Finite Group Databasebeta

← #29743141

Subgroups — #29743141

Order 15616.

932 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×#16840049C211yesmaximal, characteristic, complemented
2C2×C4×#11758199C211yesmaximal, characteristic
3#21448730C211yesmaximal, characteristic, complemented
4#21447340C211yesmaximal, characteristic, complemented
5#21447300C211yesmaximal, characteristic, complemented
6#21447193C211yesmaximal, characteristic, complemented
7#21447144C211yesmaximal, characteristic, complemented
8#21446544C211yesmaximal, 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×#2452C211yesmaximal, characteristic, complemented
16C22×#11758199C2211yescharacteristic
17C2×C4×D244C2211yescharacteristic
18C2×C4×D244C2211yescharacteristic
19C2×C4×D244C2211yescharacteristic
20#16840049—24no—
21C4×#11758199—24no—
22C2×#14519710C2211yescharacteristic
23C2×#14519710C2211yescharacteristic
24C2×#14519708C2211yescharacteristic
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×C4⋊Q8×C61C2211yescharacteristic
51C2×C22⋊Q8×C61—21no—
52C2×C4×D4×C61C2211yescharacteristic
53C61×#396—21no—
54C61×#393—21no—
55C61×#390C2211yescharacteristic
56C61×#390C2211yescharacteristic
57C61×#389C2211yescharacteristic
58C61×#389C2211yescharacteristic
59C61×#385—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×#11758199Q814yescomplemented
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—41no—
88#14519708—22no—
89#14519708—22no—
90#14519708—22no—
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×Q8×C61—21no—
104C22×D4×C61C2311yescharacteristic
105C23×C4×C61—21no—
106C4⋊Q8×C61—22no—
107C4⋊Q8×C61—41no—
108C22⋊Q8×C61—44no—
109C4×D4×C61—22no—
110C4×D4×C61—41no—
111C2×C4⋊C4×C61—21no—
112C2×C4⋊C4×C61—21no—
113C2×C4⋊C4×C61—21no—
114C2×C4⋊C4×C61D411yescharacteristic
115C2×C4⋊C4×C61D411yescharacteristic
116C2×C4⋊C4×C61C2311yescharacteristic
117C2×C4⋊C4×C61C2311yescharacteristic
118C2×C4⋊C4×C61C2311yescharacteristic
119C2×C22⋊C4×C61—21no—
120C2×C22⋊C4×C61—21no—
121C2×C22⋊C4×C61—21no—
122C2×C22⋊C4×C61—21no—
123C2×C22⋊C4×C61—21no—
124C2×C42×C61C2311yescharacteristic
125C2×C42×C61C2311yescharacteristic
126C2×C42×C61C2311yescharacteristic
127#95×C61—21no—
128#95×C61—21no—
129#95×C61D411yescharacteristic
130#95×C61D411yescharacteristic
131C24×C61—21no—
132C23×D61—21no—
133C2×Q8×C61—42no—
134C2×Q8×C61—42no—
135C2×Q8×C61—81no—
136C2×D4×C61—21no—
137C2×D4×C61—21no—
138C2×D4×C61—21no—
139C2×D4×C61—21no—
140C2×D4×C61—21no—
141C2×D4×C61C2×Q812yes—
142C22×C4×C61—21no—
143C22×C4×C61—21no—
144C22×C4×C61—21no—
145C22×C4×C61—42no—
146C22×C4×C61—42no—
147C22×C4×C61—81no—
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61C2×D411yescharacteristic
153C22×C4×C61C2×D411yescharacteristic
154C22×C4×C61C2411yescharacteristic
155C22×Dic61—21no—
156C22×Dic61—21no—
157C22×Dic61—21no—
158C2×D244—21no—
159C2×D244—21no—
160C2×D244—21no—
161C2×D244C2×D412yes—
162C2×D244C2×Q812yes—
163C2×D244#4112yes—
164C2×C4×D61—24no—
165C2×C4×D61—42no—
166C2×C4×D61—42no—
167C2×C4×D61—42no—
168C2×C4×D61—42no—
169C2×C4×D61—81no—
170C2×C4×D61—81no—
171C2×C4×D61—81no—
172C2×C4×D61—81no—
173C4⋊C4×C61—22no—
174C4⋊C4×C61—22no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—42no—
181C4⋊C4×C61—42no—
182C4⋊C4×C61—42no—
183C4⋊C4×C61C2×D412yes—
184C22⋊C4×C61—42no—
185C22⋊C4×C61—42no—
186C22⋊C4×C61—42no—
187C22⋊C4×C61—42no—
188C22⋊C4×C61—81no—
189C42×C61—21no—
190C42×C61—21no—
191C42×C61—21no—
192C42×C61—21no—
193C42×C61—21no—
194C42×C61—21no—
195#11758199—28no—
196#11758198—24no—
197#11758198—42no—
198#11758198—42no—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61—21no—
203C4⋊Dic61—21no—
204C4⋊Dic61#4112yes—
205#11758196—81no—
206#11758196—81no—
207C2×#92418—21no—
208C2×#92418—21no—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418—21no—
215C2×#92418—21no—
216C2×#92418—21no—
217C2×#92418C2×D412yes—
218C2×#92418C2×Q812yes—
219C23×C61—22no—
220C23×C61—41no—
221C23×C61—41no—
222C23×C61—41no—
223C23×C61—41no—
224C23×C61—41no—
225C23×C61—41no—
226C23×C61C22×D411yescharacteristic
227C22×D61—22no—
228C22×D61—22no—
229C22×D61—22no—
230C22×D61—41no—
231C22×D61—41no—
232C22×D61—41no—
233C22×D61—41no—
234Q8×C61—84no—
235D4×C61—22no—
236D4×C61—22no—
237D4×C61—41no—
238D4×C61—41no—
239C2×C4×C61—21no—
240C2×C4×C61—21no—
241C2×C4×C61—21no—
242C2×C4×C61—21no—
243C2×C4×C61—21no—
244C2×C4×C61—21no—
245C2×C4×C61—21no—
246C2×C4×C61—21no—
247C2×C4×C61—22no—
248C2×C4×C61—21no—
249C2×C4×C61—21no—
250C2×C4×C61—21no—
251C2×C4×C61—21no—
252C2×C4×C61—21no—
253C2×C4×C61—21no—
254C2×C4×C61—21no—
255C2×C4×C61—21no—
256C2×C4×C61—41no—
257C2×C4×C61—41no—
258C2×C4×C61—41no—
259C2×C4×C61—41no—
260C2×C4×C61—41no—
261C2×C4×C61—41no—
262C2×C4×C61—41no—
263C2×C4×C61—41no—
264C2×C4×C61—84no—
265C2×C4×C61#13512yes—
266C2×C4×C61AGL(1,ℤ/8)12yes—
267C2×C4×C61C22×D411yescharacteristic
268C2×C4×C61C22×Q811yescharacteristic
269C2×C4×C61C2×#4111yescharacteristic
270C2×C4×C61C2×#4111yescharacteristic
271C2×C4×C61H(5,𝔽2)11yescharacteristic
272C2×C4×C612−1+411yescharacteristic
273C2×Dic61—22no—
274C2×Dic61—41no—
275C2×Dic61—41no—
276C2×Dic61—41no—
277C2×Dic61—41no—
278C2×Dic61—41no—
279C2×Dic61—41no—
280C2×Dic61—41no—
281C2×Dic61—41no—
282D244—22no—
283D244—22no—
284D244—22no—
285D244C22⋊Q814yescomplemented
286C4×D61—84no—
287C4×D61—84no—
288C4×D61—84no—
289#92418—22no—
290#92418—22no—
291#92418—41no—
292#92418—41no—
293#26380—611nomaximal, Sylow
294C22×C61—42no—
295C22×C61—42no—
296C22×C61—42no—
297C22×C61—81no—
298C22×C61—81no—
299C22×C61—81no—
300C22×C61—81no—
301C22×C61#52511yescharacteristic
302C22×C61D4211yescharacteristic
303C22×C61D4×Q811yescharacteristic
304C22×C61C2×#13511yescharacteristic
305C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
306C22×C61#57611yescharacteristic
307C22×C61#57711yescharacteristic
308C2×D61—24no—
309C2×D61—42no—
310C2×D61—42no—
311C2×D61—42no—
312C2×D61—42no—
313C2×D61—42no—
314C2×D61—42no—
315C4×C61—22no—
316C4×C61—22no—
317C4×C61—41no—
318C4×C61—41no—
319C4×C61—41no—
320C4×C61—42no—
321C4×C61—41no—
322C4×C61—81no—
323C4×C61—81no—
324C4×C61C2×C22⋊Q811yescharacteristic
325C4×C61#53911yescharacteristic
326C4×C61#54711yescharacteristic
327C4×C61#55411yescharacteristic
328Dic61—42no—
329Dic61—81no—
330Dic61—81no—
331C2×#476—611no—
332C2×C4×D8—611no—
333#2452D6111yescharacteristic, complemented
334#2349—611no—
335#2092—611no—
336#2085—611no—
337#2065—611no—
338#2057—611no—
339#1941—611no—
340#1916—611no—
341#1879—611no—
342#1879—611no—
343#1864—611no—
344#1864—611no—
345#1862—611no—
346C2×C61—82no—
347C2×C61#244911yescharacteristic
348C2×C61#310611yescharacteristic
349C2×C61#311811yescharacteristic
350C2×C61#326411yescharacteristic
351C2×C61#328211yescharacteristic
352C2×C61#338011yescharacteristic
353C2×C61#338411yescharacteristic
354D61—44no—
355C22×D8—611no—
356C2×C4⋊Q8C2×D6111yescharacteristic
357C2×C22⋊Q8—21no—
358C2×C4×D4—611no—
359C2×C4×D4—611no—
360C2×C4×D4—611no—
361C2×C4×D4C2×D6111yescharacteristic
362#476—1224no—
363C4×D8—1224no—
364C2×#107—611no—
365C2×#107—611no—
366C2×#106—611no—
367C2×#106—611no—
368C2×#106—611no—
369C2×C4⋊C8—611no—
370C2×#102—611no—
371C2×#102—611no—
372C2×#102—611no—
373C2×#102—611no—
374C2×#102—611no—
375C2×#102—611no—
376C2×C4×C8—611no—
377#396—21no—
378#393—21no—
379#390C2×D6111yescharacteristic
380#390C2×D6111yescharacteristic
381#389—611no—
382#389—611no—
383#389C2×D6111yescharacteristic
384#389C2×D6111yescharacteristic
385#385—21no—
386#385—1221no—
387#385—1221no—
388#382—611no—
389#380—1221no—
390C4×C4⋊C4C2×D6111yescharacteristic
391C2×#95—1221no—
392#340—611no—
393#340—611no—
394#340—611no—
395#340—611no—
396#336—611no—
397#336—611no—
398#336—611no—
399#336—611no—
400C61#2638011yescharacteristic, complemented, Sylow
401C22×Q8—21no—
402C22×D4—611no—
403C22×D4C22×D6111yescharacteristic
404C23×C4—21no—
405C23×C4—1221no—
406C23×C4—1221no—
407C23×C4—1221no—
408C2×D8—614no—
409C2×D8—1222no—
410C2×D8—1222no—
411C22×C8—611no—
412C22×C8—611no—
413C22×C8—611no—
414C22×C8—611no—
415C4⋊Q8—22no—
416C4⋊Q8—41no—
417C22⋊Q8—44no—
418C4×D4—22no—
419C4×D4—41no—
420C4×D4—1222no—
421C4×D4—1222no—
422C4×D4—1222no—
423C4×D4—1222no—
424C4×D4—2441no—
425C4×D4—2441no—
426C2×C4⋊C4—21no—
427C2×C4⋊C4—21no—
428C2×C4⋊C4—21no—
429C2×C4⋊C4—611no—
430C2×C4⋊C4—611no—
431C2×C4⋊C4—611no—
432C2×C4⋊C4—1221no—
433C2×C4⋊C4—1221no—
434C2×C4⋊C4C22⋊D6111yescharacteristic
435C2×C4⋊C4C22⋊D6111yescharacteristic
436C2×C4⋊C4C22×D6111yescharacteristic
437C2×C4⋊C4C22×D6111yescharacteristic
438C2×C4⋊C4C22×D6111yescharacteristic
439C2×C22⋊C4—21no—
440C2×C22⋊C4—21no—
441C2×C22⋊C4—21no—
442C2×C22⋊C4—21no—
443C2×C22⋊C4—21no—
444C2×C22⋊C4—1221no—
445C2×C22⋊C4—1221no—
446C2×C22⋊C4—1221no—
447C2×C42C22×D6111yescharacteristic
448C2×C42C22×D6111yescharacteristic
449C2×C42C22×D6111yescharacteristic
450#107—1222no—
451#107—2441no—
452#106—1222no—
453#106—1222no—
454#106—1222no—
455C4⋊C8—1222no—
456#102—614no—
457#102—1222no—
458#102—1222no—
459#102—1222no—
460#102—1222no—
461#102—2441no—
462C4×C8—1222no—
463#95—21no—
464#95—21no—
465#95—1221no—
466#95—1221no—
467#95—1221no—
468#95—1221no—
469#95C22⋊D6111yescharacteristic
470#95C22⋊D6111yescharacteristic
471C24—21no—
472C24—1221no—
473C2×Q8—42no—
474C2×Q8—42no—
475C2×Q8—81no—
476C2×D4—21no—
477C2×D4—21no—
478C2×D4—21no—
479C2×D4—21no—
480C2×D4—21no—
481C2×D4—612no—
482C2×D4—612no—
483C2×D4—612no—
484C2×D4—1221no—
485C2×D4—1221no—
486C2×D4—1221no—
487C2×D4Q8×D6112yes—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—42no—
492C22×C4—42no—
493C22×C4—81no—
494C22×C4—1224no—
495C22×C4—1221no—
496C22×C4—1221no—
497C22×C4—1221no—
498C22×C4—4881no—
499C22×C4—4881no—
500C22×C4—4881no—
501C22×C4—4881no—
502C22×C4—2442no—
503C22×C4—2442no—
504C22×C4—2442no—
505C22×C4—2442no—
506C22×C4C2×C22⋊D6111yescharacteristic
507C22×C4C2×C22⋊D6111yescharacteristic
508C22×C4C2×C22⋊D6111yescharacteristic
509C22×C4C2×C22⋊D6111yescharacteristic
510C22×C4C2×C22⋊D6111yescharacteristic
511C22×C4C2×C22⋊D6111yescharacteristic
512C22×C4C23×D6111yescharacteristic
513D8—1228no—
514C2×C8—612no—
515C2×C8—612no—
516C2×C8—1221no—
517C2×C8—1221no—
518C2×C8—1221no—
519C2×C8—1221no—
520C2×C8—1221no—
521C2×C8—1221no—
522C2×C8—1221no—
523C2×C8—1221no—
524C2×C8—1221no—
525C2×C8—1221no—
526C4⋊C4—21no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—21no—
532C4⋊C4—21no—
533C4⋊C4—42no—
534C4⋊C4—42no—
535C4⋊C4—42no—
536C4⋊C4—612no—
537C4⋊C4—1221no—
538C4⋊C4—1221no—
539C4⋊C4—1221no—
540C4⋊C4—1221no—
541C4⋊C4—1221no—
542C4⋊C4—4881no—
543C4⋊C4—4881no—
544C4⋊C4D4×D6112yes—
545C22⋊C4—42no—
546C22⋊C4—42no—
547C22⋊C4—42no—
548C22⋊C4—42no—
549C22⋊C4—81no—
550C22⋊C4—1224no—
551C22⋊C4—2442no—
552C22⋊C4—2442no—
553C42—21no—
554C42—21no—
555C42—21no—
556C42—21no—
557C42—21no—
558C42—21no—
559C23—22no—
560C23—41no—
561C23—41no—
562C23—41no—
563C23—41no—
564C23—41no—
565C23—41no—
566C23—1222no—
567C23—1222no—
568C23—1222no—
569C23—2441no—
570C23—2441no—
571C23—2441no—
572C23—2441no—
573C23C22×C22⋊D6111yescharacteristic
574Q8—84no—
575D4—22no—
576D4—22no—
577D4—41no—
578D4—41no—
579D4—614no—
580D4—1222no—
581D4—1222no—
582D4—1222no—
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—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—22no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—84no—
609C2×C4—1222no—
610C2×C4—4884no—
611C2×C4—4884no—
612C2×C4—4884no—
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#1451979712yes—
622C2×C4#1451980312yes—
623C2×C4C2×D4×D6111yescharacteristic
624C2×C4C2×Q8×D6111yescharacteristic
625C2×C4#41×D6111yescharacteristic
626C2×C4#41×D6111yescharacteristic
627C2×C4#1451982211yescharacteristic
628C2×C4#1451982511yescharacteristic
629C8—1222no—
630C8—1222no—
631C8—2441no—
632C8—2441no—
633C22—42no—
634C22—42no—
635C22—42no—
636C22—81no—
637C22—81no—
638C22—81no—
639C22—81no—
640C22—1224no—
641C22—2442no—
642C22—2442no—
643C22—2442no—
644C22—2442no—
645C22—2442no—
646C22—2442no—
647C22#1684026911yescharacteristic
648C22D4×C22⋊D6111yescharacteristic
649C22Q8×C22⋊D6111yescharacteristic
650C22C2×#1451979711yescharacteristic
651C22C2×#1451980311yescharacteristic
652C22#1684056511yescharacteristic
653C22#1684057011yescharacteristic
654C4—22no—
655C4—22no—
656C4—41no—
657C4—41no—
658C4—41no—
659C4—41no—
660C4—42no—
661C4—81no—
662C4—81no—
663C4—4881no—
664C4—4881no—
665C4—2442no—
666C4C22⋊Q8×D6111yescharacteristic
667C4#1684035811yescharacteristic
668C4#1684041411yescharacteristic
669C4#1684046611yescharacteristic
670C2—82no—
671C2—2444no—
672C2#2144962511yescharacteristic
673C2#2145587311yescharacteristic
674C2#2145599711yescharacteristic
675C2#2145729511yescharacteristic
676C2#2145752711yescharacteristic
677C2#2145868211yescharacteristic
678C2#2145871411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2452C2
socleC23×C61C22×D4
prime radicalC22×#11758199C22
hypercenter#2452D61
hypocenterC61#26380
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2452

Complements

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

NH ≅ G/Ntype
1. C2×#16840049C2semidirect
3. #21448730C2semidirect
4. #21447340C2semidirect
5. #21447300C2semidirect
6. #21447193C2semidirect
7. #21447144C2semidirect
8. #21446544C2semidirect
9. #21446378C2semidirect
12. #21446080C2semidirect
13. #21446079C2semidirect
14. #21446065C2semidirect
15. C61×#2452C2semidirect
67. C2×#11758199Q8semidirect
285. D244C22⋊Q8semidirect
333. #2452D61semidirect
400. C61#26380semidirect

Maximal subgroups

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

subgroupindex
C2×#168400492
C2×C4×#117581992
#214487302
#214473402
#214473002
#214471932
#214471442
#214465442
#214463782
#214461702
#214460802
#214460792
#214460652
C61×#24522
#2638061