Finite Group Databasebeta

← #29721555

Subgroups — #29721555

Order 15616.

1050 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839975C212yesmaximal
2C2×#16839967C211yesmaximal, characteristic, complemented
3#21450811C211yesmaximal, characteristic, complemented
4D4×C4⋊Dic61C211yesmaximal, characteristic, complemented
5#21447209C212yesmaximal, complemented
6#21447059C212yesmaximal, complemented
7#21446621C211yesmaximal, characteristic, complemented
8#21446537C211yesmaximal, characteristic, complemented
9#21446151C212yesmaximal, complemented
10#21445819C211yesmaximal, characteristic, complemented
11C2×C4×D8×C61C211yesmaximal, characteristic
12C22×#11758200C2212yes—
13C2×#14519777C2211yescharacteristic
14C2×#14519764C2212yes—
15C2×D4×Dic61C2212yes—
16C22×C4⋊Dic61—21no—
17#16839975—28no—
18#16839967—24no—
19C2×C4⋊#92418C2211yescharacteristic
20C2×#14519706C2212yes—
21C2×#14519702C2212yes—
22C2×#14519702C2212yes—
23C2×#14519700C2212yes—
24C2×#14519698C2212yes—
25#16839687—22no—
26#16839667C2212yes—
27#16839661C2212yes—
28#16839637—22no—
29#16839632C2211yescharacteristic
30#16839622—21no—
31#16839622—21no—
32#16839622—21no—
33C4×C4⋊Dic61C2211yescharacteristic
34#16839524C2212yes—
35#16839516C2212yes—
36C22×D8×C61C2211yescharacteristic
37C2×C4×D4×C61C2211yescharacteristic
38C2×C4×D4×C61C2211yescharacteristic
39C4×D8×C61—24no—
40C2×#107×C61C2211yescharacteristic
41C2×#102×C61C2211yescharacteristic
42C2×#102×C61C2211yescharacteristic
43C2×C4×C8×C61C2211yescharacteristic
44C22×Dic122C2312yes—
45C23×Dic61—22no—
46C2×#11758200—24no—
47C2×#11758200—24no—
48C2×#11758200—24no—
49C22×#92418C2312yes—
50#14519777—22no—
51#14519777—22no—
52#14519764—24no—
53#14519764—42no—
54D4×Dic61—24no—
55D4×Dic61—42no—
56C2×C4⋊Dic61—21no—
57C2×C4⋊Dic61—21no—
58C2×C4⋊Dic61—21no—
59C2×C4⋊Dic61—22no—
60C2×C4⋊Dic61—21no—
61C2×C4⋊Dic61—44no—
62C2×C4⋊Dic61D411yescharacteristic
63C2×C4⋊Dic61D411yescharacteristic
64C2×C4⋊Dic61C2312yes—
65C2×C22⋊Dic61—22no—
66C2×C22⋊Dic61—22no—
67C2×C22⋊Dic61—22no—
68C2×C4×Dic61C2312yes—
69C4⋊#92418—22no—
70#14519706—24no—
71#14519702—24no—
72#14519702—42no—
73#14519700—24no—
74#14519698—24no—
75#14519684—22no—
76#14519684—22no—
77#14519684—22no—
78#14519684D412yes—
79C22×D4×C61C2311yescharacteristic
80C22×D4×C61C2311yescharacteristic
81C23×C4×C61—21no—
82C23×C4×C61—21no—
83C2×D8×C61—22no—
84C2×D8×C61—22no—
85C2×D8×C61D414yes—
86C22×C8×C61C2311yescharacteristic
87C22×C8×C61C2311yescharacteristic
88C4×D4×C61—22no—
89C4×D4×C61—22no—
90C4×D4×C61—22no—
91C4×D4×C61D414yes—
92C2×C4⋊C4×C61C2311yescharacteristic
93C2×C4⋊C4×C61C2311yescharacteristic
94C2×C22⋊C4×C61—21no—
95C2×C22⋊C4×C61—21no—
96C2×C42×C61C2311yescharacteristic
97#107×C61—22no—
98#102×C61—22no—
99#102×C61—22no—
100C4×C8×C61—22no—
101C24×C61—21no—
102C24×C61—21no—
103C2×D4×C61—21no—
104C2×D4×C61—21no—
105C2×D4×C61—21no—
106C2×D4×C61—21no—
107C2×D4×C61—21no—
108C2×D4×C61—21no—
109C2×D4×C61—21no—
110C2×D4×C61—21no—
111C2×D4×C61C2×D412yes—
112C2×D4×C61C2×D412yes—
113C2×D4×C61C2×D412yes—
114C2×D4×C61#4112yes—
115C22×C4×C61—21no—
116C22×C4×C61—24no—
117C22×C4×C61—21no—
118C22×C4×C61—42no—
119C22×C4×C61—42no—
120C22×C4×C61—42no—
121C22×C4×C61—42no—
122C22×C4×C61—42no—
123C22×C4×C61C2×D411yescharacteristic
124C22×C4×C61C2×D411yescharacteristic
125C22×C4×C61C2411yescharacteristic
126C22×Dic61—22no—
127C22×Dic61—22no—
128C22×Dic61—44no—
129C22×Dic61—44no—
130C22×Dic61—82no—
131C22×Dic61C2×D412yes—
132C22×Dic61C2×D412yes—
133C2×Dic122—22no—
134C2×Dic122—22no—
135C2×Dic122—22no—
136C2×Dic122—22no—
137C2×Dic122—22no—
138C2×Dic122—22no—
139D8×C61—28no—
140C2×C8×C61—21no—
141C2×C8×C61—21no—
142C2×C8×C61—21no—
143C2×C8×C61—21no—
144C2×C8×C61—21no—
145C2×C8×C61C2×D412yes—
146C4⋊C4×C61—21no—
147C4⋊C4×C61—21no—
148C4⋊C4×C61—21no—
149C4⋊C4×C61C2×D412yes—
150C22⋊C4×C61—24no—
151C22⋊C4×C61—42no—
152C42×C61C2×D412yes—
153C42×C61#4112yes—
154C22⋊Dic61—44no—
155C22⋊Dic61—44no—
156C22⋊Dic61—44no—
157#11758200—48no—
158C4⋊Dic61—22no—
159C4⋊Dic61—22no—
160C4⋊Dic61—22no—
161C4⋊Dic61—22no—
162C4⋊Dic61—44no—
163C4⋊Dic61—42no—
164C4⋊Dic61—42no—
165C4⋊Dic61—42no—
166C4⋊Dic61—81no—
167C4×Dic61—22no—
168C4×Dic61—22no—
169C2×#92418—22no—
170C2×#92418—22no—
171C2×#92418—22no—
172C23×C61—22no—
173C23×C61—22no—
174C23×C61—22no—
175C23×C61—22no—
176C23×C61—41no—
177C23×C61—41no—
178C23×C61—41no—
179C23×C61—41no—
180C23×C61—41no—
181C23×C61—41no—
182C23×C61—41no—
183C23×C61—41no—
184C23×C61—41no—
185C23×C61—41no—
186C23×C61C22×D411yescharacteristic
187D4×C61—22no—
188D4×C61—22no—
189D4×C61—22no—
190D4×C61—22no—
191D4×C61—22no—
192D4×C61—41no—
193D4×C61—41no—
194D4×C61#12114yes—
195C2×C4×C61—21no—
196C2×C4×C61—21no—
197C2×C4×C61—22no—
198C2×C4×C61—41no—
199C2×C4×C61—41no—
200C2×C4×C61—48no—
201C2×C4×C61—41no—
202C2×C4×C61—41no—
203C2×C4×C61—41no—
204C2×C4×C61—84no—
205C2×C4×C61#12112yes—
206C2×C4×C61#12112yes—
207C2×C4×C61#13512yes—
208C2×C4×C61AGL(1,ℤ/8)12yes—
209C2×C4×C61C22×D411yescharacteristic
210C2×C4×C61C22×D411yescharacteristic
211C2×C4×C61C2×#4111yescharacteristic
212C2×C4×C61C2×#4111yescharacteristic
213C2×C4×C61H(5,𝔽2)11yescharacteristic
214C2×C4×C612−1+411yescharacteristic
215C2×Dic61—22no—
216C2×Dic61—22no—
217C2×Dic61—22no—
218C2×Dic61—22no—
219C2×Dic61—22no—
220C2×Dic61—22no—
221C2×Dic61—42no—
222C2×Dic61—42no—
223C2×Dic61—42no—
224C2×Dic61—42no—
225C2×Dic61—42no—
226C2×Dic61—42no—
227C2×Dic61—88no—
228Dic122—42no—
229Dic122—42no—
230Dic122—42no—
231Dic122—42no—
232C8×C61—22no—
233C8×C61—41no—
234#92418—42no—
235#24727—611nomaximal, Sylow
236C22×C61—24no—
237C22×C61—42no—
238C22×C61—42no—
239C22×C61—42no—
240C22×C61—42no—
241C22×C61—42no—
242C22×C61—42no—
243C22×C61—42no—
244C22×C61—42no—
245C22×C61—42no—
246C22×C61—81no—
247C22×C61—81no—
248C22×C61—81no—
249C22×C61—81no—
250C22×C61C2×#12111yescharacteristic
251C22×C61D4211yescharacteristic
252C22×C61#54811yescharacteristic
253C22×C61C2×#13511yescharacteristic
254C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
255C22×C61#57711yescharacteristic
256C22×C61#57811yescharacteristic
257C4×C61—22no—
258C4×C61—42no—
259C4×C61—81no—
260C4×C61#46314yescomplemented
261C4×C61C2×#12111yescharacteristic
262C4×C61#53711yescharacteristic
263C4×C61#54711yescharacteristic
264C4×C61#55311yescharacteristic
265Dic61—42no—
266Dic61—42no—
267Dic61—82no—
268Dic61—82no—
269C2×#463—612no—
270C2×#461—611no—
271C2×C4×D8D6111yescharacteristic
272#2547—611no—
273D4×C4⋊C4—611no—
274#2068—612no—
275#2042—612no—
276#1956—611no—
277#1939—611no—
278#1877—612no—
279#1802—611no—
280C2×C61—44no—
281C2×C61—82no—
282C2×C61#241611yescharacteristic
283C2×C61C2×#46311yescharacteristic
284C2×C61#304611yescharacteristic
285C2×C61#327211yescharacteristic
286C2×C61#328311yescharacteristic
287C2×C61#336011yescharacteristic
288C2×C61#336711yescharacteristic
289C22×SD16—612no—
290C22×D8C2×D6111yescharacteristic
291C2×C4⋊Q8—611no—
292C2×#124—612no—
293C2×C4×D4—612no—
294C2×C4×D4C2×D6111yescharacteristic
295C2×C4×D4C2×D6111yescharacteristic
296C22×C4⋊C4—1221no—
297#463—1228no—
298#461—1224no—
299C4×D8—24no—
300C2×#107C2×D6111yescharacteristic
301C2×C4⋊C8—612no—
302C2×C4⋊C8—611no—
303C2×#103—612no—
304C2×#103—612no—
305C2×#103—612no—
306C2×#102—612no—
307C2×#102C2×D6111yescharacteristic
308C2×#102C2×D6111yescharacteristic
309C2×C4×C8C2×D6111yescharacteristic
310#397—1222no—
311#391—612no—
312#389—612no—
313#385—1222no—
314#384—611no—
315#380—1221no—
316#380—1221no—
317#380—1221no—
318C4×C4⋊C4—611no—
319#340—612no—
320#336—612no—
321C61#2472711yescharacteristic, complemented, Sylow
322C22×Q8—612no—
323C22×D4C22×D6111yescharacteristic
324C22×D4C22×D6111yescharacteristic
325C23×C4—21no—
326C23×C4—21no—
327C23×C4—1222no—
328C2×SD16—1224no—
329C2×SD16—1224no—
330C2×SD16—1224no—
331C2×D8—22no—
332C2×D8—22no—
333C2×D8D24414yes—
334C22×C8—612no—
335C22×C8C22×D6111yescharacteristic
336C22×C8C22×D6111yescharacteristic
337C4⋊Q8—1222no—
338C4⋊Q8—1222no—
339#124—1224no—
340#124—2442no—
341C4×D4—22no—
342C4×D4—22no—
343C4×D4—22no—
344C4×D4—1224no—
345C4×D4—2442no—
346C4×D4C22⋊D6114yes—
347C2×C4⋊C4—611no—
348C2×C4⋊C4—611no—
349C2×C4⋊C4—612no—
350C2×C4⋊C4—1221no—
351C2×C4⋊C4—1222no—
352C2×C4⋊C4—1221no—
353C2×C4⋊C4—1221no—
354C2×C4⋊C4—1221no—
355C2×C4⋊C4—2444no—
356C2×C4⋊C4C22×D6111yescharacteristic
357C2×C4⋊C4C22×D6111yescharacteristic
358C2×C22⋊C4—21no—
359C2×C22⋊C4—21no—
360C2×C22⋊C4—1222no—
361C2×C22⋊C4—1222no—
362C2×C22⋊C4—1222no—
363C2×C42—612no—
364C2×C42C22×D6111yescharacteristic
365#107—22no—
366C4⋊C8—1224no—
367C4⋊C8—1222no—
368#103—1224no—
369#103—1224no—
370#103—2442no—
371#102—22no—
372#102—22no—
373#102—1224no—
374C4×C8—22no—
375#95—612no—
376#95—1222no—
377#95—1222no—
378#95—1222no—
379C24—21no—
380C24—21no—
381C2×Q8—1222no—
382C2×Q8—1222no—
383C2×Q8—1222no—
384C2×Q8—1222no—
385C2×Q8—1222no—
386C2×Q8—1222no—
387C2×D4—21no—
388C2×D4—21no—
389C2×D4—21no—
390C2×D4—21no—
391C2×D4—21no—
392C2×D4—21no—
393C2×D4—21no—
394C2×D4—21no—
395C2×D4C2×D24412yes—
396C2×D4C2×D24412yes—
397C2×D4#1175822212yes—
398C2×D4C2×C22⋊D6112yes—
399C22×C4—21no—
400C22×C4—24no—
401C22×C4—21no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—612no—
408C22×C4—612no—
409C22×C4—1222no—
410C22×C4—1222no—
411C22×C4—4882no—
412C22×C4—2444no—
413C22×C4—2444no—
414C22×C4D4×D6111yescharacteristic
415C22×C4D4×D6111yescharacteristic
416C22×C4C23×D6111yescharacteristic
417SD16—2448no—
418D8—28no—
419C2×C8—21no—
420C2×C8—21no—
421C2×C8—21no—
422C2×C8—21no—
423C2×C8—21no—
424C2×C8—1222no—
425C2×C8—1222no—
426C2×C8—1222no—
427C2×C8C2×D24412yes—
428C4⋊C4—21no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—1222no—
432C4⋊C4—1222no—
433C4⋊C4—1222no—
434C4⋊C4—1222no—
435C4⋊C4—4881no—
436C4⋊C4—2444no—
437C4⋊C4—2442no—
438C4⋊C4—2442no—
439C4⋊C4—2442no—
440C4⋊C4C2×C22⋊D6112yes—
441C22⋊C4—24no—
442C22⋊C4—42no—
443C22⋊C4—2444no—
444C22⋊C4—2444no—
445C22⋊C4—2444no—
446C42—1222no—
447C42—1222no—
448C42#1175822412yes—
449C42C2×C22⋊D6112yes—
450C23—22no—
451C23—22no—
452C23—22no—
453C23—22no—
454C23—41no—
455C23—41no—
456C23—41no—
457C23—41no—
458C23—41no—
459C23—41no—
460C23—41no—
461C23—41no—
462C23—41no—
463C23—41no—
464C23C2×D4×D6111yescharacteristic
465Q8—2442no—
466Q8—2442no—
467Q8—2442no—
468Q8—2442no—
469D4—22no—
470D4—22no—
471D4—22no—
472D4—22no—
473D4—22no—
474D4—41no—
475D4—41no—
476D4#1451975114yes—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—22no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—48no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—84no—
487C2×C4—1222no—
488C2×C4—1222no—
489C2×C4—1222no—
490C2×C4—1222no—
491C2×C4—1222no—
492C2×C4—1222no—
493C2×C4—4888no—
494C2×C4—2442no—
495C2×C4—2442no—
496C2×C4—2442no—
497C2×C4—2442no—
498C2×C4—2442no—
499C2×C4—2442no—
500C2×C4#1451974912yes—
501C2×C4#1451974912yes—
502C2×C4#1451979612yes—
503C2×C4#1451980112yes—
504C2×C4C22×C22⋊D6111yescharacteristic
505C2×C4C22×D24411yescharacteristic
506C2×C4C2×#1175822411yescharacteristic
507C2×C4C2×#1175822211yescharacteristic
508C2×C4#1451982311yescharacteristic
509C2×C4#1451982411yescharacteristic
510C8—22no—
511C8—41no—
512C8—2442no—
513C22—24no—
514C22—42no—
515C22—42no—
516C22—42no—
517C22—42no—
518C22—42no—
519C22—42no—
520C22—42no—
521C22—42no—
522C22—42no—
523C22—81no—
524C22—81no—
525C22—81no—
526C22—81no—
527C22C2×#1451974911yescharacteristic
528C22D4×D24411yescharacteristic
529C22#1684042111yescharacteristic
530C22C2×#1451979611yescharacteristic
531C22C2×#1451980111yescharacteristic
532C22#1684056911yescharacteristic
533C22#1684057611yescharacteristic
534C4—22no—
535C4—42no—
536C4—81no—
537C4—4882no—
538C4—4882no—
539C4—2442no—
540C4—2442no—
541C4#1683997514yescomplemented
542C4C2×#1451975111yescharacteristic
543C4#1684034011yescharacteristic
544C4#1684041611yescharacteristic
545C4#1684045811yescharacteristic
546C2—44no—
547C2—82no—
548C2#2144925911yescharacteristic
549C2C2×#1683997511yescharacteristic
550C2#2145536711yescharacteristic
551C2#2145742011yescharacteristic
552C2#2145754011yescharacteristic
553C2#2145846611yescharacteristic
554C2#2145853511yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C4×D8×C61C2
socleC23×C61C22×D4
prime radicalC22×D8×C61C22
hypercenterC2×C4×D8D61
hypocenterC61#24727
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×D8

Complements

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

NH ≅ G/Ntype
2. C2×#16839967C2semidirect
3. #21450811C2semidirect
4. D4×C4⋊Dic61C2semidirect
5. #21447209C2semidirect
6. #21447059C2semidirect
7. #21446621C2semidirect
8. #21446537C2semidirect
9. #21446151C2semidirect
10. #21445819C2semidirect
260. C4×C61#463semidirect
321. C61#24727semidirect
541. C4#16839975semidirect

Maximal subgroups

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

subgroupindex
C2×#168399752
C2×#168399672
#214508112
D4×C4⋊Dic612
#214472092
#214470592
#214466212
#214465372
#214461512
#214458192
C2×C4×D8×C612
#2472761