Finite Group Databasebeta

← #29702027

Subgroups — #29702027

Order 15616.

870 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840060C211yesmaximal, characteristic
2#21448911C211yesmaximal, characteristic, complemented
3D4×#11758196C211yesmaximal, characteristic
4#21446786C211yesmaximal, characteristic, complemented
5#21446581C211yesmaximal, characteristic, complemented
6#21446557C211yesmaximal, characteristic, complemented
7#21446546C211yesmaximal, characteristic, complemented
8#21446404C211yesmaximal, characteristic, complemented
9#21446389C211yesmaximal, characteristic, complemented
10#21446082C211yesmaximal, characteristic
11#21446080C211yesmaximal, characteristic
12#21445695C211yesmaximal, characteristic, complemented
13#21445646C211yesmaximal, characteristic
14#102×Dic61C211yesmaximal, characteristic
15C61×#1762C211yesmaximal, characteristic
16C2×#14519768C2211yescharacteristic
17C2×D4×Dic61C2211yescharacteristic
18C2×D4×Dic61C2211yescharacteristic
19C22×#11758196—21no—
20#16840060C418yescomplemented
21C2×#14519711C2211yescharacteristic
22C2×#14519709C2211yescharacteristic
23C2×#14519708C2211yescharacteristic
24C2×#14519708C2211yescharacteristic
25C2×#14519706C2211yescharacteristic
26C2×#14519698C2211yescharacteristic
27C2×#14519698C2211yescharacteristic
28C2×#14519687C2211yescharacteristic
29C2×C8×Dic61C2211yescharacteristic
30#16839661C2211yescharacteristic
31#16839660—21no—
32#16839660C2211yescharacteristic
33#16839660C2211yescharacteristic
34#16839658C2211yescharacteristic
35#16839649—21no—
36#16839645—21no—
37#16839635—21no—
38#16839634—21no—
39#16839631C2211yescharacteristic
40#16839620—21no—
41C4⋊C4×Dic61C2211yescharacteristic
42C4⋊C4×Dic61C2211yescharacteristic
43C4×#11758196—21no—
44C4×#11758196C2211yescharacteristic
45#16839525C2211yescharacteristic
46#16839525C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839517C2211yescharacteristic
52#16839516C2211yescharacteristic
53C2×C4×D4×C61C2211yescharacteristic
54C2×#102×C61C2211yescharacteristic
55C2×#102×C61C2211yescharacteristic
56C2×#97×C61C2211yescharacteristic
57C4×C4⋊C4×C61C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C61×#340C2211yescharacteristic
60C23×Dic61—21no—
61C23×Dic61—21no—
62C22×#92418C2311yescharacteristic
63C22×#92418C2311yescharacteristic
64#14519768—22no—
65#14519768C2×C414yescomplemented
66D4×Dic61—22no—
67D4×Dic61—22no—
68D4×Dic61—41no—
69D4×Dic61C2×C414yes—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61—21no—
72C2×C4⋊Dic61C2311yescharacteristic
73C2×C4⋊Dic61C2311yescharacteristic
74C2×C4⋊Dic61C2311yescharacteristic
75C2×C4⋊Dic61C2311yescharacteristic
76C2×#11758196—21no—
77C2×#11758196—21no—
78C2×#11758196—21no—
79C2×#11758196—21no—
80C2×#11758196—21no—
81C2×#11758196—21no—
82C2×#11758196—44no—
83C2×#11758196D411yescharacteristic
84C2×#11758196D411yescharacteristic
85C2×C22⋊Dic61—21no—
86C2×C22⋊Dic61—21no—
87C2×C4×Dic61—21no—
88C2×C4×Dic61—21no—
89C2×C4×Dic61C2311yescharacteristic
90C2×C4×Dic61C2311yescharacteristic
91#14519711C2×C414yescomplemented
92#14519709—41no—
93#14519708—41no—
94#14519708C2×C414yescomplemented
95#14519706C2×C414yescomplemented
96#14519698—22no—
97#14519698C2×C414yes—
98#14519687—22no—
99C8×Dic61—41no—
100#14519684—21no—
101#14519684—21no—
102#14519684—21no—
103#14519684—21no—
104#14519684—21no—
105#14519684—21no—
106#14519684D411yescharacteristic
107#14519684D411yescharacteristic
108C22×D4×C61C2311yescharacteristic
109C23×C4×C61—21no—
110C22×C8×C61C2311yescharacteristic
111C22×C8×C61C2311yescharacteristic
112C4×D4×C61—22no—
113C4×D4×C61—41no—
114C2×C4⋊C4×C61C2311yescharacteristic
115C2×C4⋊C4×C61C2311yescharacteristic
116C2×C4⋊C4×C61C2311yescharacteristic
117C2×C22⋊C4×C61—21no—
118C2×C42×C61—21no—
119C2×C42×C61C2311yescharacteristic
120#102×C61C2×C414yes—
121#102×C61Q814yes—
122#97×C61—41no—
123#95×C61—21no—
124C24×C61—21no—
125C2×D4×C61—21no—
126C2×D4×C61—21no—
127C2×D4×C61—21no—
128C2×D4×C61C22×C412yes—
129C2×D4×C61C2×Q812yes—
130C2×D4×C61#4112yes—
131C22×C4×C61—21no—
132C22×C4×C61—21no—
133C22×C4×C61—21no—
134C22×C4×C61—42no—
135C22×C4×C61—42no—
136C22×C4×C61—81no—
137C22×C4×C61C2×D411yescharacteristic
138C22×C4×C61C2×D411yescharacteristic
139C22×C4×C61C2411yescharacteristic
140C22×Dic61—21no—
141C22×Dic61—21no—
142C22×Dic61—24no—
143C22×Dic61—21no—
144C22×Dic61—21no—
145C22×Dic61—42no—
146C22×Dic61—42no—
147C22×Dic61—42no—
148C22×Dic61—42no—
149C22×Dic61—81no—
150C22×Dic61C2×D411yescharacteristic
151C22×Dic61C2×D411yescharacteristic
152C22×Dic61C2×D411yescharacteristic
153C22×Dic61C2×D411yescharacteristic
154C2×C8×C61—21no—
155C2×C8×C61—21no—
156C2×C8×C61—21no—
157C2×C8×C61—21no—
158C2×C8×C61C22×C412yes—
159C2×C8×C61C2×Q812yes—
160C4⋊C4×C61—21no—
161C4⋊C4×C61—21no—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61C22×C412yes—
165C4⋊C4×C61C2×Q812yes—
166C22⋊C4×C61—42no—
167C42×C61—21no—
168C42×C61—21no—
169C42×C61—81no—
170C22⋊Dic61—24no—
171C22⋊Dic61—42no—
172C4⋊Dic61—21no—
173C4⋊Dic61—21no—
174C4⋊Dic61—24no—
175C4⋊Dic61—21no—
176C4⋊Dic61—21no—
177C4⋊Dic61—21no—
178C4⋊Dic61—21no—
179C4⋊Dic61—81no—
180C4⋊Dic61C22×C412yes—
181C4⋊Dic61C22×C412yes—
182#11758196—24no—
183#11758196—22no—
184#11758196—41no—
185#11758196—42no—
186#11758196—42no—
187#11758196—42no—
188#11758196—42no—
189#11758196—81no—
190C4×Dic61—21no—
191C4×Dic61—21no—
192C4×Dic61—42no—
193C4×Dic61—81no—
194C4×Dic61C22×C412yes—
195C4×Dic61#4112yes—
196C2×#92418—21no—
197C2×#92418—21no—
198C2×#92418—21no—
199C2×#92418—21no—
200C2×#92418—21no—
201C2×#92418C22×C412yes—
202C23×C61—22no—
203C23×C61—22no—
204C23×C61—22no—
205C23×C61—41no—
206C23×C61—41no—
207C23×C61—41no—
208C23×C61—41no—
209C23×C61C22×D411yescharacteristic
210D4×C61—22no—
211D4×C61—22no—
212D4×C61—22no—
213D4×C61C4×Q814yes—
214C2×C4×C61—22no—
215C2×C4×C61—21no—
216C2×C4×C61—21no—
217C2×C4×C61—22no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—41no—
223C2×C4×C61—41no—
224C2×C4×C61—41no—
225C2×C4×C61—41no—
226C2×C4×C61—41no—
227C2×C4×C61—41no—
228C2×C4×C61—41no—
229C2×C4×C61—84no—
230C2×C4×C61C23×C411yescharacteristic
231C2×C4×C61C22×Q811yescharacteristic
232C2×C4×C61C2×#4111yescharacteristic
233C2×C4×C61C2×#4111yescharacteristic
234C2×C4×C61H(5,𝔽2)11yescharacteristic
235C2×C4×C612−1+411yescharacteristic
236C2×Dic61—21no—
237C2×Dic61—21no—
238C2×Dic61—21no—
239C2×Dic61—21no—
240C2×Dic61—21no—
241C2×Dic61—21no—
242C2×Dic61—22no—
243C2×Dic61—22no—
244C2×Dic61—21no—
245C2×Dic61—21no—
246C2×Dic61—21no—
247C2×Dic61—48no—
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—
257C2×Dic61—41no—
258C2×Dic61—84no—
259C2×Dic61C4×D412yes—
260C2×Dic61C4×D412yes—
261C2×Dic61AGL(1,ℤ/8)12yes—
262C8×C61—22no—
263C8×C61—22no—
264#92418—22no—
265#92418—41no—
266#22978—611nomaximal, Sylow
267C22×C61—24no—
268C22×C61—42no—
269C22×C61—42no—
270C22×C61—42no—
271C22×C61—42no—
272C22×C61—42no—
273C22×C61—42no—
274C22×C61C2×C4×D411yescharacteristic
275C22×C61#54611yescharacteristic
276C22×C61D4×Q811yescharacteristic
277C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
278C22×C61#57511yescharacteristic
279C22×C61#57711yescharacteristic
280C22×C61#57711yescharacteristic
281C4×C61—42no—
282C4×C61—42no—
283C4×C61—41no—
284C4×C61—41no—
285C4×C61—81no—
286C4×C61C2×C4×Q811yescharacteristic
287C4×C61#52011yescharacteristic
288C4×C61#55211yescharacteristic
289C4×C61#55411yescharacteristic
290Dic61—22no—
291Dic61—22no—
292Dic61—22no—
293Dic61—42no—
294Dic61—42no—
295Dic61—41no—
296Dic61—81no—
297Dic61—81no—
298C2×#478—611no—
299#2370—611no—
300D4×C4⋊C4—611no—
301#1992—611no—
302#1948—611no—
303#1943—611no—
304#1941—611no—
305#1920—611no—
306#1918—611no—
307#1864—611no—
308#1864—611no—
309#1771—611no—
310#1762—611no—
311#1762D6111yescharacteristic
312C4×#102—611no—
313C2×C61—44no—
314C2×C61#236011yescharacteristic
315C2×C61C4×AGL(1,ℤ/8)11yescharacteristic
316C2×C61#295211yescharacteristic
317C2×C61#330711yescharacteristic
318C2×C61#331011yescharacteristic
319C2×C61#338411yescharacteristic
320C2×C61#338811yescharacteristic
321C2×#125—611no—
322C2×C4×D4—611no—
323C2×C4×D4—611no—
324C2×C4×D4C2×D6111yescharacteristic
325C22×C4⋊C4—1221no—
326#478—618no—
327C2×#107—611no—
328C2×#106—611no—
329C2×#106—611no—
330C2×#106—611no—
331C2×C4⋊C8—611no—
332C2×#102—611no—
333C2×#102—611no—
334C2×#102C2×D6111yescharacteristic
335C2×#102C2×D6111yescharacteristic
336C2×#97—611no—
337C2×#97C2×D6111yescharacteristic
338C2×C4×C8—611no—
339#389—611no—
340#389—611no—
341#389—611no—
342#389—611no—
343#389—1221no—
344#387—1221no—
345#387—1221no—
346#385—1221no—
347#385—1221no—
348#384—611no—
349#380—1221no—
350C4×C4⋊C4—611no—
351C4×C4⋊C4—611no—
352C4×C4⋊C4—611no—
353C4×C4⋊C4—1221no—
354C4×C4⋊C4C2×D6111yescharacteristic
355#340—611no—
356#340—611no—
357#340—611no—
358#340—611no—
359#340—611no—
360#340—611no—
361#340C2×D6111yescharacteristic
362#340C2×D6111yescharacteristic
363#336—611no—
364#336—611no—
365C61#2297811yescharacteristic, complemented, Sylow
366C22×D4C22×D6111yescharacteristic
367C23×C4—21no—
368C23×C4—1221no—
369C23×C4—1221no—
370C22×C8—611no—
371C22×C8—611no—
372C22×C8C22×D6111yescharacteristic
373C22×C8C22×D6111yescharacteristic
374#125—614no—
375#125—1222no—
376C4×D4—22no—
377C4×D4—41no—
378C4×D4—614no—
379C4×D4—1222no—
380C4×D4—1222no—
381C4×D4—2441no—
382C2×C4⋊C4—611no—
383C2×C4⋊C4—611no—
384C2×C4⋊C4—611no—
385C2×C4⋊C4—611no—
386C2×C4⋊C4—611no—
387C2×C4⋊C4—611no—
388C2×C4⋊C4—1221no—
389C2×C4⋊C4—1221no—
390C2×C4⋊C4—1221no—
391C2×C4⋊C4—1221no—
392C2×C4⋊C4—1221no—
393C2×C4⋊C4—1221no—
394C2×C4⋊C4—1221no—
395C2×C4⋊C4—1221no—
396C2×C4⋊C4—2444no—
397C2×C4⋊C4C22×D6111yescharacteristic
398C2×C4⋊C4C22×D6111yescharacteristic
399C2×C4⋊C4C22×D6111yescharacteristic
400C2×C22⋊C4—21no—
401C2×C22⋊C4—1221no—
402C2×C22⋊C4—1221no—
403C2×C42—21no—
404C2×C42—611no—
405C2×C42—611no—
406C2×C42—1221no—
407C2×C42—1221no—
408C2×C42C22×D6111yescharacteristic
409#107—614no—
410#106—614no—
411#106—2441no—
412#106—2441no—
413C4⋊C8—614no—
414#102—614no—
415#102—1222no—
416#102Dic12214yes—
417#102C4×D6114yes—
418#97—41no—
419#97—1222no—
420C4×C8—2441no—
421#95—21no—
422#95—611no—
423#95—611no—
424#95—1221no—
425#95—1221no—
426#95—1221no—
427#95—1221no—
428#95—1221no—
429#95—1221no—
430C24—21no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4C2×Dic12212yes—
435C2×D4C2×C4×D6112yes—
436C2×D4#1175822212yes—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—81no—
443C22×C4—611no—
444C22×C4—611no—
445C22×C4—611no—
446C22×C4—611no—
447C22×C4—1224no—
448C22×C4—1221no—
449C22×C4—1221no—
450C22×C4—1221no—
451C22×C4—1221no—
452C22×C4—4881no—
453C22×C4—2442no—
454C22×C4—2442no—
455C22×C4—2442no—
456C22×C4—2442no—
457C22×C4D4×D6111yescharacteristic
458C22×C4D4×D6111yescharacteristic
459C22×C4C23×D6111yescharacteristic
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—612no—
465C2×C8—1221no—
466C2×C8—1221no—
467C2×C8—1221no—
468C2×C8—1221no—
469C2×C8—1221no—
470C2×C8C2×Dic12212yes—
471C2×C8C2×C4×D6112yes—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—612no—
477C4⋊C4—612no—
478C4⋊C4—1222no—
479C4⋊C4—1224no—
480C4⋊C4—1224no—
481C4⋊C4—1221no—
482C4⋊C4—1221no—
483C4⋊C4—1221no—
484C4⋊C4—1221no—
485C4⋊C4—1221no—
486C4⋊C4—1221no—
487C4⋊C4—4881no—
488C4⋊C4—4881no—
489C4⋊C4—2441no—
490C4⋊C4—2442no—
491C4⋊C4—2442no—
492C4⋊C4—2442no—
493C4⋊C4—2442no—
494C4⋊C4C2×Dic12212yes—
495C4⋊C4C2×C4×D6112yes—
496C22⋊C4—42no—
497C22⋊C4—1224no—
498C22⋊C4—2442no—
499C42—21no—
500C42—21no—
501C42—81no—
502C42—612no—
503C42—612no—
504C42—1221no—
505C42—1221no—
506C42—4881no—
507C42—2442no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23C2×D4×D6111yescharacteristic
516D4—22no—
517D4—22no—
518D4—22no—
519D4C4×Dic12214yes—
520C2×C4—22no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—22no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—84no—
536C2×C4—612no—
537C2×C4—612no—
538C2×C4—612no—
539C2×C4—1221no—
540C2×C4—1221no—
541C2×C4—1221no—
542C2×C4—1221no—
543C2×C4—1221no—
544C2×C4—1221no—
545C2×C4—1222no—
546C2×C4—1222no—
547C2×C4—1221no—
548C2×C4—1221no—
549C2×C4—1221no—
550C2×C4—4884no—
551C2×C4—2448no—
552C2×C4—2441no—
553C2×C4—2441no—
554C2×C4—2441no—
555C2×C4—2441no—
556C2×C4—2441no—
557C2×C4—2441no—
558C2×C4—2441no—
559C2×C4—2441no—
560C2×C4—2441no—
561C2×C4—2441no—
562C2×C4C22×C4×D6111yescharacteristic
563C2×C4C22×Dic12211yescharacteristic
564C2×C4#41×D6111yescharacteristic
565C2×C4C2×#1175822211yescharacteristic
566C2×C4#1451982211yescharacteristic
567C2×C4#1451982411yescharacteristic
568C8—22no—
569C8—22no—
570C8—1222no—
571C8—2441no—
572C22—24no—
573C22—42no—
574C22—42no—
575C22—42no—
576C22—42no—
577C22—42no—
578C22—42no—
579C22C4×D4×D6111yescharacteristic
580C22#1684040211yescharacteristic
581C22D4×Dic12211yescharacteristic
582C22AGL(1,ℤ/8)×D6111yescharacteristic
583C22#1684056111yescharacteristic
584C22#1684057211yescharacteristic
585C22#1684057211yescharacteristic
586C4—42no—
587C4—42no—
588C4—41no—
589C4—41no—
590C4—81no—
591C4—1222no—
592C4—1222no—
593C4—1222no—
594C4—4881no—
595C4—4881no—
596C4—2442no—
597C4—2442no—
598C4—2441no—
599C4C2×C4×Dic12211yescharacteristic
600C4#1684024111yescharacteristic
601C4#1684044511yescharacteristic
602C4#1684047111yescharacteristic
603C2—44no—
604C2#2144882511yescharacteristic
605C2#2145462411yescharacteristic
606C2#2145470111yescharacteristic
607C2#2145779611yescharacteristic
608C2#2145784111yescharacteristic
609C2#2145871911yescharacteristic
610C2#2145876511yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1762C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
cosocleC22×C4×C61C24
hypercenter#1762D61
hypocenterC61#22978
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1762

Complements

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

NH ≅ G/Ntype
2. #21448911C2semidirect
4. #21446786C2semidirect
5. #21446581C2semidirect
6. #21446557C2semidirect
7. #21446546C2semidirect
8. #21446404C2semidirect
9. #21446389C2semidirect
12. #21445695C2semidirect
20. #16840060C4semidirect
65. #14519768C2×C4semidirect
91. #14519711C2×C4semidirect
94. #14519708C2×C4semidirect
95. #14519706C2×C4semidirect
365. C61#22978semidirect

Maximal subgroups

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

subgroupindex
C2×#168400602
#214489112
D4×#117581962
#214467862
#214465812
#214465572
#214465462
#214464042
#214463892
#214460822
#214460802
#214456952
#214456462
#102×Dic612
C61×#17622
#2297861