Finite Group Databasebeta

← #29698320

Subgroups — #29698320

Order 15616.

974 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839864C211yesmaximal, characteristic
2C2×#16839861C211yesmaximal, characteristic
3C2×#16839849C211yesmaximal, characteristic
4#21448738C211yesmaximal, characteristic, complemented
5#21448728C211yesmaximal, characteristic
6#21446563C211yesmaximal, characteristic, complemented
7#21446088C211yesmaximal, characteristic, complemented
8#21446086C211yesmaximal, characteristic, complemented
9#21446081C211yesmaximal, characteristic
10#21446079C211yesmaximal, characteristic
11#21445982C211yesmaximal, characteristic, complemented
12#21445833C211yesmaximal, characteristic, complemented
13#21445828C211yesmaximal, characteristic
14#21445687C211yesmaximal, characteristic, complemented
15C61×#1944C211yesmaximal, characteristic, complemented
16C22×#11758190C2211yescharacteristic
17C2×C4×Dic122C2211yescharacteristic
18C2×#14519742C2211yescharacteristic
19C2×#14519742C2211yescharacteristic
20C2×C4×D244C2211yescharacteristic
21C2×#14519739C2211yescharacteristic
22C2×#14519739C2211yescharacteristic
23#16839864—24no—
24#16839861—24no—
25#16839849C418yescomplemented
26C2×#14519711C2211yescharacteristic
27C2×#14519708C2211yescharacteristic
28C2×#14519708C2211yescharacteristic
29C2×#14519706C2211yescharacteristic
30C2×#14519702C2211yescharacteristic
31C2×#14519700C2211yescharacteristic
32C2×#14519700C2211yescharacteristic
33C2×#14519699C2211yescharacteristic
34C2×#14519697C2211yescharacteristic
35C2×#14519697C2211yescharacteristic
36C2×#14519687C2211yescharacteristic
37C2×C8×Dic61C2211yescharacteristic
38#16839661C2211yescharacteristic
39#16839661C2211yescharacteristic
40#16839659—21no—
41#16839657—21no—
42#16839636—21no—
43#16839633—21no—
44#16839627C2211yescharacteristic
45#16839627C2211yescharacteristic
46#16839626—21no—
47#16839619—21no—
48C4⋊C4×Dic61—21no—
49C4×#11758196C2211yescharacteristic
50#95×D61—21no—
51#16839524C2211yescharacteristic
52#16839524C2211yescharacteristic
53#16839517C2211yescharacteristic
54#16839516C2211yescharacteristic
55C2×#107×C61C2211yescharacteristic
56C2×#106×C61C2211yescharacteristic
57C2×#97×C61C2211yescharacteristic
58C61×#389C2211yescharacteristic
59C61×#389C2211yescharacteristic
60C61×#340C2211yescharacteristic
61C61×#340C2211yescharacteristic
62C22×Dic122C2311yescharacteristic
63C22×D244C2311yescharacteristic
64C22×C4×D61—21no—
65C22×C4×D61—21no—
66C22×C4×D61—21no—
67C2×#11758190C2×C414yes—
68C2×#11758190D414yes—
69C2×#11758190Q814yes—
70C22×#92418C2311yescharacteristic
71C22×#92418C2311yescharacteristic
72C4×Dic122—22no—
73C4×Dic122—41no—
74#14519742—22no—
75#14519742—22no—
76#14519742—41no—
77#14519742C2×C414yescomplemented
78C4×D244—22no—
79C4×D244—41no—
80#14519739—22no—
81#14519739—22no—
82#14519739—41no—
83#14519739C2×C414yes—
84C2×C4⋊Dic61—21no—
85C2×C4⋊Dic61—21no—
86C2×C4⋊Dic61C2311yescharacteristic
87C2×C4⋊Dic61C2311yescharacteristic
88C2×#11758196—21no—
89C2×#11758196—21no—
90C2×#11758196—21no—
91C2×#11758196D411yescharacteristic
92C2×#11758196D411yescharacteristic
93C2×#11758198—21no—
94C2×#11758198—21no—
95C2×#11758198—21no—
96C2×C4×Dic61—21no—
97C2×C4×Dic61—21no—
98C2×C4×Dic61—21no—
99C2×C4×Dic61C2311yescharacteristic
100C2×C4×Dic61C2311yescharacteristic
101#14519711—22no—
102#14519708—41no—
103#14519708—41no—
104#14519706—22no—
105#14519702—22no—
106#14519700—22no—
107#14519700C2×C414yescomplemented
108#14519699—22no—
109#14519697—22no—
110#14519697C2×C414yes—
111#14519687—22no—
112C8×Dic61C2×C414yescomplemented
113#14519684—21no—
114#14519684—21no—
115#14519684—21no—
116#14519684—21no—
117#14519684—21no—
118#14519684—21no—
119#14519684D411yescharacteristic
120#14519684D411yescharacteristic
121C22×C8×C61C2311yescharacteristic
122C22×C8×C61C2311yescharacteristic
123C2×C4⋊C4×C61—21no—
124C2×C4⋊C4×C61—21no—
125C2×C4⋊C4×C61C2311yescharacteristic
126C2×C4⋊C4×C61C2311yescharacteristic
127C2×C4⋊C4×C61C2311yescharacteristic
128C2×C4⋊C4×C61C2311yescharacteristic
129C2×C42×C61C2311yescharacteristic
130#107×C61—22no—
131#106×C61C2×C414yescomplemented
132#97×C61—22no—
133#95×C61—21no—
134#95×C61—21no—
135C23×D61—21no—
136C22×C4×C61—21no—
137C22×C4×C61—21no—
138C22×C4×C61—21no—
139C22×C4×C61—21no—
140C22×C4×C61C2×D411yescharacteristic
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61C2411yescharacteristic
143C22×Dic61—21no—
144C22×Dic61—21no—
145C22×Dic61—21no—
146C22×Dic61C2×D411yescharacteristic
147C22×Dic61C2×D411yescharacteristic
148C22×Dic61C2×D411yescharacteristic
149C22×Dic61C2×D411yescharacteristic
150C2×D244—21no—
151C2×D244—21no—
152C2×D244—21no—
153C2×D244C22×C412yes—
154C2×D244C2×D412yes—
155C2×D244C2×Q812yes—
156C2×C4×D61—24no—
157C2×C4×D61—42no—
158C2×C4×D61—42no—
159C2×C4×D61—42no—
160C2×C4×D61—42no—
161C2×C4×D61—81no—
162C2×C4×D61—81no—
163C2×C4×D61—81no—
164C2×C4×D61—81no—
165C2×Dic122—21no—
166C2×Dic122—21no—
167C2×Dic122—21no—
168C2×Dic122C22×C412yes—
169C2×Dic122C2×D412yes—
170C2×Dic122C2×Q812yes—
171C2×C8×C61—21no—
172C2×C8×C61—21no—
173C2×C8×C61—21no—
174C2×C8×C61C22×C412yes—
175C2×C8×C61C2×D412yes—
176C2×C8×C61C2×Q812yes—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61—21no—
183C4⋊C4×C61—42no—
184C4⋊C4×C61—81no—
185C4⋊C4×C61C22×C412yes—
186C4⋊C4×C61C22×C412yes—
187C42×C61—21no—
188C42×C61—21no—
189#11758198—24no—
190#11758198—42no—
191#11758198—42no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—42no—
197C4⋊Dic61—81no—
198#11758196—24no—
199#11758196—41no—
200#11758196—41no—
201#11758196—42no—
202#11758196—42no—
203C4×Dic61—21no—
204C4×Dic61—21no—
205C4×Dic61—24no—
206C4×Dic61—42no—
207C4×Dic61—42no—
208C4×Dic61C22×C412yes—
209C4×Dic61#4112yes—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418—21no—
215C2×#92418C22×C412yes—
216#11758190C4⋊C4116yescomplemented
217C23×C61C22×D411yescharacteristic
218C22×D61—22no—
219C22×D61—22no—
220C22×D61—22no—
221C22×D61—41no—
222C22×D61—41no—
223C22×D61—41no—
224C22×D61—41no—
225C2×C4×C61—22no—
226C2×C4×C61—22no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—41no—
234C2×C4×C61—41no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61—41no—
238C2×C4×C61—41no—
239C2×C4×C61—41no—
240C2×C4×C61—41no—
241C2×C4×C61—41no—
242C2×C4×C61—41no—
243C2×C4×C61C23×C411yescharacteristic
244C2×C4×C61C22×D411yescharacteristic
245C2×C4×C61C22×Q811yescharacteristic
246C2×C4×C61C2×#4111yescharacteristic
247C2×C4×C61H(5,𝔽2)11yescharacteristic
248C2×C4×C612−1+411yescharacteristic
249C2×Dic61—22no—
250C2×Dic61—22no—
251C2×Dic61—22no—
252C2×Dic61—21no—
253C2×Dic61—21no—
254C2×Dic61—21no—
255C2×Dic61—21no—
256C2×Dic61—21no—
257C2×Dic61—21no—
258C2×Dic61—21no—
259C2×Dic61—21no—
260C2×Dic61—21no—
261C2×Dic61—21no—
262C2×Dic61—41no—
263C2×Dic61—41no—
264C2×Dic61—41no—
265C2×Dic61—41no—
266C2×Dic61—41no—
267C2×Dic61—41no—
268C2×Dic61C4×D412yes—
269C2×Dic61C4×D412yes—
270D244—22no—
271D244—22no—
272D244—22no—
273D244C2×C4⋊C414yes—
274C4×D61—84no—
275C4×D61—84no—
276C4×D61—84no—
277Dic122—22no—
278Dic122—22no—
279Dic122—22no—
280Dic122C2×C4⋊C414yes—
281C8×C61—41no—
282C8×C61C2×C4⋊C414yes—
283#92418—22no—
284#92418—41no—
285#22673—611nomaximal, Sylow
286C22×C61C2×C4×D411yescharacteristic
287C22×C61D4211yescharacteristic
288C22×C61D4×Q811yescharacteristic
289C22×C61#57511yescharacteristic
290C22×C61#57511yescharacteristic
291C22×C61#57711yescharacteristic
292C22×C61#57711yescharacteristic
293C2×D61—24no—
294C2×D61—42no—
295C2×D61—42no—
296C2×D61—42no—
297C2×D61—42no—
298C2×D61—42no—
299C2×D61—42no—
300C4×C61—42no—
301C4×C61—42no—
302C4×C61—41no—
303C4×C61—41no—
304C4×C61—81no—
305C4×C61—81no—
306C4×C61C22×C4⋊C411yescharacteristic
307C4×C61#52011yescharacteristic
308C4×C61#54711yescharacteristic
309C4×C61#55411yescharacteristic
310Dic61—24no—
311Dic61—22no—
312Dic61—22no—
313Dic61—41no—
314Dic61—41no—
315Dic61—81no—
316Dic61—81no—
317C2×#440—611no—
318C2×#440—611no—
319C2×C4×SD16—611no—
320#2351—611no—
321#2349—611no—
322#1944—611no—
323#1944D6111yescharacteristic, complemented
324#1865—611no—
325#1865—611no—
326#1864—611no—
327#1864—611no—
328#1846—611no—
329#1805—611no—
330#1804—611no—
331#1770—611no—
332C2×C61D4×C4⋊C411yescharacteristic
333C2×C61#294311yescharacteristic
334C2×C61#294911yescharacteristic
335C2×C61#327511yescharacteristic
336C2×C61#327511yescharacteristic
337C2×C61#338811yescharacteristic
338C2×C61#338811yescharacteristic
339D61—44no—
340C22×SD16—611no—
341C2×C4×Q8—611no—
342C2×C4×Q8—611no—
343C2×C4×Q8—611no—
344C2×C4×D4—611no—
345C2×C4×D4—611no—
346C2×C4×D4—611no—
347#440—1224no—
348#440—1224no—
349C4×SD16—618no—
350C2×#107—611no—
351C2×#107C2×D6111yescharacteristic
352C2×#106—611no—
353C2×#106—611no—
354C2×#106C2×D6111yescharacteristic
355C2×C4⋊C8—611no—
356C2×#103—611no—
357C2×#103—611no—
358C2×#103—611no—
359C2×#102—611no—
360C2×#102—611no—
361C2×#102—611no—
362C2×#97—611no—
363C2×#97C2×D6111yescharacteristic
364C2×C4×C8—611no—
365#389—611no—
366#389—611no—
367#389—1221no—
368#389—1221no—
369#389C2×D6111yescharacteristic
370#389C2×D6111yescharacteristic
371#385—1221no—
372#385—1221no—
373#382—611no—
374#382—611no—
375#382—1221no—
376#380—1221no—
377C4×C4⋊C4—611no—
378C4×C4⋊C4—1221no—
379C2×#95—1221no—
380#340—611no—
381#340—611no—
382#340C2×D6111yescharacteristic
383#340C2×D6111yescharacteristic
384#336—611no—
385#336—611no—
386C61#2267311yescharacteristic, complemented, Sylow
387C22×Q8—611no—
388C22×D4—611no—
389C23×C4—1221no—
390C23×C4—1221no—
391C23×C4—1221no—
392C2×SD16—614no—
393C2×SD16—614no—
394C2×SD16—614no—
395C22×C8—611no—
396C22×C8—611no—
397C22×C8C22×D6111yescharacteristic
398C22×C8C22×D6111yescharacteristic
399C4×Q8—614no—
400C4×Q8—1222no—
401C4×Q8—1222no—
402C4×Q8—1222no—
403C4×Q8—2441no—
404C4×Q8—2441no—
405C4×D4—614no—
406C4×D4—1222no—
407C4×D4—1222no—
408C4×D4—1222no—
409C4×D4—2441no—
410C4×D4—2441no—
411C2×C4⋊C4—21no—
412C2×C4⋊C4—21no—
413C2×C4⋊C4—611no—
414C2×C4⋊C4—611no—
415C2×C4⋊C4—611no—
416C2×C4⋊C4—611no—
417C2×C4⋊C4—1221no—
418C2×C4⋊C4—1221no—
419C2×C4⋊C4—1221no—
420C2×C4⋊C4—1221no—
421C2×C4⋊C4—1221no—
422C2×C4⋊C4C22×D6111yescharacteristic
423C2×C4⋊C4C22×D6111yescharacteristic
424C2×C4⋊C4C22×D6111yescharacteristic
425C2×C4⋊C4C22×D6111yescharacteristic
426C2×C22⋊C4—1221no—
427C2×C22⋊C4—1221no—
428C2×C22⋊C4—1221no—
429C2×C42—611no—
430C2×C42—611no—
431C2×C42—1221no—
432C2×C42—1221no—
433C2×C42—1221no—
434C2×C42C22×D6111yescharacteristic
435#107—22no—
436#107—1222no—
437#106—2441no—
438#106—2441no—
439#106C4×D6114yescomplemented
440C4⋊C8—1222no—
441#103—614no—
442#103—1222no—
443#103—1222no—
444#102—614no—
445#102—1222no—
446#102—1222no—
447#97—22no—
448#97—1222no—
449C4×C8—614no—
450#95—21no—
451#95—21no—
452#95—611no—
453#95—611no—
454#95—1221no—
455#95—1221no—
456#95—1221no—
457#95—1221no—
458#95—1221no—
459#95—1221no—
460C24—1221no—
461C2×Q8—612no—
462C2×Q8—612no—
463C2×Q8—612no—
464C2×Q8—1221no—
465C2×Q8—1221no—
466C2×Q8—1221no—
467C2×D4—612no—
468C2×D4—612no—
469C2×D4—612no—
470C2×D4—1221no—
471C2×D4—1221no—
472C2×D4—1221no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—611no—
478C22×C4—611no—
479C22×C4—611no—
480C22×C4—611no—
481C22×C4—1224no—
482C22×C4—1221no—
483C22×C4—1221no—
484C22×C4—1221no—
485C22×C4—4881no—
486C22×C4—4881no—
487C22×C4—4881no—
488C22×C4—4881no—
489C22×C4—2442no—
490C22×C4—2442no—
491C22×C4—2442no—
492C22×C4—2442no—
493C22×C4D4×D6111yescharacteristic
494C22×C4D4×D6111yescharacteristic
495C22×C4C23×D6111yescharacteristic
496SD16—6116no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—612no—
501C2×C8—1221no—
502C2×C8—1221no—
503C2×C8—1221no—
504C2×C8—1221no—
505C2×C8—1221no—
506C2×C8C2×C4×D6112yes—
507C2×C8D4×D6112yes—
508C2×C8Q8×D6112yes—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—42no—
516C4⋊C4—81no—
517C4⋊C4—1224no—
518C4⋊C4—1221no—
519C4⋊C4—1221no—
520C4⋊C4—1221no—
521C4⋊C4—1221no—
522C4⋊C4—4881no—
523C4⋊C4—2441no—
524C4⋊C4—2441no—
525C4⋊C4—2442no—
526C4⋊C4—2442no—
527C4⋊C4—2442no—
528C4⋊C4C2×C4×D6112yes—
529C4⋊C4C2×C4×D6112yes—
530C22⋊C4—1224no—
531C22⋊C4—2442no—
532C22⋊C4—2442no—
533C42—21no—
534C42—21no—
535C42—612no—
536C42—612no—
537C42—1224no—
538C42—1221no—
539C42—1221no—
540C42—2442no—
541C42—2442no—
542C23—1222no—
543C23—1222no—
544C23—1222no—
545C23—2441no—
546C23—2441no—
547C23—2441no—
548C23—2441no—
549C23C2×D4×D6111yescharacteristic
550Q8—614no—
551Q8—1222no—
552Q8—1222no—
553Q8—1222no—
554D4—614no—
555D4—1222no—
556D4—1222no—
557D4—1222no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
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—41no—
575C2×C4—41no—
576C2×C4—612no—
577C2×C4—612no—
578C2×C4—1222no—
579C2×C4—1221no—
580C2×C4—1221no—
581C2×C4—1221no—
582C2×C4—1221no—
583C2×C4—1222no—
584C2×C4—1222no—
585C2×C4—1221no—
586C2×C4—1221no—
587C2×C4—1221no—
588C2×C4—1221no—
589C2×C4—1221no—
590C2×C4—1221no—
591C2×C4—4884no—
592C2×C4—4884no—
593C2×C4—4884no—
594C2×C4—2441no—
595C2×C4—2441no—
596C2×C4—2441no—
597C2×C4—2441no—
598C2×C4—2441no—
599C2×C4—2441no—
600C2×C4C22×C4×D6111yescharacteristic
601C2×C4C2×D4×D6111yescharacteristic
602C2×C4C2×Q8×D6111yescharacteristic
603C2×C4#41×D6111yescharacteristic
604C2×C4#1451982311yescharacteristic
605C2×C4#1451982511yescharacteristic
606C8—41no—
607C8—1222no—
608C8—2441no—
609C8C4⋊C4×D6114yes—
610C22—1224no—
611C22—2442no—
612C22—2442no—
613C22—2442no—
614C22—2442no—
615C22—2442no—
616C22—2442no—
617C22C4×D4×D6111yescharacteristic
618C22#1684039611yescharacteristic
619C22#1684042711yescharacteristic
620C22#1684055811yescharacteristic
621C22#1684056111yescharacteristic
622C22#1684057111yescharacteristic
623C22#1684057211yescharacteristic
624C4—42no—
625C4—41no—
626C4—41no—
627C4—42no—
628C4—81no—
629C4—81no—
630C4—1224no—
631C4—1222no—
632C4—1222no—
633C4—4881no—
634C4—4881no—
635C4—2441no—
636C4—2441no—
637C4C2×C4⋊C4×D6111yescharacteristic
638C4#1684024411yescharacteristic
639C4#1684041411yescharacteristic
640C4#1684046811yescharacteristic
641C2—2444no—
642C2#2144872011yescharacteristic
643C2#2145461411yescharacteristic
644C2#2145467511yescharacteristic
645C2#2145744711yescharacteristic
646C2#2145745711yescharacteristic
647C2#2145875911yescharacteristic
648C2#2145876311yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1944C2
socleC23×C61C22×D4
prime radicalC22×D244C23
cosocleC22×C4×C61C24
hypercenter#1944D61
hypocenterC61#22673
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1944

Complements

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

NH ≅ G/Ntype
4. #21448738C2semidirect
6. #21446563C2semidirect
7. #21446088C2semidirect
8. #21446086C2semidirect
11. #21445982C2semidirect
12. #21445833C2semidirect
14. #21445687C2semidirect
15. C61×#1944C2semidirect
25. #16839849C4semidirect
77. #14519742C2×C4semidirect
107. #14519700C2×C4semidirect
112. C8×Dic61C2×C4semidirect
131. #106×C61C2×C4semidirect
216. #11758190C4⋊C4semidirect
323. #1944D61semidirect
386. C61#22673semidirect
439. #106C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168398642
C2×#168398612
C2×#168398492
#214487382
#214487282
#214465632
#214460882
#214460862
#214460812
#214460792
#214459822
#214458332
#214458282
#214456872
C61×#19442
#2267361