Finite Group Databasebeta

← #29740209

Subgroups — #29740209

Order 15616.

818 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839986C211yesmaximal, characteristic
2#21449611C211yesmaximal, characteristic
3#21448737C211yesmaximal, characteristic, complemented
4#21447337C211yesmaximal, characteristic
5#21447313C211yesmaximal, characteristic, complemented
6#21447240C211yesmaximal, characteristic, complemented
7#21447056C211yesmaximal, characteristic
8#21446502C211yesmaximal, characteristic
9#21446376C211yesmaximal, characteristic, complemented
10#21446174C211yesmaximal, characteristic
11#21446172C211yesmaximal, characteristic
12#21446162C211yesmaximal, characteristic, complemented
13#21446087C211yesmaximal, characteristic, complemented
14#21446085C211yesmaximal, characteristic, complemented
15C2×C4×Q16×C61C211yesmaximal, characteristic, complemented
16C22×#11758201C2211yescharacteristic
17C2×#14519766C2211yescharacteristic
18C2×Q8×Dic61C2211yescharacteristic
19C2×Q8×Dic61C2211yescharacteristic
20#16839986—24no—
21C2×#14519708C2211yescharacteristic
22C2×#14519706C2211yescharacteristic
23C2×#14519701C2211yescharacteristic
24C2×#14519701C2211yescharacteristic
25C2×#14519701C2211yescharacteristic
26C2×#14519699C2211yescharacteristic
27C2×#14519699C2211yescharacteristic
28C2×#14519697C2211yescharacteristic
29#16839697—21no—
30#16839688—21no—
31#16839669C2211yescharacteristic
32#16839664C2211yescharacteristic
33#16839663C2211yescharacteristic
34#16839660—21no—
35#16839660C2211yescharacteristic
36#16839658—21no—
37#16839658C2211yescharacteristic
38#16839658C2211yescharacteristic
39#16839653—21no—
40#16839653—21no—
41#16839627—21no—
42#16839627C2211yescharacteristic
43C4×#11758196—21no—
44C4×#11758196C2211yescharacteristic
45#16839525C2211yescharacteristic
46#16839525C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839517C2211yescharacteristic
50#16839516C2211yescharacteristic
51#16839516C2211yescharacteristic
52#16839516C2211yescharacteristic
53C22×Q16×C61C2211yescharacteristic
54C2×C4×Q8×C61C2211yescharacteristic
55C2×C4×Q8×C61C2211yescharacteristic
56C4×Q16×C61—24no—
57C2×#107×C61C2211yescharacteristic
58C2×#103×C61C2211yescharacteristic
59C2×#103×C61C2211yescharacteristic
60C2×C4×C8×C61C2211yescharacteristic
61C22×D244C2311yescharacteristic
62C2×#11758201—22no—
63C2×#11758201—22no—
64C2×#11758201—22no—
65C22×#92418C2311yescharacteristic
66C22×#92418C2311yescharacteristic
67#14519766—22no—
68#14519766—41no—
69Q8×Dic61—22no—
70Q8×Dic61—22no—
71Q8×Dic61—22no—
72Q8×Dic61—41no—
73C2×C4⋊Dic61—21no—
74C2×C4⋊Dic61—21no—
75C2×C4⋊Dic61C2311yescharacteristic
76C2×C4⋊Dic61C2311yescharacteristic
77C2×C4⋊Dic61C2311yescharacteristic
78C2×#11758196—21no—
79C2×#11758196—21no—
80C2×#11758196—21no—
81C2×#11758196D411yescharacteristic
82C2×#11758196D411yescharacteristic
83C2×#11758198—21no—
84C2×#11758198—21no—
85C2×#11758198—21no—
86C2×#11758198—21no—
87C2×#11758198—21no—
88C2×#11758198—21no—
89C2×C4×Dic61—21no—
90C2×C4×Dic61—21no—
91C2×C4×Dic61C2311yescharacteristic
92C2×C4×Dic61C2311yescharacteristic
93#14519708—41no—
94#14519706—22no—
95#14519701—22no—
96#14519701—22no—
97#14519701—22no—
98#14519699—22no—
99#14519699—22no—
100#14519697—41no—
101#14519684—21no—
102#14519684—21no—
103#14519684—21no—
104#14519684—21no—
105#14519684—21no—
106#14519684—21no—
107#14519684—21no—
108#14519684D411yescharacteristic
109#14519684D411yescharacteristic
110C22×Q8×C61C2311yescharacteristic
111C22×Q8×C61C2311yescharacteristic
112C2×Q16×C61—22no—
113C2×Q16×C61—22no—
114C2×Q16×C61D414yes—
115C22×C8×C61C2311yescharacteristic
116C22×C8×C61C2311yescharacteristic
117C4×Q8×C61—22no—
118C4×Q8×C61—22no—
119C4×Q8×C61—41no—
120C4×Q8×C61—41no—
121C2×C4⋊C4×C61—21no—
122C2×C4⋊C4×C61—21no—
123C2×C4⋊C4×C61C2311yescharacteristic
124C2×C4⋊C4×C61C2311yescharacteristic
125C2×C42×C61—21no—
126C2×C42×C61—21no—
127C2×C42×C61C2311yescharacteristic
128#107×C61—22no—
129#103×C61—22no—
130#103×C61—22no—
131C4×C8×C61—22no—
132C23×D61—21no—
133C2×Q8×C61—21no—
134C2×Q8×C61—21no—
135C2×Q8×C61—21no—
136C2×Q8×C61—21no—
137C2×Q8×C61—21no—
138C2×Q8×C61—21no—
139C2×Q8×C61—21no—
140C2×Q8×C61—21no—
141C2×Q8×C61C2×D412yes—
142C2×Q8×C61C2×D412yes—
143C2×Q8×C61#4112yes—
144C2×Q8×C61#4112yes—
145C22×C4×C61—21no—
146C22×C4×C61—21no—
147C22×C4×C61—21no—
148C22×C4×C61—21no—
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2411yescharacteristic
152C22×Dic61—21no—
153C22×Dic61—21no—
154C22×Dic61—21no—
155C22×Dic61C2×D411yescharacteristic
156C22×Dic61C2×D411yescharacteristic
157C22×Dic61C2×D411yescharacteristic
158C22×Dic61C2×D411yescharacteristic
159C2×D244—21no—
160C2×D244—21no—
161C2×D244—21no—
162C2×D244—21no—
163C2×D244—21no—
164C2×D244—21no—
165Q16×C61—28no—
166C2×C8×C61—21no—
167C2×C8×C61—21no—
168C2×C8×C61—21no—
169C2×C8×C61—21no—
170C2×C8×C61—21no—
171C2×C8×C61C2×D412yes—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—42no—
177C4⋊C4×C61—42no—
178C42×C61—21no—
179C42×C61—21no—
180C42×C61—42no—
181C42×C61—42no—
182#11758201—44no—
183#11758198—42no—
184#11758198—42no—
185#11758198—42no—
186#11758198—42no—
187#11758198—81no—
188#11758198—81no—
189C4⋊Dic61—24no—
190C4⋊Dic61—21no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61—42no—
196C4⋊Dic61#4112yes—
197#11758196—22no—
198#11758196—41no—
199#11758196—42no—
200#11758196—42no—
201#11758196—81no—
202C4×Dic61—24no—
203C4×Dic61—21no—
204C4×Dic61—21no—
205C4×Dic61—21no—
206C4×Dic61—21no—
207C4×Dic61—42no—
208C2×#92418—21no—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418—21no—
213C2×#92418—21no—
214C23×C61C22×D411yescharacteristic
215C22×D61—41no—
216C22×D61—41no—
217C22×D61—41no—
218C22×D61—41no—
219C22×D61—41no—
220C22×D61—41no—
221C22×D61—41no—
222Q8×C61—22no—
223Q8×C61—22no—
224Q8×C61—22no—
225Q8×C61—22no—
226Q8×C61—22no—
227Q8×C61—41no—
228Q8×C61—41no—
229Q8×C61#12314yes—
230C2×C4×C61—22no—
231C2×C4×C61—22no—
232C2×C4×C61—21no—
233C2×C4×C61—22no—
234C2×C4×C61—21no—
235C2×C4×C61—22no—
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—41no—
246C2×C4×C61—41no—
247C2×C4×C61#13512yes—
248C2×C4×C61C22×D411yescharacteristic
249C2×C4×C61C2×#4111yescharacteristic
250C2×C4×C61C2×#4111yescharacteristic
251C2×C4×C61C2×#4111yescharacteristic
252C2×C4×C61H(5,𝔽2)11yescharacteristic
253C2×C4×C612−1+411yescharacteristic
254C2×Dic61—21no—
255C2×Dic61—21no—
256C2×Dic61—22no—
257C2×Dic61—21no—
258C2×Dic61—21no—
259C2×Dic61—21no—
260C2×Dic61—21no—
261C2×Dic61—21no—
262C2×Dic61—21no—
263C2×Dic61—21no—
264C2×Dic61—21no—
265C2×Dic61—21no—
266C2×Dic61—21no—
267C2×Dic61—41no—
268C2×Dic61—41no—
269C2×Dic61—41no—
270C2×Dic61—41no—
271C2×Dic61—41no—
272C2×Dic61—41no—
273C2×Dic61—41no—
274C2×Dic61—41no—
275D244—41no—
276D244—41no—
277D244—41no—
278D244—41no—
279C8×C61—22no—
280C8×C61—41no—
281#92418—41no—
282#92418—41no—
283#26173—611nomaximal, Sylow
284C22×C61#52511yescharacteristic
285C22×C61D4211yescharacteristic
286C22×C61#54811yescharacteristic
287C22×C61C2×#13511yescharacteristic
288C22×C61#57511yescharacteristic
289C22×C61#57711yescharacteristic
290C22×C61#57811yescharacteristic
291C2×D61—81no—
292C2×D61—81no—
293C2×D61—81no—
294C2×D61—81no—
295C2×D61—81no—
296C2×D61—81no—
297C2×D61—81no—
298C4×C61—24no—
299C4×C61—22no—
300C4×C61—42no—
301C4×C61—41no—
302C4×C61—81no—
303C4×C61—81no—
304C4×C61C2×#12311yescharacteristic
305C4×C61#53911yescharacteristic
306C4×C61#54711yescharacteristic
307C4×C61#55311yescharacteristic
308Dic61—41no—
309Dic61—42no—
310Dic61—41no—
311Dic61—41no—
312Dic61—41no—
313Dic61—81no—
314Dic61—81no—
315C2×#464—611no—
316C2×C4×Q16D6111yescharacteristic, complemented
317#2448—611no—
318#2351—611no—
319#2092—611no—
320#2087—611no—
321#2074—611no—
322#2042—611no—
323#1934—611no—
324#1916—611no—
325#1880—611no—
326#1880—611no—
327#1878—611no—
328#1865—611no—
329#1865—611no—
330C2×C61#244311yescharacteristic
331C2×C61#310211yescharacteristic
332C2×C61#312111yescharacteristic
333C2×C61#327111yescharacteristic
334C2×C61#328511yescharacteristic
335C2×C61#336511yescharacteristic
336C2×C61#337711yescharacteristic
337D61—161no—
338C22×Q16C2×D6111yescharacteristic
339C22×SD16—611no—
340C2×#124—611no—
341C2×C4×Q8—611no—
342C2×C4×Q8—611no—
343C2×C4×Q8C2×D6111yescharacteristic
344C2×C4×Q8C2×D6111yescharacteristic
345#464—1224no—
346C4×Q16—24no—
347C2×#107C2×D6111yescharacteristic
348C2×#106—611no—
349C2×C4⋊C8—611no—
350C2×#103—611no—
351C2×#103—611no—
352C2×#103—611no—
353C2×#103C2×D6111yescharacteristic
354C2×#103C2×D6111yescharacteristic
355C2×#102—611no—
356C2×#102—611no—
357C2×#102—611no—
358C2×C4×C8C2×D6111yescharacteristic
359#399—1221no—
360#397—1221no—
361#391—611no—
362#390—611no—
363#390—611no—
364#389—611no—
365#389—611no—
366#389—611no—
367#389—1221no—
368#389—1221no—
369#388—1221no—
370#388—1221no—
371#382—611no—
372#382—1221no—
373C4×C4⋊C4—611no—
374C4×C4⋊C4—1221no—
375#340—611no—
376#340—611no—
377#340—611no—
378#340—611no—
379#336—611no—
380#336—611no—
381#336—611no—
382#336—611no—
383C61#2617311yescharacteristic, complemented, Sylow
384C22×Q8C22×D6111yescharacteristic
385C22×Q8C22×D6111yescharacteristic
386C22×D4—611no—
387C2×Q16—22no—
388C2×Q16—22no—
389C2×Q16C22⋊D6114yes—
390C2×SD16—1222no—
391C2×SD16—1222no—
392C2×SD16—1222no—
393C22×C8—611no—
394C22×C8—611no—
395C22×C8C22×D6111yescharacteristic
396C22×C8C22×D6111yescharacteristic
397#124—1222no—
398#124—2441no—
399C4×Q8—22no—
400C4×Q8—22no—
401C4×Q8—41no—
402C4×Q8—41no—
403C4×Q8—1222no—
404C4×Q8—1222no—
405C4×Q8—1222no—
406C4×Q8—2441no—
407C2×C4⋊C4—21no—
408C2×C4⋊C4—21no—
409C2×C4⋊C4—611no—
410C2×C4⋊C4—611no—
411C2×C4⋊C4—611no—
412C2×C4⋊C4—611no—
413C2×C4⋊C4—611no—
414C2×C4⋊C4—1221no—
415C2×C4⋊C4—1221no—
416C2×C4⋊C4—1221no—
417C2×C4⋊C4—1221no—
418C2×C4⋊C4—1221no—
419C2×C4⋊C4C22×D6111yescharacteristic
420C2×C4⋊C4C22×D6111yescharacteristic
421C2×C22⋊C4—1221no—
422C2×C22⋊C4—1221no—
423C2×C22⋊C4—1221no—
424C2×C22⋊C4—1221no—
425C2×C22⋊C4—1221no—
426C2×C22⋊C4—1221no—
427C2×C42—21no—
428C2×C42—21no—
429C2×C42—611no—
430C2×C42—611no—
431C2×C42—1221no—
432C2×C42—1221no—
433C2×C42C22×D6111yescharacteristic
434#107—22no—
435#106—2441no—
436C4⋊C8—1222no—
437#103—22no—
438#103—22no—
439#103—1222no—
440#103—1222no—
441#103—1222no—
442#102—1222no—
443#102—1222no—
444#102—2441no—
445C4×C8—22no—
446#95—611no—
447#95—611no—
448#95—1221no—
449#95—1221no—
450#95—1221no—
451#95—1221no—
452#95—1221no—
453#95—1221no—
454#95—1221no—
455C24—1221no—
456C2×Q8—21no—
457C2×Q8—21no—
458C2×Q8—21no—
459C2×Q8—21no—
460C2×Q8—21no—
461C2×Q8—21no—
462C2×Q8—21no—
463C2×Q8—21no—
464C2×Q8#1175822212yes—
465C2×Q8#1175822212yes—
466C2×Q8C2×C22⋊D6112yes—
467C2×Q8C2×C22⋊D6112yes—
468C2×D4—1221no—
469C2×D4—1221no—
470C2×D4—1221no—
471C2×D4—1221no—
472C2×D4—1221no—
473C2×D4—1221no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—611no—
479C22×C4—611no—
480C22×C4—611no—
481C22×C4—611no—
482C22×C4—1221no—
483C22×C4—1221no—
484C22×C4—1221no—
485C22×C4D4×D6111yescharacteristic
486C22×C4D4×D6111yescharacteristic
487C22×C4C23×D6111yescharacteristic
488Q16—28no—
489SD16—2444no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—1221no—
496C2×C8—1221no—
497C2×C8—1221no—
498C2×C8—1221no—
499C2×C8—1221no—
500C2×C8—1221no—
501C2×C8C2×C22⋊D6112yes—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4—612no—
509C4⋊C4—1224no—
510C4⋊C4—1221no—
511C4⋊C4—1222no—
512C4⋊C4—1221no—
513C4⋊C4—1221no—
514C4⋊C4—1221no—
515C4⋊C4—1221no—
516C4⋊C4—4881no—
517C4⋊C4—2441no—
518C4⋊C4—2442no—
519C4⋊C4—2442no—
520C4⋊C4—2442no—
521C22⋊C4—4881no—
522C22⋊C4—4881no—
523C22⋊C4—2442no—
524C22⋊C4—2442no—
525C22⋊C4—2442no—
526C22⋊C4—2442no—
527C42—21no—
528C42—21no—
529C42—42no—
530C42—42no—
531C42—1224no—
532C42—1221no—
533C42—1221no—
534C42—1221no—
535C42—1221no—
536C42—2442no—
537C23—2441no—
538C23—2441no—
539C23—2441no—
540C23—2441no—
541C23—2441no—
542C23—2441no—
543C23—2441no—
544C23C2×D4×D6111yescharacteristic
545Q8—22no—
546Q8—22no—
547Q8—22no—
548Q8—22no—
549Q8—22no—
550Q8—41no—
551Q8—41no—
552Q8#1451976014yes—
553D4—2441no—
554D4—2441no—
555D4—2441no—
556D4—2441no—
557C2×C4—22no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—22no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—1221no—
575C2×C4—1222no—
576C2×C4—1221no—
577C2×C4—1221no—
578C2×C4—1221no—
579C2×C4—1221no—
580C2×C4—1221no—
581C2×C4—1221no—
582C2×C4—1221no—
583C2×C4—1221no—
584C2×C4—1221no—
585C2×C4—1221no—
586C2×C4—1221no—
587C2×C4—2441no—
588C2×C4—2441no—
589C2×C4—2441no—
590C2×C4—2441no—
591C2×C4—2441no—
592C2×C4—2441no—
593C2×C4—2441no—
594C2×C4—2441no—
595C2×C4#1451980012yes—
596C2×C4C22×C22⋊D6111yescharacteristic
597C2×C4#41×D6111yescharacteristic
598C2×C4C2×#1175822211yescharacteristic
599C2×C4C2×#1175822211yescharacteristic
600C2×C4#1451982211yescharacteristic
601C2×C4#1451982411yescharacteristic
602C8—22no—
603C8—41no—
604C8—2441no—
605C8—2441no—
606C22—4881no—
607C22—4881no—
608C22—4881no—
609C22—4881no—
610C22—4881no—
611C22—4881no—
612C22—4881no—
613C22#1684027111yescharacteristic
614C22D4×C22⋊D6111yescharacteristic
615C22#1684042111yescharacteristic
616C22C2×#1451980011yescharacteristic
617C22#1684055911yescharacteristic
618C22#1684056911yescharacteristic
619C22#1684057511yescharacteristic
620C4—24no—
621C4—22no—
622C4—42no—
623C4—41no—
624C4—81no—
625C4—81no—
626C4—4881no—
627C4—4881no—
628C4—2441no—
629C4—2441no—
630C4—2441no—
631C4—2442no—
632C4—2441no—
633C4C2×#1451976011yescharacteristic
634C4#1684035411yescharacteristic
635C4#1684041111yescharacteristic
636C4#1684046111yescharacteristic
637C2—9761no—
638C2#2144954811yescharacteristic
639C2#2145584311yescharacteristic
640C2#2145605411yescharacteristic
641C2#2145740411yescharacteristic
642C2#2145756011yescharacteristic
643C2#2145851311yescharacteristic
644C2#2145865711yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C4×Q16×C61C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenterC2×C4×Q16D61
hypocenterC61#26173
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×Q16

Complements

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

NH ≅ G/Ntype
3. #21448737C2semidirect
5. #21447313C2semidirect
6. #21447240C2semidirect
9. #21446376C2semidirect
12. #21446162C2semidirect
13. #21446087C2semidirect
14. #21446085C2semidirect
15. C2×C4×Q16×C61C2semidirect
316. C2×C4×Q16D61semidirect
383. C61#26173semidirect

Maximal subgroups

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

subgroupindex
C2×#168399862
#214496112
#214487372
#214473372
#214473132
#214472402
#214470562
#214465022
#214463762
#214461742
#214461722
#214461622
#214460872
#214460852
C2×C4×Q16×C612
#2617361