Finite Group Databasebeta

← #29731808

Subgroups — #29731808

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×#16840168C211yesmaximal, characteristic, complemented
2C2×#16840062C211yesmaximal, characteristic
3#21448913C211yesmaximal, characteristic, complemented
4#21447301C211yesmaximal, characteristic, complemented
5#21447301C211yesmaximal, characteristic, complemented
6#21447207C211yesmaximal, characteristic
7#21447193C211yesmaximal, characteristic
8#21446772C211yesmaximal, characteristic, complemented
9#21446585C211yesmaximal, characteristic, complemented
10#21446585C211yesmaximal, characteristic, complemented
11#21446543C211yesmaximal, characteristic, complemented
12#21446443C211yesmaximal, characteristic
13#21446436C211yesmaximal, characteristic
14#21445644C211yesmaximal, characteristic
15C61×#2563C211yesmaximal, characteristic
16C2×#14519769C2211yescharacteristic
17#16840168—24no—
18#16840062—24no—
19C2×#14519710C2211yescharacteristic
20C2×#14519710C2211yescharacteristic
21C2×#14519710C2211yescharacteristic
22C2×#14519708C2211yescharacteristic
23C2×#14519708C2211yescharacteristic
24C2×#14519708C2211yescharacteristic
25C2×C4⋊#92418C2211yescharacteristic
26C2×C4⋊#92418C2211yescharacteristic
27C2×C4⋊#92418C2211yescharacteristic
28C2×#14519698C2211yescharacteristic
29C2×#14519698C2211yescharacteristic
30C2×#14519698C2211yescharacteristic
31C2×#14519698C2211yescharacteristic
32C2×#14519688C2211yescharacteristic
33#16839660C2211yescharacteristic
34#16839660C2211yescharacteristic
35#16839657—21no—
36#16839644—21no—
37#16839643—21no—
38C4×C4⋊Dic61C2211yescharacteristic
39C4×C4⋊Dic61C2211yescharacteristic
40C4×C4⋊Dic61C2211yescharacteristic
41#16839613—21no—
42#16839524C2211yescharacteristic
43#16839524C2211yescharacteristic
44#16839524C2211yescharacteristic
45#16839524C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839517C2211yescharacteristic
49#16839517C2211yescharacteristic
50C2×C4⋊Q8×C61C2211yescharacteristic
51C2×C4×D4×C61C2211yescharacteristic
52C61×#393—21no—
53C61×#393—21no—
54C61×#390C2211yescharacteristic
55C61×#390C2211yescharacteristic
56C61×#382C2211yescharacteristic
57C61×#382C2211yescharacteristic
58C61×#380—21no—
59C61×#380—21no—
60C4×C4⋊C4×C61C2211yescharacteristic
61C22×#92418C2311yescharacteristic
62C22×#92418C2311yescharacteristic
63C22×#92418C2311yescharacteristic
64C22×#92418C2311yescharacteristic
65#14519769—22no—
66#14519769D414yes—
67C2×C4⋊Dic61C2311yescharacteristic
68C2×C4⋊Dic61C2311yescharacteristic
69C2×C4⋊Dic61C2311yescharacteristic
70C2×C4⋊Dic61C2311yescharacteristic
71C2×#11758196—21no—
72C2×#11758196—21no—
73C2×#11758196—21no—
74C2×#11758196—21no—
75C2×C4×Dic61—21no—
76C2×C4×Dic61—21no—
77C2×C4×Dic61—21no—
78#14519710—22no—
79#14519710—22no—
80#14519710—22no—
81#14519708—22no—
82#14519708—22no—
83#14519708—22no—
84C4⋊#92418—22no—
85C4⋊#92418—22no—
86C4⋊#92418—22no—
87#14519698—22no—
88#14519698—22no—
89#14519698—22no—
90#14519698Q814yescomplemented
91#14519688—22no—
92#14519684—21no—
93#14519684—21no—
94#14519684—21no—
95#14519684—21no—
96#14519684—21no—
97C22×Q8×C61—21no—
98C22×D4×C61C2311yescharacteristic
99C23×C4×C61—21no—
100C4⋊Q8×C61—22no—
101C4⋊Q8×C61—22no—
102C4×D4×C61—22no—
103C4×D4×C61—22no—
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×C22⋊C4×C61—21no—
111C2×C22⋊C4×C61—21no—
112C2×C22⋊C4×C61—21no—
113C2×C22⋊C4×C61—21no—
114C2×C22⋊C4×C61—21no—
115C2×C42×C61C2311yescharacteristic
116C2×C42×C61C2311yescharacteristic
117C2×C42×C61C2311yescharacteristic
118#95×C61—21no—
119#95×C61—21no—
120#95×C61—21no—
121#95×C61—21no—
122#95×C61D411yescharacteristic
123#95×C61D411yescharacteristic
124C24×C61—21no—
125C2×Q8×C61—42no—
126C2×Q8×C61—81no—
127C2×Q8×C61—81no—
128C2×D4×C61—21no—
129C2×D4×C61—21no—
130C2×D4×C61—21no—
131C2×D4×C61—21no—
132C2×D4×C61—21no—
133C2×D4×C61C2×Q812yes—
134C22×C4×C61—21no—
135C22×C4×C61—21no—
136C22×C4×C61—21no—
137C22×C4×C61—42no—
138C22×C4×C61—42no—
139C22×C4×C61—42no—
140C22×C4×C61C2×D411yescharacteristic
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2411yescharacteristic
147C22×Dic61—21no—
148C22×Dic61—21no—
149C22×Dic61—21no—
150C22×Dic61—21no—
151C4⋊C4×C61—21no—
152C4⋊C4×C61—21no—
153C4⋊C4×C61—21no—
154C4⋊C4×C61—21no—
155C4⋊C4×C61—21no—
156C4⋊C4×C61—21no—
157C4⋊C4×C61—41no—
158C4⋊C4×C61—41no—
159C4⋊C4×C61—42no—
160C22⋊C4×C61—42no—
161C22⋊C4×C61—81no—
162C22⋊C4×C61—81no—
163C22⋊C4×C61—81no—
164C22⋊C4×C61—81no—
165C42×C61—21no—
166C42×C61—21no—
167C42×C61—21no—
168C42×C61—21no—
169C42×C61C2×D412yes—
170C42×C61#4112yes—
171C4⋊Dic61—21no—
172C4⋊Dic61—21no—
173C4⋊Dic61—21no—
174C4⋊Dic61—21no—
175C4⋊Dic61—21no—
176C4⋊Dic61C2×D412yes—
177C4⋊Dic61C2×D412yes—
178C4⋊Dic61C2×Q812yes—
179#11758196—24no—
180#11758196—24no—
181#11758196—42no—
182#11758196—81no—
183C4×Dic61—42no—
184C4×Dic61—81no—
185C4×Dic61—81no—
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×C61—22no—
199C23×C61—41no—
200C23×C61—41no—
201C23×C61—41no—
202C23×C61—41no—
203C23×C61—41no—
204C23×C61—41no—
205C23×C61C22×D411yescharacteristic
206Q8×C61—84no—
207D4×C61—22no—
208D4×C61—22no—
209D4×C61—41no—
210D4×C61—41no—
211C2×C4×C61—21no—
212C2×C4×C61—21no—
213C2×C4×C61—21no—
214C2×C4×C61—21no—
215C2×C4×C61—21no—
216C2×C4×C61—21no—
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—21no—
226C2×C4×C61—21no—
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—41no—
235C2×C4×C61—41no—
236C2×C4×C61—84no—
237C2×C4×C61#12112yes—
238C2×C4×C61#12112yes—
239C2×C4×C61C22×D411yescharacteristic
240C2×C4×C61C22×Q811yescharacteristic
241C2×C4×C61C2×#4111yescharacteristic
242C2×C4×C61C2×#4111yescharacteristic
243C2×C4×C612−1+411yescharacteristic
244C2×C4×C612−1+411yescharacteristic
245C2×Dic61—22no—
246C2×Dic61—22no—
247C2×Dic61—22no—
248C2×Dic61—41no—
249C2×Dic61—41no—
250C2×Dic61—41no—
251C2×Dic61—41no—
252C2×Dic61—41no—
253C2×Dic61—41no—
254C2×Dic61—41no—
255C2×Dic61—41no—
256C2×Dic61—41no—
257#92418—22no—
258#92418—22no—
259#92418—41no—
260#92418—41no—
261#25466—611nomaximal, Sylow
262C22×C61—42no—
263C22×C61—42no—
264C22×C61—42no—
265C22×C61—81no—
266C22×C61—81no—
267C22×C61—81no—
268C22×C61—81no—
269C22×C61C2×#12111yescharacteristic
270C22×C61#54711yescharacteristic
271C22×C61D4×Q811yescharacteristic
272C22×C61#57511yescharacteristic
273C22×C61#57511yescharacteristic
274C22×C61#57611yescharacteristic
275C22×C61#57711yescharacteristic
276C4×C61—22no—
277C4×C61—22no—
278C4×C61—41no—
279C4×C61—41no—
280C4×C61—41no—
281C4×C61—41no—
282C4×C61—81no—
283C4×C61—81no—
284C4×C61—81no—
285C4×C61#53311yescharacteristic
286C4×C61#54111yescharacteristic
287C4×C61#54811yescharacteristic
288C4×C61D4×Q811yescharacteristic
289Dic61—42no—
290Dic61—42no—
291Dic61—42no—
292Dic61—81no—
293C2×#501—611no—
294C2×#478—611no—
295#2563D6111yescharacteristic
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—82no—
309C2×C61#242911yescharacteristic
310C2×C61#304011yescharacteristic
311C2×C61#304511yescharacteristic
312C2×C61#334711yescharacteristic
313C2×C61#334811yescharacteristic
314C2×C61#338811yescharacteristic
315C2×C61#338911yescharacteristic
316C2×C4⋊Q8C2×D6111yescharacteristic
317C2×#125—611no—
318C2×C4×D4C2×D6111yescharacteristic
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—21no—
336#393—21no—
337#390C2×D6111yescharacteristic
338#390C2×D6111yescharacteristic
339#389—611no—
340#389—611no—
341#389—1221no—
342#387—1221no—
343#387—1221no—
344#382C2×D6111yescharacteristic
345#382C2×D6111yescharacteristic
346#380—21no—
347#380—21no—
348C4×C4⋊C4—611no—
349C4×C4⋊C4—611no—
350C4×C4⋊C4—611no—
351C4×C4⋊C4—1221no—
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—21no—
363C22×D4C22×D6111yescharacteristic
364C23×C4—21no—
365C22×C8—611no—
366C22×C8—611no—
367C22×C8—611no—
368C22×C8—611no—
369C4⋊Q8—22no—
370C4⋊Q8—22no—
371#125—614no—
372#125—1222no—
373C4×D4—22no—
374C4×D4—22no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—611no—
377C2×C4⋊C4—611no—
378C2×C4⋊C4—611no—
379C2×C4⋊C4—611no—
380C2×C4⋊C4—1221no—
381C2×C4⋊C4—1221no—
382C2×C4⋊C4—1221no—
383C2×C4⋊C4—1221no—
384C2×C4⋊C4C22⋊D6111yescharacteristic
385C2×C4⋊C4C22⋊D6111yescharacteristic
386C2×C4⋊C4C22×D6111yescharacteristic
387C2×C4⋊C4C22×D6111yescharacteristic
388C2×C4⋊C4C22×D6111yescharacteristic
389C2×C22⋊C4—21no—
390C2×C22⋊C4—21no—
391C2×C22⋊C4—21no—
392C2×C22⋊C4—21no—
393C2×C22⋊C4—21no—
394C2×C42—1221no—
395C2×C42—1221no—
396C2×C42—1221no—
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—1221no—
419#95—1221no—
420#95—1221no—
421#95—1221no—
422#95—1221no—
423#95C22⋊D6111yescharacteristic
424#95C22⋊D6111yescharacteristic
425C24—21no—
426C2×Q8—42no—
427C2×Q8—81no—
428C2×Q8—81no—
429C2×D4—21no—
430C2×D4—21no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4Q8×D6112yes—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—1221no—
442C22×C4—1221no—
443C22×C4—1221no—
444C22×C4—1221no—
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—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—41no—
471C4⋊C4—41no—
472C4⋊C4—42no—
473C4⋊C4—612no—
474C4⋊C4—612no—
475C4⋊C4—612no—
476C4⋊C4—1221no—
477C4⋊C4—1221no—
478C4⋊C4—1221no—
479C4⋊C4—1224no—
480C4⋊C4—1224no—
481C4⋊C4—1221no—
482C4⋊C4—1221no—
483C4⋊C4—4881no—
484C4⋊C4—2442no—
485C22⋊C4—42no—
486C22⋊C4—81no—
487C22⋊C4—81no—
488C22⋊C4—81no—
489C22⋊C4—81no—
490C42—21no—
491C42—21no—
492C42—21no—
493C42—21no—
494C42—4881no—
495C42—4881no—
496C42—2442no—
497C42D4×D6112yes—
498C42#1175822412yes—
499C23—22no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23C22×C22⋊D6111yescharacteristic
507Q8—84no—
508D4—22no—
509D4—22no—
510D4—41no—
511D4—41no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
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—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—84no—
538C2×C4—1222no—
539C2×C4—1222no—
540C2×C4—1222no—
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#1451974812yes—
551C2×C4#1451974812yes—
552C2×C4C2×D4×D6111yescharacteristic
553C2×C4C2×Q8×D6111yescharacteristic
554C2×C4#41×D6111yescharacteristic
555C2×C4C2×#1175822411yescharacteristic
556C2×C4#1451982411yescharacteristic
557C2×C4#1451982511yescharacteristic
558C8—1222no—
559C8—1222no—
560C8—2441no—
561C8—2441no—
562C22—42no—
563C22—42no—
564C22—42no—
565C22—81no—
566C22—81no—
567C22—81no—
568C22—81no—
569C22C2×#1451974811yescharacteristic
570C22#1684041111yescharacteristic
571C22Q8×C22⋊D6111yescharacteristic
572C22#1684056011yescharacteristic
573C22#1684056011yescharacteristic
574C22#1684056511yescharacteristic
575C22#1684057311yescharacteristic
576C4—22no—
577C4—22no—
578C4—41no—
579C4—41no—
580C4—41no—
581C4—41no—
582C4—81no—
583C4—81no—
584C4—81no—
585C4—4881no—
586C4—2442no—
587C4—2442no—
588C4—2442no—
589C4#1684031411yescharacteristic
590C4#1684037211yescharacteristic
591C4#1684042211yescharacteristic
592C4#1684042511yescharacteristic
593C2—82no—
594C2#2144941411yescharacteristic
595C2#2145530911yescharacteristic
596C2#2145535411yescharacteristic
597C2#2145831911yescharacteristic
598C2#2145833411yescharacteristic
599C2#2145876411yescharacteristic
600C2#2145877311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2563C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
hypercenter#2563D61
hypocenterC61#25466
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2563

Complements

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

NH ≅ G/Ntype
1. C2×#16840168C2semidirect
3. #21448913C2semidirect
4. #21447301C2semidirect
5. #21447301C2semidirect
8. #21446772C2semidirect
9. #21446585C2semidirect
10. #21446585C2semidirect
11. #21446543C2semidirect
90. #14519698Q8semidirect
361. C61#25466semidirect

Maximal subgroups

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

subgroupindex
C2×#168401682
C2×#168400622
#214489132
#214473012
#214472072
#214471932
#214467722
#214465852
#214465432
#214464432
#214464362
#214456442
C61×#25632
#2546661