Finite Group Databasebeta

← #29684852

Subgroups — #29684852

Order 15616.

1002 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839814C211yesmaximal, characteristic
2#21448224C211yesmaximal, characteristic
3C4×C4⋊C4×D61C211yesmaximal, characteristic
4#21446578C211yesmaximal, characteristic, complemented
5#21446577C211yesmaximal, characteristic, complemented
6#21446575C211yesmaximal, characteristic, complemented
7#21446008C212yesmaximal
8#21446005C212yesmaximal
9#107×Dic61C212yesmaximal, complemented
10#21445711C212yesmaximal, complemented
11C61×#1947C211yesmaximal, characteristic, complemented
12C2×#14519735—21no—
13C2×C4⋊C4×D61C2211yescharacteristic
14C2×C4⋊C4×D61C2211yescharacteristic
15C2×C42×D61—21no—
16C2×C42×D61C2211yescharacteristic
17#16839814C418yescomplemented
18C2×#14519711C2212yes—
19C2×#14519710C2212yes—
20C2×#14519710C2212yes—
21C2×C4⋊#92418C2211yescharacteristic
22C2×#14519689C2212yes—
23C2×C8×Dic61C2212yes—
24#16839632C2211yescharacteristic
25#16839627—21no—
26#16839619—21no—
27#16839619C2212yes—
28C4⋊C4×Dic61—21no—
29C4⋊C4×Dic61C2212yes—
30C4⋊C4×Dic61C2211yescharacteristic
31C4⋊C4×Dic61C2211yescharacteristic
32C4×#11758196—21no—
33C4×C4⋊Dic61C2211yescharacteristic
34#16839613—21no—
35#95×D61—21no—
36#16839525C2212yes—
37#16839524C2212yes—
38#16839524C2212yes—
39C2×#107×C61C2212yes—
40C2×C4⋊C8×C61C2211yescharacteristic
41C61×#384C2211yescharacteristic
42C4×C4⋊C4×C61C2211yescharacteristic
43C61×#340C2212yes—
44C22×C4×D61—21no—
45C22×C4×D61—21no—
46C22×C4×D61D411yescharacteristic
47C22×C4×D61D411yescharacteristic
48C22×C4×D61C2311yescharacteristic
49C22×#92418C2312yes—
50#14519735—44no—
51C2×C4⋊Dic61—21no—
52C2×C4⋊Dic61—21no—
53C2×C4⋊Dic61C2312yes—
54C2×C4⋊Dic61C2311yescharacteristic
55C2×C4⋊Dic61C2311yescharacteristic
56C2×#11758196—21no—
57C2×#11758196—21no—
58C4⋊C4×D61—22no—
59C4⋊C4×D61—22no—
60C4⋊C4×D61—22no—
61C4⋊C4×D61—22no—
62C2×#11758198—22no—
63C2×C4×Dic61—21no—
64C2×C4×Dic61—21no—
65C2×C4×Dic61—21no—
66C2×C4×Dic61—22no—
67C2×C4×Dic61D411yescharacteristic
68C2×C4×Dic61D411yescharacteristic
69C2×C4×Dic61C2311yescharacteristic
70C42×D61—44no—
71C42×D61C2×C414yes—
72C42×D61Q814yes—
73#14519711—24no—
74#14519710—24no—
75#14519710—24no—
76C4⋊#92418C2×C414yescomplemented
77#14519689C2×C418yes—
78C8×Dic61C2×C418yescomplemented
79#14519684—21no—
80#14519684—21no—
81#14519684—21no—
82#14519684—22no—
83#14519684—22no—
84C22×C8×C61C2312yes—
85C2×C4⋊C4×C61—21no—
86C2×C4⋊C4×C61—21no—
87C2×C4⋊C4×C61C2312yes—
88C2×C4⋊C4×C61C2311yescharacteristic
89C2×C4⋊C4×C61C2311yescharacteristic
90C2×C42×C61—21no—
91C2×C42×C61C2311yescharacteristic
92#107×C61—24no—
93C4⋊C8×C61C2×C414yescomplemented
94#95×C61—21no—
95C23×D61C2×D411yescharacteristic
96C22×C4×C61—21no—
97C22×C4×C61—21no—
98C22×C4×C61—22no—
99C22×C4×C61C2×D411yescharacteristic
100C22×C4×C61C2×D411yescharacteristic
101C22×C4×C61C2411yescharacteristic
102C22×Dic61—21no—
103C22×Dic61—21no—
104C22×Dic61—22no—
105C22×Dic61C2×D411yescharacteristic
106C22×Dic61C2×D411yescharacteristic
107C22×Dic61C2×D411yescharacteristic
108C2×C4×D61—21no—
109C2×C4×D61—21no—
110C2×C4×D61—21no—
111C2×C4×D61—21no—
112C2×C4×D61—24no—
113C2×C4×D61—24no—
114C2×C4×D61—41no—
115C2×C4×D61—41no—
116C2×C4×D61—42no—
117C2×C4×D61—42no—
118C2×C4×D61—42no—
119C2×C4×D61—42no—
120C2×C4×D61C4⋊C414yes—
121C2×C4×D61C4⋊C414yes—
122C2×C4×D61D814yes—
123C2×C4×D61Q1614yes—
124C2×C4×D61C22×C412yes—
125C2×C4×D61C2×Q812yes—
126C2×C8×C61—22no—
127C2×C8×C61—22no—
128C2×C8×C61C22×C414yes—
129C4⋊C4×C61—21no—
130C4⋊C4×C61—21no—
131C4⋊C4×C61—22no—
132C4⋊C4×C61—22no—
133C4⋊C4×C61—42no—
134C4⋊C4×C61—42no—
135C4⋊C4×C61#4112yes—
136C4⋊C4×C61#4112yes—
137C42×C61—42no—
138C42×C61C22×C412yes—
139C42×C61C2×Q812yes—
140#11758198—44no—
141C4⋊Dic61—21no—
142C4⋊Dic61—21no—
143C4⋊Dic61—22no—
144C4⋊Dic61—22no—
145C4⋊Dic61—42no—
146C4⋊Dic61—42no—
147C4⋊Dic61#4112yes—
148C4⋊Dic61#4112yes—
149#11758196—24no—
150#11758196—24no—
151C4×Dic61—42no—
152C4×Dic61—42no—
153C4×Dic61—42no—
154C4×Dic61—44no—
155C4×Dic61C4⋊C414yes—
156C4×Dic61C4⋊C414yes—
157C4×Dic61C22×C412yes—
158C4×Dic61C2×Q812yes—
159C2×#92418—22no—
160C2×#92418—22no—
161C2×#92418C22×C414yes—
162C23×C61C22×D411yescharacteristic
163C22×D61—21no—
164C22×D61—21no—
165C22×D61—21no—
166C22×D61—21no—
167C22×D61C2×C4⋊C412yes—
168C22×D61C2×D812yes—
169C22×D61C2×Q1612yes—
170C2×C4×C61—21no—
171C2×C4×C61—21no—
172C2×C4×C61—22no—
173C2×C4×C61—22no—
174C2×C4×C61—41no—
175C2×C4×C61—41no—
176C2×C4×C61—41no—
177C2×C4×C61—41no—
178C2×C4×C61—42no—
179C2×C4×C61—42no—
180C2×C4×C61—42no—
181C2×C4×C61C2×C4⋊C412yes—
182C2×C4×C61C2×C4⋊C412yes—
183C2×C4×C61C2×D812yes—
184C2×C4×C61C2×Q1612yes—
185C2×C4×C61C23×C411yescharacteristic
186C2×C4×C61C22×Q811yescharacteristic
187C2×C4×C61C2×#4111yescharacteristic
188C2×C4×C61C2×#4111yescharacteristic
189C2×C4×C61C2×#4111yescharacteristic
190C2×C4×C61C2×#4111yescharacteristic
191C2×Dic61—21no—
192C2×Dic61—21no—
193C2×Dic61—21no—
194C2×Dic61—21no—
195C2×Dic61—22no—
196C2×Dic61—22no—
197C2×Dic61—41no—
198C2×Dic61—41no—
199C2×Dic61—41no—
200C2×Dic61—41no—
201C2×Dic61—42no—
202C2×Dic61—42no—
203C2×Dic61—42no—
204C2×Dic61C2×C4⋊C412yes—
205C2×Dic61C2×C4⋊C412yes—
206C2×Dic61C2×C4⋊C412yes—
207C2×Dic61C2×D812yes—
208C2×Dic61C2×Q1612yes—
209C4×D61—22no—
210C4×D61—22no—
211C4×D61—22no—
212C4×D61—22no—
213C4×D61—44no—
214C4×D61—48no—
215C4×D61—48no—
216C4×D61#107116yescomplemented
217C8×C61—24no—
218#92418—24no—
219#21358—611nomaximal, Sylow
220C22×C61C22×C4⋊C411yescharacteristic
221C22×C61#52511yescharacteristic
222C22×C61#52511yescharacteristic
223C22×C61C22×D811yescharacteristic
224C22×C61C22×Q1611yescharacteristic
225C22×C61#57511yescharacteristic
226C22×C61#57511yescharacteristic
227C2×D61—22no—
228C2×D61—22no—
229C2×D61—22no—
230C2×D61—22no—
231C2×D61—22no—
232C2×D61—22no—
233C2×D61C2×#10714yes—
234C4×C61—22no—
235C4×C61—42no—
236C4×C61—42no—
237C4×C61—82no—
238C4×C61C2×#10714yes—
239C4×C61C4×#4111yescharacteristic
240C4×C61C4×#4111yescharacteristic
241C4×C61#53311yescharacteristic
242C4×C61#53311yescharacteristic
243Dic61—22no—
244Dic61—22no—
245Dic61—42no—
246Dic61—42no—
247Dic61—82no—
248Dic61C2×#10714yes—
249C2×#431—611no—
250#2285—611no—
251C2×C4×C4⋊C4—611no—
252#1947—611no—
253#1947—611no—
254#1947—611no—
255#1947D6111yescharacteristic, complemented
256#1852—612no—
257#1852—612no—
258C4×#107—612no—
259C4×#107—612no—
260C2×C61#229611yescharacteristic
261C2×C61C22×#10711yescharacteristic
262C2×C61#291111yescharacteristic
263C2×C61#310011yescharacteristic
264C2×C61#310011yescharacteristic
265C2×C61#310411yescharacteristic
266C2×C61#310411yescharacteristic
267D61—24no—
268C2×#117—1221no—
269C22×C4⋊C4—611no—
270C22×C4⋊C4—611no—
271C22×C42—611no—
272C22×C42—1221no—
273#431—618no—
274C2×#107—612no—
275C2×#107—612no—
276C2×#107—612no—
277C2×#107C2×D6112yes—
278C2×C4⋊C8—611no—
279C2×C4⋊C8C2×D6111yescharacteristic
280C2×C22⋊C8—612no—
281C2×C4×C8—612no—
282#384—611no—
283#384C2×D6111yescharacteristic
284#382—1221no—
285#380—612no—
286#380—1221no—
287C4×C4⋊C4—611no—
288C4×C4⋊C4—611no—
289C4×C4⋊C4—612no—
290C4×C4⋊C4—611no—
291C4×C4⋊C4—1221no—
292C4×C4⋊C4—1221no—
293C4×C4⋊C4—1221no—
294C4×C4⋊C4C2×D6111yescharacteristic
295C2×#95—1221no—
296#340—612no—
297#340—612no—
298#340—612no—
299#340C2×D6112yes—
300C61#2135811yescharacteristic, complemented, Sylow
301C23×C4—611no—
302C23×C4—611no—
303C23×C4—611no—
304C23×C4—1221no—
305C23×C4—1221no—
306C22×C8—612no—
307C22×C8C22×D6112yes—
308#117—2444no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—611no—
312C2×C4⋊C4—611no—
313C2×C4⋊C4—612no—
314C2×C4⋊C4—1222no—
315C2×C4⋊C4—1222no—
316C2×C4⋊C4—1221no—
317C2×C4⋊C4—1221no—
318C2×C4⋊C4—1222no—
319C2×C4⋊C4—1222no—
320C2×C4⋊C4—1221no—
321C2×C4⋊C4—1221no—
322C2×C4⋊C4C22×D6112yes—
323C2×C4⋊C4C22×D6111yescharacteristic
324C2×C4⋊C4C22×D6111yescharacteristic
325C2×C22⋊C4—1222no—
326C2×C42—21no—
327C2×C42—611no—
328C2×C42—611no—
329C2×C42—611no—
330C2×C42—614no—
331C2×C42—614no—
332C2×C42—1222no—
333C2×C42—1221no—
334C2×C42—1221no—
335C2×C42—1221no—
336C2×C42—2444no—
337C2×C42C22×D6111yescharacteristic
338#107—24no—
339#107—1224no—
340#107—1224no—
341#107—1224no—
342C4⋊C8—614no—
343C4⋊C8C4×D6114yescomplemented
344C22⋊C8—618no—
345C4×C8—618no—
346#95—21no—
347#95—1221no—
348#95—1221no—
349#95—1222no—
350#95—1222no—
351#95—1221no—
352C24—611no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—611no—
357C22×C4—614no—
358C22×C4—614no—
359C22×C4—614no—
360C22×C4—611no—
361C22×C4—611no—
362C22×C4—612no—
363C22×C4—612no—
364C22×C4—614no—
365C22×C4—1224no—
366C22×C4—1224no—
367C22×C4—1221no—
368C22×C4—1221no—
369C22×C4—1221no—
370C22×C4—1221no—
371C22×C4—1221no—
372C22×C4—1221no—
373C22×C4—1222no—
374C22×C4—2441no—
375C22×C4—2441no—
376C22×C4—2442no—
377C22×C4—2442no—
378C22×C4—2442no—
379C22×C4—2442no—
380C22×C4D4×D6111yescharacteristic
381C22×C4D4×D6111yescharacteristic
382C22×C4C23×D6111yescharacteristic
383C2×C8—22no—
384C2×C8—22no—
385C2×C8—614no—
386C2×C8—1222no—
387C2×C8—1222no—
388C2×C8C2×C4×D6114yes—
389C4⋊C4—21no—
390C4⋊C4—21no—
391C4⋊C4—22no—
392C4⋊C4—22no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—612no—
396C4⋊C4—612no—
397C4⋊C4—1224no—
398C4⋊C4—1224no—
399C4⋊C4—1221no—
400C4⋊C4—1221no—
401C4⋊C4—1222no—
402C4⋊C4—1222no—
403C4⋊C4—2442no—
404C4⋊C4—2442no—
405C4⋊C4#1175822412yes—
406C4⋊C4#1175822612yes—
407C22⋊C4—2444no—
408C42—42no—
409C42—612no—
410C42—612no—
411C42—614no—
412C42—614no—
413C42—2442no—
414C42—2442no—
415C42—2444no—
416C42—2442no—
417C42C2×C4×D6112yes—
418C42Q8×D6112yes—
419C23—612no—
420C23—612no—
421C23—612no—
422C23—1221no—
423C23—1221no—
424C23—1221no—
425C23—1221no—
426C23C2×D4×D6111yescharacteristic
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—612no—
439C2×C4—612no—
440C2×C4—612no—
441C2×C4—6116no—
442C2×C4—612no—
443C2×C4—612no—
444C2×C4—1221no—
445C2×C4—1221no—
446C2×C4—1221no—
447C2×C4—1221no—
448C2×C4—1222no—
449C2×C4—1222no—
450C2×C4—1222no—
451C2×C4—1222no—
452C2×C4—1222no—
453C2×C4—1222no—
454C2×C4—2444no—
455C2×C4—2448no—
456C2×C4—2448no—
457C2×C4—2442no—
458C2×C4—2442no—
459C2×C4—2442no—
460C2×C4—2441no—
461C2×C4—2441no—
462C2×C4—2441no—
463C2×C4—2441no—
464C2×C4C4⋊C4×D6112yes—
465C2×C4C4⋊C4×D6112yes—
466C2×C4D8×D6112yes—
467C2×C4Q16×D6112yes—
468C2×C4C22×C4×D6111yescharacteristic
469C2×C4C2×Q8×D6111yescharacteristic
470C2×C4#41×D6111yescharacteristic
471C2×C4#41×D6111yescharacteristic
472C2×C4C2×#1175822411yescharacteristic
473C2×C4C2×#1175822611yescharacteristic
474C8—24no—
475C8—1224no—
476C22—614no—
477C22—1222no—
478C22—1222no—
479C22—1222no—
480C22—1222no—
481C22—1222no—
482C22—1222no—
483C22C2×C4⋊C4×D6111yescharacteristic
484C22#1684027211yescharacteristic
485C22#1684027511yescharacteristic
486C22C2×D8×D6111yescharacteristic
487C22C2×Q16×D6111yescharacteristic
488C22#1684055811yescharacteristic
489C22#1684056111yescharacteristic
490C4—22no—
491C4—42no—
492C4—42no—
493C4—82no—
494C4—614no—
495C4—1222no—
496C4—1222no—
497C4—4882no—
498C4—2442no—
499C4—2442no—
500C4#107×D6114yes—
501C4C4×#1175822411yescharacteristic
502C4C4×#1175822611yescharacteristic
503C4#1684031711yescharacteristic
504C4#1684031811yescharacteristic
505C2—1224no—
506C2#2144830011yescharacteristic
507C2C2×#107×D6111yescharacteristic
508C2#2145440611yescharacteristic
509C2#2145581811yescharacteristic
510C2#2145582111yescharacteristic
511C2#2145585711yescharacteristic
512C2#2145586011yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1947C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
cosocleC22×C4×C61C24
hypercenter#1947D61
hypocenterC61#21358
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1947

Complements

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

NH ≅ G/Ntype
4. #21446578C2semidirect
5. #21446577C2semidirect
6. #21446575C2semidirect
9. #107×Dic61C2semidirect
10. #21445711C2semidirect
11. C61×#1947C2semidirect
17. #16839814C4semidirect
76. C4⋊#92418C2×C4semidirect
78. C8×Dic61C2×C4semidirect
93. C4⋊C8×C61C2×C4semidirect
216. C4×D61#107semidirect
255. #1947D61semidirect
300. C61#21358semidirect
343. C4⋊C8C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168398142
#214482242
C4×C4⋊C4×D612
#214465782
#214465772
#214465752
#214460082
#214460052
#107×Dic612
#214457112
C61×#19472
#2135861