Finite Group Databasebeta

← #29684661

Subgroups — #29684661

Order 15616.

1338 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×#16840388C211yesmaximal, characteristic
2#21448431C211yesmaximal, characteristic
3#21448172C211yesmaximal, characteristic
4#21446802C218yesmaximal, complemented
5C2×#16839529C211yesmaximal, characteristic
6C2×#16839528C211yesmaximal, characteristic
7C2×#16839528C211yesmaximal, characteristic
8C2×C61×#342C211yesmaximal, characteristic
9#16840388—22no—
10#16840388—22no—
11#16840388—22no—
12#16840388—22no—
13C2×#14519756C414yescomplemented
14C2×#14519754C414yescomplemented
15C2×#14519754C2214yes—
16C22×C22⋊Dic61C2211yescharacteristic
17C22×C22⋊Dic61C2211yescharacteristic
18C22×C22⋊Dic61C2211yescharacteristic
19C22×C22⋊Dic61C2211yescharacteristic
20#16839635—22no—
21#16839634—22no—
22#16839620C2214yes—
23C22⋊C4×Dic61—22no—
24C22⋊C4×Dic61C2214yes—
25C4×C22⋊Dic61—22no—
26#16839529C2214yes—
27#16839528C2214yes—
28#16839528C2214yes—
29C22×C22⋊C4×C61C2211yescharacteristic
30C22×C22⋊C4×C61C2211yescharacteristic
31C22×C22⋊C4×C61C2211yescharacteristic
32C61×#342C2214yes—
33C22×Dic122C2×C414yescomplemented
34C23×Dic61—21no—
35C23×Dic61—21no—
36C23×Dic61C2×C412yes—
37C23×Dic61C2312yes—
38#14519756—28no—
39#14519756—28no—
40#14519754—28no—
41#14519754—28no—
42#14519754—28no—
43#14519754—28no—
44C2×C4⋊Dic61C2×C414yescomplemented
45C2×#11758196—24no—
46C2×#11758196C2×C418yescomplemented
47C2×C22⋊Dic61—21no—
48C2×C22⋊Dic61—21no—
49C2×C22⋊Dic61—21no—
50C2×C22⋊Dic61—21no—
51C2×C22⋊Dic61—21no—
52C2×C22⋊Dic61—22no—
53C2×C22⋊Dic61—22no—
54C2×C22⋊Dic61—22no—
55C2×C22⋊Dic61—21no—
56C2×C22⋊Dic61C2×C412yes—
57C2×C22⋊Dic61C2×C412yes—
58C2×C22⋊Dic61C2×C412yes—
59C2×C22⋊Dic61D412yes—
60C2×C22⋊Dic61D412yes—
61C2×C22⋊Dic61D412yes—
62C2×C22⋊Dic61D412yes—
63C2×C22⋊Dic61Q814yescomplemented
64C2×C22⋊Dic61C2312yes—
65C2×C22⋊Dic61C2312yes—
66C2×C22⋊Dic61C2312yes—
67C2×C4×Dic61—24no—
68#14519684—24no—
69#14519684—24no—
70C25×C61C2311yescharacteristic
71C23×C4×C61—21no—
72C23×C4×C61—21no—
73C23×C4×C61C2×C412yes—
74C2×C22⋊C4×C61—21no—
75C2×C22⋊C4×C61—21no—
76C2×C22⋊C4×C61—21no—
77C2×C22⋊C4×C61—21no—
78C2×C22⋊C4×C61—21no—
79C2×C22⋊C4×C61—22no—
80C2×C22⋊C4×C61—22no—
81C2×C22⋊C4×C61—22no—
82C2×C22⋊C4×C61—21no—
83C2×C22⋊C4×C61C2×C412yes—
84C2×C22⋊C4×C61Q814yescomplemented
85C2×C22⋊C4×C61C2312yes—
86C2×C22⋊C4×C61C2312yes—
87C2×C22⋊C4×C61C2312yes—
88C24×C61—82no—
89C24×C61C22×C411yescharacteristic
90C24×C61C22×C411yescharacteristic
91C24×C61C2×D411yescharacteristic
92C24×C61C2×D411yescharacteristic
93C24×C61C2×D411yescharacteristic
94C24×C61C2×D411yescharacteristic
95C24×C61C2×Q812yes—
96C24×C61C2×Q812yes—
97C24×C61#4112yes—
98C24×C61#4112yes—
99C24×C61C2411yescharacteristic
100C22×C4×C61—22no—
101C22×C4×C61—22no—
102C22×C4×C61—22no—
103C22×C4×C61—24no—
104C22×C4×C61—22no—
105C22×C4×C61—42no—
106C22×C4×C61—82no—
107C22×C4×C61—82no—
108C22×C4×C61—82no—
109C22×C4×C61C22×C414yes—
110C22×Dic61—22no—
111C22×Dic61—22no—
112C22×Dic61—22no—
113C22×Dic61—24no—
114C22×Dic61—22no—
115C22×Dic61—42no—
116C22×Dic61—82no—
117C22×Dic61—82no—
118C22×Dic61—82no—
119C22×Dic61—82no—
120C22×Dic61C22×C414yes—
121C22×Dic61C22×C414yes—
122C22×Dic61C2×D414yes—
123C22×Dic61C2×D414yes—
124C2×Dic122—28no—
125C2×Dic122—84no—
126C22⋊C4×C61—22no—
127C22⋊C4×C61—22no—
128C22⋊C4×C61—24no—
129C22⋊C4×C61—22no—
130C22⋊C4×C61—22no—
131C22⋊C4×C61—22no—
132C22⋊C4×C61—22no—
133C22⋊C4×C61—22no—
134C22⋊C4×C61—42no—
135C22⋊C4×C61C2×Q814yes—
136C22⋊Dic61—22no—
137C22⋊Dic61—22no—
138C22⋊Dic61—24no—
139C22⋊Dic61—22no—
140C22⋊Dic61—22no—
141C22⋊Dic61—22no—
142C22⋊Dic61—22no—
143C22⋊Dic61—22no—
144C22⋊Dic61—42no—
145C22⋊Dic61C22⋊C414yes—
146C22⋊Dic61C22⋊C414yes—
147C22⋊Dic61C22⋊C414yes—
148C22⋊Dic61C22⋊C414yes—
149C22⋊Dic61C2×Q814yes—
150C4⋊Dic61—28no—
151#11758196—216no—
152#11758196—84no—
153C4×Dic61—84no—
154C23×C61—42no—
155C23×C61—42no—
156C23×C61—42no—
157C23×C61—42no—
158C23×C61—42no—
159C23×C61—42no—
160C23×C61—42no—
161C23×C61—82no—
162C23×C61—82no—
163C23×C61—82no—
164C23×C61—82no—
165C23×C61C2×C22⋊C411yescharacteristic
166C23×C61C2×C22⋊C411yescharacteristic
167C23×C61C2×C22⋊C411yescharacteristic
168C23×C61C2×C22⋊C411yescharacteristic
169C23×C61C4×Q812yes—
170C23×C61C4×Q812yes—
171C23×C61C4×Q812yes—
172C23×C61C4×Q812yes—
173C23×C61C22⋊Q812yes—
174C23×C61C22⋊Q812yes—
175C23×C61C22⋊Q812yes—
176C23×C61C22⋊Q812yes—
177C23×C61#12412yes—
178C23×C61#12412yes—
179C23×C61C4⋊Q812yes—
180C23×C61C4⋊Q812yes—
181C23×C61C23×C411yescharacteristic
182C23×C61C22×D411yescharacteristic
183C23×C61C22×D411yescharacteristic
184C23×C61C22×Q811yescharacteristic
185C23×C61C22×Q811yescharacteristic
186C23×C61C2×#4111yescharacteristic
187C23×C61C2×#4111yescharacteristic
188C2×C4×C61—24no—
189C2×C4×C61—24no—
190C2×C4×C61—44no—
191C2×C4×C61—44no—
192C2×C4×C61—44no—
193C2×C4×C61—42no—
194C2×C4×C61—82no—
195C2×C4×C61—82no—
196C2×C4×C61—84no—
197C2×C4×C61—84no—
198C2×Dic61—24no—
199C2×Dic61—24no—
200C2×Dic61—44no—
201C2×Dic61—44no—
202C2×Dic61—44no—
203C2×Dic61—44no—
204C2×Dic61—44no—
205C2×Dic61—44no—
206C2×Dic61—42no—
207C2×Dic61—82no—
208C2×Dic61—82no—
209C2×Dic61—84no—
210C2×Dic61—84no—
211C2×Dic61C2×C22⋊C418yes—
212Dic122—88no—
213#21333—611nomaximal, Sylow
214C22×C61—22no—
215C22×C61—22no—
216C22×C61—22no—
217C22×C61—22no—
218C22×C61—22no—
219C22×C61—22no—
220C22×C61—22no—
221C22×C61—42no—
222C22×C61—42no—
223C22×C61—42no—
224C22×C61—42no—
225C22×C61—42no—
226C22×C61—42no—
227C22×C61—42no—
228C22×C61—42no—
229C22×C61—42no—
230C22×C61—42no—
231C22×C61—42no—
232C22×C61—42no—
233C22×C61—41no—
234C22×C61—41no—
235C22×C61—41no—
236C22×C61—41no—
237C22×C61#39112yes—
238C22×C61#39112yes—
239C22×C61#39112yes—
240C22×C61#39112yes—
241C22×C61C22×C22⋊C411yescharacteristic
242C22×C61C2×C4×Q811yescharacteristic
243C22×C61C2×C4×Q811yescharacteristic
244C22×C61C2×C22⋊Q811yescharacteristic
245C22×C61C2×C22⋊Q811yescharacteristic
246C22×C61C2×#12411yescharacteristic
247C22×C61C2×C4⋊Q811yescharacteristic
248C4×C61—44no—
249C4×C61—44no—
250C4×C61—82no—
251Dic61—28no—
252Dic61—44no—
253Dic61—44no—
254Dic61—82no—
255C2×#543—611no—
256#2315—611no—
257#2273—611no—
258#1996—618no—
259C2×#342—611no—
260C2×#342—611no—
261C2×#342—611no—
262C2×#342D6111yescharacteristic
263C2×C61—21no—
264C2×C61—21no—
265C2×C61—21no—
266C2×C61—21no—
267C2×C61—21no—
268C2×C61—21no—
269C2×C61—22no—
270C2×C61—22no—
271C2×C61—22no—
272C2×C61—22no—
273C2×C61C2×#39111yescharacteristic
274C2×C61#288312yes—
275#543—1222no—
276#543—1222no—
277#543—1222no—
278#543—1222no—
279C2×C22⋊Q8—614no—
280C2×C22⋊Q8—614no—
281C2×C22⋊Q8—614no—
282C22×C22⋊C4—611no—
283C22×C22⋊C4—611no—
284C22×C22⋊C4—611no—
285C22×C22⋊C4—611no—
286C22×C22⋊C4C2×D6111yescharacteristic
287C22×C22⋊C4C2×D6111yescharacteristic
288C22×C22⋊C4C2×D6111yescharacteristic
289#385—1222no—
290#385—1222no—
291#380—614no—
292C4×C22⋊C4—614no—
293C4×C22⋊C4—1222no—
294C4×C22⋊C4—1222no—
295#342—614no—
296#342—614no—
297#342—614no—
298#342C2×D6114yes—
299C61#2133311yescharacteristic, complemented, Sylow
300C25C22×D6111yescharacteristic
301C22×Q8—614no—
302C23×C4—21no—
303C23×C4—21no—
304C23×C4—612no—
305C23×C4—612no—
306C23×C4—1221no—
307C23×C4—1221no—
308C23×C4C4×D6112yes—
309C22⋊Q8—1228no—
310C22⋊Q8—1228no—
311C22⋊Q8—1228no—
312C22⋊Q8—1228no—
313C22⋊Q8—1228no—
314C22⋊Q8—1228no—
315C2×C4⋊C4—618no—
316C2×C4⋊C4—614no—
317C2×C4⋊C4—1224no—
318C2×C22⋊C4—21no—
319C2×C22⋊C4—21no—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—21no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—612no—
328C2×C22⋊C4—612no—
329C2×C22⋊C4—612no—
330C2×C22⋊C4—612no—
331C2×C22⋊C4—612no—
332C2×C22⋊C4—612no—
333C2×C22⋊C4—614no—
334C2×C22⋊C4—612no—
335C2×C22⋊C4—612no—
336C2×C22⋊C4—612no—
337C2×C22⋊C4—612no—
338C2×C22⋊C4—1222no—
339C2×C22⋊C4—1221no—
340C2×C22⋊C4—1221no—
341C2×C22⋊C4—1221no—
342C2×C22⋊C4—1221no—
343C2×C22⋊C4—1221no—
344C2×C22⋊C4—1222no—
345C2×C22⋊C4—1222no—
346C2×C22⋊C4—1221no—
347C2×C22⋊C4Dic12214yescomplemented
348C2×C22⋊C4C4×D6112yes—
349C2×C22⋊C4C22×D6112yes—
350C2×C22⋊C4C22×D6112yes—
351C2×C22⋊C4C22×D6112yes—
352C2×C42—1224no—
353#95—1224no—
354#95—1224no—
355C24—82no—
356C24C2×Dic12212yes—
357C24C2×C4×D6111yescharacteristic
358C24C2×C4×D6111yescharacteristic
359C24#1175822212yes—
360C24D4×D6111yescharacteristic
361C24D4×D6111yescharacteristic
362C24D4×D6111yescharacteristic
363C24D4×D6111yescharacteristic
364C24#1175822412yes—
365C24Q8×D6112yes—
366C24C23×D6111yescharacteristic
367C2×Q8—1228no—
368C2×Q8—4884no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—42no—
375C22×C4—82no—
376C22×C4—82no—
377C22×C4—82no—
378C22×C4—614no—
379C22×C4—614no—
380C22×C4—614no—
381C22×C4—614no—
382C22×C4—1222no—
383C22×C4—1222no—
384C22×C4—1224no—
385C22×C4—1222no—
386C22×C4—1222no—
387C22×C4—4882no—
388C22×C4—4882no—
389C22×C4—4882no—
390C22×C4—4882no—
391C22×C4—2442no—
392C22×C4C2×C4×D6114yes—
393C4⋊C4—12216no—
394C4⋊C4—1228no—
395C4⋊C4—4884no—
396C22⋊C4—22no—
397C22⋊C4—22no—
398C22⋊C4—22no—
399C22⋊C4—22no—
400C22⋊C4—22no—
401C22⋊C4—24no—
402C22⋊C4—22no—
403C22⋊C4—22no—
404C22⋊C4—42no—
405C22⋊C4—614no—
406C22⋊C4—614no—
407C22⋊C4—614no—
408C22⋊C4—614no—
409C22⋊C4—614no—
410C22⋊C4—1222no—
411C22⋊C4—1222no—
412C22⋊C4—1222no—
413C22⋊C4—1222no—
414C22⋊C4—1222no—
415C22⋊C4—1224no—
416C22⋊C4—1222no—
417C22⋊C4—1222no—
418C22⋊C4—2442no—
419C22⋊C4C2×Dic12214yes—
420C42—4884no—
421C23—42no—
422C23—42no—
423C23—42no—
424C23—42no—
425C23—42no—
426C23—42no—
427C23—42no—
428C23—82no—
429C23—82no—
430C23—82no—
431C23—82no—
432C23C22⋊C4×D6111yescharacteristic
433C23C22⋊C4×D6111yescharacteristic
434C23C22⋊C4×D6111yescharacteristic
435C23C22⋊C4×D6111yescharacteristic
436C23#1451974212yes—
437C23#1451974212yes—
438C23C4×Dic12212yes—
439C23C4×Dic12212yes—
440C23#1451975412yes—
441C23#1451975412yes—
442C23#1451975512yes—
443C23#1451975512yes—
444C23#1451976312yes—
445C23#1451976312yes—
446C23#1451977612yes—
447C23#1451977612yes—
448C23C22×C4×D6111yescharacteristic
449C23C2×D4×D6111yescharacteristic
450C23C2×D4×D6111yescharacteristic
451C23C2×Q8×D6111yescharacteristic
452C23C22×Dic12211yescharacteristic
453C23C2×#1175822411yescharacteristic
454C23C2×#1175822211yescharacteristic
455Q8—4888no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—42no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—84no—
465C2×C4—84no—
466C2×C4—618no—
467C2×C4—1224no—
468C2×C4—1224no—
469C2×C4—4884no—
470C2×C4—4882no—
471C2×C4—4882no—
472C2×C4—4884no—
473C2×C4—2444no—
474C2×C4—2444no—
475C2×C4—2444no—
476C2×C4—2444no—
477C2×C4—2442no—
478C2×C4—2444no—
479C2×C4—2444no—
480C22—22no—
481C22—22no—
482C22—22no—
483C22—22no—
484C22—22no—
485C22—22no—
486C22—22no—
487C22—42no—
488C22—42no—
489C22—42no—
490C22—42no—
491C22—41no—
492C22—41no—
493C22—41no—
494C22—41no—
495C22—42no—
496C22—42no—
497C22—42no—
498C22—42no—
499C22—42no—
500C22—42no—
501C22—42no—
502C22—42no—
503C22#1683966612yes—
504C22#1683966612yes—
505C22#1683966612yes—
506C22#1683966612yes—
507C22C2×C22⋊C4×D6111yescharacteristic
508C22C2×#1451974211yescharacteristic
509C22C2×C4×Dic12211yescharacteristic
510C22C2×#1451975411yescharacteristic
511C22C2×#1451975511yescharacteristic
512C22C2×#1451976311yescharacteristic
513C22C2×#1451977611yescharacteristic
514C4—44no—
515C4—44no—
516C4—82no—
517C4—1228no—
518C4—4882no—
519C4—2444no—
520C4—2444no—
521C2—21no—
522C2—21no—
523C2—21no—
524C2—21no—
525C2—21no—
526C2—21no—
527C2—22no—
528C2—22no—
529C2—22no—
530C2—22no—
531C2C2×#1683966611yescharacteristic
532C2#2145427712yes—
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC23×C61C23×C4
FrattiniC24C23×D61
FittingC2×C61×#342C2
socleC22×C61C22×C22⋊C4
prime radicalC25×C61C23
cosocleC24×C61C24
hypercenterC2×#342D61
hypocenterC61#21333
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C25 ◁ C2×#342

Complements

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

NH ≅ G/Ntype
4. #21446802C2semidirect
13. C2×#14519756C4semidirect
14. C2×#14519754C4semidirect
33. C22×Dic122C2×C4semidirect
44. C2×C4⋊Dic61C2×C4semidirect
46. C2×#11758196C2×C4semidirect
63. C2×C22⋊Dic61Q8semidirect
84. C2×C22⋊C4×C61Q8semidirect
299. C61#21333semidirect
347. C2×C22⋊C4Dic122semidirect

Maximal subgroups

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

subgroupindex
C2×#168403882
#214484312
#214481722
#214468022
C2×#168395292
C2×#168395282
C2×C61×#3422
#2133361