Finite Group Databasebeta

← #29740074

Subgroups — #29740074

Order 15616.

944 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839975C211yesmaximal, characteristic
2#21449600C211yesmaximal, characteristic, complemented
3D4×#11758196C211yesmaximal, characteristic, complemented
4#21447354C211yesmaximal, characteristic, complemented
5#21447313C211yesmaximal, characteristic, complemented
6#21447225C211yesmaximal, characteristic, complemented
7#21447058C211yesmaximal, characteristic, complemented
8#21446495C211yesmaximal, characteristic, complemented
9#21446376C211yesmaximal, characteristic, complemented
10#21446169C211yesmaximal, characteristic
11#21446153C211yesmaximal, characteristic, complemented
12#21446151C211yesmaximal, characteristic, complemented
13#21446066C211yesmaximal, characteristic, complemented
14#21446064C211yesmaximal, characteristic, complemented
15C2×C4×D8×C61C211yesmaximal, characteristic
16C22×#11758200C2211yescharacteristic
17C2×#14519764C2211yescharacteristic
18C2×D4×Dic61C2211yescharacteristic
19C2×D4×Dic61C2211yescharacteristic
20C22×#11758196—21no—
21#16839975—24no—
22C2×#14519710C2211yescharacteristic
23C2×#14519706C2211yescharacteristic
24C2×#14519702C2211yescharacteristic
25C2×#14519702C2211yescharacteristic
26C2×#14519700C2211yescharacteristic
27C2×#14519698C2211yescharacteristic
28C2×#14519698C2211yescharacteristic
29C2×#14519698C2211yescharacteristic
30#16839690—21no—
31#16839690—21no—
32#16839668C2211yescharacteristic
33#16839667C2211yescharacteristic
34#16839665C2211yescharacteristic
35#16839660C2211yescharacteristic
36#16839658C2211yescharacteristic
37#16839658C2211yescharacteristic
38#16839635—21no—
39#16839635—21no—
40#16839634—21no—
41#16839631C2211yescharacteristic
42#16839625—21no—
43#16839620—21no—
44C4×#11758196C2211yescharacteristic
45#16839525C2211yescharacteristic
46#16839525C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839517C2211yescharacteristic
50#16839516C2211yescharacteristic
51#16839516C2211yescharacteristic
52#16839516C2211yescharacteristic
53C22×D8×C61C2211yescharacteristic
54C2×C4×D4×C61C2211yescharacteristic
55C2×C4×D4×C61C2211yescharacteristic
56C4×D8×C61—24no—
57C2×#107×C61C2211yescharacteristic
58C2×#102×C61C2211yescharacteristic
59C2×#102×C61C2211yescharacteristic
60C2×C4×C8×C61C2211yescharacteristic
61C22×Dic122C2311yescharacteristic
62C23×Dic61—21no—
63C23×Dic61—21no—
64C2×#11758200—22no—
65C2×#11758200—22no—
66C2×#11758200—22no—
67C22×#92418C2311yescharacteristic
68C22×#92418C2311yescharacteristic
69#14519764—22no—
70#14519764—41no—
71D4×Dic61—22no—
72D4×Dic61—22no—
73D4×Dic61—22no—
74D4×Dic61—41no—
75C2×C4⋊Dic61C2311yescharacteristic
76C2×C4⋊Dic61C2311yescharacteristic
77C2×C4⋊Dic61C2311yescharacteristic
78C2×#11758196—21no—
79C2×#11758196—21no—
80C2×#11758196—21no—
81C2×#11758196—21no—
82C2×#11758196—21no—
83C2×#11758196—44no—
84C2×#11758196D411yescharacteristic
85C2×#11758196D411yescharacteristic
86C2×C22⋊Dic61—21no—
87C2×C22⋊Dic61—21no—
88C2×C22⋊Dic61—21no—
89C2×C22⋊Dic61—21no—
90C2×C22⋊Dic61—21no—
91C2×C22⋊Dic61—21no—
92C2×C4×Dic61C2311yescharacteristic
93C2×C4×Dic61C2311yescharacteristic
94#14519710—41no—
95#14519706—22no—
96#14519702—22no—
97#14519702—22no—
98#14519700—41no—
99#14519698—22no—
100#14519698—22no—
101#14519698—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×C61D414yes—
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#107×C61—22no—
130#102×C61—22no—
131#102×C61—22no—
132C4×C8×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×C61C2×D412yes—
144C2×D4×C61C2×D412yes—
145C2×D4×C61#4112yes—
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—24no—
159C22×Dic61—21no—
160C22×Dic61—21no—
161C22×Dic61—21no—
162C22×Dic61—21no—
163C22×Dic61—42no—
164C22×Dic61—42no—
165C22×Dic61—42no—
166C22×Dic61—42no—
167C22×Dic61—81no—
168C22×Dic61C2×D411yescharacteristic
169C22×Dic61C2×D411yescharacteristic
170C22×Dic61C2×D411yescharacteristic
171C22×Dic61C2×D411yescharacteristic
172C2×Dic122—21no—
173C2×Dic122—21no—
174C2×Dic122—21no—
175C2×Dic122—21no—
176C2×Dic122—21no—
177C2×Dic122—21no—
178D8×C61—28no—
179C2×C8×C61—21no—
180C2×C8×C61—21no—
181C2×C8×C61—21no—
182C2×C8×C61—21no—
183C2×C8×C61—21no—
184C2×C8×C61C2×D412yes—
185C4⋊C4×C61—21no—
186C4⋊C4×C61—21no—
187C4⋊C4×C61—21no—
188C4⋊C4×C61—21no—
189C22⋊C4×C61—42no—
190C22⋊C4×C61—42no—
191C42×C61—21no—
192C42×C61—21no—
193C22⋊Dic61—24no—
194C22⋊Dic61—42no—
195C22⋊Dic61—42no—
196C22⋊Dic61—42no—
197C22⋊Dic61—42no—
198C22⋊Dic61—81no—
199#11758200—44no—
200C4⋊Dic61—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61—21no—
203C4⋊Dic61—21no—
204C4⋊Dic61—21no—
205C4⋊Dic61#4112yes—
206#11758196—22no—
207#11758196—22no—
208#11758196—42no—
209#11758196—42no—
210#11758196—42no—
211#11758196—81no—
212#11758196—81no—
213C4×Dic61—21no—
214C4×Dic61—21no—
215C4×Dic61—21no—
216C4×Dic61—21no—
217C2×#92418—21no—
218C2×#92418—21no—
219C2×#92418—21no—
220C2×#92418—21no—
221C2×#92418—21no—
222C2×#92418—21no—
223C23×C61—22no—
224C23×C61—22no—
225C23×C61—22no—
226C23×C61—22no—
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
238D4×C61—22no—
239D4×C61—22no—
240D4×C61—22no—
241D4×C61—22no—
242D4×C61—22no—
243D4×C61—41no—
244D4×C61—41no—
245D4×C61#12314yes—
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—41no—
257C2×C4×C61—84no—
258C2×C4×C61—84no—
259C2×C4×C61AGL(1,ℤ/8)12yes—
260C2×C4×C61C22×D411yescharacteristic
261C2×C4×C61C2×#4111yescharacteristic
262C2×C4×C61C2×#4111yescharacteristic
263C2×C4×C61C2×#4111yescharacteristic
264C2×C4×C61H(5,𝔽2)11yescharacteristic
265C2×C4×C612−1+411yescharacteristic
266C2×Dic61—21no—
267C2×Dic61—21no—
268C2×Dic61—21no—
269C2×Dic61—22no—
270C2×Dic61—21no—
271C2×Dic61—21no—
272C2×Dic61—21no—
273C2×Dic61—21no—
274C2×Dic61—21no—
275C2×Dic61—21no—
276C2×Dic61—21no—
277C2×Dic61—21no—
278C2×Dic61—48no—
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×Dic61—41no—
288C2×Dic61—41no—
289C2×Dic61—41no—
290C2×Dic61—84no—
291C2×Dic61#13512yes—
292Dic122—41no—
293Dic122—41no—
294Dic122—41no—
295Dic122—41no—
296C8×C61—22no—
297C8×C61—41no—
298#92418—41no—
299#92418—41no—
300#26164—611nomaximal, Sylow
301C22×C61—24no—
302C22×C61—42no—
303C22×C61—42no—
304C22×C61—42no—
305C22×C61—42no—
306C22×C61—42no—
307C22×C61—42no—
308C22×C61—42no—
309C22×C61—42no—
310C22×C61—42no—
311C22×C61—81no—
312C22×C61—81no—
313C22×C61—81no—
314C22×C61—81no—
315C22×C61#52511yescharacteristic
316C22×C61D4211yescharacteristic
317C22×C61#54811yescharacteristic
318C22×C61C2×#13511yescharacteristic
319C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
320C22×C61#57611yescharacteristic
321C22×C61#57811yescharacteristic
322C4×C61—22no—
323C4×C61—41no—
324C4×C61—81no—
325C4×C61—81no—
326C4×C61C2×#12311yescharacteristic
327C4×C61#53911yescharacteristic
328C4×C61#54711yescharacteristic
329C4×C61#55311yescharacteristic
330Dic61—22no—
331Dic61—42no—
332Dic61—41no—
333Dic61—41no—
334Dic61—41no—
335Dic61—81no—
336Dic61—81no—
337Dic61—81no—
338C2×#463—611no—
339C2×C4×D8D6111yescharacteristic
340#2447—611no—
341D4×C4⋊C4—611no—
342#2095—611no—
343#2087—611no—
344#2071—611no—
345#2042—611no—
346#1933—611no—
347#1916—611no—
348#1879—611no—
349#1877—611no—
350#1877—611no—
351#1862—611no—
352#1862—611no—
353C2×C61—44no—
354C2×C61—82no—
355C2×C61#244311yescharacteristic
356C2×C61#310511yescharacteristic
357C2×C61#312411yescharacteristic
358C2×C61#326411yescharacteristic
359C2×C61#328311yescharacteristic
360C2×C61#336111yescharacteristic
361C2×C61#336711yescharacteristic
362C22×SD16—611no—
363C22×D8C2×D6111yescharacteristic
364C2×#124—611no—
365C2×C4×D4—611no—
366C2×C4×D4—611no—
367C2×C4×D4C2×D6111yescharacteristic
368C2×C4×D4C2×D6111yescharacteristic
369C22×C4⋊C4—1221no—
370#463—1224no—
371C4×D8—24no—
372C2×#107—611no—
373C2×#107C2×D6111yescharacteristic
374C2×C4⋊C8—611no—
375C2×#103—611no—
376C2×#103—611no—
377C2×#103—611no—
378C2×#102—611no—
379C2×#102—611no—
380C2×#102—611no—
381C2×#102C2×D6111yescharacteristic
382C2×#102C2×D6111yescharacteristic
383C2×C4×C8C2×D6111yescharacteristic
384#397—1221no—
385#397—1221no—
386#391—611no—
387#391—611no—
388#390—611no—
389#389—611no—
390#389—611no—
391#389—611no—
392#385—1221no—
393#385—1221no—
394#385—1221no—
395#384—611no—
396#381—1221no—
397#380—1221no—
398C4×C4⋊C4—611no—
399#340—611no—
400#340—611no—
401#340—611no—
402#340—611no—
403#336—611no—
404#336—611no—
405#336—611no—
406#336—611no—
407C61#2616411yescharacteristic, complemented, Sylow
408C22×Q8—611no—
409C22×D4C22×D6111yescharacteristic
410C22×D4C22×D6111yescharacteristic
411C23×C4—21no—
412C23×C4—21no—
413C23×C4—1221no—
414C23×C4—1221no—
415C2×SD16—1222no—
416C2×SD16—1222no—
417C2×SD16—1222no—
418C2×D8—22no—
419C2×D8—22no—
420C2×D8C22⋊D6114yes—
421C22×C8—611no—
422C22×C8—611no—
423C22×C8C22×D6111yescharacteristic
424C22×C8C22×D6111yescharacteristic
425#124—1222no—
426#124—2441no—
427C4×D4—22no—
428C4×D4—22no—
429C4×D4—41no—
430C4×D4—41no—
431C4×D4—1222no—
432C4×D4—1222no—
433C4×D4—1222no—
434C4×D4—2441no—
435C2×C4⋊C4—611no—
436C2×C4⋊C4—611no—
437C2×C4⋊C4—611no—
438C2×C4⋊C4—611no—
439C2×C4⋊C4—611no—
440C2×C4⋊C4—1221no—
441C2×C4⋊C4—1221no—
442C2×C4⋊C4—1221no—
443C2×C4⋊C4—1221no—
444C2×C4⋊C4—1221no—
445C2×C4⋊C4—2444no—
446C2×C4⋊C4C22×D6111yescharacteristic
447C2×C4⋊C4C22×D6111yescharacteristic
448C2×C22⋊C4—21no—
449C2×C22⋊C4—21no—
450C2×C22⋊C4—1221no—
451C2×C22⋊C4—1221no—
452C2×C22⋊C4—1221no—
453C2×C22⋊C4—1221no—
454C2×C22⋊C4—1221no—
455C2×C22⋊C4—1221no—
456C2×C42—611no—
457C2×C42—611no—
458C2×C42C22×D6111yescharacteristic
459#107—22no—
460#107—2441no—
461C4⋊C8—1222no—
462#103—1222no—
463#103—1222no—
464#103—2441no—
465#102—22no—
466#102—22no—
467#102—1222no—
468#102—1222no—
469#102—1222no—
470C4×C8—22no—
471#95—611no—
472#95—611no—
473#95—1221no—
474#95—1221no—
475#95—1221no—
476#95—1221no—
477#95—1221no—
478#95—1221no—
479#95—1221no—
480C24—21no—
481C24—21no—
482C2×Q8—1221no—
483C2×Q8—1221no—
484C2×Q8—1221no—
485C2×Q8—1221no—
486C2×Q8—1221no—
487C2×Q8—1221no—
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#1175822212yes—
497C2×D4#1175822212yes—
498C2×D4C2×C22⋊D6112yes—
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—1224no—
513C22×C4—1221no—
514C22×C4—1221no—
515C22×C4—1221no—
516C22×C4—1221no—
517C22×C4—4881no—
518C22×C4—2442no—
519C22×C4—2442no—
520C22×C4—2442no—
521C22×C4—2442no—
522C22×C4D4×D6111yescharacteristic
523C22×C4D4×D6111yescharacteristic
524C22×C4C23×D6111yescharacteristic
525SD16—2444no—
526D8—28no—
527C2×C8—21no—
528C2×C8—21no—
529C2×C8—21no—
530C2×C8—21no—
531C2×C8—21no—
532C2×C8—1221no—
533C2×C8—1221no—
534C2×C8—1221no—
535C2×C8—1221no—
536C2×C8—1221no—
537C2×C8—1221no—
538C2×C8C2×C22⋊D6112yes—
539C4⋊C4—21no—
540C4⋊C4—21no—
541C4⋊C4—21no—
542C4⋊C4—21no—
543C4⋊C4—612no—
544C4⋊C4—1222no—
545C4⋊C4—1221no—
546C4⋊C4—1222no—
547C4⋊C4—1221no—
548C4⋊C4—1221no—
549C4⋊C4—1221no—
550C4⋊C4—1221no—
551C4⋊C4—4881no—
552C4⋊C4—4881no—
553C4⋊C4—2442no—
554C4⋊C4—2442no—
555C4⋊C4—2442no—
556C22⋊C4—42no—
557C22⋊C4—42no—
558C22⋊C4—1224no—
559C22⋊C4—4881no—
560C22⋊C4—2442no—
561C22⋊C4—2442no—
562C22⋊C4—2442no—
563C22⋊C4—2442no—
564C42—21no—
565C42—21no—
566C42—1221no—
567C42—1221no—
568C42—1221no—
569C42—1221no—
570C23—22no—
571C23—22no—
572C23—22no—
573C23—22no—
574C23—41no—
575C23—41no—
576C23—41no—
577C23—41no—
578C23—41no—
579C23—41no—
580C23—41no—
581C23—41no—
582C23—41no—
583C23—41no—
584C23C2×D4×D6111yescharacteristic
585Q8—2441no—
586Q8—2441no—
587Q8—2441no—
588Q8—2441no—
589D4—22no—
590D4—22no—
591D4—22no—
592D4—22no—
593D4—22no—
594D4—41no—
595D4—41no—
596D4#1451976014yes—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—84no—
609C2×C4—84no—
610C2×C4—612no—
611C2×C4—1221no—
612C2×C4—1221no—
613C2×C4—1221no—
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—4884no—
624C2×C4—2448no—
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—2441no—
635C2×C4—2441no—
636C2×C4#1451980112yes—
637C2×C4C22×C22⋊D6111yescharacteristic
638C2×C4#41×D6111yescharacteristic
639C2×C4C2×#1175822211yescharacteristic
640C2×C4C2×#1175822211yescharacteristic
641C2×C4#1451982211yescharacteristic
642C2×C4#1451982411yescharacteristic
643C8—22no—
644C8—41no—
645C8—2441no—
646C8—2441no—
647C22—24no—
648C22—42no—
649C22—42no—
650C22—42no—
651C22—42no—
652C22—42no—
653C22—42no—
654C22—42no—
655C22—42no—
656C22—42no—
657C22—81no—
658C22—81no—
659C22—81no—
660C22—81no—
661C22#1684027111yescharacteristic
662C22D4×C22⋊D6111yescharacteristic
663C22#1684042111yescharacteristic
664C22#135×D6111yescharacteristic
665C22C2×#1451980111yescharacteristic
666C22#1684056311yescharacteristic
667C22#1684057611yescharacteristic
668C4—22no—
669C4—41no—
670C4—81no—
671C4—81no—
672C4—1222no—
673C4—4881no—
674C4—4881no—
675C4—4881no—
676C4—2441no—
677C4—2441no—
678C4—2441no—
679C4—2442no—
680C4C2×#1451976011yescharacteristic
681C4#1684035411yescharacteristic
682C4#1684041111yescharacteristic
683C4#1684046111yescharacteristic
684C2—44no—
685C2—82no—
686C2#2144954811yescharacteristic
687C2#2145586411yescharacteristic
688C2#2145607711yescharacteristic
689C2#2145729911yescharacteristic
690C2#2145753511yescharacteristic
691C2#2145848011yescharacteristic
692C2#2145853511yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C4×D8×C61C2
socleC23×C61C22×D4
prime radicalC22×D8×C61C22
hypercenterC2×C4×D8D61
hypocenterC61#26164
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×D8

Complements

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

NH ≅ G/Ntype
2. #21449600C2semidirect
3. D4×#11758196C2semidirect
4. #21447354C2semidirect
5. #21447313C2semidirect
6. #21447225C2semidirect
7. #21447058C2semidirect
8. #21446495C2semidirect
9. #21446376C2semidirect
11. #21446153C2semidirect
12. #21446151C2semidirect
13. #21446066C2semidirect
14. #21446064C2semidirect
407. C61#26164semidirect

Maximal subgroups

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

subgroupindex
C2×#168399752
#214496002
D4×#117581962
#214473542
#214473132
#214472252
#214470582
#214464952
#214463762
#214461692
#214461532
#214461512
#214460662
#214460642
C2×C4×D8×C612
#2616461