Finite Group Databasebeta

← #29689712

Subgroups — #29689712

Order 15616.

1138 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840291C211yesmaximal, characteristic
2C2×#16839813C211yesmaximal, characteristic, complemented
3#21448225C211yesmaximal, characteristic, complemented
4#21446784C211yesmaximal, characteristic, complemented
5#21446783C211yesmaximal, characteristic, complemented
6#21446631C211yesmaximal, characteristic, complemented
7#21446138C212yesmaximal
8#21446128C212yesmaximal, complemented
9#21445822C212yesmaximal, complemented
10#21445817C212yesmaximal, complemented
11C61×#1958C211yesmaximal, characteristic, complemented
12#16840291—24no—
13#16840291C418yescomplemented
14C2×#14519770—21no—
15C2×#14519769C2211yescharacteristic
16C2×#14519759—21no—
17C2×#14519753C2211yescharacteristic
18C2×#14519748C2211yescharacteristic
19C2×Q8×Dic61C2211yescharacteristic
20C2×D4×Dic61C2211yescharacteristic
21C2×#14519737—21no—
22C2×#14519735C2212yes—
23C2×#14519733—21no—
24C2×C42×D61C2211yescharacteristic
25#16839813—24no—
26C2×#14519706C2212yes—
27C2×#14519701C2212yes—
28C2×#14519698C2212yes—
29C2×#14519689C2212yes—
30C2×C4×#92418C2211yescharacteristic
31#16839631C2212yes—
32#16839628C2211yescharacteristic
33#16839621—21no—
34#16839621—21no—
35#16839613—21no—
36#16839613—21no—
37#16839525C2212yes—
38#16839525C2212yes—
39#16839524C2212yes—
40#16839524C2212yes—
41C2×#124×C61C2211yescharacteristic
42C2×#103×C61C2212yes—
43C2×#102×C61C2212yes—
44C2×C4×C8×C61C2211yescharacteristic
45C61×#382C2211yescharacteristic
46C22×C22⋊D61—21no—
47C23×Dic61—21no—
48C22×C4×D61D411yescharacteristic
49C22×C4×D61D411yescharacteristic
50C22×C4×D61C2311yescharacteristic
51C22×#92418C2312yes—
52#14519770—28no—
53#14519769—22no—
54#14519769C2×C414yes—
55#14519759—28no—
56#14519753—22no—
57#14519753C2×C414yes—
58#14519748—22no—
59#14519748C2×C414yes—
60Q8×Dic61—22no—
61Q8×Dic61C2×C414yes—
62D4×Dic61—22no—
63D4×Dic61C2×C414yes—
64#14519737—28no—
65#14519735—42no—
66#14519735—42no—
67#14519733—28no—
68C2×C4⋊Dic61—21no—
69C2×C4⋊Dic61C2312yes—
70C2×C4⋊Dic61C2311yescharacteristic
71C2×C4⋊Dic61C2311yescharacteristic
72C2×#11758196—21no—
73C2×#11758196—21no—
74C2×#11758196—21no—
75C2×#11758196—22no—
76C2×#11758196—22no—
77C2×C22⋊Dic61—21no—
78C2×C22⋊Dic61—21no—
79C2×#11758198—21no—
80C2×#11758198—22no—
81C2×C4×Dic61—21no—
82C2×C4×Dic61—22no—
83C2×C4×Dic61D411yescharacteristic
84C2×C4×Dic61D411yescharacteristic
85C2×C4×Dic61C2311yescharacteristic
86C42×D61C2×C414yes—
87C42×D61D414yes—
88#14519706—24no—
89#14519701—42no—
90#14519698—42no—
91#14519689—24no—
92C4×#92418—22no—
93#14519684—21no—
94#14519684—21no—
95C22×Q8×C61C2311yescharacteristic
96C22×D4×C61C2311yescharacteristic
97C22×C8×C61C2312yes—
98#124×C61—22no—
99#124×C61C2×C414yes—
100C2×C4⋊C4×C61C2312yes—
101C2×C22⋊C4×C61—21no—
102C2×C22⋊C4×C61—21no—
103C2×C42×C61C2311yescharacteristic
104#103×C61—42no—
105#102×C61—42no—
106C4×C8×C61—22no—
107#95×C61—21no—
108#95×C61—21no—
109C24×C61—21no—
110C23×D61C2×D411yescharacteristic
111C2×Q8×C61—21no—
112C2×Q8×C61—22no—
113C2×Q8×C61—22no—
114C2×Q8×C61C22×C412yes—
115C2×D4×C61—21no—
116C2×D4×C61—22no—
117C2×D4×C61—22no—
118C2×D4×C61C22×C412yes—
119C22×C4×C61—21no—
120C22×C4×C61—22no—
121C22×C4×C61C2×D411yescharacteristic
122C22×C4×C61C2×D411yescharacteristic
123C22×C4×C61C2411yescharacteristic
124C2×C22⋊D61—24no—
125C2×C22⋊D61—44no—
126C22×Dic61—21no—
127C22×Dic61—21no—
128C22×Dic61—24no—
129C22×Dic61—22no—
130C22×Dic61—44no—
131C22×Dic61C2×D411yescharacteristic
132C22×Dic61C2×D411yescharacteristic
133C22×Dic61C2×D411yescharacteristic
134C2×C4×D61—22no—
135C2×C4×D61—22no—
136C2×C4×D61—24no—
137C2×C4×D61—42no—
138C2×C4×D61C22⋊C414yes—
139C2×C4×D61C22⋊C414yes—
140C2×C4×D61C22×C412yes—
141C2×C4×D61C2×D412yes—
142C2×C8×C61—22no—
143C2×C8×C61—22no—
144C2×C8×C61—22no—
145C4⋊C4×C61—22no—
146C4⋊C4×C61—22no—
147C22⋊C4×C61—24no—
148C22⋊C4×C61—24no—
149C42×C61C22×C412yes—
150C42×C61C2×D412yes—
151C22⋊Dic61—24no—
152C22⋊Dic61—24no—
153#11758198—24no—
154#11758198—44no—
155C4⋊Dic61—21no—
156C4⋊Dic61—21no—
157C4⋊Dic61—24no—
158C4⋊Dic61—22no—
159C4⋊Dic61—22no—
160C4⋊Dic61C22×C412yes—
161C4⋊Dic61C22×C412yes—
162#11758196—24no—
163#11758196—24no—
164#11758196—24no—
165#11758196—44no—
166#11758196—44no—
167C4×Dic61—24no—
168C4×Dic61—44no—
169C4×Dic61C22⋊C414yes—
170C4×Dic61C22⋊C414yes—
171C4×Dic61C22×C412yes—
172C4×Dic61C2×D412yes—
173C2×#92418—22no—
174C2×#92418—22no—
175C2×#92418—22no—
176C23×C61—22no—
177C23×C61—41no—
178C23×C61—41no—
179C23×C61—42no—
180C23×C61—42no—
181C23×C61C22×D411yescharacteristic
182C22×D61—21no—
183C22×D61—21no—
184C22×D61—22no—
185C22×D61—22no—
186C22×D61C2×C22⋊C412yes—
187Q8×C61—24no—
188Q8×C61—42no—
189D4×C61—24no—
190D4×C61—42no—
191C2×C4×C61—22no—
192C2×C4×C61—22no—
193C2×C4×C61—22no—
194C2×C4×C61—41no—
195C2×C4×C61—41no—
196C2×C4×C61—42no—
197C2×C4×C61—42no—
198C2×C4×C61—42no—
199C2×C4×C61C2×C22⋊C412yes—
200C2×C4×C61C2×C22⋊C412yes—
201C2×C4×C61C23×C411yescharacteristic
202C2×C4×C61C22×D411yescharacteristic
203C2×C4×C61H(5,𝔽2)12yes—
204C2×C4×C612−1+412yes—
205C22⋊D61—48no—
206C2×Dic61—21no—
207C2×Dic61—21no—
208C2×Dic61—22no—
209C2×Dic61—22no—
210C2×Dic61—22no—
211C2×Dic61—42no—
212C2×Dic61—42no—
213C2×Dic61—48no—
214C2×Dic61—42no—
215C2×Dic61—42no—
216C2×Dic61—42no—
217C2×Dic61C2×C22⋊C412yes—
218C2×Dic61C2×C22⋊C412yes—
219C2×Dic61C2×C22⋊C412yes—
220C2×Dic61#13514yes—
221C4×D61—24no—
222C4×D61—24no—
223C4×D61—28no—
224C4×D61—44no—
225C8×C61—42no—
226#92418—42no—
227#21862—611nomaximal, Sylow
228C22×C61—42no—
229C22×C61—42no—
230C22×C61—42no—
231C22×C61—82no—
232C22×C61—82no—
233C22×C61C22×C22⋊C411yescharacteristic
234C22×C61#53511yescharacteristic
235C22×C61#53611yescharacteristic
236C22×C61C2×#13512yes—
237C22×C61#57512yes—
238C2×D61—22no—
239C2×D61—22no—
240C2×D61—22no—
241C2×D61—42no—
242C2×D61—42no—
243C4×C61—22no—
244C4×C61—22no—
245C4×C61—42no—
246C4×C61—82no—
247C4×C61#52012yes—
248C4×C61#53712yes—
249Dic61—22no—
250Dic61—22no—
251Dic61—42no—
252Dic61—42no—
253Dic61—82no—
254Dic61#41614yes—
255C2×#529—611no—
256C2×#431—611no—
257#2285—611no—
258#1991—611no—
259#1991—611no—
260#1958—611no—
261#1958D6111yescharacteristic, complemented
262#1875—612no—
263#1873—612no—
264#1803—612no—
265#1802—612no—
266C2×C61—82no—
267C2×C61#231111yescharacteristic
268C2×C61C2×#41611yescharacteristic
269C2×C61#289611yescharacteristic
270C2×C61#321212yes—
271C2×C61#323312yes—
272D61—42no—
273#529—618no—
274#529—1224no—
275C2×#126—1221no—
276C2×#125—611no—
277C2×#124C2×D6111yescharacteristic
278C2×#123—1221no—
279C2×C22⋊Q8—611no—
280C2×#121—611no—
281C2×C4×Q8—611no—
282C2×C4×D4—611no—
283C2×C4×D4—1221no—
284C2×#117—612no—
285C2×#117—1221no—
286C22×C42—611no—
287#431—1224no—
288C2×C4⋊C8—612no—
289C2×#103—612no—
290C2×#103C2×D6112yes—
291C2×#102—612no—
292C2×#102C2×D6112yes—
293C2×C22⋊C8—612no—
294C2×C4×C8—611no—
295C2×C4×C8C2×D6111yescharacteristic
296#384—612no—
297#382—611no—
298#382C2×D6111yescharacteristic
299#380—1221no—
300#380—1221no—
301C4×C4⋊C4—1221no—
302C4×C4⋊C4—1221no—
303#340—612no—
304#340—612no—
305#340—612no—
306#340—612no—
307C61#2186211yescharacteristic, complemented, Sylow
308C22×Q8C22×D6111yescharacteristic
309C22×D4—1221no—
310C22×D4C22×D6111yescharacteristic
311C23×C4—611no—
312C23×C4—611no—
313C23×C4—611no—
314C23×C4—1221no—
315C22×C8—612no—
316C22×C8C22×D6112yes—
317#126—1228no—
318#125—614no—
319#125—1222no—
320#124—22no—
321#124C4×D6114yes—
322#123—1228no—
323C22⋊Q8—614no—
324C22⋊Q8—1222no—
325#121—614no—
326#121—1222no—
327C4×Q8—614no—
328C4×Q8—1222no—
329C4×D4—614no—
330C4×D4—1222no—
331C4×D4—1228no—
332#117—1228no—
333#117—2442no—
334#117—2442no—
335C2×C4⋊C4—611no—
336C2×C4⋊C4—611no—
337C2×C4⋊C4—612no—
338C2×C4⋊C4—1221no—
339C2×C4⋊C4—1221no—
340C2×C4⋊C4—1221no—
341C2×C4⋊C4—1221no—
342C2×C4⋊C4—1222no—
343C2×C4⋊C4—1222no—
344C2×C4⋊C4C22×D6112yes—
345C2×C22⋊C4—21no—
346C2×C22⋊C4—21no—
347C2×C22⋊C4—1221no—
348C2×C22⋊C4—1221no—
349C2×C22⋊C4—1221no—
350C2×C22⋊C4—1222no—
351C2×C42—611no—
352C2×C42—611no—
353C2×C42—611no—
354C2×C42—614no—
355C2×C42—614no—
356C2×C42—1221no—
357C2×C42—1222no—
358C2×C42C22×D6111yescharacteristic
359C4⋊C8—1224no—
360#103—42no—
361#103—2442no—
362#102—42no—
363#102—2442no—
364C22⋊C8—1224no—
365C4×C8—22no—
366C4×C8—1222no—
367#95—21no—
368#95—21no—
369#95—1221no—
370#95—1221no—
371C24—21no—
372C24—611no—
373C2×Q8—21no—
374C2×Q8—22no—
375C2×Q8—22no—
376C2×Q8C2×C4×D6112yes—
377C2×D4—21no—
378C2×D4—22no—
379C2×D4—22no—
380C2×D4—1224no—
381C2×D4—2444no—
382C2×D4C2×C4×D6112yes—
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—611no—
386C22×C4—611no—
387C22×C4—611no—
388C22×C4—612no—
389C22×C4—612no—
390C22×C4—614no—
391C22×C4—614no—
392C22×C4—1224no—
393C22×C4—1224no—
394C22×C4—1222no—
395C22×C4—1222no—
396C22×C4—1221no—
397C22×C4—1221no—
398C22×C4—1222no—
399C22×C4—2442no—
400C22×C4—2444no—
401C22×C4D4×D6111yescharacteristic
402C22×C4D4×D6111yescharacteristic
403C22×C4C23×D6111yescharacteristic
404C2×C8—22no—
405C2×C8—22no—
406C2×C8—22no—
407C2×C8—1222no—
408C2×C8—1222no—
409C2×C8—1222no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—612no—
413C4⋊C4—612no—
414C4⋊C4—1224no—
415C4⋊C4—1224no—
416C4⋊C4—1221no—
417C4⋊C4—1221no—
418C4⋊C4—1224no—
419C4⋊C4—1224no—
420C4⋊C4—1222no—
421C4⋊C4—1222no—
422C4⋊C4—2444no—
423C4⋊C4—2444no—
424C22⋊C4—24no—
425C22⋊C4—24no—
426C22⋊C4—1224no—
427C22⋊C4—1224no—
428C22⋊C4—1224no—
429C22⋊C4—2444no—
430C42—612no—
431C42—612no—
432C42—614no—
433C42—614no—
434C42—1224no—
435C42—2444no—
436C42C2×C4×D6112yes—
437C42D4×D6112yes—
438C23—22no—
439C23—42no—
440C23—42no—
441C23—41no—
442C23—41no—
443C23—612no—
444C23—1222no—
445C23—1222no—
446C23—1221no—
447C23—1221no—
448C23C2×D4×D6111yescharacteristic
449Q8—24no—
450Q8—42no—
451D4—24no—
452D4—42no—
453D4—2448no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—612no—
463C2×C4—614no—
464C2×C4—612no—
465C2×C4—612no—
466C2×C4—1228no—
467C2×C4—1222no—
468C2×C4—1221no—
469C2×C4—1221no—
470C2×C4—1224no—
471C2×C4—1224no—
472C2×C4—1222no—
473C2×C4—1222no—
474C2×C4—2444no—
475C2×C4—2448no—
476C2×C4—2442no—
477C2×C4—2442no—
478C2×C4—2442no—
479C2×C4—2442no—
480C2×C4—2442no—
481C2×C4C22⋊C4×D6112yes—
482C2×C4C22⋊C4×D6112yes—
483C2×C4C22×C4×D6111yescharacteristic
484C2×C4C2×D4×D6111yescharacteristic
485C2×C4#1451982212yes—
486C2×C4#1451982412yes—
487C8—42no—
488C8—2442no—
489C22—42no—
490C22—42no—
491C22—42no—
492C22—82no—
493C22—82no—
494C22—1222no—
495C22—1222no—
496C22—1222no—
497C22—2442no—
498C22—2442no—
499C22C2×C22⋊C4×D6111yescharacteristic
500C22#1684032511yescharacteristic
501C22#1684033211yescharacteristic
502C22#135×D6112yes—
503C22#1684055912yes—
504C4—22no—
505C4—22no—
506C4—42no—
507C4—82no—
508C4—614no—
509C4—1222no—
510C4—1222no—
511C4—4882no—
512C4—2442no—
513C4—2442no—
514C4#1684024112yes—
515C4#1684034112yes—
516C2—82no—
517C2—2442no—
518C2#2144840911yescharacteristic
519C2#416×D6111yescharacteristic
520C2#2145433411yescharacteristic
521C2#2145679012yes—
522C2#2145699312yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1958C2
socleC23×C61C22×D4
prime radicalC2×#14519748C22
cosocleC22×C4×C61C24
hypercenter#1958D61
hypocenterC61#21862
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1958

Complements

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

NH ≅ G/Ntype
2. C2×#16839813C2semidirect
3. #21448225C2semidirect
4. #21446784C2semidirect
5. #21446783C2semidirect
6. #21446631C2semidirect
8. #21446128C2semidirect
9. #21445822C2semidirect
10. #21445817C2semidirect
11. C61×#1958C2semidirect
13. #16840291C4semidirect
261. #1958D61semidirect
307. C61#21862semidirect

Maximal subgroups

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

subgroupindex
C2×#168402912
C2×#168398132
#214482252
#214467842
#214467832
#214466312
#214461382
#214461282
#214458222
#214458172
C61×#19582
#2186261