Finite Group Databasebeta

← #29699841

Subgroups — #29699841

Order 15616.

886 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839873C211yesmaximal, characteristic
2#21448788C211yesmaximal, characteristic, complemented
3#21448764C211yesmaximal, characteristic, complemented
4#21446693C211yesmaximal, characteristic
5#21446685C211yesmaximal, characteristic
6#21446446C211yesmaximal, characteristic, complemented
7#21446444C211yesmaximal, characteristic, complemented
8#21446439C211yesmaximal, characteristic, complemented
9#21446437C211yesmaximal, characteristic, complemented
10#21445984C211yesmaximal, characteristic, complemented
11#21445828C211yesmaximal, characteristic, complemented
12#21445818C211yesmaximal, characteristic, complemented
13#21445719C211yesmaximal, characteristic, complemented
14#21445693C211yesmaximal, characteristic, complemented
15C4×#106×C61C211yesmaximal, characteristic, complemented
16C22×D488C2211yescharacteristic
17C2×#14519739C2211yescharacteristic
18C2×#14519739C2211yescharacteristic
19C2×#14519734—21no—
20#16839873C418yescomplemented
21C2×#14519711C2211yescharacteristic
22C2×#14519710C2211yescharacteristic
23C2×#14519708C2211yescharacteristic
24C2×#14519706C2211yescharacteristic
25C2×#14519699C2211yescharacteristic
26C2×#14519699C2211yescharacteristic
27C2×#14519697C2211yescharacteristic
28C2×#14519697C2211yescharacteristic
29C2×#14519697C2211yescharacteristic
30C2×#14519697C2211yescharacteristic
31C2×#14519687C2211yescharacteristic
32C2×#14519687C2211yescharacteristic
33#16839664C2211yescharacteristic
34#16839664C2211yescharacteristic
35#16839663C2211yescharacteristic
36#16839663C2211yescharacteristic
37#16839653—21no—
38#16839653—21no—
39#16839651—21no—
40#16839651—21no—
41#16839631C2211yescharacteristic
42#16839627C2211yescharacteristic
43#16839624—21no—
44#16839621—21no—
45C4⋊C4×Dic61C2211yescharacteristic
46C4⋊C4×Dic61C2211yescharacteristic
47C4×#11758196C2211yescharacteristic
48C4×#11758198—21no—
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839517C2211yescharacteristic
52#16839516C2211yescharacteristic
53C2×#106×C61C2211yescharacteristic
54C2×#106×C61C2211yescharacteristic
55C2×C4×C8×C61C2211yescharacteristic
56C4×C4⋊C4×C61C2211yescharacteristic
57C4×C4⋊C4×C61C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C61×#340C2211yescharacteristic
60C22×D244C2311yescharacteristic
61C22×D244C2311yescharacteristic
62C22×C4×D61—21no—
63C22×C4×D61—21no—
64C2×D488—22no—
65C2×D488—22no—
66C2×D488C2×C414yes—
67C22×#92418C2311yescharacteristic
68C22×#92418C2311yescharacteristic
69#14519739—22no—
70#14519739—22no—
71#14519739C2×C414yes—
72#14519739C2×C414yes—
73#14519734—44no—
74C2×C4⋊Dic61C2311yescharacteristic
75C2×C4⋊Dic61C2311yescharacteristic
76C2×#11758196—21no—
77C2×#11758196—21no—
78C2×#11758196D411yescharacteristic
79C2×#11758196D411yescharacteristic
80C2×#11758198—21no—
81C2×#11758198—21no—
82C2×#11758198—21no—
83C2×#11758198—21no—
84C2×#11758198—21no—
85C2×#11758198—21no—
86C2×#11758198—21no—
87C2×#11758198—21no—
88C2×#11758198—21no—
89C2×#11758198—21no—
90C2×C4×Dic61—21no—
91C2×C4×Dic61—21no—
92C2×C4×Dic61C2311yescharacteristic
93C2×C4×Dic61C2311yescharacteristic
94#14519711—22no—
95#14519710—41no—
96#14519708—41no—
97#14519706—22no—
98#14519699—41no—
99#14519699—41no—
100#14519697—41no—
101#14519697—41no—
102#14519697C2×C414yes—
103#14519697C2×C414yes—
104#14519687—22no—
105#14519687C2×C414yes—
106#14519684—21no—
107#14519684—21no—
108#14519684—21no—
109#14519684—21no—
110#14519684D411yescharacteristic
111#14519684D411yescharacteristic
112C22×C8×C61C2311yescharacteristic
113C22×C8×C61C2311yescharacteristic
114C2×C4⋊C4×C61C2311yescharacteristic
115C2×C4⋊C4×C61C2311yescharacteristic
116C2×C4⋊C4×C61C2311yescharacteristic
117C2×C4⋊C4×C61C2311yescharacteristic
118C2×C42×C61—21no—
119C2×C42×C61—21no—
120C2×C42×C61C2311yescharacteristic
121#106×C61—22no—
122#106×C61C2×C414yes—
123C4×C8×C61—22no—
124#95×C61—21no—
125#95×C61—21no—
126C23×D61—21no—
127C23×D61—21no—
128C22×C4×C61—21no—
129C22×C4×C61—21no—
130C22×C4×C61—21no—
131C22×C4×C61—21no—
132C22×C4×C61C2×D411yescharacteristic
133C22×C4×C61C2×D411yescharacteristic
134C22×C4×C61C2411yescharacteristic
135C22×Dic61—21no—
136C22×Dic61—21no—
137C22×Dic61C2×D411yescharacteristic
138C22×Dic61C2×D411yescharacteristic
139C22×Dic61C2×D411yescharacteristic
140C22×Dic61C2×D411yescharacteristic
141C2×D244—21no—
142C2×D244—21no—
143C2×D244—21no—
144C2×D244—21no—
145C2×D244—21no—
146C2×D244—21no—
147C2×D244—21no—
148C2×D244—21no—
149C2×D244—21no—
150C2×D244—21no—
151C2×D244C22×C412yes—
152C2×D244C22×C412yes—
153C2×C4×D61—24no—
154C2×C4×D61—24no—
155C2×C4×D61—42no—
156C2×C4×D61—42no—
157C2×C4×D61—81no—
158C2×C4×D61—81no—
159C2×C8×C61—21no—
160C2×C8×C61—21no—
161C2×C8×C61—21no—
162C2×C8×C61C22×C412yes—
163C2×C8×C61#4112yes—
164C2×C8×C61#4112yes—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—21no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61C22×C412yes—
172C4⋊C4×C61C22×C412yes—
173C42×C61—21no—
174C42×C61—21no—
175C42×C61—42no—
176C42×C61—81no—
177#11758198—24no—
178#11758198—24no—
179#11758198—42no—
180#11758198—42no—
181#11758198—42no—
182#11758198—42no—
183#11758198—81no—
184#11758198—81no—
185#11758198—81no—
186#11758198—81no—
187C4⋊Dic61—21no—
188C4⋊Dic61—21no—
189C4⋊Dic61—21no—
190C4⋊Dic61—21no—
191#11758196—22no—
192#11758196—41no—
193#11758196—42no—
194#11758196—81no—
195C4×Dic61—21no—
196C4×Dic61—21no—
197C4×Dic61—42no—
198C4×Dic61—81no—
199C4×Dic61C22×C412yes—
200C4×Dic61#4112yes—
201C2×#92418—21no—
202C2×#92418—21no—
203C2×#92418—21no—
204C2×#92418—21no—
205C2×#92418—21no—
206C2×#92418C22×C412yes—
207D488—28no—
208C23×C61C22×D411yescharacteristic
209C22×D61—22no—
210C22×D61—22no—
211C22×D61—41no—
212C22×D61—41no—
213C22×D61—41no—
214C22×D61—41no—
215C22×D61—41no—
216C22×D61—41no—
217C22×D61—41no—
218C22×D61—41no—
219C22×D61—41no—
220C22×D61—41no—
221C22×D61—41no—
222C22×D61—41no—
223C2×C4×C61—21no—
224C2×C4×C61—21no—
225C2×C4×C61—22no—
226C2×C4×C61—22no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—41no—
231C2×C4×C61—41no—
232C2×C4×C61—41no—
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×C61AGL(1,ℤ/8)12yes—
241C2×C4×C61C23×C411yescharacteristic
242C2×C4×C61C2×#4111yescharacteristic
243C2×C4×C61C2×#4111yescharacteristic
244C2×C4×C61C2×#4111yescharacteristic
245C2×C4×C61H(5,𝔽2)11yescharacteristic
246C2×C4×C612−1+411yescharacteristic
247C2×Dic61—21no—
248C2×Dic61—21no—
249C2×Dic61—21no—
250C2×Dic61—21no—
251C2×Dic61—21no—
252C2×Dic61—21no—
253C2×Dic61—21no—
254C2×Dic61—21no—
255C2×Dic61—21no—
256C2×Dic61—21no—
257C2×Dic61—41no—
258C2×Dic61—41no—
259C2×Dic61—41no—
260C2×Dic61—41no—
261C2×Dic61—41no—
262C2×Dic61—41no—
263C2×Dic61C4×D412yes—
264C2×Dic61C4×D412yes—
265D244—22no—
266D244—22no—
267D244—22no—
268D244—22no—
269D244—41no—
270D244—41no—
271D244—41no—
272D244—41no—
273C4×D61—48no—
274C4×D61—84no—
275C8×C61—41no—
276C8×C61#11714yescomplemented
277#92418—22no—
278#92418—41no—
279#22817—611nomaximal, Sylow
280C22×C61C2×C4×D411yescharacteristic
281C22×C61#54711yescharacteristic
282C22×C61#55011yescharacteristic
283C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
284C22×C61#57511yescharacteristic
285C22×C61#57611yescharacteristic
286C22×C61#57711yescharacteristic
287C2×D61—42no—
288C2×D61—42no—
289C2×D61—42no—
290C2×D61—42no—
291C2×D61—42no—
292C2×D61—42no—
293C2×D61—81no—
294C2×D61—81no—
295C2×D61—81no—
296C2×D61—81no—
297C2×D61—81no—
298C2×D61—81no—
299C2×D61—81no—
300C2×D61—81no—
301C4×C61—22no—
302C4×C61—41no—
303C4×C61—42no—
304C4×C61—42no—
305C4×C61—81no—
306C4×C61—81no—
307C4×C61C2×#11711yescharacteristic
308C4×C61#52011yescharacteristic
309C4×C61#55511yescharacteristic
310C4×C61#55611yescharacteristic
311Dic61—22no—
312Dic61—22no—
313Dic61—41no—
314Dic61—41no—
315Dic61—81no—
316Dic61—81no—
317C2×#442—611no—
318#2357—611no—
319#2354—611no—
320#1972—611no—
321#1970—611no—
322#1926—611no—
323#1926—611no—
324#1925—611no—
325#1925—611no—
326#1847—611no—
327#1804—611no—
328#1802—611no—
329#1776—611no—
330C4×#106D6111yescharacteristic, complemented
331#1771—611no—
332C2×C61#235411yescharacteristic
333C2×C61C4×AGL(1,ℤ/8)11yescharacteristic
334C2×C61#294611yescharacteristic
335C2×C61#334511yescharacteristic
336C2×C61#334711yescharacteristic
337C2×C61#339911yescharacteristic
338C2×C61#340111yescharacteristic
339D61—82no—
340D61—82no—
341C22×D8—611no—
342C2×C4×D4—611no—
343C2×C4×D4—611no—
344C2×#117—1221no—
345#442—618no—
346C2×#107—611no—
347C2×#107—611no—
348C2×#106—611no—
349C2×#106C2×D6111yescharacteristic
350C2×#106C2×D6111yescharacteristic
351C2×C4⋊C8—611no—
352C2×#102—611no—
353C2×#102—611no—
354C2×#102—611no—
355C2×#102—611no—
356C2×#102—611no—
357C2×#102—611no—
358C2×#97—611no—
359C2×#97—611no—
360C2×C4×C8C2×D6111yescharacteristic
361#390—611no—
362#390—611no—
363#390—611no—
364#390—611no—
365#388—1221no—
366#388—1221no—
367#388—1221no—
368#388—1221no—
369#384—611no—
370#382—611no—
371#381—1221no—
372#380—1221no—
373C4×C4⋊C4—611no—
374C4×C4⋊C4—611no—
375C4×C4⋊C4—611no—
376C4×C4⋊C4C2×D6111yescharacteristic
377C4×C4⋊C4C2×D6111yescharacteristic
378C4×C22⋊C4—1221no—
379#340—611no—
380#340—611no—
381#340C2×D6111yescharacteristic
382#340C2×D6111yescharacteristic
383#336—611no—
384#336—611no—
385C61#2281711yescharacteristic, complemented, Sylow
386C22×D4—611no—
387C22×D4—611no—
388C23×C4—1221no—
389C23×C4—1221no—
390C2×D8—614no—
391C2×D8—1222no—
392C2×D8—1222no—
393C22×C8—611no—
394C22×C8—611no—
395C22×C8C22×D6111yescharacteristic
396C22×C8C22×D6111yescharacteristic
397C4×D4—614no—
398C4×D4—614no—
399C4×D4—1222no—
400C4×D4—1222no—
401#117—2444no—
402C2×C4⋊C4—611no—
403C2×C4⋊C4—611no—
404C2×C4⋊C4—611no—
405C2×C4⋊C4—611no—
406C2×C4⋊C4—1221no—
407C2×C4⋊C4—1221no—
408C2×C4⋊C4C22×D6111yescharacteristic
409C2×C4⋊C4C22×D6111yescharacteristic
410C2×C4⋊C4C22×D6111yescharacteristic
411C2×C4⋊C4C22×D6111yescharacteristic
412C2×C22⋊C4—1221no—
413C2×C22⋊C4—1221no—
414C2×C22⋊C4—1221no—
415C2×C22⋊C4—1221no—
416C2×C22⋊C4—1221no—
417C2×C22⋊C4—1221no—
418C2×C22⋊C4—1221no—
419C2×C22⋊C4—1221no—
420C2×C22⋊C4—1221no—
421C2×C22⋊C4—1221no—
422C2×C42—21no—
423C2×C42—21no—
424C2×C42—611no—
425C2×C42—611no—
426C2×C42—1221no—
427C2×C42—1221no—
428C2×C42C22×D6111yescharacteristic
429#107—1222no—
430#107—2441no—
431#106—22no—
432#106—2441no—
433#106C4×D6114yes—
434C4⋊C8—1222no—
435#102—614no—
436#102—614no—
437#102—2441no—
438#102—2441no—
439#102—2441no—
440#102—2441no—
441#97—614no—
442#97—1222no—
443C4×C8—22no—
444#95—21no—
445#95—21no—
446#95—611no—
447#95—611no—
448#95—1221no—
449#95—1221no—
450#95—1221no—
451#95—1221no—
452C24—1221no—
453C24—1221no—
454C2×D4—612no—
455C2×D4—612no—
456C2×D4—1221no—
457C2×D4—1221no—
458C2×D4—1221no—
459C2×D4—1221no—
460C2×D4—1221no—
461C2×D4—1221no—
462C2×D4—1221no—
463C2×D4—1221no—
464C2×D4—1221no—
465C2×D4—1221no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—611no—
471C22×C4—611no—
472C22×C4—611no—
473C22×C4—611no—
474C22×C4—1224no—
475C22×C4—1224no—
476C22×C4—1221no—
477C22×C4—1221no—
478C22×C4—4881no—
479C22×C4—4881no—
480C22×C4—2442no—
481C22×C4—2442no—
482C22×C4D4×D6111yescharacteristic
483C22×C4D4×D6111yescharacteristic
484C22×C4C23×D6111yescharacteristic
485D8—1228no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—612no—
490C2×C8—1221no—
491C2×C8—1221no—
492C2×C8—1221no—
493C2×C8—1221no—
494C2×C8—1221no—
495C2×C8C2×C4×D6112yes—
496C2×C8#1175822212yes—
497C2×C8#1175822212yes—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—1222no—
505C4⋊C4—1221no—
506C4⋊C4—1221no—
507C4⋊C4—1221no—
508C4⋊C4—1221no—
509C4⋊C4—4881no—
510C4⋊C4—2441no—
511C4⋊C4—2442no—
512C4⋊C4C2×C4×D6112yes—
513C4⋊C4C2×C4×D6112yes—
514C22⋊C4—1224no—
515C22⋊C4—1224no—
516C22⋊C4—4881no—
517C22⋊C4—4881no—
518C22⋊C4—4881no—
519C22⋊C4—4881no—
520C22⋊C4—2442no—
521C22⋊C4—2442no—
522C22⋊C4—2442no—
523C22⋊C4—2442no—
524C42—21no—
525C42—21no—
526C42—42no—
527C42—81no—
528C42—612no—
529C42—612no—
530C42—1221no—
531C42—1221no—
532C42—4881no—
533C42—2442no—
534C23—1222no—
535C23—1222no—
536C23—2441no—
537C23—2441no—
538C23—2441no—
539C23—2441no—
540C23—2441no—
541C23—2441no—
542C23—2441no—
543C23—2441no—
544C23—2441no—
545C23—2441no—
546C23—2441no—
547C23—2441no—
548C23C2×D4×D6111yescharacteristic
549D4—1222no—
550D4—1222no—
551D4—1222no—
552D4—1222no—
553D4—2441no—
554D4—2441no—
555D4—2441no—
556D4—2441no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—41no—
565C2×C4—41no—
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—612no—
575C2×C4—612no—
576C2×C4—1221no—
577C2×C4—1221no—
578C2×C4—1221no—
579C2×C4—1221no—
580C2×C4—1221no—
581C2×C4—1221no—
582C2×C4—1221no—
583C2×C4—1221no—
584C2×C4—1221no—
585C2×C4—1221no—
586C2×C4—4884no—
587C2×C4—2448no—
588C2×C4—2441no—
589C2×C4—2441no—
590C2×C4—2441no—
591C2×C4—2441no—
592C2×C4—2441no—
593C2×C4—2441no—
594C2×C4#1451980412yes—
595C2×C4C22×C4×D6111yescharacteristic
596C2×C4#41×D6111yescharacteristic
597C2×C4C2×#1175822211yescharacteristic
598C2×C4C2×#1175822211yescharacteristic
599C2×C4#1451982311yescharacteristic
600C2×C4#1451982511yescharacteristic
601C8—41no—
602C8—1222no—
603C8—2441no—
604C8#1451973414yescomplemented
605C22—4881no—
606C22—4881no—
607C22—4881no—
608C22—4881no—
609C22—4881no—
610C22—4881no—
611C22—4881no—
612C22—4881no—
613C22—2442no—
614C22—2442no—
615C22—2442no—
616C22—2442no—
617C22—2442no—
618C22—2442no—
619C22C4×D4×D6111yescharacteristic
620C22#1684041311yescharacteristic
621C22#1684043311yescharacteristic
622C22C2×#1451980411yescharacteristic
623C22#1684055911yescharacteristic
624C22#1684056411yescharacteristic
625C22#1684056811yescharacteristic
626C4—22no—
627C4—42no—
628C4—42no—
629C4—41no—
630C4—81no—
631C4—81no—
632C4—1222no—
633C4—1222no—
634C4—4881no—
635C4—4881no—
636C4—2441no—
637C4—2441no—
638C4C2×#1451973411yescharacteristic
639C4#1684024411yescharacteristic
640C4#1684047511yescharacteristic
641C4#1684048411yescharacteristic
642C2—4882no—
643C2—4882no—
644C2#2144876411yescharacteristic
645C2C4×#1451980411yescharacteristic
646C2#2145464611yescharacteristic
647C2#2145828511yescharacteristic
648C2#2145831511yescharacteristic
649C2#2145885911yescharacteristic
650C2#2145888111yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC4×#106×C61C2
socleC23×C61C22×D4
prime radicalC22×D488C22
cosocleC22×C4×C61C24
hypercenterC4×#106D61
hypocenterC61#22817
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C4×#106

Complements

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

NH ≅ G/Ntype
2. #21448788C2semidirect
3. #21448764C2semidirect
6. #21446446C2semidirect
7. #21446444C2semidirect
8. #21446439C2semidirect
9. #21446437C2semidirect
10. #21445984C2semidirect
11. #21445828C2semidirect
12. #21445818C2semidirect
13. #21445719C2semidirect
14. #21445693C2semidirect
15. C4×#106×C61C2semidirect
20. #16839873C4semidirect
276. C8×C61#117semidirect
330. C4×#106D61semidirect
385. C61#22817semidirect
604. C8#14519734semidirect

Maximal subgroups

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

subgroupindex
C2×#168398732
#214487882
#214487642
#214466932
#214466852
#214464462
#214464442
#214464392
#214464372
#214459842
#214458282
#214458182
#214457192
#214456932
C4×#106×C612
#2281761