Finite Group Databasebeta

← #29684533

Subgroups — #29684533

Order 15616.

1250 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840318C211yesmaximal, characteristic
2C2×#16839784C211yesmaximal, characteristic
3C4×C4⋊C4×D61C211yesmaximal, characteristic
4#21446777C211yesmaximal, characteristic, complemented
5#21446776C211yesmaximal, characteristic, complemented
6#21446774C211yesmaximal, characteristic, complemented
7#21446769C212yesmaximal, complemented
8#21446765C212yesmaximal, complemented
9#21445537C212yesmaximal
10#21445537C212yesmaximal
11C61×#1989C211yesmaximal, characteristic, complemented
12C22×#11758189C2212yes—
13#16840318—24no—
14#16840318C418yescomplemented
15C2×#14519776C2211yescharacteristic
16C2×#14519776C2211yescharacteristic
17C2×#14519769C2211yescharacteristic
18C2×#14519755—21no—
19C2×#14519755—21no—
20C2×#14519742—21no—
21C2×#14519742—21no—
22C2×#14519735C2211yescharacteristic
23C2×#14519735C2211yescharacteristic
24C2×C4⋊C4×D61C2212yes—
25C2×C42×D61—21no—
26C2×C42×D61C2211yescharacteristic
27#16839784—24no—
28C2×C4⋊#92418C2211yescharacteristic
29C2×#14519706C2212yes—
30C4⋊C4×Dic61C2212yes—
31C4×#11758196—21no—
32C4×C4⋊Dic61C2211yescharacteristic
33#16839613—21no—
34#95×D61—21no—
35#16839525C2212yes—
36#16839525C2212yes—
37#16839524C2212yes—
38#16839524C2212yes—
39#16839524C2212yes—
40#16839524C2212yes—
41C2×#125×C61C2211yescharacteristic
42C2×C4⋊C8×C61C2211yescharacteristic
43C4×C4⋊C4×C61C2211yescharacteristic
44C61×#340C2212yes—
45C61×#340C2212yes—
46C22×Dic122—21no—
47C22×Dic122—21no—
48C22×C4×D61—22no—
49C22×C4×D61D411yescharacteristic
50C22×C4×D61D411yescharacteristic
51C22×C4×D61C2311yescharacteristic
52C2×#11758189—24no—
53C2×#11758189—28no—
54C22×#92418C2312yes—
55#14519776—22no—
56#14519776—22no—
57#14519776C2×C414yescomplemented
58#14519776C2×C414yescomplemented
59#14519769—22no—
60#14519769C2×C414yescomplemented
61#14519755—28no—
62#14519755—28no—
63#14519742—28no—
64#14519742—28no—
65#14519735—22no—
66#14519735—22no—
67#14519735C2×C414yes—
68#14519735C2×C414yes—
69C2×C4⋊Dic61C2311yescharacteristic
70C2×C4⋊Dic61C2311yescharacteristic
71C2×C4⋊Dic61C2312yes—
72C2×#11758196—21no—
73C2×#11758196—21no—
74C2×#11758196—22no—
75C2×#11758196—21no—
76C2×#11758196—21no—
77C4⋊C4×D61—24no—
78C4⋊C4×D61Q818yescomplemented
79C2×#11758198—21no—
80C2×#11758198—21no—
81C2×C4×Dic61—21no—
82C2×C4×Dic61—21no—
83C2×C4×Dic61—21no—
84C2×C4×Dic61—22no—
85C2×C4×Dic61D411yescharacteristic
86C2×C4×Dic61D411yescharacteristic
87C2×C4×Dic61C2311yescharacteristic
88C42×D61—44no—
89C42×D61C2×C414yes—
90C42×D61D414yes—
91C4⋊#92418—22no—
92#14519706—24no—
93#14519684—21no—
94#14519684—22no—
95C22×C8×C61C2312yes—
96#125×C61—22no—
97#125×C61C2×C414yescomplemented
98C2×C4⋊C4×C61—21no—
99C2×C4⋊C4×C61—21no—
100C2×C4⋊C4×C61C2311yescharacteristic
101C2×C4⋊C4×C61C2311yescharacteristic
102C2×C4⋊C4×C61C2312yes—
103C2×C42×C61—21no—
104C2×C42×C61C2311yescharacteristic
105C4⋊C8×C61—22no—
106#95×C61—21no—
107C23×D61C2×D411yescharacteristic
108C22×C4×C61—22no—
109C22×C4×C61—21no—
110C22×C4×C61—21no—
111C22×C4×C61C2×D411yescharacteristic
112C22×C4×C61C2×D411yescharacteristic
113C22×C4×C61C2411yescharacteristic
114C22×Dic61—22no—
115C22×Dic61—21no—
116C22×Dic61—21no—
117C22×Dic61C2×D411yescharacteristic
118C22×Dic61C2×D411yescharacteristic
119C22×Dic61C2×D411yescharacteristic
120C2×C4×D61—22no—
121C2×C4×D61—22no—
122C2×C4×D61—28no—
123C2×C4×D61—42no—
124C2×C4×D61—44no—
125C2×C4×D61—44no—
126C2×C4×D61C22⋊C414yes—
127C2×C4×D61C22⋊C414yes—
128C2×C4×D61C22×C412yes—
129C2×C4×D61C2×D412yes—
130C2×C4×D61C2×Q814yes—
131C2×C4×D61#4114yes—
132C2×Dic122—24no—
133C2×Dic122—24no—
134C2×Dic122—44no—
135C2×Dic122—42no—
136C2×Dic122—42no—
137C2×C8×C61—22no—
138C2×C8×C61—24no—
139C4⋊C4×C61—24no—
140C4⋊C4×C61—24no—
141C4⋊C4×C61—21no—
142C4⋊C4×C61—21no—
143C4⋊C4×C61—22no—
144C4⋊C4×C61C22×C412yes—
145C4⋊C4×C61C22×C412yes—
146C4⋊C4×C61C2×Q814yes—
147C42×C61—42no—
148C42×C61C22×C412yes—
149C42×C61C2×D412yes—
150#11758198—24no—
151#11758198—24no—
152C4⋊Dic61—21no—
153C4⋊Dic61—21no—
154C4⋊Dic61—22no—
155C4⋊Dic61C22×C412yes—
156C4⋊Dic61C22×C412yes—
157C4⋊Dic61C2×Q814yes—
158#11758196—24no—
159#11758196—24no—
160#11758196—24no—
161#11758196—24no—
162#11758196—28no—
163C4×Dic61—24no—
164C4×Dic61—24no—
165C4×Dic61—42no—
166C4×Dic61—44no—
167C4×Dic61C22⋊C414yes—
168C4×Dic61C22⋊C414yes—
169C4×Dic61C22×C412yes—
170C4×Dic61C2×D412yes—
171C2×#92418—22no—
172C2×#92418—24no—
173#11758189—48no—
174C23×C61C22×D411yescharacteristic
175C22×D61—21no—
176C22×D61—21no—
177C22×D61—22no—
178C22×D61C2×C22⋊C412yes—
179C22×D61C22⋊Q814yes—
180C2×C4×C61—21no—
181C2×C4×C61—21no—
182C2×C4×C61—22no—
183C2×C4×C61—22no—
184C2×C4×C61—22no—
185C2×C4×C61—24no—
186C2×C4×C61—42no—
187C2×C4×C61—41no—
188C2×C4×C61—41no—
189C2×C4×C61—42no—
190C2×C4×C61—42no—
191C2×C4×C61C2×C22⋊C412yes—
192C2×C4×C61C2×C22⋊C412yes—
193C2×C4×C61C23×C411yescharacteristic
194C2×C4×C61C22×D411yescharacteristic
195C2×C4×C61C22×Q812yes—
196C2×C4×C61C2×#4112yes—
197C2×Dic61—22no—
198C2×Dic61—22no—
199C2×Dic61—22no—
200C2×Dic61—24no—
201C2×Dic61—42no—
202C2×Dic61—41no—
203C2×Dic61—41no—
204C2×Dic61—42no—
205C2×Dic61—42no—
206C2×Dic61C2×C22⋊C412yes—
207C2×Dic61C2×C22⋊C412yes—
208C2×Dic61C2×C22⋊C412yes—
209C2×Dic61C22⋊Q814yes—
210C2×Dic61AGL(1,ℤ/8)12yes—
211C2×Dic61#13712yes—
212C4×D61—28no—
213C4×D61—44no—
214C4×D61—416no—
215C4×D61C4×Q818yes—
216C4×D61#12414yes—
217C4×D61C4⋊Q814yes—
218Dic122—48no—
219Dic122—48no—
220C8×C61—42no—
221#92418—42no—
222#21312—611nomaximal, Sylow
223C22×C61C22×C22⋊C411yescharacteristic
224C22×C61C2×C22⋊Q812yes—
225C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
226C22×C61C2×#13711yescharacteristic
227C22×C61#57512yes—
228C2×D61—24no—
229C2×D61—24no—
230C2×D61—22no—
231C2×D61—22no—
232C2×D61#39114yes—
233C4×C61—22no—
234C4×C61—22no—
235C4×C61—44no—
236C4×C61—42no—
237C4×C61—42no—
238C4×C61C2×C4×Q812yes—
239C4×C61C2×#12411yescharacteristic
240C4×C61C2×C4⋊Q811yescharacteristic
241Dic61—22no—
242Dic61—44no—
243Dic61—42no—
244Dic61—42no—
245Dic61#39114yes—
246Dic61#41714yes—
247C2×#533—611no—
248C2×#423—611no—
249C2×C4×C4⋊C4—611no—
250#1989—611no—
251#1989—611no—
252#1989—611no—
253#1989D6111yescharacteristic, complemented
254#1988—612no—
255#1987—612no—
256#1736—612no—
257#1736—612no—
258C2×C61C2×#39111yescharacteristic
259C2×C61C2×#41711yescharacteristic
260C2×C61#289611yescharacteristic
261C2×C61#307712yes—
262C2×C61#307812yes—
263D61—24no—
264C22×M16—612no—
265#533—618no—
266#533—1224no—
267C2×C4⋊Q8—611no—
268C2×C4⋊Q8—611no—
269C2×#125—611no—
270C2×#125C2×D6111yescharacteristic
271C2×C22⋊Q8—1221no—
272C2×C22⋊Q8—1221no—
273C2×C4×Q8—1221no—
274C2×C4×Q8—1221no—
275C2×#117—611no—
276C2×#117—611no—
277C22×C4⋊C4—612no—
278C22×C42—611no—
279C22×C42—1221no—
280#423—1224no—
281C2×C4⋊C8—612no—
282C2×C4⋊C8—611no—
283C2×C4⋊C8C2×D6111yescharacteristic
284C4×C4⋊C4—612no—
285C4×C4⋊C4—611no—
286C4×C4⋊C4—1221no—
287C4×C4⋊C4—1221no—
288C4×C4⋊C4C2×D6111yescharacteristic
289C2×#95—1221no—
290#340—612no—
291#340—612no—
292#340—612no—
293#340—612no—
294#340—612no—
295#340—612no—
296#340C2×D6112yes—
297#340C2×D6112yes—
298C61#2131211yescharacteristic, complemented, Sylow
299C22×Q8—1221no—
300C22×Q8—1221no—
301C23×C4—611no—
302C23×C4—611no—
303C23×C4—611no—
304C23×C4—1222no—
305C2×M16—1224no—
306C2×M16—1228no—
307C22×C8—612no—
308C22×C8C22×D6112yes—
309C4⋊Q8—614no—
310C4⋊Q8—614no—
311C4⋊Q8—1222no—
312C4⋊Q8—1222no—
313#125—22no—
314#125—614no—
315#125—1222no—
316#125C4×D6114yescomplemented
317C22⋊Q8—1228no—
318C22⋊Q8—1228no—
319C4×Q8—1228no—
320C4×Q8—1228no—
321#117—614no—
322#117—614no—
323#117—1222no—
324#117—1222no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—612no—
328C2×C4⋊C4—611no—
329C2×C4⋊C4—611no—
330C2×C4⋊C4—618no—
331C2×C4⋊C4—1224no—
332C2×C4⋊C4—1222no—
333C2×C4⋊C4—1221no—
334C2×C4⋊C4—1221no—
335C2×C4⋊C4—1221no—
336C2×C4⋊C4—1221no—
337C2×C4⋊C4C22×D6111yescharacteristic
338C2×C4⋊C4C22×D6111yescharacteristic
339C2×C4⋊C4C22×D6112yes—
340C2×C22⋊C4—1221no—
341C2×C22⋊C4—1221no—
342C2×C42—21no—
343C2×C42—611no—
344C2×C42—611no—
345C2×C42—611no—
346C2×C42—614no—
347C2×C42—614no—
348C2×C42—1221no—
349C2×C42—1221no—
350C2×C42—1222no—
351C2×C42—1221no—
352C2×C42—2444no—
353C2×C42C22×D6111yescharacteristic
354C4⋊C8—22no—
355C4⋊C8—1224no—
356C4⋊C8—1222no—
357#95—21no—
358#95—1222no—
359#95—1221no—
360C24—611no—
361C2×Q8—1224no—
362C2×Q8—1224no—
363C2×Q8—2442no—
364C2×Q8—2442no—
365C2×Q8—2444no—
366C22×C4—22no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—611no—
370C22×C4—611no—
371C22×C4—612no—
372C22×C4—612no—
373C22×C4—614no—
374C22×C4—614no—
375C22×C4—611no—
376C22×C4—614no—
377C22×C4—614no—
378C22×C4—1222no—
379C22×C4—1222no—
380C22×C4—1228no—
381C22×C4—1222no—
382C22×C4—1221no—
383C22×C4—1221no—
384C22×C4—2442no—
385C22×C4—2444no—
386C22×C4—2444no—
387C22×C4D4×D6111yescharacteristic
388C22×C4D4×D6111yescharacteristic
389C22×C4C23×D6111yescharacteristic
390M16—2448no—
391C2×C8—22no—
392C2×C8—24no—
393C2×C8—1222no—
394C2×C8—1224no—
395C4⋊C4—21no—
396C4⋊C4—21no—
397C4⋊C4—22no—
398C4⋊C4—24no—
399C4⋊C4—24no—
400C4⋊C4—612no—
401C4⋊C4—612no—
402C4⋊C4—614no—
403C4⋊C4—1228no—
404C4⋊C4—1224no—
405C4⋊C4—1224no—
406C4⋊C4—1221no—
407C4⋊C4—1221no—
408C4⋊C4—1222no—
409C4⋊C4—1224no—
410C4⋊C4—1224no—
411C4⋊C4C2×C4×D6112yes—
412C4⋊C4C2×C4×D6112yes—
413C4⋊C4Q8×D6114yes—
414C22⋊C4—1224no—
415C22⋊C4—1224no—
416C42—42no—
417C42—612no—
418C42—612no—
419C42—614no—
420C42—614no—
421C42—1224no—
422C42—1224no—
423C42—2442no—
424C42—2444no—
425C42C2×C4×D6112yes—
426C42D4×D6112yes—
427C23—612no—
428C23—614no—
429C23—1221no—
430C23—1221no—
431C23—1222no—
432C23C2×D4×D6111yescharacteristic
433Q8—2448no—
434Q8—2448no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—24no—
441C2×C4—42no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—612no—
447C2×C4—612no—
448C2×C4—614no—
449C2×C4—612no—
450C2×C4—612no—
451C2×C4—618no—
452C2×C4—614no—
453C2×C4—614no—
454C2×C4—612no—
455C2×C4—1228no—
456C2×C4—1222no—
457C2×C4—1222no—
458C2×C4—1222no—
459C2×C4—1224no—
460C2×C4—2444no—
461C2×C4—24416no—
462C2×C4—2442no—
463C2×C4—2441no—
464C2×C4—2441no—
465C2×C4—2442no—
466C2×C4—2442no—
467C2×C4C22⋊C4×D6112yes—
468C2×C4C22⋊C4×D6112yes—
469C2×C4C22×C4×D6111yescharacteristic
470C2×C4C2×D4×D6111yescharacteristic
471C2×C4C2×Q8×D6112yes—
472C2×C4#41×D6112yes—
473C8—42no—
474C8—2442no—
475C22—614no—
476C22—1224no—
477C22—1222no—
478C22—1222no—
479C22—1224no—
480C22C2×C22⋊C4×D6111yescharacteristic
481C22C22⋊Q8×D6112yes—
482C22AGL(1,ℤ/8)×D6111yescharacteristic
483C22#137×D6111yescharacteristic
484C22#1684055912yes—
485C4—22no—
486C4—22no—
487C4—42no—
488C4—42no—
489C4—44no—
490C4—614no—
491C4—614no—
492C4—1222no—
493C4—2442no—
494C4—2442no—
495C4—2444no—
496C4C4×Q8×D6112yes—
497C4#124×D6111yescharacteristic
498C4C4⋊Q8×D6111yescharacteristic
499C2—1224no—
500C2#391×D6111yescharacteristic
501C2#417×D6111yescharacteristic
502C2#2145433411yescharacteristic
503C2#2145561412yes—
504C2#2145562212yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1989C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
cosocleC22×C4×C61C24
hypercenter#1989D61
hypocenterC61#21312
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1989

Complements

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

NH ≅ G/Ntype
4. #21446777C2semidirect
5. #21446776C2semidirect
6. #21446774C2semidirect
7. #21446769C2semidirect
8. #21446765C2semidirect
11. C61×#1989C2semidirect
14. #16840318C4semidirect
57. #14519776C2×C4semidirect
58. #14519776C2×C4semidirect
60. #14519769C2×C4semidirect
78. C4⋊C4×D61Q8semidirect
97. #125×C61C2×C4semidirect
253. #1989D61semidirect
298. C61#21312semidirect
316. #125C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168403182
C2×#168397842
C4×C4⋊C4×D612
#214467772
#214467762
#214467742
#214467692
#214467652
#214455372
C61×#19892
#2131261