Finite Group Databasebeta

← #29687196

Subgroups — #29687196

Order 15616.

1048 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840076C211yesmaximal, characteristic
2C4×#14519751C211yesmaximal, characteristic
3C2×C4×C4⋊Dic61C211yesmaximal, characteristic, complemented
4#21446447C212yesmaximal, complemented
5#21446412C212yesmaximal, complemented
6#21446134C211yesmaximal, characteristic
7#21446014C211yesmaximal, characteristic, complemented
8#21445906C211yesmaximal, characteristic, complemented
9#21445716C212yesmaximal, complemented
10#21445645C212yesmaximal, complemented
11C61×#1749C211yesmaximal, characteristic, complemented
12C2×#14519751C2211yescharacteristic
13C2×C4×D244C2211yescharacteristic
14C2×C4×C22⋊D61—21no—
15C22×C4⋊Dic61C2211yescharacteristic
16C22×C4⋊Dic61C2211yescharacteristic
17C22×C4×Dic61—21no—
18#16840076C418yescomplemented
19C2×#14519711C2212yes—
20C2×#14519709C2212yes—
21C2×#14519699C2212yes—
22C2×#14519699C2212yes—
23#16839664C2211yescharacteristic
24#16839660C2211yescharacteristic
25#16839653—21no—
26#16839641—21no—
27#16839622C2211yescharacteristic
28C4×C4⋊Dic61—21no—
29C4×C4⋊Dic61—21no—
30C4×C4⋊Dic61C2212yes—
31C4×C4⋊Dic61C2212yes—
32C4×C4⋊Dic61C2211yescharacteristic
33C4×#11758198—21no—
34C2×#14519684—21no—
35#16839525C2212yes—
36#16839525C2212yes—
37#16839525C2212yes—
38#16839525C2212yes—
39C22×C42×C61C2211yescharacteristic
40C2×C22⋊C8×C61C2211yescharacteristic
41C2×C22⋊C8×C61C2211yescharacteristic
42C2×#97×C61C2212yes—
43C61×#336C2212yes—
44C22×D244C2311yescharacteristic
45C22×C22⋊D61—21no—
46C23×Dic61—21no—
47C23×Dic61—21no—
48C22×C4×D61—21no—
49#14519751—22no—
50#14519751C2×C414yes—
51C4×D244—24no—
52C4×C22⋊D61—44no—
53C2×C4⋊Dic61—21no—
54C2×C4⋊Dic61—21no—
55C2×C4⋊Dic61—22no—
56C2×C4⋊Dic61—24no—
57C2×C4⋊Dic61C2×C414yescomplemented
58C2×C4⋊Dic61C2311yescharacteristic
59C2×C4⋊Dic61C2311yescharacteristic
60C2×C4⋊Dic61C2312yes—
61C2×C4⋊Dic61C2311yescharacteristic
62C2×C4⋊Dic61C2312yes—
63C2×#11758196—21no—
64C2×C22⋊Dic61—21no—
65C2×#11758198—21no—
66C2×#11758198—21no—
67C2×#11758198—21no—
68C2×#11758198—21no—
69C2×C4×Dic61—22no—
70C2×C4×Dic61—22no—
71C2×C4×Dic61—21no—
72C2×C4×Dic61—44no—
73#14519711C2×C418yescomplemented
74#14519709—24no—
75#14519699—24no—
76#14519699C2×C418yescomplemented
77#14519684—22no—
78#14519684—22no—
79#14519684—21no—
80#14519684—21no—
81#14519684—21no—
82C23×C4×C61D411yescharacteristic
83C23×C4×C61D411yescharacteristic
84C23×C4×C61C2311yescharacteristic
85C22×C8×C61C2312yes—
86C22×C8×C61C2312yes—
87C2×C42×C61—24no—
88C2×C42×C61D411yescharacteristic
89C2×C42×C61D411yescharacteristic
90C2×C42×C61C2311yescharacteristic
91C2×C42×C61C2311yescharacteristic
92C22⋊C8×C61—22no—
93C22⋊C8×C61C2×C414yescomplemented
94#97×C61—24no—
95C24×C61C2×D411yescharacteristic
96C23×D61—21no—
97C22×C4×C61—22no—
98C22×C4×C61—22no—
99C22×C4×C61—22no—
100C22×C4×C61—22no—
101C22×C4×C61—22no—
102C22×C4×C61—22no—
103C22×C4×C61—22no—
104C22×C4×C61—41no—
105C22×C4×C61C22×C412yes—
106C22×C4×C61C2×D411yescharacteristic
107C22×C4×C61C2×D411yescharacteristic
108C22×C4×C61C2×D411yescharacteristic
109C22×C4×C61C2×D411yescharacteristic
110C22×C4×C61C2×D411yescharacteristic
111C22×C4×C61#4112yes—
112C22×C4×C61C2411yescharacteristic
113C2×C22⋊D61—24no—
114C2×C22⋊D61—42no—
115C2×C22⋊D61—42no—
116C22×Dic61—24no—
117C22×Dic61—24no—
118C22×Dic61—22no—
119C22×Dic61—22no—
120C22×Dic61—21no—
121C22×Dic61—21no—
122C22×Dic61—21no—
123C22×Dic61—42no—
124C22×Dic61—42no—
125C22×Dic61—42no—
126C22×Dic61—42no—
127C2×D244—22no—
128C2×D244—22no—
129C2×D244—21no—
130C2×D244C22×C412yes—
131C2×C4×D61—42no—
132C2×C4×D61—42no—
133C2×C4×D61—42no—
134C2×C8×C61—22no—
135C2×C8×C61—22no—
136C2×C8×C61—22no—
137C2×C8×C61—22no—
138C2×C8×C61—22no—
139C2×C8×C61C22×C414yes—
140C42×C61—22no—
141C42×C61—22no—
142C42×C61—22no—
143C42×C61#4114yes—
144C22⋊Dic61—42no—
145#11758198—24no—
146#11758198—42no—
147#11758198—42no—
148#11758198—42no—
149C4⋊Dic61—24no—
150C4⋊Dic61—24no—
151C4⋊Dic61—22no—
152C4⋊Dic61—22no—
153C4⋊Dic61—21no—
154C4⋊Dic61—22no—
155C4⋊Dic61—22no—
156C4⋊Dic61C22×C414yes—
157C4⋊Dic61C22×C412yes—
158C4⋊Dic61#4114yes—
159#11758196—42no—
160C4×Dic61—44no—
161C4×Dic61—44no—
162C4×Dic61—42no—
163C23×C61—21no—
164C23×C61—21no—
165C23×C61—21no—
166C23×C61—21no—
167C23×C61C4×D412yes—
168C23×C61AGL(1,ℤ/8)12yes—
169C23×C61AGL(1,ℤ/8)12yes—
170C23×C61C22×D411yescharacteristic
171C22×D61—22no—
172C22×D61—41no—
173C22×D61—41no—
174C22×D61—41no—
175C22×D61—41no—
176C22×D61—41no—
177C22×D61—41no—
178C2×C4×C61—28no—
179C2×C4×C61—21no—
180C2×C4×C61—21no—
181C2×C4×C61—21no—
182C2×C4×C61—21no—
183C2×C4×C61—21no—
184C2×C4×C61—21no—
185C2×C4×C61—21no—
186C2×C4×C61—21no—
187C2×C4×C61—24no—
188C2×C4×C61—24no—
189C2×C4×C61—21no—
190C2×C4×C61—21no—
191C2×C4×C61—21no—
192C2×C4×C61—44no—
193C2×C4×C61C4×D412yes—
194C2×C4×C61#12312yes—
195C2×C4×C61#12312yes—
196C2×C4×C61#13712yes—
197C2×C4×C61C23×C411yescharacteristic
198C2×C4×C61C2×#4112yes—
199C2×C4×C61C2×#4112yes—
200C2×C4×C61C2×#4111yescharacteristic
201C22⋊D61—48no—
202C2×Dic61—24no—
203C2×Dic61—22no—
204C2×Dic61—24no—
205C2×Dic61—48no—
206C2×Dic61—42no—
207C2×Dic61—42no—
208C2×Dic61—41no—
209C2×Dic61—42no—
210C2×Dic61—41no—
211C2×Dic61—42no—
212C2×Dic61—48no—
213C2×Dic61—41no—
214C2×Dic61—41no—
215C2×Dic61—41no—
216C2×Dic61—41no—
217C2×Dic61—41no—
218C2×Dic61—41no—
219D244—24no—
220D244—42no—
221C4×D61—84no—
222C8×C61—24no—
223C8×C61—42no—
224#21578—611nomaximal, Sylow
225C22×C61—22no—
226C22×C61—22no—
227C22×C61—22no—
228C22×C61—22no—
229C22×C61—22no—
230C22×C61—22no—
231C22×C61#44214yes—
232C22×C61C2×C4×D411yescharacteristic
233C22×C61C2×#12311yescharacteristic
234C22×C61#52511yescharacteristic
235C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
236C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
237C22×C61C2×#13711yescharacteristic
238C22×C61#57511yescharacteristic
239C2×D61—42no—
240C2×D61—42no—
241C2×D61—42no—
242C2×D61—81no—
243C2×D61—81no—
244C2×D61—81no—
245C2×D61—81no—
246C4×C61—22no—
247C4×C61—22no—
248C4×C61—22no—
249C4×C61—22no—
250C4×C61—41no—
251C4×C61C4×#4112yes—
252C4×C61#52912yes—
253Dic61—44no—
254Dic61—44no—
255Dic61—42no—
256Dic61—81no—
257Dic61—81no—
258C2×#480—611no—
259C4×#121—611no—
260C2×C4×C4⋊C4—611no—
261#1926—612no—
262#1921—612no—
263#1874—611no—
264#1853—611no—
265#1823—611no—
266#1776—612no—
267#1762—612no—
268#1749D6111yescharacteristic, complemented
269C2×C61—24no—
270C2×C61C4×#12311yescharacteristic
271C2×C61C2×#44211yescharacteristic
272C2×C61C4×#13711yescharacteristic
273C2×C61#309211yescharacteristic
274C2×C61#309411yescharacteristic
275C2×C61#311211yescharacteristic
276C2×C61#311411yescharacteristic
277D61—82no—
278C2×#121—611no—
279C2×C4×D4—611no—
280C2×C4×D4—1221no—
281C22×C4⋊C4—611no—
282C22×C4⋊C4—611no—
283C22×C42—1221no—
284C22×C42C2×D6111yescharacteristic
285#480—618no—
286C2×#107—612no—
287C2×#106—612no—
288C2×#102—612no—
289C2×#102—612no—
290C2×C22⋊C8C2×D6111yescharacteristic
291C2×C22⋊C8C2×D6111yescharacteristic
292C2×#97C2×D6112yes—
293#390—611no—
294#389—611no—
295#388—1221no—
296#386—1221no—
297#380—611no—
298C4×C4⋊C4—612no—
299C4×C4⋊C4—612no—
300C4×C4⋊C4—611no—
301C4×C4⋊C4—1221no—
302C4×C4⋊C4—1221no—
303C4×C22⋊C4—1221no—
304C2×#95—1221no—
305#340—612no—
306#340—612no—
307#340—612no—
308#340—612no—
309#336C2×D6112yes—
310C61#2157811yescharacteristic, complemented, Sylow
311C22×D4—611no—
312C22×D4—1221no—
313C23×C4—1221no—
314C23×C4—1221no—
315C23×C4—1221no—
316C23×C4D24411yescharacteristic
317C23×C4D24411yescharacteristic
318C23×C4C22×D6111yescharacteristic
319C22×C8C22×D6112yes—
320C22×C8C22×D6112yes—
321#121—614no—
322#121—1222no—
323C4×D4—1224no—
324C4×D4—2444no—
325C2×C4⋊C4—614no—
326C2×C4⋊C4—612no—
327C2×C4⋊C4—612no—
328C2×C4⋊C4—611no—
329C2×C4⋊C4—611no—
330C2×C4⋊C4—611no—
331C2×C4⋊C4—1224no—
332C2×C4⋊C4—1221no—
333C2×C4⋊C4—1221no—
334C2×C4⋊C4—1222no—
335C2×C4⋊C4—1221no—
336C2×C22⋊C4—1221no—
337C2×C22⋊C4—1221no—
338C2×C22⋊C4—1221no—
339C2×C22⋊C4—1221no—
340C2×C22⋊C4—1221no—
341C2×C42—24no—
342C2×C42—1221no—
343C2×C42—1222no—
344C2×C42—1222no—
345C2×C42—2444no—
346C2×C42D24411yescharacteristic
347C2×C42D24411yescharacteristic
348C2×C42C22×D6111yescharacteristic
349C2×C42C22×D6111yescharacteristic
350#107—618no—
351#106—1224no—
352#102—618no—
353#102—1224no—
354C22⋊C8—22no—
355C22⋊C8C4×D6114yescomplemented
356#97—24no—
357#95—1222no—
358#95—1222no—
359#95—1221no—
360#95—1221no—
361#95—1221no—
362C24—1221no—
363C24C2×D24411yescharacteristic
364C2×D4—612no—
365C2×D4—1224no—
366C2×D4—1222no—
367C2×D4—1222no—
368C2×D4—1221no—
369C2×D4—2442no—
370C2×D4—2442no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—41no—
379C22×C4—1224no—
380C22×C4—1222no—
381C22×C4—1222no—
382C22×C4—1221no—
383C22×C4—1224no—
384C22×C4—1221no—
385C22×C4—1221no—
386C22×C4—2442no—
387C22×C4—2442no—
388C22×C4—2442no—
389C22×C4—2442no—
390C22×C4—2442no—
391C22×C4—2442no—
392C22×C4—2442no—
393C22×C4C2×C4×D6112yes—
394C22×C4C2×D24411yescharacteristic
395C22×C4C2×D24411yescharacteristic
396C22×C4C2×D24411yescharacteristic
397C22×C4C2×D24411yescharacteristic
398C22×C4C2×D24411yescharacteristic
399C22×C4#1175822212yes—
400C22×C4C23×D6111yescharacteristic
401C2×C8—22no—
402C2×C8—22no—
403C2×C8—22no—
404C2×C8—22no—
405C2×C8—22no—
406C2×C8C2×C4×D6114yes—
407C4⋊C4—614no—
408C4⋊C4—612no—
409C4⋊C4—614no—
410C4⋊C4—1224no—
411C4⋊C4—1222no—
412C4⋊C4—1222no—
413C4⋊C4—1221no—
414C4⋊C4—1224no—
415C4⋊C4—1222no—
416C4⋊C4—1222no—
417C4⋊C4—2442no—
418C22⋊C4—1224no—
419C22⋊C4—2442no—
420C22⋊C4—2442no—
421C22⋊C4—2442no—
422C22⋊C4—2442no—
423C42—22no—
424C42—22no—
425C42—22no—
426C42—2444no—
427C42—2444no—
428C42—2442no—
429C42#1175822414yes—
430C23—21no—
431C23—21no—
432C23—21no—
433C23—21no—
434C23—1222no—
435C23—2441no—
436C23—2441no—
437C23—2441no—
438C23—2441no—
439C23—2441no—
440C23—2441no—
441C23C4×D24412yes—
442C23#1451980512yes—
443C23#1451980512yes—
444C23C22×D24411yescharacteristic
445D4—1224no—
446D4—2448no—
447D4—2442no—
448C2×C4—28no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—44no—
463C2×C4—1222no—
464C2×C4—1224no—
465C2×C4—1224no—
466C2×C4—4884no—
467C2×C4—2441no—
468C2×C4—2441no—
469C2×C4—2441no—
470C2×C4—2441no—
471C2×C4—2441no—
472C2×C4—2441no—
473C2×C4—2448no—
474C2×C4—2442no—
475C2×C4—2442no—
476C2×C4—2441no—
477C2×C4—2442no—
478C2×C4—2441no—
479C2×C4—2442no—
480C2×C4—2448no—
481C2×C4C4×D24412yes—
482C2×C4#1451976112yes—
483C2×C4#1451976112yes—
484C2×C4#1451981012yes—
485C2×C4C22×C4×D6111yescharacteristic
486C2×C4#41×D6112yes—
487C2×C4C2×#1175822412yes—
488C2×C4C2×#1175822211yescharacteristic
489C8—24no—
490C8—42no—
491C22—22no—
492C22—22no—
493C22—22no—
494C22—22no—
495C22—22no—
496C22—22no—
497C22—4881no—
498C22—4881no—
499C22—4881no—
500C22—4881no—
501C22—2442no—
502C22—2442no—
503C22—2442no—
504C22#1683987514yes—
505C22C2×C4×D24411yescharacteristic
506C22C2×#1451976111yescharacteristic
507C22#1684027411yescharacteristic
508C22C2×#1451980511yescharacteristic
509C22C2×#1451980511yescharacteristic
510C22C2×#1451981011yescharacteristic
511C22#1684056211yescharacteristic
512C4—22no—
513C4—22no—
514C4—22no—
515C4—22no—
516C4—41no—
517C4—4881no—
518C4—4881no—
519C4—2444no—
520C4—2442no—
521C4—2444no—
522C4C4×#1175822412yes—
523C4#1684029212yes—
524C2—24no—
525C2—4882no—
526C2C4×#1451976111yescharacteristic
527C2C2×#1683987511yescharacteristic
528C2C4×#1451981011yescharacteristic
529C2#2145574411yescharacteristic
530C2#2145576011yescharacteristic
531C2#2145594211yescharacteristic
532C2#2145595911yescharacteristic
subgroupG/H
centerC23C22×D244
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1749C2
socleC23×C61C22×D4
prime radicalC2×#14519751C22
cosocleC22×C4×C61C24
hypercenter#1749D61
hypocenterC61#21578
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1749

Complements

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

NH ≅ G/Ntype
3. C2×C4×C4⋊Dic61C2semidirect
4. #21446447C2semidirect
5. #21446412C2semidirect
7. #21446014C2semidirect
8. #21445906C2semidirect
9. #21445716C2semidirect
10. #21445645C2semidirect
11. C61×#1749C2semidirect
18. #16840076C4semidirect
57. C2×C4⋊Dic61C2×C4semidirect
73. #14519711C2×C4semidirect
76. #14519699C2×C4semidirect
93. C22⋊C8×C61C2×C4semidirect
268. #1749D61semidirect
310. C61#21578semidirect
355. C22⋊C8C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168400762
C4×#145197512
C2×C4×C4⋊Dic612
#214464472
#214464122
#214461342
#214460142
#214459062
#214457162
#214456452
C61×#17492
#2157861