Finite Group Databasebeta

← #29742991

Subgroups — #29742991

Order 15616.

896 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840059C211yesmaximal, characteristic, complemented
2C2×C4×#11758199C211yesmaximal, characteristic
3#21448730C211yesmaximal, characteristic, complemented
4#21447340C211yesmaximal, characteristic, complemented
5#21447300C211yesmaximal, characteristic, complemented
6#21447207C211yesmaximal, characteristic, complemented
7#21447154C211yesmaximal, characteristic, complemented
8#21446549C211yesmaximal, characteristic, complemented
9#21446378C211yesmaximal, characteristic, complemented
10#21446170C211yesmaximal, characteristic
11#21446170C211yesmaximal, characteristic
12#21446080C211yesmaximal, characteristic, complemented
13#21446079C211yesmaximal, characteristic, complemented
14#21446065C211yesmaximal, characteristic, complemented
15C61×#2454C211yesmaximal, characteristic, complemented
16C22×#11758199C2211yescharacteristic
17C2×C4×D244C2211yescharacteristic
18C2×C4×D244C2211yescharacteristic
19C2×C4×D244C2211yescharacteristic
20#16840059—24no—
21C4×#11758199—24no—
22C2×#14519710C2211yescharacteristic
23C2×#14519710C2211yescharacteristic
24C2×#14519710C2211yescharacteristic
25C2×#14519708C2211yescharacteristic
26C2×#14519708C2211yescharacteristic
27C2×C4⋊#92418C2211yescharacteristic
28C2×#14519698C2211yescharacteristic
29C2×#14519698C2211yescharacteristic
30C2×#14519698C2211yescharacteristic
31C2×#14519697C2211yescharacteristic
32C2×#14519697C2211yescharacteristic
33C2×#14519697C2211yescharacteristic
34C2×C4×#92418C2211yescharacteristic
35#16839660C2211yescharacteristic
36#16839660C2211yescharacteristic
37#16839633—21no—
38#16839633—21no—
39#16839628C2211yescharacteristic
40#16839621—21no—
41#95×D61—21no—
42#16839524C2211yescharacteristic
43#16839524C2211yescharacteristic
44#16839524C2211yescharacteristic
45#16839524C2211yescharacteristic
46#16839517C2211yescharacteristic
47#16839517C2211yescharacteristic
48#16839517C2211yescharacteristic
49#16839517C2211yescharacteristic
50C2×#125×C61C2211yescharacteristic
51C2×C4×D4×C61C2211yescharacteristic
52C61×#399—21no—
53C61×#396—21no—
54C61×#390C2211yescharacteristic
55C61×#390C2211yescharacteristic
56C61×#389C2211yescharacteristic
57C61×#389C2211yescharacteristic
58C61×#385—21no—
59C61×#380—21no—
60C4×C4⋊C4×C61C2211yescharacteristic
61C22×D244C2311yescharacteristic
62C22×C4×D61—21no—
63C22×C4×D61—21no—
64C22×C4×D61—21no—
65C2×#11758199—22no—
66C2×#11758199—22no—
67C2×#11758199Q814yes—
68C22×#92418C2311yescharacteristic
69C22×#92418C2311yescharacteristic
70C22×#92418C2311yescharacteristic
71C22×#92418C2311yescharacteristic
72C4×D244—22no—
73C4×D244—22no—
74C4×D244—22no—
75C4×D244—22no—
76C4×D244—41no—
77C4×D244—41no—
78C2×C4⋊Dic61C2311yescharacteristic
79C2×C4⋊Dic61C2311yescharacteristic
80C2×C4⋊Dic61C2311yescharacteristic
81C2×#11758196—21no—
82C2×#11758196—21no—
83C2×#11758198—21no—
84C2×#11758198—21no—
85C2×#11758198—21no—
86#14519710—22no—
87#14519710—22no—
88#14519710—22no—
89#14519708—22no—
90#14519708—41no—
91C4⋊#92418—22no—
92#14519698—22no—
93#14519698—22no—
94#14519698—41no—
95#14519697—22no—
96#14519697—22no—
97#14519697D414yes—
98C4×#92418—22no—
99#14519684—21no—
100#14519684—21no—
101#14519684—21no—
102#14519684—21no—
103C22×D4×C61C2311yescharacteristic
104C23×C4×C61—21no—
105#125×C61—22no—
106#125×C61—41no—
107C4×D4×C61—22no—
108C4×D4×C61—41no—
109C2×C4⋊C4×C61—21no—
110C2×C4⋊C4×C61—21no—
111C2×C4⋊C4×C61—21no—
112C2×C4⋊C4×C61—21no—
113C2×C4⋊C4×C61D411yescharacteristic
114C2×C4⋊C4×C61D411yescharacteristic
115C2×C4⋊C4×C61C2311yescharacteristic
116C2×C4⋊C4×C61C2311yescharacteristic
117C2×C4⋊C4×C61C2311yescharacteristic
118C2×C22⋊C4×C61—21no—
119C2×C22⋊C4×C61—21no—
120C2×C22⋊C4×C61—21no—
121C2×C22⋊C4×C61—21no—
122C2×C22⋊C4×C61—21no—
123C2×C42×C61C2311yescharacteristic
124C2×C42×C61C2311yescharacteristic
125C2×C42×C61C2311yescharacteristic
126#95×C61—21no—
127#95×C61—21no—
128#95×C61D411yescharacteristic
129#95×C61D411yescharacteristic
130C24×C61—21no—
131C23×D61—21no—
132C2×D4×C61—21no—
133C2×D4×C61—21no—
134C2×D4×C61—21no—
135C2×D4×C61—21no—
136C2×D4×C61—21no—
137C2×D4×C61C2×Q812yes—
138C22×C4×C61—21no—
139C22×C4×C61—21no—
140C22×C4×C61—21no—
141C22×C4×C61—42no—
142C22×C4×C61—42no—
143C22×C4×C61—81no—
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2411yescharacteristic
151C22×Dic61—21no—
152C22×Dic61—21no—
153C22×Dic61—21no—
154C2×D244—21no—
155C2×D244—21no—
156C2×D244—21no—
157C2×D244C2×D412yes—
158C2×D244C2×Q812yes—
159C2×D244#4112yes—
160C2×C4×D61—24no—
161C2×C4×D61—42no—
162C2×C4×D61—42no—
163C2×C4×D61—42no—
164C2×C4×D61—42no—
165C2×C4×D61—81no—
166C2×C4×D61—81no—
167C2×C4×D61—81no—
168C2×C4×D61—81no—
169C4⋊C4×C61—22no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—41no—
176C4⋊C4×C61—42no—
177C4⋊C4×C61—42no—
178C4⋊C4×C61—81no—
179C4⋊C4×C61—81no—
180C4⋊C4×C61C2×D412yes—
181C22⋊C4×C61—42no—
182C22⋊C4×C61—42no—
183C22⋊C4×C61—81no—
184C22⋊C4×C61—81no—
185C22⋊C4×C61—81no—
186C42×C61—21no—
187C42×C61—21no—
188C42×C61—21no—
189C42×C61—21no—
190C42×C61—21no—
191C42×C61—21no—
192#11758199—28no—
193#11758198—24no—
194#11758198—42no—
195#11758198—42no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201C4⋊Dic61#4112yes—
202#11758196—81no—
203#11758196—81no—
204C2×#92418—21no—
205C2×#92418—21no—
206C2×#92418—21no—
207C2×#92418—21no—
208C2×#92418—21no—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418C2×D412yes—
215C2×#92418C2×Q812yes—
216C23×C61—22no—
217C23×C61—41no—
218C23×C61—41no—
219C23×C61—41no—
220C23×C61—41no—
221C23×C61—41no—
222C23×C61—41no—
223C23×C61C22×D411yescharacteristic
224C22×D61—22no—
225C22×D61—22no—
226C22×D61—22no—
227C22×D61—41no—
228C22×D61—41no—
229C22×D61—41no—
230C22×D61—41no—
231D4×C61—22no—
232D4×C61—22no—
233D4×C61—41no—
234D4×C61—41no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61—21no—
239C2×C4×C61—21no—
240C2×C4×C61—21no—
241C2×C4×C61—21no—
242C2×C4×C61—22no—
243C2×C4×C61—21no—
244C2×C4×C61—21no—
245C2×C4×C61—21no—
246C2×C4×C61—21no—
247C2×C4×C61—21no—
248C2×C4×C61—21no—
249C2×C4×C61—21no—
250C2×C4×C61—21no—
251C2×C4×C61—21no—
252C2×C4×C61—21no—
253C2×C4×C61—41no—
254C2×C4×C61—41no—
255C2×C4×C61—41no—
256C2×C4×C61—41no—
257C2×C4×C61—41no—
258C2×C4×C61—41no—
259C2×C4×C61—41no—
260C2×C4×C61—41no—
261C2×C4×C61—84no—
262C2×C4×C61#13512yes—
263C2×C4×C61C22×D411yescharacteristic
264C2×C4×C61C22×Q811yescharacteristic
265C2×C4×C61C2×#4111yescharacteristic
266C2×C4×C61C2×#4111yescharacteristic
267C2×C4×C61H(5,𝔽2)11yescharacteristic
268C2×C4×C612−1+411yescharacteristic
269C2×Dic61—22no—
270C2×Dic61—41no—
271C2×Dic61—41no—
272C2×Dic61—41no—
273C2×Dic61—41no—
274C2×Dic61—41no—
275C2×Dic61—41no—
276C2×Dic61—41no—
277C2×Dic61—41no—
278D244—22no—
279D244—22no—
280D244—22no—
281D244C22⋊Q814yes—
282C4×D61—84no—
283C4×D61—84no—
284C4×D61—84no—
285#92418—22no—
286#92418—22no—
287#92418—41no—
288#92418—41no—
289#26370—611nomaximal, Sylow
290C22×C61—42no—
291C22×C61—42no—
292C22×C61—42no—
293C22×C61—81no—
294C22×C61—81no—
295C22×C61—81no—
296C22×C61—81no—
297C22×C61#52511yescharacteristic
298C22×C61D4211yescharacteristic
299C22×C61D4×Q811yescharacteristic
300C22×C61C2×#13511yescharacteristic
301C22×C61#57511yescharacteristic
302C22×C61#57611yescharacteristic
303C22×C61#57611yescharacteristic
304C2×D61—24no—
305C2×D61—42no—
306C2×D61—42no—
307C2×D61—42no—
308C2×D61—42no—
309C2×D61—42no—
310C2×D61—42no—
311C4×C61—22no—
312C4×C61—41no—
313C4×C61—41no—
314C4×C61—41no—
315C4×C61—41no—
316C4×C61—41no—
317C4×C61—42no—
318C4×C61—81no—
319C4×C61—81no—
320C4×C61C2×C22⋊Q811yescharacteristic
321C4×C61#53911yescharacteristic
322C4×C61#54711yescharacteristic
323C4×C61#55411yescharacteristic
324Dic61—42no—
325Dic61—81no—
326Dic61—81no—
327C2×#478—611no—
328C2×C4×D8—611no—
329#2454D6111yescharacteristic, complemented
330#2349—611no—
331#2092—611no—
332#2085—611no—
333#2067—611no—
334#2059—611no—
335#1942—611no—
336#1916—611no—
337#1879—611no—
338#1879—611no—
339#1864—611no—
340#1864—611no—
341#1862—611no—
342C2×C61—82no—
343C2×C61#244911yescharacteristic
344C2×C61#310211yescharacteristic
345C2×C61#312211yescharacteristic
346C2×C61#326811yescharacteristic
347C2×C61#328011yescharacteristic
348C2×C61#338011yescharacteristic
349C2×C61#338911yescharacteristic
350D61—44no—
351C22×D8—611no—
352C2×#125C2×D6111yescharacteristic
353C2×C4×D4—611no—
354C2×C4×D4—611no—
355C2×C4×D4—611no—
356C2×C4×D4C2×D6111yescharacteristic
357#478—1224no—
358C4×D8—1224no—
359C2×#107—611no—
360C2×#107—611no—
361C2×#107—611no—
362C2×#106—611no—
363C2×#106—611no—
364C2×C4⋊C8—611no—
365C2×#102—611no—
366C2×#102—611no—
367C2×#102—611no—
368C2×#102—611no—
369C2×#102—611no—
370C2×#102—611no—
371C2×C4×C8—611no—
372#399—21no—
373#396—21no—
374#390C2×D6111yescharacteristic
375#390C2×D6111yescharacteristic
376#389—611no—
377#389—611no—
378#389C2×D6111yescharacteristic
379#389C2×D6111yescharacteristic
380#385—21no—
381#385—1221no—
382#385—1221no—
383#382—611no—
384#380—21no—
385#380—1221no—
386C4×C4⋊C4C2×D6111yescharacteristic
387C2×#95—1221no—
388#340—611no—
389#340—611no—
390#340—611no—
391#340—611no—
392#336—611no—
393#336—611no—
394#336—611no—
395#336—611no—
396C61#2637011yescharacteristic, complemented, Sylow
397C22×D4—611no—
398C22×D4C22×D6111yescharacteristic
399C23×C4—21no—
400C23×C4—1221no—
401C23×C4—1221no—
402C23×C4—1221no—
403C2×D8—614no—
404C2×D8—1222no—
405C2×D8—1222no—
406C22×C8—611no—
407C22×C8—611no—
408C22×C8—611no—
409C22×C8—611no—
410#125—22no—
411#125—41no—
412C4×D4—22no—
413C4×D4—41no—
414C4×D4—1222no—
415C4×D4—1222no—
416C4×D4—1222no—
417C4×D4—1222no—
418C4×D4—2441no—
419C4×D4—2441no—
420C2×C4⋊C4—21no—
421C2×C4⋊C4—21no—
422C2×C4⋊C4—21no—
423C2×C4⋊C4—21no—
424C2×C4⋊C4—611no—
425C2×C4⋊C4—611no—
426C2×C4⋊C4—611no—
427C2×C4⋊C4—1221no—
428C2×C4⋊C4—1221no—
429C2×C4⋊C4C22⋊D6111yescharacteristic
430C2×C4⋊C4C22⋊D6111yescharacteristic
431C2×C4⋊C4C22×D6111yescharacteristic
432C2×C4⋊C4C22×D6111yescharacteristic
433C2×C4⋊C4C22×D6111yescharacteristic
434C2×C22⋊C4—21no—
435C2×C22⋊C4—21no—
436C2×C22⋊C4—21no—
437C2×C22⋊C4—21no—
438C2×C22⋊C4—21no—
439C2×C22⋊C4—1221no—
440C2×C22⋊C4—1221no—
441C2×C22⋊C4—1221no—
442C2×C42C22×D6111yescharacteristic
443C2×C42C22×D6111yescharacteristic
444C2×C42C22×D6111yescharacteristic
445#107—1222no—
446#107—1222no—
447#107—1222no—
448#106—1222no—
449#106—2441no—
450C4⋊C8—1222no—
451#102—614no—
452#102—1222no—
453#102—1222no—
454#102—1222no—
455#102—1222no—
456#102—2441no—
457C4×C8—1222no—
458#95—21no—
459#95—21no—
460#95—1221no—
461#95—1221no—
462#95—1221no—
463#95—1221no—
464#95C22⋊D6111yescharacteristic
465#95C22⋊D6111yescharacteristic
466C24—21no—
467C24—1221no—
468C2×D4—21no—
469C2×D4—21no—
470C2×D4—21no—
471C2×D4—21no—
472C2×D4—21no—
473C2×D4—612no—
474C2×D4—612no—
475C2×D4—612no—
476C2×D4—1221no—
477C2×D4—1221no—
478C2×D4—1221no—
479C2×D4Q8×D6112yes—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—42no—
484C22×C4—42no—
485C22×C4—81no—
486C22×C4—1224no—
487C22×C4—1221no—
488C22×C4—1221no—
489C22×C4—1221no—
490C22×C4—4881no—
491C22×C4—4881no—
492C22×C4—4881no—
493C22×C4—4881no—
494C22×C4—2442no—
495C22×C4—2442no—
496C22×C4—2442no—
497C22×C4—2442no—
498C22×C4C2×C22⋊D6111yescharacteristic
499C22×C4C2×C22⋊D6111yescharacteristic
500C22×C4C2×C22⋊D6111yescharacteristic
501C22×C4C2×C22⋊D6111yescharacteristic
502C22×C4C2×C22⋊D6111yescharacteristic
503C22×C4C2×C22⋊D6111yescharacteristic
504C22×C4C23×D6111yescharacteristic
505D8—1228no—
506C2×C8—612no—
507C2×C8—612no—
508C2×C8—1221no—
509C2×C8—1221no—
510C2×C8—1221no—
511C2×C8—1221no—
512C2×C8—1221no—
513C2×C8—1221no—
514C2×C8—1221no—
515C2×C8—1221no—
516C2×C8—1221no—
517C2×C8—1221no—
518C4⋊C4—21no—
519C4⋊C4—22no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—41no—
525C4⋊C4—42no—
526C4⋊C4—42no—
527C4⋊C4—81no—
528C4⋊C4—81no—
529C4⋊C4—612no—
530C4⋊C4—1221no—
531C4⋊C4—1221no—
532C4⋊C4—1221no—
533C4⋊C4—1221no—
534C4⋊C4—1221no—
535C4⋊C4—4881no—
536C4⋊C4—4881no—
537C4⋊C4D4×D6112yes—
538C22⋊C4—42no—
539C22⋊C4—42no—
540C22⋊C4—81no—
541C22⋊C4—81no—
542C22⋊C4—81no—
543C22⋊C4—1224no—
544C22⋊C4—2442no—
545C22⋊C4—2442no—
546C42—21no—
547C42—21no—
548C42—21no—
549C42—21no—
550C42—21no—
551C42—21no—
552C23—22no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23—1222no—
560C23—1222no—
561C23—1222no—
562C23—2441no—
563C23—2441no—
564C23—2441no—
565C23—2441no—
566C23C22×C22⋊D6111yescharacteristic
567D4—22no—
568D4—22no—
569D4—41no—
570D4—41no—
571D4—614no—
572D4—1222no—
573D4—1222no—
574D4—1222no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—22no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—84no—
602C2×C4—1222no—
603C2×C4—4884no—
604C2×C4—4884no—
605C2×C4—4884no—
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#1451979712yes—
615C2×C4C2×D4×D6111yescharacteristic
616C2×C4C2×Q8×D6111yescharacteristic
617C2×C4#41×D6111yescharacteristic
618C2×C4#41×D6111yescharacteristic
619C2×C4#1451982211yescharacteristic
620C2×C4#1451982511yescharacteristic
621C8—1222no—
622C8—1222no—
623C8—2441no—
624C8—2441no—
625C22—42no—
626C22—42no—
627C22—42no—
628C22—81no—
629C22—81no—
630C22—81no—
631C22—81no—
632C22—1224no—
633C22—2442no—
634C22—2442no—
635C22—2442no—
636C22—2442no—
637C22—2442no—
638C22—2442no—
639C22#1684026911yescharacteristic
640C22D4×C22⋊D6111yescharacteristic
641C22Q8×C22⋊D6111yescharacteristic
642C22C2×#1451979711yescharacteristic
643C22#1684056011yescharacteristic
644C22#1684056511yescharacteristic
645C22#1684056511yescharacteristic
646C4—22no—
647C4—41no—
648C4—42no—
649C4—41no—
650C4—41no—
651C4—41no—
652C4—41no—
653C4—81no—
654C4—81no—
655C4—4881no—
656C4—4881no—
657C4—2442no—
658C4C22⋊Q8×D6111yescharacteristic
659C4#1684035811yescharacteristic
660C4#1684041411yescharacteristic
661C4#1684046611yescharacteristic
662C2—82no—
663C2—2444no—
664C2#2144962511yescharacteristic
665C2#2145583111yescharacteristic
666C2#2145605711yescharacteristic
667C2#2145736011yescharacteristic
668C2#2145750011yescharacteristic
669C2#2145868211yescharacteristic
670C2#2145877311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2454C2
socleC23×C61C22×D4
prime radicalC22×#11758199C22
hypercenter#2454D61
hypocenterC61#26370
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2454

Complements

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

NH ≅ G/Ntype
1. C2×#16840059C2semidirect
3. #21448730C2semidirect
4. #21447340C2semidirect
5. #21447300C2semidirect
6. #21447207C2semidirect
7. #21447154C2semidirect
8. #21446549C2semidirect
9. #21446378C2semidirect
12. #21446080C2semidirect
13. #21446079C2semidirect
14. #21446065C2semidirect
15. C61×#2454C2semidirect
329. #2454D61semidirect
396. C61#26370semidirect

Maximal subgroups

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

subgroupindex
C2×#168400592
C2×C4×#117581992
#214487302
#214473402
#214473002
#214472072
#214471542
#214465492
#214463782
#214461702
#214460802
#214460792
#214460652
C61×#24542
#2637061