Finite Group Databasebeta

← #29730362

Subgroups — #29730362

Order 15616.

840 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840047C211yesmaximal, characteristic, complemented
2C2×#16839835C211yesmaximal, characteristic
3#21450988C211yesmaximal, characteristic
4#21450724C211yesmaximal, characteristic
5#21447258C211yesmaximal, characteristic, complemented
6#21447257C211yesmaximal, characteristic, complemented
7#21447020C211yesmaximal, characteristic, complemented
8#21447013C211yesmaximal, characteristic, complemented
9#21446792C211yesmaximal, characteristic, complemented
10#21446587C211yesmaximal, characteristic, complemented
11#21446249C211yesmaximal, characteristic, complemented
12#21446239C211yesmaximal, characteristic, complemented
13#21445851C211yesmaximal, characteristic, complemented
14#21445843C211yesmaximal, characteristic, complemented
15C61×#1763C211yesmaximal, characteristic, complemented
16C22×#11758189C2211yescharacteristic
17C22×C8×D61C2211yescharacteristic
18C2×#14519777C2211yescharacteristic
19C2×#14519753C2211yescharacteristic
20C2×#14519753C2211yescharacteristic
21C2×C4×D244C2211yescharacteristic
22#16840047—24no—
23#16839835—24no—
24C2×#14519710C2211yescharacteristic
25C2×#14519708C2211yescharacteristic
26C2×#14519708C2211yescharacteristic
27C2×#14519708C2211yescharacteristic
28C2×C4⋊#92418C2211yescharacteristic
29C2×#14519701C2211yescharacteristic
30C2×#14519701C2211yescharacteristic
31C2×#14519689C2211yescharacteristic
32C2×#14519689C2211yescharacteristic
33#16839692—21no—
34#16839679—21no—
35#16839675—21no—
36#16839675—21no—
37#16839664—21no—
38#16839653—21no—
39#16839653—21no—
40#16839628—21no—
41#16839628C2211yescharacteristic
42#16839619C2211yescharacteristic
43#16839619C2211yescharacteristic
44#16839619C2211yescharacteristic
45#16839619C2211yescharacteristic
46#16839619C2211yescharacteristic
47#16839619C2211yescharacteristic
48#16839525C2211yescharacteristic
49#16839525C2211yescharacteristic
50#16839525C2211yescharacteristic
51#16839525C2211yescharacteristic
52#16839524C2211yescharacteristic
53#16839524C2211yescharacteristic
54C2×C4×Q8×C61C2211yescharacteristic
55C2×#103×C61C2211yescharacteristic
56C2×#103×C61C2211yescharacteristic
57C2×#97×C61C2211yescharacteristic
58C4×C4⋊C4×C61C2211yescharacteristic
59C61×#340C2211yescharacteristic
60C61×#340C2211yescharacteristic
61C22×Dic122—21no—
62C22×D244D411yescharacteristic
63C22×C4×D61C2311yescharacteristic
64C22×C4×D61C2311yescharacteristic
65C2×#11758189—22no—
66C2×#11758189—22no—
67C2×#11758189—41no—
68C22×#92418C2311yescharacteristic
69C22×#92418C2311yescharacteristic
70C2×C8×D61—22no—
71C2×C8×D61—22no—
72C2×C8×D61—41no—
73#14519777—22no—
74#14519777—22no—
75#14519753—22no—
76#14519753—22no—
77#14519753—41no—
78#14519753—41no—
79C4×D244—22no—
80C4×D244—22no—
81C2×C4⋊Dic61—21no—
82C2×C4⋊Dic61D411yescharacteristic
83C2×C4⋊Dic61C2311yescharacteristic
84C2×C4⋊Dic61C2311yescharacteristic
85C2×C4⋊Dic61C2311yescharacteristic
86C2×C4⋊Dic61C2311yescharacteristic
87C2×#11758196—21no—
88C2×#11758196—21no—
89C2×#11758198—21no—
90C2×#11758198—21no—
91C2×#11758198—21no—
92C2×#11758198—21no—
93C2×#11758198—21no—
94C2×#11758198—21no—
95C2×#11758198—21no—
96C2×#11758198—21no—
97C2×#11758198D411yescharacteristic
98C2×#11758198D411yescharacteristic
99#14519710—41no—
100#14519708—22no—
101#14519708—22no—
102#14519708—41no—
103C4⋊#92418D414yes—
104#14519701—22no—
105#14519701—22no—
106#14519689—22no—
107#14519689—22no—
108#14519684—21no—
109#14519684—21no—
110#14519684—21no—
111#14519684—21no—
112#14519684—21no—
113#14519684—21no—
114#14519684—21no—
115#14519684—21no—
116C22×Q8×C61C2311yescharacteristic
117C22×C8×C61C2311yescharacteristic
118C22×C8×C61C2311yescharacteristic
119C4×Q8×C61—22no—
120C4×Q8×C61—22no—
121C2×C4⋊C4×C61—21no—
122C2×C4⋊C4×C61C2311yescharacteristic
123C2×C4⋊C4×C61C2311yescharacteristic
124C2×C4⋊C4×C61C2311yescharacteristic
125C2×C42×C61—21no—
126C2×C42×C61—21no—
127C2×C42×C61C2311yescharacteristic
128#103×C61—22no—
129#103×C61—22no—
130#97×C61—22no—
131#95×C61—21no—
132C23×D61C2×D411yescharacteristic
133C23×D61C2×D411yescharacteristic
134C2×Q8×C61—21no—
135C2×Q8×C61—21no—
136C2×Q8×C61—21no—
137C2×Q8×C61—21no—
138C2×Q8×C61—21no—
139C2×Q8×C61#4112yes—
140C22×C4×C61—21no—
141C22×C4×C61—21no—
142C22×C4×C61—21no—
143C22×C4×C61—21no—
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2411yescharacteristic
147C22×Dic61—21no—
148C22×Dic61—21no—
149C22×Dic61—21no—
150C22×Dic61—21no—
151C22×Dic61C2×D411yescharacteristic
152C22×Dic61C2×D411yescharacteristic
153C2×D244—22no—
154C2×D244—22no—
155C2×D244—41no—
156C2×C4×D61—21no—
157C2×C4×D61—21no—
158C2×C4×D61—21no—
159C2×C4×D61—21no—
160C2×C4×D61—21no—
161C2×C4×D61—21no—
162C2×C4×D61—21no—
163C2×C4×D61—21no—
164C2×C4×D61—21no—
165C2×C4×D61—21no—
166C2×C4×D61—21no—
167C2×C4×D61—21no—
168C2×Dic122—42no—
169C2×Dic122—81no—
170C2×Dic122—81no—
171C2×C8×C61—21no—
172C2×C8×C61—21no—
173C2×C8×C61—21no—
174C2×C8×C61—21no—
175C2×C8×C61—21no—
176C2×C8×C61#4112yes—
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—
184C42×C61—42no—
185C42×C61—42no—
186C42×C61C2×D412yes—
187C42×C61#4112yes—
188#11758198—22no—
189#11758198—22no—
190#11758198—42no—
191#11758198—42no—
192#11758198—42no—
193#11758198—42no—
194#11758198—81no—
195#11758198—81no—
196#11758198—81no—
197#11758198—81no—
198C4⋊Dic61—22no—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61—21no—
203C4⋊Dic61—21no—
204C4⋊Dic61—21no—
205C4⋊Dic61—21no—
206C4⋊Dic61—21no—
207C4⋊Dic61—42no—
208#11758196—42no—
209#11758196—42no—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418C2×D412yes—
215C2×#92418C2×D412yes—
216#11758189—44no—
217C8×D61—44no—
218C23×C61C22×D411yescharacteristic
219C22×D61—21no—
220C22×D61—21no—
221C22×D61—21no—
222C22×D61—21no—
223C22×D61—21no—
224C22×D61—21no—
225C22×D61—21no—
226C22×D61—21no—
227C22×D61—21no—
228C22×D61—21no—
229C22×D61—21no—
230C22×D61—21no—
231C22×D61—21no—
232C22×D61#13712yes—
233Q8×C61—22no—
234Q8×C61—22no—
235Q8×C61—41no—
236Q8×C61—41no—
237C2×C4×C61—21no—
238C2×C4×C61—21no—
239C2×C4×C61—21no—
240C2×C4×C61—21no—
241C2×C4×C61—22no—
242C2×C4×C61—41no—
243C2×C4×C61—41no—
244C2×C4×C61—41no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61—41no—
248C2×C4×C61—41no—
249C2×C4×C61—41no—
250C2×C4×C61—41no—
251C2×C4×C61—41no—
252C2×C4×C61—41no—
253C2×C4×C61#12112yes—
254C2×C4×C61#12112yes—
255C2×C4×C61C22×D411yescharacteristic
256C2×C4×C61C2×#4111yescharacteristic
257C2×C4×C61C2×#4111yescharacteristic
258C2×C4×C61C2×#4111yescharacteristic
259C2×C4×C61H(5,𝔽2)11yescharacteristic
260C2×C4×C612−1+411yescharacteristic
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—
275C2×Dic61—41no—
276C2×Dic61—41no—
277C2×Dic61—41no—
278C2×Dic61—41no—
279D244—44no—
280C4×D61—41no—
281C4×D61—41no—
282C4×D61—41no—
283C4×D61—41no—
284C4×D61—41no—
285C4×D61—41no—
286C4×D61—41no—
287C4×D61—41no—
288Dic122—84no—
289C8×C61—22no—
290C8×C61—41no—
291#92418—22no—
292#92418—22no—
293#25356—611nomaximal, Sylow
294C22×C61C2×#12111yescharacteristic
295C22×C61#54611yescharacteristic
296C22×C61#54711yescharacteristic
297C22×C61C2×#13711yescharacteristic
298C22×C61#57511yescharacteristic
299C22×C61#57711yescharacteristic
300C22×C61#57711yescharacteristic
301C2×D61—22no—
302C2×D61—22no—
303C2×D61—22no—
304C2×D61—41no—
305C2×D61—41no—
306C2×D61—41no—
307C2×D61—41no—
308C2×D61—41no—
309C2×D61—41no—
310C2×D61—41no—
311C2×D61—41no—
312C2×D61—41no—
313C2×D61—41no—
314C2×D61—41no—
315C4×C61—22no—
316C4×C61—22no—
317C4×C61—42no—
318C4×C61—81no—
319C4×C61—81no—
320C4×C61—81no—
321C4×C61#52911yescharacteristic
322C4×C61#54211yescharacteristic
323C4×C61#54611yescharacteristic
324C4×C61#54811yescharacteristic
325Dic61—41no—
326Dic61—41no—
327Dic61—81no—
328Dic61—81no—
329Dic61—81no—
330Dic61—81no—
331C2×#475—611no—
332C2×#435—611no—
333#2563—611no—
334#2536—611no—
335#2077—611no—
336#2077—611no—
337#2036—611no—
338#2035—611no—
339#1993—611no—
340#1949—611no—
341#1894—611no—
342#1892—611no—
343#1809—611no—
344#1807—611no—
345#1763D6111yescharacteristic, complemented
346C2×C61#242711yescharacteristic
347C2×C61#305611yescharacteristic
348C2×C61#305711yescharacteristic
349C2×C61#331611yescharacteristic
350C2×C61#332111yescharacteristic
351C2×C61#334411yescharacteristic
352C2×C61#334811yescharacteristic
353D61—42no—
354D61—81no—
355C22×M16—611no—
356C23×C8—611no—
357C2×C4⋊Q8—611no—
358C2×C22⋊Q8—611no—
359C2×C22⋊Q8—611no—
360C2×C4×Q8C2×D6111yescharacteristic
361C2×C4×D4—611no—
362#475—1224no—
363#435—1224no—
364C2×#107—611no—
365C2×#106—611no—
366C2×#106—611no—
367C2×#106—611no—
368C2×C4⋊C8—611no—
369C2×#103—611no—
370C2×#103—611no—
371C2×#103C2×D6111yescharacteristic
372C2×#103C2×D6111yescharacteristic
373C2×C22⋊C8—611no—
374C2×C22⋊C8—611no—
375C2×#97C2×D6111yescharacteristic
376#398—1221no—
377#394—1221no—
378#393—1221no—
379#393—1221no—
380#390—1221no—
381#388—1221no—
382#388—1221no—
383#382—611no—
384#382—1221no—
385#380—611no—
386#380—611no—
387#380—611no—
388#380—611no—
389#380—611no—
390#380—611no—
391C4×C4⋊C4C2×D6111yescharacteristic
392#340—611no—
393#340—611no—
394#340—611no—
395#340—611no—
396#340—611no—
397#340—611no—
398#340C2×D6111yescharacteristic
399#340C2×D6111yescharacteristic
400C61#2535611yescharacteristic, complemented, Sylow
401C22×Q8—1221no—
402C22×Q8C22×D6111yescharacteristic
403C22×D4—611no—
404C23×C4—611no—
405C23×C4—611no—
406C2×M16—1222no—
407C2×M16—1222no—
408C2×M16—2441no—
409C22×C8—611no—
410C22×C8—611no—
411C22×C8—1222no—
412C22×C8—1222no—
413C22×C8—2441no—
414C22×C8C22×D6111yescharacteristic
415C22×C8C22×D6111yescharacteristic
416C4⋊Q8—1222no—
417C4⋊Q8—1222no—
418C22⋊Q8—1222no—
419C22⋊Q8—1222no—
420C22⋊Q8—2441no—
421C22⋊Q8—2441no—
422C4×Q8—22no—
423C4×Q8—22no—
424C4×D4—1222no—
425C4×D4—1222no—
426C2×C4⋊C4—21no—
427C2×C4⋊C4—611no—
428C2×C4⋊C4—611no—
429C2×C4⋊C4—611no—
430C2×C4⋊C4—611no—
431C2×C4⋊C4—611no—
432C2×C4⋊C4—1221no—
433C2×C4⋊C4—1221no—
434C2×C4⋊C4—1221no—
435C2×C4⋊C4C22×D6111yescharacteristic
436C2×C4⋊C4C22×D6111yescharacteristic
437C2×C4⋊C4C22×D6111yescharacteristic
438C2×C22⋊C4—611no—
439C2×C22⋊C4—611no—
440C2×C22⋊C4—1221no—
441C2×C22⋊C4—1221no—
442C2×C22⋊C4—1221no—
443C2×C22⋊C4—1221no—
444C2×C22⋊C4—1221no—
445C2×C22⋊C4—1221no—
446C2×C22⋊C4—1221no—
447C2×C22⋊C4—1221no—
448C2×C42—21no—
449C2×C42—21no—
450C2×C42C22×D6111yescharacteristic
451#107—2441no—
452#106—1222no—
453#106—1222no—
454#106—2441no—
455C4⋊C8—614no—
456#103—22no—
457#103—22no—
458#103—1222no—
459#103—1222no—
460C22⋊C8—1222no—
461C22⋊C8—1222no—
462#97—22no—
463#95—21no—
464#95—1221no—
465#95—1221no—
466#95—1221no—
467#95—1221no—
468#95—1221no—
469#95—1221no—
470#95—1221no—
471#95—1221no—
472C24—611no—
473C24—611no—
474C2×Q8—21no—
475C2×Q8—21no—
476C2×Q8—21no—
477C2×Q8—21no—
478C2×Q8—21no—
479C2×Q8—4881no—
480C2×Q8—4881no—
481C2×Q8—2442no—
482C2×Q8#1175822212yes—
483C2×D4—1222no—
484C2×D4—1222no—
485C2×D4—2441no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—611no—
491C22×C4—611no—
492C22×C4—1221no—
493C22×C4—1221no—
494C22×C4—1221no—
495C22×C4—1221no—
496C22×C4—1221no—
497C22×C4—1221no—
498C22×C4—1221no—
499C22×C4—1221no—
500C22×C4—1221no—
501C22×C4—1221no—
502C22×C4—1221no—
503C22×C4—1221no—
504C22×C4—1221no—
505C22×C4—1221no—
506C22×C4—1221no—
507C22×C4—1221no—
508C22×C4D4×D6111yescharacteristic
509C22×C4D4×D6111yescharacteristic
510C22×C4C23×D6111yescharacteristic
511M16—2444no—
512C2×C8—21no—
513C2×C8—21no—
514C2×C8—21no—
515C2×C8—21no—
516C2×C8—21no—
517C2×C8—612no—
518C2×C8—612no—
519C2×C8—1221no—
520C2×C8—1221no—
521C2×C8—1221no—
522C2×C8—1221no—
523C2×C8—2444no—
524C2×C8#1175822212yes—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—42no—
532C4⋊C4—1222no—
533C4⋊C4—1221no—
534C4⋊C4—1221no—
535C4⋊C4—1221no—
536C4⋊C4—1221no—
537C4⋊C4—1221no—
538C4⋊C4—1221no—
539C4⋊C4—1221no—
540C4⋊C4—1221no—
541C4⋊C4—2442no—
542C4⋊C4—2442no—
543C4⋊C4—2442no—
544C22⋊C4—1222no—
545C22⋊C4—1222no—
546C22⋊C4—4881no—
547C22⋊C4—4881no—
548C22⋊C4—4881no—
549C22⋊C4—4881no—
550C22⋊C4—2442no—
551C22⋊C4—2442no—
552C22⋊C4—2442no—
553C22⋊C4—2442no—
554C42—42no—
555C42—42no—
556C42#1175822212yes—
557C42D4×D6112yes—
558C23—612no—
559C23—1221no—
560C23—1221no—
561C23—1221no—
562C23—1221no—
563C23—1221no—
564C23—1221no—
565C23—1221no—
566C23—1221no—
567C23—1221no—
568C23—1221no—
569C23—1221no—
570C23—1221no—
571C23—1221no—
572C23C2×D4×D6111yescharacteristic
573Q8—22no—
574Q8—22no—
575Q8—41no—
576Q8—41no—
577Q8—4884no—
578D4—2444no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—22no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—1221no—
596C2×C4—1221no—
597C2×C4—1221no—
598C2×C4—1221no—
599C2×C4—1221no—
600C2×C4—1221no—
601C2×C4—2441no—
602C2×C4—2441no—
603C2×C4—2441no—
604C2×C4—2441no—
605C2×C4—2441no—
606C2×C4—2441no—
607C2×C4—2441no—
608C2×C4—2441no—
609C2×C4—2441no—
610C2×C4—2441no—
611C2×C4—2441no—
612C2×C4—2441no—
613C2×C4—2441no—
614C2×C4—2441no—
615C2×C4—2441no—
616C2×C4—2441no—
617C2×C4—2441no—
618C2×C4—2441no—
619C2×C4—2441no—
620C2×C4—2441no—
621C2×C4#1451975012yes—
622C2×C4#1451975012yes—
623C2×C4C2×D4×D6111yescharacteristic
624C2×C4C2×#1175822211yescharacteristic
625C2×C4C2×#1175822211yescharacteristic
626C2×C4C2×#1175822211yescharacteristic
627C2×C4#1451982311yescharacteristic
628C2×C4#1451982411yescharacteristic
629C8—22no—
630C8—41no—
631C8—1222no—
632C8—1222no—
633C22—1222no—
634C22—1222no—
635C22—1222no—
636C22—2441no—
637C22—2441no—
638C22—2441no—
639C22—2441no—
640C22—2441no—
641C22—2441no—
642C22—2441no—
643C22—2441no—
644C22—2441no—
645C22—2441no—
646C22—2441no—
647C22C2×#1451975011yescharacteristic
648C22#1684040811yescharacteristic
649C22#1684041711yescharacteristic
650C22#137×D6111yescharacteristic
651C22#1684055811yescharacteristic
652C22#1684057111yescharacteristic
653C22#1684057111yescharacteristic
654C4—22no—
655C4—22no—
656C4—42no—
657C4—81no—
658C4—81no—
659C4—81no—
660C4—4881no—
661C4—4881no—
662C4—4881no—
663C4—4881no—
664C4—2441no—
665C4—2441no—
666C4#1684029611yescharacteristic
667C4#1684038311yescharacteristic
668C4#1684040811yescharacteristic
669C4#1684042111yescharacteristic
670C2—4881no—
671C2—2442no—
672C2#2144939611yescharacteristic
673C2#2145545211yescharacteristic
674C2#2145546111yescharacteristic
675C2#2145793511yescharacteristic
676C2#2145801011yescharacteristic
677C2#2145827911yescharacteristic
678C2#2145833911yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1763C2
socleC23×C61C22×D4
prime radicalC22×D244D4
hypercenter#1763D61
hypocenterC61#25356
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1763

Complements

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

NH ≅ G/Ntype
1. C2×#16840047C2semidirect
5. #21447258C2semidirect
6. #21447257C2semidirect
7. #21447020C2semidirect
8. #21447013C2semidirect
9. #21446792C2semidirect
10. #21446587C2semidirect
11. #21446249C2semidirect
12. #21446239C2semidirect
13. #21445851C2semidirect
14. #21445843C2semidirect
15. C61×#1763C2semidirect
345. #1763D61semidirect
400. C61#25356semidirect

Maximal subgroups

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

subgroupindex
C2×#168400472
C2×#168398352
#214509882
#214507242
#214472582
#214472572
#214470202
#214470132
#214467922
#214465872
#214462492
#214462392
#214458512
#214458432
C61×#17632
#2535661