Finite Group Databasebeta

← #29580377

Subgroups — #29580377

Order 15616.

1008 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 1008 conjugacy classes of subgroups — too many to draw. Only its 85 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455071C211yesmaximal, characteristic, complemented
2#21450750C211yesmaximal, characteristic, complemented
3#21445763C211yesmaximal, characteristic, complemented
4#21444347C211yesmaximal, characteristic, complemented
5#21444146C211yesmaximal, characteristic, complemented
6#21443671C211yesmaximal, characteristic, complemented
7C61×#2404C211yesmaximal, characteristic, complemented
8C2×#14519802—21no—
9C2×#14519760—21no—
10C2×#14519751C2211yescharacteristic
11C2×#14519746—21no—
12#16839907—22no—
13#16839901—21no—
14#16839896—21no—
15C2×#14519704—22no—
16#16839763—21no—
17#16839726C2211yescharacteristic
18#16839726C2211yescharacteristic
19#16839698—21no—
20#16839671—21no—
21#16839662—22no—
22C4×C22⋊Dic61C2211yescharacteristic
23#16839495—22no—
24#16839486—22no—
25C22×#41×C61C2211yescharacteristic
26C2×#120×C61—21no—
27C22×C22⋊C4×C61C2211yescharacteristic
28C61×#393—21no—
29C61×#392—21no—
30C61×#386—21no—
31C61×#386—22no—
32C61×#381C2211yescharacteristic
33C22×D244D411yescharacteristic, complemented
34C22×C22⋊D61—21no—
35C22×C22⋊D61—22no—
36#14519802—44no—
37C2×#11758201—22no—
38C2×#11758199—22no—
39C2×#11758194—21no—
40C2×#11758194—21no—
41#14519760—44no—
42#14519751—21no—
43#14519751—21no—
44#14519751—21no—
45#14519751—21no—
46#14519746—44no—
47C2×C4⋊Dic61D411yescharacteristic, complemented
48C2×#11758196—22no—
49C2×C22⋊Dic61—21no—
50C2×C22⋊Dic61—22no—
51C2×#11758198—22no—
52C2×#11758198—21no—
53C2×C4×Dic61D412yescomplemented
54#14519704—44no—
55#14519701—22no—
56#14519698—22no—
57C22⋊#92418—21no—
58C22⋊#92418—21no—
59C22⋊#92418—21no—
60C22⋊#92418D412yescomplemented
61#14519684—21no—
62C25×C61D411yescharacteristic
63C2×#41×C61—21no—
64C2×#41×C61—21no—
65C2×#41×C61—21no—
66C2×#41×C61—21no—
67C2×#41×C61—44no—
68C22×Q8×C61D411yescharacteristic, complemented
69C22×D4×C61—21no—
70C22×D4×C61—22no—
71C22×D4×C61D411yescharacteristic, complemented
72C23×C4×C61—21no—
73C23×C4×C61D411yescharacteristic
74C23×C4×C61C2311yescharacteristic
75#120×C61—44no—
76C2×C22⋊C4×C61—22no—
77C2×C22⋊C4×C61—21no—
78C2×C22⋊C4×C61—21no—
79C2×C22⋊C4×C61—21no—
80C2×C22⋊C4×C61—21no—
81C2×C22⋊C4×C61—21no—
82C2×C22⋊C4×C61—21no—
83C2×C22⋊C4×C61—21no—
84C2×C22⋊C4×C61—22no—
85#95×C61—22no—
86#95×C61—21no—
87#95×C61D412yescomplemented
88C24×C61—21no—
89C24×C61—21no—
90C24×C61—21no—
91C24×C61—21no—
92C24×C61—42no—
93C24×C61—42no—
94C24×C61—42no—
95C24×C61C2×D411yescharacteristic
96C23×D61—21no—
97#41×C61—24no—
98#41×C61—24no—
99#41×C61—42no—
100#41×C61—42no—
101#41×C61—42no—
102#41×C61—42no—
103#41×C61—42no—
104#41×C61—42no—
105C2×Q8×C61—21no—
106C2×Q8×C61—21no—
107C2×Q8×C61—42no—
108C2×D4×C61—24no—
109C2×D4×C61—21no—
110C2×D4×C61—21no—
111C2×D4×C61—42no—
112C2×D4×C61—42no—
113C2×D4×C61—41no—
114C2×D4×C61—41no—
115C2×D4×C61—44no—
116C2×D4×C61—44no—
117C2×D4×C61—82no—
118C22×C4×C61—21no—
119C22×C4×C61—22no—
120C22×C4×C61—21no—
121C22×C4×C61—21no—
122C22×C4×C61—21no—
123C22×C4×C61—42no—
124C22×C4×C61—42no—
125C22×C4×C61—42no—
126C22×C4×C61—41no—
127C22×C4×C61—41no—
128C22×C4×C61—42no—
129C22×C4×C61C2×D411yescharacteristic
130C22×C4×C61C2×D411yescharacteristic
131C22×C4×C61C2×D411yescharacteristic
132C22×C4×C61C2×D411yescharacteristic
133C22×C4×C61C2×D411yescharacteristic
134C22×C4×C61#4111yescharacteristic
135C2×C22⋊D61—44no—
136C2×C22⋊D61—44no—
137C2×C22⋊D61—42no—
138C2×C22⋊D61—42no—
139C2×C22⋊D61—41no—
140C2×C22⋊D61—41no—
141C2×C22⋊D61—82no—
142C22×Dic61—22no—
143C22×Dic61—21no—
144C2×D244—21no—
145C2×D244—21no—
146C2×D244—42no—
147C22⋊C4×C61—44no—
148C22⋊C4×C61—42no—
149C22⋊C4×C61—41no—
150C22⋊C4×C61—41no—
151C22⋊C4×C61—41no—
152C22⋊C4×C61—41no—
153C22⋊C4×C61—82no—
154C22⋊Dic61—24no—
155C22⋊Dic61—44no—
156#11758201—44no—
157#11758199—44no—
158#11758198—44no—
159#11758198—41no—
160#11758198—41no—
161C4⋊Dic61—21no—
162C4⋊Dic61—21no—
163#11758196—44no—
164C4×Dic61—42no—
165#11758194—42no—
166#11758194—42no—
167C2×#92418—22no—
168C2×#92418—41no—
169C23×C61—22no—
170C23×C61—22no—
171C23×C61—22no—
172C23×C61—22no—
173C23×C61—22no—
174C23×C61—22no—
175C23×C61—41no—
176C23×C61—42no—
177C23×C61—42no—
178C23×C61—41no—
179C23×C61—42no—
180C23×C61—41no—
181C23×C61—41no—
182C23×C61—41no—
183C23×C61—41no—
184C23×C61—41no—
185C23×C61—42no—
186C23×C61—42no—
187C23×C61—42no—
188C23×C61—81no—
189C23×C61—81no—
190C23×C61—88no—
191C23×C61#12011yescharacteristic
192C23×C61#12011yescharacteristic
193C23×C61#12111yescharacteristic
194C22×D61—41no—
195C22×D61—41no—
196C22×D61—42no—
197C22×D61—41no—
198C22×D61—42no—
199Q8×C61—22no—
200Q8×C61—22no—
201Q8×C61—41no—
202Q8×C61—41no—
203D4×C61—22no—
204D4×C61—22no—
205D4×C61—41no—
206D4×C61—41no—
207D4×C61—44no—
208D4×C61—44no—
209D4×C61—88no—
210C2×C4×C61—22no—
211C2×C4×C61—22no—
212C2×C4×C61—22no—
213C2×C4×C61—41no—
214C2×C4×C61—41no—
215C2×C4×C61—41no—
216C2×C4×C61—41no—
217C2×C4×C61—44no—
218C2×C4×C61—44no—
219C2×C4×C61—41no—
220C2×C4×C61—41no—
221C2×C4×C61—42no—
222C2×C4×C61—42no—
223C2×C4×C61—42no—
224C2×C4×C61—41no—
225C2×C4×C61—42no—
226C2×C4×C61—42no—
227C2×C4×C61—81no—
228C2×C4×C61#12011yescharacteristic
229C2×C4×C61#12111yescharacteristic
230C2×C4×C61#12111yescharacteristic
231C2×C4×C61#12711yescharacteristic
232C22⋊D61—88no—
233C22⋊D61—82no—
234C22⋊D61—82no—
235C2×Dic61—24no—
236C2×Dic61—42no—
237C2×Dic61—42no—
238C2×Dic61—41no—
239C2×Dic61—41no—
240C2×Dic61—41no—
241D244—42no—
242D244—81no—
243#92418—42no—
244#92418—81no—
245#9938—611nomaximal, Sylow
246C22×C61—21no—
247C22×C61—21no—
248C22×C61—21no—
249C22×C61—21no—
250C22×C61—21no—
251C22×C61—21no—
252C22×C61—21no—
253C22×C61—44no—
254C22×C61—44no—
255C22×C61—44no—
256C22×C61—44no—
257C22×C61—42no—
258C22×C61—42no—
259C22×C61—82no—
260C22×C61—82no—
261C22×C61—82no—
262C22×C61—82no—
263C22×C61—82no—
264C22×C61—81no—
265C22×C61—81no—
266C22×C61—81no—
267C22×C61—81no—
268C22×C61#39211yescharacteristic
269C22×C61#45312yes—
270C22×C61B(4,𝔽2)12yes—
271C2×D61—82no—
272C2×D61—82no—
273C2×D61—81no—
274C2×D61—81no—
275C2×D61—81no—
276C4×C61—21no—
277C4×C61—21no—
278C4×C61—21no—
279C4×C61—21no—
280C4×C61—42no—
281C4×C61—82no—
282C4×C61—81no—
283Dic61—44no—
284Dic61—81no—
285#3006—611no—
286#2539—611no—
287#2404D6111yescharacteristic, complemented
288#1789—611no—
289#1510—611no—
290#1462—611no—
291#1284—611no—
292C2×C61—22no—
293C2×C61—22no—
294C2×C61—41no—
295C2×C61—82no—
296C2×C61—82no—
297C2×C61#200211yescharacteristic
298C2×C61#201011yescharacteristic
299C2×C61#202111yescharacteristic
300D61—161no—
301C22×#41C2×D6111yescharacteristic
302C2×AGL(1,ℤ/8)—1221no—
303C2×#123—1221no—
304C2×#121—611no—
305C2×#120—21no—
306C2×#120—1221no—
307C22×C22⋊C4C2×D6111yescharacteristic
308#449—1222no—
309#448—1221no—
310#447—1221no—
311C2×#104—1222no—
312#417—1221no—
313#407—611no—
314#407—611no—
315#399—1221no—
316#393—21no—
317#392—21no—
318#392—1221no—
319#390—1222no—
320#386—21no—
321#386—22no—
322#381C2×D6111yescharacteristic
323C4×C22⋊C4—611no—
324#327—1222no—
325#323—1222no—
326C61#993811yescharacteristic, complemented, Sylow
327C25C22⋊D6111yescharacteristic
328C2×#41—21no—
329C2×#41—21no—
330C2×#41—21no—
331C2×#41—21no—
332C2×#41—44no—
333C22×Q8C22⋊D6111yescharacteristic, complemented
334C22×D4—21no—
335C22×D4—22no—
336C22×D4—611no—
337C22×D4—1222no—
338C22×D4—1221no—
339C22×D4C22⋊D6111yescharacteristic, complemented
340C23×C4—21no—
341C23×C4C22⋊D6111yescharacteristic
342C23×C4C22×D6111yescharacteristic
343AGL(1,ℤ/8)—2444no—
344C2×SD16—1222no—
345C2×D8—1222no—
346C2×M16—1221no—
347C2×M16—1221no—
348#123—2444no—
349#121—1221no—
350#121—1221no—
351#121—1221no—
352#121—1221no—
353#120—44no—
354#120—2444no—
355C2×C4⋊C4—611no—
356C2×C4⋊C4—1222no—
357C2×C22⋊C4—22no—
358C2×C22⋊C4—21no—
359C2×C22⋊C4—21no—
360C2×C22⋊C4—21no—
361C2×C22⋊C4—21no—
362C2×C22⋊C4—21no—
363C2×C22⋊C4—21no—
364C2×C22⋊C4—21no—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—1221no—
367C2×C22⋊C4—1222no—
368C2×C22⋊C4—1222no—
369C2×C22⋊C4—1221no—
370C2×C42—612no—
371#104—2444no—
372#103—1222no—
373#102—1222no—
374C22⋊C8—612no—
375C22⋊C8—1221no—
376C22⋊C8—1221no—
377C22⋊C8—1221no—
378#95—22no—
379#95—21no—
380#95—1221no—
381#95C22⋊D6112yescomplemented
382C24—21no—
383C24—21no—
384C24—21no—
385C24—21no—
386C24—42no—
387C24—42no—
388C24—42no—
389C24—1221no—
390C24C2×C22⋊D6111yescharacteristic
391#41—24no—
392#41—24no—
393#41—42no—
394#41—42no—
395#41—42no—
396#41—42no—
397#41—42no—
398#41—42no—
399C2×Q8—21no—
400C2×Q8—21no—
401C2×Q8—42no—
402C2×D4—24no—
403C2×D4—21no—
404C2×D4—21no—
405C2×D4—42no—
406C2×D4—42no—
407C2×D4—41no—
408C2×D4—41no—
409C2×D4—44no—
410C2×D4—44no—
411C2×D4—82no—
412C2×D4—1221no—
413C2×D4—1221no—
414C2×D4—4882no—
415C2×D4—2442no—
416C2×D4—2444no—
417C2×D4—2442no—
418C2×D4—2442no—
419C2×D4—2444no—
420C2×D4—2441no—
421C2×D4—2441no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—42no—
433C22×C4—1222no—
434C22×C4—1221no—
435C22×C4D4×D6111yescharacteristic
436C22×C4D4×D6111yescharacteristic
437C22×C4D4×D6111yescharacteristic
438C22×C4#1175822411yescharacteristic
439C22×C4C2×C22⋊D6111yescharacteristic
440C22×C4C2×C22⋊D6111yescharacteristic
441SD16—2444no—
442D8—2444no—
443M16—2442no—
444M16—2442no—
445C2×C8—1222no—
446C2×C8—2441no—
447C4⋊C4—1221no—
448C4⋊C4—1221no—
449C4⋊C4—2444no—
450C22⋊C4—44no—
451C22⋊C4—42no—
452C22⋊C4—41no—
453C22⋊C4—41no—
454C22⋊C4—41no—
455C22⋊C4—41no—
456C22⋊C4—82no—
457C22⋊C4—1224no—
458C22⋊C4—2444no—
459C22⋊C4—2444no—
460C22⋊C4—2441no—
461C22⋊C4—2441no—
462C42—2442no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—22no—
468C23—22no—
469C23—41no—
470C23—42no—
471C23—42no—
472C23—41no—
473C23—42no—
474C23—41no—
475C23—41no—
476C23—41no—
477C23—42no—
478C23—41no—
479C23—41no—
480C23—42no—
481C23—42no—
482C23—81no—
483C23—81no—
484C23—88no—
485C23—2441no—
486C23—2441no—
487C23—2442no—
488C23—2442no—
489C23—2441no—
490C23#1451974511yescharacteristic
491C23#1451974611yescharacteristic
492C23#1451974911yescharacteristic
493Q8—22no—
494Q8—22no—
495Q8—41no—
496Q8—41no—
497D4—22no—
498D4—22no—
499D4—41no—
500D4—41no—
501D4—44no—
502D4—44no—
503D4—88no—
504D4—4881no—
505D4—4882no—
506D4—4882no—
507D4—4888no—
508D4—2442no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—41no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—81no—
527C2×C4—1224no—
528C2×C4—2441no—
529C2×C4—2441no—
530C2×C4—2441no—
531C2×C4—2442no—
532C2×C4—2442no—
533C2×C4#1451974511yescharacteristic
534C2×C4#1451974811yescharacteristic
535C2×C4#1451974911yescharacteristic
536C2×C4#1451977311yescharacteristic
537C8—4881no—
538C8—2442no—
539C22—21no—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—21no—
544C22—21no—
545C22—21no—
546C22—44no—
547C22—44no—
548C22—44no—
549C22—44no—
550C22—42no—
551C22—42no—
552C22—82no—
553C22—82no—
554C22—82no—
555C22—82no—
556C22—82no—
557C22—81no—
558C22—81no—
559C22—81no—
560C22—81no—
561C22—4882no—
562C22—4882no—
563C22—4881no—
564C22—4881no—
565C22—4881no—
566C22#1683967011yescharacteristic
567C22#1683993012yes—
568C22#1683994812yes—
569C4—21no—
570C4—21no—
571C4—21no—
572C4—21no—
573C4—42no—
574C4—82no—
575C4—81no—
576C4—4881no—
577C4—2444no—
578C2—22no—
579C2—22no—
580C2—41no—
581C2—82no—
582C2—82no—
583C2—9761no—
584C2#2144683211yescharacteristic
585C2#2144688411yescharacteristic
586C2#2144694211yescharacteristic
subgroupG/H
centerC22#16839670
commutator, cosocleC23×C4×C61C23
FrattiniC23×C4C22×D61
FittingC61×#2404C2
socleC22×C61#392
hypercenter#2404D61
hypocenterC61#9938
LCSG ▷ C23×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2404

Complements

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

NH ≅ G/Ntype
1. #21455071C2semidirect
2. #21450750C2semidirect
3. #21445763C2semidirect
4. #21444347C2semidirect
5. #21444146C2semidirect
6. #21443671C2semidirect
7. C61×#2404C2semidirect
33. C22×D244D4semidirect
47. C2×C4⋊Dic61D4semidirect
53. C2×C4×Dic61D4semidirect
60. C22⋊#92418D4semidirect
68. C22×Q8×C61D4semidirect
71. C22×D4×C61D4semidirect
87. #95×C61D4semidirect
287. #2404D61semidirect
326. C61#9938semidirect
333. C22×Q8C22⋊D61semidirect
339. C22×D4C22⋊D61semidirect
381. #95C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
#214550712
#214507502
#214457632
#214443472
#214441462
#214436712
C61×#24042
#993861