Finite Group Databasebeta

← #29721558

Subgroups — #29721558

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×#16839980C212yesmaximal
2C2×#16839964C211yesmaximal, characteristic, complemented
3C2×C4×D488C211yesmaximal, characteristic
4#21450815C211yesmaximal, characteristic, complemented
5#21448719C211yesmaximal, characteristic, complemented
6#21447212C212yesmaximal, complemented
7#21447056C212yesmaximal, complemented
8#21446537C211yesmaximal, characteristic, complemented
9#21446150C212yesmaximal, complemented
10#21445816C211yesmaximal, characteristic, complemented
11C61×#1956C211yesmaximal, characteristic, complemented
12C22×#11758201C2212yes—
13C22×D488C2211yescharacteristic
14C2×#14519766C2212yes—
15C2×C4×D244C2211yescharacteristic
16C2×C4×D244C2211yescharacteristic
17C2×#14519739C2212yes—
18C2×C4⋊C4×D61—21no—
19#16839980—28no—
20#16839964—24no—
21C4×D488—24no—
22C2×#14519711C2211yescharacteristic
23C2×C4⋊#92418C2211yescharacteristic
24C2×#14519706C2212yes—
25C2×#14519701C2212yes—
26C2×#14519699C2211yescharacteristic
27C2×#14519699C2211yescharacteristic
28C2×#14519697C2212yes—
29#16839688—22no—
30#16839669C2212yes—
31#16839658C2212yes—
32#16839633—22no—
33#16839619—21no—
34#16839619—21no—
35#16839619—21no—
36C4×C4⋊Dic61C2211yescharacteristic
37#16839524C2212yes—
38#16839516C2212yes—
39C2×C4⋊Q8×C61C2211yescharacteristic
40C2×#103×C61C2212yes—
41C2×#103×C61C2212yes—
42C2×C4×C8×C61C2211yescharacteristic
43C61×#384C2211yescharacteristic
44C22×D244C2311yescharacteristic
45C22×D244C2311yescharacteristic
46C22×C4×D61—21no—
47C22×C4×D61—21no—
48C22×C4×D61—22no—
49C2×#11758201—24no—
50C2×#11758201—24no—
51C2×#11758201—24no—
52C2×D488—22no—
53C2×D488—22no—
54C2×D488D414yes—
55C22×#92418C2312yes—
56#14519766—24no—
57#14519766—42no—
58C4×D244—22no—
59C4×D244—22no—
60C4×D244—22no—
61C4×D244D414yes—
62#14519739—24no—
63#14519739—42no—
64C2×C4⋊Dic61D411yescharacteristic
65C2×C4⋊Dic61D411yescharacteristic
66C2×C4⋊Dic61C2311yescharacteristic
67C2×C4⋊Dic61C2311yescharacteristic
68C2×#11758196—22no—
69C4⋊C4×D61—44no—
70C2×#11758198—21no—
71C2×#11758198—21no—
72C2×#11758198—22no—
73C2×#11758198—22no—
74C2×#11758198—22no—
75C2×C4×Dic61C2312yes—
76#14519711—22no—
77C4⋊#92418—22no—
78#14519706—24no—
79#14519701—24no—
80#14519699—22no—
81#14519699—22no—
82#14519697—24no—
83#14519684—22no—
84#14519684—22no—
85#14519684—22no—
86#14519684D412yes—
87C22×Q8×C61C2312yes—
88C22×C8×C61C2311yescharacteristic
89C22×C8×C61C2311yescharacteristic
90C4⋊Q8×C61—22no—
91C4⋊Q8×C61—22no—
92C2×C4⋊C4×C61—21no—
93C2×C4⋊C4×C61—21no—
94C2×C4⋊C4×C61—21no—
95C2×C4⋊C4×C61—21no—
96C2×C4⋊C4×C61C2312yes—
97C2×C42×C61C2311yescharacteristic
98#103×C61—24no—
99#103×C61—42no—
100C4×C8×C61—22no—
101C23×D61—21no—
102C23×D61—21no—
103C2×Q8×C61—22no—
104C2×Q8×C61—22no—
105C2×Q8×C61—22no—
106C2×Q8×C61—22no—
107C2×Q8×C61—22no—
108C2×Q8×C61—22no—
109C22×C4×C61—22no—
110C22×C4×C61—22no—
111C22×C4×C61C2×D411yescharacteristic
112C22×C4×C61C2×D411yescharacteristic
113C22×C4×C61C2411yescharacteristic
114C22×Dic61—21no—
115C22×Dic61—21no—
116C22×Dic61C2×D412yes—
117C22×Dic61C2×D412yes—
118C2×D244—21no—
119C2×D244—21no—
120C2×D244—21no—
121C2×D244—21no—
122C2×D244—21no—
123C2×D244—21no—
124C2×D244—21no—
125C2×D244—21no—
126C2×D244C2×D412yes—
127C2×D244C2×D412yes—
128C2×D244C2×D412yes—
129C2×D244#4112yes—
130C2×C4×D61—24no—
131C2×C4×D61—42no—
132C2×C4×D61—42no—
133C2×C4×D61—44no—
134C2×C4×D61—44no—
135C2×C4×D61—42no—
136C2×C4×D61—42no—
137C2×C4×D61—42no—
138C2×C4×D61—82no—
139C2×C8×C61—21no—
140C2×C8×C61—21no—
141C2×C8×C61—21no—
142C2×C8×C61—21no—
143C2×C8×C61—21no—
144C2×C8×C61C2×D412yes—
145C4⋊C4×C61—22no—
146C4⋊C4×C61—22no—
147C4⋊C4×C61—42no—
148C4⋊C4×C61—42no—
149C4⋊C4×C61—42no—
150C4⋊C4×C61—81no—
151C42×C61C2×D412yes—
152C42×C61#4112yes—
153#11758201—48no—
154#11758198—24no—
155#11758198—44no—
156#11758198—44no—
157#11758198—44no—
158#11758198—42no—
159C4⋊Dic61—22no—
160C4⋊Dic61—22no—
161C4⋊Dic61—21no—
162C4⋊Dic61—21no—
163C4⋊Dic61—21no—
164C4⋊Dic61C2×D412yes—
165#11758196—44no—
166C4×Dic61—22no—
167C4×Dic61—22no—
168C2×#92418—22no—
169C2×#92418—22no—
170C2×#92418—22no—
171D488—28no—
172C23×C61C22×D411yescharacteristic
173C22×D61—22no—
174C22×D61—22no—
175C22×D61—22no—
176C22×D61—22no—
177C22×D61—41no—
178C22×D61—41no—
179C22×D61—41no—
180C22×D61—41no—
181C22×D61—41no—
182C22×D61—41no—
183C22×D61—41no—
184C22×D61—41no—
185C22×D61—41no—
186C22×D61—41no—
187Q8×C61—42no—
188Q8×C61—42no—
189Q8×C61—42no—
190Q8×C61—42no—
191C2×C4×C61—21no—
192C2×C4×C61—21no—
193C2×C4×C61—42no—
194C2×C4×C61—42no—
195C2×C4×C61—42no—
196C2×C4×C61—42no—
197C2×C4×C61—42no—
198C2×C4×C61—42no—
199C2×C4×C61#12112yes—
200C2×C4×C61#12112yes—
201C2×C4×C61#13512yes—
202C2×C4×C61AGL(1,ℤ/8)12yes—
203C2×C4×C61C22×D411yescharacteristic
204C2×C4×C61C22×D411yescharacteristic
205C2×C4×C61C2×#4111yescharacteristic
206C2×C4×C61C2×#4111yescharacteristic
207C2×C4×C61H(5,𝔽2)11yescharacteristic
208C2×C4×C612−1+411yescharacteristic
209C2×Dic61—22no—
210C2×Dic61—22no—
211C2×Dic61—22no—
212C2×Dic61—22no—
213C2×Dic61—22no—
214C2×Dic61—22no—
215C2×Dic61—22no—
216C2×Dic61—41no—
217C2×Dic61—41no—
218C2×Dic61—41no—
219C2×Dic61—41no—
220C2×Dic61—41no—
221D244—22no—
222D244—22no—
223D244—22no—
224D244—22no—
225D244—22no—
226D244—41no—
227D244—41no—
228D244#12114yes—
229C4×D61—48no—
230C4×D61—88no—
231C4×D61—84no—
232C8×C61—22no—
233C8×C61—41no—
234#92418—42no—
235#24727—611nomaximal, Sylow
236C22×C61C2×#12111yescharacteristic
237C22×C61D4211yescharacteristic
238C22×C61#54811yescharacteristic
239C22×C61C2×#13511yescharacteristic
240C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
241C22×C61#57711yescharacteristic
242C22×C61#57811yescharacteristic
243C2×D61—24no—
244C2×D61—42no—
245C2×D61—42no—
246C2×D61—42no—
247C2×D61—42no—
248C2×D61—42no—
249C2×D61—42no—
250C2×D61—42no—
251C2×D61—42no—
252C2×D61—42no—
253C2×D61—81no—
254C2×D61—81no—
255C2×D61—81no—
256C2×D61—81no—
257C4×C61—22no—
258C4×C61—82no—
259C4×C61—82no—
260C4×C61#46314yescomplemented
261C4×C61C2×#12111yescharacteristic
262C4×C61#53711yescharacteristic
263C4×C61#54711yescharacteristic
264C4×C61#55311yescharacteristic
265Dic61—42no—
266Dic61—42no—
267Dic61—42no—
268Dic61—81no—
269C2×#463—612no—
270C2×#461—611no—
271C2×C4×D8—611no—
272#2547—611no—
273D4×C4⋊C4—611no—
274#2068—612no—
275#2042—612no—
276#1956D6111yescharacteristic, complemented
277#1939—611no—
278#1877—612no—
279#1802—611no—
280C2×C61#241611yescharacteristic
281C2×C61C2×#46311yescharacteristic
282C2×C61#304611yescharacteristic
283C2×C61#327211yescharacteristic
284C2×C61#328311yescharacteristic
285C2×C61#336011yescharacteristic
286C2×C61#336711yescharacteristic
287D61—44no—
288D61—82no—
289C22×SD16—612no—
290C22×D8—611no—
291C2×C4⋊Q8C2×D6111yescharacteristic
292C2×#124—612no—
293C2×C4×D4—612no—
294C2×C4×D4—611no—
295C2×C4×D4—611no—
296C22×C4⋊C4—1221no—
297#463—1228no—
298#461—1224no—
299C4×D8—1224no—
300C2×#107—611no—
301C2×C4⋊C8—612no—
302C2×C4⋊C8—611no—
303C2×#103—612no—
304C2×#103C2×D6112yes—
305C2×#103C2×D6112yes—
306C2×#102—611no—
307C2×#102—611no—
308C2×#102—612no—
309C2×C4×C8C2×D6111yescharacteristic
310#397—1222no—
311#391—612no—
312#389—612no—
313#385—1222no—
314#384C2×D6111yescharacteristic
315#380—1221no—
316#380—1221no—
317#380—1221no—
318C4×C4⋊C4—611no—
319#340—612no—
320#336—612no—
321C61#2472711yescharacteristic, complemented, Sylow
322C22×Q8C22×D6112yes—
323C22×D4—611no—
324C22×D4—611no—
325C23×C4—1222no—
326C23×C4—1221no—
327C23×C4—1221no—
328C2×SD16—1224no—
329C2×SD16—1224no—
330C2×SD16—1224no—
331C2×D8—614no—
332C2×D8—1222no—
333C2×D8—1222no—
334C22×C8—612no—
335C22×C8C22×D6111yescharacteristic
336C22×C8C22×D6111yescharacteristic
337C4⋊Q8—22no—
338C4⋊Q8—22no—
339#124—1224no—
340#124—2442no—
341C4×D4—614no—
342C4×D4—1224no—
343C4×D4—1222no—
344C4×D4—1222no—
345C4×D4—1222no—
346C4×D4—2442no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—611no—
352C2×C4⋊C4—611no—
353C2×C4⋊C4—611no—
354C2×C4⋊C4—611no—
355C2×C4⋊C4—1222no—
356C2×C4⋊C4—2444no—
357C2×C4⋊C4C22×D6112yes—
358C2×C22⋊C4—1221no—
359C2×C22⋊C4—1222no—
360C2×C22⋊C4—1222no—
361C2×C22⋊C4—1222no—
362C2×C22⋊C4—1221no—
363C2×C42—612no—
364C2×C42C22×D6111yescharacteristic
365#107—1222no—
366C4⋊C8—1224no—
367C4⋊C8—1222no—
368#103—24no—
369#103—42no—
370#103—1224no—
371#102—1224no—
372#102—1222no—
373#102—1222no—
374C4×C8—22no—
375#95—612no—
376#95—1222no—
377#95—1222no—
378#95—1222no—
379C24—1221no—
380C24—1221no—
381C2×Q8—22no—
382C2×Q8—22no—
383C2×Q8—22no—
384C2×Q8—22no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×D4—612no—
388C2×D4—612no—
389C2×D4—612no—
390C2×D4—612no—
391C2×D4—1221no—
392C2×D4—1221no—
393C2×D4—1221no—
394C2×D4—1221no—
395C2×D4—1221no—
396C2×D4—1221no—
397C2×D4—1221no—
398C2×D4—1221no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—612no—
402C22×C4—612no—
403C22×C4—1221no—
404C22×C4—1224no—
405C22×C4—1221no—
406C22×C4—4882no—
407C22×C4—2444no—
408C22×C4—2444no—
409C22×C4—2442no—
410C22×C4—2442no—
411C22×C4—2442no—
412C22×C4—2442no—
413C22×C4—2442no—
414C22×C4D4×D6111yescharacteristic
415C22×C4D4×D6111yescharacteristic
416C22×C4C23×D6111yescharacteristic
417SD16—2448no—
418D8—1228no—
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×C8D4×D6112yes—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—42no—
431C4⋊C4—42no—
432C4⋊C4—42no—
433C4⋊C4—81no—
434C4⋊C4—612no—
435C4⋊C4—1221no—
436C4⋊C4—1222no—
437C4⋊C4—1222no—
438C4⋊C4—1221no—
439C4⋊C4—1221no—
440C4⋊C4—2444no—
441C22⋊C4—1224no—
442C22⋊C4—2444no—
443C22⋊C4—2444no—
444C22⋊C4—2444no—
445C22⋊C4—2442no—
446C42—1222no—
447C42—1222no—
448C42#1175822212yes—
449C42D4×D6112yes—
450C23—1222no—
451C23—1222no—
452C23—1222no—
453C23—1222no—
454C23—2441no—
455C23—2441no—
456C23—2441no—
457C23—2441no—
458C23—2441no—
459C23—2441no—
460C23—2441no—
461C23—2441no—
462C23—2441no—
463C23—2441no—
464C23C2×D4×D6111yescharacteristic
465Q8—42no—
466Q8—42no—
467Q8—42no—
468Q8—42no—
469D4—614no—
470D4—1222no—
471D4—1222no—
472D4—1222no—
473D4—1222no—
474D4—1222no—
475D4—2441no—
476D4—2441no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—1222no—
486C2×C4—1222no—
487C2×C4—1222no—
488C2×C4—1222no—
489C2×C4—1222no—
490C2×C4—1222no—
491C2×C4—1222no—
492C2×C4—4888no—
493C2×C4—4884no—
494C2×C4—2441no—
495C2×C4—2441no—
496C2×C4—2441no—
497C2×C4—2441no—
498C2×C4—2441no—
499C2×C4—2448no—
500C2×C4#1451975012yes—
501C2×C4#1451975012yes—
502C2×C4#1451980012yes—
503C2×C4#1451980412yes—
504C2×C4C2×D4×D6111yescharacteristic
505C2×C4C2×D4×D6111yescharacteristic
506C2×C4#41×D6111yescharacteristic
507C2×C4C2×#1175822211yescharacteristic
508C2×C4#1451982211yescharacteristic
509C2×C4#1451982411yescharacteristic
510C8—22no—
511C8—41no—
512C8—2442no—
513C22—1224no—
514C22—4881no—
515C22—4881no—
516C22—4881no—
517C22—4881no—
518C22—2442no—
519C22—2442no—
520C22—2442no—
521C22—2442no—
522C22—2442no—
523C22—2442no—
524C22—2442no—
525C22—2442no—
526C22—2442no—
527C22C2×#1451975011yescharacteristic
528C22#1684039711yescharacteristic
529C22#1684042011yescharacteristic
530C22C2×#1451980011yescharacteristic
531C22C2×#1451980411yescharacteristic
532C22#1684056811yescharacteristic
533C22#1684057511yescharacteristic
534C4—22no—
535C4—82no—
536C4—82no—
537C4—4881no—
538C4—2442no—
539C4—2442no—
540C4—2442no—
541C4#1683998014yescomplemented
542C4#121×D6111yescharacteristic
543C4#1684034111yescharacteristic
544C4#1684041711yescharacteristic
545C4#1684046211yescharacteristic
546C2—4882no—
547C2—2444no—
548C2#2144926211yescharacteristic
549C2C2×#1683998011yescharacteristic
550C2#2145537411yescharacteristic
551C2#2145741811yescharacteristic
552C2#2145753911yescharacteristic
553C2#2145847311yescharacteristic
554C2#2145854211yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1956C2
socleC23×C61C22×D4
prime radicalC22×D488C22
hypercenter#1956D61
hypocenterC61#24727
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1956

Complements

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

NH ≅ G/Ntype
2. C2×#16839964C2semidirect
4. #21450815C2semidirect
5. #21448719C2semidirect
6. #21447212C2semidirect
7. #21447056C2semidirect
8. #21446537C2semidirect
9. #21446150C2semidirect
10. #21445816C2semidirect
11. C61×#1956C2semidirect
260. C4×C61#463semidirect
276. #1956D61semidirect
321. C61#24727semidirect
541. C4#16839980semidirect

Maximal subgroups

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

subgroupindex
C2×#168399802
C2×#168399642
C2×C4×D4882
#214508152
#214487192
#214472122
#214470562
#214465372
#214461502
#214458162
C61×#19562
#2472761