Finite Group Databasebeta

← #29699365

Subgroups — #29699365

Order 15616.

838 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839863C211yesmaximal, characteristic
2#21448791C211yesmaximal, characteristic
3Q8×#11758196C211yesmaximal, characteristic, complemented
4#21446513C211yesmaximal, characteristic
5#21446505C211yesmaximal, characteristic
6#21446412C211yesmaximal, characteristic, complemented
7#21446389C211yesmaximal, characteristic, complemented
8#21446176C211yesmaximal, characteristic
9#21445951C211yesmaximal, characteristic, complemented
10#21445832C211yesmaximal, characteristic, complemented
11#21445817C211yesmaximal, characteristic
12#21445690C211yesmaximal, characteristic, complemented
13#21445651C211yesmaximal, characteristic, complemented
14C4×#14519701C211yesmaximal, characteristic, complemented
15C61×#1880C211yesmaximal, characteristic
16C2×Q8×Dic61C2211yescharacteristic
17C2×Q8×Dic61C2211yescharacteristic
18C2×D4×Dic61C2211yescharacteristic
19#16839863C418yescomplemented
20C2×#14519711C2211yescharacteristic
21C2×#14519706C2211yescharacteristic
22C2×#14519701C2211yescharacteristic
23C2×#14519701C2211yescharacteristic
24C2×#14519698C2211yescharacteristic
25C2×#14519698C2211yescharacteristic
26C2×#14519687C2211yescharacteristic
27C2×#14519687C2211yescharacteristic
28C2×C4×#92418C2211yescharacteristic
29#16839665C2211yescharacteristic
30#16839660—21no—
31#16839660C2211yescharacteristic
32#16839658—21no—
33#16839658C2211yescharacteristic
34#16839656—21no—
35#16839650—21no—
36#16839631—21no—
37#16839631C2211yescharacteristic
38#16839627C2211yescharacteristic
39#16839623—21no—
40C4⋊C4×Dic61C2211yescharacteristic
41C4×#11758196—21no—
42C4×#11758196C2211yescharacteristic
43C4×C4⋊Dic61C2211yescharacteristic
44C22⋊C4×Dic61—21no—
45#16839525C2211yescharacteristic
46#16839525C2211yescharacteristic
47#16839525C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839516C2211yescharacteristic
53C22×SD16×C61C2211yescharacteristic
54C2×C4×Q8×C61C2211yescharacteristic
55C2×#103×C61C2211yescharacteristic
56C2×#102×C61C2211yescharacteristic
57C61×#390C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C61×#336C2211yescharacteristic
60C23×Dic61—21no—
61C22×#92418C2311yescharacteristic
62C22×#92418C2311yescharacteristic
63Q8×Dic61—22no—
64Q8×Dic61—22no—
65Q8×Dic61—41no—
66Q8×Dic61C2×C414yes—
67D4×Dic61—22no—
68D4×Dic61C2×C414yes—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C4⋊Dic61C2311yescharacteristic
73C2×C4⋊Dic61C2311yescharacteristic
74C2×C4⋊Dic61C2311yescharacteristic
75C2×#11758196—21no—
76C2×#11758196—21no—
77C2×#11758196—21no—
78C2×#11758196—21no—
79C2×#11758196D411yescharacteristic
80C2×#11758196D411yescharacteristic
81C2×C22⋊Dic61—21no—
82C2×C22⋊Dic61—21no—
83C2×C22⋊Dic61—21no—
84C2×C4×Dic61—21no—
85C2×C4×Dic61—21no—
86C2×C4×Dic61—21no—
87C2×C4×Dic61—21no—
88C2×C4×Dic61C2311yescharacteristic
89C2×C4×Dic61C2311yescharacteristic
90#14519711C2×C414yes—
91#14519706—22no—
92#14519701—22no—
93#14519701C2×C414yes—
94#14519698—41no—
95#14519698C2×C414yes—
96#14519687—41no—
97#14519687C2×C414yes—
98C4×#92418—41no—
99#14519684—21no—
100#14519684—21no—
101#14519684—21no—
102#14519684—21no—
103#14519684—21no—
104#14519684—21no—
105#14519684D411yescharacteristic
106#14519684D411yescharacteristic
107C22×Q8×C61C2311yescharacteristic
108C22×D4×C61C2311yescharacteristic
109C2×SD16×C61—22no—
110C2×SD16×C61—22no—
111C2×SD16×C61C2×C414yes—
112C22×C8×C61C2311yescharacteristic
113C22×C8×C61C2311yescharacteristic
114C4×Q8×C61—22no—
115C4×Q8×C61—41no—
116C2×C4⋊C4×C61—21no—
117C2×C4⋊C4×C61C2311yescharacteristic
118C2×C4⋊C4×C61C2311yescharacteristic
119C2×C22⋊C4×C61—21no—
120C2×C22⋊C4×C61—21no—
121C2×C42×C61—21no—
122C2×C42×C61C2311yescharacteristic
123#103×C61—22no—
124#102×C61—41no—
125C24×C61—21no—
126C2×Q8×C61—21no—
127C2×Q8×C61—21no—
128C2×Q8×C61—21no—
129C2×Q8×C61C22×C412yes—
130C2×Q8×C61#4112yes—
131C2×Q8×C61#4112yes—
132C2×D4×C61—21no—
133C2×D4×C61—21no—
134C2×D4×C61—21no—
135C2×D4×C61—21no—
136C2×D4×C61—21no—
137C2×D4×C61C22×C412yes—
138C22×C4×C61—21no—
139C22×C4×C61—21no—
140C22×C4×C61—21no—
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61C2411yescharacteristic
144C22×Dic61—21no—
145C22×Dic61—21no—
146C22×Dic61—24no—
147C22×Dic61—21no—
148C22×Dic61—21no—
149C22×Dic61—81no—
150C22×Dic61—81no—
151C22×Dic61C2×D411yescharacteristic
152C22×Dic61C2×D411yescharacteristic
153C22×Dic61C2×D411yescharacteristic
154C22×Dic61C2×D411yescharacteristic
155SD16×C61—28no—
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—42no—
167C22⋊C4×C61—42no—
168C22⋊C4×C61—81no—
169C42×C61—21no—
170C42×C61—21no—
171C42×C61—42no—
172C22⋊Dic61—24no—
173C22⋊Dic61—42no—
174C22⋊Dic61—81no—
175C4⋊Dic61—21no—
176C4⋊Dic61—21no—
177C4⋊Dic61—24no—
178C4⋊Dic61—21no—
179C4⋊Dic61—21no—
180C4⋊Dic61—21no—
181C4⋊Dic61—21no—
182C4⋊Dic61—42no—
183C4⋊Dic61C22×C412yes—
184C4⋊Dic61C22×C412yes—
185#11758196—22no—
186#11758196—41no—
187#11758196—42no—
188#11758196—81no—
189#11758196—81no—
190#11758196—81no—
191C4×Dic61—21no—
192C4×Dic61—21no—
193C4×Dic61—24no—
194C4×Dic61—42no—
195C4×Dic61—81no—
196C4×Dic61—81no—
197C4×Dic61C22×C412yes—
198C4×Dic61#4112yes—
199C2×#92418—21no—
200C2×#92418—21no—
201C2×#92418—21no—
202C2×#92418—21no—
203C2×#92418—21no—
204C2×#92418C22×C412yes—
205C23×C61—22no—
206C23×C61—41no—
207C23×C61—41no—
208C23×C61—41no—
209C23×C61—41no—
210C23×C61—41no—
211C23×C61—41no—
212C23×C61C22×D411yescharacteristic
213Q8×C61—22no—
214Q8×C61—22no—
215Q8×C61—22no—
216Q8×C61#11714yes—
217D4×C61—22no—
218D4×C61—22no—
219D4×C61—41no—
220D4×C61—41no—
221C2×C4×C61—22no—
222C2×C4×C61—22no—
223C2×C4×C61—21no—
224C2×C4×C61—22no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
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×C61#13712yes—
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—22no—
248C2×Dic61—21no—
249C2×Dic61—21no—
250C2×Dic61—21no—
251C2×Dic61—21no—
252C2×Dic61—21no—
253C2×Dic61—21no—
254C2×Dic61—41no—
255C2×Dic61—41no—
256C2×Dic61—41no—
257C2×Dic61—41no—
258C2×Dic61—41no—
259C2×Dic61—41no—
260C2×Dic61—41no—
261C2×Dic61—41no—
262C2×Dic61—41no—
263C2×Dic61—41no—
264C2×Dic61—84no—
265C2×Dic61C4×D412yes—
266C2×Dic61C4×D412yes—
267C8×C61—22no—
268C8×C61—41no—
269#92418—22no—
270#92418—41no—
271#22775—611nomaximal, Sylow
272C22×C61—42no—
273C22×C61—42no—
274C22×C61—42no—
275C22×C61—81no—
276C22×C61—81no—
277C22×C61—81no—
278C22×C61—81no—
279C22×C61C2×C4×D411yescharacteristic
280C22×C61#54611yescharacteristic
281C22×C61#54811yescharacteristic
282C22×C61C2×#13711yescharacteristic
283C22×C61#57511yescharacteristic
284C22×C61#57611yescharacteristic
285C22×C61#57711yescharacteristic
286C4×C61—24no—
287C4×C61—22no—
288C4×C61—41no—
289C4×C61—81no—
290C4×C61—81no—
291C4×C61C2×#11711yescharacteristic
292C4×C61#52011yescharacteristic
293C4×C61#55211yescharacteristic
294C4×C61#55311yescharacteristic
295Dic61—22no—
296Dic61—22no—
297Dic61—42no—
298Dic61—42no—
299Dic61—41no—
300Dic61—41no—
301Dic61—81no—
302Dic61—81no—
303C2×#440—611no—
304#2357—611no—
305Q8×C4⋊C4—611no—
306#1936—611no—
307#1934—611no—
308#1921—611no—
309#1918—611no—
310#1880—611no—
311#1880D6111yescharacteristic
312#1836—611no—
313#1805—611no—
314#1802—611no—
315#1771—611no—
316#1763—611no—
317C4×#103—611no—
318C2×C61—82no—
319C2×C61#235211yescharacteristic
320C2×C61C4×#13711yescharacteristic
321C2×C61#294611yescharacteristic
322C2×C61#330911yescharacteristic
323C2×C61#332011yescharacteristic
324C2×C61#337111yescharacteristic
325C2×C61#337511yescharacteristic
326C22×SD16C2×D6111yescharacteristic
327C2×C4×Q8—611no—
328C2×C4×Q8—611no—
329C2×C4×Q8C2×D6111yescharacteristic
330C2×C4×D4—611no—
331#440—618no—
332C2×#107—611no—
333C2×C4⋊C8—611no—
334C2×#103—611no—
335C2×#103—611no—
336C2×#103C2×D6111yescharacteristic
337C2×#102—611no—
338C2×#102—611no—
339C2×#102C2×D6111yescharacteristic
340C2×#97—611no—
341C2×#97—611no—
342C2×C4×C8—611no—
343#390—611no—
344#390C2×D6111yescharacteristic
345#389—611no—
346#389—611no—
347#389—1221no—
348#389—1221no—
349#388—1221no—
350#388—1221no—
351#384—611no—
352#384—1221no—
353#382—611no—
354#381—1221no—
355C4×C4⋊C4—611no—
356C4×C4⋊C4—611no—
357C4×C4⋊C4—611no—
358C4×C4⋊C4—1221no—
359C4×C22⋊C4—1221no—
360#340—611no—
361#340—611no—
362#340—611no—
363#340—611no—
364#340—611no—
365#340—611no—
366#340—611no—
367#340C2×D6111yescharacteristic
368#336—611no—
369#336C2×D6111yescharacteristic
370C61#2277511yescharacteristic, complemented, Sylow
371C22×Q8C22×D6111yescharacteristic
372C22×D4C22×D6111yescharacteristic
373C23×C4—1221no—
374C2×SD16—22no—
375C2×SD16—22no—
376C2×SD16C4×D6114yes—
377C22×C8—611no—
378C22×C8—611no—
379C22×C8C22×D6111yescharacteristic
380C22×C8C22×D6111yescharacteristic
381C4×Q8—22no—
382C4×Q8—41no—
383C4×Q8—614no—
384C4×Q8—1222no—
385C4×Q8—1222no—
386C4×Q8—2441no—
387C4×D4—614no—
388C4×D4—1222no—
389C2×C4⋊C4—21no—
390C2×C4⋊C4—611no—
391C2×C4⋊C4—611no—
392C2×C4⋊C4—611no—
393C2×C4⋊C4—611no—
394C2×C4⋊C4—611no—
395C2×C4⋊C4—611no—
396C2×C4⋊C4—1221no—
397C2×C4⋊C4—1221no—
398C2×C4⋊C4—1221no—
399C2×C4⋊C4—1221no—
400C2×C4⋊C4—1221no—
401C2×C4⋊C4—1221no—
402C2×C4⋊C4C22×D6111yescharacteristic
403C2×C4⋊C4C22×D6111yescharacteristic
404C2×C22⋊C4—21no—
405C2×C22⋊C4—21no—
406C2×C22⋊C4—1221no—
407C2×C22⋊C4—1221no—
408C2×C22⋊C4—1221no—
409C2×C42—21no—
410C2×C42—611no—
411C2×C42—611no—
412C2×C42—1221no—
413C2×C42—1221no—
414C2×C42—1221no—
415C2×C42—1221no—
416C2×C42C22×D6111yescharacteristic
417#107—614no—
418C4⋊C8—1222no—
419#103—22no—
420#103—614no—
421#103—1222no—
422#102—41no—
423#102—614no—
424#102—2441no—
425#97—614no—
426#97—2441no—
427C4×C8—2441no—
428#95—611no—
429#95—611no—
430#95—1221no—
431#95—1221no—
432#95—1221no—
433#95—1221no—
434#95—1221no—
435#95—1221no—
436C24—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8C2×C4×D6112yes—
441C2×Q8#1175822412yes—
442C2×Q8#1175822412yes—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4C2×C4×D6112yes—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—611no—
453C22×C4—611no—
454C22×C4—611no—
455C22×C4—611no—
456C22×C4—1224no—
457C22×C4—1221no—
458C22×C4—1221no—
459C22×C4—1221no—
460C22×C4—1221no—
461C22×C4—4881no—
462C22×C4—4881no—
463C22×C4D4×D6111yescharacteristic
464C22×C4D4×D6111yescharacteristic
465C22×C4C23×D6111yescharacteristic
466SD16—28no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—612no—
473C2×C8—1221no—
474C2×C8—1221no—
475C2×C8—1221no—
476C2×C8—1221no—
477C2×C8—1221no—
478C2×C8C2×C4×D6112yes—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—42no—
484C4⋊C4—612no—
485C4⋊C4—612no—
486C4⋊C4—1224no—
487C4⋊C4—1221no—
488C4⋊C4—1221no—
489C4⋊C4—1222no—
490C4⋊C4—1221no—
491C4⋊C4—1221no—
492C4⋊C4—1221no—
493C4⋊C4—1221no—
494C4⋊C4—4881no—
495C4⋊C4—4881no—
496C4⋊C4—4881no—
497C4⋊C4—2441no—
498C4⋊C4—2442no—
499C4⋊C4—2442no—
500C22⋊C4—42no—
501C22⋊C4—81no—
502C22⋊C4—1224no—
503C22⋊C4—4881no—
504C22⋊C4—2442no—
505C42—21no—
506C42—21no—
507C42—42no—
508C42—612no—
509C42—612no—
510C42—1224no—
511C42—1221no—
512C42—1221no—
513C42—4881no—
514C42—4881no—
515C42—2442no—
516C23—22no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23C2×D4×D6111yescharacteristic
524Q8—22no—
525Q8—22no—
526Q8—22no—
527Q8#1451973314yes—
528D4—22no—
529D4—22no—
530D4—41no—
531D4—41no—
532C2×C4—22no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—612no—
547C2×C4—612no—
548C2×C4—1221no—
549C2×C4—1221no—
550C2×C4—1221no—
551C2×C4—1221no—
552C2×C4—1222no—
553C2×C4—1222no—
554C2×C4—1221no—
555C2×C4—1221no—
556C2×C4—1221no—
557C2×C4—1221no—
558C2×C4—1221no—
559C2×C4—1221no—
560C2×C4—4884no—
561C2×C4—2441no—
562C2×C4—2441no—
563C2×C4—2441no—
564C2×C4—2441no—
565C2×C4—2441no—
566C2×C4—2441no—
567C2×C4—2441no—
568C2×C4—2441no—
569C2×C4—2441no—
570C2×C4—2441no—
571C2×C4#1451980612yes—
572C2×C4C22×C4×D6111yescharacteristic
573C2×C4#41×D6111yescharacteristic
574C2×C4C2×#1175822411yescharacteristic
575C2×C4C2×#1175822411yescharacteristic
576C2×C4#1451982211yescharacteristic
577C2×C4#1451982411yescharacteristic
578C8—22no—
579C8—41no—
580C8—1222no—
581C8—2441no—
582C22—42no—
583C22—42no—
584C22—42no—
585C22—81no—
586C22—81no—
587C22—81no—
588C22—81no—
589C22C4×D4×D6111yescharacteristic
590C22#1684039911yescharacteristic
591C22#1684041811yescharacteristic
592C22C2×#1451980611yescharacteristic
593C22#1684055911yescharacteristic
594C22#1684056611yescharacteristic
595C22#1684057111yescharacteristic
596C4—24no—
597C4—22no—
598C4—41no—
599C4—81no—
600C4—81no—
601C4—1222no—
602C4—1222no—
603C4—4881no—
604C4—4881no—
605C4—2442no—
606C4—2442no—
607C4—2441no—
608C4—2441no—
609C4C2×#1451973311yescharacteristic
610C4#1684024111yescharacteristic
611C4#1684044911yescharacteristic
612C4#1684045511yescharacteristic
613C2—82no—
614C2#2144874611yescharacteristic
615C2C4×#1451980611yescharacteristic
616C2#2145465011yescharacteristic
617C2#2145782911yescharacteristic
618C2#2145799411yescharacteristic
619C2#2145859111yescharacteristic
620C2#2145863711yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1880C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
cosocleC22×C4×C61C24
hypercenter#1880D61
hypocenterC61#22775
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1880

Complements

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

NH ≅ G/Ntype
3. Q8×#11758196C2semidirect
6. #21446412C2semidirect
7. #21446389C2semidirect
9. #21445951C2semidirect
10. #21445832C2semidirect
12. #21445690C2semidirect
13. #21445651C2semidirect
14. C4×#14519701C2semidirect
19. #16839863C4semidirect
370. C61#22775semidirect

Maximal subgroups

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

subgroupindex
C2×#168398632
#214487912
Q8×#117581962
#214465132
#214465052
#214464122
#214463892
#214461762
#214459512
#214458322
#214458172
#214456902
#214456512
C4×#145197012
C61×#18802
#2277561