Finite Group Databasebeta

← #29731818

Subgroups — #29731818

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×#16840059C211yesmaximal, characteristic
2#21450993C211yesmaximal, characteristic
3#21448919C211yesmaximal, characteristic, complemented
4#21447304C211yesmaximal, characteristic, complemented
5#21447303C211yesmaximal, characteristic, complemented
6#21447202C211yesmaximal, characteristic
7#21447188C211yesmaximal, characteristic
8#21446772C211yesmaximal, characteristic, complemented
9#21446588C211yesmaximal, characteristic, complemented
10#21446584C211yesmaximal, characteristic, complemented
11#21446542C211yesmaximal, characteristic, complemented
12#21446445C211yesmaximal, characteristic
13#21446438C211yesmaximal, characteristic
14#21445645C211yesmaximal, characteristic
15C2×C61×#501C211yesmaximal, characteristic, complemented
16C2×C4×D244C2211yescharacteristic
17#16840059—24no—
18C2×#14519710C2211yescharacteristic
19C2×#14519708C2211yescharacteristic
20C2×C4⋊#92418C2211yescharacteristic
21C2×#14519706C2211yescharacteristic
22C2×#14519706C2211yescharacteristic
23C2×#14519699C2211yescharacteristic
24C2×#14519699C2211yescharacteristic
25C2×#14519697C2211yescharacteristic
26C2×#14519697C2211yescharacteristic
27#16839673—21no—
28#16839673—21no—
29#16839663C2211yescharacteristic
30#16839663C2211yescharacteristic
31#16839661—21no—
32#16839661C2211yescharacteristic
33#16839661C2211yescharacteristic
34#16839648—21no—
35#16839648—21no—
36#16839627C2211yescharacteristic
37#16839627C2211yescharacteristic
38#16839619—21no—
39#16839619—21no—
40C4⋊C4×Dic61—21no—
41C4⋊C4×Dic61C2211yescharacteristic
42C4⋊C4×Dic61C2211yescharacteristic
43C4×C4⋊Dic61C2211yescharacteristic
44C4×C4⋊Dic61C2211yescharacteristic
45#16839525C2211yescharacteristic
46#16839525C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839516C2211yescharacteristic
52#16839516C2211yescharacteristic
53C2×C4⋊Q8×C61C2211yescharacteristic
54C2×#125×C61C2211yescharacteristic
55C61×#501—24no—
56C2×#107×C61C2211yescharacteristic
57C2×#107×C61C2211yescharacteristic
58C2×#106×C61C2211yescharacteristic
59C2×#106×C61C2211yescharacteristic
60C2×#97×C61C2211yescharacteristic
61C22×D244C2311yescharacteristic
62C22×C4×D61—21no—
63C22×#92418C2311yescharacteristic
64C22×#92418C2311yescharacteristic
65C4×D244—22no—
66C4×D244—22no—
67C2×C4⋊Dic61—21no—
68C2×C4⋊Dic61—21no—
69C2×C4⋊Dic61D411yescharacteristic
70C2×C4⋊Dic61D411yescharacteristic
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C4⋊Dic61C2311yescharacteristic
73C2×C4⋊Dic61C2311yescharacteristic
74C2×#11758198—21no—
75C2×#11758198—21no—
76C2×#11758198—21no—
77C2×#11758198—21no—
78C2×#11758198—21no—
79C2×C4×Dic61—21no—
80C2×C4×Dic61—21no—
81C2×C4×Dic61—21no—
82C2×C4×Dic61C2311yescharacteristic
83C2×C4×Dic61C2311yescharacteristic
84#14519710—22no—
85#14519708—22no—
86C4⋊#92418—22no—
87#14519706—22no—
88#14519706—22no—
89#14519699—22no—
90#14519699Q814yescomplemented
91#14519697—22no—
92#14519697—22no—
93#14519684—21no—
94#14519684—21no—
95#14519684—21no—
96#14519684—21no—
97#14519684—21no—
98#14519684—21no—
99#14519684—21no—
100#14519684—21no—
101#14519684—21no—
102#14519684D411yescharacteristic
103#14519684D411yescharacteristic
104C22×Q8×C61—21no—
105C22×C8×C61C2311yescharacteristic
106C22×C8×C61C2311yescharacteristic
107C4⋊Q8×C61—22no—
108C4⋊Q8×C61—22no—
109#125×C61—22no—
110#125×C61D414yes—
111C2×C4⋊C4×C61—21no—
112C2×C4⋊C4×C61—21no—
113C2×C4⋊C4×C61—21no—
114C2×C4⋊C4×C61C2311yescharacteristic
115C2×C4⋊C4×C61C2311yescharacteristic
116C2×C4⋊C4×C61C2311yescharacteristic
117C2×C4⋊C4×C61C2311yescharacteristic
118C2×C42×C61C2311yescharacteristic
119#107×C61—22no—
120#107×C61—22no—
121#106×C61—22no—
122#106×C61—22no—
123#97×C61—22no—
124C23×D61—21no—
125C2×Q8×C61—42no—
126C2×Q8×C61—81no—
127C2×Q8×C61—81no—
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×Dic61—21no—
138C22×Dic61C2×D411yescharacteristic
139C22×Dic61C2×D411yescharacteristic
140C22×Dic61C2×D411yescharacteristic
141C22×Dic61C2×D411yescharacteristic
142C2×D244—21no—
143C2×D244—21no—
144C2×D244—21no—
145C2×D244—21no—
146C2×D244—21no—
147C2×D244C2×Q812yes—
148C2×C4×D61—42no—
149C2×C4×D61—42no—
150C2×C4×D61—42no—
151C2×C8×C61—21no—
152C2×C8×C61—21no—
153C2×C8×C61—21no—
154C2×C8×C61—21no—
155C2×C8×C61C2×Q812yes—
156C2×C8×C61#4112yes—
157C4⋊C4×C61—24no—
158C4⋊C4×C61—24no—
159C4⋊C4×C61—21no—
160C4⋊C4×C61—21no—
161C4⋊C4×C61—21no—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—42no—
166C4⋊C4×C61C2×D412yes—
167C4⋊C4×C61C2×D412yes—
168C42×C61C2×D412yes—
169C42×C61#4112yes—
170#11758198—42no—
171#11758198—81no—
172#11758198—81no—
173#11758198—81no—
174#11758198—81no—
175C4⋊Dic61—21no—
176C4⋊Dic61—21no—
177C4⋊Dic61—21no—
178C4⋊Dic61—21no—
179C4⋊Dic61—21no—
180C4⋊Dic61—41no—
181C4⋊Dic61—41no—
182C4⋊Dic61—42no—
183C4⋊Dic61—81no—
184C4⋊Dic61C2×Q812yes—
185C4×Dic61—21no—
186C4×Dic61—21no—
187C4×Dic61—21no—
188C4×Dic61—21no—
189C4×Dic61—42no—
190C4×Dic61—81no—
191C4×Dic61—81no—
192C2×#92418—21no—
193C2×#92418—21no—
194C2×#92418—21no—
195C2×#92418—21no—
196C2×#92418—21no—
197C2×#92418—21no—
198C23×C61C22×D411yescharacteristic
199C22×D61—22no—
200C22×D61—41no—
201C22×D61—41no—
202C22×D61—41no—
203C22×D61—41no—
204C22×D61—41no—
205C22×D61—41no—
206Q8×C61—84no—
207C2×C4×C61—21no—
208C2×C4×C61—21no—
209C2×C4×C61—21no—
210C2×C4×C61—21no—
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—41no—
218C2×C4×C61—41no—
219C2×C4×C61—41no—
220C2×C4×C61—41no—
221C2×C4×C61—41no—
222C2×C4×C61—41no—
223C2×C4×C61#12112yes—
224C2×C4×C61#12112yes—
225C2×C4×C61C22×D411yescharacteristic
226C2×C4×C61C22×Q811yescharacteristic
227C2×C4×C61C2×#4111yescharacteristic
228C2×C4×C61C2×#4111yescharacteristic
229C2×C4×C612−1+411yescharacteristic
230C2×C4×C612−1+411yescharacteristic
231C2×Dic61—21no—
232C2×Dic61—21no—
233C2×Dic61—21no—
234C2×Dic61—21no—
235C2×Dic61—22no—
236C2×Dic61—21no—
237C2×Dic61—21no—
238C2×Dic61—21no—
239C2×Dic61—21no—
240C2×Dic61—21no—
241C2×Dic61—21no—
242C2×Dic61—21no—
243C2×Dic61—21no—
244C2×Dic61—41no—
245C2×Dic61—41no—
246C2×Dic61—41no—
247C2×Dic61—41no—
248C2×Dic61—41no—
249C2×Dic61—41no—
250C2×Dic61—41no—
251C2×Dic61—41no—
252D244—22no—
253D244—22no—
254D244—41no—
255D244—41no—
256C4×D61—84no—
257C8×C61—22no—
258C8×C61—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—42no—
279C4×C61—42no—
280C4×C61—81no—
281C4×C61—81no—
282C4×C61#53311yescharacteristic
283C4×C61#54111yescharacteristic
284C4×C61#54811yescharacteristic
285C4×C61D4×Q811yescharacteristic
286Dic61—41no—
287Dic61—41no—
288Dic61—41no—
289Dic61—41no—
290Dic61—42no—
291Dic61—81no—
292Dic61—81no—
293C2×#501D6111yescharacteristic, complemented
294C2×#478—611no—
295#2563—611no—
296#2370—611no—
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⋊Q8C2×D6111yescharacteristic
317C2×#125C2×D6111yescharacteristic
318C2×C4×D4—611no—
319#501—24no—
320#478—1224no—
321C2×#107—611no—
322C2×#107C2×D6111yescharacteristic
323C2×#107C2×D6111yescharacteristic
324C2×#106—611no—
325C2×#106C2×D6111yescharacteristic
326C2×#106C2×D6111yescharacteristic
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×#97C2×D6111yescharacteristic
335#393—1221no—
336#393—1221no—
337#390—611no—
338#390—611no—
339#389—611no—
340#389—611no—
341#389—1221no—
342#387—1221no—
343#387—1221no—
344#382—611no—
345#382—611no—
346#380—1221no—
347#380—1221no—
348C4×C4⋊C4—611no—
349C4×C4⋊C4—611no—
350C4×C4⋊C4—611no—
351C4×C4⋊C4—611no—
352C4×C4⋊C4—1221no—
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—21no—
363C22×D4—611no—
364C23×C4—1221no—
365C22×C8—611no—
366C22×C8—611no—
367C22×C8C22×D6111yescharacteristic
368C22×C8C22×D6111yescharacteristic
369C4⋊Q8—22no—
370C4⋊Q8—22no—
371#125—22no—
372#125C22⋊D6114yes—
373C4×D4—1222no—
374C4×D4—1222no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—611no—
379C2×C4⋊C4—611no—
380C2×C4⋊C4—611no—
381C2×C4⋊C4—611no—
382C2×C4⋊C4—611no—
383C2×C4⋊C4—1221no—
384C2×C4⋊C4—1221no—
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—611no—
395C2×C42—611no—
396C2×C42—1221no—
397C2×C42—1221no—
398C2×C42—1221no—
399C2×C42C22×D6111yescharacteristic
400#107—22no—
401#107—22no—
402#107—1222no—
403#106—22no—
404#106—22no—
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—22no—
414#95—611no—
415#95—611no—
416#95—1221no—
417#95—1221no—
418#95—1221no—
419#95—1221no—
420#95—1221no—
421#95—1221no—
422#95—1221no—
423#95—1221no—
424#95—1221no—
425C24—1221no—
426C2×Q8—42no—
427C2×Q8—81no—
428C2×Q8—81no—
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—611no—
440C22×C4—611no—
441C22×C4—611no—
442C22×C4—611no—
443C22×C4—1221no—
444C22×C4—1221no—
445C22×C4—1221no—
446C22×C4—2442no—
447C22×C4—2442no—
448C22×C4—2442no—
449C22×C4D4×D6111yescharacteristic
450C22×C4D4×D6111yescharacteristic
451C22×C4C23×D6111yescharacteristic
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—1221no—
457C2×C8—1221no—
458C2×C8—1221no—
459C2×C8—1221no—
460C2×C8—1221no—
461C2×C8—1221no—
462C2×C8Q8×D6112yes—
463C2×C8#1175822612yes—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—24no—
467C4⋊C4—24no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—42no—
473C4⋊C4—612no—
474C4⋊C4—1221no—
475C4⋊C4—1221no—
476C4⋊C4—1221no—
477C4⋊C4—1221no—
478C4⋊C4—1221no—
479C4⋊C4—4881no—
480C4⋊C4—2441no—
481C4⋊C4—2441no—
482C4⋊C4—2442no—
483C4⋊C4C2×C22⋊D6112yes—
484C4⋊C4C2×C22⋊D6112yes—
485C22⋊C4—4881no—
486C22⋊C4—4881no—
487C22⋊C4—4881no—
488C22⋊C4—4881no—
489C22⋊C4—2442no—
490C42—1221no—
491C42—1221no—
492C42—1221no—
493C42—1221no—
494C42—4881no—
495C42—4881no—
496C42—2442no—
497C42#1175822412yes—
498C42C2×C22⋊D6112yes—
499C23—1222no—
500C23—2441no—
501C23—2441no—
502C23—2441no—
503C23—2441no—
504C23—2441no—
505C23—2441no—
506C23C2×D4×D6111yescharacteristic
507Q8—84no—
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—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—1221no—
529C2×C4—1221no—
530C2×C4—1221no—
531C2×C4—1221no—
532C2×C4—1221no—
533C2×C4—1221no—
534C2×C4—1221no—
535C2×C4—1221no—
536C2×C4—1221no—
537C2×C4—1221no—
538C2×C4—1221no—
539C2×C4—1221no—
540C2×C4—1222no—
541C2×C4—4884no—
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#1451974912yes—
551C2×C4#1451974912yes—
552C2×C4C22×C22⋊D6111yescharacteristic
553C2×C4C2×Q8×D6111yescharacteristic
554C2×C4C2×#1175822411yescharacteristic
555C2×C4C2×#1175822611yescharacteristic
556C2×C4#1451982511yescharacteristic
557C2×C4#1451982511yescharacteristic
558C8—22no—
559C8—22no—
560C8—2441no—
561C8—2441no—
562C22—4881no—
563C22—4881no—
564C22—4881no—
565C22—4881no—
566C22—2442no—
567C22—2442no—
568C22—2442no—
569C22C2×#1451974911yescharacteristic
570C22#1684041511yescharacteristic
571C22#1684042711yescharacteristic
572C22#1684056111yescharacteristic
573C22#1684056111yescharacteristic
574C22#1684056611yescharacteristic
575C22#1684057211yescharacteristic
576C4—22no—
577C4—22no—
578C4—42no—
579C4—42no—
580C4—81no—
581C4—81no—
582C4—4881no—
583C4—4881no—
584C4—2442no—
585C4—2441no—
586C4—2441no—
587C4—2441no—
588C4—2441no—
589C4#1684031711yescharacteristic
590C4#1684037311yescharacteristic
591C4#1684042311yescharacteristic
592C4Q8×C22⋊D6111yescharacteristic
593C2—4882no—
594C2#2144942111yescharacteristic
595C2#2145531511yescharacteristic
596C2#2145536211yescharacteristic
597C2#2145831811yescharacteristic
598C2#2145833311yescharacteristic
599C2#2145876311yescharacteristic
600C2#2145877411yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#501C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenterC2×#501D61
hypocenterC61#25466
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#501

Complements

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

NH ≅ G/Ntype
3. #21448919C2semidirect
4. #21447304C2semidirect
5. #21447303C2semidirect
8. #21446772C2semidirect
9. #21446588C2semidirect
10. #21446584C2semidirect
11. #21446542C2semidirect
15. C2×C61×#501C2semidirect
90. #14519699Q8semidirect
293. C2×#501D61semidirect
361. C61#25466semidirect

Maximal subgroups

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

subgroupindex
C2×#168400592
#214509932
#214489192
#214473042
#214473032
#214472022
#214471882
#214467722
#214465882
#214465842
#214465422
#214464452
#214464382
#214456452
C2×C61×#5012
#2546661