Finite Group Databasebeta

← #29699451

Subgroups — #29699451

Order 15616.

866 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839865C211yesmaximal, characteristic
2#21448764C211yesmaximal, characteristic
3#21448737C211yesmaximal, characteristic, complemented
4#21446514C211yesmaximal, characteristic
5#21446502C211yesmaximal, characteristic
6#21446409C211yesmaximal, characteristic, complemented
7#21446390C211yesmaximal, characteristic, complemented
8#21446174C211yesmaximal, characteristic
9#21446172C211yesmaximal, characteristic
10#21445951C211yesmaximal, characteristic, complemented
11#21445828C211yesmaximal, characteristic
12#21445822C211yesmaximal, characteristic, complemented
13#21445693C211yesmaximal, characteristic, complemented
14#21445651C211yesmaximal, characteristic, complemented
15C4×#103×C61C211yesmaximal, characteristic, complemented
16C22×#11758201C2211yescharacteristic
17C2×Q8×Dic61C2211yescharacteristic
18C2×Q8×Dic61C2211yescharacteristic
19C2×#14519739C2211yescharacteristic
20C2×#14519734—21no—
21#16839865C418yescomplemented
22C2×#14519710C2211yescharacteristic
23C2×#14519706C2211yescharacteristic
24C2×#14519701C2211yescharacteristic
25C2×#14519699C2211yescharacteristic
26C2×#14519699C2211yescharacteristic
27C2×#14519697C2211yescharacteristic
28C2×#14519687C2211yescharacteristic
29C2×#14519687C2211yescharacteristic
30#16839664C2211yescharacteristic
31#16839663C2211yescharacteristic
32#16839660—21no—
33#16839660C2211yescharacteristic
34#16839658—21no—
35#16839658C2211yescharacteristic
36#16839653—21no—
37#16839651—21no—
38#16839631C2211yescharacteristic
39#16839627—21no—
40#16839627C2211yescharacteristic
41#16839621—21no—
42C4⋊C4×Dic61C2211yescharacteristic
43C4×#11758196—21no—
44C4×#11758196C2211yescharacteristic
45#16839525C2211yescharacteristic
46#16839525C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839517C2211yescharacteristic
52#16839516C2211yescharacteristic
53C2×C4×Q8×C61C2211yescharacteristic
54C2×#103×C61C2211yescharacteristic
55C2×#103×C61C2211yescharacteristic
56C2×C4×C8×C61C2211yescharacteristic
57C4×C4⋊C4×C61C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C61×#340C2211yescharacteristic
60C22×D244C2311yescharacteristic
61C22×C4×D61—21no—
62C2×#11758201—22no—
63C2×#11758201—22no—
64C2×#11758201C2×C414yes—
65C22×#92418C2311yescharacteristic
66C22×#92418C2311yescharacteristic
67Q8×Dic61—22no—
68Q8×Dic61—22no—
69Q8×Dic61—41no—
70Q8×Dic61C2×C414yes—
71#14519739—22no—
72#14519739C2×C414yes—
73#14519734—44no—
74C2×C4⋊Dic61—21no—
75C2×C4⋊Dic61—21no—
76C2×C4⋊Dic61C2311yescharacteristic
77C2×C4⋊Dic61C2311yescharacteristic
78C2×C4⋊Dic61C2311yescharacteristic
79C2×#11758196—21no—
80C2×#11758196—21no—
81C2×#11758196—21no—
82C2×#11758196—21no—
83C2×#11758196D411yescharacteristic
84C2×#11758196D411yescharacteristic
85C2×#11758198—21no—
86C2×#11758198—21no—
87C2×#11758198—21no—
88C2×#11758198—21no—
89C2×#11758198—21no—
90C2×C4×Dic61—21no—
91C2×C4×Dic61—21no—
92C2×C4×Dic61—21no—
93C2×C4×Dic61C2311yescharacteristic
94C2×C4×Dic61C2311yescharacteristic
95#14519710C2×C414yes—
96#14519706—22no—
97#14519701—22no—
98#14519699—41no—
99#14519699C2×C414yes—
100#14519697—41no—
101#14519687—41no—
102#14519687C2×C414yes—
103#14519684—21no—
104#14519684—21no—
105#14519684—21no—
106#14519684—21no—
107#14519684—21no—
108#14519684D411yescharacteristic
109#14519684D411yescharacteristic
110C22×Q8×C61C2311yescharacteristic
111C22×C8×C61C2311yescharacteristic
112C22×C8×C61C2311yescharacteristic
113C4×Q8×C61—22no—
114C4×Q8×C61—41no—
115C2×C4⋊C4×C61—21no—
116C2×C4⋊C4×C61C2311yescharacteristic
117C2×C4⋊C4×C61C2311yescharacteristic
118C2×C4⋊C4×C61C2311yescharacteristic
119C2×C42×C61—21no—
120C2×C42×C61—21no—
121C2×C42×C61C2311yescharacteristic
122#103×C61—22no—
123#103×C61C2×C414yes—
124C4×C8×C61—41no—
125#95×C61—21no—
126C23×D61—21no—
127C2×Q8×C61—21no—
128C2×Q8×C61—21no—
129C2×Q8×C61—21no—
130C2×Q8×C61C22×C412yes—
131C2×Q8×C61#4112yes—
132C2×Q8×C61#4112yes—
133C22×C4×C61—21no—
134C22×C4×C61—21no—
135C22×C4×C61—21no—
136C22×C4×C61—21no—
137C22×C4×C61C2×D411yescharacteristic
138C22×C4×C61C2×D411yescharacteristic
139C22×C4×C61C2411yescharacteristic
140C22×Dic61—21no—
141C22×Dic61—21no—
142C22×Dic61—21no—
143C22×Dic61C2×D411yescharacteristic
144C22×Dic61C2×D411yescharacteristic
145C22×Dic61C2×D411yescharacteristic
146C22×Dic61C2×D411yescharacteristic
147C2×D244—21no—
148C2×D244—21no—
149C2×D244—21no—
150C2×D244—21no—
151C2×D244—21no—
152C2×D244C22×C412yes—
153C2×C4×D61—24no—
154C2×C4×D61—42no—
155C2×C4×D61—42no—
156C2×C8×C61—21no—
157C2×C8×C61—21no—
158C2×C8×C61—21no—
159C2×C8×C61—21no—
160C2×C8×C61—21no—
161C2×C8×C61C22×C412yes—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—42no—
168C4⋊C4×C61C22×C412yes—
169C42×C61—21no—
170C42×C61—21no—
171C42×C61—42no—
172C42×C61—42no—
173#11758201—28no—
174#11758198—24no—
175#11758198—42no—
176#11758198—42no—
177#11758198—81no—
178#11758198—81no—
179C4⋊Dic61—21no—
180C4⋊Dic61—24no—
181C4⋊Dic61—21no—
182C4⋊Dic61—21no—
183C4⋊Dic61—21no—
184C4⋊Dic61—21no—
185C4⋊Dic61—42no—
186C4⋊Dic61C22×C412yes—
187#11758196—22no—
188#11758196—41no—
189#11758196—42no—
190#11758196—42no—
191#11758196—42no—
192#11758196—81no—
193C4×Dic61—21no—
194C4×Dic61—21no—
195C4×Dic61—24no—
196C4×Dic61—42no—
197C4×Dic61—42no—
198C4×Dic61C22×C412yes—
199C4×Dic61#4112yes—
200C2×#92418—21no—
201C2×#92418—21no—
202C2×#92418—21no—
203C2×#92418—21no—
204C2×#92418—21no—
205C2×#92418C22×C412yes—
206C23×C61C22×D411yescharacteristic
207C22×D61—22no—
208C22×D61—41no—
209C22×D61—41no—
210C22×D61—41no—
211C22×D61—41no—
212C22×D61—41no—
213C22×D61—41no—
214Q8×C61—22no—
215Q8×C61—22no—
216Q8×C61—22no—
217Q8×C61#11714yescomplemented
218C2×C4×C61—22no—
219C2×C4×C61—22no—
220C2×C4×C61—21no—
221C2×C4×C61—22no—
222C2×C4×C61—21no—
223C2×C4×C61—22no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—41no—
228C2×C4×C61—41no—
229C2×C4×C61—41no—
230C2×C4×C61—41no—
231C2×C4×C61—41no—
232C2×C4×C61—41no—
233C2×C4×C61—41no—
234C2×C4×C61—41no—
235C2×C4×C61AGL(1,ℤ/8)12yes—
236C2×C4×C61C23×C411yescharacteristic
237C2×C4×C61C2×#4111yescharacteristic
238C2×C4×C61C2×#4111yescharacteristic
239C2×C4×C61C2×#4111yescharacteristic
240C2×C4×C61H(5,𝔽2)11yescharacteristic
241C2×C4×C612−1+411yescharacteristic
242C2×Dic61—21no—
243C2×Dic61—21no—
244C2×Dic61—21no—
245C2×Dic61—21no—
246C2×Dic61—22no—
247C2×Dic61—21no—
248C2×Dic61—21no—
249C2×Dic61—21no—
250C2×Dic61—21no—
251C2×Dic61—21no—
252C2×Dic61—21no—
253C2×Dic61—41no—
254C2×Dic61—41no—
255C2×Dic61—41no—
256C2×Dic61—41no—
257C2×Dic61—41no—
258C2×Dic61—41no—
259C2×Dic61—41no—
260C2×Dic61—41no—
261C2×Dic61C4×D412yes—
262C2×Dic61C4×D412yes—
263D244—22no—
264D244—22no—
265D244—41no—
266D244—41no—
267C4×D61—48no—
268C8×C61—22no—
269C8×C61—41no—
270#92418—22no—
271#92418—41no—
272#22781—611nomaximal, Sylow
273C22×C61C2×C4×D411yescharacteristic
274C22×C61#54611yescharacteristic
275C22×C61#54811yescharacteristic
276C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
277C22×C61#57511yescharacteristic
278C22×C61#57711yescharacteristic
279C22×C61#57811yescharacteristic
280C2×D61—42no—
281C2×D61—42no—
282C2×D61—42no—
283C2×D61—81no—
284C2×D61—81no—
285C2×D61—81no—
286C2×D61—81no—
287C4×C61—24no—
288C4×C61—22no—
289C4×C61—42no—
290C4×C61—41no—
291C4×C61—81no—
292C4×C61—81no—
293C4×C61C2×#11711yescharacteristic
294C4×C61#52011yescharacteristic
295C4×C61#55211yescharacteristic
296C4×C61#55311yescharacteristic
297Dic61—22no—
298Dic61—22no—
299Dic61—42no—
300Dic61—41no—
301Dic61—41no—
302Dic61—81no—
303Dic61—81no—
304C2×#440—611no—
305#2354—611no—
306#2351—611no—
307#1936—611no—
308#1934—611no—
309#1921—611no—
310#1918—611no—
311#1880—611no—
312#1880—611no—
313#1836—611no—
314#1804—611no—
315#1803—611no—
316#1771—611no—
317#1763—611no—
318C4×#103D6111yescharacteristic, complemented
319C2×C61#235211yescharacteristic
320C2×C61C4×AGL(1,ℤ/8)11yescharacteristic
321C2×C61#294711yescharacteristic
322C2×C61#330811yescharacteristic
323C2×C61#332111yescharacteristic
324C2×C61#336811yescharacteristic
325C2×C61#337711yescharacteristic
326D61—82no—
327C22×SD16—611no—
328C2×C4×Q8—611no—
329C2×C4×Q8—611no—
330C2×C4×Q8C2×D6111yescharacteristic
331C2×C4×D4—611no—
332C2×#117—1221no—
333#440—618no—
334C2×#107—611no—
335C2×C4⋊C8—611no—
336C2×#103—611no—
337C2×#103C2×D6111yescharacteristic
338C2×#103C2×D6111yescharacteristic
339C2×#102—611no—
340C2×#102—611no—
341C2×#102—611no—
342C2×#97—611no—
343C2×#97—611no—
344C2×C4×C8C2×D6111yescharacteristic
345#390—611no—
346#390—611no—
347#389—611no—
348#389—611no—
349#389—1221no—
350#389—1221no—
351#388—1221no—
352#388—1221no—
353#384—611no—
354#382—611no—
355#382—1221no—
356#380—1221no—
357C4×C4⋊C4—611no—
358C4×C4⋊C4—611no—
359C4×C4⋊C4—1221no—
360C4×C4⋊C4C2×D6111yescharacteristic
361#340—611no—
362#340—611no—
363#340—611no—
364#340—611no—
365#340—611no—
366#340—611no—
367#340C2×D6111yescharacteristic
368#340C2×D6111yescharacteristic
369#336—611no—
370#336—611no—
371C61#2278111yescharacteristic, complemented, Sylow
372C22×Q8C22×D6111yescharacteristic
373C22×D4—611no—
374C23×C4—1221no—
375C2×SD16—614no—
376C2×SD16—1222no—
377C2×SD16—1222no—
378C22×C8—611no—
379C22×C8—611no—
380C22×C8C22×D6111yescharacteristic
381C22×C8C22×D6111yescharacteristic
382C4×Q8—22no—
383C4×Q8—41no—
384C4×Q8—614no—
385C4×Q8—1222no—
386C4×Q8—1222no—
387C4×Q8—2441no—
388C4×D4—614no—
389C4×D4—1222no—
390#117—2444no—
391C2×C4⋊C4—21no—
392C2×C4⋊C4—611no—
393C2×C4⋊C4—611no—
394C2×C4⋊C4—611no—
395C2×C4⋊C4—611no—
396C2×C4⋊C4—611no—
397C2×C4⋊C4—1221no—
398C2×C4⋊C4—1221no—
399C2×C4⋊C4—1221no—
400C2×C4⋊C4—1221no—
401C2×C4⋊C4—1221no—
402C2×C4⋊C4—1221no—
403C2×C4⋊C4C22×D6111yescharacteristic
404C2×C4⋊C4C22×D6111yescharacteristic
405C2×C4⋊C4C22×D6111yescharacteristic
406C2×C22⋊C4—1221no—
407C2×C22⋊C4—1221no—
408C2×C22⋊C4—1221no—
409C2×C22⋊C4—1221no—
410C2×C22⋊C4—1221no—
411C2×C42—21no—
412C2×C42—21no—
413C2×C42—611no—
414C2×C42—611no—
415C2×C42—1221no—
416C2×C42—1221no—
417C2×C42—1221no—
418C2×C42C22×D6111yescharacteristic
419#107—614no—
420C4⋊C8—1222no—
421#103—22no—
422#103—1222no—
423#103C4×D6114yes—
424#102—614no—
425#102—2441no—
426#102—2441no—
427#97—614no—
428#97—2441no—
429C4×C8—41no—
430#95—21no—
431#95—611no—
432#95—611no—
433#95—1221no—
434#95—1221no—
435#95—1221no—
436#95—1221no—
437#95—1221no—
438C24—1221no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8C2×C4×D6112yes—
443C2×Q8#1175822212yes—
444C2×Q8#1175822212yes—
445C2×D4—612no—
446C2×D4—1221no—
447C2×D4—1221no—
448C2×D4—1221no—
449C2×D4—1221no—
450C2×D4—1221no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—611no—
456C22×C4—611no—
457C22×C4—611no—
458C22×C4—611no—
459C22×C4—1224no—
460C22×C4—1221no—
461C22×C4—1221no—
462C22×C4—1221no—
463C22×C4—2442no—
464C22×C4—2442no—
465C22×C4D4×D6111yescharacteristic
466C22×C4D4×D6111yescharacteristic
467C22×C4C23×D6111yescharacteristic
468SD16—1228no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—612no—
475C2×C8—1221no—
476C2×C8—1221no—
477C2×C8—1221no—
478C2×C8—1221no—
479C2×C8—1221no—
480C2×C8C2×C4×D6112yes—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—42no—
487C4⋊C4—612no—
488C4⋊C4—1224no—
489C4⋊C4—1221no—
490C4⋊C4—1222no—
491C4⋊C4—1221no—
492C4⋊C4—1221no—
493C4⋊C4—1221no—
494C4⋊C4—1221no—
495C4⋊C4—4881no—
496C4⋊C4—2441no—
497C4⋊C4—2442no—
498C4⋊C4—2442no—
499C4⋊C4—2442no—
500C4⋊C4—2442no—
501C4⋊C4C2×C4×D6112yes—
502C22⋊C4—1224no—
503C22⋊C4—4881no—
504C22⋊C4—4881no—
505C22⋊C4—2442no—
506C22⋊C4—2442no—
507C42—21no—
508C42—21no—
509C42—42no—
510C42—42no—
511C42—612no—
512C42—612no—
513C42—1224no—
514C42—1221no—
515C42—1221no—
516C42—2442no—
517C42—2442no—
518C23—1222no—
519C23—2441no—
520C23—2441no—
521C23—2441no—
522C23—2441no—
523C23—2441no—
524C23—2441no—
525C23C2×D4×D6111yescharacteristic
526Q8—22no—
527Q8—22no—
528Q8—22no—
529Q8#1451973414yescomplemented
530D4—1222no—
531D4—1222no—
532D4—2441no—
533D4—2441no—
534C2×C4—22no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—612no—
552C2×C4—612no—
553C2×C4—1221no—
554C2×C4—1221no—
555C2×C4—1221no—
556C2×C4—1221no—
557C2×C4—1222no—
558C2×C4—1221no—
559C2×C4—1221no—
560C2×C4—1221no—
561C2×C4—1221no—
562C2×C4—1221no—
563C2×C4—1221no—
564C2×C4—2448no—
565C2×C4—2441no—
566C2×C4—2441no—
567C2×C4—2441no—
568C2×C4—2441no—
569C2×C4—2441no—
570C2×C4—2441no—
571C2×C4—2441no—
572C2×C4—2441no—
573C2×C4#1451980412yes—
574C2×C4C22×C4×D6111yescharacteristic
575C2×C4#41×D6111yescharacteristic
576C2×C4C2×#1175822211yescharacteristic
577C2×C4C2×#1175822211yescharacteristic
578C2×C4#1451982211yescharacteristic
579C2×C4#1451982411yescharacteristic
580C8—22no—
581C8—41no—
582C8—1222no—
583C8—2441no—
584C22—4881no—
585C22—4881no—
586C22—4881no—
587C22—4881no—
588C22—2442no—
589C22—2442no—
590C22—2442no—
591C22C4×D4×D6111yescharacteristic
592C22#1684040211yescharacteristic
593C22#1684042111yescharacteristic
594C22C2×#1451980411yescharacteristic
595C22#1684055911yescharacteristic
596C22#1684056911yescharacteristic
597C22#1684057511yescharacteristic
598C4—24no—
599C4—22no—
600C4—42no—
601C4—41no—
602C4—81no—
603C4—81no—
604C4—1222no—
605C4—1222no—
606C4—4881no—
607C4—4881no—
608C4—2442no—
609C4—2441no—
610C4—2441no—
611C4C2×#1451973411yescharacteristic
612C4#1684024111yescharacteristic
613C4#1684044511yescharacteristic
614C4#1684046111yescharacteristic
615C2—4882no—
616C2#2144874211yescharacteristic
617C2C4×#1451980411yescharacteristic
618C2#2145465711yescharacteristic
619C2#2145780611yescharacteristic
620C2#2145800111yescharacteristic
621C2#2145854611yescharacteristic
622C2#2145865711yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC4×#103×C61C2
socleC23×C61C22×D4
prime radicalC22×D244C23
cosocleC22×C4×C61C24
hypercenterC4×#103D61
hypocenterC61#22781
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C4×#103

Complements

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

NH ≅ G/Ntype
3. #21448737C2semidirect
6. #21446409C2semidirect
7. #21446390C2semidirect
10. #21445951C2semidirect
12. #21445822C2semidirect
13. #21445693C2semidirect
14. #21445651C2semidirect
15. C4×#103×C61C2semidirect
21. #16839865C4semidirect
217. Q8×C61#117semidirect
318. C4×#103D61semidirect
371. C61#22781semidirect
529. Q8#14519734semidirect

Maximal subgroups

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

subgroupindex
C2×#168398652
#214487642
#214487372
#214465142
#214465022
#214464092
#214463902
#214461742
#214461722
#214459512
#214458282
#214458222
#214456932
#214456512
C4×#103×C612
#2278161