Finite Group Databasebeta

← #29687197

Subgroups — #29687197

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×#16840073C211yesmaximal, characteristic
2#121×Dic61C211yesmaximal, characteristic
3C2×C4⋊C4×Dic61C211yesmaximal, characteristic, complemented
4#21446449C212yesmaximal, complemented
5#21446414C212yesmaximal, complemented
6#21446012C211yesmaximal, characteristic, complemented
7#21445906C211yesmaximal, characteristic, complemented
8#21445715C212yesmaximal, complemented
9#21445646C212yesmaximal, complemented
10#21445583C211yesmaximal, characteristic, complemented
11C61×#1874C211yesmaximal, characteristic
12C2×D4×Dic61—21no—
13C2×D4×Dic61C2211yescharacteristic
14C22×C4⋊Dic61C2211yescharacteristic
15C22×C4×Dic61—21no—
16C22×C4×Dic61C2211yescharacteristic
17#16840073C418yescomplemented
18C2×#14519710C2212yes—
19C2×#14519708C2212yes—
20C2×#14519698C2212yes—
21C2×C22⋊#92418C2211yescharacteristic
22C2×#14519687C2212yes—
23#16839665C2211yescharacteristic
24#16839661C2211yescharacteristic
25#16839655—21no—
26#16839642—21no—
27#16839622C2211yescharacteristic
28C4⋊C4×Dic61—21no—
29C4⋊C4×Dic61—21no—
30C4⋊C4×Dic61C2212yes—
31C4⋊C4×Dic61C2212yes—
32C4⋊C4×Dic61C2211yescharacteristic
33C22⋊C4×Dic61—21no—
34C2×#14519684—21no—
35#16839525C2212yes—
36#16839525C2212yes—
37#16839524C2212yes—
38#16839516C2212yes—
39C2×#121×C61C2211yescharacteristic
40C22×C4⋊C4×C61C2211yescharacteristic
41C2×#102×C61C2212yes—
42C2×C22⋊C8×C61C2211yescharacteristic
43C61×#340C2212yes—
44C23×Dic61—21no—
45C23×Dic61—21no—
46C23×Dic61D411yescharacteristic
47C23×Dic61D411yescharacteristic
48C22×#92418C2312yes—
49D4×Dic61—24no—
50D4×Dic61—44no—
51C2×C4⋊Dic61—21no—
52C2×C4⋊Dic61—22no—
53C2×C4⋊Dic61—21no—
54C2×C4⋊Dic61C2×C414yescomplemented
55C2×C4⋊Dic61C2311yescharacteristic
56C2×C4⋊Dic61C2311yescharacteristic
57C2×C4⋊Dic61C2312yes—
58C2×C22⋊Dic61—21no—
59C2×C22⋊Dic61—21no—
60C2×C22⋊Dic61—21no—
61C2×C22⋊Dic61—21no—
62C2×C4×Dic61—24no—
63C2×C4×Dic61—22no—
64C2×C4×Dic61—22no—
65C2×C4×Dic61—21no—
66C2×C4×Dic61—44no—
67C2×C4×Dic61D411yescharacteristic
68C2×C4×Dic61D411yescharacteristic
69C2×C4×Dic61C2311yescharacteristic
70C2×C4×Dic61C2311yescharacteristic
71#14519710C2×C418yescomplemented
72#14519708—24no—
73#14519698C2×C418yescomplemented
74C22⋊#92418C2×C414yescomplemented
75#14519687—24no—
76#14519684—21no—
77#14519684—22no—
78#14519684—22no—
79#14519684—21no—
80#14519684—21no—
81C22×D4×C61—21no—
82C22×D4×C61C2311yescharacteristic
83C23×C4×C61—21no—
84C23×C4×C61C2311yescharacteristic
85C22×C8×C61C2312yes—
86#121×C61—22no—
87#121×C61C2×C414yes—
88C2×C4⋊C4×C61—21no—
89C2×C4⋊C4×C61—24no—
90C2×C4⋊C4×C61C2311yescharacteristic
91C2×C4⋊C4×C61C2312yes—
92C2×C22⋊C4×C61—21no—
93#102×C61—24no—
94C22⋊C8×C61—22no—
95C24×C61—21no—
96C24×C61C2×D411yescharacteristic
97C2×D4×C61—24no—
98C2×D4×C61—22no—
99C2×D4×C61—22no—
100C2×D4×C61—21no—
101C2×D4×C61—42no—
102C2×D4×C61—42no—
103C2×D4×C61C22×C412yes—
104C22×C4×C61—22no—
105C22×C4×C61—22no—
106C22×C4×C61—24no—
107C22×C4×C61—21no—
108C22×C4×C61—22no—
109C22×C4×C61—42no—
110C22×C4×C61—42no—
111C22×C4×C61C22×C412yes—
112C22×C4×C61C2×D411yescharacteristic
113C22×C4×C61#4112yes—
114C22×C4×C61C2411yescharacteristic
115C22×Dic61—22no—
116C22×Dic61—22no—
117C22×Dic61—22no—
118C22×Dic61—24no—
119C22×Dic61—22no—
120C22×Dic61—22no—
121C22×Dic61—22no—
122C22×Dic61—21no—
123C22×Dic61—21no—
124C22×Dic61—41no—
125C22×Dic61—42no—
126C22×Dic61—42no—
127C22×Dic61—42no—
128C22×Dic61—42no—
129C22×Dic61—42no—
130C22×Dic61C2×D411yescharacteristic
131C22×Dic61C2×D411yescharacteristic
132C22×Dic61C2×D411yescharacteristic
133C22×Dic61C2×D411yescharacteristic
134C2×C8×C61—22no—
135C2×C8×C61—22no—
136C2×C8×C61—22no—
137C4⋊C4×C61—24no—
138C4⋊C4×C61—22no—
139C4⋊C4×C61—21no—
140C4⋊C4×C61C22×C412yes—
141C4⋊C4×C61#4114yes—
142C22⋊C4×C61—24no—
143C22⋊Dic61—42no—
144C22⋊Dic61—42no—
145C22⋊Dic61—42no—
146C22⋊Dic61—42no—
147C4⋊Dic61—24no—
148C4⋊Dic61—22no—
149C4⋊Dic61—22no—
150C4⋊Dic61—22no—
151C4⋊Dic61—42no—
152C4⋊Dic61C22×C414yes—
153C4×Dic61—22no—
154C4×Dic61—22no—
155C4×Dic61—22no—
156C4×Dic61—44no—
157C4×Dic61—44no—
158C4×Dic61—42no—
159C4×Dic61#4114yes—
160C2×#92418—22no—
161C2×#92418—22no—
162C2×#92418C22×C414yes—
163C23×C61—21no—
164C23×C61—21no—
165C23×C61—21no—
166C23×C61—21no—
167C23×C61—22no—
168C23×C61—41no—
169C23×C61—41no—
170C23×C61—41no—
171C23×C61—41no—
172C23×C61—41no—
173C23×C61—41no—
174C23×C61C4×D412yes—
175C23×C61AGL(1,ℤ/8)12yes—
176C23×C61AGL(1,ℤ/8)12yes—
177C23×C61C22×D411yescharacteristic
178D4×C61—24no—
179D4×C61—48no—
180D4×C61—42no—
181C2×C4×C61—21no—
182C2×C4×C61—21no—
183C2×C4×C61—24no—
184C2×C4×C61—24no—
185C2×C4×C61—22no—
186C2×C4×C61—24no—
187C2×C4×C61—48no—
188C2×C4×C61—42no—
189C2×C4×C61—41no—
190C2×C4×C61—41no—
191C2×C4×C61—42no—
192C2×C4×C61C4×D412yes—
193C2×C4×C61C23×C411yescharacteristic
194C2×C4×C61C2×#4112yes—
195C2×C4×C61C2×#4112yes—
196C2×C4×C61C2×#4111yescharacteristic
197C2×Dic61—28no—
198C2×Dic61—21no—
199C2×Dic61—21no—
200C2×Dic61—21no—
201C2×Dic61—21no—
202C2×Dic61—21no—
203C2×Dic61—21no—
204C2×Dic61—24no—
205C2×Dic61—21no—
206C2×Dic61—21no—
207C2×Dic61—21no—
208C2×Dic61—44no—
209C2×Dic61—42no—
210C2×Dic61—42no—
211C2×Dic61—48no—
212C2×Dic61—41no—
213C2×Dic61—41no—
214C2×Dic61—41no—
215C2×Dic61—41no—
216C2×Dic61—41no—
217C2×Dic61—41no—
218C2×Dic61—84no—
219C2×Dic61#12312yes—
220C2×Dic61#12312yes—
221C2×Dic61#13712yes—
222C8×C61—42no—
223#92418—24no—
224#21578—611nomaximal, Sylow
225C22×C61—22no—
226C22×C61—22no—
227C22×C61—22no—
228C22×C61—22no—
229C22×C61—22no—
230C22×C61—22no—
231C22×C61—42no—
232C22×C61—42no—
233C22×C61—42no—
234C22×C61—81no—
235C22×C61—81no—
236C22×C61—81no—
237C22×C61—81no—
238C22×C61#44214yes—
239C22×C61C2×C4×D411yescharacteristic
240C22×C61C2×#12311yescharacteristic
241C22×C61#52511yescharacteristic
242C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
243C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
244C22×C61C2×#13711yescharacteristic
245C22×C61#57511yescharacteristic
246C4×C61—22no—
247C4×C61—44no—
248C4×C61—42no—
249C4×C61C4×#4112yes—
250C4×C61#52912yes—
251Dic61—22no—
252Dic61—22no—
253Dic61—22no—
254Dic61—41no—
255Dic61—44no—
256Dic61—81no—
257Dic61—81no—
258C2×#480—611no—
259C4×#121—611no—
260C2×C4×C4⋊C4—611no—
261#1926—612no—
262#1921—612no—
263#1874D6111yescharacteristic
264#1853—611no—
265#1823—611no—
266#1776—612no—
267#1762—612no—
268#1749—611no—
269C2×C61—24no—
270C2×C61—82no—
271C2×C61C4×#12311yescharacteristic
272C2×C61C2×#44211yescharacteristic
273C2×C61C4×#13711yescharacteristic
274C2×C61#309211yescharacteristic
275C2×C61#309411yescharacteristic
276C2×C61#311211yescharacteristic
277C2×C61#311411yescharacteristic
278C2×#121C2×D6111yescharacteristic
279C2×C4×D4—611no—
280C2×C4×D4—1221no—
281C22×C4⋊C4—611no—
282C22×C4⋊C4C2×D6111yescharacteristic
283C22×C42—611no—
284C22×C42—1221no—
285#480—618no—
286C2×#107—612no—
287C2×#106—612no—
288C2×#102—612no—
289C2×#102C2×D6112yes—
290C2×C22⋊C8—611no—
291C2×C22⋊C8C2×D6111yescharacteristic
292C2×#97—612no—
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#340C2×D6112yes—
309#336—612no—
310C61#2157811yescharacteristic, complemented, Sylow
311C22×D4—21no—
312C22×D4C22×D6111yescharacteristic
313C23×C4—21no—
314C23×C4—611no—
315C23×C4—611no—
316C23×C4—1221no—
317C23×C4—1221no—
318C23×C4C22×D6111yescharacteristic
319C22×C8—612no—
320C22×C8C22×D6112yes—
321#121—22no—
322#121C4×D6114yes—
323C4×D4—1224no—
324C4×D4—2444no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—24no—
327C2×C4⋊C4—614no—
328C2×C4⋊C4—612no—
329C2×C4⋊C4—611no—
330C2×C4⋊C4—611no—
331C2×C4⋊C4—1221no—
332C2×C4⋊C4—1222no—
333C2×C4⋊C4—1221no—
334C2×C4⋊C4C22×D6111yescharacteristic
335C2×C4⋊C4C22×D6112yes—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—1221no—
338C2×C22⋊C4—1221no—
339C2×C22⋊C4—1221no—
340C2×C22⋊C4—1221no—
341C2×C42—611no—
342C2×C42—611no—
343C2×C42—611no—
344C2×C42—611no—
345C2×C42—1224no—
346C2×C42—1221no—
347C2×C42—1222no—
348C2×C42—1222no—
349C2×C42—2444no—
350#107—618no—
351#106—1224no—
352#102—24no—
353#102—618no—
354C22⋊C8—22no—
355C22⋊C8—614no—
356#97—1224no—
357#95—1222no—
358#95—1222no—
359#95—1221no—
360#95—1221no—
361#95—1221no—
362C24—21no—
363C24D4×D6111yescharacteristic
364C2×D4—24no—
365C2×D4—22no—
366C2×D4—22no—
367C2×D4—21no—
368C2×D4—42no—
369C2×D4—42no—
370C2×D4C2×C4×D6112yes—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—21no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—611no—
379C22×C4—611no—
380C22×C4—611no—
381C22×C4—611no—
382C22×C4—1222no—
383C22×C4—1222no—
384C22×C4—1222no—
385C22×C4—1222no—
386C22×C4—1222no—
387C22×C4—1224no—
388C22×C4—1222no—
389C22×C4—1221no—
390C22×C4—1221no—
391C22×C4—2442no—
392C22×C4—2441no—
393C22×C4—2442no—
394C22×C4—2442no—
395C22×C4—2442no—
396C22×C4—2442no—
397C22×C4C2×C4×D6112yes—
398C22×C4D4×D6111yescharacteristic
399C22×C4#1175822412yes—
400C22×C4C23×D6111yescharacteristic
401C2×C8—22no—
402C2×C8—22no—
403C2×C8—22no—
404C2×C8—614no—
405C2×C8—1222no—
406C2×C8—1222no—
407C4⋊C4—24no—
408C4⋊C4—22no—
409C4⋊C4—21no—
410C4⋊C4—614no—
411C4⋊C4—1222no—
412C4⋊C4—1224no—
413C4⋊C4—1222no—
414C4⋊C4—1222no—
415C4⋊C4—2442no—
416C4⋊C4C2×C4×D6112yes—
417C4⋊C4#1175822414yes—
418C22⋊C4—24no—
419C22⋊C4—2442no—
420C22⋊C4—2442no—
421C22⋊C4—2442no—
422C22⋊C4—2442no—
423C42—614no—
424C42—1222no—
425C42—1222no—
426C42—1222no—
427C42—2444no—
428C42—2444no—
429C42—2442no—
430C23—21no—
431C23—21no—
432C23—21no—
433C23—21no—
434C23—22no—
435C23—41no—
436C23—41no—
437C23—41no—
438C23—41no—
439C23—41no—
440C23—41no—
441C23#1451973712yes—
442C23#1451980112yes—
443C23#1451980412yes—
444C23C2×D4×D6111yescharacteristic
445D4—24no—
446D4—48no—
447D4—42no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—22no—
453C2×C4—24no—
454C2×C4—48no—
455C2×C4—42no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—42no—
459C2×C4—612no—
460C2×C4—612no—
461C2×C4—612no—
462C2×C4—1228no—
463C2×C4—1221no—
464C2×C4—1221no—
465C2×C4—1221no—
466C2×C4—1221no—
467C2×C4—1221no—
468C2×C4—1221no—
469C2×C4—1224no—
470C2×C4—1221no—
471C2×C4—1221no—
472C2×C4—1221no—
473C2×C4—4884no—
474C2×C4—2444no—
475C2×C4—2441no—
476C2×C4—2441no—
477C2×C4—2441no—
478C2×C4—2441no—
479C2×C4—2441no—
480C2×C4—2441no—
481C2×C4—2442no—
482C2×C4—2442no—
483C2×C4—2448no—
484C2×C4#1451973712yes—
485C2×C4C22×C4×D6111yescharacteristic
486C2×C4#41×D6112yes—
487C2×C4C2×#1175822412yes—
488C2×C4C2×#1175822411yescharacteristic
489C8—42no—
490C8—1224no—
491C22—22no—
492C22—22no—
493C22—22no—
494C22—22no—
495C22—22no—
496C22—22no—
497C22—42no—
498C22—42no—
499C22—42no—
500C22—81no—
501C22—81no—
502C22—81no—
503C22—81no—
504C22#1683987114yes—
505C22C2×#1451973711yescharacteristic
506C22#123×D6111yescharacteristic
507C22#1684026811yescharacteristic
508C22C2×#1451980111yescharacteristic
509C22C2×#1451980411yescharacteristic
510C22#137×D6111yescharacteristic
511C22#1684055811yescharacteristic
512C4—22no—
513C4—42no—
514C4—44no—
515C4—1222no—
516C4—1222no—
517C4—1222no—
518C4—4881no—
519C4—4881no—
520C4—2444no—
521C4—2441no—
522C4C4×#1175822412yes—
523C4#1684029312yes—
524C2—24no—
525C2—82no—
526C2#2144834411yescharacteristic
527C2C2×#1683987111yescharacteristic
528C2#2145464311yescharacteristic
529C2#2145573811yescharacteristic
530C2#2145575611yescharacteristic
531C2#2145593611yescharacteristic
532C2#2145595211yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1874C2
socleC23×C61C22×D4
prime radicalC2×#121×C61C22
cosocleC22×C4×C61C24
hypercenter#1874D61
hypocenterC61#21578
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1874

Complements

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

NH ≅ G/Ntype
3. C2×C4⋊C4×Dic61C2semidirect
4. #21446449C2semidirect
5. #21446414C2semidirect
6. #21446012C2semidirect
7. #21445906C2semidirect
8. #21445715C2semidirect
9. #21445646C2semidirect
10. #21445583C2semidirect
17. #16840073C4semidirect
54. C2×C4⋊Dic61C2×C4semidirect
71. #14519710C2×C4semidirect
73. #14519698C2×C4semidirect
74. C22⋊#92418C2×C4semidirect
310. C61#21578semidirect

Maximal subgroups

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

subgroupindex
C2×#168400732
#121×Dic612
C2×C4⋊C4×Dic612
#214464492
#214464142
#214460122
#214459062
#214457152
#214456462
#214455832
C61×#18742
#2157861