×D6111yescharacteristic
436C2×C4⋊C4C22×D6111yescharacteristic
437C2×C4⋊C4C22×D6111yescharacteristic
438C2×C22⋊C4—21no—
439C2×C22⋊C4—1221no—
440C2×C22⋊C4—1221no—
441C2×C22⋊C4—1221no—
442C2×C22⋊C4—1221no—
443C2×C22⋊C4—1221no—
444C2×C22⋊C4—1221no—
445C2×C42—21no—
446C2×C42—611no—
447C2×C42—611no—
448C2×C42—1221no—
449C2×C42—1221no—
450C2×C42C22×D6111yescharacteristic
451#106—1222no—
452#106—1222no—
453C4⋊C8—1222no—
454C4⋊C8—2441no—
455C4⋊C8—2441no—
456#103—1222no—
457#103—1222no—
458#103—1222no—
459#103—2441no—
460#102—22no—
461#102—22no—
462#102—1222no—
463#102—1222no—
464#102—1222no—
465#102—1222no—
466#102—1222no—
467#102—2441no—
468#97—22no—
469#95—21no—
470#95—611no—
471#95—611no—
472#95—1221no—
473#95—1221no—
474#95—1221no—
475#95—1221no—
476#95—1221no—
477#95—1221no—
478C24—21no—
479C24—1221no—
480C2×Q8—1224no—
481C2×Q8—1221no—
482C2×Q8—1221no—
483C2×Q8—1221no—
484C2×Q8—1221no—
485C2×Q8—1221no—
486C2×Q8—1221no—
487C2×Q8—2442no—
488C2×Q8—2442no—
489C2×D4—21no—
490C2×D4—21no—
491C2×D4—21no—
492C2×D4—21no—
493C2×D4—21no—
494C2×D4—1221no—
495C2×D4—1221no—
496C2×D4—1221no—
497C2×D4—1221no—
498C2×D4—1221no—
499C2×D4—1221no—
500C2×D4—4881no—
501C2×D4—4881no—
502C2×D4—2442no—
503C2×D4—2442no—
504C2×D4—2442no—
505C2×D4—2442no—
506C2×D4#1175822212yes—
507C22×C4—21no—
508C22×C4—21no—
509C22×C4—21no—
510C22×C4—42no—
511C22×C4—81no—
512C22×C4—81no—
513C22×C4—611no—
514C22×C4—611no—
515C22×C4—611no—
516C22×C4—611no—
517C22×C4—1221no—
518C22×C4—1221no—
519C22×C4—1221no—
520C22×C4D4×D6111yescharacteristic
521C22×C4D4×D6111yescharacteristic
522C22×C4C23×D6111yescharacteristic
523C2×C8—21no—
524C2×C8—21no—
525C2×C8—21no—
526C2×C8—21no—
527C2×C8—21no—
528C2×C8—21no—
529C2×C8—1221no—
530C2×C8—1221no—
531C2×C8—1221no—
532C2×C8—1221no—
533C2×C8—1221no—
534C2×C8—1221no—
535C4⋊C4—21no—
536C4⋊C4—21no—
537C4⋊C4—21no—
538C4⋊C4—21no—
539C4⋊C4—21no—
540C4⋊C4—612no—
541C4⋊C4—612no—
542C4⋊C4—1222no—
543C4⋊C4—1221no—
544C4⋊C4—1221no—
545C4⋊C4—1224no—
546C4⋊C4—2441no—
547C4⋊C4—2442no—
548C4⋊C4—2442no—
549C4⋊C4#1175822212yes—
550C22⋊C4—42no—
551C22⋊C4—4881no—
552C22⋊C4—4881no—
553C22⋊C4—2442no—
554C22⋊C4—2442no—
555C22⋊C4—2442no—
556C22⋊C4—2442no—
557C42—42no—
558C42—1221no—
559C42—1221no—
560C42—1221no—
561C42—1221no—
562C42—2442no—
563C42—2442no—
564C42#1175822212yes—
565C42D4×D6112yes—
566C23—22no—
567C23—41no—
568C23—41no—
569C23—41no—
570C23—41no—
571C23—41no—
572C23—41no—
573C23—2441no—
574C23—2441no—
575C23—2441no—
576C23—2441no—
577C23—2441no—
578C23—2441no—
579C23—2441no—
580C23C2×D4×D6111yescharacteristic
581Q8—2448no—
582Q8—2441no—
583Q8—2441no—
584Q8—2441no—
585Q8—2441no—
586D4—22no—
587D4—22no—
588D4—41no—
589D4—41no—
590D4—4884no—
591D4—4884no—
592D4—2441no—
593D4—2441no—
594D4—2441no—
595D4—2441no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—22no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—84no—
610C2×C4—612no—
611C2×C4—1221no—
612C2×C4—1221no—
613C2×C4—1222no—
614C2×C4—1222no—
615C2×C4—1221no—
616C2×C4—1221no—
617C2×C4—1221no—
618C2×C4—1221no—
619C2×C4—1221no—
620C2×C4—1221no—
621C2×C4—1221no—
622C2×C4—1221no—
623C2×C4—1221no—
624C2×C4—2441no—
625C2×C4—2441no—
626C2×C4—2441no—
627C2×C4—2441no—
628C2×C4—2441no—
629C2×C4—2441no—
630C2×C4—2441no—
631C2×C4#1451975012yes—
632C2×C4#1451975012yes—
633C2×C4C2×D4×D6111yescharacteristic
634C2×C4C2×#1175822211yescharacteristic
635C2×C4C2×#1175822211yescharacteristic
636C2×C4C2×#1175822211yescharacteristic
637C2×C4#1451982311yescharacteristic
638C2×C4#1451982411yescharacteristic
639C8—41no—
640C8—41no—
641C8—2441no—
642C8—2441no—
643C22—42no—
644C22—42no—
645C22—42no—
646C22—81no—
647C22—81no—
648C22—81no—
649C22—81no—
650C22—4881no—
651C22—4881no—
652C22—4881no—
653C22—4881no—
654C22—4881no—
655C22—4881no—
656C22—4881no—
657C22C2×#1451975011yescharacteristic
658C22#1684040811yescharacteristic
659C22#1684042111yescharacteristic
660C22AGL(1,ℤ/8)×D6111yescharacteristic
661C22#1684056111yescharacteristic
662C22#1684057211yescharacteristic
663C22#1684057211yescharacteristic
664C4—22no—
665C4—22no—
666C4—42no—
667C4—81no—
668C4—81no—
669C4—1222no—
670C4—4881no—
671C4—2441no—
672C4—2442no—
673C4—2442no—
674C4—2441no—
675C4—2441no—
676C4#1684029611yescharacteristic
677C4#1684038311yescharacteristic
678C4#1684042111yescharacteristic
679C4#1684043311yescharacteristic
680C2—82no—
681C2—9761no—
682C2#2144938511yescharacteristic
683C2#2145528211yescharacteristic
684C2#2145533911yescharacteristic
685C2#2145778511yescharacteristic
686C2#2145786011yescharacteristic
687C2#2145859211yescharacteristic
688C2#2145861211yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1762C2
socleC23×C61C22×D4
prime radicalC2×#14519773C22
hypercenter#1762D61
hypocenterC61#25284
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1762

Complements

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

NH ≅ G/Ntype
1. C2×#16840122C2semidirect
2. C2×#16840052C2semidirect
4. #21448875C2semidirect
5. #21447201C2semidirect
6. #21447194C2semidirect
7. #21447058C2semidirect
9. #21446751C2semidirect
10. #21446750C2semidirect
12. #21446542C2semidirect
13. #21446451C2semidirect
14. #21446444C2semidirect
15. C61×#1762C2semidirect
69. #14519777D4semidirect
350. #1762D61semidirect
405. C61#25284semidirect

Maximal subgroups

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

subgroupindex
C2×#168401222
C2×#168400522
#214507862
#214488752
#214472012
#214471942
#214470582
#214470512
#214467512
#214467502
#214467442
#214465422
#214464512
#214464442
C61×#17622
#2528461