Finite Group Databasebeta

← #29731806

Subgroups — #29731806

Order 15616.

756 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840172C211yesmaximal, characteristic, complemented
2C2×#16840059C211yesmaximal, characteristic
3#21450988C211yesmaximal, characteristic
4#21447300C211yesmaximal, characteristic, complemented
5#21447300C211yesmaximal, characteristic, complemented
6#21447204C211yesmaximal, characteristic
7#21447190C211yesmaximal, characteristic
8#21446771C211yesmaximal, characteristic, complemented
9#21446587C211yesmaximal, characteristic, complemented
10#21446587C211yesmaximal, characteristic, complemented
11#21446543C211yesmaximal, characteristic, complemented
12#21446446C211yesmaximal, characteristic
13#21446439C211yesmaximal, characteristic
14#21445647C211yesmaximal, characteristic
15C61×#2370C211yesmaximal, characteristic, complemented
16C2×#14519777C2211yescharacteristic
17C2×C4×D244C2211yescharacteristic
18#16840172—24no—
19#16840059—24no—
20C2×#14519710C2211yescharacteristic
21C2×#14519710C2211yescharacteristic
22C2×#14519710C2211yescharacteristic
23C2×#14519708C2211yescharacteristic
24C2×#14519708C2211yescharacteristic
25C2×#14519708C2211yescharacteristic
26C2×C4⋊#92418C2211yescharacteristic
27C2×C4⋊#92418C2211yescharacteristic
28C2×C4⋊#92418C2211yescharacteristic
29C2×#14519697C2211yescharacteristic
30C2×#14519697C2211yescharacteristic
31C2×#14519697C2211yescharacteristic
32C2×#14519697C2211yescharacteristic
33C2×#14519688C2211yescharacteristic
34#16839675—21no—
35#16839675—21no—
36#16839664C2211yescharacteristic
37#16839664C2211yescharacteristic
38#16839628C2211yescharacteristic
39#16839628C2211yescharacteristic
40#16839619—21no—
41#16839619—21no—
42#16839524C2211yescharacteristic
43#16839524C2211yescharacteristic
44#16839524C2211yescharacteristic
45#16839524C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839517C2211yescharacteristic
49#16839517C2211yescharacteristic
50C2×#125×C61C2211yescharacteristic
51C61×#389—21no—
52C61×#389C2211yescharacteristic
53C61×#389C2211yescharacteristic
54C61×#387—21no—
55C61×#387—21no—
56C4×C4⋊C4×C61—21no—
57C4×C4⋊C4×C61C2211yescharacteristic
58C4×C4⋊C4×C61C2211yescharacteristic
59C4×C4⋊C4×C61C2211yescharacteristic
60C4×C4⋊C4×C61C2211yescharacteristic
61C22×Dic122—21no—
62C22×D244C2311yescharacteristic
63C22×C4×D61—21no—
64C22×#92418C2311yescharacteristic
65C22×#92418C2311yescharacteristic
66C22×#92418C2311yescharacteristic
67C22×#92418C2311yescharacteristic
68#14519777—22no—
69#14519777—22no—
70C4×D244—22no—
71C4×D244—22no—
72C2×C4⋊Dic61—21no—
73C2×C4⋊Dic61C2311yescharacteristic
74C2×C4⋊Dic61C2311yescharacteristic
75C2×C4⋊Dic61C2311yescharacteristic
76C2×#11758198—21no—
77C2×#11758198—21no—
78C2×#11758198—21no—
79C2×#11758198—21no—
80C2×#11758198—21no—
81#14519710—22no—
82#14519710—22no—
83#14519710—22no—
84#14519708—22no—
85#14519708—22no—
86#14519708—22no—
87C4⋊#92418—22no—
88C4⋊#92418—22no—
89C4⋊#92418—22no—
90#14519697—22no—
91#14519697—22no—
92#14519697—22no—
93#14519697Q814yescomplemented
94#14519688—22no—
95#14519684—21no—
96#14519684—21no—
97#14519684—21no—
98#14519684—21no—
99#125×C61—22no—
100#125×C61D414yes—
101C2×C4⋊C4×C61—21no—
102C2×C4⋊C4×C61—21no—
103C2×C4⋊C4×C61—21no—
104C2×C4⋊C4×C61—21no—
105C2×C4⋊C4×C61D411yescharacteristic
106C2×C4⋊C4×C61D411yescharacteristic
107C2×C4⋊C4×C61C2311yescharacteristic
108C2×C4⋊C4×C61C2311yescharacteristic
109C2×C4⋊C4×C61C2311yescharacteristic
110C2×C4⋊C4×C61C2311yescharacteristic
111C2×C42×C61—21no—
112C2×C42×C61—21no—
113C2×C42×C61—21no—
114C2×C42×C61C2311yescharacteristic
115C2×C42×C61C2311yescharacteristic
116C2×C42×C61C2311yescharacteristic
117#95×C61—21no—
118#95×C61—21no—
119#95×C61—21no—
120#95×C61—21no—
121#95×C61—21no—
122#95×C61D411yescharacteristic
123#95×C61D411yescharacteristic
124C23×D61—21no—
125C22×C4×C61—21no—
126C22×C4×C61—21no—
127C22×C4×C61—21no—
128C22×C4×C61—21no—
129C22×C4×C61C2×D411yescharacteristic
130C22×C4×C61C2×D411yescharacteristic
131C22×C4×C61C2×D411yescharacteristic
132C22×C4×C61C2×D411yescharacteristic
133C22×C4×C61C2×D411yescharacteristic
134C22×C4×C61C2×D411yescharacteristic
135C22×C4×C61C2411yescharacteristic
136C22×Dic61—21no—
137C22×Dic61—21no—
138C22×Dic61—21no—
139C2×D244—21no—
140C2×D244—21no—
141C2×D244—21no—
142C2×D244—21no—
143C2×D244—21no—
144C2×D244C2×Q812yes—
145C2×C4×D61—42no—
146C2×C4×D61—42no—
147C2×C4×D61—42no—
148C2×Dic122—42no—
149C2×Dic122—81no—
150C2×Dic122—81no—
151C4⋊C4×C61—24no—
152C4⋊C4×C61—24no—
153C4⋊C4×C61—21no—
154C4⋊C4×C61—21no—
155C4⋊C4×C61—21no—
156C4⋊C4×C61—21no—
157C4⋊C4×C61—21no—
158C4⋊C4×C61—41no—
159C4⋊C4×C61—41no—
160C4⋊C4×C61—42no—
161C4⋊C4×C61—81no—
162C4⋊C4×C61C2×D412yes—
163C4⋊C4×C61C2×D412yes—
164C4⋊C4×C61C2×Q812yes—
165C42×C61—21no—
166C42×C61—21no—
167C42×C61—21no—
168C42×C61—21no—
169C42×C61—42no—
170C42×C61—81no—
171C42×C61—81no—
172C42×C61C2×D412yes—
173C42×C61#4112yes—
174#11758198—42no—
175#11758198—81no—
176#11758198—81no—
177#11758198—81no—
178#11758198—81no—
179C4⋊Dic61—21no—
180C4⋊Dic61—21no—
181C4⋊Dic61—21no—
182C4⋊Dic61—21no—
183C4⋊Dic61—21no—
184C4⋊Dic61—21no—
185C4⋊Dic61—42no—
186C2×#92418—21no—
187C2×#92418—21no—
188C2×#92418—21no—
189C2×#92418—21no—
190C2×#92418—21no—
191C2×#92418—21no—
192C2×#92418—21no—
193C2×#92418—21no—
194C2×#92418—21no—
195C2×#92418—21no—
196C2×#92418C2×Q812yes—
197C2×#92418#4112yes—
198C23×C61C22×D411yescharacteristic
199C22×D61—22no—
200C22×D61—41no—
201C22×D61—41no—
202C22×D61—41no—
203C22×D61—41no—
204C22×D61—41no—
205C22×D61—41no—
206C2×C4×C61—21no—
207C2×C4×C61—21no—
208C2×C4×C61—21no—
209C2×C4×C61—21no—
210C2×C4×C61—22no—
211C2×C4×C61—22no—
212C2×C4×C61—21no—
213C2×C4×C61—21no—
214C2×C4×C61—21no—
215C2×C4×C61—21no—
216C2×C4×C61—22no—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—21no—
225C2×C4×C61—41no—
226C2×C4×C61—41no—
227C2×C4×C61—41no—
228C2×C4×C61—41no—
229C2×C4×C61—41no—
230C2×C4×C61—41no—
231C2×C4×C61—41no—
232C2×C4×C61—41no—
233C2×C4×C61—41no—
234C2×C4×C61#12112yes—
235C2×C4×C61#12112yes—
236C2×C4×C61C22×D411yescharacteristic
237C2×C4×C61C22×Q811yescharacteristic
238C2×C4×C61C2×#4111yescharacteristic
239C2×C4×C61C2×#4111yescharacteristic
240C2×C4×C612−1+411yescharacteristic
241C2×C4×C612−1+411yescharacteristic
242C2×Dic61—41no—
243C2×Dic61—41no—
244C2×Dic61—41no—
245C2×Dic61—41no—
246C2×Dic61—41no—
247C2×Dic61—41no—
248C2×Dic61—41no—
249C2×Dic61—41no—
250C2×Dic61—41no—
251D244—22no—
252D244—22no—
253D244—41no—
254D244—41no—
255C4×D61—84no—
256Dic122—84no—
257#92418—22no—
258#92418—22no—
259#92418—41no—
260#92418—41no—
261#25466—611nomaximal, Sylow
262C22×C61C2×#12111yescharacteristic
263C22×C61#54711yescharacteristic
264C22×C61D4×Q811yescharacteristic
265C22×C61#57511yescharacteristic
266C22×C61#57511yescharacteristic
267C22×C61#57611yescharacteristic
268C22×C61#57711yescharacteristic
269C2×D61—42no—
270C2×D61—42no—
271C2×D61—42no—
272C2×D61—81no—
273C2×D61—81no—
274C2×D61—81no—
275C2×D61—81no—
276C4×C61—22no—
277C4×C61—22no—
278C4×C61—41no—
279C4×C61—41no—
280C4×C61—41no—
281C4×C61—41no—
282C4×C61—42no—
283C4×C61—42no—
284C4×C61—42no—
285C4×C61—81no—
286C4×C61#53311yescharacteristic
287C4×C61#54111yescharacteristic
288C4×C61#54811yescharacteristic
289C4×C61D4×Q811yescharacteristic
290Dic61—81no—
291Dic61—81no—
292Dic61—81no—
293C2×#501—611no—
294C2×#478—611no—
295#2563—611no—
296#2370D6111yescharacteristic, complemented
297#2085—611no—
298#2085—611no—
299#2067—611no—
300#2065—611no—
301#1988—611no—
302#1949—611no—
303#1949—611no—
304#1940—611no—
305#1926—611no—
306#1925—611no—
307#1762—611no—
308C2×C61#242911yescharacteristic
309C2×C61#304011yescharacteristic
310C2×C61#304511yescharacteristic
311C2×C61#334711yescharacteristic
312C2×C61#334811yescharacteristic
313C2×C61#338811yescharacteristic
314C2×C61#338911yescharacteristic
315D61—82no—
316C2×C4⋊Q8—611no—
317C2×#125C2×D6111yescharacteristic
318C2×C4×D4—611no—
319#501—1224no—
320#478—1224no—
321C2×#107—611no—
322C2×#107—611no—
323C2×#107—611no—
324C2×#106—611no—
325C2×#106—611no—
326C2×#106—611no—
327C2×C4⋊C8—611no—
328C2×C4⋊C8—611no—
329C2×C4⋊C8—611no—
330C2×#102—611no—
331C2×#102—611no—
332C2×#102—611no—
333C2×#102—611no—
334C2×#97—611no—
335#393—1221no—
336#393—1221no—
337#390—611no—
338#390—611no—
339#389—21no—
340#389C2×D6111yescharacteristic
341#389C2×D6111yescharacteristic
342#387—21no—
343#387—21no—
344#382—611no—
345#382—611no—
346#380—1221no—
347#380—1221no—
348C4×C4⋊C4—21no—
349C4×C4⋊C4C2×D6111yescharacteristic
350C4×C4⋊C4C2×D6111yescharacteristic
351C4×C4⋊C4C2×D6111yescharacteristic
352C4×C4⋊C4C2×D6111yescharacteristic
353#340—611no—
354#340—611no—
355#340—611no—
356#340—611no—
357#340—611no—
358#340—611no—
359#336—611no—
360#336—611no—
361C61#2546611yescharacteristic, complemented, Sylow
362C22×Q8—1221no—
363C22×D4—611no—
364C23×C4—1221no—
365C22×C8—611no—
366C22×C8—611no—
367C22×C8—611no—
368C22×C8—611no—
369C4⋊Q8—1222no—
370C4⋊Q8—1222no—
371#125—22no—
372#125D24414yes—
373C4×D4—1222no—
374C4×D4—1222no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—611no—
380C2×C4⋊C4—611no—
381C2×C4⋊C4—611no—
382C2×C4⋊C4—1221no—
383C2×C4⋊C4C22⋊D6111yescharacteristic
384C2×C4⋊C4C22⋊D6111yescharacteristic
385C2×C4⋊C4C22×D6111yescharacteristic
386C2×C4⋊C4C22×D6111yescharacteristic
387C2×C4⋊C4C22×D6111yescharacteristic
388C2×C4⋊C4C22×D6111yescharacteristic
389C2×C22⋊C4—1221no—
390C2×C22⋊C4—1221no—
391C2×C22⋊C4—1221no—
392C2×C22⋊C4—1221no—
393C2×C22⋊C4—1221no—
394C2×C42—21no—
395C2×C42—21no—
396C2×C42—21no—
397C2×C42C22×D6111yescharacteristic
398C2×C42C22×D6111yescharacteristic
399C2×C42C22×D6111yescharacteristic
400#107—1222no—
401#107—1222no—
402#107—1222no—
403#106—1222no—
404#106—1222no—
405#106—1222no—
406C4⋊C8—1222no—
407C4⋊C8—1222no—
408C4⋊C8—1222no—
409#102—614no—
410#102—1222no—
411#102—1222no—
412#102—1222no—
413#97—1222no—
414#95—21no—
415#95—21no—
416#95—21no—
417#95—21no—
418#95—21no—
419#95—1221no—
420#95—1221no—
421#95—1221no—
422#95—1221no—
423#95C22⋊D6111yescharacteristic
424#95C22⋊D6111yescharacteristic
425C24—1221no—
426C2×Q8—4881no—
427C2×Q8—4881no—
428C2×Q8—2442no—
429C2×D4—612no—
430C2×D4—1221no—
431C2×D4—1221no—
432C2×D4—1221no—
433C2×D4—1221no—
434C2×D4—1221no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—1221no—
440C22×C4—1221no—
441C22×C4—1221no—
442C22×C4—2442no—
443C22×C4—2442no—
444C22×C4—2442no—
445C22×C4C2×C22⋊D6111yescharacteristic
446C22×C4C2×C22⋊D6111yescharacteristic
447C22×C4C2×C22⋊D6111yescharacteristic
448C22×C4C2×C22⋊D6111yescharacteristic
449C22×C4C2×C22⋊D6111yescharacteristic
450C22×C4C2×C22⋊D6111yescharacteristic
451C22×C4C23×D6111yescharacteristic
452C2×C8—612no—
453C2×C8—612no—
454C2×C8—1221no—
455C2×C8—1221no—
456C2×C8—1221no—
457C2×C8—1221no—
458C2×C8—1221no—
459C2×C8—1221no—
460C2×C8—1221no—
461C2×C8—1221no—
462C2×C8—1221no—
463C2×C8—1221no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—24no—
468C4⋊C4—24no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—41no—
472C4⋊C4—41no—
473C4⋊C4—42no—
474C4⋊C4—81no—
475C4⋊C4—1221no—
476C4⋊C4—1221no—
477C4⋊C4—1221no—
478C4⋊C4—1221no—
479C4⋊C4—1221no—
480C4⋊C4—1221no—
481C4⋊C4—2442no—
482C4⋊C4C2×D24412yes—
483C4⋊C4C2×D24412yes—
484C4⋊C4Q8×D6112yes—
485C22⋊C4—4881no—
486C22⋊C4—4881no—
487C22⋊C4—4881no—
488C22⋊C4—4881no—
489C22⋊C4—2442no—
490C42—21no—
491C42—21no—
492C42—21no—
493C42—21no—
494C42—42no—
495C42—81no—
496C42—81no—
497C42C2×D24412yes—
498C42#1175822212yes—
499C23—1222no—
500C23—2441no—
501C23—2441no—
502C23—2441no—
503C23—2441no—
504C23—2441no—
505C23—2441no—
506C23C22×C22⋊D6111yescharacteristic
507Q8—4884no—
508D4—1222no—
509D4—1222no—
510D4—2441no—
511D4—2441no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—22no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—4884no—
541C2×C4—2441no—
542C2×C4—2441no—
543C2×C4—2441no—
544C2×C4—2441no—
545C2×C4—2441no—
546C2×C4—2441no—
547C2×C4—2441no—
548C2×C4—2441no—
549C2×C4—2441no—
550C2×C4#1451975112yes—
551C2×C4#1451975112yes—
552C2×C4C22×D24411yescharacteristic
553C2×C4C2×Q8×D6111yescharacteristic
554C2×C4#41×D6111yescharacteristic
555C2×C4C2×#1175822211yescharacteristic
556C2×C4#1451982411yescharacteristic
557C2×C4#1451982511yescharacteristic
558C8—1222no—
559C8—1222no—
560C8—2441no—
561C8—2441no—
562C22—4881no—
563C22—4881no—
564C22—4881no—
565C22—4881no—
566C22—2442no—
567C22—2442no—
568C22—2442no—
569C22C2×#1451975111yescharacteristic
570C22#1684041111yescharacteristic
571C22Q8×C22⋊D6111yescharacteristic
572C22#1684056011yescharacteristic
573C22#1684056011yescharacteristic
574C22#1684056511yescharacteristic
575C22#1684057011yescharacteristic
576C4—22no—
577C4—22no—
578C4—42no—
579C4—41no—
580C4—41no—
581C4—41no—
582C4—41no—
583C4—42no—
584C4—42no—
585C4—81no—
586C4—4881no—
587C4—4881no—
588C4—4881no—
589C4#1684031611yescharacteristic
590C4#1684037011yescharacteristic
591C4#1684042411yescharacteristic
592C4Q8×D24411yescharacteristic
593C2—4882no—
594C2#2144941211yescharacteristic
595C2#2145531311yescharacteristic
596C2#2145535811yescharacteristic
597C2#2145831311yescharacteristic
598C2#2145832811yescharacteristic
599C2#2145875811yescharacteristic
600C2#2145877311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2370C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenter#2370D61
hypocenterC61#25466
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2370

Complements

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

NH ≅ G/Ntype
1. C2×#16840172C2semidirect
4. #21447300C2semidirect
5. #21447300C2semidirect
8. #21446771C2semidirect
9. #21446587C2semidirect
10. #21446587C2semidirect
11. #21446543C2semidirect
15. C61×#2370C2semidirect
93. #14519697Q8semidirect
296. #2370D61semidirect
361. C61#25466semidirect

Maximal subgroups

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

subgroupindex
C2×#168401722
C2×#168400592
#214509882
#214473002
#214472042
#214471902
#214467712
#214465872
#214465432
#214464462
#214464392
#214456472
C61×#23702
#2546661